🌀 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
• Features
• Why the Lid-Driven Cavity?
• Why the SIMPLE Algorithm?
• Governing Equations
• SIMPLE Algorithm Steps
• Numerical Methods & Boundary Conditions
• Project Roadmap
• Project Structure
• Benchmark Table
• Getting Started
• Contributing
• Code of Conduct
• Security
• Support & Discussion
• License
• References
• Citation
• 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	✅ Complete
	Vectorized NumPy solver	✅ Complete
	Parallel Python (Numba/Dask)	🚧 In Progress
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

Note: This is for a 50x50 Grid

Solver	Language	Paradigm	Elapsed Time (s)	Speedup	Status
SIMPLE2D_LidDrivenCavity	MATLAB	Serial (Loops)	1126,03	1x	✅ Complete
SimpleLidCavityVector	MATLAB	Serial (Vectorized)	1266,38	1.125x	✅ Complete
lid_cavity_serial.py	Python/NumPy	Serial (Loops)	85703.92	76x	✅ Complete
lid_cavity_vectorized.py	Python/NumPy	Serial (Vectorized)	3732.37	3.3x	✅ Complete
_lid_cavity_parallel.py_mpi	Python	Parallel	TBD	TBD	🚧 In Progress
_lid_cavity_parallel.py_openmp	Python	Parallel	TBD	TBD	🚧 In Progress
OpenFOAM Case	OpenFOAM	Industrial CFD	TBD	TBD	📋 Planned
STAR-CCM+ Case	STAR-CCM+	Commercial CFD	TBD	TBD	📋 Planned

Note: This is for a 150x150 Grid

Solver	Language	Paradigm	Elapsed Time (s)	Speedup	Status
SIMPLE2D_LidDrivenCavity	MATLAB	Serial (Loops)	TDB	TBD	🚧 In Progress
SimpleLidCavityVector	MATLAB	Serial (Vectorized)	TDB	TBD	🚧 In Progress
lid_cavity_serial.py	Python/NumPy	Serial (Loops)	TDB	TBD	🚧 In Progress
lid_cavity_vectorized.py	Python/NumPy	Serial (Vectorized)	TDB	TBD	🚧 In Progress
_lid_cavity_parallel.py_mpi	Python	Parallel	TBD	TBD	🚧 In Progress
_lid_cavity_parallel.py_openmp	Python	Parallel	TBD	TBD	🚧 In Progress
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 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. By participating, you are expected to uphold this code.

Security

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

Support & Discussion

• For bug reports or feature requests, please use our issue tracker.
• For general questions, open an issue or reach out via email.

License

This project is licensed under the MIT License.
See 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
• Reach out via email: [email protected]
• Connect on LinkedIn

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