Flow over a hump in the bed of a channel.

I want to look at the flow over a hump on the bed of a rectangular channel where the datum for specific energy changes. 

 

In figure 10-20 I have drawn a channel with a horizontal bed with a hump in the bed. The hump is an arc of a circle and is 0.5 m high in the middle and 4 m long. I have used the same flow per unit width of  by way of illustration and I intend to draw three typical surface profiles for different approach depths.

 

The presence of the hump obviously upsets the way in which we have been using specific energy until now where we used the horizontal bed as datum. The key observation is that graph 10-4 is not restricted to a particular channel but is for a particular flow per unit width of  in a rectangular channel. It relates the sum of the potential energy and the pressure energy, i.e. the depth, kinetic energy, and specific energy to depth. It can have any datum level. This means that graphs of specific energy versus depth for a given flow per unit width can be drawn for any point on the profile of the hump as the origin but the specific energy at every point must be measured from the same datum, usually the horizontal bed. Graph 10-4 also showed the critical depth to be 0.74 m and, in the channel in figure 10-20, I have shown a line representing the critical depth of 0.74 m for  in the channel with its hump. It clearly follows the profile of the channel.

 

I did not find this diagram easy to draw and it needs careful explanation. I have chosen to consider only three points on the profile of the hump. I called them A, B and C. For each point I prepared a reduced version of graph 10-1b with depth and specific energy plotted vertically and to the same vertical scale as the hump. These three graphs will have different datum levels corresponding to the elevation of points A, B and C. The graph for A has been be fitted directly on to the profile of the bed with its origin on the bed and graph C has its origin at the topmost point of the hump. There is no space for the graph for B so I have set it to the right with its horizontal axis at the level of B. and C

 

Now, as this piece of physics is dependent on ignoring friction, the specific energy is unchanged as the water flows over the hump. I have drawn three chain-dotted green lines representing lines of constant specific energy. One is arbitrarily at a level of 2 metres above the bed, a second goes through at the level of the minimum specific energy at the top of the hump and the third is part way between the other two. All three cut the specific energy lines in the three graphs. On each graph I have projected downwards from the intersections of the green lines on to the red depth line and shown the projection lines in black. Then for each graph I have projected across to the relevant vertical through A, B and C to give 3 points on each of three profiles. I have shown the projection for point C with chain dotted red lines but omitted the others because of the congestion of lines.

 

Then, in full red lines, I have drawn the three surface profiles.

 

Now we have to think about the implications of these three profiles. The upper two profiles are only possible if the channel beyond the hump is full so that the level after the highest point has been passed can rise again. Then, for this case of no friction means that the profiles are both symmetrical about the middle of the hump.

 

It is the third profile that dips down to the critical depth that is of special interest. We have to decide what happens after the water has passed the middle of the hump. If the level were to be high in the down stream channel the profile over the hump would again be symmetrical but, if the level were to be low, the flow would change to rapid flow and a hydraulic jump could appear in the channel probably on the hump.

 

But it is obvious that I could quite easily have drawn another constant specific energy line below the one that is at the minimum value of the specific energy curve for C. That line would have had no meaning because it would require the flow to have a specific energy below the minimum possible specific energy for this flow per unit width. So there is a minimum value for the upstream depth for this flow of  and that value comes when the depth over the top of the hump is the critical depth.

 

I concluded that a general exploration of the behaviour over this hump could only be made using a computer programme and that programme would require a loop and a conditional statement. I am not alone in finding that I cannot understand the programming methods used in MathCad and, if I did, I do not have sufficient use for them to be able to remember the methods. How I wish that BASIC had not been killed off. It would have been very useful to the jobbing engineer because the architecture was totally obvious and was not a memory exercise.

 

Nevertheless I think that I have done enough to come to terms with this problem.