A flat plate of chord is set at in a uniform air stream ( ) of speed

PHYSICS · Subjective · 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=2.0 mc=2.0\,\mathrm{m} is set at β=8∘\beta=8^{\circ} in a uniform air stream (ρ=1.20 kg m−3\rho=1.20\,\mathrm{kg\,m^{-3}}) of speed U=40 m s−1U=40\,\mathrm{m\,s^{-1}}. Someone works out its lift as follows: "The plate is the image, under ζ=Z+aJ2/Z\zeta=Z+a_J^2/Z, of a circle of radius a=c/4a=c/4 carrying circulation Γ\Gamma in the stream UU at incidence β\beta. Without circulation (Γ=0\Gamma=0) the flow already has a stagnation point at the circle's leading-edge image, so the flow already leaves the leading edge smoothly there. The Kutta condition only demands smooth flow at a sharp edge, and this is already satisfied at Γ=0\Gamma=0, so no circulation is required and CL=2πsin⁡βC_L=2\pi\sin\beta evaluated with Γ=0\Gamma=0, i.e. CL=0C_L=0, for a flat plate at any angle of attack." Explain what is wrong with this argument, and give the correct circulation Γ\Gamma, lift coefficient CLC_L, and lift per unit span.

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