A flat plate of chord at incidence in a stream carries the Kutta circulation

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 flat plate of chord

c

at incidence

\beta

in a stream

U

carries the Kutta circulation, clockwise, of magnitude

|\Gamma| = \pi Uc\sin\beta

(so

\Gamma=-\pi Uc\sin\beta

with anticlockwise counted positive, as in the text). Writing

a_J = c/4

, its complex potential has the large-

|\zeta|

Laurent expansion

\frac{\mathrm{d}W}{\mathrm{d}\zeta} = Ue^{-i\beta} + \frac{2iUa_J\sin\beta}{\zeta} - \frac{2iUa_J^2\sin\beta}{\zeta^2} + O(\zeta^{-3}).

Blasius's theorem, in the residue form used for the cylinder with circulation, gives the resultant force

X-iY = -\pi\rho\,C_{-1}

and the moment about the origin

M = \pi\rho\,\mathrm{Im}(D_{-1})

, where

C_{-1}

is the coefficient of

\zeta^{-1}

(\mathrm{d}W/\mathrm{d}\zeta)^2

and

D_{-1}

is the coefficient of

\zeta^{-1}

\zeta(\mathrm{d}W/\mathrm{d}\zeta)^2

. ((a)) Expand

(\mathrm{d}W/\mathrm{d}\zeta)^2

to order

\zeta^{-2}

, extract

C_{-1}

and

D_{-1}

, and show that the resultant force has magnitude

\rho U|\Gamma|

and that

M = -\tfrac{\pi}{4}\rho U^2c^2\sin\beta\cos\beta

. ((b)) For

c = 2.4\,\mathrm{m}

\beta = 9^{\circ}

U = 38\,\mathrm{m\,s^{-1}}

in air (

\rho = 1.2\,\mathrm{kg\,m^{-3}}

), evaluate

|M|

and the lever arm

|M|/(\rho U|\Gamma|)

of the lift about the mid-chord. Then find how far ahead of the mid-chord the centre of pressure lies, measured along the chord, and compare it with the quarter chord.

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