Entropy consistent formulation and numerical simulation of the BGK-Burnett equations for hypersonic flows in the continuum-transition regime.

摘要

The formulation of an extended set of hydrodynamic equations relies on the fact that these equations can be obtained by taking moments of the Boltzmann equation with the collision invariant vector. While the formulation of the Euler and Navier-Stokes equations by the moment method is relatively straightforward, the derivation of higher-order approximations is beset by two major hurdles. The formulation of the higher-order distribution functions, obtained by truncating the Chapman-Enskog expansion to the desired degree of accuracy, involves the evaluation of the highly non-linear collision integral. Further, the form of the higher-order distribution function is non-unique as it does not satisfy the moment closure property. While the difficulty of evaluating the collision integral is circumvented by approximating the same by well known model equations, the problem of moment closure plays a vital role in determining a stable set of equations that is also entropy consistent.; This dissertation presents the development of a novel set of second-order hydrodynamic equations for computing hypersonic flows in the continuum-transition regime. These equations, termed the BGK-Burnett equations, are obtained by approximating the collision integral by the Bhatnagar-Gross-Krook (BGK) model. A closed form expression for the second-order distribution function is developed by enforcing the moment closure property and solving the resulting system of algebraic equations. Subsequently, through a series of conjectures, the closure coefficients are designed to move the equations towards an entropy consistent set.; A unique feature of the higher-order hydrodynamic equations is the appearance of material derivatives in the higher-order fluxes. The approximations used to represent these derivatives are determined by applying an entropy consistent relaxation technique to the hypersonic shock structure problem. The resulting family of BGK-Burnett equations is shown to be stable to small wavelength disturbances and entropy consistent for a wide range of grid points and Mach numbers.; In order to consider a practical application, the BGK-Burnett equations are used to compute the hypersonic flow field about a blunt body for flow conditions that simulate moderately high free stream Knudsen numbers. It is shown that there are substantial differences in the solutions of the Navier-Stokes and BGK-Burnett equations as the Knudsen number increases.