A thin diamond (double-wedge) aerofoil has chord and greatest thickness at mid-chord

PHYSICS · Subjective · Fermi University Physics · Fluids & Continuum Physics · Waves, acoustics and compressible flow · Waves, acoustics and compressible flow

Signal Speeds in Fluids, Ch Steady compressible flow: energy, stagnation and the Mach number

Problem

A thin diamond (double-wedge) aerofoil has chord cc and greatest thickness t=τct=\tau c at mid-chord, so each of its four flat facets makes the small angle τ\tau (in radians) with the chord line. It flies at a small incidence α\alpha in a uniform stream at M∞>1M_\infty>1, with β=M∞2−1\beta=\sqrt{M_\infty^2-1}. (a) Using Ackeret's rule Cp=2θ/βC_p=2\theta/\beta on each facet separately, with θ\theta the facet's deflection into the stream on its own side, show that CL=4αβ,CD=4β(α2+τ2).C_L=\frac{4\alpha}{\beta},\qquad C_D=\frac{4}{\beta}\left(\alpha^2+\tau^2\right). (b) Show that the lift-to-drag ratio is greatest at α=τ\alpha=\tau, where L/D=1/(2τ)L/D=1/(2\tau) whatever the Mach number. (c) For τ=0.05\tau=0.05 at M∞=2.0M_\infty=2.0, find the best incidence in degrees, and CLC_L, CDC_D and L/DL/D there. Compare with a flat plate at the same incidence. (d) Explain why the best L/DL/D does not depend on M∞M_\infty although CLC_L and CDC_D do, and state what limits the use of these results for a thicker section or a larger incidence.

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