We perform simulations of an impulsively started, axisymmetric viscoelastic jet exiting a nozzle and entering a stagnant gas phase using the open-source code Basilisk. This code allows for efficient computations through an adaptively refined volume-of-fluid technique that can accurately capture the deformation of the liquid–gas interface. We use the FENE-P constitutive equation to describe the viscoelasticity of the liquid, and employ the log-conformation transformation, which provides stable solutions for the evolution of the conformation tensor as the jet thins down under the action of interfacial tension. For the first time, the entire jetting and breakup process of a viscoelastic fluid is simulated, including the pre-shearing flow through the nozzle, which results in an inhomogeneous initial radial stress distribution in the fluid thread that affects the subsequent breakup dynamics. The evolution of the velocity field and the elastic stresses in the nozzle are validated against analytical solutions where possible, and the early-stage dynamics of the jet evolution are compared favourably to the predictions of linear stability theory. We study the effect of the flow inside the nozzle on the thinning dynamics of the viscoelastic jet (which develops distinctive ‘beads-on-a-string’ structures) and on the spatio-temporal evolution of the polymeric stresses in order to systematically explore the dependence of the filament thinning and breakup characteristics on the initial axial momentum of the jet and the extensibility of the dissolved polymer chains.

Melt-front instabilities during the combustion of a spinning polymethylmethacrylate disk in air are investigated. Mainly straight rivulet-type flow patterns were found, though under certain conditions saw-tooth patterns were observed. The measured wavelengths of the instabilities agree with earlier theoretical predictions of driven contact-line instabilities.

Liquid bridges are formed when a flowing liquid interacts with multiple parallel fibres, as relevant to heat and mass transfer applications that utilize flow down fibre arrays. We perform a comprehensive experimental study of flowing liquid bridges between two vertical fibres whose spacing is controlled dynamically in our experimental apparatus. The bridge patterns exhibit a regular periodic spacing typical of absolute instability for low flow rates, but become spatially inhomogeneous above a critical flow rate where the base flow is convectively unstable. The shapes of individual bridges and their associated dynamics are measured, as they depend upon the liquid properties, and fibre geometry/spacing. The bridge length scales similarly to static bridges between parallel fibres. The bridge dynamics exhibits a dependence on viscosity and scale with the impedance. A simple energy balance is used to derive a scaling relationship for the bridge velocity that captures the general trend of our experimental data. Finally, we demonstrate that these scalings similarly apply when the fibres are dynamically separated or brought together.

Particles in pressure-driven channel flow are often inhomogeneously distributed. Two modes of low-Reynolds-number instability, absent in Poiseuille flow of clean fluid, are created by inhomogeneous particle loading, and their mechanism is worked out here. Two distinct classes of behaviour are seen: when the critical layer of the dominant perturbation overlaps with variations in particle concentration, the new instabilities arise, which we term overlap modes. But when the layers are distinct, only the traditional Tollmien–Schlichting mode of instability occurs. We derive the dominant critical-layer balance equations in this flow along the lines done classically for clean fluid. These reveal how concentration variations within the critical layer cause the two particle-driven instabilities. As a result of these variations, disturbance kinetic energy production is qualitatively and majorly altered. Surprisingly, the two overlap modes, although completely different in the symmetry of the eigenstructure and regime of exponential growth, show practically identical energy budgets, highlighting the relevance of variations within the critical layer. The wall layer is shown to be unimportant. We derive a minimal composite theory comprising all terms in the complete equation which are dominant somewhere in the flow, and show that it contains the essential physics. When particles are infinitely dense relative to the fluid, the volume fraction is negligible. But for finite density ratios, the volume fraction of particles causes a profile of effective viscosity. This is shown to be uniformly stabilizing in the present flow. Gravity is neglected here, and will be important to study in the future. So will the transient growth of perturbations due to non-normality of the stability operator, in a quest for the mechanism of transition to turbulence.

Evolutions of internal waves of different modes, particularly mode 1 and mode 2, passing over variable bathymetry are investigated based on a new numerical scheme. The problem is idealized as interfacial waves propagating on two interfaces of a three-layer density stratified fluid system with large-amplitude bottom topography. The Dirichlet-to-Neumann operators are introduced to reduce the spatial dimension by one and to adapt the three-layer system and significant topographic effects. However, for simplicity, nonlinear interactions between interfaces are neglected. Numerical techniques such as the Galerkin approximation, proven effective in previous works, are applied to save computational costs. Shoaling of linear waves on an uneven bottom is first studied to validate the proposed formulation and the corresponding numerical scheme. Then, for two-dimensional numerical experiments, mode transition phenomena excited by locally confined bottom obstacles and quickly varying topographies, including the Bragg resonance, mode-2 excitation, wave homogenization, etc., are investigated. In three-dimensional simulations, internal wave refraction by a Luneberg lens is considered, and good agreement is found in comparison with the ray theory. Finally, in the limiting case, when the top layer can be negligible (for example, a gas layer of extremely small density), the problem is reduced to a two-and-a-half-layer fluid system, where an interface and a surface are unknown free boundaries. In this situation, the surface signature of an internal wave is simulated and verified by introducing the realistic bathymetry of the Strait of Gibraltar and qualitatively compared with the satellite image.

We present direct numerical simulation (DNS) and modelling of incompressible, turbulent, generalized Couette–Poiseuille flow. A particular example is specified by spherical coordinates $(Re,\theta,\phi )$, where $Re = 6000$ is a global Reynolds number, $\phi$ denotes the angle between the moving plate, velocity-difference vector and the volume-flow vector and $\tan \theta$ specifies the ratio of the mean volume-flow speed to the plate speed. The limits $\phi \to 0^\circ$ and $\phi \to 90^\circ$ give alignment and orthogonality, respectively, while $\theta \to 0^\circ,\ \theta \to 90^\circ$ correspond respectively to pure Couette flow in the $x$ direction and pure Poiseuille flow at angle $\phi$ to the $x$ axis. Competition between the Couette-flow shear and the forced volume flow produces a mean-velocity profile with directional twist between the confining walls. Resultant mean-speed profiles relative to each wall generally show a log-like region. An empirical flow model is constructed based on component log and log-wake velocity profiles relative to the two walls. This gives predictions of four independent components of shear stress and also mean-velocity profiles as functions of $(Re,\theta,\phi )$. The model captures DNS results including the mean-flow twist. Premultiplied energy spectra are obtained for symmetric flows with $\phi =90^\circ$. With increasing $\theta$, the energy peak gradually moves in the direction of increasing $k_x$ and decreasing $k_z$. Rotation of the energy spectrum produced by the faster moving velocity near the wall is also observed. Rapid weakening of a spike maxima in the Couette-type flow regime indicates attenuation of large-scale roll structures, which is also shown in the $Q$-criterion visualization of a three-dimensional time-averaged flow.

The spreading of large viscous drops of density-matched suspensions of non-Brownian spheres on a smooth solid surface is investigated experimentally at the global drop scale. The focus is on dense suspensions with a solid volume fraction equal to or greater than $40\,\%$, and for drops larger than the capillary length, i.e. for which the spreading is governed by the balance of gravitational and viscous forces. Our findings indicate that all liquids exhibit a power-law behaviour typical of gravity-driven dynamics, albeit with an effective suspension viscosity that is smaller than the bulk value. When the height of the drop is of the order of the particle size, the power law breaks down as the particles freeze while the contact line continues to advance.

The dynamic behaviour of granular media can be observed widely in nature and in many industrial processes. Yet, the modelling of such media remains challenging as they may act with solid-like and fluid-like properties depending on the rate of the flow and can display a varying apparent friction, cohesion and compressibility. Over the last two decades, the $\mu (I)$-rheology has become well established for modelling granular liquids in a fluid mechanics framework where the apparent friction $\mu$ depends on the inertial number $I$. In the geo-mechanics community, modelling the deformation of granular solids typically relies on concepts from critical state soil mechanics. Along the lines of recent attempts to combine critical state and the $\mu (I)$-rheology, we develop a continuum model based on modified cam-clay in an elastoplastic framework which recovers the $\mu (I)$-rheology under flow. This model permits a treatment of plastic compressibility in systems with or without cohesion, where the cohesion is assumed to be the result of persistent inter-granular attractive forces. Implemented in a two- and three-dimensional material point method, it allows for the trivial treatment of the free surface. The proposed model approximately reproduces analytical solutions of steady-state cohesionless flow and is further compared with previous cohesive and cohesionless experiments. In particular, satisfactory agreements with several experiments of granular collapse are demonstrated, albeit with shear bands which can affect the smoothness of the surface. Finally, the model is able to qualitatively reproduce the multiple steady-state solutions of granular flow recently observed in experiments of flow over obstacles.

The gain in efficiency of the receptivity of jets to acoustic disturbances as the nozzle lip is thicker is investigated using numerical simulations. For that, axisymmetric acoustic pulses are introduced in jets with Blasius laminar boundary-layer profiles at Mach numbers $M=0.4$, 0.6, 0.9 and 1.3 for nozzle-lip thicknesses between 1 % and 93 % of the nozzle radius. They are located on the jet axis or outside the jet with incidence angles $\varphi$ between $5^{\circ }$ and $90^{\circ }$ with respect to the downstream direction. Instability waves develop in the jet shear layer after the acoustic disturbances hit the nozzle. In all cases except for $\varphi \geq 75^{\circ }$, their amplitudes and hence the efficiency of the jet receptivity to the disturbances increase with the nozzle-lip thickness. The gains in efficiency are greater for a pulse inside the jet, generating upstream-travelling pressure waves resembling guided jet waves, than for a pulse outside the jet, producing free-stream sound waves. In the second case, the gains are significant for $\varphi =5^{\circ }$ and decrease with the incidence angle, especially for $\varphi >30^{\circ }$. Moreover, the gains are stronger for a higher Mach number, and roughly double between $M=0.4$ and $1.3$, thus reaching, for a pulse inside the jet, values close to 6 between the thinnest and thickest lips. Finally, according to additional simulations for $M=0.9$, the gains in receptivity efficiency do not change appreciably when different azimuthal mode numbers of the acoustic disturbances, widths of the pulse and shapes of the boundary-layer profile are considered.

A moving static pressure distribution is commonly used to simulate a travelling ship. However, the ship movement changes the fluid velocity around the hull, inducing pressures on the hull surface that are no longer equal to the static pressure. Therefore, we introduce a dynamic pressure correction strategy, which can accurately simulate the impact of the ship movement on the hull-surface pressure and preserve the desired hull shape under both stationary and transient conditions. The strategy is applied to a high-order spectral model and used to investigate ship-induced waves and wave resistance over a both flat and variable topography. We explore various parameters in our study, including the average water depth to ship draft ratio ($h_0/d$), the channel width to ship width ratio ($W/B$), the Froude number ($Fr_0=U/\sqrt {gh_0}$) and variations in bathymetric slope. Compared with experiments on a flat bottom, the numerical results with dynamic correction show better accuracy in the simulation of ship-induced waves and wave resistance than those obtained using a static pressure distribution. The correlation coefficient for wake waves between the numerical and experimental results is improved by approximately 0.25 with the dynamic correction strategy. The amplitude and wavelength of ship-induced mini-tsunamis over a variable topography are found to be reduced when employing a dynamic correction compared with a static pressure distribution, and this effect becomes more pronounced with higher Froude number. The static pressure approach is shown to allow large deformations of the desired hull shape and changes in ship volume which are responsible for the different wave patterns from the two approaches.

We report on an experimental study of turbulent Rayleigh–Bénard convection with asymmetric top and bottom plates. The plates are covered with pyramid-shaped roughness elements whose aspect ratios are $\lambda =1$ or $\lambda =4$. In the low-Rayleigh-number regime ($Ra<1.9\times 10^9$), the heat transport efficiencies in the asymmetric cells, characterized by the Nusselt number, are smaller than those measured in a symmetric $\lambda = 1$ cell and are greater than those for a symmetric $\lambda = 4$ cell, whereas in the high-Rayleigh-number regime ($Ra>1.9\times 10^9$), the Nusselt numbers of the asymmetric cells are, in turn, greater than those for the symmetric cell with $\lambda = 1$ and smaller than those for the symmetric cell with $\lambda = 4$. In addition, the heat transports of individual plates are studied based on the temperature drops across both halves of the cell. In the low-$Ra$ regime, the $\lambda =1$ plate shows higher heat transfer than the $\lambda =4$ plate, while for the high-$Ra$ regime, the $\lambda =4$ plate shows a higher heat transport ability. In both regimes, the individual Nusselt number of the plate with lower heat transfer is insensitive to the topology of the other plate. Besides, it is found that the symmetry of the centre temperature distribution is robust to the symmetry breaking of boundary topographies. For the $Ra$ range explored, a weak temperature inversion is observed in the bulk of asymmetric rough cells. Finally, we remark that the temperature fluctuation at the cell centre and the Reynolds number associated with the large-scale circulation show universal power laws in terms of the flux Rayleigh number as $\sigma _{T_{c}}\sim Ra_F^{0.68}$ and $Re_{LSC}\sim Ra_F^{0.36}$, respectively.

A previous paper of the authors (Duck & Stephen, J. Fluid Mech., vol. 917, 2021, A56) considered the effect of three-dimensional, temporally periodic, linear and incompressible disturbances on a Blasius boundary layer, in particular when the disturbance wavelength is both comparable to and longer than the boundary-layer thickness. This previous study revealed that, unlike the two-dimensional counterpart, a mode exists that exhibits regimes of downstream spatial growth. In this paper we extend the analysis to the compressible regime, based on the boundary-region equations methodology. The aforementioned unstable mode is seen to persist into the compressible regime, and is studied using a combination of numerical and asymptotic methods. The paper adopts several approaches. First is a numerical approach in which the spatial development of the disturbances is assessed. This then leads to a consideration of the far-downstream behaviour, using (several) asymptotic limits. Of some note, in addition to unstable modes found in the incompressible case, is the existence of a further class of instability, not found in the incompressible case (which is also analysed asymptotically), corresponding to what amounts to an inviscid instability. The far-downstream analysis enables a (sub-)classification into entropy and non-entropy modes. The former, according to this analysis, are spatially damped, with one caveat, as revealed by our marching procedure, which highlights how spatial development of disturbances can be important.

The impact of shoaling on linear water waves is well known, but it has only been recently found to significantly amplify both the intensity and frequency of rogue waves in nonlinear irregular wave trains atop coastal shoals. At least qualitatively, this effect has been partially attributed to the ‘rapid’ nature of the shoaling process, i.e. shoaling occurs over a distance far shorter than that required for waves to modulate themselves and adapt to the reduced water depth. Through a theoretical model and highly accurate nonlinear simulations, we disentangle the respective effects of the length and angle of a shoal's slope. We investigate the effects of the shoaling process rapidness on the evolution of key statistical and spectral sea-state parameters. We let the wave field evolve over a slope with constant angle in all cases while we vary the slope length. Our results indicate that the non-equilibrium dynamics is not affected by the slope length, because further extending the slope length does not influence the magnitude of the statistical and spectral measures as long as the non-equilibrium dynamics dominates the wave evolution. Thus, the shoaling effect on rogue waves is deduced to be mainly driven by the slope magnitude rather than the slope length.

A macroscopic model for perfect-slip flow in porous media is derived in this work, starting from the pore-scale flow problem and making use of an upscaling technique based on the adjoint method and Green's formula. It is shown that the averaged momentum equation has a Darcy form in which the permeability tensor, $\boldsymbol{\mathsf{K}}_{ps}$, is obtained from an associated adjoint (closure) problem that is to be solved on a (periodic) unit cell representative of the structure. Similarly to the classical permeability tensor, $\boldsymbol{\mathsf{K}}$, in the no-slip regime, $\boldsymbol{\mathsf{K}}_{ps}$ is intrinsic to the porous medium under consideration and is shown to be symmetric and positive. Integral relationships between $\boldsymbol{\mathsf{K}}_{ps}$, the partial-slip flow permeability tensor, $\boldsymbol{\mathsf{K}}_{s}$, and $\boldsymbol{\mathsf{K}}$ are derived. Numerical simulations carried out on two-dimensional model porous structures, together with an approximate analytical solution and an empirical correlation for a particular configuration, confirm the validity of the macroscopic model and the relationship between $\boldsymbol{\mathsf{K}}_{ps}$ and $\boldsymbol{\mathsf{K}}$.