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

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Three Dimensional Multi-Mode Rayleigh-Taylor and Richtmyer-Meshkov Instabilities at All Density Ratios

• D. Kartoon1,2, D. Oron3, L. Arazi4, A. Rikanati1,2,
• O. Sadot1,5, A. Yosef-Hai1,5, U. Alon3, G. Ben-Dor5

and D. Shvarts1,2,5

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

IWPCTM 2001

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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.

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

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Outline

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

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Single Mode Nonlinear Stage

( )

2

) ( u S g V dt du V C V

h l h h a l

⋅ ⋅ − ⋅ − = ⋅ ⋅ ⋅ + ⋅ ρ ρ ρ ρ ρ

2

) ( ) ( u C g u C

h d l h h a l

⋅ − ⋅ − = + ρ λ ρ ρ ρ ρ &

Where Ca and Cd are geometric constants.

spike bubble

ρ ρh ρ ρl

(inertia) + (added mass) = (buoyancy) (kinematic drag) Where V/S ∝λ

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Single Mode Asymptotic Velocities

λ       + = g A 1 A 2 C 1 u

d B

t C A 1 A 1 C 1 u

a d B

λ ⋅       + + − ⋅ =

RT: RM: The geometric constants depend on the dimensionality:

π = = π = = 2 C , 1 C 6 C , 2 C

D 3 d D 3 a D 2 d D 2 a

2

) ( ) ( u C g u C

h d l h h a l

⋅ − ⋅ − = + ρ λ ρ ρ ρ ρ &

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RM Single Mode Experimental Results

3D λ≅27mm 2D λeff≅28mm

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 5 10 15 20 25 30

t [ms] hB [mm]

3D Sim. 3D Model 2D Model

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Multi Mode Drag Buoyancy Model

Generalization of the drag-buoyancy equation, using the self-similarity assumption:

( )

) ( ) ( ) ( t A b t h MM

B

λ ⋅ =

2 1

) ( ) ( gt A A b h g A c h u ⋅ ⋅ = ⋅ ⋅ = = α & ) A ( b 1 C 1 ) A 1 ( 2 1

RT d

+ = α ) A ( b 1 C 1 C A 1 A 1

RM d a

⋅ ⋅       + + − = θ

θ

t c t A b h A c h u ⋅ = ⋅ ⋅ = = 1 ) ( ) (

2

&

RT: RM:

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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:

(=0.75-1.5) (=0.25-0.5)

b

0.2 0.1(A+3)

(=0.3-0.4)

θ θ

0.05 0.05

α α 3D

(Ca=1, Cd=2π)

2D

(Ca=2, Cd=6π)

) 1 ( 2 1 + A ) 1 ( 2 3 + A

assumed

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Outline

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

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Simulation of 2D Multimode Perturbation Evolution

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Rising Bubble

Bubble Envelope (Simulation)

Multi Mode 2D Statistical Model (Alon et. Al.)

Each bubble grows with its asymptotic velocity, according to its wavelength:

( )

t , u u

(asy) 2D i i

λ =

13

t i t i+1

1

λ

2

λ

2 1

λ + λ

Multi Mode Bubble Merger

Velocity evolution of two non- identical adjacent bubbles:

( ) ( ) ( )

1 1 1

, , ,

+ + +

+ ↓

i i i i merge i i

t

λ λ λ λ ω λ λ τ 1 ~

Merger Rate:

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Multi Mode 2D Statistical Model

( ) ( ) ( ) ( ) ( ) ( ) ( )

∫ ∫

∞ ∞

′ ′ ′ − ′ ′ − + ′ ′ ′ − = ⋅ , , , , , , 2 ) , (

λ λ λ λ ω λ λ λ λ λ λ ω λ λ ∂ λ ∂

d t g t g d t g t g t t g t N death birth Where g(λ,t) is the number of bubbles with wavelength λ within interval dλ at time t, and N(t) is the total number of bubbles:

) ( ) ( t N t t N

g ⋅

− = ∂ ∂ ω

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Multi Mode 2D Statistical Model Results

• 1. The λ distribution reaches an asymptotic function:

RT simulation, A=0.5

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Multi Mode 2D Statistical Model Results

• 2. The average bubble and spike heights are obtained for both

the RT and the RM case:

0.2 b 4 . 05 .

B B

≅ ≅ θ ≅ α

RT: RM:

2 B B B asy B

gt h gh h g u

similarity self

⋅ α = → ∝     →  λ ∝

−

&

B

t c h t h h t u

B B B asy B

similarity self

θ

⋅ = → ∝     →  λ ∝

−

& 2D Statistical Model Results:

λ = =

B RM RT

h b b

similarity parameter:

• Self

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0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1

A θ

2 D M o d e l Exps

0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 . 1 2 0 . 1 4 0 . 1 6 0 . 1 8 0 . 2

A α

Exps 2 D M o d e l

0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2

A

h B/<λ >

Exps 2 D M o d e l

LEM Experimental Results

RM:

S / B

t a h

S / B θ

⋅ = RT:

2 S / B S / B

Agt h ⋅ α =

Drag-Buoyancy Model Effective 1-Mode 2D Statistical Model

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Outline

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

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<h>

αS=0.076 αB=0.046

Full 3D Numerical Simulation of the RT A=0.5 Case

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Voronoi Cell Structure of the Bubble Front Demonstrates the 3D Bubble Merger

t=0.28 n=260 t=1.4 n=112 t=2.2 n=30

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3D Simulation Results

2D Simulation 3D Simulation slice

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Outline

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

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Multi Mode 3D Statistical Model

Asymptotic velocity of each bubble 1.5-2 times higher than in the 2D case: Average number of neighbors per bubbles≈6 in 3D, rather than 2 in 2D: Dimensionality effects on the statistical model:

• No. of Neighbors Distribution

u3D=1.5-2u2D

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Bubble merging in 3D conserves area, rather than length in 2D: ti ti+1

2 2 2 1

λ λ +

λ2 λ1

Multi Mode 3D Statistical Model

Dimensionality effects on the bubble merger: The merges occur with rate ω(λ1, λ2). At first step ω3D was taken to be equal to 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 dλ λ/dt, which in turn reduces both α α and θ θ (and increases b).

( )

D 2 asy D 2 asy D 3

u u ω ⋅

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Multi Mode 3D Statistical Model Results

The segment distribution g`(d) is

• btained from the simulation:

d λ

The relation between g`(d) and g(λ) is given by:

( ) ( ) ( )

∫

− =

d

d g d g d g 2 2 2 2 ` λ λ λ

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Multi Mode 3D Statistical Model Results

0 . 5 1 1 . 5 2 2 . 5 3 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6

ξ=l/<l>

f(ξ)

A=0.9 A=0.5 A=0.2

3D Statistical model results agree well with 3D Simulations. The 3D wavelength distribution is narrower than the 2D

• distribution. The narrowing of the λ distribution is due to:

a) Reduction of dλ/dt. b) Increased number of neighbors. Simulations results indicate that the 3D statistical model may be applicable to a wide range of A:

0.67 b 18 . 055 .

B B

= = θ = α

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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:

σ σ0=0.2 σ σ0=0.1 α αasy=0.056

αΒ

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0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 . 1 2 0 . 1 4 0 . 1 6 0 . 1 8 0 . 2 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 S i m u l a t i o n R e s u l t s 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 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 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 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2 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 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 S i m u l a t i o n R e s u l t s - 3 D R T

LEM Experimental Results Vs. 3D Models and Simulations

A αΒ θΒ A hΒ/<λ>

2D+3D Model 2D Model 2D Model 3D Model 3D Model

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LEM Experimental Results Vs. 3D Simulations

αS θS A A Alpha - Spike Theta - Spike

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0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 . 1 2 0 . 1 4 0 . 1 6 0 . 1 8 0 . 2 E x p e r i m e n t a l R e s u l t s ( L E M ) S i m u l a t i o n R e s u l t s : 0 / 1 0 0 S i m u l a t i o n R e s u l t s : 5 / 9 5 S i m u l a t i o n R e s u l t s : 1 0 / 9 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 E x p e r i m e n t a l R e s u l t s ( L E M ) S i m u l a t i o n R e s u l t s : 0 / 1 0 0 S i m u l a t i o n R e s u l t s : 5 / 9 5 S i m u l a t i o n R e s u l t s : 1 0 / 9 0

Sensitivity of θS to percentage criterion

Alpha - Spike Theta - Spike αS θS A A

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θB is not sensitive to percentage criterion

0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 E x p e r i m e n t a l R e s u l t s ( L E M ) 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 : 0 / 1 0 0 S i m u l a t i o n R e s u l t s : 5 / 9 5 S i m u l a t i o n R e s u l t s : 1 0 / 9 0

θB A

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Outline

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

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Drag-Buoyancy Model for the Spike Front -

• I. Single Mode

Since the spikes develop a rounded tip, one can apply the drag-buoyancy equation to them:

2 S l d l h S l a h

u C g ) ( u ) C ( ⋅ ρ λ − ⋅ ρ − ρ = ρ + ρ &

The spikes velocity is obtained using the assumptions:

   = =    π π = = 3D 1 2D 2 ) A ( C ) A ( C 3D 2 2D 6 ) A ( C ) A ( C

aB aS dB dS

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Drag-Buoyancy Model for the Spike Front -

• I. Single Mode Shock Tube Experimental Results

t C Uasy λ ⋅ =

2 4 6 8 10 12

• 12
• 10
• 8
• 6
• 4
• 2

2 4

bubble spike

URM k t (h-h0) k

A=0.2 A=0.7

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Drag-Buoyancy Model for the Spike Front -

• II. Multi Mode

1st assumption:

Periodicity of the spikes ≡ Periodicity of the bubbles λS(t) ≡ λB(t)

0.1 0 .2 0.3 0.4

X

• 0.1

0.1 0.2

Z

0.1 0.2 0.3 0.4

X

• 0.1

0.1 0.2

Z

2D RT Simulation with A=0.9

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Na ve approach:

The ratio between the momentary velocities of the spikes and the bubbles equals the ratio between their asymptotic velocities:

l h asy B asy S B S B S B S

A 1 A 1 u u Agt 2 Agt 2 ) t ( u ) t ( u ρ ρ = − + = = α α = α α =

Drag-Buoyancy Model for the Spike Front -

• II. Multi Mode - RT

2nd assumption:

The ratio between the momentary velocities of the spikes and the bubbles equals the ratio between their velocities at the time tb in which the bubbles height reaches its self-similar value: hB(tb)=bλ

asy B b S b B b S B S B S B S

u ) t ( u ) t ( u ) t ( u Agt 2 Agt 2 ) t ( u ) t ( u ≅ = α α = α α =

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10

A

Experimental Results (LEM) 3D Drag-Bouyancy Model Asymptotic Drag-Bouyancy Model (ρh/ρl)1/3

Drag-Buoyancy Model for the Spike Front -

• II. Multi Mode - RT Results

αS / αB : Best fit by Dimonte et. Al.

( )

3 1 l h B S

~ ρ ρ α α

( ) ( )

     + π ⋅ ⋅ + − ⋅ − + = α α

−

A 1 b exp cosh A 1 A 1 tanh A 1 A 1

1 B S

3D:

na ve model

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Drag-Buoyancy Model for the Spike Front -

• II. Single Mode - RM

At late times, the bubble velocity in RM goes like λ/t. At finite time t its velocity can be expressed by a time-dependent coefficient γ(t):

1 ) t ( t u u ) t ( u

B ) t ( B

B

− = ∞ → γ       λ ∝

γ

   = < − = ∞ → γ       λ ∝

γ

1 A 1 A 1 ) t ( t u u ) t ( u

S ) t ( S

S

for all A s

B

t b ) t ( h ) t (

B θ

∝ = λ

Where, from assumption 1:

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Drag-Buoyancy Model for the Spike Front -

• II. Multi Mode - RM

( )

B b S

1 ) t ( MM S

t u

θ − ⋅ γ

∝

1 S MM S

S

t dt ) t ( dh ) t ( u

− θ

∝ =

also:

( )

B b S S

1 ) t ( 1 θ − ⋅ γ + = θ Na ve approach:

   θ = θ → < − = θ → = = γ

B S S asy S

1 A 1 1 1 A

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.5 2 2.5 3 3.5 4 4.5 5

A

Experimental Results (LEM) 3D Drag-Bouyancy Model 2D Drag-Bouyancy Model (ρh/ρl)1/5

θS / θB

Drag-Buoyancy Model for the Spike Front -

• II. Multi Mode - RM Results

( ) ( ) ( ) ( ) ( ) ( )

B B b S S

1 1 ) A 1 ( b exp A 1 A 1 1 1 ) A 1 ( b exp A 1 A 1 1 1 ) t ( 1 θ − ⋅ − + π ⋅       + − + − + π ⋅       + − − = θ − ⋅ γ + = θ

3D:

Best fit by Dimonte et. Al.

( )

5 1 l h B S

~ ρ ρ α α

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Summary

Good agreement between 3D drag-buoyancy model, statistical model, full 3D simulations and experimental results. 2D and 3D RT and RM scaling laws: Geometrical effect results in different scaling parameters: Spikes scaling laws are obtained from the drag-buoyancy model. RT: hB(S) = αB(S) A g t2 RM: hB(S) = aB(S) t θ B(S)

αB ≈ 0.05 in 2D and 3D θB ≈ 0.2-0.3 in 3D

• Vs. 0.4

in 2D b ≈ 0.75-1.5 in 3D

• Vs. 0.25-0.5

in 2D