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Highly Accurate Schemes for Wave Propagation Systems:Application in Aeroacoustics

Nathalie Bartoli Pierre MazetVincent Mouysset Francois Rogier

Onera/DTIM/M2SN

Context

propagation of aeroacoustic perturbations of an original flow

observe some non-stationary phenomena (convective orabsolute instabilities) involved by the choice of the open flows(shear flows)

long time computation

very lengthened geometry

2/25 High order schemes - JSO 2009

Objectives

to observe numerically convective/absolute instabilities

high accuracy and robustness required for the numericalschemes

need for considering increasingly powerful numericaltechniques

=⇒ consider Discontinuous Galerkin method with h− p refinement

=⇒ development of a 2D software

=⇒ test a large variety of configurations:

non-conforming gridvariable polynomial orderperfectly matched layers (PML)


Outlines

The Discontinuous Galerkin method (DG method)

The aeroacoustic problem

Numerical experiments

Conclusion


The Discontinuous Galerkin methoda generalization of the Finite Volume techniques


Properties of the DG method

1 No kind of continuity constraints across the interface betweenthe elements of the triangulation

2 Compact numerical schemes: exchange between the cells doneby the interfaces

3 Explicit time scheme easily written: local mass matrix relativeto the time discretization terms

4 Flux approach: the conservation laws of mechanics respected(Finite Volume method)

5 Flexibility:FE space approximation can change element to elementnon conforming mesh (local refinement)use suitable basis functions locally adapted

=⇒ h-p refinement

6 Complex geometries: unstructured or locally refined meshes7 Parallelization is efficient (compact nature of DG)


The aeroacoustic problem

a choice governed by 3 main reasons

a Friedrich’s system with variable coefficients

accurate numerical methods are required to capturenon-stationary phenomena

real geometries involved

=⇒ high-fidelity solutions involving low dispersive and lowdissipative numerical methods

Linearized Euler’s Equations

∂tu1 + U0∂xu1 + c0∂xρ+ ∂yU0 = 0∂tv1 + U0∂xv1 + c0∂yρ = 0∂tρ+ U0∂xρ+ c0∂xu1 + c0∂xv1 = 0,

(u1, v1) : acoustic velocity vector ρ = c0ρ1/ρ0

ρ1: acoustic density ρ0: mean flow density c0: sound speedparticular carrier flow: [U0(y),V0 = 0, ρ0 = constant]


The 2D developed software

structured grid with anisotropic refinementh = (hx , hy )

polynomial basis functions Qp of variableorder p = (px , py )

Explicit time schemes (Euler, StrongStability-Preserving Runge-Kutta methods:RK2,RK4,...)

=⇒ validation of the p order compared to the analytical solutionwith uniform carrier flow

transverse and upstream modes (0,1,2)case ∼ 1D with an initial condition depending only of x

=⇒ experiments with some shear flows


Numerical experiments

Geometry: infinite linear ductInitial condition

δ(x , y): regularisation of the Dirac distribution on a ballB(S , ε)quite difficult test case: all frequencies are excited

Boundary conditions:first order ABC at both extremitiesrigid walls (hard sound obstacle conditions)

CFL for P0 dt ≤ h/2(max(|U0(y)|+ c0))


Comparison of the results/analytical solution atconstant speedinitial condition depending only of x (∼1D)

uniform carrier flow U0 = 0.3

→ analytical solution computed

→ L2 relative error (ρ(x , t) or u1(x , t)) on 5points located along the axis

distance/Source 0.45 1.45 5.45 9.66 19.39

Q0 − h 9.47% 17.1% 36% 47.11% 60.6%

Q0 − h/2 5.66% 10.45% 23.8% 33.31% 47.14%

Q1 − h 0.29% 0.48% 0.94% 1.48% 2.04%

Q2 − h 0.04% 0.06% 0.13% 0.17% 0.30%

=⇒ strong reduction in the error with the increase in the order=⇒ loss of positivity

− with order 1 (amplitude 1/80)− with order 2 (amplitude 1/300)


Comparison of the results/analytical solution atconstant speed

transverse and upstream mode m = 2, λ = 0.62

without carrier flow U0 = 0

→ analytical solution computed

→ exact maximum amplitude of |ρ(x , t)| = 1

size mesh dof max |ρ(x , t)| CPU time

Q0 − h/4 λ/80 1 0.9320 t

Q1 − 2h λ/10 4 1.07 t/5.9

Q2 − 4h λ/5 9 1.05 t/9.5

Q3 − 6h λ/3.7 16 1.04 t/6.8

=⇒ coarse mesh used with high order=⇒ computation of derivative with sufficient accuracy


Validation of GD on non-conforming grid

• upstream mode m = 0, Q1h -Q2

2h • initial condition (∼1D), Q1h/2-Q

2h

=⇒ no dispersion observed on anisotropic meshes=⇒ validation of h − p refinement / analytical solution


Acoustic rebounds in a duct

duct geometry (40× 1)

central source point to excite ρ(x , y , t = 0)

U0(y): hyperbolic profile tangent without return

2 formulations Q0h/2 and Q1

h

=⇒ visualization of ρ over a period of time (movie)=⇒ phenomenon of acoustic rebounds

visible with Q1h , more dissipation with Q0

h/2


Acoustic rebounds in a duct

comparison of 3 formulations

Q0h/2, Q1

h and (Q0h/2,Q

1h)

=⇒ inefficiency of the hybrid approximation order 0/order 1


Convective instabilities

duct geometry 54× 1, central source point to excite u1(x , y , t = 0)

comparison of the u1 component on 3 downstream points

different approximations Q0h or Q1

h or Q22h

amplitude ratio|P3|/|P2| |P4|/|P3|

Q0h 0.98 1.14

Q1h 4.74 5.25

Q22h 4.57 5.15

−→ increase in the amplitude between two downstream points :localised disturbances disappear over time

−→ high order approximations requested to get a significantamplitude ratio between two points (4 times greater with Q1

h

or Q22h than with Q0

h)


Introduction of PML

drastic decrease of the time step with the order

rebounds on ABC can induce absolute instabilities

=⇒ reduce the computational domain by using PML

work of P. Mazet and Y. Ventribout (2006)

cartesian PML: resolution FV in space / explicit scheme in timestability proved for uniform carrier flowmathematical well-posed for frozen system

=⇒ numerical experiments for some shear flows


Convective instabilities : validation of PML

=⇒ stability preserved

=⇒ identical physical phenomena with or without PML

=⇒ time reduction Q1h : 85h instead of 137h


Convective instabilities : validation of h − prefinement

Validation with anisotropic degree : px = 1, py = 2

Validation with non-conforming grid : Q1h and Q2

2h

Validation of unrefined mesh with high order : Q24h


Absolute instabilities

duct geometry 39× 1 (1559× 40 elts)

U0(y): hyperbolic profile tangent with return

component u1 over a period oftime with Q0

h -RK2

increase in the amplitudeattributed to the phenomenon ofabsolute instabilities

=⇒ computation with Q1 and Q2


Absolute instabilities : Q0/Q1/Q2

component u1 and ρ over a periodof time with Q0

h and Q24h

an amplitude 250 times higher withorder 2 than order 0 (with refinedmesh)

=⇒ capture instabilities with qualitative information on theparameter responsible for the bifurcation (convective/absoluteinstabilities)


High order interest

unrefined mesh associated to high order

accuracy on derivative

capture instabilities

=⇒ require suitable tools for visualization−→ different criteria to find the better projection on P1 or Q1

(gmsh, paraview, tecplot,...)

(a) with refined triangles (b) with adaptive mesh


Conclusion

Increase in the order

interest of high order to capture instabilitiesdrastic decrease of the time step with the order / unrefinedmeshloss of positivity

Difficulties of the test case

transitory phenomenoncapture convective and absolute Kelvin-Helmholtz instabilities

computation over a long period of timeimportant grid (lengthened channel)

=⇒ find a theoretical solution in the unbounded domain (via theGreen dyadic function) : work of P. Delorme

=⇒ introduce local time step

=⇒ define strategy to get high accuracy with some“goal oriented”techniques (h − p refinement/unrefinement)


2D Linearized Euler Equations

hypothesis

shear carrier flow (U0(y),V0 = 0)

c0 constant sound speed∂tu1 + U0∂xu1 + c0∂xρ+ ∂yU0 = 0∂tv1 + U0∂xv1 + c0∂yρ = 0∂tρ+ U0∂xρ+ c0∂xu1 + c0∂xv1 = 0,

⇔ a symmetric linear hyperbolic system (Friedrich’s system)

∂tϕ+ Ai∂iϕ+ Bϕ = 0

Ai and B: matrices with variables coefficients depending on thesubsonic flow


System to solve

∂tϕ+ Ai∂iϕ+ Bϕ = f

Variational formulation on each local mesh element C∫C(∂tϕh + Ai∂iϕh + Bϕh, ψh) +

∫∂C\Γ

((Aini )−(ϕ+

h − ϕ−h ), ψh)

+

∫ΓMβϕ

−h ψh =

∫C(f , ψh)


System to solve

∂tϕ+ Ai∂iϕ+ Bϕ = fMβϕ = 0 in Γ× [0,T ]ϕ(x , 0) = 0 ∀x ∈ Ω,

(1)

where Mβ is the matrix

Mβ =c0

2

((β + 1)n ⊗ n (β − 1)n−(1 + β)nt (1− β)

). (2)

β = 1, hard sound obstacle (Neumann),

β = −1, soft sound obstacle (Dirichlet).

For ABC, we haveMβ = −(Aini )

−,


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