A unified analytical approach for the acoustic conceptual design of fans of modern aero-engines [original]

A unified analytical app roach fo r the acoustic
conceptual design of fans of mo dern aero-engines
vo rgelegt von
Dipl.-Ing.
Antoine Mo reau
geb. in P a ris
von der F akult¨ at V – V erk ehrs- und Maschinensysteme
der T echnischen Universit¨ at Berlin
zur Erlangung des ak ademischen Grades
Dokto r der Ingenieurwissenschaften
– Dr.-Ing. –
genehmigte Dissertation
Promotionsausschuss:
V o rsitzender: Prof. Dr.-Ing. Andreas Ba rdenhagen
Gutachter: Prof. Dr. rer. nat. La rs Engha rdt
Gutachter: Prof. Dr.-Ing. Dieter P eitsch
Gutachter: Prof. Phillip Joseph
T ag der wissenschaftlichen Aussp rache: 18. Juli 2016
Berlin 2017

’Essen tially , all mo dels are wrong, but some are useful.’
George Bo x, statistician
(quote from his b o ok
’Empirical Mo del-Building and Resp onse Surfaces’
written with Norman Drap er in 1987)

Abstract
This thesis prop oses a set of theoretical models to predict, during the conceptual design phase, the sound
emitted b y the fan stage of an aero-engine, whic h is due to the in teraction of its solid surfaces with
the flo w. The mo dels are analytical or semi-analytical in nature, and cov er a range of disciplines that
span b et w een aero dynamics and acoustics. V arious fan arc hitectures, suc h as the con v en tional ducted
turb ofan, the ducted con tra-rotating fan and the con tra-rotating op en rotors, are addressed on a common
basis, where the sensitivity of the radiated noise to the fan design is in v estigated at giv en thrust and
optimal aero dynamic p erformance.
The main part of the thesis pro vides a detailed description of the mo dels that ha v e b een selected,
adapted and implemen ted with appropriate form ulations to form a consisten t prediction en vironmen t.
F an aero dynamics is considered via a meanline approac h, where a giv en streamline is assumed to b e
represen tativ e of the whole flo w passage. A pro cedure to obtain a realistic and aero dynamically sound
fan geometry based on a reduced n um b er of parameters, t ypical of the preliminary design phase, is
prop osed. F or eac h in v estigated fan, the aero dynamic p erformance map is explored to lo cate the off-
design p oin ts relev an t for acoustic certification. The flo w p erturbations generated by the blades are
describ ed as deca ying w ak es and p oten tial fields that propagate do wn to the neigh b oring blade ro w and
in teract with it in form of acoustic w a v es.
The form ulation of the sound pressure radiated b y the blades has b een completely derived anew from
the linear theory of the Acoustic Analogy . The acoustic pressure, expressed in the frequency domain,
presen ts strong analogies in the ducted problem compared with the free-field case. The mathematical
deriv ation also sho ws that the expressions of tonal and broadband noise assume v ery similar forms.
Finally , a term common to all problems, identified as the source strength, only dep ends on the flo w
around the blades, and can therefore b e directly related to the aero dynamic quan tities deduced from the
fan design pro cedure men tioned ab o v e.
The capabilities of that unified prediction approac h are illustrated b y t w o parametric studies, whic h
deal with the acoustic implications of a mo dification of the design sp eed of the fan for the three main
arc hitectures. The motiv ation for addressing this topic is t w ofold: first, fan sp eed is kno wn to b e an
essen tial k ey driv er for noise emission; second, it is a parameter that strongly affects design and thus is a
go o d candidate to test the en tire calculation c hain. The sp eed reduction is realized either by decreasing
the fan pressure ratio, hence increasing the b ypass ratio of the engine, or b y increasing the aero dynamic
loading of the blades, if the fan diameter cannot b e further increased. It turns out that the results obtained
feature realistic, sensible, and smo oth trends that are in line with past theoretical and exp erimen tal
studies.
As a consequence, the feasibilit y of acoustic assessmen t based on analytical mo dels during a prelim-
inary design phase is demonstrated. The virtues of the con tra-rotating concepts o v er the con v en tional
turb ofan in terms of broadband and tonal self-noise can b e directly attributed to the low er aero dynamic
loading and Mac h n um b ers. The op en rotors do not b y themselv es generate more noise, the source
strength is relativ ely similar to that of ducted fans, but lac k the b enefits of a nacelle equipp ed with sound
damping liners. Finally , fan designs with a lo w pressure ratio tend to b e significantly less noisy , th us the
acoustic optim um lies far b elo w the optim um of fuel-consumption. A low-noise design alternativ e to the
reduction of the pressure ratio, is increasing the loading of the fan, whic h ho w ev er m ust result from a
careful compromise b et w een broadband and tonal noise.

Ac kno wledgemen ts
I w ould firstly lik e to thank m y colleague Dr. S ´ ebastien Gu ´ erin for his con tin uous supp ort during the
man y y ears w e ha v e b een w orking together. I often remem b er the fruitful discussions w e had in the most
v arious circumstances, esp ecially those in the evening, after a da y of business trip, in the airplane or in
the bus bringing us home. F rom the b eginning, we naturally shared a common view on what our researc h
w ork should b e and ho w to realize it. Merci ` a toi.
Muc h of the w ork presen ted in this thesis has profited from the participation and the tec hnical supp ort
of sev eral studen ts from the T ec hnical Univ ersit y of Berlin: Behnam Nouri, Nadja W agner, Rob ert Meier
zu Ummeln, Sebastian Oert wig, Arne Matsc hk e, Claas Ko ep, and last but not least, my most committed
studen t and no w colleague, Rob ert Jaron; to all of you, vielen Dank f¨ ur eure Un terst ¨ utzung!
Thanks also go to m y colleagues Dr. Rob ert Meyer and Prof. Lars Enghardt for trusting me as I
started to w ork for them, first as a studen t, then as an employ ee, at the Department of Engine Acoustics.
The kind supp ort pro vided b y Balbir Kaur, Brig Pilger, and Horst Mettc hen in ev ery-da y w ork life is
sincerely ac kno wledged, to o.
I am also grateful to the p eople that form the international comm unity of Aeroacoustics researc h.
This is a pleasan t, vivid, and h uman-scale comm unit y , full of passionate sc holars and committed engi-
neers. I ha v e learned m uc h from them, and from all the in teractions I could ha v e with the academic and
industrial partners during conferences and meetings.
Finally , I w ould lik e to sa y some w ords in F renc h to m y paren ts. Maman, Papa, v otre bienv eillance
´ eternelle ` a l’´ egard de v otre fils est inestimable et, je dois dire, m’a aid ´ e ` a surmonter la derni ` ere ligne
droite a v an t de finaliser cette th ` ese. Je v ous en suis tr` es reconnaissan t.

Nomenclature
Here are presen ted the main sym b ols and notations o ccuring in the mo dels and the corresp onding physical
quan tities they describ e. Some symbols may refer to differen t quan tities dep ending on the context.
L atin symb ols
A cross-sectional area (m 2 )
w ak e area (-)
p osition upstream of an engine comp onen t
A mn mo dal amplitude (P a)
a 0 sound sp eed (m/s)
B n um b er of blades (-)
p osition do wnstream of an engine comp onen t
C L , C D lift and drag co efficien ts (-)
C f skin friction co efficien t (-)
c blade c hord length (m)
c p sp ecific heat capacit y of air c p = 1004 . 5 k g /s
d w ak e depth (-)
D diffusion factor (-)
diameter (m)
drag (N)
f frequency (Hz)
force p er unit area (P a)
G Green’s function (1/m)
g ω
m mo dal Green’s function (1/m)
h w ak e harmonic (-)
stream tub e heigh t (m)
non-dimensional c hordwise distribution of sound sources (-)
H en thalp y (J/kg)
H 12 b oundary la y er shap e factor
i blade incidence (rad)
imaginery unit n um b er i 2 = − 1
J m Bessel function of order m (-)
k w a v en um b er (1/m)
acoustic scattering index (-)
` c hordwise or stream wise p osition (m)
turbulen t correlation length (m)
M Mac h n um b er (-)
m circumferen tial mo de order (-)
n radial mo de order (-)
p o w er split exp onen t (-)
direction normal to blade c hord (-)
N n um b er of engines, blades, or mo des (-)
o throat width (m)

7
P pressure (P a)
p fluctuation pressure (P a)
Q mass flo w rate (kg/s)
r radial p osition or direction (m)
sp ecific gas constan t of air r = 287 . 04 J /k g /K
R duct casing or blade tip radius (m)
s blade pitc h spacing (m)
subscript for source p osition
S Sears’ function (-)
non-dimensional en trop y (-)
w etted surface (m 2 )
S t Strouhal n um b er (-)
t blade thic kness (m)
subscript for total (or stagnation) thermo dynamic quantities
subscript for tangen tial or circumferen tial v elo cit y comp onen t
T temp erature (K)
time p erio d (s)
thrust (N)
u fluctuation v elo cit y in stream wise direction (m/s)
U blade rotation sp eed (m/s)
W flo w v elo cit y (m/s)
w eigh t (kg)
V flo w v elo cit y (m/s)
x axial direction (m)
y tangen tial direction (opp osed to circumferen tial direction) (m)
z s noise source p osition (m)
Gr e ek symb ols
α mn mo dal cut-on factor
β flo w angle (rad)
factor β = √ 1 − M 2
γ heat capacit y ratio of air γ = 1 . 4
Γ blade circulation (m 2 /s)
δ blade deviation angle (rad)
δ 1 , δ 2 displacemen t and momen tum thic kness of the b oundary la y er (m)
∆ difference
ζ 0 aero dynamic excitation pressure (P a)
η aero dynamic efficicency (-)
h ub-to-tip ratio (-)
θ circumferen tial p osition (rad)
Λ turbulen t in tegral length scale (m)
ν kinematic viscosit y of air (m 2 /s)
blade index (-)
Π p o w er (W)
ρ densit y (kg/m 3 )
σ pitc h-to-c hord ratio or solidit y (-)
non-dimen tional frequency (-)
throttling co efficien t (-)
source term (P a)
σ mn ( n + 1) th zero of Bessel function of order m (also called Bessel eigen v alue) (-)
τ flo w turning co efficien t (-)
φ flo w co efficien t (-)

8
phase term (-)
χ blade stagger angle (rad)
ψ acoustic emission angle (rad)
loading co efficien t (-)
Ψ acoustic c hordwise correlation function (-)
ω aero dynamic loss co efficien t (-)
pulsation frequency (rad/s)
Ω rotation frequency of rotor (rad/s)

Con ten ts
1 In tro duction 15
1 . 1 O b j e c t i v e s ............................................. 1 5
1 . 2 C h a l l e n g e s ............................................. 1 6
1 . 3 T h e a n a l y t i c a l a p p r o a c h ..................................... 1 6
1.4 State of the art on prediction mo dels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 New prediction to ol: PropNoise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Steady aero dynamics 21
2 . 1 M o t i v a t i o n a n d a p p r o a c h .................................... 2 1
2.2 Definition of parameters and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Relation b et w een pressure rise, flo w turning and lift . . . . . . . . . . . . . . . . . . . . . 25
2.4 Definition of design and off-design conditions . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Distribution of flo w v elo cities around the blades . . . . . . . . . . . . . . . . . . . . . . . . 27
2 . 6 L o s s e s ............................................... 2 8
2.6.1 Relation b et w een loss, drag and en trop y pro duction . . . . . . . . . . . . . . . . . 28
2.6.2 Blade loading, diffusion factor and stall . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6.3 Loss caused b y b oundary la y ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 . 6 . 4 L o s s c a u s e d b y s h o c k s .................................. 3 1
2 . 6 . 5 E n d w a l l l o s s ........................................ 3 1
2 . 7 F a n p e r f o r m a n c e ......................................... 3 2
2 . 8 A p p l i c a t i o n o f t h e m o d e l s .................................... 3 5
2 . 8 . 1 C a s c a d e p e r f o r m a n c e ................................... 3 6
2.8.2 F an p erformance at off-design conditions . . . . . . . . . . . . . . . . . . . . . . . . 38
2 . 9 C o n c l u s i o n ............................................ 4 0
3 Engine and fan aero dynamic design 42
3 . 1 M o t i v a t i o n a n d a p p r o a c h .................................... 4 2
3 . 2 D e s i g n c o n s t r a i n t s ........................................ 4 2
3 . 3 P r e l i m i n a r y e n g i n e d e s i g n .................................... 4 5
3 . 3 . 1 M e t h o d o l o g y ....................................... 4 5
3 . 3 . 2 E n g i n e t h r u s t ....................................... 4 6
3.3.3 Engine airflo w and fan diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.4 Engine length and nacelle dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 . 3 . 5 E n g i n e w e i g h t ....................................... 4 8
3 . 3 . 6 E n g i n e d r a g ........................................ 4 8
3 . 4 F a n d e s i g n ............................................. 4 9
3 . 4 . 1 P r i n c i p l e s ......................................... 4 9
3 . 4 . 2 E x e m p l a r y r e s u l t s .................................... 5 0
3 . 5 E n g i n e p e r f o r m a n c e ....................................... 5 4
3.6 Off-design op erating p oin ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.7 V alidation at design conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

CONTENTS 10
3 . 8 C o n c l u s i o n ............................................ 5 8
4 Unsteady aero dynamics 60
4 . 1 I n t r o d u c t i o n ............................................ 6 0
4 . 2 P o t e n t i a l fi e l d ........................................... 6 0
4.2.1 Initial strength of the p oten tial field . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.2 Circumferen tial distribution and mo des . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.3 Deca y of the p oten tial field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 . 3 M e a n - fl o w w a k e s ......................................... 6 2
4 . 3 . 1 W a k e m o d e l ........................................ 6 3
4 . 3 . 2 S p e c t r a l c o n t e n t ..................................... 6 4
4 . 3 . 3 W a k e d e c a y ........................................ 6 4
4 . 4 T u r b u l e n c e ............................................ 6 8
4.4.1 Inflo w turbulence ingested b y the fan . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4.2 W all-pressure fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 . 5 C h a n g e o f r e f e r e n c e f r a m e .................................... 7 0
4 . 6 A i r f o i l r e s p o n s e f u n c t i o n..................................... 7 1
5 Extrap olation of meanline data 74
5.1 Need for a radial extrap olation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 . 2 S t e a d y fl o w v e l o c i t i e s ....................................... 7 4
5 . 3 B l a d e g e o m e t r y.......................................... 7 5
5.4 Lift, drag and unsteady flo w v elo cities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6 Acoustics 77
6 . 1 M o d e l l i n g a p p r o a c h ........................................ 7 7
6 . 2 A s s u m p t i o n s ........................................... 7 9
6 . 3 N o i s e p r o p a g a t i o n ........................................ 8 0
6.3.1 The con v ectiv e and fly o v er problems . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6 . 3 . 2 D i s p e r s i o n r e l a t i o n .................................... 8 1
6 . 3 . 3 S o u n d p r o p a g a t i o n .................................... 8 2
6 . 3 . 4 O v e r a l l s o u n d p o w e r ................................... 8 4
6.4 Noise generated b y rotating blades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.4.1 Deriv ation of the mo dal pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.4.2 Application to the free-field and in-duct problems . . . . . . . . . . . . . . . . . . 92
6 . 4 . 3 T o n a l n o i s e ........................................ 9 4
6 . 4 . 4 B r o a d b a n d n o i s e ..................................... 9 6
6 . 4 . 5 S u m m a r y ......................................... 9 9
6.5 In terpretation of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.5.1 Classification of sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.5.2 Mo delling of the sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.5.3 Generalized cut-on criterion for efficien t radiation . . . . . . . . . . . . . . . . . . . 106
6.5.4 Effect of source non-compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.5.5 Application to a single prop eller: effect of rotation sp eed and blade coun t . . . . . 112
6.6 V alidation of the acoustic mo dels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6 . 7 J e t n o i s e .............................................. 1 1 6
6 . 8 C o n c l u d i n g r e m a r k s ....................................... 1 1 7

CONTENTS 11
7 Application of the mo dels 120
7 . 1 I n t r o d u c t i o n ............................................ 1 2 0
7.2 Assumptions done for the parametric studies . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.2.1 Aero dynamic and design assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.2.2 Noise-relev an t design assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.3 First parametric study: acoustic impact of the fan pressure ratio . . . . . . . . . . . . . . 123
7.4 Second parametric study: acoustic impact of the design rotor sp eed at constan t fan pressure
r a t i o ................................................ 1 2 9
7.5 Conclusions ab out the parameter studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8 Conclusions and outlo ok 137
8.1 F easibilit y of acoustic pre-design b y analytical metho ds . . . . . . . . . . . . . . . . . . . 137
8.2 Main conclusions on the mo del predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.2.1 Comparison of the fan concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.2.2 Choice of the design fan pressure ratio and fan loading . . . . . . . . . . . . . . . . 139
8.3 Prediction capabilit y of the mo dels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.4 F urther extensions of the metho d and outlo ok . . . . . . . . . . . . . . . . . . . . . . . . . 143
8 . 5 A f t e r w o r d s ............................................ 1 4 4

List of Figures
1.1 Mo dular structure of the prediction to ol ’PropNoise’ . . . . . . . . . . . . . . . . . . . . . 20
2 . 1 S i m p l i fi e d e n g i n e m o d e l [ 7 6 ]................................... 2 3
2.2 Flo w angles and v elo cities in the relativ e frame link ed to the blade ro w . . . . . . . . . . . 23
2.3 W orking principle of a subsonic and sup ersonic rotor blade ro w . . . . . . . . . . . . . . . 26
2.4 Comparison of the loss correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Smith Chart for CR TF(lo w er) and SR TF(upp er) . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 Cascade map represen ting the v ariation of loss as a function of the inflo w Mach n um b er
a n d i n c i d e n c e a n g l e ........................................ 3 7
2.7 V alidation of predicted pressure ratio at off-design conditions (prediction: solid gray lines,
reference: blac k lines with icons) [76] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.8 V alidation of predicted efficiency at off-design conditions (prediction: solid gra y lines,
reference: blac k lines with icons) [76] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1 sc hematic view of the ducted-fan engine configuration. . . . . . . . . . . . . . . . . . . . . 45
3.2 Principle of the engine preliminary design . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Principle of the design of a single-rotating (left) and coun ter-rotating (righ t) turb ofan stage 50
3.4 Principle of the design of rotors and stators . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 V ariation of stator p erformance for differen t geometries (solidity) and inflo w angles . . . . 51
3.6 V ariation of stator p erformance for flo w turning requiremen ts . . . . . . . . . . . . . . . . 52
3.7 V ariation of rotor–stator design: effect of inflo w Mac h n um b er at constant FPR ..... 5 3
3.8 V ariation of rotor–stator design: effect of fan pressure ratio at constan t M x ........ 5 4
3.9 T ypical results of pre-design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.10 Pro cedure to determine the fan sp eed and mass flo w rate at off-design conditions. . . . . . 56
3.11 Accuracy of the prediction of some fan design parameters [76] . . . . . . . . . . . . . . . . 58
3.12 V ariations of single- and con tra-rotating fan efficiency with pressure ratio and axial Mac h
n u m b e r [ 7 6 ] ............................................ 5 9
3.13 Results of a similar study [105] obtained after optimization with a 3D RANS flo w solv er . 59
4.1 Represen tation of a w ak e o v er a blade passage and the asso ciated mean flo w and deficit
v e l o c i t i e s ............................................. 6 4
4.2 Stream wise ev olution of w ak e parameters for differen t v alues of pressure gradien t (initial
w ak e parameters k ept constan t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Stream wise ev olution of w ak e parameters for differen t v alues of blade loading . . . . . . . 67
4.4 V ariation of w ak e width on blade drag co efficien t for the fixed streamwise position `/h = 1)
and for differen t solidit y . (solid line: correlation of Eq.(4.9), dashed line: solution of
E q . ( 4 . 1 0 ) ) ............................................. 6 8
4.5 Mo dification of a w ak e turbulence sp ectrum through transformation of the reference frame
(left: mo derate length scale S t B P F = 0 . 2 , w = 0 . 07, right: large length scale S t B P F =
0 . 6 , w = 0 . 2 ) ............................................ 7 1

LIST OF FIGURES 13
6.1 History of rotating-blade noise prediction based on the acoustic analogy . . . . . . . . . . 78
6.2 Tw o differen t p oin ts of view for form ulating the sound emission . . . . . . . . . . . . . . . 81
6.3 In tegration surface for calculation of free-field sound p o w er . . . . . . . . . . . . . . . . . 85
6.4 In tegration surface for calculation of the in-duct sound p o w er . . . . . . . . . . . . . . . . 86
6.5 airfoil system of co ordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.6 cylindrical and spherical system of co ordinates . . . . . . . . . . . . . . . . . . . . . . . . 92
6.7 Probabilit y densit y function of ∆ φ r and ( z s , z 0
s ) ......................... 9 6
6.8 Mean flo w and p erturbation v elo cities in the fixed and rotating reference frames . . . . . 103
6.9 P ath of gusts as they con v ect from rotor to stator, in the relative frame . . . . . . . . . . 106
6.10 Cut-on/cut-off domains visualized via the v ariations of the Bessel function . . . . . . . . . 107
6.11 Effect of circumferen tial correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.12 Effect of c hordwise correlation for v arious source distributions . . . . . . . . . . . . . . . . 110
6.13 Correlation effects for uniformly distributed sources in radial direction . . . . . . . . . . . 111
6.14 Effect of blade tip Mac h n um b er M tip,r el on the radiated sound p o w er (left) and on the
aero dynamic excitation term (righ t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.15 Effect of blade coun t B on the radiated sound p o w er of a subsonic (left, M tip,rel = 0 . 8)
and sup ersonic prop eller (righ t, M tip,r el = 1 . 1 ) ........................ 1 1 4
6.16 Effect of rotation sp eed (at fixed thrust) on the radiated sound p ow er (left) and on the
aero dynamic excitation term (righ t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.17 V alidation of the predicted p o w er sp ectral densit y for forw ard- and rearw ard-radiated
b r o a d b a n d n o i s e ..................................... .... 1 1 6
6.18 V ariation of jet noise with fully-expanded jet Mac h num b er in static conditions . . . . . . 117
7.1 Relation b et w een fan pressure ratio and engine b ypass ratio . . . . . . . . . . . . . . . . . 121
7.2 V ariation of fan p erformance and geometry with design fan pressure ratio [76] . . . . . . . 124
7.3 Relativ e Mac h n um b ers at blade tip of fron t and rear rotors for Sideline, Cutbac k, and
A p p r o a c h c o n d i t i o n s [ 7 6 ] ..................................... 1 2 5
7.4 W ak e size relativ e to blade spacing for Sideline, Cutbac k, and Approac h conditions [76] . 126
7.5 V ariation of expanded jet Mac h n um b er and o v erall sound p ow er of jet noise with design
fan pressure ratio at v arious op erating p oints [76] . . . . . . . . . . . . . . . . . . . . . . . 127
7.6 Ov erall sound p o w er of fan noise sources at Sideline condition [76] . . . . . . . . . . . . . 128
7.7 Ov erall sound p o w er of fan noise sources at Cutbac k condition [76] . . . . . . . . . . . . . 128
7.8 Ov erall sound p o w er of fan noise sources at Approac h condition [76] . . . . . . . . . . . . 129
7.9 V elo cit y triangles for an axial-flo w fan with differen t degree of reaction and loading [76] . 130
7.10 V ariation of fan p erformance and geometry with fan loading for tw o differen t v alues of
d e s i g n f a n p r e s s u r e r a t i o [ 7 6 ] .................................. 1 3 1
7.11 Relativ e Mac h n um b ers at blade tip for Sideline, Cutbac k, and Approac h conditions [76] . 132
7.12 W ak e size relativ e to blade spacing for Sideline, Cutbac k, and Approac h conditions [76] . 133
7.13 Ov erall sound p o w er of fan noise sources at Sideline condition [76] . . . . . . . . . . . . . 134
7.14 Ov erall sound p o w er of fan noise sources at Cutbac k condition [76] . . . . . . . . . . . . . 134
7.15 Ov erall sound p o w er of fan noise sources at Approac h condition [76] . . . . . . . . . . . . 135

List of T ables
2.1 Shap e factors of a turbulen t b oundary la y er for differen t v alues of the pressure gradien t . 31
2.2 Geometry parameters of UHBR (left) and CRISP2 (righ t) fans . . . . . . . . . . . . . . . 36
2.3 Design parameters in Cruise of the UHBR (left) and CRISP2(righ t) fans . . . . . . . . . . 36
2.4 Fligh t conditions and thrust requiremen ts common to all configurations at sp ecific op er-
a t i n g c o n d i t i o n s .......................................... 3 9
2.5 Predicted fan parameters at off-design op erating conditions in the particular case of the
D L R U H B R f a n ......................................... 4 0
3.1 Definition of the fligh t conditions for a giv en short- to medium-range airliner . . . . . . . 56
6.1 Some applications of the frequency-domain form ulation of Golstein’s v ersion of the acoustic
a n a l o g y .............................................. 7 9
6.2 F orm ula for mo dal pressure dep ending on the problem considered . . . . . . . . . . . . . . 100
6 . 3 f o r m u l a f o r s o u r c e t e r m s..................................... 1 0 0
6.4 Classification of turb ofan noise sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.5 Decomp osition of the source terms for tonal noise . . . . . . . . . . . . . . . . . . . . . . . 102
6.6 Decomp osition of the source terms for broadband noise . . . . . . . . . . . . . . . . . . . 102
6.7 Comparison of the measured and predicted tonal sound p ow er of the DLR UHBR fan . . 117
7.1 Blade coun t of fron t and rear rotors (or stator) dep ending on design fan pressure ratio [76] 124
7.2 Blade coun t of rotor and stator dep ending on design v alue of fan loading [76] . . . . . . . 130

Chapter 1
In tro duction
1.1 Ob jectiv es
Since the early da ys of civil-aircraft aero-engine turb ofans the ev olution of their design has b een charac-
terized b y a con tin uous increase in engine diameter. This developmen t w as pushed mostly b y fuel-sa ving
considerations: larger engines op erate with lo w er exhaust jet v elo cit y , so less kinetic energy is imparted
to the air for the same amoun t of energy effectiv ely used for propulsion (higher propulsiv e efficiency).
Moreo v er, the lo w jet v elo city is associated with reduced jet noise, b ecause of smaller v elo city gradien ts
and lo w er turbulence pro duction within the shear la y er b etw een the jet and the am bien t air. As a result,
increasing engine diameter has impro v ed so far b oth fuel consumption and noise emission.
No w ada ys, the gro wth rate of air traffic and the need for more stringen t noise regulations put pressure
on the civil aircraft industry to prop ose adequate tec hnological solutions. These demands can only b e
cop ed with b y steadily improving the aerodynamics and acoustics of the engines. Ho w ev er, the engine
size cannot b e muc h further increased without unacceptable w eigh t and drag p enalties resulting from the
enlarged nacelle. The strong reduction in jet noise has also b olstered the imp ortance of fan noise. As
a consequence, alternativ e approac hes m ust b e considered, which in v olv e either new engine concepts, or
other design parameters prop er to the geometry and op erating conditions of the fan. Because mo dern
fans ha v e reac hed a v ery high aero dynamic efficiency w ell ab o v e 90%, impro ving the acoustics without
degrading the aero dynamics is a really challenging task.
The main ob jective of the presen t work is to pro vide a platform for the assessmen t and reduction of
fan noise during the pre-design dev elopmen t phase of an engine. Since noise and fuel consumption ha ve
b ecome uncorrelated, and sometimes ev en comp eting asp ects, a m ulti-disciplinary approac h called the
’design-to-noise’ can in tegrate the acoustic assessmen t at an early phase of the design pro cess and result
in so-called ’quiet-b y-design’ fans. The main ob jective is articulated b y the following four aspects that
m ust b e addressed b y the platform:
• Comparison of inno v ative fan concepts : alternativ e concepts to the con v en tional rotor–stator-
stage turb ofan (TF) are currentl y considered by the industry as candidates for the future lo w-
emission engines. These are the coun ter-rotating turb ofan (CR TF) comp osed of a ducted rotor–
rotor-stage, and the coun ter-rotating op en rotors (CR OR) comp osed of t w o unducted large-diameter
prop ellers. These concepts represen t a technological c hallenge, esp ecially from the acoustic p oin t
of view.
• Iden tification of acoustically relev ant design parameters : a parametric description of the
ph ysical pro cesses is necessary at a lev el sufficien tly detailed to capture the essen tial aero dynamics-
acoustics trade-offs. The iden tification of key driv ers relies on the quan titativ e assessmen t of the
relativ e imp ortance of the differen t parameters.
• Supp ort of an analysis comp etence : the correct assessmen t of fan noise sources and their
relativ e imp ortance dep ending on the design and op erating conditions is still a strongly debated

Chapter 1. In tro duction 16
topic. The present approac h should b e p edagogical and help the user, engineer or researcher to get
more insigh t. Computational simulations suc h as CFD and CAA pro vide detailed and sometimes
v ery accurate results, but their in terpretation is difficult as they basically con tain v ast and un treated
datasets. Th us, they are often used as blac k-b o x mo dels inside whic h one cannot easily lo ok and
get insigh t in to the driving mec hanisms.
• The long-term p ersp ectiv e: represen ted b y explorativ e studies p erformed in order to iden tify
whic h configurations within the design space ha v e the b est p otential and are w orth b eing refined
and in v estigated with a more demanding and accurate approac h (for example CFD/CAA or mea-
suremen ts).
1.2 Challenges
There are t w o ma jor challenging aspects asso ciated with the ob jectiv es detailed in the previous section.
The first is the abilit y to pro vide a comparison b etw een differen t concepts and configurations whic h is
made on a fair basis: for example, the noise assessment of t w o engines has to b e carried out for the same
engine p erformance (required thrust, optimal efficiency , etc.). The second c hallenge is computation time:
the aero dynamic and acoustic calculations ha v e to b e p erformed within a reasonable amoun t of time for
parameter studies to co v er a sufficien tly large design space. More sp ecifically , the p oin ts related to these
c hallenges are:
• F an noise strongly dep ends on fan aero dynamics and should not b e interpreted alone: noise predic-
tion do es not only rely on go o d acoustic mo dels but also on a sensible description of aero dynamic
p erformance, of the steady and unsteady flow inside the fan.
• The n um b er of parameters and their range of v ariation is large: con v en tional turb ofans hav e high-
solidit y blades whereas op en rotors hav e lo w-solidit y blades, and the v ariations in pressure ratio
and relativ e flo w Mac h n um b er ma y extend from the incompressible to the sup ersonic flo w regimes.
• Some of the ph ysical problems considered are differen t in nature: the generation of tonal and
broadband noise, the generation of noise inside a duct or in the free field.
1.3 The analytical approac h
Giv en the ob jectives and the c hallenges linked to them, the prediction based on an analytical approac h
has b een c hosen. Here, the w ord ’analytical’ has to b e understo o d as opp osed to b oth ’empirical’ and
’computational’. An in tro duction to the differen t approac hes describ ed b y eac h of those adjectiv es is
giv en b y En via [1].
An empirical mo del is usually based on statistical correlations of high-lev el (global) parameters deriv ed
from exp erimen tal data. As the description of the problem is not causal, the driving mechanisms cannot
b e iden tified more than on a v ery basic lev el. F urthermore, the domain of v alidit y of the mo del is limited
and extrap olation is not p ossible a priori.
On the other side, a computational mo del is based on a low-lev el (lo cal) description of the ph ysics,
often in form of a set of differen tial equations (suc h as the Na vier-Stok es equations). The c hallenge is
then concen trated on the correct n umerical implemen tation of the mathematical description and on the
data p ost-pro cessing. Due to the lo w-lev el description, gaining ph ysical insigh t is difficult.
The analytical approac h is lo cated somewhere in b et w een: the mo dels are mathematically simple
enough to a v oid ma jor numerical problems, still they remain causal and pro vide a sufficien tly detailed
description for design studies to b e p erformed. The c hallenge of this approac h is to estimate the highest
p ossible degree of simplification allo w ed to still address the problem prop erly . The follo wing items
highligh t the main adv an tages of analytical mo dels:

Chapter 1. In tro duction 17
• Mathematical form ula of analytical mo dels already giv e indications ab out the driving mechanisms
b efore the calculation has b een p erformed. Moreo v er, they pro vide the connection b et w een the
acoustic and the aero dynamic parameters.
• As all mo dels are relativ ely simple, they can b e easily in tegrated in a single prediction platform
and more atten tion can b e paid to the deriv ation of unified form ulations of the differen t problems
considered.
• A t last, simple mo dels can b e implemen ted in few er lines of co de and their b eha viour can b e b etter
con trolled o v er a wide range of applications: this ensures fast and robust calculation routines.
1.4 State of the art on prediction mo dels
The question of fan noise prediction b y means of analytical mo dels has b een addressed by the industry
and other researc h en tities, and some w ere comm unicated in conference pap ers or public rep orts. W e will
consider t w o categories of prediction metho ds to illustrate the current state of the art.
1) The fan noise prediction to ol dev elop ed at NASA by Heidmann [2] (later impro v ed b y Kon tos,
Janardan and Glieb e [3] or adapted to small engines b y Hough and W eir [4]) is based on empirical cor-
relations deriv ed from static tests [5] on a series of eigh t differen t full-scale fans, all ha ving the same
rotor–stator arc hitecture. This mo del is part of the Aircraft Noise Prediction Program ’ANOPP’ dev el-
op ed b y NASA. Due to the relativ ely few and simple engine data required as input, this public-domain
prediction mo del is no w widely used among the industry and the universities.
Heidmann observ ed that the measured sound p o w er appro ximately scaled with the shaft p ow er
P shaf t = Q · ∆ T t , but he noticed that fans with a higher pressure ratio (lo w er mass flo w rate Q and
higher total temp erature rise ∆ T t ) emit more noise. Thus, he proposed the following scaling la w for the
o v erall sound p o w er: P sound = K · Q · ∆ T 2
t , where K is an empirical co efficient.
This scaling la w is a statistical correlation in v olving v ery high-lev el parameters of the engine. Of
course, the temp erature rise across the fan cannot b e regarded as the source of noise b y itself. Ho w ev er,
this c hoice is justified on the ground that the temp erature rise correlates well with the flo w v elo cities and
their fluctuations, whic h are the main ph ysical cause for fan noise. The empirical co efficien t K sets the
absolute noise lev el and also con tains v arious empirical corrections to accoun t for the blade count, the
axial distance b et w een the rotor and the stator, and other effects. The ranking of noise sources is p os-
sible to some exten t b ecause a distinction is made b etw een the different noise components (for example
tonal and broadband comp onen ts, rear-arc and forw ard-arc emissions) but they are neither related to
the ph ysical noise mec hanisms nor to the noise emitters, the rotor and the stator. Moreo v er, this mo del
cannot b e applied to coun ter-rotating or unducted concepts without neglecting essential aspects of the
problem.
2) The second category of prediction metho ds is formed b y the more theoretical approac hes that ha v e
b een dev elop ed as an alternativ e to Heidmann’s mo del. Some of the prediction co des and research teams
in v olv ed in this field in the USA and in Europ e are listed b elo w.
Here is a list of the co des developed in the USA:
• Co des used b y NASA and developed in collab oration with the industry or univ ersities. This list
can b e found on the w ebsites of the Glenn and Langley research cen ters:
– V072 and RSI deal with tonal and broadband mo dels for dip ole-t yp e in teraction noise. V072
fo cuses on tonal noise and is one the first analytical fan noise prediction mo dels developed at
NASA. It w as dev elop ed b y V en tres et al. [6] and enhanced b y Edmane En via [7]. It includes
a t w o-dimensional linear-cascade resp onse mo del to accoun t for scattering b y neigh b ouring
blades during the noise generation pro cess. The RSI to ol dev elop ed b y En via [8] extends the
capabilities of V072 to broadband noise prediction.

Chapter 1. In tro duction 18
– TF ANS/BF ANS are resp ectiv ely tonal and broadband cascade noise mo del used b y Pratt &
Whitney . The co de TF ANS was written b y Da vid T op ol [9, 10, 11] from Pratt & Whitney .
BF ANS is based on the w ork b y Donald Hanson [12, 13, 14] from Pratt & Whitney and
Stew art Glegg from the Florida A tlan tic Univ ersit y [15, 16, 17]. An ov erview of BF ANS is
presen ted b y Bruce Morin [18]. The to ol has the sp ecificit y to accoun t for the cascade resp onse
to a three-dimensional gust, and to mo del the transmission and reflection of acoustic mo des
through a blade ro w (and the mo de trapping phenomenon).
– F anBB is the broadband prediction co de dev elop ed b y Ramani Mani and Philip Glieb e [19, 20,
21, 22, 23] from General Electric. F anBB includes a mo del for the quadrup ole-t yp e in teraction
of incoming turbulence with the p oten tial fields of blades and a mo del for anisotropic large-
scale incoming disturbances.
– W OBBLE is a prop eller noise prediction to ol based on the w ork b y Hanson [24, 25, 26].
– LINPR OP/QPR OP is also dedicated to the prediction of dip ole and quadrup ole sources for
prop ellers; it is based on the work b y En via [27].
– ASSPIN w as dev elop ed b y F eri F arassat and Kenneth Brentner [28, 29, 30] in Langley to
pro vide a time-domain prediction of prop ellers and op en rotor noise. This is the only to ol
from the list giv en here that do es not work in the frequency domain.
• Co des dev elop ed b y American univ ersities:
– Broadband noise mo dels at the Universit y of Notre Dame: ann ular cascade resp onse mo del su-
p ervised b y Hafiz A tassi [31, 32, 33] in collab oration with Pratt & Whitney , and an anisotropic
turbulence mo del b y Da vid Stephens and Scott Morris [34].
Here is a non-exhaustiv e list of the analytical w ork p erformed in Europ e:
• F rance: a series of airfoil, prop eller and fan noise prediction mo dels w as initiated and sup ervised b y
Mic hel Roger from Ecole Cen trale de Ly on. He published in collab oration with St´ ephane Moreau on
airfoil noise and automotiv e fans [35, 36], with Y annic k Rozen b erg on trailing edge noise [37], with
H ´ el ` ene P osson on cascade resp onse [38] (w ork supp orted b y Snecma), and with Arn ulfo Carazo on
op en rotor tonal in teraction noise [39] (w ork supp orted b y Airbus). P art of this set of mo dels has
b een applied in noise assessmen t and optimization studies, for example b y P agano et al. at CIRA
(Italian aerospace researc h cen ter) [40], Marinus et al. at the V on Karman Institute in Brussels [41],
or Magne et al. at the Univ ersit y of Sherbro oke in Qu ´ eb ec [42].
• England: Rolls-Ro yce has supp orted a num b er of studies at the follo wing univ ersities:
– Cam bridge Univ ersit y: Da vid Crigh ton, Nigel P eak e w orking with An thon y P arry fo cused their
studies on the tonal self- and in teraction-noise of prop ellers and op en rotors [43, 44, 45, 46].
– Institute of Sound and Vibration in Southampton: Christopher Morfey provided essen tial
con tributions [47, 48, 49] in the 1970’s to the theory of fan noise generation. More recen tly ,
Phil Joseph, Vincen t Blandeau, and Michael Kingan [50, 51, 52, 53, 54] focused on broadband
noise of fans and op en rotors.
• The Netherlands: work on propeller tonal noise mo dels b y Sc h ulten [55] and Brou w er [56] from
NLR (national Dutc h aerospace researc h cen ter).
• German y: Klaus Heinig from MTU published a cascade mo del for compressor noise prediction [57].
These mo dels all ha v e in common a strong theoretical fundament and rely , at least for the acoustic
part of the prediction, very w eakly on empirical co efficien ts, unlik e Heidmann’s mo del. They are basically
exact analytical solutions of a simplified problem, whic h most of the time addresses one sp ecific noise
source or a limited n um b er of sources. As a result, the form ulation and program implemen tation of
eac h mo del are strongly fo cused on the problem considered and cannot b e extended easily to other noise

Chapter 1. In tro duction 19
sources or to a wider range of applications. F or example, few mo dels address the problem of tonal and
broadband noise sim ultaneously and with a similar form ulation, and no mo del deals with the ducted und
unducted problems at the same time.
In order to get the ’global picture’ (for example a source ranking) or compare differen t concepts, it is
necessary to run and analyse ev ery mo del separately , and finally cross-compare their results, whic h can
b e a delicate task due to the v ariet y of input and output data prop er to eac h mo del.
Moreo v er, these analytical mo dels cannot b e run on their o wn. They need to b e coupled to flo w
mo dels or computational solv ers since the aero dynamic input data required are m uc h more detailed than
those needed b y Heidmann’s empirical mo del. In the author’s view, these more adv anced analytical
acoustic mo dels only partly fit in to the philosoph y presen ted b efore. Esp ecially the dep endence on
external aero dynamic to ols and the need for implemen ting sev eral mo dels to obtain the o v erall picture
of fan noise generation is regarded as an obstacle to future noise assessmen t and also an obstacle to a
didactic approac h.
1.5 New prediction to ol: PropNoise
A no v el to ol for the prediction of noise pro duced b y rotating or non-rotating blade ro ws has b een dev elop ed
since 2008 [58, 59, 60, 61, 62, 63, 64, 65, 66]. It w as named ’PropNoise’ standing for Propulsion Noise.
Its description and application is the ob ject of the presen t w ork.
A tten tion has b een paid to the establishmen t of a unified approac h with strong connections b etw een
the aero dynamics and the acoustics, and a comprehensiv e description of the differen t fan noise sources for
assessing correctly the trade-offs the engineer ma y b e confronted to during the ’design-to-noise’ pro cess.
Mo derately detailed but equally accurate mo dels co v ering the main asp ects of fan noise ha v e b een prefered
to highly accurate mo dels co v ering a smaller range of asp ects. PropNoise forms a framew ork in tegrating
v arious aero dynamic and acoustic mo dels and is co ded in form of Matlab routines. The large ma jority
of the mo dels is inspired from already existing w ork av ailable in the literature but reformulated in a w a y
to pro vide homogeneit y and con tin uit y b et w een the differen t fields.
The structure of the to ol is presen ted in Figure 1.1. The three mo dules for fan design, steady ,
and unsteady aero dynamics are dedicated to the generation of input data for the acoustic calculations.
These mo dules are based on a meanline approac h, whic h means that the aero dynamic calculations are
p erformed at a single radial p osition considered to b e represen tativ e of the complete flo w. The v alidit y
of this assumption will b e discussed later. The acoustic mo dels are based on a radial-strip approach:
the resp onse of the blades to an excitation is calculated for eac h radial strip as if the problem was
t w o-dimensional, ho w ev er the emission of sound w a v es b y the differen t strips and their in terferences are
calculated with a three-dimensional mo del. An in termediate mo dule ensures the radial extrap olation of
the meanline aero dynamic data required b y the acoustic mo dels.

Chapter 1. In tro duction 20
Figure 1.1: Mo dular structure of the prediction to ol ’PropNoise’
This structure refers to the ’stand-alone’ mo de of the to ol, in which no other program needs to b e
coupled to ’PropNoise’. The acoustic-relev an t aero dynamic input data are generated inside the to ol. It
is also p ossible to disactiv ate some of the in ternal aero dynamic calculations and to pro vide the acoustic
mo dule with ﬂo w data coming from another to ol, as in the case of a RANS-informed noise prediction [67].
Ho w ev er, this asp ect is b ey ond the scop e of the presen t w ork.
The follo wing c hapters of this do cumen t will presen t eac h of the mo dules con tained in the ’stand-alone’
v ersion of the program.

Chapter 2
Steady aero dynamics
2.1 Motiv ation and approac h
In the field of aeroacoustics, the kno wledge of the steady and unsteady flo w quan tities is necessary to
determine the sound sources and the conditions of sound propagation that will affect the noise emission
p erceiv ed b y an observ er. The purp ose of this c hapter is to presen t a set of analytical mo dels for the
steady aero dynamic quan tities relev an t for aero-engine fan noise and to precise ho w these quan tities are
related to fan geometry and op erating conditions.
The mo dels for steady aero dynamics prediction pro vide t w o differen t types of information: the first
concerns the aero dynamic p erformance of the fan, in terms of mass flo w rate, pressure rise and efficiency ,
and more globally engine thrust and fuel consumption; the second type of output data delivered b y the
mo dels concerns more refined quan tities suc h as b oundary la y er thic kness and flo w v elo cities around
the blades. These quan tities are the source of flo w p erturbations within the engine, whic h propagate to
neigh b ouring blade ro ws, in teract with them, and this finally results in sound emission.
Bey ond the necessit y of generating the appropriate input data for calculating flo w p erturbations and
ev en tually noise, the mo dule for steady aero dynamics aims at establishing a link b et w een the aero dynamic
p erformance of an engine and its noise emission. A clear view on the relation b etw een those t w o essen tial
asp ects ma y help the engineer to iden tify suited configurations v ery early during the design pro cess of
an engine.
The approac h adopted here relies on simple analytical mo dels, which presen t the adv an tage of b eing fast,
robust, and easier to understand than more accurate to ols. A tten tion has b een paid to ph ysically-based
descriptiv e mo dels, whic h are able to pro vide some lev el of understanding. No claim is made here to
comp ete with mo dern flo w solv ers lik e MISES (coupled Euler-Boundary La y er cascade co de) or TRA CE
(DLR in-house co de, full 3D RANS solver). Suc h to ols pro vide no w ada ys reliable predictions for most
fan or compressor applications and are extensiv ely used among the industry to impro v e and design new
comp onen ts. Ho w ev er, the massiv e use of CFD is iden tified b y Cumpst y [68] as a p ossible cause for the
lac k of insigh t asso ciated with mo dern fan designs.
The use of simple descriptiv e mo dels is not new. The dev elopmen t of analytical fan prediction mo dels
based on empirical correlations exp erienced a significan t gro wth up to the 1980’s and suc h mo dels formed
then the basis for most designs, b efore they b ecame sup erseded by CFD. The aero-engine man ufacturers
dev elop ed eac h their o wn set of correlations, lik e those of Sw an at Bo eing [69], Grieb at MTU [70],
Ko c h & Smith at General Electric [71], or Miller, W asdell and W right at Rolls-Ro yce [72, 73]. In a
con tin uous effort to pro vide more accurate predictions, the correlations b ecame more complex while
sim ultaneously losing their link to first-principles aero dynamics. W e in tend to a v oid this fla w b y stic king
to fundamen tal theoretical studies suc h as the pioneering w ork of Lieblein [74], completed b y the more
general considerations of Den ton [75].

Chapter 2. Steady aero dynamics 22
2.2 Definition of parameters and assumptions
A n um b er of simplifications are made in order to k eep the computations rapid and robust, and to limit
the use of empirical correlations. These assumptions are listed b elo w:
• Meanline approac h: all quantities are calculated at a giv en radial station, sp ecified b y the user in
p ercen t of the duct heigh t. This radial station is assumed to b e representativ e of the fan op erating
condition and of its p erformance. It is usually referred to as the meanline radius (see Fig. 2.1).
The meanline approac h do es not allo w accoun ting for the strong radial v ariations in geometry and
flo w t ypical of fans with a lo w h ub-to-tip ratio, but w e will see on an application that the ma jor
features of the aero dynamic fan map can still b e correctly repro duced.
• Steady flo w: only the time-a veraged part of the flo w is calculated. At first, the flo w is calculated
in the relativ e reference frame (rotating with the blade ro w) and then con v erted in to the fixed
frame. Flo w p erturbations suc h as w ak es and p oten tial fields are treated in a differen t mo dule and
will b e describ ed in c hapter 4 dedicated to unsteady aero dynamics. The steady flow quan tities far
upstream and do wnstream of a blade ro w are considered axisymmetric (constan t in the circumfer-
en tial direction). As a result, the impact of the circumferen tial v ariation of incidence on rotor blade
p erformance (as migh t b e the case in the presence of a fan inflo w distortion) cannot b e addressed
b y the mo dels.
• No engine core: only the b ypass flo w of the fan is considered. Hence, the bypasss ratio of the engine
cannot b e directly determined from the geometry but ma y b e estimated at the design p oin t based
on the fan pressure ratio (see the next c hapter 3 on engine and fan design for the details).
• Axial-flo w compressor: the radial comp onen t of flo w v elo cities is neglected, as is the v ariation of
the streamline radius. These assumptions are not suited for radial compressors. The aero dynamic
influence of blade sw eep and lean is not considered.
• The axial v elo cit y comp onen t m ust b e subsonic ev erywhere: M x < 1. Sup ersonic business jets and
compressors with sup ersonic axial inflo w cannot therefore b e treated.
• No bleed air, no blade co oling: the mass of airflo w en tering eac h comp onen t is en tirely conserv ed at
the outflo w as no air extraction system is mo delled. Also, no blade co oling system is considered for
the compressors. This means that the total temp erature in the relativ e reference frame is constan t
across a blade ro w.
• The outflo w regime of eac h of the engine comp onen ts m ust b e subsonic in the rotating reference
frame link ed to the blade ro w. This condition is satisfied for most compressors, except for sup ersonic
lo w-solidit y prop ellers or axial-flo w rotors with an extremely lo w degree of reaction.
• The fan p erformance is calculated based on the blade profile loss (whic h is the sum of sho ck loss
and b oundary la y er loss) and the endw all loss (whic h is the sum of the ann ulus loss due to endw all
friction, and secondary flow loss). F or unducted configurations, the endw all loss mo del has to b e
replaced b y a tip v ortex mo del.
• Flo w conditions at whic h blade stall or c hoking within the blade passage is detected are not p er-
mitted. The routines return an error status. An exception is made for the exhaust nozzle, whic h
ma y op erate in c hok ed condition.
• The thermo dynamic prop erties of the air are considered constan t: the sp ecific gas constan t is
r = 287 . 04 J/K/kg, the heat capacit y ratio is γ = 1 . 4, and the sp ecific heat capacity is c p = γ r
γ − 1 =
1004 . 6 J/K/kg.
The follo wing part of the presen t paragraph will no w in tro duce some imp ortan t quan tities related to
the mean flo w around a blade ro w. Figure 2.2 presen ts the definition of flo w v elo cities and flo w angles

Chapter 2 . S teady a ero dynamics 23
fr ont  ro to r re a r roto r (o r sta to r )
jet
Figure 2.1: Simpliﬁed engine mo del [76]
considered in the rotating frame of reference link ed to the blades. The subscripts A and B refer to ﬂo w
conditions upstream and do wnstream of the blade ro w, resp ectiv ely . The quan tities W SS and W PS
denote the blade surface v elo cities on the suction and pressure side, resp ectiv ely . They will b e deﬁned
more precisely later in this c hapter. The quantities s , h , i and δ refer to the blade spacing, outﬂow
stream tub e heigh t, blade incidence, and deviation, resp ectiv ely .
Figure 2.2: Flo w angles and v elo cities in the relative frame link ed to the blade row

Chapter 2. Steady aero dynamics 24
As no air extraction/injection o ccurs within the engine, the mass flo w rate Q is conserv ed b et w een
p ositions A and B:
Q B = Q A (2.1)
Similarly , as the blades are not co oled, the flo w within the blade passage is considered adiabatic and
the total temp erature T t is conserv ed b et w een p ositions A and B in the rotating frame. This is v erified
b oth for rotors and stators as the radius change of the streamlines is neglected.
T tB = T tA (2.2)
The incidence is defined as the difference b et w een the inflo w angle (expressed in the rotating frame)
and the metal angle of the blade leading edge.
i = β A − χ LE
W e will define later the so-called design incidence i des , whic h is comprised b et w een the minim um-loss
incidence and the incidence of maxim um lift-to-drag ratio. The deviation angle is the difference b et w een
the outflo w angle and the blade metal angle at the trailing edge. It has a significan t effect on the
p erformance of a blade ro w at off-design incidence.
δ = β B − χ T E
The prediction of this parameter has b een sub ject to n umerous studies, whic h can b e asso ciated his-
torically to t w o differen t approac hes: the german sc ho ol, represen ted b y the w ork of W einig [77] and
Sc holz [78], dev elop ed exact theoretical mo dels to predict the lift of a cascade of arbitrary solidit y . Ho w-
ev er, the assumptions made are strong (in viscid flow, circular arc profiles, incompressible flo w). The
anglo-american sc ho ol (see Carter [79, 80] and Lieblein [81, 74, 82]) prop osed an empirical approach
based on correlations gained from lo w-sp eed cascade exp erimen ts. These correlations inheren tly feature
a lac k of univ ersalit y (they are sp ecific to certain t yp es of NA CA profiles and are not v alid for the lo w-
solidit y blade ro ws of prop ellers and op en rotors) but they were v ery p opular among the engine design
comm unit y as long as standard blade profiles w ere used. Extensions dev elop ed for transonic compres-
sors (see C ¸ etin [83]) further reduced the gap b et w een prediction and measuremen ts but they w ere not
accompanied with an increase of insigh t. F or the sake of k eeping the routines simple, fast and robust, we
c ho ose here to consider the deviation angle to b e a linear function of the incidence:
( δ ( i = i des )=0
∂ δ
∂ i →
σ → 0 1 − π σ cos χ ⇒ δ = i − i des
1 + π σ cos χ
F or simplicit y the deviation at design conditions is neglected. The presen t mo del fulfills tw o w ell-kno wn
results: at large solidities, the deviation tends to zero, whereas the v ariations of deviation and incidence
are iden tical at v anishing solidit y . The outflow angle β B can be calculated from the deviation and the
outflo w v elo cit y magnitude W B can b e directly obtained from the con tin uit y equation Eq.(2.1).
Lakshminara y ana [84] in tro duced the follo wing mean quan tities (a v erage b et w een upstream and do wn-
stream p ositions) to determine the lift and drag forces on cascade blades:
W m ≡ W A + W B
2 (v ector a v eraging)
tan β m ≡ tan β A + tan β B
2
ρ m ≡ 2 ρ A ρ B
ρ A + ρ B
(2.3)
An imp ortan t parameter for the design of fans is the so-called meridional-velocity–densit y ratio,
denoted MVDR. As w e consider axial-flo w fans without c hange of the radius through the compression

Chapter 2. Steady aero dynamics 25
pro cess, the meridional v elo cit y is iden tical to the axial v elo city , and the MVDR is equiv alen t to the
A VDR (axial-v elo city–densit y ratio). So w e ha v e:
M V D R = AV D R ≡ ρ B W xB
ρ A W xA
= A A
A B
(2.4)
If the flo w blo c k age due to endw all b oundary lay ers is neglected, the MVDR is inv ersely prop ortional to
the ratio A B / A A of the duct cross-sectional areas (duct con traction) across the blade ro w.
2.3 Relation b et w een pressure rise, flo w turning and lift
According to the Euler turb omac hinery equation (more details can b e found in the b o oks b y Laksmi-
nara y ana [84], Cumpst y [85], and Grieb [86]), the change in angular momen tum caused b y a rotor is
related to the energy receiv ed b y the fluid. The sp ecific total en thalp y rise (expressed in the fixed frame)
across an axial-flo w rotor with no meanline-radius c hange is giv en b y:
∆ H t = U · ∆ W t (2.5)
The v ariation of the circumferen tial flo w v elo cit y across the blade ro w is ∆ W t = W tA − W tB and is usually
referred to as flo w turning; its v alue is iden tical in the rotating and in the fixed reference frames. U is the
meanline circumferen tial sp eed of the rotor blades. Alternativ ely , the enthalp y rise in the absolute frame
can b e expressed in terms of the pressure ratio P R ac hiev ed b y the rotor and its isen tropic efficiency η ise :
∆ H t = c p T tA · P R γ − 1
γ − 1
η ise
The pressure ratio P R is defined as the ratio of the outflo w to the inflo w total pressures tak en in the fixed
frame. The inflo w total temp erature T tA is here expressed in the fixed reference frame, to o. F rom the last
t w o equations, w e observ e that the pressure rise accross an axial-flow compressor rotor is generated b y
turning the flo w (non-zero ∆ W t ). This is usually ac hiev ed in t w o w a ys, dep ending on the Mac h n um b er
domain for whic h the compressor is designed. In the case of subsonic compressors, the duct con tours
imp ose at the design p oin t a nearly constan t axial v elo cit y across the blade ro w (this is done to limit
diffusion and a v oid flo w separation), hence the turning can only b e obtained b y deflecting the flo w in
the rotating frame. In sup ersonic compressors, the flow angles are hardly modified in the rotating frame
b ecause the blades are thin and straigh t to a v oid the formation of strong sho cks responsible for high
losses. The turning is obtained through the reduction of the axial v elo cit y and the strong deceleration of
the flo w from sup ersonic to subsonic conditions induced b y the sho c k system. Figure 2.3 illustrates the
w orking principles of the lo w-sp eed subsonic and sup ersonic axial-flo w compressors.

Chapter 2. Steady aero dynamics 26
Figure 2.3: W orking principle of a subsonic and sup ersonic rotor blade ro w
W e will no w establish a relation b et w een the ﬂo w turning and the lift co eﬃcien t of the blades.
Lakshminara y ana [84] has sho wn that the sectional lift L can b e expressed in terms of circulation Γ:
ρ m W m Γ= L = 1
2 ρ m W 2
m C L · c
Moreo v er, the circulation around an airfoil can b e related to the ﬂo w turning of the blade ro w:
Γ ≡  Wd  =Δ W t · s (2.6)
W e obtain the follo wing relation b et w een ﬂo w turning and lift co eﬃcien t:
C L = 2Δ W t
σW m
= τ cos β m
σ ,w h e r e τ ≡ 2Δ W t
W x
= 2(tan β A − tan β B ) (2.7)
This equation indicates that the c hange in ﬂo w angles (ﬂo w turning co eﬃcient) is closely related to the
lift co eﬃcien t through the quan tit y σ = c/s called the solidit y of the cascade, whic h is the ratio of the
blade c hord c to the blade spacing s . Because the lift co eﬃcient m ust assume ﬁnite v alues, there can
b e no ﬂo w turning for an isolated airfoil (whic h is equiv alen t to a blade ro w of v anishing solidit y). F or
a blade ro w of increasing solidit y , the ﬂo w turning tends to b e solely determined by the metal angles of
the blades, so the lift co eﬃcient con tinuously decreases and tends to zero. This fundamen tal b eha viour
is describ ed in more details b y the analytical mo del of Sc holz [78] including an estimation of the in viscid
deviation. Note that the isolated airfoil can indeed generate ﬂo w turning if w e consider a ﬁnite-span wing
with tip v ortices; in that case the t w o-dimensional ﬂo w assumption is not sastisﬁed.
2.4 Deﬁnition of design and oﬀ-design conditions
A blade ro w op erates at design conditions if its incidence assumes a particular v alue, named the design
incidence. In that case, the ﬂo w o v ersp eeds around the blade (see Eq.(2.8)) are determined b y the global
lift co eﬃcien t of the airfoil without consideration of lo cal accelerations resp onsible for a loss p enalt y . This
is basically a case of near-minim um loss or near-maxim um eﬃciency for the blade. In the lo w subsonic
regime, the design incidence is zero. In the high subsonic and sup ersonic regimes, the design incidence is

Chapter 2. Steady aero dynamics 27
larger than the c hoking incidence and this has to b e mo delled b y the follo wing approac h. The c hoking
incidence primarily dep ends on the throat width o of the blade passage and on the inflo w Mac h n um b er:
o = s cos(0 . 8 χ LE + 0 . 2 χ T E ) − t
i chok = arccos  o/s
α cr  − χ LE , where α cr ≡ Q r ( M A )
Q r (1) and Q r ≡ Q · √ γ r T t
P t · A
The quan tit y Q r is the reduced mass flo w, it is solely a function of the Mac h n um b er. In subsonic flo w,
c hoking o ccurs if the throat area ratio is smaller than a critical v alue and the calculation is interrupted
in that case. With sup ersonic inflo w, the calculation is brok en up if the throat area ratio decreases b elow
unit y . In realit y , the c hoking limiting condition in sup ersonic flo w is defined b y an ev en more stringen t
case of minim um incidence called the unique incidence, whic h further reduces the range of op eration
as the Mac h n um b er is increased. The unique incidence is explained more extensiv ely b y F reeman and
Cumpst y [87], and Levine [88]. When a cascade op erates at unique incidence, it is called started and
features a sho c k system attached to the blade leading edge. Op eration at larger incidence is called
unstarted and is c haracterized b y a detac hed b o w sho c k upstream of the blade leading edge.
The design incidence is defined as:
i des = max  0 , i chok + i 0
2 + ∆ i 
The reference incidence i 0 denotes the incidence at whic h there is no flo w area con traction. The empirical
constan t ∆ i sp ecifies the distance b et w een design and c hoking in terms of incidence angle at sup ersonic
flo w conditions, a v alue of 2 degrees giv es a go o d matc h with actual fan maps.
i 0 = arccos ( o/s ) − χ LE
The stalling incidence i stall giv es a rough estimate of incidence at whic h stall is detected, it is calculated
b y:
i stall = max( i des , K of f des (1 + π σ cos χ ))
The empirical constan t K of f des has a v alue around 8 degrees.
2.5 Distribution of flo w v elo cities around the blades
As describ ed b y Den ton [75] the pro duction of loss is closely related to the distribution of flo w v elo cities
around the blades. On each side of the blade (pressure and suction sides), we will consider a single
c hordwise-a v eraged v alue of the v elo cit y , that is assumed to b e represen tativ e of the losses. The c hoice of
a scalar rather than a full c hordwise distribution w as adopted b y Lieblein [74] and Ko c h and Smith [71],
and is supp orted b y the fact that the en trop y pro duction is prop ortional to the third p ow er of velocity [75],
therefore is it essen tially determined b y the region of maxim um v elo city . W e write this c hordwise-av erage
v elo cit y in a form similar to that prop osed b y Ko c h and Smith [71] but extended to off-design incidence
conditions:
W = W A ·  1 + f ( t
h , M x ) ± 1
4 C L + ∆ of f des  (2.8)
The second term in the paren thesis corresp onds to the acceleration due to the flo w area con traction in
the blade passage, it dep ends on the chordwise-a verage blade thic kness and on the axial Mac h n um b er.
The third term represen ts the o v ersp eed due to lift: it is p ositive on the suction side and negativ e on the
pressure side. The last term is attributable to off-design incidence and proximit y to c hoking. A t design
conditions (zero incidence at lo w Mac h n um b ers, small p ositiv e v alue at high mac h n um b er dep ending
on throat width), the relation of Eq.(2.6) is retriev ed b y taking the difference b etw een the suction and
pressure side v elo cities:
Γ ≡ I W d` =  W S S − W P S  · c = 1
2 W m C L · c

Chapter 2. Steady aero dynamics 28
This equalit y is not exactly satisfied at off-design conditions b ecause ∆ of f des assumes differen t v alues on
the pressure side and suction of the blades. Physically , the off-design v elo cit y distribution departs from
the optimal distribution of v elo cities established at zero incidence, and is resp onsible for an additional
loss p enalt y . The off-design ov ersp eed factor is giv en b y:
∆ of f des = 




0 . 5 i − i des
i stall − i des , on the suction side if i>i des
max  0 . 5 i des − i
i stall − i des , 1 . 5 i des − i
i des − i chok  , on the pressure side if i<i des
0, otherwise
(2.9)
2.6 Losses
2.6.1 Relation b et w een loss, drag and en trop y pro duction
The loss co efficien t characterising the performance of a rotating blade row is usually defined as a nor-
malized total pressure loss:
ω ≡ ∆ P t
1
2 ρ m W 2
m
where ∆ P t = P tA − P tB is the difference b etw een the inflo w and outflo w total pressures in the relativ e
reference frame rotating with the blade ro w. This definition is less suited for endw all loss, whic h usually
scales with W 2
x instead of W 2
m . By considering the integral o v er the stream tub e heigh t of the momen tum
equation in c hordwise direction, w e obtain a relation b et w een the total pressure loss and the drag force
applied on the blades (see the deriv ation detailed b y Grieb [86], Cumpsty [85] and Lakshminara y ana [84]):
∆ P t · s cos β m = D = 1
2 ρ m W 2
m C D · c
By com bining these last t w o equations, w e obtain an equation linking the drag co efficien t to the loss
co efficien t:
C D = ω cos β m
σ (2.10)
Similarly to Eq.(2.7) this equation relates a quan tit y usually defined for isolated airfoils, C D , to a quan tit y
t ypical for cascades, ω . If b oth equations (2.7) and (2.10) are com bined, we can observ e that the ratio of
flo w turning to loss co efficien t, which directly affects the isen tropic efficiency of a rotor (this is sho wn in
Eq.(2.21)), is equiv alen t to the w ell-kno wn lift-to-drag ratio c haracterising airfoil p erformance:
C L
C D
= 2 ∆ W t /W x
ω = 2 tan β A − tan β B
ω = τ
ω (2.11)
It should b e noted that this relation has b een obtained after assuming a constan t axial v elo city accross
the blade ro w. This condition is usually satisfied to limit diffusion and a v oid flo w separation.
W e will no w presen t the mo dels used to calculate the losses generated b y a cascade. As stated b y
Cahill [89] and Den ton [75], the loss of total pressure is related to the pro duction of en trop y . The flo w
through a blade ro w considered in the rotating reference frame lo c k ed to the blades can b e regarded as an
adiabatic pro cess (no total temp erature c hange), so the en trop y rise is solely determined b y the v ariation
of the total pressure:
∆ S = γ
γ − 1
∆ T t
T t − ∆ P t
P t
= − ∆ P t
P t
=
1
2 ρ m W 2
m ω
P t
= − log  P tB
P tA  (2.12)
The en trop y denoted S is the usual sp ecific en trop y , non-dimensionalized by the mass flo w rate and b y
the sp ecific gas constan t r . Correlatively , the total pressure loss is link ed to the entrop y rise through:
∆ P t = P tA (1 − e − ∆ S )

Chapter 2. Steady aero dynamics 29
In the presen t mo dels, we consider that the sources of en trop y are represen ted b y the profile and endw all
losses. They are formed by the boundary lay ers developing on the blade surfaces and their con v ected
w ak es, and the sho c ks app earing within the blade passage if the inflo w Mac h n um b er is b ey ond the
critical Mac h n um b er. By sup erimp osition of the en trop y sources, the o v erall en trop y rise is simply giv en
b y the sum of three con tributions:
∆ S = ∆ S B L + ∆ S shock + ∆ S endw all
2.6.2 Blade loading, diffusion factor and stall
A fundamen tal asp ect of compressor aero dynamics is the p ositiv e stream wise pressure gradien t imp osed
to the flo w as it passes through the blade ro ws. This increases the size of the blade boundary lay er
and limits the pressure rise. The pioneering w ork of Lieblein [74] offereda a b etter understanding of
this effect and pro vided a quan titativ e prediction for it. Based on the in tegral momen tum equation of
the b oundary la y er Lieblein deriv ed a relation b etw een the momen tum thic kness at the blade trailing
edge and a parameter called the diffusion factor whic h represen ts the in tensit y of flo w deceleration (or
diffusion) o v er the blade. He applied his relation to exp erimen tal data gained during a series of lo w-sp eed
cascade measuremen ts and he obtained a strikingly go o d matc h in view of the div ersit y of the cascade
geometries considered. W e will use here a mo dified v ersion of Lieblein’s diffusion factor. F or eac h side of
the blade, w e define a diffusion factor D as:
D ≡ W − W B
W A
(2.13)
The classical diffusion factor D F in tro duced b y Lieblein [90] can b e directly retriev ed from Eq.(2.13)
with thin profiles w orking at zero-incidence and in subsonic flo w, b y appro ximating W ≈ W A (1 + 1
4 C L )
in Eq.(2.8) and C L ≈ 2∆ W t
σ W A in Eq.(2.7):
D F = 1 + 1
4 C L − W B
W A
= 1 − W B
W A
+ ∆ W t
2 σ W A
The diffusion factor D prop osed here is a more general version of DF , and is applicable for thic k airfoils
w orking at off-design subsonic or sup ersonic inflo w conditions. T urbines are characterized b y very lo w
v alues for the diffusion factor D . C ompressor blades are t ypically designed for a v alue around 0.5, where
the b oundary la y er is on the v erge of separation, whic h usually starts from the trailing edge of the blades
and mo v es upstreams as the incidence is increased. Increasing the loading well beyond this point leads
to massiv e separation; the pressure rise of the rotor abruptly drops: this is the so-called stall.
The detection of stall is still a v ery delicate problem and there is no criterion accepted univ ersally .
The diffusion factor has b een recognized as an acceptable although v ery inaccurate candidate to detect
stall. The concern of the presen t approac h is to disp ose of a robust criterion that indicates unacceptably
high losses and in terrupts the calculation to a v oid con v ergence problems. This is pro vided b y a stabilit y
criterion whic h is based on the w ork of sev eral authors lik e Greitzer [91] or Da y [92] and w as used b y
F reeman and Cumpst y [87] to predict sup ersonic compressor p erformance. This criterion stipulates that
the blades op erate in the stable domain as long as an increase in incidence at constan t rotation sp eed
(equiv alen tly , a decrease in axial flo w sp eed) leads to an increase in outflo w static pressure (or at least a
decrease small enough). The blade ro w op erates in the stable domain as long as the follo wing criterion
is satisfied: W xA
P tA
∂ P B
∂ W xA
< 0 . 2
During the dev elopmen t and testing of the aero dynamic routines, this stability criterion has pro v en to
b e more reliable than the diffusion factor, esp ecially at high sup ersonic sp eeds where the diffusion factor
do es not correlate w ell with the sho c k loss.

Chapter 2. Steady aero dynamics 30
2.6.3 Loss caused b y b oundary la y ers
The en trop y created in the b oundary la y er dev eloping on the blades is given b y Den ton [75]:
r T t · ∆ S blade = R C d ρW 3 d`
ρW x s
The en trop y rise dep ends on the third p o w er of the external flo w v elo cit y at the edge of the b oundary
la y er. As mentionned b y Denton, this term do es not include the en trop y generation during the mixing
pro cess of the wak e, whic h ma y represen t up to one third of the total loss from compressor blades with an
adv erse pressure gradien t. An alternativ e but equiv alen t approac h b y Lieblein [74] prop oses to correlate
the one-sided momen tum thic kness δ 2 with the diffusion factor defined previously .
δ 2 = C f
2 L · F ( D ), where F ( D ) = 0 . 5+0 . 5 · exp  3 D + 2 D 2  (2.14)
The airfoil semi-p erimeter L can b e appro ximated b y L = c ( χ LE − χ T E ) / 2
sin(( χ LE − χ T E ) / 2) . It roughly corresp onds
to the c hord length but ma y b e significan tly larger for highly cam b ered stator or turbine blades. The
function F is empirical and sev eral v ersions of it w ere prop osed b y Lieblein and other authors. At zero-
pressure gradien t, D = 0 and F = 1, so w e retrieve the flat-plate result δ 2 = C f
2 L . W e prop ose the
follo wing mo del for the c hordwise-a v eraged skin friction co efficien t C f for turbulen t b oundary la y er :
C f = C f 0 ·  10 6
Re  0 . 2
· 1
q 1 + γ − 1
2 M 2
(2.15)
V arious studies suggest a v alue of C f 0 = 0 . 0045 (see Sc hlic h ting [93]). Figure 2.4 presen ts a comparison
of the correlations used to estimate the momen tum thic kness. The quantit y depicted as the loss factor
is basically the term F ( D ) of Eq.(2.14) summed on the suction and pressure sides. The first correlation
w as prop osed b y Lieblein [74]; alternativ e correlations w ere later prop osed b y Grieb [86], W righ t and
Miller [73], Ko enig [94], and Ko c h and Smith [71]. The correlation presen tly implemen ted is v ery close
to the others.
Figure 2.4: Comparison of the loss correlations
The 99% b oundary lay er thic kness δ 0 , the displacemen t thic kness δ 1 and energy thic kness δ 3 are
obtained from the momen tum thic kness through the shap e factors H 12 , H 32 , and H 10 whic h are defined

Chapter 2. Steady aero dynamics 31
as follo ws:
H 12 ≡ δ 1
δ 2
, H 32 ≡ δ 3
δ 2
, H 10 ≡ δ 1
δ 0
(2.16)
W e prop ose a mo del for calculating the shap e factors as functions of the diffusion factor D only . The
dep endence on the Reynolds num b er Re is ignored here, whic h is a reasonable assumption for b oundary
la y er mostly turbulen t o v er the airfoil without laminar bubble separation. It has b een observed from
exp erimen ts that the shap e factors can b e roughly correlated to the intensit y of the pressure gradien t
o v er an airfoil. The follo wing table summarizes the exp erimen tal observ ations:
loading asymptotic v alues for accelerating BL zero-pressure gradien t near separation
( D < 0) ( D = 0) ( D ≈ 0 . 55)
H 12 1 1.3 2.7
H 32 2 1.8 1.5
H 10 0 0.13 0.45
T able 2.1: Shap e factors of a turbulent b oundary la y er for differen t v alues of the pressure gradien t
A mo del resp ecting the exp erimen tal observ ations, and the con tuinit y of the functions is giv en here:
H 12 = 1 + 2 . 795 log  1+0 . 1133 e 3 . 63 D 
H 32 = 4 H 12
3 H 12 − 1
H 10 = 0 . 5  H 12 − 1
2  0 . 7
(2.17)
According to Lieblein [74], the en trop y generated in the b oundary lay er of the blade and its w ak e reads:
∆ S B L = 2 δ 2
h · H w ak e
32 / 2
 1 − H wak e
12 δ 2
h  3 ·
1
2 ρ m W 2
m
P tA
, where H w ak e
12 = 1 . 08 and H w ak e
32 = 1 . 93
2.6.4 Loss caused b y sho c ks
There is no generally accepted metho d to describ e analytically the en trop y generated b y the complex
sho c k system formed in transonic compressors. One of the first attempts b y Miller et al. [95] simplified
the sho c k system to a single normal sho c k for whic h analytical form ula are a v ailable. This mo del fails
to describ e correctly the rapid loss rise with increasing incidence, as p oin ted out b y F reeman and Cump-
st y [87]. They also explained that sho c k loss is inheren tly a com bination of en trop y rise within the sho ck
itself and en trop y created b y the sho c k–b oundary-la y er in teraction. Here, we propose a correlation for
the sho c k-only en trop y rise :
∆ S shock = 0 . 06 · 2 γ
3( γ + 1) 2 ·  max(0 , M 2
max − 1)  2
The pre-sho c k Mac h n um b er on the surface blades is calculated based on the maximum surface v elo cities
from Eq(2.8):
M max = max( M S S , M P S )
2.6.5 Endw all loss
The ph ysics asso ciated with endw all flo ws is highly complex due to three-dimensionnal and unsteady flow.
This is still not w ell understo o d and there is no analytical or empirical mo del recognized as b eing b oth
accurate and reliable on differen t configurations. The endw all loss is usually divided in to t wo components:

Chapter 2. Steady aero dynamics 32
the ann ulus loss due to skin friction on the endw alls and secondary losses including all other loss sources.
An extensiv e review of the v arious mo dels existing is presen ted b y Grieb [86]. The mo del we propose
here is reduced to the ann ulus friction loss. The secondary losses are not mo delled b ecause we focus on
the meanline approac h and do not acccoun t for the tip flo w as a noise source.
dS endw all = 2 C f ·
1
2 W 2
x
r T t
The secondary losses dep end on the lift co efficien t of the blades and on the adv erse pressure gradien t.
It should b e also noted that the mo del do es not accoun t for the influence of blade asp ect ratio. W en-
nerstrom [96] has iden tified it is a ma jor parameter driving endw all loss and the stall limit. The author
kno ws from the literature no analytical or semi-empirical mo del that would include these aspects thor-
oughly . A diffusor analogy as prop osed by Koch [97] migh t b e implemen ted to predict the b oundary la y er
dev elopmen t on the w all. In that case, an essential parameter is the initial v alue of the endwall boundary
la y er thic kness at the b eginning of the diffusion.
2.7 F an p erformance
The follo wing quan tities are commonly used as design parameters to analyze and c haracterize the aero-
dynamic b eha viour of a compressor:
• The first parameter is the stage loading co efficien t ψ . The german definition adopted here differs
from the english one b y a factor 2. In m ultiple-stage compressors, the n um b er of stages required is
directly link ed to this parameter.
• The second parameter φ is called the flo w co efficient. It is generally desirable to maximize this
parameter as it implies a higher mass flo w p er unit area and therefore a more compact engine. The
flo w co efficien t is limited b y c hoking within the blade passage. The flo w co efficient is similar to the
adv ance ratio J defined for unducted prop ellers and op en rotors.
• The degree of reaction < is the third parameter free to b e chosen during the design phase of a
m ultiple-stage compressor. It represen ts the balance of w ork (and loss) b etw een the rotor and the
stator. It is also related to the c hoice of the flo w angle b et w een t w o successiv e stages. In the case of
purely axial inflo w t ypically encoun tered in single-stage fans, this parameter is not free but dep ends
on the stage loading co efficien t.
• The last parameter τ will b e called the turning angle co efficien t. The loss co efficient and the lift-to-
drag ratio of the blades are primarily determined b y τ . It is a measure of the aero dynamic loading
of the whole blade ro w and of the flo w turning.
• The parameter σ is called the throttling co efficien t or ’Drosselziffer’ in the german literature (see
Grieb [86]). It is an imp ortant stabilit y parameter and w e will see it is a measure of the impact of
endw all losses.
• The parameter π is the p ow er co efficien t and is an indication of the compactness of the stage rela-
tiv e to its w ork output.

Chapter 2. Steady aero dynamics 33
ψ ≡ ∆ H t
U 2 / 2 = 2∆ W t
U
φ ≡ W x
U
< ≡ (∆ H ) r otor
(∆ H ) r otor + (∆ H ) stator
= 1 − ψ
4
| {z }
axial inflo w only
(ratio of static-en thalp y rise)
τ ≡ ψ
φ = 2∆ W t
W x
= 2(tan β A − tan β B )
σ ≡ ∆ H t
W 2
x
= ψ
2 φ 2
π ≡ Q · ∆ H t
ρAU · U 2 / 2 = φ · ψ
(2.18)
The parameters presen ted here will strongly affect the p erformance of a compressor, whic h is measured
mainly through t w o figures of merit: the pressure ratio and the efficiency . The pressure ratio is defined as
the ratio of the outflo w to the inflo w total pressure expressed in the fixed frame of reference: P R ≡ P tB
P tA .
The efficiency measures the qualit y of the compression with resp ect to a thermo dynamically ideal pro cess.
Tw o differen t efficiencies are usually considered, which also ha v e a sligh tly differen t ph ysical meaning:
the isen tropic efficiency η ise and the p olytropic efficiency η pol .
η ise ≡ (∆ H t ) ise
∆ H t
= R B
A ( dT t ) ise
R B
A dT t
= T ise
tB − T tA
T tB − T tA
η pol ≡ ( dH t ) ise
dH t
= R B
A  dT t
T t  ise
R B
A
dT t
T t
=
log  T ise
tB
T tA 
log  T tB
T tA 
(2.19)
where the mass-sp ecific total en thalp y is H t = c p T t and the parameter T ise
tB is the fictiv e outflo w total
temp erature that w ould b e required if the compression pro cess w ere ideal; it is smaller than the actual
outflo w total temp erature T tB , and is giv en by:
T ise
tB = T tA ·  P tB
P tA  1 − 1 /γ
The p olytropic efficiency is defined similarly to the isen tropic efficiency but for an infinitely small com-
pression pro cess. Its dep endence on pressure ratio is m uc h w eak er, therefore it can b e used to compare
the qualit y of compressors ha ving differen t pressure ratios. The p olytropic efficiency is directly related
to the en trop y generated during an infinitesimal compression pro cess:
( dH t ) ise = dH t − γ − 1
γ H t dS
η pol = 1 − γ − 1
γ
dS
dH t /H t
= 1 −
1
2 W 2
m
dH t
ω
W e will no w deriv e a simple relation linking the p olytropic efficiency to the lift-to-drag ratio C L
C D and to
the design parameters φ and < . W e consider an incompressible flo w with constant axial v elo cit y across
the stage. The en trop y rise dS can b e form ulated in terms of loss co efficien t according to Eq.(2.12). The

Chapter 2. Steady aero dynamics 34
expressions for the en trop y rise of the first rotor, stator and second rotor, resp ectiv ely , are giv en b y:
dS r otor 1 = ω · ρ
2 P t · W 2
x +  U − ∆ W t
2  2 ! , for a TF or CR TF
dS stator = ω · ρ
2 P t · W 2
x +  ∆ W t
2  2 ! , for a TF
dS r otor 2 = ω · ρ
2 P t · W 2
x +  U + ∆ W t
2  2 ! , for a CR TF
(2.20)
The loss co efficient ω is assumed iden tical for all blade ro ws. The rotation sp eed of the second rotor is
equal in magnitude and opp osite in sign to that of the first rotor U . As the second rotor op erates at
higher flo w v elo cities than the first rotor, it pro duces more losses. This explains wh y the efficiency of the
second rotor is usually lo w er than that of the first rotor.
W e can no w further simplify the expression of the p olytropic efficiency by using Euler turbomachinery
equation dH t = U ∆ W t (see Eq.(2.5)) and iden tifying ω
2∆ W t /W x = C D
C L (see Eq.(2.11)). W e obtain for a
rotor–stator stage:
η pol = 1 − C D
C L  2 φ + < 2 + (1 − < ) 2
φ  (2.21)
The details of the deriv ation are giv en b y Grieb [86]. W e observ e from this equation that t w o terms
determine the efficiency: the lift-to-drag ratio of the blades, whic h is a decreasing function of the turn-
ing angle co efficien t τ and dep ends on the detailed blade geometry , and a second term dep ending on
global design parameters (flo w co efficien t and degree of reaction). This second term reac hes an optim um
for φ = 0 . 5 and < = 0 . 5, whic h is indeed the design p oin t usually c hosen for m ultiple-stage compres-
sors. Ho w ev er, in the case of a fan stage with purely axial inflo w, this design would imply a turning
angle co efficien t of 4 whic h is far to o large to ensure a go o d lift-to-drag ratio. Therefore, axial-inflow
fans are usually designed for a stage loading around 0.9 and a flo w co efficien t of 0.8 (see Grieb [86]),
in order to main tain an affordable v alue of τ b elo w 1.5. They op erate at a relativ ely high degree of
reaction around 0.8 whic h means that most of the static pressure rise (and most of the loss) comes from
the rotor, making the ov erall fan p erformance more sensitiv e to the rotor design than to the stator design.
A similar expression can b e derived for a rotor–rotor stage. In that case, each rotor con tributes to
the total en thalp y rise, whic h is then giv en b y dH t = 2 U ∆ W t . The degree of reaction of a rotor–rotor
stage with axial inflo w is giv en b y: < = 1
2  1 − ψ
4  . W e obtain a similar relation:
η pol = 1 − C D
C L  φ + 1 + (1 − 2 < ) 2
φ 
Here again the second term has an optim um for < = 0 . 5 but for a higher flo w co efficien t φ = 1 and
this represen ts a fundamen tal adv an tage of the con tra-rotating turb ofans compared to the single-rotating
ones. Designing at higher mass flo w rate implies a more compact engine and a lo w er stage loading.
Moreo v er, the blades can b e designed at a significan tly lo w er turning angle co efficient, whic h means a
b etter lift-to-drag ratio, and less blades, which presen ts the adv an tage of dela ying c hoking.
The qualitativ e adv an tage of the CR TF is detailed in a more quan titativ e w a y no w. Let us consider the
design space ( φ , ψ ), also called the Smith c hart [98]. Figure 2.5 depicts the v ariation of p olytropic efficiency
as a function of the the flo w and loading co efficien ts. It is a usual representation in compressor design
and is called the Smith c hart. The c hart has b een calculated based on the aero dynamic mo dels presen ted
so far and assuming incompressible flo w and constan t axial v elo cit y across the fan stage. Moreov er,
the endw all loss generated b y the mere friction has b een here additionally tak en in to accoun t (the axial
asp ect ratio of the blades is 4 and the drag co efficien t for endw all loss is assumed constan t). Over the

Chapter 2. Steady aero dynamics 35
whole design space, CR TF presen ts roughly 2% higher efficiency than a con v en tional rotor–stator stage.
Moreo v er, the region of maxim um efficiency is more flat and lo cated at higher flo w co efficien ts.
Figure 2.5: Smith Chart for CR TF(lo w er) and SR TF(upp er)
The maxim um efficiency obtained from those simplifed form ulas lies around 0.96, whic h is significan tly
b etter than the actual efficiency of rotors. As rep orted b y Cumpst y , the meanline approac h underesti-
mates the loss, b ecause a large part is attributable to endw all losses and the in teraction loss b et w een
endw all and the blades. This reminds us that the results obtained from a meanline prediction should b e
regarded cautiously .
2.8 Application of the mo dels
The aero dynamic mo dels describ ed so far will no w b e illustrated and applied to the p erformance predic-
tion of the so-called DLR UHBR fan, whic h is a researc h rotor–stator stage dev elop ed and manufactured
in 2006 b y the DLR Institute of Propulsion T ec hnology . The design pro cedure is detailed b y Kaplan [99].
The UHBR fan is designed to equip a ultra-high-b ypass ratio engine ( B P R ≈ 12). The main character-
istics of the UHBR fan geometry and p erformance in cruise are listed in T able 2.2. The aero dynamic
design p oin t (ADP) is T op of Clim b (or Max Clim b). A t T ak e-off the fan is op erated at a relativ e tip
Mac h n um b er b elo w 1.05, th us prev en ting the emission of strong sho ck-induced buzz-sa w noise. A second
fan dev elop ed b y DLR is also considered: the counter-rotating fan concept named CRISP2, whose design
is the result of an automated m ultidisciplinary optimization b y Go erke and Le Denmat [100].
The calculations are p erformed on a 1/3-scale mo del of the UHBR fan measured at the M2VP com-
pressor test facilit y of the DLR Institute of Propulsion T ec hnology in Cologne. The full-scale diameter
of the fan is 2.4 m. The p erformance is ev aluated for the meanline p osition at 70% of the blade heigh t

Chapter 2. Steady aero dynamics 36
mo del fan diameter [m] 0.8 0.8
rotor h ub-to-tip ratio 0.27 0.24
rotor blade coun t 22 / 38 12 / 10
axial asp ect ratio of rotor blades 2.5 / 2.6 3.1 / 2.6
reduced sp eed of rotors [rpm] -7300 / 0 -5045 / 3982
in cruise
reduced mass flo w [kg/s] 98 158
in cruise
T able 2.2: Geometry parameters of UHBR (left) and CRISP2 (righ t) fans
fan pressure ratio 1.38 1.30
fan isen tropic efficiency 0.90 0.93
rotor-face axial Mac h n um b er 0.63 0.66
rotor tip relativ e Mac h n um b er 1.13 1.04
flo w co efficien t φ 0.85 1.04
loading co efficien t ψ 0.99 0.61
degree of Reaction < 0.75 0.49
turning angle co efficient τ 1.16 0.59
T able 2.3: Design parameters in Cruise of the UHBR (left) and CRISP2(righ t) fans
( K M L = 0 . 7). In a first time, we will sho w the aero dynamic p erformance of the rotor blade row alone,
considered in the relativ e frame of reference. In a second time, the p erformance of the whole fan stage
will b e describ ed.
2.8.1 Cascade p erformance
The aero dynamic p erformance of a cascade is usually c haracterized b y the v ariation of losses and other
flo w quan tities dep ending on the incidence of the blades and on the inflo w Mac h n um b er in the reference
frame lo c k ed to the blade ro w. Figure 2.6 presents the t w o-dimensional map of the cascade losses of the
UHBR stator. Non-coloured domains are outside the domain of op eration of the cascade (due to stall or
c hoking). Regions of high losses are mark ed in bro wn colours.

Chapter 2. Steady aero dynamics 37
Figure 2.6: Cascade map represen ting the v ariation of loss as a function of the inflo w Mach n umber and
incidence angle
Sev eral general remarks can b e made:
• The reduction of the w orking range as Mac h n um b er is increased is clearly visible on such cascade
diagrams. F or low Mac h num b ers, the upp er and lo w er incidence limits corresp ond to stall on
the suction and pressure sides, resp ectively . They c hange w eakly with the Mac h n um b er. Bey ond
a certain v alue of Mac h n um b er related to the throat area of the cascade, the w orking range on
the negativ e incidence side starts to b e drastically reduced as a result of choking within the blade
passage.
• The loss c haracteristics presen ts a relativ ely flat minim um cen tered near the zero-incidence case.
The minim um loss incidence is not prescrib ed a priori using empirical correlations but results from
the sup erimp osition of the con tributions of the pressure and suction side b oundary la y ers, which
follo w opp osite trends as incidence is increased. The loss pro duced on the suction side of the blades
dominates at p ositiv e incidence, whereas it is mostly determined b y the pressure side at negativ e
incidence.
• Increasing the inflo w Mac h n um b er significan tly mo difies the loss c haracteristics. The exten t of the
lo w loss region (also called w orking range) is reduced as the Mac h n um b er approaches unit y . The
reduction in w orking range is more pronounced on the negativ e incidence side, which is attributable
to the shift of the c hoking limit to w ards p ositiv e incidence v alues as sup ersonic inflow conditions
are reac hed, see Cumpst y [85].
• The minim um loss incidence sligh tly increases from zero at lo w Mac h n um b er to w ards roughly 3
degrees at sonic inflo w. The c hoking incidence strongly increases and the incidence at whic h stall
is detected decreases sligh tly . These results are in agreemen t with other empirically-based mo dels
suc h as the one dev elop ed at Rolls-Royce b y W asdell, W righ t and Miller [72, 73].

Chapter 2. Steady aero dynamics 38
• Also the shap e of the loss c haracteristics is affected b y the inflo w Mac h n um b er. At lo w Mac h
n um b ers, the curves ha ve a relativ ely flat minimum. A t high Mac h n um b ers, the loss curv e is more
suitably describ ed b y a parab olic curv e of order 2. This is in agreemen t with the observ ation of
Grieb [86].
• A t sup ersonic inflo w conditions, the c hoking incidence and minim um-loss incidence are iden tical
and the loss curv e do es not feature a minimum with zero deriv ative. This was also observ ed by
Blo c h [101] when p erforming 2D-sim ulations of sup ersonic cascades. Bloch attributed the smooth
transition b et w een started and unstarted flo w observ ed in a real fan map to three-dimensional
effects and the span wise transfer of airflo w from started regions to unstarted regions.
2.8.2 F an p erformance at off-design conditions
The presen t section is dedicated to the assessmen t of the prediction at off-design conditions. The p er-
formance maps of t w o differen t fans designed b y the compressor departmen t of DLR [99, 102] ha ve been
calculated and depicted in Fig. 2.7. Lines of constan t rotor sp eed at plotted in the space fan pressure
ratio vs. mass flo w rate Q . The left side of the figure con tains the map of a rotor-stator fan stage with a
design pressure ratio of 1.38. The righ t side of the figure sho ws the map of a con tra-rotating fan with a
design pressure ratio of 1.3. The predicted iso-sp eed lines are colored in gra y and the actual p erformance
is mark ed with sym b ols and thin blac k lines. The iso-sp eed lines of the fan isen tropic efficiency are
plotted in Fig. 2.8. The follo wing conclusions are giv en:
• The prediction of the fan pressure ratio is acceptable in the vicinit y of the design p oin t and at part
sp eeds in the domain of mo derate loading. A t high sp eeds where the flo w is transonic and features
a complex system of sho c ks interacting with detac hed b oundary la y ers, the discrepancy is more
pronounced.
• A significan t o v erprediction of the pressure rise is done near the surge limit, whic h is probably
due to the strong flo w deviation caused b y thic k b oundary la y ers dev eloping on the blades. The
deviation mo del curren tly used relies on simple theoretical considerations and only accoun ts for
incidence and solidit y: deviation v anishes in the limit of high solidit y , and flo w turning v anishes
in the single-airfoil limit. Empirical corrections a v ailable in the literature migh t b e added in the
future to b etter mo del the p erformance near surge.
• The c hoking limit is sligh tly o v erpredicted at design and part sp eeds. Since choking is the conse-
quence of flo w blo c k age o v er the whole span and since its onset is v ery sensitiv e to cascade throat
width and lo cal Mac h n um b er, this limit cannot b e predicted accurately by a meanline approac h.
The author prefers a simple robust approac h o v er an empirical correlation tuned for a particular
case but that w ould lik ely b e unsuited in the more general case.
• F an efficiency is o v erestimated b y 1% to 2 % near the design p oint. This may be attributed to
the underprediction of the endw all losses, whic h do not only consist of a skin-friction comp onent
but also b oundary lay er thic k ening through diffusion and in teraction with the complex tip and hub
v ortex flo ws. The drop in efficiency at lo w er and higher sp eeds is fairly well predicted.
The off-design p oin ts particularly relev an t during the fligh t mission are the mid-cruise or aero dynamic
design p oin t (DP), max-clim b or top-of-clim b (TP), and the three acoustic certification p oin ts: tak e-off
sideline (SL), tak e-off cutbac k (CB) and approac h (AP). The fligh t conditions and thrust requiremen ts at
eac h of these p oin ts (see T able 2.4) are the same for all engine configurations considered in the follo wing
studies; in particular, the noise levels will be compared at the same thrust condition. These conditions
corresp ond to a single-aisle medium-range civil aircraft equipp ed with t w o engines (Airbus A320 or Bo eing
737). The t ypical p ositions of the off-design p oints within the fan map are mark ed in Fig. 2.7 b y blac k
circles. T ypical v alues of some flo w parameters are summarized in T able 2.5 in the particular case of the
DLR UHBR fan [99]. T op of clim b (TC) is often considered as the true dimensioning p oin t, as the fan
m ust pro vide more thrust than in cruise (hence a higher pressure ratio) with a more sev ere mass flo w

Chapter 2. Steady aero dynamics 39
50 75 100
1
1.1
1.2
1.3
1.4
1.5
1.6
FP R
Q in k g/s
TF
DP
TC
SL
CB
AP
DP
TC
SL
CB
AP
DP
TC
SL
CB
AP
DP
TC
SL
CB
AP
DP
TC
SL
CB
AP
DP
TC
SL
CB
AP
DP
TC
SL
CB
AP
DP
TC
SL
CB
AP
DP
TC
SL
CB
AP
DP
TC
SL
CB
AP
80 120 160
Q in k g/s
CR TF
DP
TC
SL
CB
AP
DP
TC
SL
CB
AP
DP
TC
SL
CB
AP
DP
TC
SL
CB
AP
DP
TC
SL
CB
AP
DP
TC
SL
CB
AP

Figure 2.7: V alidation of predicted pressure ratio at off-design conditions (prediction: solid gra y lines,
reference: blac k lines with icons) [76]
capacit y (higher axial Mac h n um b er) whic h forces the fan to op erate closer to the c hoking limit. The
three p oin ts relev an t for the noise certification are all situated at part sp eed and with a larger incidence
than in cruise esp ecially at approac h, which is due to the w eaker acceleration of the flo w o v er the spinner
as the Mac h n um b er is reduced. The sideline p oint (SL) is aerodynamically more critical as it is lo cated
at fairly high sp eed and closer to the surge limit as in cruise.
T able 2.4: Fligh t conditions and thrust requiremen ts common to all configurations at sp ecific op erating
conditions
DP TC SL CB AP
T otal net thrust in kN 37 44 170 110 38
Fligh t Mac h n um b er 0.78 0.78 0.21 0.35 0.21
Fligh t heigh t in m 10500 10500 0 500 120

Chapter 2. Steady aero dynamics 40
50 75 100
0.8
0.85
0.9
0.95
1
η ise
Q in kg/s
TF
80 120 160
Q in kg/s
CR TF

Figure 2.8: V alidation of predicted efficiency at off-design conditions (prediction: solid gra y lines, refer-
ence: blac k lines with icons) [76]
T able 2.5: Predicted fan parameters at off-design op erating conditions in the particular case of the DLR
UHBR fan
DP TC SL CB AP
N cor r in % 100 106 95 85 54
FPR 1.38 1.44 1.35 1.26 1.09
η ise 0.93 0.91 0.93 0.94 0.92
M x 0.64 0.68 0.54 0.48 0.28
i in deg 3.1 2.9 6.0 5.8 7.8
2.9 Conclusion
The presen t mo dels deliv er realistic trends for the prediction of the fan aero dynamic c haracteristics of
single- or coun ter-rotating fans. Despite of b eing a strong simplification, the meanline approach on
whic h the mo dels are based, captures sev eral fundamen tal prop erties of the flow inside a turbofan stage
and of its aero dynamic p erformance. The fact that the interpretation of the results is rendered more
straigh tforw ard b y the meanline approac h supp orts the general philosoph y presen ted in in tro duction 1:
relating the acoustic features of a fan to its aero dynamic p erformance and deriving design guidelines.
Apart from the inheren t limitations of the meanline approac h that w ere discussed b efore, some asp ects
of the mo dels could b e impro v ed while still b eing addressed on a meanline lev el:
• Deviation: this quan tit y is generally not zero at design conditions, esp ecially for lo w-solidit y blades.
These blades do not only sho w high losses at high loading, they also ha v e p o or turning capabilities,
ev en at design conditions. In tro ducing a mo del for the design deviation angle dep ending on blade
solidit y w ould mo v e the designs of loaded fans to blades with a higher solidit y .

Chapter 2. Steady aero dynamics 41
• Sho c k loss: separating sho c k-only loss from the sho c k–b oundary lay er in teraction loss is not ph ysi-
cally w ell grounded and leads to difficulties in deriving empirical prediction metho ds that repro duce
correctly the v ariations with incidence, Mach n um b er and blade geometries. The theoretical ap-
proac h prop osed b y F reeman and Cumpst y [87] could b e extended to account for airfoil cam b er,
arbitrary solidit y , duct con traction, while co v ering the subsonic and sup ersonic regimes equally . In
that case this so-called extended F reeman mo del w ould include b oth the sho c k and b oundary lay er
loss, and address the off-design flo w conditions without an y additional loss mo del.
• The effect of the duct con traction (MVDR) on the sho c k loss is not mo delled presently . It is kno wn
that increasing the MVDR reduces the c hoking limit but ma y significan tly impro v e the p erformance
at design conditions. This effect could also b e captured b y the extended F reeman mo del.

Chapter 3
Engine and fan aero dynamic design
3.1 Motiv ation and approac h
In the previous c hapter 2, aero dynamic mo dels w ere prop osed to compute the flo w and p erformance of
a giv en fan once its geometry and op erating conditions (mass flow rate and rotation speed) are known.
Ho w ev er, at an early design stage, only a few global parameters concerning the engine arc hitecture and
thermo dynamic cycle can b e sp ecified. The purp ose of the presen t mo dule is precisely to find the optimal
geometry and op erating conditions satisfying the global design sp ecifications for the engine.
These sp ecifications consist of the n um b er of engines, their t yp e (TF, CR TF, CROR), the pressure
ratio of the fan and some parameters concerning the fligh t conditions suc h as Mac h n um b er and altitude,
and the thrust required b y the aircraft to main tain its fligh t path. F or ducted configurations, the axial
Mac h n um b er upstream of the first rotor is an imp ortant additional design parameter. F or unducted
configurations, this parameter is constrained b y the thrust and cannot b e chosen freely .
The quan tities determined b y the design mo dule are essen tially the diameter and length of the engine,
as w ell as geometry parameters prop er to the fan like blade n umber, c hord length, stagger angles and the
op erating conditions.
The acoustic comparison of differen t engines is made in general difficult b y the strong discrepancy
existing in aero dynamic and flo w conditions in whic h the engines op erate. With the help of the presen t
design mo dule, a common criterion based on the selection of the b est aero dynamic design for eac h engine
configuration can b e dra wn to compare the noise emission of different engine arc hitectures.
3.2 Design constrain ts
The engine design pro cedure consists of an optimization of the engine and fan parameters asso ciated with
the lo w est fuel consumption for the sp ecified user requirements. The constrain ts listed b elow define the
design space in whic h the searc h for the optim um is carried out. The optimal solutions found dep end on
this set of constrain ts.
• Design p oin t: the design of the engine is p erformed at only one op erating p oin t, called the aero dy-
namic design p oin t (ADP), which usually corresponds to mid-cruise flight conditions. The following
design constrain ts are imp osed at ADP but may not be maintained at off-design points.
• Optimization for b est fan efficiency: the fitness function (or quan tit y to b e optimized) during the
pro cedure is basically the fuel consumption of the engine. In the case of a swirl-free exhaust flo w,
this is equiv alen t to searc hing for the engine with the b est fan efficiency . In the presen t v ersion of
the mo dule, the design is p erformed based on aero dynamic considerations solely . F uture v ersions
ma y include some acoustic criteria (for example as an additional fitness function).

Chapter 3. Engine and fan aero dynamic design 43
• Stabilit y margin: apart from being lo cated in the region of b est efficiency the design m ust also b e
robust, that means its p erformance has to remain stable ov er a range of p ositiv e incidences. The
rotation sp eed and mass flo w characterizing the stabilit y p oin t are as follo ws:
N of f = N AD P
Q of f = Q AD P · (1 − S M / 100)
The user-defined parameter S M is called the stabilit y margin and is expressed in p ercent of the
design mass flo w rate. During the design of a blade ro w (cascade design pro cedure), an av erage
efficiency based on the con tributions of the design and stabilit y p oin ts (w eigh ted b y the relativ e
loss lev el) is maximized and the corresp onding solidit y is determined. A design with a large margin
(t ypically 20%) presen ts high-solidit y blades, stable op eration ov er a wide range (well beyond the
usual off-design fligh t p oin ts), but this is comp ensated by a lo wer efficiency in cruise conditions.
Suc h a design c hoice (also called stall margin) is an essen tial asp ect of fan design and explains
for example wh y the rear-rotor of a CR TF-concept has a higher solidit y than the fron t-rotor. It
should b e noted that no stability margin is defined in that sense for a CR OR engine: the off-design
op erations are regulated b y a v ariable-pitc h system so that the blades are op erated near their
optimal incidence at all fligh t p oints.
• Ho w ev er, no syste m for adapting the blade stagger angle or the exhaust nozzle area is mo delled
here. Once determined at the design p oin t, the geometry of the fan is frozen and constan t for all
other off-design p oin ts.
• Engine arc hitecture: the engine is comp osed of fiv e comp onen ts, depicted in Figure 3.1. The
in tak e, the first rotor (R1), an in ter-stage section, the second rotor or the stator (R2), and finally
a con v ergen t exhaust nozzle. The engine can b e a con v en tional turb ofan with a rotor–stator stage
(TF), a coun ter-rotating fan (CR TF) or a coun ter-rotating op en rotor (CR OR). In the latter case,
there is no outer duct en v eloping the rotors. F or the sak e of simplicit y , neither pylons are mo delled
with the CR OR nor structural struts are mo delled with the ducted configurations TF and CR TF.
• The meanline radius R m is constan t across the fan stage. The duct con tours are designed to
imp ose streamlines parallel to the engine axis. This restriction is of course not applicable to radial
compressors.
• The v ariation of axial v elo cit y W x across a blade ro w is an essen tial asp ect of fan design. Allo wing
for constan t axial v elo cit y limits flo w diffusion and th us separation but it imp oses a mark ed duct
con traction and increases the flo w v elo cities impinging the do wnstream blade ro w. A compromise is
usually found b y selecting an appropriate v alue of the meridional-v elo cit y–densit y ratio ( M V D R ,
see Eq.(2.4)). A correlation based on the densit y ratio is used to set the design v alue of the M V D R
and finally determine the duct con tours that satisfy this condition:
( M V D R ) des =  ρ B
ρ A  n M V D R
The v alue of the user-defined exp onen t n M V D R is comprised b et w een 0 (no duct con traction) and
1 ( W x = const ). A v alue of 0.5 is represen tativ e of mo dern fans.
• The blades op erate at a sp ecific incidence pro viding the minim um loss: this is the design incidence
presen ted in the previous c hapter 2. F or small Mach n um b ers, the design incidence is zero. F or
higher Mac h n um b ers approac hing c hoking condition, the design incidence may assume moderate
p ositiv e v alues up to 5 degrees. Note that the incidence for b est efficiency is usually slightly higher
than the design incidence, ho w ev er the difference in efficiency is negligible due to the flat shap e of
the optim um.
• Swirl-free exhaust: the outflo w angle of the fan (b ehind the second rotor or the stator) is zero. In
that case, searc hing the engine with the lo w est fuel consumption is equiv alen t to searc hing the fan

Chapter 3. Engine and fan aero dynamic design 44
with the highest isen tropic efficiency . This constrain t is not applicable for the design of a single
prop eller, as the appropriate choice for the exhaust swirl is an essen tial asp ect of the optimization.
• Rotor inflo w Mac h n um b er: the optimal relativ e inflo w Mac h n um b er M r el of a rotor (at the
meanline radius) is searc hed within the range M x < M r el < 1 . 5. This corresp onds to a maxim um
relativ e tip Mac h n um b er M tip,r el of roughly 1.7, and a maximum fan pressure ratio around 1.9.
• The n um b er of blades is determined b y the axial asp ect ratio of the blades (sp ecified b y the user)
and b y the solidit y of the blade ro w whic h is found automatically b y the optimization routines. No
acoustic criterion is implemen ted in the presen t v ersion of the mo dule. The axial asp ect ratio v aries
for mo dern rotors b et w een 2 and 2.5 (b efore the 1990s the blades had higher v alues b et w een 3 and
4). Low-aspect-ratio blades are characterized b y go o d aero dynamic and aero-mechanical stabilit y
(see W ennerstrom [96]). F or a CR OR, slender blades with a high axial asp ect-ratio around 4 or 5
are imp ortan t to k eep a lo w tip v ortex drag. Ho w ev er the asso ciated loss is not mo delled in the
presen t v ersion.
• Off-design p oin ts: after the engine has b een designed at ADP , calculations are p erformed at the
follo wing off-design p oin ts: top of clim b and the three acoustic certification p oin ts tak e-off and
cut-bac k and approac h. The design is discarded if the calculation do es not con v erge or returns
an error status. This ma y b e an indication that the stabilit y margin c hosen is to o low: esp ecially
at the tak e-off and approac h p oin ts, the fan may operate at strong p ositiv e incidence and needs a
sufficien t blade surface to a v oid flo w separations and loss of p erformance. Con v ergence problems
obtained at top of clim b are usually attributable to the pro ximit y to c hoking conditions (to o large
v alues of axial Mac h n um b er or to o high flow block age inside the blade rows).
• F or the coun ter-rotating fan arc hitectures CR TF and CR OR, the ratio of rotation sp eed of the
rotors is k ept constan t for all off-design p oin ts and iden tical to that obtained at ADP . The sp eed
ratio at ADP results from the aero dynamic optimization without taking in to accoun t the constrain ts
imp osed b y the lo w-pressure turbine or a gear-b o x.
• Because only the b ypass flo w of the fan is mo delled, it is not p ossible to compute the b ypass ratio
directly . How ev er, it can b e roughly estimated at the design p oin t b y correlation with the fan
pressure ratio (FPR). The shaft p o w er on the lo w-pressure turbine (LPT) and on the fan are given
b y:
Π LP T = Q cor e · c p · (∆ T t ) LP T
Π f an = ( Q cor e + Q by pass ) · c p · (∆ T t ) f an = Π LP T
(∆ T t ) f an = T t 0 · ( FPR ( γ − 1) / ( γ η pol ) − 1)
The LPT p o w er is considered to b e en tirely transferred to the fan. The b ypass ratio B P R is defined
as the ratio of the b ypass to the core mass flo w rate. W e obtain:
B P R ≡ Q by pass
Q cor e
= (∆ T t ) LP T
T t 0 · 1
FPR ( γ − 1) / ( γ η pol ) − 1 − 1 (3.1)
The first term of the correlation is the ratio of the LPT total temp erature drop to the inflow total
temp erature and is assumed to ha v e a constan t t ypical v alue of 1.36. The p olytropic efficiency of
the fan is also assumed constan t with a v alue of 0.93.

Chapter 3. Engine and fan aero dynamic d esign 4 5
Figure 3.1: sc hematic view of the ducted-fan engine conﬁguration.
Figure 3.1 depicts sc hematically the arc hitecture of a ducted-fan engine as mo delled in the mo dule.
F or unducted conﬁgurations, the outer fan cowl is replaced b y a streamline. The diﬀerent stations are
listed from 0 to 10. The stations 0 and 10 corresp ond to the far upstream and far do wnstream p ositions,
resp ectiv ely . The engine en try plane is lo cated at p osition 1. The fan en try plane is lo cated at p osition
2. P osition 9 is the engine exhaust plane at whic h the jet exhaust pressure and the am bien t pressure are
iden tical P 9 = P 0 if the jet is subsonic M 9 < 1. F or critical con v ergen t nozzles with exactly sonic ﬂo w at
the nozzle exist, w e ha v e M 9 = 1 and P 9 >P
0 .
3.3 Preliminary engine design
3.3.1 Metho dology
Before optimizing the geometry of the fan blades, the global dimensions of the engine ha v e to b e deter-
mined during a preliminary design phase, as shown in Figure 3.2. The most imp ortan t parameter to meet
the thrust requiremen ts of the aircraft is the mass of airﬂo w en tering the engine. Based on this quan tit y ,
it is p ossible to determine the fan diameter. Subsequen tly , the nozzle diameter is determined to meet the
condition P 9 = P 0 or M 9 = 1 at the exhaust plane. All other dimensions of the engine are deriv ed b y
simple scaling. The w etted surface of the nacelle and the engine w eigh t aﬀect the drag pro duced b y the
engine itself, whic h in turn is added to the original thrust requiremen t and the airﬂo w is corrected. This
iterativ e pro cess is rep eated un til the size of the engine has con v erged.

Chapter 3. Engine and fan aero dynamic d esign 4 6
Figure 3.2: Principle of the engine preliminary design
The calculation steps presen ted in Figure 3.2 will no w b e detailed.
3.3.2 Engine thrust
The thrust required b y the aircraft for eac h engine is called the net thrust or installed thrust. It is to
b e distinguished from the uninstalled thrust, whic h is usually the quan tit y guaran teed b y the engine
man ufacturer. The uninstalled thrust, which is for example described in a b o ok b y Mattingly et al. [103],
dep ends on the airﬂo w Q ingested b y the engine and the so-called sp eciﬁc thrust V 10 − V 0 represen ting
the acceleration of the ﬂo w b y the engine:
T uninst = Q ( V 10 − V 0 ) (3.2)
The term QV 10 is usually referred to as gross thrust, and is the thrust obtained during static tests. The
term QV 0 is sometimes called the ram drag. The net thrust is the diﬀerence b et w een the uninstalled
thrust and the drag pro duced b y the engine itself. The net thrust is usually calculated b y the aircraft
designer. In the presen t mo del, w e prop ose to consider t w o terms for the engine drag: the nacelle drag
D n due to ﬂo w spillage and the friction surfaces en v eloping the engine and the additional induced drag
D i whic h is the v ortex-induced drag generated at the aircraft wing tips but caused b y the engine w eigh t.
The in teraction drag b et w een the nacelle, the p ylon and the wing is considered constan t and already
included in the net thrust requiremen ts, so it will not b e mo delled.
T net = T uninst − D n − D i (3.3)
The sp eciﬁc thrust can b e determined prior to the engine sizing as it solely dep ends on the thermo dy-
namic cycle of the engine (mostly the pressure ratio and to a lesser exten t the fan isen tropic eﬃciency).
F or the sak e of simplicit y , w e will consider here that the engine pressure ratio is the fan pressure ratio

Chapter 3. Engine and fan aero dynamic design 47
FPR sp ecified b y the user. The sp ecific thrust can b e calculated b y using the follo wing relations:
V 10 − V 0 = ( V 9 − V 0 ) + P 9 − P 0
ρ 9 V 9
V 9 = q 2 c p ( T t 9 − T 9 )
T t 9 = T t 0 · 1 + FPR γ − 1
γ − 1
η ise !
P t 9 = P t 0 · FPR
T 9 = T t 9 ·  P 9
P t 9  γ − 1
γ
(3.4)
The isen tropic efficiency of the fan η ise defined in Eq.(2.19) is a priori unkno wn and will b e obtained from
the fan design routines. Only a few iterations are necessary to conv erge as the efficiency of w ell-designed
fans is fairly high and can b e regarded from the p oin t of view of the thermo dynamic cycle as nearly
constan t.
3.3.3 Engine airflo w and fan diameter
Based on the uninstalled thrust and sp ecific thrust, Eq.(3.2) can b e reform ulated to pro vide the airflo w
Q , and subsequen tly the cross-section area in fron t of the first rotor:
A 2 = Q
ρ 2 V 2
In the case of ducted configurations the rotor-face axial v elo city V 2 is sp ecified b y the user in form of the
axial Mac h n um b er. F or the unducted CR OR configuration, this v elo cit y is not a free design parameter
but is constrained b y the pressure ratio. According to the actuator disk theory , the CR OR is mo delled
b y a disk pro ducing a static pressure jump ∆ P = P t 0 ( FPR − 1) in axial incompressible flo w, the axial
v elo cit y in the disk plane is then:
V 2 = 1
2 V 0 + s V 2
0 + 2 P t 0
ρ 0
( FPR − 1) !
Due to the incompressible flo w assumption, the axial v elo cit y is sligh tly underestimated. Moreo v er it
m ust b e b elo w the sonic limit. As a result the pressure ratio of the CR OR has to b e limited to a maxim um
v alue around 1.3 at cruise fligh t conditions (35000 ft altitude and fligh t Mac h n um b er of 0.78).
The diameter of the first rotor is:
D f an = 2 s A 2
π (1 − η )
The parameter η is the h ub-to-tip ratio at the en try of the fan and is a free design parameter sp ecified
b y the user. Similarly , the diameter of the exhaust nozzle is giv en b y:
R noz z le = r A 9
π , where A 9 = Q
ρ 9 V 9
3.3.4 Engine length and nacelle dimensions
As indicated in Figure 3.1, the axial length of the intak e and of the nozzle are considered equal. A simple
approac h could b e to scale these lengths with the fan diameter, but this would yield a too long and to o

Chapter 3. Engine and fan aero dynamic design 48
hea vy nacelle for engines with a v ery lo w fan pressure ratio. F or that reason, the length is assumed to
scale with the square ro ot of the engine thrust for dimensionality reasons:
L intake = L R E F · r T uninst
T RE F
P RE F
P 0
L noz z le = L intak e
(3.5)
The in tak e length L RE F , thrust T R E F , and pressure P R E F are those of a w ell-kno wn reference engine
used to calibrate the scaling. The axial length of the blade ro ws R1 and R2 are equal to the axial c hord,
and they are computed from the axial asp ect ratio sp ecified b y the user:
c ax = R f an (1 − η )
AR ax
The axial length of the in ter-stage duct b et w een R1 and R2 dep ends on the axial chord of the blade and
a user-defined parameter K cax :
∆ L = K cax · max( c ax,R 1 , c ax,R 2 )
This scaling is motiv ated b y acoustic considerations: the v alue of K cax should b e at least 1 to ensure a
sufficien t distance and a w eak in teraction b et w een the blade ro ws, whic h is required for lo w-noise emission.
The external surface of the nacelle is giv en b y:
S nac = 2 π · R f an · L nac
The mean thic kness of the nacelle is computed using the follo wing correlation:
t nac = t RE F · r T uninst
T RE F
P RE F
P 0
3.3.5 Engine w eigh t
The o v erall w eigh t of the engine is the sum of three terms:
W eng = W f an + W cor e + W nac
The first term is the w eigh t of the fan blades, assumed to b e made out of titanium. Lik e for the in tak e
length and nacelle thic kness, the weigh t of the engine core W cor e is assumed to scale with some p o w er
of the engine thrust. The third term is part of the engine w eigh t directly scaling with the v olume of the
external nacelle, whose a v erage densit y is estimated around 400 k g /m 3 .
W f an = ρ blade · B · V blade , where V : blade volume
W cor e = W RE F ·  T uninst
T RE F
P RE F
P 0  3 / 2
W nac = ρ nac · S nac · t nac
3.3.6 Engine drag
The drag of the engine is comp osed of differen t terms: the drag of the nacelle, whic h includes the friction
and spillage drag, and the additional aircraft induced drag caused b y the w eigh t of the engine.
The friction drag of the external nacelle is computed assuming a constan t flo w v elo cit y equal to the
fligh t v elo cit y V 0 . The skin friction co efficien t C f is obtained from Eq.(2.15) taking the fligh t Mac h
n um b er and a Reynolds n um b er based on the size of the nacelle. The in ternal friction drag of the nacelle
is already included in the isen tropic efficiency of the fan (blade ro w endw all loss).
D f r iction = 1
2 ρ 0 V 2
0 · S nac · C f ( Re nac , M 0 )

Chapter 3. Engine and fan aero dynamic design 49
The spillage drag is the result of the partial reco v ery of additiv e drag (or pre-en try drag) through
the suction force at the lip of the nacelle. Ph ysically , this drag is related to the entrop y pro duced in
the b oundary la y er and in the sho cks as a result of the flo w spillage around the lips and the asso ciated
o v ersp eeds. More details can b e found in the b o ok on aircraft engine design b y Mattingly [103] or that
of W ard on aerospace propulsion systems [104]. A reaslistic estimation for the spillage co efficien t K spill
is around 0.4.
D spillag e = K spil l ( Q ( V 1 − V 0 ) + A 1 ( P 1 − P 0 ))
The last term, the induced drag, is not directly generated on the engine, but corresp onds to the
additional induced drag of the aircraft caused b y the w eigh t of the engine added to the aircraft w eigh t.
Assuming an ideal elliptic distribution of lift L , the total induced drag of the airplane ha ving a wing span
b is giv en b y:
D i = L 2
1
2 ρ 0 V 2
0 · π b 2
In cruise, the lift exactly comp ensates the total w eigh t of the aircraft: L = ( W a + N · W eng ) g . The aircraft
w eigh t in cruise without its engines is W a and m ust b e sp ecified by the user. The acceleration constan t is
g = 9 . 81 m/s 2 . W e assume that the w eigh t of the engines is v ery small compared to the aircraft w eigh t:
( W a + N · W eng ) 2 ≈ W 2
a + 2 N · W eng · W a
As a result, the additional induced drag caused b y ev ery engine separately is giv en b y:
D i = 2 W a · W eng · g 2
1
2 ρ 0 V 2
0 · π b 2
3.4 F an design
3.4.1 Principles
As men tioned in the in tro duction of the presen t c hapter, the design of the fan relies on an iterativ e pro cess
whic h searc hes for the configuration with the highest fan efficiency . This iterativ e pro cess is presen ted in
Figure 3.3 for the con v en tional ducted turb ofan (rotor–stator stage) and the coun ter-rotating turb ofan
(ducted or unducted). The approac h consists of v arying successiv ely the meanline inflo w relativ e Mac h
n um b er M r el of the first rotor, b etw een a minim um v alue corresp onding to the axial Mach n umber
(sp ecified b y the user or imp osed) and a maxim um v alue fixed arbitrarily in the program (the maxim um
v alue is limited to 1.5, whic h is large enough to co v er fan pressure ratios up to 1.9). After systematic
v ariation of M r el , the v alue yielding the b est fan efficiency is chosen for design. In the case of coun ter-
rotating turb ofans, the p o w er split n b et w een the first and the second rotor m ust b e v aried, to o. It
increases when the pressure ratio of the first rotor increases and it is equal to 0.5 if b oth rotors ha v e the
same pressure ratio. Exp erience sho ws that the optimal p o w er split is usually lo cated b et w een 0.5 and
0.7. The p ow er split is mathematically defined as follows:
n ≡ 1
1 + log ( P R 2 )
log( P R 1 )
, where P R 1 · P R 2 = FPR

Chapter 3. Engine and fan aero dynamic d esign 5 0
Figure 3.3: Principle of the design of a single-rotating (left) and coun ter-rotating (righ t) turb ofan stage
Inside the global iteration lo op, each componen t has its own design procedure whic h is summarized
in Figure 3.4. The relativ e Mac h num b er is used to calculate the rotational sp eed of the ﬁrst rotor,
and to deduce the ﬂo w turning based on Euler’s turb omac hinery equation, see Eq.(2.5). In order to
ensure a swirl-free exhaust ﬂo w, the ﬂo w turning of the second rotor (or stator) m ust b e exactly equal in
magnitude and opp osite in sign to that of the ﬁrst rotor. The optimal lift co eﬃcien t whic h pro vides the
lo w est loss for the sp eciﬁed ﬂo w turning is found b y means of a gradien t-based searc h algorithm. Using
the relation of Eq.(2.7), the optimal solidit y can b e obtained and subsequently all parameters describing
the geometry of the blades.
Figure 3.4: Principle of the design of rotors and stators
3.4.2 Exemplary results
The design principles describ ed in the previous section will no w b e illustrated through parameter studies
p erformed on a t ypical turb ofan application.

Chapter 3. Engine and fan aero dynamic design 51
W e fo cus first on the cascade design routine which looks for the optimal lift co efficien t and cascade
solidit y for giv en inflo w Mac h n um b er and flo w turning. Figure 3.5 presen ts the results of a parame-
ter study in whic h the solidit y of a stator cascade has b een con tin uously v aried from 0 to 4 and the
p erformance and flo w quan tities calculated with the mo dule for Steady Aero dynamics describ ed in the
previous c hapter 2. The v arious coloured curv es corresp ond to differen t v alues of flo w turning v arying
from 5 degrees (blue) to 40 degrees (dark red).
• Eac h curv e presen ts an optimal solidit y for whic h the total cascade loss has a minim um. Lo w-
solidit y cascades op erate at a high lift co efficien t and consequen tly high diffusion factor asso ciated
with flo w separation o v er the blades.
• In v ersely , high-solidit y cascades w ork at a lo w er lift co efficient and diffusion factor, so separation is
a v oided but the w etted blade surfaces are larger than necessary and are resp onsible for increased
skin-friction loss. Moreo v er, the flo w at cascade throat approac hes sonic conditions (c hoking limit)
whic h is accompanied b y o v ersp eeds and an additional increase of the losses.
• As flo w turning increases, the optim um is shifted to w ards larger solidit y and diffusion factor. This
trend presen ts some analogy with the increase of wing loading observ ed on aircraft designs as the
w eigh t (hence the lift) is increased.
• These results repro duce the trends observ ed in 1959 b y Lieblein [74] as he in tro duced his concept of
diffusion factor and illustrated in a similar parameter study the implications of his new correlation
on cascade design. Ho w ev er, Lieblein did not accoun t for compressibilit y effects at the time, whic h
in particular ma y result in c hoking and increased losses as the throat flo w b ecomes sonic.
Figure 3.5: V ariation of stator p erformance for different geometries (solidit y) and inflow angles
This parameter v ariation is carried out automacially b y the cascade design routine. The optimal
solidit y corresp onding to minim um loss is found through an algorithm fitting the curv e to a p olynom
of second order whic h enables the robust determination of the optimal solidit y ev en if the curv e is not

Chapter 3. Engine and fan aero dynamic design 52
con tin uous and presen ts small jumps. Figure 3.6 presen ts the b eha viour of the optimal solution dep ending
on the flo w turning (eac h colour corresp onds to a differen t inflo w Mac h n um b er). The following remarks
can b e made:
• A t lo w v alues of the flo w turning, the optimal solidit y increases nearly linearly . The asymptotic case
of zero flo w turning th us yields a cascade of v anishing solidit y whic h corresp onds to the isolated
airfoil.
• The optimal diffusion factor increases with the flo w turning. This is in agreement with the ob-
serv ations of Den ton [75] who stated that the optimal compressor cascade should op erate at high
loading, on the v erge of separation.
• F or v ery high flo w turning angles ab o v e 40 deg, the optimal cascade is found to operate with
significan t losses and presumably some separated regions. The prediction of the correct trends is
m uc h less reliable in this domain due to the inaccuracy of the loss mo dels with detac hed flo w. The
optimal solutions found b y the routine should therefore b e considered with caution if the losses
b ecome to o large.
• A t high v alues of the inflo w Mac h n um b er, the optimal lift co efficien t b ecomes smaller in order to
a v oid to o high sup ersonic v elo cities and sho c k losses.
Figure 3.6: V ariation of stator p erformance for flow turning requiremen ts
The parameter study presen ted in Figure 3.7 illustrates ho w the optimal rotor inflo w relative Mac h
n um b er M r el is selected which is an essen tial step of the fan design routine depicted in Figure 3.3. The
sp ecifications for the rotor–stator stage were a constan t fan pressure ratio of 1.35, and constan t rotor
inflo w axial Mac h n um b er of 0.6. The rotor inflo w relativ e Mac h n um b er is con tin uously v aried from 0.6
to 1.1. The main features can b e summarized here:
• According to the Euler’s turb omac hinery equation, design at low Mac h n um b er (low rotor speed)
m ust b e comp ensated b y a high flo w turning: the flo w is strongly deflected b y the rotor and ma y
b e ev en accelerated. The rotor efficiency remains high but the strong swirl has to be recov ered b y
the stator, resulting in high stator losses and a p o or ov erall fan efficiency .

Chapter 3. Engine and fan aero dynamic design 53
• In v ersely , designing the rotor for high Mac h n um b ers allo ws to op erate the fan with a smaller in ter-
stage swirl and go o d stator efficiency . Ho w ev er, this is limited b y the loss p enalty associated with
the sho c ks forming around the rotor blades at transonic and sup ersonic sp eeds.
• The optimal v alue for M r el th us results from a balance b etw een rotor and stator losses. This
compromise is a w ell-kno wn asp ect of fan design and is related to the selection of the prop er degree
of reaction R for the stage. Rotor–stator stages op erate at relativ ely high degree of reaction around
0.8, whereas coun ter-rotationg rotors op erate around R = 0 . 5.
• F or large Mac h n um b ers ab ov e 0.9, the optimal diffusion factor deca ys b ecause the flo w turning
decreases and large v alues of p eak Mac h n um b ers asso ciated with strong sho ck loss m ust b e a v oided.
Figure 3.7: V ariation of rotor–stator design: effect of inflow Mac h n um b er at constan t FPR
The parameter study presen ted in Figure 3.8 presen ts the design parameters for increasing fan pressure
ratio. Practically , the optim um inflo w Mac h n um b er from Figure 3.7 is selected for each v alue of the
pressure ratio. The rotor inflo w axial Mac h n um b er is main tained constan t at a v alue of 0.6. The follo wing
remarks are giv en:
• The main design parameters suc h as Mac h n um b er, solidity , flo w turning, or diffusion factor tend
to increase with increasing fan pressure ratio.
• The total fan efficiency sligh tly increases at v ery lo w v alues of the pressure ratio: this is attributable
to the endw all ann ulus loss whic h do es not scale with the pressure ratio. Then, the efficiency reaches
a maxim um. A t higher pressure ratio, the efficiency decreases as a consequence of the increase in
flo w turning whic h is necessary to pro vide the desired compression. This represen ts more demanding
aero dynamic conditions.

Chapter 3. Engine and fan aero dynamic design 54
Figure 3.8: V ariation of rotor–stator design: effect of fan pressure ratio at constant M x
3.5 Engine p erformance
The ultimate figure of merit c haracterizing the aero dynamic p erformance of an engine for a giv en aircraft
application is the fuel consumption, denoted F C . W e will sho w in this section that this quan tit y is directly
related to the o v erall efficiency of the engine, whic h is defined as the ratio of output p o w er formed b y the
net propulsiv e p o w er and the input p ow er contained in the fuel injected inside the combustion c ham b er.
η eng ≡ Π out
Π in
= T net · V 0
F C · H V
The parameter H V is the heating v alue of k erozine ( H V = 40 · 10 6 J /k g ). The engine efficiency is written
as a pro duct of four terms:
η eng = Π out
Π uninst · Π uninst
Π k in · Π kin
Π th · Π th
Π in
The installation, propulsiv e, aero-thermal and com bustion efficiencies are defined as follo ws:
η inst ≡ Π out
Π uninst
= T net
T uninst
η pr op ≡ Π uninst
Π k in
= 2 V 0
V 10 + V 0
η aer other m ≡ Π k in
Π th
= ( V 2
10 − V 2
0 ) / 2
c p ( T t 10 − T t 0 )
η comb ≡ Π th
Π in
= Qc p ( T t 10 − T t 0 )
F C · H V = 1
The first term is the installation efficiency , it dep ends on the aero dynamic design of the nacelle, flo w
spillage, and the engine w eigh t relativ e to the total aircraft w eigh t. The second term is the propulsiv e
efficiency and is directly related to the exhaust v elo city of the flo w and tends to b e higher for lo w pressure
ratio fans. The aero-thermal efficiency includes the loss of efficiency due to the high temp erature of the

Chapter 3. Engine and fan aero dynamic design 55
jet and due to the transv erse kinetic energy con tained in the turbulence and the exhaust swirl (whic h ma y
b e presen t at off-design conditions). The isen tropic efficiency of the fan is part of the engine aero-thermal
efficiency . At last, the com bustion pro cess inside mo dern burners has b ecome so efficient that w e will
consider it as ideal. W e define the sp ecific fuel consumption as the fuel consumption required to pro duce
of net thrust of one Newton. It is in v ersely prop ortional to the engine efficiency:
S F C ≡ F C
T net
= V 0
η eng · H V
Note that this relation is not v alid in static conditions where the fligh t v elo city is zero (for example at
T ak e-off ). In that case the sp ecific fuel consumption reads:
S F C = V 10 / 2
η inst · η aer other m · η comb · H V
Finally the fuel consumption for eac h engine is giv en b y:
F C = S F C · T net
Figure 3.9 presen ts the v ariation with fan pressure ratio of some quan tities pro vided b y the preliminary
design routine previously describ ed in Figure 3.2. This routine is practically resp onsible for the sizing
of the engine (most imp ortantly the fan diameter and the nacelle length). There exists an optimum
fan pressure ratio at whic h the fuel consumption is minimal. Low pressure ratio engines ha ve a hi gh
propulsiv e efficiency but ha v e a large drag p enalt y due to their size. High pressure ratio engines are more
compact, but ha v e a lo w propulsiv e and aero-thermal efficiency .
Figure 3.9: T ypical results of pre-design
3.6 Off-design op erating p oin ts
After the aero dynamic design has b een p erformed, the geometry of the fan blades is fixed and it is
p ossible to calculate the p erformance of the engine at off-design fligh t conditions. The off-design p oin ts

Chapter 3. Engine and fan aero dynamic d esign 5 6
are listed in T able 3.1 with their corresp onding ﬂight Mac h n um b ers, altitudes, aircraft ov erall net thrust
requiremen ts and some appro ximate v alues of the reduced fan sp eed N re d at whic h the engine is op erated
(expressed in p ercen t of the Cruise sp eed). The oﬀ-design sp eciﬁcations listed in this table are giv en for
a t ypical commercial jet airliner with short- to medium-range.
Cruise (ADP) T o po fc l i m b Ta k e - o ﬀ Cut-back Approac h
ﬂigh t altitude [m] 10500 10500 0 500 120
ﬂigh t Mac h n um b er 0.78 0.78 0.21 0.35 0.21
required thrust [kN] 37 44 170 110 38
appro x. N re d [%] 100 110 90 80 50
T able 3.1: Deﬁnition of the ﬂigh t conditions for a giv en short- to medium-range airliner
The T op of clim b (or max clim b) ﬂigh t condition is of particular in terest for the aero dynamic p erfor-
mance of the engine: it is usually the most demanding p oint, as the engine m ust b e op erated at v ery high
sp eed (typically 5 to 10% higher than the cruise speed) with nearly chok ed fan blades. F or the acoustic
assessmen t, three other oﬀ-design p oints being part of the noise certiﬁcation are considered: T ake-oﬀ,
Cut-bac k and Approac h. The T ak e-oﬀ p oint ma y also b e c hallenging aero dynamically as the fan op erates
there near the stall limit. This is particularly the case for the CR TF-concept with lo w-solidit y blades. If
the aero dynamic p erformance cannot b e ac hiev ed (due to c hoking or stall), the subsequen t calculations
cannot carried out: an assessmen t of the fan at the corresp onding op erating p oint is not possible.
F or eac h oﬀ-design p oin t, an iterativ e pro cedure lo oks for the set of op erating conditions (fan sp eed and
airﬂo w) that deliv ers the required thrust and satisﬁes an additional criterion (the pro cedure is illustrated
in Figure 3.10). F or ducted conﬁgurations, this criterion concerns the jet expansion at the nozzle exit:
if the exit Mac h n um b er is b elo w one M 9 < 1, the exhaust static pressure m ust b e equal to the am bien t
pressure P 9 = P 0 . F or turb ofan engines with a fan pressure ratio larger than 1.3 appro ximately , the
exhaust nozzle is usually c hok ed in cruise and top of clim b conditions. In that case, the throat ﬂo w of
the nozzle is exactly sonic M 9 = 1 and the exhaust pressure is larger than the am bient pressure. F or
unducted conﬁgurations, no nozzle con trols the expansion of the jet so an alternative criterion is applied:
the axial v elo cit y m ust matc h the v elo cit y induced b y the thrust of the rotors.
Figure 3.10: Pro cedure to determine the fan sp eed and mass ﬂo w rate at oﬀ-design conditions.

Chapter 3. Engine and fan aero dynamic design 57
The configurations with coun ter-rotating rotors (CR TF and CR OR) are assumed to op erate at con-
stan t sp eed ratio (ratio of rotation sp eeds b et w een fron t- and rear-rotor) irresp ective of the off-design
p oin t considered. Thus, the speed of the rear-rotor is simply scaled from its v alue in cruise.
3.7 V alidation at design conditions
In order to v erify ho w accurate the predictions can b e in terms of design, a series of existing ducted
fans ha v e b een sim ulated for whic h the design parameters are known. The pressure ratio of these fans
ranges from 1.3 to 1.7 for the single-rotating configurations, and from 1.2 to 1.45 for the con tra-rotating
cases. Fig. 3.11 compares the solidit y and relativ e Mac h n um b er predicted b y the to ol with the actual
v alues observ ed on the real design, note that the meanline v alues are considered here again. The differen t
sym b ol st yles are related to either the fron t or rear blade ro w of the fan. P erfectly accurate predictions
are lo cated along the cen ter dashed line while the upp er resp. low er dashed lines corresp ond to an
o v erestimation resp. underestimation b y 10%.
• The first conclusion of this comparison is the fairly go o d prediction of the relativ e Mac h n um b er
o v er a broad range of v alues (righ t part of the figure): as exp ected, the stators of the con v en tional
fans ha v e a lo w inflo w Mac h n um b er and the rotors are c haracterized b y high v alues, the rotors of
the con tra-rotating concept op erate in a more mo derate Mac h n um b er domain.
• The second conclusion concerns the prediction of the blade solidit y (left side of the figure): w e
observ e a significan t disp ersion, esp ecially on the rotor-stator stages. The solidit y of the stator
tends to b e o v erestimated whereas that of the rotor is underestimated. This ma y b e explained b y
t w o imp ortan t design parameters: the stall margin and the meridional-v elo city–densit y ratio known
as MVDR. These parameters strongly affect the flo w diffusion allo w ed at design condition and th us
impact directly the solidit y; the c hoice of these parameters is part of the exp erience gained o v er
the y ears b y the engine man ufacturers, whic h mak es it a more challenging task to predict with a
simplified approac h. Stators do not deliv er p o w er input and generate lo w losses, this is probably
wh y one can afford to design them with lo w er solidities, keeping thereb y the cruise efficiency as
high as p ossible. Nevertheless, the trend of increasing solidit y with increasing fan pressure ratio is
globally resp ected; the higher solidity of the rear rotor compared to the fron t rotor of the CR TF is
also w ell repro duced (this to o is a result of designing for a sufficien t stall margin).
The next asp ect deals with the v alidation of the fan isen tropic efficiency predicted b y our aero dynamic
mo del. Fig. 3.12 sho ws the comparison b etw een the con v en tional fan TF with the con tra-rotating fan
CR TF o v er a wide range of design fan pressure ratio and fan-face axial Mac h n um b er. A similar but
m uc h more detailed study p erformed with a 3D RANS solv er has b een published b y Timea Lengy el [105]
recen tly . A result from her study is sho wn in Fig. 3.13. Although the presen t approac h relies on simple
mo dels and a one-dimensional description at meanline, the same trends are observed in both studies and
similar conclusions can b e dra wn.
• The maxim um fan efficiency is obtained for lo w pressure ratio and lo w Mac h n um b er. How ever,
the pressure ratio should not b e to o lo w, otherwise the endw all loss w ould dominate and n ullify the
aero dynamic b enefit; in the case of a v ery lo w pressure ratio, an unducted configuration (CROR)
is recommended.
• Designing for high fan-face axial Mac h n um b er reduces the fan diameter and mak es the engine more
compact, but there is an efficiency p enalty on the fan that leads to a compromise. Curren t designs
are lo cated around 0.6-0.65. F or the unducted CROR, this parameter is constrained b y the thrust
and m ust b e ab o v e the fligh t Mac h n um b er.
• The con tra-rotating ducted fan presen ts a significan t efficiency b enefit o v er the con v en tional fan b y
at least 1%. In terestingly , the higher the pressure ratio or axial Mac h n um b er is, the larger is the
b enefit, which can reac h up to 3%. This makes the CR TF a more serious competitor in the range
of mo derate b ypass ratio rather than at ultra high bypass ratio.

Chapter 3. Engine and fan aero dynamic design 58
0.5 1 1.5 2
0.5
1
1.5
2
actu a l s o lid it y σ
pr edict ed s o lid it y σ

T F rot or
T F st ator
C R T F rotor 1
C R T F rotor 2
0.5 0.8 1.1 1.4
0.5
0.8
1.1
1.4
actu a l M 1
pr edict ed M 1

T F rot or
T F st ator
C R T F rotor 1
C R T F rotor 2

Figure 3.11: Accuracy of the prediction of some fan design parameters [76]
• The reasons for the aero dynamic sup eriorit y of the CR TF o v er the TF are t w ofold: first, the w ork
input can b e split o v er b oth rotors, meaning that each rotor needs less flo w turning, a low er diffusion
factor, and a lo w er solidit y (less blades) whic h also reduces the blo ck age and delays c hoking; in all,
less friction and mixing loss is pro duced for the same ov erall w ork input. The second reason lies in
the nature of the v elo cit y triangles: let consider the Smith c hart [98] in the ( φ , ψ ) design space, the
TF has its optim um around φ = 0 . 5 and ψ = 1, a region never c hosen for compressor stages with
axial inflo w b ecause the loading is to o high and the lift-to-drag ratio is to o lo w there. The Smith
c hart of the CR TF is b etter suited for axial-inflow fans, since the optim um is at φ = 1 and ψ → 0,
a region with higher mass flo w, lo w loading and a b etter lift-to-drag ratio for the blades.
3.8 Conclusion
A pro cedure that includes the sizing of the engine nacelle and the design of the fan blades has b een
presen ted in this c hapter. The optimized engine presen ts a minimal fuel consumption for the sp ecified
thrust requiremen t, fligh t conditions, fan pressure ratio, and fan stabilit y margin. Only the b ypass flo w
of the engine is considered here, whic h is a realistic assumption for high-b ypass-ratio engines. P arameter
v ariations ha v e b een p erformed and the consiste ncy of the trends has b een discussed. The in tegration of
some acoustic design criterion, based for example on a cut-off design for the blade-passing frequencies,
ma y constitute an extension of the presen t pro cedure, in the future.

Chapter 3. Engine and fan aero dynamic design 59
M x
FP R
TF

0.88
0.89
0.9
0.91
0.91
0.92
0.92
0.92
0.93
0.93
0.94
0.94
0.95

0.5 0.6 0.7 0.8
1.1
1.2
1.3
1.4
1.5
1.6
1.7
M x
CR TF

0.92
0.93
0.93
0.94
0.94
0.94
0.95
0.95
0.95
0.96
0.96

0.5 0.6 0.7 0.8
fan is en trop c efficiency η ise
0.88
0.9
0.92
0.94
0.96

Figure 3.12: V ariations of single- and con tra-rotating fan efficiency with pressure ratio and axial Mac h
n um b er [76]
Figure 3.13: Results of a similar study [105] obtained after optimization with a 3D RANS flo w solv er

Chapter 4
Unsteady aero dynamics
4.1 In tro duction
The mo dels presen ted in this chapter establish the connection betw een the mean flow quan tities and the
acoustic calculations. T urb ofan noise basically arises from flo w p erturbations that fluctuate in time in
the reference frame of the observ er. The expression ’unsteady aero dynamics’ will b e used hereafter to
refer to the acoustically relev an t comp onen ts of the flo w. The p erturbations of t w o differen t parts of the
flo w are considered: the viscous flow (blade boundary lay ers, w akes and turbulence) and the potential
field (pressure field b ound to eac h blade ro w). The fluctuations of the temp erature field can b e neglected
in the fan and compressors of an engine. Flo w v elo city fluctuations are sometimes describ ed as gusts.
Similarly to the steady aero dynamics mo dule, the initial strength of the w ak es and p oten tial fields
and their deca y in axial direction are calculated on a meanline radius. The p e rturbations are expressed in
terms of flo w v elo cit y sp ectra dep ending on frequency and circumferen tial mo de order. It will b e sho wn
that circumferen tial v ariations of a steady v elo cit y field lo c k ed to a rotor result in time fluctuations if
observ ed in the fixed reference frame. This chapter will also presen t the lift resp onse of blades to these
fluctuations.
Generally , it should be mentioned that the domain of unsteady aerodynamics cov ers a relativ ely
wide range of problems and is one of the fundamen tal pillars for accurate noise predictions. Ho w ev er,
the dev elopmen t of theoretically-based analytical solutions has not receiv ed as m uc h atten tion in the
literature as its purely acoustic coun terpart. Therefore, unlik e the acoustic mo dels which pro vide an
exact analytical solution of a simplified problem, the mo dels for unsteady aero dynamics are partly based
on empirical correlations, inv olving constan ts that need to b e calibrated and p ossibly adapted to the
application considered.
4.2 P oten tial field
The p oten tial field is often regarded as the coun terpart of the viscous flo w field: b y definition, it is the
irrotational comp onen t of the v elo cit y field, whic h can b e written as a gradient of a scalar quan tit y called
the v elo cit y p oten tial. In a w ell-designed fan with limited separations and w eak sho c ks, the p oten tial field
is almost indep enden t of the viscous flo w. W e will consider here only the steady part of the p oten tial field
that is b ound to the blade row and rotates with it. It is generated b y the flo w displacemen t due to airfoil
thic kness and b y the blade circulation due to lift. A t subsonic mean-flo w v elo cities, the propagation tak es
place linearly but in the sup ersonic regime, sho c ks are formed and there is an energy deca y due to the
dissipation inside the sho c ks.

Chapter 4. Unsteady aero dynamics 61
4.2.1 Initial strength of the p oten tial field
As long as the p erturbations of the steady pressure field b ound to a blade ro w are small, the thickness
and lifting problems can b e treated separately . Eac h blade is then represen ted b y a distribution of steady
monop oles and dip oles, resp ectiv ely . This is the approach adopted b y P arry and Crigh ton [44]. Kemp
and Sears [106] detailed the deriv ation of the lifting mo del. Morfey [47] pro vided analytical estimations
but distinguishing b et w een the lo w-solidit y and high-solidit y limits. F or simplicity and consistency with
the mo dels for Steady Aero dynamics, w e c ho ose to ev aluate the p otential field on a meanline radius
and based on the spatial distribution of flo w o v ersp eeds around the blade. F or that purp ose, the source
strength of the quadrup ole term in tro duced b y Hanson [24] forms the starting p oin t:
ρu 2
` = 1
c 2
+ ∞
Z
` = −∞
h
Z
n =0
ρu 2
` ( `, n ) dnd` (4.1)
where u ` is the stream wise disturbance v elo cit y . The surface integral is appro ximated by:
ρu 2
` = K q uad · ρ m  ( W S S − W m ) 2 + ( W P S − W m ) 2 
The scalar quan tities W S S , W P S , are the a v eraged blade v elo cities on the suction and pressure side,
they ha v e b een already in tro duced in c hapter 2 on Steady Aero dynamics and can b e calculated based on
the thic kness, lift cofficien t and off-design incidence, see Eq.(2.8). The bac kground flo w parameters ρ m ,
W m ha v e also already b een presen ted in Eq.(2.3). The constan t K q uad m ust b e determined empirically , a
v alue around 0.5 seems appropriate. Finally , the ro ot-mean-square v alue (in the sense of circumferen tial
a v eraging) of the initial v elo cit y p erturbation is giv en b y:
u 0 = s ρu 2
`
ρ m · σ
The solidit y σ of the blade ro w app ears from a v eraging o v er a blade passage with a width equal to the
blade spacing s . The ro ot-mean-square v alue of the velocity perturbation v anishes for single airfoils and
increases with increasing solidit y , whic h agrees with the mo del of Morfey [47].
4.2.2 Circumferen tial distribution and mo des
In addition to a scalar quan tit y represen ting the energy , the prediction of noise requires to describ e
the distribution of this energy among the harmonics. Due to the 2 π -p erio dicity of the problem in
circumferen tial direction, the harmonics are in tegers denoted m and called circumferen tial or azim uthal
mo des. The p erturbation asso ciated to a mo de of order m is obtained b y F ourier decomp osition:
u ( m ) ≡ 1
2 π
2 π
Z
θ =0
u ( θ ) e imθ dθ ≡ 1
s
s
Z
y =0
u ( y ) e i m
r y dy
By definition, the ro ot-mean-square v alue of the v elo cit y p erturbation in circumferen tial direction is:
| u 0 | 2 ≡ 1
2 π
2 π
Z
θ =0
u ( θ ) 2 dθ =
+ ∞
X
m = −∞ | u ( m ) | 2
W e assume that all blades are iden tical, which implies that the mode orders are multiples of the blade
coun t: m = h · B . Moreov er, w e assume that the en tire energy is con tained solely in the first harmonic
order, which is a reasonable h yp othesis as higher harmonics deca y v ery rapidly and practically do not
sho w up in the noise sp ectra.
u ( m ) =  u 0 , if | m | = B
0 , otherwise

Chapter 4. Unsteady aero dynamics 62
The assumption of iden tical blades is less acceptable at sup ersonic sp eeds. F rom each blade a shock
is emitted, whose in tensit y and p osition dep end on the blade cam b er and thickness. The non-linear
propagation effects asso ciated with the sho c ks amplify the differences b et w een the sho cks. As a result,
minor geometry v ariations from blade to blade lead, after some distance, to a pressure pattern with a ric h
mo dal con ten t, as energy is transferred from the main harmonics to all other mode orders (also called
m ultiple pure tones). Pick ett [107] prop osed a statistical mo del to relate analytically the sp ectrum of
m ultiple pure tones to the v ariations in blade geometry .
4.2.3 Deca y of the p oten tial field
If the mean flo w is subsonic in the reference frame lo ck ed to the blade rows, the p oten tial field sources
emit w a v es in terfering destructiv ely and the resulting pressure field cannot propagate without a deca y .
The amplitude decrease of the p erturbation v elo cit y dep ends on the axial distance ∆ x to the leading edge
of the blades, the circumferen tial mo de order m and the flo w conditions. The deca y mo del presented
here is deriv ed from the analytical axial w a v en um b er inside an infinitely long duct with uniform axial
flo w in the fixed reference frame (see the c hapter 6 dedicated to Acoustics for further details).
u (∆ x ) = u 0 · exp( − α ∆ x ), where α = | m |
r mn p 1 − M 2
r el
1 − M 2
x
, for M r el < 1 (4.2)
The quan tit y r mn is the caustic radius, which will be defined later in the c hapter 6 dedicated to Acoustics.
It is appro ximated b y the meanline radius. It should b e noted that the deca y la w presen ted here is not
rigorously v alid for unducted configurations in the free field, despite its repro duction of the main qualita-
tiv e trends. The exact solution can b e obtained from the near-field theory as developed by Hanson [108]
or Sc h ulten [55] for prop eller noise prediction. The axial decay coefficient α decreases with increasing
relativ e Mac h n um b er, and v anishes as the mean flow becomes sonic: b ey ond this p oint the pressure
p erturbations can propagate in form of cut-on acoustic mo des and are c haracterized b y sho c ks. The
non-linear effects asso ciated to their large amplitudes and gradien ts result in energy dissipation whose
in tensit y can b e quan tified b y either the w eak sho c k theory or the one-dimensional non-linear acoustic
theory (see the w ork of Morfey [109]). A quan tit y called the ’time of fligh t’ is defined:
T (∆ x ) = ∆ x
s
M 2
r el
p M 2
r el − 1
1
 sin β r el − cos β r el p M 2
r el − 1  2 (4.3)
The relativ e Mac h n um b er is calculated from the axial Mac h n um b er and the rotation sp eed: M 2
r el =
M 2
x + Ω · r
a 0 . The relativ e flo w angle is giv en b y: cos β r el = M x
M r el . The deca y of sho ck pressure rise in side a
duct is giv en b y:
∆ P
P (∆ x ) = (∆ P /P ) 0
1 + γ +1
2 γ · T (∆ x ) · (∆ P /P ) 0 → 2 γ
γ + 1
1
T (∆ x ) (4.4)
A t large distances, the pressure perturbation do es not dep end on its initial v alue ∆ P 0 an ymore, and
deca ys in v ersely to the ratio of axial distance ∆ x to blade spacing s . The p erturbation v elo cit y can b e
related to the p erturbation pressure through:
∆ P = − ρW r el u (4.5)
where W r el is the circumferen tial a v erage of the stream wise v elo city component, whic h is also the v elo cit y
computed b y the Steady Aero mo dule. In the following, it will be denoted W mean or W 0 .
4.3 Mean-flo w w ak es
The b oundary la y er generated on the blade surface is con v ected in form of a w ak e from the trailing
edge of the blades. It is w ell kno wn that the impingemen t of w ak es on a blade ro w lo cated further

Chapter 4. Unsteady aero dynamics 63
do wnstream ma y b e an imp ortan t noise con tributor. The w ak e is c haracterized b y a mean v elo city deficit
and sto c hastic v elo cit y fluctuations amplified b y the v elo cit y gradien ts within the w ak e. In this part,
w e consider the mean part of the w ak e and adopt a simple mo del to describ e it: the mean w ak e has a
Gaussian shap e and is symmetric (same width on the pressure and suction sides). Therefore, the w ak e
quan tities (width, depth and area) will b e easily related to the b oundary la y er thic kness and shap e factor.
A review on existing semi-empirical mo dels has b een giv en b y Carazo [110]: the mo dels are applicable to
either isolated airfoils, cascades or rotors, and do not account for pressure-gradien t effects. T o remedy this
partly , a new mo del will b e prop osed to determine analytically the deca y of the w ak e in the stream wise
direction. The ev olution of the w ak e is considered in the reference frame lo c k ed to the blade ro w where
the w ak e is generated.
4.3.1 W ak e mo del
W e consider the steady w ak e formed b y the blades in the rotating frame of reference. The v elo cit y
p erturbation con v ected b ehind a blade ro w is mo delled b y a train of iden tical Gaussian-shap ed sym-
metrical w ak es. This mo del has b een exp erimen tally confirmed b y Ra j [111], Ra vindranath [112], and
Reynolds [113] based on t w o-dimensional cascade measuremen ts. Real wak es are non-symmetrical as the
b oundary la y er on the suction and pressure sides ha v e a differen t thic kness; the impact of non-symmetrical
w ak es has b een analyzed b y Roger [114] and sho wn to b e significan t only for higher harmonics. The
stream wise v elo cit y p erturbation expressed as a function of the circumferen tial p osition y = r θ is then
giv en b y:
u ( y ) = u max ·
+ ∞
X
k = −∞
e − π ( y − k · s
w · s ) 2 (4.6)
where u max is the p eak p erturbation v elo cit y (also called maxim um deficit v elo city), s the blade spacing
and w the non-dimensional w ak e width whic h is defined more precisely hereafter. The sum represents
the train of w ak es. The stream wise comp onen t W of lo cal flo w v elo city is:
W ( y ) = W max − u ( y )
The figure 4.1 illustrate ho w these quan tities are defined. A symmetrical Gaussian w ak e can b e c harac-
terized b y three non-dimensional parameters: the area A , depth d and width w .
A ≡ Z s
0
u ( y )
W mean
dy
s
d ≡ u max
W max
w ≡ A
d
The quan tit y W mean is the circumferen tially a v eraged v elo cit y: it is equal to the outflo w v elo cit y W B
in tro duced in the c hapter 2. The non-dimensional w ak e parameters can b e related to the displacement
and momen tum thic kness of the w ak e. These are defined as follo ws:
δ 1 = h
s Z s
y =0  1 − u ( y )
W max  dy
δ 2 = h
s Z s
y =0
u ( y )
W max  1 − u ( y )
W max  dy
The b oundary la y er thic knesses are defined in the direction normal to the direction of con v ection; this
explains the app earance of the ratio of the stream tub e heigh t h to the blade spacing s . W e can no w
express the w ak e parameters in terms of b oundary la y er quan tities:
A = δ 1
h − δ 1
d = √ 2(1 + A )  1 − 1
H 12  (4.7)

Chapter 4. Unsteady aero dynamics 64
The w ak e area dep ends on the blo c k age of the ﬂo w c hannel b y the w ak e, whic h in turn dep ends on the
viscous drag co eﬃcien t of the blade. The w ak e depth dep ends on the shap e factor H 12 and increases with
increasing blade aero dynamic loading. This is consisten t with exp erimen tal observ ations b y the researc h
team of Lakshminara y ana ([112], [113]). The initial v alue of the w ake parameters (at the trailing edge)
can therefore b e calculated using the momen tum thic kness and shap e factor of Eq.(2.14) and Eq.(2.16).
It should b e noticed that the deﬁnition of the wak e parameters is v alid for wak es clearly separated from
eac h other, if the w ak e width increases b eyond a certain v alue (t ypically around 0.4) the wak es start to
merge and the w ak e deﬁcit deca ys more rapidly as observ ed exp erimen tally by Ra j [111].
Figure 4.1: Represen tation of a w ak e o v er a blade passage and the asso ciated mean ﬂo w and deﬁcit
v elo cities
4.3.2 Sp ectral con ten t
The F ourier transform of the train of Gaussian w ak es from Eq.(4.6) is calculated analytically:
u ( h )= W mean · A · e − π · w 2 · h 2 (4.8)
T h ew a k eh a r m o n i co r d e ri sd e n o t e d h . P arsev al’s theorem (or Ra yleigh’s iden tit y) stating the conserv a-
tion of energy is v eriﬁed:
1
s  s
0
u ( y ) 2 dy =
+ ∞

h = −∞
u ( h ) 2
4.3.3 W ak e deca y
The problem of predicting the deca y rate of w ak es has b een treated b y a large n um b er of authors.
Theoretical considerations b y Prandtl and Sc hlic h ting [93] lead to a scaling la w for a t w o-dimensional
w ak e suc h that the v elo cit y defect scales with the square-ro ot of the blade drag co eﬃcien t. Due to the
strong in teraction b et w een the mean v elo city deﬁcit and the local turbulence, most approac hes rely on
empirical correlations. One of the ﬁrst quan titativ e mo dels w as prop osed b y Silv erstein [115] for a w ak e
con v ected b ehind an isolated airfoil. He correlated the w ak e deca y to the drag co eﬃcien t of the airfoil.
Wygnanski [116] rep orted on extensive v elo cit y measuremen ts b ehind isolated obstacles (airfoil, cylinder,
etc.) and form ulated the correlation in terms of the b oundary la y er thic kness. Measuremen ts ha v e b een
p erformed do wnstream of t w o-dimensional cascade blades [113] of fan rotors [112, 117]. They conﬁrm
that the w ak e deﬁcit scales fairly w ell with the square-ro ot of the blade drag co eﬃcien t if the distance
considered is large enough. In that case, the far-w ak e mo dels pro vide stream wise v ariations of the w ak e

Chapter 4. Unsteady aero dynamics 65
parameters in the follo wing form:
w ( ` ) = r w 2
T E + K · C D · `
c ,
w ( ` ) · d ( ` ) = const,
(4.9)
where ` is the distance to the trailing edge. The pro duct of w ak e width and depth (whic h is b y definition
the w ak e area) is assumed constan t, the w ak e width slo wly increases with distance. Ho w ev er this approac h
strongly relies on the exp erimen tal determination of the constan t K . Moreo v er, the rapid deca y observ ed
close to the trailing edge (near-w ak e region) is not mo delled correctly and the impact of a non-zero
pressure gradien t cannot b e included.
W e ha v e c hosen here to describ e the deca y on a more theoretical basis based on the in tegral b oundary
la y er equation. This approac h w as widely used in the 60’s and 70’s to predict the p erformance of
t w o-dimensional diffusers. In this problem, the mean flo w can b e reasonably w ell describ ed by the
one-dimensional v ersion of the con tin uit y equation in incompressible flo w, coupled to the momen tum
equation to predict the gro wth of the b oundary lay er along the diffuser w alls and the flo w blo c k age.
A third equation is usually required to close the problem, and there ha v e b een a n um b er of auxiliary
equations prop osed for this closure. The metho d of Head [118] complemen ted b y Green [119] has receiv ed
some success: the volume flo w rate (also called the en trainmen t) transp orted within the b oundary la y er
is correlated to some shap e factor represen ting the current state of the boundary lay er. W e apply this
approac h to describ e the deca y of the w ak e. W e further assume incompressible flo w, no wall, and that
the b oundary la y er is fully turbulen t.
Con tin uit y equation: d ( W ( h − δ 1 ))
d` = 0
Momen tum equation: dδ 2
d` = − ( H 12 + 2) δ 2
W
dW
d`
En trainmen t equation: 1
W
d ( W ( δ 0 − δ 1 ))
d` = 2 E
(4.10)
These equations are applied along a stream tub e, in the direction of w ak e con v ection. The v elo cit y W
denotes the v elo cit y outside of the b oundary la y er, here W max . The en trainmen t co efficien t E is related
to the shap e factor through the following empirical relation:
E = K E ( H 12 − 1)
where the constan t K E is calibrated to matc h exp erimen tal measuremen ts of a small-deficit w ak e with
zero pressure-gradien t (see T o wnsend [120]). In that case w e ha v e:
δ 2
dH 12
d` = − 0 . 468( H 12 − 1) 3 = − 2 E · ( H 21 − 1) 2
1 . 3
As explained b y Lyrio [121] and detailed more mathematically b y V eldman [122], the equations (4.10)
m ust b e solv ed sim ultaneously in order to a v oid con v ergence problems, esp ecially when the coupling
b et w een the in viscid mean flo w and the b oundary la y er is strong (t ypically near separation). The ap-
proac h presen ted here enables to accoun t for the in teraction b et w een neigh b ouring wak es and for non-zero
pressure-gradien t. Moreo v er, it repro duces the rapid deca y observ ed near the trailing edge of blades and
the more mo derate v ariations in the far-w ak e region.
W e will no w illustrate the capabilities of this mo del through three parameter studies. In Figure 4.2
is presen ted the stream wise deca y of the non-dimensional w ak e parameters. The stream wise distance
` is normalized b y the stream tub e heigh t h . Near the trailing edge of the blades ( `/h < 1), the more
in tensiv e mixing is resp onsible for rapid v ariations. Esp ecially the w ak e area deca ys v ery fast in this
region. F urther do wnstream ( `/h > 1) the deca y is mark edly slo w er. The w ak e flattens as the width

Chapter 4. Unsteady aero dynamics 66
increases and the depth decreases, but the w ak e area is nearly constan t in this region whic h is still called
the near-w ak e region. The far w ak e is not depicted here (width w > 0 . 4). It corresp onds to the region
where the w ak es of adjacen t blades start to merge; in that case the deca y accelerates and the w ak e
area decreases again. The v arious colour lines in Figure 4.2 corresp ond to a v ariation of the stream wise
pressure gradien t. The blade loading and the initial parameters of the w ak es are main tained constan t.
F or a giv en stream wise p osition, the w ak e depth and the shap e factor are larger in the presence of an
adv erse pressure gradien t. This result is in agreemen t with the exp erimen tal observ ations of Ra j [111]
made on cascade of airfoils, whic h stated that the deca y is less rapid in presence of an adv erse pressure
gradien t.
Figure 4.2: Stream wise evolution of w ake parameters for differen t v alues of pressure gradien t (initial w ak e
parameters k ept constan t)
The effect of blade loading on the stream wise ev olution of the w ak e parameters can b e examined in
Figure 4.3. F or a giv en blade geometry , the blade loading is directly related to the drag co efficien t and
to the b oundary la y er thic kness at the trailing edge of the blades. Sev eral measuremen ts on compressor
rotors carried out b y the researc h team of Lakshminara y ana [113, 112, 123] ha v e confirmed that a higher
blade loading slo ws do wn the deca y and induces a larger depth and width of the w ak es. This is also what
the presen t mo del predicts: for a given stream wise p osition, the depth and shap e factor increases with
the blade loading, ho w ev er the effect seems not to b e as pronounced as for the pressure gradien t.

Chapter 4. Unsteady aero dynamics 67
Figure 4.3: Stream wise ev olution of w ak e parameters for differen t v alues of blade loading
Finally , the scaling of the wak e width with the blade drag co efficien t is sho wn in Figure 4.4. The
solid lines depict the results of the mo del whereas the dashed lines represent the empirical scaling la w of
Eq.(4.9), with K = 0 . 3. A go o d agreemen t is observ ed for the lo w solidit y cases (blue lines σ = 0 . 1 and
green curv es σ = 0 . 2). F or the higher solidit y t ypical of a compressor blade ro w (bro wn curv es σ = 1)
the empirical scaling la w underpredicts the w ak e deca y . These results are also in agreemen t with the
measuremen ts of Ra vindranath [112] who stated that the w ak e width roughly scales with the square ro ot
of the drag co efficien t.

Chapter 4. Unsteady aero dynamics 68
Figure 4.4: V ariation of w ake width on blade drag coefficient for the fixed streamwise position `/h = 1)
and for differen t solidit y . (solid line: correlation of Eq.(4.9), dashed line: s olution of Eq.(4.10))
4.4 T urbulence
Sto c hastic v elo cit y fluctuations prop er to turbulence are the source of broadband noise. Three differen t
categories will b e considered in this section: inflo w turbulence generated in the atmosphere and ingested
b y the fan, the turbulence present in the blade w ak es, and turbulence generated inside the b oundary la y er
dev eloping on the blade and in teracting with the trailing edge. In all cases, the turbulence is assumed
isotropic with a small correlation length, and it can therefore b e fully describ ed b y the turbulen t kinetic
energy , the in tegral length scale, and a non-dimensional function for the sp ectral conten t. The expression
of turbulence in the frequency domain is preferred to that in the time domain as it enables more easily
an in terpretation of the theoretical results and suits the acoustic form ulation presen ted in c hapter 6.
Glegg [124] has recen tly p oin ted out the difficulties asso ciated to this c hoice and prop oses an alternativ e
description based on time correlations.
4.4.1 Inflo w turbulence ingested b y the fan
W e assume the turbulence ingested from the atmosphere to b e isotropic and homogeneous. The ro ot-
mean-square of the turbulen t v elo cit y fluctuations in the streamwise and transv erse direction are iden tical
and equal to 2/3 of the turbulen t kinetic energy k , i.e. u 2 = v 2 = w 2 = 2
3 k . The homogeneit y assumption
implies that the en tire energy is con tained in the circumferen tial mo de of zero order m = 0. The one-
dimensional p o w er sp ectral densit y of the v elo cit y fluctuations and the turbulence correlation length are
describ ed b y the V on Karman mo del utilized b y Amiet [125].
Φ uu ( ω ) = u 2 · Λ
W 0 · 0 . 656
(1+1 . 8 z 2 ) 5 / 6
` ( ω )=Λ · 5 . 01 z 2
√ 1+ z 2 ( 1+ 8
3 z 2 ) → 1 . 4 W 0
ω
where W 0 is the mean flo w v elo cit y , the reduced frequency is z = S t
S t 0 , the Strouhal num b er S t = f Λ
W 0
and a reference Strouhal n um b er S t 0 = 1 / 2 π ≈ 0 . 16. The quan tities u 2 and Λ are prop erties of the

Chapter 4. Unsteady aero dynamics 69
atmospheric turbulence and do not dep end on the fan considered. A t high frequencies the correlation
length do es not dep end on the in tegral length scale. Note that the p ow er sp ectral densit y Φ uu has the
dimension of a squared v elo cit y divided b y a frequency and verifies:
Z + ∞
−∞
Φ uu ( ω ) dω = u 2 (4.11)
W ak e turbulence
The c haracteristics of turbulence con v ected b ehind rotor blades has b een do cumen ted b y Reynolds [126]
and Ganz [117]. It w as sho wn in these studies that the w ak e turbulence presen ts some similarities with
isotropic free turbulence. More recen tly , hot-wire measuremen ts inside a lo w-sp eed rotor-stage [127]
at v arious op erating conditions ha v e pro vided an extensiv e set of w ak e turbulence data for the cyclo-
stationary analysis applied b y Jurdic [128]. He sho w ed that the w ak e turbulence sp ectra can b e repre-
sen ted as a pro duct of a one-dimensional frequency sp ectrum and a Gaussian-shap ed mo dulation function
dep ending on the circumferen tial mo de order m , whic h he called cyclic order. This ma y b e written math-
ematically as:
Φ uu ( ω , m )=Φ uu ( ω ) · e − 2 π · w 2 · h 2 , where h = m/B (4.12)
Note that the Gaussian function is the square of the mean-flo w Gaussian w ak e men tioned in Eq.(4.8).
The non-dimensional width w is assumed to b e identical for the mean w ak e and the turbulent w ak e. The
one-dimensional sp ectrum Φ uu ( ω ) is that utilized b y Amiet [125] for turbulence-airfoil in teraction noise.
The p eak turbulen t kinetic energy (assumed to b e on the cen ter axis of the w ake) and the in tegral
length scale are determined empirically . Exp erimental observ ations by Ganz [117] and Wygnanski [116]
indicate that the w ak e turbulen t v elo cit y scales prop ortionally to the w ak e deficit v elo cit y u ( y ) defined
in Eq.(4.6):
u 2 ( y )=( K U tur b · u ( y )) 2
After a v eraging the quan tit y u 2 ( y ) o v er a blade passage w e obtain:
u 2 =
+ ∞
X
h = −∞
( K U tur b · u ( h )) 2 , see Eq.(4.8)
The in tegral length scales with the w ak e width and the blade passage stream tub e height.
Λ = K Ltur b · w · h
Qualitativ ely , this mo del correctly repro duces the increase of turbulence in tensit y and length scale as
the blade drag co efficien t or loading increases, whic h w as observ ed b y Ganz [117] and Camp [129]. The
empirical constan ts ha v e a fairly wide range of v alues rep orted in the literature: K U tur b v aries b et w een
0.2 and 0.4, and K Ltur b v aries b et w een 0.2 and 0.6. In this mo del, the turbulence is assumed to adapt
instan taneously to c hanges in the mean-w ak e flo w, so the history of turbulence is ignored. Moreo v er, the
coupling b et w een the deca y of the mean-flo w w ak e and the turbulence pro duction is not accoun ted for.
There is a p oten tial for an impro v ed analytical description of the w ak e deca y , whic h could for example
b e based on a standard t wo-equation turbulence model coupled to the integral boundary lay er equation.
4.4.2 W all-pressure fluctuations
The prediction of trailing edge broadband noise strongly dep ends on the appropriate definition of a source
strength to serv e as an input for the acoustic mo del. The quantit y usually considered is the lo cal surface
pressure sp ectrum, whic h can b e obtained b y one-p oin t measuremen ts with a surface pressure sensor
lo cated near the trailing edge. The correlation prop osed is the follo wing:
Φ pp ( ω ) =  1
2 ρ 0 W 2
0 C D  2
· δ 1
W 0 · F ( z ) , where F ( z ) = 1
1 + z + 0 . 217 z 2 + 0 . 00562 z 4

Chapter 4. Unsteady aero dynamics 70
The reduced frequency z is defined similarly to the previous section. The Strouhal n um b er is based on
the b oundary la y er displacemen t thic kness at the blade trailing edge: S t = f · δ 1
W 0 . The non-dimensional
sp ectrum F ( z ) is that giv en b y Willmarth and Ro os [130]. In their mo del, as in the mo del of Go o dy [131],
the drag co efficien t has a constant v alue C D = 0 . 0045 or C D = C f , whic h is suited for flat plate with
zero-pressure gradien t but not for loaded fan blades. An extension of Go o dy’s mo del was proposed by
Rozen b erg [37] to accoun t for a pressure-gradien t effect. Ho w ev er, the pressure fluctuations also scale
with the lo cal skin friction co efficien t C f near the trailing edge, leading to an unrealistic noise decrease
as blade separation o ccurs (in that case C f = 0). F or this reason, we ha ve c hosen the drag co efficien t C D
as scaling parameter: it increases contin uously as blade loading increases b ey ond the separation p oin t.
The quan tit y Φ pp is expressed in squared P ascal p er Herz.
F or the span wise correlation length, w e use at high frequencies the mo del of Amiet [132] based on the
w ork of Corcos [133]:
` ( ω )=1 . 7 U c
ω (4.13)
The parameter U c is the con v ection sp eed of the eddies, it lies b et w een 60 and 80 p ercent of the external
flo w v elo cit y outside of the b oundary la y er. The co efficien t 1.7 is empirical and ma y v ary b et w een 1.4
and 2.1 dep ending on the pressure-gradien t.
A less empirical mo del migh t b e deriv ed from the stream wise dev elopmen t of the b oundary la y er along
the blade. This approac h w as adopted b y Remmler et al. [134] to calculate the pressure fluctuations from
a RANS sim ulation. They started from the Poisson equation for the fluctuating pressure. The pro cedure
is based on the in tegral within the b oundary la y er in the direction normal to the w all of the stream wise
v elo cit y profile W ( y ) and turbulen t kinetic energy (whic h b oth can b e extracted from a RANS sim ulation).
An alternativ e approac h prop osed b y Glegg [135] also pro vides the abilit y to predict trailing edge noise
from RANS calculations.
4.5 Change of reference frame
The flo w p erturbations men tioned previously were expressed in the frame of reference lock ed to the blades
that create them. How ev er the in teraction mec hanism resulting in sound emission has to b e calculated in
the reference frame lo c k ed to the blades that generate sound. F or that reason, it is necessary to conv ert
the p erturbation from one reference frame to another. Let denote p ( t, θ ) the circumferential distribution
of flo w p erturbation expressed in the time domain.
p ω
m = 1
2 π T r ef
+ ∞
Z
t = −∞
2 π
Z
θ =0
p ( t, θ ) e i ( ω t − mθ ) dθ dt
This quan tit y can b e form ulated in a differen t reference frame mo ving with a sp eed Ω relativ e to the first
one. In that case, the circumferen tial p osition is expressed as ˜
θ = θ − Ω t . The flo w p erturbation b ecomes:
p ω
m = 1
2 π T r ef
+ ∞
Z
t = −∞
− Ω t +2 π
Z
˜
θ = − Ω t
p ( t, θ ) e i (( ω − m Ω) t − m ˜
θ ) d ˜
θ dt
Hence, w e obtain b y iden tification:
p ω
m = p ˜ ω
˜ m , where  ˜ m = m
˜ ω = ω − m Ω
Changing the reference frame is therefore equiv alen t to a frequency shift similar to the Doppler shift
of a sound source in rectilinear motion. It also means that the circumferen tial mo de decomp osition of
a p erturbation is necessary to change the reference frame. T ec hnically , the transformation ma y lead

Chapter 4. Unsteady aero dynamics 71
to negativ e v alues of the pulsation frequency; in that case the negativ e part can b e mirrored in to the
p ositiv e part b y c hanging the sign of the circumferen tial mo de and taking the complex conjugate. The
sp ectrum sho ws suc h a symmetry prop ert y b ecause the signal p ( t, θ ) is real. The con v ersion from negativ e
to p ositiv e frequencies reads:
p − ω
m =  p ω
− m  ∗ ⇒   p − ω
m   2 =   p ω
− m   2
An example is no w presen ted to illustrate the effect of transforming the reference frame on the
sp ectrum of turbulence. W e consider the w ak e turbulence sp ectrum of Eq.(4.12) written as a V on Karman
sp ectrum mo dulated b y the Gaussian-shap ed mean-flo w w ak e b ehind rotor blades. Figure 4.5 sho ws
the original sp ectrum in blue dashed line as observ ed in the rotor reference frame, the sp ectrum after
transformation in to the fixed reference frame is depicted b y the thic k red line. On the left-hand side of
the figure, the integral length scale Λ is small compared to th e blade spacing and on the righ t-hand side it
is m uc h higher, corresp onding to conditions with large flo w separation on the rotor blades. Strong h umps
app ear in the sp ectrum observ ed the fixed frame for the large in tegral length scale and they are cen tered
on the harmonics of the blade passing frequency (BPF). No h ump is observ ed for the small length scale.
The formation of h umps cen tered on the BPF harmonics is driv en b y t w o non-dimensional parameters:
• The primary parameter is the Strouhal n um b er based on the BPF: S t B P F ≡ Λ f r ot B
W 0 ∝ Λ
s . This
parameter is directly prop ortional to the ratio of in tegral length scale to blade spacing. If the
Strouhal n um b er is v ery small, typically below 0.02, the sp ectrum is hardly mo dified by c hanging
the reference frame. As the Strouhal n um b er increases, energy is progressiv ely transferred from the
lo w to the high frequencies, and from the circumferen tial mo de zero to mo des rotating in the same
direction as the rotor. If the Strouhal n um b er exceeds a v alue around 0.4, h umps arise near the
BPF-harmonics.
• The width w of mean w ak e driv es the deca y of BPF harmonics. The larger the width is, the faster
is the deca y . This usually o ccurs in combination with large v alues of the in tegral length scale.
Figure 4.5: Mo dification of a wak e turbulence sp ectrum through transformation of the reference frame
(left: mo derate length scale S t B P F = 0 . 2 , w = 0 . 07, righ t: large length scale S t B P F = 0 . 6 , w = 0 . 2)
4.6 Airfoil resp onse function
The next step on the w a y to w ard noise emission is to determine ho w the blades resp ond to an incoming
v elo cit y p erturbation (also called gust). This pro cess can b e regarded as a transfer function b et w een flo w
v elo cit y fluctuations as input and the surface pressure on the blade as output. W e will see in chapter 6
dedicated to the acoustic mo dels that the source strength is formed b y the force p er unit area exerted by

Chapter 4. Unsteady aero dynamics 72
the fluid on the blades. F or the present application, this c hordwise distribution of the force is the normal
comp onen t f ω
n ( ` ), at the pulsation frequency ω . This quan tit y can b e written as a lo cal pressure jump
∆ p and is prop ortional to the inflo w dynamic pressure:
f ω
n ( ` ) = ∆ p ( ω , ` ) = 1
2 ρ 0 W 2
0 C L ( ω ) h L ( ` )
The quan tit y C L ( ω ) is the unsteady lift co efficien t and h L ( ` ) is the non-dimensional distribution of lift
along the blade c hord, indep enden t of the frequency . The original mo del dev elop ed b y Sears [136] assumes
a c hordwise distribution in the form:
h L ( ` ) = π
2 r c − `
` , note that 1
c Z c
` =0
h L ( ` ) d` = 1
This is the distribution of the resp onse to a p erturbation propagating do wnstream (the p eak loading is
lo cated at the leading edge). The unsteady lift co efficien t is giv en b y the t w o-dimensional incompressible-
flo w linear theory:
C L ( ω )=2 π u n ( ω )
W 0
S ( ω )
where u n is the v elo city fluctuation normal to the airfoil c hord and S is the Sears function [136] for
incompressible flo w.
S ( ω ) = 1
√ 1+2 π σ , where σ = ω c
2 W 0
(4.14)
This is the lo w frequency appro ximation for the incompressible Sears function as giv en in Goldstein [137]
but mo dified to tak e reference at the airfoil leading edge instead of the mid-chord (explaining wh y the
phase of S is constan t and equal to 0: the airfoil resp onse to a gust is immediate at lo w frequencies).
F urther assumptions underla y this mo del:
• The blade is appro ximated b y an infinitely thin flat plate op erating at zero mean-flow incidence. The
effects of thic kness, camber, and mean loading ha v e b een in v estigated b y Goldstein and A tassi [138]
or Kersc hen et al. [139].
• The solution prop osed here is v alid for lo w v alues of M 0 · σ (lo w Mac h n um b er and low frequencies).
This parameter can b e easily written in terms of a non-dimensional acoustic wa v en um b er: M 0 · σ =
k c/ 2. So the problem of compressibilit y is closely related to the compactness of the source. As w e
will see in the c hapter 6 presen ting the acoustic mo dels, the source non-compactness is accoun ted
for via a c hordwise correlation function that dep ends on the non-dimensional wa v en um b er k c .
• The gusts impinge the blade normally to its span or the blade is unsw ept. Solutions for oblique
gusts ha v e b een deriv ed b y Graham [140] or Adamczyk [141]. Moreov er, only harmonic gusts are
considered: this a reasonable assumption in the case of w ak es or turbulence, which deca y v ery slo wly
during their con v ection, but the exp onen tial deca y of p oten tial fields w ould require a generalized
mo del as done b y Kemp and Homicz [142].
• The presence of neigh b ouring blades and their effect on the unsteady lift resp onse function are
neglected. Significan tly more complex mo dels called cascade resp onse mo del are a v ailable in the
literature (see for example the w ork of Glegg [17] or P osson [38]).
.
The v elo cit y fluctutation normal to the airfoil is obtained b y appropriate pro jection of the stream wise
and transv erse v elo cit y comp onen ts u and w :
u n = u · sin( χ − β r el ) + w · cos( χ − β r el )
The second term is neglected during the calculation as the stream wise v elo cit y p erturbations inside a
w ak e are usually m uc h higher than the transv erse ones. The term inside the sine and cosine functions is

Chapter 4. Unsteady aero dynamics 73
the difference b et w een the stagger angle of the excited blades and the flo w angle expressed in the relativ e
system of the w ak e-generating blade ro w (whic h is appro ximately the stagger angle of the upstream blade
ro w). This difference is around 90 degrees on a coun ter-rotating fan stage and around 60 degrees on a
rotor-stator stage, th us more excitation ma y b e exp ected from a coun ter rotating stage.
F or broadband noise, the unsteady lift co efficien t is related to the sp ectral densit y of v elo cit y fluctu-
ations:
| C L | 2 ( ω ) = 4 π 2 Φ uu ( ω )
W 2
0 | S ( ω ) | 2
Note that w e assumed Φ u n u n = Φ uu whic h is a reasonable appro ximation in isotropic turbulence.

Chapter 5
Extrap olation of meanline data
5.1 Need for a radial extrap olation
As describ ed in the previous c hapters, the steady and unsteady aero dynamic calculations are based on
a meanline approac h. The information deliv ered b y this kind of approac h is sufficien t to get an estimate
of the o v erall aero dynamic p erformance of a fan, ho w ev er acoustic calculations require a more detailed
set of flo w data, ev en for the prediction of global trends. This is mainly due to the acoustic in terference
effects, whic h result from the sup erimp osition of sound w a v es emitted from sources distributed in space,
esp ecially in the radial direction. The in teference of sound w a v es ma y lead to v ery significan t reductions in
tonal noise lev els compared to the case with compact sources. This effect b ecomes more pronounced if the
amplitude of radial geometry v ariations are in the same order of magnitude as the acoustic w a v elength,
t ypically at high frequencies. F or that reason, a smo oth radial distribution of the source strength is
necessary and the flo w data obtained at meanline radius m ust b e extrap olated in radial direction. F or
broadband noise, this effect is less imp ortant due to the small correlation length of the flo w turbulence
structures, but there are still strong radial v ariations of the source strength due to the large differences
in Mac h n um b ers b et w een the tip and the h ub of a fan stage. W e will now detail the assumptions and
the mo dels used to extrap olate the meanline data.
5.2 Steady flo w v elo cities
The sound sp eed a 0 and the static air density ρ 0 are considered radially constan t. The radial v ariation of
the flo w v elo cities in the absolute reference frame is describ ed b y a p oten tial flo w mo del (see Grieb [86]).
This is also kno wn as free-v ortex design and assumes that no v ortices are shed b y the blade along the
radius. This assumption is acceptable for ducted rotors and stators with very small clearance at their
blade ends, but it may not be c orrect for op en rotors where a strong leak age of blade circulation is
observ ed to w ard the tip:
U ( r ) = Ω · r
W x = const
W t ( r ) = Γ
r
W 0 ( r ) = p W 2
x + ( W t ( r ) − U ( r )) 2
The circumferen tial v elo cit y of the blade is U and Ω is the rotation frequency in radians p er second. The
quan tit y Γ is the radially constan t blade circulation and can b e calculated from Eq.(2.6) at the meanline
radius. Com bining this equation with Eq.(2.5) sho ws that this mo del corresp onds to a radially constan t
en thalp y rise ∆ H t = Ω
2 π B Γ, whic h is an acceptable appro ximation except near the h ub and near the tip
of an unducted rotor. The h ub region is usually unloaded on purp ose to av oid to o large secondary-flo w

Chapter 5. Extrap olation of meanline data 75
disturbances, and the tip of op en rotors cannot pro duce lift. The relativ e v elo cit y magnitude W 0 will b e
used as scaling parameter to extrap olate the p erturbation v elo cities.
5.3 Blade geometry
The extrap olation of the geometry is based on the radial v ariations of the flo w v elo cities. The blade
stagger angle χ is extrap olated assuming that the blade op erates appro ximately zero incidence, i.e. the
blades are aligned with the inflo w angle:
tan χ = W t − U
W x
In practical cases, it is observ ed that rotors op erating with an axial inflow ha v e t wisted blades to satisfy
the zero-incidence conditions. Stators or rotors lo cated do wnstream of the first rotor op erate with non-
axial inflo w: in that case, w e consider un t wisted blades as the inflo w angle in the relativ e reference framce
is reasonably constan t in radial direction.
tan χ ( r ) =  tan χ M L · r
r M L , for twisted blades
tan χ M L = const , for un t wisted blades
The v ariations of the blade stagger angle determine the t wist. The axis along whic h blades are t wisted can
b e sp ecified through the quan tit y PCA, whic h stands for pitc h c hange axis and indicates the c hordwise
p osition of the axis as a fraction of the blade chord. A v alue around 0.5 is usual. The radial v ariation
of the blade c hord is linear and describ ed by the c hord tap er ratio (CTR), whic h is the ratio of the tip
c hord to the h ub c hord. The radial v ariations of solidit y are giv en b y:
σ ( r ) = B · c ( r )
2 π r
The blade thic kness is calculated similarly based on the ratio of the tip to h ub relativ e thic kness (TTR).
F or a ducted rotor-stator stage, the v alue of CTR is usually sligh tly ab ov e 1, and TTR is around 0.5. F or
a CR OR, the geometry of the blades sho ws a more pronounced tap er. The p osition of the leading and
trailing edge of eac h blade ro w is calculated from the radial t wist and the sw eep and lean angles, whose
v alues are assumed constan t in radial direction.
5.4 Lift, drag and unsteady flo w v elo cities
The lift and drag co efficien ts, and the quadrup ole term determine the rotor-alone tonal noise lev els, and
to some exten t the strength of the p oten tial field at the origin. W e prop ose the following simple models:
C L = const
C D = const
ρu 2
` ( r ) ∝ W 2
0
The quadrup ole term is assumed to scale with the square of the relative flo w velocity W 0 . The unsteady
p erturbation v elo cities and surface pressure fluctuations are also assumed to scale with a certain p o w er
of the relativ e flo w v elo cit y , w e prop ose:
Φ pp ( r ) ∝ W 4
0
Φ uu ( r ) ∝ W 2
0 · σ
u w ak e ( r ) ∝ W 0 · σ
u pf ield ( r ) ∝ W 0 · √ σ

Chapter 5. Extrap olation of meanline data 76
The flo w p erturbation also scales with some p o w er of the solidit y b ecause the drag co efficien t of blades is
assumed radially constan t. It should noted that the flo w p erturbations due to endw all losses are not tak en
in to accoun t and cannot therefore con tribute to noise emission. The amplitude of velocity fluctuations
induced in the w ak e and p oten tial field deca y axially but are assumed constant radially . Their phase,
ho w ev er, features significan t radial v ariations as a result of the tilting of the w ak es. This latter asp ect
is mo delled in more details in the next chapter dedicated to Acoustics, and is resp onsible for substan tial
sound w a v e cancellations in tonal in teraction noise.

Chapter 6
Acoustics
6.1 Mo delling approac h
The purp ose of this chapter is to presen t a fully analytical form ulation of noise generated by rotating
blades. A unified approac h is adopted to handle v arious engine configurations suc h as ducted fans
or coun ter-rotating op en rotors, and to treat the tonal and broadband noise comp onen ts on the same
mo delling basis. Apart from the abilit y of predicting noise v ery rapidly , the presen t approac h also
pro vides a theoretical description of the differen t noise generation and propagation mec hanisms with a
sp ecial atten tion giv en to their relation to the aero dynamic parameters and mean flo w quan tities. The
previous c hapters w ere dedicated to Steady and Unsteady Aero dynamics, w e no w treat and describ e the
Acoustics as a b y-pro duct of the aero dynamics of the engine.
This is essen tially the p oin t of view tak en b y the acoustic analogy of Lighthill [143]. F rom the
con tin uit y and momen tum equations, whic h basically describ e an y flo w motion, Ligh thill deriv ed a relation
equiv alen t in form to a w a v e equation in a medium at rest:
∂ 2 ρ
∂ t 2 − a 2
0 ∇ 2 ρ = ∂ 2 T ij
∂ x i ∂ x j , (6.1)
where ρ is the densit y , a 0 the propagation v elo cit y , and T ij is the Ligh thill’s stress tensor. By analogy he
iden tified the left-hand term of Eq.(6.1) as the propagation of sound and the righ t-hand term comp osed
of p erturbation flo w quan tities as the source of aero dynamic sound. His w ork marks the birth date of
Aeroacoustics as a distinct researc h domain. As explained b y Morfey [48] and Goldstein [137], the funda-
men tal asp ect of the acoustic analogy is the causal link established b et w een noise and the aero dynamic
sources, and the implicit assumption, for the theory to b e of an y practical use, that the sources can b e
ev aluated a priori indep enden tly of noise. Of course, this approac h is not rigorously exact, and represen ts
an in terpretation of the con tin uit y and momen tum equations, and there exist situations in which the
propagation and generation of sound are in terrelated without an y clear causal link. Still, the acoustic
analogy has b een v ery successful in enhancing our ph ysical understanding of aero dynamic sound and
forms the theoretical framew ork for the presen t mo delling approach.
More precisely , we adopt the solution of the acoustic analogy that w as deriv ed b y Goldstein [144, 137].
This is a generalized v ersion of the Ffo w cs-Williams & Ha wkings solution of Ligh thill’s equation with
mo ving solid b oundaries. Goldstein utilizes Green’s functions to b etter distinguish b et w een acoustic
propagation and sound sources, and to adapt the solution to the sp ecific problem considered. Through
the c hoice of the appropriate Green’s function, it is p oss ible to obtain a solution for the sound pressure
in the free space or in a duct. F urthermore, w e c ho ose a form ulation of Goldstein’s w ork in the frequency
domain, as first applied b y Hanson [25]. The frequency-domain form ulation is prefered to the time-domain
form ulation (see for example F arassat [28]) as it pro vides more insigh t in to the driving parameters of noise.
A brief history of the rotating-blade noise prediction is presen ted sc hematically in Fig. 6.1. Prop eller
noise is an old problem that has attracted atten tion since the 30’s. Gutin [145] w as the first to publish

Chapter 6. Acoustics 78
a quan titativ e prediction of prop eller noise, where he identiﬁed that sound w as pro duced b y the rotation
of steady dip oles related to the blade forces. Accompan ying the rapid gro wth of turb o jet civil aircraft,
Aeroacoustics researc h explo ded during the 50’s, relying on Lighthill’s pow erful concept of the acoustic
analogy and its extensions to mo ving ﬂo w and solid b oundaries [146]. But the ﬁrst turb o jets of that time
w ere single-stream engines with no b ypass, so the jet w as the ma jor source of noise and represented the
main fo cus of aeroacoustic researc h. As tec hnology shifted to larger, t w o-stream engines with a b ypass
(this w as an impro v emen t in terms of b oth fuel consumption and noise), the relative importance of jet
noise compared to the other sources decreased and more atten tion w as paid to rotating-blade noise. The
w ork of Ffo w cs-Williams and Ha wkings [147] marks the return to the old prop eller noise problem, no w
tac kled from the p oin t of view of the acoustic analogy . This new approach enabled to iden tify and to
separate the aero dynamic mec hanisms resp onsible for the noise emission, and under whic h conditions
some of them can b e neglected for the sak e of computational eﬃciency . Goldstein [144] extended Ffo w cs-
Williams & Ha wkings’ equation to problems with b oundary conditions like in a duct.
Figure 6.1: History of rotating-blade noise prediction based on the acoustic analogy
Sev eral studies after Hanson [25] also adopted the frequency-domain form ulation of Goldstein’s solu-
tion, they are presented in T able 6.1 and sorted according to the sp eciﬁc problem they tackle. Except for
the in-duct tonal self noise (sup ersonic rotor noise asso ciated to non-linear sho c k propagation, prediction
mo dels not based on the acoustic analogy can b e found in [107, 148]), all problems ha v e b een treated
in this w a y b y at least one researc h team. Ho w ev er, b ecause of some sp eciﬁcit y asso ciated to eac h prob-
lem, the go v erning equations are generally expressed in v ery diﬀeren t forms (including diﬀeren t notation
con v en tions), which mak es the comparison b et w een the problems as w ell as the general understanding
of rotating-blade noise more diﬃcult. The presen t w ork pro vides similar analytical form ulations for the
in-duct and free-ﬁeld, tonal and broadband, interaction and self-noise problems. Some atten tion will
b e giv en to the similarities that can b e observ ed when treating problems of op en rotor and ducted fan
noise. In the prop eller problem the sound is radiated in the free space while it propagates inside a duct
b efore radiating to the far ﬁeld in the fan noise problem. The presence of a duct mo diﬁes not only the
propagation but also the generation of sound.
F urthermore, w e will extensiv ely use the cylindrical system of co ordinates, which is w ell suited for
rotor noise problems. In t his system of co ordinates, the 2 π -p erio dicit y in circumferen tial (or azim uthal)

Chapter 6. Acoustics 79
direction allo ws to decomp ose the pressure in to a discrete sum of F ourier comp onents called azim uthal
mo des. W e will consider a purely axial uniform flo w in the reference frame fixed to the engine. The
b oundary conditions will b e either those of the free space (with the far-field appro ximation) or those of
an infinitely long ann ular duct with hard w alls. The presence of the blades as sound scatterers will b e
ignored.
problem free-field in-duct
tonal self noise Hanson(1980) [25] -
P arry&Crigh ton(1989) [44] -
Hanson(1984) [26] V en tres(1982) [6]
tonal in teraction noise P arry(1988) [43] Mey er&Envia(1996) [7]
Carazo(2011) [39]
broadband self noise Zhou(2004) [51] Glegg(1998) [16]
Blandeau(2009) [52] Glegg(1993) [15]
broadband in teraction noise Kingan(2012) [54] Joseph(2003) [50]
Lo wis(2006) [149]
V en tres(1982) [6]
Mani(1997) [150]
T able 6.1: Some applications of the frequency-domain form ulation of Golstein’s v ersion of the acoustic
analogy
As men tioned in the previous paragraph, alternative approac hes not based on the acoustic analogy
of Ligh thill ma y b e more appropriate in situations where there is a strong coupling b etw een noise propa-
gation and noise generation. F or example the w ork of A tassi [151] presen ts some general asp ects of this
approac h and a practical application. In his b o ok on Aeroacoustics, Goldstein [137] dedicates an en tire
c hapter to alternativ e theories based on the solution of the linearized v orticit y-acoustic field equations.
6.2 Assumptions
The acoustic mo dels deriv ed in this c hapter are exact solutions of a simplified problem. The assumptions
underlying the simplification of the problem are listed b elow:
• Uniform axial flo w: This assumption is linked to the lev el of accuracy used for mo delling the prop-
agation of sound in the mo ving medium, whic h is represen ted b y the Green’s function. Ph ysically ,
an y deviation from this mo del results in sound scattering, but in the acoustic analogy approac h,
this effect is in terpreted as a source (t ypically con tained in the quadrup ole term). The assumption
of uniform axial flo w neglects in particular the in terstage swirl, the tip v ortex flo w of op en rotors
and the sup ersonic regions and sho c ks on the suction side of the blades.
• F or unducted configurations, the far-field approximation of the free-space solution is tak en. F or
ducted configurations, the duct is formed b y infinitely long and hard walls with no sound absorption.

Chapter 6. Acoustics 80
As a result no reflections at the end are considered. Moreo v er, the presence of the blades as sound
scatterers is not mo delled, which means that cascade resonance and shielding effects are neglected.
• Thin and sligh tly cam b ered airfoils: the sources lo cated on the blades all ha v e the same orien tation
along the c hordline. Blade of turbines, at a rotor h ub with lo w h ub-to-tip ratio, or stators of highly
loaded stages depart significan tly from this assumption. Ho w ev er the blades ma y b e t wisted in
radial direction, and ha v e a sw eep and a lean angle.
• Small angle of attac k: the blades and the inflo w are orien ted in the same direction.
• Radial forces and radial flo w comp onen ts are neglected. This is acceptable as long as the blade
lean angle is mo derate.
• All blades of a giv en rotor or stator are geometrically iden tical and equally spaced in circumferen tial
direction and lo cated at the same axial p osition.
• Small turbulence length scale: the spanwise and circumferen tial correlation lengths are iden tical
and considered small compared to the duct heigh t and to the blade spacing, whic h means that the
blades are uncorrelated for the broadband noise comp onent.
• T a ylor’s turbulence h yp othesis: the turbulence is considered frozen as it conv ects o v er the blade,
whic h corresp onds to infinite correlation length in streamwise direction.
6.3 Noise propagation
6.3.1 The con v ectiv e and fly o v er problems
The problem of sound emitted b y an engine mo ving relativ e to a medium can b e usually form ulated from
t w o differen t p oin ts of view: W ells and Han [152] distinguish b et w een the mo ving-medium problem, which
w e call here the con v ectiv e problem, and the mo ving-observ er problem, also called flyo ver problem. As
the presen t w ork is basically dealing with the emitted sound p o w er at giv en engine op erating conditions,
w e adopt in the presen t w ork the form ulation of the con v ectiv e problem. The fly o v er problem form ulation
is more suited for describing the noise emission along a fligh t path to w ards a fixed observ er.
In the con v ectiv e problem, the observ er and the engine are steady in a mo ving medium with uniform
axial Mac h n um b er M x . In the flyo v er problem, the observ er is fixed in a steady medium while the engine
is mo ving with a constan t axial Mac h n um b er M x . Although b oth problems are equiv alen t, the con vectiv e
problem offers more simple form ulations for noise generation b ecause the emission and observ ation angles
do not dep end on time and the frequency emitted b y the source is iden tical to that p e rceiv ed b y the
observ er as the Doppler shift is remo v ed. Moreo v er, according to W ells and Han [152], this approach
presen ts some adv an tages in terms of computational efficiency . A more recen t w ork b y W ec km ueller et
al. [153] also explains wh y the con v ectiv e form ulation leads to reduced computation time and a higher
accuracy for tonal noise problems. Figure 6.2 illustrates the con v ectiv e and fly o v er problems, and precises
the ph ysical meaning of the propagation angles.

Chapter 6. Acoustics 81
Figure 6.2: Tw o diﬀeren t p oints of view for form ulating the sound emission
In the con v ectiv e problem, the emission angle ψ e and the observ ation angle ψ corresp ond to the
phase and energy propagation angles in the mo ving medium, resp ectiv ely . In the ﬂy o v er problem, as the
medium is steady , these angles are identical and a re represen ted b y ψ e . In this case, ψ is an apparent
propagation angle as w ould b e p erceiv ed b y the observ er. The relation b et w een these angles is giv en b y
Rienstra and Hirsc h b erg [154]:
sin ψ
 1 − M 2
x sin 2 ψ
= sin ψ e
D
cos ψ
 1 − M 2
x sin 2 ψ
= M x +c o s ψ e
D
where the quan tit y D is called the Doppler factor:
D =1 + M x cos ψ e (6.2)
These angles are comprised b et w een 0 (rearw ard radiation) and π (forw ard radiation), and are deﬁned
as presen ted in Figure 6.2. It should b e noted that the results given b y sev eral authors (Hanson [25],
P arry [44]) are form ulated in terms of the emission angle ψ e , whereas we c ho ose here the observ ation
angle ψ , whic h is more appropriate in the con v ectiv e problem.
6.3.2 Disp ersion relation
The disp ersion relation that describ es the sound propagation in a medium with uniform and purely axial
ﬂo w has b een deriv ed b y Morfey [155] from the linearized Euler equations applied to a harmonic sound
w a v e. In the cylindrical co ordinate system, t he follo wing relation is established b etw een the axial and
radial w a v en um b ers:
k 2 = k 2
r +( 1 − M 2
x ) k 2
x +2 M x kk x (6.3)

Chapter 6. Acoustics 82
As the disp ersion relation is a prop ert y of the medium, it is equally v alid in the free space and in a duct.
Based on the disp ersion relation in a steady medium, we define an apparen t w a v e n um b er:
˜
k 2 ≡ k 2
x + k 2
r = ( k − M x k x ) 2 = k 2
D 2 (6.4)
6.3.3 Sound propagation
Because the b oundary conditions imp ose certain w a v e forms for the problem considered, the free-space
and in-duct w a v en um b ers are usually form ulated in differen t w a ys. Ho w ev er, w e will sho w that these
form ulations presen t a strong similarit y .
F ree-field propagation
In the free space, no constrain ts are imp osed on the w a v en um b ers. The axial and radial w a v en um b ers
are giv en b y:
k r = k sin ψ
q 1 − M 2
x sin 2 ψ
= ˜
k sin ψ e
k x = k
β 2 
 − M x + cos ψ
q 1 − M 2
x sin 2 ψ 
 = ˜
k cos ψ e
(6.5)
The complex F ourier co efficien t of the pressure is written as a sum of azimuthal modes:
p ω ( x, r , θ ) =
+ ∞
X
m = −∞
p ω
m ( x, r ) e imθ
In-duct propagation
In a duct, the acoustic v elo city is specified at the duct walls depending on the wall properties. This implies
that only certain w a v e patterns are able to propagate within the duct without deca ying. These are called
mo des and are describ ed through the circumferen tial (also azim uthal) order m and the radial order n ,
whic h indicate the n um b er of zeros of the acoustic pressure in circumferential and radial directions,
resp ectiv ely . In case of hard walls, the normal comp onen t of the acoustic v elo city m ust b e zero at the
w all. This imp oses the radial and axial wa ven um b ers to assume discrete v alues in the form:
k r = σ mn
R
k ±
x = k
β 2 ( − M x ± α mn )
(6.6)
The quan tit y σ mn represen ts the ( n + 1) th zero of the first deriv ativ e of the Bessel function J m (in case of
a hollo w duct). The sup erscript sign ± ab o v e the axial w a v en um b er sp ecifies the direction of propagation
of the sound w a v e: + for do wnstream propagation, and - for upstream propagation. The quan tit y α mn
is called the mo de cut-on factor and is defined as:
α mn ≡ r 1 − (1 − M 2
x )  σ mn
k R  2 (6.7)
F or propagating mo des, the cut-on factor is a real num b er b et w een 0 and 1. F or non-propagating mo des,
the cut-on factor is purely imaginary . In that case, the axial w a v en um b er has a non-zero imaginary

Chapter 6. Acoustics 83
comp onen t whic h is resp onsible for the exp onen tial deca y of the pressure amplitude along the duct. The
F ourier co efficien t of the acoustic pressure taken at the frequency ω is usually decomposed in the form:
p ω ( x, r , θ ) =
+ ∞
X
m = −∞
+ ∞
X
n =0
A ±
mn
J m ( k r r ) + Q mn Y m ( k r r )
√ F mn
e ik ±
x x + imθ
The pressure asso ciated to a mo de of azim uthal order m (whic h will b e called mo dal pressure from no w
on) is giv en b y:
p ω
m ( x, r ) =
+ ∞
X
n =0
A ±
mn
J m ( k r r ) + Q mn Y m ( k r r )
√ F mn
e ik ±
x x (6.8)
Note that the quan tities p ω ( x, r , θ ), p ω
m ( x, r ) and A ±
mn are complex n um b ers. J m and Y m are the Bessel
and Neumann functions, resp ectiv ely . The quantit y Q mn is equal to zero for a hollo w duct (no cen ter-
b o dy) and is defined as:
Q mn ≡ − J 0
m ( σ mn )
Y 0
m ( σ mn )
The quan tities F mn are normalization factors in tro duced to v erify the follo wing equation:
1
R 2 Z R
η R
| J m ( k r r ) + Q mn Y m ( k r r ) | 2
F mn
r d r = 1 (6.9)
they are related to the factors C mn of Holste [156] b y 2 π R 2 F mn = C mn . Then:
F mn = 












1
2 [1 − η 2 ], for the plane w a v e m = n = 0
1
2  1 − m 2
σ 2
mn  ( J m ( σ mn ) + Q mn Y m ( σ mn )) 2
−  η 2 − m 2
σ 2
mn  ( J m ( η σ mn ) + Q mn Y m ( η σ mn )) 2  , otherwise
(6.10)
The w a v en um b ers presen ted in Eq.(6.6) can also b e formulated in terms of the mo dal energy propagation
angle ψ ±
mn whic h w as previously used b y the authors [61] (and denoted χ ±
x ) to compute the tra jectory of
ra ys and the n um b er of b ounces exp erienced b y a mo de propagating in a duct:
k r = k sin ψ ±
mn
q 1 − M 2
x sin 2 ψ ±
mn
k ±
x = k
β 2 
 − M x + cos ψ ±
mn
q 1 − M 2
x sin 2 ψ ±
mn 

(6.11)
The analogy with the free-field axial and radial w a v en um b ers giv en in Equation (6.5) is obvious when
iden tifying the in-duct mo dal energy propagation angle ψ ±
mn with the free-field observ ation angle ψ . The
angle ψ ±
mn is one of the t w o angles sp ecifying the propagation of a ray in a duct. The equiv alence of
mo des and ra ys, particularly relev an t at high frequencies, w as prop osed by Chapman [157]. According
to his w ork, every cut-on mode can b e describ ed b y a p olar and an azim uthal angle, a mo de ma y b e
mathematically written as a bundle of ra ys and vice v ersa. The energy propagation angle in the case of
a purely axial flo w (no swirl) can b e related to the phase propagation angle ζ ±
mn and to the mo de cut-on
factor α mn :
cos ψ ±
mn = M x + cos ζ ±
mn
p 1 + M 2
x + 2 M x cos ζ ±
mn
= ± α mn p 1 − M 2
x
p 1 − α 2
mn M 2
x
, where  0 ≤ ψ +
mn < π / 2
π / 2 ≤ ψ −
mn ≤ π

Chapter 6. Acoustics 84
This is also the angle at whic h the p eak of radiation is lo cated in the far field with the same uniform
flo w inside and outside of the duct. This equation is equiv alent to Equation 27 in the paper by Rice et
al. [158]. The phase propagation angles and their range of v alues are giv en b y:
cos ζ ±
mn = − M x ± α mn
1 ∓ α mn M x
, where  0 ≤ ζ +
mn ≤ ζ −
mn
π / 2 ≤ ζ −
mn ≤ π
It should b e noted that the case ζ +
mn > π / 2 is p ossible and corresp onds to w a v e fron ts propagating up-
streams whereas the energy propagates in the do wnstream direction. The caustic radius r mn , in tro duced
b y Chapman [157], will b e of some use:
r mn = | m |
σ mn
R (6.12)
The phase v elo cit y c ζ , and the group v elo cit y c ψ are giv en b y:
c ζ = a 0 (1 + M · n ζ ) n ζ
c ψ = a 0 ( M + n ζ )
where M = M x e x
where n ζ is the unit y v ector of the phase v elo cit y . In the outer region of the duct defined by r > r mn ,
Chapman sho w ed that the propagation of mo des can b e mathematically describ ed b y the piecewise linear
propagation of ra ys. In that region, the radial wa ven um b er k r can b e decomp osed into an azim uthal and
a purely radial comp onen t. The phase velocity v ector can then assume the form:
n ζ ( r ) = k x
˜
k e x + k θ ( r )
˜
k e θ + p k 2
r − k θ ( r ) 2
˜
k e r
where k θ ( r ) = m
r is a circumferen tial w a v en um b er. This relation is defined only for cut-on mo des and
in the outer part of the duct where r is larger than the caustic radius. The angles ψ ±
mn and ζ ±
mn can b e
calculated from:
cos ψ ±
mn = n ψ · e x
cos ζ ±
mn = n ζ · e x
6.3.4 Ov erall sound p o w er
The magnitude of the free-field and in-duct pressure cannot b e directly compared as the sound intensit y
is constan t in a hard-w alled duct while it deca ys with the square of the distance to the source in the free
space (pro vided the dissipativ e effects are negligible). An appropriate quan tit y suited for quan titativ e
comparisons is the o v erall mo dal sound p o w er Π ω
m emitted b y the source at a giv en azim uthal mo de m
and pulsation frequency ω .
Acoustic in tensit y
T o compute the acoustic in tensit y in a uniform medium at a large distance from the source, Morfey [159]
indicates that the geometric acoustic appro ximation can b e used. According to this, the acoustic in tensit y
v ector I is obtained from the Blokhin tsev in v arian t [160] which states that acoustic energy is constan t
along a ra y tub e :
I =
1
2 | p ω | 2
ρ 0 a 2
0 D c ψ
where p ω is the p eak amplitude of the pressure, D is the Doppler factor defined in Eq.( 6.2), and c ψ the
ra y or group v elo cit y v ector (p ointing in the direction of acoustic energy propagation). The sound p o w er
Π ω radiated through a surface S is giv en b y:
Π ω = Z Z
S
( I · n S ) dS
where n S is the unit y v ector normal to the surface elemen t dS .

Chapter 6. Acoustics 85
F ree-ﬁeld sound p o w er
In the free ﬁeld, the o v erall sound p ow er is obtained after in tegration of the sound in tensit y o v er a sphere
of radius ρ cen tered on the engine as sho wn in Figure 6.3.
Π ω =
2 π

θ =0
π

ψ =0
1
2 | p ω | 2
ρ 0 a 2
0 D ( c ψ · n S ) ρ 2 sin ψd ψd θ
The Doppler factor is written as D = (1 − M 2
x ) √ 1 − M 2
x sin 2 ψ
√ 1 − M 2
x sin 2 ψ − M x cos ψ .
The pro jection of the group velocity v ector onto a spherical surface is: c ψ · n S = a 0 (sin ψ e sin ψ +( M x +c o s ψ e )c o s ψ )=
a 0 · 1 − M 2
x
√ 1 − M 2
x sin 2 ψ − M x cos ψ .
Using the energy conserv ation prop ert y of the F ourier transform (kno wn as P arsev al’s theorem or Ra yleigh’s
iden tit y), the in tegral along θ can b e written as a discrete sum of circumferential modes:
2 π

θ =0
| p ω | 2 dθ =2 π
+ ∞

m = −∞ | p ω
m | 2
The ﬁnal expression for the o v erall mo dal sound p o w er emitted in the free ﬁeld is:
Π ω
m =2 π
π

ψ =0 # 1
2 | p ω
m | 2
ρ 0 a 2
0 D ( c ψ · n S ) $ ρ 2 sin ψd ψ (6.13)
Figure 6.3: In tegration surface for calculation of free-ﬁeld sound p o w er
In-duct sound p ow er
In the ducted case, the o v erall sound p o w er is calculated through in tegration o ver t wo duct cross-sections
S − and S + lo cated upstream and do wnstream of the fan, resp ectiv ely . This case is depicted in Figure 6.4.
The in tegrand con tains the amplitudes of the mo des propagating in the upstream and do wnstream direc-
tions. The sound p o w er carried b y a cut-on mo de (m,n) of amplitude A ±
mn is deriv ed b y Morfey [155]. No
sound p o w er is propagated by cut-oﬀ modes. The sound p o w er carried b y all cut-on mo des of azimuthal
order m is of the form:
Π ω
m ± =
n max

n =0
πR 2 | A ±
mn | 2
ρ 0 a 0
α mn C ±
mn (6.14)

Chapter 6. Acoustics 86
The quan tit y α mn is the mo de cut-on factor deﬁned in Equation (6.7). The co eﬃcien ts C ±
mn , called
mo dal con v ection factors, dep end on the axial Mac h n um b er and represen t the energy ampliﬁcation or
atten uation due to ﬂo w con v ection. The energy of do wnstream propagating mo des is increased due to
the presence of ﬂo w, whereas mo des propagating against the ﬂow carry less energy .
C ±
mn = (1 − M 2
x ) 2
(1 ∓ α mn M x ) 2
Figure 6.4: In tegration surface for calculation of the in-duct sound p o w er
The sound p o w er of a mo de (m,n) transmitted at the duct ends into the far ﬁeld can b e appro ximated
b y the mo dal transmission co eﬃcien t prop osed b y Morfey [155]:
 Π ω
mn ±  tr ansmitted =Π
ω
mn ± · T mn ,w h e r e T mn = 4 α mn
(1 + α mn ) 2 (6.15)
Although the exact geometry of the duct ends cannot b e considered in this formula, it pro vides a realistic
trend: the mo des well abov e cut-oﬀ ( α mn ≈ 1) are nearly completely transmitted, but the mo d es near
cut-oﬀ ( α mn << 1) - despite their strong excitation at source and large pressure lev els inside the duct -
do not particularly sho w up in the far ﬁeld due to their p o or transmission at the duct end.
6.4 Noise generated b y rotating blades
6.4.1 Deriv ation of the mo dal pressure
F undamen tal equation for noise generation
The general purp ose of this c hapter is to establish an analytical form ulation of the acoustic pressure
generated b y a rotating blades as encoun tered in op en rotors or a rotor–stator fan stage. The deriv ation
of the mo dal pressure b egins with the fundamen tal equation go v erning the generation of sound in the
presence of solid b oundaries whic h w as deriv ed b y Goldstein [137]. The sound pressure p receiv ed at time
t b y an observ er lo cated at the p oin t z is expressed in Eq.(6.16) as a sum of three terms: a monop ole
(v olume displacemen t or thic kness noise), a dip ole term (surface force noise), and a quadrup ole term
(sources lo cated in a volume around the blades). Note that the sign of the monop ole and dip ole terms is
opp osed to that of Goldstein b ecause w e ha v e c hosen here the p oin t of view of the aero dynamicist who
usually considers the action applied b y the ﬂo w on the blades.
p ( z, t )= 1
T re f
+ ∞

−∞ # −  # ρ 0 W n
DG
Dt s
+ f i
∂G
∂z si $ dz s +  # T ij
∂ 2 G
∂z si ∂z sj $ dz s $ dt s (6.16)
The v ariables z s and t s denote the p osition of the source and the time of emission, resp ectiv ely . The
term f i = f i ( z s ,t
s )i st h e i th comp onen t of force p er unit area exerted b y the ﬂuid on the blade. f i has
the dimension of a pressure, and can b e steady or unsteady in the reference frame lo c k ed to the rotor.
W n = W n ( z s ,t
s ) is the ﬂo w v elo cit y normal to the blade surface elemen t, it is indep enden t of time in

Chapter 6. Acoustics 87
the rotating reference frame if blade vibrations are neglected. T ij = T ij ( z s , t s ) is the ( i, j ) th Ligh thill’s
stress tensor elemen t, whic h is the quadrup ole term con taining steady and unsteady terms in the rotor
reference frame suc h as flo w turbulence passing o v er the blades. The Green’s function corresp onding to
the problem considered (duct or free space) is G = G ( z , t, z s , t s ) and has the dimension of an in v erse
length. The reference time T r ef is in tro duced to matc h the dimension on b oth sides of Eq.(6.16).
F orm ulation in the frequency domain and cylindrical co ordinates
The sound pressure and the Green’s function are decomp osed in F ourier series in the frequency domain
and in the azim uthal direction:
p ( z , t ) = T r ef
2 π
+ ∞
Z
−∞
+ ∞
X
m = −∞
p ω
m ( x, r ) e imθ − iω t dω
G ( z , t, z s , t s ) = T r ef
2 π
+ ∞
Z
−∞
+ ∞
X
m = −∞
G ω
m ( x, r , x s , r s ) e im ( θ − θ s ) − iω ( t − t s ) dω
The quan tities p ω
m and G ω
m ha v e the same dimension as p and G . F urthermore, w e assume that the mo dal
Green’s function can b e written as follows:
G ω
m ( x, r , x s , r s ) = g ω
m ( x, r , r s ) e − ik x x s
An auxiliary quan tit y ℵ is in tro duced to simplify the notations:
ℵ ≡ g ω
m ( x, r , r s ) e − ik x x s − imθ s + iω t s
Using these expressions, Eq.(6.16) is no w form ulated in terms of mo dal quantities:
p ω
m ( x, r ) = 1
T r ef
+ ∞
Z
−∞  − Z Z  ρ 0 W n
D ℵ
D t s
+ f i
∂ ℵ
∂ z si  dz s + Z Z Z  T ij
∂ 2 ℵ
∂ z si ∂ z sj  dz s  dt s
The forces and flo w terms can b e decomp osed in to axial, circumferen tial and radial comp onen ts. The
relativ e imp ortance of the radial comp onen t w as assessed b y Carazo [39] to b e small on the tonal noise
pro duced b y a CR OR engine. Ho w ev er, these effects may become significant on highly leaned blades.
Here w e assume that the radial terms are negligible, so the surface force and v olume stress terms reduce
to:
f i
∂ ℵ
∂ z si
= f x
∂ ℵ
∂ x s
+ f θ
1
r s
∂ ℵ
∂ θ s
T ij
∂ 2 ℵ
∂ z si ∂ z sj
= T xx
∂ 2 ℵ
∂ x 2
s
+ 2 T xθ
1
r s
∂ 2 ℵ
∂ x s ∂ θ s
+ T θ θ
1
r 2
s
∂ 2 ℵ
∂ θ 2
s
Using the assumption of uniform and purely axial flo w in the whole domain, the deriv ativ es of the
auxiliary quan tit y ℵ are giv en b y:
D ℵ
D t s
= ∂ ℵ
∂ t s
+ W x
∂ ℵ
∂ x s
= i ( ω − k x W x ) ℵ
∂ ℵ
∂ x s
= − ik x ℵ
1
r s
∂ ℵ
∂ θ s
= − i m
r s ℵ = − ik θ ℵ
After injecting these expressions, the equation for the mo dal pressure b ecomes:
p ω
m ( x, r ) = 1
T r ef
+ ∞
Z
−∞ 
 − i R R ρ 0 W n ( ω − k x W x ) ℵ dz s
+ i R R ( f x k x + f θ k θ ) ℵ dz s
− R R R ( T xx k 2
x + 2 T xθ k x k θ + T θ θ k 2
θ ) ℵ dz s 
 dt s

Chapter 6. Acoustics 88
Expression of the source term in the rotating frame of reference
As the acoustic sources are lo c k ed to the rotor, it is con v enien t to describ e their p osition in the rotating
frame of reference attac hed to it. The source p osition ˜ z s reads:
˜ z s = 

˜ x s
˜
θ s
˜ r s 
 = 

x s
θ s − Ω t s
r s 

Th us, the auxiliary quan tit y ℵ can b e written as:
ℵ = g ω
m ( x, r , r s ) e − ik x x s − im ˜
θ s + i ˜ ω t s
The quan tit y ˜ ω = ω − m Ω that app ears through this transformation is the frequency observ ed in the
rotating frame of reference lo c k ed to the rotor. Ω is the signed rotation sp eed of the rotor in [rad/s].
When the sources rotate, a shift in frequency is observ ed similarly to that o ccuring with translating
sources, the factor 1 − m Ω
ω is analog to the Doppler factor defined in Eq.(6.2). After switc hing the time
and space in tegrals, the mo dal pressure is then written as follo ws for the thic kness, surface force and
p oten tial flo w terms resp ectiv ely:
p ω
m ( x, r ) =





















− i R R " ρ 0 W n ( ˜ z s )( ω − k x W x ) g ω
m ( x, r , r s ) e − ik x x s − im ˜
θ s 1
T r ef
+ ∞
R
−∞
e i ˜ ω t s dt s # d ˜ z s
+ i R R " k i g ω
m ( x, r , r s ) e − ik x x s − im ˜
θ s 1
T r ef
+ ∞
R
−∞
f i ( ˜ z s , t s ) e i ˜ ω t s dt s # d ˜ z s
− R R R " k i k j g ω
m ( x, r , r s ) e − ik x x s − im ˜
θ s 1
T r ef
+ ∞
R
−∞
T ij ( ˜ z s , t s ) e i ˜ ω t s dt s # d ˜ z s
The time in tegrals are in terpreted in the form of F ourier co efficients of the three terms respectively:
1
T r ef
+ ∞
Z
−∞
e i ˜ ω t s dt s = δ ( ˜ ω )
1
T r ef
+ ∞
Z
−∞
f i ( ˜ z s , t s ) e i ˜ ω t s dt s = f ˜ ω
i ( ˜ z s )
1
T r ef
+ ∞
Z
−∞
T ij ( ˜ z s , t s ) e i ˜ ω t s dt s = T ˜ ω
ij ( ˜ z s )
It is w orth men tioning here again that the normal v elo city and the Ligh thill’s tensor (or Reynolds tensor
if some sources neglected) for the v elo cit y field b ound to the rotor are b oth indep enden t of time in
the rotating frame of reference. The in tegral o v er the emission time t s can b e in terpreted as a F ourier
decomp osition. F or the thickness and potential flo w terms, the F ourier transform of the unity function
is a Dirac pulse cen tered on 0. This means that only frequencies v erifying ˜ ω = 0 or ω = m Ω are excited.
In that case, the frequency and the azimuthal mode are coupled. F or the surface force term, f ˜ ω
i is the
F ourier comp onen t of the force p er unit area exerted b y the fluid on the blades, observ ed at a frequency
ω in the fixed reference frame and at a frequency ˜ ω = ω − m Ω in the rotating reference frame. Note that
lik e for the thic kness and p oten tial flo w terms, the steady force (lift and drag) terms pro duce only mo des
v erifying ω = m Ω. F or the quadrup ole term, T ˜ ω
ij is the F ourier comp onen t of the Lighthill’s tensor.
The mo dal pressure p ω
m of the azim uthal mo de m at the pulsation frequency ω reads as follo ws for
eac h source:
p ω
m ( x, r ) = 




− i R R ρ 0 W n ( ˜ z s )( ω − k x W x ) g ω
m ( x, r , r s ) e − ik x x s − im ˜
θ s d ˜ z s
+ i R R  f ˜ ω
x ( ˜ z s ) k x + f ˜ ω
θ ( ˜ z s ) k θ  g ω
m ( x, r , r s ) e − ik x x s − im ˜
θ s d ˜ z s
− R R R  T ˜ ω
xx ( ˜ z s ) k 2
x + 2 T ˜ ω
xθ ( ˜ z s ) k x k θ + T ˜ ω
θ θ ( ˜ z s ) k 2
θ  g ω
m ( x, r , r s ) e − ik x x s − im ˜
θ s d ˜ z s
(6.17)

Chapter 6. Acoustics 89
H
Figure 6.5: airfoil system of co ordinates
The surface force term of Eq.(6.17) is similar to that found b y Lo wis and Joseph [149], whic h they deriv ed
in the particular case of in-duct noise generation.
Pro jection in the system of co ordinates of the airfoil
The in tegrals o ccurring in Equation (6.17) are now form ulated along the radial, chordwise and transv erse
(normal to the c hord line) directions. W e denote  the c hordwise p osition of the source on the blades, n
the transv erse p osition, ν the blade index, and pro ject the co ordinates x s and θ s of the source p osition
in the airfoil system of co ordinates (see Figure 6.5). W e assume that the blades are thin and sligh tly
cam b ered so that they can b e appro ximated b y ﬂat plates staggered at an angle χ from the engine axis.
F urthermore, all blades of a giv en blade ro w are iden tical (but can at this p oin t still b e lo cated at diﬀeren t
axial and azim uthal p ositions).
x s = x LE ( ν, r s )+  ( r s )c o s χ ( r s ) − n ( r s )s i n χ ( r s )
˜
θ s = ˜
θ LE ( ν, r s ) −  ( r s )s i n χ ( r s )
r s
+ n ( r s )c o s χ ( r s )
r s
thic kness and surface force terms: d ˜ z s
S
= ddr s
quadrup ole term: d ˜ z s
V
= ddndr s
The c hordwise and transv erse w a v e n um b ers are obtained b y pro jecting the axial w a v en um b er and az-
im uthal w a v en um b er k θ = m/r s on to the airfoil axis:
k  = k x cos χ − k θ sin χ
k n = − k x sin χ − k θ cos χ = ˜
k · n = ˜
k sin( ζ ±
mn − χ ) (6.18)
The normal w a v en um b er k n is the scalar pro duct b et w een the apparen t w a v ev ector and the unit v ector
normal to the ﬂat plates. As w e will see in Eq.(6.22), the lift-related noise dep ends on k n , hence this
equation indicates that acoustic mo des whose w a v efron t direction of propagation is aligned with the blade
are w eakly excited, whereas mo des whose w a v efron t direction of propagation is normal to the blades are
strongly excited. That result is a direct consequence of the radiation pattern of a dip ole, whic h radiate
sound b est along its axis, and radiate no sound in the direction p erp endicular to its axis. No w, the
exp onen tial terms in Eq.(6.17) can b e written for the diﬀeren t terms:
thic kness and surface force terms: e − ik x x s − im ˜
θ s = e − ik x x LE − im ˜
θ LE · e − ik  
quadrup ole term: e − ik x x s − im ˜
θ s = e − ik x x LE − im ˜
θ LE · e − ik   − ik n n

Chapter 6. Acoustics 90
These results are inserted in Eq.(6.17) and the in tegral o v er the source domain is decomp osed in a sum
o v er B rotor blades and in tegrals for eac h blade expressed along the radial direction and the airfoil
c hordwise and transv erse directions:
p ω
m ( x, r ) =























− i
B
P
ν =1
R
R
r s = η R  g ω
m ( x, r , r s ) e − ik x x LE − im ˜
θ LE
c
R
` =0
( ω − k x W x ) ρ 0 W n · e − ik ` ` d`  dr s
+ i
B
P
ν =1
R
R
r s = η R  g ω
m ( x, r , r s ) e − ik x x LE − im ˜
θ LE
c
R
` =0  f ˜ ω
x ( ˜ z s ) k x + f ˜ ω
θ ( ˜ z s ) k θ  · e − ik ` ` d`  dr s
−
B
P
ν =1
R
R
r s = η R " g ω
m ( x, r , r s ) e − ik x x LE − im ˜
θ LE
+ ∞
R
` = −∞
+ ∞
R
n = −∞  T ˜ ω
xx k 2
x + 2 T ˜ ω
xθ k x k θ + T ˜ ω
θ θ k 2
θ  ·
e − ik ` ` − ik n n d`dn  dr s
(6.19)
Source terms and c hordwise in t egrals
W e no w ev aluate the c hordwise in tegrals of the aero dynamic excitation terms ρ 0 W n , f ˜ ω
i and T ˜ ω
ij whic h
app ear in Eq.(6.19). W e first consider the thic kness term. The lo cal v elo city normal to the blades is in
the thin blade appro ximation: W n = W 0 ∂ t
∂ ` , where t is the airfoil thickness distribution. F urthermore,
w e assume that the blades and the inflo w are orien ted in the same direction, whic h is a reasonable
appro ximation for mo derate angles of attack.
cos χ ≈ W x
W 0
and sin χ ≈ W t
W 0
W x is the mean flo w axial v elo cit y , W t = Ω r s is the tangen tial v elo cit y of the rotor blades, W 0 =
p W 2
x + W 2
t is the stream wise mean flo w v elo city in the rotating reference frame (it is also the mean flo w
c hordwise comp onen t in the h yp othesis of small angles of attac k). This form ula is not rigorously v alid
for the do wnstream rotor of a coun ter-rotating fan as a swirl comp onent is presen t b et w een the rotors.
Moreo v er, the frequency and mo de order are coupled for the thic kness term, hence: ω = m Ω. After
injecting these expressions in Eq.(6.18) w e obtain:
ω − k x W x ≈ − k ` W 0
Moreo v er, the F ourier transform of the deriv ativ e of t is giv en b y:
c
Z
` =0
∂ t
∂ ` · e − ik ` ` d` = − ik `
c
Z
` =0
t · e − ik ` ` d`
W e define a quan tit y σ T denoting the source term, whic h includes the strength of the aero dynamic exci-
tation ρ 0 W 2
0 t , the effects of acoustic radiation efficiency (through the term k 2
` ), and the non-compactness
of the sources along the c hordwise direction:
σ T = k 2
`
c
Z
` =0
ρ 0 W 2
0 t · e − ik ` ` d` (6.20)
F or the quadrup ole term, Hanson [24] prop oses to retain only the c hordwise comp onent as it is the
dominan t term for thin blades at a mo derate incidence angle. Hence w e get:
T ˜ ω
xx k 2
x + 2 T ˜ ω
xθ k x k θ + T ˜ ω
θ θ k 2
θ = T ˜ ω
`` k 2
` + 2 T ˜ ω
`n k ` k n + T ˜ ω
nn k 2
n ≈ T ˜ ω
`` k 2
` (6.21)

Chapter 6. Acoustics 91
Similarly to the thic kness term, we in tro duce the quan tit y σ Q for the quadrup ole term, σ L for the lift
noise comp onen t, and σ D for the drag noise comp onen t:
σ Q = k 2
`
+ ∞
Z
` = −∞
+ ∞
Z
n = −∞
T ˜ ω
`` · e − ik ` ` − ik n n dnd`
σ L = ik n
c
Z
` =0
f ˜ ω
L · e − ik ` ` d`
σ D = ik `
c
Z
` =0
f ˜ ω
D · e − ik ` ` d`
(6.22)
Mo delling these terms (the lift term in particular) is a cen tral problem in addressing the w ak e or turbu-
lence in teraction noise and the trailing-edge noise. This will b e done later on. The source terms σ given
b y the equations (6.20,6.22) for the differen t noise terms dep end on the radius, the blade, and on the
frequencies in the fixed and rotating frame of reference:
σ = σ ( r s , ν, ω , ˜ ω )
Final expression for the mo dal pressure.
After injecting the quan tities of the previous section in Equation (6.19), w e obtain the mo dal pressure
created b y a rotor for the thic kness, surface force (lift and drag), and quadrup ole sources:
p ω
m ( x, r ) = −
B
X
ν =1
R
Z
r s = η R
g ω
m · e − ik x x LE − im ˜
θ LE · σ · dr s (6.23)
The mo dal pressure is expressed as a sum o v er the rotor blades and an in tegral along the radius of three
terms: g ω
m represen ts the propagation of sound from the source to the observ er, e − ik x x LE − im ˜
θ LE induces
phase cancellation due to the geometrical distribution of the sources in radial and azim uthal direction,
and the term σ represen ts the sources including the amplitude of the aero dynamic excitation, the phase
shift due to flo w effects (for example w ak es tilted along the radius) and the source non-compactness in
the c hordwise direction. Eq.(6.23) is v ery similar to that derived b y Hanson. Hanson’s studies w ere
fo cused on tonal noise generated at blade passing frequencies (BPF) either b y a single rotor [25] or b y
coun ter-rotating rotors [26]. The equation presen ted here is more general and v alid for tonal noise with
non-uniform inflo w p erturbation (where tones other than the BPF-harmonics are excited) and is also
v alid for broadband noise. In the latter case, the phase of the aero dynamic excitation σ con tains a
random part whic h can b e mo delled b y a probabilit y densit y function as sho wn hereafter. Eq.(6.23) is
v alid in the free field or in a duct pro vided the correct Green’s function is used.

Chapter 6. Acoustics 92
6.4.2 Application to the free-ﬁeld and in-duct problems
F ree-ﬁeld Green’s function
Figure 6.6: cylindrical and spherical system of co ordinates
In this section, we deriv e the far-ﬁeld appro ximation of the mo dal Green’s function in the free space.
The near-ﬁeld solution is giv en b y Hanson [108], P eak e and Crigh ton [45], Sch ulten [55], or by F arassat
in the time domain [29]. The ﬂow is assumed subsonic axial and uniform in the whole space, whic h
means that sound refraction due to v elo city gradien ts (related to the thrust-induced axial velocity , or
to the p oten tial ﬁeld around the blades) or sound scattering due to solid surfaces (rotor, h ub, airplane)
cannot b e describ ed b y this solution. Blokhin tsev [161] has presen ted the free-space Green’s function
with uniform axial ﬂo w in the time domain:
G ( z, t , z s ,t
s )= 1
4 πR ∗ δ  t − t s − R e
a 0 
where R e = R ∗ − M x ( x − x s )
β 2 and R ∗ 2 =( x − x s ) 2 + β 2 ( r cos θ − r s cos θ s ) 2 + β 2 ( r sin θ − r s sin θ s ) 2 .T h e
F ourier transform of the Green’s function and Dirac impulse are giv en b y:
G ( z, t , z s ,t
s )= T re f
+ ∞

−∞
G ω ( z, z s ) e − iω ( t − t s ) dω
δ  t − t s − R e
a 0  = T re f
+ ∞

−∞
e − iω ( t − t s − R e
a 0 ) dω
Hence w e retriev e the Green’s function in the frequency domain found b y Garric k and W atkins [162]:
G ω ( z, z s )= e ik R e
4 πR ∗ (6.24)
W e deﬁne the distance b etw een the engine and the observ er ρ = √ x 2 + r 2 (see Figure 6.6) and a pseudo-
distance ρ ∗ =  x 2 + β 2 r 2 . Based on the far-ﬁeld appro ximation stating that ρ  ρ s and ρ ∗  ρ ∗
s ,t h e

Chapter 6. Acoustics 93
quan tit y R ∗ app earing in the argumen t of the exp onen tial term of Eq.(6.24) can b e simplified:
R ∗ = ρ ∗ − xx s
ρ ∗ − β 2 r r s
ρ ∗ cos( θ − θ s )
The term e ik r r s
ρ ∗ cos( θ − θ s ) whic h no w app ears in the Green’s function can b e form ulated in terms of Bessel
functions b y using the Jacobi-Anger expansion [163]:
e iz cos θ =
+ ∞
X
m = −∞
i m J m ( z ) e imθ
Hence:
e ik r r s
ρ ∗ cos( θ − θ s ) =
+ ∞
X
m = −∞
i m J m  k r r s
ρ ∗  e im ( θ − θ s )
The mo dal decomp osition in azim uthal direction yields:
G ω ( z , z s ) =
+ ∞
X
m = −∞
G ω
m ( x, r , x s , r s ) e im ( θ − θ s )
The mo dal Green’s function is then given b y:
G ω
m ( x, r , x s , r s ) = 1
4 π R ∗ i m e − ik
β 2 [ − ρ ∗ + xx s
ρ ∗ + M x ( x − x s ) ] J m  k r r s
ρ ∗ 
In the argumen ts of the exp onen tial and Bessel functions w e iden tify the radial and axial wa v en um b ers,
whic h w ere in tro duced in a previous section dedicated to in-duct propagation (see in Eq.(6.5)). They
dep end on the axial Mac h num b er and observ ation angle ψ :
k r ≡ k r
ρ ∗ = k sin ψ
q 1 − M 2
x sin 2 ψ
k x ≡ k
β 2  − M x + x
ρ ∗  = k
β 2 
 − M x + cos ψ
q 1 − M 2
x sin 2 ψ 

F urthermore, the quan tit y R ∗ is appro ximated b y R ∗ ≈ ρ ∗ in the denominator of Eq.(6.24). Finally w e
obtain:
G ω
m ( x, r , x s , r s ) = i m e ik r r + ik x x
4 π ρ
J m ( k r r s )
q 1 − M 2
x sin 2 ψ
e − ik x x s
And:
g ω
m ( x, r , r s ) = i m e − ik r r − ik x x
4 π ρ
J m ( k r r s )
q 1 − M 2
x sin 2 ψ
This quan tit y can b e also written as:
g ω
m ( x, r , r s ) = e ik r r + ik x x · ˆ g ω
m ( ρ, ψ , r s )
where
ˆ g ω
m ( ρ, ψ , r s ) = i m
4 π ρ · J m ( k r r s )
q 1 − M 2
x sin 2 ψ
(6.25)

Chapter 6. Acoustics 94
In-duct Green’s function
The Green’s function v alid for an infinitely long hard-w all circular duct with flo w is written b y Gold-
stein [137] as a sum of propagating radial mo des of order n . The result of Goldstein is adapted to the
case of an ann ular duct with a cen ter-b o dy , w e obtain for the mo dal Green’s function:
G ω
m ( x, x s , r , r s ) = i
4 π R
+ ∞
X
n =0
( J m ( k r r ) + Q mn Y m ( k r r )) · ( J m ( k r r s ) + Q mn Y m ( k r r s ))
k R · α mn · F mn
e ik ±
x ( x − x s )
where the quan tities F mn and α mn are resp ectively the normalization and cut-on factors defined in
Equations (6.7) and(6.9). A more general solution v alid for an ann ular duct with uniform mean flo w and
noise damping w alls (liner) is giv en b y Rienstra [164]. The in-duct axial and radial wa ven um b ers are
detailed in Eq.(6.6). The quan tit y g ω
m ( x, r , r s ) is written:
g ω
m ( x, r , r s ) =
+ ∞
X
n =0
J m ( k r r ) + Q mn Y m ( k r r )
√ F mn
e ik ±
x x · ˆ g ω
mn ( r s )
where
ˆ g ω
mn ( r s ) = i
4 π R · J m ( k r r s ) + Q mn Y m ( k r r s )
k R α mn √ F mn
(6.26)
Expressions for the mo dal pressure
F or the free-fied problem, w e in tro duce the quan tit y ˆ g m of Eq.(6.25) in to Eq.(6.23) and obtain:
p ω
m ( ρ, ψ ) = − i m e ik r r + ik x x
B
X
ν =1
R
Z
r s = η R
ˆ g ω
m · e − ik x x LE − im ˜
θ LE · σ · dr s (6.27)
F or the in-duct problem, the sound field is most suitably describ ed by a sum of radial mo des of orders
m and n whic h are c haracterized b y their complex amplitude A ±
mn :
p ω
m ( x, r ) =
+ ∞
X
n =0
A ±
mn
J m ( k r r ) + Q mn Y m ( k r r )
√ F mn
e ik ±
x x
As for the free-field case, w e in tro duce the quan tit y ˆ g mn of Eq.(6.26) in to Eq.(6.23), and by iden tification
with Eq.(6.8) w e obtain an expression for the mo dal amplitude:
A ±
mn = − i
B
X
ν =1
R
Z
r s = η R
ˆ g ω
mn · e − ik x x LE − im ˜
θ LE · σ · dr s (6.28)
This expression is v alid for all mo des: for cut-on mo des, the cut-on ratio and the axial w a v en umbers
are real, for cut-off mo des these quan tities ha v e a non-zero imaginary part asso ciated to the axial decay
of the mo de along the axis.
6.4.3 T onal noise
General result
W e will denote φ exc the phase of the source term σ app earing in Eq.(6.23). As all blades are assumed
iden tical, lo cated at the same axial p osition and equally spaced in azim uthal directions, the phase φ exc
is the only term that dep ends on the blade index ν ; it can b e written as the sum of t w o deterministic

Chapter 6. Acoustics 95
terms, one only dep ending on the blade index, the other only dep ending on the radius, and an additional
term relev an t for broadband noise represen ting the randomness of the excitation.
φ exc ( r s , ν ) = − m 0
2 π
B ν + φ 0 ( r s ) + φ r and ( ν , r s , $ )
m 0 denotes the azim uthal mo de order of the incoming p erturbation. φ 0 is the phase of the incoming
aero dynamic excitation. $ denotes the random even t considered in the broadband noise case. F or tonal
noise, the sources are assumed to b e fully correlated whic h means that the phase of the excitation is fully
deterministic, the random part φ r and is equal to zero. The azim uthal p osition of the leading edge is given
b y:
˜
θ LE ( r s , ν ) = 2 π
B ν + θ LE ( r s )
The quan tit y θ LE denotes the shift angle of the blade leading edge in azim uthal direction due to lean.
F or tonal noise w e assume that the sources are fully correlated, all blades are identical and equally spaced
in the circumferen tial direction and lo cated at the same axial p osition. Based on these assumptions, it
is p ossible to extract the term dep ending on the blade index from the in te gral along the blade radius in
Equation (6.23). W e obtain the follo wing expression:
p ω
m ( x, r ) = −
B
X
ν =1
e i ( m − m 0 ) 2 π
B ν ·
R
Z
r s = η R
g ω
m · e − ik x x LE − imθ LE · σ · dr s
The summation term represen ts the blade-to-blade in terference effects whic h can b e destructiv e or con-
structiv e dep ending on the order m of the acoustic mo de and on the order m 0 of the incoming aero dynamic
p erturbation:
B
X
ν =1
e i ( m − m 0 ) 2 π
B ν =  B , if m − m 0
B is an in teger
0, else (6.29)
Here, we ha v e deriv ed a more general form of the famous result of T yler and Sofrin [165] stating that only
sp ecific acoustic mo des (so-called T yler & Sofrin mo des) can b e excited. F or iden tical incoming blade
w ak es (or p oten tial field) generated b y a blade ro w with N blades, m 0 = hN , where h is the harmonic
index of the w ak e. F or the self-noise terms (thic kness, steady lift and drag, and quadrup ole terms), no
incoming excitation is defined. The p erturbation is lo calized at eac h blade and should then b e mo delled
b y a train of Dirac functions with m 0 = hN , one can sho w ho w ev er that for these terms ( f 0 = 0), this is
equiv alen t to taking m 0 = 0, whic h represen ts a simpler and more rapid alternativ e.
F ree-field solution
Using this result, w e no w obtain a simplified form of Eq.(6.27) v alid for fully correlated sources and for
the so-called T yler & Sofrin mo des:
p ω
m = i m e ik r r + ik x x · B
R
Z
r s = η R
ˆ g ω
m · e − ik x x LE − imθ LE · σ · dr s (6.30)
In-duct solution
In the in-duct case w e obtain for T yler & Sofrin mo des:
A ±
mn = i · B
R
Z
r s = η R
ˆ g ω
mn · e − ik x x LE − imθ LE · σ · dr s (6.31)

Chapter 6. Acoustics 96
Figure 6.7: Probabilit y densit y function of ∆ φ r and ( z s , z 0
s )
6.4.4 Broadband noise
General deriv ation
In the broadband noise problem the mo dal pressure p ω
m is a random quan tit y whic h is not con v enien t
to w ork with. The deterministic quan tit y to consider is the exp ectation of the squared magnitude of
the mo dal pressure, | p ω
m | 2 . T o deriv e an expression for this quan tit y , we start from Eq.(6.23) presen ted
in a simplified form for the sak e of clarit y . The mo dal pressure results from an in tegral of sources of
magnitude A and phase Φ.
p ω
m = Z
z s
A ( z s ) e i Φ( z s ) dz s
The conjugate v alue of the mo dal pressure is giv en b y:
p ω
m ∗ = Z
z 0
s
A ( z 0
s ) e − i Φ( z 0
s ) dz 0
s
Hence the squared magnitude of the mo dal pressure is:
| p ω
m | 2 = p ω
m ∗ · p ω
m = Z
z s Z
z 0
s
A ( z s ) A ( z 0
s ) e i (Φ( z s ) − Φ( z 0
s )) dz 0
s dz s (6.32)
F or symmetry reasons, this quan tit y can also b e written in this wa y:
| p ω
m | 2 = Z
z 0
s Z
z s
A ( z 0
s ) A ( z s ) e i (Φ( z 0
s ) − Φ( z s )) dz s dz 0
s = Z
z s Z
z 0
s
A ( z s ) A ( z 0
s ) e − i (Φ( z s ) − Φ( z 0
s )) dz 0
s dz s (6.33)

Chapter 6. Acoustics 97
After summation of the equations (6.32) and (6.33), and iden tifying cos( x ) = 1
2 ( e + ix + e − ix ), we obtain:
| p ω
m | 2 = Z
z s Z
z 0
s
A ( z s ) A ( z 0
s ) cos(Φ( z s ) − Φ( z 0
s )) dz 0
s dz s
A statistical a v eraging is no w p erformed b y computing the exp ectation v alue of | p ω
m | 2 . The exp ectation
v alue of random v ariable X dep ending on the index of the ev en t $ is giv en b y:
X = E ( X ) = lim
Ω → + ∞
1
Ω
Ω
X
$ =1
X ( $ )
If X is a con tin uous real v ariable, it can b e describ ed by a probabilit y density function f and its expec-
tation v alue is calculated through the follo wing in tegral:
X =
+ ∞
Z
−∞
X · f ( X ) · dX (6.34)
F urthermore, if w e assume that the pro duct of amplitudes and the cosine term are uncorrelated, then
the exp ectation v alue of their pro duct is equal to the pro duct of their resp ective expectation v alues,
whic h yields:
| p ω
m | 2 = Z
z s Z
z 0
s
A ( z s ) A ( z 0
s ) · cos(Φ( z s ) − Φ( z 0
s )) dz 0
s dz s (6.35)
F rom this relation, it turns out that t w o-p oin t correlation statistics o v er the whole source domain are
needed to compute broadband noise in the general case. The direct computation of these quan tities is
v ery demanding and their measuremen t in a real fan or compressor en vironmen t is also v ery c hallenging.
F or that reason, there exist a n um b er of mo dels for the t w o-p oin t correlations, see for example the w ork
of T am and Auriault [166] or Morse and Ingard [167]. W e prop ose here a simple mo del for the phase
statistics. The phase Φ is first decomp osed as a sum of a deterministic and a sto c hastic or random term:
Φ( z s ) − Φ( z 0
s ) = φ ( z s ) − φ ( z 0
s )+∆ φ r and ( z s , z 0
s , $ )
The con tin uous random v ariable ∆ φ rand ( z s , z 0
s ) is mo delled by a probabilit y densit y function f in the
form of a Gaussian-shap ed (or normal) distribution, whose mean v alue is zero and its v ariance is equal
to σ 2 = 2 π | z s − z 0
s | 2
` 2 . The smaller the distance b et w een t w o sources, the higher is the probabilit y that they
ha v e the same phase.
f (∆ φ r and ) = 1
√ 2 π σ 2 · e − (∆ φ r and ) 2
2 σ 2 (6.36)
where ` is a turbulence correlation length whic h dep ends on the p osition considered and on the frequency
of the aero dynamic excitation ˜ ω . The question of anisotropic turbulence can b e handled here b y consid-
ering a differen t correlation length for eac h direction of space. This problem has b een treated b y A tassi
[168] and Stephens [34] in the case of turbulence ingestion b y a fan with strongly axially stretc hed eddies.
Previous mo dels based on exp erimen tal studies (see the NASA rep orts b y Ganz et al. [117] and Glieb e et
al. [23]) ha v e also sho wn that the turbulence b ehind a fan rotor exhibits some similarities with isotropic
turbulence. W e will assume here that the turbulence is isotropic. A mo del for ` is prop osed in c hapter 4.
The t w o-p oin t correlation of the phase term of Eq.(6.35) is given b y:
cos(Φ( z s ) − Φ( z 0
s )) = cos( φ ( z s ) − φ ( z 0
s ))cos(∆ φ r and ) − sin( φ ( z s ) − φ ( z 0
s ))sin(∆ φ r and )

Chapter 6. Acoustics 98
The quan tities sin(∆ φ r and ) and cos(∆ φ r and ) are the exp ectation v alues of the random v ariables sin(∆ φ r and )
and cos(∆ φ r and ). After calculation, w e obtain using Eq.(6.34) and (6.36):
sin(∆ φ r and ) =
+ ∞
Z
−∞
sin(∆ φ r and ) · f (∆ φ r and ) · d ∆ φ r and = 0
cos(∆ φ r and ) =
+ ∞
Z
−∞
cos(∆ φ r and ) · f (∆ φ r and ) · d ∆ φ r and = e − σ 2
2 = e − π | z s − z 0
s | 2
` 2
This result for the t w o-p oin t correlation of the phase term presen ts a form very similar to the models
prop osed b y T am and Auriault [166], and Morse and Ingard [167]. The final expression for the mo dal
pressure is:
| p ω
m | 2 = Z
z s Z
z 0
s
A ( z s ) A ( z 0
s ) cos( φ ( z s ) − φ ( z 0
s )) e − π | z s − z 0
s | 2
` 2 dz 0
s dz s (6.37)
This form ula is v alid for an y v alue of the correlation length ` pro vided the turbulence is isotropic. In
the case of a v ery large turbulence correlation length, the exp onential term tends to 1 o ver the whole
source domain, which leads to an expression for the mo dal pressure in the case of purely tonal noise
(fully correlated sources). In the case of a very small turbulence correlation length compared to b oth the
c haracteristic dimensions of the rotor and the acoutic w a v elength, then the amplitude A and deterministic
phase φ are nearly constan t in the domain where the exp onen tial term of Eq.(6.37) is non-negligible. After
setting η = z 0
s − z s , Eq.(6.37) b ecomes:
| p ω
m | 2 = Z
z s

 A 2 ( z s ) · Z
η
e − π | η | 2
` 2 dη 
 dz s
The in tegral of the Gaussian function is:
Z
η
e − π | η | 2
` 2 dη = `
Finally , w e obtain with the h yp othesis of small-scale turbulence:
| p ω
m | 2 = Z
z s
A 2 ( z s ) · ` ( z s ) · dz s (6.38)
W e assume here that the turbulence resp onsible for broadband noise is con v ected o v er a rotor c hord length
with no significan t mo dification of its prop erties. This assumption is kno wn as the frozen turbulence
h yp othesis (also kno wn as T a ylor’s h yp othesis) and leads the sources to b e fully correlated along the
stream wise direction (also c hordwise direction in the small cam b er and incidence h yp othesis). The
sources are fully uncorrelated if they are lo cated on different blades, and correlated in radial direction
only if their distance is of the same of magnitude as ` .
F ree-field solution
The general equation of the mo dal pressure form ulated in the free-field giv en in Eq.(6.27) is simplified
according to the approac h of the previous section with the assumption of small scale turbulence. The
equiv alen t of Eq.(6.38) applied to the free-field case then b ecomes:
| p ω
m ( ρ, ψ ) | 2 = B
R
Z
r s = η R
| ˆ g ω
m | 2 · | σ | 2 · ` · dr s (6.39)

Chapter 6. Acoustics 99
Note that the thic kness noise and quadrup ole terms generate no broadband noise as their are related
to the steady flo w around the blades, whic h is purely deterministic. It should also b e noted that all
azim uthal mo des carry energy con trary to the tonal noise case for whic h only the T yler & Sofrin mo des
are excited. This is due to the small turbulence scale assumption, that leads the sources from different
blades to b e uncorrelated.
In-duct solution
In the in-duct case, the adaptation of Eq.(6.28) to the broadband formulation yields under the same
assumptions as for the free-field case:
| A ±
mn | 2 = B
R
Z
r s = η R
| ˆ g ω
mn | 2 · | σ | 2 · ` · dr s (6.40)
This expression is similar to that giv en b y Glegg [15] in the case of noise generated from stator v anes.
Broadband noise with partly coheren t blades
W e can express the relation of Eq.(6.37) if the turbulen t correlation length in radial and circumferen tial
directions are differen t.
| p ω
m | 2 =
B
X
ν =1
B
X
ν 0 =1 Z Z
r s ,r 0
s
A ( r s ) A ( r 0
s ) cos [ φ ( r s , ν ) − φ ( r 0
s , ν 0 )] e − π  r s − r 0
s
` r  2
+  r s θ s − r 0
s θ 0
s
` θ  2  dr 0
s dr s (6.41)
The o v erlined quan tities are statistical a v erages. The deterministic part of the phase φ ( r s , ν ) is
assumed to dep end only on the blade count and not on the radius. F rom the relations dev elop ed in
the section 6.4.3, w e ha v e: φ ( r s , ν ) = 2 π · m − m 0
B · ν , where m 0 is the circumferen tial mo de order of the
incoming turbulence (obtained after F ourier transforming of the statistically a v eraged turbulen t kinetic
energy in circumferen tial direction). By keeping the assumption of small span wise correlation length,
whic h is still applicable in the case of axially stretc hed eddies, w e can decouple the terms that dep end
on the radius and those that dep end on the blade index ν . The term r s θ s − r 0
s θ 0
s in the exp onen tial is
written s ( ν − ν 0 ) with s b eing the radially av eraged blade spacing. W e obtain:
| p ω
m | 2 = Σ( m ) · Z
r s
A 2 · ` r · dr s , where Σ( m ) =
B
X
ν =1
B
X
ν 0 =1
cos  2 π · m − m 0
B · ( ν − ν 0 )  e − π  s
` θ  2 ( ν − ν 0 ) 2
Σ( m ) represen ts the circumferen tial correlation function of the blades. In the case of a circumferen tial
correlation length v ery small compared to the spacing ( ` θ << s ), w e retriev e the result of Eq.(6.39) and
(6.40): all acoustic mo des are equally excited and the squared mo dal pressure scales with the n um b er
of blades B . In the opp osite case, where the circumferen tial correlation length is large compared to the
blade spacing ( ` θ >> s ), we retriev e the T yler and Sofrin mo de excitation rule of Eq.(6.29) for tonal
noise: only mo des v erifying m − m 0
B an in teger (or equiv alen tly m = m 0 + k B ) are excited and their squared
mo dal pressure scales with B 2 .
The circumferen tial correlation length is giv en b y:
` θ = p ( ` x tan β r el ) 2 + ` 2
r
6.4.5 Summary
T ables 6.2 and 6.3 presen t a summary of the results obtained so far.

Chapter 6. Acoustics 100
problem free-field in-duct
Green’s function ˆ g ω
m = i m
4 π ρ · J m ( k r r s ))
√ 1 − M 2
x sin 2 ψ ˆ g ω
mn = i
4 π R · J m ( k r r s )+ Q mn Y m ( k r r s )
k R α mn √ F mn
tonal noise p ω
m = e ik r r + ik x x B · A ±
mn = B ·
R
R
r s = η R
ˆ g ω
m e − ik x x LE − imθ LE σ dr s
R
R
r s = η R
ˆ g ω
mn e − ik x x LE − imθ LE σ dr s
broadband noise | p ω
m | 2 = B
R
R
r s = η R | ˆ g ω
m | 2 | σ | 2 `dr s | A ±
mn | 2 = B
R
R
r s = η R | ˆ g ω
mn | 2 | σ | 2 `dr s
T able 6.2: F orm ula for mo dal pressure dep ending on the problem considered
source σ
thic kness noise k 2
`
c
R
` =0
ρ 0 W 2
0 te − ik ` ` d`
quadrup ole noise k 2
`
+ ∞
R
` = −∞
h
R
n =0
T ˜ ω
`` e − ik ` ` − ik n n dnd`
lift noise ik n
c
R
` =0
f ˜ ω
L e − ik ` ` d`
drag noise ik `
c
R
` =0
f ˜ ω
D e − ik ` ` d`
T able 6.3: form ula for source terms
6.5 In terpretation of the results
6.5.1 Classification of sources
This section presen ts a summary of the differen t noise mec hanisms and sources whic h are mo delled in
the frame of the presen t study . The first t yp e of classification w e will consider is that obtained naturally
from the mathematical deriv ation according to T able 6.3. The follo wing mec hanisms are distinguished:
• thic kness noise is asso ciated to the v olume displacemen t induced b y the rotation of the blades.
This term is usually referred to as a monop ole term, but as noted b y Hanson [24], it scales lik e
the quadrup ole term and has a similar radiation pattern. The thic kness term is indeed formed b y
a con tin uous c hordwise distribution of monop oles, but this is not equiv alen t to a single monop ole.
The name giv en to sources in terms of m ultip oles should therefore b e considered v ery carefully .
• quadrup ole noise: this name is c hosen here for the sak e of simplicit y but the name steady o v ersp eed
noise w ould b e more appropriate, b ecause this term is reduced here to the noise generated b y
the steady v elo cit y gradien ts around the blades and do es not include the in teraction of incoming
turbulence with the p oten tial field (this source is part of the fan broadband noise mo del dev elop ed
b y Mani [20, 150] and ma y b ecome significan t at high Mac h n um b ers and inflow turbulence lev els).
Again, the name quadrup ole should b e considered with caution.

Chapter 6. Acoustics 101
• drag noise is less significan t for w ell-designed blades as the drag co efficien t is usually smaller than
the lift co efficien t b y one or t w o orders of magnitude.
• lift noise is the mec hanism common to a n um b er of differen t sources dep ending on the steady or
unsteady nature of the p erturbation observ ed in the frame lo c k ed to the blades:
– steady lift noise is the first source to ha v e b een considered b y Gutin [145] in the prop eller
noise problem. As for the thic kness, quadrup ole and drag terms, noise is generated here by
the mere rotation of steady m ultip oles.
– unsteady lift noise is asso ciated to time v ariations of the pressure distribution on the blades.
In the presen t study , four sources are based on that mec hanism:
∗ tonal p oten tial field in teraction noise
∗ tonal w ak e in teraction noise
∗ broadband w ak e in teraction noise
∗ broadband trailing edge self noise
The sources can also b e classified according to considerations more related to mo delling asp ects.
T able 6.4 distinguishes b et w een tonal and broadband noise, and self and interaction noise on the other
side. The prediction of interaction noise is usually more prone to inaccuracies than self noise, b ecause
not only the blade ro w generating sound (and its flo w c haracteristics) has to b e mo delled, the blade row
generating the aero dynamic p erturbation and the propagation of that p erturbation ha v e to b e repro duced
correctly .
t yp e tonal noise broadband noise
self noise thic kness & quadrup ole noise trailing edge noise
steady drag & lift noise
in teraction noise w ake in teraction noise turbulence in teraction noise
p oten tial field in teraction noise
T able 6.4: Classification of turb ofan noise sources
The turbulence in teraction noise comprises t w o differen t sources that rely on the same mec hanism:
the in teraction with a turbulen t w ak e generated b y a blade ro w lo cated upstream, and the interaction
with fan inflo w turbulence generated upstream of the engine (for example in the atmosphere). The next
T ables 6.5 and 6.6 summarize the mathematical form ulation of the differen t comp onen ts of the source
terms σ (see Eq.(6.42)) for the tonal and broadband noise, resp ectively:
6.5.2 Mo delling of the sources
Decomp osition of the source terms
The source term σ summarized in T able 6.3 for the differen t mec hanisms can b e written in a more general
form:
σ ( r s , ω , ˜ ω ) = R ( r s , ω ) · Ψ( r s , ω ) · ζ 0 ( r s , ˜ ω ) · e iφ 0 ( r s , ˜ ω ) (6.42)
The source is decomp osed in to acoustic and aero dynamic terms. Tw o non-dimensional acoustic terms
dep end on the radius r s and on the acoustic frequency ω expressed in the fixed frame: an acoustic radiation
term R and a c hordwise correlation term Ψ originally in tro duced by Hanson [25], whic h is mathematically
defined as the c hordwise F ourier decomp osition of the source distribution. The normalized chordwise
distributions (denoted h ) of blade thic kness, surface force and Ligh thill’s stress are assumed each to be
constan t along the blade radius. They are normalized suc h as 1
c
c
R
` =0
h ( ` ) d` = 1. The acoustic terms v erify
at zero frequency: R ( r s , 0) = 0 and Ψ( r s , 0) = 1. The correlation function determines the directivity (or

Chapter 6. Acoustics 102
source R Ψ ζ 0 φ 0
thic kness noise k 2
` c 2 1
c
c
R
` =0
h T ( ` ) e − ik ` ` d` ρ 0 W 2
0 t
c 0
quadrup ole noise k 2
` c 2 1
c 2
+ ∞
R
` = −∞
h
R
n =0
h Q ( `, n ) e − ik ` ` − ik n n dnd` ρu 2
` 0
steady-drag noise ik ` c 1
c
c
R
` =0
h D ( ` ) e − ik ` ` d` 1
2 ρ 0 W 2
0 C D 0
steady-lift noise ik n c 1
c
c
R
` =0
h L ( ` ) e − ik ` ` d` 1
2 ρ 0 W 2
0 C L 0
unsteady-lift ik n c 1
c
c
R
` =0
h L ( ` ) e − ik ` ` d` 1
2 ρ 0 W 2
0 C L ( ˜ ω ) m 0 · ( θ g ust − θ 1 )
noise
T able 6.5: Decomp osition of the source terms for tonal noise
source |R| 2 | Ψ | 2 | ζ 0 | 2
trailing edge noise ( k n c ) 2    
1
c
c
R
` =0
h L ( ` ) e − ik ` ` d`    
2
| S ( ˜ ω ) | 2 Φ pp ( ˜ ω )
turbulence in teraction noise ( k n c ) 2    
1
c
c
R
` =0
h L ( ` ) e − ik ` ` d`    
2
π 2 ρ 2
0 W 2
0 | S ( ˜ ω ) | 2 Φ uu ( ˜ ω )
T able 6.6: Decomp osition of the source terms for broadband noise
the mo dal distribution) and the deca y of the frequency sp ectrum. The other term ζ 0 e iφ 0 is called the
aero dynamic excitation pressure and dep end on the aero dynamic p erturbation frequency ˜ ω expressed in
the rotating frame. The phase term inside the exp onen tial mo dels the dela y of excitation due to the
incoming p erturbation (due to tilted w akes, or the sk ewness of the incoming p ertubation).
F or the broadband noise sources, the source term is solely mo delled through the exp ectation v alue of
its squared magnitude and its phase is ignored. So Eq.(6.42) b ecomes:
| σ | 2 = |R| 2 · | Ψ | 2 · | ζ 0 | 2
Amplitude of the aero dynamic excitation
F or the thic kness noise the source term of Eq.(6.20) is written:
σ T = k 2
` c 2 · ρ 0 W 2
0 t · Ψ T
where
Ψ T = 1
c
c
Z
` =0
h T ( ` ) e − ik ` ` d`
Ψ T is a non-dimensional c hordwise correlation function, whic h w as originally iden tified b y Hanson [25]
as a quan tit y represen ting the effect of source non-compactness along the blade c hord. The function
h T represen ts the non-dimensional c hordwise distribution of thic kness and v erifies h T (0) = h T (1) = 0.

Chapter 6. Acoustics 103
Figure 6.8: Mean ﬂo w and p erturbation velocities in t he ﬁxed and rotating reference frames
F or the quadrup ole noise source term of Eq.(6.22), w e will consider here only the part that is steady
in the rotating reference frame, and the Ligh thill’s tensor elemen t is reduced to the Reynolds stress
T ij = ρu i u j ,w h e r e u i represen ts the p erturbation v elo city expressed in the ﬁxed frame of reference
(denoted v 
i b y Goldstein [144]). Hence the lo cal ﬂo w v elo cit y is represen ted b y the axial comp onent
W x + u x and the circumferen tial comp onent u θ . According to Eq.(6.21) the quadrup ole source term then
reduces to T  = ρu 2
 ,w h e r e u  = u x cos χ − u θ sin χ = W  − W 0 denotes the c hordwise comp onen t of the
p erturbation v elo cit y observ ed in the rotating reference frame as sp eciﬁed by Hanson [24] and P eake [45]
in the case of prop eller blades (see ﬁgure 6.8). In the case of a compressor fan stage, the mean ﬂo w v e-
lo cit y W 0 in the rotating reference frame is diﬀeren t upstream and do wnstream of the blade ro w, leading
to a non-zero source term far from the blades. W e are confron ted here with an inheren t limitation of
the acoustic analogy: if the left-hand side of Eq.(6.1) misses to repro duce signiﬁcan t propagation eﬀects
within the ﬂo w, these are i ncluded in the righ t-hand side and are falsely in terpreted as a source. P eake [46]
suggests that the calculation of the quadrup ole ﬁelds may simply correspond to distortion eﬀects in the
propagation of sound (esp ecially due to large sup ersonic p ortions and sho cks around the blades). In the
case where w e ha v e diﬀeren t c hordwise v elo cities W A and W B , w e can force the quadrup ole source to
v anish far upstream and do wnstream of the blade ro w b y setting: u 2
 = | W  − W A |·| W  − W B | .
T h es o u r c et e r mi sw r i t t e n :
σ Q = k 2
 c 2 · ρu 2
 · Ψ Q
where
Ψ Q = 1
c 2
+ ∞

 = −∞
h

n =0
h Q ( , n ) e − ik   − ik n n dnd
Ψ Q is the correlation function similar to Ψ T . The function h Q represen ts the non-dimensional distribu-
tion of lo cal ﬂo w v elo cit y around the airfoil (only the c hordwise comp onen t) and v anishes upstream and
do wnstream of the airfoil. The term ρu 2
 is the mean c hordwise stress a v eraged o v er the ﬂo w domain
capted b y a blade passage. It is strong at transonic sp eed where the ﬂo w p erturbations are large as
sho wn b y Hanson [24]. The presence of strong sho cks leads to large regions of o v er- and under-sp eed
around the airfoil. A t higher ﬂo w v elo cities, the sho c ks b ecome oblique and w eak er, leading to smaller
ﬂo w p erturbations around the airfoil. The ev aluation of the quadrup ole term is a delicate task for whic h
no analytical mo dels exist to the kno wledge of the author, other than that presen ted in the previous
c hapter (see Unsteady Aero dynamics 4). This term can b e ev aluated more precisely using a full 3D
RANS sim ulation.
F or the surface force noise, the source term of Eq.(6.22) is form ulated in diﬀeren t w a ys dep ending on
the problem considered. F or tonal noise the form ulation b y Hanson [26] is used. He writes the c hordwise
distribution of lift f ˜ ω
L (  ) (also called lo cal pressure jump) as a pro duct of a chordwise-a v eraged lift and

Chapter 6. Acoustics 104
a non-dimensional distribution assumed to b e indep enden t of the frequency of the impinging gust, from
a mathematical p ersp ectiv e, this is a mere separation of the v ariables ˜ ω and ` :
f ˜ ω
L ( ` ) = 1
2 ρ 0 W 2
0 C L ( ˜ ω ) · h L ( ` )
It follo ws for the source term:
σ L = ik n c · 1
2 ρ 0 W 2
0 C L ( ˜ ω ) · Ψ L ( ω ) (6.43)
where
Ψ L = 1
c
c
Z
` =0
h L ( ` ) e − ik ` ` d`
Ψ L is the c hordwise F ourier transform of the non-dimensional distribution of loading. h L ( ` ) is the non-
dimensional c hordwise distribution of loading whic h v erifies h L (0) = h L (1) = 0 and 1
c
c
R
` =0
h L ( ` ) d` = 1,
see previous section 4.6. The unsteady lift co efficien t is giv en b y:
C L ( ˜ ω )=2 π u n ( ˜ ω )
W 0
S ( ˜ ω )
where S is the Sears function [136], for which a model is prop osed in chapter 4, Eq.(4.14), v alid in in-
compressible flo w and at lo w frequencies. This form ulation presen ts the adv an tage of separating the term
C L that dep ends on the frequency of the excitation ˜ ω from the c hordwise correlation function Ψ whic h
is link ed to the frequency of acoustic radiation ω . Moreov er, the unsteady lift co efficien t is a widely used
quan tit y for whic h substan tial researc h has b een done. F or the drag noise term, an equiv alen t formulation
is used.
Other authors adopt a differen t approac h than that of Hanson to mo del the source term σ L . A more
general alternativ e, where no separation of v ariables is imp osed, is prop osed b y Amiet [125]:
σ L = ik n c · π ρ 0 W 0 u n ( ˜ ω ) · L ( ˜ ω , ω ), where L =
1
c
c
R
` =0
f ˜ ω
L e − ik ` ` d`
π ρ 0 W 0 u n
(6.44)
The quan tit y L is called b y Amiet c hordwise in tegral of the surface loading, or chordwise aeroacoustic
transfer function b y Roger and Moreau [36]. The main difference to Hanson’s approac h is the fact that
the distribution of loading ma y dep end on the frequency and the acoustic w a v es generated along the
airfoil ma y in teract with the loading fluctuations (suc h as the bac k-scattering of w a v es coming from the
trailing edge).
In broadband noise problems, the relev an t quan tit y is | σ L | 2 , the statistical a v eraging of the squared
magnitude of the source term, whic h is mo delled b y the pro duct of a p o w er sp ectral upw ash velocity
sp ectrum Φ uu (denoted S w w in the original pap ers) for inciden t turbulence noise and b y a w all-pressure
sp ectrum Φ pp for trailing edge noise. More details are found in the studies of Amiet [125, 132], and Roger
and Moreau [36]:
inciden t turbulence noise: | σ L | 2 = ( k n c ) 2 · π 2 ρ 2
0 W 2
0 · Φ uu ( ˜ ω ) · |L ( ˜ ω , ω ) | 2
trailing edge noise: | σ L | 2 = ( k n c ) 2 · Φ pp ( ˜ ω ) · |I ( ˜ ω , ω ) | 2
The transfer functions L and I include the c hordwise correlation function but assume a more complex
form b ecause they also include bac k-scattering corrections and accoun t to some exten t for the coupling
b et w een sources and the sound field they generate. They are basically high-frequency solutions, that
can b e extended to lo w frequencies when considering bac k-scattering corrections. These functions are
originally expressed for a non-rotating single airfoil for whic h ω = ˜ ω , but an extension of this approac h to

Chapter 6. Acoustics 105
rotating blades is prop osed b y Rozen b erg [37]. In our approac h, the question of rotating and non-rotating
sources is treated in a unified w a y from the b eginning. The effects captured b y the quan tities L and I
can b e alternativ ely describ ed through Ψ and the Sears function S :
inciden t turbulence noise: | σ L | 2 =( k n c ) 2 · | Ψ( ω ) | 2 ·  1
2 ρ 0 W 2
0  2
· | C L | 2 ( ˜ ω )
, where | C L | 2 ( ˜ ω )=4 π 2 Φ uu ( ˜ ω )
W 2
0 | S ( ˜ ω ) | 2
trailing edge noise: | σ L | 2 =( k n c ) 2 · | Ψ( ω ) | 2 · Φ pp ( ˜ ω ) | S ( ˜ ω ) | 2
By comparison of b oth form ulations and considering a non-rotating blade ro w generating noise ( ˜ ω =
ω ), w e obtain the follo wing equiv alence relations:
|L| 2 ≈ |I | 2 ≈ | Ψ | 2 · | S | 2
An asymptotic expansion at high frequencies yields:
| Ψ | 2 → 1
k c
| S | 2 → M 0
k c  ⇒ |L| 2 → M 0
( k c ) 2 and |I | 2 → M 0
( k c ) 2
whic h is in agreemen t with the results of Roger and Moreau [36] (although they differ b y a multiplicativ e
factor around 2 dB in amplitude). The phase of L tends to + π / 4, whic h is also the case for the non-
compactness term Ψ (ph ysically represen ting the time dela y b etw een gust excitation and airfoil resp onse).
A t lo w frequencies, the aeroacoustic transfer functions L and I ha v e unit y amplitude and zero phase
(considering an infinite series of bac k-scattering corrections), as do es the pro duct Ψ · S .
In summary , the Hanson’s approac h adopted here is not equiv alen t to Amiet’s approach, b ecause
the non-dimensional distribution of loading is the Sears one and do es not dep end on the gust frequency .
Ho w ev er Hanson’s approac h mo dels the non-compactness of the sources and it pro vides a similar lo w-
and high-frequency b eha viour as in Amiet’s work.
Phase of the aero dynamic excitation
The phase of the aero dynamic excitation pressure is giv en b y φ 0 ( r s ) and represen ts the phase of the
aero dynamic excitation (inciden t wak e or p oten tial field) and the time dela ys of excitation that can b e
induced b y the blade geometry (for example sw eep and lean). If the quan tit y resp onsible for the excitation
is con v ected with the flo w (i.e. a w ak e generated b y a blade ro w or p ylon lo cated upstreams as depicted
in Figure 6.9), the term φ 0 is expressed as follo ws:
φ 0 ( r s ) = m 0 · ( ˜
θ g ust − θ 1 ), where ˜
θ g ust = ˜
θ 0 − tan β g ust
x 1 − x 0
r s
The quan tities θ 1 and x 1 are resp ectiv ely the azim uth and the axial p osition of the blade edge in teracting
with the gust (leading edge for a do wnstream propagating gust, trailing edge otherwise), ˜
θ 0 and x 0 are
resp ectiv ely the azim uth and the axial p osition at whic h the gust is generated, and ˜
θ g ust is the azim uth
of the incoming gust cen terline as it impinges on to the blade. The tilde sign refers to the relativ e frame
link ed to the blade ro w creating the gust. The circumferential mode order of the gust is m 0 . Finally , the
quan tit y β g ust represen ts the gust propagation angle and is giv en b y:
β g ust = 












β r el , con v ected w ak e
arctan  M 2
x
1 − M 2
x
tan β r el  , p oten tial field in subsonic flo w
arctan M 2
x
1 − M 2
x
tan β r el ± p M 2
r el − 1
1 − M 2
x ! , p oten tial field in sup ersonic flo w

Chapter 6. Acoustics 106
Figure 6.9: P ath of gusts as they con v ect from rotor to stator, in the relativ e frame
A w ak e is con v ected b y the ﬂo w, so the propagation angle in that case is the ﬂo w angle in the
reference lo c k ed to the blade ro w generating the w ak e. A p otential ﬁeld is not con vected but propagated
at a diﬀeren t angle, whic h dep ends strongly on the axial Mac h n um b er M x . Note that for low subsonic
ﬂo w where M x << 1, the p oten tial ﬁeld propagation is nearly axial, irresp ective of the ﬂow angle. A
p oten tial ﬁeld propagating in sup ersonic ﬂo w ( M re l > 1) is actually constituted of sho cks; in that case
the quan tit y β gu s t represen ts the angle normal to the sho c k fron t for upstream (-) resp. do wnstream (+)
propagation. In the limit case of sonic axial inﬂo w ( M x = 1), it can b e sho wn via an asymptotic expansion
that the sho c ks propagate axially irresp ectiv e of the rotor rotation sp eed and other ﬂo w parameters.
As En via [169] and Gu ´ erin [62] p oin ted out, the radial v ariations of the quantities θ gu st and θ 1 , hence
of the phase φ 0 ( r s ), hav e an essential impact on tonal noise and ma y lead to v ery substan tial cancellations
of the pressure w a v es emitted from the blades. F or example, the w ak e generated b y rotor blades is tilted
increasingly as it con v ects do wn to the stator, this is due to the stronger swirl v elo city at the h ub (which
is t ypical of free-v ortex swirl distributions). This eﬀect can b e enhanced b y a bac kw ard sw eep of the
stator v anes and com bined with v anes leaned in the direction of rotation to ac hiev e tonal noise reduction.
6.5.3 Generalized cut-on criterion for eﬃcien t radiation
The existence of a strict cut-on condition is t ypical of in-duct noise propagation. This condition can b e
presen ted in the form:
σ mn < kR
 1 − M 2
x
It states that at a giv en frequency , mo des of large azim uthal and radial order cannot propagate an y
acoustic energy in the duct. This criterion is related to the propagation only , but it pro vides no further
indication on the abilit y of a mo de to b e eﬃciently excited or not. In the free-ﬁeld case, no strict cut-
on/cut-oﬀ condition exists. F rom the expressions for the mo dal Green’s functions in the free-ﬁeld and
in-duct, giv en resp ectiv ely in Eq.(6.25) and (6.26), we propose to deﬁne a common criterion based on
the Bessel function term J m ( k r r s ). Figure 6.10 sho ws the v alues tak en b y this term as a function of
the reduced frequency k r r s for m = 20 on the left, and of the azim uthal mo de order m of the right for
k r r s = 20.
On b oth sides of the ﬁgure, t w o domains can b e distinguished. A domain of large mo de order (resp.
lo w frequency) where the Bessel function assumes small v alues. In that domain, the radiation eﬃciency
decreases rapidly with increasing mo de order (resp. with decreasing frequency). A t some critical v alue

Chapter 6. Acoustics 107
Figure 6.10: Cut-on/cut-off domains visualized via the v ariations of the Bessel function
roughly corresp onding to | k r r s | ≈ | m | , the Bessel function reac hes its global maxim um v alue, and its
v alues remain large in a v erage but sho ws significan t oscillations b ey ond that p oin t. As a result, there is a
mark ed b oundary b et w een efficien t and inefficien t radiation, and the cut-on criterion can b e form ulated
as:
| k r r s | > | m |
If w e replace the radial w a v en um b er b y its expression giv en in Eq.(6.5), w e obtain:
s  k r s
m  2
+ M 2
x · sin ψ > 1 (6.45)
The term k r s
m can b e related to the azim uthal phase v elo city . The phase of a harmonic w a v e is φ = ω t − mθ ,
its azim uthal phase v elo cit y is giv en b y v φ = dθ
dt | φ = const · r s = ω
m · r s , hence k r s
m = v φ
a 0 . So the left-hand
term of Eq.(6.45) can b e in terpreted as the comp onen t of the w a v e fron t Mach n umber p oin ting to w ards
the observ er. In other words, a pressure p erturbation can radiate efficien tly sound if its w a v e fron t
v elo cit y p oin t to w ards the observ er is sup ersonic. In the particular case of noise generated b y pressure
patterns rotating with the rotor (whic h v erify ω = m Ω, lik e for thic kness or steady loading noise), the
Mac h n um b er of the w a v e fron ts is equiv alen t to the inflo w Mac h n um b er, Eq.(6.45) b ecomes:
M r el ( r s ) > 1
sin ψ
P arry and Crigh ton [170] concluded that for a giv en observ er angle ψ , there exist a radius (so-called
Mac h radius) on the rotor blade ab o v e whic h the sources radiate efficien tly . The Mach radius v erifies
M r el ( r s ) = 1
sin ψ and presen ts the highest radiation efficiency (global maxim um of the Bessel function)
Except for the case of sideline radiation ( ψ = π
2 ), the Mach radius is larger than the sonic radius (ra-
dius ab o v e whic h the lo cal inflo w of the rotor blades is sup ersonic). This result can also b e related to
the cut-off criterion of the Bessel function in Figure 6.10. Note also that in the ducted case the Mac h

Chapter 6. Acoustics 108
radius can b e iden tified with the caustic radius of Eq.(6.12). The criterion deriv ed here is v alid b oth for
free-field and in-duct sound emission, as it is essen tially related to the flo w Mac h n um b er rather than to
the b oundary conditions.
Some care m ust b e taken while considering the axial flo w Mach n um b er utilized in the cut-on criterion.
In accordance with the acoustic analogy approac h, this should b e the con v ection Mac h n um b er of the
mo ving medium in the fixed frame of reference. How ev er, in some cases such as a CR OR engine, there
is no unique con v ection Mac h n um b er b ecause the fligh t Mac h n um b er and the lo cal axial Mac h n um b er
ahead of the rotors are differen t. So the assumption of the acoustic analogy is violated and the c hoice of
the prop er Mac h n um b er is not clear: c ho osing the lo cal Mac h n um b er resp ects the conditions of noise
generation, while c ho osing the flight Mac h n um b er, as done b y P eak e [46], is more represen tativ e of the
far-field propagation.
6.5.4 Effect of source non-compactness
This effect b ecomes significan t when the spatial extent of the source domain is similar to or larger than the
w a v elength of the sound generated. Interferences o ccur b et w een the sound w a v es emitted b y correlated
sources and this usually leads to v ery substan tial cancellations at high frequencies. This illustrates the
effects of source correlation that tak e place in eac h direction of space.
Circumferen tial correlation
The correlation of sources in circumferen tial direction is related to blade-to-blade in terferences and will
therefore affect only tonal noise in the presen t study . Tw o differen t t yp es of b eha viour ha v e to b e distin-
guished dep ending on the cut-on criterion dev elop ed in the previous section. If the reduced frequency of
the source is not high enough to excite mo des that radiate efficiently , the noise lev el is very sensitiv e to
the azim uthal order of the excited mo de and will strongly decrease with increasing mo de order, as sho wn
in Figure 6.10. This effect is particularly obvious on single-rotating prop ellers op erating at subsonic con-
ditions: adding one blade to a prop eller may result in noise reduction of up to 5 dB. The rapid decrease
of acoustic radiation efficiency is analogous to the exp onen tial deca y of the p oten tial field presen ted in a
previous c hapter 4 (see Eq.(4.2)) where the impact of blade coun t is similar.
If the cut-on criterion is resp ected, the radiation efficiency of the mo des is high and presen t some
fluctuations dep ending on the mo de order. The sole mo des that do not result in destructiv e in terferences
are the so-called T yler and Sofrin mo des (see Eq.(6.29)). The ratio of sound p ow er emitted b y B correlated
p oin t sources to that emitted b y the same n um b er of uncorrelated p oin t sources is determined b y the
n um b er of T yler & Sofrin mo des N T S and the total n um b er of mo des N tot that radiate efficien tly (the
criterion for efficien t radiation is presen ted in the previous section). Figure 6.11 shows the asymptotic
b eha viour of that sound p o w er ratio as frequency is increased. The ratio N T S
N tot asymptotically tends the
constan t v alue B irresp ectiv e of the mo de order of the incoming p erturbation m 0 . F or broadband noise,
the correlation length is assumed small so that neigh b ouring blades are uncorrelated. In that case, the
T yler and Sofrin rule do es not hold and all mo des can b e excited.
Chordwise correlation
The c hordwise correlation function, denoted Ψ in Eq.(6.42), is illustrated in Figure 6.12 for v arious source
distributions, corresp onding resp ectiv ely to the t ypical thic kness distribution of a NA CA profile and t w o
loading distributions (one flat distribution and one with a strong p eak at the leading edge). These results
are presen ted b y Hanson [171] in a more detailed manner. The c hordwise correlation function presen ts
a w a vy shap e that is visible in airfoil broadband noise measurements and predictions (see Amiet [132]
and Moreau&Roger [35]). W e also observ e that smo oth distributions presen t a more rapid deca y at high
frequencies. The high-frequency asymptotic b eha viour of | Ψ | is giv en b y Crigh ton&P arry [170] in the

Chapter 6. Acoustics 109
Figure 6.11: Effect of circumferen tial correlation
form:
| Ψ | ∝    cos  | k ` c |
2 − (1 + ν ) π
2    
| k ` c | 1+ ν
The quan tit y ν is the p o w er exp onen t of the p olynomial asymptotic expansion of the distribution to-
w ards the leading and trailing edges. It represen ts the smo othness of the distribution. The cosine term
describ es the w a vy shap e of the sp ectrum. Lo cal maxima are found at particular v alues of the chordwise
w a v en um b er k ` , whic h corresp ond to the resonance of the sound w a v e o v er the blade c hord (the c hord
length is then a m ultiple of the half w a v elength). The blac k dashed line in Figure 6.12 represen ts the
high-frequency appro ximation with ν = 0 . 5. The kno wledge of the exact source distribution is necessary
to predict accurately noise lev els at high frequencies. F or broadband noise sources, the c hordwise non-
compactness is also mo delled b y the function Ψ, as the turbulent eddies are assumed frozen while they
are con v ected o v er the blade.
Radial correlation
The effect of radially distributed sources has b een studied in t w o previous publications b y Moreau and
Gu ´ erin [60, 64] and w e prop ose to summarize here the results of these studies. At first glance, the problem
of radial correlation differs from those of circumferen tial and c hordwise correlation as the b oundary
conditions in the free space and in a duct are not iden tical. Ho w ev er, the previous paragraph has
demonstrated the analogy existing b et w een the free-field and in-duct form ulations and a consequence is
that the ducted and unducted solutions assume the same asymptotic b ehaviour at high frequencies, as
sho wn in Figure 6.13. The sound p o w er Π radiated b y uniformly distributed sources in radial direction

Chapter 6. Acoustics 110
Figure 6.12: Effect of c hordwise correlation for v arious source distributions
is related to the sound p ow er Π r ef radiated by an equiv alent point source ha ving the same strength σ :
Π r ef = σ 2
ρ 0 c 0
R 2
8 π
W e consider here a single blade to isolate the effect of radial non-compactness from the circumferen tial
one. In Figure 6.13, the ratio Π
Π r ef is represen ted b y the blue solid line in the free-field case and b y red
circles in the in-duct case. The tonal and broadband noise cases are represen ted on the upp er and lo w er
part of the figure, resp ectiv ely . A t high frequencies (t ypically for k R > 20), the ducted and unducted
solutions are nearly iden tical. A t lo w frequencies, significan t differences in lev els can b e observed due to
the reduced n um b er of radial mo des and more pronounced resonances observ ed in the ducted case. It is
remark able to note that the tonal and broadband noise sources ha v e a similar high-frequency b eha viour
despite their differen t mathematical mo delling. Referring to equations of T able 6.2 with radially constan t
source strength σ , the sound p ow er Π is prop ortional to:
tonal noise: Π ∝  Z ˆ g ω
m dr s  2
broadband noise: Π ∝ ` · Z | ˆ g ω
m | 2 dr s
The radial v ariation of the Green’s function ˆ g ω
m is c haracterized b y oscillations around the zero v alue,
whic h increase with frequency . As a result, in the case of tonal sources, the radial in tegral of this function
decreases to zero with increasing frequency , whic h ma y b e in terpreted as sound w a v e in terferences leading
to noise cancellations. The asymptotic law of deca y is π /k R and is depicted by the blac k dashed line in
Figure 6.13. F or the broadband noise, the in tegral of the square magnitude of the Green’s function tends
to a constan t finite v alue, so the radial correlation is not mo delled through the oscillations of the Green’s
function but through the turbulence correlation length ` , whose dep endency to frequency is describ ed b y
Eq.(4.13).

Chapter 6. Acoustics 111
Figure 6.13: Correlation effects for uniformly distributed sources in radial direction

Chapter 6. Acoustics 112
These conclusions hold for a source strength σ constan t o v er the radius, which represen ts the case
where radial noise cancellations are maxim um. On a real application, the effect of radial non-compactness
will dep end on the radial distribution of sources and will b e less pronounced if the sources are concen trated
around a giv en radial p ositions. This is precisely the case for subsonic rotors whose noise is dominated
b y the blade tip as stated b y P arry and Crigh ton [44]. In sup ersonic rotors, on the con trary , the sources
are more smo othly distributed in the outer region b et w een the tip and the sonic radius.
6.5.5 Application to a single prop eller: effect of rotation sp eed and blade
coun t
In this section w e prop ose to illustrate the effect of rotation sp eed and blade count of a prop eller, and
to consider only the tonal self noise sources (thic kness, quadrup ole, steady lift and drag noise). The
application considered is a single-rotating prop eller lo cated in the free space with uniform axial flo w.
Realistic v alues ha v e b een c hosen for it geometric and aero dynamic c haracteristics, these are listed b elo w:
• tip radius R = 1, h ub-to-tip ratio η = 0 . 3
• prop eller solidit y at meanline radius σ = 0 . 2
• t wisted but straigh t blades with zero sw eep and lean angles
• radially constan t blade c hord length
• radially constan t mean relativ e thic kness t
c = 0 . 03
• radially constan t lift and drag co efficients C L = 0 . 5, C D = 0 . 012
• radially constan t critical Mac h n um b er M cr = 0 . 87
Assuming a constan t lift co efficient along the radius is not realistic especially near the hub and the tip
where lift should v anish. Therefore, the noise levels presen ted hereafter are ov erestimated with resp ect
to real noise lev els. Ho w ev er, they presen t realistic trends.
Effect of rotation sp eed at fixed stagger angle
The relev an t parameter to describ e the effect of rotation sp eed is the inflo w Mac h n um b er of the blade
tip observ ed in the rotating frame, whic h w e call relativ e tip Mac h n um b er M tip,r el . This quantit y has
b een v aried from the lo w subsonic to high sup ersonic v alues while k eeping the blade angle of attac k and
stagger angle constan t; th us the axial v elo cit y and the thrust steadily increase with the rotation sp eed.
Figure 6.14 sho ws the v ariations of o v erall sound p o w er (left-hand side) and of the radially a v eraged
aero dynamic excitation pressure ζ 0 (right-hand side) for each of the sources considered. The results are
sho wn for a prop eller with 10 blades.

Chapter 6. Acoustics 113
Figure 6.14: Effect of blade tip Mac h n um b er M tip,r el on the radiated sound p o w er (left) and on the
aero dynamic excitation term (righ t)
The follo wing remarks can b e made:
• Lift noise is the source dominating the lo w subsonic regime. This is mainly due to the muc h higher
v alues of the excitation pressure ζ 0 for the lift comp onen t than for the thic kness, quadrup ole and
drag comp onen t. The drag noise lev els are lo w compared to the other sources o v er the whole Mac h
n um b er range, whic h is explained b y the lo w v alue of drag co effic ien t.
• Approac hing sonic conditions at the blade tip, thic kness noise increases more rapidly and b ecomes
dominan t in the transonic domain around M tip,r el = 1. This is related to the v ariations of the
radiation term R app earing in Eq.(6.42): the normal w a v en um b er k n decreases whereas the squared
c hordwise w a v en um b er k 2
` increases rapidly .
• Quadrup ole noise is negligible at lo w subsonic sp eeds but exp eriences, lik e thic kness noise, a sudden
rise in lev el as transonic conditions are passed (b eyond the critical Mac h n um b er). The effect is
ev en more pronounced than for thic kness noise b ecause of the increase of the excitation pressure at
transonic conditions.
Effect of blade coun t
In a second approac h, the blade coun t of the prop eller has b een v aried while k eeping the rotation sp eed
and other aero dynamic parameters constant. The solidit y has also b een kept constan t, so increasing
the blade coun t is accompanied b y an equiv alen t decrease in blade c hord length. The v ariations of
o v erall sound p o w er lev el are sho wn in Figure 6.15, separating the subsonic case (left-hand side) from
the sup ersonic case (righ t-hand side).

Chapter 6. Acoustics 114
Figure 6.15: Effect of blade coun t B on the radiated sound p ow er of a subsonic (left, M tip,r el = 0 . 8) and
sup ersonic prop eller (righ t, M tip,r el = 1 . 1)
The blade coun t affects the radiated sound p o w er of a prop eller v ery differen tly if the blade tip is
subsonic or sup ersonic. In the subsonic case, represen ted on the left side of Figure 6.15, noise is extremely
sensitiv e to the blade coun t: adding one blade decreases noise b y appro ximately 5 dB in the example
giv en here. This is a well-kno wn result, observed b y sev eral authors in the past (see the mo del scale
tests on the General Electric CR OR named ’UDF’ [172, 173]) and whic h w e prop ose to explain from
theoretical considerations. P arry and Crigh ton [44] demonstrated that the noise of subsonic prop ellers is
dominated b y the tip region. Moreo v er, most of the noise is radiated in the sideline direction ( ψ = π / 2).
The sound p o w er therefore scales as the squared amplitude of the Bessel term ev aluated at the tip:
Π ∝ | J m ( k r R ) | 2 , where k r R = mM tip
p 1 − M 2
x
and m = hB
This appro ximation is depicted in Figure 6.15 b y the dashed blac k line and presen ts the same dep endency
on blade coun t as do es the exact solution. The mo des excited b y a subsonic prop eller are cut-off (they do
not radiate sound efficien tly , but this is comp ensated by a large excitation pressure), and similarly to the
exp onen tial axial deca y of a cut-off p oten tial field, more blades lead to more rapid near-field cancellations.
In the case of a sup ersonic prop eller, the blade coun t has a m uc h less pronounced impact on noise. This
is b ecause the cut-on limit is passed at some radial station, the mo des excited are cut-on and it is not
decisiv e ho w high ab o v e the cut-on limit they are.
Effect of rotation sp eed at fixed thrust
The thrust of the prop eller, the blade coun t and the c hord length are main tained constan t, whereas
the lift co efficien t, the rotation sp eed and the blade stagger angle are adjusted to satisfy the follo wing
relation: 1
2 ρ 0 W 2
0 · C L · sin χ = const . This quan tit y is the axial comp onen t of the lift force p er unit surface,
it scales prop ortionally to the thrust. The result of this parameter study is presen ted in Figure 6.16

Chapter 6. Acoustics 115
Figure 6.16: Effect of rotation sp eed (at fixed thrust) on the radiated sound p ow er (left) and on the
aero dynamic excitation term (righ t)
There are some notable differences with the results of Fig.(6.14), esp ecially concerning the steady-
lift and quadrup ole noise sources. Here, their ae ro dynamic excitation terms decreases with increasing
rotation sp eed. How ev er, the sound p o w er lev els still sho w a mark ed increase with rotation sp eed in the
subsonic regime. Again, this can b e explained b y the tip-dominated Bessel term | J m ( k r R ) | 2 men tioned
in the previous section. This term is v ery sensitiv e to the Mac h n um b er for cut-off mo des v erifying
| k r R | << | m | . In the sup ersonic regime, the increase is m uc h less pronounced (there is ev en a decrease
of quadrup ole noise lev els). The general increase of noise levels with increasing rotation speed at fixed
thrust has b een observ ed exp erimen tally on the ’UDF’ CR OR b y Janardan and Glieb e [172].
6.6 V alidation of the acoustic mo dels
Tw o examples of v alidation of the noise prediction are no w presented. A v alidation of broadband noise
could b e p erformed during a to ol b enc hmarking w orkshop organized during the last AIAA Aviation
conference in 2014 [174]. Flo w and noise data obtained from a past comprehensiv e exp eriment of NASA
on a rotor-stator fan stage (Source Diagnostic T est [175]) w ere pro vided to the w orkshop participan ts.
Therefore the acoustic mo del of PropNoise could b e informed b y radial distributions of the mean flo w and
turbulence measured b y hot-wire prob es. The acoustic mo del sim ulated the broadband w ak e in teraction
b et w een the rotor w ak es and the stator v anes. The comparison of the prediction sound p ow er with the
measured noise data at approac h condition is sho wn in Fig. 6.17. The discrepancy in sound p o w er lev els
is less than 3 dB o v er a wide range of frequencies, the predicted ov erall sound p o w er agrees with the
measuremen t within less than 1 dB, whic h is a v ery go o d result.
The other case of v alidation concerns the prediction of in teraction tones on a rotor-stator fan stage
designed b y DLR and measured in 2012 [66]. The comparison of the measured and predicted sound p o w er
at approac h condition is detailed in T able 6.7. A t this op erating condition, the first BPF harmonic is cut-
off. F or this calculation, the exact mean flow and turbulence flo w data w ere not a v ailable (unlik e the SDT
case), so PropNoise was started in its stand-alone mode (introduced in Fig. 1.1) and the aero dynamic
input to the acoustic mo del had to b e estimated. That may explain the larger inaccuracy of the prediction,
whic h nev ertheless remains within a reasonable range of 5 dB for the second and third BPF harmonics;
at the fourth BPF, the acoustic resp onse starts to b ecome highly sensitiv e to the flo w and geometry
details of the blades, whic h explains probably the larger discrepancy with the measuremen ts. Another
v alidation exercise on op en rotor tones p erformed with the RANS-informed mo de [67] sho ws that a v ery
go o d noise estimation can b e ac hiev ed, at least for subsonic op erating conditions lik e approac h.

Chapter 6. Acoustics 116
1 10 50
50
55
60
65
70
75
80
85
f i n k Hz
P SD in dB (r e . 10 − 1 2 W )
F o rwa r d -r a d ia te d s o u n d
1 10 50
f i n k Hz
Rea rwa rd -r a d ia t ed s o u n d

P r e di cti on
Me as u r e me n t

Figure 6.17: V alidation of the predicted p o w er sp ectral density for forw ard- and rearw ard-radiated broad-
band noise
6.7 Jet noise
During the design pro cess of an engine, the c hoice of the prop er thermo dynamic cycle is related to the fan
pressure ratio and will determine the relativ e imp ortance of fan noise compared to other engine sources,
esp ecially jet noise. In this section a simple estimation of jet noise is prop osed based on an empirical
scaling la w for t w o broadband jet noise mec hanisms: the turbulen t-mixing noise and the sho c k-asso ciated
noise. F or a cold jet expanding in a non-mo ving medium, the turbulent-mixing noise scales appro ximately
with the eigh t p o w er of the jet v elo city according to Ligh thill [176]. The effect of temp erature is neglected
b ecause only the cold jet expanding from the b ypass is considered here. The mec hanism underlying sho c k-
asso ciated noise is still a sub ject of con tro v ersy; one explanation prop osed by Christopher T am is the
distortion of turbulen t eddies as they con v ect across the sho c ks and expansion wa v es of the jet, th us
generating sound. Another explanation attributable to Ulf Michel is the modulation of the turbulent-
mixing noise sources b y the w a vy flo w pattern induced b y the sho c ks and expansion w a v es. The fact
that sho c k-asso ciated noise decreases when turbulen t-mixing noise is reduced (for example via a c hevron
nozzle) w ould supp ort the latter explanation. Moreo v er, flo w and sound visualizations on a sup ersonic
jet sho w that the sound radiates from a p ortion of the jet lo cated further do wnstream where the sho c ks
are oblique and w eak and ough t not cause a strong in teraction with the turbulence. W e prop ose here a

Chapter 6. Acoustics 117
T able 6.7: Comparison of the measured and predicted tonal sound p o w er of the DLR UHBR fan
O APWL in dB 2-BPF 3-BPF 4-BPF
measured 102 88 88
predicted 106 85 95
simple scaling la w for the o v erall sound p o w er of jet noise:
Π = K J N · ρ 0 c 3
0 · A j ·  V j + V 0
2  4 · ( V j − V 0 ) 4
c 8
0 ·  1 + K B B S N · β 4  , where β = q M 2
j − 1
Basically , this mo del states that the sho ck-associated noise is prop ortional to turbulen t-mixing noise;
moreo v er, it dep ends on the quantit y β 4 whic h follo ws the observ ations of Fisher [177] and Harp er
Bourne [178]. The empirical constan t K J N = 10 − 5 is detailed b y Goldstein [137]. The constan t
K B B S N = 36 is calibrated to pro vide the same lev el as turbulen t-mixing noise for M j = 1 . 08, ac-
cording to exp erimen tal results of Fisher [177]. The quan tit y A j is the cross-section area of the expanded
jet. Note that the mo del is v alid for a con v ergen t nozzle only . In that case, the jet Mac h n um b er M j
is indeed represen tativ e of the sho c k strength and flow expansion inside the jet. F or a well-designed
con v ergen t-div ergen t nozzle ho w ev er, the jet ma y b e sup ersonic and fully expanded at the nozzle exit,
th us free of sho c ks.
The v ariation of jet noise is sho wn in Figure 6.18 for a nozzle of unit y cross-section area and zero
fligh t v elo cit y V 0 = 0.
Figure 6.18: V ariation of jet noise with fully-expanded jet Mac h n um b er in static conditions
6.8 Concluding remarks
A set of mo dels based on a unified application of the acoustic analogy has b een deriv ed for rotating-blade
noise. The analytical formulations expressed in the frequency domain enable to separate and iden tify the

Chapter 6. Acoustics 118
v arious con tributors to noise generation and propagation, and to relate them to more in tuitiv e quan tities
suc h as mean-flo w and geometry parameters. Beyond the assumptions that m ust b e made at some p oint,
and whic h w ere detailed at the b eginning of this chapter, the acoustic mo dels may in the future be
extended in the follo wing w a ys:
• The in tegration of noise-damping structures in the nacelle of an engine (also called liners) is a
v ery common and efficien t measure to reduce fan noise emissions. Considering liners as part of the
acoustic design of an engine th us requires a mo del for the prediction of the so-called insertion loss
(damping co efficien t of sound w a v es). Such a model was deriv ed and published b y the author [61];
it is based on the ra y structure of acoustic duct mo des and necessitates the frequency-dep endent
imp edance of the liner. This quantit y must be determined via a liner design mo dule that relates the
geometry of the nacelle (esp ecially its length and thickness) to its acoustic damping c haracteristics.
Bram bley [179] giv es an o v erview on differen t liner imp edance mo dels a v ailable in the literature; the
extended Helmholtz resonator mo del dev elop ed b y Rienstra [180] is widely used for lo cally reacting
liners.
• The tonal self noise of sup ersonic rotors is presen tly o v erestimated b ecause the propagation of
the sound w a v es is describ ed b y a linear mo del that do es not accoun t for the energy dissipation
o ccurring inside sho c ks. The application of the sho c k deca y mo del presen ted in Eq.(4.4) for a
p oten tial field should b e extended to all acoustic w a v es. Moreo v er the scattering of energy from
the blade-passing frequencies in to other engine orders (whic h is t ypical of buzz-sa w noise) should
b e tak en in to accoun t. A semi-empirical approac h w as prop osed by Pic k ett [107]; it is based on the
statistical mo delling of the sho c k amplitude and sho c k p osition.
• Rotor shielding: the transmission of sound through a blade ro w ma y b e of relev ance. In particular
transonic rotors are c haracterized b y large sup ersonic domains inside their blade passage that
prev en t the sound w a v es emitted b y the stator v anes to propagate upstream and to radiate in the
forw ard arc. Sev eral t w o-dimensional cascade mo dels w ere dev elop ed in the 1970s: see for example
the w ork of Smith [181], Ka ji and Ok azaki [182]. An application of these mo dels on a realistic fan
has b een presen ted b y Jenkins [183] in 2012. A metho d based on the ra y theory ma y represen t here
again a simple alternativ e.
• The steady increase of fan diameter observ ed on mo dern and future engine designs must be com-
p ensated b y a decrease of the nacelle length relative to its diameter in order to main tain the engine
w eigh t in a reasonable range. Suc h short-nacelle designs do not only imply less liner surface and
less fan noise damping, they are more prone to inlet flow distortion, whic h ma y in teract with the
fron t rotor and constitute an additional noise source. The mec hanism underlying this source is ba-
sically unsteady lift noise (tonal or broadband) and w as already treated in this c hapter. Ho wev er,
a flo w distortion mo del describing the amplitude and sp ec tral con ten t of the v elo cit y p erturbation
in relation to the in tak e length, engine incidence, and flo w deceleration inside the in tak e has to b e
dev elop ed completely as no analytical mo del a v ailable in the literature is kno wn to the author.
• Moreo v er the short in tak e cannot prev en t completely the acoustic cut-off mo des generated b y the
fan stage to radiate in to the far field. T o accoun t for that, the decay of cut-off modes should b e
implemen ted in a w a y similar to Eq.(4.2). An in-duct to far-field radiation mo dule should also b e
included, for example inspired by the w ork of Morfey [184] or Homicz and Lordi [185] for simple
in tak e resp. nozzle geometries.
• The in teraction b et w een neigh b oring blades of a cascade is neglected in the curren t mo del. As a
result, eac h blade reacts to a gust as if it w ere isolated; correlativ ely no scattering of the generated
pressure w a v es is considered. Several models hav e b een published to accoun t for the cascade effect,
see for example the 3D approac h of P osson [38]; suc h mo dels add a substan tial complexit y to the
analytical description and are highly time consuming. Blandeau et al. [186] stated that the sound
p o w er emitted is sufficien tly w ell predicted b y the isolated-airfoil resp onse as long as the reduced
frequency ˜ ω s
a 0 related to the blade passage is larger than one. This corresp onds to the minim um

Chapter 6. Acoustics 119
frequency at whic h the first in ter-blade acoustic mo de b ecomes propagating inside the blade passage.
The isolated-airfoil assumption is therefore acceptable at high frequencies or for lo w-solidit y blade
ro ws t ypical for the op en rotors.
• The imp ortance of turbulence-asso ciated quadrup ole noise relativ e to the other broadband fan noise
sources still has to b e clarified. This mechanism describes basically the interaction of turbulen t
eddies with the p oten tial field surrounding blades: for example the rotor tip self noise or the
in teraction of incoming turbulence with sho c ks. The latter source is describ ed as a significan t one
b y Glieb e [187]. The impact of tip noise has b een assessed b y Ganz et al. [117]. Other authors
claim ho w ev er that the dip ole-scaling mec hanisms as w ak e turbulence in teraction are dominan t.
F uture theoretical and exp erimen tal studies will hav e to remedy this ob vious lac k of consense on
the comm unit y .
A thorough and careful v alidation of the mo dels will pro vide the insigh t necessary to select the most
relev an t impro v emen ts. In the next c hapter, the first step of that v alidation will b e addressed by applying
the mo dels in the frame of global trend studies. The predictions pro vided b y the mo dels will b e discussed
with regard to their implications in terms of engine and fan design c hoices.

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