Page 116 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 101

In particular, along a wall the normal component of the relative veloc-

ity of the fluid (versus the wall) should vanish. Concerning the pressure,
it can be calculated by the Bernoulli theorem which, in this case, states
that

2
p+ 3 ~ Um +p Bb C(t),
Ot

where the “constant” C(t), depending on time, will be determined with
the initial conditions.
An important case is when there is only one mobile body (obstacle)
in the mass of the fluid, which allows a simple formulation of the initial
and boundary conditions (on the body surface). More precisely, by con-

sidering a mobile reference frame (system of coordinates) Axy, rigidly
linked to the obstacle (body), and by using the linear expression of Z as
function of z (with the coefficients depending on time, in fact a change
of variables, the flow being watched within the fixed frame OXY), we
get first f (z,t) = F (Z,t) which represents the complex potential of the
flow expressed in the variables z and tf.

Hence for the components u and wv of the velocity vector, we have
U—W = a (here u and v are the components of the absolute fluid velocity
versus the fixed system OXY, these components being expressed in the
variables x and y).
Let us now denote by a(t) and {(t) the components on Az and Ay

respectively, of the vector vy, the velocity of the point A belonging to
the body, and by 2 (t) the magnitude of the body rotation; the contour
(surface) of the obstacle being then defined by the time free parametric
equations z = 2(s), y = y(s), the velocity vp of a point P(s), belonging
to this contour, is vp = v4 + 2k x AP whose components are a — Qy,
B+Qx«.

Then, the normal component of the relative velocity at the point P,
belonging to the obstacle contour, is (u — a+ Dy) ay —(v—B-Ox)
such that the slip-condition can be written, for any fixed t#, in the form

dy _ dy _ ,de d (x? +y?

ds ds ds ds

This last expression determines, to within an additive function of time,
the value of 7 along the contour, precisely

Vip = a(t)y—B(ta— 7 (2? +42).