Nils Berglund | Vortices on a sphere like octahedral symmetry @NilsBerglund | Uploaded April 2024 | Updated October 2024, 4 minutes ago.
Like the video youtu.be/igIhR6Tbba4 , this one shows a simulation of the incompressible Euler equations on a sphere, with an initial state consisting of eight vortices. The difference is that the vortices are no spread evenly along the equator of the sphere. Their direction of rotation alternates from one to the next. The vortices tend to repel each other, in such a way that their centers arrange near the vertices of a regular octahedron. Note that there is some numerical instability causing oscillations where vortices meet shortly after the beginning of the simulation. Since they disappear again after a while, I decided to keep the simulation.
The video has two parts, showing the same simulation with two representations:
3D: 0:00
2D: 1:29
The 2D part uses an equirectangular projection of the sphere.
The color hue and radial coordinate depend on the vorticity of the fluid, which measures its quantity of rotation. In green areas, the vorticity is zero, meaning that the speed of the fluid does not change when moving laterally with respect to its velocity. In yellow and blue areas, the vorticity is large. The point of view of the observer is rotating around the polar axis of the sphere at constant latitude.
The incompressible Euler equations describe the velocity field of a non-viscous, incompressible fluid. They are strongly non-linear, because of the presence of a so-called advection term in what is known as the material derivative. Another difficulty in solving these equations numerically is the incompressibility condition, which requires the velocity to remain divergence-free. Since most numerical schemes will not conserve the divergence, a common approach is to "project" the solution on the space of divergence-free flows after each integration step.
I used here a different approach, based on the so-called stream function. By writing the velocity as the curl (rotational) of a stream function, directed along the z-axis perpendicular to the plane of the two-dimensional fluid, one ensures that the velocity is always divergence-free. The incompressible Euler equations are then equivalent to a partial differential equation for the stream function, and a Poisson equation linking stream function and vorticity. Since I had no solver for the Poisson equation at hand, I used a forced heat equation for the evolution of the vorticity, which admits the solution of the Poisson equation as a stationary solution. This induces additional errors, which are hopefully not too important. There is also a weak dissipation term, proportional to the Laplacian of the stream function, to prevent numerical blow-up, so the governing system is in fact closer to the Navier-Stokes equations with small viscosity.
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 - 2 hours 18 minutes
Part 2 - 2 hours 20 minutes
Compression: crf 23
Color scheme: Parula, originally from Matlab
mathworks.com/help/matlab/ref/colormap.html
Music: "We Lucky Few" by Hainbach@Hainbach
The simulation solves the incompressible Euler equation by discretization.
C code: github.com/nilsberglund-orleans/YouTube-simulations
#Euler_equation #fluid_mechanics #vortex
Like the video youtu.be/igIhR6Tbba4 , this one shows a simulation of the incompressible Euler equations on a sphere, with an initial state consisting of eight vortices. The difference is that the vortices are no spread evenly along the equator of the sphere. Their direction of rotation alternates from one to the next. The vortices tend to repel each other, in such a way that their centers arrange near the vertices of a regular octahedron. Note that there is some numerical instability causing oscillations where vortices meet shortly after the beginning of the simulation. Since they disappear again after a while, I decided to keep the simulation.
The video has two parts, showing the same simulation with two representations:
3D: 0:00
2D: 1:29
The 2D part uses an equirectangular projection of the sphere.
The color hue and radial coordinate depend on the vorticity of the fluid, which measures its quantity of rotation. In green areas, the vorticity is zero, meaning that the speed of the fluid does not change when moving laterally with respect to its velocity. In yellow and blue areas, the vorticity is large. The point of view of the observer is rotating around the polar axis of the sphere at constant latitude.
The incompressible Euler equations describe the velocity field of a non-viscous, incompressible fluid. They are strongly non-linear, because of the presence of a so-called advection term in what is known as the material derivative. Another difficulty in solving these equations numerically is the incompressibility condition, which requires the velocity to remain divergence-free. Since most numerical schemes will not conserve the divergence, a common approach is to "project" the solution on the space of divergence-free flows after each integration step.
I used here a different approach, based on the so-called stream function. By writing the velocity as the curl (rotational) of a stream function, directed along the z-axis perpendicular to the plane of the two-dimensional fluid, one ensures that the velocity is always divergence-free. The incompressible Euler equations are then equivalent to a partial differential equation for the stream function, and a Poisson equation linking stream function and vorticity. Since I had no solver for the Poisson equation at hand, I used a forced heat equation for the evolution of the vorticity, which admits the solution of the Poisson equation as a stationary solution. This induces additional errors, which are hopefully not too important. There is also a weak dissipation term, proportional to the Laplacian of the stream function, to prevent numerical blow-up, so the governing system is in fact closer to the Navier-Stokes equations with small viscosity.
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 - 2 hours 18 minutes
Part 2 - 2 hours 20 minutes
Compression: crf 23
Color scheme: Parula, originally from Matlab
mathworks.com/help/matlab/ref/colormap.html
Music: "We Lucky Few" by Hainbach@Hainbach
The simulation solves the incompressible Euler equation by discretization.
C code: github.com/nilsberglund-orleans/YouTube-simulations
#Euler_equation #fluid_mechanics #vortex
