Report

Skew-symmetric matrices and accurate simulations of compressible turbulent flow Wybe Rozema Johan Kok Roel Verstappen Arthur Veldman 1 A simple discretization +1 −1 ℎ −1 +1 − −1 = + (ℎ2 ) 2ℎ +1 The derivative is equal to the slope of the line 2 The problem of accuracy −1 +1 How to prevent small errors from summing to complete nonsense? 3 Compressible flow shock wave turbulence acoustics Completely different things happen in air 4 It’s about discrete conservation Skew-symmetric matrices Simulations of turbulent flow 5 Governing equations + ∙ = 0 + ∙ ⊗ + = ∙ + ∙ + ∙ = ∙ ∙ − ∙ convective transport pressure forces viscous friction heat diffusion Convective transport conserves a lot, but this does not end up in standard finite-volume method 6 Conservation and inner products Square root variables density 2 kinetic energy internal energy Inner product , = ⅆ Physical quantities , mass , , internal energy 2 2 momentum , 2 kinetic energy Why does convective transport conserve so many inner products? 7 Convective skew-symmetry Convective terms Inner product evolution + = … , = , + , = − , − , () +... = 0 +... 1 1 = ∙ + ∙ 2 2 Skew-symmetry , = − , Convective transport conserves many physical quantities because () is skew-symmetric 8 Conservative discretization Computational grid Ω Discrete skew-symmetry () 1 = Ω 0 Discrete inner product , = Ω 1 −1 − 1 = Ω − 2 2 () ∙ 2 1 −2 0 − 1 +2 + 0 1 2 The discrete convective transport () should correspond to a skew-symmetric operator 9 Matrix notation Matrix equation Discrete conservation + = ⋯ , + , = Ω + Ω = (Ω) +Ω = 0 + = ⋯ Discrete inner product , = Ω The matrix Ω should be skew-symmetric (Ω) = −Ω 10 Is it more than explanation? A conservative discretization can be rewritten to finite-volume form Energy-conserving time integration requires squareroot variables Square-root variables live in L2 11 Application in practice NLR ensolv ∆ξ multi-block structured curvilinear grid collocated 4th-order skew-symmetric spatial discretization explicit 4-stage RK time stepping Skew-symmetry gives control of numerical dissipation 12 Delta wing simulations transition coarse grid and artificial dissipation outside test section test section Preliminary simulations of the flow over a simplified triangular wing 13 It’s all about the grid conical block structure fine grid near delta wing Making a grid is going from continuous to discrete 14 The aerodynamics bl sucked into the vortex core α suction peak in vortex core The flow above the wing rolls up into a vortex core 15 Flexibility on coarser grids skew-symmetric no artificial dissipation sixth-order artificial dissipation LES model dissipation (Vreman, 2004) Artificial or model dissipation is not necessary for stability 16 The final simulations preliminary Δx = const. Δy = k x Δy preliminary final M 0.3 0.3 75° 85° α 25° 12.5° 5 x 104 1.5 x 105 2.7 x 107 1.4 x 108 5 x 105 3.7 x 106 Δx Rec # cells final (isotropic) CHs Δx = Δy y x 23 weeks on 128 cores 17 The glass ceiling what to store? post-processing 18 Take-home messages The conservation properties of convective transport can be related to a skew-symmetry We are pushing the envelope with accurate delta wing simulations [email protected] [email protected] 19