PROFILE THEORY BY THE METHOD OF CONFORMAL MAPPING

2- 3-1 Complex Presentation

Complex stream function Computation of a plane inviscid flow requires much less effort than that of three-dimensional flow. The reason lies not so much in the fact that the plane flow has only two, instead of three, local coordinates as that it can be treated by means of analytical functions. This is a mathematical discipline, developed in great detail, in which the two local coordinates (x, у) of two-dimensional flow can be combined to a complex argument. A plane, frictionless, and incompressible flow can, therefore, be represented as an analytical function of the complex argument z = x + iy:

F{z) = F{x + iy) = Ф(х, у) + і ¥{x, у) (2-16)

where Ф and У7, the potential and stream functions, are real functions of x and y. The curves Ф = const (potential lines) and lF = const (streamlines) form two families of orthogonal curves in the xy plane. By taking a suitable streamline as a solid wall, the other streamlines then form the flow field above this wall. The velocity components in the x and у directions, that is, и and v, are given by

_ дФ _ dW _ дФ _ 0 XF

U dx dy V dy Ox

The function F(z) is called a complex stream function. From this function, the velocity field is obtained immediately by differentiation in the complex plane, where

— = u —iv= vv(z) (2-17)

Here, w = u—iv is the conjugate complex number to w—uFiv, which is obtained by reflection of w on the real axis. In words, Eq. (2-17) says that the derivative of the complex stream function with respect to the argument is equal to the velocity vector reflected on the real axis.

The superposition principle is valid for the complex stream function precisely as for the potential and stream functions, because F(z’)= c1Fi(z) + c2F2(z) can be considered to be a complex stream function as well as F1(z’) and F2(z).

For a circular cylinder of radius a, approached in the x direction by the undisturbed flow velocity u00, the complex stream function is

F{z)=uO0(z + ^j (2-18)

For an irrotational flow around the coordinate origin, that is, for a plane potential vortex, the stream function is

*■<*) = (2-19)

where Г is a clockwise-turning circulation.

Conformal mapping First, the term conformal mapping shall be explained (see [6]). Consider an analytical function of complex variables and split it into real and imaginary components:

/(2) = IH І У) = С = £ (Ж, у) + і V (х> у) (2-20)

The relationship between the complex numbers z = x 4- iy and f = % + irj in Eq. (2-20) can be interpreted purely geometrically. To each point of the complex z plane a point is coordinated in the f plane that can be designated as the mirror image of the point in the z plane. Specifically, when the point in the z plane moves along a curve, the corresponding mirror image moves along a curve in the f plane. This curve is called the image curve to the curve in the z plane. The technical expression of this process is that, through Eq. (2-20), the z plane is conformally mapped on the £ plane. The best known and simplest mapping function is the Joukowsky mapping function,

(2-21)

It maps a circle of radius a about the origin of the z plane into the twice-passed straight line (slit) from -2a to +2a in the f plane.

For the computation of flows, this purely geometrical process of conformal mapping of two planes on each other can also be interpreted as transforming a certain system of potential lines and streamlines of one plane into a system of those in another plane. The problem of computing the flow about a given body can then be solved as follows. Let the flow be known about a body with a contour A in the z plane and its stream function F(z), for which, usually, flow about a circular cylinder is assumed [see Eq. (2-18)]. Then, for the body with contour В in the f plane, the flow field is to be determined. For this purpose, a mapping function

£ = /(*) (2-22)

must be found that maps the contour A of the z plane into the contour В in the f plane. At the same time, the known system of potential lines and streamlines about the body A in the z plane is being transformed into the sought system of potential lines and streamlines about the body В in the f plane. The velocity field of the body В to be determined in the f plane is found from the formula

F(z) and w(z) are known from the stream function of the body A in the z plane (e. g., circular cylinder). Here dz/a= 1 jfz) is the reciprocal differential quotient of the mapping function f = /(z). The sought velocity distribution about body В can be computed from Eq. (2-23) after the mapping function /(z) that maps body A into body В has been found. The computation of examples shows that the major task of this method lies in the determination of the mapping function f = f(z), which maps the given body into another one, the flow of which is known (e. g., circular cylinder).

Applying the method of complex functions, von Mises [71] presents integral formulas for the computation of the force and moment resultants on wing profiles in frictionless flow. They are based on the work of Blasius [71].