Question 10 of 13: Velocity and Flow-Rate Response of a Wide River in Flood

Question 10: Velocity and Flow-Rate Response of a Wide River in Flood (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

For a wide, shallow channel of fixed width, the wetted perimeter is dominated by the bottom (the two side walls contribute negligibly as the width-to-depth ratio is large), so the hydraulic radius R_h=A/P ≈ y (the flow depth itself). Combined with the Manning/Chezy uniform-flow relation V ∝ R_h^2/3 (slope and roughness unchanged), this lets both ratios be found from the single depth ratio alone.

1. (a) Velocity ratio. With R_h≈y and V ∝ R_h^2/3: $$\frac{V_{flood}}{V_{normal}} = \left(\frac{y_{flood}}{y_{normal}}\right)^{2/3} = 2^{2/3} = \boxed{1.587}$$ The flood velocity is therefore greater than the normal velocity — but (since the exponent 2/3 < 1) less than double it.
2. (b) Flow-rate ratio. Flow rate Q=VA=V(y·width), and since the width is fixed: $$\frac{Q_{flood}}{Q_{normal}} = \frac{V_{flood}}{V_{normal}}\times\frac{y_{flood}}{y_{normal}} = 2^{2/3}\times 2 = 2^{5/3} = \boxed{3.17}$$ The flood flow rate is therefore greater than two times the normal flow rate.
3. (c) Justification. Both results follow from the Manning-type relation V=(1/n)R_h^2/3S^1/2: doubling the depth doubles the driving cross-section directly (the "A" in Q=VA") AND increases the hydraulic radius (hence velocity, via reduced relative wall friction) by a further factor of 2^2/3, so the two effects compound and Q rises faster than linearly with depth.