2011 DSMC Workshop

Improvements to the Discrete Velocity Method for the Boltzmann Equation

Peter Clarke

D. Hegermiller, A.B. Morris , P.T. Bauman, P. L. Varghese, D. B. Goldstein

University of Texas at AustinDepartment of Aerospace Engineering

DSMC Workshop September 2011

Funding: Some of this material is based upon work supported by the DOE [National Nuclear Security Administration] under Award Number [DE-FC52-08NA28615] and NASA’s NSTRF Fellowship program

2011 DSMC Workshop Outline

• Motivation

• The Discrete Velocity Method

• Previous work

• Variance reduction using an interpolation scheme

• Non-uniform grids in velocity space

• Application of VHS and VSS collision models

• Future Work• Inclusion of internal energy in DVM

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2011 DSMC Workshop Motivation

• Discrete velocity methods are comparable to DSMC

• But discrete velocity methods have several traditional problems

• High Mach number flows and other flows that require large velocity bounds.

• The inclusion of physics in the model such as varying types of molecular potentials, multi-species flow, internal

energy, and chemical reactions are often neglected in preliminary

DVM investigations.

• We wish to solve the first problem with the eventual application of

adaptive velocity grids.

• The first step towards this goal is the implementation of non uniform grids in velocity space.

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2011 DSMC Workshop DVM Formulation

ˆˆˆ])ˆ(ˆ)ˆ(ˆ)'ˆ(ˆ)'ˆ(ˆ[

1ˆ

ˆˆ

ˆˆ

dVgKnxt T

depletingngreplenishi

depl Trepl T dVgdVgxt

ˆˆ ˆˆ)]ˆ(ˆ)ˆ(ˆ[ˆˆ)]'ˆ(ˆ)'ˆ(ˆ[ˆ

ˆˆ

ˆˆ

The collision integral is split into replenishing and depleting parts:

We begin with the scaled Boltzmann equation:

To solve the Boltzmann equation using DVM, we must discretize the integro-differential equation.

L

xx ˆ

r ˆ

m

Tk rbr

22

3ˆ

r

rn

Ltt r

ˆr

ˆ

r

gg

ˆ

Scaling Factors:

1, pseudo-maxwell 1, pseudo-maxwell

4

2011 DSMC Workshop DVM Formulation

collisionsconvectiont

ˆˆ'ˆ

ijk ijk

tt 33 )ˆ(ˆ)ˆ(ˆˆ)'ˆ(ˆ)'ˆ(ˆˆˆ'ˆ

We separate the convection and collision parts of the equation

4th order convection: 22111122' nnnonnn

We then approximate the collision integral with finite summations:

β

5Bobylev, A.V., 1976, Soviet Phys. Dokl., 20, 822-824.Krook, M., and Wu, T.T., 1977, Phys. Fluid, 20, 1589-1595

2011 DSMC Workshop DVM Formulation

DSMC – “Fixed mass, variable velocity particles.”

ηi

ηj

ϕ

ηi

ηj

DVM – “Fixed velocity, variable mass quasi-particles.”

ϕ

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2011 DSMC Workshop Variance Reduction

ˆˆˆ EQ

Decompose ϕ into an equilibrium part and a deviation from equilibrium part

As has been previously presented by A. Morris we use a stochastic discrete velocity model:

Baker, L.L. and N.G. Hadjiconstantinou, "Variance Reduction for Monte Carlo Solutions of the Boltzmann Equation," Physics of Fluids, 17, 2005Morris, A.B., “Variance Reduction for a Discrete Velocity Gas” 2011

7

δ

2011 DSMC Workshop

ˆ

ˆ

ˆ

ˆˆ)]ˆ(ˆ)ˆ(ˆ)'ˆ(ˆ)'ˆ(ˆ[

ˆˆ)]ˆ(ˆ)ˆ(ˆ)'ˆ(ˆ)'ˆ(ˆ[2

ˆˆ)]ˆ(ˆ)ˆ(ˆ)'ˆ(ˆ)'ˆ(ˆ[

dVdg

dVdg

dVdgI

EQEQ

EQEQEQEQ

Variance Reduction

As has been previously presented by A. Morris we use a stochastic discrete velocity model:

Baker, L.L. and N.G. Hadjiconstantinou, "Variance Reduction for Monte Carlo Solutions of the Boltzmann Equation," Physics of Fluids, 17, 2005Morris, A.B., “Variance Reduction for a Discrete Velocity Gas” 2011

0

8

δ

2011 DSMC Workshop Variance Reduction

Calculate depletion mass:

Select random collision partners, either two from the deviation distribution or one from the deviation and one from the equilibrium distribution.

1; 0sgn( )

1; 0

xx

x

31,

ˆˆ)ˆsgn(ˆˆ

c

EQ

N

nnt

32,2

ˆˆ)ˆsgn()ˆsgn(ˆˆ

cN

nnt

or

β

9

spacingmesh

densitynumbern

collisionsofnumberNc

ˆ

2011 DSMC Workshop Interpolation

06/)(

)(

)(

)(

01

222

222

cbacbaf

fcf

fbf

faf

cbaf

e

extiz

extiy

extix

o

Begin with conservation equations:

MassMomentumEnergy

The system of equations is solved:

iz

ez

ex

ey

iy

xyz

ix

o

ab

c

Δϕ

Varghese, P.L., “Arbitrary Post-Collision Velocities in a Discrete Velocity Scheme for the Boltzmann Equation.” 2007 10

2011 DSMC Workshop Code Development

DVM has been implemented using modern software engineering principles that enhance maintainability and ease of testing to allow more thorough verification of the software and, thus, increases confidence that the implementation is correct. These practices include:

• Object-oriented code style to enhance encapsulation and minimize code duplication.

• Source code revision control using svn.

• Build system (Autotools) for portability between computing systems (code currently tested on Linux and Mac OS X environments)

• Build system also enables easy addition of unit and regresssion tests. Current suite is at 42 tests.

• Full documentation of code and algorithms

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2011 DSMC Workshop Variance Reduction with Interpolation

We combine the variance reduction technique with the interpolation scheme that has been developed:

4th, 6th, and 8th moments of the relaxation of the BKW distribution

- Interpolation

- analytic

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- No interpolation

No Interpolation

Interpolation

2011 DSMC Workshop Variance Reduction with Interpolation

- No interpolation

- Interpolation

Mach 2 Shock density profile:

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2011 DSMC Workshop Non-uniform grids in velocity space

Additions to the Discrete Velocity Method:

Due to the interpolation scheme we can relax the requirement that β be a constant number

ijk ijk

kjikji tt )ˆ(ˆ)ˆ(ˆ)'ˆ(ˆ)'ˆ(ˆ'

βi

βj

βk

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2011 DSMC Workshop Non-uniform grids in velocity space

3D homogeneous relaxation with variable grid:

The optimal configuration for the velocity grid is an area of active research15

2011 DSMC Workshop VHS and VSS

/1

1cos2d

b

deplrepldVgdVg

xt

ˆ)1(

ˆ)1( ˆ)]ˆ(ˆ)ˆ(ˆ[ˆ)]'ˆ(ˆ)'ˆ(ˆ[

ˆ

ˆˆ

ˆˆ

VSS is similar to VHS except the scattering is no longer isotropic .

When picking post-collision velocities, sample from the scattering distribution.

VHS VSS

refrefT g

g

Collision probability depends on relative speed.

The amount depleted during a collision is now proportional to collision probability.

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2011 DSMC Workshop VHS Example

DVM

DSMC

A Mach 2 shock with VHS:

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2011 DSMC Workshop

ηk

ηj

ηi

Erot Evib

Future Work

A major addition to the Discrete Velocity Method that allows for more accurate physics is the inclusion of internal energy

We assign a single internal energy to each location in velocity space.

Future work will allow a distribution of energies at every velocity location

Internal Energy:

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2011 DSMC Workshop Future Work

An exchange in energy between translation and internal energy is calculated using a Landau-Teller-like equation.

The exchange changes the magnitude of the post-collision relative velocity vector as well as adding or subtracting from the internal energy distributions.

Interpolation allows for any post collision relative velocity vector length.

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2011 DSMC Workshop Summary

We showed: • Comparison between DSMC and DVM

• Combination of Variance Reduction with Interpolation

• Non-uniform velocity grids

• Application of VHS and VSS collision models

Future Work:• Full implementation of internal energy including distributions of energy at every point in velocity space.

• Adaptable velocity grids.

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