We investigate cryogenic flows of liquid 4He between two grids oscillating in phase, at temperatures ranging from approximately 1.3 to 2.5 K, resulting in suitably defined Reynolds numbers up to $10^{5}$. We specifically study the flow-induced motions of small particles suspended in the fluid by using the particle tracking velocimetry technique. We focus on turbulent flows of superfluid 4He that occur below approximately 2.2 K and are known to display, in certain conditions, features different from those observed in flows of classical viscous fluids, such as water. We find that, at large enough length scales, larger than the mean distance between quantized vortices, representing the quantum length scale of the flow, the shapes of the velocity and velocity increment statistical distributions are very similar to those obtained in turbulent flows of viscous fluids. The experimental outcome strongly supports the view that, in the range of investigated parameters, particles probing flows of superfluid 4He behave as if they were tracking classical flows.

The effect of base roughness on the transition and dynamics of a dense granular flow down an inclined plane is examined using particle based simulations. Different types of base topographies, rough bases made of frozen particles in either random or hexagonally ordered configurations, as well as sinusoidal bases with height modulation in both the flow and the spanwise directions, are examined. The roughness (characteristic length of the base features scaled by the flowing particle diameter) is defined as the ratio of the base amplitude and particle diameter for sinusoidal bases, and the ratio of frozen and moving particle diameters for frozen-particle bases. There is a discontinuous transition from an ordered to a disordered flow at a critical base roughness for all base topographies studied here, indicating that it is a universal phenomenon independent of base topography. The transition roughness does depend on the base configuration and the height of the flow, but is independent of the contact model and is less than 1.5 times the flowing particle diameter for all of the bases considered here. The bulk rheology is independent of the base topography, and follows the Bagnold law for both the ordered and the disordered flows. The base topography does have a dramatic effect on the flow dynamics at the base. For flows over frozen-particle bases, there is ordering down to the base for ordered flows, and the granular temperature is comparable to that in the bulk. There is virtually no velocity slip at the base, and the mean angular velocity is equal to one-half of the vorticity down to the base. For flows over sinusoidal bases, there is significant slip at the base, and the mean angular velocity is approximately an order of magnitude higher than that in the bulk within a region of height approximately one particle diameter at the base. This large particle spin results in a disordered and highly energetic layer of approximately 5–10 particle diameters at the base, where the granular temperature is an order of magnitude higher than that in the bulk. Thus, this study reveals the paradoxical result that gentler base topographies result in large slip and large agitation at the base, whereas rougher topographies such as frozen-particle bases result in virtually no slip and no agitation at the base for both ordered and disordered flows.

The unified transform, also known as the Fokas method, provides a powerful methodology for studying boundary value problems. Employing this methodology, we analyse inviscid, irrotational, two-dimensional water waves in a bounded domain, and in particular we study the generation of waves by a moving piecewise horizontal bottom, as it occurs in tsunamis. We show that this problem is characterised by two equations which involve only first-order derivatives. It is argued that under the assumptions of ‘small amplitude waves’ but not of ‘long waves’, the above two equations can be treated numerically via a recently introduced numerical technique for elliptic partial differential equations in a polygonal domain. In the particular case that the moving bottom is horizontal and under the assumption of ‘small amplitude waves’, but not of ‘long waves’, these equations yield a non-local generalisation of the Boussinesq system. Furthermore, under the additional assumption of ‘long waves’ the above system yields a Boussinesq-type system, which however includes the effect of the moving boundary.

We describe a mathematical model of double-diffusive (thermosolutal) convection in a saturated porous layer, when the solubility of the solute depends on the temperature, and the porosity and permeability of the porous medium evolve through dissolution and precipitation. We present the results of linear and weakly nonlinear stability analyses and explore the longer-term development of the system numerically. When the solutal concentration gradient is destabilising, the dynamics are somewhat similar to those previously found for single-species convection (Ritchie & Pritchard, J. Fluid Mech., vol. 673, 2011, pp. 286–317), including the occurrence of subcritical instabilities driven by a reaction–diffusion mechanism. However, when the solutal concentration gradient is stabilising and the thermal gradient is destabilising, novel dynamics emerge. These include a vertical segregation of circulation cells and porosity perturbations near the onset of convection, and over longer time scales the formation of a low-permeability region in the middle of the layer, pierced by occasional high-permeability channels. Under these conditions, convection may die away to nearly zero for extended periods before resuming vigorously in localised regions at later times.

We examine the dynamics of the leading-edge vortex (LEV) on a rapidly pitching plate with the aim of elucidating the underlying flow physics that dictates the stability and circulation of the LEV. A wide variety of flow conditions is considered in the present study by systematically varying the leading-edge sweep angle ($\unicode[STIX]{x1D6EC}=0^{\circ }$, $11.3^{\circ }$, $16.7^{\circ }$) and the reduced frequency ($f^{\ast }=0.064{-}0.151$), while keeping the pitching amplitude and the Reynolds number fixed. Tomographic particle image velocimetry is used to characterise the three-dimensional fluid motion inside the vortex core and its relation to the LEV stability and growth. A series of control volume analyses are performed to quantify the relative importance of the vorticity transport phenomena taking place inside the LEV to the overall vortex development. We show that, near the wing apex where tip effects can be neglected, the vortex develops in a nominally two-dimensional manner, despite the presence of inherently three-dimensional vortex dynamics such as vortex stretching and compression. Furthermore, we demonstrate that the vortex formation time and circulation growth are well-described by the principles of optimal vortex formation number, and that the occurrence of vortex shedding is dictated by the relative energetics of the feeding shear layer and the resulting vortex.

We perform an experimental study of turbulent Rayleigh–Bénard convection up to very high Rayleigh number, $10^{8}<Ra<10^{14}$, in a cylindrical aspect ratio one cell, 30 cm in height, filled with cryogenic helium gas. We monitor temperature fluctuations in the convective flow with four small (0.2 mm) sensors positioned in pairs 1.5 cm from the sidewalls and 2.5 cm vertically apart and symmetrically around the mid-height of the cell. Based on one-point and two-point correlations of the temperature fluctuations, we determine different types of Reynolds numbers, $\mathit{Re}$, associated with the large-scale circulation (LSC). We observe a transition between two types of $\mathit{Re}(\mathit{Ra})$ scaling around $\mathit{Ra}=10^{10}{-}10^{11}$, which is accompanied by a scaling change of the skewness of the probability distribution functions (PDFs) of the temperature fluctuations. The $\mathit{Re}(\mathit{Ra})$ dependencies measured near the sidewall at Prandtl number $\mathit{Pr}\sim 1$ are consistent with the $\mathit{Ra}^{4/9}\mathit{Pr}^{-2/3}$ scaling above the transition, while for $\mathit{Ra}<10^{10}$, the $\mathit{Re}(\mathit{Ra})$ dependencies are steeper. It seems likely that this change in $\mathit{Re}(\mathit{Ra})$ scaling is linked to the previously reported change in the Nusselt number $\mathit{Nu}(\mathit{Ra})$ scaling. This behaviour is in agreement with independent cryogenic laboratory experiments with $\mathit{Pr}\sim 1$, but markedly different from the $\mathit{Re}$ scaling obtained in water experiments ($\mathit{Pr}\sim 3.3{-}5.6$). We discuss the results in comparison with different versions of the Grossmann–Lohse theory.

Droplet-laden flows with phase change are common. This study brings to light a mechanism by which droplet inertial dynamics and local phase change, taking place at sub-Kolmogorov scales, affect vortex dynamics in the inertial range of turbulence. To do this we consider vortices placed in a supersaturated ambient initially at constant temperature, homogeneous vapour concentration and uniformly distributed droplets. The droplets also act as sites of phase change. This allows the time scales associated with particle inertia and phase change, which could be significantly different from each other and from the time scale of the flow, to become coupled, and for their combined dynamics to govern the flow. The thermodynamics of condensation and evaporation have a characteristic time scale $\unicode[STIX]{x1D70F}_{s}$. The water droplets are treated as Stokesian inertial particles with a characteristic time scale $\unicode[STIX]{x1D70F}_{p}$, whose behaviour we approximate using an $O(\unicode[STIX]{x1D70F}_{p})$ truncation of the Maxey–Riley equation for heavy particles. This inertia leads the water droplets to vacate the vicinity of vortices, leaving no nuclei for the vapour to condense. The condensation process is thus spatially inhomogeneous, and leaves vortices in the flow colder than their surroundings. The combination of buoyancy and vorticity generates a lift force on the vortices perpendicular to their velocity relative to the fluid around them. In the case of a vortex dipole, this lift force can propel the vortices towards each other and undergo collapse, a phenomenon studied by Ravichandran et al. (Phys. Rev. Fluids, vol. 2, 2017, 034702). We find, spanning the space of the two time scales, $\unicode[STIX]{x1D70F}_{p}$ and $\unicode[STIX]{x1D70F}_{s}$, the region in which lift-induced dipole collapse can occur, and show numerically that the product of the time scales is the determining parameter. Our findings agree with our results from scaling arguments. We also study the influence of varying the initial supersaturation, and find that the strength of the lift-induced mechanism has a power-law dependence on the phase-change time scale $\unicode[STIX]{x1D70F}_{s}$. We then study systems of many vortices and show that the same coupling between the two time scales alters the dynamics of such systems, by energising the smaller scales. We show that this effect is significantly more pronounced at higher Reynolds numbers. Finally, we discuss how this effect could be relevant in conditions typical of clouds.

Guided by computation, we theoretically calculate the steady flow driven by the Marangoni stress due to a surfactant introduced on a fluid interface at a constant rate. Two separate extreme cases, where the surfactant dynamics is dominated by the adsorbed phase or the dissolved phase, are considered. We focus on the case where the size of the surfactant source is much smaller than the size of the fluid domain, and the resulting Marangoni stress overwhelms the viscous forces so that the flow is strongest in a boundary layer close to the interface. We derive the resulting flow in a region much larger than the surfactant source but smaller than the domain size by approximating it with a self-similar profile. The radially outward component of fluid velocity decays with the radial distance $r$ as $r^{-3/5}$ when the surfactant spreads in an adsorbed phase, and as $r^{-1}$ when it spreads in a dissolved phase. Universal flow profiles that are independent of the system parameters emerge in both the cases. Three hydrodynamic signatures are identified to distinguish between the two cases and verify the applicability of our analysis with successive stringent tests.

A fundamental study has been conducted to understand the front characteristics and the mixing in the flow of density currents over rough surfaces. A large-eddy simulation (LES) has been performed for lock-exchange release density currents over rough walls to shed light on the unsteady mixing processes. A volume-penalization method, which is a special case of the immersed-boundary method, has been implemented to realize the bottom-mounted rough topology. In this study, the LES has been conducted in a channel with a lower wall covered with three-dimensional cube- and pyramid-shaped roughness elements, such that all cases have the same base area, but differences in the roughness solidity and volume fraction of roughness. Both cases of identical roughness elements and those with randomness in height have been considered. The maximum roughness height for all cases is kept at a constant fraction (10 %) of the total channel height. The study focuses on the instantaneous mixing processes in lock-exchange release currents over rough surfaces. An important contribution of the work is that qualitative and quantitative analysis has been conducted to demonstrate additional mixing mechanisms due to the presence of surface roughness that enhances dilution of the current. Enhanced mixing due to roughness is related to the strength of the shear layer resulting from the roughness, and hence depends on friction Reynolds number ($Re_{\unicode[STIX]{x1D70F}}$). The combined role of current characteristics and $Re_{\unicode[STIX]{x1D70F}}$ together dictate the mixing processes and extent of dilution in density currents over surface roughness.

We consider turbulence driven by a large-scale horizontal shear in Kolmogorov flow (i.e. with sinusoidal body forcing) and a background linear stable stratification with buoyancy frequency $N_{B}^{2}$ imposed in the third, vertical direction in a fluid with kinematic viscosity $\unicode[STIX]{x1D708}$. This flow is known to be organised into layers by nonlinear unstable steady states, which incline the background shear in the vertical and can be demonstrated to be the finite-amplitude saturation of a sequence of instabilities, originally from the laminar state. Here, we investigate the next order of motions in this system, i.e. the time-dependent mechanisms by which the density field is irreversibly mixed. This investigation is achieved using ‘recurrent flow analysis’. We identify (unstable) periodic orbits, which are embedded in the turbulent attractor, and use these orbits as proxies for the chaotic flow. We find that the time average of an appropriate measure of the ‘mixing efficiency’ of the flow $\mathscr{E}=\unicode[STIX]{x1D712}/(\unicode[STIX]{x1D712}+{\mathcal{D}})$ (where ${\mathcal{D}}$ is the volume-averaged kinetic energy dissipation rate and $\unicode[STIX]{x1D712}$ is the volume-averaged density variance dissipation rate) varies non-monotonically with the time-averaged buoyancy Reynolds numbers $\overline{Re}_{B}=\overline{{\mathcal{D}}}/(\unicode[STIX]{x1D708}N_{B}^{2})$, and is bounded above by $1/6$, consistently with the classical model of Osborn (J. Phys. Oceanogr., vol. 10 (1), 1980, pp. 83–89). There are qualitatively different physical properties between the unstable orbits that have lower irreversible mixing efficiency at low $\overline{Re}_{B}\sim O(1)$ and those with nearly optimal $\mathscr{E}\lesssim 1/6$ at intermediate $\overline{Re}_{B}\sim 10$. The weaker orbits, inevitably embedded in more strongly stratified flow, are characterised by straining or ‘scouring’ motions, while the more efficient orbits have clear overturning dynamics in more weakly stratified, and apparently shear-unstable flow.

We theoretically investigate the pattern formation observed when a fluid flows over a solid substrate that can dissolve or melt. We use a turbulent mixing description that includes the effect of the bed roughness. We show that the dissolution instability at the origin of the pattern is associated with an anomaly at the transition from a laminar to a turbulent hydrodynamic response with respect to the bed elevation. This anomaly, and therefore the instability, disappears when the bed becomes hydrodynamically rough, above a threshold roughness-based Reynolds number. This suggests a possible mechanism for the selection of the pattern amplitude. The most unstable wavelength scales as observed in nature on the thickness of the viscous sublayer, with a multiplicative factor that depends on the dimensionless parameters of the problem.