Flow Net and Streamlines Through a Conical Converging Nozzle: 04-BS-7, May 2016, Q12

Question 12: Flow Net and Streamlines Through a Conical 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.

Fig. Q12 — flow net through the converging nozzle: ~10 streamlines crowd together through the cone, continue to converge slightly past the exit (inertia), reach the vena contracta, then run parallel in the free jet.

Reasoning behind the sketch. Inside the large-diameter approach section the streamlines are widely spaced and nearly parallel (low velocity, by continuity). As they approach the conical convergence they crowd together smoothly — streamline spacing is inversely proportional to local velocity for an incompressible flow net, so converging streamlines directly indicate accelerating flow. Because the flow has significant inertia, the streamlines nearest the wall cannot follow the sharp geometric corner at the nozzle exit instantaneously; their momentum carries them slightly inward past the physical exit plane, so the jet continues to contract for a short distance AFTER leaving the nozzle, reaching its minimum area — the vena contracta — a little downstream of the exit, before the streamlines finally straighten and run parallel in the free jet. Roughly 10 streamlines are drawn (a mix of near-axis and near-wall lines) so that their converging spacing itself illustrates the velocity increase from the approach pipe through the throat and into the vena contracta, after which the equally-spaced parallel lines in the jet show the (now constant) jet velocity.

Feature	Streamline behaviour
Approach pipe	Wide, evenly spaced, parallel — low, uniform velocity
Converging cone	Streamlines crowd together smoothly — accelerating flow
Just past the nozzle lip	Wall streamlines curve slightly INWARD past the geometric exit (inertia)
Vena contracta	Minimum jet area, a short distance beyond the nozzle exit
Free jet	Streamlines parallel again, constant (maximum) velocity

Topic: Flow nets and the vena contracta at a sharp-edged / high-velocity nozzle exit.

Key relations: incompressible continuity ⇒ streamline spacing ∝ 1/V; coefficient of contraction C_c = A_jet/A_nozzle (typically 0.6–0.9 depending on the approach geometry).

Why this works. A flow net is a graphical solution of the (irrotational, incompressible) flow equations: orthogonal streamlines and equipotentials that must crowd together wherever the geometry accelerates the flow; the vena contracta is a direct, purely geometric consequence of fluid inertia being unable to turn a sharp corner instantaneously.

Common pitfall. Drawing the streamlines as if they reach their minimum spacing exactly AT the nozzle exit plane; inertia carries the contraction a short distance past the physical exit before the jet stabilizes.

Source: White, Fluid Mechanics, Ch. 8 (flow nets, potential flow); Munson et al., Fundamentals of Fluid Mechanics, Ch. 8 (orifice/vena-contracta discharge).

Question 12 of 13: Flow Net and Streamlines Through a Conical Converging Nozzle

Question 12: Flow Net and Streamlines Through a Conical Converging Nozzle (5 marks)