The formation of inhomogeneities within fluidized beds, both in terms of the particle configurations and flow structures, have a pronounced effect on the interaction force between the fluid and particles. While recent numerical studies have begun to probe the effects of inhomogeneities on the drag force at the particle scale, the applicability of prior microscale constitutive drag relations is still limited to random, homogeneous distributions of particles. Since an accurate model for the drag force is needed to predict the fluidization behaviour, the current study utilizes the lessons of prior inhomogeneity studies in order to derive a robust drag relation that is both able to account for the effect of inhomogeneities and applicable as a constitutive closure to larger-scale fluidization simulations. Using fully resolved lattice Boltzmann simulations of systems composed of fluid and monodisperse spherical particles in the low-Reynolds-number (Re) regime, the fluid–particle drag force, normalized by the ideal Stokes drag force, is found to significantly decrease, over a range of length scales, as the extent of inhomogeneities increases. The extent of inhomogeneities is found to most effectively be quantified through one of two subgrid-scale quantities: the scalar variance of the particle volume fraction or the drift flux, which is the correlation between the particle volume fraction and slip velocity. Scale-similar models are developed to estimate these two subgrid measures over a wide range of system properties. Two new drag constitutive models are proposed that are not only functions of the particle volume fraction and the Stokes number ($St$), but also dependent on one of these subgrid measures for the extent of inhomogeneities. Based on the observed, appreciable effect of inhomogeneities on drag, these new low-Re drag models represent a significant advancement over prior constitutive relations.

A high-order spectral element method is used to perform direct numerical simulations of flow past a square cylinder at the transition from the two-dimensional von Kármán vortex street in the wake of the cylinder to three-dimensionality. A Reynolds number range between 100 and 300 has been considered and good agreement with previous numerical and experimental results is obtained. At transition, the spanwise perturbation observed in the cylinder wake occurs before the onset of the streamwise vortex separation, while the former mechanism is commonly shown, in the literature, to originate from the latter by an instability of the cores or interactions of shed vortices in the vicinity of the cylinder. It is shown that the first three-dimensional unstable mode, the mode $A$, originates from an axial stretching of the upstream perturbed vorticity, existing on the braids, due to the strain field created by the spanwise vortices which evolve under a shear instability of the wake when viscous effects are small. On the other hand, the mode $B$ is observed to arise only after spanwise vortices are shed downstream of the cylinder.

Vortical structures and behaviour associated with vortex-ring collisions upon round cylinders with different cylinder-to-vortex-ring diameter ratios were studied using laser-induced fluorescence and time-resolved particle-image velocimetry techniques. Circular vortex rings of Reynolds number 4000 and three diameter ratios of $D/d=1$, 2 and 4 were considered in the present investigation. Results reveal that the collision behaviour is very different from that associated with flat surfaces, in which vortex disconnection and reconnection processes caused by the strong interactions between primary and secondary vortex rings produce small-scale vortex ringlets that eject away from the cylinders. For the cylinder with the largest diameter ratio used here, these vortex ringlets move towards each other along the collision axis, where they eventually collide to produce a vortex dipole that propagates upstream. However, as the diameter ratio decreases, these vortex ringlets are produced further away from the collision axis, which results in them ejecting away from the cylinder at increasingly larger angles relative to the collision axis. Trajectories of key vortex cores were extracted from the experimental results to demonstrate quantitatively the strong sensitivity of these vortical motions upon the diameter ratio. Furthermore, significant differences in the primary vortex-ring circulation along convex surfaces and straight edges after the collisions are observed. In particular, vortex flow models are presented here to better illustrate the highly three-dimensional flow dynamics of the collision behaviour, as well as highlighting the strong dependency of the secondary vortex-ring formation, vortex disconnection/reconnection processes, and ejection of the resulting vortex ringlets upon the diameter ratio. As such, these results are expected to shed more light on the more general scenario of vortex-ring collisions upon arbitrarily contoured solid boundaries.

The point of this short paper is to provide a useful set of equations governing stratified convection, expressed solely in terms of two thermodynamic variables, i.e. the pressure and the entropy, and the velocity field of the flow, free from any additional assumptions about the properties of turbulence. The pressure fluctuation is entirely eliminated from the energy equation and it appears only in the momentum balance, easily removable by taking its curl. This goal is achieved through the well-known anelastic approximation and an assumption of constant thermal diffusivity coefficient. The rigorously derived system of anelastic liquid equations constitutes a useful tool for modelling the dynamics of stellar interiors.

Simulations of homogeneously sheared turbulence (HST) are conducted until a universal self-similar state is established at the long non-dimensional time $\unicode[STIX]{x1D6E4}t=20$, where $\unicode[STIX]{x1D6E4}$ is the shear rate. The simulations are enabled by a new robust and discretely conservative algorithm. The method solves the governing equations in physical space using the so-called shear-periodic boundary conditions. Convection by the mean homogeneous shear flow is treated implicitly in a split step approach. An iterative Crank–Nicolson time integrator is chosen for robustness and stability. The numerical strategy captures without distortion the Kelvin modes, rotating waves that are fundamental to homogeneously sheared flows and are at the core of rapid distortion theory. Three direct numerical simulations of HST with the initial Taylor scale Reynolds number $Re_{\unicode[STIX]{x1D706}0}=29$ and shear numbers of $S_{0}^{\ast }=\unicode[STIX]{x1D6E4}q^{2}/\unicode[STIX]{x1D716}=3$, 15 and 27 are performed on a $2048\times 1024\times 1024$ grid. Here, $\unicode[STIX]{x1D716}$ is the dissipation rate and $1/2q^{2}$ is the turbulent kinetic energy. The long integration time considered allows the establishment of a self-similar state observed in experiments but often absent from simulations conducted over shorter times. The asymptotic state appears to be universal with a long time production to dissipation rate ${\mathcal{P}}/\unicode[STIX]{x1D716}\sim 1.5$ and shear number $S^{\ast }\sim 10$ in agreement with experiments. While the small scales exhibit strong anisotropy increasing with initial shear number, the skewness of the transverse velocity derivative decreases with increasing Reynolds number.

While Stokes’ hypothesis of neglecting bulk viscous effects is exact for monatomic gases and unlikely to strongly affect the dynamics of fluids whose bulk-to-shear viscosity ratio is small and/or of weakly compressible turbulence, it is unclear to what extent this assumption holds for compressible, turbulent flows of gases whose bulk viscosity is orders of magnitude larger than their shear viscosities (e.g. $\text{CO}_{2}$). Our objective is to understand the mechanisms by which bulk viscosity and the associated phenomena affect moderately compressible turbulence, in particular energy transfer and dissipation. Using direct numerical simulations of the compressible Navier–Stokes equations, we study the decay of compressible, homogeneous, isotropic turbulence for ratios of bulk-to-shear viscosity ranging from 0 to 1000. Our simulations demonstrate that bulk viscosity increases the decay rate of turbulent kinetic energy; whereas enstrophy exhibits little sensitivity to bulk viscosity, dilatation is reduced by over two orders of magnitude within the first two eddy-turnover times. Via a Helmholtz decomposition of the flow, we determine that the primary action of bulk viscosity is to damp the dilatational velocity fluctuations and reduce dilatational–solenoidal exchanges, as well as pressure–dilatation coupling. In short, bulk viscosity renders compressible turbulence incompressible by reducing energy transfer between translational and internal degrees of freedom. Our results indicate that for gases whose bulk viscosity is of the order of their shear viscosity (e.g. hydrogen) the turbulence is not significantly affected by bulk viscous dissipation, in which case neglecting bulk viscosity is acceptable in practice. However, in problems involving compressible, turbulent flows of gases like $\text{CO}_{2}$ whose bulk viscosities are thousands of times greater than their shear viscosities, bulk viscosity cannot be ignored.

Turbulence in the atmospheric boundary layer (ABL) is usually measured using sonic anemometers (sonics), but coarse spatial (${\sim}10$  cm) and temporal (${\sim}32$  Hz) resolutions of sonics preclude direct measurement of fine-scale parameters such as the turbulent kinetic energy (TKE) dissipation rate $\unicode[STIX]{x1D700}$. Instead, $\unicode[STIX]{x1D700}$ is estimated using techniques based on Kolmogorov theory. Fine-scale measurements of ABL turbulence down to Kolmogorov scale were made with a sonic and hot-film anemometer dyad (a ‘combo’ probe) during the field campaigns of the Mountain Terrain Atmospheric Modeling and Observations (MATERHORN) programme. The hot-film probe was located on a gimbal within the sonic probe volume, and was automated to rotate in the horizontal plane to align with the mean flow measured by sonic. This procedure not only helped satisfy the requirement of hot-film alignment with the mean flow, but also allowed in situ calibration of hot-film probes. This paper analyses a period of nocturnal flow that was similar to a stratified parallel shear flow. The combo-probe measurements showed an interesting phenomenon – the occurrence of strong bursts, characterized by short-term increase of velocity fluctuations and simultaneous increase of TKE dissipation rate by orders of magnitude. These bursts were indicative of unusual turbulence activity at finer (${\sim}0.1$–0.4 m) scales that are not captured by sonics since the smallest scales resolved by the latter are greater than 0.6 m. With bursting present, the spectra exhibited bumps at scales intermediate to inertial and dissipation subranges, resembling a bottleneck phenomenon. Its manifestation, although unequivocally related to bursts, may not convincingly fit into the framework of previous bottleneck-effect theories that allude to either viscous effects or buoyancy effects modifying the local energy cascade via non-local effects. The origins of burst are yet to be identified. Stratified ABL with bursts exhibits non-Kolmogorov behaviour, and hence should be modelled with caution.

The theory of Faltinsen et al. (J. Fluid Mech., vol. 289, 1995, pp. 179–198; FNV) for calculation of higher-order wave loads in deep water on a vertical free-surface-piercing circular bottom-mounted non-moving cylinder, based on potential flow of an incompressible fluid, is generalized to finite water depth. Systematic regular wave experiments are carried out, and the harmonics of the horizontal wave loads are compared with the generalized FNV theory. The horizontal force and mudline overturning moment are studied. The main focus is on the third harmonic of the loads, although all harmonics from one to five are considered. The theoretically predicted third harmonic loads are shown to agree well with the experiments for small to medium wave steepnesses, up to a rather distinct limiting wave steepness. Above this limit, the theory overpredicts, and the discrepancy in general increases monotonically with increasing wave steepness. The local Keulegan–Carpenter ($KC$) number along the axis of the cylinder indicates that flow separation will occur for the wave conditions where there are discrepancies. The assumption of $KC$-dependent added mass coefficients and the addition of a drag term in the FNV model, as is done in Morison’s equation, do not explain the discrepancies. A distinct run-up at the rear of the cylinder is observed in the experiments. A 2D Navier–Stokes simulation is carried out, and the resulting pressure, due to flow separation, is shown to qualitatively explain the local rear run-up.

The flow in a decelerating turbulent round jet is investigated using direct numerical simulation. The simulations are initialised with a flow field from a statistically stationary turbulent jet. Upon stopping the inflow, a deceleration wave passes through the jet, behind which the velocity field evolves towards a new statistically unsteady self-similar state. Assumption of unsteady self-similar behaviour leads to analytical relations concerning the evolution of the centreline mean axial velocity and the shapes of the radial profiles of the velocity statistics. Consistency between these predictions and the simulation data supports the use of the assumption of self-similarity. The mean radial velocity is predicted to reverse in direction near to the jet centreline as the deceleration wave passes, contributing to an approximately threefold increase in the normalised mass entrainment rate. The shape of the mean axial velocity profile undergoes a relatively small change across the deceleration transient, and this observation provides direct evidence in support of previous models that have assumed that the mean axial velocity profile, and in some cases also the jet spreading angle, remain approximately constant within unsteady jets.

We explore the dynamics of starting plumes by analysis of a series of new small-scale laboratory experiments combined with a theoretical model for mass, momentum, and buoyancy conservation. We find that the head of the plume ascends with a speed which is approximately 0.6 times the characteristic speed of the fluid in the following steady plume, in accord with Turner (J. Fluid Mech., vol. 13 (03), 1962, pp. 356–368), and so the fluid released from the source eventually catches the head of the flow. On reaching the top of the plume it recirculates and mixes in the plume head. We estimate that approximately $0.61\pm 0.04$ of the total buoyancy released from the source accumulates in the plume head, with the remainder in the following steady plume. Using measurements of the volume of the head, we estimate that a fraction $0.16\pm 0.08$ of the volume of the head is entrained directly from the ambient, with the remainder of the fluid in the head being supplied by the following steady plume. These results imply that the buoyancy force exerted on the plume head plus the momentum flux supplied by the following plume exceeds the rate of change of momentum of the plume head even including the added mass of the plume head. We propose that the difference is associated with a drag force resulting from the displacement of ambient fluid around the plume head. Using our experimental data, we estimate that the drag coefficient $C_{d}$ has a value $4.2\pm 1.4$, with the range in values associated with the uncertainty in our estimate of entrainment of fluid directly into the plume head. As a test, the proposed model is shown to provide a reasonable description of a starting plume rising through a stratified environment in the region below the maximum height of rise of the associated steady plume, although, above this point, the shape of the plume head changes and the model breaks down.

We present the results of a combined experimental and theoretical study of drop coalescence in the presence of an initial temperature difference $\unicode[STIX]{x0394}T_{0}$ between a drop and a bath of the same liquid. We characterize experimentally the dependence of the residence time before coalescence on $\unicode[STIX]{x0394}T_{0}$ for silicone oils with different viscosities. Delayed coalescence arises above a critical temperature difference $\unicode[STIX]{x0394}T_{c}$ that depends on the fluid viscosity: for $\unicode[STIX]{x0394}T_{0}>\unicode[STIX]{x0394}T_{c}$, the delay time increases as $\unicode[STIX]{x0394}T_{0}^{2/3}$ for all liquids examined. This observed dependence is rationalized theoretically through consideration of the thermocapillary flows generated within the drop, the bath and the intervening air layer.

Fundamental solutions (Green’s functions) are derived for the regularised 13-moment system (R13) of rarefied gas dynamics, for small departures from equilibrium; these solutions show the presence of Knudsen layers, associated with exponential decay terms, that do not feature in the solution of lower-order systems (e.g. the Navier–Stokes–Fourier equations). Incorporation of these new fundamental solutions into a numerical framework based on the method of fundamental solutions (MFS) allows for efficient computation of three-dimensional gas microflows at remarkably low computational cost. The R13-MFS approach accurately recovers analytic solutions for low-speed flow around a stationary sphere and heat transfer from a hot sphere (for which a new analytic solution has been derived), capturing non-equilibrium flow phenomena missing from lower-order solutions. To demonstrate the potential of the new approach, the influence of kinetic effects on the hydrodynamic interaction between approaching solid microparticles is calculated. Finally, a programme of future work based on the initial steps taken in this article is outlined.