A high-order gas-kinetic Navier-Stokes flow solver
- Department of Engineering Mechanics, Tsinghua University, Beijing 100084 (China)
- Mathematics Department, Hong Kong University of Science and Technology (Hong Kong)
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.
- OSTI ID:
- 21417244
- Journal Information:
- Journal of Computational Physics, Vol. 229, 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.; ISSN 0021-9991
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
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