Lid-Driven Cavity Flow Simulation

This project numerically solves the 2D lid-driven cavity flow using the finite difference method, with RK3/RK4 (Runge-Kutta 3rd/4th order) for time integration and a projection method to enforce incompressibility through the pressure Poisson equation.

Overview

The lid-driven cavity problem is a classic benchmark problem in computational fluid dynamics (CFD). It involves a square cavity where the top lid moves with a constant velocity, while all other walls remain stationary. The goal is to solve the incompressible Navier-Stokes equations to obtain the velocity and pressure fields.

Key Features

• Finite difference discretization
• RK3/RK4 for time integration of intermediate velocities
• Projection method for incompressible flow
• Numba-accelerated performance
• NPZ binary file format for efficient data saving
• Visualization using Matplotlib
• Animated GIF creation of the velocity field

Mathematical Formulation

Equations Solved

• Momentum Equation (Intermediate velocity)

  $$\frac{\partial \vec{u}}{\partial t} + (\vec{u} \cdot \nabla)\vec{u} = -\nabla p + \nu \nabla^2 \vec{u}$$

• Continuity Equation (Incompressibility Constraint)

  $$\nabla \cdot \vec{u} = 0$$

• Projection Method

  Intermediate velocity is projected onto a divergence-free space by solving the Pressure Poisson Equation:

  $$\nabla^2 p = \frac{\rho}{\Delta t} \nabla \cdot \vec{u}^*$$

Boundary Conditions

• Top wall (moving lid): $(u = 1)$, $( v = 0)$
• Other walls: $(u = 0)$, $(v = 0)$

References

• Jofre, L., Abdellatif, A., & Oyarzun, G. (2023). RHEA: an open-source Reproducible Hybrid-architecture flow solver Engineered for Academia. "Journal of Open Source Software", 13 Gener 2023, vol. 8(81), núm. 4637, p. 1-6.
• Moin, P. Fundamentals of Engineering Numerical Analysis. "Cambridge University Press", 2010.
• Pope, S. B. Turbulent Flows. "Cambridge University Press", 2000.
• Vermeire, B. C., Pereira, C. A., & Karbasian, H. Computational Fluid Dynamics: An Open-Source Approach. "Concordia University", 2020.
• Hager, G., & Wellein, G. Introduction to High Performance Computing for Scientists and Engineers. "CRC Press", 2011.
• Manneville, P. Instabilities, Chaos and Turbulence: An Introduction to Nonlinear Dynamics and Complex Systems. "Imperial College Press", 2004.