Mean velocity profiles and total shear stress profiles in adverse-pressure-gradient turbulent boundary layers considering history effect

TL;DR

This paper addresses the challenge of predicting mean velocity and total shear-stress profiles in adverse-pressure-gradient turbulent boundary layers, where history effects degrade traditional local-parameter scalings. It introduces a β-based outer scaling with $u_β = u_τ \sqrt{1+β}$ and corresponding starred variables to recover streamwise self-similarity, and develops an estimation–correction method that integrates a history descriptor $γ = (ν/u_β)\,dβ/dx$ into a four-term decomposition of the total stress $τ^* = τ_0^* + τ_β^* + τ_e^* + τ_γ^*$; the velocity and stress fields are iteratively corrected via a RANS-based framework with a universal mixing-length form $λ^*$ determined by a joint optimization. Validation against eight DNS/LES datasets demonstrates improved mean-velocity and total-stress predictions across a wide range of β, $Re_θ$, and γ, and shows that history terms can contribute up to roughly half of the total shear stress, especially in downstream regions. The approach provides a physically transparent pathway to incorporate history effects into wall-models and turbulence closures for complex boundary-layer flows, with potential implications for LES modeling under APG conditions. The framework also opens avenues for higher-order history corrections and broader applicability beyond APG, including potential extensions to FPG cases with careful consideration of the scaling limits.

Abstract

This study focuses on developing a predictive model for mean velocity profiles and total shear stress profiles in turbulent boundary layers subjected to adverse pressure gradients, especially with history effects. A new scaling using friction velocity modified by Clauser pressure gradient parameter is introduced to restore streamwise self-similarity. Furthermore, an estimation-correction model is developed, explicitly incorporating a streamwise derivative of pressure gradient, which effectively captures history effect beyond the reach of Reynolds-averaged Navier-Stokes equations. With the help of the model, the total shear stress is decomposed into four parts, representing respectively the Reynolds number effects, equilibrium pressure gradient effects, the coupling between free-stream velocity and pressure gradient, and local non-equilibrium pressure gradient effects. The latter two are considered first-order history effects, and can account for up to approximately half of the total stress. Validation against multiple DNS/LES datasets across a wide range of pressure gradients and Reynolds numbers demonstrates the model's accuracy in predicting both mean velocity profiles and total shear stress profiles.

Figures (9)

• Figure 1: An overall description of seven flow fields along streamwise direction. The data are obtained directly through DNS or LES Bobke_Vinuesa_Örlü_Schlatter_2017Yoon_Hwang_Sung_2018Pozuelo_Li_Schlatter_Vinuesa_2022. The horizontal axis denotes the streamwise coordinate $\hat{x}=x/\langle \delta_{99}\rangle$, where $x$ is non-dimensionalized by the nominal boundary layer thickness $\delta_{99}$ averaged over the computational domain. (a) Clauser pressure gradient $\beta$; (b) Streamwise derivative of pressure gradient $\gamma$ as defined in \ref{['eq:Gamma Definition']}; (c) Friction Reynolds number $Re_\tau$; (d) Momentum thickness Reynolds number $Re_\theta$; The markers in subfigure (a) indicate the locations of the profiles used for validation in Section \ref{['subsec:Iteration Model']}.
• Figure 2: Profile of non-dimensional mixing length $\lambda^*$ divided by the friction Reynolds number $Re_\tau^*$ at different streamwise locations, in different flow fields. The data are obtained directly through DNS or LES Bobke_Vinuesa_Örlü_Schlatter_2017Lee_2017Yoon_Hwang_Sung_2018. The result has been processed with Gaussian smoothing, and the critical ratio $n$ in criteria \ref{['eq:diagnostic_function']} is set to $2$. Within the same flow field, lines of lighter color indicate locations further downstream.
• Figure 3: Profiles of $u^+$ and $u^*$ at different streamwise locations. (a),(b) denotes different profiles in flow b1n, and (c),(d) denotes different profiles in flow m16n. The data are obtained directly through LES Bobke_Vinuesa_Örlü_Schlatter_2017. (a),(c) shows the profile of $u^+$ versus $y^+$; (b),(d) shows the profile of $u^*$ versus $y^*$.
• Figure 4: Velocity and total shear stress profiles in flow field b1n Bobke_Vinuesa_Örlü_Schlatter_2017. (a)(b), (c)(d), (e)(f), (g)(h) show different streamwise locations respectively. (a)(c)(e)(g) are the profiles of mean velocity. The black solid line represents the reference LES data, the green solid line represents the original model by Subrahmanyam_Cantwell_Alonso_2022, the blue dashed line represents the estimation $\widetilde{u}^*$ obtained from equation \ref{['eq:u1']}, and the red solid line represents the correction $u^*$ obtained from equation \ref{['eq:u2(y)']}. (b)(d)(f)(h) are the profiles of total shear stress. The black line represents the reference LES data, the green line represents the original model for ZPG TBLs by Subrahmanyam_Cantwell_Alonso_2022, the red line represents the present model $\tau^*$ from \ref{['eq:tau1_y']}, and the blue line represents its decomposition $\tau_{0}^*+\tau_{\beta}^*$, from equation \ref{['eq:Understanding']}. The specific flow parameters are listed below: (a)(b) $\hat{x}=21, \beta = 1.1, \gamma = -1.4\times 10^{-5}, Re_\tau = 448$, (c)(d) $\hat{x}=29, \beta = 1.0, \gamma = -1.2\times 10^{-5}, Re_\tau = 568$, (e)(f) $\hat{x}=41, \beta = 0.9, \gamma = -0.9\times 10^{-5}, Re_\tau = 716$, (g)(h) $\hat{x}=52, \beta = 0.6, \gamma = -14.7\times 10^{-5}, Re_\tau = 851$.
• Figure 5: Velocity and total shear stress profiles in flow field m16n Bobke_Vinuesa_Örlü_Schlatter_2017. (a)(b), (c)(d), (e)(f), (g)(h) show different streamwise locations respectively. The legend follows the same convention as in Figure \ref{['fig:b1n']}. The specific flow parameters are listed below: (a)(b) $\hat{x}=17, \beta = 2.8, \gamma = -1.1\times 10^{-5}, Re_\tau = 422$, (c)(d) $\hat{x}=23, \beta = 2.5, \gamma = -4.0\times 10^{-5}, Re_\tau = 539$, (e)(f) $\hat{x}=32, \beta = 2.1, \gamma = -4.8\times 10^{-5}, Re_\tau = 704$, (g)(h) $\hat{x}=41, \beta = 0.9, \gamma = -50.0\times 10^{-5}, Re_\tau = 861$.