Specific force and its minimum at critical flow

Specific force (momentum function)

Specific force $F$ (or specific momentum $M$) is the sum of the momentum flux and the hydrostatic pressure force per unit weight of water at a section of the channel:

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

where $A$ is the flow area and $\bar y$ is the depth of the centroid of the area below the free surface. Its unit is m³.

Derivation (momentum equation)

Apply the momentum equation between two sections 1 and 2 over a short horizontal reach of a prismatic channel (friction and the weight component neglected, as in a hydraulic jump):

$$\gamma A_1\bar y_1-\gamma A_2\bar y_2=\rho Q(V_2-V_1)=\rho Q^2\left(\frac1{A_2}-\frac1{A_1}\right)$$

Dividing by $\gamma$ and rearranging:

$$\frac{Q^2}{gA_1}+A_1\bar y_1=\frac{Q^2}{gA_2}+A_2\bar y_2\;\Rightarrow\;F_1=F_2$$

Proof that $F$ is minimum at critical flow

For a given discharge, differentiate $F$ with respect to depth $y$:

$$\frac{dF}{dy}=-\frac{Q^2}{gA^2}\frac{dA}{dy}+\frac{d(A\bar y)}{dy}$$

With top width $T$, $dA=T\,dy$. Also $\dfrac{d(A\bar y)}{dy}=A$ (when the depth increases by $dy$, the first moment of area about the new surface increases by $A\,dy$). Therefore

$$\frac{dF}{dy}=-\frac{Q^2T}{gA^2}+A$$

For minimum $F$, $\dfrac{dF}{dy}=0$:

$$\frac{Q^2T}{gA^3}=1\;\Rightarrow\;\frac{Q^2}{g}=\frac{A^3}{T}$$

This is exactly the condition for critical flow ($Fr=V/\sqrt{gA/T}=1$). The second derivative $\dfrac{d^2F}{dy^2}=\dfrac{2Q^2T^2}{gA^3}-\dfrac{Q^2}{gA^2}\dfrac{dT}{dy}+T$ is positive for usual channel sections, so $F$ is a minimum there. Hence, for a given discharge, the specific force is minimum when the flow is critical.

  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

For each $F>F_{min}$ there are two depths (the conjugate depths) with the same specific force.

Uses of the concept

1. Hydraulic jump: the conjugate (sequent) depths $y_1$ and $y_2$ have equal specific force, so $F_1=F_2$ gives the depth after the jump.
2. Calculation of the force on structures (sluice gate, weir, baffle blocks, stilling basin) from the difference in specific force.
3. Locating the jump position in a stilling basin.
4. Analysis of flows where the energy loss is not known (jump, abrupt expansion, flow below gates).
5. Checking the flow type: the lower limb of the F-curve is supercritical and the upper limb is subcritical.