Serrations are commonly employed to mitigate the turbulent boundary layer trailing-edge noise. However, significant discrepancies persist between model predictions and experimental observations. In this paper, we show that this results from the frozen turbulence assumption. A fully developed turbulent boundary layer over a flat plate is first simulated using the large-eddy simulation method, with the turbulence at the inlet generated using the digital filter method. The space–time correlations and spectral characteristics of wall pressure fluctuations are examined. The simulation results demonstrate that the coherence function decays in the streamwise direction, deviating from the constant value of unity assumed in the frozen turbulence assumption. By considering an exponential decay function, we relax the frozen turbulence assumption and develop a prediction model that accounts for the intrinsic non-frozen nature of turbulent boundary layers. To facilitate a direct comparison with frozen models, a correction coefficient is introduced to account for the influence of non-frozen turbulence. The comparison between the new and original models demonstrates that the new model predicts lower noise reductions, aligning more closely with the experimental observations. The physical mechanism underlying the overprediction of the noise model assuming frozen turbulence is discussed. The overprediction is due to the decoherence of the phase variation along the serrated trailing edge. Consequently, the ratio of the serration amplitude to the streamwise frequency-dependent correlation length is identified as a crucial parameter in determining the correct prediction of far-field noise.

Large amounts of small inertial particles embedded in a turbulent flow are known to modify the turbulent statistics and structures, a phenomenon referred to as turbulence modulation. While particle electrification is ubiquitous in particle-laden turbulence and significantly alters particle behaviour, the effects of inter-particle electrostatic forces on turbulence modulation and the underlying physical mechanisms remain unclear. To fill this gap, we perform a series of point-particle direct numerical simulations of turbulent channel flows at a friction Reynolds number of approximately 540, laden with uncharged and charged bidisperse particles. The results demonstrate that, compared to flows laden with uncharged particles, the presence of inter-particle electrostatic forces leads to substantial changes in both turbulent intensities and structures. In particular, the inner-scaled mean streamwise fluid velocity is found to shift towards lower values, indicating a noticeable increase in fluid friction velocity. Turbulent intensities appear to be further suppressed through facilitating the particles to extract momentum from the fluid phase and increasing extra turbulent kinetic dissipation by particles. Importantly, the overall drag is enhanced by indirectly strengthening the contribution of particle stress, even though the contribution of the total fluid stress is decreased. On the other hand, the magnitude of the large-scale motions is weakened by simultaneously reducing turbulent production and increasing particle feedback around the scales of the large-scale motions. Meanwhile, the average streaky fluid structures in the streamwise–spanwise planes and inclined fluid structures in the streamwise–wall-normal planes become expanded and flattened, respectively.

We consider the dynamic wetting and dewetting processes of films and droplets of complex liquids on planar surfaces, focusing on the case of colloidal suspensions, where the particle interactions can be sufficiently attractive to cause agglomeration of the colloids within the film. This leads to an interesting array of dynamic behaviours within the liquid and of the liquid–air interface. Incorporating concepts from thermodynamics and using the thin-film approximation, we construct a model consisting of a pair of coupled partial differential equations that represent the evolution of the liquid film and the effective colloidal height profiles. We determine the relevant phase behaviour of the uniform system, including finding associated binodal and spinodal curves, helping to uncover how the emerging behaviour depends on the particle interactions. Performing a linear stability analysis of our system enables us to identify parameter regimes where agglomerates form, which we independently confirm through numerical simulations and continuation of steady states, to construct bifurcation diagrams. We obtain various dynamics such as uniform colloidal profiles in an unstable situation evolving into agglomerates and thus elucidate the interplay between dewetting and particle aggregation in complex liquids on surfaces.

High-dimensional dynamical systems projected onto a lower-dimensional manifold cease to be deterministic and are best described by probability distributions in the projected state space. Their equations of motion map onto an evolution operator with a deterministic component, describing the projected dynamics, and a stochastic one representing the neglected dimensions. This is illustrated with data-driven models for a moderate-Reynolds-number turbulent channel. It is shown that, for projections in which the deterministic component is dominant, relatively ‘physics-free’ stochastic Markovian models can be constructed that mimic many of the statistics of the real flow, even for fairly crude operator approximations, and this is related to general properties of Markov chains. Deterministic models converge to steady states, but the simplified stochastic models can be used to suggest what is essential to the flow and what is not.

A simulation method has been developed to efficiently evaluate the motion of colloidal particles in a low-Reynolds-number confined microchannel flow using a Lagrangian-based approach. In this method, the background velocity within the channel, in the absence of suspended particles, is obtained from a fluid dynamics solver and is used to update the velocity at the particle centres using the Stokesian dynamics (SD) method, which incorporates multi-body hydrodynamic interactions. As a result, instead of computing the momentum of both the fluid and particles throughout the entire computational domain, the microscopic balance equation is solved only at the particle centres, increasing the computational efficiency. To accommodate complex boundary conditions within the SD framework, imaginary particles are placed on the channel walls, allowing the mobility relation to be reformulated to apply velocity constraints to immobilized wall particles. By employing this constrained SD approach, global mobility interactions that need to be computed at each time step are limited to the interior particles, resulting in a significant reduction in computational cost. The efficiency of this study is demonstrated through case studies on particulate flows in contraction and cross-flow microchannels. By using colloidal particles that incorporate Brownian motion and inter-particle attraction, observations through the entire stages of fouling dynamics are possible, from particle inflow to channel blockage. The fouling patterns observed in the simulations are consistent with experiments conducted under the same flow conditions. This study provides an efficient approach for analysing the effect of hydrodynamic interactions on particle dynamics in microfluidics and materials processing fields while allowing for predictions about structural changes over long-time scales, including complex phenomena such as clogging.

The clustering of debris floating on liquid interfaces such as water surfaces is a complex phenomenon that finds its applications in numerous examples from industrial processing and environmental systems. The recent paper by Shin & Coletti (J. Fluid Mech., vol. 984, 2024, R7) presents an experimental campaign investigating the three effects of turbulence, particle interactions and interfacial effects, to elucidate how the three force scales drag, capillary forces and lubrication give rise to three distinct regimes of clustering in dense suspensions. The study, hence, provides a useful systematic to categorize the clustering mechanisms. As an important finding, it is shown that, depending on volume fraction and non-dimensional turbulent shear, particles either tend to cluster into aggregate sizes larger than the Kolmogorov scale or can break into pieces that are as small as the primary particle size.

Shock–droplet interaction in the early stage involves intricate wave structures. Investigating this phenomenon is inherently challenging due to the fine spatial and temporal scales involved. Past research has suggested that the occurrence of cavitation, marked by a negative peak pressure, is linked to the focus of the reflected expansion wave. In this study, a high-fidelity compressible numerical approach is utilized to replicate the initial phase of shock–droplet interactions. The location of the negative peak pressure is meticulously documented and compared with experimental measurement and numerical results. Results indicate a strong alignment between the negative peak pressure positions identified through numerical simulations and the focal points identified in theoretical models for low gas–liquid wave velocity ratios. However, this alignment is notably disrupted when dealing with higher gas–liquid wave velocity ratios. Further enhancements are made to the theoretical model, enabling a more precise depiction of internal wave structures and focus points, particularly under conditions of high gas–liquid wave velocity ratios. The study delves into the various factors influencing internal pressure fluctuations within the liquid droplet, categorizing them into four phases: the shock wave effect, relaxation effect, fluctuation effect, and expansion wave effect. Analysing the pressure decrease portion reveals that while the converging of the reflected expansion wave leads to a substantial pressure drop, it accounts for only a fraction of the total pressure variation. Consequently, any model predicting negative peak pressure positions must comprehensively consider all contributing factors.

The influence of free-stream conicity on the various aspects of the flow over a spherical test model is examined using both analytical and numerical methods. For the analytical method, a simple closed-form analytical model is assembled. Six different free-stream conditions with different Mach numbers, Reynolds numbers and thermochemistry are tested at four different degrees of conicity corresponding to those which can realistically be encountered in experiments. It is found that the results around the stagnation point are mostly insensitive to the flow condition and gas type, except for some mild non-equilibrium effects, and excellent agreement between the analytical and numerical results exists. The shock stand-off distance on the stagnation streamline is shown to decrease with increasing conicity. This decrease increases the tangential velocity gradient at the stagnation point, increasing the stagnation point heat flux and decreasing the stagnation point boundary layer thickness. The free-stream conicity is also found to alter the normalized distributions of the shock stand-off distance, heat flux, surface pressure and boundary layer thickness with the angle from the stagnation point. In general, increasing the conicity magnifies the slope of these distributions. Regarding the boundary layer transition, it is found that, if it occurs in a uniform free stream, it would also occur in a conical free stream, albeit with the transition point shifted upstream closer to the stagnation point due to the increase in the boundary layer edge tangential velocity. Overall, considering the relevant experimental uncertainties, corrections for free-stream conicity are generally recommended when larger test models are used.

Particle segregation in dense flowing size-disperse granular mixtures is driven by gravity and shear, but predicting the associated segregation force due to both effects has remained an unresolved challenge. Here, a model of the combined gravity- and kinematics-induced segregation force on a single intruder particle is integrated with a model of the concentration dependence of the gravity-induced segregation force. The result is a general model of the net particle segregation force in flowing size-bidisperse granular mixtures. Using discrete element method simulations for comparison, the model correctly predicts the segregation force for a variety of mixture concentrations and flow conditions in both idealized and natural shear flows.

To date, a growing body of literature has documented the existence and impacts of coherent structures known as large- and very-large-scale motions within wall-bounded turbulent flows under neutral and unstable thermal stratification. These coherent structures can account for a considerable fraction of the overall turbulent transport and have been found to modulate small-scale turbulent fluctuations near the wall. In the context of stably stratified flows, however, the examination of such coherent structures has garnered relatively little attention. Stable stratification limits vertical transport and turbulent mixing within flows, which makes it unclear the extent to which previous findings on coherent structures under unstable and neutral stratification are applicable to stably stratified flows. In this study, we investigate the existence and characteristics of coherent structures under stable stratification with a wide range of statistical and spectral analyses. Outer peaks in premultiplied spectrograms under weak stability indicate the presence of large-scale motions, but these peaks become weaker and eventually vanish with increasing stability. Quadrant analysis of turbulent transport efficiencies (the ratio of net fluxes to their respective downgradient components) demonstrates dependencies on both stability and height above ground, which is evidence of morphological differences in the coherent structures under increasing stability. Amplitude modulation by large-scale streamwise velocity was found to decrease with increasing gradient Richardson number, whereas modulation by large-scale vertical velocity was approximately zero across all stability ranges. For sufficiently stable stratification, large eddies are suppressed enough to limit any inner–outer scale interactions.

Both experimental and theoretical studies of fast and microscale physical phenomena occurring during the growth of vapour bubbles in nucleate pool boiling are reported. The focus is on the liquid film of micrometric thickness (a ‘microlayer’) that can form between the heater and the liquid–vapour interface of a bubble. The microlayer strongly affects the macroscale heat transfer and is thus important to be understood. The microlayer appears as a result of the inertial forces that cause the hemispherical bubble shape. It is shown that the microlayer can be seen as the Landau–Levich film deposited by the bubble foot edge during its receding. Paradoxically, the deposition is controlled by viscosity and surface tension. The microlayer profile measured with white-light interferometry, the temperature distribution over the heater, and the bubble shape are observed with synchronised high-speed cameras. According to the numerical simulations, the microlayer consists of two regions: a dewetting ridge near the contact line, and a longer and flatter bumped part. It is shown that the ridge cannot be measured by interferometry because of its intrinsic limitation on the interface slope. The ridge growth is linked to the contact line receding. The simulated dynamics of both the bumped part and the contact line agrees with the experiment. The physical origin of the bump in the flatter part of microlayer is explained.

We study the behaviour of the streamwise velocity variance in turbulent wall-bounded flows using a direct numerical simulation (DNS) database of pipe flow up to friction Reynolds number ${{Re}}_{\tau } \approx 12000$. The analysis of the spanwise spectra in the viscous near-wall region strongly hints to the presence of an overlap layer between the inner- and the outer-scaled spectral ranges, featuring a $k_{\theta }^{-1+\alpha }$ decay (with $k_{\theta }$ the wavenumber in the azimuthal direction, and $\alpha \approx 0.18$), hence shallower than suggested by the classical formulation of the attached-eddy model. The key implication is that the contribution to the streamwise velocity variance $(\langle{u}^2\rangle)$ from the largest scales of motion (superstructures) slowly declines as ${{Re}}_{\tau }^{-\alpha }$, and the integrated inner-scaled variance follows a defect power law of the type $\langle u^2 \rangle ^+ = A - B \, {{Re}}_{\tau }^{-\alpha }$, with constants $A$ and $B$ depending on $y^+$. The DNS data very well support this behaviour, which implies that strict wall scaling is restored in the infinite-Reynolds-number limit. The extrapolated limit distribution of the streamwise velocity variance features a buffer-layer peak value of $\langle u^2 \rangle ^+ \approx 12.1$, and an additional outer peak with larger magnitude. The analysis of the velocity spectra also suggests a similar behaviour of the dissipation rate of the streamwise velocity variance at the wall, which is expected to attain a limiting value of approximately $0.28$, hence slightly exceeding the value $0.25$ which was assumed in previous analyses (Chen & Sreenivasan, J. Fluid Mech., vol. 908, 2021, R3). We have found evidence suggesting that the reduced near-wall influence of wall-attached eddies is likely linked to the formation of underlying turbulent Stokes layers.

Immersed nonlinear elements are prevalent in biological systems that require a preferential flow direction, such as the venous and the lymphatic system. We investigate here a certain class of models where the fluid is driven by peristaltic pumping and the nonlinear elements are ideal valves that completely suppress backflow. This highly nonlinear system produces discontinuous solutions that are difficult to study. We show that, as the density of valves increases, the pressure and flow are well approximated by a continuum of valves which can be analytically treated, and we demonstrate through numeric simulation that the approximation works well even for intermediate valve densities. We find that the induced flow is linear in the peristaltic amplitude for small peristaltic forces and, in the case of sinusoidal peristalsis, is independent of pumping direction. Despite the continuum approximation used, the physical valve density is accounted for by modifying the resistance of the fluid appropriately. The suppression of backflow causes a net benefit in adding valves when the valve density is low, but once the density is high enough, valves predominately suppress forward flow, suggesting there is an optimum number of valves per wavelength. The continuum model for peristaltic pumping through an array of valves presented in this work can eventually provide insights about the design and operating principles of complex flow networks with a broad class of nonlinear elements.

Three-dimensional direct numerical simulations of rotating Rayleigh–Bénard convection in the planar geometry with no-slip top and bottom and periodic lateral boundary conditions are performed for a broad parameter range with the Rayleigh number spanning in $5\times 10^{6}\leq Ra \leq 5\times 10^{13}$, Ekman number within $5\times 10^{-9}\leq Ek \leq 5\times 10^{-5}$ and Prandtl number $Pr=1$. The thermal and Ekman boundary layer (BL) statistics, temperature drop within the thermal BL, interior temperature gradient and scaling behaviours of the heat and momentum transports (reflected in the Nusselt $Nu$ and Reynolds numbers $Re$) as well as the convective length scale are investigated across various flow regimes. The global and local momentum transports are examined via the $Re$ scaling derived from the classical theoretical balances of viscous–Archimedean–Coriolis (VAC) and Coriolis–inertial–Archimedean (CIA) forces. The VAC-based $Re$ scaling is shown to agree well with the data in the cellular and columnar regimes, where the characteristic convective length scales as the onset length scale ${\sim } Ek^{1/3}$, while the CIA-based $Re$ scaling and the inertia length scale $\sim (ReEk)^{1/2}$ work well in the geostrophic turbulence regime for $Ek\leq 1.5\times 10^{-8}$. The examinations of $Nu$, global and local $Re$, and convective length scale as well as the temperature drop within the thermal BL and its thickness scaling behaviours, indicate that for extreme parameters of $Ek\leq 1.5\times 10^{-8}$ and $80\lesssim RaEk^{4/3}\lesssim 200$, we have reached the diffusion-free geostrophic turbulence regime.

Two-dimensional (2-D) and three-dimensional (3-D) direct numerical simulations are conducted for flow past rectangular cylinders with various cross-sectional aspect ratios. The primary focuses are the interactions between the 2-D wake transitions in the spanwise vortex street (with distance downstream) and the 3-D wake transitions in the streamwise vortices, and the influence of both 2-D and 3-D wake transitions on the hydrodynamic forces on the cylinder. The 2-D wake transitions generally move upstream with increasing Reynolds number and decreasing aspect ratio. The corresponding reasons are explained. The 2-D wake transitions emerging close to the cylinder may directly alter the hydrodynamic forces on the cylinder, e.g. the Strouhal number, time-averaged drag coefficient and root-mean-square lift coefficient. By using specifically designed numerical cases to decompose the effects of the two 2-D transitions, it is found that the first 2-D transition from the primary to the two-layered vortex street results in reductions in the hydrodynamic forces, while the second 2-D transition to the secondary vortex street results in increases in the forces. The reduction/increase in the hydrodynamic forces becomes more significant when the transition location moves closer to the cylinder. The physical mechanisms for the influence on the hydrodynamic forces are elucidated. The upstream movement of the 2-D wake transitions also induces complex interactions between the 2-D and 3-D wake transitions (which also depends on the type of the 3-D mode). Correspondingly, the 3-D hydrodynamic forces may be governed by both 2-D and 3-D wake transitions (and their mutual influence).

Flexible canopy flows are often encountered in natural scenarios, e.g. when crops sway in the wind or when submerged kelp forests are agitated by marine currents. Here, we provide a detailed characterisation of the turbulent flow developed above and between the flexible filaments of a fully submerged dense canopy and we describe their dynamical response to the turbulent forcing. We investigate a wide range of flexibilities, encompassing the case in which the filaments are completely rigid and standing upright as well as that where they are fully compliant to the flow and deflected in the streamwise direction. We are thus able to isolate the effect of the canopy flexibility on the drag and on the inner–outer flow interactions, as well as the two flapping regimes of the filaments already identified for a single fibre. Furthermore, we offer a detailed description of the Reynolds stresses throughout the wall-normal direction resorting to the Lumley triangle formalism, and we show the multi-layer nature of turbulence inside and above the canopy. The relevance of our investigation is thus twofold: the fundamental physical understanding developed here paves the way towards the investigation of more complex and realistic scenarios, while we also provide a thorough characterisation of the turbulent state that can prove useful in the development of accurate turbulence models for RANS and LES.

The scale-dependent variability of convective velocities and structure inclination angles in wall-bounded turbulence was studied experimentally via space–time energy spectrum measurements. We found that the variability of convection velocities for large-scale motions (LSMs) decreased inversely with streamwise wavenumbers, and that the variability trend was not fully explained by earlier applications of Kraichnan's ‘random-sweeping’ model of turbulence that assumed perfect scale separation. By analytically extending the random-sweeping model to allow for a dominant large scale in the random-sweeping signal that can interact with other LSMs, we showed how scale interactions can explain the variability trend in convection velocities for LSMs. The variability in convection velocities was also shown to correlate with the scale-dependent inclination angles of coherent structures that were obtained via cross-spectral analysis. Large-scale motions tended to exhibit shallower inclination to the wall with increasing convection velocity, while small-scale motions and those far from the wall exhibited the reverse behaviour. We proposed that these two opposite relationships between inclination angle and convection velocity can be explained in terms of a balance between opposing effects of the mean shear and the coherent structure geometry. Descriptions and models of convection velocity variability effects are useful both for modelling turbulence spectra and explaining the geometry of coherent structures.

We study the stationary Navier–Stokes equations in the region between two rotating concentric cylinders. We first prove that, for a small Reynolds number, if the fluid flow is axisymmetric and if its velocity is sufficiently small in the $L^\infty$-norm, then it is necessarily the Taylor–Couette–Poiseuille flow. If, in addition, the associated pressure is bounded or periodic in the $z$ axis, then it coincides with the well-known canonical Taylor–Couette flow. We discuss the relation between uniqueness and stability of such a flow in terms of the Taylor number in the case of narrow gap of two cylinders. The investigation in comparison with two Reynolds numbers based on inner and outer cylinder rotational velocities is also conducted. Next, we give a certain bound of the Reynolds number and the $L^\infty$-norm of the velocity such that the fluid is, indeed, necessarily axisymmetric. As a result, it is clarified that smallness of Reynolds number of the fluid in the two rotating concentric cylinders governs both axisymmetry and the Taylor–Couette–Poiseuille flow with the exact form of the pressure.

We investigate Reynolds number effects in strong shock-wave/turbulent boundary-layer interactions (STBLI) by leveraging a new database of wall-resolved and long-integrated large-eddy simulations. The database encompasses STBLI with massive boundary-layer separation at Mach $2.0$, impinging-shock angle $40^{\circ }$ and friction Reynolds numbers ${\textit {Re}}_\tau$ $355$, $1226$ and $5118$. Our analysis shows that the shape of the reverse-flow bubble is notably different at low and high Reynolds number, while the mean-flow separation length, separation-shock angle and incipient plateau pressure are rather insensitive to Reynolds number variations. Velocity statistics reveal a shift in the peak location of the streamwise Reynolds stress from the separation-shock foot to the core of the detached shear layer at high Reynolds number, which we attribute to increased pressure transport in the separation-shock excursion domain. Additionally, in the high Reynolds case, the separation shock originates deep within the turbulent boundary, resulting in intensified wall-pressure fluctuations and spanwise variations associated with the passage of coherent velocity structures. Temporal spectra of various signals show energetic low-frequency content in all cases, along with a distinct peak in the bubble-volume spectra at a separation-length-based Strouhal number $St_{L_{sep}}\approx 0.1$. The separation shock is also found to lag behind bubble-volume variations, consistent with the acoustic propagation time from reattachment to separation and a downstream mechanism driving the shock motion. Finally, dynamic mode decomposition of three-dimensional fields suggests a Reynolds-independent statistical link among separation-shock excursions, velocity streaks and large-scale vortices at low frequencies.