[mczero57]

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 ?

Thanks

[FMDenaro]

Originally Posted by mczero57

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 ?

Thanks

Take the divergence of the momentum equation and see what happens

[mczero57]

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.

[FMDenaro]

Indeed pressure obeys a Poisson not a Laplace equazioni
However, that is not the real pressure in thermodynamics meaning

[sbaffini]

For inviscid, steady, irrotational flows it is the total pressure, static + dynamic, that obeys a laplace equation

[FMDenaro]

Originally Posted by sbaffini

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, isn’t that?

[mczero57]

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.

Thanks

Sam.

[FMDenaro]

Originally Posted by mczero57

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.

Thanks

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

[mczero57]

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.

[FMDenaro]

Originally Posted by mczero57

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