Page 27 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

12 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

tremely important case, of the plane and axially symmetric (revolution)

motions.
A continuum motion is said to be plane, parallel with a fixed plane
(P), if, at any moment t, the velocity vector (together with other vectors
which characterize the motion) is parallel with the plane (P) and all the
mechanical (scalar or vectorial) parameters of the motion are invariant
vs. a translation normal to (P). We denote by z and y the Cartesian

coordinates in (P) so that 21 = @, zz = y, the variable x3 not playing a
role. In the same way, we denote v; = u, v2 = v, (v3 = 0), k being the
unit vector normal to (P) and oriented as «3 axis.
One says that a motion is axially symmetric vs. the fixed axis Oz,
if, at any moment ¢, the velocity vector’s supports (and of supports of

other vectors characterizing the motion) intersect the Oz axis and all the
mechanical parameters associated to the motion are rotation (vs. Oz)
invariants. We denote by Oz and Oy the orthogonal axes in a merid-
ian half-plane (bounded by Oz), by k the unit vector which is directly
orthogonal to Ox and Oy and by wu and v the respective components of
the vectors v obviously located in this half-plane.

Now let be, at a fixed instant t, a contour (C) drawn in Oxy and let
(Xc) be the corresponding surface generated by:
a) a translation motion, parallel to k and of unit amplitude, in the
case of plane motions or
b) an Oz-rotation motion of a 27-amplitude, in the case of revolution
(axially symmetric) motions.

Let m be a number which equals O, in the case of a plane motion and
equals 1, in the case of a revolution motion. Hence

// pv -ndo = / (2ry)” pv -nds = / (2ry)” p(udy — vdz),

(Zc) (Cc) (C)

(with the remark that do = 2myds), the (C) orientation being that

obtained by a rotation from n with +5 and ds is the elemental arc
length on (C).
If the motion is steady’ and (C) is a closed curve bounding the area
(co) from Oxy, the above expressions vanish® and, by using the divergence
(Green) theorem, we get

"The result keeps its validity even for unsteady motion provided that the continuum is incom-
pressible; in these hypotheses the function yw which will be introduced in the sequel, depends
on the time ¢ too.
“We have an exact total differential due to the condition div (pv) =0.