Page 67 - Mechanical Engineers' Handbook (Volume 4)

56 Fluid Mechanics

j
k
i
1 1
curl V i j k
y
x
z
2 2 x y z
u v w
If the flow is irrotational, these quantities are zero.
4.4 Vorticity and Circulation
Vorticity is defined as twice the angular velocity, and thus is also zero for irrotational flow.
Circulation is defined as the line integral of the velocity component along a closed curve
and equals the total strength of all vertex filaments that pass through the curve. Thus, the
vorticity at a point within the curve is the circulation per unit area enclosed by the curve.
These statements are expressed by

V dl (udx v dy wdz) and lim
A
A→0 A
Circulation—the product of vorticity and area—is the counterpart of volumetric flow
rate as the product of velocity and area. These are shown in Fig. 15.
Physically, fluid rotation at a point in a fluid is the instantaneous average rotation of
two mutually perpendicular infinitesimal line segments. In Fig. 16 the line x rotates posi-
tively and y rotates negatively. Then ( v/ x u/ y)/2. In natural coordinates (the
x
n direction is opposite to the radius of curvature r) the angular velocity in the s-n plane is
1 V
V
1 V
1
V

2 A 2 r n 2 r r
This shows that for irrotational motion V/r V/ n and thus the peripheral velocity V
increases toward the center of curvature of streamlines. In an irrotational vortex, Vr C
and in a solid-body-type or rotational vortex, V r.
A combined vortex has a solid-body-type rotation at the core and an irrotational vortex
beyond it. This is typical of a tornado (which has an inward sink flow superimposed on the
vortex motion) and eddies in turbulent motion.

4.5 Continuity Equations
Conservation of mass for a fluid requires that in a material volume, the mass remains con-
stant. In a control volume the net rate of influx of mass into the control volume is equal to

Figure 15 Similarity between a stream filament and a vortex filament.