Spectral NS Solver 1

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Overview

This library numerically solves the incompressible Navier-Stokes equations with a scalar field in two- and three-dimensional Cartesian domains using the spectral method. This is developed for a self-study purpose to experience by myself the difference between the finite-difference methods and the spectral methods.

Feature

• Fourier-Galerkin method.
• Orszag–Patterson algorithm.
• 2/3 de-aliasing.
• Pencil-based MPI parallelisation for more than 10^4 process.
• Fourth-order Runge-Kutta method (non-linear terms) combined with the integrating-factor technique (linear terms) for temporal integration.

Please refer to the documentation for details (under construction).

Dependency

• C compiler
• GNU Make
• MPI
• FFTW3

To initialise the flow field easily, I recommend

• Python with NumPy (see initial_condition/main.py)

Quick start

1. Prepare workplace

   mkdir -p /path/to/your/directory
   cd       /path/to/your/directory

2. Get source

   git clone --recurse-submodules https://github.com/NaokiHori/SpectralNSSolver1
   cd SpectralNSSolver1

3. Set initial condition

   Although the scalar field can be arbitrary, the velocity field should be solenoidal. main.py offers several examples.

   cd initial_condition
   python3 main.py 0
   cd ..

4. Build

   make clean
   make output
   make all

5. Execute

   Parameters are defined in exec.sh.

   bash exec.sh

   This may take a few minutes, depending on your machine spec.

The flow fields are saved under output/save/ in NPY format. Note that these velocities are in the spectral domain; you need to perform the inverse Fourier transform (and the normalisation) to recover the velocities in the physical domain.

If proper Python libraries are installed, you can visualise the flow fields by

python3 visualise/2d.py

3D simulation

Please checkout 3d branch and re-compile the whole source files. You also need to re-generate the initial flow field.

Reference

• Canuto et al., Spectral Methods - Fundamentals in Single Domains, Springer
• Canuto et al., Spectral Methods - Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer

Acknowledgement

I would like to thank Dr. Chris Howland for fruitful discussions.