A New Two- -Scale Mix Model: Towards Scale Mix Model: Towards a Multi a Multi- - A New Two A New Two-Scale Mix Model: Towards a Multi- Component Model of Turbulent Mixing* 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. 8th IWPCTM 1
Abstract Abstract Abstract velocities for each component of the fluid 2 . Turbulent mixing of the fluids in a multi- component system is of interest in situations However, the necessity to carry separate such as inertial confinement fusion (ICF) velocities for each component of the fluid and core-collapse supernovae 1 . We report greatly increases the memory requirements results of a project to include a model of and complexity of the computer turbulent mixing in a multi-component implementation. In contrast, we present a new two-scale formulation of the K- ε hydrodynamics and physics model called KULL, which is used for ICF. Because turbulent mixing model, with production KULL is a complex, multi-dimensional terms based on a recent scaling analysis 3 , model, we have developed a simplified, one- which treats all components of the fluid as if dimensional version called sKULL to speed- they had the same velocity. We also show up the development of the turbulent mixing that our new method for the initial model. conditions of the uncoupled two-scale K- ε model yields asymptotic growth. Future Of primary interest in the development of a work will compare the results of using this turbulent mixing model for a multi- single velocity model with those from a more component fluid is the question of whether it complete multi-velocity formulation of is necessary to allow each component of the turbulent mixing, to decide whether the fluid to retain its own velocity. Generally a multi-velocity formulation needs to be used multi-component, multi-velocity turbulent in KULL. mixing model should allow separate 8th IWPCTM 2
The goal of this work is to develop a turbulent mixing 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 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 supernovae 1 � 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 1 Remington, B.A., D. Arnett, R.P. Drake, and H. Takabe, Modeling Astrophysical Phenomena in the Laboratory with Intense Lasers, Science 284 , 1488 (1999). 8th IWPCTM 3
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 4
We have an appropriate path to develop a turbulent 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 mixing model for KULL Classic KULL: ALE Hydrodynamics Single sKULL: Duplicates KULL’s ALE Hydrodynamics Multi-sKULL: Multi- The most general The most general model of turbulen model o turbulent Component mixing mixing is is mu multi lti- and component an component and d multi-velocity multi locity Multi-Velocity 8th IWPCTM 5
sKULL is the right platform in which to develop a 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 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 8th IWPCTM 6
Side- Side-by-side runs of KULL and sKULL on the Sod -by by- -side runs of KULL and sKULL on the Sod side runs of KULL and sKULL on the Sod Side shock problem produce the same results shock problem produce the same results shock problem produce the same results 1 � Duplication of KULL results on selected problems w/ sKULL verifies 0.8 that we’ve duplicated KULL’s numerics Velocity Sod Shock Tube 0.6 � Sod (1978) shock tube problem: 0.4 Kull_Eul ρ = 1 ρ = 0.125 0.2 sKull-Eul 0 p = 1 p = 0.1 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.5 u = 0 u = 0 ρ Pressure, Density 0.4 � Standard test problem � Compared Lagrangian, Eulerian, and 0.3 ALE results to ensure that the results p from the two codes agreed 0.2 0.1 -0.1 0 0.1 0.2 0.3 0.4 0.5 x 8th IWPCTM 7
sKULL MC- -1V’s simulation of the Benjamin air 1V’s simulation of the Benjamin air- -SF SF 6 sKULL MC sKULL MC-1V’s simulation of the Benjamin air-SF 6 6 shock tube agrees well with the exact solution shock tube agrees well with the exact solution shock tube agrees well with the exact solution � Benjamin et al. (1993) air-SF 6 shock tube: Air* Air* SF 6 SF ) r Result of artificial viscous stress a b M ρ = 1.27 × 10 -3 ρ = 4.85 × 10 -3 ( e r u = 1.05 × 10 4 u u = 0 s s e p = 1.21 × 10 6 p = 8.00 × 10 5 r P Shock ( Shock (Ma=1.2 a=1.2) � Pressure results from the Distance (cm) MC-1V Lagrangian simulation versus exact solution at time 232 µ s show good agreement 8th IWPCTM 8
A multi- -component, multi component, multi- -velocity (MC velocity (MC- -MV) approach MV) approach A multi A multi-component, multi-velocity (MC-MV) approach needs to be considered for the turbulent mixing model 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 model 2 � 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 2 Youngs, D.L., Laser & Particle Beams 12 , 725 (1994). 8th IWPCTM 9
sKULL will be used to test multi- -velocity versus single velocity versus single sKULL will be used to test multi sKULL will be used to test multi-velocity versus single velocity- -based turbulent mixing models based turbulent mixing models velocity velocity-based turbulent mixing models Because of sKULL’s simplified nature it is faster and cheaper than KULL � 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) Faster and cheaper makes sKULL th Faster and cheape 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 8th IWPCTM 10
2 ) add a great deal The MC-MV equations (Youngs 2 ) add a great deal The MC- -MV equations (Youngs MV equations (Youngs 2 ) add a great deal The MC of complexity of complexity of complexity u r , Lagrangian “ALE”-like, ALE: “Grid Velocity” u = g = 0 if shock tube, 0, Eulerian Interactions, ∂ x = u Turbulence Transport ∂ t ( ) ∂ f r ρ r ∂ ∂ u [ ] − f r ρ r ( ) = − f r ρ r u r − u ∂ t ∂ x ∂ x ( ) ∂ f r ρ r u r = − ∂ ∂ u ∂ P [ ] − f r ρ r u r ( ) f r ρ r u r u r − u ∂ x − f r ∂ x + f r ρ r g ∂ t ∂ x ∂ τ ( ) + D rs + M rs − m r � ∂ x s ( ) ∂ f r ρ r e r = − ∂ ∂ u ∂ u [ ] − f r ρ r e r ( ) f r ρ r e r u r − u − h r P r ∂ t ∂ x ∂ x ∂ x � � + ∂ ∂ e r � f r ρ r ν r � + f r ε � � ∂ x ∂ x 8th IWPCTM 11
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