Critical Flow Defined

Critical flow in an open channel occurs under specific conditions related to the flow's depth, velocity, and energy. It represents a transition state between subcritical (tranquil) and supercritical (rapid) flow.

Statements Analysis for Critical Flow

Let's analyze each statement regarding critical flow:

• A. The specific energy is minimum for a given discharge.

  Specific energy ($E$) is defined as $E = y + \frac{V^2}{2g}$, where $y$ is the flow depth and $V$ is the average velocity. For a constant discharge ($Q$), $E$ is a function of $y$. Differentiating $E$ with respect to $y$ and setting the derivative to zero ($dE/dy = 0$) yields the condition for minimum specific energy. This condition simplifies to $V^2/g = A/T$, where $A$ is the cross-sectional area and $T$ is the top width. This is equivalent to the Froude number ($Fr$) being 1. Thus, specific energy is indeed minimum for a given discharge at critical depth. Statement A is true.

• B. The discharge is maximum for a given specific energy.

  For a constant specific energy ($E$), the discharge ($Q$) can be expressed as a function of depth ($y$). Maximizing $Q$ (or $Q^2$) with respect to $y$ by setting $dQ/dy = 0$ leads to the condition $E - y = \frac{A}{2(dA/dy)}$. Substituting $E-y = V^2/(2g)$ and $dA/dy = T$, we get $V^2/(2g) = A/(2T)$, or $V^2/g = A/T$. This is the condition $Fr = 1$. Therefore, the discharge is maximum for a given specific energy at critical depth. Statement B is true.

• C. The specific force is minimum for a given discharge.

  The specific force (often represented by the momentum function $M$) is given by $M = \frac{Q^2}{gA} + \frac{A \bar{y}}{1}$, where $\bar{y}$ is the depth of the centroid of the flow area from the free surface. For a given discharge ($Q$), minimizing $M$ with respect to depth ($y$) leads to the condition $V^2/g = A/T$, which is equivalent to $Fr = 1$. Thus, the specific force is minimum for a given discharge at critical depth. Statement C is true.

• D. Froude number of the flow is equal to unity.

  The Froude number ($Fr$) is defined as the ratio of the flow velocity ($V$) to the celerity of a small surface wave ($\sqrt{gy_h}$), where $y_h = A/T$ is the hydraulic depth. Critical flow is precisely defined as the condition where the Froude number equals 1 ($Fr = 1$). Statement D is true.

Final Answer Derivation

All four statements (A, B, C, and D) accurately describe conditions associated with critical flow in open channels. Therefore, the correct option includes all these statements.

The analysis confirms that critical flow is characterized by:

• Minimum specific energy for a given discharge.
• Maximum discharge for a given specific energy.
• Minimum specific force for a given discharge.
• Froude number equal to unity.

Consequently, the option stating A, B, C, and D are all correct is the valid choice.