Specific energy, minimum value and critical flow

Specific energy

Specific energy $E$ is the energy per unit weight of water measured with the channel bed as datum:

$$E=y+\frac{\alpha V^2}{2g}\approx y+\frac{V^2}{2g}=y+\frac{Q^2}{2gA^2}$$

It has the unit of length and, unlike the total head, it can increase or decrease along the flow.

Proof of minimum energy at critical flow

For constant $Q$, $E$ is a function of $y$ only. Differentiate with respect to $y$ ($dA=T\,dy$):

$$\frac{dE}{dy}=1-\frac{Q^2}{gA^3}\frac{dA}{dy}=1-\frac{Q^2T}{gA^3}$$

For minimum $E$, $dE/dy=0$:

$$\frac{Q^2T}{gA^3}=1\;\Rightarrow\;\frac{V^2}{g(A/T)}=1\;\Rightarrow\;Fr=\frac{V}{\sqrt{gD_h}}=1$$

Since $\dfrac{d^2E}{dy^2}=\dfrac{3Q^2T^2}{gA^4}-\dfrac{Q^2}{gA^3}\dfrac{dT}{dy}>0$ for usual sections, this is a minimum. The condition $Fr=1$ is critical flow. Hence the specific energy is minimum at critical flow.

For a rectangular channel: $y_c=\sqrt[3]{q^2/g}$ and $E_{min}=\dfrac32y_c$.

Specific energy curve

  y ^
    |                    .   upper limb: subcritical (Fr < 1),
    |                  .     approaches the line E = y
 y1 |----------------*
    |             .
    |         .
 yc |-------*  <- critical point: y = yc, E = E_min
    |         .
    |             .
 y2 |----------------*       lower limb: supercritical (Fr > 1),
    |                    .   approaches the E-axis
    +-------+--------+-------> E
          E_min      E

• The curve has two branches for a given $E>E_{min}$ and a single point at $E=E_{min}$.
• Alternate depths: the two depths $y_1$ (subcritical, upper limb) and $y_2$ (supercritical, lower limb) have the same specific energy $E$.
• Critical depth $y_c$ and $E_{min}$ are at the nose of the curve.
• Subcritical zone: $y>y_c$, $V<\sqrt{gD_h}$, upper limb (asymptotic to the 45° line $E=y$).
• Supercritical zone: $y<y_c$, $V>\sqrt{gD_h}$, lower limb (asymptotic to the $E$-axis).
• For a rectangular section the curve is a cubic: $y^3-Ey^2+\dfrac{q^2}{2g}=0$.