Page 95 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

80 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

oo R nr
—}.
z =Z + po tian +3 (on + in) (F)

The main problem is the effective calculation of thecoefficients po, go,
> Pny Qn. To do that, we will consider the previous development at the
point Z = Re” ,0 < 6 < 2r, of the circumference (C) and then we will
separate the real and the imaginary parts, thus obtaining

£(0) = po + (R+ pi) cosé sin# + S~ (pn. cos nO + gn sinn6),
qy
+
n=2

oO
+
6
+
y (9) = go + (R — p,) sin8 q, cos S| (dn cos nO — pp sind).
n=2
Although the coordinates (x,y) of the points of the contour (c) are
known, either in a tabular or in a functional form, the functions (0)

and y(@) are still unknown. That is why an iterative method to calculate
z(@) and y(9) must use the coefficients po, go,-.-) Pny Gn-
First, due to the orthogonality conditions for the trigonometric func-
tions, we have

1 27
po ~ on Jp x(0)dé,

1 20 1 20
R+pi= - | z(8) cos 0d0, R-—p, = “| y(0) sin 0d6,

Qn
20
pn = —= | y(@) sinn6d0,n > 1 an = = [ y(0)cosnddb,n > 0
0 0

and, from here, we could write that

20
R= = 5 [x(8) cos 8 + y(9) sin 8] dd,

1 27
P=>z- [x(0) cos 9 — y(@) sin 6] dé.
20 0
Then we choose for x (@) its “initial” (of order zero) approximation
z° (0) = a+ Bcos@where a and £ are arbitrary. From the expression
of po and p; + R, we have ps =a, p”) + RO = £B.