Determining physically admissible boundary conditions for higher moments in an extended continuum model is recognised as a major obstacle. Boundary conditions for the regularised 26-moment (R26) equations obtained using Maxwell's accommodation model do exist in the literature; however, we show in this article that these boundary conditions violate the second law of thermodynamics and the Onsager reciprocity relations for certain boundary value problems, and, hence, are not physically admissible. We further prove that the linearised R26 (LR26) equations possess a proper $H$-theorem (second-law inequality) by determining a quadratic form without cross-product terms for the entropy density. The establishment of the $H$-theorem for the LR26 equations in turn leads to a complete set of boundary conditions that are physically admissible for all processes and comply with the Onsager reciprocity relations. As an application, the problem of a slow rarefied gas flow past a spherical droplet with and without evaporation is considered and solved analytically. The results are compared with the numerical solution of the linearised Boltzmann equation, experimental results from the literature and/or other macroscopic theories to show that the LR26 theory with the physically admissible boundary conditions provides an excellent prediction up to Knudsen number $\lesssim 1$ and, consequently, provides transpicuous insights into intriguing effects, such as thermal polarisation. In particular, the analytic results for the drag force obtained in the present work are in an excellent agreement with experimental results even for very large values of the Knudsen number.

Lee waves generated by stratified flow over rough bottom topography in the ocean extract momentum and energy from the geostrophic flow, causing drag and enhancing turbulence and mixing in the interior ocean when they break. Inviscid linear theory is generally used to predict the generation rate of lee waves, but the location and mechanism of wave breaking leading to eventual dissipation of energy and irreversible mixing are poorly constrained. In this study, a linear model with viscosity, diffusivity and an upper boundary is used to demonstrate the potential importance of the surface in reflecting lee wave energy back into the interior, making the case for treating lee waves as a full water-column process. In the absence of critical levels, it is shown that lee waves can be expected to interact with the upper ocean, resulting in enhanced vertical velocities and dissipation and mixing near the surface. The impact of the typical oceanic conditions of increasing background velocity and stratification with height above bottom are investigated and shown to contribute to enhanced upper ocean vertical velocities and mixing.

We propose a semi-analytical solution for the advection–diffusion equation in cylindrical domains, with an aim towards extracting blood flow rates from contrast variations in a coronary computed tomography angiography image. The solution proposed in this work, in contrast with existing methods, which only consider advection, incorporates both radial velocity variation and diffusion. By means of a Galerkin approach using Bessel functions, a solution for a three-dimensional concentration field at a single time point is obtained after a Laplace transformation. This semi-analytical solution forms the basis for a novel advection–diffusion flow estimation (ADFE) method. The ADFE is derived, validated through numerical spectral-element method computations, and shown to exhibit improved accuracy against the state-of-the-art method for image-based blood flow extraction.

We use direct numerical simulations to study the two-dimensional flow of a rotating, half soap bubble that is heated at its equator. The heating produces buoyancy and rotation generates Coriolis forces in the fluid. However, owing to the curved surface of the bubble, the buoyancy and Coriolis forces vary with latitude on the bubble, giving rise to rich flow behaviour. We first explore the single-point properties of the flow, including the Reynolds and Nusselt numbers, mean fields and Reynolds stresses, all as a function of latitude. For a given Rayleigh number, we observe a non-monotonic dependence on the Rossby number $Ro$, and large-scale mean circulations that are strongly influenced by rotation. We then consider quantities that reveal the multiscale nature of the flow, including spectra and spectral fluxes of kinetic and thermal energy, enstrophy and structure functions of velocity and temperature. The fluxes show that just as for non-buoyant two-dimensional turbulence on a flat surface, there is an upscale flux of kinetic energy at larger scales (fed by buoyancy injection of turbulent kinetic energy at smaller scales), and a downscale flux of enstrophy at smaller scales. The kinetic energy spectrum and velocity structure functions are well described by Bolgiano–Obukhov (BO) scaling at scales where the effects of rotation are weak. The temperature structure functions do not, however, satisfy BO scaling in general, owing to strong intermittency in the temperature field.

Highly turbulent free-surface flows are characterised by complex and rapidly varying air–water surface features, leading to enhanced surface roughness, breakup and disintegration processes. Such a strong free-surface turbulence has an impact on a number of environmental flows, and a deeper understanding of its physical nature is fundamental. Unsteady breaking bores are of particular interest because of their recirculating motion, with large air entrainment and splashes, resulting in highly fluctuating and rapidly varying free-surface flows. Herein, new methodologies and innovative approaches are used in support of a deeper understanding of the physical processes within a breaking roller, inclusive of a comprehensive assessment of its free-surface dynamics. Because of the unsteadiness of the flow, multiple repetitions were necessary and all results were based upon an ensemble statistical analysis. Ultra-high-speed videos recorded from both top and side views allowed for a detailed characterisation of the roller's free surface, providing a description and classification of the most recurring air–water surface features. A quantification of their main properties in terms of geometry, duration and frequency of appearance revealed an evolution of these features during their lifespans. In parallel, the use of optical flow techniques provided a characterisation of the surface velocity fields, yielding information on the free-surface kinematic properties and revealing a strong link between air–water surface features, energy dissipation and time/length scales.

The fluid mechanics of hurricanes strongly depends on boundary layer energetics due to the warm-core nature of the system with peak velocities located at lower levels. One barrier that has inhibited a more complete characterization of energy transfer in the boundary layer is a lack of observations that resolve large, turbulent eddies. In particular, the occurrence and structure of upscale energy transfer (backscatter) in the hurricane boundary layer as well as the effects of backscatter on the vortex intensity are unknown. The analysis presented here of very high-resolution, three-dimensional wind observations from Hurricane Rita (2005) at peak intensity reveals large regions of organized backscatter in the boundary layer associated with coherent, turbulent eddies. Strong forwardscatter is also found next to the backscatter regions due to the interaction between adjacent eddies. Two components of the stress tensor are primarily responsible for this alternating scatter structure, as shown by large correlation coefficients between the fields: the radial–vertical component ($\tau _{13}$) and the azimuthal–vertical component ($\tau _{23}$) with average correlations of 79 % and 49 %, respectively. The Leonard, Reynolds and cross-term stress components are also provided. The impact of the sub-filter-scale energy transfer is estimated by computing the kinetic energy budget for the resolved-scale and eddy-scale motions. The results show that the sub-filter-scale energy transfer term is of the same order as the other terms in the eddy-scale budgets, contributing between 16 % and 40 % to the local time tendency with an average contribution of approximately 30 %. These results indicate that the coherent turbulent eddies can affect the vortex dynamics through wave–wave nonlinear interactions, which can subsequently influence the wave–mean flow interactions. This is the first study to examine the full sub-filter-scale energy transfer and its impact on the kinetic energy budget in the hurricane boundary layer. These findings emphasize the importance of coherent turbulence in the energy cascade and have the potential to improve turbulence closure schemes used in numerical simulations.

The ignition of a self-sustained premixed expanding flame constitutes a crucial problem in fundamental combustion research. In this work, a transient formulation on the forced ignition of a premixed expanding spherical flame in a quiescent mixture is proposed under the framework of the thermal-diffusive model. The present theory considers the unsteady evolution of the temperature and fuel mass fraction distributions subject to finite duration central heating. It can determine both critical heating power and minimum ignition energy for successful ignition. The transient flame initiation process is found to consist of four stages, including fast establishment of the ignition kernel, ignition-energy-supported flame kernel propagation, unsteady transition of the flame kernel, and quasi-steady spherical flame propagation. The unsteady effects lead to the observation of flame kernel establishing stage and considerably affect the subsequent flame kernel development by altering the flame propagation speed. Time scale analysis indicates that the transient formulation completely degenerates to the quasi-steady theory in the limits of both stationary flame ball and planar flame. Previous quasi-steady theory shows that the critical heating power for successful ignition is proportional to the cube of the critical flame radius. However, that scaling relation shall be revised in the transient formulation due to the unsteady thermal conduction from heating centre to flame front. The memory effect that persistently supports flame propagation subsequent to switching off the central heating is examined. It is found that as the heating power grows, the memory effect becomes increasingly important, and it can greatly reduce the predicted minimum ignition energy.

The removal of a liquid blockage from a tube is of importance in many processes. If the Bond number (which measures the relative size of the gravitational force by comparison with the surface tension force on the blockage plug) is large enough, then the tube will become non-occluding automatically. If not, then other measures are required to remove the blockage and the insertion of a rod is one such measure. We investigate this situation in a horizontal capillary in a downward gravity field. Theoretical results are obtained and compared with experiments. We observe that a rod insertion can cause a change from liquid plug to non-occlusion in a horizontal capillary. For uniform inner and outer contact angles, compared with the case without an inside rod, the maximum of the critical emptying line decreases significantly, but the minimum decreases a little only for a large enough value of the ratio of inner radius to outer radius (χ). We find that changing the contact angles of the inserted tube can significantly affect the non-occluding of the tube. The minimum of critical emptying line can be lowered clearly, and the minimum for a large enough value of χ is much lower than that reached for a circular tube. The insertion of a hydrophobic (hydrophilic) rod with a large enough radius can make the liquid emptying easier in a horizontal hydrophilic (hydrophobic) capillary. This provides an effective method of triggering drainage of a fluid from a capillary in applications such as optofluidics and microfluidics.

Mechanical degradation of macromolecules in strong flows is encountered in many industrial processes spanning from biopharmaceutics manufacturing to enhanced oil recovery. In spite of extensive research, from molecular studies to large experiments, unifying scaling laws and design rules to harness this phenomenon are still at an early stage. Some of the current modelling approaches predict the onset of flow-induced degradation only, leaving out quantitative calculations of scission events, while others are restricted to a particular process or the materials they have been empirically developed for. In this work we re-examine a previously published constitutive equation for the scission kinetics of polymers and implement the model using the finite volume library OpenFoam. We test and validate this model using experimental degradation measurements of aqueous poly(ethylene oxide) solutions flowing through narrow constrictions. Three polymer molecular weights and three constriction geometries are investigated. For each molecular weight, experimental degradation data of one geometry is used to calibrate the model. Following this calibration step, the level of polymer degradation as a function of flow rate can be predicted for the two other geometries, suggesting that mechanisms linking single molecule scission to macroscopic chemical reaction rate are accurately captured by the model. Although the focus of this work is on flexible linear polymers in dilute concentrations and laminar flow conditions, we discuss how to alleviate these assumptions and extend the applicability of the model to a broader range of materials and industrially relevant flow conditions.

Turbulent plane Poiseuille and Couette flows share the same geometry, but produce their flow rate owing to different external drivers: pressure gradient and shear, respectively. By looking at integral energy fluxes, we pose and answer the question as to which flow performs better at creating flow rate. We define a flow efficiency, which quantifies the fraction of power used to produce flow rate instead of being wasted as a turbulent overhead; effectiveness, instead, describes the amount of flow rate produced by a given power. The work by Gatti et al. (J. Fluid Mech., vol. 857, 2018, pp. 345–373), where the constant power input concept was developed to compare turbulent Poiseuille flows with drag reduction, is here extended to compare different flows. By decomposing the mean velocity field into a laminar contribution and a deviation, analytical expressions are derived which are the energy-flux equivalents of the FIK identity. These concepts are applied to literature data supplemented by a new set of direct numerical simulations, to find that Couette flows are less efficient but more effective than Poiseuille flows. The reason is traced to the more effective laminar component of Couette flows, which compensates for their higher turbulent activity. It is also observed that, when the fluctuating fields of the two flows are fed with the same total power fraction, Couette flows dissipate a smaller percentage of it via turbulent dissipation. A decomposition of the fluctuating field into large and small scales explains this feature: Couette flows develop stronger large-scale structures, which alter the mean flow while contributing less significantly to dissipation.

We explore flow of a completely wetting fluid in a funnel, with particular focus on contact line instabilities at the fluid front. While the flow in a funnel may be related to a number of other flow configurations as limiting cases, understanding its stability is complicated due to the presence of additional azimuthal curvature, as well as due to convergent flow effects imposed by the geometry. The convergent nature of the flow leads to thickening of the film, therefore influencing its stability properties. In this work, we analyse these stability properties by combining physical experiments, asymptotic modelling, self-similar type of analysis and numerical simulations. We show that an appropriate long-wave-based model, supported by the input from experiments, simulations and linear stability analysis that originates from the flow down an incline plane, provides a basic insight allowing an understanding of the development of contact line instability and emerging length scales.

As a first step towards the description of coherent structures in compressible shear flows, we present an asymptotic description of nonlinear travelling-wave solutions of the Navier–Stokes equations in the compressible asymptotic suction boundary layer (ASBL). We consider free-stream Mach numbers $M_\infty$ in the subsonic and moderate supersonic regime so that $0\leqslant M_\infty \leqslant 2$. We extend the large-Reynolds-number asymptotic theory of Deguchi & Hall (J. Fluid Mech., vol. 752, 2014, pp. 602–625) describing ‘free-stream’ coherent structures in incompressible ASBL flow to describe a nonlinear interaction in a thin layer situated just below the free stream. Crucially, the nonlinear interaction equations for the velocity field in this layer are identical to those obtained in the incompressible problem, and thus the asymptotic analysis supporting free-stream coherent structures in compressible ASBL is easily deduced from its incompressible counterpart. The nonlinear interaction produces streaky disturbances to both the velocity and temperature fields, which can grow exponentially towards the wall. We complete the description of the growth of the velocity and thermal streaks throughout the flow by solving the compressible boundary-region equations numerically. We show that the velocity and thermal streaks obtain their maximum amplitude in the unperturbed boundary layer. Increasing the free-stream Mach number enhances the thermal streaks and suppresses the velocity streaks, whereas varying the Prandtl number suppresses the velocity streaks, and can either enhance or suppress the thermal streaks depending on whether the flow is in the subsonic or moderate supersonic regime. Such nonlinear equilibrium states have been implicated in shear transition in incompressible flows; therefore, our results indicate that a similar mechanism may also be present in compressible flows.

The time-averaged strength $\langle \delta \rangle /\varDelta$ of a convective cellular pattern and large-scale circulation (LSC) in the turbulence regime of turbulent Rayleigh–Bénard convection exhibits a sequence of sharp changes with the Rayleigh number $Ra$. Changes occur when $Ra$ reaches transition values between the conduction, convection, chaotic, transition, soft turbulence and hard turbulence regimes. Measurements were taken from two cylindrical cells with Plexiglas walls and nitrogen gas as the working fluid. The data cover the range $10^{3} \lesssim Ra \lesssim 10^{9}$ at $Pr = 0.72$, where $Pr$ is the Prandtl number and $\varGamma \equiv D/H=1.00$ is the aspect ratio (diameter over height). The cellular pattern strength $\delta$ grows continuously as $Ra$ exceeds the critical value $Ra_c=7300$ for the wall admittance $C=2.02$ in the convection regime. In the oscillation regime, the temperature power spectra at the sidewall show an oscillatory frequency peak. In the chaotic regime, δ is diminished as $Ra$ increases. In the transition regime, $\langle \delta \rangle /\varDelta$ continues to decrease, nearly to 0. Under soft turbulence where the LSC is formed, $\langle \delta \rangle$ grows with $Ra$ as a cellular pattern in the convection regime, suggesting that LSC reflects a cellular pattern. Under hard turbulence, the LSC flow strength decreases as $Ra$ increases. The Reynolds number $Re$ was also measured based on the LSC turnover time, and it was found that two power laws, $\langle \delta \rangle /\varDelta \times Ra/Pr = 0.007Re^{3.0}$ and $\langle \delta \rangle /\varDelta \times Ra/Pr = 15.4Re^{1.76}$, fit the data for $Re<400$ and $Re>400$, respectively.

The flow of an electrically conducting fluid in a thin disc under the action of an azimuthal Lorentz force is studied experimentally. At small forcing, the Lorentz force is balanced by either viscosity or inertia, yielding quasi-Keplerian velocity profiles. For very large current I and moderate magnetic field B, we observe a new regime, fully turbulent, which exhibits large fluctuations and a Keplerian mean rotation profile $\varOmega \sim {\sqrt {IB}}/{r^{3/2}}$, where r is the distance from the axis. In this turbulent regime, the dynamics is typical of thin layer turbulence, characterized by a direct cascade of energy towards the small scales and an inverse cascade to large scales. Finally, at very large magnetic field, this turbulent flow bifurcates to a quasi-bidimensional turbulent flow involving the formation of a large scale condensate in the horizontal plane. These results are well understood as resulting from an instability of the Bödewadt–Hartmann layers at large Reynolds number and discussed in the framework of similar astrophysical flows.

Translation of a non-spherical particle trapped at a membrane or at a complex interface between fluids is a relevant situation occurring in biological, technological and everyday life systems. Here, we consider prolate spheroidal particles, translating at a clean or (insoluble) surfactant-laden planar interface located between a viscous fluid and air, protruding into the surrounding subphase. Both the subphase and monolayer contribute to the total resistance experienced by the particle, which in turn is a function of interface and bulk viscosities, particle size and aspect ratio as well as the immersion depth of the particle. We explore how the drag on a spheroidal particle at a viscous interface can both rise or decrease with particle size depending on the dimensionless Boussinesq and Marangoni numbers. For a surfactant-laden interface, the surfactant distribution in the vicinity of the moving spheroid is significantly affected by the particle's immersion depth. When a particle sinks more in the viscous fluid, as determined by the involved surface tensions, the difference in surfactant concentration between front and rear of the particle decreases. For the drag coefficient of a spherical particle at an incompressible interface at low shear Boussinesq numbers, we propose a correction to previously reported analytic expressions. We probe both parallel and perpendicular friction coefficients as they are significantly different depending on particle shape, qualitatively different depending on surface shear viscosity, and we resolve the full three-dimensional distortion of the flow field around the moving particle.

Bubble-laden thermals provide a formidable gas transport mechanism responsible, for instance, for the explosive foaming-up process during the beer tapping prank, or the infamous gas eruption of Lake Nyos in 1986. In this work we investigate experimentally the growth and motion of laser-induced turbulent thermals in a carbonated water solution with surfactants. One of the novelties of this study is that we are able to quantify with high temporal resolution the rate at which the gas volume contained in the bubbles grows. After an initial transient stage, the gas bubble and entrained liquid volumes of the thermal both grow as a cubic power of time. The buoyancy generation rate is well explained by the mass transfer scaling expected for individual bubbles. In contrast, the thermal rise velocity does not adhere to any particular scaling law. These facts lie in qualitative agreement with a phenomenological model, based on classical models for turbulent thermals, that takes into account buoyancy generation.

We experimentally investigate the interfacial instabilities governing the dynamics of an interface between two superposed immiscible liquids (oil and water) in a cylindrical container oscillating about its axis. The viscosity and density contrasts are $100$ and $0.968$, respectively. Depending on the vibrational Froude number, the evolution of interfacial wave is categorized into single-droplet (SD) formation (at the core region) and multiple/emulsion-droplet formation (at the near-wall region), and the breakage of the deformed interface into a SD is analysed for the first time. The thresholds for the onset of different instabilities responsible for each regime are presented by the amplitude and frequency of rotation, of which the boundaries predicted through the inviscid theory and scaling arguments are in good agreement with measurement. For SD formation, in particular, it is related to the critical rise velocity of the interface, represented by the vibrational Froude number. We emphasize the opposing contributions between (i) the viscous effect, i.e. the dimensionless thickness of the Stokes boundary layer, and (ii) the inviscid effect, i.e. the dimensionless maximum interface rise at the centre region (inviscid core), promoting and preventing the formation of a falling jet, respectively, which is necessary for SD formation. Our results indicate that viscosity plays an important role in shaping the boundary of SD and multiple-droplet regimes, especially at a relatively small (high) oscillating amplitude (frequency). When the amplitude is small, the enhanced viscous effect forces the deformed interface to migrate to multiple-droplet formation, skipping SD formation, with increasing frequency.

We perform computations of wake transition behind a pitching panel using an immersed boundary-based method. At a Reynolds number $Re=1000$, simulations are carried out over a wide range of pitching frequencies in the Strouhal number ($St$) range $0.2\leqslant St \leqslant 2$, for two different aspect ratio panels, given by 0.54 and 2.38. Our results reveal appearances of a reverse von Kármán vortex street at a moderate pitching frequency, which bifurcates into two distinct branches connected by strands at higher Strouhal number. In addition, the larger-aspect-ratio panel exhibits an avalanche of large, tangled three-dimensional vortex structures for $St\geqslant 1.5$. We observe strong spanwise wake compression, for the larger-aspect-ratio panel, greatly influencing the secondary instabilities at increased pitching frequency. However, no direct relationship between the growth of secondary instabilities and the spanwise pressure gradient can be established. We show that an increase in $St$ leads to a stable central region and renders a larger concentration of small-scale structures at the jet plane. The continuous wavelet transform shows complete synchronisation of the shed structure with the pitching Strouhal number of the low-aspect-ratio panel. The spectral characteristic for the high-aspect-ratio panel at $St=1$ is found to be dominated by a narrow band of low-frequency cells. We observed the period-doubling phenomena in the spanwise cells during space–time reconstruction of the velocity signals. For the present ranges of $St$, both thrust and lift signals reveal a constant amplitude oscillation, the latter being double, in magnitude, the former. Both root-mean-squared thrust and lift coefficients grow with $St$ and aspect ratio.

The impact of a forward-facing step (FFS) on the development of stationary crossflow instability is investigated on a swept wing model in a low-turbulence wind tunnel at chord Reynolds number of $2.3 \times 10^{6}$. Infrared thermography and particle image velocimetry measurements are used to quantify the transition location and growth of the crossflow instability under the influence of FFSs with different heights. Forced monochromatic stationary crossflow vortices experience an abrupt change in their trajectory as they interact with the step geometry. As the boundary layer intercepts the step an increase in the vertical velocity component and an amplification of the crossflow vortices is observed. Near the step, the vortices reach maximum amplification, while dampening downstream. The smaller FFS cases, show a local stabilising effect on the primary stationary mode and its harmonics, while in the higher step cases transition occurs. The analysis of the temporal velocity fluctuations shows a reduction in the region associated with the type-III travelling crossflow modes downstream of the step. In contrast, the velocity fluctuations in the region associated with type-I secondary instabilities increase past the FFS edge. Nonetheless, in the shortest FFS cases, these velocity fluctuations eventually decay below the clean configuration (i.e. without an FFS) levels. This behaviour is linked to a novel transition delay effect for the shortest step height investigated. The findings highlight new physical aspects driving the interaction between an amplified stationary crossflow vortex and an FFS and provide insight into possible transition delay mechanisms using such geometries.

We investigate experimentally the outflow that occurs when a plume of dense fluid strikes a sloping permeability jump in a saturated two-layered porous medium of finite extent. Contact of the plume with the jump results in a pair of leaky (primary) gravity currents that propagate up- and downdip. Previous studies have considered a thin or thick lower layer; here, we identify the non-trivial dynamics that arises for the intermediate thickness case. For instance, the primary gravity currents become arrested when the volume supplied by the plume matches that lost by basal draining. However, fluid draining into the lower layer will eventually contact the bottom boundary leading to the formation of secondary gravity currents. As these secondary gravity currents propagate left and right, they ‘tug’ upon the primary gravity currents causing remobilization. Of course, the motion of the primary and secondary gravity currents must eventually become impeded by the sidewall boundaries leading, thereafter, to layer filling. We categorize the overall filling process as either simultaneous (upper and lower layers fill concurrently) or sequential (lower layer fills first). Factors influencing the filling mode include the permeability jump angle and the plume source conditions/location. Owing to the asymmetric manner in which the upper layer fills, the plume, originally vertical, must become deflected. We categorize this deflection and identify scenarios where, somewhat counterintuitively, the plume bends towards the lower (lower permeability) layer. The significance of our measurements relative to real geophysical flows is highlighted with particular attention paid to geological sequestration and storage.