Thin Airfoil Theory for Planar Inviscid Shear Flow

Abstract

Steady, inviscid, planar and constant density shear flows past thin airfoils have strongly coupled thickness and camber flowfields which can be simply analyzed using some invariant properties of Euler’s equations. In this paper, both the nonlinear vorticity generation term in Poisson’s disturbance stream-function equation and the Bernoulli constant (which varies from streamline to streamline) are explicitly evaluated in terms of the known plane parallel flow far upstream, recognizing that vorticity convects unstretched and unchanged along streamlines; this produces a governing differential equation with an explicitly available right side plus a Bernoulli integral with a true constant fixed throughout the entire flow field. Using these results, the inverse problem, which solves for the shape that induces a prescribed pressure, is formulated as a nonlinear boundary value problem with mixed Dirichlet and Neumann surface conditions; here, trailing edge shape constraints are directly enforced by controlling jumps in the stream-function or its streamwise derivative through the downstream wake (analogies with potential flows past axisymmetric ringwings showing similar source and vortex interactions are also described). For flows with arbitrary profile curvature, we show that the linearized stream-function equation can be put in conservation form, thus leading to new and more general Cauchy-Riemann conditions which extend the notion of the velocity potential; the analysis problem, which determines the flow past a known shape subject to Kutta’s condition, is shown to satisfy a simple boundary value problem for a “super-potential” identical to the irrotational planar formulation except for an axisymmetric-like change to the Cartesian form of Laplace’s equation. For flows with uniformity vorticity, closed-form solutions are given which clearly illustrate the interaction between thickness and camber; also, for arbitrary thin airfoils, a particularly simple expression shows that the lift due to thickness varies directly as the product of vorticity and enclosed area. With more general shears, numerical approaches are especially convenient; these require only simple code changes to existing algorithms, for example, established potential function methods readily available for analysis problems, or “direct stream-function methods” for inverse problems recently developed by this author. Numerical methods and results that cross-check and extend derived analytical solutions are given for both analysis and design problems with vorticity. Also, extensions of the basic theory to three-dimensional constant density and planar compressible shear flows are given.