In this paper we are interested in the mathematical and numerical analysis of the time-dependent Galbrun equation in a rigid duct. This equation models the acoustic propagation in the presence of a flow [1]. We propose a regularized variational formulation of the problem, in the subsonic case, suitable for an approximation by Lagrange finite elements, and corresponding absorbing boundary conditions. 

A. -s. Bonnet-ben Dhia

K. Berriri

P. Joly

arXiv

International audienceIn this paper we are interested in the mathematical and numerical analysis of the time-dependent Galbrun equa- tion in a rigid duct. This equation models the acoustic propagation in presence of flow [1]. We propose a regu- larized variational formulation of the problem, in the sub- sonic case, suitable for an approximation by Lagrange finite elements, and corresponding absorbing boundary conditions

Berriri, Kamel

Bonnet-Ben Dhia, Anne-Sophie

Joly, Patrick

INRIA a CCSD electronic archive server

HAL Id: hal-00021587https://hal.archives-ouvertes.fr/hal-00021587Submitted on 22 Mar 2006HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.NUMERICAL ANALYSIS OF TIME-DEPENDENTGALBRUN EQUATION IN AN INFINITE DUCTKamel Berriri, Anne-Sophie Bonnet-Ben Dhia, Patrick JolyTo cite this version:Kamel Berriri, Anne-Sophie Bonnet-Ben Dhia, Patrick Joly. NUMERICAL ANALYSIS OF TIME-DEPENDENT GALBRUN EQUATION IN AN INFINITE DUCT. WAVES 2005, 2005, France. pp.6.￿hal-00021587￿ccsd-00021587, version 1 - 22 Mar 2006NUMERICAL ANALYSIS OF TIME-DEPENDENT GALBRUN EQUATION IN AN INFINITE DUCTK. Berriri†,∗, A.-S. Bonnet-Ben Dhia†, P. Joly††Laboratory POems, UMR 2706 CNRS-INRIA-ENSTA, Paris, FRANCE∗Email: Kamel.Berriri@inria.frAbstractIn this paper we are interested in the mathematical andnumerical analysis of the time-dependent Galbrun equa-tion in a rigid duct. This equation models the acousticpropagation in the presence of a flow [1]. We propose aregularized variational formulation of the problem, in thesubsonic case, suitable for an approximation by Lagrangefinite elements, and corresponding absorbing boundaryconditions.1 IntroductionSound propagation in a flow is a subject of great inter-est for numerical analysis. The main application concernsnoise reduction in aeronautics. The understanding of phe-nomena of interaction between acoustic waves and flowsis a crucial feature to find the components that efficientlyreduce the sound.Most of the mathematical and numerical studies fortransient linearized aeroacoustics are based on linearizedEuler equations, whose unknowns are the perturbationsof velocity and pressure. Nevertheless, Galbrun equa-tions, whose unknown is the lagrangian perturbation ofdisplacement, are an attractive alternative to model thephenomenon of acoustic propagation in a flow. Indeed,these equations seem to have a structure similar to equa-tions in electromagnetism and elastodynamics and theyallow a simple treatment of boundary conditions.Recent studies on Galbrun equations have focused ontime harmonic dependence. We are interested here instudying the transient case.This problem raises many theoretical and numericaldifficulties. The major one is the lack of a natural func-tional frame for a variational setting of the problem.Moreover, a naive numerical resolution of the problem us-ing standard nodal finite elements for the space discretiza-tion is unstable (see Figure 1). In order to overcome thesedifficulties , we propose a regularization method analo-gous to the one developed for Maxwell’s equations [3].This allows to derive a variational formulation suitablefor an approximation by Lagrange finite elements.2 Statement of the problemLetΩ = R×]−h, h[ be a two-dimensional rigid duct con-taining a compressible fluid. We suppose that this fluid isin motion and that the flow can be described by a profileof Mach’s numberM(y) (M ∈] − 1, 1[). If f denotesthe source of acoustic waves, we have to solve the tran-sient Galbrun equation in dimensionless form (ξ is theperturbation of lagrangian displacement):D2ξDt2− ∇(div ξ) = f in Ω × R∗+ (1)ξ · n = 0 on ∂Ω × R∗+ (2)whereDDt= ∂∂t+M(y) ∂∂xis the material derivative andn the unit outward normal vector to∂Ω. We complete theequations (1)-(2) by adding initial conditions.3 The Lagrangian vorticity equationWhenf ∈ H(curl, Ω), applying the curl operator to (1),we prove that thecurl ξ = ψ satisfies:D2ψDt2= 2M ′(y)DDt(∂ξx∂x) + curlf in Ω × R∗+ (3)Note that when the flow is uniform (M(y) is constant),the vorticityψ can be computed independently ofξ andthe solution is given by :ψ(x, y, t) = α(x−Mt, y) + xβ(x−Mt, y) +1M2∫ x0(x− a)(curlf)(a, y, t− x−aM)dawhereα etβ are two functions that depend only of initialconditions of the problem.4 Regularized Formulation Galbrun equationThe idea of the regularization, initially introduced forMaxwell’s equations, was extended to the time har-monic Galbrun equation by A. S Bonnet-Bendhia and al(2001). The idea consists in adding the artificial terms curl (curl−ψ) to Galbrun equation.We replace the initial value problem (1)-(2) by the equiv-alentregularized problem :D2ξDt2− ∇(div ξ) + s curl (curl ξ − ψ) = f in Ω × R+DDt(DψDt− 2M ′(y) ∂ξx∂x)= curl f in Ω × R∗+ξ · n = 0, curl ξ = ψ on ∂Ω × R+(4)wheres is a non negative parameter.In this paper we restrict to the case of uniform flow(M ′(y) = 0 ∀y ∈] − h, h[) for which the problem inψandξx are decoupled. The study of the coupled systemwill be a subject of future works5 Mathematical analysis of the regularized problemWhen the Mach number is constant the vorticityψ isknown and the problem inξ is written :{D2ξDt2− ∇(div ξ) + s curl (curl ξ) = fs in Ω × R+ξ · n = 0, curl ξ = ψ on ∂Ω × R+(5)wheref s = f + s curl ψ.The second boundary condition(curl ξ = ψ) of system(5) is necessary for the equivalence with the initial prob-lem (1)-(2). The property of ellipticity of the spatial op-erator−∇(div ) + s curl (curl) (which equals−∆ ifs = 1) allows to carry out the mathematical and numeri-cal study of this problem in a classical frame.We considerH := H0(Ω) × L2(Ω)2 whereH0(Ω) :={ξ ∈ (H1(Ω))2/ ξ · n = 0, on∂Ω}.To apply Hille-Yosida theorem, we introduce a new un-known ζ = Dξ/Dt. If we poseU = (ξ, ζ)t, then wecan rewrite (5) under the following form :dUdt+AsU = FsU(0) = U0(6)AsU =−ζ +M∂ξ∂x−∇(div ξ) + s curl (curl ξ) +M∂ζ∂x,Fs =(0fs).The domain of the unbounded operatorAs is defined by :D(As) ={U = (ξ, ζ)t ∈ H tel que AsU ∈ H}.Using Costabel’s identity [3] and Hille-Yosida’s theory,we prove that the operatorAs is maximal monotoneTheorem 5.1 : If min(1, s) > M2, then for fs ∈C1(R+; L2(Ω)2) and sufficiently regular initial data,problem(5) has a unique solution which satisfies:ξ ∈ C1(R+; H0) ∩ C2(R+; L2(Ω)2)6 The absorbing boundary conditionsFor solving numerically the problem, we need to truncatethe unbounded domainΩ. The truncated domainΩb :=]−R; R[×[−h; h[ requires the introduction of absorbingboundary conditions (ABCs) on the artificial boundariesΓ± := {(x, y) ∈ Ω, x = ±R} (R > 0). The difficultyis to find appropriate boundary conditions adapted for theregularized formulation. Fors = 1, the boundary condi-tions that we propose are the following :DξDt+∂ξ∂n= 0 onΓ± (7)wheren = (±1, 0) is the unit outward normal vector ofΓ± andτ = (0, ±1). The main properties of conditions(7) arei) These are exact conditions fory-independent solu-tions (plane waves) : this is why we speak of firstorder conditions.ii) Well-posedness: the truncated problem is wellposed and forf = 0 we have the energy decay re-sult:ddtE(t) +∫Γ−∪Γ+∣∣∣∣∂ξ∂t∣∣∣∣2= 0whereE(t) =12∫Ωb∣∣∣∣∂ξ∂t∣∣∣∣2+ |∇ξ|2 −M2∣∣∣∣∂ξ∂x∣∣∣∣2iii) These are compatible with a variational formula-tion of the regularized problem, namely:d2dt2(ξ, η) +ddt(b(ξ, η) + cΓ±(ξ, η))+a(ξ, η) + dΓ±(ξ, η) = (f ,η)(8)with (·, ·) is theL2-inner product anda(ξ, η) =∫Ωbdiv ξ div η + curl ξ curl η −M2∂ξ∂x·∂η∂x,b(ξ, η) =∫Ωb2Mξ ·∂η∂x, cΓ±((ξ,η) =∫∪Γ∓(1 ±M)ξ · η dγ,dΓ±(ξ,η) =∫Γ−∪Γ+R∂ξ∂τ· η dγ, R =(0 −11 0)Remark 6.1 The construction of good absorbing bound-ary conditions is not trivial. For example the followingnatural ABC’s :(1 −M2)∂ξ1∂x+∂ξ2∂y± (1 ∓M)∂ξ1∂t∣∣∣∣Γ±= 0 (9)(1 −M2)∂ξ2∂x−∂ξ1∂y± (1 ∓M)∂ξ2∂t∣∣∣∣Γ±= 0 (10)are still exact for plane waves and variational, but notstable.7 Discretization of Galbrun equationsThe Lagrange finite element approximation of (8) leadsto the following ordinary differential system :Mhd2ξhdt2+ ((Bh + CΓ±h )dξhdt+ (Ah + DΓ±h )ξh = F h (11)whereMh is the mass matrix, andAh, Bh, CΓ±h andDΓ±hare the matrices respectively associated to the bilinearformsa(., .), b(., .), cΓ±(., .) anddΓ±(., .).For the time discretization of (11) we use a centeredsecond order finite difference scheme :Mhξn+1h − 2ξnh + ξn−1h∆t2+ (Bh + CΓ±h )ξn+1h − ξn−1h2∆t+(Ah + DΓ±h )ξnh = Fnh8 Numerical SimulationIn this first experiment, we simulate a wave initially ex-cited by a rotational source located in the center of thedomain, in the presence of a horizontal uniform flow withM = 0.5. The first simulation shows that the method isnot stable if the equation is not regularized (s = 0).Figure 1: Euclidean norm ofξ at att1 = 1.5s andt2 =1.75s andt3 = 2s. Cases = 0The second simulation corresponds to the regularizedcase (s = 1). When can distinguish the two parts ofthe Lagrangian displacement : the irrotational part corre-sponds to the outer circular wavefront (whose radius in-creases with time) while the rotational part corresponds tothe inner circular wavefront (whose radius remains con-stant). Both are centered at a point which is convected bythe flow.Figure 2: Euclidean norm ofξ at t1 = 1.5s andt2 =1.75s andt3 = 2s. Cases = 1In the second experiment, we simulate Galbrun wavepropagation in an infinite rigid duct. The wave is excitedby a Gaussian signal in time and a quasi-punctual irrota-tionnal source in space in the uniform flow withM = 0.5.We remark on this result that the ABC prevents (partially)the unphysical reflexion.Figure 3: Evolution of the Euclidean norm of the dis-placementξReferences[1] H. Galbrun, ”Propagation d’une onde sonore dansl’atmosphère terrestre et théorie des zones de si-lence”, Gauthier-Villars, Paris, 1931.[2] G. Legendre, ”Rayonnement acoustique dans unfluide en écoulement : analyse mathématique etnumérique de l’équation de Galbrun, PHD thesis2003.[3] M. Costabel, ”A coercive bilinear form forMaxwell’s equations”, J. Math. Anal. Appl, 157,pp. 527-541, 1991

NUMERICAL ANALYSIS OF TIME-DEPENDENT GALBRUN EQUATION IN AN INFINITE DUCT

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NUMERICAL ANALYSIS OF TIME-DEPENDENT GALBRUN EQUATION IN AN INFINITE DUCT

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