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Tracking Brownian fluid particles in large eddy simulations

Zihao Guo, Zhongmin Qian

TL;DR

The paper addresses the challenge of simulating wall-bounded turbulent flows by integrating the random vortex method with large-eddy simulation through a filtering operator and a stochastic integral representation using Brownian fluid particles. It develops a forward-in-time Monte Carlo scheme for the filtered velocity and provides explicit representations for the pressure gradient in wall-bounded domains via Neumann Green functions and half-space Biot-Savart kernels, including wall reflections. Numerical experiments in two and three dimensions, covering laminar and turbulent regimes, demonstrate numerical stability, moderate computational cost, and the ability to reveal wall-layer flow mechanisms. The approach closes the Navier–Stokes equations without additional subgrid-stress models and offers a repeatable, mesh-flexible framework for studying near-wall turbulence.

Abstract

In this paper, we propose an approach for simulating wall-bounded incompressible turbulent flows by integrating the technology of random vortex method with the core principles of large-eddy simulations (LES). In particular, we employ the filtering function, interpreted as a spatial averaging operator, together with the integral representation theorem for parabolic equations, to construct a closed numerical scheme suitable for computing solutions to the Navier-Stokes equations. This framework numerically overcomes the difficulties associated with the non-locally integrable three-dimensional kernel inherent in the random vortex method, enabling efficient computation of flow fields via the Monte Carlo method. Several numerical experiments are presented for both laminar and turbulent flows in wall-bounded domains, to thereby reveal the underlying flow mechanisms near the wall boundary. The experimental results and systematic comparisons with alternative numerical approaches consistently demonstrate that the proposed method is numerically stable, possesses low theoretical complexity, and achieves acceptable computational efficiency.

Tracking Brownian fluid particles in large eddy simulations

TL;DR

The paper addresses the challenge of simulating wall-bounded turbulent flows by integrating the random vortex method with large-eddy simulation through a filtering operator and a stochastic integral representation using Brownian fluid particles. It develops a forward-in-time Monte Carlo scheme for the filtered velocity and provides explicit representations for the pressure gradient in wall-bounded domains via Neumann Green functions and half-space Biot-Savart kernels, including wall reflections. Numerical experiments in two and three dimensions, covering laminar and turbulent regimes, demonstrate numerical stability, moderate computational cost, and the ability to reveal wall-layer flow mechanisms. The approach closes the Navier–Stokes equations without additional subgrid-stress models and offers a repeatable, mesh-flexible framework for studying near-wall turbulence.

Abstract

In this paper, we propose an approach for simulating wall-bounded incompressible turbulent flows by integrating the technology of random vortex method with the core principles of large-eddy simulations (LES). In particular, we employ the filtering function, interpreted as a spatial averaging operator, together with the integral representation theorem for parabolic equations, to construct a closed numerical scheme suitable for computing solutions to the Navier-Stokes equations. This framework numerically overcomes the difficulties associated with the non-locally integrable three-dimensional kernel inherent in the random vortex method, enabling efficient computation of flow fields via the Monte Carlo method. Several numerical experiments are presented for both laminar and turbulent flows in wall-bounded domains, to thereby reveal the underlying flow mechanisms near the wall boundary. The experimental results and systematic comparisons with alternative numerical approaches consistently demonstrate that the proposed method is numerically stable, possesses low theoretical complexity, and achieves acceptable computational efficiency.
Paper Structure (18 sections, 10 theorems, 96 equations, 16 figures, 3 tables)

This paper contains 18 sections, 10 theorems, 96 equations, 16 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Theoretical aspect of random LES
  3. Computation of the pressure gradient
  4. Representing the velocity in terms of fluid particles
  5. Random LES method
  6. Random LES for flows in $\mathbb{R}^{d}$
  7. Random LES for flows past a wall
  8. Numerical experiments and analysis
  9. Numerical schemes and variables settings
  10. Experiments for random LES method
  11. Experiment 1: 2-dimensional advection through a wall (laminar flows)
  12. Experiment 2: 2-dimensional advection through a wall (turbulent flows)
  13. Experiment 3: 3-dimensional initially stationary fluid
  14. Experiment 4: 3-dimensional advection through a wall (turbulent flows)
  15. Analysis and comparison
  16. ...and 3 more sections

Key Result

Lemma 1

(2D flows). For a 2D incompressible viscous flow the gradient of the pressure at any instance has the following singular integral representation for $x\in\mathbb{R}_{+}^{2}$, where the last singular integral kernel is given by for $x_{2}>0$.

Figures (16)

  • Figure 1: Velocity fields of wall-bounded laminar flows on $\mathbb{R}_{+}^{2}$
  • Figure 2: Vorticity of wall-bounded laminar flows on $\mathbb{R}_{+}^{2}$
  • Figure 3: Velocity fields of wall-bounded turbulent flows on $\mathbb{R}_{+}^{2}$
  • Figure 4: Vorticity of wall-bounded turbulent flows on $\mathbb{R}_{+}^{2}$
  • Figure 5: Velocity fields wall-bounded flows on $\mathbb{R}^{3}_{+}$
  • ...and 11 more figures

Theorems & Definitions (15)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 5 more