The forced vortex

A real fluid in a suitable container can be made to rotate like a wheel. When it rotates steadily like a wheel the relationship between tangential speed and radius is simple. It is  where  is the angular velocity and  is the tangential velocity.

If  it can be re-written . This can be  where  is the pressure at radius .

 

This is not very suitable for use by engineers who are more accustomed to work in terms of head so it can be rewritten:-

         .

This can be plotted for some values of ,  and .

 

I have plotted graph 15-1 for a maximum radius of 250 mm and a speed of 250 rpm. The relationship between pressure head and radius is parabolic. At a radius of 250 mm the pressure head is about 2.2 m of fluid.

 

I think that a reader will find it to be valuable to explore this graph for other values of speed and radius.

 

It is worth looking at a worked example of a real system that could be made to flow with a forced vortex.

 

Figure 15-4 represents a cylindrical tank that is fitted with a top and is mounted so that it can be rotated about a vertical axis at a steady speed. The tank is completely filled with water. The top has a small hole in its centre so that the pressure of the water adjacent to the hole is atmospheric.

 

When the tank is rotated at a steady speed of 120 rpm the water will ultimately rotate with the tank as if the tank and the water formed a solid. The water then rotates as a forced vortex. One tends to associate swirling flow with free surface effects but the presence of the top on the tank has prevented free surface effects. However we can predict the pressure distribution through the water and the obvious thing to do is to predict the pressure distribution over the lower face of the top so that we can see what happens when the top is removed. We can also predict the pressure distribution over the upper face of the bottom.

We have  so the pressure head can be calculated using, say, Mathcad. It is shown in graph 15-2 where the scale of the pressure head is the same as the linear scale of the tank. The red line is clearly a parabola. The pressure head acts vertically upwards on the lower face of the top and will produce a force on the top tending to lift it.

 

This force can be calculated. It is:-

 

                                 .

The distribution of pressure head over the bottom of the tank is that for the top plus the depth of the tank, that is, 0.1 metres. The force exerted on the bottom is equal to the upward force on the top plus the weight of water. Therefore:-

                               force = .

 

This tank has one more thing to tell us. Suppose that the top were to be removed and the tank extended to a depth of 200 mm. If the tank is once again rotated at 120 rpm the water could only be in equilibrium if the surface took on the shape of this parabola and formed a paraboloid of revolution. Necessarily the middle would drop and the sides rise and, if water were to be added, the apex of the paraboloid could be brought back to its original position at a depth of 0.1 metres.

 

 I have shown the result in figure 15-5 where the red lines show the surface as a paraboloid of revolution.

 

It is most unlikely that anyone would see such a surface under normal day-to-day activity because it clearly requires some mechanical device in which to produce it. I wondered about this surface as I had seen one created in a rig filled with water made by someone else and it was very fragile in that it was very easily disturbed. Thumping the bench on which it stood produced a fascinating array of surface waves on the paraboloidal surface.

 

It seemed to me, as an engineer, that the whole process of bringing such a tank to a state of equilibrium is not as simple as it sounds and there is some value in looking more closely at this process.