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21ème Congrès Français de Mécanique Bordeaux, 26 au 30 août
2013
Tonal noise generation in the flow around an aerofoil:
a global stability analysis
M. Fosas de Pandoa, Peter J. Schmida, Denis Sippb
a. LadHyX, CNRS–École Polytechnique, route de Saclay, 91128
Palaiseau (France)b. ONERA DAFE, 8 rue des vertugadins, 92190
Meudon (France)
Résumé :
Sous certaines conditions, le spectre acoustique de
l’écoulement compressible autour d’un profil aéro-dynamique est
caractérisé par un ensemble de fréquences discrètes
équiréparties. De précédentesexpériences et simulations
numériques ont montré que l’origine de ce phénomène est due à
l’interactiondes instabilités de couche limite avec le rayonnement
acoustique du bord de fuite. Cependant, l’analysede ce phénomène
a jusqu’à présent été limitée par le champ d’application de la
théorie classiqued’instabilité hydrodynamique qui ne peut pas
prendre en compte des effets de couplage de longuedistance.
Notamment, l’origine physique des fréquences discrètes est encore
mal comprise. Nousnous intéressons ici à ce phénomène en
étudiant la réponse fréquentielle sur l’ensemble du
domaine.L’écoulement autour d’un profil aérodynamique
bidimensionnel est d’abord analysé au moyen de simu-lations
non-linéaires. Nous montrons ensuite que la dynamique linéaire
autour de l’écoulement moyenprésente une amplification importante
aux fréquences discrètes qui dominent le spectre acoustique.
En-fin, les structures spatiales de la réponse optimale et du
forçage optimal sont brièvement présentées.
Abstract :
At certain flow conditions, the acoustic spectrum of the
compressible flow around an aerofoil is char-acterized by a set of
discrete equally-spaced tones. Previous experiments and numerical
simulationshave shown that at an interplay between boundary-layer
instabilities and acoustic radiation at the trail-ing edge is at
the origin of this phenomenon. However, the analysis of the tonal
noise is at presentlimited by the shortcomings of classical
hydrodynamic stability theory, which can not account for
longdistance feedback effects. At present, the physical origin of
discrete tones is under debate. In thiswork, we address the tonal
noise phenomenon on aerofoils by means of a global frequency
responseanalysis. The flow around a two-dimensional aerofoil is
first computed using a nonlinear simulationcode; it is then showed
that the linearized dynamics about the mean flow display strong
amplification atthe frequencies of the acoustic tones. Finally, the
spatial features of the optimal response and optimalforcing are
briefly described.
Mots clefs : hydrodynamic stability; computational
aeroacoustics;
1 Introduction
Under certain parameter regime, aerofoil flows can display
substantial levels of noise radiation. In thecase of
moderate-to-high Reynolds number (105 < Re < 2 · 106), small
angles of incidence (α < 7deg)and low-turbulence incoming flow
(smaller than 0.1%), experiments have shown that the
acousticspectrum is characterized by strong equally-spaced discrete
tones. This phenomenon, known as tonalnoise, is associated with the
ringing of coherent structures near the trailing edge, and involves
complexinteraction between hydrodynamic features of separated
boundary layers and acoustic radiation in thenear wake[7]. At
present, a complete description of the tonal noise phenomenon is
still missing, partlyowing to the limitations and shortcomings of
classical hydrodynamic stability theory. For instance,
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21ème Congrès Français de Mécanique Bordeaux, 26 au 30 août
2013
spatial stability analyses correctly predict the frequency of
the most dominant tone[3], but they areunable to predict the
occurrence of multiple tones and to quantify long-distance feedback
effects.
The aim of the present work is to provide further insight into
the tonal noise phenomenon with the aidof numerical calculations
and global stability theory[2]. For this task, an optimal frequency
responseanalysis is carried out, and the obtained gain is compared
to the acoustic spectrum obtained fromnonlinear calculations.
2 Numerical set-upA direct numerical simulation code based on
the compressible Navier–Stokes equations has beenused for the
present study. The governing equations have been implemented using
the so-calledpseudo-characteristics formulation[8]; the
discretization in space and in time has been performed
usinghigh-order compact schemes and a low-storage Runge–Kutta
scheme, respectively. In addition, thisnumerical code features an
efficient extraction technique for gaining access to the linearized
direct andadjoint operators[4] provided that a suitable base flow
is given.
(a)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2x
−0.15−0.10−0.05
0.000.050.100.15
y
−200−160−120−80−4004080120160200
(b)
−1 0 1 2x
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
y
−0.5−0.4−0.3−0.2−0.10.0
0.1
0.2
0.3
0.4
0.5
Figure 1: Instantaneous flow field showing (a) vorticity levels
and (b) instantaneous dilatation contoursonce a quasi-periodic
regime is established.
The response of the linearized operator A to a harmonic forcing
fe−iωt and zero initial condition isgiven by the solution of the
following system of ordinary differential equations:
dv
dt= (A + iωI)v + f with v(0) = 0, (1)
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21ème Congrès Français de Mécanique Bordeaux, 26 au 30 août
2013
where the quantity e−iωt has been factored out. At time t, the
response reads
v(t;ω)e−iωt with v(t;ω) = R(t;ω)f , and R(t;ω) = (A +
iωI)−1(e(A+iωI)t − I
). (2)
If the operator A is stable, the asymptotic response to a given
forcing is given by v = − (A + iωI)−1 f .However, the evaluation of
the latter expression requires solving a large-scale linear system,
which isuntractable with typical available computational resources.
Instead, we opt for a matrix-free iterativeapproach, and
approximate the asymptotic response by the long-time integration of
equation (2). Fora sufficiently large integration time T , we have
v ≈ R(T ;ω)f .Our aim is to compute for a given angular frequency
ω, the forcing that maximizes the flow response.In detail, we seek
for a forcing fmax(ω) such that
Gmax(ω) = maxf
‖R(T ;ω)f‖‖f‖
. (3)
The optimal forcing fmax, the optimal response vmax and the
optimal gain Gmax are given, respectively,by the leading singular
value, the right singular vector and left singular vector of the
operator R(T ;ω).Their calculation has been performed using the
iterative Lanczos technique as implemented in thesoftware package
SLEPc[6], which requires the evaluation of matrix-vector products
involving theoperator R(T ;ω) and its adjoint R∗(T ;ω).It is
important to note that, the evaluation of equation (2) has been
performed using the exponentialKrylov time-integration
technique[9]. In this case, the adjoint operator satisfies the
duality rela-tion 〈R∗(T ;ω)w,v〉 = 〈w,R(T ;ω)v〉 up to machine
precission, which is crucial for fast convergence ofthe Lanczos
algorithm[5].
In the following, we focus on the two-dimensional flow around a
NACA0012 aerofoil section at 2deg ofincidence. The governing
equations are considered in non-dimensional form; the aerofoil
chord lengthis taken as the reference length, and the reference
quantities for the flow variables are taken as those inthe
unperturbed free-stream. The Reynolds number Re is 2 · 105, the
Mach number M is 0.4, and theheat capacity ratio γ is 1.4.
3 Results
In figure 1, we display a representative snapshot of the
instantaneous flow field from the nonlinearcalculations, showing
separated boundary-layer dynamics and acoustic radiation into the
far-field.The separation of the boundary layer on the suction
surface of the aerofoil (upper surface) extendsover 0.41 < x
< 0.66; on the pressure surface of the aerofoil (lower surface),
the separated regionextends over 0.72 < x < 0.97.
In order to analyse the frequency content of the acoustic
radiation, the pressure signal at x = 1and y = 0.5 has been
extracted, and its Fourier transform has been computed. The result,
shown infigure 2a, confirms that the acoustic spectrum consists of
a set of discrete tones. The angular frequencyof the tones is given
in table 1.
The attention is now focused on the optimal frequency response
of the linear dynamics of the meanflow computed from the nonlinear
simulation. Using the procedure sketched in section 2, the
optimalfrequency response has been computed for several angular
frequencies. In our case, an integrationtime T = 15 has been chosen
due to restrictions in the CPU time for the calculations; with this
choice,the asymptotic response is converged up to 10−2. The
calculation of the leading singular values usingKrylov spaces of
size m = 8 yielded converged leading singular values within a
relative error � ≈ 10−6.The optimal gain for selected frequencies
is given in figure 2b. It is readily noticed that the optimal
gaindisplays multiple local maxima in the frequency range of the
acoustic tones. Despite the reduced numberof calculations, the
frequency of the acoustic tones and the frequencies at which the
amplification bythe linearized dynamics is maximum agree within
remarkable agreement (see table 1). This observation
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21ème Congrès Français de Mécanique Bordeaux, 26 au 30 août
2013
(a)
40 41 42 43 44 45 46 47 48ω
0.0
0.5
1.0
1.5
2.0
2.5
p
×10−4
(b)
40 41 42 43 44 45 46 47 48ω
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
σ1
×107
Figure 2: (a) Magnitude of the DFT of the pressure signal
extracted from the nonlinear simulation atthe probe point, x = 1
and y = 0.5. (b) Frequency response of the linearized operator
about the meanflow showing gain in amplitude (also the leading
singular value of the resolvent) versus frequency. Notethat the
continuous curve has been obtained by a piecewise interpolant
constructed with sensitivityinformation.
Frequency peaks
nonlinear simulations 39.46 41.84 44.30 —linear: frequency
response — 41.75 44.00 46.15
Table 1: Comparison between the tones in the acoustic spectrum
(nonlinear simulation) and the localmaxima of the amplitude gain
(frequency response of the linearized operator).
suggests that the linearized dynamics account for the physical
mechanisms for the occurrence ofmultiple discrete tones.
The spatial structure of the optimal flow response for ω = 44 is
presented in figure 3a–b. The stream-wise velocity component (see
figure 3a) displays the growth of instability waves near the
reattachmentpoint of the separated boundary layer of the suction
surface of the aerofoil section, and in the nearwake region. As the
near-wake instability waves interact with the trailing edge of the
aerofoil, acousticradiation at a characteristic wavelength and in
counterphase between the upper and lower half of theaerofoil is
observed in the far-field (see figure 3b).
In figure 3c, the optimal forcing is represented by the
stream-wise velocity component. Contrarilyto the optimal response,
the optimal forcing displays spatial support on the pressure
surface of theaerofoil upstream the separation point, and decays
exponentially towards the free-stream. This resultindicates that
the maximum response of the flow is achieved by forcing the flow at
this location. It isimportant to remark that in early experiments
and recent numerical simulations it has been reportedthat the
frequency of the most dominant tone was selected by the instability
characteristics of thepressure surface boundary layer at this
precise location[3].
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21ème Congrès Français de Mécanique Bordeaux, 26 au 30 août
2013
(a)
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1x
−0.2
−0.1
0.0
0.1
0.2y
ω = 44.0, u
(b)
−0.5 0.0 0.5 1.0 1.5x
−0.6−0.4−0.2
0.0
0.2
0.4
0.6
y
ω = 44.0, p
(c)
0.0 0.2 0.4 0.6 0.8 1.0x
−0.25−0.20−0.15−0.10−0.05
0.000.050.10
y
ω = 44.0, |u∗|
10−4
10−3
10−2
10−1
100
100
0.1 0.2 0.3 0.4 0.5 0.6−6.4−6.2−6.0−5.8−5.6−5.4−5.2−5.0
×10−2
Figure 3: Spatial structure of the optimal response and optimal
forcing (ω = 44). The amplitude gainis 3.48 · 107. (a) Stream-wise
velocity component of the optimal response. (b) Pressure component
ofthe optimal response (c) Stream-wise component of the optimal
forcing.
4 Conclusions
In this work, we have presented a frequency response analysis of
the tonal noise generated by theflow around an aerofoil. For the
chosen flow case, the nonlinear calculations display typical
featuresof the tonal noise phenomenon. The linear optimal response
analysis reveals large amplification ofdisturbances in the flow,
and the optimal gain displays local maxima near the frequency of
acoustictones, which indicates that the occurrence of discrete
tones is represented by the linear dynamics of theflow. The main
features of the optimal response and the optimal forcing have been
presented: whilethe optimal response displays instability
mechanisms in the suction-surface separation bubble, thenear wake
and acoustic radiation in the far-field, the optimal forcing is
located on the pressure surface,upstream the reattachment
point.
Although in our study the linearized operator has been found
stable, previous experiments[7] andnumerical simulations[3] have
shown that the tonal noise phenomenon occurs even in the absence
ofexternal forcing. For these reasons, further analysis is required
to gain insight into the onset of aself-sustained process.
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Fluid Mech. 591 155–182
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21ème Congrès Français de Mécanique Bordeaux, 26 au 30 août
2013
[4] M. Fosas de Pando, D. Sipp, P.J. Schmid 2012 Efficient
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