A laminar flow over an immersed dodecahedron @NilsBerglund

Nils Berglund | A laminar flow over an immersed dodecahedron @NilsBerglund | Uploaded August 2024 | Updated October 2024, 39 seconds ago.
Like the video youtu.be/Vb-S2YrZpH4 , this one shows a shallow water flow over a dodecahedron. The difference is the initial state, which is a slightly perturbed laminar flow, flowing eastward in the northern hemisphere and westward in the southern hemisphere.
The depth in the shallow water equation has been computed from the distance between the sphere and an embedded dodecahedron 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 dodecahedron.
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 - Parula, originally from Matlab
mathworks.com/help/matlab/ref/colormap.html
Parts 2 and 4 - Twilight by Bastian Bechtold
github.com/bastibe/twilight

Music: "Moonrise" by Reed Mathis@reedmathis4623

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

$Waves of two different frequencies crossing a hexagonal lattice Several recent videos on this channel showed waves from two different sources, with different frequencies, crossing a diffraction grating. This simulation is similar, except that there are now 24 columns of obstacles, so maybe this should not be called a grating anymore. Nevertheless, it is interesting to see what kind of interference and diffraction patterns are created. One thing that stands out here is how more energy is able to cross certain channels, in particular from the lower source. The frequency of the lower source is three times the frequency of the upper one. The resulting wavelengths are such that the open intervals in the grating are between both wavelengths. As a result, waves of the lower source pass the grating more easily, as can be best seen on the energy plot. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:07 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. The contrast has been enhanced by a shading procedure, similar to the one I have used on videos of reaction-diffusion equations. The process is to compute the normal vector to a surface in 3D that would be obtained by using the third dimension to represent the field, and then to make the luminosity depend on the angle between the normal vector and a fixed direction. There are absorbing boundary conditions on the borders of the simulated rectangle. The display at the right shows a time-averaged version of the signal near the right boundary of the simulated rectangular area. More precisely, it shows the square root of an average of squares of the respective field value (wave height or energy). Render time: 29 minutes 52 seconds Compression: crf 23 Color scheme: Part 1 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing Part 2 - Plasma by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: March of the Hares by Nathan Moore@NathanMooreSongs 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 #diffraction #grating$

Growth of a mono-quasicrystal from pentagons

Waves in shallow water with cube-like symmetry on the sphere

Vortex dynamics on the sphere with weaker Coriolis force

$Waves of two different frequencies crossing a sunflower lattice Some recent simulations on this channel, such as https://youtu.be/Y9Zn8T8dYgg , showed waves of two different frequencies, produced by two different sources, crossing a regular lattice of scatterers. Here the regular lattice has been replaced by a sunflower pattern of discs. This pattern is obtained by staring with a disc at the center of the display, and then gradually adding shifted discs, each time turning by a golden angle of (phi - 1) times 360°, where phi = 1.618... is the golden ratio or golden mean. The distance to the original circle is gradually increased, in such a way that the density of circles remains constant. This arrangement has appeared before on this channel, see for instance https://youtu.be/fQaVL3mE6VY or https://youtu.be/c8sKkM8ZDTs . The frequency of the lower source is three times the frequency of the upper one. The resulting wavelengths are such that the open intervals in the grating are between both wavelengths. As a result, waves of the lower source pass the grating more easily, as can be best seen on the energy plot. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:19 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. The contrast has been enhanced by a shading procedure, similar to the one I have used on videos of reaction-diffusion equations. The process is to compute the normal vector to a surface in 3D that would be obtained by using the third dimension to represent the field, and then to make the luminosity depend on the angle between the normal vector and a fixed direction. There are absorbing boundary conditions on the borders of the simulated rectangle. The display at the right shows a time-averaged version of the signal near the right boundary of the simulated rectangular area. More precisely, it shows the square root of an average of squares of the respective field value (wave height or energy). Render time: 35 minutes 28 seconds Compression: crf 23 Color scheme: Part 1 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: Raw Deal by Gunnar Olsen 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 #diffraction #grating$

Using six sources to excite resonant modes in a circle

Vortices in the compressible Euler equation on the sphere with Coriolis force

The shallow water equation on the sphere

$Symmetric low-frequency pulses in a magnetron-shaped resonator This simulation of a magnetron-shaped resonator uses pulses separated by relatively long time intervals as a source, with the aim of creating long wavelengths, that are closer to what occurs for real magnetrons. The anode is slightly refracting, with a high damping coefficient. The reason for this choice is that I was afraid that perfectly reflecting boundary conditions would prevent the waves from entering the smaller cavities. Neumann boundary conditions would probably be more appropriate, but they are harder to simulate, and making the walls slightly refracting is an easier surrogate for that. Magnetrons were used in early radars, and are still used in microwave ovens, as sources of the microwaves. Cavity magnetrons use a resonating cavity with several chambers, together with a magnetic field. In this simulation, there is no magnetic field, and waves are produced as pulses in the center of the device, so it does not represent a real magnetron. However, I still found it interesting to look at the effect of the resonating cavities on the output. Since this simulation is in 2D, I added a channel at one side to act as an outlet for the waves. In 3D, my understanding is that the outlet would rather be along the axis of the device. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:52 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 absorbing boundary conditions on the borders of the simulated rectangle. The graph at the right shows a slightly time-averaged version of the signal. Render time: 43 minutes 52 seconds 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: Blue Macaw by Quincas Moreira@/UCL1zFMJb0sthwdAlGjGbdyg 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 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 #resonator #magnetron$