We examine the flow resulting from a vertically distributed wall-source plume in both an unventilated and ventilated space. First, we present experimental ambient buoyancy measurements for an unventilated ‘filling box’ where the developing ambient buoyancy profiles are successfully modelled using an adapted ‘peeling’ model which incorporates results presented in Part 1 of this work. We then present steady-state ambient buoyancy measurements for a ventilated box. Using dye visualisation, it is observed that, in the steady state, negligible ambient vertical transport occurs within the stratified region, implying a linear ambient buoyancy stratification within this region, and we predict the gradient of this linear stratification. Finally, we apply our experimental results to two practical examples. We present a methodology to create a given linear ambient temperature stratification within a room via a prescribed uniform wall heat flux and consider the resulting temperature stratification within a large ventilated atrium with a wall heated by solar radiation.

The fluid dynamics of a bubble collapsing near an elastic or viscoelastic material is coupled with the mechanical response of the material. We apply a multiphase fluid–solid coupled computational model to simulate the collapse of an air bubble in water induced by an ultrasound shock wave, near different types of materials including metals (e.g. aluminium), polymers (e.g. polyurea), minerals (e.g. gypsum), glass and foams. We characterize the two-way fluid–material interaction by examining the fluid pressure and velocity fields, the time history of bubble shape and volume and the maximum tensile and shear stresses produced in the material. We show that the ratio of the longitudinal acoustic impedance of the material compared to that of the ambient fluid, $Z/Z_0$, plays a significant role. When $Z/Z_0<1$, the material reflects the compressive front of the incident shock into a tensile wave. The reflected tensile wave impinges on the bubble and decelerates its collapse. As a result, the collapse produces a liquid jet, but not necessarily a shock wave. When $Z/Z_0>1$, the reflected wave is compressive and accelerates the bubble's collapse, leading to the emission of a shock wave whose amplitude increases linearly with $\log (Z/Z_0)$, and can be much higher than the amplitude of the incident shock. The reflection of this emitted shock wave impinges on the bubble during its rebound. It reduces the speed of the bubble's rebound and the velocity of the liquid jet. Furthermore, we show that, for a set of materials with $Z/Z_0\in [0.04, 10.8]$, the effect of acoustic impedance on the bubble's collapse time and minimum volume can be captured using phenomenological models constructed based on the solution of Rayleigh–Plesset equation.

This study examines the flow of dense granular materials under external shear stress and pressure using discrete element method simulations. In this method, the material is allowed to strain along all periodic directions and adapt its solid volume fraction in response to an imbalance between the internal state of stress and the external applied stress. By systematically varying the external shear stress and pressure, the steady rheological response is simulated for: (1) rate-independent quasi-static flow; and (2) rate-dependent inertial flow. The simulated flow is viscometric with non-negligible first and second normal stress differences. While both normal stress differences are negative in inertial flows, the first normal stress difference switches from negative to slightly positive, and second normal stress difference tends to zero in quasi-static flows. The first normal stress difference emerges from a lack of coaxiality between a second-rank contact fabric tensor and strain rate tensor in the flow plane, while the second normal stress difference is linked to an excess of contacts in the shear plane compared with the vorticity direction. A general rheological model of second order (in terms of strain rate tensor) is proposed to describe the two types of flow, and the model is calibrated for various values of interparticle friction from simulations on nearly monodisperse spheres. The model incorporates normal stress differences in both regimes of flow and provides a complete viscometric description of steady dense granular flows.

We perform direct numerical simulation and large-eddy simulation of an initially spherical region of turbulence evolving in free space. The computations are performed with a lattice Green's function method, which allows the exact free-space boundary conditions to be imposed on a compact vortical region. Large-eddy simulations are conducted with the stretched vortex subgrid stress model. The initial condition is spherically windowed, isotropic homogeneous incompressible turbulence. We study the spectrum and statistics of the decaying turbulence and compare the results with decaying isotropic turbulence, including cases representing different low-wavenumber behaviour of the energy spectrum (i.e. $k^2$ versus $k^4$). At late times the turbulent sphere expands with both mean radius and integral scale showing similar timewise growth exponents. The low-wavenumber behaviour has little effect on the inertial scales, and we find that decay rates follow the predictions of Saffman (J. Fluid Mech., vol. 27, 1967, pp. 581–593) in both cases, at least until approximately $400$ initial eddy turnover times. The boundary of the spherical region develops intermittency and features ejections of vortex rings. These are shown to occur at the integral scale of the initial turbulence field and are hypothesized to occur due to a local imbalance of impulse on this scale.

Laboratory experiments are conducted to examine the evolution of and sedimentation from a particle-bearing jet advancing along a horizontal or downward-sloping boundary underlying a uniform-density ambient fluid. The jet front advances along the bottom while exhibiting a self-similar profile. As the jet propagates downstream, particles settle out, resulting in a teardrop-shaped sediment bed whose geometric parameters are measured non-intrusively using a light attenuation technique. The bed shape is well represented by the theory that assumes a Gaussian radial profile of velocity within the jet and accounts for the bedload transport of particles after they settle. In particular, the bed length is given by $l_0 = (1.8 \pm 0.4) [M_0/(g^\prime d_p)]^{1/2}$, in which $M_0$ is the source momentum flux, $g^\prime$ is the reduced gravity of the particles and $d_p$ is the particle diameter. The corresponding scale for the sediment depth captures the anticipated order of magnitude for the maximum depth of deposit, but the measurements indicate additional dependence upon $M_0$, suggesting that the morphology of the bed non-negligibly influences particle settling.

We consider the Brownian motion of a rigid spherical particle in a binary fluid mixture, which lies in the homogeneous phase near the demixing critical point, assuming that neither component is more attracted by the particle surface. In a recent study, it was experimentally shown that the self-diffusion coefficient first decreases and then reaches a plateau as the critical point is approached. The decrease should reflect the critical enhancement of the viscosity, while the plateau was interpreted as representing the suppression of the enhancement due to the shear around the particle. To examine this interpretation, we take into account the local shear rate to calculate the dependence of the drag coefficient on the particle speed, and then utilize a Langevin equation to calculate the self-diffusion coefficient.

The dynamics of thin volatile droplets comprising of binary mixtures deposited on a heated substrate are investigated. Using lubrication theory, we develop a novel one-sided model to predict the spreading and retraction of an evaporating sessile axisymmetric droplet formed of a volatile binary mixture on a substrate with high wettability. A thin droplet with a moving contact line is considered, taking into account the variation of liquid properties with concentration as well as the effects of inertia. The parameter space is explored and the resultant effects on wetting and evaporation are evaluated. Increasing solutal Marangoni stress enhances spreading rates in all cases, approaching those of superspreading liquids. To validate our model, experiments are conducted with binary ethanol–water droplets spreading on hydrophilic glass slides heated from below. The spreading rate is quantified, revealing that preferential evaporation of the more volatile component (ethanol) at the contact line drives superspreading, leading in some cases to a contact line instability. Good qualitative agreement is found between our model and experiments, with quantitative agreement being achieved in terms of spreading rate.

We show that, in addition to providing effective and competitive closures, when analysed in terms of the dynamics and physically relevant diagnostics, artificial neural networks (ANNs) can be both interpretable and provide useful insights into the on-going task of developing and improving turbulence closures. In the context of large-eddy simulations (LES) of a passive scalar in homogeneous isotropic turbulence, exact subfilter fluxes obtained by filtering direct numerical simulations are used both to train deep ANN models as a function of filtered variables, and to optimise the coefficients of a turbulent Prandtl number LES closure. A priori analysis of the subfilter scalar variance transfer rate demonstrates that learnt ANN models outperform optimised turbulent Prandtl number closures and Clark-type gradient models. Next, a posteriori solutions are obtained with each model over several integral time scales. These experiments reveal, with single- and multi-point diagnostics, that ANN models temporally track exact resolved scalar variance with greater accuracy compared to other subfilter flux models for a given filter length scale. Finally, we interpret the artificial neural networks statistically with differential sensitivity analysis to show that the ANN models feature a dynamics reminiscent of so-called ‘mixed models’, where mixed models are understood as comprising both a structural and functional component. Besides enabling enhanced-accuracy LES of passive scalars henceforth, we anticipate this work to contribute to utilising neural network models as a tool in interpretability, robustness and model discovery.

In resolvent analyses of turbulent channel flows it has been common practice to neglect or model the nonlinear forcing term that forms the input of the resolvent. However, the spatiotemporal structure of this term is mostly unknown. Here, this nonlinear forcing term is quantified. The Fourier transform of its two-point space–time correlation, its cross-spectral density (CSD), is computed. The CSD is evaluated for two channel flows at friction Reynolds numbers $Re_{\tau } =179$ and $Re_{\tau } =543$ via direct numerical simulations (DNS). The CSDs are computed for energetic structures typical of buffer-layer and large-scale motions, for different temporal frequencies. It is found that the forcing is structured and that its solenoidal part, which is the only one affecting the velocity field, is the combination of an oblique streamwise vortical forcing and a streamwise component that counteract each other, as in a destructive interference. It is shown that a rank-2 approximation of the forcing, with only the most energetic spectral proper orthogonal decomposition (SPOD) modes, leads to the bulk of the response. Moreover, it is found that the nonlinear forcing term has a non-negligible projection onto the linear sub-optimal forcings of resolvent analysis, which demonstrates that the linear optimal forcing is not representative of the nonlinear forcing. Finally, it is clarified that the Cess eddy-viscosity-modelled forcing improves the accuracy of resolvent analysis prediction because the modelled forcing projects onto the linear sub-optimal forcings similarly to DNS data.

The lattice Boltzmann method (LBM), which was originally designed for near-incompressible Navier–Stokes flows, has been extended to rarefied gas flows with high-order quadrature in recent years. Although the ability of the high-order LBM to capture rarefaction effects has been demonstrated by many authors, its accuracy and efficiency are often undermined by numerical dissipation introduced by the off-lattice abscissas in Gauss–Hermite quadrature. Here, using the spontaneous Rayleigh–Brillouin scattering problem as the benchmark, we assess the accuracy and efficiency of the high-order LBM with on-lattice quadrature rules up to 39th order. The numerical error comprises two parts, one due to the rarefaction effect and the other due to temporal-spatial discretization, and we find that the former depends not only on the number of discrete velocities, but also on their distribution in velocity space. With a quadrature of 29th order, the error between the LBM and the discrete velocity method is found to be below 1 % up to $Kn=2.0$. Compared with a finite-volume Bhatnagar–Gross–Krook solver using Gauss–Hermite quadrature, the on-lattice LBM has a numerical dissipation several orders of magnitude lower, and achieves the same accuracy with fewer discrete velocities.

This paper characterizes the nature of velocity fluctuations in swimmer suspensions by determining the fluid velocity variance and the diffusivity of immersed passive tracers in dilute suspensions of hydrodynamically interacting slender microswimmers. The swimmers considered include straight-swimmers whose orientations change only on account of hydrodynamic interactions, and run-and-tumble particles (RTPs) whose orientations change in addition due to tumble events obeying Poisson statistics. In a dilute non-interacting swimmer suspension, the fluid velocity variance is finite and the covariance is short ranged, decaying for distances larger than the swimmer length. In contrast, we show, for a suspension of interacting straight-swimmers, that pair interactions lead to a non-decaying velocity covariance, and a variance that diverges logarithmically with system size. For suspensions of RTPs, the aforementioned divergence is arrested due to tumbling. While the variance remains finite, and the covariance short ranged, for suspensions of interacting rapid tumbling RTPs (short run lengths), the underlying straight-swimmer divergence manifests as a logarithmic increase of the variance with the swimmer run length for persistent RTPs (long run lengths), with a correspondingly long-ranged covariance. The tracer mean squared displacement undergoes an increasingly broad crossover from the ballistic to the diffusive regime for persistent RTPs, with the tracer diffusivity exhibiting a stronger linear increase with the swimmer run length. Our analysis explains the bifurcation of the velocity variance and tracer diffusivities between pusher and puller suspensions, as well as numerous observations of a volume-fraction-dependent crossover time for passive tracer dynamics.

The influence of the different thermal boundary conditions at the bottom and top plates on the dynamics and statistics of a turbulent Rayleigh–Bénard convection flow is studied in three-dimensional direct numerical simulations. The flow evolves in a closed cylinder with an aspect ratio of $\varGamma =1/2$ in air for a Prandtl number $Pr=0.7$ and a Rayleigh number $Ra=10^7$ and in the liquid metal alloy GaInSn at $Pr=0.033$ and $Ra=10^7$, $10^8$. We apply for each case three different thermal boundary conditions at the top and bottom of the fluid volume while leaving the solid sidewall thermally insulated: (i) fixed temperature, (ii) fixed heat flux and (iii) conjugate heat transfer which couples the temperature and heat flux in the working fluid to that of the finitely thick, solid plates enclosing the turbulent flow. The global heat transfer is enhanced by up to 19 % for the conjugate heat transfer case in comparison to that of isothermal plates. The differences decrease for the lower of the two Prandtl numbers; they remain generally smaller for the global turbulent momentum transfer. Mean temperature profiles and root mean square velocity fluctuations are surprisingly weakly affected. The largest difference appears for the distribution of local thermal boundary scales when the cases of fixed temperature and of conjugate heat transfer are compared. We also discuss our results in view to experimental uncertainties in liquid metal experiments.

Cross-streamline migration of deformable capsules is studied in three-dimensional curved vessels using a numerical model. Two geometries are chosen: torus vessels to study the effect of constant vessel curvature, and U-shaped vessels to study the effect of change in curvature. A wide range of inertia is considered to include a broad range of applications, from the microcirculation to inertial microfluidics. Vessel curvature and change in curvature are shown to affect capsule migration because of the way they affect the undisturbed flow. In toroidal vessels at negligible inertia, no secondary flow exists and the axial fluid velocity is shifted toward the inner surface of the vessel. In this limit, capsules settle at a location that is away from the vessel centreline, and between the location of the maximum fluid velocity and the innermost surface. Increasing the vessel curvature results in the equilibrium position being increasingly closer to the inner surface. The results suggest the presence of a curvature-induced migration that drives the capsule continuously toward the region of higher streamline curvature. The equilibrium locations are on the symmetry plane of the torus, and are stable equilibria when inertia is negligible. At finite inertia, capsules released on the symmetry plane first settle at an equilibrium located on this plane, which may either be stable or unstable. This equilibrium location depends on both capsule deformability (characterised by capillary number, $Ca$) and the tube Reynolds number $R{e_t}$. It is closer to the inner edge of the torus for smaller $R{e_t}$ and larger $Ca$, and toward the outer edge for larger $R{e_t}$ and smaller $Ca$. This dependence of the equilibria arises due to the secondary flow (or Dean's vortex), which opposes the curvature-induced inward migration. The equilibrium locations on the symmetry plane are unstable if the secondary flow is strong, in which case capsules depart the symmetry plane, and set on to a spiralling motion, eventually settling near the centres of the Dean's vortices. For the U-shaped vessels, the curvature change is shown to create a radial velocity component, even at small inertia, that can be of comparable magnitude to the axial velocity, and that increases with increasing curvature. This radial velocity causes a large and abrupt change in capsule trajectory toward the inner side of the vessel where the curvature increases, but toward the outer side where the curvature decreases.

We study the dynamic wetting of a self-propelled viscous droplet using the time-dependent lubrication equation on a conical-shaped substrate for different cone radii, cone angles and slip lengths. The droplet velocity is found to increase with the cone angle and the slip length, but decrease with the cone radius. We show that a film is formed at the receding part of the droplet, much like the classical Landau–Levich–Derjaguin film. The film thickness $h_f$ is found to decrease with the slip length $\lambda$. By using the approach of matching asymptotic profiles in the film region and the quasi-static droplet, we obtain the same film thickness as the results from the lubrication approach for all slip lengths. We identify two scaling laws for the asymptotic regimes: $h_fh''_o \sim Ca^{2/3}$ for $\lambda \ll h_f$ and $h_f h''^{3}_o\sim (Ca/\lambda )^2$ for $\lambda \gg h_f$; here, $1/h''_o$ is a characteristic length at the receding contact line and $Ca$ is the capillary number. We compare the position and the shape of the droplet predicted from our continuum theory with molecular dynamics simulations, which are in close agreement. Our results show that manipulating the droplet size, the cone angle and the slip length provides different schemes for guiding droplet motion and coating the substrate with a film.

We have investigated steep three-dimensional surface gravity wave groups formed by dispersive focusing using a fully nonlinear potential flow solver. We find that third-order resonant interactions result in rapid energy transfers to higher wavenumbers and reduced directional spreading during focusing, followed by spectral broadening during defocusing, forming steep wave groups with augmented kinematics and a prolonged lifespan. If the wave group is initially narrow-banded, quasi-degenerate interactions arise, characterised by energy transfers along the resonance angle, ${\pm }35.26^{\circ }$, of the Phillips ‘figure-of-eight’ loop. Spectral broadening due to the quasi-degenerate interactions facilitates non-degenerate interactions, characterised by oblique energy transfers at approximately ${\pm }55^{\circ }$ to the spectral peak. We consider the influence of steepness, finite depth, directional spreading and the high-wavenumber tail on spectral evolution. Steepness is found to augment both the quasi-degenerate and non-degenerate interactions similarly. However, a reduction in depth is found to weaken the quasi-degenerate interactions more severely than the non-degenerate interactions. We observe that increased directional spreading reduces spectral evolution, partially because wave groups with more spreading focus for a shorter duration due to linear dispersion. However, we also find that directional spreading reduces the peak rates of energy transfer. Inclusion of the high-wavenumber tail of the Joint North Sea Wave Project spectrum further reduces rates of energy transfer compared with a Gaussian wavenumber spectrum. Thus, directional spreading and the high-wavenumber tail may be integral to a form of spectral equilibrium that reduces rapid energy transfers during a steep wave event.

Two important issues that arise when using an Eulerian–Eulerian two-phase model to simulate submarine granular flows have not been addressed well: the shear-induced volume change and the resultant pore-pressure feedback. This study develops a multiphase model with a novel evolution equation to determine the static solid pressure resulting from prolonged contact between particles. The evolution equation can effectively describe the relaxation process of the static solid pressure and the shear-induced volume change in plane-shear configurations. Additionally, the evolution equation allows the present model to capture two typical phenomena associated with pore-pressure feedback: the time delay of the initiating submarine granular flows and the different collapse processes for differently packed columns.

A model problem for analysing the interaction between coherent structures in shear flows with the presence of a convective instability is proposed in this work. Starting from Couette flow, a permanent forcing in the shape of a hyperbolic tangent is introduced in the laminar equations, leading to a wall-bounded flow with an inflection point, which triggers a hydrodynamic instability. Temporal linear stability analysis applied to this new flow model shows that this flow is unstable at low Reynolds numbers, giving rise to Kelvin–Helmholtz-like vortices. Due to the presence of shear, streaks and rolls (streamwise vortices), predicted by resolvent analysis, are also present in the flow, and these structures will interact with vortices via oblique waves. Results of locally parallel analysis inspired the design of a computational box for a direct numerical simulation of such flow and the numerical results exhibit a limit cycle involving streaks, vortices, rolls, oblique waves and the mean flow, so that the flow becomes periodically unstable for the present case. The flow dynamics is shown to reproduce some of the features of jets and mixing layers, such as jitter and translational instability, showing that the present model can potentially clarify some of the phenomena involved in the turbulent dynamics of such flows.

We consider a spun-up system of an inner cylinder of fluid surrounded by an outer fluid layer within a rotating cylindrical container, in the absence of gravity. The outer layer may be of differing density and viscosity to the inner layer. If the inner layer is denser than the outer layer then the effect of rotation, in the presence of a perturbation to the interface between the two layers, is to force the inner fluid outwards and the outer fluid inwards, subject to possible surface tension stabilisation. The relative importance of viscosity to rotation is described by an Ekman number. We investigate the behaviour of perturbations to the interface in the inviscid limit and low and high viscosity limits. In the low viscosity limit, perturbations grow as an $O(Ek^{1/2})$ correction to the inviscid growth rate. In the high viscosity limit, perturbations grow as $O(Ek^{-1})$. In the absence of surface tension, the preferred mode of growth is independent of the layer density difference and depends only upon the domain aspect ratio, initial position of the interface, and viscosity contrast. Numerical simulations of the flow are carried out using a volume-of-fluid formulation. The growth rates from these simulations are compared with the theoretical predictions in both low and high viscosity limits and the agreement is seen to be good. Finally, we examine the special case of a single-layer rotating viscous column and describe the preferred-mode boundary between a varicose mode and a spiral mode in the high viscosity, high surface tension limit.

We present simulations of a circular cylinder undergoing vortex-induced vibration in-line with a free stream in conjunction with a theory for the fluid dynamics. Initially, it is shown that increasing the Reynolds number from 100 to 250 results in a 12-fold increase of the peak response amplitude at a fixed mass ratio of $m^*=5$. Subsequently, $m^*$ is varied from 2 up to 20 at a fixed Reynolds number of 180. The response amplitude as a function of the reduced velocity $U^*$ displays a single excitation region with peak amplitudes of approximately 1% of the cylinder diameter, irrespective of the $m^*$ value. The vibration is always excited by the alternating shedding of single vortices. We develop a new model for the in-line fluid force, which comprises an inviscid inertial force, a quasi-steady drag and a wake drag induced by vortex shedding. Our analysis shows that the wake drag appropriately captures a gradual shift in the timing of vortex shedding in its phase variation as a function of $U^*$ while the magnitude of the wake drag displays a resonant amplification within the excitation region. We use the theory to illustrate why peak amplitudes, which occur when the vibration frequency is equal to the structural frequency in still fluid, do not depend on $m^*$, in agreement with our simulations as well as previous experiments at Reynolds numbers higher than considered here. This new theory provides physical insight which could not be attained heretofore by employing semi-empirical approaches in the literature.

We consider the evolution of a shock wave generated by an impulsively accelerated, two-dimensional, almost planar piston with a sinusoidally corrugated surface of amplitude $\epsilon$. We develop a complex-variable formulation for a nonlinear theory of generalized geometrical shock dynamics (GGSD) (Best, Shock Waves, vol. 1, issue 4, 1991, pp. 251–273; Best, Proc. R. Soc. Lond. A, vol. 442, 1993, pp. 585–598) as a hierarchical expansion of the Euler equations that can be closed at any order. The zeroth-order truncation of GGSD is related to the equations of Whitham's geometrical shock dynamics (GSD), while higher-order corrections incorporate non-uniformity of the flow immediately behind the piston-driven shock. Numerical solutions to GGSD systems up to second order are coupled to an edge-detection algorithm in order to investigate the hypothesized development of a shock-shape curvature singularity as the rippled shock evolves. This singular behaviour, together with the simultaneous development of a Mach-number discontinuity, is found at all orders of the GGSD hierarchy for both weak and strong shocks. The critical time at which a curvature singularity occurs converges as the order of the GGSD system increases at fixed $\epsilon$, and follows a scaling inversely proportional to $\epsilon$ at sufficiently small values. This result agrees with the weakly nonlinear GSD analysis of Mostert et al. (J. Fluid Mech., vol. 846, 2018, pp. 536–562) for a general Mach-number perturbation on a planar shock, and suggests that this represents the universal behaviour of a slightly perturbed, planar shock.