04-BS-7 · December 2019
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
04-BS-7 Mechanics of Fluids — December 2019 (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. Constants used throughout (from the paper's own Constants page): g = 9.81 m/s², patm = 100 kPa, ρwater = 1000 kg/m³, SGglycerine = 1.26, SGmercury = 13.56, ρconcrete = 2400 kg/m³, ρair = 1.19 kg/m³ (20°C) / 1.21 kg/m³ (15°C), μwater = 1.0×10⁻³ N·s/m², μair = 1.8×10⁻⁵ N·s/m², Rair = 287 J/kg·K.
Reference texts: F. M. White, Fluid Mechanics, 8th ed. (McGraw-Hill) — fluid statics and manometry (Ch. 2), hydrostatic forces and the middle-third rule (Ch. 2), buoyancy and equilibrium (Ch. 2), dimensional analysis and drag (Ch. 5, 7), pipe friction and the Moody/Colebrook relation (Ch. 6), control-volume momentum and propeller/actuator-disk theory (Ch. 3, 11); J. D. Anderson, Fundamentals of Aerodynamics — wave/compressibility drag divergence (Ch. 5) for the Boeing 747 wind-tunnel chart used in Question 9.
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. Two geometrically identical 90° rectangular duct bends carrying the same high-velocity airflow: Duct A is a plain (unvaned) bend; Duct B has a set of curved turning vanes installed across the bend.
Find. Which duct has the lesser head loss, and why.
Duct B (with curved turning vanes) has the lesser head loss. In Duct A, high-velocity air approaching the sharp 90° bend cannot follow the abrupt change in direction, particularly along the INNER wall of the turn; the flow separates from the wall there, and a large, energetic recirculating eddy (secondary flow) forms in the separated region. This separation is the dominant source of loss in an unvaned bend — the kinetic energy trapped in the recirculating eddy is continuously dissipated into heat by turbulent mixing rather than being usefully carried around the bend, and the effective flow area is also reduced (the eddy partially blocks the duct), increasing the velocity, and hence the dynamic loss, in the remaining clear passage.
In Duct B, the curved vanes divide the single sharp 90° bend into several smaller, more gently-curved passages in parallel. Each individual passage has a much SMALLER turning radius-to-width ratio than the single unvaned bend, so the flow within each passage can negotiate its share of the turn without separating from either wall — the vanes essentially guide the flow smoothly around the corner instead of asking it to turn sharply on its own. With no separation and no recirculating eddy, the loss coefficient for a vaned bend is substantially lower than for an equivalent sharp, unvaned bend at the same velocity and duct size.
Why this justifies the sketch. The streamline sketches above show the mechanism directly: Duct A's streamlines cannot hug the inner corner and peel away into a dashed recirculation zone, while Duct B's streamlines (guided by each vane) remain smooth and attached along the entire length of the bend — attached, laminar-following streamlines correspond to a much smaller pressure loss than a separated, eddying wake.
