Mean Velocity Scaling and Friction Coefficient Estimation for Rough Surface Turbulent Boundary Layers in Turbomachines

Abstract

Based on flat plate results, mean velocity and friction coefficient estimation methods are proposed for rough surface turbulent boundary layers on axial compressor and turbine blades. The ratio of the displacement thickness to boundary layer thickness (δ*/δ) was first suggested by Zagarola and Smits (1998) for smooth pipe flows. The same parameter is proposed in this paper to scale the normalized mean velocity defect of smooth and rough surface flat plate turbulent boundary layers with zero, favorable, and adverse pressure gradients. The available mean velocity defect profiles of smooth and rough surface boundary layers from axial compressor and turbine blades are also scaled and compared to the flat plate results. Irrespective of the Reynolds number (Reθ), pressure gradient (K), and roughness (k), δ*/δ provides appropriate scaling for collapsing the flat plate and turbomachinery data. From the results, a new one-variable power law based on δ*/δ is proposed to estimate the mean velocity profile. The proposed power law can accurately estimate boundary layers on flat plates, compressor blades, and turbine blades. Finally, a new empirical Cf correlation is proposed for rough surface turbulent boundary layers under pressure gradients. The proposed Cf correlation is based on that of Bergstrom et al. (2005) and newly incorporates the acceleration parameter K. It can accurately estimate Cf in turbulent boundary layers of rough surface flat plates as well as those of smooth turbine blades.