On scaling the mean momentum balance and its solutions in turbulent Couette-Poiseuille flow

The statistical properties of fully developed planar turbulent Couette-Poiseuille flow result from the simultaneous imposition of a mean wall shear force together with a mean pressure force. Despite the fact that pure Poiseuille flow and pure Couette flow are the two extremes of Couette-Poiseuille flow, the statistical properties of the latter have proved resistant to scaling approaches that coherently extend traditional wall flow theory. For this reason, Couette-Poiseuille flow constitutes an interesting test case by which to explore the efficacy of alternative theoretical approaches, along with their physical/mathematical ramifications. Within this context, the present effort extends the recently developed scaling framework of Wei et al. (2005a) and associated multiscaling ideas of Fife et al. (2005a, b) to fully developed planar turbulent Couette-Poisettille flow. Like Poiseuille flow, and contrary to the structure hypothesized by the traditional inner/outer/overlap-based framework, with increasing distance from the wall, the present flow is shown in some cases to undergo a balance breaking and balance exchange process as the mean dynamics transition from a layer characterized by a balance between the Reynolds stress gradient and viscous stress gradient, to a layer characterized by a balance between the Reynolds stress gradient (more precisely, the sum of Reynolds and viscous stress gradients) and mean pressure gradient. Multiscale analyses of the mean momentum equation are used to predict (in order of magnitude) the wall-normal positions of the maxima of the Reynolds shear stress, as well as to provide an explicit mesoscaling for the profiles near those positions. The analysis reveals a close relationship between the mean flow structure of Couette-Poiseuille flow and two internal scale hierarchies admitted by the mean flow equations. The averaged profiles of interest have, at essentially each point in the channel, a characteristic length that increases as a well-defined 'outer region' is approached from either the bottom or the top of the channel. The continuous deformation of this scaling structure as the relevant parameter varies from the Poiseuille case to the Couette case is studied and clarified.