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NASA Technical Reports Server (NTRS) 20150010386: Solutions of the Taylor-Green Vortex Problem Using High-Resolution Explicit Finite Difference Methods PDF

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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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