River Flow Velocity Under Backwater Effect Tool

River Flow Velocity & Backwater Profile Calculator

This tool evaluates river flow velocity under backwater effects using Manning’s Equation, Continuity Equation, and Gradually Varied Flow (GVF) profile analysis. Comparison with HEC-RAS expected behavior is also provided.

Step-by-Step Solved Example

Problem Statement: A rectangular river channel with a bottom width $b = 20\text{ m}$ and a bed slope $S_0 = 0.001$ has a Manning's roughness coefficient $n = 0.03$. Normal depth before backwater effects is $h_1 = 5\text{ m}$. A downstream dam raises the backwater depth near the dam to $h_2 = 10\text{ m}$ where the energy slope drops to $S_f = 0.0002$. Determine initial discharge, backwater velocities, and the water surface slope ($\frac{dy}{dx}$).

Parameter Symbol & Formula Calculated Value
1. Initial Normal Flow Area $A_1 = b \times h_1 = 20 \times 5$ $100.00\text{ m}^2$
2. Initial Wetted Perimeter $P_1 = b + 2h_1 = 20 + 2(5)$ $30.00\text{ m}$
3. Initial Hydraulic Radius $R_1 = \frac{A_1}{P_1} = \frac{100}{30}$ $3.333\text{ m}$
4. Normal Flow Velocity $V_1 = \frac{1}{n} R_1^{2/3} S_0^{1/2} = \frac{1}{0.03} (3.333)^{2/3} (0.001)^{0.5}$ $2.352\text{ m/s}$
5. Total Discharge $Q = A_1 \times V_1 = 100 \times 2.352$ $235.20\text{ m}^3/\text{s}$
6. Backwater Flow Area (Near Dam) $A_2 = b \times h_2 = 20 \times 10$ $200.00\text{ m}^2$
7. Backwater Velocity (Continuity) $V_{2,\text{continuity}} = \frac{Q}{A_2} = \frac{235.20}{200}$ $1.176\text{ m/s}$
8. Froude Number (Near Dam) $\text{Fr}_2 = \frac{V_2}{\sqrt{g \cdot h_2}} = \frac{1.176}{\sqrt{9.81 \times 10}}$ $0.119$ (Subcritical)
9. Water Surface Slope (GVF $\frac{dy}{dx}$) $\frac{dy}{dx} = \frac{S_0 - S_f}{1 - \text{Fr}_2^2} = \frac{0.001 - 0.0002}{1 - 0.0142}$ $8.115 \times 10^{-4}\text{ m/m}$
Key Takeaway: Due to backwater pooling behind the dam, flow area doubles ($100 \rightarrow 200\text{ m}^2$), reducing velocity by half ($2.352 \rightarrow 1.176\text{ m/s}$). The Froude number $\text{Fr}_2 < 1$ confirms an M1-type gradually varied backwater profile.

Interactive Backwater Calculator