Numerical simulations of the shallow water equations on rotating spheres produce mixtures of robust vortices and alternating zonal jets, as seen in the atmospheres of the gas giant planets. However, simulations that include Rayleigh friction invariably produce a sub-rotating (retrograde) equatorial jet for Jovian parameter regimes, whilst observations of Jupiter show a super-rotating (prograde) equatorial jet that has persisted over several decades. Super-rotating equatorial jets have recently been obtained in shallow water simulations that include a Newtonian relaxation of perturbations to the layer thickness to model radiative cooling to space, and in simulations of the thermal shallow water equations that include a similar relaxation term in their temperature equation. Simulations of global quasigeostrophic forms of these different models produce equatorial jets in the same directions as the parent models, suggesting that the mechanism responsible for setting the direction lies within quasigeostrophic theory. We provide such a mechanism by calculating the effective force acting on the thickness-weighted zonal mean flow due to the decay of an equatorially trapped Rossby wave. Decay due to Newtonian cooling creates an eastward zonal mean flow at the equator, consistent with the formation of a super-rotating equatorial jet, while decay due to Rayleigh friction leads to a westward zonal mean flow at the equator, consistent with the formation of a sub-rotating equatorial jet. In both cases the meridionally integrated zonal mean of the absolute zonal momentum is westward, consistent with the standard result that Rossby waves carry westward pseudomomentum, but this does not preclude the zonal mean flow being eastward on and close to the equator.

Isotropic turbulence is typically studied numerically through direct numerical simulations (DNS). The DNS flows are described by the Navier–Stokes equation in a ‘box’, defined through periodic boundary conditions. Ideal isotropic turbulence lives in infinite space. The DNS flows live in a compact space and they are not isotropic in their large scales. Hence, the investigation of important phenomena of isotropic turbulence, such as anomalous scaling, through DNS is affected by large-scale effects in the currently available Reynolds numbers. In this work, we put isotropic turbulence – or better, the associated formal theory – in a ‘box’, by imposing periodicity at the level of the correlation functions. This is an attempt to offer a framework where one may investigate isotropic theories/models through the data of DNS in a manner as consistent with them as possible. We work at the lowest level of the hierarchy, which involves the two-point correlation functions and the Karman–Howarth equation. Periodicity immediately gives us the discrete wavenumber space of the theory. The wavenumbers start from 1.835, 2.896, 3.923, and progressively approach integer values, in an interesting correspondence with the DNS wavenumber shells. Unlike the Navier–Stokes equation, infinitely smooth periodicity is obstructed in this theory, a fact expressed by a sequence of relations obeyed by the normal modes of the Karman–Howarth equation at the endpoints of a unit period interval. Similar relations are imparted to the two-point functions under the condition that the energy spectrum and energy transfer function are realizable. Hence, these relations are necessary conditions for realizability in this theory. Naturally constructed closure schemes for the Karman–Howarth equation do not conform to such relations, thereby destroying realizability. A closure can be made to conform to a finite number of them by adding corrective terms, in a procedure that possesses certain analogies with the renormalization of quantum field theory. Perhaps the most important one is that we can let the spectrum be unphysical (through sign-changing oscillations of decreasing amplitude) for the infinitely large wavenumbers, as long as we can controllably extend the regime where the spectrum remains physical, deep enough in the dissipation subrange so as to be realistically adequate. Indeed, we show that one or two such ‘regularity relations’ are needed at most for comparisons of the predictions of the theory with the current resolution level results of the DNS. For the implementation of our arguments, we use a simple closure scheme previously proposed by Oberlack and Peters. The applicability of our ideas to more complex closures is also discussed.

The application of local periodic heating for control of a spatially developing shear layer downstream of a finite-thickness splitter plate is examined by numerically solving the two-dimensional Navier–Stokes equations. At the trailing edge of the plate, an oscillatory heat flux boundary condition is prescribed as the thermal forcing input to the shear layer. The thermal forcing introduces a low level of oscillatory surface vorticity flux and baroclinic vorticity at the actuation frequency in the vicinity of the trailing edge. The vortical perturbations produced can independently excite the fundamental instability that accounts for shear layer roll-up as well as the subharmonic instability that encourages the vortex pairing process farther downstream. We demonstrate that the nonlinear dynamics of a spatially developing shear layer can be modified by local oscillatory heat flux as a control input. We believe that this study provides a basic foundation for flow control using thermal-energy-deposition-based actuators such as thermophones and plasma actuators.

Spherical shell dynamo models based on rotating convection show that the flow within the tangent cylinder is dominated by an off-axis plume that extends from the inner core boundary to high latitudes and drifts westward. Earlier studies explained the formation of such a plume in terms of the effect of a uniform axial magnetic field that significantly increases the length scale of convection in a rotating plane layer. However, rapidly rotating dynamo simulations show that the magnetic field within the tangent cylinder has severe lateral inhomogeneities that may influence the onset of an isolated plume. Increasing the rotation rate in our dynamo simulations (by decreasing the Ekman number $E$) produces progressively thinner plumes that appear to seek out the location where the field is strongest. Motivated by this result, we examine the linear onset of convection in a rapidly rotating fluid layer subject to a laterally varying axial magnetic field. A Cartesian geometry is chosen where the finite dimensions $(x,z)$ mimic $(\unicode[STIX]{x1D719},z)$ in cylindrical coordinates. The lateral inhomogeneity of the field gives rise to a unique mode of instability where convection is entirely confined to the peak-field region. The localization of the flow by the magnetic field occurs even when the field strength (measured by the Elsasser number $\unicode[STIX]{x1D6EC}$) is small and viscosity controls the smallest length scale of convection. The lowest Rayleigh number at which an isolated plume appears within the tangent cylinder in spherical shell dynamo simulations agrees closely with the viscous-mode Rayleigh number in the plane layer linear magnetoconvection model. The lowest Elsasser number for plume formation in the simulations is significantly higher than the onset values in linear magnetoconvection, which indicates that the viscous–magnetic mode transition point with spatially varying fields is displaced to much higher Elsasser numbers. The localized excitation of viscous-mode convection by a laterally varying magnetic field provides a mechanism for the formation of isolated plumes within the Earth’s tangent cylinder. The polar vortices in the Earth’s core can therefore be non-axisymmetric. More generally, this study shows that a spatially varying magnetic field strongly controls the structure of rotating convection at a Rayleigh number not much different from its non-magnetic value.

This theoretical and numerical study presents three-dimensional boundary-layer solutions for laminar incompressible flow adjacent to a semi-infinite flat plate, subject to a uniform free-stream speed and injection through the plate surface. The novelty in this case arises from a fully three-dimensional formulation, which also allows for slot injection over a spanwise length scale comparable to the boundary-layer thickness. This approach retains viscous effects in both the spanwise and transverse directions, and effectively results in a parabolised Navier–Stokes system (sometimes referred to as the ‘boundary-region equations’). Any injection profile can be described in this approach, but we restrict attention to three-dimensional states driven by a finite-width slot aligned with the flow direction and self-similar in their downstream development. The classical two-dimensional states are known to only exist up to a critical (‘blow off’) injection amplitude, but the three-dimensional solutions here appear possible for any injection velocity. These new states take the form of low-speed streamwise-aligned streaks whose geometry depends on the amplitude of injection and the spanwise width of the injection slot; intriguingly, although very low wall shear is typically obtained, streamwise flow reversal is not observed, however hard the blowing. Asymptotic descriptions are provided in the limit of increasing slot width and fixed injection velocity, which allow for classification of the solutions according to two bounding injection rates.

Three-dimensional particle tracking velocimetry is applied to particle-laden turbulent pipe flows at a Reynolds number of 10 300, based on the bulk velocity and the pipe diameter, for developed fluid flow and not fully developed flow of inertial particles, which favours assessment of the radial migration of the inertial particles. Inertial particles with Stokes number ranging from 0.35 to 1.11, based on the particle relaxation time and the radial-dependent Kolmogorov time scale, and a ratio of the root-mean-square fluid velocity to the terminal velocity of order 1 have been used. Core peaking of the concentration of inertial particles in up-flow and wall peaking in down-flow have been found. The difference in mean particle and Eulerian mean liquid velocity is found to decrease to approximately zero near the wall in both flow directions. Although the carrier fluid has all of the characteristics of the corresponding turbulent single-phase flow, the Reynolds stress of the inertial particles is different near the wall in up-flow. These findings are explained from the preferential location of the inertial particles with the aid of direct numerical simulations with the point-particle approach.

A suspended ellipsoidal particle inside a Poiseuille flow with Reynolds number up to 360 is studied numerically. The effects of tube diameter ($D$), inertia of the particle and the flow, and the particle geometry (both prolate and oblate ellipsoids) are considered. When a prolate particle with $a/b=2$ is inside a wider tube (e.g. $D/A>1.9$), where $A=2a$ is the length of the major axis of the particle, the terminal stable state is tumbling. When the prolate particle is inside a narrower tube ($1.0<D/A<1.9$), log-rolling or kayaking modes may appear. Which mode occurs depends on the competition between fluid and particle inertia. When the fluid inertia is dominant, the log-rolling mode appears, otherwise, the kayaking mode appears. Inclined and spiral modes may appear when $D/A<1$ and $D/A=1$, respectively. For a prolate ellipsoid with $a/b=4$, if $1<D/A<1.9$, there is only the kayaking mode and the log-rolling mode is not observed. When an oblate particle is inside a wider tube (e.g. $D/A>3.5$), it may adopt the log-rolling mode. Inclined and intermediate modes are firstly identified in narrower tubes. The phase diagram of the modes is also provided. The modes in the phase diagrams were not found to be affected by the initial state of the particle based on limited observation.

We present an extensive numerical comparison of a family of balance models appropriate to the semi-geostrophic limit of the rotating shallow water equations, and derived by variational asymptotics in Oliver (J. Fluid Mech., vol. 551, 2006, pp. 197–234) for small Rossby numbers $Ro$. This family of generalized large-scale semi-geostrophic (GLSG) models contains the $L_{1}$-model introduced by Salmon (J. Fluid Mech., vol. 132, 1983, pp. 431–444) as a special case. We use these models to produce balanced initial states for the full shallow water equations. We then numerically investigate how well these models capture the dynamics of an initially balanced shallow water flow. It is shown that, whereas the $L_{1}$-member of the GLSG family is able to reproduce the balanced dynamics of the full shallow water equations on time scales of $O(1/Ro)$ very well, all other members develop significant unphysical high wavenumber contributions in the ageostrophic vorticity which spoil the dynamics.

A numerical method is developed to study the dynamic behaviour of an encapsulated bubble when the viscous forces of the surrounding liquid are accounted for. The continuity and Navier–Stokes equations are solved for the liquid, whereas the coating is described as a viscoelastic shell with bending resistance. The Galerkin Finite Element Methodology is employed for the spatial discretization of the flow domain surrounding the bubble, with the standard staggered grid arrangement that uses biquadratic and bilinear Lagrangian basis functions for the velocity and pressure in the liquid, respectively, coupled with a superparametric scheme with $B$-cubic splines as basis functions pertaining to the location of the interface. The spine method and the elliptic mesh generation technique are used for updating the mesh points in the interior of the flow domain as the shape of the interface evolves with time, with the latter being distinctly superior in capturing severely distorted shapes. The stabilizing effect of the liquid viscosity is demonstrated, as it alters the amplitude of the disturbance for which a bubble deforms and/or collapses. For a step change in the far-field pressure the dynamic evolution of the microbubble is captured until a static equilibrium is achieved. Static shapes that are significantly compressed are captured in the post-buckling regime, leading to symmetric or asymmetric shapes, depending on the relative dilatation to bending stiffness ratio. As the external overpressure increases, shapes corresponding to all the solution families that were captured evolve to exhibit contact as the two poles approach each other. Shell viscosity prevents jet formation by relaxing compressive stresses and bending moments around the indentation generated at the poles due to shell buckling. This behaviour is conjectured to be the inception process leading to static shapes with contact regions.

A number of micro-scale biological flows are characterized by spatio-temporal chaos. These include dense suspensions of swimming bacteria, microtubule bundles driven by motor proteins and dividing and migrating confluent layers of cells. A characteristic common to all of these systems is that they are laden with active matter, which transforms free energy in the fluid into kinetic energy. Because of collective effects, the active matter induces multi-scale flow motions that bear strong visual resemblance to turbulence. In this study, multi-scale statistical tools are employed to analyse direct numerical simulations (DNS) of periodic two-dimensional (2-D) and three-dimensional (3-D) active flows and to compare the results to classic turbulent flows. Statistical descriptions of the flows and their variations with activity levels are provided in physical and spectral spaces. A scale-dependent intermittency analysis is performed using wavelets. The results demonstrate fundamental differences between active and high-Reynolds-number turbulence; for instance, the intermittency is smaller and less energetic in active flows, and the work of the active stress is spectrally exerted near the integral scales and dissipated mostly locally by viscosity, with convection playing a minor role in momentum transport across scales.

We derive a new expression for the entrainment coefficient in a turbulent plume using an equation for the squared mean buoyancy. Consistency of the resulting expression with previous relations for the entrainment coefficient implies that the turbulent Prandtl number in a pure plume is equal to 3/5 when the mean profiles of velocity and buoyancy have a Gaussian form of equal width. Entrainment can be understood in terms of the volume flux, the production of turbulence kinetic energy or the production of scalar variance for either active or passive variables. The equivalence of these points of view indicates how the entrainment coefficient and the turbulent Prandtl and Schmidt numbers depend on the Richardson number of the flow, the ambient stratification and the relative widths of the velocity and scalar profiles. The general framework is valid for self-similar plumes, which are characterised by a power-law scaling. For jets and pure plumes it is shown that the derived relations are in reasonably good agreement with results from direct numerical simulations and experiments.

We present scaling laws for the jet velocity resulting from bubble collapse at a liquid surface which bring out the effects of gravity and viscosity. The present experiments conducted in the range of Bond numbers $0.004<Bo<2.5$ and Ohnesorge numbers $0.001<Oh<0.1$ were motivated by the discrepancy between previous experimental results and numerical simulations. We show here that the actual dependence of $We$ on $Bo$ is determined by the gravity dependency of the bubble immersion (cavity) depth which has no power-law variation. The power-law variation of the jet Weber number, $We\sim 1/\sqrt{Bo}$, suggested by Ghabache et al. (Phys. Fluids, vol. 26 (12), 2014, 121701) is only a good approximation in a limited range of $Bo$ values ($0.1<Bo<1$). Viscosity enters the jet velocity scaling in two ways: (i) through damping of precursor capillary waves which merge at the bubble base and weaken the pressure impulse, and (ii) through direct viscous damping of the jet formation and dynamics. These damping processes are expressed by a dependence of the jet velocity on Ohnesorge number from which critical values of $Oh$ are obtained for capillary wave damping, the onset of jet weakening, the absence of jetting and the absence of jet breakup into droplets.

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

Direct numerical simulations of turbulent pipe flow of power-law fluids at $Re_{\unicode[STIX]{x1D70F}}=323$ are analysed in order to understand the way in which shear thinning or thickening affects first- and second-order flow statistics including turbulent kinetic energy production, transport and dissipation in such flows. The results show that with shear thinning, near-wall streaks become weaker and the axial and azimuthal correlation lengths of axial velocity fluctuations increase. Viscosity fluctuations give rise to an additional shear stress term in the mean momentum equation which is negative for shear-thinning fluids and which increases in magnitude as the fluid becomes more shear thinning: for an equal mean wall shear stress, this term increases the mean velocity gradient in shear-thinning fluids when compared to a Newtonian fluid. Consequently, the mean velocity profile in power-law fluids deviates from the law of the wall $U_{z}^{+}=y^{+}$ in the viscous sublayer when traditional near-wall scaling is used. Consideration is briefly given to an alternative scaling that allows the law of wall to be recovered but which results in loss of a common mean stress profile. With shear thinning, the mean viscosity increases slightly at the wall and its profile appears to be approximately logarithmic in the velocity log layer. Through analysis of the turbulent kinetic energy budget, undertaken here for the first time for generalised Newtonian fluids, it is shown that shear thinning decreases the overall turbulent kinetic energy production but widens the wall-normal region where it is generated. Additional dissipation terms in the mean flow and turbulent kinetic energy budget equations arise from viscosity fluctuations; with shear thinning, these result in a net decrease in the total viscous dissipation. The overall effect of shear thinning on the turbulent kinetic energy budget is found to be largely confined to the inner layers, $y^{+}\lesssim 60$.

We analyse numerically the linear stability of fully developed liquid metal flow in a square duct with insulating side walls and thin, electrically conducting horizontal walls. The wall conductance ratio $c$ is in the range of 0.01 to 1 and the duct is subject to a vertical magnetic field with Hartmann numbers up to $\mathit{Ha}=10^{4}$. In a sufficiently strong magnetic field, the flow consists of two jets at the side walls and a near-stagnant core with relative velocity ${\sim}(c\mathit{Ha})^{-1}$. We find that for $\mathit{Ha}\gtrsim 300,$ the effect of wall conductivity on the stability of the flow is mainly determined by the effective Hartmann wall conductance ratio $c\mathit{Ha}.$ For $c\ll 1$, the increase of the magnetic field or that of the wall conductivity has a destabilizing effect on the flow. Maximal destabilization of the flow occurs at $\mathit{Ha}\approx 30/c$. In a stronger magnetic field with $c\mathit{Ha}\gtrsim 30$, the destabilizing effect vanishes and the asymptotic results of Priede et al. (J. Fluid Mech., vol. 649, 2010, pp. 115–134) for ideal Hunt’s flow with perfectly conducting Hartmann walls are recovered.

Using the energy method we investigate the stability of pure conduction in Pearson’s model for Bénard–Marangoni convection in a layer of fluid at infinite Prandtl number. Upon extending the space of admissible perturbations to the conductive state, we find an exact solution to the energy stability variational problem for a range of thermal boundary conditions describing perfectly conducting, imperfectly conducting, and insulating boundaries. Our analysis extends and improves previous results, and shows that with the energy method global stability can be proven up to the linear instability threshold only when the top and bottom boundaries of the fluid layer are insulating. Contrary to the well-known Rayleigh–Bénard convection set-up, therefore, energy stability theory does not exclude the possibility of subcritical instabilities against finite-amplitude perturbations.

We develop a generalized Rotne–Prager–Yamakawa approximation for the dipolar components of the inverse friction matrix and use it for calculating the intrinsic viscosity of rigidly connected bead conglomerates. Such bead models are commonly used in the calculation of hydrodynamic properties of macromolecules. We consider both the case of non-overlapping constituent beads as well as overlapping beads of different sizes. We demonstrate the accuracy of the approximation in two test cases and show that it performs well even if the distances between the beads are small or if the beads overlap. Robust performance of this approximation in the case of overlapping beads stems from its correct limiting behaviour at a complete overlap, with one sphere fully immersed in the other. The generalized Rotne–Prager–Yamakawa approximation is thus well suited for evaluation of intrinsic viscosity, which is a key quantity in characterizing molecular conformations of biological macromolecules.

We demonstrate that the critical magnetic Reynolds number $Rm_{c}$ for a turbulent non-helical dynamo in the limit of low magnetic Prandtl number $Pm$ (i.e. $Pm=Rm/Re\ll 1$ ) can be significantly reduced if the flow is subjected to global rotation. Even for moderate rotation rates the required energy injection rate can be reduced by a factor of more than $10^{3}$ . This strong decrease in the onset is attributed to the transfer of energy to the large scales, forming a large-scale condensate, and the reduction in the turbulent fluctuations that cause the flow to have a much larger cutoff length scale than in a non-rotating flow of the same Reynolds number. The dynamo thus behaves as if it is driven just by the large scales that act as a laminar flow (i.e. it behaves as a high $Pm$ dynamo) even though the actual Reynolds number is much higher than the magnetic Reynolds number (i.e. low $Pm$ ). Our finding thus points to a new paradigm for the design of new experiments on liquid metal dynamos.

Low-Reynolds-number polymer solutions exhibit a chaotic behaviour known as ‘elastic turbulence’ when the Weissenberg number exceeds a critical value. The two-dimensional Oldroyd-B model is the simplest constitutive model that reproduces this phenomenon. To make a practical estimate of the resolution scale of the dynamics, one requires the assumption that an attractor of the Oldroyd-B model exists; numerical simulations show that the quantities on which this assumption is based are bounded. We estimate the Lyapunov dimension of this assumed attractor as a function of the Weissenberg number by combining a mathematical analysis of the model with direct numerical simulations.

A comparison between swimming by flapping and by periodic contractions is conducted. Swimming by flapping is approximated as a pitching plate while swimming by periodic contractions is approximated as clapping plates. A direct comparison is made between the two propulsion mechanisms by utilizing a machine that can operate in either a flapping or a clapping mode between Reynolds numbers of 1880 and 11 260 based on the average plate tip velocity and span. The average thrust generated and the average input power required per cycle are compared between cases where the total sweep angle and the total sweep time are identical. Variation of the kinematics results in a similar thrust between the two mechanisms, but a greater power is required for clapping. Variation of the flexibility results in a consistent decrease in the required power for clapping and a decrease in thrust at high flexibility. Variation of the duty cycle for clapping rigid plates results in a significant increase in thrust and a significant decrease in the required power. Overall, the results suggest that flapping propulsion is the more effective propulsion mechanism within the range of Reynolds numbers tested.