Log-linear law of the mean streamwise velocity in turbulent boundary layers with moderate adverse pressure gradients

TL;DR

The classical log law for mean velocity in wall-bounded turbulence does not hold under moderate adverse pressure gradients. The authors derive a log--linear law from the total shear-stress balance and a β-dependent eddy-viscosity scaling, yielding a closed-form expression for $U^+$ that includes both logarithmic and linear terms and recovers the traditional log law as $\beta \to 0$. Validation against diverse incompressible APG TBL datasets with $0.73 \le \beta \le 9.0$ shows substantially better agreement than the classical log law, with the diagnostic function $\Phi_2$ remaining near unity across cases. The approach preserves the classical Kármán constant $\kappa \approx 0.41$, introduces no extra empirical coefficients, and offers a simple, robust enhancement for predicting APG TBLs in RANS/WMLES applications.

Abstract

An essential feature of canonical zero-pressure-gradient (ZPG) turbulent boundary layers (TBLs) is that the mean streamwise velocity exhibits a logarithmic dependence on the wall-normal distance, known as the log law. In this study, we demonstrate that this conventional log law is not suitable for turbulent boundary layers subjected to pressure gradients (PGs). Instead, a log--linear law is theoretically derived for TBLs under moderate adverse pressure gradients (APGs), based on the total shear-stress balance and a rescaled eddy-viscosity model. The validity of the proposed log--linear law is assessed using available datasets of incompressible APG TBLs with the Clauser pressure-gradient parameter $β$ ranging from 0.73 to 9.0. Compared with the conventional log law, the present log--linear formulation shows significantly improved agreement with the measured mean velocity profiles. In the limiting case of $β\to 0$, the proposed law naturally recovers the classical log law.

Figures (4)

• Figure 1: Flow configurations and diagnostic functions for case B10B2017 (a--c), B20B2017 (d--f), M13B2017 (g--i), M16B2017 (j--l), and M18B2017 (m--o). The flow configurations, characterized by the Clauser parameter $\beta$ and friction Reynolds number $Re_\tau$, are presented by the markers in the left column. The diagnostic functions for the log law ($\varPhi_{1}$) and the log--linear law ($\varPhi_{2}$) are presented in the middle and right columns, respectively. Zoomed views of the diagnostic functions are provided in the insets in the middle and right columns. The horizontal grey dashed line indicates the plateau of $\varPhi \approx 1$, with two black dash-dotted lines representing a tolerance of $\pm 20\%$ of the plateau.
• Figure 2: Flow configurations and diagnostic functions for cases B14Y2018 (a--c) and B14P2022 (d--f). The flow configurations, characterized by the Clauser parameter $\beta$ and friction Reynolds number $Re_\tau$, are presented by the markers in the left column. The diagnostic functions for the log law ($\varPhi_{1}$) and the log--linear law ($\varPhi_{2}$) are presented in the middle and right columns, respectively. Zoomed views of the diagnostic functions are provided in the insets. The horizontal grey dashed line indicates the plateau of $\varPhi \approx 1$, with two black dash-dotted lines representing a tolerance of $\pm 20\%$ regarding the plateau.
• Figure 3: Diagnostic functions for cases B07L2017 (a), B22L2017 (b), and B90L2017 (c). Blue lines and red lines represent the diagnostic functions for the log law and the log--linear law, respectively. The horizontal grey dashed line indicates the plateau of $\varPhi \approx 1$, with two black dash-dotted lines representing a tolerance of $\pm 20\%$ regarding the plateau.
• Figure 4: Effective lengths for different cases. Circle, diamond, square, right-triangle, left-triangle, up-triangle, and down-triangle markers correspond to cases B10B2017, B20B2017, M13B2017, M16B2017, M18B2017, B14Y2018, and B14P2022, respectively. The marker colors correspond to the flow configurations in figures \ref{['fig:1']} and \ref{['fig:2']}. The blue, green, and red hexagrams correspond to cases B07L2017, B22L2017, and B90L2017, respectively. The open and filled markers represent the effective lengths of the log law and the log--linear law, respectively.