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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. [ABSTRACT FROM AUTHOR]

This paper presents an outer velocity scale and an inner scaling law of Reynolds shear stresses that are valid for the entire adverse pressure gradient (APG) region before separation, based on asymptotic expansions in turbulent boundary layers (TBLs). By reexamining the velocity defect and Reynolds shear stress in the outer region of TBLs, it is demonstrated that the pressure gradient effect can be included in asymptotic expansions. The mathematical requirements to be satisfied by the outer velocity scale are given. On this basis, we define a new outer velocity scale Uo, which is used in the mean momentum balance analysis over the entire APG region. The results show that there is a slowly evolving region of the Reynolds shear stress gradient in APG TBLs. We obtain a logarithmic decay law for the Reynolds stress gradient by asymptotic matching in this region, thus providing functional expressions for the total shear stress and total shear stress gradient in the inner region of TBLs. These expressions are valid for the entire APG region. Furthermore, we use the new velocity scale to determine three self-similar parameters in the momentum equation to discuss the conditions for approximate equilibrium. The changes in parameters point to changes in two physical properties of APG TBLs, the inertial force and turbulence production. [ABSTRACT FROM AUTHOR]

This paper presents an attempt to improve the predictive accuracy of the Menter shear stress transport (SST) turbulence model for adverse pressure gradient (APG) flows. The foundation of this work lies in the newly modified law of the wall and turbulent kinetic energy (TKE) transport equation within the APG turbulent boundary layer. The modification of the law of the wall is achieved through a newly proposed variable λ, which is derived theoretically and calibrated using direct numerical simulation data. Based on the new law of the wall, we analyze the TKE transport equation of the SST model at APG and demonstrate that the analytical solution satisfying the law of the wall cannot achieve equilibrium in the current SST model. Further theoretical analysis reveals that this imbalance is caused by the TKE production term Pk, and the decomposition of the wall friction coefficient Cf indicates that Pk directly affects the prediction of flow separation. Based on the analysis and the new law of the wall, an additional term for Pk is suggested to correct Pk and improve the model's predictive accuracy for APG flows. Numerical validation results show that the correction leads to an increase in Pk within the APG turbulent boundary layer, resulting in more accurate predictions of Cf and the mean velocity profile in the inner layer of the turbulent boundary layer. [ABSTRACT FROM AUTHOR]

In this study, we investigate the distribution of turbulent kinetic energy (TKE) in the near wake of a circular cylinder in a turbulent flow. Numerical calculations were performed using the delayed detached eddy simulation method, incorporating two different production terms for TKE: one in its original form and the other with the Kato–Launder correction term. Our results demonstrate that the turbulence model utilizing the Kato–Launder correction term exhibits a strong correlation between TKE and the vorticity field, which is related to the calculations of mean velocity, velocity fluctuation, and other parameters that are in closer agreement with direct numerical simulation and experimental values. By employing the proper orthogonal decomposition technique, we extract and reconstruct three significant modes within the flow: the shear layer mode, vortex shedding mode, and near-wake bubble mode. The findings reveal that the Kato–Launder correction term offers a more detailed portrayal closer to the real flow physics. Conversely, the original form of the TKE production term exhibits an uneven energy distribution among the three modes and affects the role of the diffusion term within the flow. This leads to a less accurate representation of the vortex shedding mode and an excessive diffusion effect in the near-wake bubble mode. Finally, possible modifications of the turbulence model in this problem are given to enhance the portrayal of these characteristics. This work presents an analytical framework that enables a comprehensive analysis of turbulence models, providing valuable physical insights and guidance for improvement. [ABSTRACT FROM AUTHOR]