fluid_mechanics

176 Chapter 4 ■ Fluid Kinematics
Moving vane
Nozzle
V CV = V 0
Control volume
V 1
V 0 moves with speed V 0
F I G U R E 4.20
Example of a moving control volume.
The absolute and
relative velocities
differ by an amount
equal to the control
volume velocity.
4.4.6 Moving Control Volumes
For most problems in fluid mechanics, the control volume may be considered as a fixed volume
through which the fluid flows. There are, however, situations for which the analysis is simplified
if the control volume is allowed to move or deform. The most general situation would involve a
control volume that moves, accelerates, and deforms. As one might expect, the use of these control
volumes can become fairly complex.
A number of important problems can be most easily analyzed by using a nondeforming
control volume that moves with a constant velocity. Such an example is shown in Fig. 4.20 in
which a stream of water with velocity V 1 strikes a vane that is moving with constant velocity V 0 .
It may be of interest to determine the force, F, that the water puts on the vane. Such problems
frequently occur in turbines where a stream of fluid 1water or steam, for example2 strikes a series
of blades that move past the nozzle. To analyze such problems it is advantageous to use a moving
control volume. We will obtain the Reynolds transport theorem for such control volumes.
We consider a control volume that moves with a constant velocity as is shown in Fig. 4.21.
The shape, size, and orientation of the control volume do not change with time. The control volume
merely translates with a constant velocity, V cv , as shown. In general, the velocity of the control
volume and the fluid are not the same, so that there is a flow of fluid through the moving control
volume just as in the stationary control volume cases discussed in Section 4.4.2. The main difference
between the fixed and the moving control volume cases is that it is the relative velocity, W, that
carries fluid across the moving control surface, whereas it is the absolute velocity, V, that carries
the fluid across the fixed control surface. The relative velocity is the fluid velocity relative to the
moving control volume—the fluid velocity seen by an observer riding along on the control volume.
The absolute velocity is the fluid velocity as seen by a stationary observer in a fixed coordinate
system.
The difference between the absolute and relative velocities is the velocity of the control
volume, V cv V W, or
V W V cv
(4.22)
Since the velocity is a vector, we must use vector addition as is shown in Fig. 4.22 to obtain the
relative velocity if we know the absolute velocity and the velocity of the control volume. Thus, if
the water leaves the nozzle in Fig. 4.20 with a velocity of V 1 100î fts and the vane has a velocity
of V 0 20î fts 1the same as the control volume2, it appears to an observer riding on the vane that
the water approaches the vane with a velocity of W V V cv 80î fts. In general, the absolute
Particle B at t 0
Particle A at t 0
At t 1
V A
Control volume and system
at time t 0
Control volume
at time t 1 > t 0
System at time t 1 > t 0
At t 1
V B
V CV = Control volume velocity
F I G U R E 4.21
Typical moving control volume and system.