# A high-order gas-kinetic Navier-Stokes flow solver

## Abstract

The foundation for the development of modern compressible flow solver is based on the Riemann solution of the inviscid Euler equations. The high-order schemes are basically related to high-order spatial interpolation or reconstruction. In order to overcome the low-order wave interaction mechanism due to the Riemann solution, the temporal accuracy of the scheme can be improved through the Runge-Kutta method, where the dynamic deficiencies in the first-order Riemann solution is alleviated through the sub-step spatial reconstruction in the Runge-Kutta process. The close coupling between the spatial and temporal evolution in the original nonlinear governing equations seems weakened due to its spatial and temporal decoupling. Many recently developed high-order methods require a Navier-Stokes flux function under piece-wise discontinuous high-order initial reconstruction. However, the piece-wise discontinuous initial data and the hyperbolic-parabolic nature of the Navier-Stokes equations seem inconsistent mathematically, such as the divergence of the viscous and heat conducting terms due to initial discontinuity. In this paper, based on the Boltzmann equation, we are going to present a time-dependent flux function from a high-order discontinuous reconstruction. The theoretical basis for such an approach is due to the fact that the Boltzmann equation has no specific requirement on the smoothness of the initialmore »

- Authors:

- Department of Engineering Mechanics, Tsinghua University, Beijing 100084 (China)
- Mathematics Department, Hong Kong University of Science and Technology (Hong Kong)

- Publication Date:

- OSTI Identifier:
- 21417244

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 229; Journal Issue: 19; Other Information: DOI: 10.1016/j.jcp.2010.05.019; PII: S0021-9991(10)00280-9; Copyright (c) 2010 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; BOLTZMANN EQUATION; COMPRESSIBLE FLOW; EVALUATION; FUNCTIONS; INTERPOLATION; MATHEMATICAL EVOLUTION; NAVIER-STOKES EQUATIONS; NONLINEAR PROBLEMS; RUNGE-KUTTA METHOD; SMOOTH MANIFOLDS; TIME DEPENDENCE; VISCOUS FLOW; CALCULATION METHODS; DIFFERENTIAL EQUATIONS; EQUATIONS; EVOLUTION; FLUID FLOW; INTEGRO-DIFFERENTIAL EQUATIONS; ITERATIVE METHODS; KINETIC EQUATIONS; MATHEMATICAL MANIFOLDS; MATHEMATICAL SOLUTIONS; NUMERICAL SOLUTION; PARTIAL DIFFERENTIAL EQUATIONS

### Citation Formats

```
Li Qibing, E-mail: lqb@tsinghua.edu.c, Xu Kun, E-mail: makxu@ust.h, and Fu Song, E-mail: fs-dem@tsinghua.edu.c.
```*A high-order gas-kinetic Navier-Stokes flow solver*. United States: N. p., 2010.
Web.

```
Li Qibing, E-mail: lqb@tsinghua.edu.c, Xu Kun, E-mail: makxu@ust.h, & Fu Song, E-mail: fs-dem@tsinghua.edu.c.
```*A high-order gas-kinetic Navier-Stokes flow solver*. United States.

```
Li Qibing, E-mail: lqb@tsinghua.edu.c, Xu Kun, E-mail: makxu@ust.h, and Fu Song, E-mail: fs-dem@tsinghua.edu.c. 2010.
"A high-order gas-kinetic Navier-Stokes flow solver". United States.
```

```
@article{osti_21417244,
```

title = {A high-order gas-kinetic Navier-Stokes flow solver},

author = {Li Qibing, E-mail: lqb@tsinghua.edu.c and Xu Kun, E-mail: makxu@ust.h and Fu Song, E-mail: fs-dem@tsinghua.edu.c},

abstractNote = {The foundation for the development of modern compressible flow solver is based on the Riemann solution of the inviscid Euler equations. The high-order schemes are basically related to high-order spatial interpolation or reconstruction. In order to overcome the low-order wave interaction mechanism due to the Riemann solution, the temporal accuracy of the scheme can be improved through the Runge-Kutta method, where the dynamic deficiencies in the first-order Riemann solution is alleviated through the sub-step spatial reconstruction in the Runge-Kutta process. The close coupling between the spatial and temporal evolution in the original nonlinear governing equations seems weakened due to its spatial and temporal decoupling. Many recently developed high-order methods require a Navier-Stokes flux function under piece-wise discontinuous high-order initial reconstruction. However, the piece-wise discontinuous initial data and the hyperbolic-parabolic nature of the Navier-Stokes equations seem inconsistent mathematically, such as the divergence of the viscous and heat conducting terms due to initial discontinuity. In this paper, based on the Boltzmann equation, we are going to present a time-dependent flux function from a high-order discontinuous reconstruction. The theoretical basis for such an approach is due to the fact that the Boltzmann equation has no specific requirement on the smoothness of the initial data and the kinetic equation has the mechanism to construct a dissipative wave structure starting from an initially discontinuous flow condition on a time scale being larger than the particle collision time. The current high-order flux evaluation method is an extension of the second-order gas-kinetic BGK scheme for the Navier-Stokes equations (BGK-NS). The novelty for the easy extension from a second-order to a higher order is due to the simple particle transport and collision mechanism on the microscopic level. This paper will present a hierarchy to construct such a high-order method. The necessity to couple spatial and temporal evolution nonlinearly in the flux evaluation can be clearly observed through the numerical performance of the scheme for the viscous flow computations.},

doi = {},

url = {https://www.osti.gov/biblio/21417244},
journal = {Journal of Computational Physics},

issn = {0021-9991},

number = 19,

volume = 229,

place = {United States},

year = {2010},

month = {9}

}