A steady vortex is maintained in a large tank of water of uniform density and is well…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow

Fluids and Continuum Physics, Ch Vorticity, circulation and vortex motion

Problem

A steady vortex is maintained in a large tank of water of uniform density

\rho = 1000\ \mathrm{kg\,m^{-3}}

and is well modelled as a Rankine vortex about a vertical axis, viewed from above with

z

upward and anticlockwise positive. Inside a core of radius

a = 5.0\ \mathrm{cm}

the water turns anticlockwise as a rigid body, and at the edge of the core the tangential speed is

0.60\ \mathrm{m\,s^{-1}}

. Outside the core the flow is the line vortex

u_\theta = \Gamma/(2\pi r)

, which has zero vorticity for

r > a

. The flow is steady, incompressible and may be treated as inviscid, and the velocity is horizontal and tangential everywhere. Two small pressure tappings are set flush in the flat, horizontal tank floor, on the same horizontal level, at radii

0.15\ \mathrm{m}

and

0.30\ \mathrm{m}

from the axis, and are connected to a manometer. ((a)) Calculate the circulation round the horizontal circle of radius

0.15\ \mathrm{m}

centred on the axis. ((b)) Calculate the water speed at

r = 0.15\ \mathrm{m}

and at

r = 0.30\ \mathrm{m}

. ((c)) Calculate the pressure difference

p(0.30\ \mathrm{m}) - p(0.15\ \mathrm{m})

registered by the manometer, stating the four conditions under which you use Bernoulli's theorem and why here it may be applied between the two radii even though they lie on different streamlines. ((d)) Using the material derivative, calculate the magnitude of the acceleration of a water particle at

r = 0.15\ \mathrm{m}

and state its direction.

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