Nils Berglund | A laminar flow over an immersed icosahedron @NilsBerglund | Uploaded August 2024 | Updated October 2024, 8 minutes ago.
The video youtu.be/Y1wyZ-FM_l4 showed a solution of the shallow water equations on a sphere, with a depth given by an embedded regular dodecahedron. This video shows an analogous situation, with an icosahedron instead of the dodecahedron. The water depth influences the wave speed, and thereby the water height, which reveals the presence of the icosahedron. The initial state is a slightly perturbed laminar flow, flowing eastward in the northern hemisphere and westward in the southern hemisphere.
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: 1:41
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 57 minutes
2D part - 1 hour 55 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 - HSL/Jet
Music: "Awake" by Emmit Fenn@emmitfenn
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 #icosahedron
The video youtu.be/Y1wyZ-FM_l4 showed a solution of the shallow water equations on a sphere, with a depth given by an embedded regular dodecahedron. This video shows an analogous situation, with an icosahedron instead of the dodecahedron. The water depth influences the wave speed, and thereby the water height, which reveals the presence of the icosahedron. The initial state is a slightly perturbed laminar flow, flowing eastward in the northern hemisphere and westward in the southern hemisphere.
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: 1:41
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 57 minutes
2D part - 1 hour 55 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 - HSL/Jet
Music: "Awake" by Emmit Fenn@emmitfenn
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 #icosahedron
