Inverse Mapping from the w Plane to the Physical Plane

Eq. (15.17) gives a direct mapping of the airfoil from the physical plane to the w plane. After the computation, it will be necessary to reverse the procedure to find the point z corresponding to a point w in the transformed plane. For the mapping function under consideration, it is easy to see that away from the airfoil z = w – iQ. Therefore, for these points, the inverse mapping may be found by simple Newton iteration.

Подпись: N w0 = zn + Azn + J2 Qj ln j=1 Подпись: zn + Azn z j zn + Azn - z0 Подпись: + iQ. Подпись: (15.24)

Let zn +1 = zn + Azn be the point in the z plane corresponding to the point w = w0 in the w plane after the (n + 1)th iteration. Eq. (15.17) gives

Подпись: (15.25)

By expanding the terms on the right side of Eq. (15.24) for small Azn, and ignoring all higher-order terms, it is straightforward to find that

To start the iteration, the obvious choice is to take z1 = w0 – iQ.

For points closer to the airfoil, it is advantageous to keep the second-order terms in expanding the right side of Eq. (15.24). Let


where 7 = d/dZ.

Points on the slit in the w plane are more difficult to invert. To invert these points, the following numerical method may be used. Let ws be the point of which the inverse, say zs, is to be found. Suppose the inverse of a point w denoted by z near ws is already found. w may or may not be on the slit. Let w be a point on the line joining w to ws so that

Подпись: (15.28)

Подпись: -[1 + f (Zn )] + [1 + 2 f (Z„ ) + ( f (Z„ ))2 + 2 f" (Z„ ){WQ - Z„ f'' (z„)
Подпись: f (Z„ ) - ІЙ)]2
Подпись: Az

w — w = reie.

Подпись: dZ dw Подпись: N 1 + £ Qj j=1 Подпись: 1 z — z0_ Подпись: (15.29)

Now, by Eq. (15.17) it is readily found that

Подпись: dz dr Подпись: 1 z—z0_ Подпись: (15.30)

On combining Eqs. (15.28) and (15.29), along the line joining w and ws, the following differential equation is established:

zs may now be determined by integrating Eq. (15.30) from r = 0to r = |ws – w| with starting condition r = 0, z = z.