Recent Topics in Mathematical Fluid Mechanics from Turbulence Model to Topological Fluid Dynamics

Takashi Sakajo
Kyoto University

2017-06-26 ~ 2017-06-26


Science Building No. 1 1303


Mathematical fluid mechanics is a field of applied mathematics where many researchers try to describe complex fluid phenomena with using modern mathematical theories. It includes not only fundamental but also interdisciplinary research topics from turbulence theory to engineering and medical applications. In the seminar, I would like to introduce my recent research topics in this field.The first topic is motivated by fluid turbulence with cascade phenomenon. One way to tackle the problem of fluid turbulence is to study its model with drastically reduced degrees of freedom. These models are made in mostly phenomenological ways to qualitatively have a few selected aspects of turbulence. As a minimal mathematical model, we propose a one-dimensional partial differential equation that conserves the integral of the squared vorticity analog (enstrophy) in the inviscid case. With a large-scale random forcing an small viscosity, we find numerically that the model exhibits the enstrophy cascade, the broad energy spectrum with a sizable correction to the dimensional-analysis prediction, peculiar intermittency, and self-similarity in the dynamical system structure. This is the joint work with Dr. Takeshi Matsumoto in Kyoto University.The second topic is coming from topological fluid dynamics. We are concerned with streamline topologies of incompressible fluid flows, which are the level sets of the Hamiltonian. We provide a classification procedure to assign a unique sequence of letters, called word representation and tree representations to every structurally stable streamline pattern. Owing to this procedure, we can identify any streamline pattern with those symbolic representations uniquely topologically. In addition, based this theory, we propose a combinatorial method to provide a list of possible transient structurally unstable streamline patterns between two different structurally stable patterns by simply comparing those representations. We also demonstrate how the present theory is applied to fluid flow problems in real worlds. This is the joint work with Tomoo Yokoyama in Kyoto University of Education.