Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An efficient fully decoupled finite element method with second-order accuracy for the micropolar Rayleigh-Benard convection system

Ming Cui, Akang Hou, Xiaoyu Dong

TL;DR

This work develops a fully decoupled, linear, second-order-in-time finite element method for the nonlinear micropolar Rayleigh-Bénard convection system. By leveraging a second-order pressure projection approach, the velocity, pressure, angular velocity, and temperature are updated through independent linear solves at each time step, enabling significant computational efficiency. The authors prove unconditional stability and derive optimal error estimates using a dual-norm framework with inverse Stokes operators. Numerical experiments in 2D and 3D, including accuracy tests, passive-scalar stirring, and lid-driven cavity flows, validate the method’s accuracy, stability, and robustness, highlighting its potential for complex microstructured fluid applications.

Abstract

The micropolar Rayleigh-B{é}nard convection system, which consists of Navier-Stokes equations, the angular momentum equation, and the heat equation, is a strongly nonlinear, coupled, and saddle point structural multiphysics system. A second-order pressure projection finite element method, which is linear, fully decoupled, and second-order accurate in time, is proposed to simulate the system. Only a few decoupled linear elliptic problems with constant coefficients are solved at each time step, simplifying calculations significantly. The stability analysis of the method is established and the optimal error estimates are derived rigorously with the negative norm technique. Extensive numerical simulations, including 2D and 3D accuracy tests, the lid-driven cavity flow, and the passive-scalar mixing experiment, are carried out to illustrate the effectiveness of the method.

An efficient fully decoupled finite element method with second-order accuracy for the micropolar Rayleigh-Benard convection system

TL;DR

This work develops a fully decoupled, linear, second-order-in-time finite element method for the nonlinear micropolar Rayleigh-Bénard convection system. By leveraging a second-order pressure projection approach, the velocity, pressure, angular velocity, and temperature are updated through independent linear solves at each time step, enabling significant computational efficiency. The authors prove unconditional stability and derive optimal error estimates using a dual-norm framework with inverse Stokes operators. Numerical experiments in 2D and 3D, including accuracy tests, passive-scalar stirring, and lid-driven cavity flows, validate the method’s accuracy, stability, and robustness, highlighting its potential for complex microstructured fluid applications.

Abstract

The micropolar Rayleigh-B{é}nard convection system, which consists of Navier-Stokes equations, the angular momentum equation, and the heat equation, is a strongly nonlinear, coupled, and saddle point structural multiphysics system. A second-order pressure projection finite element method, which is linear, fully decoupled, and second-order accurate in time, is proposed to simulate the system. Only a few decoupled linear elliptic problems with constant coefficients are solved at each time step, simplifying calculations significantly. The stability analysis of the method is established and the optimal error estimates are derived rigorously with the negative norm technique. Extensive numerical simulations, including 2D and 3D accuracy tests, the lid-driven cavity flow, and the passive-scalar mixing experiment, are carried out to illustrate the effectiveness of the method.

Paper Structure

This paper contains 14 sections, 5 theorems, 68 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The micropolar Rayleigh-Bénard convection system
  3. Numerical scheme and Stability analysis
  4. Second-order pressure projection scheme
  5. Stability analysis
  6. Error estimates
  7. The suboptimal error estimate
  8. The optimal error estimate
  9. Numerical examples
  10. Accuracy tests in 2D
  11. Accuracy tests in 3D
  12. Stirring of a passive scalar
  13. Driven cavity flows in 2D
  14. Conclusions

Key Result

Lemma 2.1

Let $\Omega$ be a bounded Lipschitz domain such that for all $\boldsymbol{v}\in X$,

Figures (9)

  • Figure 1: Numerical distributions of the velocity and pressure.
  • Figure 2: Numerical distributions of the angularr velocity and temperature.
  • Figure 3: Distribution map of numerical solutions for velocity $\mathbf{u}$, angular velocity $w$, pressure $p$ and temperature $\theta$ at $t=50$.
  • Figure 4: Mixing evolution of a convected passive scalar $\phi$ over time. Each subfigure shows the scalar field at a specific time $t$. For better visual comparison, the color axis is fixed to [-1,1], although the actual scalar field $\phi$ is constrained in [0, 1]
  • Figure 5: Visualization of the numerical solution distribution for streamlines.
  • ...and 4 more figures

Theorems & Definitions (8)

  • Lemma 2.1: div-grad relation inequality Pyo2009Pyo2013
  • Lemma 2.2: He2015
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof