PECOSPredictive Engineering and

Computational Sciences

TOWARDS A WENO-BASED CODE FOR INVESTIGATINGRANS MODEL CLOSURES FOR HYDRODYNAMIC INSTABILITIES

Rhys Ulerich1 Oleg Schilling2

1University of Texas at Austin 2Lawrence Livermore National Laboratory

Background

Rayleigh–Taylor, Richtmyer–Meshkov, and Kelvin–Helmholtzhydrodynamic instabilities impact applications ranging from inertialconfinement fusion (ICF) to supernovae dynamics. Though theNavier–Stokes equations can exactly capture the physics, directnumerical simulation (DNS) is prohibitively expensive. Instead, the flowphysics can be efficiently approximated statistically usingReynolds-averaged Navier–Stokes (RANS) models. However, theseinstabilities are challenging because the models must accommodatevariable density, inhomogeneity, nonstationarity, and anisotropy.Additionally, the simultaneous presence of shocks and turbulencerequires sophisticated numerical techniques.

1

1.2

1.4

1.6

1.8

2

2.2

0.3

0.5

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0 0.125 0.25

∇ρ

∇p

∇ρ

∇p

Misalignment of ∇ρ and ∇p in anAt = ρh−ρl

ρh+ρl= 1/3 configuration

initiates the Rayleigh–Taylorinstability via baroclinic vorticityproduction.

Objectives

To develop a nonoscillatory, shock-capturing gasdynamics code designed to• simulate multi-species hydrodynamic instabilities and• facilitate N -equation RANS model closure evaluation and development.

To investigate the Rayleigh–Taylor instability and mixing, including• comparing RANS models with self-similar solutions,•measuring mixing statistics and equation budgets, and• quantifying sensitivities relative to initial perturbations and model coefficient choices.

High-Order Numerics and Code

Numerics allow hybrid upwind/central difference shock-capturing RANS & DNS:• Inviscid fluxes computed using• 9th-, 5th, or 3rd-order weighted essentially nonoscillatory (WENO) reconstruction [4],• Roe’s approximate Riemann solver [3], and• global Lax–Friedrichs flux splitting

• Viscous terms use 8th, 4th-, or 2nd-order centered finite differences• Total variation diminishing explicit Runge–Kutta time stepping• Selectable orders allow isolating numerical viscosity effects

New, modular Fortran 95 code designed for flexibility:• Equation-agnostic driver handles all MPI and IO considerations• Equation- and problem-specific modules provide relevant physics• Currently supports:• Single species Navier–Stokes with constant gamma, viscosity• Rayleigh–Taylor and Richtmyer–Meshkov initial conditions

• New equations and problems easily added by implementing:• Equation of state and any unique transport equations• Roe-averaged eigenvectors from system’s inviscid limit

• Supports single or multimode initial perturbations• Includes serial and parallel regression test suite• Flexible restart handling and statistics output• Features to simplify batch runs and parameter sweeps• Reasonable scalability and performance for effort to date• Doxygen-based documentation evolves with code

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0 0.250.10.20.15

1/480

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1/240

5,4

1/240

9,8

1/240

At ∆x = 1/240, the (9,8)th

order method fullyresolves At = 1/3 flowstructures while lowerorders do not. It requiresonly 30% of the (3,2)th

order’s compute time toobtain a convergedsolution.

0.1

1

10

1 10 100

Wal

l tim

e pe

r tim

este

p (s

)

Number of MPI ranks

WENO3, FD2WENO5, FD4WENO9, FD8

Strong, parallel scaling forAt = 1/3 at ∆x = 1/480

DNS of Large-Atwood-Number, Single-Mode Rayleigh–Taylor Instability

Preliminary results from single species, γ = 5/3, µ = 10−5 2D Navier–Stokes simulations of theRayleigh–Taylor instability at several large Atwood numbers.

Gravity is downward. Dirichlet boundary conditions are used at the top and bottom of the domain. Thehorizontal direction is periodic. At t = 0 a non-diffuse interface is established at the domain midpoint.The initial velocity perturbation is −(c/40) cos (8πx) where c is the local speed of sound.

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At 0.5

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At 0.6 1

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At 0.7

10

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2

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At 0.8

5

10

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25

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35

At 0.9

Density fields for the indicated Atwood numbers at t = 1.95 s. Fields for At > 0.5 show “inviscid” flow structures suggestingthe 768 points per wavelength resolution was insufficient to fully resolve dissipative effects. Prior runs at 512 points perwavelength (not shown) indicate adequate resolution for At = 0.5.

0

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bubb

le a

mpl

itude

time

At 0.5At 0.6At 0.7At 0.8At 0.9

0

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1

0 0.5 1 1.5 2

spik

e am

plitu

de

time

At 0.5At 0.6At 0.7At 0.8At 0.9

Bubble and spike amplitudes were found by thresholding the heavy-fluid mass fraction m1 = ρ1(ρ−ρ2)ρ(ρ1−ρ2)

∈ [0, 1] at 0.99 and0.01. The spike amplitudes increase smoothly with Atwood number while the bubble amplitudes remain clustered forAt = 0.8, 0.9.

Work in Progress

Implement two-component, ideal perfect gas formulation:• Adds mass fraction transport equation [2]• Use Sutherland’s Law for species molecular transport coefficients

Implement 2-, 3-, and 4-equation Rayleigh-Taylor mixing-optimized RANS models for transport of:• turbulent kinetic energy (K)• turbulent kinetic energy dissipation rate (ε)• density variance (ρ′2)• density variance dissipation rate (ε′ρ)

APS DFD 2010: Comparisons of a Reynolds-Averaged Navier-Stokes Model with Self-SimilarSolutions for Large Atwood Number Rayleigh–Taylor Mixing

Near-Term Plans

Investigate RANS closure budgets against averaged DNS fields:• Add support for perturbation with prescribed energy spectra• Add output of energy spectra, PDFs, and other statistics

Extend work to other instabilities:• Add inflow/outflow characteristic boundary conditions• Study Richtmyer–Meshkov and Kelvin–Helmholtz instabilities

Other related items:• Improve serial code performance• Hybrid WENO/finite differencing using Ducros sensor [1]• Add option for implicit time-evolution• Extend code to three dimensions• Implement a thermodynamically consistent and fully conservative formulation [5]• Add subgrid-scale models for large-eddy simulation (LES)

2.5 2.75 3

1

3

5

7

9

11

t=1 t=2

The Richtmyer–Meshkov instability generated by a Ma = 1.5 shock interacting with a perturbed At = 1/3, γ = 7/2interface. Density fields show a reshock event and post-reshock mixing layer growth at the indicated times.

References

[1] F. DUCROS, Large-eddy simulation of the shock/turbulence interaction, Journal of ComputationalPhysics, 152 (1999), pp. 517–549.

[2] D. J. HILL, C. PANTANO, AND D. I. PULLIN, Large-eddy simulation and multiscale modelling of aRichtmyer–Meshkov instability with reshock, Journal of Fluid Mechanics, 557 (2006), pp. 29–61.

[3] P. L. ROE, Approximate Riemann solvers, parameter vectors, and difference schemes, Journal ofComputational Physics, 43 (1981), pp. 357–372.

[4] C.-W. SHU, High order weighted essentially nonoscillatory schemes for convection dominatedproblems, SIAM Review, 51 (2009), pp. 82–126.

[5] S.-P. WANG, M. H. ANDERSON, J. G. OAKLEY, M. L. CORRADINI, AND R. BONAZZA, Athermodynamically consistent and fully conservative treatment of contact discontinuities forcompressible multicomponent flows, Journal of Computational Physics, 195 (2004), pp. 528–559.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Thismaterial is also based in part upon work supported by the Department of Energy National Nuclear Security Administration under Award Number DE-FC52-08NA28615.

1rhys@ices.utexas.edu, 2schilling1@llnl.gov LLNL-POST-448252