HYPERSONIC AERO-THERMO-DYNAMIC HEATING PREDICTION WITH HIGH-ORDER DISCONTINOUS GALERKIN SPECTRAL ELEMENT METHODS

Transcription

1 1 / 36 HYPERSONIC AERO-THERMO-DYNAMIC HEATING PREDICTION WITH HIGH-ORDER DISCONTINOUS GALERKIN SPECTRAL ELEMENT METHODS Jesús Garicano Mena, E. Valero Sánchez, G. Rubio Calzado, E. Ferrer Vaccarezza Universidad Politécnica de Madrid jesus.garicano.mena@upm.es III/04/2015

2 2 / 36 Outline 1 Introduction 2 Goal of this presentation 3 Shocked flows Perfect Gas flow over Adiabatic Flat Plate Inviscid Perfect Gas flow around Cylinder 4 Conclusions and Future Work

3 3 / 36 Outline Introduction 1 Introduction 2 Goal of this presentation 3 Shocked flows Perfect Gas flow over Adiabatic Flat Plate Inviscid Perfect Gas flow around Cylinder 4 Conclusions and Future Work

4 4 / 36 Objective Introduction Can Discontinuous Galerkin Spectral Element Methods predict accurately hypersonic stagnation heating?

5 5 / 36 Objective Introduction Can Discontinuous Galerkin Spectral Element Methods predict accurately hypersonic stagnation heating?

6 Introduction Stagnation heating 6 / 36 The stagnation heating problem Requirements Robust Conservative Monotonicity respecting Accurate (boundary layer) V > 5a

7 Introduction Stagnation heating 7 / 36 Governing Equations Conservation form Conserved variables U t + F c = F d U = [ ρ, ρu j, ρe ] t Continuous regime Kn 1 Perfect Gas model F c = Convective flux tensor ρ u j ρ u i u j + pδ i,j ρ H u j = F c x j 1 t x j F d = Diffusive flux tensor 0 τ i,j τ i,j u j q i = F d x j 1 t x j

8 8 / 36 Introduction Discontinuous Galerkin Spectral Element Methods Discontinuous Galerkin Spectral Element Method [1] Galerkin: weak formulation ( U ) t, ϕ + ( F c F d ), ϕ = 0, ϕ V }{{} F L 2 projection (u, v) = Ω u v dω [1] Implementing Spectral Methods for Partial Differential Equations, D.A. Kopriva, 2009.

9 Introduction Discontinuous Galerkin Spectral Element Methods 9 / 36 Discontinuous Galerkin Spectral Element Method Domain Ω and a possible tesselation Ω h. n Elem Ω Ω h = Ω k Ω i Ω j = if i j. k=1 Physical and computational domains: iso-parametric mapping

10 Introduction Discontinuous Galerkin Spectral Element Methods Discontinuous Galerkin Spectral Element Method Lagrange polynomial expansion - 1D n U (x, t) U DoF N (x, t) = U J (t) l J (x), l α(x β ) = δ α,β J=1 Spectral magic Choose as nodes the zeros of orthogonal polynomials 1.0 Approximation is optimal Exact quadrature for polynomials up to degree 2N T 3 T 4 T L 2 L 3 L 4 L 5 T k (x) 0 L k (x) T x Chebyshev polynomials x Legendre Polynomials 10 / 36

11 11 / 36 Introduction Discontinuous Galerkin Spectral Element Methods Discontinuous Galerkin Spectral Element Method Discrete form ( U ) ( t, ϕn + F, ) ϕ N = 0, ϕ N V h 0 Ωk Approximate solution and fluxes U U N N (ξ, η) = U µ,νφ N µ,ν ; µ,ν=0 N F F N (ξ, η) = µ,ν=0 F N µ,νφ µ,ν Basis functions 2D - tensor product φ µ,ν = l µ (ξ)l ν (η) P N P N

12 12 / 36 Introduction Discontinuous Galerkin Spectral Element Methods Discontinuous Galerkin Spectral Element Method ( ϕ N = φ N = Test function N µ,ν=0 ϕ N µ,νφ µ,ν Linearity of projection operator U ) N ), φ i,j + ( F N, φ i,j = 0, i, j = 0,..., N t

13 13 / 36 Introduction Discontinuous Galerkin Spectral Element Methods Discontinuous Galerkin Spectral Element Method ( Departing point U ) N ), φ i,j + ( F N, φ i,j = 0, i, j = 0,..., N t ( Integration by parts U ) N ), φ i,j + F N φ i,j 1 ext dσ ( F N, φ i,j = 0, i, j = 0,..., N t δω k Integration Numerical Quadrature 1 1 v(ξ, η)dξdη = 0 0 n,m=0 N v(ξ n, η m)w nw m v P 2N+1,2N+1

14 Introduction Discontinuous Galerkin Spectral Element Methods Discontinuous Galerkin Spectral Element Method ( Time derivative term U ) N d, φ i,j = w i w j U t dt i,j N Space operator - edge (surface) contribution F N φ i,j 1 ext dσ = F N (1, η j )φ i,j (1, η j )w j F N (0, η j )φ i,j (0, η j )w j δω k + F N (ξ i, 1)φ i,j (ξ i, 1)w i F N (ξ i, 0)φ i,j (ξ i, 0)w i Space operator - area (volume) contribution ) ( F N, φ i,j = N i,j=0 F N i,j φ i,j w i w j 14 / 36

15 15 / 36 Introduction Discontinuous Galerkin Spectral Element Methods Discontinuous Galerkin Spectral Element Method d w i w j U dt i,j N + δω k Assembled discrete ode F N, φ i,j 1 ext dσ N i,j=0 w i w j F N i,j φ i,j = 0, i, j = 1,..., N; k = 1,..., n Elem R N ( U) = Discrete partial differential operator δω k F 1 ext φ i,j d σ }{{} Information propagation N i,j=0 w i w j F N i,j φ i,j i, j = 0,..., N

16 Introduction Discontinuous Galerkin Spectral Element Methods Discontinuous Galerkin Spectral Element Method d w i w j U dt i,j N + δω k Assembled discrete ode F N, φ i,j 1 ext dσ N i,j=0 w i w j F N i,j φ i,j = 0, i, j = 1,..., N; k = 1,..., n Elem Boundary Conditions [1] Weakly enforced, exploiting characteristic information [1] Implementing Spectral Methods for Partial Differential Equations, D.A. Kopriva, / 36

17 17 / 36 Outline Goal of this presentation 1 Introduction 2 Goal of this presentation 3 Shocked flows Perfect Gas flow over Adiabatic Flat Plate Inviscid Perfect Gas flow around Cylinder 4 Conclusions and Future Work

18 18 / 36 Goal of this presentation Goal of this presentation Evaluate DGSEM performance for Perfect Gas (PG) shocked flows Two configurations: Flat plate Straight cylinder

19 19 / 36 Goal of this presentation Handling shock waves Shock capturing term (Euler) [1] U t + F c = F d + F ad F ad = ε µ F d [1] Sub-cell Shock Capturing for DG Methods, P.-O. Persson and J. Peraire, AIAA

20 Goal of this presentation Handling shock waves Shock capturing term (Euler) U t + F c = F d + F ad F ad = ε µ F d Artificial viscosity 0 if ( s e s 0 κ ) ε ε = sin( π(se s 0) ) 2κ ε 0 if s 0 + κ s e ε 0 h N, s 0 if s 0 κ s e s 0 + κ 1, κ empirically chosen N4 s e = lg S e, with shock sensor S e = ( p N p N 1, p N p N 1) (p N, p N ) 20 / 36

21 21 / 36 Outline Shocked flows 1 Introduction 2 Goal of this presentation 3 Shocked flows Perfect Gas flow over Adiabatic Flat Plate Inviscid Perfect Gas flow around Cylinder 4 Conclusions and Future Work

22 22 / 36 Flat plate in PG flow Shocked flows Perfect Gas flow over Adiabatic Flat Plate Ma Re Ld Flat plate. Definition of the domain. Surface y = 0, x (0, L d ): no-slip adiabatic wall Surface x = L d is a supersonic outlet Surfaces x = L u and y = H: supersonic inlets, with the upstream state U Surface y = 0, x ( L u, 0): slip wall

23 23 / 36 Shocked flows Perfect Gas flow over Adiabatic Flat Plate Viscous Flow over Flat Plate - Ma = 2.0, Re = 10 4 N = 2 N = 3 Non-dimensional pressure field. O2 vs O3.

24 24 / 36 Shocked flows Perfect Gas flow over Adiabatic Flat Plate Viscous Flow over Flat Plate - Ma = 2.0, Re = 10 4 N = 2 N = 3 Non-dimensional temperature field. O2 vs O3.

25 25 / 36 Shocked flows Perfect Gas flow over Adiabatic Flat Plate Viscous Flow over Flat Plate - Ma = 2.0, Re = 10 4 N = 2 N = 3 Ma field. O2 vs O3.

26 Shocked flows Perfect Gas flow over Adiabatic Flat Plate Viscous Flow over Flat Plate - Ma = 2.0, Re = 10 4 Wall quantities - C p. Reference value from [1]. [1] A second-order weak interaction expansion for moderately hypersonic flow past a flat plate, Ko and Kubota, AIAAJ, Vol. 5, No. 10 (1967). 26 / 36

27 Shocked flows Perfect Gas flow over Adiabatic Flat Plate Viscous Flow over Flat Plate - Ma = 2.0, Re = 10 4 Wall quantities - C f. Reference value from [1]. [1] A second-order weak interaction expansion for moderately hypersonic flow past a flat plate, Ko and Kubota, AIAAJ, Vol. 5, No. 10 (1967). 27 / 36

28 28 / 36 Shocked flows Inviscid Perfect Gas flow around Cylinder Inviscid Flow around Cylinder - Ma = 1.2 N = 3 Non-dimensional pressure field.

29 29 / 36 Shocked flows Inviscid Perfect Gas flow around Cylinder Inviscid Flow around Cylinder - Ma = 1.2 N = 3 Non-dimensional temperature field.

30 30 / 36 Shocked flows Inviscid Perfect Gas flow around Cylinder Inviscid Flow around Cylinder - Ma = 1.2 N = 3 Ma field.

31 31 / 36 Outline Conclusions and Future Work 1 Introduction 2 Goal of this presentation 3 Shocked flows Perfect Gas flow over Adiabatic Flat Plate Inviscid Perfect Gas flow around Cylinder 4 Conclusions and Future Work

32 32 / 36 Conclusions and Future Work Conclusions and Future Work Can Discontinuous Galerkin Spectral Element Methods predict accurately hypersonic stagnation heating? Not yet...

33 Conclusions and Future Work 33 / 36 Conclusions and Future Work Can Discontinuous Galerkin Spectral Element Methods predict accurately hypersonic stagnation heating? Not yet... but now shortcomings have been identified cures are known (from RDS, FV, SUPG)

34 34 / 36 Conclusions and Future Work Conclusions and Future Work Can Discontinuous Galerkin Spectral Element Methods predict accurately hypersonic stagnation heating? To do list Improve robustness Turn implicit [1,2] Shock handling Explictly enthalpy-preserving [3] PDE-based artificial viscosity [4] [1] Jacobian-free Newton-Krylov methods, Knoll and Keyes, JCP, Vol.193, [2] Ph.D. dissertation, Universidad Politécnica de Madrid (in preparation), C. Redondo Calle. [3] Ph.D. dissertation, University of Texas at Austin, Kirk, [4] Shock capturing with PDE-based artificial viscosity for DGFEM, Barter and Darmofal, JCP, Vol.225, 2010.

35 35 / 36 Acknowledgements Conclusions and Future Work This research has been funded by Universidad Politécnica de Madrid

36 Conclusions and Future Work 36 / 36 Thanks for your attention Questions?