Maximum Entropy Closure of Boltzmann-Equation Moment-Hierarchy

使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分子运动距离与物体尺寸相当 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，航天器重返地球和地外大气层，惯性约束聚变中发生的气体流动和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/气体混合中的一个或两个活跃的情况下。 本项目开发了一种新的方法，用于工程利益的气体动力流的计算模拟。 这是基于考虑广义的，聚合的气体属性称为时刻，结合概率论和统计学的思想。预计这项研究将提供一个新的和可行的方法，以提高预测能力，目前可用的方法，真实气体流动的数值模拟。该研究提出了一种方法，用于数值求解玻尔兹曼方程描述气体的动力学理论在平均自由程水平。这是基于一个新的方法来解决的时刻关闭的问题，投在格拉德13+ 9 N-时刻的玻尔兹曼方程的扩展，其中N是一个任意的整数。对于任何特定的N = 1，2..，可以用公式表示一组用于力矩的时间-空间微分-积分方程。 这些不是封闭的，因为每个方程都包含位于保留时间导数集合之外的两个矩，并且碰撞积分项包含分布函数本身。新的方法是通过构造局部分布函数的解析形式来在每个时间步关闭系统，该局部分布函数最大化单粒子熵的标准度量，同时满足已知矩作为给定的一组约束。 这可以作为约束优化问题在数值上完成。 其结果是一个本地的分析形式的本地分布函数，满足积极性，然后允许数值计算的未封闭的时刻和碰撞条款，实施的边界条件，然后更新的保留力矩方程的时间。 所提出的方法已被测试与一个已知的精确解的松弛时间的一个给定的初始分布函数的平衡状态进行比较。 建议将该方法应用于一系列的流动与增加的空间维度，包括稀薄和过渡Couette流，内部结构的冲击波和自由分子连续过渡流的机构在两个和三个空间维度。

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.