Law-of-the-wall analytical formulations for Type-A turbulent boundary layers

Abstract

In-depth analyses of existing direct numerical simulations (DNS) data led to a logical and important classification of generic turbulent boundary layers (TBLs), namely Type-A, -B and -C TBL, based on the distribution patterns of the time-averaged wall-shear stress. Among these types, Type-A TBL and its related law formulations were investigated in terms of the analytical velocity profiles independent on Reynolds number (Re). These formulations were benchmarked by the DNS data of turbulence on a zero-pressure-gradient flat-plate (ZPGFP). With reference to the analysis from von Karman in developing the traditional law-of-the-wall, the current study first physically distinguished the time-averaged local scale used by von Karman from the time-space-averaged scale defined in the current paper, and then derived the governing equations with the Re — independency under the time-space-averaged scales. Based on the indicator function (IDF) and TBL thickness, the sublayer partitions were quantitatively defined. The analytical formulations for entire ZPGFP TBL were derived, including the formula in the inner, buffer, semi-logarithmic (semi-log) and wake layers. The research profoundly understood the damping phenomenon and its controlling mechanism in the TBL with its associated mathematical expressions, namely the damping function under both linear and logarithmic coordinates. Based on these understandings and the quantified TBL partitions, the analytical formulations for the entire ZPGFP TBL were established and were further proved being uniform and consistent under both the time-averaged local and the time-space-averaged scales. Comparing to the traditional law, these formulations were validated by the existing DNS data with more accuracy and wider applicability. The findings advance the current understandings of the conventional TBL theory and its well-known foundation of law-of-the-wall.