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High-Reynolds-number turbulent boundary layers under adverse pressure gradients. Part 2. A composite mean velocity profile

Ahmad Zarei, Mitchell Lozier, Rahul Deshpande, Ivan Marusic

Abstract

A robust composite mean velocity profile is developed for turbulent boundary layers (TBLs) subjected to adverse pressure gradients (APGs), extending the composite formulation for generic pressure-gradient TBLs proposed by \citeauthor{nickels} (\textit{J.\ Fluid Mech.}, vol.\ 521, 2004). Several modifications are introduced to capture key features of APG flows. A new parameter accounts for pressure-gradient history effects in the wake region, a velocity-overshoot function is incorporated in the inner region, and the wake function is reformulated using an independent, physically motivated definition of boundary-layer thickness. A compilation of APG TBL datasets from the literature, including the new dataset presented in Part~1, is used to assess and refine the formulation. The resulting composite profile contains three physically meaningful parameters that capture pressure-gradient effects on the mean velocity profile, determined through nonlinear curve fitting. These parameters provide a framework for identifying `well-behaved' APG TBLs and quantifying the strength of pressure-gradient history effects. The profile also enables reliable estimation of the friction velocity and boundary-layer thickness in well-behaved APG TBLs, providing a practical tool for scaling analyses when these quantities are not directly measurable. Its analytical form yields improved estimates of mean velocity gradients, facilitating evaluation of the indicator function and identification of inflection points. Finally, the formulation predicts both the coefficients and spatial extent of the logarithmic region of the mean streamwise velocity profile, enabling assessment of its universality in high-Reynolds-number APG TBLs. This shows that the von K'arm'an coefficient approaches an invariant value of $κ\approx 0.39$ at sufficiently high Reynolds numbers, independent of pressure-gradient effects.

High-Reynolds-number turbulent boundary layers under adverse pressure gradients. Part 2. A composite mean velocity profile

Abstract

A robust composite mean velocity profile is developed for turbulent boundary layers (TBLs) subjected to adverse pressure gradients (APGs), extending the composite formulation for generic pressure-gradient TBLs proposed by \citeauthor{nickels} (\textit{J.\ Fluid Mech.}, vol.\ 521, 2004). Several modifications are introduced to capture key features of APG flows. A new parameter accounts for pressure-gradient history effects in the wake region, a velocity-overshoot function is incorporated in the inner region, and the wake function is reformulated using an independent, physically motivated definition of boundary-layer thickness. A compilation of APG TBL datasets from the literature, including the new dataset presented in Part~1, is used to assess and refine the formulation. The resulting composite profile contains three physically meaningful parameters that capture pressure-gradient effects on the mean velocity profile, determined through nonlinear curve fitting. These parameters provide a framework for identifying `well-behaved' APG TBLs and quantifying the strength of pressure-gradient history effects. The profile also enables reliable estimation of the friction velocity and boundary-layer thickness in well-behaved APG TBLs, providing a practical tool for scaling analyses when these quantities are not directly measurable. Its analytical form yields improved estimates of mean velocity gradients, facilitating evaluation of the indicator function and identification of inflection points. Finally, the formulation predicts both the coefficients and spatial extent of the logarithmic region of the mean streamwise velocity profile, enabling assessment of its universality in high-Reynolds-number APG TBLs. This shows that the von K'arm'an coefficient approaches an invariant value of at sufficiently high Reynolds numbers, independent of pressure-gradient effects.
Paper Structure (20 sections, 23 equations, 17 figures, 1 table)

This paper contains 20 sections, 23 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Composite mean velocity profile formulations for the TBL
  3. Datasets
  4. Formulation of a new composite profile for APG TBLs
  5. The new composite profile formulation
  6. Addition of a wake-stretching parameter
  7. Addition of an overshoot function
  8. Linking local and upstream PGs to $z^{+}_{c}$
  9. Note for high-Reynolds-number TBLs
  10. Fitting method
  11. Quality and interpretation of the new composite profile
  12. Overlap region
  13. Wake region
  14. Utilities of the new composite profile
  15. Revisiting previous datasets
  16. ...and 5 more sections

Figures (17)

  • Figure 1: Contributions of the inner, overlap and wake expressions from nickels as a function of (a) $p_x^{+}$ and (b) $z^{+}_{c}$. The composite mean velocity profile (solid lines) is the sum of the three individual expressions. A baseline dataset with $Re_{\tau}=5000$ was used to generate these profiles.
  • Figure 2: Mean velocity data and corresponding composite profile \ref{['eq:TotPressureG']} for (a) a well-resolved LES ZPG dataset, (b) an APG dataset influenced by history effects, and (c) a high-$Re_{\tau}$ APG dataset with minimal PG history.
  • Figure 3: Contributions of the inner \ref{['eq:PresentInner']}, overlap \ref{['eq:Presentoverlap']} and wake \ref{['eq:Presentwake']} expressions to the present composite profile as a function of (a) the wake history factor $C_{H_w}$, (b) $z^{+}_{c}$, and (c) $\Pi$. A reference dataset with $Re_{\tau}=5000$ was used to generate these profiles.
  • Figure 4: Validation of the new composite profile for ZPG TBLs with minimal PG history from (a) the low-$Re_{\tau}$ LES dataset of eitel2014simulation, and (b) the high-$Re_{\tau}$ dataset of marusic2015. Individual profiles are offset by five units in the vertical direction. Dashed lines represent \ref{['eq:closeWall']} and dash-dotted lines represent \ref{['eq:loglaw']}.
  • Figure 5: Validation of the new composite profile for high-$Re_{\tau}$ moderate-APG TBLs with minimal PG history effects from Part 1. The mean velocity profiles are plotted in both (a) logarithmic and (b) linear scaling to confirm the quality of the composite profile across all regions of the TBL. Dashed lines represent \ref{['eq:closeWall']} and dash-dotted lines represent \ref{['eq:loglaw']}.
  • ...and 12 more figures