The regularized versions of extended-hydrodynamic equations for a dilute granular gas, in terms of 10-, 13- and 14-moments, are derived from the inelastic Boltzmann equation. The regularization/parabolization is achieved by adding gradient terms that are derived following a Chapman-Enskog like gradient-expansion (H. Struchtrup, 2004, Phys. Fluids). For both granular and molecular gases, the resulting moment equations are found to be free from the well-known finite Mach-number singularity (that occurs in the Riemann problem of planar shock-waves) since the regularized gradient terms yield parabolic equations in contrast to the hyperbolic nature of original moment equations. In order to clarify the advantage of these regularized equations, the plane shock-wave problem is solved numerically for both molecular and granular gases; the calculated hydrodynamic profiles compare favourably with previous simulation results for molecular gases. For a granular gas, both regularized and non-regularized moment models predict asymmetric density and temperature profiles, with the maxima of both density and temperature occurring within the shock-layer, and the hydrodynamic fields are found to be smooth for the regularized models for all Mach-numbers studied. It is demonstrated that, unlike in the case of molecular gases, a "second" regularization of the regularized moment equations must be carried out in order to arrest the unbounded growth of density within the shock-layer in a granular gas.

中文翻译：

---

稀疏颗粒气体和平面冲击波的正则化扩展水动力方程

稀颗粒气体的扩展流体力学方程的正则化形式，从10阶，13阶和14阶矩出发，是从非弹性Boltzmann方程导出的。通过添加梯度项来实现正则化/代谢，该梯度项是根据类似于Chapman-Enskog的梯度扩展（H. Struchtrup，2004，Phys.Fluids）导出的。对于颗粒气体和分子气体，由于正则化的梯度项产生的抛物线方程与之相反，因此发现的矩方程没有众所周知的有限马赫数奇点（在平面激波的黎曼问题中出现）。原始矩方程的双曲性质。为了阐明这些正则方程的优势，对分子气体和颗粒气体都用数值方法解决了平面激波问题。计算出的流体动力学曲线与分子气体的先前模拟结果相比具有优势。对于粒状气体，正则和非正则矩模型都可以预测密度和温度分布的不对称性，密度和温度的最大值都出现在冲击层内，并且对于所有形式的正则模型，流体动力学场都是平滑的马赫数研究。已经证明，与分子气体的情况不同，必须进行正则矩方程的“第二”正则化，以阻止颗粒气体中冲击层内密度的无限增长。正规化和非正规化矩模型都可以预测密度和温度分布的不对称性，密度和温度的最大值都出现在激波层内，并且对于所有研究的马赫数的正规化模型，流体动力学场均是平滑的。已经证明，与分子气体的情况不同，必须进行正则矩方程的“第二”正则化，以阻止颗粒气体中冲击层内密度的无限增长。正则化和非正则化的矩模型都可以预测密度和温度分布的不对称性，密度和温度的最大值都出现在激波层内，并且对于所有研究的马赫数的正则化模型，流体动力学场均很平滑。已经证明，与分子气体的情况不同，必须进行正则矩方程的“第二”正则化，以阻止颗粒气体中冲击层内密度的无限增长。