Hello,

Does anyone know if the exact solution for the potential flow about a zero-thickness hemisphere (a bowl) is available in the literature?

Thanks in advance

Adrin Gharakhani

Is the hemisphere normal to the flow? (like the nose of an aircraft for e.g.) or is it sitting on its diameter in the middle of the flow? For the first case, R.L. Panton and I.G. Currie (separately) outline methods to obtain useful quantities of flow (pr. coeff etc., I'm not sure if they provide full vel field solns.)

Jay,

I need the info to verify the accuracy of a new 3-D Galerkin Boundary Element Method that I have developed for potential flow across zero-thickness bodies. So to answer your question, it doesn't matter (to me) what direction the flow is (normal, parallel or inclined). All these cases would be useful references.

As for the data I need most: I need the potential (jump) distribution on the surface, because that is the primary unknown in BEM. The velocity field is post-processed info!

Currie talks about 2D potential flow, so do most others!

Thanks for the reply and info

Adrin Gharakhani

It was a long time ago that I took my grad course in fluids, but I remember that Panton's book "Incompressible Flow" that we used devoted a full chapter to 3-D potential flows; so you may benefit from taking a look at it. I also have seen some books by F.A Williams and (separately) Chorlton, both of who discussed potential flows. It may be that they too discuss only 2-D flows although I think Chorlton's book may give further references for 3D topics (I don't remember, but it may in fact also discuss 3D flows.)

I can't offer you any other references, but I'm curious: why only a hemisphere, why not a sphere? The potential/stream-function solution for a sphere is rather well known. Can't you use this? And if you have already used the results for a sphere, why are you still looking for resuts for a hemisphere, how (and why) is it substantially different from that of a sphere; and also why not other geometries.

Chorlton's book, for example, provides, an analytic expression for the laminar flow of fluid thru a triangular c/s pipe. It is a rather clever work of algebra of the manner in which he specifies the BCs and obtains the soln. Perhaps you can test the velocity field obtained from your method (derived as it may be) against the velocity prediction from his expression. May be I don't know what I'm suggesting and so perhaps I should shut up too, correct ?

If you do come across some reference of use to you, I too will be glad to know of it; I have an abiding interest in Potential methods in general.

> why only a hemisphere, why not a sphere?

The formulation I have developed should in principle work with spheres as well (I will use the sphere as last resort), but a hemisphere is fundamentally different from a sphere (analogous to the difference between a flat zero-thickness plate and cube or thick plate)

The surface of a sphere closes on itself; whereas a hemisphere has an outer edge or perimeter. We solve for the surface potential distribution on the sphere, but we need to solve for the surface potential JUMP distribution for the hemisphere; i.e., the difference in the potential distributions on the two sides of the zero-thickness body. The latter implies that this jump is zero at the outer edges, but that it's derivative is unbounded.

So, briefly the velocity field for a sphere is well behaved whereas for the hemisphere it is infinite at the skirt! I need to know how well the method can handle these singularities.

> how (and why) is it substantially different from that of a
> sphere; and also why not other geometries.

See above for the first part. As for other geometries, the hemisphere is the simplest (non-flat) zero-thickness geometry I could think of. If you know of others with exact solutions, by all means, please recommend them.

Adrin Gharakhani

Aah! All this while I thought that the hemisphere also closed in on itself except at its diameter! Obviously this is not the case?! (Or does it, but the sharp corner makes a difference?)

....just thinking aloud....I wonder if its possible to make up such arbitrarily shaped "open" geometries w/ combinations of sources, sinks, doublets, and uniform flow....hmm...will be some interesting exercises....

In my original message I brought the "bowl" example since I expected some people may think of the hemisphere as half a sphere (though I had explained the "zero-thickness" constraint)

Adrin Gharakhani

<The latter implies that this jump is zero at the outer edges... the velocity field for a sphere is well behaved whereas for the hemisphere it is infinite at the skirt! I need to know how well the method can handle these singularities.>

By introducing vortex (dipole) sheets trailing from the edges -- longitudinal for stationary cases and longitudinal+transversal for nonsteady. Strength of discontinuity (vorticity) of these vortex sheets at the trailing edge is determined by (non-zero in common case) jump of potential at the edges.

<the hemisphere is the simplest (non-flat) zero-thickness geometry I could think of>

A hemisphere is a bluff body, and also not simple. I should try i. g. a parabolic plate of small curvature.

With best regards

p. s. See any book on potential wing theory -- these vortex methods are used there since Prandl's works

Best wishes

Yes, I re-read it...sorry my bad...