The critical depth, tranquil flow and rapid flow.

In the creation of a science the existence of some special value of a variable leads us to give it a name. The depth at which the specific energy takes its minimum is called the critical depth and denoted . There is nothing critical about it but some name has to be used to identify it and this is the one that is in common use. It becomes familiar with use. Clearly our graph indicates that we shall need to think of the flow as being able to occur in one of two regimes, that is, above and below the critical depth. As usual we want a short name to distinguish between these two modes of flow. Some call flow above the critical depth tranquil and below the critical depth rapid, others use super-critical and sub-critical probably because these names sound dramatic. In this text I will use tranquil and rapid as being much more descriptive of what we see when we go out to look at these two modes of flow. However these descriptive terms do not mean that the mode of flow is instantly recognisable, only flow at or near the critical depth is easily recognised from the disorganised undulations of the free surface but you will not see it very often.

 

It is of interest to find out how  is related to the minimum specific energy. This can be done if we differentiate to find the conditions for the minimum ;-

    and . Now if we put  at the critical depth it follows that:-                                        .

 From this it follows that the critical value of the specific energy is,  but as the value of  we get .

In other words, at the critical depth in this rectangular channel, the kinetic energy is 1/3 of the specific energy or the critical depth is 2/3 of the total head. For   =0.741 m and  =1.112 m which confirms the readings from the graph. This result gives us an insight to the behaviour of water flowing in rectangular channels but it must also be broadly the same for other practical sections.

 

It will be useful later to note that, if  is the velocity at the critical depth, . Then for tranquil flow  and for rapid flow . [1]

 

The line of d versus E in figure 12-4 is just one of a family of curves and graph 10-3 shows curves for values of flow per unit width of 3, 2.5, 2, 1.5, 1, and 0.5 m³/s/m.

 

This graph tells us that the behaviour of water flowing in a channel is the same regardless of the flow and that two regimes of flow can occur. The area on the graph between the line of critical depth and the maximum depth is the region of tranquil flow and the region between the line of critical depth and the x-axis is the region of rapid flow. I have used the presentation with the depth plotted vertically so that the depths for tranquil flow appear to be greater than the depths for rapid flow as, indeed, they are.