The animation is slowed down by a factor of three. See [ Ссылка ] for a real-time version.

The formation of dunes is one of the prime examples of self-organized pattern formation. The underlying mechanism is based on erosion and deposition of granular material submerged in a flowing fluid. The video shows the result of a simulation where a layer of granular matter is subject to a stream of air inside a tube. At first, (possibly due to the eddy formation caused by the boundary conditions) an instability towards a small amplitude-high frequency-pattern occurs. The amplitude of this primary pattern grows until the hills start to drift in flow direction and neighbors merge. The mechanism of growing pattern amplitudes can be explained by the observation that granular matter is pushed onto and along the flow-facing side of the hills. This causes more matter to be piled up at initial inhomogeneities. The drag along the flow-facing side of the hill then results in matter being eroded and pushed over the top of the hill, after which it slides down on the other side of the hill. Over time this leads to a drifting pattern.

In the center panel one can see the absolute value of the velocity field in the heatmap (range up to 7 m/s) and the drag force/acceleration experienced by the individual grains (cut off at one multiple of gravity).

The bottom panel illustrates the flow velocity relative to the average flow along the tube in the heatmap. The color-coding of the grains shows the recent displacement (cut off at 20cm) and shows where grains which have been transported over a larger distance are deposited.

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