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04-BS-7 · May 2014

Question 10 of 13: Flow Net for a Converging Nozzle

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

Notes on this paper

04-BS-7 Mechanics of Fluids — National Examinations, May 2014. Three (3) hours, closed book. Section A: Calculative (9 questions, do 7); Section B: Analytical/Graphical (4 questions, do 3). Ten questions constitute a complete paper (50 marks). Every printed question is solved below, including the two "extra" questions in each Section beyond the minimum required.

Reference texts: White, Fluid Mechanics, 8th ed. (fluid statics & capillarity Ch.2; Bernoulli/energy equation Ch.3; pipe friction & the Moody chart Ch.6; drag on immersed bodies Ch.7; buoyancy Ch.2; momentum & jet propulsion Ch.3).

Question 10 — Flow Net for a Converging Nozzle (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.

converging nozzle — streamlines crowd together (accelerating flow) vena contracta, just past the exit
Fig. Q10 — flow net for the conical converging nozzle: streamlines converge smoothly inside the cone (inertia-dominated, near-uniform velocity profile) and continue to contract slightly past the exit lip before the jet stabilises.

At high Reynolds number the flow through a smoothly converging cone is effectively inviscid outside a thin wall boundary layer, so the streamlines are drawn as a family of smooth, non-crossing curves that converge steadily toward the axis as the section area shrinks — by continuity, V·A is constant along a stream tube, so streamlines must crowd closer together (higher local velocity) wherever the tube narrows. Because real fluid has inertia, the converging streamlines do not stop contracting exactly at the physical nozzle lip: momentum carries the outermost streamlines slightly further inward just past the exit, producing the classic vena contracta — a jet cross-section a little smaller than the nozzle exit itself, typically reached within about one nozzle-exit-diameter downstream, after which the streamlines run parallel and the jet diameter stabilises.

The construction rule for the flow net itself is that streamlines are drawn to divide the flow into equal-discharge stream tubes, so they space out where the passage is wide (upstream, low velocity) and pack together where it is narrow (through the throat and at the vena contracta, high velocity) — sketching roughly ten such curves, symmetric about the centreline, from the wide inlet, converging smoothly through the cone, necking slightly past the exit lip at the vena contracta, and then running parallel in the free jet, satisfies the construction the question asks for.