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Skew-symmetric matrices and accurate simulations of compressible turbulent flow

Skew-symmetric matrices and accurate simulations of compressible turbulent flow. Wybe Rozema Johan Kok Roel Verstappen Arthur Veldman. A simple discretization. The d erivative is equal to the slope of the line. The problem of accuracy. exact. 2 nd order.

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Skew-symmetric matrices and accurate simulations of compressible turbulent flow

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  1. Skew-symmetric matrices and accurate simulations of compressible turbulent flow WybeRozema Johan Kok RoelVerstappen Arthur Veldman

  2. A simple discretization The derivative is equal to the slope of the line

  3. The problem of accuracy exact 2ndorder How to prevent small errors from summing to complete nonsense?

  4. Compressible flow shock wave turbulence acoustics Completely different things happen in air

  5. It’s about discrete conservation Skew-symmetric matrices Simulations ofturbulent flow &

  6. Governing equations convective transport viscous friction heat diffusion pressure forces Convective transport conserves a lot, but this does not end up in standard finite-volume method

  7. Conservation and inner products Inner product Physical quantities Square root variables mass internal energy density kinetic energy internal energy momentum kinetic energy Why does convective transport conserve so many inner products?

  8. Convective skew-symmetry Skew-symmetry Inner product evolution Convective terms +... = 0 +... Convective transport conserves many physical quantities because is skew-symmetric

  9. Conservative discretization Discrete skew-symmetry Computational grid Discrete inner product The discrete convective transport should correspond to a skew-symmetric operator

  10. Matrix notation Discrete conservation Matrix equation Discrete inner product The matrix should be skew-symmetric

  11. Is it more than explanation?  A conservative discretization can be rewritten to finite-volume form  Energy-conserving time integration requires square-root variables  Square-root variables live in L2

  12. Application in practice NLR ensolv • multi-block structured curvilinear grid • collocated 4th-orderskew-symmetric spatial discretization • explicit 4-stage RK time stepping Skew-symmetry gives control of numerical dissipation

  13. Delta wing simulations coarse grid and artificial dissipation outside test section transition test section Re = 5·104 27M cells M = 0.3  = 75° α = 25° α Preliminary simulations of the flow over a simplified triangular wing

  14. It’s all about the grid conical block structure fine grid near delta wing Making a grid is going from continuous to discrete

  15. The aerodynamics bl sucked into the vortex core suction peak in vortex core α The flow above the wing rolls up into a vortex core

  16. 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

  17. The final simulations preliminary Δx Δx = const. Δy = k x Δy final (isotropic) Δx = Δy y 23 weeks on 128 cores x

  18. The glass ceiling what to store? post-processing

  19. 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 wyberozema@gmail.com w.rozema@rug.nl

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