For a thin flat-plate wing section of chord at angle of attack in a steady

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

For a thin flat-plate wing section of chord bcb_c at angle of attack β\beta in a steady, incompressible, inviscid, irrotational stream of speed UU, the Kutta condition (the real fluid leaves the sharp trailing edge smoothly rather than whipping round it at enormous speed) selects the circulation of magnitude ∣Γ∣=πUbcsin⁡β|\Gamma| = \pi U b_c \sin\beta. The lift coefficient per unit span is defined as CL=L′/(12ρU2bc)C_L = L'/(\tfrac12\rho U^{2}b_c). ((a)) Starting from the Kutta-Joukowski result ∣L′∣=ρU∣Γ∣|L'| = \rho U|\Gamma|, show that CL=2πsin⁡βC_L = 2\pi\sin\beta. ((b)) Obtain the lift-curve slope dCL/dβ\mathrm{d}C_L/\mathrm{d}\beta at β=0\beta = 0 and express it per degree of angle of attack. A wind-tunnel test of a thin section measures 0.0980.098 per degree; state the percentage by which the ideal result exceeds it. ((c)) Such a wing of chord bc=1.8b_c = 1.8 m flies at U=55U = 55 m s−1^{-1} in air of density ρ=1.2\rho = 1.2 kg m−3^{-3} at β=5.0∘\beta = 5.0^\circ. Find the selected circulation magnitude and the lift per unit span. ((d)) In one sentence, say what the Kutta condition supplies that ideal-flow theory on its own does not.

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