Nils Berglund | Adding tracers in 3D to vortex simulations on a rotating sphere @NilsBerglund | Uploaded May 2024 | Updated October 2024, 3 minutes ago.
The new feature of this video is that tracers have been added to the 3D visualizations. In previous videos such as youtu.be/mSCI7UmYZ7E , the tracers were only present in 2D. Due to the way I handle 3D plots, adding the tracers in 3D required a little work on the code. Apart from that, as the previous video, this one shows a solution of the compressible Euler equations on the sphere, with an initial state made of four vortices. The direction of rotation of two vortices has been changed compared to the previous simulation, which appears to make them less stable.
The video has four parts, showing the same simulation with two different color gradients and two different representations:
Speed, 3D: 0:00
Vorticity, 3D: 0:56
Speed, 2D: 1:58
Vorticity, 2D: 2:54
The 2D parts use an equirectangular projection of the sphere. The velocity field is materialized by 1000 tracer particles that are advected by the flow.
In parts 1 and 3, the color hue depends on the speed of the fluid, as does the radial coordinate in part 1. In parts 2 and 4, they depend 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 - 55 minutes 48 seconds
Part 2 - 55 minutes 1 second
Compression: crf 23
Color scheme: Parts 1 and 3 - Plasma by Nathaniel J. Smith and Stefan van der Walt
github.com/BIDS/colormap
Parts 2 and 4 - Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
Music: "Pure Potentiality" by Benjamin Martins@soundtherapy-bestsleepandr8945@TrackTribe
The simulation solves the compressible Euler equation by discretization.
C code: github.com/nilsberglund-orleans/YouTube-simulations
#Euler_equation #fluid_mechanics #vortex
The new feature of this video is that tracers have been added to the 3D visualizations. In previous videos such as youtu.be/mSCI7UmYZ7E , the tracers were only present in 2D. Due to the way I handle 3D plots, adding the tracers in 3D required a little work on the code. Apart from that, as the previous video, this one shows a solution of the compressible Euler equations on the sphere, with an initial state made of four vortices. The direction of rotation of two vortices has been changed compared to the previous simulation, which appears to make them less stable.
The video has four parts, showing the same simulation with two different color gradients and two different representations:
Speed, 3D: 0:00
Vorticity, 3D: 0:56
Speed, 2D: 1:58
Vorticity, 2D: 2:54
The 2D parts use an equirectangular projection of the sphere. The velocity field is materialized by 1000 tracer particles that are advected by the flow.
In parts 1 and 3, the color hue depends on the speed of the fluid, as does the radial coordinate in part 1. In parts 2 and 4, they depend 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 - 55 minutes 48 seconds
Part 2 - 55 minutes 1 second
Compression: crf 23
Color scheme: Parts 1 and 3 - Plasma by Nathaniel J. Smith and Stefan van der Walt
github.com/BIDS/colormap
Parts 2 and 4 - Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
Music: "Pure Potentiality" by Benjamin Martins@soundtherapy-bestsleepandr8945@TrackTribe
The simulation solves the compressible Euler equation by discretization.
C code: github.com/nilsberglund-orleans/YouTube-simulations
#Euler_equation #fluid_mechanics #vortex
