The free spiral vortex.

In describing the system shown in figure 15-8 I said that the tank was filled using some arrangement that minimises the disturbance in the water and this produces radial inward flow. Now suppose that a ring of guide blades were to be fitted as shown in figure 15-9 on the edge of the plate and fitting between the plate and the bottom of the tank.. These blades will impart angular momentum to the water and the flow between the bottom and the plate will not only be inwards but it will also rotate. A free vortex and steady inward radial flow will co-exist. This arrangement is used to create a swirling flow in water turbines and the runner would fit inside the ring of guide blades.

 

For the arrangement shown the flow pattern will change from the simple radial lines and there will be flow lines that curve from entry to the plate and exit to the nozzle. We need to know what these flow lines will look like.

 

If this is a combination of inward radial flow and a free vortex we can say for the inward radial flow that  where  is the radial velocity at radius  and  is the volume flow. For the free vortex the tangential velocity  at radius  is given by . If the flow at any point is a combination of inward radial flow and a free vortex the absolute velocity  at that point is the vector sum of  and .

In figure 15-10 I have drawn a flow line in red and the radial velocity  and the tangential velocity  at radius  are shown. The vector sum of these velocities is , the absolute velocity, and this makes an angle  with the tangent to the circle at radius .

 

We know that both  and  are inversely proportional to , so  must be the same at every radius. This is the characteristic feature of a logarithmic spiral that, for a polar plot, has the equation  where  are constants and  is the independent variable.  It will be clear that constant  simply increases the radius for any combination of values of  and . Constant  changes the angle and when  =1 the angle is 45°.

Graphs 15-5 shows the effect of changing the value of  which is to change the angle and graph 15-6 shows the effect of changing .

 

If the guide blades are all mounted so that they can be rotated together they will alter angle  and change the free spiral vortex.

 

I have drawn a set of flowlines for a combination of inward flow and a free vortex for the case of  = 45°. It is figure 15-10. Patterns like this occur in nature in the atmosphere and in spiral galaxies on a large scale, and in eddies on a small scale. I think that all that this tells us is that the logarithmic spiral occurs in many guises not necessarily that we are looking at fre spiral vortices.

 

We see immediately that this is a pattern for steady flow and the each flow line starts at one point and finishes at another unlike the flow lines of a vortex that are constrained to circles. The shearing of the free vortex has gone and this free spiral vortex has flow lines that have no velocity change across the lines.

 

The rotation reduces the rate of flow because whilst we can still write for point B in figure 15-8  the relevant velocity is the absolute velocity and not the radial one. It follows that the radial velocity at B is smaller and this means that the flow is reduced.