Unified Flow and Thermal Law-of-the-wall for Type-A and B Turbulent Boundary Layers

Abstract

Systematic study of the existing statistical data from direct numerical simulations (DNS) lead to a logical and important classification of generic turbulent boundary layers (TBL), namely Type A,B and C TBL, based on the distribution patterns of time averaged wall shear stress. Among these three types, Type A and B TBLs are investigated in terms of its universal statistical laws of velocity and temperature with independency on both Reynolds (Re) and Prandtl (Pr) numbers. With reference to the analysis by von Karman in developing the conventional law of the wall for Type A TBL, the current study first physically distinguishes the time averaged local velocity scale from the ensemble averaged velocity scale, and properly defines the TBL temperature scale based on wall heat flux, similar to the well known frictional velocity scale based on wall shear stress. The unified law of the wall formulations are then derived for the inner layer of Type A and B thermal TBLs using these scales. The formulations are constructed by introducing the general damping and enhancing functions in inner layer, and further, these functions are extended to logarithmic coordinate based on the multiscale characteristics of TBL velocity profiles. The validations of the law are conducted both by the DNS guided near wall integration of Type B TBL governing equations, which uncovered the Re and Pr independency mechanisms of the law. The research advances the current understandings of the conventional TBL theory and develop the complete analytical forms of law of the wall for Type A and B TBLs, including inner layer, transition layer, logarithmic layer and wake layer.