Nils Berglund | A shallower laminar flow over an immersed icosahedron @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/dP_XVvwxmD0 , but with a shallower 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 icosahedron 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 icosahedron.
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: 0:43
Wave height, 2D: 1:33
Velocity (norm and directions), 2D: 2:18
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 16 minutes
2D part - 1 hour 18 minutes
Color scheme: Parts 1 and 3 - Twilight by Bastian Bechtold
github.com/bastibe/twilight
Parts 2 and 4 - Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
Music: "Half Day" by Bad Snacks@badsnacks
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/dP_XVvwxmD0 , but with a shallower 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 icosahedron 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 icosahedron.
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: 0:43
Wave height, 2D: 1:33
Velocity (norm and directions), 2D: 2:18
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 16 minutes
2D part - 1 hour 18 minutes
Color scheme: Parts 1 and 3 - Twilight by Bastian Bechtold
github.com/bastibe/twilight
Parts 2 and 4 - Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
Music: "Half Day" by Bad Snacks@badsnacks
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
![[Flash warning] A planar wave crossing a gradient index lens at an angle](https://i.ytimg.com/vi/vZ_pKzjFXvo/hqdefault.jpg "[Flash warning] A planar wave crossing a gradient index lens at an angle
The first part of this video shows some flashing due to the formation of standing waves.
This is a variant of the simulation https://youtu.be/CdpTq3dSrXA , in which the incoming planar wave makes a small angle with the axis of symmetry of the lens. The videos are inspired by Huygens Optics recent short https://youtube.com/shorts/VGd3Ajnp6e0 showing the principle of a gradient index lens.
Lenses focus incoming rays of light by delaying them more near the center of the lens than at its periphery. This is often done with a material of constant index of refraction, by making the lens thicker near the center, as shown for instance in the simulation https://youtu.be/rrJJBh9ubUE . However, one can also build lenses of constant thickness, by making the index of refraction of their material depend on the location in the lens.In this simulation, the index decreases like sqrt(n0² - a*r²), where r is the distance to the axis of symmetry. This results in the incoming planar wave being focused at a point in the focal plane, marked by a vertical line. The plot to the right shows a time-averaged value of the field along that plane.
This video has two parts, showing the same evolution with two different color gradients:
Wave height: 0:00
Averaged wave energy: 1:03
In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, averaged over a sliding time window.
There are periodic boundary conditions between the top and bottom boundaries, absorbing boundary conditions on the right boundary, and a time-periodic signal is imposed on the left boundary. The display at the right shows the signal along the focal plane, which is indicated by a vertical line.
Render time: 30 minutes 1 second
Compression: crf 23
Color scheme: Part 1 - Twilight by Bastian Bechtold
https://github.com/bastibe/twilight
Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt
https://github.com/BIDS/colormap
Music: Chicago by Joe Bagale
See also https://images.math.cnrs.fr/Des-ondes-dans-mon-billard-partie-I.html for more explanations (in French) on a few previous simulations of wave equations.
The simulation solves the wave equation by discretization. The algorithm is adapted from the paper https://hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf
C code: https://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!
#wave #lens #gradient_index")
