Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Random large eddy simulation for 3-dimensional incompressible viscous flows

Zihao Guo, Zhongmin Qian

TL;DR

The paper addresses instability in 3D random vortex methods by merging LES filtering with an integral representation for parabolic equations to produce a closed, Monte Carlo solvable scheme for the incompressible Navier–Stokes equations. It develops a random-LES framework that uses a Gaussian filter to obtain a locally averaged velocity and derives a closed system for the filtered velocity $\hat{u}$ and pressure $\hat{p}$, enabling practical 3D simulations without subgrid-stress modelling. An integral representation theorem in the Appendix underpins the theoretical foundation by linking forward and backward parabolic operators through time-reversed diffusion. Numerical experiments demonstrate the method's capability to capture laminar and turbulent flows, validating accuracy and computational viability for 3D incompressible viscous flows.

Abstract

We develop a numerical method for simulation of incompressible viscous flows by integrating the technology of random vortex method with the core idea of Large Eddy Simulation (LES). Specifically, we utilize the filtering method in LES, interpreted as spatial averaging, along with the integral representation theorem for parabolic equations, to achieve a closure scheme which may be used for calculating solutions of Navier-Stokes equations. This approach circumvents the challenge associated with handling the non-locally integrable 3-dimensional integral kernel in the random vortex method and facilitates the computation of numerical solutions for flow systems via Monte-Carlo method. Numerical simulations are carried out for both laminar and turbulent flows, demonstrating the validity and effectiveness of the method.

Random large eddy simulation for 3-dimensional incompressible viscous flows

TL;DR

The paper addresses instability in 3D random vortex methods by merging LES filtering with an integral representation for parabolic equations to produce a closed, Monte Carlo solvable scheme for the incompressible Navier–Stokes equations. It develops a random-LES framework that uses a Gaussian filter to obtain a locally averaged velocity and derives a closed system for the filtered velocity and pressure , enabling practical 3D simulations without subgrid-stress modelling. An integral representation theorem in the Appendix underpins the theoretical foundation by linking forward and backward parabolic operators through time-reversed diffusion. Numerical experiments demonstrate the method's capability to capture laminar and turbulent flows, validating accuracy and computational viability for 3D incompressible viscous flows.

Abstract

We develop a numerical method for simulation of incompressible viscous flows by integrating the technology of random vortex method with the core idea of Large Eddy Simulation (LES). Specifically, we utilize the filtering method in LES, interpreted as spatial averaging, along with the integral representation theorem for parabolic equations, to achieve a closure scheme which may be used for calculating solutions of Navier-Stokes equations. This approach circumvents the challenge associated with handling the non-locally integrable 3-dimensional integral kernel in the random vortex method and facilitates the computation of numerical solutions for flow systems via Monte-Carlo method. Numerical simulations are carried out for both laminar and turbulent flows, demonstrating the validity and effectiveness of the method.
Paper Structure (6 sections, 1 theorem, 51 equations, 10 figures)

This paper contains 6 sections, 1 theorem, 51 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Random large eddy simulation method
  3. Numerical schemes and numerical experiments
  4. Numerical experiments -- laminar flows
  5. Turbulence flows
  6. Appendix

Key Result

Theorem 4.1

Suppose $b(x,t)$ is divergence-free, that is, $\nabla\cdot b=0$ in the sense of distribution on $\mathbb{R}^d$. Then for $\xi\in\mathbb{R}^{d}$ and $T>0$.

Figures (10)

  • Figure 1: Velocity fields of incompressible viscous flows on $\mathbb{R}^{3}$
  • Figure 2: Vorticity fields of incompressible viscous flows flows on $\mathbb{R}^{3}$
  • Figure 3: Section Velocity fields of incompressible viscous flows on plane $z=0$
  • Figure 4: Section Velocity fields of incompressible viscous flows on plane $x=0$
  • Figure 5: Section Velocity fields of incompressible viscous flows on plane $y=0$
  • ...and 5 more figures

Theorems & Definitions (2)

  • Theorem 4.1
  • proof