Create Your Own Navier-Stokes Spectral Method Fluid Simulation (With Python)

For today’s recreational coding exercise, we solve the Navier-Stokes equations for an incompressible viscous fluid. To do so, we will implement a spectral method. The Navier-Stokes equations approximately describe the motions of a fluid such as water. The equations can capture the phenomenon of turbulence. The system is also of interest in mathematics as a general proof for the existence and smoothness of solutions in 3D is one of the open <a href="https://en.wikipedia.org/wiki/Millennium_Prize_Problems" rel="noopener ugc nofollow" target="_blank">Millenium Prize Problems</a>. You may find the accompanying <a href="https://github.com/pmocz/navier-stokes-spectral-python" rel="noopener ugc nofollow" target="_blank">Python code on GitHub</a>. Before diving in, below is a gif of what running our simulation looks like: <img alt="Navier-Stokes simulation" src="https://miro.medium.com/v2/resize:fit:270/1*aSTHQxFaXKuKtYyB7ybEIQ.gif" style="height:300px; width:300px" /> <h2>The Navier-Stokes Equations</h2> We will consider the evolution of the velocity field v(x) of an incompressible fluid with viscosity ν, governed by the Navier-Stokes equations: <img alt="Navier-Stokes equations" src="https://miro.medium.com/v2/resize:fit:508/1*w4PDPX5rUyOkhbYfmjkg8w.png" style="height:227px; width:565px" /> where P is the fluid pressure. The pressure field in the case of incompressible fluids is special: it is not a separate variable that is evolved in time (unlike in the case of <a href="https://levelup.gitconnected.com/create-your-own-finite-volume-fluid-simulation-with-python-8f9eab0b8305" rel="noopener ugc nofollow" target="_blank">compressible fluids</a>). Instead, the pressure is the solution that keeps the velocity field divergence-free (the second equation). The velocity field experiences advection — the non-linear term on the left-hand side of the first equation — which can cause compression. The fluid also experiences diffusion due to the viscosity coefficient ν. The larger the viscosity coefficient, the ‘stickier’ the fluid becomes, like honey. <a href="https://levelup.gitconnected.com/create-your-own-navier-stokes-spectral-method-fluid-simulation-with-python-3f37405524f4">Click Here</a>