Transformation of Flow Pattern

4.1 Introduction

Any flow pattern can be considered to consist of a set of streamlines and potential lines (ф and ф lines). Thus, transformation of a flow pattern essentially amounts to the transformation of a set of streamlines and potential lines, whilst the transformation of individual lines implies the transformation of a number of points.

4.2 Methods for Performing Transformation

Choose a transformation function Z = f (z) to transform the points specified by the Cartesian coordinates x and y, in the physical plane, given by z = x + iy, to a transformed plane given by Z = Z + in. To carry out this transformation, we need to expand the transformation function Z = f (z) = Z + in, equate the real and imaginary parts and find the functional form of Z and n, in terms of x and y, that is, find:

Z = f1 (x, y) n = f2(x, y).

Thus, any point p(x, iy) in the physical plane (z-plane) gets transformed to point P(Z, in) in the trans­formed plane (Z-plane).

Example 4.1

Transform a point p(x, iy) in the physical plane to Z-plane, with the transformation function Z = 1/z.

Solution

Given, Z = 1/z. Also, z = x + iy.

Theoretical Aerodynamics, First Edition. Ethirajan Rathakrishnan.

© 2013 John Wiley & Sons Singapore Pte. Ltd. Published 2013by John Wiley & Sons Singapore Pte. Ltd.

Therefore, from the transformation function Z = 1/z, we get:

Using these expressions for § and n, any point in the physical plane, with coordinates (x, y), can be transformed to a point, with coordinates (§, n), in the transformed plane. That is, from any point of the given flow pattern in the original plane, values of the coordinates x and y can be substituted into the expressions of § and n to get the corresponding point (§, n), in the transformed plane.