Abstract
In this paper, we propose a simple yet powerful vortex method to numerically approximate the dynamics of an incompressible flow. The idea is to sample the distribution of the initial vortices of the fluid flow in question and then follow vortex dynamics along Taylor's Brownian fluid particles. The weak convergences of this approximation scheme are obtained for both two-dimensional (2D) and three-dimensional (3D) fluid flows, though only for small time in 3D case. Based on our method, the simulation results are quite attracting.
Highlights
Numerical methods have become important components in the study of fluid dynamics, in particular for modeling turbulence flows
The vortex dynamics is replaced by a random dynamical system in which a Brownian motion term is added to the equation of motion of the fluid particles, and the fluid particles become Brownian fluid particles
We propose a simple method of tracking the vortex dynamics of an incompressible fluid which gives surprisingly satisfactory simulations for both inviscid and viscous fluid flows
Summary
Numerical methods have become important components in the study of fluid dynamics, in particular for modeling turbulence flows. We do not evolve the Taylor diffusion by mollifying the Biot-Savart kernel as in the traditional vortex methods, but instead we sample the distribution of the initial vortices and develop the initial distribution according to the SDEs determined by the vorticity equation Let us describe this approach in more detail and at the same time establish the notations we will use throughout the paper. The main contribution of the present paper is to show that the previous simple vortex dynamics gives rise to good approximations to the motions of vortex dynamics We demonstrate this by showing several theoretical results about the approximation solutions to the vorticity equations constructed in terms of Xn and An, and by simulations based on this simple vortex method.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
