Simulation of Navier-Stokes Equations by Pseudo-Spectral Method

Introduction

This project implements a numerical simulation of the 2D Navier-Stokes equations in the $\omega-\psi$ formulation using the pseudo-spectral method. This approach enables efficient resolution of fluid flows in the spectral domain (Fourier space).

Fr En

Main Features

• 2D simulation of Navier-Stokes equations in the $\omega-\psi$ formulation
• Use of the pseudo-spectral method (Fourier Transforms with FFTW)
• GTK graphical interface for parameter control
• Parallelization with OpenMP for better performance
• Real-time visualization of results
• Export of results to video via ffmpeg

Mathematical Formulation and Pseudo-Spectral Method

A detailed demonstration of the $\omega-\psi$ formulation is available in docs/English/demonstration_NS.md. This formulation transforms the Navier-Stokes equations into a coupled system that is simpler to solve numerically.

The pseudo-spectral method combines the advantages of spectral methods and physical space methods:

1. Fourier Transforms: Spatial derivatives are computed in spectral space, where they become simple multiplications.
2. Non-linear terms: Computed in physical space to avoid costly convolutions.

For more details on the implementation, see docs/English/Pseudo_Spectral_method.md.

Project Structure

navier-stokes-spectral/
├── app/                    # Main Fortran code
├── src/                    # Fortran source code
├── docs/                   # Documentation
│   ├── French/             # Documentation in French
│   └── English/            # Documentation in English
├── data/                   # Folder for results
└── fpm.toml                # Project configuration

Types of Simulated Flows

The program allows simulation of three classic flow types in fluid mechanics:

1. Co-rotating and Counter-rotating Vortex Simulation

This simulation shows the interaction of several vortices that can rotate in the same direction (co-rotating) or in opposite directions (counter-rotating). This phenomenon is particularly interesting in aerodynamics and meteorology.

More details

2. Kelvin-Helmholtz Instability

This instability occurs at the interface between two fluids moving at different speeds. It manifests as the formation of characteristic vortices.

More details

3. Taylor-Green Vortex

This classic test case in fluid mechanics allows the study of the transition to turbulence.

More details

Prerequisites

The following dependencies are required:

• Fortran Compiler (gfortran recommended)
• GTK (version 4.x)
• fpm (version 0.9.0 or higher)
• FFTW (version 3.x)
• ffmpeg (for video export)
• OpenMP (for parallelization)

Installation

Linux (Debian/Ubuntu)

# Install system dependencies
sudo apt-get update
sudo apt-get install gfortran libgtk-3-dev libfftw3-dev ffmpeg libomp-dev

# Install fpm
curl -Lo fpm https://github.com/fortran-lang/fpm/releases/download/v0.11.0/fpm-0.11.0-linux-x86_64-gcc-12
chmod +x fpm
sudo mv fpm /usr/local/bin

Windows (MSYS2)

# Install dependencies
pacman -Syu
pacman -S mingw-w64-x86_64-gcc-fortran mingw-w64-x86_64-gtk3 mingw-w64-x86_64-fftw mingw-w64-x86_64-ffmpeg

# Install fpm
pacman -S git mingw-w64-x86_64-gcc-fortran mingw-w64-x86_64-fpm

macOS (with Homebrew)

# Install dependencies
brew install gcc gtk+3 fftw ffmpeg libomp

# Install fpm
brew tap fortran-lang/homebrew-fortran
brew install fpm

Usage

Compilation and Execution

# Clone the repository
git clone https://github.com/Minard-Jules/navier-stokes-spectral
cd navier-stokes-spectral

# Compile and run
fpm run

Simulation Configuration

1. Open the graphical interface
2. Set the parameters :
   • Spatial resolution (Nx, Ny)
   • Reynolds number
   • Time step
   • Simulation duration
3. Select the type of flow
4. Start the simulation

Visualization

Available Visualization Types

• Velocity fields
• Vorticity
• Stream function

Colormap Options

Blue Orange Colormap (divergent)

vorticity_mag.mp4

'jet' Colormap

vorticity_mag_jet.mp4

Exporting Results

Results are automatically saved in the data/ folder in the following formats :

• Data files (.vtk)
• Videos (.mp4)

License

This project is licensed under the MIT License - see the LICENSE file for details.

Credits

• Minard Jules: Creator and main maintainer of the project