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Part 16 Aeroacoustics
Part 16
Aeroacoustics
Euler - Navier-Stokes coupling for Aeroacousticsproblems
Michel Borrel, Laurence Halpern & Juliette Ryan
1 Introduction
In the last few years numerical aeroacoustics have undergone a very rapid evolutiondue to the emergence of direct simulations of sound emitted by flows, and in partic-ular turbulent flows. However, these computations raise the question of the couplingbetween LES and acoustics or in other words between the Navier-Stokes and the Eu-ler computations, especially if we want to take into account the multiscale aspect,both in space and time, of the problem.
What we propose here is a new method of coupling between aerodynamics andacoustics . It is based on earlier work on Schwarz waveform relaxation methods , de-veloped for time-dependent advection - diffusion - reaction. These methods are do-main decomposition algorithms. They consist in solving the equations alternativelyin each subdomain, and transmitting the necessary information through differentialspace-time transmission conditions, see for example [2]. They allow for differentdiscretization in different subdomains, see [1], even in a nonconformal manner [7].They can potentially extend to coupling different models in different zones, see [5],and are therefore well-adapted to our purpose.
A first step, presented in this paper, is to test this method , for reasons of simplic-ity, with a Discontinuous Galerkin scheme, for a) the CFD where the Navier-Stokesare solved and b) the CAA where Euler , perturbed Euler or linearized Euler equa-tions, according to test cases are resolved. Borrel et al. [3] have formulated a newDiscontinuous Galerkin scheme (EDG) for the viscous term that easily applies toeither structured or non structured discretizations. Special attention must be paidto the multi-scale aspect requiring highly non conforming space-time discretizationfor which the discontinuous Galerkin approach is particularly well adapted.
The present method will be evaluated on a 2D laminar configuration: the low-Reynolds subsonic flow around a cylinder.
In Section 2 we recall the Discontinuous Galerkin algorithm we use. In Section3 we present the Schwarz waveform relaxation algorithm followed by a descriptionin section 4 of the numerical coupling algorithm. Then in Section 5 we present two
Michel Borrel e-mail: [email protected], Juliette Ryan e-mail: [email protected], BP72 - 29 avenue de la Division Leclerc, FR-92322, CHATILLON CEDEX, FRANCE,
Laurence Halpern e-mail: [email protected], Universite Paris XIII and CNRS, 99 avenue J.B. Clement, 93430 Villetaneuse, FRANCE
1
2 Michel Borrel, Laurence Halpern & Juliette Ryan
sets of experiments: we first test the implementation of Discontinuous Galerkin intothe Schwarz waveform relaxation with Robin transmission condition on the heatequation. Then we present the coupling of Navier-Stokes with Euler via the classicalSchwarz waveform relaxation (i.e. exchange of Dirichlet data on the interfaces).
2 Discontinuous Galerkin applied to the Navier-Stokes equations
DG formulation: The 2D time-dependent dimensionless Navier-Stokes equations
∂tW + ∇ ⋅F(W )− ∇ ⋅D(W ,∇W ) = 0 (1)
where W = (ρ, ρ−→U , ρE) is the conservation variable vector with classical no-
tations, F and D are the convective and diffusive flux vectors. These are solvedin a domain Ω discretized by either a Cartesian or an unstructured triangular gridTh =
∪Ωi and the associated function space Vh,
Vh = φ ∈ L2(Ω) ∣ φ/Ωi ∈ Pk (2)
where Pk is the space of polynomials of degree k.The DG formulation based on a weak formulation after a first integration by parts
is of the form : find W h in (Vh)4 such that for all Ωi in Th ,
∀φ ∈Vh,∫
Ωi∂tWh φ dx =
∫Γi(Fh−Dh) φ dγ−
∫Ωi(Fh−Dh)∇φ dx. (3)
Here, the numerical fluxes Fh,Dh and Wh are approximations of F, D and W .The inviscid fluxes F are classically determined using the HLLC ([13]) or LLFtechniques; the viscous flux is computed with the EDG [3] method.
Time is discretized with Shu-Osher’s [12] explicit TVD time stepping RK3.
11
1p
kW P
eR
2p
kW Pp
r kW P
2
1
Fig. 1 Definition of the elastoplast ele-ment
Note on EDG: The simple idea of the EDGmethod is to regularize locally the discontinu-ous solution Wh over each edge using a recon-struction in a rectangular element RE overlap-ping this edge (see figure 1). Reconstruction isdone through an L2 projection on a DG basisin E of the same order k as the DG basis de-fined in the elements. For k=2, the order of themethod varies from 3.5 on a regular mesh to2.1 on quite heterogeneous meshes (see [4]).
Euler - Navier-Stokes coupling for Aeroacoustics problems 3
This idea of reconstruction in a rectangular element will be used for the Euler(structured) - Navier-Stokes (unstructured) coupling.
3 Schwarz waveform relaxation methods
These methods are based on Schwarz domain decomposition algorithms, inventedby H.A. Schwarz in 1870 [11]. In order to solve a Laplace equation in the domainΩ , it is split into two subdomains with overlap Ω1 and Ω2, in which the equationis solved alternatively. The exchange of informations is made on the boundariesby exchange of Dirichlet values. This algorithm has been extended by P.L. Lionsto nonoverlapping subdomains by using different transmission conditions, such asRobin conditions [9]. For an extension to evolution problem, we couple it to a wave-form relaxation algorithm, which is an extension both of the Picard’s ”approxima-tions successives” and relaxation methods for algebraic systems, due to Lelarasmee[8]. Versions with Dirichlet or optimized Robin transmission conditions have beendesigned in [2]. Consider for instance the unsteady heat equation with prescribedDirichlet boundary conditions and initial data:⎧⎨⎩
∂tu − ν u = f in Ω × [0,T ]u(x,0) = u0(x) in Ω
u = g on ∂Ω
(4)
where u is the temperature, ν is the constant diffusive coefficient and representsthe Laplace operator. A parallel version of the domain Schwarz waveform relaxationalgorithm can be written for Ω = Ω1∪Ω2 as
Ω1 Ω2Γ1Γ2
t
x
y
⎧⎨⎩∂t uk
1−νuk1 = f in Ω1× (0,T ),
uk1(⋅,0) = u0 in Ω1,
uk1 = uk−1
2 on Γ1× (0,T ),uk
1 = g on (∂Ω1−Γ1)× (0,T ),⎧⎨⎩∂t uk
2−νuk2 = f in Ω2× (0,T ),
uk2(⋅,0) = u0, in Ω2,
uk2 = uk−1
1 on Γ2× (0,T ),uk
2 = g on (∂Ω2−Γ2)× (0,T ).
In the Robin case, define ni the unit normal exterior to Ωi. The transmissionconditions become (ν∂ni + p)uk+1
i = (ν∂ni + p)ukj for (i, j) = (1,2) or (2,1). The
parameter p is determined asymptotically as a function of the physical parameters,the size of the space-time domains, and the mesh parameters (see [2]).
4 Michel Borrel, Laurence Halpern & Juliette Ryan
4 Numerical Coupling
Coupling discontinuous Galerkin andSchwarz Waveform Relaxation, explicit unsteady scheme
Dirichlet or Robin transmission condition
1u û1
( )Sl 2
1
2uû1
1( )j 2( )jS11
200
9 -L
ille
1 2
TSA
GI
41
1 2S
Fig. 2 Definition of the coupling
Coupling technique: The unstruc-tured solution in Ω1 is recon-structed on a rectangle ΩS adja-cent to the common edge witha rectangular element Ω2 in thesame way as the EDG technique.All the reconstructed DG coeffi-cients are sent to the other do-main so that either a Dirichlet or aRobin transmission condition canbe applied. On the structured side(Ω2), all coefficients are sent di-rectly without reconstruction.
Domains proceed in time independently, using at first predefined interface values.At the end of the time window, domains exchange their newly computed boundarycells values for all time steps (including sub times for the Runge Kutta scheme)and a new time march is carried out with updated interface values. This iterativeprocedure is repeated till solution ceases to vary. This method allows for differenttime steps and different space interface discretization as received values from otherdomains can be interpolated and projected on the local time-space grid.
5 Numerical Results
All computations are DG-P2 and no limiters were used. For both Cartesian and un-structured computations, we used the same functional space Vh. All triangular gridshave been obtained with the freeware mesh generator Gmsh [6],
Unsteady Heat Equation The first test case concerns the scalar heat equation(4) and is chosen to assess the performance in terms of precision and stability ofthe present coupling procedure of unstructured triangular and Cartesian grids in 2D.Robin type transmission conditions are imposed at each time step as presented in[4]. The unsteady Navier-Stokes solver is downgraded to solve an unsteady 2D heatequation.
The steady numerical solution is compared with the analytic solution for apure heat diffusion problem on the square unit [0,1] x [0,1]. The exact steadysolution chosen is the same as in [10] defining the Dirichlet boundary conditions
T exact(x,y) =1
sinhπ
[sinh(πx)sin(πy)+ sinh(πy)sin(πx)
]. The initial field is set to
zero T 0 = 0 and the numerical error between exact and approximated solutions at
Euler - Navier-Stokes coupling for Aeroacoustics problems 5
convergence is measured in the L∞ norm which is probably the worst norm for DGbut gives a good evaluation of diffusion for wave phenomena. The domain is splitinto 2 subdomains (one unstructured, the other structured), with an overlap of sizeh as in Fig. 2, with h the length of a boundary edge.
Fig. 3 Coupling accuracy
Three computations were made forvalues of h=1/10, 1/20, 1/40. Timewindows of 5 time steps have beenused with 3 Runge Kutta sub timesteps. Convergence of the Schwarz(Robin) scheme is obtained in 2 to4 iterations. On Fig. 3, convergenceslope (”Robin”) is shown togetherwith those of an ideal 3rd orderscheme (”order 3”) and a 4th orderone (”order 4”).
The TVD RK time scheme is not adapted to the computation of a steady prob-lem, so the steady solution is obtained with about 10 000 time steps or 2000 timewindows. An average order of 3.3 is obtained showing that the coupling procedurepreserves the order of the original DG schemes.
Low-Reynolds Number Flow around a Cylinder We have chosen this basicunsteady vortical flow as a first coupling illustration : reducing the Navier-Stokescomputation to a small computational domain containing the main noise sourcesand coupling it with an Euler computation to take into account sound propagation.This test case studies the Von Karman vortex shedding in the wake behind a cylinderin 2D. Within this configuration, supersonic pockets will grow alternatively up anddown, which result in generating vortices. To simplify, only two domains are con-sidered with, for the moment, no different grid size at the interface, but of course,our iterative method will be completely justified when the space-time multi-scaleaspect will be taken into account. The computation has been run with three Schwarzsub-iterations, which allow us to converge to sixth order for each time window prob-lem. First results on the mesh (Fig. 4(a) - 5000 triangles, 2400 rectangles) shown inFigs. 4(b),4(c),4(d) present the computed vortex shedding seen through entropy iso-values. The interface introduces no obvious spurious perturbations, which is verypromising for further computations using finer grids overlapping the far-field do-main.
References
1. D’Anfray P. , Halpern L., Ryan J.: New trends in coupled simulations featuring domain de-composition and metacomputing. M2AN 36 (5), 953–970 (2002)
2. Bennequin D., Gander M.J., Halpern L: A homographic best approximation problem withapplication to optimized Schwarz waveform relaxation. Math. Comp. 78, 185–223 (2009).
6 Michel Borrel, Laurence Halpern & Juliette Ryan
3. Borrel M, Ryan J.: A new discontinuous Galerkin method for the Navier-Stokes equations,Proceedings of the ICOSAHOM09, Trondheim (Norway),June 22-26 2009. To appear in Lec-ture Notes in Comp. Sci. and Eng. Springer Ed..
4. Borrel M, Ryan J.: The Elastoplast Discontinuous Galerkin (Edg) Method For The Navier-Stokes Equations. Submitted to J. Comp. Phys. 2010
5. Gander M.J., Halpern L., Japhet C., Martin V: Viscous Problems with Inviscid Approxima-tions in Subregions: a New Approach Based on Operator Factorization. ESAIM Proc 27,272–288 (2009).
6. Geuzaine C., Remacle J.F.: Gmsh: a three-dimensional finite element mesh generator withbuilt-in pre- and post-processing facilities, Int. J. for Num. Meth. in Eng. 79-11, 1309–1331(2009)
7. Halpern L., Japhet C., Szeftel J.: Discontinuous Galerkin and nonconforming in time opti-mized Schwarz waveform relaxation. In: Proceedings of the Eighteenth International Con-ference on Domain Decomposition Methods, http://numerik.mi.fu-berlin.de/DDM/DD18/(2009)
8. Lelarasmee E., Sangiovanni-Vincentelli A.L., Ruehli A.E.: The waveform relaxation methodfor time-domain analysis of large scale integrated circuits. IEEE Trans. on Computer-AidedDesign of Integrated Circuits and Systems, vol. CAD-1, 131–145 (1982)
9. Lions P.-L.. On the Schwarz alternating method. I. In Roland Glowinski, Gene H. Golub,Gerard A. Meurant, and Jacques Periaux, editors, First International Symposium on DomainDecomposition Methods for Partial Differential Equations, SIAM, 1–42 (1988)
10. Puigt G., Auffray V., J.D. Mueller: Discretisation of diffusive fluxes on hybrid grid. J. Com-put. Phys. 229, 1425–1447 (2010)
11. Schwarz H. A. Uber einen Grenzubergang durch alternierendes Verfahren. Vierteljahrsschriftder Naturforschenden Gesellschaft in Zurich, 15, 272–286 (1870)
12. Shu C.-W. ,Osher S.: Efficient implementation of essentially non-oscillatory shock-capturingschemes. J. Comput. Phys. 77 , 439-471 (1988)
13. Toro E. F. , Spruce M., Speares W.: Restoration of the contact surface in the HLL-Riemannsolver. Shock Waves 4, , 2534 (1994)
(a) Mesh (b) Entropy
(c) Entropy (d) Entropy
Fig. 4 Vortex shedding behind a cylinder: Mesh and Entropy time evolution
Computational Aeroacoustics on aSmall Flute Using a Direct Simulation
Yasuo Obikane1
Abstract This research pursues a numerical simulation of sounds generatedby musical instruments. We selected a flute blown from the side of the cylin-der with the preliminary tests of a recorder, a kind of flute that is blownfrom the front of the cylinder. Both two and three-dimensional aeroacousticsimulations were direct in the near fields by using the compressible Navier-Stokes equations, with the exception of the boundary conditions. We aimedto see the flow details where the theoretical approach was omitted [Ref.2]. Inthe last work presented in ICCFD5 [Ref.1], we computed a two-dimensionalrecorder, but were unable to confirm the theoretical results, since the direc-tion the recorder’s blow was orthogonal to the direction of the real flute.In the present work, 1) We used the same flute shape that was used in thetheoretical analysis [Ref.2]. 2) In 2007, we utilized the weighted directionalscheme, which was calibrated in the high Reynolds number. However, in thepresent case the flow speed was low, so we dropped the weighted scheme inthe inner domain. Then, we used the non-weighted scheme in the inner do-main, and a new isotropic weighted scheme at the outer boundaries. 3) Inaddition, we made a theoretical interpretation of the T.Poinsot and S.K.Lele’souter boundary condition [PL condition] [Ref.3], and proposed a new formfor damping terms in their equation with a turbulence modeling equation. Asthe preliminary computational results in the two-dimensional recorder indi-cate, we obtained a clear wave pattern in a larger domain than that obtainedin [Ref.1]. As the flute’s results demonstrate, the streamlines showed verticaloscillations near the mouth, i.e. the result implies that a vertical velocity oscil-lation exists. This is consistent with the main assumption of the flute theoryby M.S.Howe. For three-dimensional flute computations, we confirmed theaeroacoustics coupling frequency with a guided jet’s frequency and the basic
Institute of Computational Fluid Dynamics, 1-16-5 Haramachi, Meguroku-ku, Tokyo
152-0011/ Sophia University , Mechanical Engineering, 7-1 Kioichou,Chiyouda-ku,Tokyo,Japan 102-8554, [email protected]
1
2 Yasuo Obikane
frequency of the cylinder. The realizations of the main geometrical featurewere successful, since small, spherical sound patterns were clearly observed.
1 Introduction
It is widely known that a wind brow makes sounds, but few understand howto create mild sounds with the instruments aeroacoustically. Aeroacousticalstudies were conducted for several decades, mainly for high speed flows, butfew have dealt with the low speed that is known as a resonance oscillationby the global instability and local instability of vorticities. In the presentanalysis, we take steps toward the realization of the three-dimensional soundof a small size flute at a low speed. In addition, we interpret the role of eddies,which control the mode of sound and the instability to understand the detailmechanism.
2 Scheme for the low speed
There is no finite difference scheme that can cover every speed of flows.The scheme in the 2007 project used for the prediction of a recorder wasweighted to adjust for the inertial subrange’s cutting frequency. The mainflow direction took the derivatives as 2/3, and the derivatives taken along theoblique direction were weighted as 1/3. The new representation for the presentanalysis is as follows: if we treat the wave propagating at outer boundarypoints, the waves will propagate spherically, and all directional derivativesshould be equally weighted. Thus, the representation becomes as follows:
Fig.1. Oblique coordinate system
u∂
∂xϕ = c1Lx(u, ϕ) + c2Lζ(u, ϕ) + c3Lη(u, ϕ) (1)
where Lη(u, ϕ) is defined as
Lη(u, ϕ) = u(−ϕ(i+ 2, j − 2) + 8(ϕ(i+ 1, j − 1)− ϕ(i− 1, j + 1))
Acs Near Fields Using the Comp.NS.eq. 3
+ ϕ(i− 2, j)) + 3abs(u)(ϕ(i− 2, j + 2)− 4ϕ(i− 1, j + 1)
+ 6ϕ(i, j)− 4ϕ(i+ 1, j − 1) + ϕ(i+ 2, j − 2))/(12h) (2)
where the coordinates ζ and η are sketched in Fig.1. The test computationwave patterns in 20x20 meshes and 50x50 meshes are shown in Fig.2. Themodified isotropic scheme proved to be superior to the inertial calibratedscheme in the pattern in Fig.3.
Fig.2. Test wave patterns
Fig.3. Computational results (Rectangular mark:new scheme)
3 An interpretation of the Poinsot and Lele’s condition(PL condition)
The Poinsot and Lele’s damping condition can be modified to satisfy thedynamic balance, as in Fig.4. Since the force acting on any closed surfaceis given by the surface integration with the momentum equation, we have adrag force if the integral is taken on the body where the velocity becomeszero.
Drag(t) =
∮∂
∂n(−p+ τi, j +Πi,j)dn (3)
If we choose the surface at infinity, the meaning of the integral will be thesame.
4 Yasuo Obikane
Fig.4 Acting and reacting force diagram
If we treat the turbulent flows, the turbulence modeling equation must beadjusted to satisfy to balance of the dynamics (i.e. the Reynolds stress equa-tion must satisfy the same equilibrium condition at a far field). Thus, themomentum equation can be evaluated as
∂mi
∂t+
∂(mimj/ρ)
∂xj= −mimj/ρ
∂xj(4)
The symmetry form of the model is written as
∂mpmi
∂t≃ µ < k > uiΨ1p(far) + λ < k > uiΨ2p(far)
+µ < k > upΨ1i(far) + λ < k > upΨ2i(far) (5)
By taking the integration over the outer boundaries, we have
∆Ms(t) = − 1
Uinf< k >
∮(uj(µα1 + λα2) + µα3 + λα3)dσ (6)
where alphas are some invariant functions of the isotropy tensor of mpmi.Since the amplitude L1 of the wave given by the Poinsot and Lele’ s noreflective condition is given as
L1 = σ(p− pinf )c/XL, (7)
and if the accuracy is sufficiently high, both the drag force and the momentumdeficit at the outer boundary are balanced. The sum of extremely low speedmomentum at the outer boundary must yield the equal force. If the Reynoldsstress equation is adjusted at the outer boundary, the PL condition should beidentical in form to the turbulence model equation in the asymptotic form.Thus, the sigma in the outer condition of the PL condition is proportional tothe wave number that is the form of the modeling equation.
Acs Near Fields Using the Comp.NS.eq. 5
4 Generation of Musical Sound
4.1 The boundary conditions
The blowing velocity at the outlet of the guide to the opening of the recorderwas 5 to 10m/sec. The boundary conditions were as follows: The velocityon the body was zero, and the pressure and temperature had symmetricalconditions on the body. The outlet condition was defined as the Poinsotand Lele’s condition (i.e. the damping term was adjusted for the referencepressure). If we set the no-reflective condition, the acoustic energy producedby the jet was not accumulated, and was released at the outlet.
4.2 Two-dimensional cases
For the recorder’s preliminary computation, we took 2000x1000 meshes,which had sufficient points to produce the spherical waves shown in Fig.6.They also depicted the waves more clearly than that of the 2007’s result il-lustrated in Fig.5. In the 2007 results for the two-dimensional recorder, theamplitude of the vortices center was 1mm in a small domain, as indicated inFig.4. In the present case, two vorticities were also observed for the largerdomain. However, the resolution became poor, so we were unable to show the1mm vertical oscillation visually. For the two-dimensional flute, we observedthat vertical oscillations initiated at an extremely early stage of the compu-tation, as shown in Fig.7. Thus, we can infer that the flute can more easilyproduce a sound than the recorder.
Fig.5. Result in 2007 recorder coarse Fig.6. Present fine mesh case
6 Yasuo Obikane
4.3 Three-dimensional cases
For the three-dimensional preliminary computation results of the recorder,the volume of the air in the interface chamber was so small that the sounddid not project clearly. However, for the three-dimensional flute, the interfacevolume of the air chamber was equal to the volume of the cylinder, whichhds a mass sufficiently large enough to produce a stable vortex rotation. Thepressure pattern is very clear in Fig.8. Because of the three-dimensionality,we can observe the spherical pattern even though the computational domainwas small.
5 Conclusion
1)The two-dimensional recorder was actually a projection model of a three-dimensional flute. Thus, the sound eventually projected. 2)For the three-dimensional recorder, the interface volume was so small that it was unableto produce a strong resonance among the components (a guided jet, a pipebody, and an opening). 3) The small, dimensional flute may be easier to beplayed as the result of this computation. Thus, if the shape of the recorderis modified to create an increased interface volume, it will generate a soundmore easily.
Fig.7. Two dimensional flute Fig.8.Three dim flute (Pressure pattern)
References
[Ref1] Y.Obikane, K.Kuwahara, Direct Simulation for Acoustic near Fields Using the Com-pressible Navier-Stokes Equation”, ICCDF5 (2008) pp. 85-91.
[Ref2] M.S.Howe, ”Contributions to the theory of aerodynamics sound, with application
to excess jet noise and the theory of the flute”, J.Fluid Mech,(1975),vol 71,part 4, pp.625-673.
[Ref3] T.Poinsot and S.K.Lele: Boundary conditions for direct simulation of compressibleviscous flows. J.Compu.Phys.101(1992), pp. 104-129.
An iterative procedure for the computation ofacoustic fields given by retarded-potentialintegrals
Florent Margnat
Abstract An optimized procedure for the computation of acoustic fields given byretarded-time intgrals is provided. It is written in the time-domain and for fixedsources. It is devoted to applications in which there is a large amount of source data.Thus, as many observer points are required to build the acoustic image, the result-ing number of source-observer pair may cause an issue in accessing to the source-observer distance, if this quantity can not be stored in a global variable. The algo-rithm is validated through comparisons with reference data in the case of a simpleharmonic source and in the case of the aerodynamic noise generated by the cylinderflow. The implementation of the convected Green function is also presented.
1 Introduction
Flow-generated acoustic fields are often predicted by computing retarded-potentialintegrals which appear in the formalism of aeroacoustic analogies or wave extrapo-lation methods. Such procedures return the acoustic emission of unsteady flows bytwo steps: firstly, the flow is simulated, giving access to source quantities; secondly,those quantities are propagated until observer/listener locations.
In the present contribution, the numerical implementation of a retarded-potentialintegral is adressed in the time-domain. An optimised method is provided for thecomputation of the acoustic quantity on an observer grid in order to build an acousticpicture - field. It is very useful when large size source data are considered, thusinvolving a large amount of source-observer distances which cannot be stored in aglobal variable.
Florent MargnatArts et Metiers Paristech, DynFluid, 151 bd de l’Hopital, 75013 PARIS - FRANCE, e-mail: [email protected]
1
2 Florent Margnat
2 Principle of the procedure
2.1 Problem formulation
A general form of aeroacoustic integrals can be written, considering a source quan-tity S to be integrated over a source domain D in order to compute the acousticquantity, pressure, at an observer position x and time ta, as:
pa(x, ta) =1
4π
∫D
S(
y, ta −|x−y|
c0
)dy
|x−y|(1)
where y denotes the position over the source domain and c0 is the ambient soundspeed. Such expression is usually called a retarded potential integral, since, unlessthe source is compact, the propagation distance deviation between two source pointsimplies their respective contributions must be collected at different emission timesin order to reach the observer point at the same time.
The following discrete expression is considered:
4π pa(xi, t la) =
Nys
∑j
S(
y j, t la −
|xi −yj|c0
) ∆Vy j
|xi −yj|(2)
where ∆Vy j is the elementary volume attached to the jth source element located atyj, and Nys is the number of source elements. If it is assumed that Nts datafiles ofthe source term S can be computed and sampled at ∆ ts, one have to interpolate thesource quantity at the time ta −
|xi−yj|c0
in order to compute the acoustic quantity ongiven x and ta grids.
One key point is how to organise the loops on the source elements, source fields,and in the present case, observer points. The advanced time principle (Casalino [1],Kessler & Wagner [2]) consists in fixing the source time first, and then radiating thecontribution at the observer points at a reception time which is determined by thesource-observer distance. The present algorithm is dedicated to configurations forwhich the source-observer distances cannot be stored in a local variable. This hap-pens when a 2D acoustic field is computed (not only a signal at a couple of observerpoints) using 3D or large 2D source grids (e. g. volumic source distributions).
2.2 An optimized algorithm
Let l be fixed such as t la = l∆ ts, thus defining the reception time when the acoustic
picture is to be computed. Following the advanced time principle, let a source timeinterval be fixed, which is bounded by k∆ ts and (k+1)∆ ts, where the source quantityis available. Consequently, the [source-observer] pairs involved at this step of theaccumulation process are such that:
An iterative procedure for retarded-potential integrals 3
(l − (k+1))c0∆ ts < ri j ≤ (l − k)c0∆ ts (3)
where ri j = |xi −yj|. If the source location is fixed as well, (3) defines two spherescentred on it of radii r1 = (l − (k+1))c0∆ ts and r2 = (l − k)c0∆ ts. Thus, in the ob-server plane, defined by y3 = 0, observer points verifying (3) are within two circles
centred on (x1 = y1, x2 = y2) of radii r,1 =√
r21 − y2
3 and r,2 =√
r22 − y2
3These geometric relations are sketched in figure 1-left. If the grid in the spanwise
Fig. 1 Left: sketch of the configuration for the implementation in the spanwise direction. M is thesource point, located in (y1,y2,y3) ; (x1,x2) are the coordinates in the observer plane ; r1 and r2are the propagation distances defined by (3) ; r,1 and r,2 are the corresponding radii of the circlesobtained where the spheres centred on the source point cut the observer plane. Right: Principleof the extraction of the observer points located on the ring. ∆x1 and ∆x2 are the grid steps in theobserver plane.
direction is basically an extrusion of the grid in the (y1,y2) plane, the procedure isthe following:
1. fix (loop on) the source field(s)2. fix the reception time(s); the propagation distances r1 and r2 are computed.3. fix (loop on) the source location(s) in the spanwise direction; the radii r,1 and r,2
in the observer plane are computed.4. fix (loop on) the source coordinates (y1, y2), search the observer points located
within the two circles, and add the interpolated source contribution to the acousticpressure at them.
The search procedure at the fourth step is performed as follows: splitting the ringin 4 arcs, and for each grid step in the x1 direction, finding the grid steps in the x2direction satisfying (3). As sketched in figure 1, for a given step i1 such as x1 =(i1 − 1)∆x1, a loop on the second observer coordinate can be introduced, bounded
by i(1)2 = int
[√(r,1)
2 +(x1 − y1)2
∆x2
]+2 and i(2)2 = int
[√(r,2)
2 +(x1 − y1)2
∆x2
]+1
4 Florent Margnat
3 Test-case: monopole source
The implementation of the procedure is validated through the case of a harmonicsource located at the origin in a two-dimensional configuration. In the time domain,the general solution of the 2D inhomogeneous wave equation is:
p(x1,x2, t) =− 12π
∫ +∞
−∞
∫ +∞
−∞
∫ +∞
−∞
H (t − τ − r/c0)√(t − τ)2 − r2/c2
0
S (y1,y2,τ)dτ
dy1dy2
(4)where H is the Heavyside function and r = |x−y|=
√(x1 − y1)2 +(x2 − y2)2.
It is relatively straightforward to convert the time integral into an integration overa third spatial direction, noted y3. One obtains:
p(x1,x2,0, t) =1
2π
∫ +∞
−∞
∫ +∞
−∞
∫ 0
−∞
S(
y1,y2,0, t −√
r2+y23
c0
)√
r2 + y23
dy3
dy1dy2 (5)
This formulation shows how the procedure developed here can be validated orapplied in a 2D configuration, provided that a 3D Green function is combined witha space integration over the spanwise direction and a replication of the source dataknown in the (y1,y2) plane.
The acoustic pressure computed with the algorithm presented above is comparedto the reference solution obtained by computing directly (5). This allows to validatethe implementation of the various loops, of the interpolation and of the integrationoperation. As plotted in figure 2-left an excellent agreement is found. 30 points byperiod were used to discretize the source signal, and the interpolation is linear.
0 1 2 3 4 5
−0.1
−0.05
0
0.05
0.1
0.15
x1/λ
p / (
Adσ
)
−4 −3 −2 −1 0 1 2 3 4
−0.1
−0.05
0
0.05
0.1
0.15
x1/λ
p / (
Adσ
)
Fig. 2 Acoustic pressure field for a monopole source in 2D. Solid line: reference solution; Sym-bols: algorithm solution; dashed line: r−0.5 laws. Left: quiescent medium Green’s function; Right:convected case.
An iterative procedure for retarded-potential integrals 5
For the propagation in a uniformly moving flow at the subsonic Mach numberM0 in the direction y1, the 3D integral solution for the acoustic pressure is:
pa(x, ta) =1
4π
∫D
S(
y, ta −rβ −M0(x1 − y1)
c0β 2
)dyrβ
(6)
where β 2 = 1−M20 , rβ =
√(x1 − y1)2 +β 2 [(x2 − y2)2 +(x3 − y3)2]
and τ∗ =rβ −M0(x1 − y1)
c0β 2
According to the present methodology, one have to fix the reception time t and apair of emission times τ and τ +dτ . Then one have to find the source-observer pairsfor which the propagation time τ∗ satisfies:
(l − (k+1))c0∆ ts <rβ −M0(x1 − y1)
β 2 ≤ (l − k)c0∆ ts (7)
It is relatively straigthforward to show that for a fixed y, the following equation:√(x1 − y1)2 +β 2 [(x2 − y2)2 +(x3 − y3)2]−M0(x1 − y1) = Rβ 2 (8)
where R is a constant, is the equation of a sphere centred on (y1 +RM0,y2,y3) ofradius R. Consequently, the algorithm presented in section 2 can be applied usingthe convected Green function, provided that the centre of the circles are moveddownstream the source point by the distance RM0, and that |x−y| is replaced by rβat the denominator in the integration.
The acoustic pressure computed with the algorithm adapted to the convectedpropagation case is compared to the reference solution obtained by computing di-rectly (6). As plotted in figure 2-right for M0 = 0.5, an excellent agreement is found.The propagation in a moving flow leads to a different wavelength between observerslocated upstream and observers located downstream.
4 Application to the aeolian tone prediction
The aeolian tone prediction illustrates well the interest of the present algorithm de-velopment. The source data, as written in Curle’s analogy [3], are provided by a2D simulation using a tool designed by Gloerfelt et al. [4]. The flow is at the Machnumber M0 = 0.3 and the Reynolds number Re = 136. The reference computationis performed in the Fourier domain, using the 2D spectral convected Green func-tion. The acoustic pressure field generated by lift fluctuations is plotted in figure3. The directivity is modified by convection effects, and this is well tracked by thealgorithm.
6 Florent Margnat
Fig. 3 Acoustic pressure field generated by lift fluctuation on a cylinder in a 2D uniform flow atMach number = 0.3. Left: computation with the spectral convected Green function; right: compu-tation with the temporal convected Green function using the present algorithm.
5 Concluding remarks
An iterative ring-guided procedure is proposed which optimises the search of ob-server points where the contribution of a given source point must be added onceboth the emission and reception time are fixed. Such a procedure appeared veryfaster than an intuitive search method in that context. It can be used for all kindsof retarded-time integrals, such as volume sources in Lighthill’s formalisms as wellas surface sources in Kirchhoff’s methods. It keeps the advanced time principle al-lowing an acoustic prediction parallel to the flow simulation and an easy connectionwith usual CFD tools.
Acknowledgements The author gratefully acknowledges X. Gloerfelt for sharing source data andreference solution for the cylinder flow.
References
1. D. Casalino, “An advanced time approach for acoustic analogy predictions”, J. Sound Vib.,261(4),583-612 (2003).
2. M. Kessler, and S. Wagner, ”Source-time dominant aeroacoustics”,3. N. Curle, ”The influence of solid boundaries upon aerodynamic sound”, Proc. Royal Soc. A,
231,505-514, (1955).4. Gloerfelt X., Bailly C. and D. Juve, ”Direct computation of the noise radiated bya sub-
sonic cavity flow and application of integral methods”, Journal of Sound and Vibration, 266,119146, (2003).
Innovative tool for realistic cavity flow analysis :global stability
Fabien Mery1 & Gregoire Casalis1
1 Introduction
Cavity noise has several issues for the aeronautical industry. While for takeoff theengine jet is the main source of the airframe noise, for a landing, the airframe noisemainly arises from the landing gear, the slat and flap systems. Flow past an open cav-ity has been studied for decades. Cavity oscillations in compressible flows are nowdescribed as a flow acoustic resonance phenomenon. The first detailed descriptionis credited to Rossiter [1]. This paper presents results of compressible and turbulentflows Large Eddy Simulations (LES) over a rectangular cavity and uses global linearstability approach as an analysis tool in order to reveal the mode selection mecha-nism for different inflow velocities. The basic flow for the stability analysis corre-sponds to the time averaged part of the LES. As the flow is fully two-dimensionalin the median plane, the classical local stability theory is not appropriate. We mustuse the method first developed by Tatsumi et al. [2], which can be called a globalstability theory. Recent studies using global stability theory on a compressible highReynolds number cylindrical cavity flow have given good results, see Mery et al. [3]In the following part of the paper, we provide first a description of the chosen config-urations. The global stability approach is then presented. Finally, a section dedicatedto a LES computation with a variable inflow velocity is presented in order to put thestrike on the mode selection phenomenon.
Fabien MryONERA, Avenue E.Belin, BP 4025-31055 Toulouse Cedex 4, France, e-mail: [email protected]
Grgoire CasalisONERA ,Avenue E.Belin, BP 4025-31055 Toulouse Cedex 4, France, e-mail: [email protected]
1
2 Fabien Mery1 & Gregoire Casalis1
2 Chosen configurations
The configuration retained for this study is a rectangular cavity, with a length L =0.1 m and a depth D = 0.1 m, in the spanwise direction the length W = 0.25 m.The flow is computed by LES, using the ONERA in-house code called Flu3M. Theinflow is an ideal gas with a constant specific heat. The upstream boundary layeris a mean turbulent profile and its characteristics are based on experimental results.Table 1 presents the numerical characteristics and the momentum thickness δ 0
θ atthe upstream edge of the cavity. The wall boundary layer is discretized with about
U∞ (m/s) Mach Number ReD50 0.14 330 00070 0.2 460 00090 0.257 591 000110 0.314 723 000120 0.34 789 000
Table 1 Numerical characteristics
40 cells in δ99. The near wall mesh size is about 10−4m and the growth ratio 1.05.Cavity vertical walls are meshed with the same two parameters. The cell size at thecavity bottom wall is 10−4 m. Concerning the shear layer, in order to have a goodresolution, there are more than 10 cells in δ 0
ω . This minimum is recommended byseveral former simulations [4, 5, 6].
In the spanwise direction, periodic boundary conditions are imposed in order toensure the computation to be a 3D simulation for the turbulent variables whereas themean flow in the spanwise direction is expected to be roughly null. For the flat plateand cavity inner walls, standard no slip boundary conditions are imposed. The usedprocedure has already been validated on a cylindrical cavity with the same aspectratio by Mincu et al. [7].
3 Global stability approach
The linear stability theory (LST) is used considering the linearized inviscid equa-tions in the compressible form for the flow in the streamwise plane z = 0. In thisplane, the mean flow is assumed to be two-dimensionnal in the sense that thereare two non zero velocity components: the streamwise and the normal velocity(u(x,y), v(x,y)) which depend on x and y only. According to the small perturbationmethod, a small fluctuation is superimposed on the mean flow, i.e each variable isdecomposed as : q(x,y, t) = q(x,y)+ q(x,y, t) with q the mean part which is actuallythe mean part of the LES and q the perturbation to be determined. The perturbationis written in the form of normal modes : q(x,y, t) = q(x,y)e−iωt with ω the complexeigenvalue and q(x,y) the corresponding complex eigenfunction.
Innovative tool for realistic cavity flow analysis : global stability 3
Introducing the previous decomposition into the linearized inviscid equationsleads to an eigenvalue problem via a partial derivative equation system. The lat-ter is discretized using a finite element method. Finally, the problem is expressed asa generalized eigenvalue problem in the form of
Aq = ωBq (1)
where A and B are matrices and q = p, u, v the eigenfunction vector. Each modeis characterized by the complex number ω = (ωr,ωi) where ωr gives the circularfrequency of the perturbation and ωi the temporal growth rate. We define a Strouhalnumber as follows : St = ωrD
2πU∞.
The results are reported in Table 2. The classical Rossiter modes are predicted byBlock’s relationship [8].
St =fnLU∞
=n
M(1+ 0.514L/D )+ 1
κ(2)
There is a very good agreement between the LES Strouhal number and the globalstability Strouhal number. The perturbated flow field is actually comparable to theinstantaneous LES flow field as can be seen in figure 1.
I II III IV50 m/s St of Rossiter mode 0.54 1.08 1.62 2.16
St of KH mode (LST) 0.51 1.02 1.31 2.08St of the LES 0.97
70 m/s St of Rossiter mode 0.51 1.02 1.53 2.04St of KH mode (LST) 0.51 0.94 1.32 1.97St of the LES 0.90 1.34 2.11
90 m/s St of Rossiter mode 0.49 0.98 1.47 1.96St of KH mode (LST) 0.50 0.93 1.31 1.83St of the LES 0.52
110 m/sSt of Rossiter mode 0.47 0.94 1.41 1.88St of KH mode (LST) 0.49 0.93 1.30 1.67St of the LES 0.45 1.71
120 m/sSt of Rossiter mode 0.46 0.92 1.38 1.84St of KH mode (LST) 0.46 0.92 1.29 1.63St of the LES 0.46 1.59
Table 2 Comparison between the Strouhal numbers of the Kelvin-Helmholtz (KH) modes calcu-lated by the present LST, Strouhal numbers predicted by Block’s relationship and the resonatingfrequencies from the LES for different inflow velocities, I,II,III and IV refer to the mode number
4 Fabien Mery1 & Gregoire Casalis1
(a) stability, Mode KH II (b) LES, DFT at the resonating fre-quency
Fig. 1 Real part of the vertical velocity fluctuation
4 LES with time variable inflow velocity
This section emphasizes the mode selection. A LES is driven on the same cavitywith an inflow velocity which varies with time. The simulation starts at 50 m/s andfor each time step, the inflow velocity is increased until 122 m/s. The computingtime is 2.9s. Sensors have been placed above the shear layer, near the downstreamedge of the cavity and at the bottom of the cavity. A sequential least square algorithmis applied for the decomposition and the analysis of the sensor pressure signals. Thisalgorithm has been proposed by David et al [9]:
[h]p(t) =r−1
∑k=0
bke−δk+i2πνkt +w(t) (3)
with bk complex amplification, δk damping coefficient for mode k, νk frequency formode k. A sufficient window and a sufficient number of mode to describe the sig-nal are chosen. The signal is decomposed on 11 modes and it leads to HR-ogrampresented in fig 2 : for low velocities, the dominant mode is KH II, for higherones, the dominant mode is KH I and KH IV appear. These modes correspond tothe static results coming from the global stability approach. Thanks to this analy-sis, the behaviour of the cavity is underlined. Originally, some experiments drivenby East [10] on rectangular cavity put the stress on a mechanism which explainsthe mode selection in rectangular caivites. Actually, East observed a special tun-ing between the cavity box mode proposed in equation 4 by East and the classi-cal Rossiter mode. When the corrected quarter wavalength is closed to a Rossitermode, the Rossiter mode is excited. This mechanism has been numerically provedby our slowly varying inflow velocity simulation on the rectangular cavity. AroundM = 0.25 to M = 0.31, as mode I seems to be particularly excited, around a Strouhalnumder St = 1.5, non-linear modes seem to be relevant for this configuration. Thequestion of the presence of non-linear modes is open...
Innovative tool for realistic cavity flow analysis : global stability 5
f1/4 =c0
4D(1+0.65( LD )
0.75)(4)
Fig. 2 HR-ogram of the sensor signal over the shear layer function of the inflow velocity from50 m/s to 122 m/s
5 Conclusion
This study is an analysis of global instabilities, at a high Reynolds number, of com-pressible flow over a cylindrical cavity using linearized inviscid equations. This pa-per shows the efficiency of a simple method (global instability theory) for complexflow configurations (high Reynolds number, complex geometry...). The results en-able a new interpretation of the Rossiter formula, particularly, for modes I and II.Moreover, the slowly varying inflow velocity simulation enables to give a completemap of the behaviour of a rectangular cavity: East mechanism is validated.
6 Fabien Mery1 & Gregoire Casalis1
Acknowledgements This work is a part of AEROCAV project which is supported by FRAE (Fon-dation de la Recherche pour lAeronautique et l’Espace). Fabien Mery’s Phd is supported by DGA,the french armament agency.
References
1. J.E. Rossiter. Wind-tunnel experiments on the flow over rectangular cavities at subsonic andtransonic speeds. Reports and memoranda n 3438, Aeronautical research council, 1964.
2. T. Tatsumi and T. Yoshimura. Stability of the laminar flow in a rectangular duct. Journal ofFluid Mechanics, 212:437–449, 1990.
3. F.Mery, G. Casalis, D. Mincu, and A . Sengissen. Noise generation analysis of a cylindricalcavity by les and global instability. AIAA Paper, 2009.
4. X. Gloerfelt, C. Bailly, and D. Juv. Direct computation of the noise radiated by a subsoniccavity flow and application of integral methods. Journal of Sound and Vibration, 266:119–146, 2003.
5. L. Larcheveque. Simulation des grandes chelles de l’coulement au-dessus d’une cavit. PhDthesis, Paris VI, 2003.
6. L. Larcheveque, P. Sagaut, I. Mary, and O. Labbe. Large-eddy simulation of a compressibleflow past a deep cavity. Phys. Fluids, 15, 2003.
7. D.C. Mincu, I. Mary, S. Redonnet, L. Larcheveque, and J-P. Dussauge. Numerical simulationsof the unsteady flow and radiated noise over a cylindrical cavity. Vancouver, May 2008. 14thAIAA/CEAS Aeroacoustics Conference.
8. P. Block. Noise response of cavities of varying dimensions at subsonic speeds. Nasa TechnicalNote, NASA TN D-8351, 1976.
9. R. Badeau. Mathodes haute rsolution pour l’estimation et le suivi de sinusodes modules.Application aux signaux de musique. PhD thesis, ENST, 2005.
10. L.F. East. Aerodynamically induced resonance in rectangular cavities. Journal of Sound Vib.,3:277–287, 1966.
