# A coupled ordinates method for solution acceleration of rarefied gas dynamics simulations

## Abstract

Non-equilibrium rarefied flows are frequently encountered in a wide range of applications, including atmospheric re-entry vehicles, vacuum technology, and microscale devices. Rarefied flows at the microscale can be effectively modeled using the ellipsoidal statistical Bhatnagar–Gross–Krook (ESBGK) form of the Boltzmann kinetic equation. Numerical solutions of these equations are often based on the finite volume method (FVM) in physical space and the discrete ordinates method in velocity space. However, existing solvers use a sequential solution procedure wherein the velocity distribution functions are implicitly coupled in physical space, but are solved sequentially in velocity space. This leads to explicit coupling of the distribution function values in velocity space and slows down convergence in systems with low Knudsen numbers. Furthermore, this also makes it difficult to solve multiscale problems or problems in which there is a large range of Knudsen numbers. In this paper, we extend the coupled ordinates method (COMET), previously developed to study participating radiative heat transfer, to solve the ESBGK equations. In this method, at each cell in the physical domain, distribution function values for all velocity ordinates are solved simultaneously. This coupled solution is used as a relaxation sweep in a geometric multigrid method in the spatial domain. Enhancementsmore »

- Authors:

- NNSA PRISM: Center for Prediction of Reliability, Integrity and Survivability of Microsystems (United States)
- (United States)

- Publication Date:

- OSTI Identifier:
- 22465625

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 289; Other Information: Copyright (c) 2015 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; BOLTZMANN EQUATION; BOUNDARY CONDITIONS; CONVERGENCE; DISCRETE ORDINATE METHOD; DISTRIBUTION FUNCTIONS; HEAT TRANSFER; NUMERICAL SOLUTION; RAREFIED GASES; SIMULATION; VELOCITY

### Citation Formats

```
Das, Shankhadeep, E-mail: shankhadeep@utexas.edu, Department of Mechanical Engineering, The University of Texas at Austin, TX 78712, Mathur, Sanjay R., Department of Mechanical Engineering, The University of Texas at Austin, TX 78712, Alexeenko, Alina, School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907, Murthy, Jayathi Y., and Department of Mechanical Engineering, The University of Texas at Austin, TX 78712.
```*A coupled ordinates method for solution acceleration of rarefied gas dynamics simulations*. United States: N. p., 2015.
Web. doi:10.1016/J.JCP.2015.02.035.

```
Das, Shankhadeep, E-mail: shankhadeep@utexas.edu, Department of Mechanical Engineering, The University of Texas at Austin, TX 78712, Mathur, Sanjay R., Department of Mechanical Engineering, The University of Texas at Austin, TX 78712, Alexeenko, Alina, School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907, Murthy, Jayathi Y., & Department of Mechanical Engineering, The University of Texas at Austin, TX 78712.
```*A coupled ordinates method for solution acceleration of rarefied gas dynamics simulations*. United States. doi:10.1016/J.JCP.2015.02.035.

```
Das, Shankhadeep, E-mail: shankhadeep@utexas.edu, Department of Mechanical Engineering, The University of Texas at Austin, TX 78712, Mathur, Sanjay R., Department of Mechanical Engineering, The University of Texas at Austin, TX 78712, Alexeenko, Alina, School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907, Murthy, Jayathi Y., and Department of Mechanical Engineering, The University of Texas at Austin, TX 78712. Fri .
"A coupled ordinates method for solution acceleration of rarefied gas dynamics simulations". United States. doi:10.1016/J.JCP.2015.02.035.
```

```
@article{osti_22465625,
```

title = {A coupled ordinates method for solution acceleration of rarefied gas dynamics simulations},

author = {Das, Shankhadeep, E-mail: shankhadeep@utexas.edu and Department of Mechanical Engineering, The University of Texas at Austin, TX 78712 and Mathur, Sanjay R. and Department of Mechanical Engineering, The University of Texas at Austin, TX 78712 and Alexeenko, Alina and School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907 and Murthy, Jayathi Y. and Department of Mechanical Engineering, The University of Texas at Austin, TX 78712},

abstractNote = {Non-equilibrium rarefied flows are frequently encountered in a wide range of applications, including atmospheric re-entry vehicles, vacuum technology, and microscale devices. Rarefied flows at the microscale can be effectively modeled using the ellipsoidal statistical Bhatnagar–Gross–Krook (ESBGK) form of the Boltzmann kinetic equation. Numerical solutions of these equations are often based on the finite volume method (FVM) in physical space and the discrete ordinates method in velocity space. However, existing solvers use a sequential solution procedure wherein the velocity distribution functions are implicitly coupled in physical space, but are solved sequentially in velocity space. This leads to explicit coupling of the distribution function values in velocity space and slows down convergence in systems with low Knudsen numbers. Furthermore, this also makes it difficult to solve multiscale problems or problems in which there is a large range of Knudsen numbers. In this paper, we extend the coupled ordinates method (COMET), previously developed to study participating radiative heat transfer, to solve the ESBGK equations. In this method, at each cell in the physical domain, distribution function values for all velocity ordinates are solved simultaneously. This coupled solution is used as a relaxation sweep in a geometric multigrid method in the spatial domain. Enhancements to COMET to account for the non-linearity of the ESBGK equations, as well as the coupled implementation of boundary conditions, are presented. The methodology works well with arbitrary convex polyhedral meshes, and is shown to give significantly faster solutions than the conventional sequential solution procedure. Acceleration factors of 5–9 are obtained for low to moderate Knudsen numbers on single processor platforms.},

doi = {10.1016/J.JCP.2015.02.035},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = ,

volume = 289,

place = {United States},

year = {2015},

month = {5}

}