An extended gas-kinetic scheme for shock structure calculations

Abstract

In super/hyper-sonic flows in near space and low speed flows around MEMS, the gas distribution function deviates from the equilibrium, because the molecular mean free path is comparable with the characteristic length of the flow in certain part of the flow flied. The non-equilibrium distribution function makes the constitutive relations of stress and heat flux deviate from the linear form used in NS equation. Methods for rarefied flow (DSMC and DVM) and unified methods (UGKS, GKUA and DUGKS) are able to simulate non-equilibrium flows, while, for flows in near continuum regimes, their computational costs are much larger than that of the continuum NS method. For the flows in near space and around MEMS, the rarefaction and non-equilibrium effects are not too strong as these in free molecular flows. Therefore, an Extend Gas-Kinetic Scheme (EGKS) is proposed in the present article for near continuum flows and weak non-equilibrium flows where certain amount of non-equilibrium exists. As an extension of the Gas-Kinetic Scheme (GKS), by using a perturbation of the distribution function that has the maximum entropy under the constrains of conservation variables, stress and heat flux, the nonlinear constitutive relations can be directly introduced into the EGKS. There are many nonlinear constitutive relations so far, whose numerical predictions are conducted in their own frameworks. There is a huge demand for a unified and physical framework in order to investigate their validity and accuracy. Since all these constitutive relations are derived from the gas-kinetic theory, a GKS framework is chosen in consideration of physical accommodation. Burnett, SCB, RCE and NCCR nonlinear constitutive relations are used in the EGKS along with linear Stokes-Fourier relations. Using the shock structure cases in a wide range of Mach number from weak non-equilibrium to high non-equilibrium (Mach number from 1.2 to 20), the EGKS scheme and the nonlinear constitutive relations are verified. The density, temperature, stress and heat flux profiles along with shock thickness and density-temperature separation, are systematically examined. Since the accuracy of the EGKS depends on the model of stress and heat flux used, the EGKS with nonlinear constitutive relations gives much better predictions than that with linear constitutive relations in high Mach number cases. Benefit from the present framework, the RCE and NCCR constitutive relations are also modified using the concept and property of the entropy generation of collision term, the modified constitutive relations provide good predictions in high Mach number shock structure calculations.