Round Cavity Noise Simulations using Lattice-Boltzmann

Round Cavity Noise Simulations using
Lattice-Boltzmann Solver
Christophe Coreixas∗
CERFACS, 42 Avenue G. Coriolis, 31507 Toulouse, France
A new three-dimensional CFD solver, named LaBS, based on Lattice-Boltzmann Methods (LBM) has been developed within an academic and industrial consortium. In this
paper, Wall-Modelled Large-Eddy Simulations (WMLES) of a grazing flow over a round
cavity are performed. The latter features an aspect ratio depth-diameter H/D = 1 with
a freestream Mach number M∞ = 0.20. A subgrid scale model belonging to the turbulent
eddy-viscosity family is applied: the Shear-Improved Smagorinsky Model (SISM). No turbulence injection processes are employed here. Instead, an approximated Blasius laminar
solution is used as an inlet condition, and the boundary layer is left free to develop. Aerodynamic and aeroacoustic results are directly computed from simulations and show good
agreements with respect to experiments carried out during the AEROCAV project.
Nomenclature
Theory
c
Mesoscopic velocity
f
Velocity distribution function (VDF)
f eq
Equilibrium VDF
τ
Relaxation time
ρ
Macroscopic density
ρu
Macroscopic momentum
θ
Molecular agitation temperature
r
Perfect gas constant
T
Macroscopic temperature
∆t
Time step
I.
Numerical Simulations
H
Cavity depth
D
Cavity diameter
δ
Boundary layer thickness
x, y, z Cartesian coordinates
u
Longitudinal mean velocity
p
Pressure
Subscript
α
Discrete velocity index
∞
Freestream condition
Introduction
Being an important part of noise radiation during approach and landing flight phases, airframe noise is
an active research field in Aeronautics. Among the different contributions encountered, such as wing high-lift
devices or landing gears, highly tonal noise can be emitted from cavities including anti-icing and kerosene
over-pressure vents, tube openings or burst-disk cavities. For a long time, less attention has been paid to
this component of airframe noise since its impact on acoustic radiation was not, or only bearly, noticeable
on noise certification levels. Recent improvements, in reducing the other contributions, have forced aircraft
manufacturers to seriously account for noise generated by flow grazing over round cavities.
Over the past fifty years, acoustic noise radiated by cavities has been widely studied. It can be considered as a multi-faceted configuration. Indeed, cavity noise is the result of several phenomena such as
turbulent boundary layer development into a shear layer, mixing layer impingement, complex recirculation
and aeroacoustic feedback. Their coupling can be explained thanks to Vortex Sound Theory:1
1. Flow interactions with solid surfaces create vorticity which is self-sustained by the flow itself, vortex
interactions or acoustic energy conversion into vortical structures.
∗ PhD
Student, CFD Team, [email protected]
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2. In turn, acoustic waves are generated through either the impact of the mixing layer with the leading
edge of the cavity, or resonant cavity mode excitation.
In order to estimate the influence of key geometrical and flow parameters, windtunnel experiments have
been conducted and the acquired data have been studied, as part of a French research program named
AEROCAV (AEROacoustics for cylindrical CAVities), by the French office ONERA2 and Ecole Centrale de
Lyon.3 From these studies, two main configurations emerged as great candidates for numerical comparisons
both including round cavities with H/D = 1 but with a freestream Mach number of M∞ = 0.20 and 0.26
respectively. Several numerical computations, aiming to identify underlying mecanisms of round cavity noise
generation, have been recently published.4–7 None of them included direct measurement of far-field noise,
instead integral formulations were used.
The aim of the present work is to provide a first aeroacoustic validation, of the brand new solver LaBS, on a
configuration including complex noise radiation phenomena. Thus, to accurately judge the capability of the
software to provide realistic cavity noise generation and propagation, it has been chosen to not apply any
kind of integral formulation. Instead, pressure fluctuations in the far-field region will be directly simulated
through Wall-Modelled Large-Eddy-Simulations (WMLES) based on Lattice-Boltzmann Methods (LBM).
This paper is organized as follows: the numerical approach is briefly recalled in section II. Then, the
investigated configuration and numerical setup are adressed in section III. The computational results are
presented and analysed in section IV.
II.
II.A.
Numerical approach
Lattice-Boltzmann Method
Lattice Boltzmann Methods (LBM) originate from Kinetic Theory of Gas. They consider the spatiotemporal evolution of the velocity distribution function f (x, c, t), the number of fictitious particles at point
(x, t) travelling at speed c, through Boltzmann equation Eq. (1). From the macroscopic point of view,
collisions can be associated to a relaxation process that makes f tend to its thermodynamic equilibrium
f eq . Then, the RHS of Eq. (1) can be substituted by the Bhatnagar-Gross-Krook (BGK8 ) collision operator,
which leads to Eq. (2):
∂f
∂f
∂f
+ ci
=
(1)
∂t
∂xi
∂t coll
2
f − f eq
=−
τ
with
f eq
−
ρ
=
e
3/2
(2πθ)
(c-u)
2θ
(2)
where τ is the relaxation time after which f is supposed to have recovered thermodynamic equilibrium f eq ,
u the macroscopic velocity and θ = rT the molecular agitation temperature.
An efficient way to solve this equation is to consider a discrete set of speeds (cα )α instead of all possible
particle speeds in R3 . Using only 19 speeds for velocity space discretization in the three dimensional case
(D3Q19 model), isothermal Navier-Stokes set of equations can be recovered in the hydrodynamic (Kn 1)
and weakly compressible (M 1) limits.9 Further spatial/temporal discretization of derivatives eventually
leads to the famous second-order explicit scheme defined by:
fα (x + cα ∆t, t + ∆t) = fα (x, t) −
∆t
[fα (x, t) − fαeq (x, t)]
τg
(3)
with τg = τ + ∆t/2. Then, in order to recover macroscopic behaviors of the flow, it is necessary to compute
stochastic moments of f with respect to mesoscopic velocity c:
Z

X


ρ(x,
t)
=
f
(x,
c,
t)
dc
=
fα (x, t)


3


α
ZR
X
(4)
ρ(x, t)u(x, t) =
cf (x, c, t) dc =
cα fα (x, t)


3

R

α


θ(x, t) =
θ0
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This numerical scheme yields several advantages for Computational Fluid Dynamics (CFD). Among them,
low-dissipation10 and intrinsic parallelizable properties make LBM serious candidates for aeroacoustic computations, where density fluctuation waves orginating from turbulence need to be perfectly convected over
long distances with respect to grid cell size.
II.B.
Numerical Method
In the present work, the newly developed Lattice-Boltzmann Solver (LaBS) is used to directly compute
aerodynamic (density ρ, momentum ρu) and aeroacoustic (pressure p) properties of the flow. This code has
been developed within a consortium of industrial companies (Renault, Airbus, CS), academic laboratories
(UPMC, ENS Lyon) and strong partnerships with other entities (Onera, Gantha, Paris Sud University, Alstom, Matelys) from 2010 to 2014.
The discretization of the velocity space is based on a D3Q19 lattice and a Two-Relaxation-Time (TRT) collision operator.11 The latter adds a regularizion step of pre-collision distribution functions, allowing better
stability/accuracy properties of the numerical scheme through rotational invariance recovery.12, 13
Furthermore, to properly recover the behaviors of turbulent flow without solving all vortical structures
down to Kolmogorov scale, a subgrid scale model based on turbulent eddy-viscosity is used. Thus, LES are
performed using a Shear-Improved Smagorinsky Model (SISM).14 Due to high-Reynolds characteristics of
the flow encountered, wall-resolved meshes are hardly achievable within industry perspectives. Consequently,
a wall model including pressure gradient contribution15 is used to correctly recover flow behaviors (friction
coefficient and velocity profile) at the first fluid node above walls.
III.
III.A.
Investigated Configuration and Simulation Setup
Configuration of Interest
As a highly resonant case, the configuration including a round cavity with an aspect ratio H/D = 1 and
a free-stream velocity of u∞ = 70 m/s (M∞ = 0.20 at ISA conditions) is considered. For this particular
configuration, experiments conducted at Ecole Centrale de Lyon provide measurements of mean/rms velocity
profiles, steady/unsteady wall pressure and far-field noise. Data are obtained via Laser-Doppler-Anemometry
(LDA), pressure taps and a revolvable arc of microphones respectively.16
III.B.
III.B.1.
Mesh Strategy & Boundary Conditions
Grid Generation
In this paper, aeroacoustic radiations emitted from the cavity will be computed directly: acoustic analogy
formulations are not employed here. Thus, a 25D × 25D × 15D multi-block mesh is built, with the cavity
diameter D = 10 cm . The revolvable arc of microphones, where pressure data are compiled, is located
1m above the cavity center and is directly simulated in the present paper. Besides, 2.5-meter thick sponge
layers, based on Perfectly Match Layer formulations,17 have been added to all boundary conditions but flat
plate.
Ultimately, octree-based18 grid refinements are used to:
• Allow a realistic development of the boundary layer from the inlet to the cavity.
• Capture phenomena occuring above (shear layer formation / impingement) and inside (strong recirculation) the cavity.
The global view of the computational domain and a longitudinal cut of Medium case mesh are illustrated in
Fig. 1.
III.B.2.
Boundary Conditions
No turbulence injection process is used in this study. Consequently, there is no need to introduce a
turbulent mean velocity profile at the fluid domain inlet since its slope is directly linked to molecular
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(a)
(b)
Figure 1. Mesh strategy: Simulation domain (a) and Y=0 m mesh cut (b). The cavity center is the origin of the
reference frame.
viscosity, which is itself reinforced by turbulent mecanisms. Instead, a laminar mean velocity profile is
applied as inflow condition, and the boundary layer is left free to develop in order to provide a realistic
behavior of the flow. The polynomial approximated solution of Blasius problem enables the introduction of
a laminar boundary layer with a thickness δ for various orders (see Fig. 2). Being the most accurate, the
6th-order approximated solution of Blasius, defined by Eq. (5), is then chosen as inflow condition.
z 5
z 6
z 4
u
z
+6
−2
=2 −5
u∞
δ
δ
δ
δ
(5)
where u is the longitudinal mean velocity, and z is the coordinate normal to the flat plate. Regarding
the outlet and the upperside of the simulation domain, Dirichlet pressure conditions are applied. Finally,
periodicity conditions are employed for both lateral sides.
Figure 2. Longitudinal mean velocity profile u, with δ = 15 mm et u∞ = 70 m.s−1 . Black: Blasius solution, blue: 4th
order polynomial approximation, green: 6th order polynomial approximation (from Desvigne5 ).
III.C.
Simulation Matrix
The mesh convergence has been analysed through two successive grid refinements of Coarse case. This
methodology allows a statistical convergence of aerodynamic and aeroacoustic variables. All computations,
including geometrical and physical parameters of interest, are summarized in Tab. 1.
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Mesh
SGS
Coarse
Medium
Fine
SISM
SISM
SISM
Voxels
Points/D
6
5 · 10
40 · 106
140 · 106
100
200
300
∆xmin (m)
−3
1.0 · 10
5.0 · 10−4
3.3 · 10−4
∆tmin (s)
Mean z +
TS (s)
TCP U (h)
200
100
75
0.22
0.22
0.22
1.25 · 103
4.34 · 104
5.26 · 104
−6
1.68 · 10
8.4 · 10−7
5.6 · 10−7
Table 1. Computations and numerical/physical parameters review. The minimal time step is computed through the
unity sound-based CFL condition inherent in conventional LBM.
IV.
Results
Before analysing the simulation results, two main discrepancies with respect to the experimental setup
are to be pointed out. Indeed, numerical computations were done assuming a cylindrical cavity over an
infinite flat plate and the windtunnel geometry is, for conveniency, not included into the simulation. Thus,
3D installation effects are not taking into account here.
IV.A.
IV.A.1.
Aerodynamic Results
Velocity Profiles
Mean and rms velocity profiles have been obtained through LDA measurements at several locations (see
Fig. 3). In the present work, computation results are compared to these experimental data for Y = 0 m, and
only longitudinal velocity is investigated. Comparisons are plotted in Fig. 4 and show satisfactory agreement
for longitudinal mean velocity profiles. Regarding the longitudinal velocity fluctuations, good results are
obtained even if a lack of turbulent structures is observed in the upstream boundary layer. Nevertheless, it
is worth noting that recovering a boundary layer with specific thickness, mean and rms velocity profiles at a
given location, remains a tedious issue in CFD. Above all, when a transition from laminar to turbulent flow
is expected.
Figure 3. Sketch of experimental measurement locations (from Desvigne5 ).
IV.A.2.
Cavity Wall Pressure
After developping above the round cavity, the shear layer originated from the upstream boundary layer
impacts the dowstream corner of the cavity. This leads to a strong recirculation which can be divided into
three phenomena: (i) part of the mixing layer enters inside the cavity, (ii) inflow attains the cavity bottom,
(iii) it goes up and interacts with the shear layer at the upstream corner (see sketches of Figs 5, 6 and 7
respectively).
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Figure 4. Longitudinal velocity profile evolution at Y = 0 m: mean (up) u1 and rms (down) urms . Fine case results
(green) are compared with LDA measurements (· · · ) at locations : x = −55, x = −45, x = −35, x = 0, x = 35, x = 50,
x = 55 et x = 90 mm.
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(a) Sketch of some measurement locations.
(b)
P1 : (x, y, z) = (D/2, 0, −D/10)
(c)
P2 : (x, y, z) = (D/2, 0, −D/4)
(d)
P3 : (x, y, z) = (D/2, 0, −D/2)
(e)
P4 : (x, y, z) = (D/2, 0, −3D/4)
(f)
P5 : (x, y, z) = (3D/8, 0, −D)
Figure 5. Configuration sketch (a) and cavity wall pressure fluctuations at P1 (b), P2 (c), P3 (d), P4 (e), P5 (f ):
Experimental measurement (black cross), Coarse case (green), Medium case (blue) and Fine case (red).
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(a) Sketch of some measurement locations.
(c)
(e)
(b)
P7 : (x, y, z) = (D/8, 0, −D)
(d)
P9 : (x, y, z) = (−D/8, 0, −D)
(f)
P6 : (x, y, z) = (D/4, 0, −D)
P8 : (x, y, z) = (0, 0, −D)
P10 : (x, y, z) = (−D/4, 0, −D)
Figure 6. Configuration sketch (a) and cavity wall pressure fluctuations at P6 (b), P7 (c), P8 (d), P9 (e), P10 (f ):
Experimental measurement (black cross), Coarse case (green), Medium case (blue) and Fine case (red).
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(a) Sketch of some measurement locations.
(b)
P11 : (x, y, z) = (−3D/8, 0, −D)
(c)
P12 : (x, y, z) = (−D/2, 0, −3D/4)
(d)
P13 : (x, y, z) = (−D/2, 0, −D/2)
(e)
P14 : (x, y, z) = (−D/2, 0, −D/4)
(f)
P15 : (x, y, z) = (−D/2, 0, −D/10)
Figure 7. Configuration sketch (a) and cavity wall pressure fluctuations at P11 (b), P12 (c), P13 (d), P14 (e), P15 (f ):
Experimental measurement (black cross), Coarse case (green), Medium case (blue) and Fine case (red).
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Wall pressure fluctuations, corresponding to each part of the recirculation, are compared with experimental data in Figs 5, 6 and 7. Agreement between curves increases when recirculation strength decreases.
This is essentially due to very scattered hydrodynamic pressure fluctuations. In addition, these discrepancies
seem to be unchanged even with finer grids. The remaining differences between experiments and numerical
computations are also pointed out in several other simulations5, 7, 19, 20 and are still not well understood.
Nevertheless, one possible explanation could lie in the fact that hydrodynamic pressure fluctuations are in
the range of 100−1000 Pa, whereas an overestimation of 10 dB leads to an overprediction of about 1 to 20 Pa
(for decibel ranges encountered). Thus, achieving a reasonnable fitting between experiments and numerical
simulations is, a priori, out of reach for WMLES. Finally, the pressure fluctuation peaks at f ' 650 Hz
(resulting from the coupling between the second shear layer mode and the first depth mode of the cavity5 )
and higher frequencies (attributable to cavity modes5 ) are properly recovered.
IV.B.
Aeroacoustic Results: Far-Field
As a reminder, far-field noise measurements are taken through a revolvable arc of microphones located one
meter above the cavity center (see Fig. 8 (a)). Moreover, without taking into account windtunnel geometry,
installation effects have been ignored whereas they does have an impact on far-field noise measurements.4
Consequently, only data obtained exactly one meter above the cavity center (microphone 4) are compared
to experiments (see Fig. 8 (b)).
Simulations show good agreement for frequencies higher than 400 Hz, and noise generated by the aeroacoustic
feedback is well recovered. It must be noted that acoustic radiations at low frequencies are directly linked to
the windtunnel background noise.4 Thus, it is a priori very unlikely to recover, at this range of frequencies,
good pressure fluctuations without including installation geometries. Regarding frequencies higher than
1500 Hz, it seems that parasitic noise is generated inside the simulation domain. It induces an emitted
noise overstatement of less than 10 dB at high frequencies. Origins of this parasitic noise are still under
investigation.
(a) Experimental sketch
(b) Data comparison at microphone 4
Figure 8. Far-field noise measurements.16 Experimental (black cross) and numerical results are compared: Coarse case
(green), Medium case (blue) and Fine case (red).
V.
Conclusion
A new three-dimensional CFD solver, LaBS, based on Lattice-Boltzmann Methods (LBM) has been developed in the framework of an university and industry consortium. For the first time, direct noise radiations
of a grazing flow over a round cavity, with an aspect ratio depth-diameter H/D = 1 and a freestream Mach
number M∞ = 0.20, have been computed using this solver.
Whereas a simple laminar mean velocity profile was imposed as inflow conditions and led to an underestimation of the turbulence intensity inside the upstream boundary layer, satisfactory aerodynamic results
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have been obtained inside and above the cavity. Regarding wall pressure fluctuations, uncertainties remain
although aeroacoustic feedback and cavity modes are well recovered. Finally, installation effects coverage is
found to be crucial to properly predict far-field noise propagation in the low frequency range. Although parasitic noise leads to an overestimation of high frequency cavity acoustic radiation, the aeroacoustic feedback
is well simulated. Similar conclusions can be found in Sengissen et al .21
Though young, LaBS software has provided promising aerodynamic and aeroacoustic results that are a
first step towards further improvements.
Acknowledgments
The author wants to thank J.-C. Jouhaud & J.-F Boussuge from CERFACS and N. Gourdain from ISAE
for their useful pieces of advice. A. Sengissen from Airbus Operations SAS, J.-C. Giret & R. Cuidard from
CS and D. Ricot from Renault are gratefully acknowledged for their technical and theoretical support on
Lattice Boltzmann Solver: LaBS. All numerical simulations have been conducted on Airbus’ HPC resources.
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