Monday, August 30, 2010

Flow over bumps

As Fluid flows over a bump surface disturbances, such as waves, develop potentially extending both up and downstream. The form these disturbances take depends on fluid velocity and fluid depth which can be combined together in the Froude (F) number. The Froude number equals the fluid velocity over the square root of gravity times fluid depth. Because it has no dimensions, the Froude number allows flow in dramatically different circumstances to be compared, for example, atmospheric flow over a mountain range could be compared to tap water flow from your kitchen sink.

Naval architecture provided the original application for the Froude number. Here the hull length of a ship is used instead of fluid depth. It's important because, the Froude number relates to the ship's drag or resistance to moving through the water. This number is named after William Froude (1810-1879) who experimented on ship's hulls in his large fluid tank. William developed towing tank techniques in his efforts to model frictional drag on ships. However, he didn't discover the dynamical relationship between the fluid velocity and hull length. It was Ferdinand Reech (1805-1880) who first described this relationship and used it for testing ships and propellers around 1852. It's likely that this relationship's roots reside with even earlier French mathematicians.

So what does the Froude number tell us? When F is smaller than one, flow over the bump is 'subcritical'. Waves on the surface can travel upstream, meaning that downstream conditions affect the flow upstream. For example, when a pebble is tossed into the water of a flowing stream, the resulting ripples propagate both upstream and downstream. When F is larger than 1, flow is 'supercritical'. In this case, no surface disturbance can travel upstream. The ripples created by a pebble tossed in downstream cannot overcome the speed of the water. The flow upstream is not changed. When F is equal to one the flow is 'critical'. This is the point of transition from subcritical to supercritical effects.

Now, back to flow over a bump. As subcritical water is pushed over the bump, squeezing takes place because the water is now shallower and the same amount of water is flowing through. This forces the water to speed up over the bump and transition to supercritical. This faster water crosses over to the other side of the bump, where it's again deeper and slower moving. When the fast flowing water reaches the slower water it abruptly slows and waves form. Since the water is moving too quickly to allow waves to propagate upstream, (because it is supercritical) these waves build up, forming a sudden water level increase that can be standing still in the flowing water. This is called a hydraulic jump, a non-linear effect and can be observed in a kitchen sink or in water passing over a weir. Mathematically, a hydraulic jump is a discontinuity, however in the real world viscosity makes it a region of rapid change instead.

The greater the Froude number is, the more pronounced the jump will be. For initial flow speeds slightly above the critical speed, the transition appears as an undulating wave. As flow speed increases, the Froude number also increases and the transition becomes stronger eventually developing a more abrupt shape. When the speed is high enough, the transition front will break and curl back upon itself. At this point, the jump may contain violent turbulence, eddying, air entrainment, and surface waves. Turbulence removes the extra energy, allowing the flow to transition from supercritical back to subcritical.

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