A rigid sphere of radius is held fixed in an otherwise uniform stream of speed far…

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

Fluid Dynamics, Ch Potential flow, conformal mapping and the finite wing

Problem

A rigid sphere of radius

a

is held fixed in an otherwise uniform stream of speed

U

far upstream, in an incompressible, irrotational, inviscid flow. The flow is axisymmetric about the stream direction, with

\theta

measured from the front stagnation point, and may be built from a uniform stream plus an axisymmetric potential dipole, using the Stokes stream function of

\Psi_{\text{stream}} = \tfrac12 U r^2\sin^2\theta

for the stream and

\Psi_{\text{dipole}} = -\dfrac{k\sin^2\theta}{r}

for a dipole of strength

k

. ((a)) Find

k

by requiring

r=a

to be a streamline (

\Psi = 0

there for all

\theta

), and hence write

\Psi(r,\theta)

for the whole flow. ((b)) Use

u_r = (r^2\sin\theta)^{-1}\partial\Psi/\partial\theta

and

u_\theta = -(r\sin\theta)^{-1}\partial\Psi/\partial r

to find the surface velocity at

r=a

, and the pressure coefficient

C_p = 1-(u_\theta/U)^2

on the surface. ((c)) For

a = 0.40\,\mathrm{m}

and

U = 12\,\mathrm{m\,s^{-1}}

, evaluate the surface speed and

C_p

\theta = 50^{\circ}

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