Direct Numerical Simulation of Rayleigh-Taylor Instability [PDF]

Lit. Search - Background• Stage 1– Linear growth– Amplitudes of 0.4λ• Stage 2– Nonlinear growth– Amplitudes of λ– Formation ofbubbles and spikes– 3-D D effects– Atwood numberA =ρHρH− ρL+ ρSpikeLBubble6/22/2009 Prelimary Examination - Research Presentation p. 5

Lit. Search - BackgroundHydrodynamics simulation of the Rayleigh-Taylor instability.(Li, Shengtai and Hui Li. Parallel AMR Code for Compressible MHD or HDEquations. Los Alamos National Laboratory.)• Stage 3– Nonlinear interactionamong perturbations– Mushrooming from Kelvin-Helmholtz instability– Bubble amalgamation• Stage 4– Spike breakup– Bubble penetration– Leading to turbulentmixing6/22/2009 Prelimary Examination - Research Presentation p. 6

Lit. Search - Background• As light fluid penetrates, the amplitude ofthe bubbles is described as:• With terminal velocity:ρh− ρlgDbVb= Frρ 2• So,h = α Agt bαbb2Fr= 8Dhhbb2ρh+ ρlρhA ≡ Atwood number (density ratio)bubble diameter6/22/2009 Prelimary Examination - Research Presentation p. 7hρDh,lb≡ bubble amplitudeα ≡ bubble acceleration constantVbbb≡ bubble terminal velocityFr ≡ Froude number≡ density of≡heavy/light fluid

Lit. Search - Current Codes• Key details for simulations:– Single-mode or multi-mode mode perturbation– 2-D D or 3-D3– Compressible or incompressible– Miscible or immiscible6/22/2009 Prelimary Examination - Research Presentation p. 8

Lit. Search - Current CodesInstitutionAWEU. ChicagoLLNLLLNLTexas A&MLLNLSandia NL• Most current simulations of RTI areforms of MILES (monotone-integratedlarge-eddy eddy simulation)FLASHCodeTURMOIL3DWP/PPMNAV/STKRTI-3DHYDRAALEGRAMethodEulerianPPMPPMNSEulerianALEALECompressibleYesYesYesYesNoYesYesPerturbationDensityVelocityDensityDensityDensityDistorted meshDensityAdaptiveMeshAWE: Atomic Weapons Establishment; LLNL: Los Alamos National Laboratory; PPM: Piecewise ParabolicMethod; NS: Navier-Stokes; ALE: Arbitrary Lagrangian Eulerian6/22/2009 Prelimary Examination - Research Presentation p. 9NoYesNoNoNoNoNoInterfaceReconstructionNoNoNoNoNoYesYes

Lit. Search - Current Codes• Experimental value α b is 0.057• Evolution of is examined: Steadyvalue is found to be 0.025 (40% less)ExperimentalValue6/22/2009 Prelimary Examination - Research Presentation p. 10

Lit. Search - Current Codes• Approaches– MILES approach– Direct Numerical Simulation (DNS) forincompressible and weakly compressible case• Issues– Broadband spectrum– Resolution issues for large Atwood numbers• No DNS for compressible case to date!6/22/2009 Prelimary Examination - Research Presentation p. 11

Research ProjectProblem Statement• Wavelet-Based Direct NumericalSimulation of Rayleigh-Taylor Instability– Utilize the localized nature of wavelets toperform highly adaptive simulations whilestill resolving all scales involved– Explore the effects that the parameters,even in extreme limits, have on the growthof the instability6/22/2009 Prelimary Examination - Research Presentation p. 12

Wavelets• Dyadic Grid:Each valuationpoint isassigned aresolutionlevelj + 1x = j kx2kxj+12k+ 16/22/2009 Prelimary Examination - Research Presentation p. 13

Wavelets• Wavelet Decompositionf( x)=∞ ∑∑j=0 k∈Κjdjkψjk( x)j− resolution levelk − location– ψ j(x) -wavelet interpolating functions: setkof basis functionsj– -wavelet coefficientsd k6/22/2009 Prelimary Examination - Research Presentation p. 14

DAWC MethodWaveletcoefficientsRepresentationat j-level• Forward WaveletTransformdcjkjk1 ⎛= ⎜c2 ⎝= cj+12kj+12k+ 1∑∑jk + l• Inverse WaveletTransform+l−w~ljk , ldwjk , lcj+12k+ 2lInterpolatingweightscc⎞⎟⎠j+12kj+12k+ 1= c=6/22/2009 Prelimary Examination - Research Presentation p. 15jk−2djk∑l+w~∑lj + 1x = j kx2kjk,ldwjk + ljk , lcj+12k+ 2lxj+12k+ 1

DAWC Methodf( x)=∞∑∑j= 0 k∈Κjdψjkjk( x)6/22/2009 Prelimary Examination - Research Presentation p. 16

DAWC Method - Resultsf( x,t = t0)⇒Adaptive Non-Grid AdaptiveGriddjk≥εf(x,t)=−tanh⎛⎜⎝x−x20−6/22/2009 Prelimary Examination - Research Presentation p. 17vt⎞⎟⎠+e3x0= −2( − 64 ( x + x )20 + vt )v = 1υ −2υ = 1012

DAWC Method - Results6/22/2009 Prelimary Examination - Research Presentation p. 18

DAWC Method• Spatial derivatives6/22/2009 Prelimary Examination - Research Presentation p. 19

DAWC Method - Results• Use DAWC Method to solve Burgersequation.• PDE:∂u∂t+u∂u∂x2∂ u= ν2∂x,x ∈(−1,1),t>0• IC and BCs:u( x,0)= −sin(πx),u(± 1, t)=0• Solved for:ν = 1 and 0 ≤ t ≤ 2 /π6/22/2009 Prelimary Examination - Research Presentation p. 20

DAWC Method - Results6/22/2009 Prelimary Examination - Research Presentation p. 21

Future Work• Use code with 2-D 2 D and 3-D 3 D capabilities tosimulate RTI for compressible misciblefluids– Take equations and nondimensionalize– Set ICs and apply perturbation– Explore the influence of the following on thegrowth rate of the instability• Compressibility• Sc (viscosity/mass diffusivity)• Stratification (Fr=inertial/gravitational)• Initial conditions6/22/2009 Prelimary Examination - Research Presentation p. 22

Questions?Picture taken for Flow Visualization course at CU by Laurel Swift.6/22/2009 Prelimary Examination - Research Presentation p. 23

Blooper6/22/2009 Prelimary Examination - Research Presentation p. 25