Page 105 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

90 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

We remark that for a body of a complicated shape the calculation
of the normals to the panels at control points is not always easily per-

formed. We can modify the above algorithm by choosing the boundary
points (X;,Y;) to be on the surface of the body and the control points
(x3, yi) to be the midpoints of the panels. The panel orientation is given

by
0; = Arctg (ea=}) jt =1,...,m

where Arctg takes its values on [—7, x]. This technique is easier to apply

but it is not as accurate as the previous method. Now the control points
are located near the surface of the body and they will approach the
surface if the number of panels increases.
Other remark is that the panels could be of different sizes. It is useful
to take small panels in a part of the body of large curvature, in order to
increase the accuracy of the method.

After the calculation of the dimensionless strengths Nj the velocity
potential ®(z;,y;) may be written. The velocities at the control points
are tangent to the panels and thus at these points

d
V(2i, yi) = a (xi, Yi)

where ¢; 1s a tangent vector to the surface of the 7-th panel.
Taking the derivative of ® with respect to n; we also obtain

m
V (ri, yi) _ cos 6; + S> Hi
jj

Here Jj, 1s given by

S?42AS;
I}, = —§ cos (6; — 8;) In }1+ ad

— sin (6; — 0;) larctg ( 53) —arctg (4)]

for i#j and Jj, = Ofor every i.
Finally, the pressure on the surface of the body could be described by

the pressure coefficient (2.9)

p—P V\?
C= pa = 1l- —_ .

We will illustrate this method with the following problem. Let us
consider two circular cylinders of radius 1m, placed in a uniform flow of