fluid_mechanics

6.5 Some Basic, Plane Potential Flows 291
B
A
v
1_ θ ~ r
B
A
v ~ r
θ
r
r
(a)
(b)
F I G U R E 6.19 Motion of fluid element from A to B: (a) for
irrotational (free) vortex; (b) for rotational (forced) vortex.
Vortex motion can
be either rotational
or irrotational.
direction. The other stick would rotate in a clockwise direction due to the nature of the flow
field—that is, the part of the stick nearest the origin moves faster than the opposite end. Although
both sticks are rotating, the average angular velocity of the two sticks is zero since the flow is
irrotational.
If the fluid were rotating as a rigid body, such that v u K 1 r where K 1 is a constant, then
sticks similarly placed in the flow field would rotate as is illustrated in Fig. 6.19b. This type of
vortex motion is rotational and cannot be described with a velocity potential. The rotational vortex
is commonly called a forced vortex, whereas the irrotational vortex is usually called a free vortex.
The swirling motion of the water as it drains from a bathtub is similar to that of a free vortex,
whereas the motion of a liquid contained in a tank that is rotated about its axis with angular velocity
v corresponds to a forced vortex.
A combined vortex is one with a forced vortex as a central core and a velocity distribution
corresponding to that of a free vortex outside the core. Thus, for a combined vortex
v u vr r r 0
(6.87)
and
v u K r
r 7 r 0
(6.88)
where K and v are constants and r 0 corresponds to the radius of the central core. The pressure
distribution in both the free and forced vortex was previously considered in Example 3.3. (See Fig.
E6.6a for an approximation of this type of flow.)
F l u i d s i n t h e N e w s
Some hurricane facts One of the most interesting, yet potentially
devastating, naturally occurring fluid flow phenomenan is a hurricane.
Broadly speaking a hurricane is a rotating mass of air circulating
around a low pressure central core. In some respects the motion
is similar to that of a free vortex. The Caribbean and Gulf of Mexico
experience the most hurricanes, with the official hurricane season
being from June 1 to November 30. Hurricanes are usually 300 to
400 miles wide and are structured around a central eye in which
the air is relatively calm. The eye is surrounded by an eye wall
which is the region of strongest winds and precipitation. As one
goes from the eye wall to the eye the wind speeds decrease sharply
and within the eye the air is relatively calm and clear of clouds.
However, in the eye the pressure is at a minimum and may be 10%
less than standard atmospheric pressure. This low pressure creates
strong downdrafts of dry air from above. Hurricanes are classified
into five categories based on their wind speeds:
Category one—74–95 mph
Category two—96–110 mph
Category three—111–130 mph
Category four—131–155 mph
Category five—greater than 155 mph.
(See Problem 6.58.)