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04-BS-7 · December 2013

Question 10 of 13: Spillway Flow — Streamlines, EGL and HGL

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 10: Spillway Flow — Streamlines, EGL and HGL (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.

EGLHGL (free surface)spillwayhydraulic jump / tailwaterstreamlines, turbulence, EGL and HGL
Reservoir → accelerating flow down the spillway face → turbulent hydraulic jump → subcritical tailwater; EGL (red, dashed) and HGL/free surface (blue) sketched over the same reach.

Over the reservoir the water is nearly still and both grade lines sit close together, high above the bed — almost all of the available energy is potential (elevation) head, with negligible velocity head and negligible friction loss over the short reservoir reach. As the flow accelerates down the sloped spillway face, the free surface (HGL) drops sharply and the streamlines converge and steepen, tracking the spillway profile closely with essentially parallel, energy-conserving flow (only a modest EGL drop from spillway-surface friction, since the flow is fast but the wetted length is short); nearly all of the elevation drop reappears as velocity head, so the gap between EGL and HGL widens steadily down the chute — that gap IS the velocity head, $V^2/2g$.

At the toe of the spillway the flow is shooting (supercritical, high V, shallow y) and meets the slower, deeper tailwater; it cannot smoothly decelerate, so it does so abruptly through a hydraulic jump. This is the one region with visible, concentrated turbulence: eddies and surface rollers churn within the jump as kinetic energy is violently converted to heat and re-mixed turbulence rather than recovered as useful head. Because this dissipation is real energy loss (not merely traded for elevation or velocity head), the EGL drops sharply and irreversibly across the jump — a much steeper EGL drop, over a much shorter distance, than anywhere else in the reach — while the HGL actually rises abruptly (the free surface jumps up from the shallow supercritical depth to the deeper subcritical tailwater depth). Downstream of the jump, in the tailwater channel, the flow is slow and deep again; EGL and HGL run nearly parallel and close together, both dropping gently with ordinary channel friction, mirroring the calm reservoir condition upstream but at a lower total energy level than before the jump.

Conclusion: velocity head is largest right at the spillway toe (fast, shallow flow, EGL and HGL far apart) and collapses through the jump as the flow slows and deepens; friction losses are minor and gradual everywhere except at the jump itself, where the single, concentrated, irreversible loss produces the one sharp step down in the EGL.