Convergent, divergent and curving flow within a flowing fluid.

Figure 4-5 is a representation of a tank like the Perspex tank but fitted with a nozzle and not an orifice. (This makes it easier for me to draw.) I have shown the tank to be fitted with a sprinkler system so that water can be fed into the tank without seriously disturbing the water as a jet would do. If water is supplied to the tank at a steady rate, a jet, flowing vertically downwards, will emerge from the nozzle. The level in the tank will rise until the outflow from the nozzle is just equal to the inflow.

 

We have seen that the main body of the water will flow downwards quite slowly in an orderly manner and then turn towards the nozzle and eventually make a rapid exchange of pressure energy and potential energy for kinetic energy in the vicinity of the nozzle. Figure 3-5 shows one of the paths as water flows from the free surface to the exit plane of the nozzle. I have called such a path a flow line. This flow line is one of an infinite number of such lines. The flow pattern for this tank is shown in figure 4-6.

 

This flow pattern is, of course, a cross-section of a three-dimensional flow pattern and figure 4-7 shows two surfaces that contain the flow lines which start in concentric circles at the surface. The water that starts at the surface between these concentric circles must subsequently flow between the two surfaces.

 

It is useful to try to decide what factors determine a flow pattern. Clearly the flow is caused by the gravitational attraction on the water. There are two special flow lines, the one that follows the long path down the wall and across the bottom, and the central one that coincides with the axis. A more typical flow line is that shown in figure 4-5. There must be a reason for the shape of this path.

 

A flow pattern can be thought of as being made up of many tubes of more or less square section that all fit together to fill the space occupied by the flow. Every tube is bounded by an infinite number of flow lines. I have drawn one of these “square” sectioned tubes in figure 4-7.  It is evident that the water changes its direction of flow as it flows through the tube. It also changes its velocity in the direction of flow. Now we can look at what goes on around a short length of this tube to find out what forces are acting on it and what effect these forces produce.

 

 

 

 

 

 

 

 

 

Figure 4-10 shows water flowing through a short length of one of these imaginary tubes, which is of small cross-sectional area, and through which the rate of flow is . Figure 4-10a shows the front view of the elemental volume and shows water flowing into the elemental volume at section 1 where the area is  with a velocity  and out at section 2 at  and .

 

The water in the elemental volume is subjected to a pressures  and  on sections 1 and 2 shown in Figure 4-10b and these pressures produce forces of  and  acting on the water. All four sides of the element are subjected to non-uniform pressures and, because the elemental volume is tapered, this forces produced by all of these pressures produce a resultant force on the water.

 

The four side surfaces are also surrounded by water that will be moving at different velocities to that of the element. We have already acknowledged the fact that fluids resist forces producing a change in shape and so we must expect there to be tangential shearing forces, which could be in either direction, exerted on the water. The forces on two of the sides are shown in figure 4-10c.

 

Finally the mass,  of water in the elemental volume has a gravitational force exerted on it equal to  shown in figure 4-10d.

 

All these forces combine to produce an acceleration acting along the path and a centripetal acceleration to give the change in the direction of flow of the water as it flows through the elemental volume. The whole flow is made up of many elemental volumes between which normal pressures and tangential forces are exerted. The flow will become steady when, in every elemental volume the water moves at the right speed, and the volume has the right shape and is subject to the right system of forces to move in equilibrium with all the other elemental volumes. In the limit these elements tend to become just the path line and differential equations can be set up to describe the flow.

 

In the past all this was recognised and the possibility of setting up mathematical equations that might be solved to describe any flow must have been evident. Navier and Stokes chose to model the shear forces by treating the flow as laminar (See later) and both contributed to the derivation of the partial differential equations, now called the Navier-stokes equations, that might produce the desired result if they could be solved but that must have been a far-off dream for them. Now they can be solved to various levels of complexity by powerful, fast computers but it is not easy and very expensive. Engineers will mostly have to soldier on with ordinary methods. They should recognise that Computational Fluid Dynamics, the name for the application of the Navier-Stokes equations, is still just a tool albeit an expensive one, for exploring the way in which some fluid systems work. It is only as good as the input to it. I watched a television programme on the building of the cruise liner Queen Mary 2 and it reported that the model testing showed that the hull would not meet the contracted speed. We had a sequence of the head-man being towed along beside the very large model of the ship in a tiny pram and studying the flow over the bulbous bow. He came back and ordered the length of the bulb to be increased by one metre. The fact that the change was a nominal amount tells us that he had backed his experience and intellect against whatever design method had been used and, it seems, the necessary improvement was achieved. So, as an engineer, keep the initiative over the CFD and remember, “slaves do arithmetic”. [1]