Efficient algorithm on a nonstaggered mesh for simulating Rayleigh-Bénard convection in a box.

Area of Science:

• Computational fluid dynamics
• Heat transfer and fluid flow
• Numerical analysis

Background:

• Rayleigh-Bénard convection is a fundamental phenomenon in fluid dynamics and heat transfer.
• Accurate and efficient numerical methods are crucial for studying complex fluid flow problems.
• Existing methods may face challenges with stability or computational cost for specific boundary conditions.

Purpose of the Study:

• To develop and describe an efficient semi-implicit second-order-accurate finite-difference method.
• To investigate incompressible Rayleigh-Bénard convection in a box with various sidewall conditions.
• To assess the numerical stability, efficiency, and accuracy of the proposed method.

Main Methods:

• Employs operator-splitting and a projection method to simplify the algorithm.
• Reduces each time step to solving four Helmholtz equations and one Poisson equation.
• Utilizes fast direct solvers for efficient computation.
• Implements a single nonstaggered mesh compatible with boundary conditions.

Main Results:

• The developed method demonstrates numerical stability across different boundary conditions.
• The algorithm is efficient, reducing computational complexity per time step.
• Second-order accuracy is achieved, ensuring reliable simulation results.
• Characterization of efficiency and accuracy for representative convection problems.

Conclusions:

• The proposed finite-difference method offers an efficient and stable approach for simulating Rayleigh-Bénard convection.
• The method's versatility in handling periodic, insulated, and conducting sidewalls makes it broadly applicable.
• The numerical stability on a single nonstaggered mesh is a key advantage.