Condition for a free vortex
Suppose that in some way we could produce a free vortex in an extensive mass of fluid and rotating about a vertical axis. If the fluid were to be real, viscosity would soon alter the flow pattern so let us suppose that the fluid is frictionless so that we can see what might happen. The important feature would be that the flow had no torque exerted on it anywhere because the presence of a torque is not consistent with steady flow. As torque equals the rate of change of angular momentum the necessary condition for zero torque is that the rate of change of angular momentum with radius is the same everywhere in the vortex and is zero. If we consider the flow to be made up of many particles of fluid each with the same mass every particle must have the same angular momentum about the centre of rotationof the ho vortex.
For any particle the angular momentum is given by :-
where is the mass of the particle.
It follows that the condition for a free vortex is that where is a constant or of course that . So where is a new constant.
Then :- .
This can be integrated between two radii, both of which must be finite because the condition of uniform angular momentum leads to infinite tangential velocity at zero radius and zero tangential velocity at infinite radius.
. Then:-
and this can be put in Newton metre/Newton :-
This result shows that not only is the angular momentum of the particles of fluid uniform throughout the vortex but the sum of the pressure energy and the tangential kinetic energy, , is the same everywhere in the vortex as well[1].
It is now possible to draw a graph of pressure head aganst radius for a free vortex with, say, the pressure at some suitably large radius being atmospheric. Then the pressure throughout the vortex will be less than atmospheric. We can put:-
and this can be evaluated as a graph.
I have plotted a graph of pressure head against radius to suitable values in graph 15-3. It is the red line. It tells us the character of the free vortex. All the interesting things happen in the middle where the tangential velocity tends to infinity as the radius tends to zero.
Any real free vortex will occur in a real fluid and real fluids have viscosity. This means that energy will be lost in the fluid wherever there is a velocity gradient with the inevitable shearing. It is pertinent to plot a graph of the velocity gradient versus radius to see how the rate of change of shearing strain varies with radius.
We have for a free vortex :- and . This can be plotted for the same value of the constant as was used in graph 15-3 and I have added it in blue to graph 15-3 to give 15-4. Note the dividing factor of 100 to get the two graphs on the same pair of axes. In the middle the velocity gradient is very large indeed.
So a free vortex in a real fluid would lose energy to viscous friction very quickly in the centre but very slowly in the rest of the vortex. It might exist for a short time in a fluid of low viscosity but is unlikely ever to come into existence for fluids with high viscosity. A real fluid, even if it has low viscosity, cannot flow with a free vortex in the centre. Some change in the flow pattertn must occur. It looks to an observer with a knowledge of the forced vortex as though a forced vortex forms in the centre of the free vortex and this is a very plausible interpretation of this mode of flow.
The size of the forced vortex will depend on both the viscocity and density of the fluid and may be quite small for mercury and very large for thick oil. But there is nothing practical about it and in truth we are only interested in the character of this notional flow so that we can understand it when the free vortex is combined with inward radial flow and the flow pattern ceases to be imaginary and becomes real in the sense that the energy equation can be applied toit.