A spectral solution of the Boltzmann equation for the infinitely strong shock
A spectral solution of the Boltzmann equation for the infinitely strong shock
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无限强激波玻尔兹曼方程的谱解
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发表时间：
1990
期刊：
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通讯作者：
P. Das
中科院分区：
文献类型：
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作者：
R. Narasimha;P. Das
我们制定和实施一个新的谱方法的玻尔兹曼方程的解决方案，广泛使用的不可约张量理论与狄拉克符号。这些工具提供了一个透明的组织代数的方法和有效的自动化相关的计算。I他的权力所提出的方法证明了应用程序的高度非线性问题的无限强冲击。它表明，在这个极限的分布函数可以分解成一个奇异的部分，对应的分子束，这代表了超音速一侧的冲击，和一个规则的部分，提供了不断发展的'背景气体，并覆盖其余的速度空间。单独的控制方程的奇异性和定期的部分，推导出，并解决了后者在无限系列的正交函数的扩展。这种扩展的基础与Burnett（Proc. Lond. 39，385-430（1935）），但以（固定的）下游麦克斯韦为中心。由于球谐函数的存在，提供了SO（3）群的不可约表示，因此该基础有助于利用强大的群论工具。目前的扩展，不是关于局部平衡，并不意味着任何本构关系;相反，它减少了玻尔兹曼方程一个等效的无限阶非线性动力系统。一个有六种模式的解决方案显示出令人鼓舞的收敛的密度分布，对约6.7热侧平均自由程的冲击厚度。
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.