NASA Technical Reports Server (NTRS) 20150010386: Solutions of the Taylor-Green Vortex Problem Using High-Resolution Explicit Finite Difference Methods PDF
Preview NASA Technical Reports Server (NTRS) 20150010386: Solutions of the Taylor-Green Vortex Problem Using High-Resolution Explicit Finite Difference Methods
Solutions of the Taylor-Green Vortex Problem Using High-Resolution Explicit Finite Difference Methods! Jim DeBonis! Inlet and Nozzle Branch! NASA Glenn Research Center! Cleveland, OH 44135! [email protected]! Taylor-Green Vortex! • Simple benchmark case to study vortical flow, transition and turbulence! • Test case from 1st International Workshop on High- Order CFD Methods at 2012 AIAA Aerospace Science Meeting! • Time accurate! • Incompressible! • Flow Conditions! – Re = 1600! – M = 0.1! • Periodic domain! – Simple cartesian grids! – No complicated boundary conditions! Z-Vorticity Evolution! t*<3! t*=5! t*=7! inviscid! vortex roll-up! structure changes! t*=9! t*=11! t*>11! coherent breakdown! fully turbulent! turbulent decay! WaveResolvingLES! • Compressible Navier-Stokes equations! • Generalized curvilinear coordinates! • Temporal discretization! – Low Dispersion Runge-Kutta! • Spatial discretization! – Standard central differencing, 2nd - 12th order! – Dispersion relation preserving (DRP)! • Tam & Webb’s 7-point scheme! • Bogey & Bailley’s 9-, 11- and 13-point schemes! – Solution filtering for stability! • Kennedy & Carpenter filters, 2nd – 12th order! • DRP filters! – 4th–order viscous terms! • Sub-grid models! – Smagorinsky! – Dynamic Smagorinsky! Fourier Analysis of the Schemes! 1 BB13 1 KC04 KC08 0.8 KC12 0.8 0.6 c BB13 *c/ St04 x)(cid:1) 0.6 St08 (cid:2) 0.4 St12 D( 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 (cid:2)(cid:1)x (cid:2)(cid:1)x Error in phase speed for cent. diff.! Damping function for filters! Kinetic Energy Dissipation Rate (KEDR)! • Directly computed KEDR! 1 v ⋅ v dE ∫ E = ρ dΩ ε(E ) = − k k ρ Ω 2 k dt 0 Ω • Enstrophy based KEDR! 1 ω⋅ω µ ζ = ∫ ρ dΩ ε(ζ) = 2 ζ € ρ Ω 2 ρ 0 Ω 0 • Comparison to ref. solution! – 5123 grid, spectral method! – van Rees et al, v. 230, J. Comp. Phys., 2011! Computing! • 643 and 1283 cases! – Six core single processor desktop! – 1 grid block! – 6 OpenMP processes! • 2563 and 5123 cases! – NASA Pleiades system! – 2563 – 8 nodes! – 5123 – 46 nodes! – 8 processors/OpenMP processes per node! grid size! time step! machine! cores! wallclock time! 643! 3.385•10-3! desktop! 6! .5! 1283! 1.693•10-3! desktop! 6! 9! 2563! 8.463•10-4! Pleiades! 64! 40! 5123! 4.231•10-4! Pleiades! 368! 130! Baseline Case! • Numerical Scheme! – Temporal Discretization! • Carpenter and Kennedy’s 4-stage, 3rd-order! – Spatial Discretization! • Bogey & Bailly’s 13-point DRP scheme, BB13! – Filter! • Bogey & Bailly’s 13-point filter, BB13! • Filter coefficient halved until minimum stable value was found! • Min. stable coefficient, σ = 0.05! • Grids! – 643, 1283, 2563 and 5123! BB13 – 4 grid resolutions! 0.14 0.08 64 64 128 128 0.12 256 256 512 512 ref. soln. 0.06 ref. soln. 0.1 0.08 k k E E 0.04 0.06 0.04 0.02 0.02 0 0 0 5 10 15 20 10 12 14 16 18 20 t* t* Evolution of kinetic energy! Closep-up of! Evolution of kinetic energy! BB13 – 4 grid resolutions! 0.016 0.016 64 64 0.014 128 0.014 128 256 256 512 512 0.012 ref. soln. 0.012 ref. soln. 0.01 0.01 )k )(cid:1) E 0.008 (0.008 ((cid:1) (cid:2) 0.006 0.006 0.004 0.004 0.002 0.002 0 0 0 5 10 15 20 0 5 10 15 20 t* t* Directly computed KEDR! Enstrophy based KEDR!
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