Page 79 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

64 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

constants, will lead to a uniform (constant velocity) flow while the log-

arithmic functions f(z) = # logz and f (z) = os log z, defined on the
whole plane without its origin (D and I being real constants) correspond
respectively to a source (sink) — according to the sign of flow rate D
— and to a point vortex of circulation I’, all of them being located at
the origin. For practical applications one considers also the so-called
doublet (dipole) of axis Ox and strength (moment) K, located at the
origin, whose complex potential is f(z) = —g75.

Of course all these singular flows could be shifted to another location
zo of the plane (and even with an axis making an angle @ with Oz) by
considering the change of coordinates

z= 2+ Ze.

Properties of the above elementary flows as well as a set of additional

examples of such simple flows one finds, for instance, in Caius Iacob’s
book “Introduction mathématique a la mécanique de fluides’’, chapter
VII, page 407 [69].
We now remark that any linear combination of the complex potentials
fi(z) 1s still a complex potential in the common definition domain where
the analytic functions f;(z) satisfy the uniformity requirements stated
above. Consequently, starting with some given fluid flows, it is always

possible to set up, by superposition, new flows, that means to consider
linear combinations of the respective complex potentials.
For instance by superposition of a uniform flow parallel to the Oz
axis, of complex potential Voz, and of a doublet placed at the origin of
. R2 . we
complex potential Vo=- (Vo and R being positive real constants), one
gets the complex potential of the fluid flow past a circular disk (cylinder)
of radius R without circulation. If we superpose on the previous flow a
point vortex located at the origin, which leads to the complex potential

R? c
f(z)=Vo (2+ =) + — log
z,
1
z 24
we obtain the fluid flow past the same disk of radius R but this time
with circulation YT.
Detailed considerations on the steady, plane, potential, incompressible
flows past a circular obstacle can be found, for instance, in the same [69]

or in [52].

4. Conformal Mapping and its Applications
within Plane Hydrodynamics

In the previous section we mentioned the technique to build up fluid
flows by considering elementary analytic functions. But it will be im-