Assessment of active flow control techniques to prevent airfoil stall

BACHELOR THESIS
TITLE: Assessmen o Ac i e Flow Con ol echniques o p e en ai oil s all
DEGREE: Bachelo 's Deg ee in Ae ospace Sys ems Enginee ing
AUTHOR: Ma í A mengol Roig
DIRECTOR: Fe nando Pablo Mellibo sky Els ein
SUBMISSION DATE: 07/02/2025
TITLE: Assessmen o Ac i e Flow Con ol echniques o p e en ai oil s all
DEGREE: Bachelo 's Deg ee in Ae ospace Sys ems Enginee ing
AUTHOR: Ma í A mengol Roig
DIRECTOR: Fe nando Pablo Mellibo sky Els ein
SUBMISSION DATE: 07/02/2025
Abs ac
Ac i e Flow Con ol (AFC) echniques a e cu en ly being in he spo ligh o esea che s
because o i s capabili ies o imp o e signi ican ly he ae odynamic pe o mance o ai oils.
This bachelo hesis ocuses on he use o Syn he ic Je Ac ua o s (SJA) o imp o e he
pe o mance o he NACA 0012 p o ile a pos -s all angles o a ack and high Reynolds
numbe s. This pu pose was ca ied on by pe o ming Compu a ional Fluid Dynamics (CFD)
simula ions using pisoFoam, a ansien Reynolds-A e aged Na ie -S okes (RANS) sol e ,
and using he Spala -Allma as (S-A) u bulence model. The mesh consis ed in a hyb id
(con aining uns uc u ed and s uc u ed g ids) C- ype mesh, widely used o ai oil simula ions.
The s udied ai oil was he NACA 0012 p o ile in he pos -s all angle o a ack alpha=19º a a
Reynolds numbe Re=2e6. A e pe o ming a mesh dependence s udy, compa ing he esul s
wi h expe imen al wind unnel li e a u e, he SJA ac ua ed simula ions we e pe o med. The
esul s concluded ha he use o SJA could signi ican ly imp o e he ae odynamics o he
non-ac ua ed case, inding ha he non-dimensional equency F+ and he momen um
coe icien C_mu o he je had a majo impac on he beha io o he ai low, specially
in luenced by he la e .
Als meus pa es, que m’han dona l’educació
i els alo s que m’han pe mès a iba
a se la pe sona que sóc a dia d’a ui.
Table o con en s
1 INTRODUCTION.................................................................................................................. 8
1.1 Flow Con ol Techniques ........................................................................................... 8
1.1.1 Passi e Flow Con ol ............................................................................................. 8
1.1.2 Ac i e low con ol ................................................................................................. 8
1.2 Objec i e and Me hodology o he s udy ................................................................... 9
1.2.1 Objec i e ............................................................................................................... 9
1.2.2 Me hodology ......................................................................................................... 9
2 FLUID DYNAMICS BACKGROUND .................................................................................. 12
2.1 Go e ning equa ions o Fluid Dynamics ................................................................. 12
2.1.1 Reynolds and Mach numbe s .............................................................................. 13
2.2 Tu bulence and i s modelling .................................................................................. 14
2.2.1 O e iew on u bulence ....................................................................................... 14
2.2.2 Types o simula ions depending on how u bulence is ea ed .............................. 15
2.2.3 Tu bulence modelling .......................................................................................... 16
2.3 Bounda y laye heo y .............................................................................................. 17
2.3.1 Lamina and u bulen bounda y laye s ................................................................ 18
2.3.2 Bounda y laye sepa a ion and SJAs ................................................................... 19
2.4 CFD impo an pa ame e s ....................................................................................... 20
2.4.1 Wall 𝒚+ and Law o he wall................................................................................ 20
2.4.2 Cou an numbe .................................................................................................. 21
2.5 Mesh design concep s ............................................................................................. 21
2.5.1 S uc u ed and uns uc u ed g ids ........................................................................ 22
2.5.2 Mesh quali y me ics ............................................................................................ 22
3 NUMERICAL SETUP ......................................................................................................... 24
3.1 De ini ion o he baseline and ac ua ed scena ios .................................................. 24
3.1.1 S udied scena ios ................................................................................................ 25
3.2 Sol e and bounda y condi ions .............................................................................. 28
3.2.1 Flow sol e .......................................................................................................... 28
3.2.2 Bounda y condi ions ............................................................................................ 28
3.3 Mesh dependence s udy .......................................................................................... 30
3.3.1 Time-s ep con igu a ion ....................................................................................... 30
3.3.2 Meshes wi hou he je ......................................................................................... 30
3.3.3 Meshes wi h he je .............................................................................................. 34
3.3.4 Baseline scena io o he implemen a ion o he je .............................................. 35
4 RESULTS .......................................................................................................................... 37
4.1 O e iew o all he scena ios ................................................................................... 37
4.2 Analysis o each case ............................................................................................... 37
4.2.1 Case 1. 𝑭+=𝟓𝟎,𝑪𝝁=𝟏𝟎−𝟒 ............................................................................ 38
4.2.2 Case 2. 𝑭+=𝟓𝟎,𝑪𝝁=𝟏𝟎−𝟑 ............................................................................ 38
4.2.3 Case 3. 𝑭+=𝟓𝟎,𝑪𝝁=𝟏𝟎−𝟐 ............................................................................ 39
4.2.4 Case 4. 𝑭+=𝟏𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟒 .......................................................................... 39
4.2.5 Case 5. 𝑭+=𝟏𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟑 .......................................................................... 40
4.2.6 Case 6. 𝑭+=𝟏𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟐 .......................................................................... 40
4.2.7 Case 7. 𝑭+=𝟓𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟒 .......................................................................... 41
4.2.8 Case 8. 𝑭+=𝟓𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟑 .......................................................................... 41
4.2.9 Case 9. 𝑭+=𝟓𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟐 .......................................................................... 42
4.2.10 Case 10. 𝑭+=𝟏𝟎𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟒 ...................................................................... 42
4.2.11 Case 11. 𝑭+=𝟏𝟎𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟑 ...................................................................... 43
4.2.12 Case 12. 𝑭+=𝟏𝟎𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟐 ...................................................................... 43
4.3 Resul s discussion ................................................................................................... 44
4.3.1 Mos li coe icien imp o emen .......................................................................... 44
4.3.2 Mos d ag coe icien imp o emen ...................................................................... 45
4.3.3 Mos ae odynamic e iciency imp o emen ........................................................... 45
4.3.4 Final commen s ................................................................................................... 46
5 SUSTAINABILITY ANALYSIS ........................................................................................... 47
5.1 En i onmen al impac .............................................................................................. 47
5.2 Economic impac ...................................................................................................... 47
5.3 Social impac ............................................................................................................ 48
6 CONCLUSIONS AND FURTHER INVESTIGATION .......................................................... 49
6.1 Conclusions .............................................................................................................. 49
6.2 Fu he in es iga ions .............................................................................................. 49
BIBLIOGRAPHY .................................................................................................................... 51

Index o igu es
Figu e 1.1. Wingle o an Ai bus a 320. [1] ................................................................................. 8
Figu e 2.1. Rep esen a ion o he cho d o an ai oil. [10] ......................................................... 13
Figu e 2.2. Wake c ea ed by he wing ip o ices o a comme cial ai plane. [11] ..................... 14
Figu e 2.3. Typical u bulen ene gy spec um. [12] ................................................................. 15
Figu e 2.4. Bounda y laye g aphical ep esen a ion. [19] ........................................................ 18
Figu e 2.5. Lamina (a) and u bulen (b) bounda y laye s. [20] ................................................ 18
Figu e 2.6. Bounda y laye sepa a ion. [19] ............................................................................. 19
Figu e 2.7. P essu e o a NACA 0012 ai oil a 𝐴𝑜𝐴=10.15°. .................................................. 19
Figu e 2.8. Schema ic o a syn he ic je ac ua o . [21] .............................................................. 20
Figu e 2.9. Law o he wall. Dimensionless eloci y p o ile in he p oximi ies o a solid ( ed line).
[22] ......................................................................................................................................... 21
Figu e 2.10. Example o a s uc u ed g id (le side) and an uns uc u ed g id ( igh side). [23] . 22
Figu e 2.11. Mesh non-o hogonali y. [24]................................................................................ 22
Figu e 2.12. Visual ep esen a ion o he skewness ec o s. [24] ............................................. 23
Figu e 2.13. Visual ep esen a ion o he smoo hness concep . [25] ......................................... 23
Figu e 3.1. Nume ical domain (a). ........................................................................................... 24
Figu e 3.2. Nume ical domain (b). ........................................................................................... 25
Figu e 3.3. Visual ep esen a ion o he 𝐶𝑙 and 𝐶𝑑 da a ga he ed by Ladson. ........................... 26
Figu e 3.4. Schema ic o an SJA. ............................................................................................ 27
Figu e 3.5. Mesh nea he ai oil. ............................................................................................. 30
Figu e 3.6. Bounda y laye a he leading edge. ....................................................................... 31
Figu e 3.7. S udy o di e en alues o 𝜈. ................................................................................. 32
Figu e 3.8. Compa ison o he non-ac ua ed scena ios wi hou je implemen a ion. .................. 33
Figu e 3.9. Image o he leading edge o he inal je mesh. Je si ua ed a he high densi y cell
egion. .................................................................................................................................... 34
Figu e 3.10. Compa ison o he non-ac ua ed scena ios wi h je implemen a ion. ..................... 35
Figu e 3.11. Time e olu ion o he baseline scena io. .............................................................. 35
Figu e 3.12. Fou ie ans o m o he 𝐶𝑙 a 𝛼=19°. ................................................................. 36
Figu e 3.13. S eamlines and eloci y ield o he non-ac ua ed, 𝛼=19° scena io. 𝑡=3.6 𝑠. .... 36
Figu e 4.1. Time e olu ion and Fou ie ans o m o case 1. ..................................................... 38
Figu e 4.2. Time e olu ion and Fou ie ans o m o case 2. ..................................................... 38
Figu e 4.3. Time e olu ion and Fou ie ans o m o case 3. ..................................................... 39
Figu e 4.4. Time e olu ion and Fou ie ans o m o case 4. ..................................................... 39
Figu e 4.5. Time e olu ion and Fou ie ans o m o case 5. ..................................................... 40
Figu e 4.6. Time e olu ion and Fou ie ans o m o case 6. ..................................................... 40
Figu e 4.7. Time e olu ion and Fou ie ans o m o case 7. ..................................................... 41
Figu e 4.8. Time e olu ion and Fou ie ans o m o case 8. ..................................................... 41
Figu e 4.9. Time e olu ion and Fou ie ans o m o case 9. ..................................................... 42
Figu e 4.10. Time e olu ion and Fou ie ans o m o case 10. ................................................. 42
Figu e 4.11. Time e olu ion and Fou ie ans o m o case 11. ................................................. 43
Figu e 4.12. Time e olu ion and Fou ie ans o m o case 12. ................................................. 43
Figu e 4.13. S eamlines and eloci y magni ude o case 5 a 𝑡=4 𝑠. ...................................... 44
Figu e 4.14. S eamlines and eloci y magni ude o case 8 a 𝑡=4 𝑠. ...................................... 45
Figu e 4.15. S eamlines and eloci y magni ude o case 9 a 𝑡=4 𝑠. ...................................... 45
Index o ables
Table 3.1. 𝐶𝑙, 𝐶𝑑, and 𝑙/𝑑 da a ga he ed by Ladson. [8] .......................................................... 26
Table 3.2. Baseline scena io a e age 𝐶𝑙 and 𝐶𝑑. ..................................................................... 26
Table 3.3. Non-ac ua ed cases bounda y condi ions. ............................................................... 29
Table 3.4. Ac ua ed cases bounda y condi ions. ...................................................................... 29
Table 3.5. S udy o he non-ac ua ed case a α=10.15°.......................................................... 31
Table 3.6. S udy o di e en alues o 𝜈. .................................................................................. 32
Table 3.7. 𝑝𝑖𝑠𝑜𝐹𝑜𝑎𝑚 and 𝑠𝑖𝑚𝑝𝑙𝑒𝐹𝑜𝑎𝑚 compa ison a 𝛼=14.25°. ........................................... 33
Table 3.8. Non-ac ua ed scena ios wi hou je implemen a ion. ................................................ 33
Table 3.9. Non-ac ua ed scena ios wi h je implemen a ion. ..................................................... 34
Table 4.1. O e iew o he con igu a ion o all ac ua ed cases. ................................................ 37
Table 4.2. Impo an da a o case 1. ........................................................................................ 38
Table 4.3. Impo an da a o case 2. ........................................................................................ 38
Table 4.4. Impo an da a o case 3. ........................................................................................ 39
Table 4.5. Impo an da a o case 4. ........................................................................................ 39
Table 4.6. Impo an da a o case 5. ........................................................................................ 40
Table 4.7. Impo an da a o case 6. ........................................................................................ 40
Table 4.8. Impo an da a o case 7. ........................................................................................ 41
Table 4.9. Impo an da a o case 8. ........................................................................................ 41
Table 4.10. Impo an da a o case 9. ...................................................................................... 42
Table 4.11. Impo an da a o case 10. .................................................................................... 42
Table 4.12. Impo an da a o case 11. .................................................................................... 43
Table 4.13. Impo an da a o case 12. .................................................................................... 43
Table 4.14. Summa y o all he cases. ..................................................................................... 44
Glossa y
CFD Compu a ional Fluid Dynamics
AFC Ac i e Flow Con ol
AoA Angle o A ack
M Mach numbe
Re Reynolds numbe
PFC Passi e Flow Con ol
FA Fluidic Ac ua o s
ZNMFA Ze o Ne Mass Flow Ac ua o
GUI G aphical Use In e ace
DNS Di ec Nume ical Simula ions
LES La ge Eddy Simula ions
RANS Reynolds-A e aged Na ie -S okes
S-A Spala -Allma as
SST Shea S ess T anspo
14 Assessmen o Ac i e Flow Con ol echniques o p e en ai oil s all
This is e y use ul o wind unnel es ing, as he models ha a e es ed a e usually much smalle
han he p o o ypes. In addi ion, his can also be in e es ing in CFD simula ions, as he esul s
ob ained o a de e mina e ai oil will hold o di e en si ua ions (e.g. di e en ai oil sizes).
The e a e se e al dimensionless numbe s: e.g. Reynolds (𝑅𝑒), Mach (𝑀), Webe (𝑊𝑒), F oude
(𝐹𝑟), e c. Depending on he applica ion, i is e y impo an o keep some o hem cons an , while
he o he s may no be as c i ical.
In his s udy, he impo an numbe s o conside a e he Reynolds and Mach numbe s. A cons an
Reynolds numbe implies a simila low wi h espec o he ela i e impo ance o he ine ial and
iscous e ec s. This is he eason why he esul s p esen ed in his documen can be applied in
mul iple scena ios, bu only hose in which 𝑅𝑒=2·106 and 𝑀<0.3. The Mach numbe is also
ele an , as o bigge Mach numbe s he low would be comp essible.
2.2 Tu bulence and i s modelling
Tu bulence is a ascina ing physical phenomenon ha occu s in all kinds o scena ios, and i can
happen in bo h liquid and solid lows. In ae odynamics i appea s e e ywhe e, om he low inside
a je engine o he wake c ea ed when an ai c a is mo ing. Mo e con en ional si ua ions in which
u bulence can be ound could be wa e lowing ou o a ap, smoke lea ing ou o a chimney, o
wind passing h ough he buildings o a ci y. E e ywhe e he e is a luid, he e is u bulence.
Figu e 2.2. Wake c ea ed by he wing ip o ices o a comme cial ai plane. [11]
2.2.1 O e iew on u bulence
A u bulen low consis s o eddies ( egions o luid whose di ec ion di e s om ha o he gene al
low, c ea ing swi ling mo ions) o di e en sizes all occu ing a he same ime in all di ec ions.
These can a y in ange om being compa able o he cha ac e is ic leng h o he sys em ( he
cho d in he case o an ai oil) o many o de s o magni ude smalle [12].
Tu bulence is known o p esen uns eady and chao ic low, which is cha ac e ized by ha ing a
la ge amoun o eddies. Chao ic phenomena a e cha ac e ized by being ex emely sensi i e o
small pe u ba ions, changing comple ely he ou come om he ini ial scena io o he pe u bed
one. The e a e wo ways o dealing wi h his:
 Sol e u bulence by ha ing a e y ine mesh so ha all eddies, including he smalle
ones, a e simula ed.
 Model u bulence comple ely o pa ially (depending on he size o he eddies)
s a is ically.
Each o hese me hods has i s ad an ages and incon eniences. This will be elabo a ed in sec ion
2.2.2.

Fluid Dynamics backg ound 15
A e y impo an pa ame e ha cha ac e izes u bulence is u bulen kine ic ene gy. Figu e 2.3
shows ha u bulen kine ic ene gy decays as he wa e numbe (in e se o he eddy size)
diminishes.
Figu e 2.3. Typical u bulen ene gy spec um. [12]
Eddies can be classi ied acco ding o hei size and he s age o hei “li e”, i ing in o h ee
ca ego ies:
 Ene gy con aining ange: I con ains he eddies ha a e ha ing an injec ion o ene gy,
which end o be he bigge ones.
 Ine ial sub- ange: Eddies ha a e ans e ing u bulen ene gy a e con ained in he
ine ial sub- ange. These a e usually mid-sized.
 Dissipa ion ange: This ange is o med by eddies ha a e dissipa ing, momen s p io o
hei “dea h”. These a e he smalles ones, and di ec ly sol ing hem equi es e y big
compu a ional powe . I is also known as Kolmogo o scale.
2.2.2 Types o simula ions depending on how u bulence is ea ed
The e a e h ee di e en ways o sol ing lows in CFD depending on how u bulence is ea ed.
2.2.2.1 Di ec Nume ical Simula ions (DNS)
The ull kine ic ene gy spec um is sol ed by he CFD simula ion in his app oach, including he
Kolmogo o leng h scale. I gi es he mos accu a e esul s ha can be achie ed by CFD, bu i
needs e y ine meshes and small ime-s eps. These equi emen s make DNS he mos high
compu a ional cos pa h a ailable, so much ha i is used in e y conc e e scena ios whe e he
o he op ions may no be sui able, o ex eme p ecision is equi ed. Some examples could be
u bulence in bounda y laye s a low o mode a e Reynolds numbe s.
2.2.2.2 La ge Eddy Simula ions (LES)
LES is a me hod ha was p oposed by Joseph Smago insky in 1963. The key concep is ha only
la ge-sized eddies a e sol ed ( ypically hese belong o he ene gy con aining ange o pa o he
ine ial sub- ange), while he smalle ones ( he es , mainly in he dissipa ion ange) a e modelled.
Since sol ing he smalle eddies equi es mos o he compu a ional cos , he cos s o he
simula ions ( ime, size o he iles and ene gy consump ion o he compu e ) is diminished
signi ican ly and he accu acy o he esul s is s ill qui e p ecise.
2.2.2.3 Reynolds-A e aged Na ie S okes (RANS)
Mos o he u bulen ene gy spec um is sol ed using ime-a e aged Na ie -S okes equa ions
( ha is, only he la ge eddies a e sol ed di ec ly). In o de o do so, many di e en u bulence
models ha e been de eloped o decades. These can be one equa ion models (e.g. Spala -
Allma as and P and l’s one equa ion model) o wo equa ion models (e.g. k-epsilon and k-omega).
Each model has i s ad an ages and disad an ages. This s udy has been done wi h he model
16 Assessmen o Ac i e Flow Con ol echniques o p e en ai oil s all
Spala -Allma as, as i is speci ically de eloped o ae odynamic applica ions and i is known o
ha e good pe o mance compa ed o o he s.
2.2.3 Tu bulence modelling
I is usually he case ha he compu a ional cos equi ed by he complica ed scena ios ha mus
be aced in enginee ing is ex emely big i he simula ions a e DNS o LES. Fo his eason,
s a is ical ea men o he low becomes a e y in e es ing op ion, as i signi ican ly educes
compu a ional cos s associa ed o hose compu a ions.
Reynolds was he i s pe son o ake his app oach, and de eloped he Reynolds-A e aged
Na ie -S okes equa ions. The idea behind hese equa ions was o apply Reynolds decomposi ion,
whe eby an ins an aneous quan i y is decomposed in o a ime-a e aged and a luc ua ing quan i y
[13]:
𝑈(𝑥,𝑡)=〈𝑈(𝑥,𝑡)〉+𝑢(𝑥,𝑡)
(2.7)
Whe e 〈𝑈(𝑥,𝑡)〉 is he ime-a e aged eloci y ield, and 𝑢(𝑥,𝑡) is he luc ua ing quan i y.
Now, applying Eins ein no a ion, he RANS equa ions a e:
𝜕〈𝑈𝑖〉
𝜕𝑥𝑖=0
(2.8)
𝜕〈𝑈𝑗〉
𝜕𝑡 +〈𝑈𝑖〉𝜕〈𝑈𝑗〉
𝜕𝑥𝑖=𝜈𝜕2〈𝑈𝑗〉
𝜕𝑥𝑖2−1
𝜌𝜕⟨𝑝⟩
𝜕𝑥𝑗−𝜕〈𝑢𝑖𝑢𝑗〉
𝜕𝑥𝑖
(2.9)
RANS equa ions in oduce he eloci y co a iance e m 〈𝑢𝑖𝑢𝑗〉, o en e e ed o as Reynolds
s esses. No ice ha i hese we e o be ze o, he RANS equa ions and he Na ie -S okes
equa ions would be he same. Thus, he di e ence o he beha io o 𝑈(𝑥,𝑡) wi h espec o
〈𝑈(𝑥,𝑡)〉 is caused due o he p esence o he Reynolds s esses.
The RANS equa ions can be e o mula ed o in oduce he Reynolds s esses as:
𝜌𝜕〈𝑈𝑗〉
𝜕𝑡 +𝜌〈𝑈𝑖〉𝜕〈𝑈𝑗〉
𝜕𝑥𝑖=𝜕
𝜕𝑥𝑖[𝜇(𝜕〈𝑈𝑖〉
𝜕𝑥𝑗+𝜕〈𝑈𝑗〉
𝜕𝑥𝑖)−〈𝑝〉𝛿𝑖𝑗−𝜌〈𝑢𝑖𝑢𝑗〉]
(2.10)
In his o m o he equa ion, he o al s ess is composed by he iscous o ces s ess, he iso opic
s ess caused by he mean p essu e ield, and ha om he luc ua ing eloci y ield. Mo eo e ,
he Reynolds s esses de ine he u bulen kine ic ene gy (𝑘), which is he mean kine ic ene gy
pe uni mass associa ed o he luc ua ing eloci y ield, 𝑢(𝑥,𝑡).
𝑘=1
2〈𝑢𝑖𝑢𝑖〉
(2.11)
No ice ha , o a wo-dimensional u bulen low, he e a e h ee equa ions go e ning he mean
eloci y ield ( he wo ime-a e aged Na ie -S okes and he mean con inui y equa ion), bu he e
a e ou unknowns since he Reynolds s esses also appea . Thus, he sys em needs ano he
equa ion o be closed, which is p o ided by physical models.
2.2.3.1 Tu bulen - iscosi y hypo hesis
The u bulen - iscosi y hypo heses is used as a ool o close he RANS equa ions sys em in
u bulen lows. In oduced by Boussinesq in 1877, i ela es he Reynolds s esses o he mean
low in o de o close he sys em o equa ions.
This hypo hesis will no be explained in his s udy, as i is conside ed a complex subjec in luid
mechanics ha is ou o he scope o his p ojec . I he eade s a e in e es ed in such opic, i is
ecommended ha hey look a e e ences [14], [15].
Fluid Dynamics backg ound 17
Se e al u bulence models a e based on he u bulen - iscosi y hypo hesis, some examples being
Spala -Allma as (S-A), 𝑘−𝜀, 𝑘−𝜔, and Men e ’s Shea S ess T anspo (SST). As men ioned
in p e ious sec ions, he Spala -Allma as model has been chosen o pe o m he simula ions
p esen ed in Sec ion 4.
2.2.3.2 Spala -Allma as u bulence model
The Spala -Allma as is a one-equa ion u bulence (see Equa ion 2.12) model ha sol es one
modelled anspo equa ion o he kinema ic eddy u bulen iscosi y, 𝜈, also called he Spala -
Allma as a iable. I was speci ically de eloped o ae ospace applica ions, showing good esul s
o bounda y laye s subjec ed o ad e se p essu e g adien s.
𝜕𝜈
𝜕𝑡+𝑢𝑗𝜕𝜈
𝜕𝑥𝑗=𝐶𝑏,1(1−𝑓𝑡,2)𝑆󰆹𝜈−[𝑣𝑤,1𝑓𝑤−𝐶𝑏,1
𝑘2𝑓𝑡,2](𝜈
𝑑)2+1
𝜎[𝜕
𝜕𝑥𝑗((𝜈+𝜈)𝜕𝜈
𝜕𝑥𝑗)+𝐶𝑏,2𝜕𝜈
𝜕𝑥𝑖𝜕𝜈
𝜕𝑥𝑖]
(2.12)
Equa ion 2.12 [16] and he ma hema ical backg ound o he S-A model will no be explained in
his s udy, as hey ha e also been conside ed ad anced knowledge ou o he scope o his
p ojec , bu i is encou aged by he au ho o del e in o e e ences [16], [17], [18] o ha e a deepe
unde s anding o his subjec .
2.3 Bounda y laye heo y
When a luid mo es pas an objec (o an objec mo es inside a luid), he molecules nex o he
su ace s ick o i . This causes he molecules jus abo e he su ace o slow down, as hey collide
wi h he ones ha a e s uck on he su ace. A he same ime, hese molecules slow down he
low nea hem, and so on. As he dis ance o he walls o he su ace inc eases, he e ec o he
wall is diminished, un il a heigh a which he objec does no ha e any e ec is eached. Thus,
he e is a laye o luid cha ac e ized by a eloci y g adien , going om ze o a he su ace o he
objec (no-slip condi ion) o he ee s eam eloci y in i s ou e bo de .
This laye is known as bounda y laye , and inside i s domain, iscosi y o ces ha e a majo impac
on he beha io o he luid. This opic is o majo ele ance in his s udy, as he bounda y laye
is c i ical o he ae odynamics o he ai oil bu , since AFC consis s in exchanging momen um
wi h he bounda y laye , i is e en mo e impo an in his conc e e scena io.
Ou side o he bounda y laye , he iscosi y o he luid plays no ole, and in iscid low can be
assumed.
The hickness o he bounda y laye , 𝛿, is ma hema ically de ined as:
𝛿=𝑦(𝑈=0.99𝑈∞)
(2.13)
Tha is, he bounda y laye hickness is he dis ance no mal o he wall a which he eloci y is
99% o ha a he ee s eam (in which iscous e ec s a e negligible).
Figu e 2.4. Bounda y laye g aphical ep esen a ion. shows a g aphical ep esen a ion o a
bounda y laye and i s hickness.
18 Assessmen o Ac i e Flow Con ol echniques o p e en ai oil s all
Figu e 2.4. Bounda y laye g aphical ep esen a ion. [19]
I is impo an o men ion ha because o he loss o eloci y, a shea s ess appea s on he
su ace o he objec , acco ding o he ma hema ical exp ession:
𝜏𝑠=𝜇𝜕𝑈
𝜕𝑦|𝑦=0
(2.14)
I is s aigh o wa d o see ha s eep eloci y g adien s will cause high alues o shea s ess,
inc easing he skin d ag.
2.3.1 Lamina and u bulen bounda y laye s
No all bounda y laye s p esen he same beha io , and hey a e usually classi ied in wo
ca ego ies: lamina and u bulen bounda y laye s.
On one hand, lamina bounda y laye s a e cha ac e ized by ha ing o de ed, smoo h and laye ed
low, ee o any mixing be ween successi e laye s, and hey a e associa ed wi h low Reynolds
numbe s (see Figu e 2.5. Lamina (a) and u bulen (b) bounda y laye s.
On he o he hand, u bulen bounda y laye s (see Figu e 2.5. Lamina (a) and u bulen (b)
bounda y laye s. ) ha e lo s o mixing, causing a mo e uni o m low ou side o he immedia e
egion a he wall, and a e associa ed o high Reynolds numbe s. Because o he o a ional mo ion
o he luid, o ices and eddies a e o med, making hem hicke han lamina bounda y laye s
because o he mixing. This ac makes hem mo e esis an o ad e se p essu e g adien s, bu i
also con ibu es o ha ing s eepe eloci y g adien s, and hus, causing bigge alues o shea
s ess a he wall, inc easing he skin d ag.
Figu e 2.5. Lamina (a) and u bulen (b) bounda y laye s. [20]
When he low s a s o go h ough he objec , he bounda y laye is lamina , and he e is no
u bulence. As he low ad ances, some u bulence s a s o appea , c ea ing a ansi ion be ween
Fluid Dynamics backg ound 19
a lamina low and a u bulen low. Finally, he e is a poin a which he low can al eady be
conside ed ully u bulen .
2.3.2 Bounda y laye sepa a ion and SJAs
The bounda y laye sepa a ion is de ined as he de achmen o he bounda y laye om he
su ace o he objec . This leads o he c ea ion bigge o ices in he wake. P essu e g adien s
ha e a majo impac on he de achmen o a bounda y laye , as hey p omo e o complica e he
mo emen o he luid:
 Nega i e p essu e g adien s enable he low, as hey “push” he luid in he di ec ion o
he low.
 Posi i e p essu e g adien s (usually e e ed o as ad e se p essu e g adien s) ha e he
opposi e e ec , as he low is mo ing om a egion o low p essu e o a egion o high
p essu e.
Figu e 2.6. Bounda y laye sepa a ion. [19]shows a isual ep esen a ion o he de achmen o
he low as he luid su e s he e ec s o a posi i e p essu e g adien :
Figu e 2.6. Bounda y laye sepa a ion. [19]
2.3.2.1 Why Syn he ic Je Ac ua o s a e used o delaying bounda y laye sepa a ion
Ai oils su e an ad e se p essu e g adien in he ex ados once he luid has passed he low
p essu e bubble. As al eady men ioned, his ac p omo es he de achmen o he bounda y laye ,
which is undesi able, since he li diminishes and he o e all d ag is inc eased. Figu e 2.7Figu e
2.7. P essu e o a NACA 0012 ai oil a 𝐴𝑜𝐴=10.15°. shows a ypical p essu e dis ibu ion a he
ex ados o a NACA 0012 ai oil a 𝐴𝑜𝐴=10.15°. No ice how he p essu e diminishes ab up ly a
he leading edge, and soon s a s o inc ease, gene a ing an ad e se p essu e g adien and
con ibu ing o he de achmen o he bounda y laye .
Figu e 2.7. P essu e o a NACA 0012 ai oil a 𝐴𝑜𝐴=10.15°.

20 Assessmen o Ac i e Flow Con ol echniques o p e en ai oil s all
SJA a e memb anes ha , by ib a ing, hey cause oscilla o y p essu e and eloci y ields. Thus,
hey beha e as u bulence gene a o s, inducing o ices and eddies in he low, ene gizing he
bounda y laye . This changes he eloci y p o ile a he bounda y laye o a mo e u bulen and
“ ulle ” eloci y p o ile, making i mo e esis an o ad e se p essu e g adien s. The e o e, he
sepa a ion o he bounda y laye is delayed, ea aching he low o he objec ’s su ace and
subs an ially imp o ing i s ae odynamics (inc easing he li and dec easing he o e all d ag).
Figu e 2.8 shows a schema ic o a ypical SJA a an incidence angle o 90º.
Figu e 2.8. Schema ic o a syn he ic je ac ua o . [21]
2.4 CFD impo an pa ame e s
The e a e some impo an pa ame e s o be conside ed when pe o ming CFD simula ions. Wall
y+ is c i ical o sol ing co ec ly he bounda y laye , and he Cou an numbe , 𝐶𝑜, is necessa y
o ensu e he selec ion o an app op ia e ime-s ep.
2.4.1 Wall 𝒚+ and Law o he wall
When dimensional analysis is applied o he low a he p oximi ies o an objec , i can be obse ed
ha i s beha io depends only on he cha ac e is ics o he su ace, no ma e he cha ac e is ics
o he ee-s eam low. This disco e y, in oduced by Niku adse and P and l, is called he Law o
he wall, and i is e y use ul when dealing wi h u bulen lows.
The wo a iables ha a e used o o mula e his law a e he dimensionless leng h, 𝑦+, and
dimensionless eloci y, 𝑢+, which a e de ined by he ollowing exp essions:
𝑦+=𝑦
𝜈(𝜏𝑤
𝜌)1
2
(2.15)
𝑢+=𝑢(𝜏𝑤
𝜌)−1
2
(2.16)
Wi h his wo a iables de ined, he e ical eloci y p o ile, 𝑢(𝑦), can be no malized, becoming
𝑢+(𝑦+). This no malized eloci y p o ile, despi e i s chao ic na u e, is almos always he same o
any u bulen low a he su ace o a solid. The law o he wall is cons i u ed by h ee di e en
egions, which a e he iscous sublaye , he bu e laye , and he log-law egion. The iscous
sublaye is cha ac e ized by 𝑢+ being p opo ional o 𝑦+, he log-law egion by a loga i hmic
dependence (o 𝑢+ on 𝑦+) , and he bu e laye is a ansi ion laye be ween he o he wo.
Fluid Dynamics backg ound 21
Figu e 2.9. Law o he wall. Dimensionless eloci y p o ile in he p oximi ies o a solid ( ed line). [22]
Figu e 2.9 shows in blue he di e en ends ha he dimensionless eloci y p o iles ollows in he
iscous sublaye and he log-law egion, while he ed line is he ac ual dimensionless eloci y
p o ile.
The main eason why he law o he wall is so impo an o RANS CFD simula ions is ha i can
be used as a model o he eloci y p o ile nea a wall. This modelling is known as wall unc ions,
and hey a e di e en depending on he model o u bulence ha is used. Usually, he model
equi es a speci ic ange o alues o 𝑦+ o he i s cell o wo k co ec ly. In OpenFOAM, Spala -
Allma as can wo k wi h 𝑦+∈[1,300], which is known as insensi i e wall unc ions.
2.4.2 Cou an numbe
The Cou an numbe (also known as Cou an -F ied ichs-Le y numbe ), 𝐶, is a dimensionless
alue ha is used in CFD simula ions o ensu ing ha he ime s ep is in he igh ange o alues
o ha ing accu a e esul s. When a ime s ep is oo la ge, i does no cap u e p ope ly he
beha io o he low, as i exceeds i s physical imescales. The Cou an condi ion s a es ha he
Cou an numbe should be equal o smalle han one. The Cou an numbe is ma hema ically
de ined as:
𝐶= ∆𝑡
∆𝑥/𝑢≤1
(2.17)
Whe e ∆𝑡 is he ime s ep, ∆𝑥 is he dis ance be ween he wo closes mesh nodes, and 𝑢 is he
eloci y a hose cells. This condi ion is imposing is ha he low canno go ac oss mo e han one
cell in each ime s ep, and hus, he in o ma ion in each ime s ep is only being ansmi ed o
neighbo nodes.
2.5 Mesh design concep s
Ha ing a p ope mesh o he applica ion ha is being modeled is ex emely impo an in CFD, as
i is in o he enginee ing a eas, such as s uc u al analysis o elecommunica ions.
This sec ion will gi e some impo an concep s o be conside ed when designing a 2D mesh o
an ai oil.
22 Assessmen o Ac i e Flow Con ol echniques o p e en ai oil s all
2.5.1 S uc u ed and uns uc u ed g ids
These wo di e en app oaches can p oduce h ee ypes o meshes: s uc u ed g id meshes,
uns uc u ed g id meshes, and a hyb id, con aining egions wi h s uc u ed g ids and o he s wi h
uns uc u ed g ids.
S uc u ed g ids a e cha ac e ized by using quad ila e al and hexahed al cells o 2D and 3D
geome ies, and hey ollow a clea o de . They p oduce he bes accu acy when s udying lamina
lows, and he con e gence CPU ime is usually as e , as he sol ing algo i hms a e mo e
e icien . I can be complex o use his ype o elemen s when dealing wi h complex geome ies.
In he opposi e side, uns uc u ed g ids a e o med by iangula and e ahed al cells o 2D and
3D geome ies, and unlike s uc u ed g ids, hese do no ollow a clea o de . One o i s main
ad an ages is ha hey can adap o complex geome ies. In luid dynamics, hey can o e be e
esul s o u bulen lows, as hey a e mo e chao ic and less s uc u ed.
Bounda y laye s usually bene i om ha ing s uc u ed g ids, as e en i he low is u bulen ,
s uc u ed geome ies allow he use o geome ical p og essions o size he cells, p o ing o be
use ul o sol e he s eep eloci y g adien s ha a e p esen in his egion o he low.
Figu e 2.10. Example o a s uc u ed g id (le side) and an uns uc u ed g id ( igh side). [23]
2.5.2 Mesh quali y me ics
The e a e ou impo an conside a ions ha ha e o be aken in o accoun when designing a
mesh o CFD applica ions:
 Non-o hogonali y is he angula de ia ion be ween he ec o o wo adjacen cell cen e s
and he no mal o he ace sha ed by he cells. The mo e o hogonal i is (angles close o
ze o), he be e he quali y o he mesh, as high alues o his angle cause nume ical
ins abili y.
Figu e 2.11. Mesh non-o hogonali y. [24]
 Aspec a io is he a io o a cell’s longes ace o he sho e one. The smalle i is (close
o one), he be e he quali y o he elemen . This me ic usually canno accomplished in
he bounda y laye .
 Skewness is he no malized dis ance be ween a line going om he cen e o wo adjacen
cells o he cen e o he ace hey sha e.
Fluid Dynamics backg ound 23
𝑆𝑘𝑒𝑤𝑛𝑒𝑠𝑠=|𝑠|
|𝑑|
(2.14)
Figu e 2.12. Visual ep esen a ion o he skewness ec o s. [24]
The ideal alue is ze o (cen e o he aces con ained in ec o 𝑑).
 Smoo hness, o expansion a e, de ines he ansi ion in size be ween wo con iguous
cells. La ge alues o smoo hness add di usion o he solu ion, and ideally i should no
be bigge han a 20%.
Figu e 2.13. Visual ep esen a ion o he smoo hness concep . [25]
I mus be men ioned ha hese quali y me ics canno be always ul illed. The e may be a ew
egions o he mesh in which one o wo o he me ics a e no me because o geome ical
easons, bu i is impo an o conside hem and ensu e hese a e con olled excep ions.
30 Assessmen o Ac i e Flow Con ol echniques o p e en ai oil s all
3.3 Mesh dependence s udy
The mesh is esponsible o disc e izing he low domain ha is o be simula ed, making i s design
a e y sensi i e ask.
Mesh dependence s udies a e c i ical in any ield ha in ol es pe o ming nume ical simula ions.
They a e used o s udy he in luence ha meshes ha e on he low, make changes o op imize
he simula ions, and ge mo e accu a e esul s.
The mesh dependence s udy was implemen ed o ob ain he mesh ha bes i ed he
expe imen al da a ga he ed by Ladson, which was p e y accu a ely accomplished o he li
coe icien , bu no as much wi h he d ag. The e was also a disc epancy wi h he s all angle o
a ack, as he simula ed alue was highe han he expe imen al.
3.3.1 Time-s ep con igu a ion
All he simula ions pe o med in subsec ion 3.3.2 we e con igu ed wi h a ime-s ep ∆𝑡=0.0004,
while he simula ions in subsec ion 3.3.3 had a ∆𝑡=0.0001. This was because he je mesh
con ains smalle cells, needing a smalle ime-s ep o co ec ly sol e he low.
This was, un o una ely, a mis ake. As explained in sec ion 2.4.2, he Cou an numbe (Equa ion
2.17) de e mines he maximum alue o he ime-s ep o a co ec ly con igu ed simula ion. The
e o was o use 𝑢=𝑈∞. Since he je is loca ed a he suc ion bubble, he ampli ude o he
eloci y ield was 𝑢>𝑈∞, leading o Cou an numbe s 𝐶>1.
This mis ake was no iced oo la e o change i . The e o e, i will be men ioned as u he wo k, as
i could mean ha he bounda y laye is no co ec ly sol ed, leading o w ong alues o he li
and d ag coe icien s.
3.3.2 Meshes wi hou he je
Se e al i e a ions we e pe o med o op imize he mesh a mul iple angles o a ack. These angles
we e 𝛼=[10.15,14.15,15.25,16.25,17.25].
A e some es s wi h di e en mesh con igu a ions, as explained in 3.1, and s ablishing he inal
ype o mesh (s uc u ed bounda y laye , uns uc u ed inne egion, and uns uc u ed ou e egion,
see Figu e 3.5 and Figu e 3.6), he angle o a ack 𝛼=10° was chosen o pe o m he ini ial
adjus men s.
Figu e 3.5. Mesh nea he ai oil.

Nume ical Se up 31
Figu e 3.6. Bounda y laye a he leading edge.
The eason o choosing his angle and no a highe alue, which could seem mo e in ui i e a
i s because o he p oximi y o he angles ha we e o be es ed wi h he SJA in he u u e, is
ha s all is a complica ed phenomenon o s udy. I happens wi h some equency ha academic
s udies show di e en esul s on s all, as i may be hea ily in luenced by many ac o s ha a e
no easy o con ol, bu a lowe angles o a ack hey usually ha e e y simila esul s. The e o e,
i was assessed ha designing a mesh a a mode a ely high angle o a ack would be a be e
op ion ha a nea -s all condi ions.
3.3.2.1 Scena ios a 𝛼=10.15°
The i s hing ha was ound is ha inc easing he numbe o cells o he ou e su ace (see
Figu e 3.1 and Figu e 3.2) did no ha e an impac i he chosen alues we e easonable (inle ,
uppe and lowe bounda ies disc e ized a ∆𝑥≤2 𝑚), so only he a ia ions ha we e applied
we e hose o he inne , bounda y laye , and wake su aces. The esul s in his subsec ion we e
ob ained using he sol e 𝑠𝑖𝑚𝑝𝑙𝑒𝐹𝑜𝑎𝑚.
Numbe o
nodes
𝑪𝒍𝒔𝒊𝒎
𝑪𝒅
𝒔𝒊𝒎
𝑪𝒍 absolu e
e o
𝑪𝒅 absolu e
e o
𝑪𝒍 ela i e
e o (%)
𝑪𝒅 ela i e
e o (%)
152876
1.1538
0.02315
0.1088
0.00965
10.41
71.48
307541
1.0897
0.02153
0.0447
0.00803
4.28
59.48
394262
1.0541
0.01980
0.0091
0.00630
0.87
46.67
434266
1.0452
0.01945
0.0002
0.00595
0.02
44.07
514389
1.0452
0.01943
0.0002
0.00593
0.02
43.92
Table 3.5. S udy o he non-ac ua ed case a α=10.15°.
Analyzing he esul s o his s udy, p esen ed a Table 3.5, i is s aigh o wa d o see ha he
simula ions p e y much con e ged a a ound 400000 nodes. Fo his eason, he 434266 node
mesh was chosen o pe o m he nex simula ions.
Wi h ega ds o he analysis o he esul s, i is e y easy o see ha once he solu ion con e ged,
he ob ained 𝐶𝑙 alues we e ex emely p ecise, wi h he chosen mesh p esen ing only a ela i e
e o o 0.02%. Rega ding he 𝐶𝑑, he esul was ex emely di e en , as i p esen ed a ela i e
e o o 40.81%.
In o de o educe he e o o he d ag coe icien , se e al alues o 𝜈 we e ied, as modi ying
his pa ame e by some o de s o magni ude can some imes imp o e he calcula ion o he d ag
32 Assessmen o Ac i e Flow Con ol echniques o p e en ai oil s all
coe icien . A his poin , i s ill had no been no iced ha he ime-s ep con igu a ion was no
adequa e, so changing o smalle ime-s eps was no conside ed.
The se o alues ha we e es ed we e 𝜈=[5·10−3,5·10−4,5·10−5,5·10−6,5·10−7,5·
10−8,5·10−9,5·10−10,5·10−11,5·10−12] ( he ini ial alue was 𝜈= 5 · 10−7, which is
speci ied in sec ion 3.2.2. The esul s o hese simula ions a e p esen ed in Table 3.1 and Figu e
3.7.
𝝂
𝑪𝒅
𝒔𝒊𝒎
𝑪𝒅 ela i e e o
𝟓·𝟏𝟎−𝟑
0.04891
262.28
𝟓·𝟏𝟎−𝟒
0.03012
123.09
𝟓·𝟏𝟎−𝟓
0.02125
57.38
𝟓·𝟏𝟎−𝟔
0.01961
45.28
𝟓·𝟏𝟎−𝟕
0.01945
44.07
𝟓·𝟏𝟎−𝟖
0.01944
44.04
𝟓·𝟏𝟎−𝟗
0.01944
43.98
𝟓·𝟏𝟎−𝟏𝟎
0.01946
44.16
𝟓·𝟏𝟎−𝟏𝟏
0.01942
43.87
𝟓·𝟏𝟎−𝟏𝟐
0.01942
43.84
Table 3.6. S udy o di e en alues o 𝜈.
Figu e 3.7. S udy o di e en alues o 𝜈.
Analyzing he da a p esen ed in Table 3.6 and Figu e 3.7, i is clea ha no signi ican
imp o emen is obse ed in he alues o 𝐶𝑑, which ge much wo se o alues bigge han 𝜈>
5·10−6. Fo his eason, i is concluded ha his e o does no depend on he alues o 𝜈.
F om now on, he e o o he simula ed 𝐶𝑑 was conside ed a sys ema ic e o , and i was
assumed o he es o he simula ions. Two possible explana ions, e en hough he e may be
mo e, a e p oposed in o de o explain his phenomenon:
 The i s hypo hesis is he a o emen ioned possibili y o ha ing luid egions wi h
comp essibili y e ec s in he expe imen al wind unnel es s, bu his is only a heo y ha
should be es ed by ga he ing new expe imen al da a a lowe Mach numbe s.
 The second one is ha i was p oduced because o he oo la ge ime-s eps p oblem,
explained in subsec ion 3.3.1.
Nume ical Se up 33
3.3.2.2 O he alues o angle o a ack
The ollowing esul s ha e been compu ed wi h he sol e 𝑝𝑖𝑠𝑜𝐹𝑜𝑎𝑚, wi h he only excep ion o
he case a 𝛼=10.15°, which has been exposed in he p e ious subsec ion. Since he ac ua ed
scena ios we e o be simula ed using he ansien sol e , changing o 𝑝𝑖𝑠𝑜𝐹𝑜𝑎𝑚 was conside ed
app op ia e.
𝑝𝑖𝑠𝑜𝐹𝑜𝑎𝑚 p oduced he same exac a e age ae odynamic esul s as 𝑠𝑖𝑚𝑝𝑙𝑒𝐹𝑜𝑎𝑚, which was
e i ied wi h he case a 𝛼=14.25° (see Table 3.7).
𝒔𝒐𝒍𝒗𝒆𝒓
𝑪𝒍𝒔𝒊𝒎
𝑪𝒅
𝒔𝒊𝒎
𝒔𝒊𝒎𝒑𝒍𝒆𝑭𝒐𝒂𝒎
1.3870
0.03058
𝒑𝒊𝒔𝒐𝑭𝒐𝒂𝒎
1.3870
0.03058
Table 3.7. 𝑝𝑖𝑠𝑜𝐹𝑜𝑎𝑚 and 𝑠𝑖𝑚𝑝𝑙𝑒𝐹𝑜𝑎𝑚 compa ison a 𝛼=14.25°.
Once i was seen ha he sol e was no a ec ing he esul s, he ollowing simula ions we e
pe o med wi h 𝑝𝑖𝑠𝑜𝐹𝑜𝑎𝑚. The esul s can be obse ed in Table 3.8.
𝜶
𝑪𝒍,𝒂𝒗
𝒔𝒊𝒎
𝑪𝒅,𝒂𝒗
𝒔𝒊𝒎
𝑪𝒍 absolu e
e o
𝑪𝒅 absolu e
e o
𝑪𝒍 ela i e e o
(%)
𝑪𝒅 ela i e e o
(%)
10.15
1.0452
0.01945
0.0002
0.00595
0.02
44.07
14.25
1.3870
0.03058
0.0250
0.00778
1.84
34.12
15.25
1.4793
0.03666
0.0713
0.00756
5.06
25.98
16.25
1.6119
0.04651
0.8589
-
114.06
-
17.25
1.7207
0.06505
-
-
-
-
18.25
1.2248
0.22203
-
-
-
-
Table 3.8. Non-ac ua ed scena ios wi hou je implemen a ion.
And i s isual ep esen a ion, wi h a compa ison o he expe imen al esul s by Ladson.
Figu e 3.8. Compa ison o he non-ac ua ed scena ios wi hou je implemen a ion.
No ice how hese a e a e age 𝐶𝑙 and 𝐶𝑑 alues, as o ex shedding s a ed o appea a 𝛼=15°.
Table 3.8 and Figu e 3.8 clea ly show wo ele an esul s:
34 Assessmen o Ac i e Flow Con ol echniques o p e en ai oil s all
 The e o in he d ag coe icien seems indeed o be a sys ema ic e o , as i s absolu e
alue is p e y simila o all he angles o a ack ha can be compa ed, educing i s
ela i e alue o he bigge AoA.
 S all occu s la e in he simula ions. As al eady men ioned, i is qui e usual o no ha e
he same s all AoA, as his phenomenon is e y sensi i e o many ac o s.
This comple es he s udy o he non-ac ua ed meshes wi hou a speci ic su ace o he je (un il
now, he ai oil was di ided jus in o wo di e en su aces, which we e he ex ados and he
in ados).
3.3.3 Meshes wi h he je
Wi h an al eady es ed mesh, i was inally possible o implemen he je . To do so, a small pa o
he ex ados nea o he leading edge was selec ed and he je was implemen ed ollowing he
me hod explained in sec ion 3.1.1.3 (see Figu e 3.9).
Figu e 3.9. Image o he leading edge o he inal je mesh. Je si ua ed a he high densi y cell egion.
The je mesh was designed wi h he same bounda y laye hickness cha ac e is ics o ob ain a
simila solu ion o he meshes wi hou he je . Rega ding he disc e iza ion o he su ace o he
ai oil, i was needed o e ine he mesh o co ec ly cap u e he beha io o he luid a he je and
i s icini ies. The esul s go en o his mesh a e p esen ed in Table 3.9 and Figu e 3.10. In his
case, he e o s a e exp essed wi h espec o he p e ious simula ions and no he expe imen al
da a, as hese ha e been aken as he baseline scena ios o designing his mesh.
𝜶
𝑪𝒍,𝒂𝒗
𝒔𝒊𝒎
𝑪𝒅,𝒂𝒗
𝒔𝒊𝒎
𝑪𝒍 absolu e
e o
𝑪𝒅 absolu e
e o
𝑪𝒍 ela i e e o
(%)
𝑪𝒅 ela i e e o
(%)
14.25
1.3877
0.03064
0.0007
0.00006
0.05
0.20
15.25
1.4733
0.03648
-0.006
-0.00018
0.41
0.49
16.25
1.6321
0.04742
0.0202
0.00091
1.25
1.96
17.25
1.8480
0.31682
0.1273
0.25177
7.40
387.04
18.25
1.8683
0.34580
0.6435
0.12377
52.54
55.74
19
1.4802
0.42733
-
-
-
-
20
1.3630
0.42841
-
-
-
-
Table 3.9. Non-ac ua ed scena ios wi h je implemen a ion.
Nume ical Se up 35
Figu e 3.10. Compa ison o he non-ac ua ed scena ios wi h je implemen a ion.
As i can be obse ed in Table 3.9 and Figu e 3.10, his mesh p esen s quasi-iden ical beha io
compa ed o he non-je mesh a 𝛼=[14.25,15.25,16.25]. E en i his is ue, i di e s again in
he s all condi ions, which happens o highe alues o bo h 𝐶𝑙 and 𝛼.
3.3.4 Baseline scena io o he implemen a ion o he je
Wi h his, knowing ha 𝛼𝑚𝑎𝑥 =18.25°, he baseline o he implemen a ion o he je has been se
a 𝛼=19°. This scena io p esen ed he ollowing 𝐶𝑙 and 𝐶𝑑 ime e olu ion:
Figu e 3.11. Time e olu ion o he baseline scena io.
Figu e 3.11 can be di icul o in e p e , as wi h such a de ached bounda y laye , o ex shedding
p esen s a complex beha io . In o de o be e unde s and wha is happening, i can be use ul o
compu e he Fou ie T ans o m o he 𝐶𝑙 o see which a e he main equencies o his mode. In
o de o do his, he ma hema ical algo i hm 𝑓𝑓𝑡 is used ( o mo e in o ma ion, see [36]):

36 Assessmen o Ac i e Flow Con ol echniques o p e en ai oil s all
Figu e 3.12. Fou ie ans o m o he 𝐶𝑙 a 𝛼=19°.
Obse ing Figu e 3.12, i can be seen ha he main equencies o his mode a e a ound 𝑓1=
5 𝐻𝑧 and 𝑓2=4 𝐻𝑧, being 𝑓1 sligh ly mo e dominan . F om now on, 𝑓1 will be conside ed he o ex
shedding equency, 𝑓𝑣𝑠.
To add some s a is ical da a, he main de ia ion o 𝐶𝑙 and 𝐶𝑑 we e compu ed. The main de ia ion,
𝜎, is de ined as:
𝜎=√∑(𝑥𝑖−𝑥)2
𝑛
𝑖=1𝑛−1
(3.5)
Whe e 𝑥𝑖 is he alue o he 𝑖𝑡ℎ poin , 𝑥 is he mean alue, and 𝑛 is he numbe o poin s o he
da a se .
All o hese in o ma ion is summa ized in , which de ines he baseline o he implemen a ion o
he je .
𝜶(°)
𝑪𝒍,𝒂𝒗
𝒏𝒐 𝒋𝒆𝒕
𝑪𝒅,𝒂𝒗
𝒏𝒐 𝒋𝒆𝒕
𝜼
𝒇𝒗𝒔 (𝑯𝒛)
𝝈𝑪𝒍
𝝈𝑪𝒅
19
1.4802
0.42733
3.4638
5
1.0013
0.30371
In o de o illus a e he sepa a ion o he bounda y laye in his scena io, was gene a ed using
Pa aView:
Figu e 3.13. S eamlines and eloci y ield o he non-ac ua ed, 𝛼=19° scena io. 𝑡=3.6 𝑠.
Resul s 37
4 Resul s
In his sec ion, he eade will ind an o e iew, an analysis case by case, and a discussion abou
he esul s o he ac ua ed cases o his s udy.
4.1 O e iew o all he scena ios
As explained in 3.1.1.3, wi h he posi ion o he je and he incidence angle comple ely ixed, he e
a e wo a iables ha ully de e mine he beha io o a SJA. These a e he non-dimensional
equency, 𝐹+, and he momen um coe icien , 𝐶𝜇. The ange o alues o non-dimensional
equencies a e 𝐹+=[50,100,500,1000], and he momen um coe icien s 𝐶𝜇=
[10−4,10−3,10−2].
No ice how he non-dimensional equencies, which co espond o je equencies 𝑓𝑗=
[50,100,500,1000], as 𝑐=1 𝑚 and 𝑈∞=1 𝑚/𝑠, ha e been chosen o ul ill 𝑓𝑗
𝑓𝑣𝑠 =
[10,20,100,200]. These equencies we e pu posely chosen o see he e ec s ha he non-
dimensional equency had a alues ep esen ing an inc ease o one and wo o de s o
magni ude (𝐹+=[50,500]), and hei i s ha monics (𝐹+=[100,1000]).
Table 4.1 is a synopsis o all he cases ha we e analyzed in he nex subsec ion, 4.2.
Case
𝑭+
𝒇𝒋
𝒇𝒗𝒔
𝑪𝝁
1
50
10
10−4
2
10−3
3
10−2
4
100
20
10−4
5
10−3
6
10−2
7
500
100
10−4
8
10−3
9
10−2
10
1000
200
10−4
11
10−3
12
10−2
Table 4.1. O e iew o he con igu a ion o all ac ua ed cases.
4.2 Analysis o each case
This subsec ion p o ides de ailed ae odynamic da a and g aphical esou ces o all he ac ua ed
cases. The pa ame e s ha we e s udied we e: li and d ag coe icien s, ae odynamic e iciency,
o ex shedding equency and i s ampli ude.
38 Assessmen o Ac i e Flow Con ol echniques o p e en ai oil s all
4.2.1 Case 1. 𝑭+=𝟓𝟎, 𝑪𝝁=𝟏𝟎−𝟒
Figu e 4.1. Time e olu ion and Fou ie ans o m o case 1.
𝑪𝒂𝒔𝒆
𝑪𝒍,𝒂𝒗
𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅
𝑪𝒅,𝒂𝒗
𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅
𝜼
∆𝑪𝒍
∆𝑪𝒅
∆𝜼
∆𝑪𝒍
𝑪𝒍𝒃𝒍(%)
∆𝑪𝒅
𝑪𝒅
𝒃𝒍 (%)
∆𝜼
𝜼𝒃𝒍(%)
1
1.7114
0.4254
4.0230
0.2312
-0.00193
0.5592
15.62
-0.45
16.14
Table 4.2. Impo an da a o case 1.
4.2.2 Case 2. 𝑭+=𝟓𝟎, 𝑪𝝁=𝟏𝟎−𝟑
Figu e 4.2. Time e olu ion and Fou ie ans o m o case 2.
𝑪𝒂𝒔𝒆
𝑪𝒍,𝒂𝒗
𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅
𝑪𝒅,𝒂𝒗
𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅
𝜼
∆𝑪𝒍
∆𝑪𝒅
∆𝜼
∆𝑪𝒍
𝑪𝒍𝒃𝒍(%)
∆𝑪𝒅
𝑪𝒅
𝒃𝒍 (%)
∆𝜼
𝜼𝒃𝒍(%)
2
1.706
0.27908
6.1129
0.2258
-0.14825
2.6491
15.25
-34.69
76.48
Table 4.3. Impo an da a o case 2.
Resul s 39
4.2.3 Case 3. 𝑭+=𝟓𝟎, 𝑪𝝁=𝟏𝟎−𝟐
Figu e 4.3. Time e olu ion and Fou ie ans o m o case 3.
𝑪𝒂𝒔𝒆
𝑪𝒍,𝒂𝒗
𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅
𝑪𝒅,𝒂𝒗
𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅
𝜼
∆𝑪𝒍
∆𝑪𝒅
∆𝜼
∆𝑪𝒍
𝑪𝒍𝒃𝒍(%)
∆𝑪𝒅
𝑪𝒅
𝒃𝒍 (%)
∆𝜼
𝜼𝒃𝒍(%)
3
1.9755
0.03665
53.9018
0.4953
-0.39068
50.4379
33.46
-91.42
1456.13
Table 4.4. Impo an da a o case 3.
4.2.4 Case 4. 𝑭+=𝟏𝟎𝟎, 𝑪𝝁=𝟏𝟎−𝟒
Figu e 4.4. Time e olu ion and Fou ie ans o m o case 4.
𝑪𝒂𝒔𝒆
𝑪𝒍,𝒂𝒗
𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅
𝑪𝒅,𝒂𝒗
𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅
𝜼
∆𝑪𝒍
∆𝑪𝒅
∆𝜼
∆𝑪𝒍
𝑪𝒍𝒃𝒍(%)
∆𝑪𝒅
𝑪𝒅
𝒃𝒍 (%)
∆𝜼
𝜼𝒃𝒍(%)
4
1.5004
0.38373
3.9100
0.0202
-0.0436
0.4462
1.36
-10.20
12.88
Table 4.5. Impo an da a o case 4.
46 Assessmen o Ac i e Flow Con ol echniques o p e en ai oil s all
4.3.4 Final commen s
I could be a gued ha he mos impo an ae odynamic pe o mance pa ame e in his s udy was
he ae odynamic e iciency. The only alid solu ions we e hose ha hugely imp o ed his alue,
and hese we e, o de ing hem om he bes o he wo s :
 Case 9: 𝐹+=500, 𝐶𝜇=10−2.
 Case 8: 𝐹+=500, 𝐶𝜇=10−3.
 Case 6: 𝐹+=100, 𝐶𝜇=10−2.
 Case 12: 𝐹+=1000, 𝐶𝜇=10−2.
 Case 3: 𝐹+=50, 𝐶𝜇=10−2.
I is easy o see ha e e y single case wi h 𝐶𝜇=10−2 p oduced a alid solu ion. In addi ion, he e
we e solu ions wi h all he alues o he non-dimensional equency.
The e o e, i is concluded ha , o a NACA 0012 p o ile a pos -s all condi ions and 𝑅𝑒=2·106,
he momen um coe icien was a mo e impo an pa ame e han he non-dimensional equency
in he domain ha was s udied (𝐹+=[50,100,500,1000] and 𝐶𝜇=[10−4,10−3,10−2]). I was also
no iced ha 𝐶𝜇=10−4 may be oo low, as no a single scena io wi h his momen um coe icien
p o ided a good enough esul .
As o he non-dimensional equency, i seems sa e o s a e ha he bes es ed op ion was 𝐹+=
500, and hus, 𝑓𝑗=500 𝐻𝑧 and 𝑓𝑗
𝑓𝑣𝑠 =100.

Sus ainabili y analysis 47
5 Sus ainabili y analysis
In he cu en con ex o inno a ion in ae onau ics, imp o ing ae odynamic e iciency is no only
assessed om a echnical poin o iew, bu also om a sus ainabili y s andpoin . Syn he ic Je
Ac ua o s o e a po en ial solu ion o con olling ai low and imp o ing ae odynamic pe o mance
in pos -s all condi ions, wi h possible implica ions o educing uel consump ion, lowe ing
ope a ional cos s, and imp o ing ligh sa e y.
This chap e p esen s a sus ainabili y s udy abou he elabo a ion o his p ojec , aking in o
accoun h ee key dimensions: en i onmen al, economic, and social impac .
This p ojec equi ed o use se e al esou ces, di ided in echnological and human esou ces:
 Technological esou ces:
o Ma e ial: one lap op, one ex e nal 2 TB ha d d i e, and a mouse.
o So wa e: GMSH, OpenFOAM, Pa aView, Py hon, Visual S udio Code,
Mic oso Excel 2016, Mic oso Wo d 2016.
 Human esou ces: one enginee ing s uden , dedica ion ime o 300 ℎ.
5.1 En i onmen al impac
I is es ima ed ha an ac i e, high p o ile lap op consumes app oxima ely 170 W o powe . The
lap op has been wo king app oxima ely 300 h wi h asks such as mesh design, w i ing, and o he
asks in ol ing human ac i i y. In addi ion i is es ima ed ha 150 h o simula ions ha e been
pe o med wi h he au ho no being ac i ely wo king. This gi es a o al o app oxima ely 450 h o
use o he lap op o his speci ic p ojec , gi ing an ene gy consump ion o 𝐸=170 𝑊· 450 ℎ=
76.5 𝑘𝑊ℎ.
Acco ding o he mos ecen da a (2023) in e e ence [37], he ca bon dioxide emissions
associa ed o he elec ic ene gy consump ion we e 260 𝑔 𝐶𝑂2/𝑘𝑊ℎ. Thus, he ca bon dioxide
emissions p oduced by his p ojec ha e been: 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑠 (𝑔 𝐶𝑂2)=260 𝑔
𝑘𝑊ℎ·76.5 𝑘𝑊ℎ=
19.89 𝑘𝑔 𝐶𝑂2.
I is conside ed ha he e was no en i onmen al impac associa ed wi h he human esou ces.
5.2 Economic impac
The lap op ha has been used has a cu en ma ke alue o 500 €, he ex e nal ha d d i e bough
exclusi ely o he simula ions (which occupied 946 GB o space) was bough o 75 €, and he
cos o he mouse was 10 €. As o he so wa e, Mic oso O ice 2016 can be cu en ly bough
o 10 €. Thus, he economic cos associa ed o he echnology used in his p ojec was 𝐶𝑜𝑠𝑡𝑡𝑒𝑐ℎ=
500+75+10·2=595 €.
Rega ding he human esou ces, acco ding o [38], he a e age sala y o a junio enginee in
Spain is 13.33€
ℎ. This gi es a human esou ce cos o 𝐶𝑜𝑠𝑡ℎ𝑢𝑚𝑎𝑛 =13.33€
ℎ·300 ℎ=4000 €.
Finally, he ene gy cos s ha e o be conside ed oo. Acco ding o [39], elec ical ene gy had an
associa ed cos o 0.2436 €/𝑘𝑊ℎ in June 2024, which will be aken as a e e ence o he yea .
The e o e, he ene gy cos s we e 𝐶𝑜𝑠𝑡𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑖𝑡𝑦 =0.2436 €
𝑘𝑊ℎ·76.5 𝑘𝑊ℎ=18.64 €.
This ga e a o al cos 𝐶𝑜𝑠𝑡𝑜𝑣𝑒𝑟𝑎𝑙𝑙 =595+4000+18.64=4613.64 €.
48 Assessmen o Ac i e Flow Con ol echniques o p e en ai oil s all
5.3 Social impac
The social impac o his p ojec is almos non-exis ing, and canno be p ope ly es ima ed.
Ne e heless, he e is he possibili y ha SJA a e implemen ed in comme cial and ci il a ia ion in
he u u e. They could con ibu e o enhance sa e y o ai ehicles and p o ide new employmen
posi ions in uni e si ies and he ae onau ic indus y. Since hese would be used o imp o e he
ae odynamic pe o mance o ehicles and his would educe he uel consump ion, i would be
bene icial o he clien s ha use ai anspo , making i mo e accessible.
Conclusions and u he in es iga ion 49
6 Conclusions and u he in es iga ion
This chap e p o ides wo sec ions wi h he inal conclusions and u he a eas o in es iga ion
and imp o emen .
6.1 Conclusions
A e he ini ial bibliog aphical esea ch and he mesh dependence s udy, in ending o p o ide he
maximum amoun o accu acy possible, di e en disco e ies we e encoun e ed du ing he
elabo a ion o his bachelo inal p ojec .
As o he mesh, i was ound ha he bes design o his conc e e applica ion consis ed in di iding
he nume ical domain in ou egions:
 The ou e egion, o med by an uns uc u ed g id and coa se elemen s.
 The inne egion, also cons uc ed wi h an uns uc u ed g id, bu his ime wi h ine
elemen s.
 The wake, made wi h he same s uc u e as he ou e and he inne egions, bu wi h a
ine mesh nea he ai oil.
 The bounda y laye , being he only s uc u ed-g id su ace, cha ac e ized by a e y ine
mesh, and being s ongly a ec ed depending on he u bulence model ha is used.
Rega ding he simula ions, i was imp essi e o see he accu acy on he li coe icien coe icien s,
as he e we e angles wi h a ela i e e o smalle han 1%. Un o una ely, he same hing could
no be said abou he d ag coe icien s, which ca ied a sys emic e o ha could be p oduced by
he inaccu acy in he Cou an numbe , al hough i is no clea .
I was in e es ing o see oo he di icul y o he p edic ion o he s all condi ions, which ended up
being signi ican ly highe in he simula ions compa ed o he expe imen al da a by Ladson.
The implemen a ion o he je was de ini ely he mos in e es ing pa o he hesis, as he e we e
no ewo hy imp o emen s, eaching inc emen s in he ae odynamic e iciency up o ∆𝜂9
𝜂𝑏𝑙 (%)=
1598.57 % due o achie ing he ea achmen o he bounda y laye , which could be e i ied using
Pa aView. In he end, i was ound ha he alues ha p esen ed he bes imp o emen we e
𝐹+=500 (which co esponded wi h 𝑓𝑗
𝑓𝑣𝑠 =100) and 𝐶𝜇=10−2. Also, i is wo h men ioning ha all
he simula ions wi h 𝐶𝜇=10−2 accomplished he objec i e o his p ojec , which was o ea ach
he bounda y laye o he ai oil a pos -s all angles o a ack.
Fo hese easons, i could be said ha he main objec i e o his p ojec was achie ed: he
ae odynamic pe o mance o he NACA 0012 p o ile a pos -s all angles o a ack and high
Reynolds numbe s has been es ed wi h sa is ying esul s. E en so, he e has been one
signi ican mis ake wi h he Cou an numbe s, making hem bigge han one, ha could be he
cause o ha ing he sys emic e o o he d ag coe icien . Since his mis ake was no iced oo la e,
he e was no enough ime o make all he simula ions again and, o his eason, i will be
p oposed as an imp o emen o u he in es iga ions.
6.2 Fu he in es iga ions
This p ojec has been a i s app oach in o he use o SJA a high Reynolds numbe s. F om now
on, he e a e se e al challenges o ake in his in e es ing subjec .
 Mo e app op ia e Cou an numbe s could be used o ind ou i he e o o he d ag
coe icien is ela ed o he inaccu acy o hese being o la ge, and maybe e en achie e
50 Assessmen o Ac i e Flow Con ol echniques o p e en ai oil s all
be e s all condi ions in he simula ions. In addi ion, he p oblem o he wall shea s ess
could be add essed in o de o ha e be e knowledge o he sepa a ion o he bounda y
laye .
 An in e es ing possibili y would be o y his echnology wi h 3D lows, making hem mo e
ealis ic, as 2D lows a e equi alen o ha ing an in ini e 3D wing. Those esul s could be
used as a baseline o implemen SJA in eal models and es hem in wind o wa e
unnels
 Ano he op ion could be o make 2D CFD simula ions, as in his s udy, abou je s wi h
di e en incidence angles, o e en angen ial je s. This would be in e es ing, as angen ial
je s ha e been s udied al eady in less u bulen lows, wi h lowe Reynolds numbe s, and
ha e gi en e y in e es ing esul s, some imes p o ing o be e en be e han
pe pendicula je s.
 Finally, i is p oposed o combine wo syn he ic je s a di e en cho d posi ions. This would
open he doo o combining se e al con igu a ions, modi ying pa ame e s such as 𝐹+, 𝐶𝜇,
and 𝜃𝑗 ( he incidence angle). Since his would imply ha ing almos in ini e possibili ies,
he inal s ep could be o use op imiza ion algo i hms (e.g. gene ic o a i icial in elligence
algo i hms) o ind he alues ha would gi e he bes ae odynamic pe o mance.
Bibliog aphy 51
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