Page 113 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

98 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

which the problem of the fluid flow past a profile is reduced, is a classical

issue in the literature’”.
The solution of this problem, applied to the function < =V+w
Zz
whose real part is known on the boundary AB, leads to

wel PeO-4O, pe f bOt+hO ,
v= af z-€ 1 @-) VFO

where k is a real constant, P(z) = (z — a) (z— 6) while the chosen de-

termination for ,/P(z) equals to +,/P(z) at z=2z > b.
Unfortunately, this bounded solution of the proposed Dirichlet prob-
lem does not satisfy yet the condition expressing the rest of fluid at far
. . dF
distances i1.e., { — ) = 0. To satisfy this condition too we will add
dz

; ; ., /2—6 ;
to the previous solution a term of the type 7A, /——, where 4 is a real
z-—a
constant (not chosen yet) and the determination of the squared root is
the same as the previous one (i.e., it is positive at z = x > b)’*. Since
in the neighborhood of infinity we have

2
I
VPG = 241-2884 Sh, =2(1+84+54--),
z z-E€ 2Z

z—-b. a—b 1
ia tach,
oe
SE = aa{is
we could write

wai /ft-8 4 Pe Otho
U iv +inf2—$ = x | Pe d&é+k

i fh h®+h® (,_ a+b lL fhe

s/f |P(é) € 2 )ac+ 5 f [lo () ty (€)] dé

tae yta{is S24 Sey}.

z Qz 2?

12 A direct and elegant manner for solving this problem, even in the more general case of a
boundary formed by n distinct segments on Oz, can be found, starting from page 201, in the
book of C. Iacob [69].
Az + bt
Really, by adding to U —iV a term in the form i , where A, yw € IR, the values of V
VP(z)
on AB will not be modified.