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This question on condensation and wingtip vortices, What is the mechanism by which condensation forms above wings?, attributes the reduction in temperature found in wingtip vortices to Adiabatic cooling.

The reasoning for this, as quoted from Wikipedia, is given as:

Adiabatic cooling occurs when the pressure on an adiabatically isolated system is decreased, allowing it to expand, thus causing it to do work on its surroundings. When the pressure applied on a parcel of air is reduced, the air in the parcel is allowed to expand; as the volume increases, the temperature falls as internal energy decreases

Emphasis mine

My question is: why are we assuming that a parcel of air—which, according to Bernoulli's principle has a lower internal pressure—would be allowed to expand against the external pressure which would be higher?

enter image description here

This reasoning to me would suggest that the parcel would compress (if it was compressible) and that its temperature would increase.

Also, why are we assuming that air is now compressible below Mach?

Wouldn't it make more sense to attribute the reduction in temperature to an isochoric cooling process? This would only require the internal pressure to decrease, and would not require a change in density.

enter image description here

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  • $\begingroup$ @frederico I have asked a follow up question to your answer , hope you can help $\endgroup$ – Quentin Chester Oct 25 '16 at 7:40
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The vortex is rotating. This by itself should suffice to explain the expansion at its core. The mechanism is less Bernoulli (which applies to straight flows) but centrifugal force. Compare it to the decreasing height at the center of the bathtub vortex when water flows out the drain and starts to rotate. The height is proportional to the pressure in the fluid. The same happens with air - the pressure change is highest at the core because the angular velocity decreases with the distance from the core.

And, yes, air is compressible below Mach 1. It is only incompressible ideally and at Mach 0. The density changes are minuscule at low speed, but they are there. The thermodynamic process is an adiabatic one (no heat flow) with changes to pressure, volume and density (a tiny bit).

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  • $\begingroup$ Thanks for your very good reply Peter , a few questions I always understood that the radial force of the pressure gradient is what keeps the flow in uniform rotation against the inertial centrifugal force . Are you saying that in the core the centrifugal if greater than the centripetal and this is what allows the expansion? Also when we allow expansion in the core we must have also compression outside the core? So outside the core heats up as the core cools down? $\endgroup$ – Quentin Chester Oct 26 '16 at 1:44
  • $\begingroup$ @QuentinChester: The flow is in an equilibrium between centrifugal (= inertial) and centripetal (= pressure) forces. Only the expansion creates a centripetal force which balances the centrifugal force of the rotating flow. Technically, there is a compression outside, but it has a near-infinite volume to work on, so effectively only an expansion can be observed which dies down with increasing distance from the vortex core. $\endgroup$ – Peter Kämpf Oct 26 '16 at 9:38
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The air already inside the vortex is not expanding. It is already expanded and is being slowly recompressed again as it's kinetic energy is dissipated and the vortex slows. The expansion occurs when the air enters the vortex.

The oncoming air is being sucked into low-pressure pocket created by the wing and it expands to fill it. Afterwards it is just the centrifugal force of the rotation opposing the pressure gradient and preventing collapse of that low pressure pocket, but the pressure gradient is not increasing any more. It is slowly decreasing as internal friction slows the rotation.

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