Page 108 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 93

“nearly potential”. Such situations occur in particular when it con-
siders “streamlined” bodies, that is bodies so shaped as to reduce their

drag.
Now we shall analyze the model of incompressible inviscid fluid flow
due to the presence of N (point) vortices, located at the points rj,ro,
.. ,rny in the plane and of strength [,,Ts, ... ,[n, respectively. The
stream function joined to the j-th vortex, ignoring the other vortices for
a moment, is given by

T;
W, (r) =~ 52 n[r — 45].

The vorticity associated to the same vortex will be given by

wi = ~Ayp; =1;d(r —x;),

where 6 is the Dirac function while the corresponding velocity field (ig-
noring again the influence of the other vortices) is

vi (Oy%;, —Ozp;) = (- 244 sa

with r = lr —r,|.
Obviously, due to the interaction of vortices, the points where the
vortices are centered (located) start to move. More precisely, taking

into account the superposed interaction of all the vortices, rj (xj, y;)
move according to the differential equations

dt; 1 ~ Tilyj-—yi) dy; 1 S TP; (xj — 2)

dt - ~ Oar r2. ’ dt 7 Qn re.
tj 9 tAj ‘J

where rj; = |r;—r;| .
Then, if we retake the previous way in a reverse sense, we conclude
that:
Let a system of constants T, , .... fy and a system of points (initial

positions) r; (21,41) , -.-, rw (cn, yn) be in the plane. Suppose we allow
these points to move according to the above equations whose solutions
could be written in theform 2; = z; (t) and y; = y; (t). Definethen v; =
. N
(-pee ri oe ) and let v(r,t) = 5°) vj (r,t). This last equality
12n
j=1
provides a solution of Euler’s equations, a solution which preserves the
circulation. Really, if C is a contour encircling k vortices rj,ro,..., x
k
then lg = 5° T; and T¢ is flow invariant (constant).
t==1