PECOSPredictive Engineering and

Computational Sciences

A WENO-Based Code for Investigating RANS Model Closures forMulticomponent Hydrodynamic Instabilities

1Rhys Ulerich 2Oleg Schilling

1 Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin

2Lawrence Livermore National Laboratory

63rd Annual Meeting of the APS Division of Fluid DynamicsLong Beach, California

21 November 2010

Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344and by the DOE National Nuclear Security Administration under Award Number DE-FC52-08NA286.

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 1 / 9

Background and Objectives

A WENO-based code was developed to aid Reynolds-averagedNavier–Stokes (RANS) model assessment for hydrodynamic instabilities

Rayleigh–Taylor instability impacts many applications such asinertial confinement fusion and supernovae

• DNS of Navier–Stokes equations is accurate but expensive• RANS inexpensively describes statistical moment evolution• Instabilities challenging for RANS models due to variable

density, inhomogeneity, nonstationarity, and anisotropy

Develop a nonoscillatory, shock-capturing gasdynamics code to• simulate multi-species hydrodynamic instabilities• facilitate N -equation RANS model closure evaluation and

development

Investigate Rayleigh–Taylor instability and mixing, including• comparing RANS models with self-similar solutions• measuring mixing statistics and DNS, RANS equation budgets• assessing advanced model closures

∇ρ

∇p

∇ρ

∇p

Baroclinic vorticity productioninitiates the Rayleigh–Taylorinstability shown here forAt =

ρh−ρlρh+ρl

= 1/3

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 2 / 9

Numerics and Code

Code uses high-order numerics for their efficiency and resolving capability

Split system according to ~φt + ~F (~φ)~x = ~V(~φ)

Inviscid fluxes ~F computed using

• Roe approximate Riemann solver [Roe, 1981]

• global Lax–Friedrichs (LF) flux splitting

• 9th-, 5th-, or 3rd-order weighted essentiallynonoscillatory (WENO) reconstruction [Shu, 2009]

Viscous and diffusive terms ~V use 8th-, 4th-, or 2nd-ordercentered finite differences

Total variation diminishing (TVD) 3rd- or 4th-order explicitRunge–Kutta time stepping

Choices allow shock-capturing DNS, RANS, and LES

New, modular, parallel Fortran 95 code designed forflexibility to allow rapid closure prototyping

1

1.2

1.4

1.6

1.8

2

2.2

0.2

0.4

0.6

0.8

0 0.250.10.20.15

1/480

3,2

1/240

5,4

1/240

9,8

1/240

(9,8) order method resolves sample flowat 30% of cost of (3,2) order

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 3 / 9

Large Atwood Number Single-Mode Rayleigh–Taylor Instability

DNS of single-mode Rayleigh–Taylor instability for γ = 5/3, µ = 10−4

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

At .6

0

50

100

150

200

0 25

0.001

0.0015

0.002

0.0025

0.003

At .5 0.001

0.002

0.003

0.004

0.005

0.006

0.007

At .7

0.002

0.004

0.006

0.008

0.01

0.012

At .8

0.005

0.01

0.015

0.02

0.025

0.03

0.035

At .9

Non-diffuse (sharp) initialization using velocity perturbation. No filtering necessary. 8192× 1024 grid.

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 4 / 9

Argon–Air Multiple Mode Rayleigh–Taylor Instability

DNS of At = 0.16 Argon–Air multimode Rayleigh–Taylor instability

Multicomponent Navier–Stokes simulation [Hill et al., 2006]:

• Power law viscosity for each species

• Constant Prandtl and Schmidt numbers

• Reference properties from 1 atm and 298 K

Initial velocity perturbation:

• Normally distributed amplitudes

• Uniformly distributed phases

• 44 points per minimum wavelength

Interface pressure is 1/1000 atm

m2 = 1−m1

γ =m1cp,1 +m2cp,2

m1cv,1 +m2cv,2

µr = µ0r

(T/T

0r

)βr

∀φ ∈ {µ, κ,D} φ =

m1√M1

φ1 +m2√M2

φ2

m1√M1

+m2√M2

94

98

102

106

0 10 20 30 40 50

0

0.2

0.4

0.6

0.8

1

Heavy species mass fraction at t = 0.5 s

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 5 / 9

Argon–Air Multiple Mode Rayleigh–Taylor Instability

Evolution of Argon–Air multimode Rayleigh–Taylor instability showsprogressive development of small-scale structures as mixing layer grows

0

50

100

150

200

0 25 50

0

50

100

150

200

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150

200

0 25 50

0

50

100

150

200

0 25 50

Heavy species mass fraction at t = 0.5, 1, 1.5, 2, 2.5, 3 s

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 6 / 9

Multicomponent RANS for Rayleigh–Taylor Instability

Multicomponent RANS formulation

∂tρ+ ∂xj (ρ vj) = 0

∂t(ρ vi) + ∂xj (ρ vi vj + p δij) = ∂xjσij − ∂xj τij

∂t(ρ e) + ∂xj [(ρ e+ p) vj ] = ∂xj (σij vi) + ∂xj

(χ∂xj T −

∑2

r=1hr Jr,j

)− ∂xj

(τij vi +

p v′′j

γ − 1+ρ v′′2 v′′j

2+

γ

γ − 1p′ v′′j +���σ′′ij v

′′i

)

+ ∂xj∑2

r=1Dr[ρr������˜(

U ′′r ∂xj∂m′′r

)+����p′r ∂xjm

′′r

]∂t (ρ m1) + ∂xj (ρ m1 vj) = ∂xj

(ρD ∂xj m1

)− ∂xj

(ρm′′1 v

′′j

)∂t(ρ E′′

)+ ∂xj

(ρ E′′ vj

)= . . .

∂t(ρ ε′′

)+ ∂xj

(ρ ε′′ vj

)= . . .

• Inviscid treatment of turbulent pressure 23ρE′′δij following [Siikonen, 2005]

• Closure v′′j = −ρ′ v′jρ

≈ νtσρρ

∂xjρ yields problematic diffusive term −∂xj[eintνtσρ

∂xjρ]

• Using v′′j =T ′′ v′′jT

−p′ v′′jp

[Lele, 1994] and assumingT ′′ v′′jT

≈ 0 yields ∂xj[

Cpuµt(γ−1)σK

∂xj E′′]

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 7 / 9

Multicomponent RANS for Rayleigh–Taylor Instability

As expected, in comparison to DNS, RANS field shows similar structuresbut is significantly more diffusePresent simulation turbulent quantities initialized in spirit of [Banerjee et al., 2010]

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98

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0

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1

DNS (2048x512) versus RANS (480x120) heavy species mass fraction at t = 0.5 s

Even with T ′′ v′′j /T ≈ 0, diffusive explicit time step issues arise at middle timesUlerich, Schilling LLNL-PRES-462652 21 November 2010 8 / 9

Ongoing Work and Future Directions

Ongoing Work and Future Directions

Mitigate diffusive time step restrictions:

• Quantify acceptability of numerics-driven v′′j assumption• If possible, augment numerics to avoid such issues while keeping explicit evolution• Otherwise, move to semi-implicit time evolution [Shen et al., 2007, Yang, 1998]

Compare simulated layer evolution with analytical, self-similar solutions to transport equations

Investigate RANS closure budgets with DNS field statistics

Add 3- and 4- equation capabilities to describe scalar turbulence physics:

• density variance (ρ′2)• density variance dissipation rate (ε′ρ)

Future directions:• Improve code’s range-of-applicability [Poinsot and Lele, 1992, Hu et al., 2010, Wang et al., 2004]

• Investigate RANS model closures for Richtmyer–Meshkov instability

2.5 2.75 3

1

3

5

7

9

11

t=1 t=2

Richtmyer–Meshkov instability due to aMa = 1.5 shock interacting with aperturbedAt = 1/3, γ = 7/2interface

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 9 / 9

References

Banerjee, A., Gore, R. A., and Andrews, M. J. (2010).

Development and validation of a turbulent-mix model for variable-density and compressible flows.Physical Review E, 82(4):046309+.

Hill, D. J., Pantano, C., and Pullin, D. I. (2006).

Large-eddy simulation and multiscale modelling of a Richtmyer–Meshkov instability with reshock.Journal of Fluid Mechanics, 557:29–61.

Hu, X. Y., Wang, Q., and Adams, N. A. (2010).

An adaptive central-upwind weighted essentially non-oscillatory scheme.Journal of Computational Physics.

Lele, S. K. (1994).

Compressibility Effects on Turbulence.Annual Review of Fluid Mechanics, 26(1):211–254.

Poinsot, T. and Lele, S. (1992).

Boundary conditions for direct simulations of compressible viscous flows.Journal of Computational Physics, 101(1):104–129.

Roe, P. L. (1981).

Approximate Riemann solvers, parameter vectors, and difference schemes.Journal of Computational Physics, 43:357–372.

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 1 / 8

References

Shen, Y. Q., Wang, B. Y., and Zha, G. C. (2007).

Implicit WENO scheme and high order viscous formulas for compressible flows.AIAA Paper, 4431:2007.

Shu, C.-W. (2009).

High order weighted essentially nonoscillatory schemes for convection dominated problems.SIAM Review, 51:82–126.

Siikonen, T. (2005).

An application of Roe’s flux-difference splitting for k-epsilon turbulence model.International Journal for Numerical Methods in Fluids, 21(11):1017–1039.

Wang, S.-P., Anderson, M. H., Oakley, J. G., Corradini, M. L., and Bonazza, R. (2004).

A thermodynamically consistent and fully conservative treatment of contact discontinuities for compressiblemulticomponent flows.Journal of Computational Physics, 195(2):528–559.

Yang, J. (1998).

Implicit weighted ENO schemes for the three-dimensional incompressible Navier-Stokes equations.Journal of Computational Physics, 146(1):464–487.

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 2 / 8

Backup

The Roe solver reduces system to characteristic waves

Approximate ∂t~φ+ ∂x ~F (~φ) = 0 by linearized, localproblems via

∂x ~F (~φ) =(∂~φ~F)∂x~φ ≈ A

(~φL, ~φR

)∂x~φ

where A is the Roe-averaged matrix satisfying

~φL, ~φR → ~φ =⇒ A(~φL, ~φR

)→ ∂~φ

~F

~FL − ~FR = A(~φL, ~φR

)(~φL − ~φR

)Using eigendecomposition A = RΛR−1 givesdecoupled characteristic space wave equations

∂t(R−1~φ

)(k)

+ λ(k)∂x(R−1~φ

)(k)

= 0

Example: 1D Euler flux

~φ = [ρ, ρu, ρeT ]T

~F =[ρu, p + ρu

2, u (p + ρeT )

]Tp = (γ − 1)

(ρeT − ρu

2/2)

h = (ρeT + p) /ρ

uRL =

√ρLuL +

√ρRuR

√ρL +

√ρR

hRL = . . .

cRL = c(uRL, hRL)

R =

1 1 1u + c u− c u

h + cu h− cu u2/2

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 3 / 8

Backup

LF + WENO propagates characteristic waves

Consider scalar problem ∂tφ+ ∂xf(φ) = 0:

• Compute global Lax–Friedrichs flux split f(φ) = f+(φ) + f−(φ)where f±(φ) = 1

2[f(φ)± αφ] and α = maxφ |∂φf |

• Observe ddtφi + 1

∆x

[(f+

i+ 12

− f+

i− 12

)+(f−i+ 1

2

− f−i− 1

2

)]= 0

• Perform biased WENO reconstruction on inputs f± to find f±,a Lipschitz continuous, consistent numerical flux

5th order WENO interpolation

xi+1/2xi-2 xi-1 xi xi+1 xi+2

S1

S2

S3

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 4 / 8

Backup

Inviscid treatment for each spatial direction

Simple procedure provides a robust, system-agnostic solver:

1 Compute global maximum eigenvalues and Roe eigenvectors

2 Project physical state and flux to characteristic space

3 Perform Lax–Friedrichs flux splitting for each characteristic field

4 Reconstruct the numerical flux in each field using WENO

5 Project characteristic numerical fluxes back to physical space

System information enters only through implementations of

• System Flux

• System Eigenvalues

• System Roe Eigenvectors

Applicable to other hyperbolic systems (e.g. magnetohydrodynamics, combustion)

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 5 / 8

Backup

New, modular Fortran 95 code designed for flexibility

• Equation-agnostic driver handles all MPI and IOconsiderations

• Equation- and problem-specific modules providerelevant physics

• New equations and problems easily added byimplementing:

I Equation of state and any unique transportequations

I Roe-averaged eigenvectors from system’s inviscidlimit

• Batch-friendly restart handling and statisticsoutput

• Reasonable performance and scalability foreffort-to-date

1

10

100W

all

tim

e p

er

tim

este

p (

s)

Number of MPI ranks

Scaling on 8192 x 2048 grid

32 MF54 MF98 MF

98

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 6 / 8

Backup

Attention to code verification and documentation

• Full serial and parallel regression test suite• Tests to ensure correct convergence order for manufactured fields

• Eigen-analysis and manufactured fields captured within MathematicaTM

• Doxygen-based documentation evolves with code

1e-16

1e-12

1e-08

0.0001

5 10 15 20

l 1 a

bsolu

te e

rror

h = 2-x

inviscid term convergence (WENO + Roe + LF)

x 3y 5x 5y 5x 9y 9

1e-16

1e-12

1e-08

0.0001

5 10 15 20

l 1 a

bsolu

te e

rror

h = 2-x

viscous term convergence (centered FD)

248

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 7 / 8

Backup

Asymmetry in bubble, spike amplitudes increases with At

0

10

20

30

40

0 0.5 1 1.5 2

bubble

am

plit

ude

time

At 0.5At 0.6At 0.7At 0.8At 0.9

0

20

40

60

80

100

0 0.5 1 1.5 2

spik

e a

mplit

ude

time

At 0.5At 0.6At 0.7At 0.8At 0.9

Mean heavy-fluid mass fraction m1 = ρ1(ρ−ρ2)ρ(ρ1−ρ2)

thresholded at 0.01, 0.99

Spike amplitudes more At-dependent than bubble amplitudes

Non-smooth behavior for At = 0.9 likely due to insufficient resolution

Ulerich, Schilling LLNL-PRES-462652 21 November 2010 8 / 8