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04-BS-7 · May 2018

Question 11 of 13: The Laminar-to-Turbulent Friction-Factor Jump on the Moody Diagram

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

Notes on this paper

04-BS-7 Mechanics of Fluids — National Examination, 2018-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 manometry (Ch. 2), Bernoulli and the energy equation (Ch. 3), viscous flow in ducts and the Moody chart (Ch. 6), flow past immersed bodies and drag (Ch. 7), potential flow and the Magnus effect (Ch. 8), open-channel flow and the hydraulic jump (Ch. 10), turbomachinery and jet propulsion (Ch. 11).

Check — assumptions used across this paper:
  • Q1's manometer chain is read off the extraction as a two-stage water–mercury–glycerine–(air)–glycerine–mercury system. The enclosed air pocket between the two glycerine columns is treated as weightless (uniform pressure), so only the one described open end is needed to close the hydrostatic chain back to pipe P; the second "opening" is not load-bearing for this calculation.
  • Q5's wave/hydraulic-jump analysis takes the depth "in front of the wave" (0.15 m, undisturbed, at rest) as the upstream state and "behind the wave" (0.75 m) as the downstream state, per the question's own prose (the raw figure-label ordering in the extraction is a reconstruction and is not used to override the stated text). The classic hydraulic-jump head-loss formula is applied to the given depths, and the swept flow rate uses the measured wave celerity directly — a standard engineering estimate, not a fully momentum-self-consistent bore solution.
  • Q6's air properties are taken at 20°C (domestic ambient, ρ=1.19 kg/m³) since no duct-air temperature is stated.
  • Q7(c)'s "discharged at right angles to the initial direction (20° becomes 0°)" is read as: the reverser's exhaust jet is normally angled 20° forward of the fully-radial (right-angle) direction; part (c) removes that forward lean entirely, leaving a purely radial (90° to the engine axis) discharge with zero axial velocity component.
  • Q8's terminal velocity and Q9's cable drag coefficient are obtained from the Reynolds-number relations the attached charts themselves plot (Morrison's sphere-drag correlation for Q8; the flat subcritical Cd≈1.2 plateau of the smooth-cylinder curve for Q9, since Re≈3×104 falls solidly within it).

Question 11: The Laminar-to-Turbulent Friction-Factor Jump on the Moody Diagram (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.

Given. Moody diagram behaviour: laminar branch $f=64/Re$ (a smooth, continuous curve); turbulent branch begins abruptly near $Re\approx4000$ at roughly double the laminar-extrapolated value.

Find. The implication for pipe flow at fixed flow rate, and the fundamental energy-based cause of the discontinuity.

At the critical Reynolds number ($Re\approx2300$), laminar theory gives $f=64/2300=0.0278$; the turbulent correlation just past transition (Blasius, $f=0.316/Re^{0.25}$) gives roughly $f\approx0.046$ at the same Reynolds number — a jump of about 65–100%, matching the "nearly double" description. Implication at fixed flow rate. Since head loss is $h_L=f(L/D)(V^2/2g)$ and $V$ (hence $Re$) is fixed by the flow rate, a system operating right at the critical zone that is nudged from laminar into turbulent flow — by a small disturbance, a fitting, or surface roughness — suffers a near-instantaneous doubling of frictional head loss and pumping power requirement, with no change whatsoever in the delivered flow rate. A pump or pipe network sized assuming laminar operation near $Re_{crit}$ can therefore be significantly under-powered the moment the flow trips into turbulence.

Fundamental cause, in terms of energy. Laminar flow dissipates energy through an entirely orderly mechanism: adjacent fluid layers slide past one another with a smooth, parabolic velocity profile, and viscous shear stress $\tau=\mu\,du/dy$ converts kinetic energy to heat at a rate set purely by that gentle velocity gradient (Poiseuille's law, giving the smooth $f=64/Re$ curve). Once the flow becomes unstable to disturbances beyond $Re_{crit}$, this ordered structure collapses over a short transition zone and a second, much more effective dissipation channel switches on: turbulent eddies continuously exchange momentum between fluid layers (Reynolds stresses), and the kinetic energy of large eddies cascades down through progressively smaller eddies until it is finally dissipated as heat at the smallest (Kolmogorov) scales. This eddy-cascade mechanism is fundamentally additional to, not a smooth continuation of, laminar viscous shear — because the instability that triggers it is itself a threshold phenomenon (small disturbances grow only above $Re_{crit}$), the extra dissipation appears suddenly rather than building up gradually, which is exactly the discontinuity the Moody chart displays.