Vortex dynamics on the sphere with weaker Coriolis force
Like the video • Vortex dynamics on the sphere in a st... , this one shows a simulation of the compressible Euler equations on the sphere. The initial state is given by two counter-rotating vortices at antipodal points on the sphere. The difference is that the Coriolis force is ten times smaller than in the previous simulation.
The video has two parts, showing the same simulation with two representations:
2D: 0:00
3D: 0:58
The 2D part uses an equirectangular projection of the sphere. The velocity field is materialized by 1000 tracer particles that are advected by the flow.
The color hue depends on the speed of the fluid. The radial coordinate depends on the vorticity of the fluid, which measures its quantity of rotation. The point of view of the observer is rotating around the polar axis of the sphere at constant latitude. The white bar above the sphere points away from the polar axis in a fixed direction, to indicate the position of points with constant longitude on the sphere.
In a sense, the compressible Euler equations are easier to simulate than the incompressible ones, because one does not have to impose a zero divergence condition on the velocity field. However, they appear to be a bit more unstable numerically, and I had to add a smoothing mechanism to avoid blow-up. This mechanism is equivalent to adding a small viscosity, making the equations effectively a version of the Navier-Stokes equations. The equation is solved by finite differences, where the Laplacian and gradient are computed in spherical coordinates. Some smoothing has been used at the poles, where the Laplacian becomes singular in these coordinates.
Render time: Part 1 - 54 minutes 55 seconds
Part 2 - 56 minutes 24 seconds
Compression: crf 25
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-wor...
Music: "Ringside" by Dyalla@Dyalla
The simulation solves the compressible Euler equation by discretization.
C code: https://github.com/nilsberglund-orlea...
#Euler_equation #fluid_mechanics #vortex
