A tall brick chimney of circular cross-section

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 Potential flow, complex potentials and lift

Problem

A tall brick chimney of circular cross-section, radius

a = 0.60

m and vertical axis, stands in a horizontal wind of speed

U = 12

m s

^{-1}

. Take the air to be of uniform density

\rho = 1.2

kg m

^{-3}

and the flow in the horizontal plane through a point half way up the chimney to be steady, incompressible, inviscid and irrotational, with the stream along

+x

. All points considered lie at the same height, so the

\rho g z

term is the same everywhere. Far from the chimney the absolute pressure is

p_\infty = 1.013\times10^{5}

Pa. The polar angle

\theta

is measured from the

+x

axis, so the stagnation points of the ideal flow lie at

\theta = 0

and

\theta = 180^{\circ}

and the shoulders at

\theta = \pm 90^{\circ}

. ((a)) Very close to the upstream stagnation point, take local coordinates

x

y

in metres centred on that point, with

x

along the chimney surface and

y

pointing upstream, away from the surface. There the flow is described by the velocity potential

\phi = \tfrac12 S(x^{2} - y^{2})

with

S = 2U/a = 40

^{-1}

and

\phi

in m

^{2}

^{-1}

. Verify that

\phi

satisfies Laplace's equation, write down the velocity field, and find the flow speed at the point

(x, y) = (0.020, 0.030)

m. ((b)) Find the absolute pressure on the surface of the chimney at the shoulder

\theta = 90^{\circ}

. ((c)) The chimney is fitted with a device that gives the flow round it a circulation

\Gamma = -80

^{2}

^{-1}

(that is, clockwise). Find the transverse force per unit length of chimney and say whether it acts along

+y

-y

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