Original publication

Erik A. Martens, Shinya Watanabe, Tomas Bohr
Model for polygonal hydraulic jumps

Complex Systems

Liquid with corners

A new model explains why a fluid spreading out over a surface can form a polygonal hydraulic jump

April 05, 2012

Complex phenomena can also be observed in the kitchen sink: When a jet of water hits the bottom of a kitchen sink, the water first quickly flows away from the point of impact. At a certain distance, however, the water level rises suddenly and the flow speed decreases abruptly. This so-called hydraulic jump assumes the form of a perfect circle when the bottom of the sink is horizontal and even. A few years ago, researchers discovered that under certain circumstances, complex shapes such as for example triangles, pentagons or octagons, can also emerge from the circular jump. Now scientists at the Max Planck Institute for Dynamics and Self-Organization, together with colleagues from Denmark and Japan, have developed a theoretical model which describes the fundamental mechanisms of this surprising flow phenomenon.

<p>Polygonal hydraulic jump with eight corners: a jet of fluid hits a level, horizontal base plate. Around the impact point, the liquid first spreads radially in all directions at high speed. Further away from the point of impact, the flow velocity decreases due to viscosity and the divergence of the flow. This eventually leads to a sudden increase in the fluid level. The resulting step in height, i.e.  kink-like flow structure, is known as the hydraulic jump. Normally this jump assumes the shape of a circle. But if the level of fluid downstream of the hydraulic jump is sufficiently high, the jump may instead assume the shape of a regular polygon.<i> </i></p> Zoom Image

Polygonal hydraulic jump with eight corners: a jet of fluid hits a level, horizontal base plate. Around the impact point, the liquid first spreads radially in all directions at high speed. Further away from the point of impact, the flow velocity decreases due to viscosity and the divergence of the flow. This eventually leads to a sudden increase in the fluid level. The resulting step in height, i.e.  kink-like flow structure, is known as the hydraulic jump. Normally this jump assumes the shape of a circle. But if the level of fluid downstream of the hydraulic jump is sufficiently high, the jump may instead assume the shape of a regular polygon.

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Everyday physics often starts in such innocuous places as the kitchen sink. Probably everybody is familiar with the following phenomenon: When one turns on the tap, a water jet falls from the tap and hits the bottom of the kitchen sink; at the point of impact, the liquid first spreads at high speed from the jet in a very thin layer. At a certain distance from the impact point, the water level rises suddenly and a jump is formed; behind this kink, the water continues to flow more slowly. This effect is called a hydraulic jump. If the base is horizontal and completely even, and if a fluid more viscous than water, such as anti-freeze, is used, the hydraulic jump attains the shape of a perfect circle. It is unlikely, however, that anyone looking down in their kitchen sink has observed that the circular jump can turn into an octagon. Yet, in 1997 scientists discovered this surprising phenomenon in the laboratory: in their experiments, the fluid jet first fell on a round glass plate; far away from the hydraulic jump, the researchers mounted a small weir across which the fluid has to flow after it has passed the jump. The height of this weir can be adjusted and thereby one may control how high the liquid backs up downstream from the hydraulic jump.

The hydraulic jump forms triangles, pentagons or octagons

When the liquid level reaches a certain height, the hydraulic jump may lose its circular form: the jump develops corners and, surprisingly, attains the shape of a regular polygon. Experts call this phenomenon a polygonal hydraulic jump. The number of corners depends on the downstream liquid level and the flow rate – so for instance, the researchers have observed triangles, pentagons octagons, and have discovered that as many as 14 corners are possible.

Although this flow phenomenon may at first appear relatively simple, to date there has been no theoretical model which accurately describes the experimental observations. Now a team of physicists at the Max Planck Institute for Dynamics and Self-Organization in Göttingen, the Technical University of Denmark, and Ibaraki University in Japan have developed a comprehensive model for polygonal hydraulic jumps.

In the normal circular hydraulic jump, the liquid initially spreads out when the jet impacts with the bottom at a so-called super-critical velocity: this means that the bulk flow moves faster than perturbations can travel as surface waves; smaller perturbations move downstream and quickly die away.

<p>How a polygonal hydraulic jump is formed: the normal hydraulic jump is generated by an annular vortex which is located downstream of the jump and rotates clockwise. When the fluid downstream of the jump is set to a certain height, a second vortex is formed which lies above the first vortex and rotates in the opposite direction (left image). When this second vortex is present, polygons can form (right image).</p> Zoom Image

How a polygonal hydraulic jump is formed: the normal hydraulic jump is generated by an annular vortex which is located downstream of the jump and rotates clockwise. When the fluid downstream of the jump is set to a certain height, a second vortex is formed which lies above the first vortex and rotates in the opposite direction (left image). When this second vortex is present, polygons can form (right image).

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With increasing distance from the impact point of the jet, the velocity of the flow decreases, and at a certain distance the fluid level rises suddenly where the hydraulic jump appears. Right after this region, a vortex is formed which surrounds the hydraulic jump like a ring and rotates in clockwise direction. The liquid has to flow across this vortex, resulting in the hydraulic jump due to the  change in liquid height.

On the downstream side of the jump, the speed of the flow is sub-critical which means that perturbations can spread as surface waves in both upstream and downstream directions.

 
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