Procedia IUTAM 14 ( 2015 ) 141 – 146
Available online at www.sciencedirect.com
2210-9838 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering)
doi: 10.1016/j.piutam.2015.03.034
ScienceDirect
IUTAM ABCM Symposium on Laminar Turbulent Transition
Stability analysis of time-averaged jet flows:
fundamentals and application
Kilian Oberleithnera,∗, Christian Oliver Paschereita, Julio Soriab
aInstitut f¨ur Str¨omungsmechanik und Technische Akustik
Technische Universit¨at Berlin
bLaboratory for Turbulence Research in Aerospace &Combustion
Department of Mechanical and Aerospace Engineering
Monash University, Melbourne
Abstract
We report on experimental and theoretical investigations of shear flow instabilities in jet flows. Linear stability analysis is applied to
the time-averaged flow taken from experiments, contrasting the ’classic’ stability approach that is based on a stationary base flow.
To some extend, mean flow stability eigenmodes may deal as a model for instability waves at their nonlinearly saturated state, which
is typically encountered in experiments. The capability of mean flow stability models is first demonstrated on laminar oscillating
jets where the primary interaction takes place between the mean flow and the instability wave. We then focus on turbulent swirling
jets where additional interactions occur between the fine-scale turbulence and the instability waves. Swirling flows are widely used
in combustion applications where the associated high turbulence levels and internal recirculation zones (vortex breakdown bubble)
are exploited for flame stabilization. We demonstrate the application of mean flow stability analysis on the flow field of a industry-
relevant swirl-stabilized flame. We show that the flame response to acoutstic perturbations is closely linked to the flow receptivity
predicted from linear stability analysis, which suggests that the adopted theoretical framework is very useful for thermoacoustic
modeling.
c
2014 The Authors. Published by Elsevier B.V.
Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering).
Keywords: linear stability analysis, jet flows, convective/global instability, coherent structures, thermoacoustic instability
1. Theoretical concept of mean flow stability wave models
1.1. Governing equations
The stability equations of the mean (time-averaged) flow are derived from the classic triple decomposition. The
instantaneous flow field vector u(x,t) is decomposed into a time-averaged part u(x), a periodic (coherent) part ˜
u(x,t),
∗Corresponding author
E-mail address: oberleithner@tu-berlin.de
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering)
142 Kilian Oberleithner et al. / Procedia IUTAM 14 ( 2015 ) 141 – 146
and a randomly fluctuating (turbulent) part u(x,t), reading
u(x,t)=u(x)+˜
u(x,t)+u(x,t).(1)
The phase-average u(x,t)is used to separate the fine-scale turbulent fluctuations from the coherent motion such that
˜
u(x,t)=u(x,t)−u(x).
The triple decomposition is substituted into the incompressible Navier–Stokes equation and the continuity equation,
and, after some manipulations, the governing equations for each of the three parts can be formulated10.Themean
flow equations are
u·∇u=−∇p+1
Re∇2u−∇·uu+˜
u˜
u(2a)
∇·u=0,(2b)
indicating how the mean flow is modified through the generation of turbulent and coherent Reynolds stresses. The
equations for the coherent motion are given as
∂˜
u
∂t+˜
u·∇
u+u·∇˜
u=−∇ ˜p+1
Re∇2˜
u−∇·τN+˜τ(3a)
∇·˜
u=0,(3b)
where the nonlinear terms τN=˜
u˜
u−˜
u˜
uare neglected in the following. The terms ˜τ=uu−uu=
uurepresent
the modification of the turbulent field during the passage of a coherent structure. These turbulent-coherent interactions
are unknown and must be modeled appropriately. The mean-coherent and mean-turbulent interactions are reflected in
the actual mean flow shape and are implicitly accounted for in the perturbation equations (3). The turbulent-coherent
interactions are modeled through a Newtonian eddy viscosity model
˜τij =−
u
iu
j=νt∂˜ui
∂xj
+∂˜uj
∂xi,(4)
where νtis the eddy viscosity of the undisturbed flow and the indices i,j=1,2,3 indicate the three velocity compo-
nents.
1.2. Solution for weakly nonparallel flows
Equations (3) are first solved for a parallel flow u0=(f(r),0,0)T. In the presented studies, our interest lies in the
spatial growth and decay of instabilities. Therefore, the perturbations have the form
˜
u(x,t)=ˆ
u0(r)ei(αx+mθ−ωt)+c.c., (5)
with complex spatial wavenumber α=αr+iαi, integer real azimuthal wavenumber m, and real temporal oscillation
frequency ω. The conjugate complex of the perturbation isindicated by ’c.c.’. The imaginary part of αcorresponds to
the spatial growth rate of the parallel flow and determines whether a perturbation of a given mand ωgrows (−αi>0)
or decays (−αi<0) in the streamwise direction. Substituting the ansatz (5) and the equivalent for the pressure into
(3) leads to the eigenvalue problem
D(ω)ψ0=αE(ω)ψ0,(6)
with the eigenvalue αand the eigenfunction ψ0=(ˆu0,ˆv0,ˆw0,ˆp0)T, and the matrices Dand Econtaining the parallel
flow profiles u0.
The stability analysis is extended to weakly nonparallelflows by adopting the correction scheme developed by
Crighton & Gaster2. To account for a slow streamwise jet divergence, we introduce a slow axial scale X=x,where
1, and a radial component v1=v/. The global perturbation field is given as
˜
u(X,r,θ,t;)=N(X)ˆ
u0(X,r;)expi
X
α(ξ)dξ+imθ−iωt+c.c., (7)
Kilian Oberleithner et al. / Procedia IUTAM 14 ( 2015 ) 141 – 146 143
with the amplitude factor N(X) given as
dN(X)
dXG(X)+N(X)K(X)=0.(8)
The expressions Gand Kin (8) are derived from the solvability condition ofthe first order problem, and they contain
the radial and streamwsie derivatives of eigenfunctions and their adjoints of the (zero order) parallel flow solution.
The derivations of the multiple-scale approximation for swirling jets is given in Oberleithner et al.7.
2. Results from recent laminar, turbulent, and reacting jet studies
In the following, we briefly describe the results from mean flow stability analysis of three different flow configura-
tions. The first deals with a laminar oscillating jet. This study is aimed to asses the accuracy of the mean flow model
in predicting the nonlinear fluctuations at strongly forced conditions. Details to this study are given in Oberleithner et
al.7. The second configuration is a turbulent swirling jet at pre-breakdown conditions. In this study, we are interested
in the helical modes that develop in the highly turbulent, konvectively unstable shear layer. Details to this study are
given in Oberleithner et al.6. The third study combines the findings of the first and second investigation. We ap-
ply mean flow stability concepts to a turbulent swirling jet at post-breakdown conditions with an anchored V-shaped
flame. This configuration is typically found in gas turbine combustion chambers. We compute the flow receptivity
to acoustic perturbations at low and high amplitude acoustic forcing and show correlations to the respective flame
response. Details to this study are given in Oberleithner et al.8.
2.1. Laminar oscillating jets
The axisymmetric laminar jet was generated by a piston-cylinder-type arrangement and released into a large water
tank. The jet’s axisymmetric mode was excited by imposing a sinusoidal motion onto the piston’s mean motion at
relative amplitudes ranging from 0.1 % to 100 %. The flow field was obtained using high-resolution, low-speed,
PIV. The oscillations corresponding to the fundamental and higher harmonics of the forcing frequency were extracted
from the uncorrelated PIV snapshots using a POD-based phase reconstruction scheme. The experimental results are
compared to the stability wave model, including amplitude and phase distribution, growth rates, and phase velocities.
Figure 1 shows the results for the jet forced sinusoidally at amplitudes of 5 % of the mean flow rate. The overall
agreement between the stability wave model and the measured flow field fluctuations is excellent. The analysis of the
flow forced at a wide range of amplitudes further reveals that the stability wave model captures well the growing phase
and precisely predicts the neutral point and its upstream displacement with increasing forcing amplitudes. Nonlinear
interactions, indicated by the higher harmonics seem not to affect the accuracy of the prediction. The decay phase
of the stability wave is well predicted up to a streamwise location where the Reynolds stresses of the fundamental
wave change their sign. The study suggests that the flow model fails once energy is transferred from the coherent
fluctuations back to the mean flow.
The study gives credibility to mean flow stability analyses conducted recently, such as on the cylinder wake vortex
shedding1,4, on the vortex breakdown bubble9, or on turbulent jets3,6. Inaccuracies of the model due to the interaction
of the mean flow with the first harmonic, as observed for the cavity flow11, have not been observed presently.
2.2. Turbulent swirling jets
The nearfield of a turbulent swirling jet is investigated at a Reynolds number of ReD=20000. Four different swirl
intensities were investigated, ranging from zero swirl to intensities that are just below the onset of vortex breakdown.
The time-averaged flow field was measured via PIV and fed into the stability solver. The computed eigenmodes were
then compared to hot-wire measurements that were phase-locked to low amplitude external forcing
Two excitation modes were considered. First the flow was excited a single frequency for different azimuthal modes.
The phase-averaged velocities were then compared to the corresponding mean flow eigenmodes. This investigation
reveals that the growth rate of the excited wave train is very well predicted of one accounts for the coherent-turbulent
interactions (eddy viscosity model)6.
144 Kilian Oberleithner et al. / Procedia IUTAM 14 ( 2015 ) 141 – 146
x/D
|ˆv|
x/D
r/D
arg(ˆv)
r/D
(ˆv)
r/D
(ˆv)
r/D
|ˆu|
r/D
arg(ˆu)
r/D
(ˆu)
r/D
Linear stability analysis
(ˆu)
PIV measurements
r/D
012345678910012345678910
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
Fig. 1. Laminar axisymmetric jet forced at 5 % of the mean flow rate: Overall comparison of the flow oscillations measured via PIV (left)
and the perturbation field obtained from the linear stability analysis (right). Displayed are the real and imaginary part, phase, and magnitude of
the streamwise and radial component of the velocity fluctuations at the fundamental frequency. Nomenclature: streamwise coordinate x;radial
coordinate r; nozzle diameter D. See Oberleithner et al. 7for further information.
In a second experiment, the flow was excited by a time- and space-discrete pulse localized at the nozzle lip. The
streamwise development of the wavepacket that was generated by this pulse was tracked by means of ensemble-
averaged hot-wire measurements. Figure 2 shows results from these measurements. The modal content of the
wavepacket measured at different streamwise positions is compared to the prediction from mean flow stability anal-
ysis. Results are shown for the non-swirling jet and the swirling jet. Evidently, the stability wave model is capably
to predict the mode selection that occurs naturally in the flow for both, the swirling and non-swirling jet. The figure
reveals that the non-swirling jet prefers the axisymmetric mode near the nozzle and the bending single-helical modes
(m=±1)further downstream. In contrast, the swirling jet prefers helical modes everywhere with strong amplification
of weakly rotating and steady modes (ω=0) further downstream.
2.3. Reacting turbulent swirling jets
This study focuses on the formation of coherent flow structures in reacting combustor flows and its impact on
the global heat release rate fluctuations. We consider a perfectly premixed swirl-stabilized flame that is acoustically
forced axially mimicking the dynamics associated with thermoacoustic instability. Phase-averaged flow field and
OH∗-chemiluminescence measurements show that heat release fluctuations are generated through the roll-up of the
flame tip due to large-scale coherent flow structures imping the flame front. Details to the flame shape and dynamics
are given in Oberleithner et al.8.
The flame’s response to the forcing is first characterized by a flame describing function (FDF) by essentially
treating the flow field and the flame in a black box manner5. We obtain the global heat release rate fluctuations from
a photomultiplier behind a OH∗filter. The left frame of figure 3 shows such a FDF measured in the laboratory at the
TU Berlin.
Kilian Oberleithner et al. / Procedia IUTAM 14 ( 2015 ) 141 – 146 145
x/D=3
x/D=2.5
x/D=2
x/D=1.5
x/D=1
mm
ωδ/Vωδ/Vωδ/Vωδ/V
strong swirlno swirl
ωδ/V
−4−3−2−101234
−4−3−2−101234
0
0.5
1
0
0.1
0.2
0
0.1
0.2
0
0.1
0.2
0
0.1
0.2
0
0.1
0.2
(a) Hot-wire measurements
x/D=3
x/D=2.5
x/D=2
x/D=1.5
x/D=1
m
m
ωδ/Vωδ/Vωδ/Vωδ/V
strong swirlno swirl
ωδ/V
−4−3−2−101234
−4−3−2−101234
0
0.5
1
0
0.1
0.2
0
0.1
0.2
0
0.1
0.2
0
0.1
0.2
0
0.1
0.2
(b) Linear stability analysis
Fig. 2. Modal amplitude distribution of a wavepacket traveling in the shear layer of a turbulent non-swirling and swirling jet. Swirl tends to break
symmetry and destabilizes modes at low and zero frequencies (steady modes). Nomenclature: azimuthal wave number m; frequency ω; axial shear
layer thickness δ; bulk velocity V; streamwise coordinate x; radial coordinate r; nozzle diameter D. See Oberleithner et al.6for further information.
The FDF is the main empirical input for thermoacoutic network models. However, the measurement of FDFs is
very costly and time consuming and the demand for FDF prediction models is very high in the gas turbine industry.
In order to develop prediction models we need to depart from this FDF-type black-box approach and develop specific
flow and flame models. We suggest to employ hydrodynamic linear stability analysis based on the mean flow, which
provides detailed information about the formation of the coherent flow structures excited by the inflow perturbations.
In analogy to the FDF, the analytic approach provides a flow describing function, which characterizes the flow’s
response to the inflow perturbations.
The right frame in figure 3 shows such a flow transfer function derived from a local linear stability analysis. It
reveals that the receptivity of the shear flow is highest for the unforced case with a significant dependence of the gain
on the perturbation frequency. This dependence is also found in the FDF revealing that the maximum gain in the flame
response is associated with maximum growth of the shear layer instability. With stronger forcing, the receptivity of
the shear layer decreases and so does the gain in the corresponding FDF. At sufficiently strong forcing, the shear
layer is saturated and the inflow perturbations are transported to the flame front without any amplification. A further
discussion of the results is given in Oberleithner et al.8
3. Conclusion
The different studies mentioned here demonstrate the high accuracy and great potential of linear hydrodynamic
stability analysis applied to time-averaged turbulent flows as encountered in many experimental and industrial envi-
146 Kilian Oberleithner et al. / Procedia IUTAM 14 ( 2015 ) 141 – 146
Fig. 3. left: Flame describing function (FDF) of a swirl-stabilized combustor. The graphs show the gain in the heat release rate fluctuations
(OH∗emmision) versus the acoustic excitation frequency. Each graph corresponds to a different acoustic forcing amplitude. This figure discribes
the saturation of the flame response at increasing excitation amplitude. right: Results from linear hydrodynamic stability analysis of the mean
combustor flow. The graphs represent the gain of the axisymmetric Kelvin-Helmholtz mode that couples with the acoustic forcing. Apparently,
the flow response shows the same qualitative behavior as the flame response suggesting that the flame’s gain and saturation is directly linked to
hydrodynamic instability.
ronments. The laminar jet experiments show that the type of interaction between the wave train and the mean flow
is crucial for the linear model to hold. The turbulent jet experiments show that the interactions between the fine-
scale structures and the modeled coherent structures must be accounted for accurate prediction of growth rates. The
discussion of the swirling flame configuration shows an application of these flow models to a highly complex multi-
shear-layered reacting flow, which demonstrates their robustness and ability to solve industry relevant problems.
Acknowledgements. The authors acknowledge the financial support of the Australian Research Council (ARC), and
the German Research Foundation (DFG), and the Research Association for Combustion Engines (FVV). Kilian Ober-
leithner was supported by a fellowship within the Postdoc-Program of the German Academic Exchange Service
(DAAD).
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