AeroFoam: an accurate inviscid compressible solver for aerodynamic applications

Transcription

1 3 rd OpenFOAM Workshop th July 2008 AeroFoam: an accurate inviscid compressible solver for aerodynamic applications Giulio Romanelli Elisa Serioli Paolo Mantegazza Aerospace Engineering Department Politecnico di Milano

2 Table of contents Introduction 2 Numerical scheme 3 Test problems 4 Work in progress 5 Future work

3 Outline Objective Develop a new accurate inviscid compressible Godunov-type coupled solver for aerodynamic (and aeroelastic) applications in transonic and supersonic regime To begin with OpenFOAM built-in inviscid compressible segregated (with PISO correction) solvers rhosonicfoam, rhopsonicfoam and sonicfoam Previous similar work centralfoam by L. Gasparini: inviscid compressible central (or central-upwind) coupled solver, 2 nd order formulation of Kurganov, Noelle and Petrova (KNP) GASDYN+OpenFOAM D/3D formulation by Dipartimento di Energetica, Politecnico di Milano: inviscid compressible Godunov-type coupled solver, st order Harten, Lax and vanleer (HLL/C) Riemann solver

4 Euler equations Governing equations for time-dependent, compressible, ideal (inviscid µ = 0 and nonconducting κ = 0) fluid flows in coupled, integral, conservative form: β n S = V R Nd V R Nd S inflow S β n n d u dv + f(u) n ds = 0 dt V S u(x, 0) = u 0 (x) u(x S inflow, t) = u b (t) Conservative variables vector and inviscid flux function tensor: ρ ρ m u = m = ρ v f = E t ρe + 2 ρ v 2 m ρ m + P [ I ] m ρ (Et + P ) Polytropic Ideal Gas (PIG) thermodynamic model: γ = C p /C v, R = R u /M

5 FV Framework S h = V h Ω i Ω j Γ ij n ij du i dt + Ω i j= On each Ω i cell: averaged conservative variables vector (2 volumescalarfield, volumevectorfield) U i (t) = u(x, t) dv Ω i Ω i On each Γ ij interface: numerical fluxes vector (2 surfacescalarfield, surfacevectorfield) F ij (t) = Γ ij Γ ij F ij = 0 Targets:? Γ ij f(u) n(x, t) ds On each Ω i cell: cell-centered spatially discretized Euler equations ODE system Monotone and sharp solution N f near discontinuities 2 nd order of accuracy in space in smooth flow regions

6 Monotone numerical fluxes Several Godunov-type monotone st order expressions for the numerical fluxes vector presented in Literature in the form: F ij = F ij (U i, U j ) Approximate Riemann Solver (ARS) Convective Upwind and Split Pressure (CUSP) Implemented in AeroFoam: Harten-Lax-vanLeer (HLL/C) Osher-Solomon (OS) Euler equations rotational invariance From a 3D problem in global reference frame G (X Y Z) to an D equivalent problem in local reference frame L (x y z) R( n G ij) y, v L x, u local solutions U L i = R(bn ij) U G i Y, v Y Ωi z, w nij local fluxes F L ij = F L ij (U L i, U L j ) G X, v X Γij Z, v Z back to global fluxes F G ij = R(bn ij) F L ij

7 Approximate Riemann Solver Roe (98): local linearization at each interface Γ ij of the governing equations and exact solution of the resulting Riemann problem (e.g. D) U j U i u t +  u x = 0 x i Γ ij x j Projected Jacobian matrix evaluated at Roe s intermediate state  = f u bu n ij satisfying the following properties: b A A(U) smoothly as Ui, U j U 2 b A = b R b Λ b R diagonalizable with real eigenvalues and orthogonal eigenvectors 3 consistency: b A (U j U i) = ˆ f(u j) f(u i) bn ij Monotone st order numerical fluxes vector (generalization of upwind method): F ARS ij = f(u i) + f(u j ) n ij 2 2 R Λ R (U j U i )

8 High resolution numerical fluxes Idea: combine a monotone st order numerical flux F I ij (works fine near shocks) and a 2 nd order numerical flux F II ij (works fine in smooth flow regions) by means of a flux-limiter function Φ F HR ij = F I ij + Φ ( F II ij F I ij ) = F I ij + A ij, Implemented in AeroFoam Lax-Wendroff (LW) Jameson-Schmidt-Turkel (JST) Remark To build the antidissipative numerical fluxes vector A ij (U i, U j ; U i, U j ) solutions U i and U j on extended cells Ω i and Ω j are also needed

9 Extended cells connectivity Idea: continue Ω i and Ω j cells along n ij, e.g. Ω j : Ω q B(P j ) = {Ω q P j Ω q } such that = (x q x ij ) (x q x ij ) n ij n ij is minimum Extended cells connectivity data structures (2 labelfield) are initialized in the pre-processing stage with the following algorithms (meshsearch library is used): Ω P i i P Ω i Ω j j 0 Ω j Γ n ij ij 0 A. Incremental search algorithm (works fine on structured meshes) initial guess x A = s A bn ij Ω j : Ω q such that x A Ω q if Ω j Ω j update s A = 2 s A B. Nonincremental search algorithm (works fine on unstructured meshes) x B = s B bn ij where s B = 4 Ω j / Γ ij Ω j : Ω q such that x B Ω q

10 Lax-Wendroff Lax-Wendroff (96): the following antidissipative numerical fluxes vector A ij is added to ARS monotone st order numerical fluxes vector F ARS ij A ij = ( ) 2 R t Λ x j x i Λ 2 W Φ Characteristic variables jump vector W = R (U j U i ) is suitably limited as follows (e.g. vanleer flux limiter): W Φ = W Q + Q W Q + W + ε where the r th element of characteristic variables upwind jump vector reads: Q = Q r = { R r (U j U j ) if λr > 0 R r (U i U i ) if λr 0.

11 Im Introduction Numerical scheme Test problems Work in progress Future work Time discretization Explicit Runge-Kutta method of order p = 2, 3, 4 as a compromise between: computational efficiency memory requirements RK4 RK2 RK Re Remarks only 2 time levels must be stored CFL stability constraint Co max 2.8 CFL stability condition order of accuracy in time 8 {U} (k+) = {U} (k) + t K 6 + K K K 4 6 K = R( t (k), {U} (k) ) >< RK4: K 2 = R t (k) + t 2, {U}(k) + t 2 K K 3 = R t (k) + t 2, {U}(k) + t 2 K 2 >: K 4 = R( t (k) + t, {U} (k) + t K 3 ) O( t 2 ) minimum order of accuracy in time is granted also for nonlinear problems

12 Boundary conditions At each boundary interface Γ ij S h = V h a suitable numerical solution must be set on the fictitious or ghost cells Ω GC j and Ω GC j Ω GC j v ij u ij Ω GC j Ω Ω i n i ij Γ ij S 0 h = V h Characteristic splitting the number of physical boundary conditions N bc equals the number of negative eigenvalues λ r < 0 Physical boundary conditions: N bc primitive variables assigned Numerical boundary conditions: N d +2 N bc primitive variables extrapolated

13 Riemann boundary conditions At asymptotic/external boundary subset S Boundary type Flow type N bc Assigned variables Extrapolated variables S inflow v ij bn ij < 0 S outflow v ij bn ij > 0 Supersonic flow v ij bn ij > c ij ( SupersonicInlet) Subsonic flow v ij bn ij < c ij ( Inlet) Supersonic flow v ij bn ij > c ij ( ExtrapolatedOutlet) Subsonic flow v ij bn ij < c ij ( Outlet) N d + 2 T, v, P N d + T, v P 0 T, v, P T v, P

14 Slip boundary conditions At solid/impermeable boundary subset S body linear extrapolation of solution U GC j and U GC j on ghost cells ΩGC j and Ω GC j set to zero normal velocity component and update conservative variables ṽ j = v j (v j n ij ) n ij Ẽ t j = E t j 2 ρ j v j ρ j ṽ j 2 Generalization: transpiration boundary conditions The geometric and kinematic effects of a given body displacement law s(x, t) (rigid and deformative) can be simulated by means of a transpiration velocity: ṽ j = v j (v j n ij ) n ij + Vn n ij V n = v j n + ṡ n 0 + ṡ n }{{}}{{}}{{} geometric kinematic mixed Mesh is not deformed runtime (expensive) but only in the post-processing stage with the implemented utility showdisplacement (motionsolver library is used)

15 Introduction Numerical scheme Test problems Work in progress Future work Comparison with existing solvers 2D Oblique shock reflection M = 2.9, α = 29 2 M2, P2, T2 v θ θ v2 M, P, T v2 v3 h 3 M3, P3, T3 α α L 4 hexahedral meshes (blockmesh) from Nv = 40 0 to Nv = comparison with exact solution PC AMD GHz, Gbyte RAM rhosonicfoam rhopsonicfoam sonicfoam AeroFoam

16 Comparison with existing solvers Least Squares fit for e h L h e h L [ ] CPUtime [ s ] O(h) rhosonicfoam rhopsonicfoam sonicfoam AeroFoam O(h 2 ) h [ m ] 0 00 rhosonicfoam rhopsonicfoam 0 sonicfoam AeroFoam O(N e ) 0. 2 O(N e ) N e Speedup [ ] Solver A p R h rhosonicfoam rhopsonicfoam sonicfoam AeroFoam rhosonicfoam 6 rhopsonicfoam 5 sonicfoam N e

17 Incompressible limit M = 0.05! 2D Fixed cylinder 5K triangular mesh (Gmsh) comparison with exact solution (potential theory) single iteration CPUtime = 0.09 s 3 2 AeroFoam Exact C p [ ] 0 C p Exact = 4 sin 2 ϑ x/r [ ]

18 NACA 02 airfoil 2D NACA 02 airfoil M = 0.75, α = 4 7K triangular mesh (Gmsh) comparison with FLUENT and experimental data (AGARD 38) single iteration CPUtime = 0.04 s.5 Upper Surface C p [ ] Lower Surface WT -0.5 FLUENT AeroFoam x/c [ ]

19 Thin airfoil 2D Thin airfoil (t/c = 0.) M = 0.5, 0.7,.0,.2 vertical step gust v g/v = tan( ) 7K triangular mesh (Gmsh) comparison with exact solution (Bisplinghoff) single iteration CPUtime = 0.04 s.6.6 C L/α /2π [ ] M =0.7 M =0.5 C L/α /2π [ ] M =.0 M = Exact AeroFoam τ [ ] 0.6 Exact AeroFoam τ [ ]

20 ONERA M6 wing () cr = m 0.44 cr y x ΛLE = 30 b =.96 m M = ΛTE = 5.8 c ct = m 3D ONERA M6 wing M = 0.84, α = K tetrahedral mesh (GAMBIT) comparison with FLUENT and experimental data (AGARD 38) single iteration CPUtime = 3.48 s

21 ONERA M6 wing (2).5 y/b = y/b = 0.95 C p [ ] Upper Surface Lower Surface -0.5 WT FLUENT AeroFoam x/c [ ] C p [ ] Upper Surface Lower Surface -0.5 WT FLUENT AeroFoam x/c [ ]

22 RAE A wing + body () L =.928 m xw = m xo = m cr = m R(x) Ro = m Λ c/2 = 30 x ϕ c xb =0.760 m Rb=Ro/2 b = m 3D RAE A wing + body M = 0.9, α = 500K tetrahedral mesh (GAMBIT) comparison with FLUENT and experimental data (AGARD 38) single iteration CPUtime = 4.52 s ct = m

23 RAE A wing + body (2) ϕ = Upper Surface y/b = C p [ ] ϕ = 5 C p [ ] 0 Lower Surface -0.6 WT -0.8 FLUENT Body AeroFoam x/l [ ] -0.5 WT FLUENT Wing AeroFoam x/c [ ]

24 Pitching NACA 64A0 airfoil 0.5 2D NACA 64A0 airfoil M = 0.796, α = ±., k = ωc/2v = K triangular mesh (Gmsh) comparison with Flo3xx and experimental data (AGARD 702) single iteration CPUtime = 0.02 s Cycling to limit cycle C L [ ] WT Flo3xx AeroFoam α [ ]

25 Introduction Numerical scheme Test problems Work in progress Future work YF-7 fighter High AoA LEX vortex flow M = 0.28, α = K tetrahedral mesh (Gmsh) comparison with numerical and experimental data q/q

26 AGARD wing Aeroelastic transonic flutter boundary fluid-structure interaction (FSI) M = 0.678, 0.960,.40, α = 0 50K tetrahedral mesh (GAMBIT) comparison with numerical and experimental data (Langley TDT) Flutter index [ ] Experimental CFL3D EDGE FLUENT AeroFoam M = Mode n M [ ]

27 Future work Numerical scheme fully implicit time integration scheme (template for LDUmatrix class needed) runtime mesh deformation and ALE formulation (dynamicmesh library) parallelization Physical models thermodynamic model of real reacting gas mixture in thermo-chemical equilibrium (important for hypersonic flows) viscous numerical fluxes and turbulence models (e.g. Spalart-Allmaras) More 3D test problems (e.g. Piaggio P-80, Apollo reentry capsule)

28 References Numerical scheme M. Feistauer, J. Felcman and I. Straškraba. Mathematical and Computational Methods for Compressible Flow. Oxford University Press, 2003 R. J. LeVeque. Numerical Methods for Conservation Laws. Birkhäuser Verlag, 992 Test problems R. L. Bisplinghoff, H. Ashley and R. L. Halfman. Aeroelasticity. Dover, 996 Various Authors. Experimental Data Base for Computer Program Assessment. AGARD Advisory Report 38, 979 S. S. Davies and G. N. Malcolm. Experimental Unsteady Aerodynamics of Conventional and Supercritical Airfoils. NASA Tech. Memorandum 822, 980