An inviscid incompressible irrotational flow can be described in terms of a potential. The velocity u(x,y) is given by
If we define a complex potential
then the imaginary part of the complex function is called the stream function. The fluid flows along the lines of constant streamfunction.
First we consider the flow in the positve x-direction, say. This has complex potential
where U0 is some real constant. This flow looks like
where we will take the red lines to be the stream lines, and the yellow lines are lines or equipotential. In this case a pretty dull flow from left to right.
If we take the above flow to be in the w-plane then we can use the map
to map this flow onto the region outside a circle. Note: in the far field (i.e. z and w both large) we have that the transformation is, at leading order, the identity transformation. Thus the flow far from the obstruction is unaltered. The above transformation gives the flow from left to right past a circlular obstruction. The potential for the flow is
This flow looks like
We can then rotate the flow by any angle by replacing z by
Why is there a minus sign in the exponential? Here is the flow rotated by 0.2 radians:
Mapping this back to the original geometry gives a flow that looks like
Here the blue line represents the flat wing going from -2 to +2. This is not the flow that would be seen! This flow has infinite velocities at the leading and trailing edges. What tends to happen in reality is that there would be an eddy shed by the wing (a starting vortex) which would result in a circulation around the body (see x3 Fluids for more details). It is found that the best model of a real flow is obtained by using a circulation which makes the velocity at the trailing edge finite.
To add circulation to the plate we return to the flow past the circle. We find a suitable potential for the flow around a cylinder with circulation by adding the potential for a vortex at the origin to the potential shown in the third picture. In the z-plane this gives a potential of
The flow looks like
Note that because of the logarithm in the potential we now have a branch cut (the dashed green line), but the red lines are continuous across the cut (can you see why?). Although the potential is not continuous, its gradient (the velocity) is! The value of the circulation has been chosen so that the red line joins the circle at z=1, otherwise there would still be a singularity in the flow at the trailing edge. This occurs when the derivative of f(z) is zero, i.e. when
If we then map this flow back into the orgininal plane we get
We could rotate this back so that the distant flow was again horizontal, but I will leave this as a mental exercise fot the reader (or alternatively tilt your head). From this flow we can find the lift (see X3 Fluids). However, this is not a perfect model of fluid flow. There are always boundary layer effects (thin regions near the wall where viscosity is important). This model also still has infinite velocities ot the leading edge. It will only give a good representation of the flow when the angle of attack (theta) is small, probably smaller than the value used here.
It is not much more difficult to modify this approach to generate flows around more realistically shaped aerofoils which do not have any points with infinite velocities. For example the Joukowski aerofoil:
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