The mechanisms of destabilisation of the flow through soft-walled channels/tubes are qualitatively different from those in rigid-walled conduits. The stability depends on two dimensionless parameters, the Reynolds number $(\rho _f V_f h_f/ \mu _f)$ and $\varSigma = (\rho _f G h_f^{2}/\mu _f^{2})$, where $\rho _f$ and $\mu _f$ are the fluid density and viscosity, $h_f$ and $V_f$ are the fluid length and velocity scale and $G$ is the wall elasticity modulus. There is an instability at zero Reynolds number when the dimensionless parameter $\varGamma = (\mu _f V_f / h_f G)$ exceeds a critical value. The low-Reynolds-number instability of the Couette flow past a compliant surface is well understood, and has been confirmed in experiments, but that in a pressure-driven flow is not completely understood. Two modes of instability at high Reynolds number have been predicted: the inviscid mode with an internal viscous layer, for which the transition Reynolds number scales as ${Re}_t \propto \varSigma ^{1/2}$; and the wall mode instability with a viscous layer at the wall, for which ${Re}_t \propto \varSigma ^{3/4}$. The wall mode instability has been observed in experiments at Reynolds number as low as 300 in a soft-walled tube and as low as 100 in a channel with one compliant wall, though the scaling of the transition Reynolds number differs from the theoretical prediction due to substantial wall deformation. Though the flow after transition shares many of the characteristics of hard-wall turbulence, it differs in significant ways, suggesting that soft-wall turbulence is a separate class distinct from hard-wall turbulence.

This work examines consequences of modelling approximation errors made within the context of the Navier–Stokes–Boussinesq system. Starting from a canonical Boussinesq model, where density fluctuations are allowed to interact with all accelerative terms of the incompressible Navier–Stokes equations in arbitrarily accelerating reference frames, a unified treatment is developed that provides a straightforward way to identify buoyancy forcing associated with gravitational effects, centrifugal forcing associated with frame rotation, as well as centrifugal-type forcing due to variations in flow kinetic energy. The results of the cases studied in inertial, rotating and mixed reference frames demonstrate that in general it may be important to apply buoyancy effects to all non-local accelerative terms, including non-gradient terms such as Coriolis acceleration. Additionally, it is shown that the common practice of ignoring terms representing interaction between density fluctuation and local fluid acceleration can lead to non-negligible error in Boussinesq modelling of highly unsteady flows. These findings have special significance for accurate simulation of flows with density variations in which there may be both background rotation and localised regions of strong swirl, but are also relevant for studies conducted in the inertial frame of reference.

We experimentally demonstrate quadrupolar electro-osmotic flows around charged dielectric microspheres immersed in an electrolyte when subjected to an alternating current electric field. We present an electrokinetic model that predicts the flow characteristics based on the phenomena of surface conductance and polarization of the electrolyte concentration around the particles. We refer to these flows as concentration polarization electro-osmosis. We anticipate that these flows may play a major role in the electric-field-induced assembly of colloids and on the electrokinetic manipulation of dielectric micro- and nanoparticles.

This paper reports a new phenomenon in the near-free-surface region of open channel flows (OCFs): namely that higher relative population densities of spanwise vortices are observed than are seen at equivalent positions in turbulent boundary layers. The presence of additional retrograde spanwise vortices in OCFs throughout the region $y/h>0.2$ (where $y$ is the distance to the wall and $h$ is the water depth) prompts us to re-examine the wall-normal extent that the free-surface effect can reach, which is classically expected to be limited within the surface and blockage layers. Finally, possible mechanisms for the phenomenon of additional spanwise vortices in OCFs are provided.

The hypothesis in the classical Kolmogorov picture of turbulence with perhaps the most far-reaching consequences is that of universality, the notion that small-scale turbulence dynamics are independent of the way the turbulence was generated. The assumption of universality can be evaluated by comparing measurements taken in many kinds of flows. However, up to now the range of flows that can be used to study universality from a Lagrangian viewpoint has been highly constrained, because large-scale Eulerian inhomogeneity manifests as Lagrangian non-stationarity. The recent work of Viggiano et al. (J. Fluid Mech., vol. 918, 2021, A25) significantly extends this range by showing how the dynamics along Lagrangian trajectories can be continuously renormalised using local Eulerian scales, at least in flows whose development is self-similar. They demonstrate their results on a turbulent jet, a classical flow that is well studied from the Eulerian perspective, though not in a Lagrangian sense. Their work provides an exciting roadmap for expanding the scope of Lagrangian analysis of turbulent flows.

A depth-averaged, convection-dispersion equation is derived for the particle volume fraction distribution in the pressure-driven flow of a concentrated suspension of neutrally buoyant, non-colloidal particles between two parallel plates, by implementing a two-time-scale perturbation expansion of the suspension balance model (Nott & Brady, J. Fluid Mech., vol. 275, 1994, pp. 157–199) coupled with the constitutive equations of Zarraga et al. (J. Rheol., vol. 52, issue 2, 2000, pp. 185–220). The Taylor-dispersion coefficient in the macrotransport equation scales as $U_c^{\prime}B^{\prime 3}/a^{\prime 2}$, where $U_c^{\prime}$ is the characteristic depth-averaged velocity, $B^{\prime}$ is the half-depth of the channel and $a^{\prime}$ is the particle radius. Taylor dispersion relaxes gradients in the depth-averaged volume fraction along the local velocity vector. Perpendicular to the flow, however, only shear-induced migration can cause particle redistribution, leading to fluxes down gradients in volume fraction, shear rate and streamline curvature that scale as $U_c^{\prime}a^{\prime 2}/B^{\prime}$. To determine the velocity and particle distributions in Hele-Shaw suspension flows, one only needs to solve two coupled partial differential equations in the pressure and the depth-averaged volume fraction, achievable on commercially available solvers. Analogous to the macrotransport equation for suspension flow through a circular tube (Ramachandran, J. Fluid Mech., vol. 734, 2013, pp. 219–252), the evolution of the particle volume fraction distribution is dependent only on the total strain experienced by the suspension, and is independent of the suspension velocity scale. However, unlike the tube problem, a positive concentration gradient along the flow direction is susceptible to viscous miscible fingering. A linear stability analysis performed for a step increase in the volume fraction in the direction of flow with a velocity $U^{\prime}$ reveals that the growth rate and wavenumber corresponding to fastest growing mode scale as $U^{\prime}a^{\prime 2}/B^{\prime 3}$ and $a^{\prime 2/3}/B^{\prime 5/3}$, respectively.

We investigate how turbulence in liquid affects the rising speed of gas bubbles within the inertial range. Experimentally, we employ stereoscopic tracking of bubbles rising through water turbulence created by the convergence of turbulent jets and characterized with particle image velocimetry performed throughout the measurement volume. We use the spatially varying, time-averaged mean water velocity field to consider the physically relevant bubble slip velocity relative to the mean flow. Over a range of bubble sizes within the inertial range, we find that the bubble mean rise velocity $\left \langle v_z \right \rangle$ decreases with the intensity of the turbulence as characterized by its root-mean-square fluctuation velocity, $u'$. Non-dimensionalized by the quiescent rise velocity $v_{q}$, the average rise speed follows $\left \langle v_z \right \rangle /v_{q}\propto 1/{\textit {Fr}}$ at high ${\textit {Fr}}$, where ${\textit {Fr}}=u'/\sqrt {dg}$ is a Froude number comparing the intensity of the turbulence to the bubble buoyancy, with $d$ the bubble diameter and $g$ the acceleration due to gravity. We complement these results by performing numerical integration of the Maxey–Riley equation for a point bubble experiencing nonlinear drag in three-dimensional, homogeneous and isotropic turbulence. These simulations reproduce the slowdown observed experimentally, and show that the mean magnitude of the slip velocity is proportional to the large-scale fluctuations of the flow velocity. Combining the numerical estimate of the slip velocity magnitude with a simple theoretical model, we show that the scaling $\left \langle v_z \right \rangle /v_{q}\propto 1/{\textit {Fr}}$ originates from a combination of the nonlinear drag and the nearly isotropic behaviour of the slip velocity at large ${\textit {Fr}}$ that drastically reduces the mean rise speed.

Inter-species drag forces in granular flows play a central role in setting the speed and extent of segregation, a process that separates grains of different size or density. Here, we study this drag force in detail, using a novel configuration of discrete element simulations that allows us to completely characterise the drag in inertial granular flows by studying it in a uniform environment. By applying opposing forces to grains in monodisperse and size-bidisperse shear flows, we show that the strength of the drag force scales as $I^{-7/4}$, where $I$ is the granular inertial number, and propose a model that explains this scaling by relating the strength of drag to grain velocity fluctuations. These findings suggest that much of the previously observed dependence of the segregation rate on the local shear rate and pressure in dense free-surface flows is due to variation in the strength of the inter-species drag, rather than the strength of forces that drive segregation.

We experimentally study how the turbulent energy dissipation rate scales in the cross-stream direction of turbulent wake flows generated by two side-by-side square prisms. We consider three different such turbulent flows with gap ratios $G/H=1.25$, $2.4$ and $3.5$, where $G$ is the distance between the prisms and $H$ is the prism width. These three flows have a very different dynamics, inhomogeneities and large-scale features. The measurements were taken with a multi-camera particle image velocimetry system at several streamwise locations between $2.5H$ and $20H$ downstream of the prisms. After removing the large-scale most energetic coherent structures, the normalised turbulence dissipation coefficient $C'_{\epsilon }$ of the remaining incoherent turbulence is found to scale as $C'_{\epsilon } \sim (\sqrt {Re_{L}}/Re'_{\lambda })^{3/2}$ along the highly inhomogeneous cross-stream direction for all streamwise locations tested in all three flows and for all three inlet Reynolds numbers considered; $Re'_{\lambda }$ and $Re_{L}$ are, respectively, a Taylor length-based and an integral length-based Reynolds number of the remaining incoherent turbulence.

We present a comprehensive study of the mesoscale eddy forcing in the ocean by proposing spatially local filtering of the high-resolution double-gyre ocean circulation solution into its large- and small-scale (eddy) components. The large-scale component is dominated by the mid-latitude gyres, the western boundary currents and their highly transient eastward jet extension; the eddy component is concentrated around the eastward jet and strongly interacts with it. The proposed decomposition method achieves flow filtering based on the local spatial correlations. This is different from the existing decomposition methods, e.g. classical Reynolds decomposition and moving-average (spatial) filtering with a constant filter size based on the first baroclinic Rossby deformation radius. Next, we characterize the dynamical impacts of the resulting eddy forcing on the large-scale flow in terms of their mutual time-lagged spatial correlations, formulated as product integral characteristics. Its temporal statistics uncover robust causality between the eddy forcing and the induced large-scale potential vorticity anomalies – referred to as the eddy backscatter. The results also prove the significance of the transient eddy forcing and the time lag dependence of the eddy backscatter. We argue that these properties are to be considered by eddy parametrization schemes. We further used the decomposed eddy fields to augment a coarse-resolution ocean model. The augmented solution statistically reproduces the missing eastward jet extension, enhances the eddy activities around it and recovers the essential large-scale low-frequency variability. This justifies a reduced-order statistical emulation of the eddies – an emerging methodology for including eddy effects in non-eddy-resolving ocean models.

This work investigates how turbulence in open-channel flows is altered by the passage of surface waves by using experimental data collected with laboratory tests in a large-scale flume facility, wherein waves followed a current. Flow velocity data were measured with a laser Doppler anemometer and used to compute profiles of mean velocity and Reynolds stresses, and pre-multiplied spectra. The velocity signal containing contributions from the mean flow, wave motion and turbulence was decomposed using the empirical mode decomposition (EMD), which is considered a promising tool for the analysis of velocity time series measured in complex flows. A novel outer length scale $h_{0}$ is proposed which separates the flow into two regions depending on the competition between the vertical velocities associated with the wave motion and the turbulent velocities imposed by the current. This outer length scale allows for the identification of a genuine overlap layer and an insightful scaling of turbulent statistics in the current-dominated flow region (i.e. $y/h_{0} < 1$). As the wave contribution to the vertical velocity increases, the pre-multiplied spectra reveal two intriguing features: (i) in the current-dominated flow region, the very large-scale motions (VLSMs) are progressively weakened but attached eddies are still present; and (ii) in the wave-dominated flow region (i.e. $y/h_{0} > 1$), a new spectral signature associated with long turbulent structures (approximately 6 and 25 times the flow depth $h$) appears. These longitudinal structures present in the wave-dominated flow region seem to share many features with Langumir-type cells.

Motivated by biological systems, such as human hearts and the propulsors of aquatic creatures, the interaction between deformable structures and fluid jets has drawn considerable attention to understand the mechanism of effective fluid transport through such jets. In this study, the formation of a starting jet through a novel eversion process is investigated experimentally using a simple vortex generator model with everted sheets. The ends of two everted sheets are clamped at either side of a rectangular flow channel, with the other free ends in contact with each other in the middle of the channel. Geometric and kinematic parameters, such as the length and bending rigidity of the everting sheets and the speed of the piston, are varied to examine their effects on the deformation of the sheets and the formation of the jet. By introducing a dimensionless bending rigidity, the behaviour of the sheets during the eversion process can be correlated with jet characteristics such as the velocity profile and hydrodynamic impulse. The interaction between the starting jet and the everting sheets enables a notably faster jet with an improved hydrodynamic impulse to be developed within a shorter stroke time.

A large-Reynolds-number asymptotic reduction of the Navier–Stokes equations capable of describing a locally periodic vortex-wave array and the associated large-scale variation of the mean-shear velocity field first suggested in Hall (J. Fluid Mech., vol. 850, 2018, pp. 46–82) is extended. The sustaining process of the locally periodic coherent structures is based on the vortex-wave interaction theory of Hall & Smith (J. Fluid Mech., vol. 227, 1991, 641–666), wherein two-dimensional roll–streak fields are supported by localised nonlinear self-interactions of three-dimensional waves that are largest in size within critical layers of the streak field. The variation of the mean velocity is made possible by incorporating a slow change to the mean profile using a Wentzel–Kramers–Brillouin-type approach. As the first extension, we demonstrate that the local structure corresponds to the asymptotic limit of computations in a shearing box. A variety of solutions with different symmetry properties are found via the hybrid numerical asymptotic approach of Blackburn, Hall & Sherwin (J. Fluid Mech. vol. 721, 2013, 58–85). Moreover, some solutions show generic flow features such as uniform momentum zones and spatial intermittency known to occur in near-wall turbulent boundary layers. We extend the vortex-wave interaction array theory to show that, in addition to a Reynolds-averaged-Navier–Stokes-type relationship between the large-scale vertical variation of the mean flow and local roll–streak scale, a higher-order analysis gives a second constraint on the slow-scale dynamics. Those constraints are used for the first time to derive the logarithmic law of the wall through a closed asymptotic analysis of self-similar local coherent structures, consistent with the attached eddy hypothesis.

We present an optimization-based method to efficiently calculate accurate nonlinear models of Taylor vortex flow. Through the resolvent formulation of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), we model these Taylor vortex solutions by treating the nonlinearity not as an inherent part of the governing equations but rather as a triadic constraint which must be satisfied by the model solution. We exploit the low-rank linear dynamics to calculate an efficient basis, the coefficients of which are calculated through an optimization problem to minimize the triadic consistency of the solution with itself as well as the input mean flow. Our approach constitutes, what is to our knowledge, the first fully nonlinear and self-sustaining, resolvent-based model in the literature. Our results compare favourably with the direct numerical simulation (DNS) of Taylor–Couette flow at up to five times the critical Reynolds number. Additionally, we find that as the Reynolds number increases, the flow undergoes a fundamental transition from a classical weakly nonlinear regime, where the forcing cascade is strictly down scale, to a fully nonlinear regime characterized by the emergence of an inverse (up scale) forcing cascade. Triadic contributions from the inverse and traditional cascade destructively interfere implying that the accurate modelling of a certain Fourier mode requires knowledge of its immediate harmonic and sub-harmonic. We show analytically that this is a direct consequence of the structure of the quadratic nonlinearity of the Navier–Stokes equations formulated in Fourier space. Finally, using our model as an initial condition to a higher Reynolds number DNS significantly reduces the time to convergence.

Turbulent mixing layers in nature are often characterised by the presence of a mean shear and an unstable buoyancy gradient between two streams of different velocities. Depending on the relative strength of shear versus buoyancy, either the former or the latter may dominate the turbulence and mixing between the two streams. In this paper, we present a phenomenological theory that leads to the identification of two distinct turbulent regimes: an early regime, dominated by mean shear, and a later regime dominated by buoyancy. The main theoretical result consists of the identification of a cross-over timescale that distinguishes between the shear- and the buoyancy-dominated turbulence. This cross-over time depends on three large-scale constants of the flow, namely, the buoyancy difference, the velocity difference between the two streams and the gravitational acceleration. We validate our theory against direct numerical simulations of a temporal turbulent mixing layer compounded with an unstable stratification. We observe that the cross-over time correctly predicts the transition from shear- to buoyancy-driven turbulence, in terms of turbulent kinetic energy production, energy spectra scaling and mixing layer thickness.

To mimic the unsteady vortex–wall interaction of animal propulsion in a canonical test case, a vorticity-annihilating boundary layer was examined through the spin-down of a vortex from solid-body rotation. A cylindrical, water-filled tank was rapidly stopped, and the decay of the vortex from solid-body rotation was observed by means of planar and stereo particle image velocimetry. High Reynolds-number ($Re$) measurements were achieved by combining a large-scale facility (diameter, $D=13\ \textrm {m}$) with a novel approach to reduce end-wall effects. The influence of the boundary-layer formation at the tank's bottom wall was minimised by introducing a saturated salt-water layer. The experimental efforts have allowed us to assess the $Re$ dependency of the laminar–turbulent transition of the vorticity-annihilating side-wall boundary layer at scales similar to large cetaceans. The scaling of the transition mechanism and its onset time were found to agree with predictions from linear stability analysis. Furthermore, the growth rate of the curved turbulent boundary layer was also in good agreement with an empirical scaling formulated in the literature for much smaller $Re$. Eventually, the scaling of vorticity annihilation was addressed. The earlier onset of transition at high $Re$ compensates for the reduced effects of viscosity, leading to similar vorticity annihilation rates during the early stages of the spin-down for a wide $Re$ range.

The statistical relation between ocean wave geometry and water particle movements can be formulated in the stochastic Gauss–Lagrange model. In this paper we use Slepian models to obtain detailed information of the sea surface elevation in the neighbourhood of local maxima in a Gaussian wave model and of the movements of the top particle of the waves. We present full conditional distributions of the Gaussian vertical and horizontal movements in the Gauss–Lagrange model, and represent them as one regression component depending on the height and curvature at the wave maxima and one residual component. These conditional distributions define the explicit vertical and horizontal Slepian models. The Slepian models are used to simulate individual min–max–min waves in space, in particular their front–back asymmetry, and the velocity vector of the particle at the wave maximum. We find that there is a strong relation between the degree of front–back wave asymmetry and the direction of the particle movement. We discuss the role of second-order corrections to the Gaussian components and find only minor effects for the sea states studied. The Slepian model is shown to be an efficient tool to obtain detailed information about Gaussian and related models in the neighbourhood of critical points, without the need for time and space consuming simulations. In particular, they permit easy simulation of shape and kinematics of rare extreme waves.

The local streamline topology classification method of Chong et al. (Phys. Fluids A: Fluid Dyn., vol. 2, no. 5, 1990, pp. 765–777) is adapted and extended to describe the geometry of infinitesimal vortex lines. Direct numerical simulation (DNS) data of forced isotropic turbulence reveals that the joint probability density function (p.d.f.) of the second ($q_\omega$) and third ($r_\omega$) normalized invariants of the vorticity gradient tensor asymptotes to a self-similar bell shape for $Re_\lambda > 200$. The same p.d.f. shape is also seen at the late stages of breakdown of a Taylor–Green vortex suggesting the universality of the bell-shaped p.d.f. form in turbulent flows. Additionally, vortex reconnection from different initial configurations is examined. The local topology and geometry of the reconnection bridge is shown to be nearly identical in all cases considered in this work. Overall, topological characterization of the vorticity field provides a useful analytical basis for examining vorticity dynamics in turbulence and other fluid flows.

This paper characterises the steady and time-periodic behaviour of swirling jets using numerical bifurcation analysis. Its objective is to elucidate the dynamics of fully developed, unconfined, laminar swirling jets under variations in the Reynolds number $\textit {Re}$ and swirl ratio $S$. Within the $(0,0)\leq (\textit {Re},S)\leq (300,3)$ range, the steady, axisymmetric flow exhibits several distinct patterns ranging from a quasi-columnar jet along the central axis at low $S$ to a radial jet attached to the containing wall at high $S$ with various forms of vortex breakdown in between. A cusp bifurcation appears in the steady solution manifold which triggers bistable behaviour due to a competition between inner and outer low pressure regions associated with vortex breakdown and entrainment of the ambient fluid, respectively. Instability of the steady flow is linked to eigenmodes which are singly ($|m|=1$) or doubly ($|m|=2$) azimuthally periodic, although additional instabilities with other azimuthal wavenumbers occur at $(\textit {Re},S)$-values beyond the leading neutral curves. The various branches of limit cycle solutions stemming from these neutral curves are associated with both super- and sub-critical Hopf bifurcations. The resulting unsteady flow fields exhibit a wide array of rotating, three-dimensional flow structures, and comparisons between the time-averaged and steady flow patterns highlight the role of these unsteady nonlinear interactions on the overall behaviour of swirling jets. Similarities and differences between this laterally unconfined jet and broader classes of swirling flows, including confined swirling jets and unconfined vortex models, are also discussed.

Through laboratory experiments and numerical simulations, we examine the evolution of buoyant plumes as they are influenced by background rotation in a uniform density ambient fluid. The source Rossby number is sufficiently large that rotation does not directly affect the plume at early times. However, on a time scale of the order of half a rotation period, the plume becomes deflected from the vertical axis. For some experiments and simulations, the deflection persists and the flow precesses about the vertical axis. In other cases, shortly after being deflected, the plume laminarizes near the source to form a near-vertical columnar vortex, which we refer to as a ‘tornado’. Tornado formation occurs in some experiments and not in others even if the source and background rotation parameters are identical. However, their formation is more likely if the plumes are ‘lazy’. Simulations reveal that this is a consequence of the competing dynamics that occurs on comparable time scales. As a consequence of entrainment, vertical vorticity builds up within the plume reducing the Rossby number and suppressing vertical motion at distances progressively closer to the source. Meanwhile, the swirl (the ratio of the azimuthal to vertical flow) around the vicinity of the source increases, which tends to suppress three-dimensional turbulence in the near-source flow. Although the former process ultimately acts to deflect the plume off axis, in some instances, the swirl around the source succeeds in laminarizing the flow, resulting in tornado formation.