BGK-Burnett equations: A new set of second-order hydrodynamic equations for flows in continuum-transition regime

This paper presents a review of the development of a novel set of second-order hydrodynamic equations, designated as the BGK-Burnett equations for computing flows in the continuum-transition regime.. The second-order distribution function that forms the basis of this formulation is obtained by the first three terms of the Chapman-Enskog expansion applied to the Boltzmann equation with Bhatnagar-Gross-Krook (BGK) approximation to the collision terms. Such a distribution function, however, does not readily satisfy the moment closure property. Hence, an exact closed form expression for the distribution function is obtained by enforcing moment closure and solving a system of algebraic equations to determine the closure coefficients. Through a series of conjectures, the closure coefficients are designed to move the resulting system of hydrodynamic equations towards an entropy consistent set. An important step in the formulation of the higher-order distribution functions is the proper representation of the material derivatives in terms of the spatial derivatives. While the material derivatives in the first-order distribution function are approximated by Euler Equations, proper representations of these derivatives in the second-order distribution function are determined by an entropy consistent relaxation technique. The BGK-Burnett equations, obtained by taking moments of the Boltzmann equation with the second-order distribution function, are shown to be stable to small wavelength disturbances and entropy consistent for a wide range of grid points and Mach numbers. The paper also describes other equations of Burnett family namely the original, conventional and augmented Burnett equations for the purpose of comparison with BGK-Burnett equations and discusses their shortcomings. The relationship between the Burnett equations and the Grad's 13 moment equations as shown by Struchtrup by employing the Maxwell-Truesdell-Green iteration is also presented.