Fluid Dynamics Simulation of Rayleigh-Taylor Instability Xinwei Li, - PowerPoint PPT Presentation

Fluid Dynamics Simulation of Rayleigh-Taylor Instability Xinwei Li, Xiaoyi Xie, Yu Guo 1

Rayleigh-Taylor Instability n Introduction q Instability of an interface between two fluids of different densities q Instability is initialized by pertubations n Cause q The dense fluid is pushed by the dilute fluid q Both fluids are subject to the gravity. The dense fluid is placed on top of the dilute fluid 2

Rayleigh-Taylor Instability n RT instability evident in Crab Nebula 3

Rayleigh-Taylor Instability n RT fingers 4

Euler Equations ∂ t + ∇⋅ ( ρ  ∂ ρ Conservation of Mass u ) = 0 ∂ ρ   ∂ t + ∇⋅ (  u ⊗ ( ρ  u Conservation of Momentum u )) + ∇ p = 0 ∂ t + ∇⋅ (  ∂ E u ( E + p )) = 0 Conservation of Energy  u = ( u , v , w ) E = ρ e + 1 2 ρ ( u 2 + v 2 + w 2 ) 5

Euler Equations Rewrite Euler equations in conservative form    ∂  f y ∂ x + ∂ f z w f x ∂ y + ∂ ∂ t + ∂ ∂ z = 0 " % " % " % " % ρ v ρ u ρ w ρ $ ' $ ' $ ' $ ' ρ uv p + ρ u 2 ρ uw ρ u $ ' $ ' $ '   $ '   $ ' $ ' $ ' p + ρ v 2 f y = ρ vw f z = $ ' f x = ρ uv m = ρ v $ ' $ ' $ ' $ ' p + ρ w 2 ρ vw ρ uw $ ' $ ' $ ' ρ w $ ' $ ' $ ' $ ' $ ' v ( E + p ) w ( E + p ) u ( E + p ) # & # & E # & # &

Simulation Result 7

PDE Solver n Introduction q Finite Difference Method q Finite Element Method q Finite Volume Method IAE University of Toulouse 2010-2011 8

PDE SolverScreen Shot 2012-12-19 at 6.58.14 PM n Introduction q Finite Difference Method q Finite Element Method q Finite Volume Method IAE University of Toulouse 2010-2011 9

Approximate Riemann Solver n HLL( Harten-Lax-Van Leer) riemann solver ∂ U ∂ t + ∂ F ∂ t + ∂ G ∂ t = 0 dU i , j = L ( U ) = − F i + 1/2, j − F − G i , j + 1/2 − G i , j − 1/2 i − 1/2, j dt Δ x Δ y 10

Flux Calculation F HLL = α + F L + α − F R − α + α − ( U R − U L ) α + + α − α ± = MAX {0, ± λ ± ( U L ), ± λ ± ( U R )} λ ± = υ ± c s Δ t < Δ x / MAX ( α ± ) c s = γ P / ρ 11

High Resolution Schemes n High-order in time ( Runge-Kutta) U (1) = U n + Δ tL ( U n ) U (2) = 3 4 U n + 1 4 U (1) + 1 4 Δ tL ( U (1) ) U n + 1 = 1 3 U n + 2 3 U (2) + 2 3 Δ tL ( U (2) ) High-order in space( PLM ) R c i + 1/2 = c i + 1 + 0.5minmod( θ ( c i + 1 − c i ),0.5( c i + 2 − c i ), θ ( c i + 2 − c i + 1 )) L c i + 1/2 = c i − 0.5minmod( θ ( c i − c i − 1 ),0.5( c i + 1 − c i − 1 ), θ ( c i + 1 − c i )) 12

Flux Limiter n Avoid the spurious oscillations in HRS minmod( x , y , z ) = 1 |sgn( x ) + sgn( y )|(sgn( x ) + sgn( z ))min(| x |,| y |,| z |) 4 13

Demo 14