Nils Berglund | A laminar flow over an immersed octahedron @NilsBerglund | Uploaded August 2024 | Updated October 2024, 1 minute ago.
This is a simulation of a the shallow water equation similar to the one in the video youtu.be/7OdRw0kuQ0I , but with a shallow layer of water, and with an initial state flowing eastwards.
The depth in the shallow water equation has been computed from the distance between the sphere and an embedded octahedron of smaller volume, completely contained in the sphere. The water depth influences the wave speed, and thereby the water height, which reveals the presence of the octahedron.
The shallow water equations are nonlinear equations, which give a better description of the motion of water than the linear wave equation. In particular, unlike the linear wave equation, they conserve the total volume of water. The linear equation gives an approximation of the solutions, when the wave height remains close to its average over space.
The equations used here include viscosity and dissipation, as described for instance in
en.wikipedia.org/wiki/Shallow_water_equations#Non-conservative_form , including the Coriolis force.
The video has four parts, showing the same simulation with two different color schemes and two different visualizations:
Wave height, 3D: 0:00
Velocity (norm and direction), 3D: 0:48
Wave height, 2D: 1:42
Velocity (norm and directions), 2D: 2:30
In the first and third part, the color hue and radial coordinate show the height of the water. In the second and fourth part, the color hue shows the direction of the flow, and the radial coordinate shows its speed (the norm of the velocity). The 2D parts use a projection in equirectangular coordinates. In the 3D parts, the point of view is slowly rotating around the sphere in a plane containing its center. The position of the sphere is represented by a line starting above the north pole, and pointing in a fixed direction.
The velocity field is materialized by 2000 tracer particles that are advected by the flow.
Render time: 3D part - 1 hour 22 minutes
2D part - 1 hour 44 minutes
Color scheme: Parts 1 and 3 - Magma by Nathaniel J. Smith and Stefan van der Walt
github.com/BIDS/colormap
Parts 2 and 4 - Turbo, by Anton Mikhailovgist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
Music: "Lusciousness" by Asher Fulero@AsherFulero
See also
https://images.math.cnrs.fr/des-ondes-dans-mon-billard-partie-i/ for more explanations (in French) on a few previous simulations of wave equations.
The simulation solves the 2D shallow water equation by discretization (finite differences).
C code: github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code!
#shallowwater #waves #wave
This is a simulation of a the shallow water equation similar to the one in the video youtu.be/7OdRw0kuQ0I , but with a shallow layer of water, and with an initial state flowing eastwards.
The depth in the shallow water equation has been computed from the distance between the sphere and an embedded octahedron of smaller volume, completely contained in the sphere. The water depth influences the wave speed, and thereby the water height, which reveals the presence of the octahedron.
The shallow water equations are nonlinear equations, which give a better description of the motion of water than the linear wave equation. In particular, unlike the linear wave equation, they conserve the total volume of water. The linear equation gives an approximation of the solutions, when the wave height remains close to its average over space.
The equations used here include viscosity and dissipation, as described for instance in
en.wikipedia.org/wiki/Shallow_water_equations#Non-conservative_form , including the Coriolis force.
The video has four parts, showing the same simulation with two different color schemes and two different visualizations:
Wave height, 3D: 0:00
Velocity (norm and direction), 3D: 0:48
Wave height, 2D: 1:42
Velocity (norm and directions), 2D: 2:30
In the first and third part, the color hue and radial coordinate show the height of the water. In the second and fourth part, the color hue shows the direction of the flow, and the radial coordinate shows its speed (the norm of the velocity). The 2D parts use a projection in equirectangular coordinates. In the 3D parts, the point of view is slowly rotating around the sphere in a plane containing its center. The position of the sphere is represented by a line starting above the north pole, and pointing in a fixed direction.
The velocity field is materialized by 2000 tracer particles that are advected by the flow.
Render time: 3D part - 1 hour 22 minutes
2D part - 1 hour 44 minutes
Color scheme: Parts 1 and 3 - Magma by Nathaniel J. Smith and Stefan van der Walt
github.com/BIDS/colormap
Parts 2 and 4 - Turbo, by Anton Mikhailovgist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
Music: "Lusciousness" by Asher Fulero@AsherFulero
See also
https://images.math.cnrs.fr/des-ondes-dans-mon-billard-partie-i/ for more explanations (in French) on a few previous simulations of wave equations.
The simulation solves the 2D shallow water equation by discretization (finite differences).
C code: github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code!
#shallowwater #waves #wave
