GATE CE 2022 Set 2 — Question 47

Let's analyze each statement regarding flow over a hump in a horizontal, frictionless, rectangular channel:Choked Condition: Choking occurs when the flow over the hump reaches the critical depth ($y_c$). At this point, the specific energy is minimum for the given discharge, and any further increase in hump height or decrease in specific energy is not possible without altering the upstream flow conditions.Statement B: "In choked condition, $y_2$ is equal to the critical depth if the flow is supercritical or subcritical."
This statement is correct. By definition, the choked condition at the hump means that the flow depth $y_2$ over the hump becomes equal to the critical depth $y_c$. This holds true irrespective of whether the upstream flow ($y_1$) was initially subcritical or supercritical.Statement A: "In choked condition, $y_1$ decreases if the flow is supercritical and increases if the flow is subcritical."
This statement is incorrect. Let's consider the behavior of $y_1$ as the hump height increases towards choking:

• Subcritical upstream flow: As the hump height increases, the upstream depth $y_1$ decreases to provide the necessary specific energy to pass over the hump. If choking occurs, $y_1$ would have decreased from its initial value.
• Supercritical upstream flow: As the hump height increases, the upstream depth $y_1$ increases (or a hydraulic jump forms upstream, raising the effective upstream depth) to provide the necessary specific energy. If choking occurs, $y_1$ would have increased from its initial value.

So, the statement describes the opposite behavior for both cases.Statement D: "In choked condition, $y_1$ increases if the flow is supercritical and decreases if the flow is subcritical."
This statement is correct in describing the behavior of $y_1$ as the flow approaches choking. As explained for statement A, if the upstream flow is subcritical, $y_1$ decreases, and if it's supercritical, $y_1$ increases (or a jump forms). However, statement B is a more direct and fundamental definition of the choked condition at the hump itself.Statement C: "In unchoked condition, $y_1$ remains unaffected when the flow is supercritical or subcritical."
This statement is incorrect. In an unchoked condition, if the hump height changes, the upstream depth $y_1$ will generally be affected to accommodate the change in specific energy required over the hump. For example, if the hump height increases, $y_1$ will decrease for subcritical flow and increase for supercritical flow (or a jump will form upstream). $y_1$ does not remain unaffected.Comparing B and D, statement B provides the defining characteristic of the choked condition at the hump ($y_2 = y_c$), which is a direct consequence of choking. Statement D describes the response of the upstream depth $y_1$ as the flow transitions to a choked state. In the context of "the correct statement(s)", B is a more precise and fundamental description of the choked state at the hump.The final answer is $\boxed{B}$