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8th IWPCTM 1

A New Two-Scale Mix Model: Towards a Multi- Component Model of Turbulent Mixing* A New Two A New Two-

• Scale Mix Model: Towards

Scale Mix Model: Towards a Multi a Multi-

• Component Model of Turbulent Mixing*

Component Model of Turbulent Mixing*

Presented to the 8th International Workshop on the Physics of Compressible Turbulent Mixing Donald E. Eliason, William H. Cabot, and Ye Zhou AX-Division, Lawrence Livermore National Laboratory 11 December 2001

*This work was performed under the auspices of the U.S. Department of Energy by the University of California Lawrence

Livermore National Laboratory under contract No. W-7405-Eng-48.

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Abstract Abstract Abstract

Turbulent mixing of the fluids in a multi- component system is of interest in situations such as inertial confinement fusion (ICF) and core-collapse supernovae1. We report results of a project to include a model of turbulent mixing in a multi-component hydrodynamics and physics model called KULL, which is used for ICF. Because KULL is a complex, multi-dimensional model, we have developed a simplified, one- dimensional version called sKULL to speed- up the development of the turbulent mixing model. Of primary interest in the development of a turbulent mixing model for a multi- component fluid is the question of whether it is necessary to allow each component of the fluid to retain its own velocity. Generally a multi-component, multi-velocity turbulent mixing model should allow separate velocities for each component of the fluid2. However, the necessity to carry separate velocities for each component of the fluid greatly increases the memory requirements and complexity of the computer

• implementation. In contrast, we present a

new two-scale formulation of the K-ε turbulent mixing model, with production terms based on a recent scaling analysis3, which treats all components of the fluid as if they had the same velocity. We also show that our new method for the initial conditions of the uncoupled two-scale K-ε model yields asymptotic growth. Future work will compare the results of using this single velocity model with those from a more complete multi-velocity formulation of turbulent mixing, to decide whether the multi-velocity formulation needs to be used in KULL.

8th IWPCTM 3

The goal of this work is to develop a turbulent mixing model for the ICF code called KULL The goal of this work is to develop a turbulent mixing The goal of this work is to develop a turbulent mixing model for the ICF code called KULL model for the ICF code called KULL

Turbulent mixing of the fluids in a multi-component system is of

interest in situations such as inertial confinement fusion (ICF) and core-collapse supernovae1

We report results of a project to include a model of turbulent

mixing in a multi-component hydrodynamics and physics model called KULL, which is used for ICF

Because KULL is a complex, multi-dimensional code, we have

developed a simplified, one-dimensional version called sKULL to speed-up the development of the turbulent mixing model

1Remington, B.A., D. Arnett, R.P. Drake, and H. Takabe, Modeling Astrophysical

Phenomena in the Laboratory with Intense Lasers, Science 284, 1488 (1999).

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Three areas of this research are highlighted Three areas of this research are highlighted Three areas of this research are highlighted

sKULL reproduces KULL’s multi-component hydrodynamics and

numerics

A single velocity, multi-component, two-scale K-ε turbulent

mixing model has been developed within sKULL

A new method for the uncoupled two-scale K-ε initial conditions

yields asymptotic growth

8th IWPCTM 5

We have an appropriate path to develop a turbulent mixing model for KULL We have an appropriate path to develop a turbulent We have an appropriate path to develop a turbulent mixing model for KULL mixing model for KULL

Classic KULL: ALE Hydrodynamics Multi-sKULL: Multi- Component and Multi-Velocity Single sKULL: Duplicates KULL’s ALE Hydrodynamics The most general The most general model o model of turbulen turbulent mixing mixing is is mu multi lti- component an component and d multi multi-velocity locity

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sKULL is the right platform in which to develop a turbulent mixing model for KULL sKULL is the right platform in which to develop a sKULL is the right platform in which to develop a turbulent mixing model for KULL turbulent mixing model for KULL

sKULL duplicates KULL’s hydrodynamics Side-by-side runs of KULL and sKULL on the Sod shock

produce the same results

We tested the Lagrangian, Eulerian,and ALE capabilities of

sKULL to ensure they matched KULL’s

The simplified nature of sKULL, due both to 1-D and no addi-

tional physics, allows it to run more quickly

Faster run times lead to shorter turn-around times for testing

turbulent mixing models

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Duplication of KULL results on

selected problems w/ sKULL verifies that we’ve duplicated KULL’s numerics

Sod (1978) shock tube problem: Standard test problem Compared Lagrangian, Eulerian, and

ALE results to ensure that the results from the two codes agreed

Side-by-side runs of KULL and sKULL on the Sod shock problem produce the same results Side Side-

• by

by-

• side runs of KULL and sKULL on the Sod

side runs of KULL and sKULL on the Sod shock problem produce the same results shock problem produce the same results

ρ = 1 p = 1 u = 0 ρ = 0.125 p = 0.1 u = 0

0.2 0.4 0.6 0.8 1

• 0.1

0.1 0.2 0.3 0.4 0.5 Velocity Kull_Eul sKull-Eul 0.1 0.2 0.3 0.4 0.5

• 0.1

0.1 0.2 0.3 0.4 0.5 Pressure, Density x p ρ Sod Shock Tube

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sKULL MC-1V’s simulation of the Benjamin air-SF6 shock tube agrees well with the exact solution sKULL MC sKULL MC-

• 1V’s simulation of the Benjamin air

1V’s simulation of the Benjamin air-

• SF

SF6

6

shock tube agrees well with the exact solution shock tube agrees well with the exact solution

P r e s s u r e ( M b a r ) Distance (cm) Result of artificial viscous stress

Benjamin et al. (1993) air-SF6

shock tube:

Pressure results from the

MC-1V Lagrangian simulation versus exact solution at time

232 µs show good agreement

Air* Air* SF SF6 Shock ( Shock (Ma=1.2 a=1.2) ρ = 1.27×10-3

u = 1.05×104 p = 1.21×106

ρ = 4.85×10-3

u = 0 p = 8.00×105

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A multi-component, multi-velocity (MC-MV) approach needs to be considered for the turbulent mixing model A multi A multi-

• component, multi

component, multi-

• velocity (MC

velocity (MC-

• MV) approach

MV) approach needs to be considered for the turbulent mixing model needs to be considered for the turbulent mixing model

In RTI/RMI, zones may contain more than one component, each

with its own velocity

Component interactions (e.g., drag) can lead to mixing From the rocket rig experiments, this led David Youngs (AWE)

to create his MC-MV mixing model2

The MC-MV equations add a great deal of complexity Carrying separate velocities increases the memory

requirement

The drag term may require an implicit treatment

2Youngs, D.L., Laser & Particle Beams 12, 725 (1994).

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sKULL will be used to test multi-velocity versus single velocity-based turbulent mixing models sKULL will be used to test multi sKULL will be used to test multi-

• velocity versus single

velocity versus single velocity velocity-

• based turbulent mixing models

based turbulent mixing models

Because of sKULL’s simplified nature it is faster and cheaper than KULL Faster and cheape Faster and cheaper makes sKULL th r makes sKULL the ideal e ideal plat platform to to test test whether MC-MV whether MC-MV mi might be needed ght be needed in in K KULL LL

The extra memory requirement of MC-MV will be manageable Additional computation for interactions will be do-able Different numerical treatments of the drag term can be tested

(explicit vs. implicit vs. iterated)

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The MC-MV equations (Youngs2 ) add a great deal

• f complexity

The MC The MC-

• MV equations (Youngs

MV equations (Youngs2

2 ) add a great deal

) add a great deal

• f complexity
• f complexity

∂ x ∂ t = u ∂ f rρ r

( )

∂ t = − ∂ ∂ x fr ρ r u r − u

( )

[ ]− f r ρ r

∂ u ∂ x ∂ fr ρ ru r

( )

∂ t = − ∂ ∂ x frρ r u r u r − u

( )

[ ] − fr ρ ru r

∂ u ∂ x − fr ∂ P ∂ x + fr ρ r g + D rs + M rs

( )

s

• − m r

∂ τ ∂ x ∂ fr ρ re r

( )

∂ t = − ∂ ∂ x frρ r e r u r − u

( )

[ ]− f r ρ re r

∂ u ∂ x − h r Pr ∂ u ∂ x + ∂ ∂ x fr ρ rν r ∂ e r ∂ x

• + f rε

ALE: “Grid Velocity” u = ur, Lagrangian 0, Eulerian

“ALE”-like, g = 0 if shock tube, Interactions, Turbulence Transport

8th IWPCTM 12

Because of the MC-MV equations’ complexity, we’ve first developed a single velocity version, MC-1V Because of the MC Because of the MC-

• MV equations’ complexity, we’ve

MV equations’ complexity, we’ve first developed a single velocity version, MC first developed a single velocity version, MC-

• 1V

1V

Rey Reynol nolds ds st stress ess

Turbul Turbulent di ent dissipati pation

Ki Kinemat nematic v viscosi scosity/Schmi /Schmidt number number

dx dt = u dV

r

dt ≈ h

r

dV dt ,V = V

r r

• ρ Du

Dt = −∂P ∂x −∂τ ∂x frρr De

r

Dt =−hrP

r

∂u ∂x + frρrεt + ∂ ∂x frρr νr σe ∂e

r

∂x

• sKULL MC-1V

Lagrangian equations with a new two-scale K-ε mixing model

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The use of the compressibility in the effective pressure allows the simplification to single velocity The use of the compressibility in the effective The use of the compressibility in the effective pressure allows the simplification to single velocity pressure allows the simplification to single velocity

P= P

r frK r r

• frK

r r

• ,P

r =p r +q r

K

r −1 =ρr

∂P

r

∂ρr e

r

+ P

r

ρr ∂P

r

∂e

r ρr

,h

r = frK r

fsK

s s

• Ef

Effecti ective pr ve pressur essure e (incl ncludes art udes artificial l vi visco scosi sity)

In Invers rse e effe ffectiv ive compression; compression; Relative com Relative com- pression pression For an i For an ideal gas a deal gas and q nd qr = 0, K = 0, Kr

• 1
• 1 =

= γrpr (adiaba (adiabatic co tic compre mpres-s s-sibility bility), , and h and hr = [f = [fr/( /(ρr cr

2)]

)]/[ /[s fs/( /(ρs cs

2)]

(Youn

• ungs

gs2)

8th IWPCTM 14

The viscosity for energy diffusion and Reynolds stress comes from the two-scale K-ε model The viscosity for energy diffusion and Reynolds The viscosity for energy diffusion and Reynolds stress comes from the two stress comes from the two-

• scale K

scale K-

• ε

ε model

model

Kinemati Kinematic visc. visc. Reynolds stres Reynolds stress Equations f Equations for Kα and nd εα are needed are needed for closure for closure

Turbul Turbulent ent Mol Molecul cular ar

τ =

2 3 ρ K p − 4 3 ρ ν p

∂ u ∂ x ν α = ν 0 + ν T α ν T α = C µ K α

2

ε α

8th IWPCTM 15

The two-scale K-ε equations describe evolution of the production and turbulence scales The two The two-

• scale K

scale K-

• ε

ε equations describe evolution of the

equations describe evolution of the production and turbulence scales production and turbulence scales

Production Production scale scale Turbulence Turbulence scale scale

Pr Product

• duction

DKp Dt = P

R* − εp + ∂

∂x νp σ K ∂Kp ∂x − τ ρ ∂u ∂x DKt Dt = ε p − εt + ∂ ∂x νt σK ∂Kt ∂x Dε p Dt = Cp1 ε p Kp P

R* − Cp2

εp

2

Kp + ∂ ∂x νp σ ε ∂ε p ∂x Dεt Dt = Ct 1 εpεt Kt −Ct 2 εt

2

Kt + ∂ ∂x νt σε ∂εt ∂x

8th IWPCTM 16

The production terms PR* for the two-scale K-ε equations parameterize mixing caused by RTI or RMI The production terms P The production terms PR*

R* for the two

for the two-

• scale K

scale K-

• ε

ε

equations parameterize mixing caused by RTI or RMI equations parameterize mixing caused by RTI or RMI

P

RT =4C RTεp 1/2(gA

)

3/4(k − 1/ 4 −k 1 −1/4)

P

RM =2C RMεp 1/2(A∆u) 3/2(k 1 1/2 −k0 1/2)

Rayl Raylei eigh- gh-Tayl Taylor

• r

Ri Richt chtmyer myer-Meshko

• Meshkov

Based on a recent scaling analysis3 of RT and RM instabilities, the production term may be written as

3Zhou, Y., A scaling analysis of turbulent flows driven by Rayleigh-Taylor and

Richtmyer-Meshkov instabilities, Phys. Fluids 13, 538–543 (2001).

8th IWPCTM 17

Wave numbers k0 and k1 for the production terms evolve with the flow Wave numbers k Wave numbers k0

0 and k

and k1

1 for the production terms

for the production terms evolve with the flow evolve with the flow

k 0 =

4 7 C RT ε p 1 / 2( gA ) 1 / 4 4 7 C RT ε p 1 / 2( gA ) 1 / 4 k 1 − 3 / 4 + K p

• 4 / 3

Rayl Raylei eigh- gh-Tayl Taylor

• r

Initially k0 and k1 are set by the initial perturbation scales, but thereafter evolve according to the computed production and turbulence scales

k1 = ε t( 3

2 CK / Kt ) 3/ 2 Production scale Turbulence scale RT RT o

• r R

RM

k 0 = 4 C RM

2

ε p A ∆ u 2 C RM ε p A ∆ u

( )

1 / 2 k1 − 1 / 2 + K p

[ ]

2 Ri Richt chtmyer myer-Meshko

• Meshkov

8th IWPCTM 18

The change in total energy is due to production minus dissipation and surface fluxes The change in total energy is due to production minus The change in total energy is due to production minus dissipation and surface fluxes dissipation and surface fluxes

• D

Dt ρu

2/2+e+K

( )

• dV
• = ρP

R *−ε t

( )

• dV
• −uP+τ

( )+F

e+F K

[ ]

S

• Total Energy

Total Energy Change Change Production - Production - Dis Dissipat ipation ion Surf Surface Fl ace Fluxes uxes

F

e = − ν

σ K ∂e ∂x ,F

Kα = −να

σK ∂Kα ∂x F

K =F Kp + F Kt

Diff Diffusive f usive fluxes uxes of internal internal and and turbulent turbulent kinet kinetic ic energies energies

8th IWPCTM 19

Results from Orszag and Speziale will be used to provide ICs for the two-scale turbulent mixing model Results from Orszag and Speziale will be used to Results from Orszag and Speziale will be used to provide ICs for the two provide ICs for the two-

• scale turbulent mixing model

scale turbulent mixing model

St

Steve Or eve Orszag’ szag’s wo s work f for t r the ASC he ASCI Tur Turbul bulence G Group:

• up:

If v viol

• lat

ated: t ed: too

• o much

much turbul bulence i ence init itially, , interf rface di ace dies out es out If v viol

• lat

ated: no ed: no turbulent bulent vi visco scosi sity devel develops,

• ps, Or

Orszag’ szag’s s hi high Re r gh Re run bl un blew ew up up

K p < ˜ P

R 0

ν 0 / C µ , ˜ P

R 0 = PR 0 / ε p 0 1 / 2

C µ K p 0

2

ν 0 < ε p 0 < ˜ P

R 0 2

In Init itia ial R l RT o

• r R

RM P Productio ion

Is this resul

esult consi consistent ent w w/ Spezi Speziale’s fixed poi ed point nt anal analysi ysis? s?

8th IWPCTM 20

The result using Orszag’s approach is consistent with Speziale’s fixed point analysis The result using Orszag’s approach is consistent with The result using Orszag’s approach is consistent with Speziale’s fixed point analysis Speziale’s fixed point analysis

The “f

The “fixed poi xed point nts” from Spezi

• m Speziale’s anal

analysi ysis4 act act as att as attractor

• rs

Init

itiali lize wit ze with fixed xed poi point nts t s that ar hat are consist e consistent ent w with desir desired l ed long- ng-term be behavi havior

• r

Leads mor Leads more qui quickl ckly y to t the desir he desired l ed long-ter

• ng-term st

m state

Spezi

Speziale’s anal analysi ysis s yi yiel elds t ds the he following fi ng fixed poi xed point nts: s:

εpf = fpP

R0,εtf = ftε pf

fp = (Cp1 −1) /(Cp2 −1), ft = (Ct1 −1) /(Ct 2 −1)

su suggest ggesting

ε p 0 = fp

2 ˜

P

R0 2

consi consist stent ent w with O Orszag’s appr approach if

• ach if ƒp

p < 1 4Speziale, C.G., and N. Mac Giolla Mhuiris, On the prediction of equilibrium

states in homogeneous turbulence, J. Fluid Mech., 209, 591-615 (1989).

8th IWPCTM 21

The Orszag-Speziale ICs yield asymptotically growing solutions for the two-scale turbulent mixing model The Orszag The Orszag-

• Speziale ICs yield asymptotically growing

Speziale ICs yield asymptotically growing solutions for the two solutions for the two-

• scale turbulent mixing model

scale turbulent mixing model

Cur

Current recommended val ecommended values C ues Cp1

p1 = 1

= 1.5, Cp2

p2 = 2

2 gi give ve ƒp = 1/ = 1/2 The r The resul esult usi t using Or ng Orszag’s appr approach i

• ach is co

consi nsist stent ent w with Spezi Speziale’ le’s fixed poi xed point anal nt analysi ysis Al Also consi so consist stent ent with const with constraints C C*2

*2 > 1

> 1 an and C d C*2

*2>C

>C*1

*1 >1/2 nee

neede ded d to get get asympt asymptotically gr cally growi

• wing sol

ng soluti utions

• ns

Sol

Solution o

• n of the two-sc

he two-scal ale e turbul bulent mi ent mixing model ng model equat equations

• ns as ODEs

as ODEs usi using t ng the Or he Orszag- szag-Spezi zial ale I e ICs pr s produced

• duced the expect

he expected asympt ed asymptot

• ticall

cally gr growi

• wing result

esults The Or The Orszag- szag-Spezi peziale I ICs will be used i be used in the t the two-

• -scal

scale e turbul bulent mi ent mixing mo ng model del f for sKULL MC- r sKULL MC-1V

8th IWPCTM 22

A test of the Orszag-Speziale ICs w/ the uncoupled two-scale K-ε model yields asymptotic growth A test of the Orszag A test of the Orszag-

• Speziale ICs w/ the uncoupled

Speziale ICs w/ the uncoupled two two-

• scale K

scale K-

• ε

ε model yields asymptotic growth

model yields asymptotic growth

Rayleigh-Taylor Test

gA = 4000 k1(0) = 10k0(0) = 20π CK = CRT = 1.5 Cp1 = 1.5, Cp2 = 2.0 Ct1 = 1.08, Ct2 = 1.15 Constraints for asymptotic

growth are satisfied, and asymptotic growth achieved Consistency with a stand-alone ODE solution demonstrates that the uncoupled two-scale K-ε equations have been correctly implemented in sKULL

Time P r

• d

u c t i

• n

, K ,

ε ε

PRT, εp, εt Kp, Kt

8th IWPCTM 23

Summary and Future Work Summary and Future Work Summary and Future Work

A simplified version of the ICF code KULL has been developed which

reproduces KULL’s multi-component hydrodynamics

The purpose of simplified KULL, sKULL, is to serve as a test-bed for

implementation of multi-component turbulent mixing models

Tests show that sKULL faithfully duplicates KULL’s numerics A single velocity, multi-component, two-scale K-ε turbulent mixing

model has been developed within sKULL

A new method for the uncoupled two-scale K-ε initial conditions

yields asymptotic growth

Future work will compare these single velocity results with those from a

more complete multi-velocity formulation of turbulent mixing, to decide whether the multi-velocity formulation needs to be used in KULL

8th IWPCTM 24

Table of Symbols #1 Table of Symbols #1 Table of Symbols #1

Quantity Description

D

rs Drag force on fluid r due to fluid s

er

Specific internal energy of fluid r

ε p

Dissipation at production scale

εt

Dissipation at turbulence scale

f r

Volume fraction of fluid r

g

Acceleration (e.g., gravitational)

hr

Relative compressibility of fluid r

Kp

Turbulence kinetic energy at production scale

Kt

Turbulence kinetic energy at turbulence scale

m

r

Mass fraction of fluid r

8th IWPCTM 25

Table of Symbols #2 Table of Symbols #2 Table of Symbols #2

Quantity Description

Mrs

Added mass effect f

• r fluid r due to fluid s

ν

0r

Molecular viscosity of fluid r

ν

T =νp+νt

Total turbulent viscosity

νp

Turbulent viscosity, production scale

ν

t

Turbulent viscosity, turbu lence scale

ν

r =ν0r+ν T

Total viscosity of fluid r

P

r =p r +q r

Effective p ress ure of fluid r

P=p+q

Effective mea n pressure

p= h

rp r r

• Me

an pressure

p

r = p r ρ r,e r

( )

Pressure of fluid r from EO S

8th IWPCTM 26

Table of Symbols #3 Table of Symbols #3 Table of Symbols #3

Quant ity Desc ription

q= frq

r r

• Me

an a rtif icial viscous s tre ss

q

r

Artif icial visc

• u

s stre ss of f luid r

ρ= frρ

r r

• Me

an density

ρ

r

Density of fluid r

t

Tim e

u

Me an v elocity

u = fru

r r

• Vo

lume-w eigh te d m ea n ve locity

u

r = ur + ν r

ρ r ∂ρr ∂x

Vo lume-w eigh te d velocity of fluid r

ur

Velo city

• f

fluid r

x

Po sition at tim e t