![](../images/spacer.gif) |
![](../images/spacer.gif) |
A wall in which there is a small nozzle, or hole,
of radius R, separates two large compartments which contain the
same fluid (e.g air or water). The pressure p1
in the left-hand compartment far from the nozzle is greater than the pressure
p2 in the right-hand compartment, and a steady
volume flow Q takes place from 1 to 2.
The flow through the nozzle is an incompressible,
high Reynolds number flow typical of the ones termed "inviscid".
However, as in all such flows, viscous forces are responsible for the
phenomenon of flow separation, which gives rise to a profound difference
between the inflow and outflow regions of the nozzle flow field.
On the inflow side, in compartment 1, the flow
is directed radially inward toward the nozzle entrance until one gets
close to the nozzle, and is essentially inviscid. The pressure decreases
and the velocity increases as one approaches the nozzle.
In compartment 2, on the other hand, the flow
separates from the wall and emerges as a jet with approximately horizontal
streamlines. This is due to boundary layer separation, the momentum-induced
inability of real viscous flows to follow sharply curved walls like the
lip of the nozzle at the exit plane. Inside compartment 2, viscous forces
slow the jet and drag the surrounding fluid with it, and at high Reynolds
numbers the combined flow becomes a turbulent jet which gradually broadens
and slows down with distance x. The process whereby the jet drags some
of the ambient fluid along with it is call "entrainment," and
gives rise in compartment 2 to a secondary bulk flow that is directed
approximately radially inward toward the axis of the jet, as sketched
(the radial referring now to a cylindrical coordinate system). The velocities
associated with this secondary flow are small compared with the jet velocity,
however, and the pressure in compartment 2 can be taken as being essentially
uniform, inside the jet as well as outside it.
(a) Consider a disc-shaped portion of the wall
extending a radial distance r from the nozzle centerline. Using
a control volume whose left side is a hemisphere of radius r, where
r >> R, show that the x-component of force exerted by the flow
on this portion of the wall is given by
Gravity is negligible in this problem.
|
![](../images/spacer.gif) |
![](../images/spacer.gif) |