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Extension to moving grids P. Lafon 1, F. Crouzet 2 & F. Daude 1 1 LaMSID - UMR EDF/CNRS 2832 2 EDF R&D, AMA April 3, 2008

1 Governing equations Physical coordinates Generalized coordinates Geometrical conservation 2 Numerical procedure Fluid dynamics Geometrical motion Moving overlapping grids approach 3 Validation Strategy Single domain Multi-domain 4 STURM4 Project Partners Context Application to non-cartesian bodies

Physical coordinates 3-D compressible unsteady Navier-Stokes equations: t U + x E + y F + z G = S ν Physical coordinates (t, x, y, z), Conservative variables U = (ρ, ρu, ρv, ρw, ρe), Inviscid flux-vectors E, F et G : ρu ρu 2 + p E = ρuv ρuw (ρe + p) u Viscous source term S ν.

Generalized coordinates Equations rewritten in time-dependent generalized coordinates (Visbal & Gaitonde JCP 2002, Hixon AIAA J. 2000): ( ) 1 τ J U + ξ Ê + η ˆF + ζ Ĝ = Ŝ ν (t, x, y, z) (τ, ξ, η, ζ), J Jacobian of the transformation (x, y, z) (ξ, η, ζ), Inviscid terms: Ê = 1 J ( ) ξ t U + ξ x E + ξ y F + ξ z G ˆF = 1 J Ĝ = 1 J ( ) η t U + η x E + η y F + η z G ( ) ζ t U + ζ x E + ζ y F + ζ z G Viscous terms (Marsden et al. JCA 2005).

Geometrical conservation ( ) 1 J ξ x ( ) 1 J ξ y ( ) 1 J ξ z ξ ξ ξ ( ) 1 + J η x ( ) 1 + J η y ( ) 1 + J η z η η η ( ) 1 + J ζ x ( ) 1 + J ζ y ( ) 1 + J ζ z ( ) ( ) ( ) ( ) 1 1 1 1 + J τ J ξ t + ξ J η t + η J ζ t ζ ζ ζ ζ = 0 = 0 = 0 = 0 Geometrical Conservation Law (GCL) nds = 0 Ω(t) Notations: ˆξ x = 1 d J ξ x, ˆξt = 1 J ξ t... dv = 0 dt Ω(t)

Fluid dynamics Approach for moving/deforming meshes (Visbal & Gaitonde JCP 2002): Time derivative in NS splitted in two parts: ( ) 1 τ J U = 1 ( ) 1 J τ U + U τ J Use of the GCL for the second term in the right-hand side: ( ) [ ( ) ( ) ( ) ] 1 1 1 1 τ = J J ξ t + ξ J η t + η J ζ t ζ Final equations to be solved: [ )] τ U + J ξ Ê + η ˆF + ζ Ĝ Ŝ ν U ( ξ ˆξt + η ˆη t + ζ ˆζt = 0 } {{ } R

Fluid dynamics Spatial discretization: 11-point optimized centered finite difference scheme (Bogey & Bailly JCP 2004) d dτ U i,j,k + R i,j,k (U, J, ξ, ) ξ t,... = 0 Time integration: explicit 4-stage Runge-Kutta scheme Stability requirement: CFL = max(cfl ξ, CFL η, CFL ζ ) 1 with CFL ξ = ( ξt + V. ξ + c ξ ξ ) τ Selective filtering: 11-point optimized centered low-pass filter (Bogey & Bailly JCP 2004)

Fluid dynamics Shock-capturing filtering (1/2) Non-linear shock-capturing filter: ( ) 1 n+1 ( ) 1 (5) ( ) J U = i,j,k J U + τ D ξ i,j,k + Dη i,j,k + Dζ i,j,k i,j,k with the dissipation term: D ξ i,j,k = 1 ξ ( ) D ξ i+1/2 Dξ i 1/2 where D ξ i+1/2 is the numerical dissipation flux-vector: D ξ i+1/2 = λ ξ ( ) i+1/2 ɛ (2) J i+1/2 U (5) i+1,j,k U(5) i,j,k i+1/2

Fluid dynamics Shock-capturing filtering (2/2) with: the stencil eigenvalue: λ ξ i+1/2 = 3 max m= 2 ) ( λ ξ i+m,j,k min 3 ) ( λ ξ i+m,j,k m= 2 with the eigenvalue: λ ξ = ξ t + V. ξ + c ξ the midpoint Jacobian: J i+1/2 = 1 2 (J i+1,j,k + J i,j,k ) the non-linear dissipation function: ɛ (2) i+1/2 = κ j,k 3 max m= 2 ( ) ν ξ i+m,j,k

Geometrical motion Metrics expressions (1/2) spatial: conservative form (Thomas & Lombard AIAA J. 1979) Jacobian: standard manner 1 J ξ x = (y η z) ζ (y ζ z) η 1 J η x = (y ζ z) ξ (y ξ z) ζ 1 J ζ x = (y ξ z) η (y η z) ξ 1 J = x ξy η z ζ + x η y ζ z ξ + x ζ y ξ z η x ξ y ζ z η x η y ξ z ζ x ζ y η z ξ

Geometrical motion Metrics expressions (2/2) temporal: standard manner (Visbal & Gaitonde JCP 2002): ( ) 1 J ξ 1 t = x τ J ξ 1 x + y τ J ξ 1 y + z τ J ξ z ξ t = V e. ξ ( ) 1 J η 1 t = x τ J η 1 x + y τ J η 1 y + z τ J η z ( ) 1 J ζ 1 t = x τ J ζ 1 x + y τ J ζ 1 y + z τ J ζ z with V e = (x τ, y τ, z τ ) T the mesh velocity. Similar to ALE (Arbitrary Lagrangian Eulerian) approach in the finite-volume framework: V V e. η t = V e. η ζ t = V e. ζ

Geometrical motion Boundary conditions (1/2) Figure: Sketch of the overlapping grids around an airfoil Non-reflective conditions: fixed domains far-field radiation (Bogey & Bailly Acta Acoustica 2002)...

Geometrical motion Boundary conditions (2/2) Figure: Wall surface in the physical domain Wall conditions: slip condition (Euler) : ξ t + V. ξ = 0 in the ξ direction no-slip condition (Navier-Stokes) : ξ t + V. ξ = 0 η t + V. η = 0 in all directions ζ t + V. ζ = 0

Geometrical motion Updating of the grid coordinates: Use of the RK4 scheme: ( ) x (l) (l 1) i,j,k = xi,j,k n + τβ(l) x τ i,j,k y (l) i,j,k = y n i,j,k + τβ(l) ( y τ ) (l 1) i,j,k ( ) z (l) (l 1) i,j,k = zi,j,k n + τβ(l) z τ i,j,k New stability requirement linked to the grid motion: In 1-D: dx dt = 0 τ x + ξ t ξ x = 0

Geometrical motion Algorithm At each stage of the RK4 scheme: ( U (l) = U n + τβ (l) R U (l 1), J (l 1), ξ (l 1) t, ξ ) (l 1)... x (l) = x n + τβ (l) (x τ ) (l 1)

Geometrical motion Algorithm At each stage of the RK4 scheme: ( U (l) = U n + τβ (l) R U (l 1), J (l 1), ξ (l 1) t, ξ ) (l 1)... x (l) = x n + τβ (l) (x τ ) (l 1) Known variables: U (l 1) and x (l 1),

Geometrical motion Algorithm At each stage of the RK4 scheme: ( U (l) = U n + τβ (l) R U (l 1), J (l 1), ξ (l 1) t, ξ ) (l 1)... x (l) = x n + τβ (l) (x τ ) (l 1) Known variables: U (l 1) and x (l 1), Computation of J (l 1), ξ (l 1),... Evaluation of V (l 1) e, Computation of ξ (l 1) t,...

Moving overlapping grids approach Updating of interpolations data (1/2) η P Nη P δ η ξ y η δ ξ N ξ x ξ Figure: Example of a 2-D interpolation stencil (Emmert 2007) with L ξ i = φ P N ξ 1 m=0,m i N ξ 1 i=0 N η 1 j=0 L ξ i Lη j φ IQ +i,j Q +j δ ξ m i m and Lη j = N η 1 m=0,m j δ η m j m

Moving overlapping grids approach Updating of interpolations data (2/2) Figure: As the annular component grid moves inactive points become active (Henshaw & Schwendeman JCP 2006) Interpolation stencils (donor and receiver points), Holes cut by the overset-grid approach, Lagrangian coefficients,

Moving overlapping grids approach Updating of interpolations data (2/2) Figure: As the annular component grid moves inactive points become active (Henshaw & Schwendeman JCP 2006) Interpolation stencils (donor and receiver points), Ogen Holes cut by the overset-grid approach, Ogen Lagrangian coefficients,

Strategy 2-D inviscid cases are considered: Simplicity (2-D interpolation, spatial metrics evaluation...), Only inviscid terms are modified by the grid motion. Validation procedure is performed on: Single domain (validation of the numerical algorithm, the metrics evaluation, the grid updating...), Multi-domain (coupling the updating of the interpolation data and the numerical algorithm).

Single domain Vortex advection on single dynamically deforming domain Figure: Inviscid vortex advection Periodic boundary conditions, Analytic grid velocity as time-dependent sinus function is imposed, Supposition: V (l) e = V n e.

Figure: Comparison between static and deforming mesh computations with CFL=0.5: velocity fields ; mesh deformation and swirl velocity Single domain Vortex advection on single dynamically deforming domain

Multi-domain Flow past a moving cylinder Figure: View of the global mesh ; Streamwise velocity CFL = 0.5 M e 0.3 d = 0.08 x

Multi-domain Multiple moving bodies y 2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5 4 3 2 1 0 1 2 3 4 x 50 0 50 Figure: Streamwise velocity CFL = 0.5 d = 0.04 x M e 0.13

Partners Simulation de la TUrbulence à haut Reynolds Multi-Maillage et Multi-Modèles (STURM4) EDF R&D (D. Laurence, S. Benhamadouche, F. Crouzet) Valeo (S. Moreau, M. Henner, B. Demory,...), IJLRDA (ex LMM) (P. Sagaut,...) LaMSID (P. Lafon, F. Daude,...) ECL (C. Bailly)

Context Objectifs Étendre la LES aux applications industrielles via : un couplage entre les modélisations statistique (RANS) et déterminitiste (DNS/LES) de la turbulence, des maillages composites superposés. Applications au bruit et aux vibrations induits par les écoulements turbulents. S-P 4 : Simulations Aéroacoustiques Directes sur des maillages composites structurés en calcul parallèle S-P 6 : Application au rayonnement acoustique de pale de ventilateur VALEO

Application to non-cartesian bodies The rod-airfoil interaction (Greschner et al. C&F 2008) Geometrical aspects: Symmetric NACA0012 airfoil (chord c = 0.1 m), Circular rod (d/c = 0.1). Physical aspects: U = 72 m.s 1 Re d = 48, 000, Re c = 480, 000 M 0.2 Direct computation of the rod-airfoil interaction noise via LES

Application to non-cartesian bodies Figure: Streamwise velocity

Application to non-cartesian bodies Thank you For your attention!!

Numerical procedure Validation 5 Numerical procedure Fluid dynamics 6 Validation Single domain Multi-domain

Numerical procedure Validation Fluid dynamics Approach for moving/deforming meshes (Visbal & Gaitonde JCP 2002): Inconvenients: does not satisfy strong conservation in time (split of the time derivative), temporal metrics evaluated in the standard manner. Advantages: use of the GCL compensates errors introduced in evaluating temporal metrics, CPU time affordable. Conservative approach (Hixon AIAA J. 2000) time consuming

Numerical procedure Validation 5 Numerical procedure Fluid dynamics 6 Validation Single domain Multi-domain

Numerical procedure Validation Single domain Vortex advection on single dynamically deforming domain Analytic grid velocity: (x τ ) i,j = 2πωA x x 0 cos(2πωt) sin ( n x π y ) i,j(0) y min α x y max y min ( (y τ ) i,j = 2πωA y y 0 cos(2πωt) sin n y π x ) i,j(0) x min α y x max x min with α x = exp ( 4 log(2) x i,j(0) 2 + y i,j (0) 2 ) (x max x min ) 2 α y = exp ( 4 log(2) x i,j(0) 2 + y i,j (0) 2 ) (y max y min ) 2

Numerical procedure Validation Multi-domain Flow past a moving cylinder Figure: Mach number CFL = 1 d = 0.2 x M e 1.2

Numerical procedure Validation Multi-domain Publications Conference proceedings: Emmert, Lafon & Bailly, AIAA, Vancouver, 2008, Daude, Emmert, Lafon, Crouzet & Bailly, AIAA, Vancouver, 2008, Daude, Emmert, Lafon, Crouzet & Bailly, ETMM7, Limassol, 2008, Daude, Emmert, Lafon, Crouzet & Bailly, Acoustics 08, Paris, 2008. Articles: Emmert, Lafon & Bailly, in preparation, 2008,