Page 75 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

60 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

so that ®,(P) > ®,(M), which means v?(P) > v?(M). In other words

the unique possibility for the velocity to get a maximum value is only
on the boundary. Concerning the minimum value of the velocity this
could be reached inside the domain, namely in the so-called stagnation
points (with zero velocity). If the fluid flow is steady and the external
forces can be neglected, from the second Bernoulli theorem (integral)

it comes that, at a such stagnation point, the pressure has a maximum
while at boundary points of maximum velocity, the pressure should have
a minimum.
Let us now consider the case of an incompressible irrotational plane
(2-dimensional) fluid flow.
Let Oxy be the plane where we study the considered fluid flow, u

and v being the velocity vector components on Oz and Oy respectively,
and gq the magnitude of this vector. The fluid being incompressible, the
equation of continuity can be written gu + oe = 0, such that udx — vdy
is, for every fixed t, an exact total differential in « and y. Consequently,
there is a function w (z,y,t) , defined to within an additive function of
time by the equality udy —vudz = dy, where ¢ is seen as a parameter and
not as an independent variable.

This function w(z,y,t) is the stream function of the flow since the
curves ~ = constant, at any fixed moment t, define the streamlines
of the flow that has been shown. On the other side, the flow being
irrotational, we also have Su — ge = 0 which proves the existence of a

second function ® (z,y,t), the velocity potential, defined also to within
an additive function of time, such that udz + vdy = d® where again t is
considered a parameter and not an independent variable. Hence

ya 22 _ Oh _ OY
Ox = Oy’ Oy Ox

or, under vectorial form

v = grad® = —k x grady,

k being the unit vector of the axis directly perpendicular on the plane

Oxy.
But these equalities show that the two functions ® and w satisfy
the classical Cauchy-Riemann system and, consequently, the function
f = &@+ ty is a monogenic (analytic) function of the complex variable
z= 2+ty which could depend, eventually, on the parameter t. This
function is called the complex potential of the flow and it is obviously
defined to within an additive function of time. The real and imaginary

part of f(z), which means the velocity potential and the stream function
of the flow, are two conjugate harmonic functions; the equipotential lines