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Is pressure a harmonic function in ideal fluid flow ?

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Old   June 3, 2019, 15:17
Default Is pressure a harmonic function in ideal fluid flow ?
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For an incompressible, irrotational flow in three dimensions, does the pressure satisfy Laplace's equation ? I know the velocity potential does. What about the pressure field itself ?

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Old   June 3, 2019, 15:46
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For an incompressible, irrotational flow in three dimensions, does the pressure satisfy Laplace's equation ? I know the velocity potential does. What about the pressure field itself ?

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Take the divergence of the momentum equation and see what happens
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Old   June 3, 2019, 18:24
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Filippo,

Thanks.

Thinking about it some more, the answer has to be - not in general. We can show this by counterexample - take the point source which has the potential 1/r. Its velocity field is 1/r^2. Since p + 1/2 rho v^2 = constant, its pressure field is thus 1/r^4 which is not harmonic.
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Old   June 4, 2019, 02:34
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Indeed pressure obeys a Poisson not a Laplace equazioni
However, that is not the real pressure in thermodynamics meaning
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Old   June 4, 2019, 05:33
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For inviscid, steady, irrotational flows it is the total pressure, static + dynamic, that obeys a laplace equation
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Old   June 4, 2019, 06:15
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For inviscid, steady, irrotational flows it is the total pressure, static + dynamic, that obeys a laplace equation
Actually some specific conditions could be possibile, for example a total pressure constant only along the streamline but changing from streamline to streamline if the flow has not upward homogenous conditions. Then the total pressure could not have a vanishing laplacian, isnít that?
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Old   June 4, 2019, 13:00
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Paolo,

For inviscid, steady, incompressible, irrotational flows, the total pressure p +1/2 rho v^2 is spatially constant. So, it satisfies Laplace's equation trivially.

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Old   June 4, 2019, 13:08
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Paolo,

For inviscid, steady, incompressible, irrotational flows, the total pressure p +1/2 rho v^2 is spatially constant. So, it satisfies Laplace's equation trivially.

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Sam.



homogeneous in space provided that the upward conditions are homogeneous in space... the total pressure is integrated along a streamline, but could change from a streamline to a different streamline
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Old   June 4, 2019, 13:40
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If the flow is irrotational everywhere, the total pressure is a constant. You are thinking of the case where there is vorticity, in which case it is constant along a streamline.
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Old   June 4, 2019, 13:52
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If the flow is irrotational everywhere, the total pressure is a constant. You are thinking of the case where there is vorticity, in which case it is constant along a streamline.

Yes, of course one of the hypotheses in Bernoulli is the irrotational flow constraint. Removing this constraint is possible in the general Crocco relation from which you can deduce the laplacian of the pressure.
However, the general answer is that that the pressure equation for irrotational, inviscid, incompressible flow is



Lap p =-rho*(Lap |v|^2/2 + Lap psi)


psi being the gravitational potential function
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