The effect of rotation on horizontal convection in a cylindrical enclosure is investigated numerically. The thermal forcing is applied radially on the bottom boundary from the coincident axes of rotation and geometric symmetry of the enclosure. First, a spectral element method is used to obtain axisymmetric basic flow solutions to the time-dependent incompressible Navier–Stokes equations coupled via a Boussinesq approximation to a thermal transport equation for temperature. Solutions are obtained primarily at Rayleigh number $\mathit{Ra}=10^{9}$ and rotation parameters up to $Q=60$ (where $Q$ is a non-dimensional ratio between thermal boundary layer thickness and Ekman layer depth) at a fixed Prandtl number $\mathit{Pr}=6.14$ representative of water and enclosure height-to-radius ratio $H/R=0.4$. The axisymmetric solutions are consistently steady state at these parameters, and transition from a regime unaffected by rotation to an intermediate regime occurs at $Q\approx 1$ in which variation in thermal boundary layer thickness and Nusselt number are shown to be governed by a scaling proposed by Stern (1975, Ocean Circulation Physics. Academic). In this regime an increase in $Q$ sees the flow accumulate available potential energy and more strongly satisfy an inviscid change in potential energy criterion for baroclinic instability. At the strongest $Q$ the flow is dominated by rotation, accumulation of available potential energy ceases and horizontal convection is suppressed. A linear stability analysis reveals several instability mode branches, with dominant wavenumbers typically scaling with $Q$. Analysis of contributing terms of an azimuthally averaged perturbation kinetic energy equation applied to instability eigenmodes reveals that energy production by shear in the axisymmetric mean flow is negligible relative to that produced by conversion of available potential energy from the mean flow. An evolution equation for the quantity that facilitates this exchange, the vertical advective buoyancy flux, reveals that a baroclinic instability mechanism dominates over $5\lesssim Q\lesssim 30$, whereas stronger and weaker rotations are destabilised by vertical thermal gradients in the mean flow.

Pleated membrane filters are widely used in many applications, and offer significantly better surface area to volume ratios than equal-area unpleated membrane filters. However, their filtration characteristics are markedly inferior to those of equivalent unpleated membrane filters in dead-end filtration. While several hypotheses have been advanced for this, one possibility is that the flow field induced by the pleating leads to spatially non-uniform fouling of the filter, which in turn degrades performance. In this paper we investigate this hypothesis by developing a simplified model for the flow and fouling within a pleated membrane filter. Our model accounts for the pleated membrane geometry (which affects the flow), for porous support layers surrounding the membrane, and for two membrane fouling mechanisms: (i) adsorption of very small particles within membrane pores; and (ii) blocking of entire pores by large particles. We use asymptotic techniques based on the small pleat aspect ratio to solve the model, and we compare solutions to those for the closest-equivalent unpleated filter.

Vapour bubbles nucleating at micro-cavities etched into the silicon bottom plate of a cylindrical Rayleigh–Bénard sample (diameter $D=8.8$  cm, aspect ratio ${\it\Gamma}\equiv D/L\simeq 1.00$ where $L$ is the sample height) were visualized from the top and from the side. A triangular array of cylindrical micro-cavities (with a diameter of $30~{\rm\mu}\text{m}$ and a depth of $100~{\rm\mu}\text{m}$) covered a circular centred area (diameter of 2.5 cm) of the bottom plate. Heat was applied to the sample only over this central area while cooling was over the entire top-plate area. Bubble sizes and frequencies of departure from the bottom plate are reported for a range of bottom-plate superheats $T_{b}-T_{on}$ ($T_{b}$ is the bottom-plate temperature, $T_{on}$ is the onset temperature of bubble nucleation) from 3 to 12 K for three different cavity separations. The difference $T_{b}-T_{t}\simeq 16$  K between $T_{b}$ and the top plate temperature $T_{t}$ was kept fixed while the mean temperature $T_{m}=(T_{b}+T_{t})/2$ was varied, leading to a small range of the Rayleigh number $Ra$ from $1.4\times 10^{10}$ to $2.0\times 10^{10}$. The time between bubble departures from a given cavity decreased exponentially with increasing superheat and was independent of cavity separation. The contribution of the bubble latent heat to the total enhancement of heat transferred due to bubble nucleation was found to increase with superheat, reaching up to 25 %. The bubbly flow was examined in greater detail for a superheat of 10 K and $Ra\simeq 1.9\times 10^{10}$. The condensation and/or dissolution rates of departed bubbles revealed two regimes: the initial rate was influenced by steep thermal gradients across the thermal boundary layer near the plate and was two orders of magnitude larger than the final condensation and/or dissolution rate that prevailed once the rising bubbles were in the colder bulk flow of nearly uniform temperature. The dynamics of thermal plumes was studied qualitatively in the presence and absence of nucleating bubbles. It was found that bubbles enhanced the plume velocity by a factor of four or so and drove a large-scale circulation (LSC). Nonetheless, even in the presence of bubbles the plumes and LSC had a characteristic velocity which was smaller by a factor of five or so than the bubble-rise velocity in the bulk. In the absence of bubbles there was strongly turbulent convection but no LSC, and plumes on average rose vertically.

The early stages of drop impact onto a solid surface are considered. Detailed numerical simulations and detailed asymptotic analysis of the process reveal a self-similar structure both for the velocity field and the pressure field. The latter is shown to exhibit a maximum not near the impact point, but rather at the contact line. The motion of the contact line is furthermore shown to exhibit a ‘tank-treading’ motion. These observations are apprehended with the help of a variant of Wagner theory for liquid impact. This framework offers a simple analogy where the fluid motion within the impacting drop may be viewed as the flow induced by a flat rising expanding disk. The theoretical predictions are found to be in very close agreement both qualitatively and quantitatively with the numerical observations for approximately three decades in time. Interestingly, the inviscid self-similar impact pressure and velocities are shown to depend solely on the self-similar variables $(r/\sqrt{t},z/\sqrt{t})$. The structure of the boundary layer developing along the wet substrate is investigated as well. It is found to be in first approximation analogous to the boundary layer growing in the trail of a shockwave. Interestingly, the corresponding boundary layer structure only depends on the impact self-similar variables. This allows us to construct a seamless uniform analytical approximation encompassing both impact and viscous effects. The depiction of the different dynamical fields enables to quantitatively predict observables of interest, such as the evolution of the integral viscous shearing force and of the net normal force.

Rotationally coherent Lagrangian vortices are formed by tubes of deforming fluid elements that complete equal bulk material rotation relative to the mean rotation of the deforming fluid volume. We show that the initial positions of such tubes coincide with tubular level surfaces of the Lagrangian-averaged vorticity deviation (LAVD), the trajectory integral of the normed difference of the vorticity from its spatial mean. The LAVD-based vortices are objective, i.e. remain unchanged under time-dependent rotations and translations of the coordinate frame. In the limit of vanishing Rossby numbers in geostrophic flows, cyclonic LAVD vortex centres are precisely the observed attractors for light particles. A similar result holds for heavy particles in anticyclonic LAVD vortices. We also establish a relationship between rotationally coherent Lagrangian vortices and their instantaneous Eulerian counterparts. The latter are formed by tubular surfaces of equal material rotation rate, objectively measured by the instantaneous vorticity deviation (IVD). We illustrate the use of the LAVD and the IVD to detect rotationally coherent Lagrangian and Eulerian vortices objectively in several two- and three-dimensional flows.

A mechanical system consisting of water covered by brash ice and a body freely floating near equilibrium is considered. The water occupies a half-space into which an infinitely long surface-piercing cylinder is immersed, thus allowing us to study two-dimensional modes of the coupled motion, which is assumed to be of small amplitude. The corresponding linear setting for time-harmonic oscillations reduces to a spectral problem whose parameter is the frequency. A constant that characterises the brash ice divides the set of frequencies into two subsets and the results obtained for each of these subsets are essentially different. For frequencies belonging to a finite interval adjacent to zero, the total energy of motion is finite and the equipartition of energy holds for the whole system. For every frequency from this interval, a family of motionless bodies trapping waves is constructed by virtue of the semi-inverse procedure. For sufficiently large frequencies outside of this interval, all solutions of finite energy are trivial.

The instability of the Kármán vortex street is revisited under a spatio-temporal perspective that allows the taking into account of the advection of the vortices by the external flow. We analyse a simplified point vortex model and show through numerical simulations of its linear impulse response that the system becomes convectively unstable above a certain critical advection velocity. This critical velocity decreases as the aspect ratio approaches its specific value for temporal stability, and increases with the confinement induced by lateral walls. In the limiting unconfined case, direct application of the Briggs–Bers criterion to the dispersion relation gives results in excellent agreement with the numerical simulations. Finally, a direct numerical simulation of the $Re=100$ flow past a confined cylinder is performed, and the actual advection velocity of the resulting vortex street is found to be much larger than the critical advection velocity for convective instability given by our model. The Kármán vortex street is therefore strongly convectively unstable.

Turbulent boundary layer measurements above a smooth wall and sandpaper roughness are presented across a wide range of friction Reynolds numbers, ${\it\delta}_{99}^{+}$, and equivalent sand grain roughness Reynolds numbers, $k_{s}^{+}$ (smooth wall: $2020\leqslant {\it\delta}_{99}^{+}\leqslant 21\,430$, rough wall: $2890\leqslant {\it\delta}_{99}^{+}\leqslant 29\,900$; $22\leqslant k_{s}^{+}\leqslant 155$; and $28\leqslant {\it\delta}_{99}^{+}/k_{s}^{+}\leqslant 199$). For the rough-wall measurements, the mean wall shear stress is determined using a floating element drag balance. All smooth- and rough-wall data exhibit, over an inertial sublayer, regions of logarithmic dependence in the mean velocity and streamwise velocity variance. These logarithmic slopes are apparently the same between smooth and rough walls, indicating similar dynamics are present in this region. The streamwise mean velocity defect and skewness profiles each show convincing collapse in the outer region of the flow, suggesting that Townsend’s (The Structure of Turbulent Shear Flow, vol. 1, 1956, Cambridge University Press.) wall-similarity hypothesis is a good approximation for these statistics even at these finite friction Reynolds numbers. Outer-layer collapse is also observed in the rough-wall streamwise velocity variance, but only for flows with ${\it\delta}_{99}^{+}\gtrsim 14\,000$. At Reynolds numbers lower than this, profile invariance is only apparent when the flow is fully rough. In transitionally rough flows at low ${\it\delta}_{99}^{+}$, the outer region of the inner-normalised streamwise velocity variance indicates a dependence on $k_{s}^{+}$ for the present rough surface.

Application of the scaled fundamental equation of state of Balfour et al. (Phys. Lett. A, vol. 65, 1978, pp. 223–225) based upon universal critical exponents, demonstrates that there exists a bounded thermodynamic domain, located within the vapour–liquid equilibrium region and close to the critical point, featuring so-called negative nonlinearity. As a consequence, rarefaction shock waves with phase transition are physically admissible in a limited two-phase region in the close proximity of the liquid–vapour critical point. The boundaries of the admissibility region of rarefaction shock waves are identified from first-principle conservation laws governing compressible flows, complemented with the scaled fundamental equations. The exemplary substances considered here are methane, ethylene and carbon dioxide. Nonetheless, the results are arguably valid in the near-critical state of any common fluid, namely any fluid whose molecular interactions are governed by short-range forces conforming to three-dimensional Ising-like systems, including, e.g. water. Computed results yield experimentally feasible admissible rarefaction shock waves generating a drop in pressure from 1 to 6 bar and pre-shock Mach numbers exceeding 1.5.

The stability of a horizontal boundary layer flow on a vertical wall in a viscous stably stratified fluid is considered in this work. A temporal stability analysis is performed for a tanh velocity profile as a function of the Reynolds number $Re=UL/{\it\nu}$ and the Froude number $F=U/(LN)$ where $U$ is the main stream velocity, $L$ the boundary layer thickness, $N$ the buoyancy frequency and ${\it\nu}$ the kinematic viscosity. The diffusion of density is neglected. The boundary layer flow is found to be unstable with respect to two instabilities. The first one is the classical viscous instability which gives rise to Tollmien–Schlichting (TS) waves. We demonstrate that, even in the presence of stratification, the most unstable TS wave remains two-dimensional and therefore independent of the Froude number. The other instability is three-dimensional, inviscid in nature and associated with the stratification. It corresponds to the so-called radiative instability. We show that this instability appears first for $Re\geqslant Re_{c}^{(r)}\approx 1995$ for a Froude number close to 1.5 whereas the viscous instability develops for $Re\geqslant Re_{c}^{(v)}\approx 3980$. For large Reynolds numbers, the radiative instability is also shown to exhibit a much larger growth rate than the viscous instability in a large Froude number interval. We argue that this instability could develop in experimental facilities as well as in geophysical situations encountered in ocean and atmosphere.

We consider the incompressible Navier–Stokes equations with periodic boundary conditions and time-independent forcing. For this type of flow, we derive adjoint equations whose trajectories converge asymptotically to the equilibrium and travelling-wave solutions of the Navier–Stokes equations. Using the adjoint equations, arbitrary initial conditions evolve to the vicinity of a (relative) equilibrium at which point a few Newton-type iterations yield the desired (relative) equilibrium solution. We apply this adjoint-based method to a chaotic two-dimensional Kolmogorov flow. A convergence rate of $100\,\%$ is observed, leading to the discovery of $21$ new steady-state and travelling-wave solutions at Reynolds number $Re=40$. Some of the new invariant solutions have spatially localized structures that were previously believed to exist only on domains with large aspect ratios. We show that one of the newly found steady-state solutions underpins the temporal intermittencies, i.e. high energy dissipation episodes of the flow. More precisely, it is shown that each intermittent episode of a generic turbulent trajectory corresponds to its close passage to this equilibrium solution.

Simultaneous density and velocity turbulence statistics for Rayleigh–Taylor-driven flows at a moderately high Atwood number ($A_{t}$) of $0.73\pm 0.02$ are obtained using a new convective type or statistically steady gas tunnel facility. Air and air–helium mixture are used as working fluids to create a density difference in this facility, with a thin splitter plate separating the two streams flowing parallel to each other at the same velocity ($U=3~\text{m}~\text{s}^{-1}$). At the end of the splitter plate, the two miscible fluids are allowed to mix and the instability develops. Visualization and Mie-scattering techniques are used to obtain structure shape, volume fraction profile and mixing height growth information. Particle image velocimetry (PIV) and hot-wire techniques are used to measure planar and point-wise velocity statistics in the developing mixing layer. Asymmetry is evident in the flow field from the Mie-scattering images, with the spike side showing a more gradual decline in volume fraction than the bubble side. The spike side of the mixing layer grows 50 % faster than the bubble side. PIV is implemented for the first time in these moderately high-Atwood-number experiments ($A_{t}>0.1$) to obtain root-mean-square velocities, anisotropy tensor components and Reynolds stresses across the mixing layer. Overall, the turbulence statistics measured have shown different scaling compared to small-Atwood-number experiments. However, the total probability density functions for the velocities and turbulent mass fluxes exhibit behaviour similar to small-Atwood-number experiments. Conditional statistics reveal different values for turbulence statistics for spikes and bubbles, unlike small-Atwood-number experiments.

Turbulence in the asymptotic suction boundary layer is investigated numerically at the verge of laminarisation using direct numerical simulation. Following an adiabatic protocol, the Reynolds number $Re$ is decreased in small steps starting from a fully turbulent state until laminarisation is observed. Computations in a large numerical domain allow in principle for the possible coexistence of laminar and turbulent regions. However, contrary to other subcritical shear flows, no laminar–turbulent coexistence is observed, even near the onset of sustained turbulence. High-resolution computations suggest a critical Reynolds number $Re_{g}\approx 270$, below which turbulence collapses, based on observation times of $O(10^{5})$ inertial time units. During the laminarisation process, the turbulent flow fragments into a series of transient streamwise-elongated structures, whose interfaces do not display the characteristic obliqueness of classical laminar–turbulent patterns. The law of the wall, i.e. logarithmic scaling of the velocity profile, is retained down to $Re_{g}$, suggesting a large-scale wall-normal transport absent in internal shear flows close to the onset. In order to test the effect of these large-scale structures on the near-wall region, an artificial volume force is added to damp spanwise and wall-normal fluctuations above $y^{+}=100$, in viscous units. Once the largest eddies have been suppressed by the forcing, and thus turbulence is confined to the near-wall region, oblique laminar–turbulent interfaces do emerge as in other wall-bounded flows, however only transiently. These results suggest that oblique stripes at the onset are a prevalent feature of internal shear flows, but will not occur in canonical boundary layers, including the spatially growing ones.

We examine the extension of an axisymmetric viscous thread that is pulled at both ends with a prescribed speed such that the effects of inertia are initially small. After neglecting surface tension, we derive a particularly convenient form of the long-wavelength equations that describe long and thin threads. Two generic classes of initial thread shape are considered as well as the special case of a circular cylinder. In these cases, we determine explicit asymptotic solutions while the effects of inertia remain small. We further show that inertia will ultimately become important only if the long-time asymptotic form of the pulling speed is faster than a power law with a critical exponent. The critical exponent can take two possible values depending on whether or not the initial minimum of the thread radius is located at the pulled end. In addition, we obtain asymptotic expressions for the solution at large times in the case in which the critical exponent is exceeded and hence inertia becomes important. Despite the apparent simplicity of the problem, the solutions exhibit a surprisingly rich structure. In particular, in the case in which the initial minimum is not at the pulled end, we show that there are two very different types of solution that exhibit very different extension mechanics. Both the small-inertia solutions and the large-time asymptotic expressions compare well with numerical solutions.

We have discovered a peculiar behaviour of the interface between two miscible liquids placed in a finite-size container under horizontal vibration. We provide evidence that periodic wave patterns created by the Kelvin–Helmholtz instability and Faraday waves simultaneously exist in the same system of miscible liquids. We show experimentally in reduced and normal gravity that large-scale frozen waves yield Faraday waves with a smaller wavelength on a diffusive interface. The emergence of the different scale patterns observed in the experiments is confirmed numerically and explained theoretically.

This paper studies the large-scale low-frequency variability of the wind-driven midlatitude ocean gyres and their western boundary currents, such as the Gulf Stream or Kuroshio, simulated with the eddy-resolving quasi-geostrophic model. We applied empirical orthogonal functions analysis to turbulent flow solutions and statistically extracted robust and significant large-scale decadal variability modes concentrated around the eastward jet extension of the western boundary currents. In order to interpret these statistical modes dynamically, we linearized the governing quasi-geostrophic equations around the time-mean circulation and solved for the corresponding full set of linear eigenmodes with their eigenfrequencies. We then projected the extracted decadal variability on the eigenmodes and found that this variability is a multimodal coherent pattern phenomenon rather than a single mode or a combination of several modes as in the flow regimes preceding developed turbulence.

We theoretically investigate the lift force on spherical nanoparticles in a shear flow of a dilute gas, wherein the non-rigid-body collision between the particle and the gas molecules is considered. The analytical formula of the lift force is derived based on the gas kinetic theory. In the limit of rigid-body collision, the formula is consistent with the theoretical results in the literature (Liu & Bogy, Phys. Fluids, vol. 20, 2008, 107102), which predicts that the lift force is in the opposite direction to the fluid velocity gradient (negative lift force). However, by taking into account gas–particle intermolecular interactions, the direction of the lift force on the nanoparticle is found to be dependent on temperature, i.e. both positive and negative lift forces exist in a certain temperature range. An explanation for the direction change of the lift force is given based on the analysis of the scattering angle under non-rigid-body particle–molecule collisions.

A nonlinear stability analysis of the viscous circular Couette flow to axisymmetric finite-amplitude perturbations under axial periodic boundary conditions is developed. The analysis is based on investigating the properties of a reduced Arnol’d energy-Casimir function $\mathscr{A}_{rd}$ of Wang (Phys. Fluids, vol. 2, 2009, 084104). A weighted kinetic energy of the perturbation, which has a form of ${\rm\Delta}\mathscr{A}_{rd}$, the difference between the reduced Arnol’d function and its base flow value, is used as a Lyapunov function. We show that all the inviscid flow effects as well as all the viscous-dependent terms that are related to the flow boundaries vanish. The evolution of ${\rm\Delta}\mathscr{A}_{rd}$ depends only on the viscous effects of the perturbation’s dynamics inside the flow domain. The requirement for the temporal decay of ${\rm\Delta}\mathscr{A}_{rd}$ leads to two novel sufficient conditions for the nonlinear stability of the circular Couette flow in response to axisymmetric perturbations. The linearized version of these conditions for infinitesimally small perturbations recovers the recent linear stability results by Kloosterziel (J. Fluid Mech., vol. 652, 2010, pp. 171–193). By examining the nonlinear stability conditions, we establish a definite operational region of the viscous circular Couette flow that is independent of the fluid viscosity. In this region of operation, the flow is nonlinearly stable in response to perturbations of any size, provided that the initial total circulation function is above a minimum level determined by the operational conditions of the base flow. Comparisons with historical studies show that our results shed light on the experimental measurements of Wendt (Ing.-Arch., vol. 4, 1933, pp. 577–595) and extend the classical nonlinear stability results of Serrin (Arch. Rat. Mech. Anal., vol. 3, 1959, pp. 1–13) and Joseph & Hung (Arch. Rat. Mech. Anal., vol. 44, 1971, pp. 1–22). When the flow is nonlinearly stable and evolves axisymmetrically for all time, then it always decays asymptotically in time to the circular Couette flow determined uniquely by the set-up of the rotating cylinders. Finally, we derive upper-bound estimates on the decay rate of finite-amplitude perturbations for the solid-body rotation flow between two coaxial rotating cylinders and for the circular Couette flow. We demonstrate via numerical simulations that the theoretical upper bound is relevant to the dynamics of various axisymmetric perturbations tested, where it is strictly obeyed. This present study provides new physical insights into a classical flow problem that was studied for many decades.

In this study, the development of a boundary layer past a change in surface roughness (from rough to smooth, $\text{R}\rightarrow \text{S}$) is examined. Measurements of the flow were made by hot wires, whereas the friction velocity was estimated by Preston tube measurements. By means of a diagnostic plot of the turbulence intensity, it is shown that above the internal layer the flow exhibits characteristics of a rough, wall-bounded flow, whereas near the wall the turbulence intensity is similar to that of an isolated smooth wall. Similarly, viscous scaling of the mean streamwise velocity shows an excessive wake region downstream of the $\text{R}\rightarrow \text{S}$ wall surface change that diminishes with the fetch from the surface change. Above the internal layer a second peak in the streamwise Reynolds stress was associated with the upstream rough-wall flow. Examination of the turbulent spectra revealed the presence of large-scale motions within this region that gradually diminish in strength with increasing distance from the change in surface roughness. The magnitude of the near-wall peak failed to collapse to that of a comparable smooth-wall boundary layer under viscous scaling, however, the wall-normal location of the peak appears to be at $y^{+}\approx 15$ at all downstream distances. A new mixed scaling is proposed for the near-wall peak based on the corrected wake deficit and the friction velocity. This shows the importance of outer region to the growth of near-wall peak in this non-equilibrium boundary layer.

Large-scale transport is investigated in terms of new explicit expressions for eddy diffusivities and effective advection obtained from asymptotic perturbative methods. The carrier flow is formed by a large-scale component plus a small-scale contribution mimicking a turbulent flow. The scalar dynamics is observed in its pre-asymptotic regimes (i.e. on scales comparable to those of the large-scale velocity). The resulting eddy diffusivity is thus a tensor field which explicitly depends on the large-scale velocity. Small-scale interactions also cause the emergence of an effective large-scale (compressible) advection field which, as a result of the present study however, turns out to be of negligible importance. Two issues are addressed by means of Lagrangian simulations: quantifying the possible deterioration of the eddy-diffusivity/effective advection description by reducing to zero the spectral gap separating the large-scale velocity component from the small-scale component; comparing the accuracy of our closure against other simple, reasonable, options. Answering these questions is important in view of possible applications of our closure to tracer dispersion in environmental flows.