An Inverse Inner-Variable Theory for Separated Turbulent Boundary Layers

Abstract

An integral method is presented for computing separated and reattached turbulent boundary layers for incompressible two-dimensional flows. This method is a substantial improvement over the inner-variable approach of Das and White (1986), which was based on a direct boundary layer scheme that had several shortcomings. In this new approach, the integral equations have been completely reformulated so that the theory now proceeds in an inverse mode using displacement thickness as input. This new formulation eliminates the need for the second derivative of velocity distribution, which in the past has always been a source of error in all previous inner-variable approaches. Other significant additions are: (a) a single pressure gradient-wake correlation from a large amount of experimental data; and (b) replacement of the wake parameter from the final equations with a more stable parameter, wake velocity. Derivations of integral equations and their final working expressions, in both dimensional and nondimensional forms, are presented in detail. Predictions by this theory for skin friction, freestream velocity, momentum thickness, velocity profile and separation, and reattachment points agree well with experimental data. Sensitivity studies display that the theory is stable against variations in initial conditions, input distributions, and the pressure gradient-wake correlation.