A Viscous Flow Analog to Prandtl’s Optimized Lifting Line Theory Utilizing Rotating Biquadratic Bodies of Revolution

Prandtl’s lifting line theory expanded the Kutta-Joukowski theorem to calculate the lift and induced drag of finite wings. The circulation distribution about a real wing was represented by a superposition of infinitesimal vortex filaments. From this theory, the optimum distribution of circulation was determined to be elliptical. A consequence of this theory led to the prediction that the elliptical chord distribution on a real fixed wing would provide the elliptical circulation distribution. The author applied the same line of reasoning to lift-producing rotating cylinders in order to determine the cylindrical geometry that would theoretically produce an elliptical circulation distribution. The resulting geometry was the biquadratic body of revolution (BBOR). Water tunnel testing was conducted to compare force coefficients and ratios between a lifting arrangement incorporating BBORs and a lifting arrangement incorporating a more traditional cylindrical arrangement, the constant diameter circular cylinder (CDCC). As directed by the Navier-Stokes equation, testing was conducted at low Re, 10² [100] ≤ Re ≤ 10⁴ [10,000], where viscous effects would become more pronounced. Results showed the BBOR arrangement to produce the highest lift to drag ratio within specific ranges of α [alpha], surface velocity to free stream velocity. Lift coefficients were shown to increase with α [alpha] and approach values an order of magnitude larger than known fixed wing lift coefficients at low Re.