Uniform inward radial flow

This is important because, in nature, rotating flow often occurs where there is an inward radial flow. This inward radial flow provides the energy to sustain the rotating flow with its continuous loss of energy to friction. Inward radial flow can be produced quite easily in engineering so we deal with it like any other flow using the steady flow energy equation and the continuity equation because this is a normal flow in which energy can be lost steadily and made up externally whereas, as we have seen, the circulating flow lacks any mechanism by which energy can be imparted to the flow. 

 

Figure 15-8 represents a simple real system in which some form of radial flow must exist. It is an open-topped tank with a radiused nozzle fitted centrally in its base and with a circular plate set symmetrically above the nozzle at a fixed distance  above the base. When water is supplied steadily to the tank, in a way that minimises the disturbance caused by the filling, it will flow between the base and the plate and out of the nozzle. Eventually the level in the tank will become constant. What has been made is really a convergent nozzle that converts potential energy to kinetic energy and like all other convergent flows it will be stable and involve little loss of energy. Such a system might produce rotation as well as flowing radially but, if we treat the flow between the base and the plate as wholly radial, we can use our normal methods of ignoring friction and regarding the flow as one-dimensional and relate pressure, velocity and radius.

 

 is a point in the free surface and its total energy relative to the level midway between the plate and the base is  and  and   and both are equal to zero. The total energy at  = .

 

I have drawn a single flow line in blue that starts somewhere in the free surface and goes through the gap between the plate and the base to emerge in the jet. I have shown another identical flow line  in the the plan view to show that it is radial. The flow line passes through point  at radius  between the plates. Having chosen to ignore friction we can say that :-  where  is the pressure  and  is the velocity of the water at  and is the radial direction. 

 

The same equation applies to every other point between the plate and the base.

 

We need a relationship between the velocity and the radius and that will come from the continuity equation. The area of crossection of the flow at radius  is  and so the inward velocity  at radius  is  where  is the volume flow between the plates. It follows that:-

                                          and this is of the same form as the expression for pressure for the free vortex. It means that inward radial flow and the free vortex can co-exist.