  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.
