# 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.

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.

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.

#### Polygons are caused by effects of surface tension

If the liquid level is increased even further, above a certain value, the fluid begins to flow in reverse direction above the vortex. The fluid sloshes inwards, and as a consequence, a second annular vortex is generated but it rotates in the opposite direction compared to the first vortex below. It is in this situation, where the second vortex has appeared, that the jump can become polygonal.

In early approaches to explaining the phenomenon, researchers attempted to account for the formation of the polygonal hydraulic jumps as the result of an interplay between internal friction, i.e. viscosity, and gravitational force of the water towards the inside, i.e. hydraulic pressure, acting on the secondary vortex. However, these forces alone could not explain why polygons are formed. A further piece of the puzzle was provided by the effect of surface tension, i.e. by taking into account the binding forces between the molecules on the fluid surface. The scientists discovered in particular that the mechanism which is responsible for the polygonal form is similar to a phenomenon which is known as the Rayleigh-Plateau instability.

This instability is an everyday phenomenon: when a jet of water flows from a tap, provided it is long enough, the jet breaks up into a chain of drops at a certain distance from the tap. The chain is formed because perturbations, which are not initially visible in the top part of the water jet, are amplified further down the jet due to surface tension effects. This amplification ultimately leads to the breaking up of the jet and thus to the formation of water drops. To explain the formation of the polygons, the researchers consider a similar scenario in their model: small perturbations of the second circular vortex near the surface can be amplified by surface tension, which in the end leads to the formation of the corners.

#### Pressure, viscosity and surface tension together provide the explanation

The new model developed by the team around Erik A. Martens at the Max Planck Institute for Dynamics and Self-Organization combines the aspects of earlier model approaches: hydraulic pressure, viscosity and surface tension. “We can now describe the basic mechanisms that lead to the instability of the circular hydraulic jump and the formation of a polygon,” says Erik A. Martens.

The flow profile in a polygonal hydraulic jump is much more complicated than in a circular jump. Visualizations of the flow structures (see Figure 3) reveal that very strong flow jets emerge from the polygon corners – it looks almost as if the second vortex has broken apart. This is where the researchers come up against one of the limits of their modeling approach. “In order to develop better models for the complicated polygonal flows, we would need more experimental knowledge on the velocity and height profiles of the polygonal patterns,” explains Erik A. Martens. “It is indeed one of the objectives of our work to further stimulate the research in this field.” The researchers envisage that more detailed studies of these flow processes could possibly lead to the observation of novel and unexpected phenomena.

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