Unsteady Thin-Airfoil Theory Revisited: Application of a Simple Lift Formula

A = heaving amplitude, m Cl = lift coefficient c = wing chord, m Dbl = boundary-layer domain Dout = outer flow domain F = aerodynamic force, N f = flapping frequency, s−1 k = πfc∕U, reduced frequency k = unit vector normal to the freestream velocity L, L 0 = lift or sectional lift, N or N · m−1 La, L 0 a = lift or sectional lift associated with the fluid acceleration, N or N · m−1 Lvor, L 0 vor = vortex lift or sectional vortex lift, N or N · m −1 l = Lamb vector, m · s−2 n = unit normal vector pointing outward from control boundary n 0 = −n p = pressure, Pa q∞ = dynamic pressure, Pa Re = Reynolds number S = wing area, m T = flapping period, s t = time, s U = incoming flow velocity, m · s−1 u = fluid velocity, m · s−1 Vf = control volume x = coordinate along the airfoil cord, m X; Y; Z = coordinates in direct numerical simulation, m xref = reference location, m x = x − xLE ∕c, normalized coordinate zc = vertical position of airfoil center, m α = angle of attack, deg Γ = circulation, m · s−1 γ = vortex sheet strength, m · s−1 γ1 = unsteady vortex sheet strength, m · s −1 γw = wake vortex sheet strength, m · s −1 γ0 = quasi-steady vortex sheet strength, m · s −1 ∂B = solid boundary of the body (wing) domain ∂Bbl = boundary-layer edge ρ = fluid density, kg · m−3 Σ = outer surface of a control volume τ = skin friction, N · m−2 φ = velocity potential, m · s−1 ω = vorticity, s−1