Session F16: Convection and Buoyancy-Driven Flows: Free-Convection and Rayleigh-Benard (3:55pm - 4:40pm CST)

Turbulent convection driven by evaporation in water pools

Turbulent natural convection in open cavities is encountered in numerous industrial applications and natural phenomena. In this presentation we report on direct numerical simulations of the problem in hand at different Rayleigh numbers. A shear-free boundary on top of a cubic domain approximates a free surface. At the same location we estimate realistic evaporative and convective heat losses, forming the basis of a non-zero Neumann boundary condition for the temperature. Our simulations predict that the shear-free surface increases heat transfer within the domain, however the exponent in the power-law relation between the Nusselt and Rayleigh numbers, is similar to that of classical turbulent Rayleigh-B\'{e}nard Convection. The effects of the shear-free surface on the large-scale circulation, the thermal boundary layers and the flow statistics are also analyzed herein. Overall, this configuration of turbulent convection shows unique characteristics, borrowing from both turbulent Rayleigh-B\'{e}nard convection and evaporative cooling.   

Large-scale cell formation in turbulent Rayleigh-B\'enard convection

The gradual aggregation of turbulent plumes and circulation rolls to a large-scale cell which eventually fills the whole periodic, horizontally extended layer of aspect ratio $\Gamma = 60$ is reported in high-resolution spectral element simulations of three-dimensional turbulent Rayleigh-B\'enard convection. It is shown that this final state of the flow is reached in extraordinary long simulations of the order of $10^{4}$ convective time units and proceeds only when the turbulence in the convection layer is driven by a constant heat flux at the bottom and top boundaries, independently of velocity boundary conditions. The formation takes place for fixed Prandtl number $Pr = 1$ and varying Rayleigh number in a range from $Ra \sim 10^{4}$ to $Ra \sim 10^{7}$. This implies that even though the convection is in the (fully) turbulent regime -- far beyond the linear stability threshold -- the most unstable mode at onset of convection with a critical wave number of $k = 0$ still seems to dominate the long-term dynamics. Our present study might have interesting implications for atmospheric and stellar convection processes where heat fluxes are typically prescribed at the boundaries of the convection zone.   

Dynamics of the large-scale circulation of turbulent Rayleigh Benard convection in cubic containers

We report experimental results on the dynamics of the large-scale circulation (LSC) of turbulent Rayleigh-Benard convection in cubic containers. A new oscillation in the shape of the temperature profile of the LSC is found in cubic (but not cylindrical) cells. This can be explained by assuming the heat transported by the LSC is proportional to the pathlength of the LSC along the thermal boundary layers at the top and bottom plates. In a non-cylindrical cell, the pathlength oscillates as the LSC oscillates around a corner. We also test the forcing on the LSC orientation $\theta_0$ in a cell tilted relative to gravity. Using a low-dimensional model of diffusion of $\theta_0$ in a potential, the probability distribution of $\theta_0$ is used to obtain a potential acting on $\theta_0$. This potential is used to successfully predict changes in the barrier-crossing rate for $\theta_0$ to escape different corners of the cell. The shape of the tilt-induced potential is due to the vector direction of the buoyancy force acting on the LSC. However, the magnitude of this forcing is found to be two orders of magnitude larger than previously predicted.   

Reservoir Computing of Dry and Moist Turbulent Convection

Reservoir computing is one efficient implementation of a recurrent neural network that can describe the evolution of a dynamical system by supervised machine learning without solving the underlying nonlinear partial differential equations. We apply reservoir computing to approximate the large-scale evolution and the resulting low-order turbulence statistics of two-dimensional dry and moist Rayleigh-Bénard convection. This approach is motivated by the potential to parametrize subgrid-scale turbulence in larger scale atmospheric circulation models. We acquire training and test data by long-term direct numerical simulations (DNS). Post-processing is done by a Proper Orthogonal Decomposition (POD) with the snapshot method. The training data comprise time series of the first 150 POD modes, which are associated with the largest total energy amplitudes and thus the large-scale structure of the flows. Feeding the data to the reservoir computing model (RCM) and optimizing the its parameters results in predictions for the evolution of the dry and moist convection flows. Amongst others we find good agreement of the vertical profiles of mean buoyancy, mean temperature as well as mean convective heat and buoyancy fluxes. This opens new avenues for modeling meso-scale convection processes.   

The Cooling Box Problem: Numerical and Laboratory Fluxes

In very cold freshwater, density increases with temperature so that the surface water will actually be colder than the water below. This counter-intuitive temperature structure results from the density of freshwater depending nonlinearly with temperature. Through surface cooling and mixing, many lakes transition between the intuitive hot-over-cold temperature stratification in the summer, and the reverse cold-over-warm stratification in the winter. The division between these two regimes occurs at the temperature of maximum density $T_{MD}$ (for fresh water, $T_{MD}\approx 4 \ ^\circ C$). We want to understand how the transport of heat is changed during this transition period, near $T_{MD}$. We perform a set of numerical and laboratory experiments by taking a body of warm water $T>T_{MD}$ and instantaneously cooling its surface temperature. We are interested in quantifying the rate of change in bulk water temperature as it approaches $T_{MD}$. We develop a model for the rate of cooling within the domain and demonstrate its dependancy on the surface water temperature. The model agrees well with the simulations and laboratory measurements. We highlight the key parameters of interest in this problem, and elaborate on how these results may be applicable to field measurements.   

Symmetry-breaking transition in natural convection flow between parallel vertical walls maintained at different temperatures

The purpose of this study is to observe transitions in natural convection flow of water in the vertical channel as the Rayleigh number increases. The Navier-Stokes equation and energy equation were solved under The Boussinesque approximation using direct numerical simulation. Aspect ratio of the domain is 12:2:6 (streamwise: wall-normal: spanwise). The range of Rayleigh number is from 50,000 to 2,000,000. Various different flow regimes such as laminar steady flow, time-periodic unsteady flow, chaotic unsteady flow and turbulent flow were observed. Most interestingly, the symmetry between up flow near the hot wall and down flow near the cold wall is broken intermittently as Rayleigh number increases so that the mean flow is upward for a while, then it quickly switches to the downward flow, and it repeats temporally alternating pattern randomly. As Rayleigh number further increases, transitions between the up and down flows becomes less frequent. Detailed statistics and flow patterns will be presented in the meeting.   

Turbulent Rayleigh-B\'{e}nard convection in a strong vertical magnetic field

Turbulent Rayleigh-B\'{e}nard convection in a vertical cylinder of aspect ratio 1 with imposed vertical magnetic field is analyzed in direct numerical simulations. The flow structure and transport properties with the Prandtl number 0.025 and the Rayleigh and Hartmann numbers up to 10$^{\mathrm{9}}$ and 1400, respectively, are considered. Increase in the strength of the magnetic field has the anticipated effects of suppression of the heat transfer rate and flow's kinetic energy. At the same time, growth of these characteristics with the Rayleigh number is found to be faster in flows at high Hartmann numbers. This behavior is consistent with earlier experimental data. The simulations allow us to attribute it to the newly discovered flow regime characterized by prominent quasi-two-dimensional structures reminiscent of vortex sheets extending into the core. Rotating tongue-like wall modes qualitatively similar to those in the Rayleigh-B\'{e}nard convection with rotation are found in flows near the Chandrasekhar linear stability limit. A detailed analysis of the spatial structure of the flows and its effect on global transport properties is reported.   

Examining departures from the Boussinesq approximation in chaotic Rayleigh-Benard convection using persistent homology

Persistent homology is a data analysis technique that can be used to quantify the topological information of image data. In the spatio-temporally chaotic flow known as spiral defect chaos in Rayleigh-Benard convection, we explore the connection between convective plumes detected by persistent homology and the extent that the flow deviates from the Boussinesq approximation, a common simplification used to study convective flows. Simulations of the Boussinesq approximation predict that hot and cold plumes occur at roughly equal rates over time scales comparable to the horizontal diffusion time. However, using the same mean values of physical parameters, we demonstrate that rates of hot and cold plume formation differ in simulations and experiments that deviate from the Boussinesq approximation.   

Lagrangian Coherent Sets as transport barriers in convection

The analysis of the turbulent heat transfer in thermal convection flow is mostly done in the Eulerian frame of reference. Here, we use an ensemble of more than half a million individual Lagrangian tracer trajectories, which are advected together with the evolving turbulent flow, to identify subsets that contribute least to the heat transfer. These trajectories probe Lagrangian Coherent Sets — sets that remain spatially connected for a finite time and evolve in time. The analysis is based on three-dimensional direct numerical simulations at three different Prandtl numbers. Our Lagrangian analysis results provide a complementary view on the transport and are found to be consistent with the standard Eulerian results.   

On the Dynamics of air bubbles in Rayleigh-Benard Convection at various aspects ratios

A laboratory investigation was carried out to uncover the distinct dynamics of air bubbles in Rayleigh-Benard (RB) convection in a rectangular cell at aspect ratios of 1.25, 1.5, 2.0, and 2.5. The experiments were performed at Rayleigh numbers on the order of Ra$\sim $ 10$^{\mathrm{10}}$. Streams of 1-mm bubbles were released from the bottom of the RB tank along the path of the roll structure at a fixed location of s/D$=$1/2, where s is the distance along the diagonal with respect to the center of the tank and D is the half diagonal distance. Three-dimensional particle tracking velocimetry was used to track simultaneously a relatively large number of bubbles, and to quantify the associated pair dispersion for various initial separations in the range of 20$\le \eta \le $200, where $\eta $ is the local Kolmogorov length scale. We will discuss distinct effects of the aspect-ratio dependency on the dynamics of the bubbles, path instability, pair dispersion, and the relation with the large-scale roll structures.   

Boundary layer precursors of extreme events in the bulk of turbulent Rayleigh-B\'{e}nard convection

We study the connection between extreme events of thermal and kinetic dissipation rates in the bulk of three-dimensional Rayleigh-B\'{e}nard convection and the wall-shear-stress at the top and the bottom boundary layers of the cell. Local minima in the vicinity of zero points of the wall shear-stress vector field are connected to the smallest magnitudes of the vertical temperature derivative in the boundary layer, and identify the locations where thermal plumes rise from the boundary layers [Schumacher and Scheel, 2016]. Our direct numerical simulations (DNS) at Rayleigh number of $Ra = 5\times10^5$ and $Ra = 1.5\times10^4$, and Prandtl number $Pr=1$ in a Cartesian domain of aspect ratio $\Gamma=8$ cell show that local maxima of the vertical temperature derivative which appear simultaneously at similar horizontal positions in both boundary layers can be considered as a precursor for an extreme event of thermal and kinetic energy dissipation in the bulk of the cell. We show from DNS and stereo PIV measurements that a typical example of extreme event of energy dissipation in the bulk is a front that forms between two colliding thermal plumes. Such collision can be predicted by the boundary layer dynamics up to a few free fall times in advance.   

New Bounds on Convective Heat Transport in Internally Heated Convection

We use quadratic auxiliary functions, which are equivalent to the classical background method, to prove bounds on the mean convective heat transport \textless wT\textgreater in internally heated (IH) convection. Bounds for Rayleigh-B\'{e}nard convection have been extensively studied, yet an extension of the same analysis to IH convection is not complete. The change in mechanism driving convection presents a unique problem, which is of importance in geophysical and astrophysical applications such as convection in the mantle. Bounds that depend explicitly on the Rayleigh number Ra have not yet been proved for IH convection. This talk will demonstrate that such Ra-dependent bound on \textless wT\textgreater can be obtained using quadratic auxiliary functions. Our bound grows as Ra$^{\mathrm{1/5}}$ and improves the previously known uniform bound \textless wT\textgreater $\le $ 1/2 over a finite range of Rayleigh numbers. Our bounds exceeds 1/2 when the corresponding critical temperature field violates the physical constraint of pointwise positivity. When this constraint is enforced by means of a Lagrange multiplier, numerical bounds appear to approach the value of 1/2 asymptotically.   

Periodically modulated thermal convection

Many natural and industrial turbulent flows are subjected to time-dependent boundary conditions. Here, we perform numerical simulations of Rayleigh-B\'enard (RB) convection with time periodic modulation in the temperature boundary condition and report how this modulation can lead to a significant heat flux (Nusselt number Nu) enhancement. Using the concept of Stokes thermal boundary layer, we can explain the onset frequency of the Nu enhancement and the optimal frequency at which Nu is maximal, and how they depend on the Rayleigh number Ra and Prandtl number Pr. From this, we construct a phase diagram in the 3D parameter space (f, Ra, Pr) and identify: (i) a regime where the modulation is too fast to affect Nu; (ii) a moderate modulation regime, where Nu increases with decreasing f and (iii) slow modulation regime, where Nu decreases with further decreasing f. Our findings provide a framework to study other types of turbulent flows with time-dependent forcing.   

What rotation rate maximizes heat transport in rotating Rayleigh–Bénard convection

The heat transfer and flow structure in rotating Rayleigh–Bénard convection are strongly influenced by the Rayleigh (Ra), Prandtl (Pr), and Rossby (Ro) number. For $Pr>1$ and intermediate rotation rates, the heat transfer is increased compared to the non-rotating case. We find that the regime of increased heat transfer is subdivided into a low and a high Ra number regime. For $Ra<5 \times 10^{8}$ the heat transfer at a given Ra and Pr is highest at an optimal rotation rate, at which the thickness of the viscous and thermal boundary layer is about equal. From the scaling relations of the thermal and viscous boundary layer thicknesses, we derive that the optimal rotation rate scales as $1/Ro_{opt} \approx 0.12 Pr^{1/2}Ra^{1/6}$. For $Ra > 5 \times 10^{8}$ the above scaling for the optimal rotation rate does not hold anymore. It turns out that in the high Ra regime, the flow structures at the optimal rotation rate are very different than for lower Ra. Surprisingly, the heat transfer in the high $Ra$ regime differs significantly for a periodic domain and cylindrical cells with different aspect ratios, which originates from the sidewall boundary layer dynamics and the corresponding secondary circulation.   

Regime transitions in high-Rayleigh number vertical convection

Vertical convection (VC) is investigated using direct numerical simulations over wide range of Rayleigh numbers $10^5\le Ra\le10^{14}$, in a two-dimensionl convection cell with unit aspect ratio. The Prandtl numbers $Pr$ vary from 0.71 to 30. The dependence of the mean vertical center temperature gradient [$S=\left<\partial T/\partial z\right>_{c,t}$] on $Ra$ shows three different regimes. In Regime I where $Ra \le 5\times10^{10}$, $S$ hardly depends on $Ra$. In the newly-identified Regime II ($5\times10^{10} \le Ra \le 5\times10^{12}$), $S$ first increases with increasing $Ra$ and then reaches its maximum as function of $Ra$, before decreasing again. In Regime III where $Ra\ge5\times10^{12}$, $S$ again becomes weakly dependent on $Ra$, with a smaller value than that of Regime I. It is further found that the change of $S$ is closely related to the change of global flow organization: The flow in Regime III is characterized by well-mixed bulk region due to continuous ejection of plumes over large fraction of the plate, thus $S$ is smaller than that of the first regime. The scaling exponent $\beta$ in the effective scaling $Nu\sim Ra^\beta$ reaches a value very close to 1/3 in Regime II and III, which is larger than the value around 1/4 in Regime I.   

Exact relations between Rayleigh-B\'enard and rotating plane Couette flow in 2D

Relying on an exact relationship between Rayleigh-B\'enard Convection (RBC) and Rotating Plane Couette Flow (RPC) restricted to two spatial independent variables, we deduce several relations between both flows: (i) Heat and angular momentum transport differ by $(1-R_\Omega)$, explaining why angular momentum transport is not symmetric around $R_\Omega=1/2$ even though the relation between $Ra$ and $R_\Omega$ has this symmetry. This relationship leads to a predicted value of $R_\Omega$ that maximizes the angular momentum transport that agrees remarkably well with existing numerical simulations of the full 3D system. (ii) One variable in both flows satisfy a maximum principle i.e., the fields' extrema occur at the walls. Accordingly, backflow events in shear flow \emph{cannot} occur in this quasi two-dimensional setting. (iii) For free slip boundary conditions on the axial and radial velocity components, previous rigorous analysis for RBC implies that the azimuthal momentum transport in RPC is bounded from above by $Re_S^{5/6}$ with a scaling exponent smaller than the anticipated $Re_S^1$.   

Anisotropy turbulent Rayleigh--B\'{e}nard convection in spherical shell due to shifting of gravity center

we use direct numerical simulation to simulate the thermal convection for a fluid with Prandtl number of unity in spherical shells with an aspect ratio of 0.3 between the inner and outer sphere radius. We study the influence of the gravity center location by shifting it from the geometrical center to 0.8 of inner sphere radius, for Rayleigh number up to 3E$+$7. When the gravity center is moved to the south, a convective jet is generated in the opposite northern direction. Besides, a large-scale vortex is formed, while the flow in the southern hemisphere becomes stratified. However, surprisingly, the global heat transfer seems relatively insensitive to the shifting gravity center, even though the flow pattern is completely changed. On the outer sphere, the heat transfer is highest near the north pole while on the inner sphere, it is highest around the south pole. We use quadrant analysis and model analysis to investigate the inhomogeneous flow structures when the gravity center is not located at the geometrical center. Our results indicate that it is essential to consider density heterogeneities when modeling convection in the Earth's outer core.   

Two-layer Thermally Driven Turbulence: Mechanisms for Interface Breakup

It is commonly accepted that the breakup criteria of drops or bubbles in turbulence is governed by surface tension and inertia. However, also buoyancy can play an important role at breakup. In order to better understand this role, here we numerically study (two-dimensional) Rayleigh-B\'{e}nard convection for two immiscible fluid layers. We explore the parameter space spanned by the Weber number $5 \le We \le 5000$ and the density ratio between the two fluids $0.001 \le \Lambda \le 1$, at fixed Rayleigh number $Ra= 10^8$ and Prandtl number $Pr = 1$. At low $We$, the interface undulates due to plumes. When $We$ is larger than a critical value, the interface eventually breaks up. Depending on $\Lambda$, two breakup types are observed: The first type occurs at small $\Lambda \ll 1$ (e.g. air-water systems) when local filament thicknesses exceed the Hinze length scale. The second, strikingly different, type occurs at large $\Lambda$ with roughly $0.5 < \Lambda < 1$ (e.g. oil-water systems): The layers undergo a periodic overturning caused by buoyancy overwhelming surface tension. For both types the breakup criteria can be derived from force balance arguments and show good agreement with the numerical results.   

Effect of gravity profiles on Rayleigh-Benard convection in spherical shells.

Rayleigh-Benard convection of flows confined by spherical boundaries is analysed by three-dimensional direct numerical simulations in spherical coordinates. The dynamics under different radial gravity profiles have been explored: the different gravity laws can often be absorbed by the introduction of an effective Rayleigh number $Ra_e$, although this is not true for a few particular cases with non-monotonic gravity. Two different fluids have been studied: air (Prandtl number $Pr=0.71$) and water ($Pr=7.1$), and in both cases, onset of convection is at $Ra_e=1800$ for a domain aspect ratio $\eta=0.71$. On the other hand unsteady convection, occurring when the inertial terms overcome the viscous terms, has a clear dependence on $Pr$ and is thus different between the two fluids. In between these two regimes, a series of different quasi-stable states, with a non trivial dependence on the Prandtl number, appear as $Ra$ is increased: this behaviour can induce hysteresis in the system and initial conditions are crucial to determine the final flow configuration.   

Combining confinement and rotation to improve heat transport in Rayleigh-Bénard convection

It has been shown by Chong et al., (2017) that different stabilizing mechanisms similary increase heat transport in Rayleigh-Bénard convection (RBC) compared to the standard non-stabilized system. Here, we study the combination of two stabilizing mechanisms, namely rotation, and confinement, by performing direct numerical simulations of confined rotating RBC in a cylindrical cell for Prandtl number $Pr=4.38$ and various Rayleigh numbers $Ra\in[2\cdot10^8,7\cdot10^9]$. In the absence of rotation, flow confinement can result in a higher heat transport as it makes the large scale circulation more effective in transporting heat, while system rotation can increase heat transport due to a process known as Ekman pumping. We find that Ekman pumping becomes more effective at a certain aspect ratio, i.e. diameter over the height of the sample, when the flow forms a stable structure of two vortices. In addition, we find that for high Ra and very low aspect ratio cells, a third distinct maximum evolves, caused by the formation of a singular strong vortex. We show that for relatively low $Ra<2\cdot10^9$ the combination of confinement and rotation results in the highest heat transport. For $Ra>2\cdot10^9$ the use of confinement is the most effective way to increase heat transport.   

Similarity solutions of natural convection boundary layers with transpiration

We present the asymptotic, similar solutions of concentration driven, natural convection boundary layer equations with a spatially uniform transpiration $(V_{i})$ at the wall.The characteristic scales for the species and velocity boundary layer thicknesses ($\delta_{dc}$ and $\delta_{vc}$),horizontal velocity ($u_c$) and vertical velocity($v_c$) are first obtained from an order of magnitude analysis of the integral boundary layer equations. We define a pseudo similarity variable $\eta =y/\delta_{vc}$ and a non-similarity variable $\xi=Re_x^{5/4}/Gr_x^{1/4}$, which is also a dimensionless blowing parameter.Considering the dimensionless stream function $f$ and the dimensionless concentration function $\theta$ as functions of $\eta$ and $\xi$, and normalizing the variables in the boundary layer equations with the characteristic variables, we obtain the dimensionless boundary layer equation.We find complete similarity for two asymptotic cases, namely (a) when $\xi\to 0$ i.e. small blowing velocity (b) when $Sc\to \infty$ with small blowing velocity so that $\theta\to 1$ inside the species boundary layer.For the all other cases non-similar solutions have to be obtained numerically to obtain the horizontal velocity profile and concentration profile over wide range of $\xi$ and $Sc$.   

Large-scale Natural Convection to Validate Models of Passive Cooling Systems of Large Structures

Passive cooling of large, vertical structures is a design feature that can increase safety and decrease cost associated with heat removal. Preliminary studies of the heat transfer rate, velocity and temperature profiles from a large heated wall with Rayleigh number (based on height) of 7x10$^{\mathrm{12\thinspace }}$to 1.5x10$^{\mathrm{13}}$ are presented. The heat flux and temperature of the wall are measured continuously, along with the temperature profile of the air along positions normal to the heated wall, the inlet and outlet air temperature and humidity, and the velocity and turbulence profiles measured with laser Doppler velocimetry. The trends generally agree with a similar study in the literature, but the scale and aspect ratio of the flow channel are unprecedented in the open literature. The data show that there are unique aspects to the development of the flow field turbulence that are not well characterized for this scale of natural convection.   

Strain Rate Signatures of Plumes in Rayleigh B\'{e}nard Convection

We study the distribution of eigenvalues of the 2D strain rate tensor in a horizontal plane close to the hot plate in Rayleigh B\'{e}nard Convection (RBC). The regions of negative dominant eigenvalues are shown as attracting Lagrangian Coherent structures that govern material folding in the instantaneous limit. Deriving the relation between horizontal divergence, total strain and eigenvalues, we show that these regions are also the regions with negative horizontal divergence. We propose these regions to be thermal plumes, arguing based on earlier visualisations that fluid elements transform from dominant extension to dominant compression in a horizontal plane as they traverse from boundary layers to plumes. We thereby propose two alternate, equivalent, velocity based criteria - namely regions with negative dominant eigenvalues and regions with negative horizontal divergence - to detect thermal plumes in a horizontal plane close to the hot plate in RBC. We then compare the extent of these regions with available theoretical relations and with the regions detected by different temperature based plume detection criteria.   

Saturation Ratio Fluctuations in Rayleigh-Benard Convection: Measurements in the MTU $\Pi$-Chamber

The saturation ratio is incredibly important for the behavior of atmospheric clouds as it controls the diffusional growth rate of cloud droplets. Turbulent fluctuations in the saturation ratio could induce broadening of the cloud droplet size distribution, leading to precipitation. We report 1Hz measurements of the temperature, water vapor concentration and saturation ratio (ratio of vapor pressure to the saturated vapor pressure) in the Michigan Tech $\Pi$-chamber (aspect ratio=2 and Pr=0.7). These experiments were conducted in moist (in the absence of cloud droplets) and cloudy convection. The $\Pi$ chamber operates as a Rayleigh-Benard convection cell, driven by an unstable temperature and vapor pressure gradient, and is designed to study microphysical cloud processes in a turbulent environment. In moist (cloudy) Rayleigh-Benard convection the fluctuations in water vapor and temperature and their covariance determines the distribution of the saturation ratio. Significant fluctuations were found to be present in both moist and cloudy convection.   

Velocity measurements in rotating Rayleigh-B\'enard convection and the Boundary Zonal Flow

Rotating turbulent thermal convection is of great importance in various astro- and geophysical systems, where the buoyancy driven flow strongly influenced by Coriolis forces due to rotation of the celestial bodies. It has been studied for decades in the so-called Rayleigh-B\'{e}nard setup, where a horizontal fluid layer is heated at the bottom and cooled at the top and rotated around the vertical axis. We investigate the horizontal velocity field using 2D-particle image velocimetry (PIV) in a cylindrical cell ($H=196\,$mm high) with aspect ratio $\Gamma=D/H=1$. We use water and various water-glycerol mixtures as working fluid resulting in a Prandtl number (Pr) in the range $6\leq Pr\leq70$ and Rayleigh numbers $10^8 < Ra < 2\times 10^9$. With our rotating table we reach $Ek$ as low as $10^{-5}$. We are mainly interested in studying the recently discovered {\em Boundary Zonal Flow} (BZF, see Zhang et al., Phys.Rev.Lett. 2020). The BZF is observed in a region close to the lateral sidewall with a cyclonic flow, i.e, a positive mean azimuthal velocity that is separated from and anticyclonic bulk, with negative mean azimuthal velocity. We measure the size of the BZF as a function of $Ek$ and $Ra$, and compare the results with DNS (Zhang and Shishkina, 2020).   

Global stability map of the flow in a horizontal concentric cylinder forced by natural convection

There are a large number of studies in the literature on natural convection in the annular region between horizontal concentric cylinders. However, not many publications dealing with global stability analysis in this kind of flow have been published in the literature. For a fixed diameter ratio $A \equiv D_i/L=2\,R_i/(R_o-R_i)$, being $R_i$ and $R_o$ the inner and outer cylinder radii respectively, and assuming Boussinesq approximation, the solution only depends on Prandtl ($Pr\equiv \nu/\alpha$) and Rayleigh ($Ra\equiv g\,\beta\,L^3(T_i-T_o)/(\nu\,\alpha)$) numbers where $T_i-T_o$ is the temperature difference between the inner and outer cylinders. A spectral collocation code has been developed to solve the problem by means of Chebyshev and Fourier differentiation matrices for $A=1.25$ and it has been validated with both classical experimental results and numerical simulations. Steady solutions have been sought within the range $Pr \in [10^{-2} ,1]$ and $Ra \in [10^{2} ,5\times 10^{6}]$. As a result, a steady solution $Pr$-$Ra$ map (consisting of 149 x 75 points) has been traced, where the different families of similar solutions found are detailed, mainly characterized by presenting single or multiple plumes. In addition, two main double-solution regions have been found.   

Bounding flow quantities in modulated convection

For a given configuration in convection, it is of interest to find bounds on physically important quantities related to the temperature gradient, such as heat transfer. The motivation for the convection modeled in the present work comes from observations of springtime warming in a lake, which show convection occurring in a diurnal cycle. Analytical bounds on flow quantities can be found, and have been found, in many configurations. Here we model springtime warming in a lake by considering modulated convection with zero-mean sinusoidal forcing at the boundary. With zero-mean forcing, no net heat transfer occurs, and therefore the quantity of interest becomes the squared and averaged magnitude of the temperature gradient, also referred to as the temperature dissipation. We use the background method and the Boussinesq equations to establish for the first time an exact, analytical bound on the temperature dissipation for this configuration.   

The Effect of Tidal Force and Topography in Horizontal Convection

We present numerical study of horizontal convection system with tidal force and topography. In our study, amplitude of tidal force is fixed and simulations with a series of tidal frequency $\omega$ are conducted. Our results demonstrate that local dissipation near topography will be enhanced when tide is sufficiently strong. Such enhancement is related to the height of topography and increases as $\omega$ decreases. For smaller $\omega$, the global dissipation becomes less sensitive to $\omega$ and asymptotically approaches to a constant. We interpret the behavior of dissipation as a result of the change of control mechanism. According to the control mechanism, this system can be divided into three regimes as $\omega$ changes, which are buoyancy-control, tide-control and viscosity-control regime respectively.We further show that mixing efficiency can be represented as a function of a single parameter, which is the ratio between tidal and potential energy injection. According to the global energy input to the ocean interior, we provide our estimation of global mixing efficiency being 0.17-0.3, which agrees with the empirical value 0.2.