Symmetry relations are derived for the added mass and damping of structures where the shape is unchanged by rotation about the vertical axis through an angle $\theta = 2 {\rm \pi}/N$ with the integer $N\ge 3$. For this type of structure, the added mass and damping for horizontal translation are the same for all directions, as in the case of axisymmetric structures. The same symmetry applies to rotations about horizontal axes. The principal application is to offshore structures and other bodies floating on the free surface or submerged, but the same symmetry relations apply more generally to unsteady body motions in an ideal fluid.

Statistically homogeneous flows obey exact kinematic relations. The Betchov homogeneity constraints (Betchov, J. Fluid Mech., vol. 1, 1956, pp. 497–504) for the average principal invariants of the velocity gradient are among the most well-known and extensively employed homogeneity relations. These homogeneity relations have far-reaching implications for the coupled dynamics of strain and vorticity, as well as for the turbulent energy cascade. Whether the Betchov homogeneity constraints are the only possible ones or whether additional homogeneity relations exist has not been proven yet. Here we show that the Betchov homogeneity constraints are the only homogeneity constraints for incompressible and statistically isotropic velocity gradient fields. Our analysis also applies to compressible/perceived velocity gradients, and it allows the derivation of homogeneity relations involving the velocity gradient and other dynamically relevant quantities, such as the pressure Hessian and viscous stresses.

Viscoplastic materials cannot flow if the local stress falls below the yield stress of the material. Such materials tend to ‘clog up’ geometrical features like corners or side branches by forming rigid plugs, because the stress becomes insufficient to yield them. In this volume, Taylor-West and Hogg (J. Fluid Mech., vol. 941, 2022, A64) consider the problem of flow of an idealised viscoplastic fluid in a wedge forced by a disturbance far from the corner; this is the viscoplastic analogue of Moffatt's famous corner eddies. They demonstrate and describe the details of how the fluid clogs up the corner of the wedge with a rigid plug bounded by a viscoplastic eddy. More generally, this study provides a way to explain some of the details of cloggings in different geometries that have previously been observed in viscoplastic materials.

We re-examine the model proposed by Alexander et al. (Phys. Fluids, vol. 18, 2006, 062103) for the closing of a circular hole in a molecularly thin incompressible Langmuir film situated on a Stokesian subfluid. For simplicity their model assumes that the surface phase is inviscid which leads to the result that the cavity area decreases at a constant rate determined by the ratio of edge tension to subfluid viscosity. We reformulate the problem, allowing for a regularising monolayer viscosity. The viscosity-dependent corrections to the hole dynamics are analysed and found to be non-trivial, even when the monolayer viscosity is small; these corrections may explain the departure of experimental data from the theoretical prediction when the hole radius becomes comparable to the Saffman–Delbrück length. Through fitting, under these relaxed assumptions, we find the edge tension could be as much as six times larger ($\sim$4.0 pN) than reported previously.

The flow of polymer solutions past solid obstacles or through porous media gives rise to rich physical phenomena over a wide range of spatial and temporal scales. Viscoelasticity, in particular, can induce a strong nonlinear response with an increase of flow resistance even for a solution whose viscosity decreases in simple shear flow. Various hypotheses have been proposed to explain this phenomenon but a clear picture of the pore-scale mechanisms involved and their impact upon larger scales is still lacking. Here, we show that localized zones of large polymeric stress, known as birefringent strands, drive the flow of an Oldroyd-B fluid through two-dimensional arrays of cylinders. Combining a recently developed numerical scheme with high performance computing, we find that these strands generate a complete reorganization of the flow with an increase of stagnation zones, a reinforcement of preferential paths and a splitting of flow channels. Furthermore, we show that this reorganization is the source of an increase in the viscous dissipation of the solvent and also that the stretching of polymer molecules in the strands is associated with entropy production. Both these phenomena yield a global increase in dissipation that can be directly linked to the increase of flow resistance. Our results demonstrate that the birefringent strands – not the elongational viscosity – drive the flow of viscoelastic fluids through porous media and that the increase of flow resistance can occur even at steady state, before the transition to elastic turbulence.

The unsteadiness of a turbulent separation bubble (TSB) formed close to the trailing edge of a two-dimensional wing was investigated using time-resolved particle image velocimetry. The angle of attack was set to 9.7° and the chord-based Reynolds number was 720 000. The TSB consisted of two shear layers and formed a triangular shape in the streamwise–wall-normal plane. The vertices of this triangle consisted of an intermittent detachment point, a fixed corner close to the airfoil trailing edge and an intermittent endpoint in the wake region. The velocity field had three energetic regions each with different Strouhal numbers (Stl): (a) an upstream turbulent boundary layer (TBL) with Stl = 0.1 to 4, (b) a TSB with Stl = 0.03 to 0.08 and (c) two shear layers with Stl = 0.4 to 0.8. The low-frequency motions in the TSB consisted of large zones of positive and negative streamwise velocity fluctuation that were several times wider than the large-scale structures of the upstream TBL. These zones forced an undulation of the separation line and were attributed to Görtler structures. They were also correlated with the velocity fluctuations between the two shear layers. The breathing motion of the TSB occurred at Stl = 0.05. This breathing correlated with the location of the TSB endpoint and the flapping of the upper shear layer. The detachment point of the TSB featured broad fluctuations and did not demonstrate a strong correlation with the breathing motion.

Many complex flows such as those arising from the collective motion of ocean plastics in geophysics or motile cells in biology are characterized by sparse and noisy trajectory datasets. We introduce techniques for identifying Lagrangian coherent structures (LCSs) of hyperbolic and elliptic nature in such datasets. Hyperbolic LCSs, which represent surfaces with maximal attraction or repulsion over a finite amount of time, are computed through a regularized least-squares approximation of the flow map gradient. Elliptic LCSs, which identify regions of coherent motion such as vortices and jets, are extracted using DBSCAN – a popular data clustering algorithm – combined with a systematic parameter selection strategy. We deploy these methods on various benchmark analytical flows and real-life experimental datasets ranging from oceanography to biology and show that they yield accurate results, despite sparse and noisy data. We also provide a lightweight computational implementation of these techniques as a user-friendly and straightforward Python code.

Production and inter-component redistribution of turbulence in adverse pressure gradient (APG) turbulent boundary layers (TBLs) with small and large velocity defects are investigated, along with the structures playing a role in these energy transfer mechanisms. We examine the wall-normal and spectral distributions of energy, production and pressure-strain in APG TBLs, and compare these distributions with those in canonical flows. It is found that the spectral distributions of production and pressure-strain are not affected profoundly by an increase of the velocity defect, although the energy spectra change drastically in the inner layer of the large-defect APG TBL. In the latter, the signature of the inner-layer streaks is absent from the energy spectra. In the outer layer, energetic, production and pressure-strain structures appear to change from wall-attached to wall-detached structures with increasing velocity defect. Despite this, the two-dimensional spectral distributions have similar shapes and wavelength aspect ratios of the peaks in all these flows. Therefore, the conclusion is that the mechanisms responsible for turbulence production and inter-component energy transfer may remain the same within each layer in all these flows. It is the intensity of these mechanisms within one layer that changes with velocity defect, because of the local mean shear variation.

The jetting behaviour and migratory characteristics of a laser-induced cavitation bubble in a rectangular channel are investigated both experimentally and numerically, for various combinations of the geometric and physical parameters of the system. High-speed photography is used to visualize the temporal development of the bubble shape, the formation of liquid jets during bubble collapse, and the bubble displacement in contact with the sidewalls of the channel during two oscillation cycles of the bubble. The bubble profiles, pressure contours and velocity vectors ambient to the bubble are obtained through numerical simulation results by using an Eulerian finite element method with a compressible liquid impact model. The jetting behaviour of the bubble varies between single jet formation and the formation of three liquid jets directed towards each wall of the channel. The numerical calculations indicate that the liquid jets directed towards the sidewalls of the channel reach maximum velocities of 100 m s−1 while the peak velocity of the liquid jet directed towards the channel endwall is about 55 m s−1. A small bubble generated close to a sidewall of the channel develops only a single inclined jet during collapse. Such jets can reach velocities of up to 110 m s−1. A bubble displacement in contact with the sidewalls of the channels of 350 μm was observed during the first two oscillation cycles for a bubble with a maximum diameter slightly smaller than the height of the channel. The results of our investigations are compared to previous results obtained in similar configurations.

We investigate the large-scale influence of numerical noises as tiny artificial stochastic disturbances on a sustained turbulence. Using two-dimensional (2-D) turbulent Rayleigh–Bénard convection (RBC) as an example, we solve numerically the Navier–Stokes equations, separately, by means of a traditional algorithm with double precision (denoted RKwD) and the so-called clean numerical simulation (CNS). The numerical simulation given by RKwD is a mixture of the ‘true’ physical solution and the ‘false’ numerical noises that are random and can be regarded as a kind of artificial stochastic disturbances; unfortunately, the ‘true’ physical solution is mostly at the same level as the ‘false’ numerical noises. By contrast, the CNS can greatly reduce the background numerical noise to any a required level so that the ‘false’ numerical noises are negligible compared with the ‘true’ physical solution, thus the CNS solution can be used as a ‘clean’ benchmark solution for comparison. It is found that the numerical noises as tiny artificial stochastic disturbances could indeed lead to large-scale deviations of simulations not only in spatio-temporal trajectories but also even in statistics. In particular, these numerical noises (as artificial stochastic disturbances) even lead to different types of flows. The shearing convection occurs for the RKwD simulations, and its corresponding flow field turns to a kind of zonal flow thereafter; however, the CNS benchmark solution always sustains the non-shearing vortical/roll-like convection during the whole process of simulation. Thus we provide rigorous evidence that numerical noises as a kind of small-scale artificial stochastic disturbances have quantitatively and qualitatively large-scale influences on a sustained turbulence, i.e. the 2-D turbulent RBC considered in this paper.

We discuss (i) the measurement, (ii) interpretation and (iii) technical impact of temperature-dependent wall slip and its activation energy for polar and non-polar hydrocarbon Newtonian fluids moving relative to machined metal surfaces. A newly developed apparatus, the slip length tribometer (SLT), overcomes the drawbacks of existing measurement devices in terms of characterising relevant rough surfaces by measuring the slip length at different temperatures over a sufficiently large wetted area. The experimental data show that the bulk viscosity and slip length at the fluid–metal interface is independent of the shear rate up to $10^5\,\mathrm {s}^{-1}$, being consistent with recent results from molecular dynamics simulations by Mehrnia & Pelz (J. Mol. Liq., vol. 336, 2021, 116589). Furthermore, the activation energies for wall slip and bulk shear determined by means of the SLT differ by a factor of two, i.e. $E_{\lambda }\approx 0.5 E_\mu$ for non-polar hydrocarbon molecules sliding relative to metal walls. This difference is explained by a generalised Eyring model applied to wall slip. The paper closes with the impact of wall slip on Sommerfeld's similarity theory of tribology and the resulting Stribeck curve (Sommerfeld, Z. Math. Phys., vol. 50, 1904, pp. 97–155; Z. Tech. Phys., vol. 2. Jahrg., 1921; Mechanik der deformierbaren Medien, 1944, Akademische Verlagsgesellschaft Becker & Erler). For this purpose, Reynolds’ equation generalised for wall slip is solved in combination with the experimentally determined constitutive relations for bulk shear and wall slip to predict typical characteristics of journal bearings. The results show that, for a typical journal bearing, where the ratio of slip length to average bearing clearance is of the order of $10^{-2}$, the influence of wall slip on both load-carrying capacity and dissipation is not negligible. This work combines nanofluidics and tribology in order to provide methods and knowledge for e.g. tailor-made fluids and interfaces.

Resolvent analysis is a powerful tool for modelling and analysing transitional and turbulent flows and, in particular, for approximating coherent flow structures. Despite recent algorithmic advances, computing resolvent modes for flows with more than one inhomogeneous spatial coordinate remains computationally expensive. In this paper we show how efficient and accurate approximations of resolvent modes can be obtained using a well-posed spatial marching method for flows that contain a slowly varying direction, i.e. one in which the mean flow changes gradually. First, we derive a well-posed and convergent one-way equation describing the downstream-travelling waves supported by the linearized Navier–Stokes equations. The method is based on a projection operator that isolates downstream-travelling waves. Integrating these one-way Navier–Stokes (OWNS) equations in the slowly varying direction, which requires significantly less CPU and memory resources than a direct solution of the linearized Navier–Stokes equations, approximates the action of the resolvent operator on a forcing vector. Second, this capability is leveraged to compute approximate resolvent modes using an adjoint-based optimization framework in which the forward and adjoint OWNS equations are marched in the downstream and upstream directions, respectively. This avoids the need to solve direct and adjoint globally discretized equations, therefore bypassing the main computational bottleneck of a typical global resolvent calculation. The method is demonstrated using the examples of a simple acoustics problem, a Mach 1.5 turbulent jet and a Mach 4.5 transitional zero-pressure-gradient flat-plate boundary layer. The optimal OWNS results are validated against corresponding global calculations, and the close agreement demonstrates the near-parabolic nature of these flows.

We employ a recently introduced structured input–output analysis (SIOA) approach to analyse streamwise and spanwise wavelengths of flow structures in stably stratified plane Couette flow. In the low-Reynolds-number ($Re$) low-bulk Richardson number ($Ri_b$) spatially intermittent regime, we demonstrate that SIOA predicts high amplification associated with wavelengths corresponding to the characteristic oblique turbulent bands in this regime. SIOA also identifies quasi-horizontal flow structures resembling the turbulent–laminar layers commonly observed in the high-$Re$ high-$Ri_b$ intermittent regime. An SIOA across a range of $Ri_b$ and $Re$ values suggests that the classical Miles–Howard stability criterion ($Ri_b\leq 1/4$) is associated with a change in the most amplified flow structures when the Prandtl number is close to one ($Pr\approx 1$). However, for $Pr\ll 1$, the most amplified flow structures are determined by the product $PrRi_b$. For $Pr\gg 1$, SIOA identifies another quasi-horizontal flow structure that we show is principally associated with density perturbations. We further demonstrate the dominance of this density-associated flow structure in the high $Pr$ limit by constructing analytical scaling arguments for the amplification in terms of $Re$ and $Pr$ under the assumptions of unstratified flow (with $Ri_b=0$) and streamwise invariance.

This paper is concerned with the optimal upper bound on mean quantities (torque, dissipation and the Nusselt number) obtained in the framework of the background method for the Taylor–Couette flow with a stationary outer cylinder. Along the way, we perform the energy stability analysis of the laminar flow, and demonstrate that below radius ratio $0.0556$, the marginally stable perturbations are not the axisymmetric Taylor vortices but rather a fully three-dimensional flow. The main result of the paper is an analytical expression of the optimal bound as a function of the radius ratio. To obtain this bound, we begin by deriving a suboptimal analytical bound using analysis techniques. We use a definition of the background flow with two boundary layers, whose relative thicknesses are optimized to obtain the bound. In the limit of high Reynolds number, the dependence of this suboptimal bound on the radius ratio (the geometrical scaling) turns out to be the same as that of numerically computed optimal bounds in three different cases: (1) the perturbed flow only satisfies the homogeneous boundary conditions but need not be incompressible; (2) the perturbed flow is three-dimensional and incompressible; (3) the perturbed flow is two-dimensional and incompressible. We compare the geometrical scaling with the observations from the turbulent Taylor–Couette flow, and find that the analytical result indeed agrees well with the available direct numerical simulations data. In this paper, we also dismiss the applicability of the background method to certain flow problems and therefore establish the limitation of this method.

In the present study, turbulent particle-laden boundary layer combustion over a flat plate is investigated using direct numerical simulation (DNS). A two-way coupled Eulerian–Lagrangian point particle method is used for the solid phase. The effects of particle Stokes numbers, mass loadings and chemical reactions on the interactions between particles and boundary layer turbulence in the near-wall region are explored. It was found that particle heat transfer is dominant over wall heat transfer in the reacting case with heavy particles and large mass loadings, resulting in a lower fluid temperature. Particle accumulation due to the turbophoresis effect in the near-wall region is observed, which is more prominent in the cases with a large Stokes number. The turbophoresis effect is examined via the magnitude of streamwise vorticity $\varOmega _x$. It is shown that $\varOmega _x$ is attenuated by heavy particles, and the attenuation increases with increasing mass loadings. Therefore, particle wall–accumulation is less prominent in the cases with large mass loadings. Compared with the non-reacting cases, the distribution of particles is more inhomogeneous for the reacting cases, where the particles move faster due to intense reactions with increasing wall-normal distance. Finally, the flow topologies and the Reynolds stress anisotropy are examined to understand turbulence modulation by combustion and particles. It was suggested that light particles augment the vortex-dominant topologies, whereas heavy particles have an opposite effect. The anisotropy mapping of the Reynolds stress shows that the flows become more one-dimensional in the near-wall region for the cases with combustion and/or large mass loadings.

Liquid spreading through a randomly packed particle-resolved bed influenced by capillary or inertial ($AB_s \sim 1$), and gravitational force (moderately ($AB_s \sim 0.1$) and strongly ($AB_s \sim 0.01$)) is investigated using the volume-of-fluid simulations. The relative contribution of governing forces at different stages of spreading is analysed using the time evolution of Weber ($We_I$) and $AB_I$ numbers. We show that the dynamics of liquid spreading at $AB_s \sim 1$ is primarily governed by the inertial force in the beginning ($AB_I > 1$, $We_I > 1$) followed by the capillary force at $t/t^* \sim 1$. This interplay of governing forces leads to inertia- and capillary-induced bubble entrapments at the void scale and promote lateral liquid spreading. When the $AB_s \sim 0.1$, the $t/t^*$ for which the flow is governed by inertial ($AB_I > 1$, $We_I > 1$) and capillary forces ($AB_I > 1$, $We_I < 1$) decreases and the relative contribution of gravitational force is substantial at large $t/t^*$ ($AB_I < 1$). This force balance leads to unified-void filling characterised by negligible bubble trapping and results in a decrease in the lateral spreading. Further decrease in the $AB_s$ to ${\sim } 0.01$ results in liquid spreading primarily governed by gravitational force ($AB_I < 1$) with small contribution of inertial and capillary forces at the very beginning leading to trickling flow and a further decrease in lateral spreading. Finally, a regime map is proposed, which provides the relationship between different forces, void-scale events, and the resultant liquid spreading at $t/t^* \sim 1$.

When a freely suspended liquid film ruptures, it retracts spontaneously under the action of surface tension. If the film is surrounded by air, the retraction velocity is known to approach the constant Taylor–Culick velocity. However, when surrounded by an external viscous medium, the dissipation within that medium dictates the magnitude of the retraction velocity. In the present work, we study the retraction of a liquid (water) film in a viscous oil ambient (two-phase Taylor–Culick retractions), and that sandwiched between air and a viscous oil (three-phase Taylor–Culick retractions). In the latter case, the experimentally measured retraction velocity is observed to have a weaker dependence on the viscosity of the oil phase as compared with the configuration where the water film is surrounded completely by oil. Numerical simulations indicate that this weaker dependence arises from the localization of viscous dissipation near the three-phase contact line. The speed of retraction only depends on the viscosity of the surrounding medium and not on that of the film. From the experiments and the numerical simulations, we reveal unprecedented regimes for the scaling of the Weber number ${We}_{f}$ of the film (based on its retraction velocity) or the capillary number ${Ca}_{s}$ of the surroundings versus the Ohnesorge number ${Oh}_{s}$ of the surroundings in the regime of large viscosity of the surroundings (${Oh}_{s} \gg 1$), namely ${We}_{f} \sim {Oh}_{s}^{-2}$ and ${Ca}_{s} \sim {Oh}_{s}^{0}$ for the two-phase Taylor–Culick configuration, and ${We}_{f} \sim {Oh}_{s}^{-1}$ and ${Ca}_{s} \sim {Oh}_{s}^{1/2}$ for the three-phase Taylor–Culick configuration.

The linear dynamics of perturbations developing in an infinite channel with compliant walls is investigated for pulsatile flow conditions. Two-dimensional modal perturbations are considered for Womersley-type pulsating base flows and the wall motion is only allowed in the normal direction. It is found that the flow dynamics is mainly governed by four control parameters: the Reynolds number $Re$, the reduced velocity $V_R$, the Womersley number $Wo$ and the amplitude of the base-flow modulation $\tilde Q$. Linear stability analyses are carried out within the framework of Floquet theory, implementing an efficient approach for removing spurious eigenmodes. The characteristics of flow-based (Tollmien–Schlichting) and wall-based (both travelling-wave flutter and divergence) modes are investigated over a large control-parameter space. It is shown that travelling-wave flutter (TWF) modes are predominantly influenced by the reduced velocity and that the Reynolds number has only a marginal effect. The critical reduced velocity (corresponding to onset of linear instability) is demonstrated to depend both on the Womersley number and modulation amplitude for a given set of wall parameters. Similarly to the steady flow case, the Tollmien–Schlichting (TS) mode is also found to be only weakly affected by the flexibility of the wall in pulsatile flow conditions. Finally, the classification given by Benjamin (J. Fluid Mech., vol. 16, 1963, pp. 436–450) is found to be too restrictive in the case of pulsatile base flows. In particular, a new type of mode is identified that shares characteristics of two distinct Floquet eigenmodes: TS and TWF modes. Due to coupling of the different Floquet harmonics, a phenomenon specific to time-periodic base flows, this two-wave mode exhibits a beating over the intracyclic dynamics.

Hydrodynamic stability of compressible boundary layers is strongly influenced by the Mach number ($M$), Prandtl number ($Pr$) and thermal wall boundary condition. These effects manifest on the flow stability via the flow–thermodynamics interactions. Comprehensive understanding of stability flow physics is of fundamental interest and important for developing predictive tools and closure models for integrated transition-to-turbulence computations. The flow–thermodynamics interactions are examined using linear analysis and direct numerical simulations in the following parameter regime: $0.5 \leq M \leq 8$; and $0.5 \leq Pr \leq 1.3$. For the adiabatic wall boundary condition, increasing Prandtl number has a destabilizing effect. In this work, we characterize the behaviour of production, pressure–strain correlation and pressure dilatation as functions of the Mach and Prandtl numbers. First and second instability modes exhibit similar stability trends but the underlying flow physics is shown to be diametrically opposite. The Prandtl number influence on instability is explicated in terms of the base flow profile with respect to the different perturbation mode shapes.

The influence of a magnetic field on the pattern of stationary waves formed on the surface of a magnetic fluid (ferrofluid) when an obstacle moves has been studied both theoretically and experimentally. It is found that a vertical magnetic field narrows the cone of stationary waves and increases their amplitude. In the wake region, the peaks of the Rosensweig instability appear in a magnetic field that is smaller than the critical field that determines this instability occurrence. A horizontal magnetic field parallel to the obstacle velocity expands the cone of waves but reduces their amplitude up to the suppression of stationary waves. A horizontal field perpendicular to the obstacle velocity also expands the cone of waves and stabilizes their amplitude.