2.18 The Complex Potential Function

A potential flow can be described in terms of either a stream function or a velocity potential function . These two functions are very closely related - lines of constant cross lines of constant orthogonally. Given this relationship, it should be possible to combine the two functions into a single potential function. The orthogonality suggests that it might be necessary to use a complex function.

The Derivative of a Complex Function

Define a complex variable z=x + iy and a complex function w= f (z) = f (x + iy)

At a point P in a complex plane, z=x + iy and w(z) = f (x + iy) . At a point Q some distance away, , and

Similarly,

The derivative of the complex function is:

divide top and bottom by :

Because the value of the expression is dependent on the angle at which Q approaches P, there is no single value for the derivative , unless

Substituting in :

Thus: The derivative of a complex function is only defined if or , in which case

A complex function which has a unique derivative is called a monogenic function.

The Cauchy-Riemann Relations

Assume that w is a complex monogenic function that consists of real and imaginary parts:

The functions and are both real functions. w is a monogenic function, so that

However:

Both real and imaginary parts of these equations must be equal, so that:

and

2.19 The Complex Potential Function

The Cauchy-Riemann relations and determine whether or not a complex function has a unique derivative. For the stream function and velocity potential functions we know that

and

If we set and , the complex velocity potential function defined as

satisfies the Cauchy-Riemann conditions, and is thus a uniquely differentiable function. The derivative is given by

The magnitude of the velocity vector is

Exercise:  Prove that the following expressions are valid:

For a source at the origin:

For a vortex at the origin:

For a uniform flow at angle of attack :

For a doublet at the origin:

2.20 Conformal Mapping

Describing a potential flow field with a complex velocity potential doesn't really make it any easier, simpler or more intuitive to analyse. However, it does allow us to use a very powerful mathematical technique: conformal transformation or mapping.

We would like to achieve

Principles of conformal mapping

1. Each point on the z-plane is uniquely transformed or mapped to the -plane.
2. Lines should intersect each other at the same angle on the -plane as on the z-plane.
3. In the limit, elements will retain their similar geometric form.
4. Streamlines and equi-potential lines, which intersected each other orthogonally in the z-plane, will still intersect orthogonally in the -plane.
5. A transformation function is used to perform the mapping. For example will simply double the size of the body when transformed to the -plane.

Relations between the planes - Ratio of lengths:

Thus:

Hence:

Example:  Using the transformation function , (ie. double the size), the size ratio is

Ratio of velocities: In the z-plane the velocity is given by:

Similarly, in the -plane the velocity is given by:

The ratio of velocity magnitudes is then:

The velocity ratio is the inverse of the size ratio, which is logical, as continuity is still satisfied.

Example:   Still using the simple function , the velocity ratio , hence the velocity is halved, and the volume of fluid crossing a surface remains the same.

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