Effect of increasing hump height on water surface

Set-up

Consider a rectangular channel with discharge $q$ per unit width, approach depth $y_1$ (subcritical) and specific energy $E_1=y_1+\dfrac{q^2}{2gy_1^2}$. A hump of height $\Delta z$ is placed on the bed. With no energy loss, the specific energy over the hump is

$$E_2=E_1-\Delta z$$

On the specific energy diagram, moving onto the hump shifts the state to the left by $\Delta z$. The critical hump height is $\Delta z_c=E_1-E_{min}=E_1-\tfrac32y_c$.

  y ^
    |                       .
 y1 |----------------------*   (1) upstream, E1
    |                  .
 y2 |-------------*            (2) over hump, case (i): E2 = E1 - dz
    |        .
 yc |-----*                    critical point, E_min (case ii: E2 = E_min)
    |        .
    |             .
    |                   .      supercritical limb
    +-----+-------+--------+-----> E
        E_min     E2       E1
                  |<- dz ->|

(i) Hump height less than critical ($\Delta z<\Delta z_c$)

• $E_2=E_1-\Delta z>E_{min}$, so flow over the hump is possible with the same discharge.
• The upstream depth $y_1$ is unchanged.
• The depth over the hump falls along the subcritical limb (it is lower than $y_1$ but still greater than $y_c$). The water surface drops over the hump.
• After the hump the depth rises again to $y_1$.

(ii) Hump height equal to critical ($\Delta z=\Delta z_c$)

• $E_2=E_{min}$, so the depth over the hump is exactly critical ($y=y_c$).
• The upstream depth is still $y_1$ (unchanged), but this is the limit: the hump now acts as a control (like a broad-crested weir).
• The discharge is the maximum that can pass for this upstream energy.

(iii) Hump height greater than critical ($\Delta z>\Delta z_c$)

• $E_1-\Delta z<E_{min}$: the energy available over the hump is less than the minimum required for $q$, so the flow cannot pass with the given upstream energy.
• The flow backs up: the upstream depth rises to $y_1'$ so that the upstream energy becomes $E_1'=E_{min}+\Delta z$.
• Critical depth occurs over the hump, and downstream of the hump the flow becomes supercritical (and may end in a hydraulic jump).
• Increasing the hump height further raises the upstream depth further.

If the hump is combined with a constriction of constant width, the same logic applies, with $q$ increased in the contracted section: the critical condition is $E_1-\Delta z=\tfrac32y_c$, with $y_c=\sqrt[3]{q_2^2/g}$ and $q_2=Q/b_2$.