# Unified Flow and Thermal Law-of-the-wall Complete Formulations for Type-A Turbulent Boundary Layers

## TL;DR

This paper develops unified analytical formulations for velocity and temperature profiles in Type-A turbulent boundary layers, validated by DNS and experimental data, advancing the classical law-of-the-wall theory.

## Contribution

It introduces a complete, Reynolds-number independent law-of-the-wall formulation for Type-A TBL, including damping and enhancement functions, applicable to velocity and temperature profiles.

## Key findings

- Unified law-of-the-wall formulations validated by DNS and experiments.
- Introduction of general damping and enhancement functions for TBL modeling.
- Applicability of formulations to both velocity and temperature in Type-A TBL.

## Abstract

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.

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Source: https://tomesphere.com/paper/1904.08606