How does knowing the 'specific energy' of water flowing in a channel help you tell if the flow is 'critical,' 'subcritical,' or 'supercritical'?

Knowing the 'specific energy' of water flowing in a channel helps classify the flow as 'critical,' 'subcritical,' or 'supercritical' by providing a direct comparison point based on the flow's energy state relative to its depth and velocity. Specific energy (E) is defined as the total energy per unit weight of water relative to the channel bottom. It is the sum of the flow depth (y) and the velocity head (V^2 / 2g), where 'V' is the average flow velocity and 'g' is the acceleration due to gravity. So, E = y + V^2 / 2g. For a given discharge (Q), there is a unique relationship between specific energy and flow depth. This relationship shows that there is a minimum specific energy possible for that discharge, which occurs at a specific depth called the 'critical depth' (y_c). This state of flow is known as 'critical flow.'

'Critical flow' occurs when the Froude number (Fr) is exactly 1. The Froude number is a dimensionless quantity that compares inertial forces to gravitational forces, and it is calculated as V / sqrt(g*D), where 'D' is the hydraulic depth. At critical flow, the flow velocity is equal to the speed of a shallow water wave. This condition represents a balance where the specific energy is at its minimum for a given discharge. Thus, if the calculated specific energy (E) for the flow equals this minimum specific energy (E_min) for the given discharge, the flow is critical, and its depth (y) is the critical depth (y_c).

'Subcritical flow' occurs when the Froude number is less than 1 (Fr < 1). In this condition, the flow is relatively deep and slow. The actual flow depth (y) is greater than the critical depth (y_c) for the given discharge. For subcritical flow, the specific energy (E) is always greater than the minimum specific energy (E_min) required for critical flow. In this regime, disturbances can propagate upstream.

'Supercritical flow' occurs when the Froude number is greater than 1 (Fr > 1). Here, the flow is relatively shallow and fast. The actual flow depth (y) is less than the critical depth (y_c) for the given discharge. Similar to subcritical flow, the specific energy (E) is also greater than the minimum specific energy (E_min) required for critical flow. In this regime, disturbances cannot propagate upstream, only downstream.

To determine the flow type, one first calculates the actual specific energy (E) of the flowing water using its depth and velocity. Then, for the same discharge, one calculates the critical depth (y_c) and the corresponding minimum specific energy (E_min). By comparing the actual flow depth (y) to the critical depth (y_c), or by observing the specific energy diagram (which plots specific energy against depth for a constant discharge), the flow regime can be identified:

1. If the actual flow depth (y) equals the critical depth (y_c), or if the calculated specific energy (E) equals the minimum specific energy (E_min) for that discharge, the flow is 'critical'.
2. If the actual flow depth (y) is greater than the critical depth (y_c), the flow is 'subcritical'. Although its specific energy (E) is greater than E_min, it occupies the upper limb of the specific energy diagram.
3. If the actual flow depth (y) is less than the critical depth (y_c), the flow is 'supercritical'. Its specific energy (E) is also greater than E_min, but it occupies the lower limb of the specific energy diagram.

Therefore, knowing the specific energy, along with the discharge and actual flow depth, allows for a precise classification of the flow regime by relating the flow's current energy state to the critical energy condition.