New lattice Boltzmann model for simulating compressible flows
Prof. Takeshi Kataoka (Department of Mechanical Engineering, Kobe University, Japan) 片岡 武 准教授 （神戸大学大学院 工学研究科 機械工学専攻）
We have developed a new type of simple lattice Boltzmann (LB) model for the compressible Euler and NS equations based on the kinetic-equation approach proposed by Sone in 2002. The model uses the kinetic equation of the free-molecular type in the streaming process, and modifies the distribution function to its Chapman-Enskog type in the collision process. Compared with the conventional LB models which solve the kinetic equation of the BGK type, the proposed model is superior in the following two points: (i) any flow parameters, including the specific-heat ratio and three transport coefficients, can be chosen freely according to our convenience; (ii) there are no inherent errors associated with the Knudsen number. Numerical tests and error estimates confirm these merits.
Mathematical analysis of Chorin’s projection method
Prof. Kohei Soga (Department of Mathematics, Faculty of Science and Technology, Keio University, Japan) 曽我 幸平 准教授 （慶應義塾大学 理工学研究科 基礎理工学専攻）
Chorin’s projection method, originally introduced by Alexandre Joel Chorin in 1969, is a numerical technique of computational fluid dynamics. The method can be seen as the most elementary and direct approach to solve the incompressible Navier-Stokes equations in a general setting. Nowadays, many versions of the original method are known. In this talk, we revisit Chorin’s original projection method and discuss its potential to be a mathematical tool beyond a computational technique for smooth solutions. We first observe that the method yields a Leray-Hopf weak solution. Then, we apply the method to investigate time-periodic solutions, which is an attempt to capture qualitative features of flows through discrete approximation. Finally, we investigate the accuracy of the method.
Recent Developments of Nonequilibrium Thermodynamic Theories of Gases
Prof. Takashi Arima (Department of Engineering for Innovation, National Institute of Technology, Tomakomai College, Japan) 有馬 隆司 准教授 （苫小牧工業高等専門学校 創造工学科 総合自然科学系）
Nonequilibrium thermodynamic theories of continuous media of which applicable range goes beyond the local thermodynamic equilibrium have been developed. Starting from the pioneering works of Grad in the context of kinetic theory, of Cattaneo for a rigid heat conductor, and of Müller for the first phenomenological version of extended thermodynamics, several attempts have been made, for example, Rational Extended Thermodynamics, Extended Irreversible Thermodynamics, General Equation for the Nonequilibrium Reversible–Irreversible Coupling, the regularized moment approach, and others.
In this talk, we present the state of the art on these modern nonequilibrium theories focusing on Rational Extended Thermodynamics. In particular, we consider rarefied gases and discuss the linkage with the kinetic theory. Conceptual discussions of the differences among these nonequilibrium theories are also summarized.
Some formulations of the volume force in the immersed boundary method and a new approach in combination with the lattice Boltzmann method
Prof. Kosuke Suzuki (Institute of Engineering, Academic Assembly, Shinshu University, Japan) 鈴木 康祐 准教授 （信州大学 学術研究院 (工学系) ）
One of the important issues in computational fluid dynamics is to simulate moving-boundary flows efficiently. The immersed boundary method (IBM), which was proposed by Peskin in 1970s in order to simulate blood flows in the heart, has been reconsidered as an efficient method for simulating moving-boundary flows on a fixed Cartesian grid. In the IBM, it is assumed that the boundary is regarded as an infinitely thin shell, an incompressible viscous fluid fills in both inside and outside of the boundary, and the no-slip condition on the boundary is satisfied by volume force applied only near the boundary. The way to determine the volume force is the key concept of the IBM. In this talk, I introduce some formulations of the volume force in the IBM. Then, I present a new approach in combination with the lattice Boltzmann method (LBM). In this approach, the volume force of the IBM is regarded as the discontinuity of the stress tensor, and the stress tensor is calculated from the desired particle distribution functions of the LBM. This approach enables us to calculate the stress tensor on the boundary which is blurred by the volume force.
Mathematical Analysis of Moving Boundary Problems in the Kinetic Theory of Gases
Dr. Kai Koike (Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Japan) 小池 開 氏 （日本学術振興会 特別研究員(PD)）
Moving boundary problems for kinetic equations have become an active area of research mainly due to its importance in MEMS applications. It has also proved to be a source of interesting mathematical problems. Despite this, it’s mathematical theory has not developed to a satisfactory level although there are some recent progresses. In this talk, I would like to review these results hoping to stir interaction between the engeneering and the mathematical community further, which has always been an important element in this field.