Efficient Simulation of Nonlinear Flows in Rarefied Gases

气体流动的计算通常基于速度场的Navier-Stokes方程，结合能量平衡和傅里叶定律(如果感兴趣的温度场)。然而，这些经典模型只适用于接近热平衡的流动。对于稀薄气体或微观环境中的情况，已知纳维-斯托克斯-傅立叶系统(NSF)会产生物理错误的结果，因此不能在这些情况下使用。近年来，在Boltzmann方程的矩方程的基础上发展了新的连续介质模型，扩展了传统流体力学的适用范围。其中一个模型是正则化的13矩系统(R13)，它形成了一组关于密度、速度、温度以及应力张量和热流的偏微分方程组。在早期的DFG项目中，R13系统已经成功地发展成为一种成熟的模拟工具，可以用于慢速稀薄气体系统的微观模拟，但经典的矩方程(如R13系统)的主要缺点之一是对小扰动的限制，如慢速、低马赫数过程。数学上的原因可以从用来逼近气体颗粒速度分布函数的展开技术中找到。目前的项目探索了将矩方程的使用扩展到强非线性过程的新方法，这些过程表现出速度和温度的强烈变化，如高超声速中的冲击波。在这个项目中，我们将结合两种现代方法从矩重建分布：统计中的最大似然/熵估计和多相/多分散流中引入的矩的求积方法。一个成功和有效的重建将使处理边界条件和实现数值方法变得容易。一个受应用启发的测试案例将考虑卫星推进器的气流撞击。

The computation of gas flows is typically based on the Navier-Stokes equations for the velocity field combined with the energy balance and Fourier's law if the temperature field is of interest. However, these classical models are valid only for flows close to thermal equilibrium. For situations in rarefied gases or in microscopic settings the Navier-Stokes-Fourier system (NSF) is known to produce physically wrong results, so that they can not be used in these cases. Over the past years new continuum models have been developed on the bases of moment equations of the Boltzmann equation which extend the range of applicability of conventional fluid dynamics. One of these models is the Regularized 13-Moment system (R13) which forms a set of partial differential equations for density, velocity and temperature, as well as stress tensor and heat flux. In an earlier DFG project the R13 system has been successfully developed to a mature simulation tool for slow rarefied gas systems, like in microscopic settings.However, one of the main drawbacks of classical moment equations like the R13 system is the restriction to small perturbation like in slow, low Mach number processes. The mathematical reason can be found in the expansion technique used to approximate the velocity distribution function of the gas particles. The current project explores new ways to extend the use of moment equations to strongly nonlinear processes, exhibiting strong variations in velocity and temperature, like shock waves in hypersonics. During the project we will combine two modern approaches to reconstruct a distribution from moments: maximum likelihood/entropy estimation from statistics and the quadrature method of moments, introduced in multi-phase/multi-disperse flows. A successful and efficient reconstruction will allow to handle boundary conditions and to implement a numerical method easily. One application-inspired test case will consider flow impingement of satellite thrusters.