Page 81 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

66 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

THEOREM 2.2. If (d) is a simply-connected domain from the plane

(z), bounded by a simply closed curve c, and if Z = h(z), a holomorphic
function in ( d ), has the additional property that when z is deplaced
along the contour c in a certain sense, its image Z describes a simply
closed curve C— delimiting a domain (D) from the plane (Z), in such
a way that the correspondence between c and C is a bijection, then the
correspondence between (d) and (D) will also be a bijection and, conse-

quently, the function Z = h(z) will be a conformal mapping of (d) onto
(D).
Let now F(Z) be the complex potential of a given fluid flow defined
in a domain (D) of the plane (Z); we suppose as known the function
Z = h(z) and its inverse z = H(Z) which establish a conformal map-

ping between the domain (D) of the plane (Z) and a domain (d) of the
plane (z). Then the function f(z) = F(h(z)), with the same regularity
properties as F(Z), will be the complex potential of a new fluid flow
defined in (d) and called the associated (transformed) flow of the given
fluid flow by the above mentioned conformal mapping.
Really f(z) could be considered as a complex potential because

dF
_
df _dPdZ
dh
dz dZdz dZdz

and so f(z) will be a uniform function in (d) together with F’(Z) in
(D),as well as h'(z) is also uniform together with A(z).
We also remark that in two homologous points z and Z of the con-
sidered conformal mapping, we have f(z) = F(Z). But then the values

of the velocity potential and of the stream function are equal at such
homologous points; consequently, the streamlines and the equipotential
lines of the two flows are also homologous within the considered confor-
mal mapping. More, the circulations along two homologous arcs and the
rates of the flow across two homologous arcs are equal. Particularly, if
a fluid flow defined by F(Z) has a singularity at Z) € D (source, point

vortex, etc.), the associated flow will have at the point zg, the homol-
ogous of Zp, a singularity of the same nature and even strength. Of
course, at two homologous points the fluid velocities are not (in general)
the same, which comes out from the above equalities for the complex
velocities.
Concerning the kinetic energy this will be preserved too, as from
the relation between the surface elements dA = |Z’|* da it results that

pu’*da = pVdA, v and V being the velocities magnitude in the associ-
ated flows of the same fluid density p.