Random Problems Problem (3 pts) An untwisted wing with an elliptical planform has an aspect ratio of 6 and a span of m. The wing loading (defined as the lift per unit area of the wing) is 9N/m when flying at a speed of 5km/hr (4.67m/s) at sea level where the density is.6kg/m 3. A. Compute C L. B. Compute C Di. C. Compute the induced drag, D i, in Newtons. D. Compute the induced angle of attack φ i in degrees. E. Compute the lift curve slope, dc L /dφ, using angles in radians and assuming the -D lift slope is π.
Problem (4 pts) The horseshoe vortex system for a bi-plane with wing span b and separation distance h is shown above. Assume that the induced drag coefficient for single wing of the bi-plane is given by, C Dib v b C L b vsolo πar =, where v b is the downwash of a single wing of the bi-plane, v solo is the downwash of a wing in a solo configuration (not a bi-plane), C Lb is the lift coefficient for a single wing of the bi-plane, and AR is the aspect ratio of a single wing of the bi-plane. A. Prove that the downwash at the wing root for a wing in the bi-plane configuration is, v b = +. v solo + (h/b) Does a wing in the bi-plane configuration experience more or less downwash than a solo wing? B. Consider a monoplane which has the same wing as a single wing of the bi-plane. Assume the monoplane lift distribution is elliptic. When the planes are producing the same lift, prove that the ratio, D ib /D i, is, D ib v b =, D i v solo where D ib is the induced drag of the bi-plane, and D i is the induced drag of the monoplane. Does the bi-plane or the monoplane produce the larger induced drag? Which configuration would you expect to stall first as the total amount of lift required increases: the bi-plane or the monoplane and why? C. Consider a monoplane which has a wing with twice the aspect ratio and twice the surface area as a single wing of the bi-plane. Assume the monoplane lift distribution is elliptic. When the planes are producing the same lift, prove that the ratio, D ib /D i, is, D ib D i v b =, where D i is the induced drag of the monoplane. Does the bi-plane or the monoplane produce the larger induced drag? Which configuration would you expect to stall first as the total amount of lift required increases: the bi-plane or the monoplane and why? v solo
Problem 3 (3 pts) A. The figure contains lift curves for: () -D cambered airfoil, () elliptic planform, airfoil from above, no geometric twist, AR = 5, (3) elliptic planform, airfoil from above, no geometric twist, AR =,.5.5.5 8 6 4 4 6 8 angle of attack B. The figure contains lift curves for for three wings all with the same airfoil cross-section, AR =, and: () elliptic planform, no geometric twist, () rectangular planform, no geometric twist, (3) rectangular planform, geometric twist..5.5.5 8 6 4 4 6 8 angle of attack 3
C. The figure contains polars of C Di versus C L for three wings all with the same airfoil crosssection, no geometric twist, and: () elliptic planform, AR =, () rectangular planform, AR =, (3) rectangular planform, AR =...9.8.7.6 CDi.5.4.3...5.5.5.5 4
Extra Credit Problem (5 pts) In this problem, we will find the flow around a sphere of radius, R. To do this, we combine a 3-D doublet and a freestream to give the potential, φ = φ D + φ where the doublet potential is, θ cos θ φ D =, 4π r and freestream in the negative z direction has a potential, φ = U z, Note, the spherical coordinate system, (r, θ, Φ), is related to the Cartesian coordinate system, (x, y, z), by, x = r sin θ cos Φ, y = r sin θ sin Φ, z = r cos θ. Also, the gradient in the spherical coordinate system is given by, πφ πφ πφ φ = e r + e θ + e Φ. πr r πθ r sin θ πφ A. Find the value of the doublet strength, θ, which is needed to satisfy all boundary conditions for the flow around a sphere of radius, R. B. Proof that the velocity on the surface of the sphere is, u r =, u θ = 3 U sin θ, u Φ = C. Proof that the pressure coefficient on the surface of the sphere is, 9 C p = sin θ. 4 5