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

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 du...

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Veröffentlicht in:	Journal of computational physics Jg. 229; H. 19; S. 6715 - 6731
Hauptverfasser:	Li, Qibing, Xu, Kun, Fu, Song
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Kidlington Elsevier Inc 20.09.2010 Elsevier
Schlagworte:	BOLTZMANN EQUATION CALCULATION METHODS COMPRESSIBLE FLOW Computation Computational techniques DIFFERENTIAL EQUATIONS EQUATIONS EVALUATION EVOLUTION Exact sciences and technology FLUID FLOW Flux FUNCTIONS Gas-kinetic scheme High-order method INTEGRO-DIFFERENTIAL EQUATIONS INTERPOLATION ITERATIVE METHODS KINETIC EQUATIONS Mathematical analysis MATHEMATICAL EVOLUTION MATHEMATICAL MANIFOLDS MATHEMATICAL METHODS AND COMPUTING Mathematical methods in physics Mathematical models MATHEMATICAL SOLUTIONS NAVIER-STOKES EQUATIONS Navier–Stokes eqautions NONLINEAR PROBLEMS NUMERICAL SOLUTION PARTIAL DIFFERENTIAL EQUATIONS Physics Reconstruction RUNGE-KUTTA METHOD SMOOTH MANIFOLDS Temporal logic TIME DEPENDENCE VISCOUS FLOW Gas-kinetic scheme Boltzmann equation High-order method Navier–Stokes eqautions Discontinuity Second order Euler equations Particle collision Calculation methods Hierarchy Interpolation Divergences Decoupling Kinetic equations Navier-Stokes eqautions Dynamics First order Calculation Particle transport Runge-Kutta methods Performance Compressible flow Stokes flow Viscous flow Navier-Stokes equations
ISSN:	0021-9991, 1090-2716
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Zusammenfassung:	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.
Bibliographie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23
ISSN:	0021-9991 1090-2716
DOI:	10.1016/j.jcp.2010.05.019