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04-BS-7 · May 2017

Question 10 of 13: Significance of the Laminar Sublayer

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

Notes on this paper

04-BS-7 Mechanics of Fluids — National Examination, 2017-May. Three (3) hours duration, closed book. Section A (Calculative, 9 questions, do 7) and Section B (Graphical & 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: White, F.M., Fluid Mechanics (8th ed.) — fluid statics and capillarity (Ch. 2), integral analysis and buoyancy (Ch. 3), viscous flow in ducts, pipe friction and the Moody chart (Ch. 6), flow past immersed bodies and drag (Ch. 7), open-channel flow and the hydraulic jump (Ch. 10).

Check — assumptions used across this paper:
  • Q2's figure shows the vertical leaf of the gate rising above the free surface, so water acts over the full depth x = 2.0 m; both "0.3 m" dimensions locate the centre of gravity (0.3 m right of the vertical leaf, 0.3 m above the arm). Reservoir water also fills the space beneath the 1.2 m arm (the vertical lines below O are dimension extension lines, not a wall), so the arm carries uplift at pressure ρgx, and the arm tip bears up against a lip on the spillway crest — the gate can only open by rotating clockwise, vertical leaf toward the spillway.
  • Q4's flow coefficient K ≈ 0.69 is read from the attached VDI chart on the Do/D1 = 0.50 curve (third from the top, the one that levels off at 0.62) at the approach Reynolds number R ≈ 2550. The printed curve gives 0.695 at R = 2000 and 0.684 at R = 3000; a reading of 0.68–0.70 moves the answer by only ±0.3 kPa.
  • Q7 and Q8 friction factors are obtained from the Colebrook–White/Haaland equation the Moody chart itself plots (smooth-wall case, as both problems specify or imply smooth surfaces).
  • Q9's sphere drag coefficient vs. Reynolds number is obtained from the Schiller–Naumann correlation $C_D=\tfrac{24}{Re}\left(1+0.15\,Re^{0.687}\right)$, which reproduces the published sphere-drag curve (the same curve reprinted in the attachment) to within a few percent for $Re<1000$ — used here because the resulting Reynolds number (≈19–20) is well above the range where the simple Stokes'-law formula ($C_D=24/Re$) alone is valid.

Question 10: Significance of the Laminar Sublayer (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.

In turbulent pipe flow, the bulk of the cross-section is dominated by chaotic, momentum-exchanging eddies, but immediately adjacent to the wall, the no-slip condition forces velocity to zero and viscous shear (not turbulent mixing) must carry the momentum across a thin film called the laminar (or viscous) sublayer. Its significance is that it is this thin film — not the bulk turbulence — that actually "feels" the wall's surface texture, and so it is the laminar sublayer that decides whether a given pipe behaves as hydraulically smooth or hydraulically rough.

If $\delta>\epsilon$, the roughness elements (bumps of physical height $\epsilon$ on the pipe wall) are entirely buried within the laminar sublayer. The turbulent core "sees" a wall that is effectively smooth, because the viscous film absorbs the small perturbations the roughness would otherwise create; the friction factor in this regime depends only on Reynolds number, exactly the "hydraulically smooth pipes" curve on the Moody chart. If instead $\delta<\epsilon$, the roughness elements poke up through the thin sublayer directly into the turbulent core. Each protruding bump now sheds its own turbulent wake, generating additional form drag on top of ordinary viscous shear; the friction factor becomes essentially independent of Reynolds number and depends only on the relative roughness $\epsilon/D$ — the "fully rough" region at the top of the Moody chart.

The laminar sublayer's thickness is not fixed: it scales inversely with the flow's shear velocity, roughly $\delta\approx5\nu/u^{*}$, where the friction (shear) velocity $u^{*}=\sqrt{\tau_w/\rho}$ increases as the bulk velocity (and hence Reynolds number) increases. So as the flow velocity rises, $\delta$ shrinks. Since $\epsilon$ is a fixed physical property of the pipe wall, the ratio $\delta/\epsilon$ falls as velocity increases, meaning a pipe that behaves as hydraulically smooth at low velocity can progressively expose its own roughness and drift into the transitional, then fully rough, regime purely by speeding up the flow — with no change to the pipe itself. This is why the same physical pipe can trace a path across several zones of the Moody chart (smooth → transition → fully rough) as $Re$ increases at fixed relative roughness $\epsilon/D$; the pipe's "apparent" roughness, in the sense of how much it actually affects the friction factor, grows with velocity even though its true (measured) roughness never changes.