Three Dimensional Multi-Mode Rayleigh-Taylor and Richtmyer-Meshkov - PowerPoint PPT Presentation

IWPCTM 2001 Three Dimensional Multi-Mode Rayleigh-Taylor and Richtmyer-Meshkov Instabilities at All Density Ratios D. Kartoon 1,2 , D. Oron 3 , L. Arazi 4 , A. Rikanati 1,2 , O. Sadot 1,5 , A. Yosef-Hai 1,5 , U. Alon 3 , G. Ben-Dor 5 and D. Shvarts 1,2,5 1. Dept. of Physics, Nuclear Research Center Negev, Israel 2. Dept. of Physics, Ben-Gurion University, Beer-Sheva, Israel 3. Fac. of Physics, The Weizmann Institute of Science, Rehovot, Israel 4. School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, Israel 5. Dept. of Mech. Eng., Ben-Gurion University, Beer-Sheva, Israel 1

Relevant Publications Oron et al. PoP 2001. Alon et al, PRL 1994, 1995. Hecht et al. PoF 1994. Oron et al. PoP 1998. Rikanati et al. PRE 1998. Sadot et al. PRL 1998. Layzer APJ 1955. Dimonte PoF, PoP 2000. 2

Shock-Tube Statistical experiments model Theoretical Drag-Buoyancy prediction model Full 3D simulations LEM experiments 3

Outline The Drag-Buoyancy Model for Bubbles 2D Statistical Model Full 3D Numerical Simulations 3D Statistical Model The Drag-Buoyancy Model for Spikes Conclusion 4

Single Mode Nonlinear Stage du ( ) ρ ⋅ + ⋅ ρ ⋅ ⋅ = ρ − ρ ⋅ − ρ ⋅ ⋅ 2 V C V V g S u ( ) l a h h l h dt (inertia) + (added mass) = (buoyancy) (kinematic drag) ρ h ρ Where V/S ∝λ bubble ρ l ρ spike C ρ + ρ = ρ − ρ ⋅ − ρ ⋅ C u g d u & 2 ( ) ( ) l a h h l λ h Where C a and C d are geometric constants. 5

Single Mode Asymptotic Velocities C ρ + ρ = ρ − ρ ⋅ − ρ ⋅ C u g u d & 2 ( ) ( ) l a h h l λ h   1 2 A = λ   RT: u g + B   C 1 A d − λ   1 1 A = ⋅ + ⋅   RM: u C + B a   C 1 A t d The geometric constants depend on the dimensionality: = = π 2 D 2 D C 2 , C 6 a d = = π 3 D 3 D C 1 , C 2 a d 6

RM Single Mode Experimental Results 3D 2D λ eff ≅ 28mm λ≅ 27mm 30 25 3D Model 20 3D Sim. h B [mm] 15 10 2D Model 5 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t [ms] 7

Multi Mode Drag Buoyancy Model Generalization of the drag-buoyancy equation, using the self-similarity assumption: ( ) h MM = ⋅ λ t b A t ( ) ( ) ( ) B h & = = ⋅ ⋅ = α ⋅ ⋅ 2 u h c A g A gt ( ) RT: b A 1 ( ) 1 θ h & = = ⋅ ⋅ = ⋅ u h c A c t RM: ( ) b A 2 ( ) t −   1 1 1 1 A 1 1 α = θ = + ⋅ ⋅   C + + RT a   RM 2 ( 1 A ) C b ( A ) 1 A C b ( A ) d d 8

3D Multi Mode the α θ b Relations Using the 3D coefficients in the expressions relating α , θ and b, and assuming that b (RM) =b (RT) , the differences between 2D and 3D bubble front growth are obtained: 2D 3D (C a =2, C d =6 π ) (C a =1, C d =2 π ) assumed α α 0.05 0.05 θ θ (=0.3-0.4) 0.2 0.1(A+3) 1 3 b (=0.25-0.5) (=0.75-1.5) + + A A 2 ( 1 ) 2 ( 1 ) 9

Outline The Drag-Buoyancy Model for Bubbles 2D Statistical Model Full 3D Numerical Simulations 3D Statistical Model The Drag-Buoyancy Model for Spikes 10

Simulation of 2D Multimode Perturbation Evolution 11

Multi Mode 2D Statistical Model (Alon et. Al.) Bubble Envelope (Simulation) Each bubble grows with its asymptotic velocity, according to its wavelength: ( ) = λ (asy) u u , t i i 2D Rising Bubble 12

Multi Mode Bubble Merger Velocity evolution of two non- t i+1 λ + λ identical adjacent bubbles: 1 2 t i λ λ 2 1 Merger Rate: ( ) λ λ , + i i 1 ( ) τ ↓ ω λ λ t ~ 1 , , + merge i i 1 ( ) λ + λ + i i 1 13

Multi Mode 2D Statistical Model ∞ ∂ λ g t ( , ) ( ) ( ) ( ) ( ) ∫ ′ ′ ′ ⋅ = − λ λ ω λ λ λ N t g t g t d death 2 , , , ∂ t 0 ∞ ( ) ( ) ( ) ∫ ′ ′ ′ ′ ′ + λ − λ λ ω λ − λ λ λ g t g t d birth , , , 0 Where g( λ ,t) is the number of bubbles with wavelength λ within interval d λ at time t, and N(t) is the total number of bubbles: ∂ N t ( ) = − ω g ⋅ N t ( ) ∂ t 14

Multi Mode 2D Statistical Model Results 1. The λ distribution reaches an asymptotic function: RT simulation, A=0.5 15

Multi Mode 2D Statistical Model Results 2. The average bubble and spike heights are obtained for both the RT and the RM case: − & ∝ λ      → ∝ → = α ⋅ self similarity RT: asy 2 u g h gh h gt B B B B λ h − & ∝      → ∝ → = ⋅ θ self similarity RM: asy B u h h c t B B B B t t h = = RT RM B b b Self - similarity parameter: λ 2D Statistical Model Results: α ≅ θ ≅ ≅ 0 . 05 0 . 4 b 0.2 16 B B

LEM Experimental Results 1 0 . 2 0 . 9 0 . 1 8 0 . 8 0 . 1 6 0 . 7 0 . 1 4 2 D M o d e l 0 . 6 0 . 1 2 2 D M o d e l 0 . 5 0 . 1 α θ 0 . 4 0 . 0 8 0 . 3 0 . 0 6 0 . 2 0 . 0 4 Exps Exps 0 . 1 0 . 0 2 0 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 A A θ = ⋅ = α ⋅ 2 h a t RT: h Agt RM: B / S 2 B / S 0 B / S B / S 1 . 8 1 . 6 1 . 4 Exps Drag-Buoyancy Model 1 . 2 h B /< λ > Effective 1-Mode 1 0 . 8 2D Statistical Model 0 . 6 2 D M o d e l 0 . 4 0 . 2 17 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 A

Outline The Drag-Buoyancy Model for Bubbles 2D Statistical Model Full 3D Numerical Simulations 3D Statistical Model The Drag-Buoyancy Model for Spikes 18

Full 3D Numerical Simulation of the RT A=0.5 Case <h> α S =0.076 α B =0.046 19

Voronoi Cell Structure of the Bubble Front Demonstrates the 3D Bubble Merger t=0.28 t=1.4 n=260 n=112 t=2.2 n=30 20

3D Simulation Results 2D Simulation 3D Simulation slice 21

Outline The Drag-Buoyancy Model for Bubbles 2D Statistical Model Full 3D Numerical Simulations 3D Statistical Model The Drag-Buoyancy Model for Spikes 22

Multi Mode 3D Statistical Model Dimensionality effects on the statistical model: Asymptotic velocity of each bubble 1.5-2 times higher than in the 2D case: u 3D =1.5-2u 2D Average number of neighbors per bubbles ≈ 6 in 3D, rather than 2 in 2D: No. of Neighbors Distribution 23

Multi Mode 3D Statistical Model Dimensionality effects on the bubble merger: Bubble merging in 3D conserves area, rather than length in 2D: λ + λ λ 1 λ 2 2 2 1 2 t i t i+1 The merges occur with rate ω ( λ 1 , λ 2 ). At first step ω 3D was ( ) ⋅ ω taken to be equal to asy asy 2 D u u 3 D 2 D Because of the area conservation, a 3D bubble has to merge with more of its neighbors in order to reach the same λ λ . This effect reduces d λ λ /dt, which in turn reduces both α α and θ θ (and increases b). 24

Multi Mode 3D Statistical Model Results The segment distribution g`(d) is obtained from the simulation: d λ The relation between g`(d) and g( λ ) is given by: d ( ) ( ) ( ) ∫ = − λ λ λ g d g d g d ` 2 2 2 2 0 25

Multi Mode 3D Statistical Model Results 3D Statistical model results agree well with 3D Simulations. α = θ = = 0 . 055 0 . 18 b 0.67 B B The 3D wavelength distribution is narrower than the 2D distribution. The narrowing of the λ distribution is due to: a) Reduction of d λ /dt. 1 . 6 b) Increased number of neighbors. 1 . 4 1 . 2 A=0.9 1 A=0.5 f( ξ ) 0 . 8 A=0.2 Simulations results indicate that the 0 . 6 3D statistical model may be 0 . 4 applicable to a wide range of A: 0 . 2 26 0 0 0 . 5 1 1 . 5 2 2 . 5 3 ξ =l/<l>

3D Statistical Model Dependence on Initial Distribution Using the initial wavelength distribution derived from the voronoi diagram in the statistical model gives the α dependence on the generation number: α Β σ 0 =0.2 σ α asy =0.056 α σ 0 =0.1 σ 27

LEM Experimental Results Vs. 3D Models and Simulations 0 . 2 E x p e r i m e n t a l R e s u l t s ( L E M ) α Β 3 D D r a g - B o u y a n c y M o d e l 0 . 1 8 S i m u l a t i o n R e s u l t s 0 . 1 6 0 . 1 4 h Β /< λ > 0 . 1 2 0 . 1 2 0 . 0 8 1 . 8 0 . 0 6 1 . 6 0 . 0 4 2D+3D Model 1 . 4 0 . 0 2 1 . 2 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 1 3D Model E x p e r i m e n t a l R e s u l t s ( L E M ) 0 . 8 3 D D r a g - B o u y a n c y M o d e l 0 . 9 θ Β 2 D D r a g - B o u y a n c y M o d e l 0 . 6 S i m u l a t i o n R e s u l t s 0 . 8 E x p e r i m e n t a l R e s u l t s ( L E M ) 3 D D r a g - B o u y a n c y M o d e l 2D Model 0 . 4 0 . 7 2 D D r a g - B o u y a n c y M o d e l S i m u l a t i o n R e s u l t s - 3 D R M 0 . 2 S i m u l a t i o n R e s u l t s - 3 D R T 0 . 6 0 0 . 5 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 2D Model 0 . 4 A 0 . 3 0 . 2 3D Model 0 . 1 0 28 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 A

LEM Experimental Results Vs. 3D Simulations Alpha - Spike Theta - Spike α S θ S A A 29