Lid Cavity MATLAB

A MATLAB CFD benchmark project for the two-dimensional incompressible lid-driven cavity problem using a SIMPLE-style pressure-correction solver.

This repository contains only the MATLAB implementation.

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Overview

This project solves the 2D incompressible Navier-Stokes equations for the classical lid-driven cavity benchmark.

The solver includes:

• full 2D incompressible momentum equations,
• pressure-correction / SIMPLE-style coupling,
• loop-based MATLAB implementation,
• vectorized MATLAB implementation,
• upwind and central convection schemes,
• red-black Gauss-Seidel pressure solver,
• red-black SOR pressure solver,
• mesh and Reynolds-number studies,
• validation against Ghia et al. benchmark data,
• automatic generation of plots and summary tables.

The purpose of the project is to understand CFD from the numerical-method level rather than relying only on black-box CFD software.

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Features

• MATLAB implementation of a 2D incompressible Navier-Stokes solver.
• SIMPLE-style pressure correction.
• Collocated structured Cartesian grid.
• Vectorized and loop-based solver comparison.
• Upwind and central differencing comparison.
• RBGS and RBSOR pressure solver comparison.
• Mesh study with N = 32, 64, 128.
• Reynolds-number study with Re = 100, 400, 1000.
• Ghia benchmark validation.
• Residual, velocity, pressure, streamline, vorticity, and validation plots.

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Governing Equations

The continuity equation is:

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

The incompressible Navier-Stokes equations are:

$$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \frac{1}{Re}\nabla^2 \mathbf{u}$$

where:

• u and v are velocity components,
• p is pressure,
• Re is the Reynolds number.

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SIMPLE-Style Pressure Correction

The solver uses a pressure-correction procedure:

1. Initialize velocity and pressure.
2. Predict intermediate velocities from the momentum equations.
3. Solve the pressure correction Poisson equation.
4. Correct velocity using the pressure correction.
5. Update pressure with under-relaxation.
6. Compute residuals and repeat until convergence or maximum iteration count.

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Study Matrix

The full study runs:

Parameter	Values
Mesh sizes	32, 64, 128
Reynolds numbers	100, 400, 1000
Convection schemes	upwind, central
Pressure solvers	RBGS, RBSOR
Implementations	loop, vectorized

Total simulations:

3 x 3 x 2 x 2 x 2 = 72 simulations

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Validation

The solver is validated against the benchmark centerline velocity data from:

Ghia, U., Ghia, K. N., and Shin, C. T.
"High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method."
Journal of Computational Physics, 48(3), 387-411, 1982.

The validation compares:

• vertical centerline velocity u(y) at x = 0.5,
• horizontal centerline velocity v(x) at y = 0.5.

Validation data is included for:

• Re = 100,
• Re = 400,
• Re = 1000.

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Example Results

Residual History

Velocity Magnitude

Streamlines

Ghia Validation: u Centerline

Ghia Validation: v Centerline

Runtime Comparison

Pressure Solver Comparison

Mesh and Scheme Validation Error

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Project Structure

Lid-Cavity-Evolution/
│
├── README.md
├── LICENSE
├── .gitignore
│
├── main.m
├── main_quick.m
├── main_medium.m
├── default_config.m
│
├── run.sh
├── run_quick.sh
├── run_medium.sh
│
├── core/
├── studies/
├── validation/
├── post/
├── docs/
├── assets/
└── results/

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Important Files

Main scripts

File	Purpose
main.m	Runs the full 72-case study
main_quick.m	Runs a smaller quick study
main_medium.m	Runs an intermediate study
default_config.m	Main configuration file

Core solver

Folder	Purpose
core/	Solver, pressure correction, residuals, boundary conditions
studies/	Study automation scripts
validation/	Ghia benchmark validation
post/	Plotting and result export
assets/	Selected figures for GitHub
results/	Generated results, ignored by Git

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How to Run

Quick Run

chmod +x run_quick.sh
./run_quick.sh

or in MATLAB:

main_quick

Medium Run

chmod +x run_medium.sh
./run_medium.sh

or in MATLAB:

main_medium

Full Run

chmod +x run.sh
./run.sh

or in MATLAB:

main

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Output

Generated output is saved in:

results/data/
results/figures/

Typical generated files include:

• study_summary.csv,
• residual plots,
• velocity magnitude contours,
• pressure contours,
• streamlines,
• vorticity plots,
• Ghia validation plots.

The full generated results/ folder is not intended to be committed to GitHub.

Only selected figures are stored in:

assets/figures/

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Summary Table Columns

The generated study_summary.csv contains:

Column	Meaning
Implementation	Loop or vectorized
N	Mesh size
Re	Reynolds number
Scheme	Upwind or central
PressureSolver	RBGS or RBSOR
FinalRu	Final u-velocity residual
FinalRv	Final v-velocity residual
FinalRcMass	Normalized continuity residual
FinalRcDiv	Raw divergence residual
Runtime_s	Runtime in seconds
AvgPoissonIterations	Average pressure iterations
Ghia_u_L2	L2 error for u centerline
Ghia_v_L2	L2 error for v centerline
ValidationPass	Benchmark validation status

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Notes on MATLAB .fig Files

.fig files are MATLAB-native editable figures.

They are useful if you want to reopen a plot in MATLAB and edit:

• axes,
• labels,
• legends,
• colors,
• line widths,
• annotations.

They are not needed for GitHub and are ignored by .gitignore.

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Known Limitations

This is an educational and research-style MATLAB CFD solver.

Current limitations:

• The solver uses a collocated finite-difference formulation.
• It is not a commercial finite-volume CFD solver.
• Rhie-Chow interpolation is not implemented.
• Turbulence modeling is not included.
• High-Reynolds-number cases require sufficiently fine meshes.
• Coarse high-Reynolds-number cases can be under-resolved.
• The code prioritizes clarity and numerical comparison over maximum performance.

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Repository Hygiene

The following generated files are ignored:

*.mat
*.fig
results/data/*
results/figures/*

Only selected plots are included under:

assets/figures/

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References

1. Ghia, U., Ghia, K. N., and Shin, C. T.
   High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method.
   Journal of Computational Physics, 48(3), 387-411, 1982.

2. Patankar, S. V.
   Numerical Heat Transfer and Fluid Flow.
   Hemisphere Publishing, 1980.

3. Ferziger, J. H., Peric, M., and Street, R. L.
   Computational Methods for Fluid Dynamics.
   Springer, 2002.

4. Versteeg, H. K., and Malalasekera, W.
   An Introduction to Computational Fluid Dynamics: The Finite Volume Method.
   Pearson, 2007.

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License

This project is licensed under the MIT License.

See LICENSE for details.

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Contact

Ahmed Kandil

• GitHub: Kandil2001
• LinkedIn: Ahmed Kandil
• Email: a.akandil@outlook.com