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04-BS-7 · December 2013

Question 13 of 13: Flow Characteristics — Sink/Jet Streamlines and Orifice Contraction

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 13: Flow Characteristics — Sink/Jet Streamlines and Orifice Contraction (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.

Aconverging, sink-like-> flow ENTERS pipeBdiverging, jet mixing-> flow LEAVES pipe
(a) Converging streamlines drawn smoothly from every direction into the pipe (A) vs. a diverging, spreading jet issuing from the pipe into the tank (B).

(a) Figure A shows streamlines converging smoothly toward the pipe mouth from every direction — this is flow entering the pipe (a sink). A sink flow accelerates continuously as it approaches the opening under a favourable pressure gradient, so it stays attached and nearly irrotational; its shape is set almost entirely by the geometry and the (inviscid) pressure field, essentially independent of viscosity. Figure B shows the streamlines diverging/spreading after leaving the pipe — this is flow leaving the pipe as a discharging jet. A jet decelerates and spreads because it is a shear layer: turbulent mixing entrains the surrounding stagnant tank fluid at the jet's edges, and this entrainment—a genuinely VISCOUS/turbulent effect—is what makes the emerging flow diverge; an idealised inviscid jet would instead remain a fixed-diameter stream.

Astrong contraction: higher ReBlittle contraction: lower Re
(b) A sharp-edged orifice jet with a pronounced vena contracta (A) vs. one with comparatively little contraction (B).

(b) A sharp-edged orifice naturally produces a contracted jet (a "vena contracta") because fluid approaching the opening from all directions cannot turn the sharp corner instantaneously; its own inertia carries the streamlines past the edge before curving back in, pinching the jet to roughly 60–65% of the orifice area in the idealised (high-Re, near-inviscid) limit. At LOWER Reynolds number, viscous effects thicken the boundary layer along the orifice edge and damp out this inertia-driven overshoot, so the jet stays closer to the full orifice diameter with comparatively little contraction. Figure A, showing the pronounced vena contracta, therefore corresponds to the greater Reynolds number; Figure B, with little contraction, corresponds to the lower-Re case.

Conclusion: (a) A = flow entering (sink, inertia/pressure-gradient controlled), B = flow leaving (jet, viscosity/turbulence-controlled spreading). (b) Figure A (strong contraction) has the greater Reynolds number.

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