Improvements to the Discrete Velocity Method for the Boltzm - PowerPoint Presentation

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Improvements to the Discrete Velocity Method for the Boltzmann EquationPeter Clarke D. Hegermiller, A.B. Morris , P.T. Bauman, P. L. Varghese, D. B. GoldsteinUniversity of Texas at AustinDepartment of Aerospace EngineeringDSMC 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 programSlide2

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 DVM2Slide3

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 preliminaryDVM 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.3Slide4

DVM FormulationThe 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.

Scaling Factors:

1,

pseudo-

maxwell

1,

pseudo-

maxwell

4Slide5

DVM FormulationWe separate the convection and collision parts of the equation4th order convection:We then approximate the collision integral with finite summations:β

5

Bobylev

, A.V., 1976,

Soviet Phys.

Dokl

., 20, 822-824.

Krook

, M., and Wu, T.T., 1977,

Phys. Fluid, 20, 1589-1595Slide6

DVM FormulationDSMC – “Fixed mass, variable velocity particles.”ηiηjϕ

η

i

η

j

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

ϕ

6Slide7

Variance ReductionDecompose ϕ into an equilibrium part and a deviation from equilibrium partAs 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” 20117δSlide8

Variance ReductionAs 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” 201108δSlide9

Variance ReductionCalculate depletion mass:Select random collision partners, either two from the deviation distribution or one from the deviation and one from the equilibrium distribution.orβ9Slide10

InterpolationBegin with conservation equations:MassMomentumEnergy The system of equations is solved:izezexeyiy

xyz

ix

o

a

b

c

Δϕ

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

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Code DevelopmentDVM 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 algorithms11Slide12

Variance Reduction with InterpolationWe 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

InterpolationSlide13

Variance Reduction with Interpolation- No interpolation- InterpolationMach 2 Shock density profile:13Slide14

Non-uniform grids in velocity spaceAdditions to the Discrete Velocity Method:Due to the interpolation scheme we can relax the requirement that β be a constant numberβiβj

β

k

14Slide15

Non-uniform grids in velocity space3D homogeneous relaxation with variable grid:The optimal configuration for the velocity grid is an area of active research15Slide16

VHS and VSSVSS is similar to VHS except the scattering is no longer isotropic .When picking post-collision velocities, sample from the scattering distribution.VHSVSSCollision probability depends on relative speed. The amount depleted during a collision is now proportional to collision probability.16Slide17

VHS ExampleDVMDSMCA Mach 2 shock with VHS:17Slide18

ηkηjηiErotEvibFuture 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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Future WorkAn 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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SummaryWe showed: Comparison between DSMC and DVM Combination of Variance Reduction with Interpolation Non-uniform velocity grids Application of VHS and VSS collision modelsFuture Work: Full implementation of internal energy including distributions of energy at every point in velocity space. Adaptable velocity grids.20