Numerical and asymptotic analyses of the Boltzmann equation-Their engineering and mathematical aspects-

玻尔兹曼方程的数值和渐近分析-它们的工程和数学方面-

• 批准号: 10045040
• 负责人: AOKI Kazuo
• 金额: $ 2.82万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B).
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-10045040/
• 关键词: Boltzmann equation; Euler equation; boundary-layer equation; Liouville equation; Knudsen layer; rarefied gas flows; flow bifurcation; flow stability; 多成分混合気体; 速度分布関数; 輸送方程式; 希薄化効果

1. With the help of the French group (FG), the Japanese group (JG) has established the asymptotic theory (the fluid-dynamic equations, the slip boundary conditions, and the Knudsen-layer corrections near the boundary) describing the general behavior of slightly rarefied gas flows with finite Mach numbers by means of a systematic asymptotic analysis of the Boltzmann equation. This theory also contains the systematic derivation of the viscous boundary-layer equation in fluid dynamics from the Boltzmann equation. The results inspired the mathematical study of FG on the boundary-layer equation.2. JG and FG have jointly worked on the direct derivation of the fluid-dynamic equations (the Euler system of equations) from the many-particle system (the Liouville equation). Using different approaches, each group has succeeded in the formal derivation. In addition, JG reflected on the formal derivation of the Boltzmann equation from the many-particle system and provided FG with a clear and systema … More tic derivation. This leads to a possibility of a new mathematical proof of the derivation of the Boltzmann equation.3. Cooperating with FG, JG investigated the rarefied gas flow induced by the discontinuity in the boundary temperature by means of a numerical analysis and clarified the behavior of the flow for a wide range of gas rarefaction. In this flow, the discontinuity of the molecular velocity distribution function propagates from the boundary into the gas. As the first step of mathematical study, on the basis of a linear transport equation that possesses a similar property to the Boltzmann equation concerning the propagation of discontinuity but has a much simpler structure, JG and FG jointly analyzed, with mathematical rigor, the propagation of discontinuity in the medium.4. JG constructed a systematic theory for the kinetic solution scheme of the general conservation equations including fluid-dynamic equations (e.g., the standard Euler and Navier-Stokes equations) and provided FG with the theory for further mathematical study. In addition, JG, with the help of FG, succeeded in obtaining mathematically rigorous bounds of the boundary conditions for the fluid-dynamic equations on the interface where evaporation or condensation is taking place.5. JG provided FG with the system of fluid-dynamic type equations that describe the ghost effect for gaseous mixtures, and FG investigated its mathematical properties. Less

1.在法国小组（FG）的帮助下，日本小组（JG）建立了描述有限马赫数的轻度稀薄气体流动的一般行为的渐近理论（流体动力学方程、滑动边界条件和边界附近的努森层修正）通过对波尔兹曼方程的系统渐进分析。该理论还包含了从玻尔兹曼方程系统推导流体动力学中的粘性边界层方程。这一结果对FG在边界层方程上的数学研究具有启发意义. JG和FG共同致力于从多粒子系统（刘维尔方程）直接推导流体动力学方程（欧拉方程组）。使用不同的方法，每个小组都成功地进行了形式推导。此外，JG还对从多粒子系统出发推导玻尔兹曼方程进行了反思，为FG提供了一个清晰而系统的理论框架。 ...更多信息 tic派生这导致了一个新的数学证明的玻尔兹曼方程的推导的可能性。JG与FG合作，通过数值分析研究了边界温度不连续性引起的稀薄气体流动，并阐明了气体稀薄度范围很宽时的流动行为。在这种流动中，分子速度分布函数的不连续性从边界传播到气体中。作为数学研究的第一步，JG和FG基于一个与Boltzmann方程具有类似性质但结构更简单的线性输运方程，以数学的严谨性共同分析了介质中不连续性的传播. JG为包括流体动力学方程（例如，标准的欧拉和纳维-斯托克斯方程），并为FG提供了进一步数学研究的理论。此外，在FG的帮助下，JG成功地获得了蒸发或冷凝发生界面上流体动力学方程边界条件的数学严格边界. JG为FG提供了描述气体混合物鬼效应的流体动力学型方程组，FG研究了其数学性质。少

项目成果

Kazuo Aoki: "The behavior of a vapor-gas mixture in the continuum limit! Asymptotic analysis based on the Boltzmann equation"Rarefied Gas Dynamics (AIP,Melville,edited by T.J.Bartel et.al.). (印刷中). (2001)

Kazuo Aoki：“连续极限中蒸气-气体混合物的行为！基于玻尔兹曼方程的渐近分析”Rarefied Gas Dynamics（AIP，Melville，T.J.Bartel 等人编辑）（2001 年出版）。 ）

Y.Sone: "Asymptotic theory of the Boltzmann system, for a steady flow of a slightly rarefied gas with a finite Mach number : General theory"Eur.J.Mech, B/Fluids. 19. 325-360 (2000)

Y.Sone：“玻尔兹曼系统的渐近理论，对于具有有限马赫数的稍微稀薄的气体的稳定流动：一般理论”Eur.J.Mech，B/Fluids。

Kazuo Aoki: "Numerical analysis of the Taylor-Couette problem for a rarefied gas by the direct simulation Monte Carlo method"Rarefied Gas Dynamics (Cepadues,Toulouse,France edited by R.Brun et.al.). Vol.2. 109-116 (1999)

Kazuo Aoki：“通过直接模拟蒙特卡罗方法对稀薄气体的 Taylor-Couette 问题进行数值分析”Rarefied Gas Dynamics（Cepadues，图卢兹，法国，R.Brun 等人编辑）。

Shingo Kosuge: " Shock-wave structure for a binary gas mixture : finite-difference analysis of the Boltzmann equation for hard-sphere molecules"European Journal of Mechanics, B/Fluids. 20(1). 87-126 (2001)

Shingo Kosuge：“二元气体混合物的冲击波结构：硬球分子玻尔兹曼方程的有限差分分析”欧洲力学杂志，B/流体。

Kazuo Aoki: "Numerical analysis of the Taylor-Couette problem for a rarefied gas by the direct simulation Monte CaHo method"Rarefied Gas Dynamics (edited by R. Brun. Et al. Cepadues,Toulouse, France.). Vol.2. 109-116 (1999)

Kazuo Aoki：“通过直接模拟 Monte CaHo 方法对稀薄气体的 Taylor-Couette 问题进行数值分析”Rarefied Gas Dynamics（由 R. Brun. 等人编辑。 Cepadues，图卢兹，法国）。