A temporal complex network-based approach is proposed as a novel formulation to investigate turbulent mixing from a Lagrangian viewpoint. By exploiting a spatial proximity criterion, the dynamics of a set of fluid particles is geometrized into a time-varying weighted network. Specifically, a numerically solved turbulent channel flow is employed as an exemplifying case. We show that the time-varying network is able to clearly describe the particle swarm dynamics, in a parametrically robust and computationally inexpensive way. The network formalism enables us to straightforwardly identify transient and long-term flow regimes, the interplay between turbulent mixing and mean flow advection and the occurrence of proximity events among particles. Thanks to their versatility and ability to highlight significant flow features, complex networks represent a suitable tool for Lagrangian investigations of turbulent mixing. The present application of complex networks offers a powerful resource for Lagrangian analysis of turbulent flows, thus providing a further step in building bridges between turbulence research and network science.

Nominally zero-pressure-gradient fully developed flat-plate turbulent boundary layers with heated and unheated isothermal walls at supercritical pressures are studied by solving the full compressible Navier–Stokes equations using direct numerical simulation. With a heated isothermal wall, the wall temperature sets such that the flow temperature varies through the pseudo-critical temperature, and thus pseudo-boiling phenomena occur within the boundary layers. The pseudo-boiling process induces strongly nonlinear real-fluid effects in the flow and interacts with near-wall turbulence. The peculiar abrupt density variations through the pseudo-boiling process induce significant near-wall density fluctuations $\sqrt{\overline{\unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D70C}^{\prime }}}/\overline{\unicode[STIX]{x1D70C}}\approx 0.4{-}1.0$ within the heated transcritical turbulent boundary layers. The large near-wall density fluctuations induce a turbulent mass flux $\unicode[STIX]{x1D70C}^{\prime }u_{i}^{\prime }$, and the turbulent mass flux amplifies the Favre-averaged velocity fluctuations $u_{i}^{\prime \prime }$ in the near-wall predominant structures of streamwise low-speed streaks that are associated with the ejection (where $u^{\prime \prime }<0$ and $v^{\prime \prime }>0$), while reducing the velocity fluctuations in the high-speed streaks associated with the sweep ($u^{\prime \prime }>0$ and $v^{\prime \prime }<0$). Although the near-wall low-speed and high-speed streak structures dominate the Reynolds-shear-stress generation, the energized Favre-averaged velocity fluctuations in the low-speed streaks enhance both the mean-density- and density-fluctuation-related Reynolds shear stresses ($-\overline{\unicode[STIX]{x1D70C}}\overline{u^{\prime \prime }v^{\prime \prime }}$ and $-\overline{\unicode[STIX]{x1D70C}^{\prime }u^{\prime \prime }v^{\prime \prime }}$) in the ejection event and, as a result, alter the Reynolds-shear-stress profile. The large density fluctuations also alter the near-wall viscous-stress profile and induce a near-wall convective flux $-\overline{\unicode[STIX]{x1D70C}}\widetilde{u}\widetilde{v}$ (due to non-zero $\widetilde{v}$). The changes in the contributions in the stress-balance equation result in a failure of existing velocity transformations to collapse to the universal law of the wall. The large density fluctuations also greatly contribute to the turbulent kinetic energy budget, and especially the mass flux contribution term becomes noticeable as one of the main positive terms. The unheated non-transcritical turbulent boundary layers show a negligible contribution of the real-fluid effects, and the turbulence statistics agree well with the statistics of an incompressible constant-property turbulent boundary layer with a perfect-gas law.

The role of a transversal concentration gradient in detonation propagation in a two-dimensional channel filled with an $\text{H}_{2}{-}\text{O}_{2}$ mixture is examined by high-resolution simulation. Results show that, compared to propagation in homogeneous media, a concentration gradient reduces the average detonation velocity because of the delay in reaching downstream reaction equilibrium, leading to a large amount of unreacted $\text{H}_{2}$ and hence significant species fluctuations. The transversal concentration gradient also enhances the cellular detonation instability. Steepening it reduces considerably the number of triple points on the front, lengthens the global detonation front structure on average and consequently increases the deficit of the average detonation velocity. It is further found that the interaction of the leading shock with the transversal concentration gradient influences the formation of local $\text{H}_{2}$ bump and thus the unreacted pocket behind the front, while the transverse wave causes mixing and burning of the residue fuel downstream. Nevertheless, for the steepened concentration gradient, a transverse detonation is present and consumes the fuel in the compressed and preheated zone by the leading shock; consequently, the detonation velocity deficit is not increased significantly for detonation with the single-head propagation mode close to the limit.

The dynamics of spreading of a macroscopic liquid droplet over a wetting surface is often described by a power-law relaxation, namely, the droplet radius increases as $t^{m}$ for time $t$, which is known as Tanner’s law. Here we show, by both experiments and theory, that when the liquid spreading takes place between a thin soap film and a glass fibre penetrating the film, the spreading is significantly slowed down. When the film thickness $\ell$ becomes smaller than the fibre diameter $d$, the strong hydrodynamic confinement effect of the soap film gives rise to a logarithmic relaxation with fibre creeping time $t$. Such a slow dynamics of spreading is observed for hours both in the measured time-dependent height of capillary rise $h(t)$ on the fibre surface and viscous friction coefficient $\unicode[STIX]{x1D709}_{s}(t)$ felt by the glass fibre in contact with the soap film. A new theoretical approach based on the Onsager variational principle is developed to describe the dynamics of thin film spreading along a fibre. The newly derived equations of motion provide the analytical solutions of $h(t)$ and contact angle $\unicode[STIX]{x1D703}(t)$, which are found to be in good agreement with the experimental results. Our work thus provides a common framework for understanding the confinement effect of thin soap films on the dynamics of spreading along a fibre.

Direct numerical simulations are used to investigate the nature of fully resolved small-scale convection and its role in large-scale circulation in a rotating $f$-plane rectangular basin with imposed surface temperature difference. The large-scale circulation has a horizontal geostrophic component and a deep vertical overturning. This paper focuses on convective circulation with no wind stress, and buoyancy forcing sufficiently strong to ensure turbulent convection within the thermal boundary layer (horizontal Rayleigh numbers $Ra\approx 10^{12}{-}10^{13}$). The dynamics are found to depend on the value of a convective Rossby number, $Ro_{\unicode[STIX]{x0394}T}$, which represents the strength of buoyancy forcing relative to Coriolis forces. Vertical convection shifts from a mean endwall plume under weak rotation ($Ro_{\unicode[STIX]{x0394}T}>10^{-1}$) to ‘open ocean’ chimney convection plus mean vertical plumes at the side boundaries under strong rotation ($Ro_{\unicode[STIX]{x0394}T}<10^{-1}$). The overall heat throughput, horizontal gyre transport and zonally integrated overturning transport are then consistent with scaling predictions for flow constrained by thermal wind balance in the thermal boundary layer coupled to vertical advection–diffusion balance in the boundary layer. For small Rossby numbers relevant to circulation in an ocean basin, vertical heat transport from the surface layer into the deep interior occurs mostly in ‘open ocean’ chimney convection while most vertical mass transport is against the side boundaries. Both heat throughput and the mean circulation (in geostrophic gyres, boundary currents and overturning) are reduced by geostrophic constraints.

The present study deals with a theoretical investigation of a close-contact melting (CCM) process involving a vertical cylinder on a horizontal isothermal surface, where the liquid phase is a non-Newtonian viscoplastic fluid that behaves according to the Bingham model. Accordingly, a new approach is formulated based on the thin layer approximation and different quasi-steady process assumptions. By analytical derivation, an algebraic equation that relates the molten layer thickness and the solid bulk height is developed. The problem is then solved numerically, coupled with another equation for the melting rate. The new model shows that as the yield stress increases the melting rate decreases and the molten layer thickness increases. It is found that under certain conditions, the model can be reduced to a form that allows an analytical solution. The approximate model predicts an exponential dependence of both the melt fraction and the molten layer thickness. Comparison between the numerical and analytical solutions shows that the analytical approximation provides an excellent estimation for sufficiently large values of the yield stress. Dimensional analysis, which is supported by the analytical model results, reveals the dimensionless groups that govern the problem. For the general case, the melt fraction is a function of two dimensionless groups. For the analytical approximation, it is shown that the melt fraction is governed by a single dimensionless group and that the molten layer thickness is governed by two dimensionless groups.

Numerous experimental, numerical and theoretical studies have shown that core annular flows can be unstable. This instability can be convective or absolute in different situations: miscible fluids with matched density but different viscosities, creeping flow of two immiscible fluids or buoyant flow along a fibre. The analysis of the linear stability of the flow equation of two fluids injected in a co-current and concentric manner into a cylindrical tube leads to a rather complex eigenvalue problem. Until now, all analytical solution to this problem has involved strong assumptions (e.g. lack of inertia) or approximations (e.g. developments at long or short wavelengths) even for axisymmetric disturbances. However, in this latter case, following C. Pekeris, who obtained, almost seventy years ago, an elegant explicit solution for the dispersion relationship of the flow of a single fluid, we derive an explicit solution for the more general case of two immiscible fluids of different viscosity, density and inertia separated by a straight interface. This formulation is well adapted to commercial software. First, we review the creeping flow limit (zero Reynolds number) of two immiscible fluids as it is used in microfluidics. Secondly, we consider the case of two fluids of different viscosities but of the same density in the absence of surface tension and also without diffusion (i.e. miscible fluids with infinite Schmidt number). In both cases, we study the transition from convective to absolute instability according to the different control parameters.

We consider here three-dimensional water flows in the $\unicode[STIX]{x1D6FD}$-plane approximation. In a quite general setting we show that the only flow exhibiting a constant vorticity vector is the stationary flow with vanishing velocity field and flat free surface.

We discuss a fluid dynamic variant of the classical Bernoulli’s brachistochrone problem. The classical brachistochrone for a non-dissipative particle is governed by maximization of the particle’s kinetic energy, resulting in a cycloid. We consider a variant where the particle is replaced by a cylinder (bottle) filled with a viscous fluid and attempt to identify the shape of the curve connecting two points along which the bottle would move in the shortest time. We derive the system of integro-differential equations governing system dynamics for a given shape of the curve. Using these equations, we pose the brachistochrone problem by invoking an optimal control formalism and show that (in general) the curve deviates from a cycloid. This is due to the fact that increasing the rate of change of the bottle’s kinetic energy is accompanied by increased viscous dissipation. We show that the bottle motion is governed by a balance between the desire to minimize travel time and the need to reach the end point in the face of increased dissipation. The trade-off between these two physical forces plays a vital role in determining the brachistochrone of a fluid-filled cylinder. We show that in the two limits of either vanishing or high viscosity, the brachistochrone for this problem reduces to a cycloid. An intermediate viscosity range is identified where the fluid brachistochrone is non-cycloidal. Finally, we show the relevance of these results to the dynamics of a rolling liquid marble.

Fast tidal streams are a promising source of clean, predictable power, but the task of arranging tidal turbines for maximum power capture is complicated. Actuator disc models, such as the two-scale actuator disc theory, have proven useful in seeking optimal turbine arrangements, yet these models assume flows that are frictionless and steady, and thus quite unlike the channel flow conditions that actual tidal turbines experience. In this paper, we use numerical methods to relax these assumptions and explore how optimal turbine arrangements change as the flow transitions from frictionless and steady to rough and oscillatory. In so doing, we show that, under certain conditions, the assumption of quasi-steady flow in models of tidal turbines may neglect leading-order physics. When the ratio of drag to inertial forces in the unexploited channel is very low, for instance, the optimal turbine arrangements are found to be quite different, and the potential for enhanced power capture is found to be much greater than predicted by two-scale actuator disc theory.

We report an abrupt change in the diffusive transport of inertial objects in wave-driven turbulence as a function of the object size. In these non-equilibrium two-dimensional flows, the turbulent diffusion coefficient $D$ of finite-size objects undergoes a sharp change for values of the object size $r_{p}$ close to the flow forcing scale $L_{f}$. For objects larger than the forcing scale ($r_{p}>L_{f}$), the diffusion coefficient is proportional to the flow energy $U^{2}$ and inversely proportional to the size $r_{p}$. This behaviour, $D\sim U^{2}/r_{p}$ , observed in a chaotic macroscopic system is reminiscent of a fluctuation–dissipation relation. In contrast, the diffusion coefficient of smaller objects ($r_{p}<L_{f}$) follows $D\sim U/r_{p}^{0.35}$. This result does not allow simple analogies to be drawn but instead it reflects strong coupling of the small objects with the fabric and memory of the out-of-equilibrium flow. In these turbulent flows, the flow structure is dominated by transient but long-living bundles of fluid particle trajectories executing random walk. The characteristic widths of the bundles are close to $L_{f}$. We propose a simple phenomenology in which large objects interact with many bundles. This interaction with many degrees of freedom is the source of the fluctuation–dissipation-like relation. In contrast, smaller objects are advected within coherent bundles, resulting in diffusion properties closely related to those of fluid tracers.

We develop a bifurcation theory describing the conditions under which vortices are created or destroyed in a two-dimensional incompressible flow. We define vortices using the $Q$-criterion and analyse the vortex structure by considering the evolution of the zero contours of $Q$. The theory identifies topological changes of the vortex structure and classifies these as four possible types of bifurcations, two occurring away from boundaries, and two occurring near no-slip walls. Our theory provides a description of all possible codimension-one bifurcations where time is treated as the bifurcation parameter. To illustrate our results, we consider the early stages of boundary layer eruption at moderate Reynolds numbers in the range from $Re=750$ to $Re=2250$. By analysing numerical simulations of the phenomenon, we show how to describe the eruption process as sequences of the four possible bifurcations of codimension one. Our simulations show that there is a single codimension-two point within our parameter range. This codimension-two point arises at $Re=1817$ via the coalescence of two codimension-one bifurcations which are associated with the creation and subsequent destruction of one of the vortices that erupt from the boundary layer. We present a theoretical description of this process and explain how the occurrence of this phenomenon separates the parameter space into two regions with distinct evolution of the topology of the vortices.

We consider the response to periodic forcing between 5 Hz and 50 Hz of an interface separating immiscible fluids under the microgravity conditions of a parabolic flight. Two pairs of liquids with viscosity ratios differing by one order of magnitude are investigated. By combining experimental data with numerical simulations, we describe a variety of dynamics including harmonic and subharmonic (Faraday) waves, frozen waves and drop ejection, determining their thresholds and scaling properties when possible. Interaction between these various modes is facilitated in microgravity by the relative ease with which the interface can move, altering its orientation with respect to the forcing axis. The effects of key factors controlling pattern selection are analysed, including vibrational forcing, viscosity ratio, finite-size effects and residual gravity. Complex behaviour often arises with features on several spatial scales, such as Faraday waves excited on the interface of a larger columnar structure that develops due to the frozen wave instability – this type of state was previously seen in miscible fluid experiments but is described for the first time here in the immiscible case.

An inviscid analytical theory of a slow steady liquid mass rotation during the swirl-type sloshing in a vertical circular cylindrical tank with a fairly deep depth is proposed by utilising the asymptotic steady-state wave solution by Faltinsen et al. (J. Fluid Mech., vol. 804, 2016, pp. 608–645). The tank performs a periodic horizontal motion with the forcing frequency close to the lowest natural sloshing frequency. The azimuthal mass transport (first observed in experiments by Prandtl (Z. Angew. Math. Mech., vol. 29(1/2), 1949, pp. 8–9)) is associated with the summarised effect of a vortical Eulerian-mean flow, which, as we show, is governed by the inviscid Craik–Leibovich equation, and an azimuthal non-Eulerian mean. Suggesting the mass-transport velocity tends to zero when approaching the vertical wall (supported by existing experiments) leads to a unique non-trivial solution of the Craik–Leibovich boundary problem and, thereby, gives an analytical expression for the summarised mass-transport velocity within the framework of the inviscid hydrodynamic model. The analytical solution is validated by comparing it with suitable experimental data.

Particle loading affects the dynamics of buoyant plumes, since the difference between particle and fluid densities is much greater than that in the fluid alone. In stratified environments, plume rise is density limited; after initial overshoot, the plume reaches a terminal level and spreads radially. Particles dropping from this horizontal intrusion may be re-entrained. This recycling of dense matter reduces plume buoyancy and intrusion height and, for sufficient load, can lead to plume collapse. Entrainment-based formulae yield a steady-state plume rise. We identify a new conserved quantity for such plumes. Integrating paths of particles dropping from the intrusion yields the fraction re-entrained. A simple mathematical model predicts from buoyancy ratio at source ($P=$ negative particle buoyancy divided by positive fluid buoyancy) whether a particle-laden plume will collapse. Under this model, for small settling velocity, a particle-laden plume will not collapse if $P<0.368$. Above this, collapse depends also on the amount of particle-free ambient fluid entrained in the overshoot region. For pure plumes, experimental evidence suggests that this is small. For forced plumes, more substantial overshoot and entrainment is shown to increase the critical ratio. An extension, based on successive recycling, estimates time to collapse. To investigate further we develop a simple computational model, coupling a ‘top-hat’ plume model, an analytical formula for radially decaying concentrations in the intrusion and an axisymmetric finite-volume solution for time-dependent settling and entrainment. The model can predict the impact of particle load on final rise, as well as the occurrence and time scales of plume collapse.

Plasma synthetic jet actuators (PSJAs) are particularly suited for high-Reynolds-number, high-speed flow control due to their unique capability of generating supersonic pulsed jets at high frequency (${>}5$  kHz). Different from conventional synthetic jets driven by oscillating piezoelectric diaphragms, the exit-velocity variation of plasma synthetic jets (PSJs) within one period is significantly asymmetric, with ingestion being relatively weaker (less than $20~\text{m}~\text{s}^{-1}$) and longer than ejection. In this study, high-speed phase-locked particle image velocimetry is employed to investigate the interaction between PSJAs (round exit orifice, diameter 2 mm) and a turbulent boundary layer at constant Strouhal number (0.02) and increasing mean velocity ratio ($r$, defined as the ratio of the time-mean velocity over the ejection phase to the free-stream velocity). Two distinct operational regimes are identified for all the tested cases, separated by a transition velocity ratio, lying between $r=0.7$ and $r=1.0$. At large velocity and stroke ratios (first regime, representative case $r=1.6$), vortex rings are followed by a trailing jet column and tilt downstream initially. This downstream tilting is transformed into upstream tilting after the pinch-off of the trailing jet column. The moment of this transformation relative to the discharge advances with decreasing velocity ratio. Shear-layer vortices (SVs) and a hanging vortex pair (HVP) are identified in the windward and leeward sides of the jet body, respectively. The HVP is initially erect and evolves into an inclined primary counter-rotating vortex pair ($p$-CVP) which branches from the middle of the front vortex ring and extends to the near-wall region. The two legs of the $p$-CVP are bridged by SVs, and a secondary counter-rotating vortex pair ($s$-CVP) is induced underneath these two legs. At low velocity and stroke ratios (second regime, representative case $r=0.7$), the trailing jet column and $p$-CVP are absent. Vortex rings always tilt upstream, and the pitching angle increases monotonically with time. An $s$-CVP in the near-wall region is induced directly by the two longitudinal edges of the ring. Inspection of spanwise planes ($yz$-plane) reveals that boundary-layer energization is realized by the downwash effect of either vortex rings or $p$-CVP. In addition, in the streamwise symmetry plane, the increasing wall shear stress is attributed to the removal of low-energy flow by ingestion. The downwash effect of the $s$-CVP does not benefit boundary-layer energization, as the flow swept to the wall is of low energy.

In this paper, we formulate the dynamic equations of radial and surface modes for a cylindrical cavitation bubble subject to a prescribed uniform rectilinear vortex flow. The potential flow in the bulk volume of the external flow is modelled using the general mode decomposition approach. The stability of surface modes is investigated under linear analysis. The effects of confinement due to a limited flow domain in a water tunnel and viscosity at the bubble surface are evaluated, which can be fairly neglected for the cylindrical cavitation bubbles discussed. Our model is capable of predicting the developments of surface modes, which agrees well with experimental observations reported in the literature. We derive the Mathieu structure in the dynamic equation of the surface oscillation and the associated instability condition of the surface mode oscillations. The numerical results confirm that the Mathieu-type instability controls the stability diagrams and the emergence of surface modes under specific radial oscillation.

We present the turbulent kinetic energy (t.k.e.) budget of a dilute bubble plume in its asymptotic state. The budget is derived from an experimental dataset of bubble plumes formed inside an unstratified water tank. The experiments cover both the adjustment phase and asymptotic state of the plume. The diameters $d$ of air bubbles are in the range 1–4 mm and the air void fraction $\unicode[STIX]{x1D6FC}_{g}$ is between 0.7 % and 1.8 %. We measured the three components of the instantaneous liquid velocity vector with a profiling acoustic Doppler velocimeter. From the experiments, we found the following inside the heterogeneous bubble core of the plume: (i) the probability density functions of the standardized liquid fluctuations are very similar to those of homogeneous bubble swarms rising with and without background liquid turbulence; (ii) the characteristic temporal frequency $f_{cwi}$ at which bubbles inject t.k.e. into the liquid agrees with the prediction $f_{cwi}=0.14u_{s}/d$ observed and theoretically derived for homogeneous bubble swarms ($u_{s}$ is the bubble slip velocity); (iii) the liquid turbulence is anisotropic with the ratio of turbulence intensities between the vertical and horizontal components in the range 1.9–2.1; (iv) the t.k.e. production by air bubbles is much larger than that by liquid mean shear; and (v) an increasing fraction of the available work done by bubbles is deposited into liquid turbulence as one moves away from the plume centreline. Together with the existing knowledge of homogeneous bubble swarms, our results of the heterogeneous bubble plume support the view that millimetre-sized bubbles create specific patterns of liquid fluctuations that are insensitive to flow conditions and can therefore be possibly modelled by a universal form.

Direct numerical simulations of turbulent channels with rough walls are conducted in the transitionally rough regime. The effect that roughness produces on the overlying turbulence is studied using a modified triple decomposition of the flow. This decomposition separates the roughness-induced contribution from the background turbulence, with the latter essentially free of any texture footprint. For small roughness, the background turbulence is not significantly altered, but merely displaced closer to the roughness crests, with the change in drag being proportional to this displacement. As the roughness size increases, the background turbulence begins to be modified, notably by the increase of energy for short, wide wavelengths, which is consistent with the appearance of a shear-flow instability of the mean flow. A laminar model is presented to estimate the roughness-coherent contribution, as well as the displacement height and the velocity at the roughness crests. Based on the effects observed in the background turbulence, the roughness function is decomposed into different terms to analyse different contributions to the change in drag, laying the foundations for a predictive model.

Stuart vortices are among the few known smooth explicit solutions of the planar Euler equations with a nonlinear vorticity, and they have a counterpart for inviscid flow on the surface of a fixed sphere. By means of a perturbative approach we adapt the method used to investigate Stuart vortices on a fixed sphere to provide insight into some large-scale shallow-water flows on a rotating sphere that model the dynamics of ocean gyres.