Specific force and conjugate depths

Specific force (momentum function)

Specific force $F$ at a section is the sum of the momentum flux and the hydrostatic pressure force per unit weight of water:

$$F=\frac{Q^2}{gA}+A\bar y$$

where $\bar y$ = depth of the centroid of the flow area below the water surface.

Development (from the momentum equation)

For a short horizontal reach (friction and the weight component negligible), the net force equals the change in momentum flux:

$$\gamma A_1\bar y_1-\gamma A_2\bar y_2=\rho Q(V_2-V_1)=\rho Q^2\left(\frac{1}{A_2}-\frac{1}{A_1}\right)$$ $$\frac{Q^2}{gA_1}+A_1\bar y_1=\frac{Q^2}{gA_2}+A_2\bar y_2\;\Rightarrow\;F_1=F_2$$

For a rectangular channel of width $b$ ($q=Q/b$, $A\bar y=by^2/2$), the specific force per unit width is

$$F=\frac{q^2}{gy}+\frac{y^2}{2}$$

Conjugate (sequent) depths

For a given $Q$ and a value of $F>F_{min}$, the equation $F(y)=$ const has two positive roots, $y_1$ (supercritical) and $y_2$ (subcritical). These are the conjugate depths: they have the same specific force. They are the depths before and after a hydraulic jump, where energy is lost but specific force is conserved. For a rectangular channel:

$$\frac{y_2}{y_1}=\frac12\left(\sqrt{1+8Fr_1^2}-1\right)$$

Specific force curve

  y ^
    |                    .   upper limb: subcritical
 y2 |----------------*       conjugate depth y2
    |             .
    |         .
 yc |-------*  <- F_min, critical depth
    |         .
    |             .
 y1 |----------------*       conjugate depth y1
    |                    .   lower limb: supercritical
    +-------+--------+-------> F
          F_min   F1 = F2

• Lower limb ($y<y_c$): supercritical flow.
• Nose of the curve: $F=F_{min}$ at $y=y_c$, where $\dfrac{Q^2T}{gA^3}=1$ (critical flow).
• Upper limb ($y>y_c$): subcritical flow.
• A vertical line at a given $F$ cuts the curve at the two conjugate depths. The difference in specific energy between these depths is the energy loss in the jump.