A wing of span carries the elliptic circulation distribution

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 wing of span bb carries the elliptic circulation distribution Γ(θ)=Γ0sin⁡θ\Gamma(\theta) = \Gamma_0\sin\theta, where the spanwise coordinate is y=−(b/2)cos⁡θy=-(b/2)\cos\theta, θ∈[0,π]\theta\in[0,\pi]. The downwash at station θ0\theta_0 from the trailing vortex sheet, counted positive downward, is w(θ0)=14π∫−b/2b/2dΓ/dyy0−y dy,w(\theta_0) = \frac{1}{4\pi}\int_{-b/2}^{b/2}\frac{\mathrm{d}\Gamma/\mathrm{d}y}{y_0-y}\,\mathrm{d}y, and Glauert's integral gives ∫0πcos⁡nθcos⁡θ−cos⁡θ0 dθ=πsin⁡nθ0sin⁡θ0\int_0^\pi \dfrac{\cos n\theta}{\cos\theta-\cos\theta_0}\,\mathrm{d}\theta = \pi\dfrac{\sin n\theta_0}{\sin\theta_0}. ((a)) Change variables to θ\theta and use Glauert's integral with n=1n=1 to show that w=Γ0/(2b)w=\Gamma_0/(2b), independent of θ0\theta_0, and hence derive CD,i=CL2/(πAR)C_{D,i}=C_L^2/(\pi AR) for elliptic loading, given ∫−b/2b/2Γ(y) dy=πbΓ0/4\int_{-b/2}^{b/2}\Gamma(y)\,\mathrm{d}y = \pi b\Gamma_0/4 and the definition AR=b2/SAR = b^2/S. ((b)) A sailplane wing has span b=16 mb = 16\,\mathrm{m}, area S=17 m2S = 17\,\mathrm{m^2}, and flies at U=25 m s−1U = 25\,\mathrm{m\,s^{-1}} in air (ρ=1.2 kg m−3\rho = 1.2\,\mathrm{kg\,m^{-3}}) at CL=0.65C_L = 0.65. Find the downwash speed ww and the induced drag force.

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