Scaling laws for the mean velocity in adverse pressure gradient turbulent boundary layers based on asymptotic expansions

In this paper, the mean velocity profiles and the multi-layer division of the turbulent boundary layer (TBL) are studied for the entire adverse pressure gradient (APG) region before separation. By asymptotic expansions of the mean velocity, the difference in the APG effect between the inner and outer regions and the problems of directly matching the mean velocity are demonstrated. The wall law of the overlapping region, i.e., the log-pressure law, is derived from the perspective of turbulent transport in combination with the results of matching the Reynolds shear stress. The results show that it is applicable in the entire APG region, thus answering the questions of the existence of the log law, the slope of the log law, and the breakdown of the log law. A new half-power law is then proposed after reconsidering the self-similarity of the outer region. This law is believed to be valid only in the outer region of larger APGs and is only an approximate self-similar law compared to the wall law in the inner region. Furthermore, a multi-layer division of APG TBLs is presented to form a complete framework for describing the mean flow of APG TBLs. In particular, the basis of the division and the specific range of each layer are specified, where the range of the overlapping region involves the definition of a critical Reynolds number.