Direct Numerical Simulations (DNS) of MicroPolar Fluids.

The micropolar equations are given as

$$ \begin{align} \frac{\partial \vec{u}}{\partial t} + (\vec{u} \cdot \nabla)\vec{u} &= -\nabla p + \frac{1}{Re}\nabla^2 \vec{u} + \frac{m}{Re}\nabla \times \vec{w} \\ \frac{JN}{m}\left(\frac{\partial \vec{w}}{\partial t} + (\vec{u} \cdot \nabla)\vec{w}\right) &= \frac{1}{Re}\nabla^2 \vec{w} + \frac{N}{Re}\nabla \times \vec{u} - \frac{2N}{Re}\vec{w} \end{align} $$

where $\vec{u} \text{ and } \vec{w}$ are the linear and angular velocity vectors, respectively, $p$ is pressure and $m, J, N \text{ and } Re$ are parameters. The equations are solved in a three-dimensional channel configuration, with orthogonal polynomials (Chebyshev) in the wall-normal direction and Fourier exponentials in the streamwise and spanwise directions. For time-integration we can choose between several IMEX Runge Kutta methods of different order and accuracy. More information about micropolar fluid flow can be found in

George Sofadis and Ioannis Sarris "Microrotation viscosity effect on turbulent micropolar fluid channel flow", 
Phys. Fluids 33, 095126 (2021); doi: 10.1063/5.0063591

The DNS solver is based on shenfun. Shenfun is a high performance computing platform for solving partial differential equations (PDEs) by the spectral Galerkin method. Shenfun has quite a few dependencies

• mpi4py
• FFTW
• mpi4py-fft
• cython
• numpy
• sympy
• scipy
• h5py
• pyyaml

that are mostly straight-forward to install, or already installed in most Python environments. The first two are usually most troublesome. Basically, for mpi4py you need to have a working MPI installation, whereas FFTW is available on most high performance computer systems. If you are using conda, which is strongly reccommended, then all you need to install a fully functional shenfun, with all the above dependencies, is

conda install -c conda-forge shenfun

You probably want to install into a fresh environment, though, which can be achieved with

conda create --name shenfun -c conda-forge shenfun
conda activate shenfun

Note that this gives you shenfun with default settings. This means that you will probably get the openmpi backend. To make sure that shenfun is is installed with mpich instead do

conda create --name shenfun -c conda-forge shenfun mpich

If you do not use conda, then you need to make sure that MPI and FFTW are installed by some other means. You can then install any version of shenfun hosted on pypi using pip

pip install shenfun

whereas the following will install the latest version from github

pip install git+https://github.com/spectralDNS/shenfun.git@master

Note that a common approach is to install shenfun from conda-forge to get all the dependencies, and then build a local version by (after cloning or forking to a local folder) running from the top directory

pip install .

or

python setup.py build_ext -i

This is required to build all the Cython dependencies locally. To use the local version instead of the one installed through conda-forge, you need to add the folder where shenfun lives to the PYTHONPATH

export PYTHONPATH='local folder for shenfun':$PYTHONPATH

This is the approach used by the main developer of shenfun:-)