Simulating complex gas flows from turbulent to rarefied regimes is a long-standing challenge, since turbulence and rarefied flow represent contrasting extremes of computational aerodynamics. We propose a multiscale method to bridge this gap. Our method builds upon the general synthetic iterative scheme for the mesoscopic Boltzmann equation, and integrates the $k$–$\omega$ model in the macroscopic synthetic equation to address turbulent effects. Asymptotic analysis and numerical simulations show that the macroscopic–mesoscopic coupling adaptively selects the turbulence model and the laminar Boltzmann equation. The multiscale method is then applied to opposing jet problems in hypersonic flight surrounding by rarefied gas flows, showing that the turbulence could cause significant effects on the surface heat flux, which cannot be captured by the turbulent model nor the laminar Boltzmann solution alone. This study provides a viable framework for advancing understanding of the interaction between turbulent and rarefied gas flows.

We investigate the turbulence below a quasi-flat free surface, focusing on the energy transport in space and across scales. We leverage a large zero-mean-flow tank where homogeneous turbulence is generated by randomly actuated jets. A wide range of Reynolds number is spanned, reaching sufficient scale separation for the emergence of an inertial sub-range. Unlike previous studies, the forcing extends through the source layer, although the surface deformation remains millimetric. Particle image velocimetry along a surface-normal plane resolves from the dissipative to the integral scales. The contributions to turbulent kinetic energy from both vertical and horizontal components of velocity approach the prediction based on rapid distortion theory as the Reynolds number is increased, indicating that discrepancies among previous studies are likely due to differences in the forcing. At odds with the theory, however, the integral scale of the horizontal fluctuations grows as the surface is approached. This is rooted in the profound influence exerted by the surface on the inter-scale energy transfer: along horizontal separations, the direct cascade of energy in horizontal fluctuations is hindered, while an inverse cascade of that in vertical fluctuations is established. This is connected to the structure of upwellings and downwellings. The former, characterized by somewhat larger spatial extent and stronger intensity, are associated with extensional surface-parallel motions. They thus transfer energy to the larger horizontal scales, prevailing over downwellings which favour the compression (and concurrent vertical stretching) of the eddies. Both types of structures extend to depths between the integral scale and the Taylor microscale.

Stall cells are transverse cellular patterns that often appear on the suction side of airfoils near stalling conditions. Wind-tunnel experiments on a NACA4412 airfoil at Reynolds number ${Re}=3.5 \times 10^5$ show that they appear for angles of attack larger than $\alpha = 11.5^{\circ }\ (\pm 0.5^{\circ })$. Their onset is further investigated based on global stability analyses of turbulent mean flows computed with the Reynolds-averaged Navier–Stokes (RANS) equations. Using the classical Spalart–Allmaras turbulence model and following Plante et al. (J. Fluid Mech., vol. 908, 2021, A16), we first show that a three-dimensional stationary mode becomes unstable for a critical angle of attack $\alpha = 15.5^{\circ }$ which is much larger than in the experiments. A data-consistent RANS model is then proposed to reinvestigate the onset of these stall cells. Through an adjoint-based data-assimilation approach, several corrections in the turbulence model equation are identified to minimize the differences between assimilated and reference mean-velocity fields, the latter reference field being extracted from direct numerical simulations. Linear stability analysis around the assimilated mean flow obtained with the best correction is performed first using a perturbed eddy-viscosity approach which requires the linearization of both RANS and turbulence model equations. The three-dimensional stationary mode becomes unstable for angle $\alpha = 11^{\circ }$ which is in significantly better agreement with the experimental results. The interest of this perturbed eddy-viscosity approach is demonstrated by comparing with results of two frozen eddy-viscosity approaches that neglect the perturbation of the eddy viscosity. Both approaches predict the primary destabilization of a higher-wavenumber mode which is not experimentally observed. Uncertainties in the stability results are quantified through a sensitivity analysis of the stall cell mode's eigenvalue with respect to residual mean-flow velocity errors. The impact of the correction field on the results of stability analysis is finally assessed.

Direct numerical simulations of the turbulence of a Herschel–Bulkley (HB) fluid in a rough channel are performed at a shear Reynolds number $Re_{\tau } \approx 300$ and a Bingham number ${Bn} \approx 0.9$. For the type of rough surface used in this study, the results indicate that Townsend's wall similarity hypothesis also holds for HB fluids. However, there are notable differences compared with the effect of roughness on Newtonian fluids. More specifically, the effect of roughness appears to be slightly stronger for HB fluids, in the sense that the bulk Reynolds number, based on the viscosity at the wall, is reduced further due to the increase in viscosity in the troughs of the roughness surface induced by the low shear. At the same time, for the simulated rough surface, the contribution of form drag to the total pressure drop is reduced from 1/4 to about 1/5 due to the persistence of viscous shear in the boundary layer, reducing its shielding effect. As for the friction factor, due to the nonlinearity of the HB constitutive relation, its use with the wall shear rate from the mean wall shear stress underpredicts the minimum viscosity at the wall by up to 18 %. This inevitably leads to uncertainties in the prediction of the friction factor. Finally, it is observed that the rough surface is unable to break the peculiar near-wall flow structure of HB fluids, which consists of long persistent low-speed streaks occupying the entire domain. This means that the small-scale energy is significantly reduced for HB fluids, even in rough channels, with the energy more concentrated in the lower wavenumber range, implying an increase in the slope of the power spectrum to $-7/2$ in the inertial range, as shown by Mitishita et al. (J. Non-Newtonian Fluid Mech., vol. 293, 2021, 104570).

In this paper it is shown that a modal detuned instability of periodic near-wall streaks originates a large-scale structure in the bulk of the turbulent channel flow. The effect of incoherent turbulent fluctuations is included in the linear operator by means of an eddy viscosity. The base flow is an array of periodic two-dimensional streaks, extracted from numerical simulations in small domains, superposed to the turbulent mean profile. The stability problem for a large number of periodic units is efficiently solved using the block-circulant matrix method proposed by Schmid et al. (Phys. Rev. Fluids, vol. 2, 2017, 113902). For friction Reynolds numbers equal or higher than $590$, it is shown that an unstable branch is present in the eigenspectra. The most unstable eigenmodes display large-scale modulations whose characteristic wavelengths are compatible with the large-scale end of the premultiplied velocity fluctuation spectra reported in previous computational studies. The wall-normal location of the large-wavelength near-wall peak in the spanwise spectrum of the eigenmode exhibits a power-law dependence on the friction Reynolds number, similarly to that found in experiments of pipes and boundary layers. Lastly, the shape of the eigenmode in the streamwise-wall-normal plane is reminiscent of the superstructures reported in the recent experiments of Deshpande et al. (J. Fluid Mech., vol. 969, 2023, A10). Therefore, there is evidence that such large-wavelength instabilities generate large-scale motions in wall-bounded turbulent flows.

An experimental investigation is conducted to study the flow patterns, spectral properties and energy fluxes in thin-layer turbulence with varying system sizes and damping rates. It is found that although a system-size vortex (an indicator of spectral condensation) occurs for small system sizes and does not for large ones, the spectra for different system sizes consistently exhibit a scaling close to $k^{-3}$ in inverse cascade (another indicator of spectral condensation). On the other hand, under a fixed system size larger than the friction-dominated length scale, the energy spectrum in the inverse cascade range changes from $k^{-3}$ to $k^{-5/3}$ as the damping rate increases, suggesting that the friction-dominated length scale may not be a suitable parameter for predicting spectral transition. At lower damping rates and large system sizes, turbulent structures grow larger via inverse cascade, manifesting as long streamers, and the small-scale vortices are suppressed. This suppression leads to a reduction of energy flux at intermediate scales and a change in the spectral shape. The dimensionless Taylor microscale is found to exhibit a monotonic dependence on the damping rate. With the reduction in the damping rate, the Taylor microscale increases to become comparable with the forcing scale, and the spectrum in inverse cascade transits to a steeper scaling, $k^{-3}$, indicating that the dimensionless Taylor microscale may be used as a diagnostic parameter for spectral transition.

Predicting the magnitude of the slip velocity of non-tracer particles with respect to the surrounding fluid is crucial to address both fundamental and practical questions involving dispersed turbulent flows. Here we derive an analytical model to predict the slip velocity of spherical particles in homogeneous isotropic turbulence. We modulate the particle equation of motion according to the inertial filtering framework, and obtain closed-form expressions for the mean slip velocity magnitude as a function of the governing parameters. These are compared against laboratory measurements and direct numerical simulations, demonstrating close agreement for both light and heavy particles, both smaller and larger than the Kolmogorov scales. The predictive value of the model and its implications are discussed, as well as the range of validity of the underlying assumptions.

Tom et al. (J. Fluid Mech., vol. 947, 2022, p. A7) investigated the impact of two-way coupling (2WC) on particle settling velocities in turbulence. For the limited parameter choices explored, it was found that (i) 2WC substantially enhances particle settling compared with the one-way coupled case, even at low mass loading $\varPhi _m$ and (ii) preferential sweeping remains the mechanism responsible for the particles settling faster than the Stokes settling velocity in 2WC flows. However, significant alterations to the flow structure that can occur at higher mass loadings mean that the conclusions from Tom et al. (J. Fluid Mech., vol. 947, 2022, p. A7) may not generalise. Indeed, even under very low mass loadings, the influence of 2WC on particle settling might persist, challenging the conventional assumption. We therefore explore a much broader portion of the parameter space, with simulations covering cases where the impact of 2WC on the global fluid statistics ranges from negligible to strong. We find that, even for $\varPhi _m=7.5\times 10^{-3}$, 2WC can noticeably increase the settling for some choices of the Stokes and Froude numbers. When $\varPhi _m$ is large enough for the global fluid statistics to be strongly affected, we show that preferential sweeping continues to be the mechanism that enhances particle settling rates. Finally, we compare our results with previous numerical and experimental studies. While in some cases there is reasonable agreement, discrepancies exist even between different numerical studies and between different experiments. Future studies must seek to understand this before the discrepancies between numerical and experimental results can be adequately addressed.

Many turbulent flows exhibit time-periodic statistics. These include turbomachinery flows, flows with external harmonic forcing and the wakes of bluff bodies. Many existing techniques for identifying turbulent coherent structures, however, assume the statistics are statistically stationary. In this paper, we leverage cyclostationary analysis, an extension of the statistically stationary framework to processes with periodically varying statistics, to generalize the spectral proper orthogonal decomposition (SPOD) to the cyclostationary case. The resulting properties of the cyclostationary SPOD (CS-SPOD for short) are explored, a theoretical connection between CS-SPOD and the harmonic resolvent analysis is provided, simplifications for the low and high forcing frequency limits are discussed, and an efficient algorithm to compute CS-SPOD with SPOD-like cost is presented. We illustrate the utility of CS-SPOD using two example problems: a modified complex linearized Ginzburg–Landau model and a high-Reynolds-number turbulent jet.

The wake of two tandem square cylinders of identical width (d) is experimentally studied, with a view to understanding the dependence of the flow structure, aerodynamics forces and Strouhal number on the centre-to-centre spacing ratio L/d and Reynolds number Re, where L is the distance between the cylinder centres. Extensive measurements are carried out, using hot-wire, particle imaging velocimetry, laser-induced fluorescence flow visualization, surface-oil-flow visualization and surface pressure scanning techniques, for L/d = 1.0 ~ 5.0 and Re ≡ U∞d/ν = 2.8 × (103 ~ 104), where U∞ is the free-stream velocity and ν is the kinematic viscosity of the fluid. The flow is classified into four regimes, i.e. the extended-body (L/d ≤ 1.5–2.0), reattachment (1.5–2.0 < L/d < 2.7–3.2), co-shedding (L/d ≥ 3.0–3.4) and transition (2.7 ≤ L/d ≤ 3.3) where both reattachment and co-shedding phenomena may take place. The mean drag and fluctuating drag and lift exhibit distinct features for different flow regimes, which is fully consistent with the proposed flow classification. Comparison is made between this flow and the wake of two tandem circular cylinders, which provides valuable insight into the profound effect of the flow separation point and the presence of sharp corners on the flow development and classification.

Despite decades of research, a universal method for prediction of roughness-induced skin friction in a turbulent flow over an arbitrary rough surface is still elusive. The purpose of the present work is to examine two possibilities; first, predicting equivalent sand-grain roughness size $k_s$ based on the roughness height probability density function and power spectrum (PS) leveraging machine learning as a regression tool; and second, extracting information about relevance of different roughness scales to skin-friction drag by interpreting the output of the trained data-driven model. The model is an ensemble neural network (ENN) consisting of 50 deep neural networks. The data for the training of the model are obtained from direct numerical simulations (DNS) of turbulent flow in plane channels over 85 irregular multi-scale roughness samples at friction Reynolds number $Re_\tau =800$. The 85 roughness samples are selected from a repository of 4200 samples, covering a wide parameter space, through an active learning (AL) framework. The selection is made in several iterations, based on the informativeness of samples in the repository, quantified by the variance of ENN predictions. This AL framework aims to maximize the generalizability of the predictions with a certain amount of data. This is examined using three different testing data sets with different types of roughness, including 21 surfaces from the literature. The model yields overall mean error 5 %–10 % on different testing data sets. Subsequently, a data interpretation technique, known as layer-wise relevance propagation, is applied to measure the contributions of different roughness wavelengths to the predicted $k_s$. High-pass filtering is then applied to the roughness PS to exclude the wavenumbers identified as drag-irrelevant. The filtered rough surfaces are investigated using DNS, and it is demonstrated that despite significant impact of filtering on the roughness topographical appearance and statistics, the skin-friction coefficient of the original roughness is preserved successfully.

Proper orthogonal decomposition (POD) is applied to three-dimensional (3-D) velocity fields collected from large-eddy simulations (LES) of a baffled stirred tank. In the LES, the tank operates with a Rushton-type impeller under turbulent conditions (at least in the near-impeller region) and the working fluid exhibits either Newtonian or shear-thinning rheology. The most energetic POD modes are analysed, and a POD reconstruction based on the higher modes is proposed to approximate the fluctuating component of the velocity field. Subsequently, the POD reconstruction is used to identify vortical structures and characterise them in terms of their shape. The structures are identified by considering a frame-invariant formulation of a popular, Eulerian, local-region-type method: the $Q$-criterion. Statistics of shape-related parameters are then investigated to address the morphology of the structures. It is found that: (i) regardless of the working fluid rheology, it seems feasible to decompose the 3-D field into its mean, most energetic periodic and fluctuating components using POD, allowing, for instance, reduced-order modelling of the energetic periodic motions for mixing enhancement purposes, and (ii) vortical structures related to turbulence are mostly tubular. Finding (ii) implies that, as starting point, phenomenological models for the interaction between fluid particles (drops and bubbles) and vortices should consider the latter as cylindrical structures rather than of spherical shape, as classically assumed in these models.

In this paper, we study the aeroacoustic instability which occurs in a deep axisymmetric cavity in a turbulent pipe flow. This phenomenon is the axisymmetric counterpart of the classical whistling of a rectangular deep cavity subject to a grazing flow. The whistling of such axisymmetric cavity originates from the interaction of the coherent fluctuations of the vorticity at the cavity's opening with one of its trapped azimuthal or radial acoustic modes. We focus here on the situation involving the first pure azimuthal mode, which is trapped in the cavity. As a consequence of the rotational symmetry of the configuration, azimuthal modes are actually pairs of degenerate eigenmodes, or almost degenerate in the presence of small asymmetries. Therefore, the aeroacoustic instabilities exhibit more complex mechanisms than in the case of a rectangular deep cavity. In particular, we show that self-sustained spinning modes induce a symmetry breaking of the mean flow and we will elucidate the details of this phenomenon. To that end, simultaneous acoustic and time-resolved stereoscopic particle image velocimetry (PIV) measurements are performed. They reveal that when large-amplitude aeroacoustic waves spin around the cavity, a quasi-steady mean flow starts whirling slowly in the opposite direction to the wave propagation. A linear perturbation analysis around an axisymmetric mean flow confirms the experimental observations: although the incoming pipe flow is not swirling, the hydrodynamic component of the aeroacoustic wave induces such whirling motion of the mean flow because of the forcing from the steady part of the coherent Reynolds stress tensor.

We propose an experimental study on the gravitational settling velocity of dense, sub-Kolmogorov inertial particles under different background turbulent flows. We report phase Doppler particle analyser measurements in a low-speed wind tunnel uniformly seeded with micrometre scale water droplets. Turbulence is generated with three different grids (two consisting of different active-grid protocols while the third is a regular static grid), allowing us to cover a very wide range of turbulence conditions in terms of Taylor-scale-based Reynolds numbers ($Re_\lambda \in [30\unicode{x2013}520]$), Rouse numbers ($Ro \in [0\unicode{x2013}5]$) and volume fractions ($\phi _v \in [0.5\times 10^{-5}\unicode{x2013}2.0\times 10^{-5}]$). We find, in agreement with previous works, that enhancement of the settling velocity occurs at low Rouse number, while hindering of the settling occurs at higher Rouse number for decreasing turbulence energy levels. The wide range of flow parameters explored allowed us to observe that enhancement decreases significantly with the Taylor–Reynolds number and is significantly affected by the volume fraction $\phi _v$. We also studied the effect of large-scale forcing on settling velocity modification. The possibility of changing the inflow conditions by using different grids allowed us to test cases with fixed $Re_\lambda$ and turbulent intensity but with different integral length scale. Finally, we assess the existence of secondary flows in the wind tunnel and their role on particle settling. This is achieved by characterising the settling velocity at two different positions, the centreline and close to the wall, with the same streamwise coordinate.

We use experiments to explore the effect of surfactants on bubble-induced turbulence (BIT) at different scales, considering how the bubbles affect the flow kinetic energy, anisotropy and extreme events. To this end, high-resolution particle shadow velocimetry measurements are carried out in a bubble column in which the flow is generated by bubble swarms rising in water for two different bubble diameters (3 and 4 mm) and moderate gas volume fractions (0.5 %–1.3 %). We use tap water as the base liquid and add 1-Pentanol as an additional surfactant with varying bulk concentration, leading to different bubble shapes and surface boundary conditions. The results reveal that with increasing surfactant concentration, the BIT generated increases in strength, even though bubbles of a given size rise more slowly with surfactants. We also find that the level of anisotropy in the flow is enhanced with increasing surfactant concentration for bubbles of the same size, and that for the same surfactant concentration, smaller bubbles generate stronger anisotropy in the flow. Concerning the intermittency quantified by the normalized probability density functions of the fluid velocity increments, our results indicate that extreme values in the velocity increments become more probable with decreasing surfactant concentration for cases with smaller bubbles and low gas void fraction, while the effect of the surfactant is much weaker for cases with larger bubble and higher void fractions.

This work aims at studying the mechanisms behind the occurrence of extreme dissipation events in a channel flow, identifying nonlinear optimal perturbations as potential precursors of these events. Nonlinear optimal perturbations with respect to a generic turbulent instantaneous snapshot are computed for the first time using a direct-adjoint algorithm in the channel flow at $Re_{\tau }\approx 180$. The resulting initial perturbation displays the upstream tilting characteristic of Orr's mechanism and is positioned along the interfaces between two opposite-sign velocity streaks of the pre-existing turbulent field. Such a perturbation induces a sudden breakdown of the pre-existing structures and a heavier tail in the dissipation probability density function distribution. Different mechanisms are at play during this process: the high shear present at the interface between coherent low- and high-momentum regions is exploited to break down the larger structures and drive energy to small scales. This energy cascade is fed by an enhanced lift-up effect that produces intense streaks near the wall. It is found that the optimal perturbation grows exponentially during the first phase of its evolution reflecting the existence of a secondary modal instability of the streaks. To corroborate the results, the conditional spatiotemporal proper orthogonal decomposition (POD) analysis of Hack & Schimdt (J. Fluid Mech., vol. 907, 2021, A9) is performed both in the perturbed and in the unperturbed flow, showing a clear agreement between the two cases and with the reference study. Thus, the optimal perturbation at initial time can be considered as a precursor of extreme events.

The equation for the fluid velocity gradient along a Lagrangian trajectory immediately follows from the Navier–Stokes equation. However, such an equation involves two terms that cannot be determined from the velocity gradient along the chosen Lagrangian path: the pressure Hessian and the viscous Laplacian. A recent model handles these unclosed terms using a multi-level version of the recent deformation of Gaussian fields (RDGF) closure (Johnson & Meneveau, Phys. Rev. Fluids, vol. 2 (7), 2017, 072601). This model is in remarkable agreement with direct numerical simulations (DNS) data and works for arbitrary Taylor Reynolds numbers $\textit {Re}_\lambda$. Inspired by this, we develop a Lagrangian model for passive scalar gradients in isotropic turbulence. The equation for passive scalar gradients also involves an unclosed term in the Lagrangian frame, namely the scalar gradient diffusion term, which we model using the RDGF approach. However, comparisons of the statistics obtained from this model with DNS data reveal substantial errors due to erroneously large fluctuations generated by the model. We address this defect by incorporating into the closure approximation information regarding the scalar gradient production along the local trajectory history of the particle. This modified model makes predictions for the scalar gradients, their production rates, and alignments with the strain-rate eigenvectors that are in very good agreement with DNS data. However, while the model yields valid predictions up to $\textit {Re}_\lambda \approx 500$, beyond this, the model breaks down.

Turbulence in the restricted nonlinear (RNL) dynamics is analysed and compared with direct numerical simulations (DNS) of Poiseuille turbulence at Reynolds number $R=1650$. The structures are obtained by proper orthogonal decomposition (POD) analysis of the two components of the flow partition used in RNL dynamics: the streamwise mean flow and fluctuations. POD analysis of the streamwise mean flow indicates that the dominant POD modes, in both DNS and RNL dynamics, are roll-streaks harmonic in the spanwise direction. However, we conclude that these POD modes do not occur in isolation but rather are Fourier components of a coherent roll-streak structure. POD analysis of the fluctuations in DNS and RNL dynamics reveals similar complex structures consisting in part of oblique waves collocated with the streak. The origin of these structures is identified by their correspondence to POD modes predicted using a stochastic turbulence model (STM). These predicted POD modes are dominated by the optimally growing structures on the streak, which the STM predicts correctly to be of sinuous oblique wave structure. This close correspondence between the roll-streak structure and the associated fluctuations in DNS, RNL dynamics and the STM implies that the self-sustaining mechanism operating in DNS is essentially the same as that in RNL dynamics, which has been associated previously with optimal perturbation growth on the streak.

Despite elegant theoretical descriptions and numerous potential applications in nature, the various features of wave turbulence and its associated transfers across scales are still debated. One reason is the difficulty, both experimentally and numerically, of studying a chaotic nonlinear wavy system with sufficient precision, over a sufficient duration, and with sufficient separation between the injection scale, the system scale and the dissipation scale, to dwell in detail on the description of a statistically converged state. The numerical study of Thomas & Ding (J. Fluid Mech., 2023, this issue) sheds new light on one key aspect of this global problem: the upscale transfer of waves in rotating shallow water equations, which is of relevance to atmosphere and ocean dynamics. Their results first confirm the theoretical predictions, with a robust inverse wave cascade, a predominant role of quartic resonances and a slope value of the wave spectrum in agreement with the expected range. But their deeper analysis of the results also highlights some weakness in the theoretical foundations, exhibiting strong intermittency and departure from ideal Gaussian statistics. This work thus calls for improved modelling, acknowledging that important large-scale climatic features could actually result from the piling up of small-scale waves, unresolved by typical Earth system models. Beyond rotating shallow water systems, this study more generically resonates with open questions in the field of wave turbulence and its geophysical applications, including recent research in deep rotating and stably stratified systems.

Exact budget equations are derived for the coherent and stochastic contributions to the second-order structure function tensor. They extend the anisotropic generalised Kolmogorov equations (AGKE) by considering the coherent and stochastic parts of the Reynolds stress tensor, and are useful for the statistical description of turbulent flows with periodic or quasi-periodic features, like, for example, the alternate shedding after a bluff body. While the original AGKE describe production, transport, inter-component redistribution and dissipation of the Reynolds stresses in the combined space of scales and positions, the new equations, called $\varphi$AGKE, contain the phase $\varphi$ as an additional independent variable, and describe the interplay among the mean, coherent and stochastic fields at the various phases. The newly derived $\varphi$AGKE are then applied to a case where an exactly periodic external forcing drives the flow: a turbulent plane channel flow modified by harmonic spanwise oscillations of the wall to reduce drag. The phase-by-phase action of the oscillating transversal Stokes layer generated by the forcing on the near-wall turbulent structures is observed, and a detailed description of the scale-space interaction among mean, coherent and stochastic fields is provided thanks to the $\varphi$AGKE.