Determining River Flow Condition Using Froude Number

The flow condition in an open channel, like a river, is determined by comparing the flow velocity to the speed of a small gravity wave. This comparison is quantified using the Froude number ($Fr$).

Froude Number Calculation

The Froude number ($Fr$) is calculated as:

$ Fr = \frac{V}{\sqrt{g \cdot L_h}} $

• $V$ is the average flow velocity.
• $g$ is the acceleration due to gravity ($ \approx 9.81 \text{ ms}^{-2}$).
• $L_h$ is the hydraulic depth.

Hydraulic Depth

For a wide rectangular channel, the hydraulic depth ($L_h$) is approximately equal to the flow depth ($D$). In this problem:

• Flow velocity ($V$) = $2 \text{ ms}^{-1}$
• Channel depth ($D$) = $10 \text{ m}$
• We approximate $L_h \approx D = 10 \text{ m}$.

The flood plain width ($2 \text{ km}$) is significantly larger than the channel depth and typically does not directly influence the Froude number calculation for the main channel flow unless specified otherwise.

Calculating the Froude Number

Substitute the values into the formula:

$ Fr = \frac{2 \text{ ms}^{-1}}{\sqrt{9.81 \text{ ms}^{-2} \cdot 10 \text{ m}}} $

$ Fr = \frac{2}{\sqrt{98.1}} $

$ Fr \approx \frac{2}{9.9045} $

$ Fr \approx 0.2019 $

Interpreting the Froude Number

The calculated Froude number determines the flow condition:

• If $Fr < 1$, the flow is sub-critical (streaming).
• If $Fr = 1$, the flow is critical.
• If $Fr > 1$, the flow is super-critical (shooting).

Since $Fr \approx 0.2019$, which is less than 1, the flow condition is sub-critical.