Reynolds-number independent inner-layer analytical formulation for type-A turbulent boundary layer based on DNS data

Abstract

Searching for universal law of turbulent boundary layer (TBL) has been one of the persistent efforts of turbulence research community for the last century. Based on the direct numerical simulation (DNS) data for a zero-pressure-gradient flat-plate TBL from Schlatter and Orlu, the paper pointed out the necessity to reunderstand the conventional TBL theory based on the recent TBL classification proposed by Cao and Xu. Type-A TBL in the classification, as represented by the DNS data, was thoroughly investigated in the inner layer using the velocity scales of both conventional time-averaged local frictional velocity and newly-defined time-space or ensemble-averaged frictional velocity. With the ensemble-averaged frictional velocity as scale, the new mathematical expressions for the inner-layer law were derived by introducing the general damping and enhancing functions. The control parameters in the expressions were found independent on Reynolds number based on the universal governing equation under the ensemble-averaged scales. The physical meanings of the parameters in the law formulation were analyzed and clearly demonstrated that the parameter Δ represented the incremental local wall-shear stress and the parameter D stood for the charateristic length within which the linear viscous law relation was applicable. Comparing to the conventional law-of-the-wall in inner layer, the current law formulation significantly improved the law’s predictive accuracy and applicable range. Particularly, when the time-averaged wall-shear stress gets away from the ensmeble-averaged wall shear stress, the new inner-layer law is applicable up to a near-wall range of d * ≅ 10.0 , while the conventional inner-layer law can only perform well within a wall distance of d * ≅ 6.0 . With these studies, the properties of the general damping and enhancing functions are well understood, which provides a solid physical and mathematical foundation for developing the complete analytical formulations for law-of-the-wall, which are expected to include the inner layer, the transition layer, the semi-log linear layer and the wake layer.