Direct Numerical Simulation of Rayleigh-Taylor Instability [PDF]
Direct Numerical Simulation of Rayleigh-Taylor Instability [PDF]
Direct Numerical Simulation of Rayleigh-Taylor Instability [PDF]
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Lit. Search - Background• Stage 1– Linear growth– Amplitudes <strong>of</strong> 0.4λ• Stage 2– Nonlinear growth– Amplitudes <strong>of</strong> λ– Formation <strong>of</strong>bubbles and spikes– 3-D D effects– Atwood numberA =ρHρH− ρL+ ρSpikeLBubble6/22/2009 Prelimary Examination - Research Presentation p. 5
Lit. Search - BackgroundHydrodynamics simulation <strong>of</strong> the <strong>Rayleigh</strong>-<strong>Taylor</strong> 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 <strong>of</strong>the 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 <strong>of</strong>≡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 <strong>of</strong> RTI areforms <strong>of</strong> 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 <strong>of</strong> 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– <strong>Direct</strong> <strong>Numerical</strong> <strong>Simulation</strong> (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 <strong>Direct</strong> <strong>Numerical</strong><strong>Simulation</strong> <strong>of</strong> <strong>Rayleigh</strong>-<strong>Taylor</strong> <strong>Instability</strong>– Utilize the localized nature <strong>of</strong> wavelets toperform highly adaptive simulations whilestill resolving all scales involved– Explore the effects that the parameters,even in extreme limits, have on the growth<strong>of</strong> 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: setk<strong>of</strong> 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 <strong>of</strong> the following on thegrowth rate <strong>of</strong> 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