A fundamental study has been conducted to understand the mean and turbulence characteristics and their role in mixing processes of starting turbulent plumes initiated by a heated area source and released into a neutral atmosphere. Large eddy simulation has been performed in a domain of size 7 km (vertical) and $4~\text {km} \times 4~\text {km}$ (horizontal) for five different cases with surface heat flux varying from 0.5–2.5 km s$^{-1}$. With increasing buoyancy flux, the plume Reynolds number $Re$ (based on the maximum centreline velocity) increases from $6 \times 10^{7}\text {--}10^{8}$; however, the Froude number $Fr$ does not increase monotonically with increasing $B_o$, it varies from 1.091–1.149. Key physics that contributes to mixing and entrainment of ambient fluid into the plume has been investigated. The buoyancy- and shear-production of turbulence kinetic energy contributes to the mixing processes within the plume. The characteristic features of a starting plume are: the presence of a spiralling columnar neck region surrounding the central core that extends to $5D$ ($D$ is the source diameter), and which is capped with an unsteady head region that actively entrains fluid due to instabilities at the plume interface. Formation of vortex rings and instability mechanisms are linked to the entrainment process. Unsteady entrainment has been calculated using first principles, and the results revealed that entrainment, though negligible initially, initiates with the growth of the instabilities in the head region of the plume, and it correlates with the local Reynolds number in the head region.

We investigate the validity and accuracy of two commonly used two-dimensional models of a rotating, incompressible, shallow fluid layer: the shallow-water (SW) model and the Serre/Green–Naghdi (GN) model. The models differ in just one respect: the SW model imposes the hydrostatic approximation while the GN model does not. Both models assume the horizontal velocity is independent of height throughout the layer. We compare these models with their parent three-dimensional (3-D) model, initialised with a height-independent horizontal velocity, and with a mean height small compared to the domain width. For small to moderate Rossby and Froude numbers, we verify that both models well approximate the vertically averaged 3-D flow. Overall, the GN model is found to be substantially more accurate than the SW model. However, this accuracy is only achieved using an explicit reformulation of the equations in which the vertically integrated non-hydrostatic pressure is found from a novel linear elliptic equation. This reformulation is shown to extend straightforwardly to multi-layer flows having uniform density layers.

We present results on the effect of dispersed droplets in vertical natural convection (VC) using direct numerical simulations based on a two-way fully coupled Euler–Lagrange approach with a liquid phase and a dispersed droplets phase. For increasing thermal driving, characterised by the Rayleigh number, Ra, of the two analysed droplet volume fractions, $\alpha = 5\times 10^{-3}$ and $\alpha = 2\times 10^{-2}$, we find non-monotonic responses to the overall heat fluxes, characterised by the Nusselt number, Nu. The Nu number is larger when the droplets are thermally coupled to the liquid. However, Nu values remain close to the 1/4-laminar VC scaling, suggesting that the heat transport is still modulated by thermal boundary layers. Local analyses reveal the non-monotonic trends of local heat fluxes and wall-shear stresses: whilst regions of high heat fluxes are correlated to increased wall-shear stresses, the spatio-temporal distribution and magnitude of the increase are non-monotonic, implying that the overall heat transport is obscured by competing mechanisms. Most crucially, we find that the transport mechanisms inherently depend on the dominance of droplet driving to thermal driving that can quantified by (i) the bubblance parameter $b$, which measures the ratio of energy produced by the dispersed phase and the energy of the background turbulence, and (ii) ${\textit {Ra}}_d/{\textit {Ra}}$, where ${\textit {Ra}}_d$ is the droplet Rayleigh number, which we introduce in this paper. When $b \lesssim O(10^{-1})$ and ${\textit {Ra}}_d/{\textit {Ra}} \lesssim O(100)$, the Nu scaling is expected to recover to the VC scaling without droplets, and comparison with $b$ and ${\textit {Ra}}_d/{\textit {Ra}}$ from our data supports this notion.

How quickly does an ice cube melt or a lump of sugar dissolve? We address the open problem of the shapes of solids left to melt or dissolve in an ambient fluid driven by stable natural convection. The theory forms a convective form of a Stefan problem in which the evolution is controlled by a two-way coupling between the shape of the body and stable convection along its surface. We develop a new model describing the evolution of such bodies in two-dimensional or axisymmetric geometries and analyse it using a combination of numerical and analytical methods. Different initial conditions are found to lead to different fundamental shapes and descent rates. For the cases of initially linear surfaces (wedges or cones), the model admits similarity solutions in which the tip descends from its initial position as $t^{4/5}$, where t is time. It is determined that the evolving shape always forms a parabola sufficiently near the tip. For steeply inclined bodies, we establish a general two-tiered asymptotic structure comprising a broad $4/3$-power intermediate near-tip region connected to a deeper parabolic region at the finest scale. The model results apply universally for any given relationship between density, viscosity, diffusivity and concentration, including two-component convection. New laboratory experiments involving the dissolution of cones of sugar candy in water are found to collapse systematically onto our theoretically predicted shapes and descent rates with no adjustable parameters.

The unsteady forces acting on a body depend strongly on the local flow structures such as vortices. A quantitative understanding of the contribution of these structures to the instantaneous overall force is of fundamental significance. In the present study, a three-dimensional (3-D) vortex force map (VFM) method, extended from a two-dimensional (2-D) one, is used to provide better insight into the complex 3-D flow dynamics. The VFM vectors are obtained from solutions of potential equations and used to build the 3-D VFMs where the critical regions and directions associated with significant positive or negative contributions to the forces are identified. Using the existing velocity/vorticity field near the body, these VFMs can be used to obtain the body forces. A decomposed form of the force formula is also derived to separate the correction term contributed from the uncaptured vortices (close to or far away from the body). The present method is applied to the starting flow of a delta wing at high angle of attack, where LEVs are enhanced and stabilized by an axial flow effect. The analogy between the normal force of a slender delta wing and that of a 2-D flat plate with a steadily growing span is demonstrated via the VFM analysis. We find, for this application, that the force evolution exhibits some similar behaviour to a 2-D airfoil starting flow and, surprisingly, the force contribution mainly comes from the conical vortex sheet rather than the central core. Moreover, a quantitative understanding of the influence of LEVs in different evolution regimes on the body force is demonstrated.

In the present work, the influence of the gas–surface interaction law on the classical problems of viscous drag and thermophoresis on a spherical particle with high thermal conductivity immersed in a monatomic rarefied gas is investigated on the basis of the solution of a kinetic model to the linearized Boltzmann equation. The scattering kernel proposed by Cercignani and Lampis is employed to model the gas–surface interaction law via the setting of two accommodation coefficients, namely the tangential momentum accommodation coefficient and the normal energy accommodation coefficient. The viscous drag and thermophoretic forces acting on the sphere are calculated in a range of the rarefaction parameter, defined as the ratio of the sphere radius to an equivalent free path of gaseous particles, which covers the free molecular, transition and continuum regimes. In the free molecular regime the problem is solved analytically via the method of the characteristics to solve the collisionless kinetic equation, while in the transition and continuum regimes the discrete velocity method is employed to solve the kinetic equation numerically. The numerical calculations are carried out in a range of accommodation coefficients which covers most situations encountered in practice. The macroscopic characteristics of the gas flow around the sphere, namely the density and temperature deviations from thermodynamic equilibrium far from the sphere, the bulk velocity and the heat flux are calculated and their profiles as functions of the radial distance from the sphere are presented for some values of rarefaction parameter and accommodation coefficients. The results show the appearance of the negative thermophoresis in the near-continuum regime and the dependence of this phenomenon on the accommodation coefficients. To verify the reliability of the calculations, the reciprocity relation between the cross phenomena which is valid at an arbitrary distance from the sphere was found and then verified numerically within an accuracy of 0.1 %. The results for the thermophoretic force are compared to the more recent experimental data found in the literature for a copper sphere in argon gas.

The dynamics of the velocity gradient tensor in turbulence is governed in part by the anisotropic pressure Hessian, which is a non-local functional of the velocity gradient field. This anisotropic pressure Hessian plays a key dynamical role, for example in preventing finite-time singularities, but it is difficult to understand and model due to its non-locality and complexity. In this work a symmetry transformation for the pressure Hessian is introduced, such that when the transformation is applied to the original pressure Hessian, the dynamics of the invariants of the velocity gradients remains unchanged. We then exploit this symmetry transformation to perform a dimensional reduction on the three-dimensional anisotropic pressure Hessian, which, remarkably, is possible everywhere in the flow except on zero-measure sets. The dynamical activity of the newly introduced dimensionally reduced anisotropic pressure Hessian is confined to two-dimensional manifolds in the three-dimensional flow, and exhibits striking alignment properties with respect to the strain-rate eigenframe and the vorticity vector. The dimensionality reduction, together with the strong preferential alignment properties, leads to new dynamical insights for understanding and modelling the role of the anisotropic pressure Hessian in three-dimensional turbulent flows.

In a standard Rayleigh–Bénard experiment, a layer of fluid is confined between two horizontal plates and the convection regime is controlled by the temperature difference between the hot lower plate and the cold upper plate. The effect of direct heat injection into the fluid layer itself, for example by light absorption, is studied here theoretically. In this case, the Nusselt number ($Nu$) depends on three non-dimensional parameters: the Rayleigh ($Ra$) and Prandtl ($Pr$) numbers and the ratio between the spatial extension of the heat source ($l$) and the height of the fluid layer ($h$). For both the well-known classical and ultimate convection regimes, the theory developed here gives a formula for the variations of the Nusselt number as a function of these parameters. For the classical convection regime, by increasing $l/h$ from 0 to 1/2, $Nu$ gradually changes from the standard scaling $Nu \sim Ra^{1/3}$ to an asymptotic scaling $Nu \sim Ra^{{\theta }}$, with $\theta =2/3$ or $\theta =1$ by adopting, respectively, the Malkus (Proc. R. Soc. A, vol. 225, 1954, pp. 196–212) theory or the Grossmann & Lohse (J. Fluid Mech., vol. 407, 2000, pp. 27–56) theory. For the ultimate convection regime, $Nu$ gradually changes from $Nu \sim Ra^{1/2}$ scaling to an asymptotic behaviour seen only at very high $Ra$ for which $Nu \sim Ra^{2}$. This theory is validated by the recent experimental results given by Bouillaut et al. (J. Fluid Mech., vol. 861, 2019, R5) and also shows that for these experiments, $Ra$ and $Re$ numbers were too small to observe the ultimate regime. The predictions for the ultimate regime cannot be confirmed at this time due to the absence of experimental or numerical work on convection driven by internal sources and for very large $Ra$ numbers.

The linear stability of a liquid film falling down an inclined flexible plane under the influence of gravity is investigated using analytical and computational techniques. A general model for the flexible substrate is used leading to a modified Orr–Sommerfeld problem addressed numerically using a Chebyshev tau decomposition. Asymptotic limits of long waves and small Reynolds numbers are addressed analytically and linked to the computations. For long waves, the flexibility has a destabilising effect, where the critical Reynolds number decreases with decreasing stiffness, even destabilising Stokes flow for sufficiently small stiffness. To pursue this further, a Stokes flow approximation was considered, which confirmed the long-wave results, but also revealed a short wave instability not captured by the long-wave expansions. Increasing the surface tension has little effect on these instabilities and so they were characterised as wall modes. Wider exploration revealed mode switching in the dispersion relation, with the wall and surface mode swapping characteristics for higher wavenumbers. The zero-Reynolds-number results demonstrate that the long-wave limit is not sufficient to determine instabilities so the numerical solution for arbitrary wavenumbers was sought. A Chebyshev tau spectral method was implemented and verified against analytical solutions. Short wave wall instabilities persist at larger Reynolds numbers and destabilisation of all Reynolds numbers is achievable by increasing the wall flexibility, however increasing the stiffness reverts back to the rigid wall limit. An energy decomposition analysis is presented and used to identify the salient instability mechanisms and link them to their physical origin.

Laboratory experiments were carried out in an open-channel recirculating water flume for different bottom treatments and a variety of flow conditions. We acquired overhead images of the free surface downstream of the bottom features and used these to train convolutional neural network based classifiers. Using these classifiers, we demonstrate that information acquired at the surface alone can be used to differentiate between the physical features that lie at the bottom boundary. We show that although external physical processes, such as winds, can modulate the free surface, they do not necessarily eliminate the free-surface signature of the submerged bottom features. Our results provide strong motivation for future studies that probe the physical processes responsible for transporting information about the bottom of the flow to the surface.

The boundary conditions at a liquid–gas interface can be modified by the presence of pollutants. This can in turn affect the stability of the associated flow. We consider this issue in the case of a simple open cylindrical cavity flow where a liquid is set in motion by the rotation of the bottom. The problem is addressed using an experimental set-up, a linear stability code and direct numerical simulation. A robust mismatch between numerical and experimental predictions of the onset of instability is found. We model the possible effect of unidentified pollutants at the interface using an advection–diffusion equation and a closure equation linking the surface tension to the superficial pollutant concentration. The chosen closure is inspired by studies of free-surface flows with surfactants. Numerical stability analysis reveals that the base flow and its linear stability threshold are strongly affected by the addition of pollutants. Pollutants tend to decrease the critical Reynolds number; however, the nonlinear dynamics is less rich than without pollutants. For sufficiently high pollution levels, the most unstable mode belongs to a different family, in agreement with experimental findings.

We perform large-eddy simulations of turbulent flow in a channel constricted by streamwise periodically distributed hill-shaped protrusions. Two Reynolds number cases, i.e. $Re_h=10\,595$ (Fröhlich et al., J. Fluid Mech., vol. 526, 2005, pp. 19–66) and $Re_h=33\,000$ (Kähler et al., J. Fluid Mech., vol. 796, 2016, pp. 257–284), are repeated and utilized to verify and validate our numerical results, including the pressure and skin friction coefficients on bottom and top walls of the channel, mean velocity profiles and Reynolds stresses. All comparisons show reasonable agreement, providing a measure of validity that enables us to further probe simulation results at higher Reynolds number ($Re_h=10^5$) into aspects of flow physics that are not available from experiments. Effects of variation of Reynolds number are studied, with emphasis on the mean skin friction coefficients, separation bubble size and pressure fluctuations that are related to separation and reattachment. In addition, the main large-scale features of the separation behind the hill, including the scaling of the mean velocity profiles, are discussed. Furthermore, the instantaneous near-wall flow field is analysed in terms of skin friction portraits, and we confirm the existence of the local very small separation bubble on the hill crest as observed in experimental and numerical investigations. The flow field at the top wall, which is generally not given sufficient attention, is evaluated with the empirical friction law and universal logarithmic law as in planar channel flows. It is found that these empirical laws compare well with the large-eddy simulation results, although the hill constrictions behave as a perturbation source and the developed shear layer has some effects on the flow field near the top wall.

During its collapse a bubble can draw and concentrate energy from its surroundings. In the present work, we investigate the behaviour of certain multibubble configurations that have the potential for achieving significant levels of energy focusing. The dynamics of these configurations are studied for the first time in three dimensions, and are shown to be significantly different from those in two dimensions. Novel observations regarding focusing regimes in collapsing arrays are presented. Finally, through a series of numerical experiments on previously unexplored arrangements, we demonstrate that substantially enhanced levels of energy concentration can be achieved.