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04-BS-7 · December 2013

Question 12 of 13: Flood Velocity and Flow Rate in a Wide Shallow River

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

04-BS-7 Mechanics of Fluids — National Examination, 2013-Dec. Three (3) hours duration, closed book. Section A (Calculative, 9 questions, do 7) and Section B (Analytical, 4 questions, do 3); every question is answered below regardless of the exam's "do N of M" instruction, so the set is a complete study resource.

Reference texts: Crowe, C.T., Elger, D.F. & Roberson, J.A., Engineering Fluid Mechanics; Douglas, J.F., Gasiorek, J.M., Swaffield, J.A. & Jack, L.B., Fluid Mechanics; White, F.M., Fluid Mechanics.

Check — assumptions used across this paper:
  • Air density is taken from the paper's own Constants table at the temperature each question states: 1.19 kg/m³ at 20°C (Q5's wind, Q7's inlet air).
  • Q1's touching-rod array is modelled as a repeating square unit cell of four mutually tangent rods (pitch = rod diameter, per the question's own "closely packed" wording), giving a curvilinear-square pore whose perimeter/area ratio drives the capillary rise.
  • Q8's Moody diagram and Q9's drag-coefficient diagram are supplied as attachments. Both are solved via the equations the charts themselves plot: the Colebrook–White equation for Q8 and the Morrison (2013) curve-fit for sphere drag vs. Reynolds number for Q9.

Question 12: Flood Velocity and Flow Rate in a Wide Shallow River (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 river that is much wider than it is deep, the hydraulic radius $R=A/P$ (flow area over wetted perimeter) is very well approximated by the depth itself, $R\approx y$, because the wetted perimeter is dominated by the (constant) bed width and barely changes with depth. Manning's equation, $V=\tfrac{1}{n}R^{2/3}S^{1/2}$, with slope S and roughness n both unchanged, then gives velocity scaling purely with depth: $V\propto y^{2/3}$.

Since the banks hold the width constant, flow area scales as $A\propto y$ (width × depth, width fixed), so flow rate scales as $Q=VA\propto y^{2/3}\times y = y^{5/3}$.

(a) Velocity. With flood depth twice normal, $V_{flood}/V_{normal}=2^{2/3}=1.587$: the flood velocity is greater than the normal velocity (by about 59%), never doubling since velocity depends only on the two-thirds power of depth.

(b) Flow rate. $Q_{flood}/Q_{normal}=2^{5/3}=3.17$: the flood flow rate is greater than two times the normal flow rate — substantially more than double, because the extra depth increases both the cross-sectional area (linearly) and the velocity (via the higher hydraulic radius) at the same time.

QuantityValue
Velocity ratio, flood:normal1.587 (greater than normal)
Flow-rate ratio, flood:normal3.17 (greater than 2×)