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

Narasimha, Roddam and Das, P (1988) A spectral solution of the Boltzmann equation for the infinitely strong shock. Technical Report. National Aeronautical Laboratory, Bangalore, India.

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    Abstract

    Two key features of the spectral method formulated and implemented here for the solution of the Boltzmann equation are the extensive use of the theory of irreducible tensors together with the symbolic notation of Dirac. These tools enable a transparent organisation of the algebra of the method and the efficient automation of the associated calculations. The power of the proposed13; spectral technique is demonstrated by application to the highly nonlinear problem of the infinitely strong shock. It is shown that the distribution function can in this case be taken as the sum of a singular part corresponding13; to the molecular beam that represents the supersonic side of the shock, and a regular part that provides the evolving 'background' gas and covers the rest of velocity space. Separate governing equations for the singular and13; 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 (1935), but is centred around the (fixed)13; downstream Maxwellian. The well-known failure of the classical Burnett expansion in the strong-shock problem is attributable, in the present view, to its inability to remove the singularity that develops in the distribution as the approach Mach number tends to infinity, and its resort to a non-convergent expansion around local equilibrium. The Burnett basis itself (suitably centred) is most appropriate, especially because its use of spherical harmonics, which provide an irreducible representation of the group S0(3), enables economy in calculation using group-theoretic tools.13; 13; The present expansion reduces the Boltzmann equation to an equivalent infinite-order nonlinear dynamical system. Solution with six Burnett modes shows encouraging convergence in the density profile, towards a shock thickness of 6.7 hot-side mean free paths.

    Item Type: Proj.Doc/Technical Report (Technical Report)
    Uncontrolled Keywords: Spectral solution;Boltzmann equation;Shock structure;Irreducible tensors;Group theory;Hypersonic flow;13; Rarefied gas dynamics
    Subjects: ENGINEERING > Fluid Mechanics and Thermodynamics
    Division/Department: Fluid Dynamics, Other
    Depositing User: Mr. Ravikumar R
    Date Deposited: 26 Sep 2006
    Last Modified: 24 May 2010 09:51
    URI: http://nal-ir.nal.res.in/id/eprint/2823

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