Geometries of potential theory (fluid mechanics)

[Nikitin]

Hello! My book (fluid mechanics by White) doesn't explain the formulas it uses for finding geometric information about a potential field. For instance, sometimes if a stream-function is kept constant it, will form a figure like the one in this picture.

https://scontent-b-lhr.xx.fbcdn.net/hphotos-ash3/1379638_10201611755512363_1918244517_n.jpg
Its stream-function: $\psi = U_{\infty} r \sin(\theta) + m(\theta_1 - \theta_2)$ (8.34), where $\theta_1$ and $\theta_2$ are the angles relative to the source and sink.

What is the general way of thinking for extracting information (like in this case, L and h) about the geometries of potential-functions? Why is L and h that weird formula in 8.35, in this case?

PS: Sorry for flooding this forum with so many questions, lately. I'll make up for it in a few years when I'm done with my studies.

 

[Nikitin]

Perhaps I should use the formula for volume flow along two streamlines to find h?
$$d \psi_2 - d \psi_1 = Q = h \cdot V$$

 

[SteamKing]

It's called potential theory because the stream functions, etc. are derived from the solutions to the Laplace equation:

$\nabla^{2}\phi$ = 0

http://en.wikipedia.org/wiki/Potential_theory

http://en.wikipedia.org/wiki/Laplace's_equation

In order to derive the stream functions, you'll need to know partial differential equations and vector calculus, among other topics. If you are studying intro fluid mechanics, you probably are only encountering PDEs and vector calculus for the first time, if at all.

 

[AlephZero]

There is a lot of math theory behind this, though its practical use been replaced by other types of computer modelling. The math applies to other areas of physics, for example modelling electromagnetic fields.

A math-based method is "conformal mapping". http://en.wikipedia.org/wiki/Conformal_map. The basic idea is to take a known solution for a simple geometry, then create mathematical transformations that change the shape of the boundary. Applying the same transformation to the streamlines gives the (exact) solution of the more complicated problem. For example that was how the flow pattern round the Joukowski aerofoil shape was first derived.

Another method is to invent fictitious sources and sinks for the fluid flow, such that they produce a flow pattern normal to the boundary of the real problem. (The diagram at the top of your image might be taking that approach). In the general case, this links up with methods of solving partial differential equations or integral equations using Green's functions. It was developed into a numerical method where "dipoles" were placed around the boundary of the component instead of using arbitrary pairs of sources and sinks, and the computer calculated the strength of each dipole to match the boundary conditions. This was (and still is) the basic idea of "panel methods" for computing external flows round objects - there are some more techniques required to approximate the effect of the boundary layer, etc, but that's going beyond your question.

 

[Nikitin]

It's called potential theory because the stream functions, etc. are derived from the solutions to the Laplace equation:

$\nabla^{2}\phi$ = 0

http://en.wikipedia.org/wiki/Potential_theory

http://en.wikipedia.org/wiki/Laplace's_equation

In order to derive the stream functions, you'll need to know partial differential equations and vector calculus, among other topics. If you are studying intro fluid mechanics, you probably are only encountering PDEs and vector calculus for the first time, if at all.

Well, it is indeed only my 2nd year and the fluid mechanics course is just an intro. However we're working on our fourth math-class out of a total of five, and we have gone through much of the formalities behind potential theory. So go ahead and fire with the big guns, as I am familiar with the laplace equation and the derivation of both the stream and velocity functions.

There is a lot of math theory behind this, though its practical use been replaced by other types of computer modelling. The math applies to other areas of physics, for example modelling electromagnetic fields.

A math-based method is "conformal mapping". http://en.wikipedia.org/wiki/Conformal_map. The basic idea is to take a known solution for a simple geometry, then create mathematical transformations that change the shape of the boundary. Applying the same transformation to the streamlines gives the (exact) solution of the more complicated problem. For example that was how the flow pattern round the Joukowski aerofoil shape was first derived.

Another method is to invent fictitious sources and sinks for the fluid flow, such that they produce a flow pattern normal to the boundary of the real problem. (The diagram at the top of your image might be taking that approach). In the general case, this links up with methods of solving partial differential equations or integral equations using Green's functions. It was developed into a numerical method where "dipoles" were placed around the boundary of the component instead of using arbitrary pairs of sources and sinks, and the computer calculated the strength of each dipole to match the boundary conditions. This was (and still is) the basic idea of "panel methods" for computing external flows round objects - there are some more techniques required to approximate the effect of the boundary layer, etc, but that's going beyond your question.

Thanks for the reply! But how do do I calculate the geometries when using the second method?

I intuively thouht about using the formula
$$d \psi_2 - d \psi_1 = Q = \int_1^2 \vec{V} \cdot \vec{n} \cdot dh = h \cdot V$$ to find the top height where Q is the volume flow per width and $\vec{n}$ is the normal-vector of h. Because I mean, in the middle (x=0) h is perpendicular to both the streamline $\psi_1$ and $\psi_2$.

Anyway the thing is that the fluid mechanics book we use is very bad as it is unclear and often assumes lots of things "go without saying". So I just need some way of knowing how to handle this stuff.

 

[Nikitin]

Ah, so maybe I should find the stagnation points (for L) and the maximum velocity points for h? However, why would the velocity be at a max on the top of the rankine oval in the picture in the OP?