A cylindrical bridge pier of radius m and vertical axis stands in a river whose…

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 cylindrical bridge pier of radius

a = 0.40

m and vertical axis stands in a river whose current far from the pier has speed

U = 2.5

m s

^{-1}

along

+x

. The water has uniform density

\rho = 1000

kg m

^{-3}

. Treat the flow in a horizontal plane below the free surface as steady, incompressible, inviscid and irrotational; all points considered lie at the same depth, so the

\rho g z

term is the same everywhere. The polar angle

\theta

is measured from the

+x

axis, so the upstream stagnation point of the ideal flow is at

\theta = 180^{\circ}

and the shoulders are at

\theta = \pm 90^{\circ}

. ((a)) A model flow is given by the velocity potential

\phi = A(x^{3} - 3xy^{2})

with

A = 5

^{-1}

^{-1}

x

and

y

in metres and

\phi

in m

^{2}

^{-1}

. Verify that

\phi

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

(x, y) = (0.10, 0.20)

m. ((b)) For the ideal flow past the pier, find the difference between the surface pressure at the upstream stagnation point and the surface pressure at the shoulder

\theta = 90^{\circ}

. ((c)) A slow anticlockwise eddy is trapped around the pier, giving the flow a circulation

\Gamma = +3.0

^{2}

^{-1}

. Find the transverse force per unit length of pier, stating whether it acts along

+y

-y

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