Question 13 of 13: Streamline Patterns for Tank Inflow/Outflow and Sharp-Edged Orifice Jets

Question 13: Streamline Patterns for Tank Inflow/Outflow and Sharp-Edged Orifice Jets (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.

(a) Tank inflow vs. outflow.

Sketch A (streamlines splaying out in a wide fan the instant they leave the pipe mouth) shows flow entering the tank: a submerged jet discharging into a large body of still fluid is a free shear flow, and the high-velocity jet boundary is turbulently unstable — it entrains and mixes with the surrounding still water, spreading (diffusing) rapidly in every direction away from the pipe. Sketch B (streamlines gathering smoothly from a broad area, curving gently in only close to the pipe) shows flow leaving the tank (being drawn down into the pipe, like a drain): a converging (sink-like) flow accelerates smoothly toward the outlet, which is a FAVOURABLE pressure gradient that suppresses separation and turbulence, so the streamlines stay smooth and well-ordered almost all the way to the pipe. The fluid characteristic responsible is the stability of shear layers under a favourable vs. an adverse/free-shear condition: a converging (sink) flow is inertia-stable and nearly irrotational, while a diverging free jet is unstable to turbulent entrainment and mixing.

(b) Which orifice jet has the greater Reynolds number.

Sketch B (pronounced, sharply curved vena contracta, jet narrowing noticeably beyond the orifice) has the greater Reynolds number. At high Re, viscosity plays almost no role in guiding the fluid near the sharp edge; streamlines converging radially toward the orifice from all directions carry their inward (radial) momentum PAST the edge before inertia allows them to straighten out, over-shooting inward and producing a pronounced contraction (C_c well below 1) a short distance downstream. Sketch A (straighter, only gently narrowed streamlines) is at LOWER Reynolds number: viscous forces are more significant relative to inertia, damping out the radial overshoot and letting the streamlines hug the boundary more closely, so the jet contracts much less. The fluid characteristic that determines the jet shape is therefore the ratio of inertial to viscous forces (Reynolds number) controlling how strongly the fluid's own momentum resists turning the sharp corner at the orifice.