On exactly self-similar turbulent boundary layers and single-layer scaling

Abstract

Turbulent boundary layers are traditionally thought to obey two layer scaling. However, it was shown by Dixit and Ramesh [J. Fluid Mech. 615, 445–475 (2008)] that the so-called sink flow turbulent boundary layer obeys a Blasius-like scaling all through the boundary layer. We make this single layer scaling as an operational definition for perfectly equilibrium flow and derive equations for a general class of equilibrium flows with permeable walls allowing suction and blowing of fluid. Using the above definition, constraints required to be satisfied by such equilibrium turbulent boundary layers are derived and discussed in detail. Using direct numerical simulations (DNS), we demonstrate the existence of such turbulent boundary layers, and various properties of these boundary layers are studied in detail. In particular, it is shown that the mean velocity profiles and other turbulent statistics display excellent similarity, and various boundary layer parameters in such boundary layers vary systematically with the magnitude of blowing/suction parameter. By extension, it is also demonstrated by DNS that higher order moments (up to third order moments) also display streamwise self-similarity. We believe that this is an important result where self-similarity is demonstrated for the first time for higher moments of fluctuating velocity. The non-universality of the log-law coefficients in favorable pressure gradient flows is already known [S. A. Dixit and O. N. Ramesh, J. Fluid Mech. 615, 445–475 (2008) and Chauhan et al., in Proceedings of 11th EUROMECH European Turbulence Conference on Turbulence, Porto, Portugal (Springer Proceedings in Physics, 2007), pp. 239–241]. We demonstrate their non-universal nature in equilibrium boundary layers with suction and blowing using the present direct numerical simulations. Remarkably, the ratio of Reynolds shear stress (u′v′¯) to streamwise (u′2¯) and spanwise turbulent intensity (w′2¯) in these boundary layers is constant over the log-region in the mean velocity profile even though u′2¯, w′2¯, and u′v′¯ by themselves show variation with y+ in the log region, indicative of the absence of inactive motion in these flows. It is shown that the dominance of active motion and more importantly the absence of inactive motion leads to a universality of scaling that manifests as a single layer Blasius scaling with self-similarity in such flows. Equivalently, it could also be seen as a single layer scaling with all variables all through the boundary layer. This would mean that the friction velocity scales the entire boundary layer, which could be taken to suggest equivalence of attached eddies with the active component of motion.