  In-depth analyses of existing direct numerical simulations (DNS) data from
various sources supported a logical and important classification of generic
turbulent boundary layers (TBL), namely Type-A, -B and -C TBL, based on
distribution patterns of time-averaged wall-shear stress. Among these types,
Type-A TBL and its related law, as represented by the DNS data of turbulence on
a zero-pressure-gradient semi-infinite flat-plate, was investigated in terms of
analytical formulations of velocity independent on Reynolds ( ) number. With
reference to the analysis from von Karman in developing the conventional
law-of-the-wall, the current study first physically distinguished the
time-averaged local scale used by von Karman from the ensemble-averaged scale
defined in the paper, and then derived the governing equations with the
-independency under the ensemble-averaged scales. Based on indicator function
(IDF) and TBL thickness, the sublayer partitions were rigorously defined. The
analytical formulations for entire TBL, namely the complete law-of-the-wall,
were established, including the formula in inner, buffer, semi-logarithmic
(semi-log) and wake layer. The researches were featured by introducing the
general damping and enhancing functions (GDF and GEF) and applying these
functions to both linear and logarithmic coordinates. These law formulations
were proved uniform and consistent in time-averaged local and ensemble-averaged
scales, which were validated by the existing DNS and experiment data. Based on
the similarity of relevant properly-scaled governing equations, the law
formulations were logically reasoned being applicable to the temperature in
Type-A thermal TBL. The findings advance the current understandings of the
conventional TBL theory and its well-known foundations of law-of-the-wall.
