Maximum Entropy Closure of Boltzmann-Equation Moment-Hierarchy

利用先进的计算平台，对高速和低速气体流动进行数值模拟，其中分子碰撞之间的距离与身体尺寸相当，这在空气动力学和气体流体力学中仍然是一个具有挑战性的学科。重要的实际例子是低地球轨道卫星的运动、重返地球和地外大气层的空间飞行器、在惯性约束聚变中发生的气体流动以及纳米装置的流体动力学行为，特别是在分子-表面相互作用和流体/气体混合活跃的情况下。本项目开发了一种具有工程意义的气体动力流动计算模拟的新方法。这是基于考虑广义的、聚集的气体性质，即矩，并结合了概率论和统计学的思想。该研究将为实际气体流动的数值模拟提供一种新的可行方法，并具有比现有方法更高的预测能力。本研究提出了一种在平均自由程水平上描述气体动力学理论的玻尔兹曼方程的数值解方法。这是基于一种矩闭问题的新方法，用Boltzmann方程的Grad 13+ 9n矩展开来表达，其中N是一个任意整数。对于任意特定的N =1,2..，一组时间-空间，微分-积分方程的矩可以表述。这些不是封闭的，因为每个方程都包含在保留时间导数集合之外的力矩，而且碰撞积分项也包含分布函数本身。新的方法是通过构建局部分布函数的解析形式来关闭系统，该函数可以最大化单粒子熵的标准测量，同时满足已知矩作为给定约束集。这可以作为一个约束优化问题在数值上完成。结果是满足正性的局部分布函数的局部解析形式，然后允许对未闭矩和碰撞项进行数值计算，实现边界条件，然后及时更新保留的矩方程。通过与已知的初始分布函数向平衡态随时间弛豫的精确解进行比较，验证了所提出的方法。提出了将该方法应用于空间维数增加的一系列流动，包括稀化和过渡库埃特流、激波内部结构和自由分子到二维和三维空间中关于物体的连续过渡流。

The numerical simulation, using advanced computational platforms, of both high and low-speed flows of gases where the distance that molecules travel between molecular collisions is comparable with a body dimension remains a challenging discipline in both aerodynamics and gas-fluid mechanics in general. Important practical examples are the motion of low-earth orbit satellites, space-vehicle re-entry into both earth and extra-terrestrial atmospheres, gas-flows that occur in inertial-confinement fusion and the fluid-dynamical behavior of nano-devices, particularly where either or both molecule-surface interactions and fluid/gas mixing are active. The present project develops a novel methodology for the computational simulation of gas-dynamic flows of engineering interest. This is based on consideration of generalized, aggregated gas properties known as moments, combined with ideas from the theory of probability and statistics. It is expected that this research will provide a new and viable approach to the numerical simulation of real-gas flows with improved predictive power over presently available methods. The research proposes a methodology for the numerical solution of the Boltzmann equation describing the kinetic theory of gases at the mean-free path level. This is based on a new approach to the moment-closure problem, cast in terms of the Grad 13+9N-moment expansion of the Boltzmann equation, where N is an arbitrary integer. For any specific N =1,2.., a set of time-space, differential-integral equations for moments can be formulated. These are not closed because each equation contains both moments that lie outside the set of retained time derivatives, and also collision-integral terms contain the distribution function itself. The new approach is to close the system at each time step by constructing an analytic form of the local distribution function that maximizes a standard measure of the single-particle entropy, while satisfying the known moments as a given set of constraints. This can be done numerically as a constrained optimization problem. The result is a local analytic form for the local distribution function that satisfies positivity, and which then allows numerical evaluation of unclosed moments and collision terms, implementation of boundary conditions followed by an update of the retained moment equations in time. The proposed method has been tested by comparison with a known exact solution of the relaxation in time of a given initial distribution function towards an equilibrium state. It is proposed to apply the method to a sequence of flows with increasing space dimensionality, including rarefied and transition Couette flow, the internal structure of shock waves and free-molecule to continuum transition flows about bodies in two and three space dimensions.