The entrainment hypothesis states that the mean inflow velocity across the boundary of a turbulent flow is proportional to a characteristic velocity of the flow. Proposed by G. I. Taylor approximately 80 years ago, it is still a common model of turbulence closure widely used in environmental engineering and geophysical fluid mechanics. Although it is a very simple concept and mathematical model, it has proven to be able to predict the entrainment in a variety of geophysical flows, e.g. convective clouds and plumes from erupting volcanoes in the atmosphere; dense water overflows and turbidity currents in the ocean; magma injection in a magma chamber in the interior of the Earth, to name just a few. In a seminal paper, Turner (J. Fluid Mech., vol. 173, 1986, pp. 431–471) presents a variety of laboratory and geophysical flows to illustrate the success of the entrainment hypothesis and discusses why such a simple hypothesis works so well even when the original assumptions are no longer valid.

Radial unstable stratification is a potential source of turbulence in the cold regions of accretion disks. To investigate this thermal effect, here we focus on two-dimensional Rayleigh–Bénard convection in an annulus subject to radially dependent gravitational acceleration $g \propto 1/r$. Next to the Rayleigh number $Ra$ and Prandtl number $Pr$, the radius ratio $\eta$, defined as the ratio of inner and outer cylinder radii, is a crucial parameter governing the flow dynamics. Using direct numerical simulations for $Pr=1$ and $Ra$ in the range from $10^7$ to $10^{10}$, we explore how variations in $\eta$ influence the asymmetry in the flow field, particularly in the boundary layers. Our results show that in the studied parameter range, the flow is dominated by convective rolls and that the thermal boundary-layer (TBL) thickness ratio between the inner and outer boundaries varies as $\eta ^{1/2}$. This scaling is attributed to the equality of velocity scales in the inner ($u_i$) and outer ($u_o$) regions. We further derive that the temperature drops in the inner and outer TBLs scale as $1/(1+\eta ^{1/2})$ and $\eta ^{1/2}/(1+\eta ^{1/2})$, respectively. The scalings and the temperature drops are in perfect agreement with the numerical data.

The development of thermal boundary layers and plume near a section-triangular roof under different isothermal heating conditions has been the focus of numerous numerical studies. However, flow transition in this type of flow has never been observed experimentally. Here, phase-shifting interferometry and thermistor measurements are employed to experimentally observe and quantify the flow transitions in a buoyancy-driven flow over an isothermal section-triangular roof. Visualisation of temperature contours is conducted across a wide range of Rayleigh numbers from laminar at 103 to chaotic state at 4 × 106. Power spectral density of the temperature measurements reveals the type of bifurcations developing as the Rayleigh number is increased. This flow transition is characterised as a complex bifurcation route with the presence of two fundamental frequencies, a low and a high frequency. We found that the thermal stratification in the environment plays a significant role in the flow transition. The spatial development of flow is also quantitatively and qualitatively described. In addition to clarifying flow transition in experiments, the work demonstrates the implementation of phase-shifting interferometry and punctual temperature measurements for characterisation of near-field flow over a heated surface.

We use large-eddy simulations to study the penetration of a buoyant plume carrying a passive tracer into a stably stratified layer with constant buoyancy frequency. Using a buoyancy-tracer volume distribution, we develop a method for objectively partitioning plume fluid in buoyancy-tracer space into three regions, each of which corresponds to a coherent region in physical space. Specifically, we identify a source region where undiluted plume fluid enters the stratified layer, a transport region where much of the transition from undiluted to mixed fluid occurs in the plume cap and an accumulation region corresponding to a radially spreading intrusion. This method enables quantification of different measures of turbulence and mixing within each of the three regions, including potential energy and turbulent kinetic energy dissipation rates, an activity parameter and the instantaneous mixing efficiency. We find that the most intense buoyancy gradients lie in a thin layer at the cap of the penetrating plume. This provides the primary stage of mixing between plume and environment and exhibits a mixing efficiency around 50 %. Newly generated mixtures of environmental and plume fluid join the intrusion and experience relatively weak turbulence and buoyancy gradients. As the intrusion spreads radially, environmental fluid surrounding the intrusion is mixed into the intrusion with moderate mixing efficiency. This dominates the volume of environmental fluid entrained into the region containing plume fluid. However, the ‘strongest’ entrainment, as measured by the specific entrainment rate, is largest in the plume cap, where the most buoyant environmental fluid is entrained.

This study presents direct numerical simulation results of two-layer Rayleigh–Bénard convection, investigating the previously unexplored Rayleigh–Weber parameter space $10^6\leq Ra\leq 10^8$ and $10^2\leq We\leq 10^3$. Global properties, such as the Nusselt and Reynolds numbers, are compared against the extended Grossmann–Lohse theory for two fluid layers, confirming a weak Weber number dependence for all global quantities and considerably larger Reynolds numbers in the lighter fluid. Statistics of the flow reveal that the interface fluctuates more intensely for larger Weber and smaller Rayleigh numbers, something also reflected in the increased temperature root mean square values next to the interface. The dynamics of the deformed two-fluid interface is further investigated using spectral analysis. Temporal and spatial spectrum distributions reveal a capillary wave range at small Weber and large Rayleigh numbers, and a secondary energy peak at smaller Rayleigh numbers. Furthermore, the maxima of the space–time spectra lie in an intermediate dispersion regime, between the theoretical predictions for capillary and gravity-capillary waves, showing that the gravitational energy of the interfacial waves is strongly altered by temperature gradients.

In this experimental study, we explore the dynamics of the thermal boundary layer in liquid metal Rayleigh–Bénard convection, covering the parameter ranges of $0.026 \leq$ Prandtl numbers $(Pr) \leq 0.033$ and Rayleigh numbers ($Ra$) up to $2.9\times 10^9$. Our research focuses on characterising the thermal boundary layer near the top plate of a cylindrical convection cell with an aspect ratio of 0.5, distinguishing between two distinct regions: the shear-dominated region around the centre of the top plate and a location near the side wall where the boundary layer is expected to be affected by the impact or ejection of thermal plumes. The dependencies of the boundary layer thickness on $Ra$ at these positions reveal deviating scaling exponents with the difference diminishing as $Ra$ increases. We find stronger fluctuations in the boundary layer and increasing deviation from the Prandtl–Blasius–Pohlhausen profile with increasing $Ra$, as well as in the measurements outside the centre region. Our data illustrate the complex interplay between flow dynamics and thermal transport in low-$Pr$ convection.

Emptying–filling boxes have been studied in a wide range of configurations for decades, but the flow created in the box by two plumes rising from sources of arbitrary strength and elevation was previously unsolved. Guided by experiments and simplified analytical modelling, we reveal a rich array of two- and three-layer stratifications across seven possible flow regimes. The governing equations for these regimes show how the prevailing regime and stratification properties vary with three key parameters: the relative strength of the plumes, the height difference between their sources and a parameter characterising the resistance of the box to emptying. We observe and explain new behaviours not described in previous studies that are crucial to understanding emptying–filling boxes with multiple plumes. In particular, we demonstrate that the oft-assumed premise that $n$ plumes leads to a stratification with $n+1$ layers is not necessarily true, even in the absence of mixing. Two emptying–filling box models are developed: an analytical model addressing all combinations of the governing parameters and an extended model for three-layer stratifications that incorporates two mixing processes observed in the experiments. The predictions of these two models are in generally excellent agreement with measurements from the experimental campaign covering 69 combinations of the governing parameters. This study improves our understanding of emptying–filling boxes and could facilitate improvements to natural ventilation building design, as demonstrated by an example scenario in which occupants feel cooler upon the addition of a second source of heat.

The transition route and bifurcations of the buoyant flows developing on a heated horizontal circular surface are elaborated using direct numerical simulations and direct stability analysis. A series of bifurcations, as a function of Rayleigh numbers ($Ra$) ranging from $10^6$ to $6.0\times 10^7$, are found on the route to chaos of the flows at $Pr=7$. When $Ra<1.0\times 10^3$, the buoyant flows above the heated horizontal surface are dominated by conduction, because of which the distinct thermal boundary layer and plume are not present. At $Ra=1.1\times 10^6$, a Hopf bifurcation occurs, resulting in the flow transition from a steady state to a periodic puffing state. As $Ra$ increases further, the flow enters a periodic rotating state at $Ra=1.9\times 10^6$, which is a unique state that was rarely discussed in the literature. These critical transitions, leaving from a steady state and subsequently entering a series of periodic states (puffing, rotating, flapping and period-doubling) and finally leading to chaos, are diagnosed using two-dimensional Fourier transforms. Moreover, direct stability analysis is conducted by introducing random numerical perturbations into the boundary condition of the surface heating. We find that when the state of a flow is in the vicinity of critical values (e.g. $Ra=2.0\times 10^6$), the flow is conditionally unstable to perturbations, and it can bifurcate from the rotating state to the flapping state in advance. However, for relatively stable flow states, such as at $Ra=1.5\times 10^6$, the flow remains in its periodic puffing state even though it is being perturbed.

We report an experimental study of the formation and evolution of laminar thermal structures generated by a small heat source, with a focus on their correlation to the thermal boundary layer and effects of heating time $t_{heat}$. The experiments are performed over the flux Rayleigh number ($Ra_f$) range $2.1\times 10^6 \leq Ra_f \leq 3.6\times 10^{7}$ and the Prandtl number ($Pr$) range $28.6 \leq Pr \leq 904.7$. The corresponding Rayleigh number ($Ra= t_{heat}\,Ra_{f}/\tau _0\,Pr$) range is $900 \leq Ra \leq 4\times 10^{4}$, where $\tau _0$ is a diffusion time scale. For thermal structures generated by continuous heating (i.e. starting plumes), their formation process exists three characteristic times that are well reflected by changes in the thermal boundary layer thickness. These characteristic times, denoted as $t_{emit}$, $t_{recover}$ and $t_{static}$, correspond to the moments when the plume emission begins and completes, and when the thermal boundary layer becomes quasi-static, respectively. Their $Ra_f$–$Pr$ dependencies are found to be $t_{emit}/\tau _0\sim Ra_f^{-0.41}\,Pr^{0.41}$, $t_{recover}/\tau _0\sim Ra_f^{-0.48}\,Pr^{0.48}$ and $t_{static}/\tau _0\sim Ra_f^{-0.49}\,Pr^{0.33}$, respectively. Thermal structures generated by finite $t_{heat}$ exhibit similar evolution dynamics once $t_{heat} \ge t_{emit}$, with the accelerating stage behaving like starting plumes and the decay stage like thermals (i.e. a finite amount of buoyant fluids). It is further found that their maximum rising velocity experiences a transition in the $Ra$-dependence from $Ra$ to $(Ra\ln Ra)^{0.5}$ at $Ra \simeq 6000$; and their maximum acceleration reaches the value of starting plumes at $t_{heat}\simeq t_{recover}$, and remains unchanged for larger $t_{heat}$. In particular, the maximum rising velocity for the cases with $t_{heat} = t_{recover}$ follows a scaling relation $Ra_f^{0.37}\,Pr^{-0.37}$, in contrast to the relation $Ra_f^{0.48}\,Pr^{-0.48}$ for starting plumes. This study provides a more comprehensive understanding of laminar thermal structures, which are relevant to a range of processes in nature and laboratory systems such as Rayleigh–Bénard convection.

We present the results from a series of experiments investigating the dynamics of gravity currents which form when a dense saline or particle-laden plume issuing from a moving source interacts with a horizontal surface. We define the dimensionless parameter $P$ as the ratio of the source speed, $u_a$, to the buoyancy speed, $(B_0/z_0)^{1/3}$, where $B_0$ and $z_0$ are the source buoyancy flux and height above the horizontal surface, respectively. Using our experimental data, we determine that the limiting case in which $P=P_c$ the gravity current only spreads downstream of the initial impact point occurs when $P_c=0.83\pm 0.02$. For $P< P_c$, from our experiments we observe that the plume forms a gravity current that spreads out in all directions from the point of impact and the propagation of the gravity current is analogous to a classical constant-flux gravity current. For $P>P_c$, we observe that the descending plume is bent over and develops a pair of counter-rotating line vortices along the axis of the plume. The ensuing gravity current spreads out downstream of the source, normal to the motion of the source. Analogous processes occur with particle-laden plumes, but there is a second dimensionless parameter $S$, the ratio of the particle fall speed, $v_s$, to the vertical speed of a plume in a crossflow, $(B_0/u_a z_0)^{1/2}$. For $S\ll 1$, particles remain well mixed in the plume and a particle-driven gravity current develops. For $S\gg 1$, particles separate from the plume prior to impacting the boundary which leads to a fall deposit and no gravity current. We discuss these results in the context of deep-sea mining.

We report an experimental study about the effect of an obstructed centre on heat transport and flow reversal by inserting an adiabatic cylinder at the centre of a quasi-two-dimensional Rayleigh–Bénard convection cell. The experiments are carried out in a Rayleigh number ($Ra$) range of $2\times 10^7 \leq Ra \leq 2\times 10^9$ and at a Prandtl number ($Pr$) of $5.7$. It is found that for low $Ra$, the obstructed centre leads to a heat transfer enhancement of up to 21 $\%$, while as $Ra$ increases, the magnitude of the heat transfer enhancement decreases and the heat transfer efficiency ($Nu$) eventually converges to that of the unobstructed normal cell. Particle image velocimetry measurements show that the heat transfer enhancement originates from the change in flow topology due to the presence of the cylindrical obstruction. In the low-$Ra$ regime the presence of the obstruction promotes the transition of the flow topology from the four-roll state to the abnormal single-roll state then to the normal single-roll state with increasing obstruction size. While in the high-$Ra$ regime, the flow is always in the single-roll state regardless of the obstruction size, although the flow becomes more coherent with the size of the obstruction. We also found that in the presence of the cylindrical obstruction, the stability of the corner vortices is significantly reduced, leading to a large reduction in the frequency of flow reversals.

Local instantaneous exchanges of volume, momentum and buoyancy across turbulent/non-turbulent interfaces (TNTIs) and turbulent/turbulent interfaces (TTIs) are studied using data from direct numerical simulations of a turbulent forced fountain. We apply a novel algorithm that enables independent calculation of the instantaneous local entrainment and detrainment fluxes, and therefore, for the first time, the entrainment and detrainment coefficients according to the fountain model (Bloomfield & Kerr, J. Fluid Mech., vol. 424, 2000, pp. 197–216) are determined explicitly. Across the interface between the fountain and the ambient fluid, which is a TNTI, only volume entrainment occurs, and it is well predicted by the fountain model. Across the interface between the rising upflow and falling downflow within the fountain, which is a TTI, both entrainment and detrainment of volume, momentum and buoyancy occur – with the magnitude of both entrainment and detrainment typically being large compared with the net for all exchanges. However, the model seems to be unable to capture the momentum exchanges due to its ignorance of the pressure. We find that each conditional entrainment and detrainment rate, of volume, momentum and buoyancy, can be described accurately by Gaussian profiles, while the net exchange that is the superposition of the entrainment and detrainment cannot. Moreover, the entrainment exchange rate has its maximum closer to the fountain centreline than that of detrainment, explaining the tendency for net entrainment closer to the fountain centreline and net detrainment further away.

In this study, we performed a set of experiments of the thermal plume on an open cylinder heated from below and a simple scaling analysis. The flow structure of the plume was visualized using the shadowgraph technique, and the temperature at specific points was measured using a thermistor. Transient plumes in a developing stage, an equilibrating stage and a fully developed stage were described. The new scaling laws of the stem radius and the velocity of the rising plume were presented, which are different from those on a two-dimensional heated plate due to the cylindrical effect and agree with experimental results. In the fully developed stage, there is a transition route of the plume from a steady to chaotic state with an increase in the Rayleigh number, which involves a series of bifurcations. A reverse bifurcation from a periodic (Ra = 1.17 × 106) back to a steady (Ra = 1.30 × 106) state has been observed in the experiment, which is referred to as the period bubbling bifurcation. Thus, the present experimental results validate the previous numerical results presented by Zhang et al. (Phys. Fluids, vol. 33 (6), 2021, 064110). The bifurcation diagram, spectral analysis and attractor are used to characterize the transitional plume. In addition, the heat and mass transfer have also been quantified.

A series of laboratory experiments are reported in which a continuous stream of bubbles rise from a small source at the base of a tank of water. Using different nozzles, bubble sizes $d$ ranging from 1.2 to 11.6 mm were produced for a number of gas volume fluxes, $Q_b$, ranging between 1.1 and $21.1\times 10^{-6}\ {\rm m}^3\ {\rm s}^{-1}$. Within a small distance from the source, the slip speed of these bubbles exceeds the speed of the equivalent single-phase plume with the same buoyancy flux, leading to formation of what we refer to as the ‘slip plume’ regime. Through a combination of high-speed photography, coupled with flow visualisation in the plume and the ambient fluid using dye, we find that the bubble speed and the fluid speed remain nearly constant with height, with the maximum fluid speed being of order $0.30\pm 0.03$ of the bubble speed. Using the filling box method, we also find that the net fluid volume flux in the slip plume increases linearly with distance from the source at a rate $Q_l = \lambda Bz/v_s^2$, where $B$ is the buoyancy flux of the gas, $v_s$ the rise speed of the gas bubbles, $z$ the distance above the source and $\lambda$ is a constant related to the dimensionless volume of fluid in the wake of each bubble. This slip-dominated flow regime can be understood in terms of kinetic energy imparted to the fluid as the bubbles rise and release potential energy. Further experiments with particle-laden plumes illustrate similar scalings for the volume flux in a particle-driven slip plume once the slip speed of the particles exceeds the bulk speed of the equivalent single-phase buoyant plume with the same buoyancy flux. Near the source the slip speed may be smaller than the plume speed, and the flow follows the classical model for a turbulent buoyant plume, with the transition to the slip regime occurring at a distance $z^*\approx (32\pm 5)\lambda ^{3/2} B/v_s^3$ from the source, where the dimensionless parameter $\lambda$ relates to the dimensionless volume of the fluid wake, which we find varies with the Reynolds number of the particles.

This article presents a theoretical modelling framework for the previously unconsidered case of turbulent wall plumes that detrain continually with height in stably stratified environments. Built upon the classic turbulent plume model, our approach incorporates turbulent detrainment with a variable coefficient of detrainment. Based on a linear constitutive relationship between the ratio of the detrainment to entrainment coefficients and the ambient buoyancy gradient, it is found that for linear ambient stratifications, a dynamic quasi-equilibrium region, characterised by a near invariant local plume Richardson number, is achieved, downstream of which this equilibrium rapidly breaks down. With increasing ambient buoyancy gradient, while the plume becomes increasingly slender with weaker vertical motions, the level at which the plume breaks down to form a horizontal intrusion first decreases and then increases. Moreover, distinct from classic purely entraining plumes, a detraining wall plume can swell within the pre-equilibrium adjustment stage provided the local Richardson number is sufficiently low $({Ri\ll 6})$, behaviour which is in accordance with observations made in filling-box experiments.

Previous mathematical models of quasi-steady turbulent plumes and fountains have described the flow that results from a prescribed input of buoyancy. We offer a new perspective by asking, what input of buoyancy would give rise to a plume, or fountain, with given properties? Addressing this question by means of an analytical framework, we take a first step toward enabling a plume with specific characteristics, i.e. a synthetic plume, to be designed. We develop analytical solutions to the conservation equations that describe four kinds of turbulent flow: axisymmetric plume, starting fountain, line plume and wall plume. Crucially, our solutions do not require the buoyancy distribution to be specified, whether this be the source or off-source distribution. Key to our approach, we specify a function for the volume flux, $Q=f(z)$, and take advantage of the weak coupling between the conservation equations to uniquely express general solutions in terms of $f$. We show that any analytic function $f$ can form the basis for a set of solutions for the fluxes, local variables and local Richardson number, though $f/({\rm d}f/{\rm d}z)>0$ is a necessary condition for physically realistic solutions. As an example of plume synthesis, we show that an axisymmetric plume can have an invariant radius if there is an exponentially increasing input of buoyancy to the plume centreline. We also consider how plume synthesis could be achieved practically.

We investigate the coupling effect of buoyancy and shear based on an annular centrifugal Rayleigh–Bénard convection (ACRBC) system in which two cylinders rotate with an angular velocity difference. Direct numerical simulations are performed in a Rayleigh number range $10^6\leq Ra\leq 10^8$, at fixed Prandtl number $Pr=4.3$, inverse Rossby number $Ro^{-1}=20$, and radius ratio $\eta =0.5$. The shear, represented by the non-dimensional rotational speed difference $\varOmega$, varies from $0$ to $10$, corresponding to an ACRBC without shear and a radially heated Taylor–Couette flow with only the inner cylinder rotating, respectively. A stable regime is found in the middle part of the interval for $\varOmega$, and divides the whole parameter space into three regimes: buoyancy-dominated, stable and shear-dominated. Clear boundaries between the regimes are given by linear stability analysis, meaning the marginal state of the flow. In the buoyancy-dominated regime, the flow is a quasi-two-dimensional flow on the $r\varphi$ plane; as shear increases, both the growth rate of instability and the heat transfer are depressed. In the shear-dominated regime, the flow is mainly on the $rz$ plane. The shear is so strong that the temperature acts as a passive scalar, and the heat transfer is greatly enhanced. The study shows that shear can stabilize buoyancy-driven convection, makes a detailed analysis of the flow characteristics in different regimes, and reveals the complex coupling mechanism of shear and buoyancy, which may have implications for fundamental studies and industrial designs.

The linear temporal and absolute/convective stability characteristics of a thermal plume generated along a heated vertical cylinder are investigated theoretically under the Boussinesq approximation. Special focus is given to the uniform-wall-buoyancy-flux case whereby the cylinder surface sustains the same linear temperature gradient as the environment. A competition between the axisymmetric and helical modes is a remarkable feature of the instability, distinguishing these ‘annular plumes’ from free plumes/jets for which the helical mode is generally dominant. It is found that higher surface curvature stabilises the temporal axisymmetric mode significantly, but only has moderate effects on the helical mode. The most temporally unstable perturbation mode switches from a helical into an axisymmetric mode when the Prandtl number increases beyond a critical value. Both the roles of shear and buoyancy during the destabilisation are identified through an energy analysis which indicates that, while the shear work is usually a major source of perturbation energy, the buoyancy work manifests for long-wave axisymmetric perturbation modes, and for thin cylinders and high Prandtl numbers. For the specific temperature configuration considered herein, an annular plume is always convectively unstable whereas decreasing the cylinder radius from the planar limiting case first decreases and then increases the tendency of the flow towards being absolutely unstable. The helical mode is especially susceptible to being absolutely unstable on very thin cylinders. Several conditions for the onset of cellular thermal convection and plume detrainment are proposed based on our results and a hypothesis which connects the absolute instability to the detrainment phenomenon.

The flow in and around a pair of plumes from sources that are vertically and horizontally offset is investigated. An analytical potential flow model is developed using an adapted version of the Milne–Thomson circle theorem to represent the flow due to adjacent circular and point sinks. This approximates the horizontal section through offset plumes, which have differing radii at the same vertical position. The predictions of this model are compared against Reynolds-averaged Navier–Stokes (RANS) simulations of the same system. These yield time-averaged results and so are particularly suitable for investigating the relatively weak entrainment field. Single and double plume results from RANS are also presented to compare with known results. Good agreement is found between the features in the analytical model and in the numerical solutions including the location of the stagnation point between the two plumes.

We investigate the ventilation conditions required to control the propagation of smoke, produced by a tunnel fire, in the presence of two inertial forcings: a transverse extraction system and a longitudinal flow. For that purpose, we performed a series of experiments in a reduced-scale tunnel, using a mixture of air and helium to simulate the release of hot smoke during a fire. Experiments were designed to focus on the ventilation flows that allow the buoyant release to be confined between two adjacent extraction vents. Different source conditions, in terms of density and velocity of the buoyant release, were analysed along with different vent configurations. Experiments allowed us to quantify the increase of the extraction velocity needed to confine the buoyant smoke, overcoming the effect of an imposed longitudinal velocity. Vents with a rectangular shape, and spanning over the whole tunnel width, provide the best performance. Finally, we studied the stratification conditions of the flow, individuating four regimes. Interestingly, when the stratification conditions fade out, as both the longitudinal flow and vertical extraction flows increase, the flow dynamics becomes almost independent of the forcing induced by the presence of buoyant smoke, which eventually acts as a passive scalar transported by the flow.