Specific energy and the critical depth.
Now we can look at just one section in a rectangular channel. This is comparable with the treatment of flow in pipes where we related flow, area, kinetic energy and power using only continuity and the energy equation. Here the specific energy is the same for all the water at a given section and we can explore the physics of this specific energy. We have and if we suppose that the width of the channel is we can say, for a flow of , that . If now we put , where is the flow per unit width, and we can say that :-
and this is an equation in and constants. Now there is a choice to be
made. In my view it is best for me to draw graphs of the way and kinetic energy vary with depth because I
cannot convert mathematical symbols into spatial mental models. Generally I
like graphs to have the independent variable plotted as the x-axis and I try
never to suppress the zero but, in this case, the independent variable is the
depth and plotting this horizontally is counter-intuitive. I think that, in
this text, I will plot it both ways and my reader can choose which presentation
suits best. I need figures in order to plot a graph and, as a velocity of 1 m/s
is high enough to keep silt in suspension it must be a useful velocity to
consider. For this velocity and a depth of 2 m the flow/metre width =2m/s/m. Taking 2 m as the starting value for
depth we can use Mathcad to plot graphs of kinetic head, the sum of the
pressure energy and potential energy relative to the bed and of the specific
energy .
Graph 10-1a gives depth plotted horizontally and Graph 10-1b gives it plotted vertically. Neither is a wholly satisfactory presentation but I get on best with 10-1a.
The figures are plotted in graph 10-1a or 10-1b and the graphs give us a first insight to the flow patterns that are observable in real channels. The first observation must be that for any value of the specific energy there can be two depths. It is useful to give these two depths a common name and they are often called either conjugate depths or alternative depths. I shall use alternate depths.[1] The second observation is the existence of a minimum value of the specific energy for the selected value of flow/unit width. For our value of this minimum appears to be 1.15 J/N and it occurs at a depth of about 0.74 m. In addition, for every depth above 0.74 m there is a depth below 0.74 m that has the same specific energy. They are alternative depths. The shape of the graph at the minimum has implications. The minimum is not sharply defined and a relatively large change in depth can be associated with a small change in specific energy. This means that flow at the minimum specific energy will never be clearly defined. All this comes from a graph of quite simple quantities.
There is one important graph that must be plotted with depth horizontal and that is the graph of velocity versus depth. We shall need it later in this text. It is graph 10-2. It tells us that quite high velocities might be possible but engineers view high velocities with extreme caution. There is always a snag.