In the presence of inertia-gravity waves, the geostrophic and hydrostatic balance that characterises the slow dynamics of rapidly rotating, strongly stratified flows holds in a time-averaged sense and applies to the Lagrangian-mean velocity and buoyancy. We give an elementary derivation of this wave-averaged balance and illustrate its accuracy in numerical solutions of the three-dimensional Boussinesq equations, using a simple configuration in which vertically planar near-inertial waves interact with a barotropic anticylonic vortex. We further use the conservation of the wave-averaged potential vorticity to predict the change in the barotropic vortex induced by the waves.

Regulation of fluid flow by deformations of the surrounding elastic structure is observed in many natural and artificial system, such as in the cardiovascular system. As the first step to study the regulation of oscillating flows, we consider synchronization of vortex shedding past a cylinder within an elastic structure with a sinusoidal external forcing. We use phase-reduction theory to evaluate the synchronization characteristics of the oscillating fluid–structure coupled dynamics. We find that the phase-sensitivity function, which characterizes the phase-response of the oscillation, is significantly affected by the Cauchy number and slightly affected by the fluid-to-structure density ratio and Poisson's ratio of the structure material, for fixed model configuration and Reynolds number. The predicted synchronization characteristics are in close agreement with results from direct numerical simulations. The synchronization region is maximized when the sinusoidal perturbation is applied near the downstream end of the cylinder. These findings open further possibility for the utilization of phase-reduction theory to characterize synchronization in other practical problems exhibiting fluid–structure coupled dynamics, such as in biological systems and the control of microfluidics.

The effect of a small background constant horizontal mean flow on the parametric subharmonic instability (PSI) of locally confined internal wave beams is discussed for the case where the beam frequency is close to twice the inertial frequency due to background rotation. Under this condition, PSI is particularly potent because of the vanishing of the group velocity at the inertial frequency, which prolongs contact of near-inertial subharmonic perturbations with the primary wave. The mean flow generally stabilizes the very short-scale limit of such perturbations. By contrast, the stability of longer-scale perturbations hinges on the strength and the direction of the mean flow; particularly, a negative mean flow (antiparallel to the horizontal projection of the beam group velocity) can extend the sub-inertial range of PSI. However, a large enough mean flow of either sign ultimately weakens PSI.

Numerical simulations are used to probe Rayleigh–Darcy convection in fluid-saturated porous media towards the ultimate regime. The present three-dimensional dataset, up to Rayleigh–Darcy number $\textit {Ra}=8\times 10^4$, suggests that the appropriate scaling of the Nusselt number is $\textit {Nu}=0.0081\textit {Ra}+0.067\textit {Ra}^{0.61}$, fitting the computed data for $\textit {Ra}\gtrsim 10^3$. Extrapolation of current predictions to the ultimate linear regime yields the asymptotic law $\textit {Nu}=0.0081 \textit {Ra}$, about $16\,\%$ less than indicated in previous studies. Upon examination of the flow structures near the boundaries, we confirm previous indications of small flow cells hierarchically nesting into supercells, and we show evidence that the supercells at the boundary are the footprints of the megaplumes that dominate the interior part of the flow. The present findings pave the way for more accurate modelling of geophysical systems, with special reference to geological $\textrm {CO}_2$ sequestration.

A surface roughness from a recently cleaned and painted ship's hull was scanned, scaled and replicated for laboratory testing to systematically investigate the influence of the ratio of in-plane roughness wavelength, $\lambda$, with respect to the boundary layer thickness $\delta$. The experiments were performed by geometrically scaling the surface which maintains a constant effective slope $ES_x$ and solidity $\varLambda$, while the ratio of $\lambda /\delta$ is varied. Here we scale the scanned roughness topography by a factor of 2.5 and 15, and measure the mean velocity profiles in the turbulent boundary layers developing over these surfaces at a range of free stream velocities and streamwise measurement locations. The results show that the $2.5\times$ scaled roughness, which has $\lambda /\delta \ll 1$, behaves in the expected $k$-type manner, with a roughness function ${\rm \Delta} U^+$ that is proportional to the viscous-scaled roughness height. The $15\times$ surface, however, which has $\lambda /\delta \approx 1$, exhibits very different non-$k$-type behaviour. This larger surface does not approach the fully rough asymptote and also exhibits a drag penalty that is comparable to the $2.5\times$ case despite the sixfold increase in the roughness height. Measurements on a spanwise–wall-normal plane reveal that the $15\times$ surface has introduced a large-scale spanwise variation in mean streamwise velocity (dispersive stresses) that extend far beyond the logarithmic region. Together this evidence suggests that a demarcation between $k$-type and non-$k$-type behaviour can occur in situations where the in-plane roughness wavelength approaches the boundary layer thickness. This finding has important implications to how we scale small-scale roughness from high Reynolds number (Re) large-scale applications for testing in low Re small-scale laboratory facilities or simulations.

The presence of large-scale coherent structures in various wall bounded turbulent flows, often called superstructures in turbulent boundary layers (TBLs), has been of great interest in recent years. These meandering high- and low-momentum structures can extend up to several boundary layer thicknesses in the streamwise direction and contain a relatively large portion of the layer's turbulent kinetic energy. Therefore, studying these features is important for understanding the overall dynamics of turbulent boundary layers and for the development of flow control strategies or near-wall flow modifications. However, compared to the extensive number of incompressible investigations, much less is known about the structural characteristics for compressible turbulent boundary layer flows. Therefore, in this investigation turbulent boundary layers developing on a flat plate with zero pressure gradient (ZPG) over a range of Reynolds numbers and Mach numbers are considered in order to examine the effect of compressibility on superstructures. More specifically, measurements are performed on a flat plate model in the Trisonic Wind Tunnel Munich (TWM) for the Mach number range $0.3 \leq Ma \leq 3.0$ and a friction Reynolds number range of $4700 \leq Re_{\tau } \leq 29\,700$ or $11\,730 \leq Re_{\delta _2} = \rho _e u_e \theta ^*/\mu _{w} \leq 74\,800$. Velocity fields are recorded using planar particle image velocimetry methods (PIV and stereo-PIV) in three perpendicular planes. Using multi-point correlation and spectral analysis methods it was found that the most energetic frequencies have slightly longer streamwise wavelengths for the supersonic case when compared to the subsonic case. Furthermore, a distinct increase in the spanwise spacing of the superstructures was found for the supersonic cases when compared to the subsonic and transonic turbulent boundary layers.

We analyse the inter-scale transfer of energy for two types of plane Poiseuille flow: the P4U exact coherent state of Park & Graham (J. Fluid Mech., vol. 782, 2015, pp. 430–454) and turbulent flow in a minimal channel. For both flows, the dominant energy-producing modes are streamwise-constant streaks with a spanwise spacing of approximately 100 wall units. Since the viscous dissipation for these scales is not sufficient to balance production, the nonlinear terms redistribute the excess energy to other scales. Spanwise-constant scales (that is, Tollmien–Schlichting-like modes with zero spanwise wavenumber), in particular, account for a significant amount of net energy gain from the nonlinear terms. We compare the energy balance to predictions from resolvent analysis, and we show that it does not model energy transfer well. Nevertheless, we find that the energy transferred from the streamwise-constant streaks can be predicted reasonably well by a Cess eddy viscosity profile. As such, eddy viscosity is an effective model for the nonlinear terms in resolvent analysis and explains good predictions for the most energetic streamwise-constant streaks. It also improves resolvent modes as a basis for structures whose streamwise lengths are greater than their spanwise widths by counteracting non-normality of the resolvent operator. This is quantified by computing the inner product between the optimal resolvent forcing and response modes, which is a metric of non-normality. Eddy viscosity does not respect the conservative nature of the nonlinear energy transfer, which must sum to zero over all scales. Since eddy viscosity tends to remove energy, it is less effective in modelling nonlinear transport for scales that receive energy from the nonlinear terms.

The spatial development of a turbulent boundary layer (TBL) subjected to freestream turbulence (FST) is investigated experimentally in a water channel for friction Reynolds numbers up to $Re_{\tau }=5060$. Four different FST intensities are generated with an active grid, ranging from a low-turbulence reference case to $u^{\prime }_{\infty }/U_{\infty }=12.5\,\%$. Wall-normal velocity scans are performed with laser doppler velocimetry at three positions downstream of the grid. There are two combating influences as the flow develops: the TBL grows while the FST decays. Whilst previous studies have shown the wake region of the TBL is suppressed by FST, the present measurements demonstrate that the wake recovers sufficiently far downstream. For low levels of FST, the near-wall variance peak grows as one moves downstream, whereas high FST results in an initially high variance peak that decays with streamwise position. These results are mirrored in the evolution of the spectrograms, where low FST results in the emergence of an outer spectral peak as the flow evolves, while high FST sees an initially high outer spectral peak decay in space. This finding is significant as it suggests the FST does not permanently mature the TBL ahead of its natural evolution. Finally, it is explicitly demonstrated that it is not sufficient to characterize the TBL solely by conventional parameters such as $Re_{\tau }$, but that the level of FST and the evolution of the two flows must also be considered.

We present the first detailed thermal and velocity field characterization of convection in a rotating cylindrical tank of liquid gallium, which has thermophysical properties similar to those of planetary core fluids. Our laboratory experiments, and a closely associated direct numerical simulation, are all carried out in the regime prior to the onset of steady convective modes. This allows us to study the oscillatory convective modes, sidewall modes and broadband turbulent flow that develop in liquid metals before the advent of steady columnar modes. Our thermo-velocimetric measurements show that strongly inertial, thermal wind flows develop, with velocities reaching those of non-rotating cases. Oscillatory bulk convection and wall modes coexist across a wide range of our experiments, along with strong zonal flows that peak in the Stewartson layer, but that extend deep into the fluid bulk in the higher supercriticality cases. The flows contain significant time-mean helicity that is anti-symmetric across the midplane, demonstrating that oscillatory liquid metal convection contains the kinematic components to sustain system-scale dynamo generation.

In this work, Joukowski's rotor wake model is considered for a two-blade rotor of radius $R_b$ rotating at the angular velocity $\varOmega _R$ in a normal incident velocity $V_{\infty }$. This model is based on a description of the wake by a limited number of vortices of core size $a$: a tip vortex of constant circulation $\varGamma$ for each blade and a root vortex of circulation $-2\varGamma$ on the rotation axis. Using a free-vortex method, we obtain solutions matching uniform interlaced helices in the far field that are steady in the frame rotating with the rotor for a large range of tip-speed ratios $\lambda = R_b \Omega _R/V_{\infty }$ and vortex strengths $\eta = \varGamma /(R_b^{2} \Omega _R)$. Solutions are provided for a two-bladed rotor for both helicopters and wind turbines. Particular attention is brought to the study of the solutions describing steep-descent helicopter flight regimes and large tip-ratio wind turbine regimes, for which the vortex structure is strongly deformed in the near wake and crosses the rotor plane. Both the geometry of the structure and its induced velocity field are analysed in detail. The thrust and the power coefficient of the solutions are also provided and compared to the momentum theory. The stability of the solutions is studied by monitoring the linear spatio-temporal development of a localized perturbation placed at different locations. Good agreements with the theoretical predictions for uniform helices and for point vortex arrays are demonstrated for the stability properties in the far wake. However, a more complex evolution is observed for the more deformed solutions when the perturbation is placed close to the rotor.

The present study focuses on acoustically induced thermal effects on Rayleigh streaming inside a resonator. Firstly, we consider the effect of the transverse (or wall-normal) mean temperature gradient on the acoustic streaming flow generated by a standing wave between two parallel plates. Analytical expressions for acoustic quantities are developed and used to express the sources of linear streaming. The influence of a transverse temperature variation on the streaming velocity is clearly identified through a term proportional to the temperature difference and to the square of the half-width of the guide. This term modifies the Rayleigh streaming pattern and may generate an additional vortex. On the other hand, the longitudinal (or wall-parallel) temperature difference is calculated as a cumulated effect of thermoacoustic heat transport in the fluid, heat conduction in the wall and heat convection of the air outside the resonator. At high acoustic levels, heat is significantly convected by the streaming flow and the resulting transverse temperature difference is proportional to the longitudinal temperature difference. Combining these expressions brings out a new criterion parameter for the nonlinear Reynolds number ($Re_{NL}$) characterizing the transition in streaming patterns at high acoustic levels. This result explains previous experimental and numerical observations of the streaming flow dynamics at high acoustic amplitudes, under different temperature boundary conditions, and can provide a powerful prediction tool for streaming pattern transitions.

Thermal counterflow of superfluid $^4$He is investigated experimentally, by employing the particle tracking velocimetry technique. A flat heater, located at the bottom of a vertical channel of square cross-section, is used to generate this unique type of thermally driven flow. Micronic solid particles, made in situ, probe this quantum flow and their time-dependent positions are collected by a digital camera, in a plane perpendicular to the heat source, away from the channel walls. The experiments are performed at relatively large heating powers, resulting in fluid velocities exceeding $10\ \textrm {mm}\,\textrm {s}^{-1}$, to ensure the existence of sufficiently dense tangles of quantized vortices. Within the investigated parameter range, we observe that the particles intermittently switch between two distinct motion regimes, along their trajectories, that is, a single particle can experience both regimes while travelling upward. The regimes can be loosely associated with fast particles, which are moving away from the heat source along almost straight tracks, and to slow particles, whose erratic upward motion can be said to be significantly influenced by quantized vortices. We propose a separation scheme to study the properties of these regimes and of the corresponding transients between them. We find that particles in both regimes display non-classical, broad distributions of velocity, which indicate the relevance of particle–vortex interactions in both cases. At the same time, we observe that the fast particles move along straighter trajectories than the slow ones, suggesting that the strength of particle–vortex interactions in the two regimes is notably different.

We employ direct numerical simulations of salt fingering engendered turbulent mixing to derive a parameterization scheme for the representation of this physical process in low-resolution ocean models and compare the results with those previously suggested on empirical grounds. In this analysis we differentiate between the reversible and irreversible contributions to diapycnal diffusivity associated with the turbulence generated by this mechanism. The necessity of such a distinction has been clearly recognized in connection with shear-driven density stratified turbulence processes: only irreversible processes can contribute to the effective turbulent diapycnal diffusivity. We expand the formalism herein to the more complicated salt-fingering case as a first step towards analysis of the general case. The irreversible fluxes are determined in the case of salt fingering related turbulence by examining high-resolution direct numerical simulation (DNS)-derived turbulence data sets based upon two different models: namely the ‘unbounded gradient model’ and the ‘interface model’ with depth-dependent gradients of temperature and salinity. By fitting the irreversible diapycnal fluxes in the unbounded gradient model (for equilibrium states) as a function of density ratio (the governing non-dimensional parameter), we derive a functional form that can be used as a basis for a next generation salt-fingering parametrization scheme. By applying this scheme to the interface model, we demonstrate that the local fluxes predicted agree well with those obtained from the numerical simulations based upon this more complex model. We compare this new DNS-derived turbulence parameterization with those that have been derived empirically.

We present a numerical study analysing the breakup of a single cohesive particle aggregate in turbulence. Solid particles with diameters smaller than the Kolmogorov length scale ($d_p<\eta$) are initially aggregated into a spherical ‘clump’ of diameter $D>\eta$ and placed in homogeneous isotropic turbulence. Parameters are chosen relevant to dust or powder suspended in air such that cohesion due to van der Waals is important. Simulations are performed using an Eulerian–Lagrangian framework that models two-way coupling between the fluid and solid phases and resolves particle–particle interactions. Aggregate breakup is investigated for different adhesion numbers ${Ad}$, Taylor microscale Reynolds numbers ${Re}_\lambda$ and non-dimensional clump sizes $D/d_p$. The intermittency of turbulence is found to play a key role on the early stage breakup process, which can be characterized by a turbulent adhesion number ${Ad}_\eta$ that relates the potential energy of the van der Waals force to turbulent shear stresses. A scaling analysis shows that the time rate of breakup for each case collapses when scaled by ${Ad}_\eta$ and an aggregate Reynolds number proportional to $D$. A phenomenological model of the breakup process is proposed that acts as a granular counterpart to the Taylor analogy breakup (TAB) model commonly used for droplet breakup. Such a model is useful for predicting particle breakup in coarse-grained simulation frameworks, such as Reynolds-averaged Navier–Stokes, where relevant spatial and temporal scales are not resolved.

We investigate the rheology of strain-hardening spherical capsules, from the dilute to the concentrated regime under a confined shear flow using three-dimensional numerical simulations. We consider the effect of capillary number, volume fraction and membrane inextensibility on the particle deformation and on the effective suspension viscosity and normal stress differences of the suspension. The suspension displays a shear-thinning behaviour that is a characteristic of soft particles such as emulsion droplets, vesicles, strain-softening capsules and red blood cells. We find that the membrane inextensibility plays a significant role on the rheology and can almost suppress the shear-thinning. For concentrated suspensions a non-monotonic dependence of the normal stress differences on the membrane inextensibility is observed, reflecting a similar behaviour in the particle shape. The effective suspension viscosity, instead, grows and eventually saturates, for very large inextensibilities, approaching the solid particle limit. In essence, our results reveal that strain-hardening capsules share rheological features with both soft and solid particles depending on the ratio of the area dilatation to shear elastic modulus. Furthermore, the suspension viscosity exhibits a universal behaviour for the parameter space defined by the capillary number and the membrane inextensibility, when introducing the particle geometrical changes at the steady state in the definition of the volume fraction.

Heating a quiescent fluid from below gives rise to cellular convective motion as the temperature gradient becomes sufficiently steep. Typically, this transition increases heat transfer. Differentially rotating spherical shells also generate a state of cellular motion, which in this case transports angular momentum. When both effects are present, it is often assumed that the fluid adopts a configuration which maximises the transfer of angular momentum and heat. Depending on how the equilibrium is reached, however, this maximisation may not always be achieved, with two different stable equilibria often co-existing for the same heating and rotation strengths. We want to understand why the fluid motion in a spherical shell is bistable, and how this scenario might arise. We consider a deep, highly viscous fluid layer, of relevance to the ice shells of Saturn's and Jupiter's moons. We find that bistability depends largely on the relative strength of heating and differential rotation, as characterised by the Rayleigh number $Ra$ and inner sphere Reynolds number $Re_1$, and that the nature of the transition between bistable states depends strongly on the ratio of momentum diffusivity $\nu$ to thermal diffusivity $\kappa$ defined by the Prandtl number ${\textit {Pr}} = \nu /\kappa$. In particular, we find that the transition between solutions at large ${\textit {Pr}}$, depends on the strength of thin thermal layers and can occur either due to the destabilisation of an equatorial jet by buoyancy forces, or alternatively of a polar thermal plume by differential rotation. Our results demonstrate that, although bistability in this system cannot be simply explained by the flow maximising its torque or heat transfer, the polar and equatorial regions are of particular significance.

Flow in thin films is highly dependent on the boundary conditions. Here, we study the capillary levelling of thin bilayer films composed of two immiscible liquids. Specifically, a stepped polymer layer is placed atop another, flat polymer layer. The Laplace pressure gradient resulting from the curvature of the step induces flow in both layers, which dissipates the excess capillary energy stored in the stepped interface. The effect of different viscosity ratios between the bottom and top layers is investigated. We invoke a long-wave expansion of the low-Reynolds-number hydrodynamics to model the energy dissipation due to the coupled viscous flows in the two layers. Good agreement is found between the experiments and the model. Analysis of the latter further reveals an interesting double cross-over in time, from Poiseuille flow, to plug flow and finally to Couette flow. The cross-over time scales depend on the viscosity ratio between the two liquids, allowing for the dissipation mechanisms to be selected and finely tuned by varying this ratio.

Understanding the physics of viscoelastic liquid jets is relevant to jet-based printing and deposition techniques. In this paper we study the behaviour of jets induced from viscoelastic liquid films, using the mechanical impulse provided by a laser pulse to actuate jet formation. We present direct numerical simulations of viscoelastic liquid jets solving the two-phase flow problem, accounting for the Oldroyd-B rheology. We describe how the jet extension time and length are controlled by the Deborah number (ratio of the elastic and inertia-capillary time scales), the viscous dissipation described by the Ohnesorge number (ratio of the viscous-capillary and inertia-capillary time scales), as well as the ratio of laser impulse energy to the energy required to create free surface during jet formation and propagation. Using the droplet ejection laser threshold energy of a Newtonian liquid, we investigate the influence of increasing viscoelastic effects. We show that viscoelastic effects can modify the effective drop size at the tip of the jet, while the maximum jet length increases with increasing Deborah number. Using the simulations, we identify a high-Deborah-number regime, where the time of maximum jet extension can be described as $t_{max} = c_1 De^{1/4}$, with $c_1$ depending on the Ohnesorge number and blister geometry, while the length of maximum extension reaches an asymptotic value $L_{max}^{\infty }$ for $De>100$, $L_{max}^{\infty }$ depending on the Ohnesorge number and laser energy. The observed asymptotic relationships are in good agreement with experiments performed at much higher Deborah numbers.

We extend linear input/output (resolvent) analysis to take into account nonlinear triadic interactions by considering a finite number of harmonics in the frequency domain using the harmonic balance method. Forcing mechanisms that maximise the drag are calculated using a gradient-based ascent algorithm. By including nonlinearity in the analysis, the proposed frequency-domain framework identifies the worst-case disturbances for laminar-turbulent transition. We demonstrate the framework on a flat-plate boundary layer by considering three-dimensional spanwise-periodic perturbations triggered by a few optimal forcing modes of finite amplitude. Two types of volumetric forcing are considered, one corresponding to a single frequency/spanwise wavenumber pair, and a multi-harmonic where a harmonic frequency and wavenumber are also added. Depending on the forcing strategy, we recover a range of transition scenarios associated with $K$-type and $H$-type mechanisms, including oblique and planar Tollmien–Schlichting waves, streaks and their breakdown. We show that nonlinearity plays a critical role in optimising growth by combining and redistributing energy between the linear mechanisms and the higher perturbation harmonics. With a very limited range of frequencies and wavenumbers, the calculations appear to reach the early stages of the turbulent regime through the generation and breakdown of hairpin and quasi-streamwise staggered vortices.

We investigate aspects of the spherical squirmer model employing both large-scale numerical simulations and asymptotic methods when the squirmer is placed in weakly elastic fluids. The fluids are modelled by differential equations, including the upper-convected Maxwell (UCM)/Oldroyd-B, finite-extensibility nonlinear elastic model with Peterlin approximation (FENE-P) and Giesekus models. The squirmer model we examine is characterized by two dimensionless parameters related to the fluid velocity at the surface of the micro-swimmer: the slip parameter $\xi $ and the swirl parameter $\zeta $. We show that, for swimming in UCM/Oldroyd-B fluids, the elastic stress becomes singular at a critical Weissenberg number, Wi, that depends only on $\xi$. This singularity for the UCM/Oldroyd-B models is independent of the domain exterior to the swimmer, or any other forces considered in the momentum balance for the fluid – we believe that this is the first time such a singularity has been explicitly demonstrated. Moreover, we show that the behaviour of the solution at the poles is purely extensional in character and is the primary reason for the singularity in the Oldroyd-B model. When the Giesekus or the FENE-P models are utilized, the singularity is removed. We also investigate the mechanism behind the speed and rotation rate enhancement associated with the addition of swirl in the swimmer's gait. We demonstrate that, for all models, the speed is enhanced by swirl, but the mechanism of enhancement depends intrinsically on the rheological model employed. Special attention is paid to the propulsive role of the pressure and elucidated upon. We also study the region of convergence of the perturbation solutions in terms of Wi. When techniques that accelerate the convergence of series are applied, transformed solutions are derived that are in very good agreement with the results obtained by simulations. Finally, both the analytical and numerical results clearly indicate that the low-Wi region is more important than one would expect and demonstrate all the major phenomena observed when swimming with swirl in a viscoelastic fluid.