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Random vortex and expansion-rate model for Oberbeck-Boussinesq fluid flows

Zihao Guo, Zhongmin Qian, Zihao Shen

TL;DR

This work develops a vorticity–expansion-rate framework to simulate compressible Oberbeck–Boussinesq-type flows using a random vortex method. It combines a new parabolic integral representation with Biot–Savart constructions to obtain stochastic representations for velocity, temperature, and density, and extends the approach to wall-bounded domains via reflection and cut-off techniques. The method yields implicit McKean–Vlasov–type representations and decoupled SDEs in a constant-density limit, enabling Monte Carlo simulations that capture Benard-like convection and density–temperature coupling. Numerical experiments in a half-space demonstrate boundary-driven vortex dynamics and small density variations consistent with near-incompressible OB physics, supported by error-term analysis in the Appendix.

Abstract

By using a formulation of a class of compressible viscous flows with a heat source via vorticity and expansion-rate, we study the Oberbeck-Boussinesq flows. To this end we establish a new integral representation for solutions of parabolic equations subject to certain boundary condition, which allows us to develop a random vortex method for certain compressible flows and to compute numerically solutions of their dynamical models. Numerical experiments are carried out, which not only capture detailed Bénard convection but also are capable of providing additional information on the fluid density and the dynamics of expansion-rate of the flow.

Random vortex and expansion-rate model for Oberbeck-Boussinesq fluid flows

TL;DR

This work develops a vorticity–expansion-rate framework to simulate compressible Oberbeck–Boussinesq-type flows using a random vortex method. It combines a new parabolic integral representation with Biot–Savart constructions to obtain stochastic representations for velocity, temperature, and density, and extends the approach to wall-bounded domains via reflection and cut-off techniques. The method yields implicit McKean–Vlasov–type representations and decoupled SDEs in a constant-density limit, enabling Monte Carlo simulations that capture Benard-like convection and density–temperature coupling. Numerical experiments in a half-space demonstrate boundary-driven vortex dynamics and small density variations consistent with near-incompressible OB physics, supported by error-term analysis in the Appendix.

Abstract

By using a formulation of a class of compressible viscous flows with a heat source via vorticity and expansion-rate, we study the Oberbeck-Boussinesq flows. To this end we establish a new integral representation for solutions of parabolic equations subject to certain boundary condition, which allows us to develop a random vortex method for certain compressible flows and to compute numerically solutions of their dynamical models. Numerical experiments are carried out, which not only capture detailed Bénard convection but also are capable of providing additional information on the fluid density and the dynamics of expansion-rate of the flow.
Paper Structure (6 sections, 10 theorems, 112 equations, 6 figures)

This paper contains 6 sections, 10 theorems, 112 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The vorticity and expansion-rate function formulation
  3. An integral representations for parabolic equations
  4. Random vortex method for compressible flows
  5. Wall-bounded vortex system
  6. Numerical experiments of Oberbeck-Boussinesq flows in .

Key Result

Lemma 2.1

If $\rho,\phi,\omega,\theta$ and $u$ are solutions to the above initial and boundary problem, then

Figures (6)

  • Figure 1: Velocity fields of Oberbeck-Boussinesq flows on $\mathbb{R}_{+}^{2}$
  • Figure 2: Boundary velocity fields of Oberbeck-Boussinesq flows on $\mathbb{R}_{+}^{2}$
  • Figure 3: Temperature of Oberbeck-Boussinesq flows on $\mathbb{R}_{+}^{2}$
  • Figure 4: Density of Oberbeck-Boussinesq flows on $\mathbb{R}_{+}^{2}$
  • Figure 5: Vorticity of Oberbeck-Boussinesq flows on $\mathbb{R}_{+}^{2}$
  • ...and 1 more figures

Theorems & Definitions (15)

  • Lemma 2.1
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • Theorem 5.3
  • Definition 6.1
  • Lemma 6.2
  • ...and 5 more