inviscid-flow

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Crash Course Physics returns to the subject of fluids with their video on fluid dynamics. They stick with ideal fluids (i.e. incompressible, inviscid, laminar flows) for simplicity and cover some of the basics by discussing conservation of mass (also called continuity) and a simple form of Bernoulli’s equation. Despite keeping things basic, the video does a nice job introducing these topics; I especially like that they explain Bernoulli’s equation as a form of conservation of energy. Sometimes it’s easy to let the terminology in fluid dynamics mask the fact that the equations we use are just alternative forms of the classical equations for conserving mass, momentum, and energy. As with their fluids at rest video, the information is densely packed, so expect to pause and rewind. (Video credit: Crash Course)

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Reader junolivi asks: 

When shallow water (like runoff from melting snow) flows across pavement, it creates small repeated wave-like ripples. What creates that texture and why isn’t it just a steady flow?

This is a great question that’s probably crossed the mind of anyone who’s seen water running down the gutter of a street after a storm. The short answer is that this gravity-driven flow is becoming unstable. 

Fluid dynamicists often like to characterize flows into two main types: laminar and turbulent. Most flows in nature are turbulent, like the wild swirls you see behind cars driving in the rain. But there are laminar flows in nature as well. Often flows that begin as laminar will become turbulent. This happens because those laminar flows are unstable to disturbances.

The classic example of stability is a ball on a hill. If the ball is at the top of the hill and you disturb it, it will roll down the hill because its original position was unstable. If, on the other hand, the ball is in a depression, then you can prod the ball and it will eventually settle back down into its original place because that position was stable. Another way of looking at it is that, in the unstable case, the disturbance–how far the ball is from its original position–grows uncontrollably. In the stable case, on the other hand, the disturbance can be initially large but eventually decays away to nothing.

There are many ways to disturb a laminar flow–surface roughness, vibrations, curvature, noise, etc., etc. These disturbances enter the flow and they can either grow (and become unstable) or decay (because the flow is stable to the disturbance). Just as one can look at the stability of a pendulum, one can mathematically examine the stability of a fluid flow. When one does this for water flowing down an incline, one finds that the flow is quite unstable, even in the ideal case of a pure, inviscid fluid flowing down a smooth wall. 

The reason that one sees distinctive waves with a particular wavelength (assuming that they aren’t caused by local obstructions) is directly related to this idea of instability. Essentially, the waves are the disturbance, having grown large enough to see. One could imagine that any wavelength disturbance is possible in a flow, but mathematically, what one finds, is that different wavelengths have different growth rates associated with them. The wavelength we observe is the most unstable wavelength in the flow. This is the wavelength that grows so much quicker than the others that it just overwhelms them and trips the flow to turbulence. This is very common. For example, you can see distinctive waves showing up before the flow goes turbulent in both this mixing layer simulation and this boundary layer flow. (Image credits: anataman, mo_cosmo; also special thanks to Garth G. who originally asked a similar question via email)

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On a cold and windy day, the plume from a smokestack sometimes sinks downstream of the stack instead of immediately rising (Figure 1). This isn’t an effect of temperature–after all, the exhaust should be warm compared to the ambient, which would make it rise. It’s actually caused by vorticity.

In Figure 2, we see a simplified geometry. The wind is blowing from right to left, and its velocity varies with height due to the atmospheric boundary layer. Mathematically, vorticity is the curl of the velocity vector, and because we have a velocity gradient, there is positive (counterclockwise) vorticity generated.

According to Helmholtz, we can imagine this vorticity as a bunch of infinite vortex lines convecting toward the smokestack, shown in Figure 3. Those vortex lines pile up against the windward side of the smokestack–Helmholtz says that vortex lines can’t end in a fluid–and get stretched out in the wake of the stack. If we could stand upstream of the smokestack and look at the caught vortex line, we would see a downward velocity immediately behind the smokestack and an upward velocity to either side of the stack. It’s this downward velocity that pulls the smokestack’s plume downward.

Now Helmholtz’s theories actually apply to inviscid flows and the real world has viscosity in it–slight though its effects might be–and that’s why this effect will fade. The vortex lines can’t sit against the smokestack forever; viscosity dissipates them.