Wave propagation in channels with area changes is a topic of significant practical interest that involves a rich set of coupled physics. While the acoustic wave problem has been studied extensively, the shock propagation problem has received less attention. In addition to its practical significance, this problem also introduces deep fundamental issues associated with how energy in propagating large-amplitude disturbances is redistributed upon interaction with inhomogeneities. This paper presents a study of shock scattering and entropy and vorticity coupling for shock wave propagation through discrete area changes. It compares results from computational fluid dynamics to those of one-dimensional quasi-steady calculations. The solution space is naturally divided into five ‘regimes’ based upon the incident shock strength and area ratio. This paper also presents perturbation methods to quantify the dimensionless scaling of physical effects associated with wave reflection/transmission and energy transfer to other disturbances. Finally, it presents an analysis of the ‘energetics’ of the interaction, quantifying how energy that initially resides in dilatational disturbances and propagates at the shock speed is redistributed into finite-amplitude reflected and transmitted waves as well as convecting vortical and entropy disturbances.

We introduce a wall model for large-eddy simulation (WMLES) applicable to rough surfaces with Gaussian and non-Gaussian distributions for both the transitionally and fully rough regimes. The model is applicable to arbitrary complex geometries where roughness elements are assumed to be underresolved, i.e. subgrid-scale roughness. The wall model is implemented using a multi-hidden-layer feedforward neural network, with the mean geometric properties of the roughness topology and near-wall flow quantities serving as input. The optimal set of non-dimensional input features is identified using information theory, selecting variables that maximize information about the output while minimizing redundancy among inputs. The model also incorporates a confidence score based on Gaussian process modelling, enabling the detection of potentially low model performance for untrained rough surfaces. The model is trained using a direct numerical simulation (DNS) roughness database comprising approximately 200 cases. The roughness geometries for the database are selected from a large repository through active learning. This approach ensures that the rough surfaces incorporated into the database are the most informative, achieving higher model performance with fewer DNS cases compared with passive learning techniques. The performance of the model is evaluated both apriori and aposteriori in WMLES of turbulent channel flows with rough walls. Over 550 channel flow cases are considered, including untrained roughness geometries, roughness Reynolds numbers and grid resolutions for both transitionally and fully rough regimes. Our rough-wall model offers higher accuracy than existing models, generally predicting wall shear stress within an accuracy range of 1%–15 %. The performance of the model is also assessed on a high-pressure turbine blade with two different rough surfaces. We show that the new wall model predicts the skin friction and the mean velocity deficit induced by the rough surface on the blade within 1%–10 % accuracy except the region with transition or shock waves. This work extends the building-block flow wall model (BFWM) introduced by Lozano-Durán & Bae (2023. J. Fluid Mech. 963, A35) for smooth walls, expanding the BFWM framework to account for rough-wall scenarios.

This paper presents a numerical investigation of the turbulence transition phenomenon in the wake of wall-mounted prisms. Large-eddy simulations are performed at $Re = 1\times 10^3 {-}5\times 10^3$ for prisms with a range of aspect ratio (height to width) from $0.25$ to $1.5$, and depth ratios (length to width) between $1$ and $4$. The results show that the wake irregularity is enhanced with increasing depth ratio, evidenced by higher turbulent kinetic energy (${\approx}90\,\%$) near the leading edge, and the onset of irregular, unsteady vortex shedding. This is attributed to interactions between Kelvin–Helmholtz instability (KHI) of the shear layer and large-scale vortex shedding, and it is induced by an unsteady shear layer, resembling flapping-like motion. These interactions elevate the flow momentum due to increased turbulence intensity and mixing, contributing to the wake transition phenomenon. To this end, this study defines the role of depth ratio in the transition phenomenon by showing that increasing depth ratio (e.g. from $1$ to $4$) leads to earlier onset of KHIs in the shear layer. These instabilities intensify with depth ratio, resulting in stronger interactions between shear layer and large-scale vortex shedding. Specifically, KHI-induced vortices interact more frequently with large-scale wake structures for higher depth ratio prisms, exciting larger flow fluctuations and irregular wake patterns. This interaction alters the frequency and coherence of vortex shedding, revealing a complex coupling mechanism that drives the transition to turbulence.

Drag reduction induced by a polydisperse solution of polyethylene oxide is investigated by direct numerical simulations of the Navier–Stokes equations coupled with the Lagrangian evolution of the polymers, modelled as dumbbells. Simulation parameters are chosen to match the experimental conditions of Berman (1977), who measured the polymer molecular weight distribution. Drag reduction is induced only by the few high molecular weight polymers fully stretched by the turbulent flow, whilst the hundreds of parts per million of low molecular weight chains are ineffective.

Experiments are presented to explore the non-axisymmetric instabilities of spreading films of aqueous suspensions of Carbopol and Xanthan gum floating on a bath of perfluoropolyether oil. The experimental observations are compared against theoretical predictions exploiting a shallow-film model in which the viscoplastic rheology is captured by the Herschel–Bulkley constitutive law. With this model, we construct axisymmetric base states that evolve from the moment that the film floats onto the bath, out towards long times at which spreading becomes self-similar, and then test their linear stability towards non-axisymmetric perturbations. In the geometry of a thinning expanding film, we find that shear thinning does not drive a loss of axisymmetry at early times (when the degree of expansion is small), but when the film has expanded in radius by a factor of two or so, shear-thinning hoop stresses drive non-axisymmetric instabilities. Unstable modes possess relatively low angular wavenumber, and the loss of symmetry is not particularly dramatic. When the oil in the bath is replaced by salty water, the experiments are completely different, with dramatic non-axisymmetric patterns emerging from interfacial effects.

We investigate the natural oscillations of sessile drops with a central trapped bubble on a plane using linear potential flow theory, considering both free and pinned contact lines. The system is governed by the contact angle $\alpha$ and the ratio $\tau$ of inner to outer contact line radii. For bubble-containing (BC) hemispherical drops with free contact lines (referred to as free BC semi-drops), the modes mirror half of those in concentric spherical BC drops due to plane symmetry. These modes are labelled ‘plus’ (with greater inner surface deformation) and ‘minus’ (with greater outer surface deformation). As $\tau \to 0$, minus modes converge to those of bubble-free drops. Results show that varying $\alpha$ from $90^\circ$ or pinning the contact line in free BC semi-drops alters the topology of spectral lines, turning original crossings of spectral lines between minus and plus modes into avoided crossings. This shift causes minus and plus modes to form spectral trends with avoided crossings, maintaining their original spectral shapes. In an avoided crossing, two coupled modes cannot be classified as plus or minus due to their comparable inner and outer surface deformations, resulting in mode beating when both are excited, as confirmed by our direct numerical simulations. This study on the impact of inner bubbles on the spectrum may help in predicting bubble size in opaque sessile drops.

The formation process of a vortex pair generated by a two-dimensional starting jet has been investigated numerically over a range of Reynolds numbers from 500 to 2000. The effects of stroke ratio and nozzle configuration are examined. Only a single vortex pair can be observed in the vorticity field generated by small stroke ratios less than 10 while the leading vortex pair formed by larger stroke ratios eventually disconnects from the trailing jet. The formation numbers (13.6 and 9.3) for a straight nozzle and an orifice nozzle have been identified by the circulation criterion and they are further analysed by four other criteria. Using the contraction coefficient, formation numbers can be transformed into a universal value at about 16.5 for both nozzles. The effect of Reynolds number on the formation number is found to be within 12 % for parallel flow cases but it will increase up to 27 % for non-parallel flow cases due to shear-layer instability. A modified contraction-based slug model is proposed, and it can accurately predict the total invariants (e.g. circulation, hydrodynamic impulse and kinetic energy) shedding from the nozzle edge. Analytical estimation of the formation number is further conducted by matching the predicted total invariants to the Pierrehumbert model of steady vortex pairs. By assuming that pinch-off starts when the vortex pair achieves the steady state, two analytical models are proposed in terms of vortex impulse and translational velocity. The latter appears to be more appropriate to predict the formation number for two-dimensional flows.

Turbulent transonic buffet is an aerodynamic instability causing periodic (albeit, often irregular) oscillations of lift/drag in aerospace applications. Involving complex coupling between inviscid and viscous effects, buffet is characterised by shock wave oscillations and flow separation/reattachment. Previous studies have identified both two-dimensional (2-D) chordwise shock-oscillation and three-dimensional (3-D) buffet-/stall-cell modes. While the 2-D instability has been studied extensively, investigations of 3-D buffet have been limited to only low-fidelity simulations or experiments. Due to computational cost, almost all high-fidelity studies to date have been limited to narrow span-widths around 5 % of aerofoil chord length (aspect ratio, ), which is insufficiently wide to observe large-scale three-dimensionality. In this work, high-fidelity simulations are performed up to , on an infinite unswept NASA Common Research Model (CRM) wing profile at $Re=5\times 10^{5}$. At , intermittent 3-D separation bubbles are observed at buffet conditions. While previous Reynolds-averaged Navier–Stokes (RANS)/stability-based studies predict quasi-simultaneous onset of 2-D- and 3-D-buffet, a case that remains essentially 2-D is identified here. Strongest three-dimensionality was observed near low-lift phases of the buffet cycle at maximum flow separation, reverting to essentially 2-D behaviour during high-lift phases. Buffet was found to become 3-D when extensive mean flow separation was present. At , multiple 3-D separation bubbles form in a spanwise wavelength range $\lambda =1c$ to $1.5c$. Spectral proper orthogonal decomposition (SPOD) was applied to analyse the spatio/temporal structure of 3-D buffet-cells. In addition to the 2-D chordwise shock-oscillation mode (Strouhal number $St \approx 0.07-0.1$), 3-D modal structures were observed at the shock wave/boundary layer interaction at $St \approx 0.002-0.004$.

Two-dimensional gaseous detonations near critical propagation state were studied numerically in a channel with stoichiometric H$_2$/air and H$_2$/O$_2$ mixtures. Detonation waves exhibit a mode-locking effect (MLE) in a single-headed mode regime. Increasing the channel width alters the strength and propagation period of the single transverse wave. This leads to MLE failure and the occurrence of the single-dual-headed critical mode, featuring the emergence of a new transverse wave. For a stoichiometric H$_2$/air mixture, generation of the new transverse wave is due to interactions between the detonation front and the local explosion wave originating from interactions between the transverse wave and unreacted gas pocket downstream. Whereas, for a stoichiometric H$_2$/O$_2$ mixture, a transverse wave interacting with the wall produces Mach reflection bifurcation, causing MLE failure and generation of the new transverse wave. Our results show that all transverse waves manifest as strong transverse wave (STW) structures, with most belonging to the second kind, and an acoustic coupling exists between the typical second kind of STW structure and the acoustic wave in the induction zone behind the Chapman–Jouguet detonation front. A high-pressure region close to the STW structure plays a crucial role in exploring the transverse dynamics of this structure. Shock polars with rational assumptions are adopted to predict flow states in this region. The roles of pivotal factors in influencing the flow states and wave structure are clarified, and characteristic pressure values derived adequately represent the STW structure’s transverse dynamic behaviours. Lastly, the relationship between the kinematics and kinds of STW structures is unveiled.

The broad-band direct combustion noise is an important problem for industrial and domestic burners. The power spectral density (PSD) of this noise is related to the local spectral density of fluctuating heat release rate (HRR) ($\psi _{\dot {q}}$), which is challenging to measure but is readily available from large eddy simulations (LES) results. The behaviour of $\psi _{\dot {q}}$ for a wide range of thermochemical and turbulence conditions is investigated. Three burners are studied, namely a dual-swirl burner, a bluff-body burner and a jet in cross-flow burner, operating at atmospheric conditions with $\textrm {CH}_4$–air and $\textrm {H}_2$–air mixtures. In contrast to the classical $f^{-5/2}$ scaling, the far-field sound pressure level and volume-integrated HRR ($\psi _{\dot {Q}}$) spectra reveal a universal $f^{-5}$ scaling for high frequencies. This differing spectral decay rate for $\psi _{\dot {Q}}$ compared to the classical scaling is due to multi-regime combustion, related to either partial premixing or the local turbulence intensity. The dependence of $\psi _{\dot {q}}$ on the chosen spatial locations, flame configuration and its relation to velocity spectra are studied. A simple model for $\psi _{\dot {q}}$ involving the velocity spectra is found that compares well with LES results. The characteristic frequency involved in this model is related to the time scale of the coherent structures of the flow.

Three-dimensional eddy-resolving simulations are used to study the structure of turbulent boundary layers developing in an open channel where an array of sparse roughness elements in the form of partially burrowed mussels is placed on the smooth, flat channel bed starting at a certain streamwise location. The identical mussels are oriented with their major axis parallel to the mean flow. Their positions are randomised while making sure that the mussels are close to uniformly distributed inside the array. The turbulent flow approaching the leading edge of the array is fully developed. The protruding parts of the mussels play the role of sparse roughness elements and generate a rough-bed, internal boundary that is characterised by zero net flow exchange but non-zero local flow exchange due to active filtering. A double-averaging technique is used to obtain an equivalent, width- and time-averaged, boundary layer over a ‘flat’ rough surface containing no mussels. The paper discusses the effects of varying the mussel array density, protruding height of the mussels and filtering discharge on the spatial growth of the two-dimensional boundary layer. With proper scaling, the profiles of the (double-averaged) streamwise velocity are close to self-similar inside the inertial layer (e.g. for h $\lt$ z $\lt$ $\delta$, where h is the height of the protruding part of the mussels and $\delta$ is the boundary layer thickness) starting some distance from the leading edge of the array. The scaled turbulent kinetic energy and concentration profiles associated with the scalar advected through the excurrent syphons are also found to be self-similar above the vertical location where the maximum value is reached. An analytical model containing three subzones is proposed for the streamwise velocity in between the bed (z $=$ 0) and z $=$ $\delta$. The velocity profile inside the inertial region contains a law-of-the-wall component supplemented by a law-of-the-wake component. The scaling coefficient of the law-of-the-wake component is found to be larger than typical values used to describe velocity variation in turbulent boundary layers developing in a surrounding flow with close-to-uniform free stream velocity. The equivalent roughness height for this particular type of boundary layers developing over sparse roughness elements increases monotonically with h and the mussel array density, $\rho$N. The paper also discusses the effect of the mussel bed density on the average refiltration fraction and the phytoplankton removal efficiency of the mussel bed.

Large-eddy simulations are analysed to determine the influence of suspended canopies, such as those formed in macroalgal farms, on ocean mixed layer (OML) deepening and internal wave generation. In the absence of a canopy, we show that Langmuir turbulence, when compared with wind-driven shear turbulence, results in a deeper OML and more pronounced internal waves beneath the OML. Subsequently, we examine simulations with suspended canopies of varying densities located in the OML, in the presence of a background geostrophic current. Intensified turbulence occurs in the shear layer at the canopy’s bottom edge, arising from the interaction between the geostrophic current and canopy drag. Structures resembling Kelvin–Helmholtz (KH) instability emerge as the canopy shear layer interacts with the underlying stratification, radiating internal waves beneath the OML. Both intensified turbulence and lower-frequency motions associated with KH-type structures are critical factors in enhancing mixing. Consequently, the OML depth increases by up to a factor of two compared with cases without a canopy. Denser canopies and stronger geostrophic currents lead to more pronounced KH-type structures and internal waves, stronger turbulence and greater OML deepening. Additionally, vertical nutrient transport is enhanced as the OML deepens due to the presence of the canopy. Considering that the canopy density investigated in this study closely represents offshore macroalgal farms, these findings suggest a mechanism for passive nutrient entrainment conducive to sustainable farming. Overall, this study reveals the intricate interactions between the suspended canopy, turbulent mixing and stratification, underscoring their importance in reshaping OML characteristics.

Linear and nonlinear mechanisms governing the growth of second-mode waves are analysed using a newly derived disturbance energy conservation equation that highlights the physical processes responsible for fluctuation energy production, flux-based transport and destruction. Axisymmetric direct numerical simulations (DNS) data from a Mach 6 hypersonic boundary layer, simulated over a $3^\circ$ half-angle sharp cone at zero angle of attack, is used as a reference. A Legendre polynomial-based forcing methodology is used to trigger transition in the DNS over a range of various amplitude levels and different forcing frequency content. Closure of the disturbance energy budgets is demonstrated numerically using the DNS data. The terms responsible for the amplification of the disturbance are identified, and nonlinear attenuation effects are discussed. We show that the interaction between the entropy/velocity fluctuations and the base temperature gradient governs the second-mode growth in the linear regime. Energy production occurs in the critical layer due to non-isentropic processes and accumulates in acoustic form below the relative sonic line through downward transport. At higher forcing amplitudes, nonlinear spectral broadening is observed, with simultaneous thermoviscous diffusion attenuating the disturbance energy growth. This effect is responsible for the non-monotonic streamwise variation of the wall-pressure spectrum. Phase speed and growth rate analyses, informed by linear stability theory (LST), reveal wave steepening effects preceding this nonlinear attenuation effect. The disturbance energy is observed to match the LST predictions at lower forcing amplitudes, deviating, as expected, at higher amplitudes.

The transport of particles in elastoviscoplastic (EVP) fluids is of significant interest across various industrial and scientific domains. However, the physical mechanisms underlying the various particle distribution patterns observed in experimental studies remain inadequately understood in the current literature. To bridge this gap, we perform interface-resolved direct numerical simulations to study the collective dynamics of spherical particles suspended in a pressure-driven EVP duct flow. In particular, we investigate the effects of solid volume fraction, yield stress, inertia, elasticity, shear-thinning viscosity, and secondary flows on particle migration and formation of plug regions in the suspending fluid. Various cross-streamline migration patterns are observed depending on the rheological parameters of the carrier fluid. In EVP fluids with constant plastic viscosity, particles aggregate into a large cluster at the duct centre. Conversely, EVP fluids with shear-thinning plastic viscosity induce particle migration towards the duct walls, leading to formation of particle trains at the corners. Notably, we observe significant secondary flows ($O(10^{-2})$ compared to the mean velocity) in shear-thinning EVP suspensions, arising from the interplay of elasticity, shear-thinning viscosity and particle presence, which further enhances corner-ward particle migration. We elucidate the physical mechanism by which yield stress augments the first normal stress difference, thereby significantly amplifying elastic effects. Furthermore, through a comprehensive analysis of various EVP suspensions, we identify critical thresholds for elasticity and yield stress necessary to achieve particle focusing at the duct corners.

In binary mixtures, the lifetimes of surface bubbles can be five orders of magnitude longer than those in pure liquids because of slightly different compositions of the bulk and the surfaces, leading to a thickness-dependent surface tension of thin films. Taking advantage of the resulting simple surface rheology, we derive the equations describing the thickness, flow velocity and surface tension of a single liquid film, using thermodynamics of ideal solutions and thin film mechanics. Numerical resolution shows that, after a first step of tension equilibration, the Laplace-pressure-driven flow is associated with a flow at the interface driven by an induced Marangoni stress. The resulting parabolic flow with mobile interfaces in the film further leads to its pinching, eventually causing its rupture. Our model paves the way for a better understanding of the rupture dynamics of liquid films of binary mixtures.

We investigate fully developed turbulent flow in curved channels to explore the interaction between turbulence and curvature-driven coherent structures. By focusing on two cases of mild and strong curvature, we examine systematically the effects of the Reynolds number through a campaign of direct numerical simulations, spanning flow regimes from laminar up to the moderately high Reynolds number – based on bulk velocity and channel height – of $87\,000$. Our analysis highlights the influence of curvature on the friction coefficient, showing that flow transition is anticipated by concave curvature and delayed by convex curvature. In the case of mild curvature, a frictional drag reduction compared with plane channel flow is found in the transitional regime. Spectral analysis reveals that the near-wall turbulence regeneration cycle is maintained in mildly curved channels, while it is absent or severely inhibited on the convex wall of strongly curved channels. Streamwise large-scale structures resembling Dean vortices are found to be weakly dependent on the Reynolds number and strongly affected by curvature: increasing curvature shifts these vortices towards the outer wall and reduces their size and coherence, limiting their contribution to streamwise velocity fluctuations and momentum transport. In the case of strong curvature, spanwise large-scale structures are also detected. These structures are associated with large pressure fluctuations and the suppression of turbulent stresses near the convex wall, where a region with negative turbulence production is observed and characterised via quadrant analysis.

The influence of irregular three-dimensional rough surfaces on the displacement of the logarithmic velocity profile relative to that of a smooth wall in turbulent flow, known as the roughness function, is studied using direct numerical simulations. Five different surface power spectral density (PSD) shapes were considered, and for each, several rough Gaussian surfaces were generated by varying the root mean square ($k_{rms}$) of the surface heights. It is shown that the roughness function ($\Delta U^{+}$) depends on both the PSD and $k_{rms}$. For a given $k_{rms}$, $\Delta U^{+}$ increases as the wavenumbers of the PSD expand to large values, but at a rate that decreases with the magnitude of the wavenumbers. Although $\Delta U^{+}$ generally does not scale with either $k_{rms}$ or the effective slope $ES$ when these variables are considered singularly, for PSDs with low wavenumbers, $\Delta U^{+}$ tends to scale with $ES$, whereas as wavenumbers increase, $\Delta U^{+}$ tends to scale with $k_{rms}$. An equivalent Nikuradse sand roughness of about eight times $k_{rms}$ is found, which is similar to that observed in previous studies for a regular three-dimensional roughness. Finally, it is shown that $k_{rms}$ and the effective slope are sufficient to describe the roughness function in the transitional rough regime.

We report the first measurement of turbulent mixing developing from the convergent Richtmyer–Meshkov (RM) instability using time-resolved planar laser-induced fluorescence in a semi-annular convergent shock tube. A membraneless yet sharp interface with random short-wavelength perturbations, but controllable long-wavelength perturbations, is created by an automatically retractable plate, enhancing the reproducibility and reliability of RM turbulence experiments. The cylindrical air/SF$_6$ interface formed is first subjected to a convergent shock, then to its reflected shock and subsequently transits to turbulent mixing. It is found that the mixing width after reshock has a linear growth rate more than twice the rate in planar geometry. Also, the mixing width does not present power-law growth at late stages as in a planar geometry. However, the scalar spectrum and transition criterion obtained are similar to their planar counterparts. These findings indicate that the geometric constraint greatly affects the large scales of the flow, while having a weaker effect on the small scales. It is also found that the reflected shock significantly increases both scale separation and Reynolds number, explaining the rapid transition to turbulence following reshock.

Morphodynamic descriptions of fluid deformable surfaces are relevant for a range of biological and soft matter phenomena, spanning materials that can be passive or active, as well as ordered or topological. However, a principled, geometric formulation of the correct hydrodynamic equations has remained opaque, with objective rates proving a central, contentious issue. We argue that this is due to a conflation of several important notions that must be disambiguated when describing fluid deformable surfaces. These are the Eulerian and Lagrangian perspectives on fluid motion, and three different types of gauge freedom: in the ambient space; in the parameterisation of the surface; and in the choice of frame field on the surface. We clarify these ideas, and show that objective rates in fluid deformable surfaces are time derivatives that are invariant under the first of these gauge freedoms, and which also preserve the structure of the ambient metric. The latter condition reduces a potentially infinite number of possible objective rates to only two: the material derivative and the Jaumann rate. The material derivative is invariant under the Galilean group, and therefore applies to velocities, whose rate captures the conservation of momentum. The Jaumann derivative is invariant under all time-dependent isometries, and therefore applies to local order parameters, or symmetry-broken variables, such as the nematic $Q$-tensor. We provide examples of material and Jaumann rates in two different frame fields that are pertinent to the current applications of the fluid mechanics of deformable surfaces.

Manipulation of small-scale particles across streamlines is the elementary task of microfluidic devices. Many such devices operate at very low Reynolds numbers and deflect particles using arrays of obstacles, but a systematic quantification of relevant hydrodynamic effects has been lacking. Here, we explore an alternative approach, rigorously modelling the displacement of force-free spherical particles in vortical Stokes flows under hydrodynamic particle–wall interaction. Certain Moffatt-like eddy geometries with broken symmetry allow for systematic deflection of particles across streamlines, leading to particle accumulation at either Faxen field fixed points or limit cycles. Moreover, particles can be forced onto trajectories approaching channel walls exponentially closely, making possible quantitative predictions of particle capture (sticking) by short-range forces. This rich, particle-size-dependent behaviour suggests the versatile use of inertia-less flow in devices with a long particle residence time for concentration, sorting or filtering.