https://github.com/kandil2001/lid-cavity-evolution

Lid Cavity Evolution is an open-source CFD suite for the lid-driven cavity problem, featuring MATLAB, Python, and parallel solvers with benchmark comparisons.
https://github.com/kandil2001/lid-cavity-evolution

benchmarking cfd cfd-simulation lid-driven-cavity matlab mpi numpy open-source openfoam openmp parallel-computing python scientific-computing simple-algorithm star-ccm

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Lid Cavity Evolution is an open-source CFD suite for the lid-driven cavity problem, featuring MATLAB, Python, and parallel solvers with benchmark comparisons.

Host: GitHub
URL: https://github.com/kandil2001/lid-cavity-evolution
Owner: Kandil2001
License: mit
Created: 2025-09-07T18:34:52.000Z (4 months ago)
Default Branch: main
Last Pushed: 2025-09-15T17:58:15.000Z (4 months ago)
Last Synced: 2025-09-15T19:23:35.144Z (4 months ago)
Topics: benchmarking, cfd, cfd-simulation, lid-driven-cavity, matlab, mpi, numpy, open-source, openfoam, openmp, parallel-computing, python, scientific-computing, simple-algorithm, star-ccm
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Open Issues: 0
Metadata Files:
- Readme: README.md
- Contributing: CONTRIBUTING.md
- License: LICENSE
- Code of conduct: CODE_OF_CONDUCT.md
- Security: SECURITY.md

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🌀 Lid Cavity Evolution

A benchmark suite for unsteady incompressible CFD: From MATLAB fundamentals to industrial applications




  

    

  

  

    

  

  

    

  

  

    

  

  

    

  

  

    

  

  

    

  

  

    

  

  

    

  

  

  





  





# Lid Cavity Evolution

_A benchmark suite for unsteady incompressible CFD: From MATLAB fundamentals to industrial applications_

## Table of Contents

- [Introduction](#introduction)

- [Features](#features)

- [Why the Lid-Driven Cavity?](#why-the-lid-driven-cavity)

- [Why the SIMPLE Algorithm?](#why-the-simple-algorithm)

- [Governing Equations](#governing-equations)

- [SIMPLE Algorithm Steps](#simple-algorithm-steps)

- [Numerical Methods & Boundary Conditions](#numerical-methods--boundary-conditions)

- [Project Roadmap](#project-roadmap)

- [Project Structure](#project-structure)

- [Benchmark Table](#benchmark-table)

- [Getting Started](#getting-started)

- [Contributing](#contributing)

- [Code of Conduct](#code-of-conduct)

- [Security](#security)

- [Support & Discussion](#support--discussion)

- [License](#license)

- [References](#references)

- [Citation](#citation)

- [Contact](#contact)

## Introduction

**Lid Cavity Evolution** is an open-source benchmark suite that chronicles the development of the classic lid-driven cavity CFD problem—from foundational MATLAB scripts to industrial-grade solvers. The project emphasizes accuracy, performance, and reproducibility for unsteady incompressible flow simulation.

## Why the Lid-Driven Cavity?

- **Standard Test Case:** Simple geometry, well-defined boundaries, and established reference solutions make it ideal for CFD code verification.

- **Rich Physics:** Captures vortex formation, boundary layers, and evolving flow structures.

- **Unsteady Simulation:** Tracks time evolution of flow fields, including transient and nonlinear phenomena.

## Why the SIMPLE Algorithm?

Simulating incompressible flows is numerically challenging due to tight coupling of velocity and pressure. The **SIMPLE (Semi-Implicit Method for Pressure-Linked Equations)** algorithm is widely used because it:

- Decouples momentum and continuity equations for robust convergence.

- Uses a staggered grid to prevent checkerboard pressure artifacts.

- Applies under-relaxation for improved stability and efficiency.

## Governing Equations

The solver models unsteady, incompressible, two-dimensional flow in a square cavity with a moving lid.

### 1. Continuity Equation (Incompressibility)

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

- $\mathbf{u}$: Velocity vector, $\mathbf{u} = (u, v)$

  - $u$: velocity in $x$-direction

  - $v$: velocity in $y$-direction

This equation enforces conservation of mass for incompressible flow: the net flow into any control volume is zero.

### 2. Momentum Equations (Navier-Stokes, Unsteady, 2D)

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

Where each term means:

- $\frac{\partial \mathbf{u}}{\partial t}$: **Unsteady term** — Time rate of change of velocity.

- $(\mathbf{u} \cdot \nabla)\mathbf{u}$: **Convection term** — Transport of momentum by fluid motion.

- $- \nabla p$: **Pressure gradient term** — Acceleration caused by pressure differences.

- $\frac{1}{Re}\nabla^2 \mathbf{u}$: **Diffusion term** — Viscous spreading of momentum.

- $Re = \frac{UL}{\nu}$: **Reynolds number**  

    - $U$: Lid velocity  

    - $L$: Cavity length  

    - $\nu$: Kinematic viscosity

#### Expanded in 2D Components

- **x-Momentum:**

  $\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = -\frac{\partial p}{\partial x} + \frac{1}{Re} \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right )$

  - $\frac{\partial u}{\partial t}$: Time derivative of $u$

  - $u \frac{\partial u}{\partial x}$: Convection of $u$ in $x$

  - $v \frac{\partial u}{\partial y}$: Convection of $u$ in $y$

  - $-\frac{\partial p}{\partial x}$: Pressure gradient in $x$

  - $\frac{1}{Re} \frac{\partial^2 u}{\partial x^2}$: Viscous diffusion in $x$

  - $\frac{1}{Re} \frac{\partial^2 u}{\partial y^2}$: Viscous diffusion in $y$

- **y-Momentum:**

  $\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = -\frac{\partial p}{\partial y} + \frac{1}{Re} \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right )$

  - $\frac{\partial v}{\partial t}$: Time derivative of $v$

  - $u \frac{\partial v}{\partial x}$: Convection of $v$ in $x$

  - $v \frac{\partial v}{\partial y}$: Convection of $v$ in $y$

  - $-\frac{\partial p}{\partial y}$: Pressure gradient in $y$

  - $\frac{1}{Re} \frac{\partial^2 v}{\partial x^2}$: Viscous diffusion in $x$

  - $\frac{1}{Re} \frac{\partial^2 v}{\partial y^2}$: Viscous diffusion in $y$

## SIMPLE Algorithm Steps

The SIMPLE algorithm solves these equations with the following procedure:

1. **Predictor Step:**  

   - Solve momentum equations for an intermediate velocity $\mathbf{u}^*$ using the current pressure estimate.

2. **Pressure Correction:**  

   - Solve the pressure correction Poisson equation:  

     $\nabla^2 p' = \frac{1}{\Delta t} \nabla \cdot \mathbf{u}^*$  

     where $p'$ is the pressure correction and $\Delta t$ is the time step.

3. **Corrector Step:**  

   - Update velocities and pressure:  

     $\mathbf{u}^{n+1} = \mathbf{u}^* - \Delta t \nabla p'$  

     $p^{n+1} = p^{n} + \alpha p'$  

     where $\alpha$ is the under-relaxation factor ($0 < \alpha \leq 1$).

## Numerical Methods & Boundary Conditions

- **Spatial Discretization:** Second-order central differencing

- **Time Discretization:** First-order implicit Euler

- **Grid:** Staggered (pressure at cell centers, velocities at faces)

- **Boundary Conditions:**

    - **Top lid:** $u = 1$, $v = 0$ (moving wall)

    - **Other walls:** $u = v = 0$ (no-slip)

    - **Pressure:** Neumann ($\frac{\partial p}{\partial n} = 0$) at boundaries

## Project Roadmap

| Phase | Description | Status |

|-------|-------------|--------|

| **Phase 1** | MATLAB loop-based SIMPLE solver   | ✅ Complete |

|             | Vectorized MATLAB implementation  | ✅ Complete |

| **Phase 2** | Python/NumPy serial port          | 🚧 In Progress |

|             | Vectorized NumPy solver           | 📋 Planned |

|             | Parallel Python (Numba/Dask)      | 📋 Planned |

| **Phase 3** | OpenFOAM case setup               | 📋 Planned |

|             | STAR-CCM+ case setup              | 📋 Planned |

| **Phase 4** | Validation & analysis             | 📋 Planned |

## Project Structure

```

main/

├── matlab/

│ ├── iterative-solver/

│ │ ├── IterativeSolver.m

│ │ └── README.md

│ ├── vectorized-solver/

│ │ ├── VectorizedSolver.m

│ │ └── README.md

│ └── README.md # MATLAB-specific overview

├── python/

│ ├── serial/

│ │ ├── iterative/

│ │ │ ├── IterativeSolver.py

│ │ │ └── README.md

│ │ ├── vectorized/

│ │ │ ├── VectorizedSolver.py

│ │ │ └── README.md

│ │ └── README.md # Serial solvers overview

│ ├── parallel/

│ │ ├── mpi/

│ │ │ ├── MPISOlver.py

│ │ │ └── README.md

│ │ ├── openmp/

│ │ │ ├── OpenMPSolver.py

│ │ │ └── README.md

│ │ └── README.md # Parallel solvers overview

│ └── README.md # Python-specific overview

├── logos/ # Technology logos

├── assets/

├── .github/ISSUE_TEMPLATE/ # GitHub issue templates

├── CODE_OF_CONDUCT.md

├── CONTRIBUTING.md

├── LICENSE

├── SECURITY.md

└── README.md # Main project documentation

```

## Benchmark Table

| Solver                                   | Language      | Paradigm                | Elapsed Time (s) | Speedup | Status         |

|-------------------------------------------|--------------|-------------------------|------------------|---------|---------------|

| SIMPLE2D_LidDrivenCavity                  | MATLAB       | Serial (Loops)          | 36435            | 1x      | ✅ Complete    |

| _SimpleLidCavityVector_                   | MATLAB       | Serial (Vectorized)     | TBD              | TBD     | 🚧 In Progress |

| _lid_cavity_serial.py_                    | Python/NumPy | Serial (Loops)          | TBD              | TBD     | 📋 Planned     |

| _lid_cavity_vectorized.py_                | Python/NumPy | Serial (Vectorized)     | TBD              | TBD     | 📋 Planned     |

| _lid_cavity_parallel.py_                  | Python       | Parallel                | TBD              | TBD     | 📋 Planned     |

| _OpenFOAM Case_                           | OpenFOAM     | Industrial CFD          | TBD              | TBD     | 📋 Planned     |

| _STAR-CCM+ Case_                          | STAR-CCM+    | Commercial CFD          | TBD              | TBD     | 📋 Planned     |

_Hardware: 

## Getting Started

### Prerequisites

- **MATLAB:** R2020a or later (for initial implementations)

- **Python:** 3.8+ (NumPy, planned)

- **OpenFOAM:** (planned)

- **STAR-CCM+:** (planned)

## Contributing

We welcome all contributions! Please see [CONTRIBUTING.md](CONTRIBUTING.md) for guidelines on:

- Adding new solver implementations

- Improving code or documentation

- Adding validation cases

- Reporting issues or suggesting enhancements

Feel free to open an issue or submit a pull request!

## Code of Conduct

Please note that this project follows a [Code of Conduct](CODE_OF_CONDUCT.md). By participating, you are expected to uphold this code.

## Security

If you discover a vulnerability, please see our [SECURITY.md](SECURITY.md) for instructions on reporting it.

## Support & Discussion

- For bug reports or feature requests, please use our [issue tracker](https://github.com/Kandil2001/Lid-Cavity-Evolution/issues).

- For general questions, open an issue or reach out via [email](mailto:kandil.ahmed.amr@gmail.com).

## License

This project is licensed under the MIT License.  

See [LICENSE](LICENSE) for details.

## References

1. Ghia, U., Ghia, K. N., & Shin, C. T. (1982). _High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method_. J. Comput. Phys., 48(3), 387-411.

2. Patankar, S. V. (1980). _Numerical Heat Transfer and Fluid Flow_. Hemisphere Publishing.

3. Ferziger, J. H., Perić, M., & Street, R. L. (2002). _Computational Methods for Fluid Dynamics_. Springer.

## Contact

For questions, suggestions, or collaboration inquiries, feel free to:

- Open an issue on [GitHub](https://github.com/Kandil2001/Lid-Cavity-Evolution/issues)

- Reach out via email: **kandil.ahmed.amr@gmail.com**

- Connect on [LinkedIn](https://www.linkedin.com/in/ahmed-kandil01)

> **Note:** This project is under active development. Check back for updates and new solver releases!

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