Non-uniform discharge in a rectangular flume
In many cases the discharge Q in a flume is not uniform. We distinguish between gradually and rapidly varying discharge.
gradually varying discharge: the discharge depth h varies, the discharge Q or type of flow itself is (initially) subcritical. Gradually varying discharge occurs for example, in a slightly sloping flume with considerable surface roughness.
rapidly varying discharge occurs for example during fl ow over weirs. In many cases the discharge is supercritical.
Subcritical discharge has a large discharge depth h at smaller fl ow velocity v. In supercritical discharge the opposite is true: small discharge depth h and large flow velocity v. The flow transition from subcritical to supercritical discharge occurs with a continuous change of discharge depth h, flow velocity v and specific energy E, for example with an increase in the slope.
The flow transition from supercritical to subcritical discharge, on the other hand, always occurs with an abrupt change in the discharge depth h and a loss of specific energy ΔE, such as in a hydraulic jump.
Relationship between discharge Q, specific energy E and discharge depth h
Energy heads of a control volume
velocity head (v²/2g)
pressure head (h)
specific energy (E)
Considerations of the energy head at the control volume result in a third-order equation for the discharge depth h. The discharge depth h depends on the specific energy E and the discharge Q. A specific energy diagram shows the discharge depth h graphically as a function of the specific energy E at constant discharge Q. The minimum specific energy Emin only has one possible discharge depth, which is known as the critical depth hc. Critical discharge prevails at the critical depth hc. For all other specific energies there are two alternative depths that are relevant from a physics point of view (see diagram with hydraulic jump). The correct one of the two discharge depths has to be calculated in each case (is there subcritical or supercritical discharge?).
The maximum discharge Q at a given specific energy E can also be determined.
Relationship between momentum equation, specific force F and discharge depth h
The third important equation after Bernoulli and the conservation of mass is the momentum equation. The equilibrium of forces is established at the control volume. In many cases, the influence of the weight and the friction force is negligible. Therefore only the forces acting on the flow areas come into play: the static pressure force and the dynamic motive force. The specific force F is the sum of these two forces and is determined by the momentum equation.
The specific force can also be represented in a diagram. The specific force diagram plots the discharge depth h over specific force F at constant discharge Q. Similar to the specific energy diagram, there is the minimum specific force Fmin at critical depth hc. For all other specific forces there are two sequent depths.