Recovery of turbulent boundary layers from pressure gradient history effects

TL;DR

This work addresses how smooth-wall turbulent boundary layers recover after non-equilibrium pressure-gradient history (PGH) events. Using controlled PGH sequences imposed by a NACA0012 wing and hot-wire measurements at matched $Re_\tau$ and Clauser parameter $β$, the authors dissect mean-flow, turbulence intensities, and energy spectra to isolate history effects. They demonstrate that PGH leaves a pronounced imprint on outer-layer turbulence via a distinct PG peak and modified VLSM, with energy amplification scaling with PGH strength, while the inner-layer behavior largely follows canonical ZPG dynamics. The findings emphasize scale interactions as central to recovery under complex PGH and offer guidance for modeling TBLs in engineering applications where upstream history influences downstream turbulence.

Abstract

The present study experimentally investigates the recovery of smooth-wall turbulent boundary layers (TBLs) following non-equilibrium pressure gradients (PGs). The imposed pressure gradient history (PGH) comprises favourable-adverse pressure gradient (FAPG) sequences of varying strength, followed by recovery to zero-pressure-gradient (ZPG) conditions. Hot-wire anemometry measurements were obtained at multiple downstream stations in the recovery region, with friction Reynolds numbers $Re_τ$ ranging from 2000 to 6000 depending on downstream development. Comparative analysis at matched $Re_τ$ and Clauser pressure gradient parameter $β$ enables clear assessment of history effects on TBL behaviour. Results show that increasing PGH strength enhances the wake in mean velocity profiles and amplifies turbulence intensities across the boundary layer, including the inner peak, logarithmic region, and outer peak (a signature of APG). Downstream, the mean flow gradually recovers toward a ZPG-like state, but turbulence in the outer region retains a lasting impact of PGH. Spectral analysis indicates that PGH primarily affects outer-layer scales, introducing a distinct PG peak and modifying the VLSM peak - with energy amplification dependent on PGH strength and spatial characteristics governed by history effects. Downstream recovery involves merging of large-scale wavelengths and the reorganisation of turbulence structures toward a ZPG-like state - although the `recovered' VLSM streamwise length becomes shortened due to the mixing of lengthscales with the PG peak. These results demonstrate that even under matched local parameters, TBLs retain a clear imprint of their upstream history, consistent with the findings of Preskett et al. (2025); moreover, this study provides new insights regarding the central role of scale interactions in the recovery mechanism of TBL subjected to complex PGH.

Figures (8)

• Figure 1: Schematic of the BLWT test section. The black dashed line indicates the upstream reference measurement location, where the reference boundary-layer thickness upstream of the wing was measured as $\delta_0 = 35.23$ mm (not to scale in the schematic). The red dashed lines denote downstream measurement locations, with lighter shades corresponding to further downstream positions. All locations are normalised by $\delta_0$, corresponding to $x/\delta_0 =$ 41 (upstream), and 109, 122, 133, 147, 177, 191, and 201 (downstream). Numbered markers indicate key components: ① boundary-layer trip; ② pressure tap array (starting 0.12 m from the test section start); ③ upstream Pitot tube used to set $U_{\infty,0}$; ④ NACA 0012 aerofoil generating the PG; ⑤ local Pitot tube measuring $U_{\infty,l}$ and serving as hot-wire calibration reference; ⑥ hot-wire probe.
• Figure 2: Streamwise pressure coefficient gradient $\mathrm{d}C_p/\mathrm{d}x$, normalised by the upstream boundary layer thickness $\delta_0$, for all three cases. , , and denote the symbols and colours used for the $-4^\circ$ (weak), $-6^\circ$ (mild), and $-8^\circ$ (strong) PGH cases, respectively. The grey shaded region indicates the presence of the wing, and vertical dashed lines mark the streamwise measurement locations described previously.
• Figure 3: Grouping of experimental cases based on matched $Re_\tau$ for all three PGH conditions. (a) Streamwise evolution of experimentally obtained $Re_\tau$ values. Circles indicate selected cases grouped by similar $Re_\tau$, with corresponding baseline ZPG values from published datasets shown as horizontal dashed lines. (b) Clauser PG parameter, $\beta$, for the grouped cases, highlighted by shaded rectangles. Dashed lines represent $\beta$ for all other measurements, omitted for clarity. Full numerical values are provided in Table \ref{['Table:1']}.
• Figure 4: Inner-scaled mean velocity profiles for matched friction Reynolds number of $Re_\tau\approx$ (a) 2300, (b) 3000, and (c) 5500, with the corresponding inner-scaled turbulence intensity profiles shown in (d), (e), and (f). , , and denote the reference ZPG data at $Re_\tau=$ 2030, 2998, and 5710, respectively. The magnitude $|\alpha|$ serves as a measure of PGH strength, with larger $|\alpha|$ producing higher FAPG amplitudes and thus stronger PGH cases. The coloured symbols correspond to the test cases listed in Table \ref{['Table:1']}.
• Figure 5: Premultiplied one-dimensional energy spectra of streamwise velocity fluctuations at $Re_\tau \approx 2300$ for (a) the ZPG case and (b) the weak PGH case ($-4^\circ$). Both cases exhibit the inner-site ('$\times$') and VLSM ('$+$') peaks, while the weak PGH case additionally shows a PG peak ('$\circ$'), highlighting the effect of PGH on the spectra.