Wall-resolved large-eddy simulations of flow over an axisymmetric body of revolution (DARPA SUBOFF bare model) at $ \it{Re}_L=1.1\times 10^6$ are performed to investigate wall pressure fluctuations under the combined effects of transverse curvature and varying pressure gradients. Due to the coexistence of convex and concave streamwise curvatures, the flow in the stern region features alternating zones of favourable and adverse pressure gradients (APGs). The simulation validates experimental findings by Balantrapu et al. (2023, J. Fluid Mech., vol. 960, A28), confirming that in APG-dominant axisymmetric boundary layers without streamwise curvatures, the root mean square wall pressure fluctuations ($p_{w,rms}$) decrease downstream alongside the wall shear stress ($\tau _w$), maintaining a constant ratio $p_{w,rms}/\tau _{w}$. This study further finds that when streamwise curvatures and strong streamwise pressure gradient variations present, this relationship breaks down, suggesting that $\tau _w$ is not the dominant contributor to wall pressure fluctuations. Instead, the local maximum Reynolds shear stress $-\rho \langle u_su_n\rangle _{max }$ emerges as a more robust pressure scaling parameter. Normalising the wall pressure spectra by $-\rho \langle u_su_n\rangle _{max }$ yields better collapse across the entire stern region compared to conventional inner or mixed scaling methods. The magnitude and location of $-\rho \langle u_su_n\rangle _{max }$ significantly influence the spectral levels of wall pressure fluctuations across different frequency bands. As the turbulence intensity and $-\rho \langle u_su_n\rangle _{max }$ shift away from the wall, outer-layer structures – with larger spatial and temporal scales – dominate the coherence of wall pressure fluctuations. This mechanism drives sustained attenuation of high-frequency pressure fluctuations and a simultaneous increase in both the streamwise and transverse correlation lengths of wall pressure fluctuations over the stern region.

Recently, Nagib et al. (Phys. Fluids, vol. 36, no. 7, 2024, 075145) used indicator functions of streamwise normal stress profiles to identify the valid wall-distance and Reynolds number ranges for two models in direct numerical sumulation (DNS) of channel and pipe flows. Since such functions are challenging to construct from experimental data, we propose a simpler, more robust method better suited to experiments. Applied to the two leading models – logarithmic and power-law – for normal stresses in the ‘fitting region’ of wall-bounded flows, this method is tested on prominent experimental data sets in zero-pressure-gradient (ZPG) boundary layers and pipe flows across a wide Reynolds number range ($Re_\tau$). Valid regions for the models appear only for $Re_\tau \gtrapprox 10{\,}000$, with a lower bound $y^+_{in} \sim (Re_\tau )^{0.5}$ and $y^+_{in} \gtrapprox 400$. The upper bound is a fixed fraction of the boundary layer thickness or pipe radius, independent of $Re_\tau$. The power-law model is found to hold over a broader range, up to $Y \approx 0.4$ in ZPG and $Y \approx 0.5$ in pipe flows, compared with the logarithmic trend, which is formulated to be coincident with the classical logarithmic region for the mean flow ($Y \lessapprox 0.15$). A slightly higher exponent ($0.28$) than that of Chen & Sreenivasan (J. Fluid Mech. vol. 933, 2022, A20; J. Fluid Mech. vol. 976, 2023, A21) extends the power-law model’s validity and correcting for outer intermittency in ZPG flows further broadens it. Projections to the near-wall region of both models yield nearly identical predictions of near-wall peak stress across the highest available $Re_\tau$. These findings, alongside results from Monkewitz & Nagib (J. Fluid Mech. vol. 967, 2023, A15) and Baxerras et al. (J. Fluid Mech. vol. 987, 2024, R8), highlight the importance of nonlinear eddy growth and residual viscous effects in wall-bounded flow modelling, informing potential refinements to the logarithmic model, such as those proposed by Deshpande et al. (J. Fluid Mech. vol. 914, 2021, A5).

This study uses the diffusion analogy (Miyake, Sci. Rep., 5R-6, 1965, Univ. of Washington, Seattle, USA) to predict the full growth behaviour of internal boundary layers (IBLs) induced by a roughness change for neutrally – and especially stably – stratified boundary layers with finite thickness. The physics of the diffusion analogy shows that the streamwise variation of the IBL thickness is dictated by $\sigma _w/U$ at the interface, where $\sigma _w$ and $U$ represent wall-normal Reynolds stress and mean streamwise velocity, respectively. The existing variants of the model, summarised by Savelyev & Taylor (2005, Boundary-Layer Meteorol., vol. 115, pp. 1–25), are tailored to IBLs confined within the constant shear stress layer. To extend the applicability of the model to the outer region, we investigate the relation between $\sigma _w/U$ and $U/U_\infty$ in the outer region across varying stratification, where $U_\infty$ is the free-stream velocity. Our analysis reveals that wind tunnel data from a number of facilities collapse onto a master curve when $\sigma _w/U$ is premultiplied by a height-independent parameter, which is a function of the ratio of Monin–Obukhov length to the boundary layer thickness. The scaled $\sigma _w/U$ decreases inversely with $U/U_\infty$ in the surface layer, transitioning to a linear decrease as $U/U_\infty$ increases. The new model, which integrates these findings, along with the effects of streamline displacement and acceleration, captures the complete characteristics of IBLs as they develop within turbulent boundary layers of finite thickness.

A model is proposed for the one-dimensional spectrum and streamwise Reynolds stress in pipe flow for arbitrarily large Reynolds numbers. Constructed in wavenumber space, the model comprises four principal contributions to the spectrum: streaks, large-scale motions, very-large-scale motions and incoherent turbulence. It accounts for the broad and overlapping spectral content of these contributions from different eddy types. The model reproduces well the broad structure of the premultiplied one-dimensional spectrum of the streamwise velocity, although the bimodal shape that has been observed at certain wall-normal locations, and the $-5/3$ slope of the inertial subrange, are not captured effectively because of the simplifications made within the model. Regardless, the Reynolds stress distribution is well reproduced, even within the near-wall region, including key features of wall-bounded flows such as the Reynolds number dependence of the inner peak, the formation of a logarithmic region, and the formation of an outer peak. These findings suggest that many of these features arise from the overlap of energy content produced by both inner- and outer-scaled eddy structures combined with the viscous-scaled influence of the wall. The model is also used to compare with canonical turbulent boundary layer and channel flows, and despite some differences being apparent, we speculate that with only minor modifications to its coefficients, the model can be adapted to these flows as well.

The paper presents a simulation of the turbulent flow over and through a submerged aquatic canopy composed of 672 long, slender ribbons modelled as Cosserat rods. It is characterized by a bulk Reynolds number of 20 000, and a friction Reynolds number of 2638. Compared with a smooth turbulent channel at the same bulk Reynolds number, the canopy increases drag by a factor of 12. The ribbons are highly flexible, with a Cauchy number of 25 000, slightly buoyant, and densely packed. Their length exceeds the channel height by a factor of 1.6, while their average reconfigured height is only a quarter of the channel height. Different from lower-Cauchy-number cases, the movement of the ribbons, characterized by the motion of their tips, is very pronounced in the vertical direction, and even more in the spanwise direction, with root-mean-square fluctuations of the spanwise tip position 1.5 times the vertical ones. A canopy hull is defined to analyse the collective motion of the canopy and its interaction with the outer flow. Dominant spanwise wavelengths at this interface measure approximately one channel height, corresponding to twice the spacing of adjacent high- and low-speed streaks identified in two-point correlations of fluid velocity fluctuations. Conditional averages associated with troughs and ridges in the topography of the hull reveal streamwise-oriented counter-rotating vortices. They are reminiscent of the head-down structures related to the monami phenomenon in lower-Cauchy-number cases.

In this study, changes in the mean flow of a compressible turbulent boundary layer spatially evolving from low to ‘moderate’ Reynolds numbers are examined. All discussions are based on literature data and a direct numerical simulation (DNS) of a supersonic boundary layer specifically designed to be effectively free of spurious inflow effects in the range $4000 \lessapprox Re_\theta \lessapprox 5000$, which enables discussion of sensitive properties such as the turbulent wake. Most noticeably, the DNS data show the formation of a distinct ‘bend’ in the friction coefficient distribution reflected in sudden deviation from established low-Reynolds-number correlations. As will be shown, the bend is related to the surprisingly abrupt saturation of the turbulent wake, marking the change from low- to moderate-Reynolds-number behaviour; in previous studies, this trend was potentially obscured by data scatter in experiments and/or insufficient domain length in DNS. Moreover, the influence of the wake saturation on the formation of the early logarithmic overlap layer is assessed, which, if fully developed, leads to the onset of high-Reynolds-number behaviour further downstream.

This study presents an experimental investigation on the drag reduction (DR) over air-fed hydrophobic surfaces (AFHS) with longitudinal grooves in a turbulent boundary layer (TBL). The AFHS, designed with longitudinal grooves and air supplement channels, enables active maintenance and reversible restoration of the plastron in TBL. The shear stress sensor, particle image velocimetry (PIV) and interfacial visualization are applied for simultaneous measurement of the skin friction drag, TBL velocity profiles and plastron coverage. The AFHS demonstrated the ability to control plastron shape and enhance its sustainability with friction Reynolds numbers up to 1723. Drag reductions ranging from 14.8–35.8 % are obtained over the AFHS. At same designed air fraction, the AFHS exhibits higher DR than the conventional hydrophobic surface. By minimizing influences of the degradation of plastron coverage and the shape, the monotonic increase in DR and slip velocity with Reynolds number is confirmed, which corroborates trends from direct numerical simulations. Turbulence statistics measured by PIV reveal an apparent decrease in near-wall viscous shear stress, and corresponding slip velocities both in the viscous sublayer and log-law region. The Reynolds shear stress and streamwise velocity fluctuations over the AFHS are larger than those over a smooth wall, where near-wall vortex cores of the AFHS are found to be shifted 10 % towards the wall. This study presents the first simultaneous experimental quantification of skin friction, plastron coverage and turbulence statistics under sustained plastron conditions in TBL. The results demonstrate the efficacy of the plastron control strategy on hydrophobic surfaces and address a critical gap in validating numerical predictions for turbulent flows in practical applications.

Modelling the nonlinear forcing is critical for linear models based on resolvent or input–output analyses. For compressible wall-bounded turbulence, little is known on what the real forcing looks like due to limited data, so the prediction agrees more qualitatively than quantitatively with direct numerical simulations (DNSs). Here, we present detailed forcing statistics of stochastic linear models, derived from elaborate DNS datasets for channel flows with bulk Mach number reaching 3. These statistics directly explain the success and failure of current models and provide guidance for further improvements. The benchmark linearised Navier–Stokes (LNS) and eLNS models are considered; the latter is assisted by eddy-viscosity-related terms. First, we prove the self-consistency of the models by using DNS-computed forcing as the input. Second, we present the spectral distributions of the forcing and its components. Third, we quantify the acoustic components, absent in incompressible cases, within the linear models. We reveal that the LNS forcing can exhibit relatively high coherence and low rank, very different from the modelled diagonal full-rank forcing. The eddy-viscosity-related term is not partial modelling of the LNS forcing; contrarily, the former is much larger than the latter, serving to disrupt the low-rank feature, enhance diagonal dominance and increase robustness across scales. The scales narrow in either horizontal direction are most susceptible to acoustic modes, while the others are little affected (${\lt}2\,\%$ in energy). Furthermore, the extended strong Reynolds analogy is assessed in predicting the density and temperature components.

Synthetic-aperture radar images and mesoscale models show that wind-farm wakes differ from single-turbine wakes. For instance, wind-farm wakes often narrow and do not disperse over long distances, contrasting the broader and more dissipating wakes of individual turbines. In this work, we aim to better understand the mechanisms that govern wind-farm wake behaviour and recovery. Hence we study the wake properties of a $1.6$ GW wind farm operating in conventionally neutral boundary layers with capping-inversion heights $203$, $319$, $507$ and $1001$ m. In shallow boundary layers, we find strong flow decelerations that reduce the Coriolis force magnitude, leading to an anticlockwise wake deflection in the Northern Hemisphere. In deep boundary layers, the vertical turbulent entrainment of momentum adds clockwise-turning flow from aloft into the wake region, leading to a faster recovery rate and a clockwise wake deflection. To estimate the wake properties, we propose a simple function to fit the velocity magnitude profiles along the spanwise direction. In the vertical direction, the wake spreads up to the capping-inversion height, which significantly limits vertical wake development in shallow-boundary-layer cases. In the horizontal direction and for shallow boundary layers, the wake behaves as two distinct mixing layers located at the lateral wake edges, which expand and turn towards their low-velocity side, causing the wake to narrow along the streamwise direction. A detailed analysis of the momentum budget reveals that in deep boundary layers, the wake is predominantly replenished through turbulent vertical entrainment. Conversely, in shallow boundary layers, wakes are mostly replenished by mean flow advection in the spanwise direction.

Direct numerical simulations are performed to explore the impact of surface roughness on inter-scale energy transfer and interaction in a turbulent open-channel flow over differently arranged rough walls. With friction Reynolds number approximately 540, six distinct configurations of roughness arrangements are examined. The results show that the clustered roughness arrangements yield notable changes in large-scale secondary-flow structures, which manifest in the profiles of dispersive stresses, predominantly near the roughness elements. They are marked by the presence of spanwise alternating high-momentum pathways and low-momentum pathways. From the outer peak in the spanwise energy spectra, the size and intensity of turbulent secondary flows are shown to be related to the spanwise spacing of the roughness heterogeneity. The emergence of turbulent secondary flows serves to suppress the original large-scale structures in the outer region of smooth-wall turbulence, paving the way for the development of new turbulent structures at the second harmonic scale. Furthermore, the spanwise triadic interaction analysis reveals the mutual energy exchange between the secondary harmonic scale and the secondary-flow scale. These findings elucidate the underlying mechanisms behind the attenuation of large-scale structures in the outer region influenced by roughness, offering new insights into the dynamic interplay of scale interactions in rough-wall turbulence.

The physical fidelity of turbulence models can benefit from a partial resolution of fluctuations, but doing so often comes with an increase in computational cost. To explore this trade-off in the context of wall-bounded flows, this paper introduces a framework for turbulence-resolving integral simulations (TRIS) with the goal of efficiently resolving the largest motions using a two-dimensional, three-component representation of the flow defined by instantaneous wall-normal integrals of velocity and pressure. Self-sustaining turbulence with qualitatively realistic large-scale structures is demonstrated for TRIS on an open-channel (half-channel) flow configuration using moment-of-momentum integral equations derived from Navier–Stokes with relatively simple closure approximations. Evidence from direct numerical simulations (DNS) suggests that TRIS can theoretically resolve $35\,\%{-}40\,\%$ of the turbulent skin friction enhancement for friction Reynolds numbers between $180$ and $5200$, without a noticeable decrease or increase as a function of Reynolds number. The current implementation of TRIS can match this resolution while simulating one flow through time in ${\sim}1$ minute on a single processor, even for very large Reynolds numbers. The framework facilitates a detailed apples-to-apples comparison of predicted statistics against data from DNS. Comparisons at friction Reynolds numbers of $395$ and $590$ show that TRIS generates a relatively accurate representation of the flow, while highlighting discrepancies that demonstrate a need for improving the closure models. The present results for open-channel flow represent a proof of concept for TRIS as a new approach for wall-bounded turbulence modelling, motivating extension to more general flow configurations such as boundary layers on immersed objects.

Sub-convective wall pressure fluctuations play a critical role in vibroacoustic and noise analyses of vehicle structures as they serve as the primary forcing function. However, measuring these fluctuations is challenging due to their weak pressure magnitudes, typically $10^{-3}{-}10^{-5}$ of convective fluctuations. This study introduces a non-intrusive measurement technique using an array of multi-pore Helmholtz resonator sensors to capture sub-convective fluctuations with high resolution. The array features large-area, spanwise-oriented sensors arranged linearly for optimal sampling. Results provide a continuous streamwise wavenumber–frequency spectrum, resolving sub-convective fluctuations with sufficient range and accuracy. Convergence analysis indicates that long sampling durations, $\mathcal{O}(10^6 \delta ^*/U_\infty )$, $\delta^*$ is the displacement thickness of the boundary layer. $U_\infty$ is the freestream velocity are necessary to capture true sub-convective levels. Comparisons with four existing wall pressure models, which account for sensor area averaging, reveal discrepancies in predicted levels, convection speed relations and convective ridge characteristics. Notably, the measured data align most closely with the Chase (1980, J. Sound Vib., vol.70, pp. 29–67) model at convective peak levels and in the sub-convective domain. However, the observed roll-off at wavenumbers exceeding the convective wavenumber decays more slowly than predicted, giving the convective ridge an asymmetric profile about the convective line. These findings underscore the need for improved wall pressure models that incorporate frequency-dependent convective speed relations, ridge asymmetry, and more accurate sub-convective levels. Further validation using a microphone array from Farabee & Geib (1991) confirms the accuracy of our measurements, which indicate sub-convective pressure levels lower than reported previously.

Wall-resolved large-eddy simulation (LES) of a non-equilibrium turbulent boundary layer (TBL) is performed. The simulations are based on the experiments of Volino (2020a J. Fluid Mech. 897, A2), who reported profile measurements at several streamwise stations in a spatially developing zero pressure gradient TBL evolving through a region of favourable pressure gradient (FPG), a zero pressure gradient recovery and subsequently an adverse pressure gradient (APG) region. The pressure gradient quantified by the acceleration parameter $K$ was held constant in each of these three regions. Here, $K = -(\nu /\rho U_e^{3}) {\textrm d}P_e/{\textrm d}x$, where $\nu$ is the kinematic viscosity, $\rho$ is density, $U_e$ is the free stream velocity and ${\textrm d}P_e/{\textrm d}x$ is the streamwise pressure gradient at the edge (denoted by the subscript ‘$e$’) of the TBL. The simulation set-up is carefully designed to mimic the experimental conditions while keeping the computational cost tractable. The computational grid appropriately resolves the increasingly thinning and thickening of the TBL in the FPG and APG regions, respectively. The results are thoroughly compared with the available experimental data at several stations in the domain, showing good agreement. The results show that the computational set-up accurately reproduces the experimental conditions and the results demonstrate the accuracy of LES in predicting the complex flow field of the non-equilibrium TBL. The scaling laws and models proposed in the literature are evaluated and the response of the TBL to non-equilibrium conditions is discussed.

Based on data from pore-resolved direct numerical simulation of turbulent flow over mono-disperse random sphere packs, we evaluate the budgets of the double-averaged turbulent kinetic energy (TKE) and the wake kinetic energy (WKE). While TKE results from temporal velocity fluctuations, WKE describes the kinetic energy in spatial variations of the time-averaged flow field. We analyse eight cases which represent sampling points within a parameter space spanned by friction Reynolds numbers $Re_\tau \in [150, 500]$ and permeability Reynolds numbers $Re_K \in [0.4, 2.8]$. A systematic exploration of the parameter space is possible by varying the ratio between flow depth and sphere diameter $h/D \in \{ 3, 5, 10 \}$. With roughness Reynolds numbers of $k_s^+ \in [20,200]$, the simulated cases lie within the transitionally or fully rough regime. Revisiting the budget equations, we identify a WKE production mechanism via viscous interaction of the flow field with solid surfaces. The scaling behaviour of different processes over $Re_K$ and $Re_\tau$ suggests that this previously unexplored mechanism has a non-negligible contribution to the WKE production. With increasing $Re_K$, progressively more WKE is transferred into TKE by wake production. A near-interface peak in the TKE production, however, primarily results from shear production and scales with interface-related scales. Conversely, further above the sediment bed, the TKE budget terms of cases with comparable $Re_\tau$ show similarity under outer-scaling. Most transport processes relocate energy in the near-interface region, whereas pressure diffusion propagates TKE and WKE into deeper regions of the sphere pack.

Experiments are carried out in a smooth-wall turbulent boundary layer (TBL) ($\textit{Re}_\tau \geq 3500$) subjected to different pressure gradient (PG) histories. Oil-film interferometry is used to measure the skin friction evolution over the entire history while wide-field particle image velocimetry captures the mean flow field. This data are used to demonstrate the influence of PG history on skin friction as well as other integral quantities such as displacement ($\delta ^*$) and momentum thickness ($\theta$). Based on observations from the data, a new set of ordinary differential equations are proposed to model the streamwise evolution of a TBL subjected to different PG histories. The model is calibrated using a limited number of experimental cases and its utility is demonstrated on other cases. Moreover, the model is applied to data from large-eddy simulations of flows in adverse PG conditions (Bobke et al. 2017, J. Fluid Mech. 820, 667–692). The model is subsequently used to identify the impact of PG history length on the boundary layer. This can also be interpreted as determining the spatial frequency response of the boundary layer to PG disturbances. Results suggest that short spatial variations in PGs primarily affect a small portion of the TBL evolution, whereas longer-lasting ones have a more extensive impact.

The accurate quantification of wall-shear stress dynamics is of substantial importance for various applications in fundamental and applied research, spanning areas from human health to aircraft design and optimization. Despite significant progress in experimental measurement techniques and postprocessing algorithms, temporally resolved wall-shear stress fields with adequate spatial resolution and within a suitable spatial domain remain an elusive goal. Furthermore, there is a systematic lack of universal models that can accurately replicate the instantaneous wall-shear stress dynamics in numerical simulations of multiscale systems where direct numerical simulations (DNSs) are prohibitively expensive. To address these gaps, we introduce a deep learning architecture that ingests wall-parallel streamwise velocity fields at $y^+ \approx 3.9 \sqrt {Re_\tau }$ of turbulent wall-bounded flows and outputs the corresponding two-dimensional streamwise wall-shear stress fields with identical spatial resolution and domain size. From a physical perspective, our framework acts as a surrogate model encapsulating the various mechanisms through which highly energetic outer-layer flow structures influence the governing wall-shear stress dynamics. The network is trained in a supervised fashion on a unified dataset comprising DNSs of statistically one-dimensional turbulent channel and spatially developing turbulent boundary layer flows at friction Reynolds numbers ranging from $390$ to $1500$. We demonstrate a zero-shot applicability to experimental velocity fields obtained from particle image velocimetry measurements and verify the physical accuracy of the wall-shear stress estimates with synchronized wall-shear stress measurements using the micro-pillar shear-stress sensor for Reynolds numbers up to $2000$. In summary, the presented framework lays the groundwork for extracting inaccessible experimental wall-shear stress information from readily available velocity measurements and thus, facilitates advancements in a variety of experimental applications.

An experimental study was conducted in the CICLoPE long-pipe facility to investigate the correlation between wall-pressure and turbulent velocity fluctuations in the logarithmic region, at high friction Reynolds numbers ($4794 \lesssim Re_\tau \lesssim 47\,015$). Hereby, we explore the scalability of employing wall-pressure to effectively estimate off-the-wall velocity states (e.g. to be of use in real-time control of wall-turbulence). Coherence spectra for wall-pressure and streamwise (or wall-normal) velocity fluctuations collapse when plotted against $\lambda _x/y$ and thus reveals a Reynolds-number-independent scaling with distance-from-the-wall. When the squared wall-pressure fluctuations are considered instead of the linear wall-pressure term, the coherence spectra for the wall-pressure-squared and velocity are higher in amplitude at wavelengths corresponding to large-scale streamwise velocity fluctuations (e.g. at $\lambda _x/y = 60$, the coherence value increases from roughly 0.1 up to 0.3). This higher coherence typifies a modulation effect, because low-frequency content is introduced when squaring the wall-pressure time series. Finally, quadratic stochastic estimation is employed to estimate turbulent velocity fluctuations from the wall-pressure time series only. For each $Re_\tau$ investigated, the estimated time series and a true temporal measurement of velocity inside the turbulent pipe flow yield a normalised correlation coefficient of $\rho \approx 0.6$ for all cases. This suggests that wall-pressure sensing can be employed for meaningful estimation of off-the-wall velocity fluctuations and thus for real-time control of energetic turbulent velocity fluctuations at high-$Re_\tau$ applications.

The turbulent boundary layer (TBL) development over an air cavity is experimentally studied using planar particle image velocimetry. The present flow, representative of those typically encountered in ship air lubrication, resembles the geometrical characteristics of flows over solid bumps studied in the literature. However, unlike solid bumps, the cavity has a variable geometry inherent to its dynamic nature. An identification technique based on thresholding of correlation values from particle image correlations is employed to detect the cavity. The TBL does not separate at the leeward side of the cavity owing to a high boundary layer thickness to maximum cavity thickness ratio ($\delta /t_{max}= 12$). As a consequence of the cavity geometry, the TBL is subjected to alternating streamwise pressure gradients: from an adverse pressure gradient (APG) to a favourable pressure gradient and back to an APG. The mean streamwise velocity and turbulence stresses over the cavity show that the streamwise pressure gradients and air injection are the dominant perturbations to the flow, with streamline curvature concluded to be marginal. Two-point correlations of the wall-normal velocity reveal an increased coherent extent over the cavity and a local anisotropy in regions under an APG, distinct from traditional APG TBLs, suggesting possible history effects.

This study proposes a machine-learning-based subgrid scale (SGS) model for very coarse-grid large-eddy simulations (vLES). An issue with SGS modelling for vLES is that, because the energy-containing eddies are not accurately resolved by the computational grid, the resolved turbulence deviates from the physically accurate turbulence. This limits the use of supervised machine-learning models commonly trained using pairs of direct numerical simulation (DNS) and filtered DNS data. The proposed methodology utilises both unsupervised learning (cycle-consistency generative adversarial network (GAN)) and supervised learning (conditional GAN) to construct a machine-learning pipeline. The unsupervised learning part of the proposed method first transforms the non-physical vLES flow field to resemble a physically accurate flow field. The second supervised learning part employs super-resolution of turbulence to predict the SGS stresses. The proposed pipeline is trained using a fully developed turbulent channel at the friction Reynolds number of approximately 1000. The a priori validation shows that the proposed unsupervised–supervised pipeline successfully learns to predict the accurate SGS stresses, while a typical supervised-only model shows significant discrepancies. In the a posteriori test, the proposed unsupervised–supervised-pipeline SGS model for vLES using a progressively coarse grid yields good agreement of the mean velocity and Reynolds shear stress with the reference data at both the trained Reynolds number 1000 and the untrained higher Reynolds number 2000, showing robustness against varying Reynolds numbers. A budget analysis of the Reynolds stresses reveals that the proposed unsupervised–supervised-pipeline SGS model predicts a significant amount of SGS backscatter, which results in the strengthened near-wall Reynolds shear stress and the accurate prediction of mean velocity.

Görtler vortices induced by concave curvature in supersonic turbulent flows are investigated using resolvent analysis and large-eddy simulations at Mach 2.95 and Reynolds number $ Re_{\delta }=63\,500$ based on the boundary-layer thickness $ \delta$. Resolvent analysis reveals that the most amplified coherent structures manifest as streamwise counter-rotating vortices with optimal spanwise wavelength $ 2.4\delta$ at cut-off frequency $f\delta /{u}_{\infty } =0.036$, where $ {u}_{\infty }$ is the freestream velocity. The leading spectral proper orthogonal decomposition modes with spanwise wavelength approximately $ 2\delta$ align well with the predicted coherent structures from resolvent analysis at $f\delta /{u}_{\infty } =0.036$. These predicted and extracted coherent structures are identified as Görtler vortices, driven by the Görtler instability. The preferential spanwise scale of the Görtler vortices is further examined under varying geometric and freestream parameters. The optimal spanwise wavelength is insensitive to the total turning angle beyond a critical value, but sensitive to the concave curvature $ K$ at the same turning angle. A limit spanwise wavelength $ 1.96\delta$, corresponding to an infinite concave curvature as $ K\rightarrow \infty$, is identified and validated. Increasing the freestream Mach number or decreasing the ratio of wall temperature to freestream temperature reduces the optimal wavelength normalised by $ \delta$, while variations in freestream Reynolds number have negligible impact. Additionally, a modified definition of the turbulent Görtler number $ G_{T}$ based on the peak eddy viscosity in boundary layers is proposed and employed to assess the occurrence of Görtler instability.