Application of Karman's Logarithmic Law to Prediction of Airship Hull Drag

Abstract

A F T E R experiments had shown that the i power law theory of von Karman and Prandtl, for the skin friction of a pointed flat plate in a two-dimensional flow parallel to the surface and perpendicular to the leading edge, could be used satisfactorily in determining the skin friction at the curved surface of a streamline body in a two-dimensional flow when the radius of curvature is large compared to the boundary layer thickness, Jones, Dryden and Kuethe extended the theory so as to give an account of the skin friction acting on a dirigible model in an axial flow. In both these discussions the dirigible was replaced by an ''equivalent flat plate. ' ' Although Boltze had given the equations for the laminar boundary layer for a body with axial symmetry about an axis in the direction of flow, and Levi-Civita had attacked the general problem of the turbulent boundary layer for a body of any shape, it remained for Millikan to give a satisfactory analysis applicable to dirigible models. Millikan derived the approximate form of the boundary layer equations in the neighborhood of the surface of a figure of revolution from the Navier-Stokes and continuity equations for the steady motion of a viscous incompressible fluid, letting the kinematic viscosity go to zero and neglecting powers of the square root of the kinematic viscosity higher than the first. The resulting equations are valid when the boundary layer thickness is small compared to its distance from the axis. In the turbulent boundary layer, Millikan assumed the one-seventh power law for the velocity profile. However, von Karman has shown that the power law is merely an interpolation of or approximation to the logarithmic law; and experiments, especially those of Nikuradse, show that the l /7 th power does indeed become a l /8 th power, then a l /9 th power, etc., as the Reynolds' Number increases. Both Millikan and von Karman have pointed out the desirability, where the analysis is to be used to predict drag coefficients at large Reynolds' Numbers, of extending Millikan's work by introduction of the logarithmic law. As a preliminary step, Millikan's equations were generalised to hold for a 1/n-th power law for the velocity distribution, and drag coefficients were calculated for several values of n for the N. P. L. Long Model at a large Reynolds' Number. The differences in the drag coefficients for the various power laws were found to be sufficient to justify the further refinement of introducing the logarithmic law. It was then assumed that the logarithmic velocity distribution, as developed by von Karman for the boundary layer on the flat plate, holds also for the boundary layer on the axially symmetric body. Aqcordingly we introduce the logarithmic velocity distribution