A spectral solution of the Boltzmann equation for the infinitely strong shock

Abstract

We formulate and implement a new spectral method for the solution of the Boltzmann equation, making extensive use of the theory of irreducible tensors together with the symbolic notation of Dirac. These tools are shown to provide a transparent organization of the algebra of the method and the efficient automation of the associated calculations. I he power of the proposed method is demonstrated by application to the highly nonlinear problem of the infinitely strong shock. It is shown that the distribution function can in this limit be decomposed into a singular part corresponding to the molecular beam, which represents the supersonic side of the shock, and a regular part, which provides the evolving ‘ background gas and covers the rest of velocity space. Separate governing equations for the singular and regular parts are derived, and solved by an expansion of the latter in an infinite series of orthogonal functions. The basis for this expansion is the same set that was used by Burnett ( Proc. Lond. math. Soc . 39, 385-430 (1935)), but is centred around the (fixed) downstream maxwellian. This basis, because of the presence of spherical harmonics which provide an irreducible representation of the group SO (3), lends itself to the utilization of powerful group-theoretic tools. The present expansion, not being about local equilibrium, does not imply any constitutive relations; instead it reduces the Boltzmann equation to an equivalent infinite-order nonlinear dynamical system. A solution with six modes shows encouraging convergence in the density profile, towards a shock thickness of about 6.7 hot-side mean free paths.