Introduction
The Ideal Flow Mapper is designed for students learning ideal flow theory. The term 'Ideal flow' describes the way in which a fluid (liquid or gas) moves when the effects of compressibility and viscosity are negligible. Ideal flow is often the first type of fluid motion that student engineers and scientists study, because it is the simplest. Large parts of the flows past ships, submarines, cars and light aircraft are closely ideal.

This applet is designed to give students an environment where they may create and visualize elementary two-dimensional ideal flows and then examine the effects of various conformal mappings upon them. Conformal mapping is an advanced topic, usually taught to seniors or graduates. If you are not a fluid dynamicist, or are rusty on ideal flow theory you may find it more satisfying to run the Ideal Flow Machine - an applet dealing more with the fundamentals of ideal flow. Alternatively, you can dive right into this one. In that case you may wish to skip the technical details and go straight to the applet. You can start by clicking anywhere on left hand grid (this will start the flow moving at speed 1 from left to right - just like a real air or water flow you can't see the flow until you put dye or some other marker in it). Then press the button 'New Mapping'. Choose "s = az^b" and enter 2 for 'a' and 0.5 for 'b' and then press OK. Click anywhere towards the left hand side of the left hand grid. Keep on clicking at a range of different points within the grid and you will see a flow pattern emerge on the right hand grid that looks like this. Note that the color of the streamlines depend on the speed of the flow. You may then add any number of other flows (sources, sinks, doublets, source panels and vortex panels) and select any other kind of mapping and see what they do by drawing more streamlines. In case you find this exercise a little frustrating (there are a lot of ugly 'crossed-out' flow patterns that can be produced by a misused mapping) check out the examples.

Technical details/Specific Instructions
Conformal mapping follows from the description of two-dimensional ideal flows in terms of complex numbers (described for example by Karamcheti K, " Priciples of Ideal Fluid Aerodynamics", 2nd edition, Kreiger, 1980). This description is a very natural one. Suppose you use a complex number to represent positions in two dimensions, e.g. z = x + iy. Then, by definition, any analytic (differentiable) function of the complex variable 'z' is a solution to Laplace's equation - the governing equation of ideal flow. We can therefore describe any 2D flow as a function of z.

The problem with this result is that one does not know a priori which function of z will produced the specific flow of interest. One approach to finding the solution to complex flow problems is therefore to begin with very simple ideal flows, that are easily understood and described, and then to add them together to produce the complex flow patterns desired. This is the process modelled in The Ideal Flow Machine.

The description of ideal flows in terms of complex numbers, however, allows for an additional route - the distortion of one flow into another by conformal mapping. Conformal mapping involves the transformation of the complex coordinate 'z' into a different coordinate, say, s = p+iq via a function, i.e. s=s(z). This process also transforms the flow, described say by a function F(z), to a new flow F₁(s) = F(z(s)). As long as the derivative of the mapping function s=s(z) is not zero, then it turns out that the mapped flow is a solution to Laplace's equation and is thus valid. At points where the derivative is zero, termed critical points, the mapped flow is not valid. Paradoxically, these points turn out to be particularly useful when trying to create sharp corners in a flow, such as at the trailing edge of the airfoil.

This process of constructing a flow in the complex plane and then mapping it is what is illustrated by the Ideal Flow Mapper.

When you go to the applet page, you will see a window like that shown below -

Constructing a Flow
The window shows two grids (square regions containing the gray '+'s). The left hand grid represents the z-plane in which you can construct and visualize your ideal flow. To construct the flow you use the mouse, the text fields labeled 'Strength' and 'Angle', and a choice menu with the selections 'Freestream', 'Source', 'Vortex', 'Doublet', 'Source Sheet', 'Vortex Sheet', 'Circle', 'Circle with K.c.' and 'Draw Streamline'. With the exception of 'Circle', 'Circle with K.c.' and 'Draw Streamline' these selections are elementary ideal flows from which can build your own complex fluid flow.

The operation of this part if the applet is very similar to the Ideal Flow Machine where the methods used to build a flow are described in detail. In short, what you do is select the type of flow you want to add, type in the strength and angle, as appropriate, and then click the mouse on the grid at the point where you want that flow to appear. You may add any number of sources, vortices, doublets, source sheets or vortex sheets. Then you select 'Draw Streamline' and the computer will draw streamlines starting wherever you click your mouse on the grid.

The choice-menu selections 'Circle' and 'Circle with K.c.' allow you to apply the Milne Thompson Circle Theorem to a flow. This theorem adds a circular streamline in any flow, simply by manipulating the function that describes that flow. To use this feature, first put together the flow in which you want to create a circular streamline using the tools described above. Then select 'Circle'. Click the mouse at a point within the grid where you want your circular streamline to be centered and then drag the mouse to select the radius. Note that you can only add one circular streamline at a time (adding a second simply overwrites the first). The option 'Circle with K.c.' functions in the same way, except that it allows you to specify additionally the point at which the flow detaches (or attaches) to the circular streamline (i.e. you get to specify a Kutta Condition). This point is specified by the location where you release the mouse after dragging it to create the circle. A good way to examine the operation of both of these options is to try them on a uniform flow. Note, however, that they will work on any flow, however complex.

The basic effects of most of the mappings are illustrated in the examples and are described in many standard texts (e.g. CRC Standard Mathematical Tables, 27th Edition, CRC Press, 1986). Somewhat unusual is the mapping 's = (z-a)/(az-1) + b' which maps the space in an annulus to the space outside two circles.

One important effect to watch out for is branching. A single flow in the z-plane can map to several different, and overlapping, flows in the s-plane (branches). This is entirely correct, but it can be very ugly when you get overlapping streamline patterns. The number of possible branches can be trimmed somewhat by restricting the range of angles in the z-plane to 0 to 2Pi (as is done here). There are still plenty of combinations of mapping and flow that will produced multiple branches though, so if you just want to see one branch be careful where you click that mouse!

William Devenport