Von Karman re-visited

A number of authors have presented extended versions of the integral momentum equation, allowing for perturbations or fluctuations in the boundary layer. ``Conventional wisdom'' is that these added terms can be neglected and one can apply the von Karman version directly. For two-dimensional turbulent boundary layers at high Reynolds numbers, experience shows this assumption to be reasonable. However, recent examination of entropy generation in bypass transition with zero pressure gradient shows a term for turbulence energy convection can be important in determining the energy dissipation coefficient [Walsh et al., JFE 2011]. The present study employs the direct numerical simulations of Zaki and Durbin [JFM 2006] for bypass transition with streamwise pressure gradients to quantify the additional normal stress term when estimating the skin friction coefficient via a momentum balance. It is found that this term becomes noticeable in the pre-transitional laminar boundary layer and can exceed forty per cent of C$_{f}$ in the transition region. Thus, it should be included in such calculations.