Page 119 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

104. =BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

the obstacle — but watching the absolute flow (that is versus the fixed
system OX ,X2X3) we set again

23,t)
£2,
p (£1, = ®(X1, X2, X3,t).
To determine this function y, the velocity potential of the absolute
flow but expressed in the variables of the mobile system Az,2973 (a
function which is also harmonic and with zero gradient at infinity), we

should write the slip-condition on the surface (%) of the obstacle. Let
then v4 and 2 be the velocity of the point A, belonging to the obstacle,
and, respectively, the obstacle rotation; these are known vectorial func-
tions of time. At a point P of the contour (3), ifn is the unit outward
normal drawn to (%) at P, we have for the function y the condition

U-n= <e =vpn= (v4+2k
AP)-n,
x
i.e., the projection of the relative velocity U — v, on nis zero.
We denote now by Vj, V2, V3 the components of v4 on the Azr,, Azo,
Az3 axes and by V4, V5, Ve those of 92 on the same axes; let also n 1,2, 73
be the components of n while 14,75, ng are those of AP x n on the same
axes of the reference frame Az,z273. With this notation, the above

condition is

While np are geometric entities depending only on P from (%) and
not on ¢, vp are known functions of time, independent of P from (%).

Let us admit that there are the functions y!) (x1, x2, 23) harmonic

outside of (D) so that agit = Mp On (%) and whose grad y?) vanish at
far distances. In fact the existence of these functions comes from the
solving of a Neumann problem for the exterior of the domain (D), with
the additional requirements that the first order partial derivative of yp
tends to zero when the point P tends to infinity.
It is known that such Neumann problems, in quite general conditions,
admit one unique solution and only one [52].
Setting then

6
y (x1, 22,03,t) = D> vp (t) y (21, 22,23) ,
p=l1

this function y satisfies all the conditions of the problem and defines the
searched velocity potential for fluid flow outside the obstacle. Once the