04-BS-7 · May 2015
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
04-BS-7 Mechanics of Fluids — May 2015 (National Examinations, three hours, closed book). Section A (Calculative) offers 9 questions and instructs "do seven"; Section B (Analytical) offers 4 questions and instructs "do three." Every question is answered below (13 of 13), so students can use the full paper as a study resource.
Reference texts: F. M. White, Fluid Mechanics, 8th ed. (McGraw-Hill) — fluid statics (Ch. 2), the linear-momentum and energy equations (Ch. 3), pipe friction and the Moody chart (Ch. 6), and drag on immersed bodies (Ch. 7).
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 fully developed turbulent pipe flow, the flow near the wall organizes itself into a very thin viscous sublayer (thickness δ, typically well under a millimetre) in which viscous shear dominates and the velocity rises almost linearly from zero at the wall, followed by a logarithmic "law-of-the-wall" region, and finally the turbulent core. When the physical roughness height ε of the pipe wall is much smaller than δ ("hydraulically smooth"), the roughness elements sit entirely inside the viscous sublayer and never disturb the outer flow at all — the velocity profile and the friction factor are then essentially identical to a perfectly smooth pipe, independent of ε.
Once ε becomes comparable to, and then much larger than, δ ("fully rough" flow, the condition stated in the question), the roughness elements physically project through the viscous sublayer and shed their own turbulent wakes directly into the log-law region. This destroys the coherent viscous sublayer, so the wall shear stress — and hence the friction factor — becomes independent of the Reynolds number and depends only on the relative roughness ε/D (exactly the flat, horizontal region at the right-hand side of every curve on the Moody chart used in Questions 7 and 8). Because the extra wall drag removes more momentum from the fluid nearest the wall, continuity (same total flow rate Q in every case, per the question) forces the velocity profile to become blunter: the near-wall velocity gradient steepens sharply right at the wall, while the profile flattens out across most of the core, and the ratio of mean velocity to centreline velocity (Vavg/Vmax) rises toward 1 as roughness increases. A smooth pipe, by contrast, keeps a fuller logarithmic profile with a gentler velocity gradient spread more evenly across the pipe radius.
