Is there a mathematical analysis of the submerged hydraulic jump, such as under what condition it is formed, the speed and trajectory of the circular flow therein, etc.? Is there a good mathematically sophisticated reference on this topic?
In open channels when the location of supercritical flow and hydraulic jump are very close, or restriction in down stream encourages higher elevation, submerged or partially submerged hydraulic jump happen.
For example, when they use flumes to measure the flow rate of an irrigation open channel because of the downstream conditions or obstacles the hydraulic jump may build up and partially or totally submerge the supercritical flow up-stream.
In closed pipe flow such as sewer system, the incidental blockage before an elbow or other constrain will cause submerged jump leading to deposit of more solids, that's why they design those locations with access openings, clean-outs.
because of wide range of conditions leading to this situation, I think the only practical way of detailing and calculation the submerged jump is using FEM and simulation programs.