Page 78 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 63

the initial and the final value is again m(['+iD). Consequently the

function f(z) — rip log(z — a) is uniform, that is holomorphic in (D).
We conclude that afunction f(z), in the case of a doubly connected
domain, could be considered a complex potential if it admits the repre-
sentation TP log(z — a) plus a holomorphic function of z.
More generally, the following result holds:
Let (A;), (Ae), ...,(A,) be the connected components of the com-
plement of a bounded domain (DP), and let A,(a,) be a set of internal

points of (Aq), respectively (q = 1,p). An analytic function f(z) can be
considered a complex potential of a fluid flow in (DP), if and only if there
are a set of real numbers Ty and D, (q = 1,p) such that

p .
Tg +iD
fa-> <a log(z — aq)
q=1

is a holomorphic function in (D).

Case of steady flows. Ifthe flow is steady u and v will be free of ¢ (they
do not depend explicitly on time) and consequently we may suppose that
® , w and f(z) have the same property.
Concerning the effective determination of the complex potential for a
certain plane flow, it could be done taking into account the boundary

conditions. In the particular case when the fluid past a fixed wall, this
wall, due to the slip condition v -n = 0 = dy, is a streamline of our flow
and consequently, along this curve, y = Im f(z) is constant. Conversely,
if a plane fluid flow is known (given), we could always suppose that a
streamline is a “solid wall’, because the slip condition is automatically

fulfilled (¢ = a = 0); shortly, we could say that it is possible to so-
lidify (materialize) the streamlines of a given flow (under the above

assumption).
Finally, supposing that f(z) and implicitly the velocity field are deter-
mined, it will always be possible to calculate the pressure at any point
of the fluid flow by using the second Bernoulli theorem which can be
, 2 ; Le
written as K = £ +8 —U = constant. To assess this constant it is suf-

ficient to have both the magnitude of the velocity q; and the pressure p;
at a point M; belonging to the flow domain. Additionally, if f = 0, we
also have Te =1- Bry Each of the two sides of the previous equality
ZFS 1
is non-dimensional. The first one, denoted by C5, is called the pressure
coefficient.
Starting from some analytical functions f(z) satisfying the unifor-
mity properties stated above, it could always build up corresponding
fluid flows. For instance a linear function f (z) = az + b, a and 6b being