We investigate the convective cooling of a fluid with a quadratic equation of state by performing three-dimensional direct numerical simulations of a flow with a fixed top-boundary temperature, which is lower than the initial fluid temperature. We consider fluid temperatures near the density maximum, where the nonlinearity is expected to be important. When the equation of state is nonlinear, the resultant vertical transport of heat is fundamentally different and significantly lower than the predictions derived for a linear equation of state. Further, three dimensionless groups parameterize the convective system: the Rayleigh number ($\hbox{Ra}_0$), the Prandtl number (Pr), and the dimensionless bottom water temperature $(T_B)$. We further define an effective Rayleigh number ($\hbox{Ra}_{eff} = Ra_0 \ T_B^2$), which is equivalent to the traditional Rayleigh number used with a linear equation of state. We present a predictive model for the vertical heat flux, the top boundary-layer thickness, and the turbulent kinetic energy of the system. We show that this model agrees well with the direct numerical simulations. This model could be used to understand how quickly freshwater lakes cool in high latitude environments.
We generalize a recent method for computing optimal 2D convection cooling flows in a horizontal layer to a wide range of geometries, including those relevant for technological applications. We write the problem in a conformal pair of coordinates which are the pure conduction temperature and its harmonic conjugate. We find optimal flows for cooling a cylinder in an annular domain, a hot plate embedded in a cold surface, and a channel with hot interior and cold exterior. With a constraint of fixed kinetic energy, the optimal flows are all essentially the same in the conformal coordinates. In the…
A fully non-dimensional correlation for the Nusselt number associated to cooling processes in vertical storage tanks under laminar natural convection is presented. The derived correlation depends on the average temperature and can be used for the improvement of temperature profiles found from global methods. A Finite Volume method is employed using a transient bi-dimensional model in cylindrical coordinates and is experimentally validated. Forty cases of cooling with four aspect ratios, five overall heat transfer coefficients and two different volumes are considered. Thermal losses for the env…
In the present work, the effects of diffuse and collimated radiation on the symmetrical cooling case of natural convection in a two-dimensional cavity heated from the bottom have been investigated, numerically. The cavity is convectively heated from the bottom with heat transfer coefficient of 50 W/m2K and free stream temperature 305 K, while both vertical walls of cavity are isothermal at temperature of 296 K. The top wall is adiabatic and all walls are opaque for radiation heat transfer. For collimated case, a small semitransparent window of non-dimensional width of 0.05 at height of 0.7 has…
We examine the role of small-scale granulation in helping to drive supergranulation and even larger scales of convection. The granulation is modeled as localized cooling events introduced at the upper boundary of a 3-D simulation of compressible convection in a rotating spherical shell segment. With a sufficient number of stochastic cooling events compared to uniform cooling, we find that supergranular scales are realized, along with a differential rotation that becomes increasingly solar-like.
Transition from steady to oscillatory buoyancy convection of air in a laterally heated cubic box is studied numerically by straight-forward time integration of Boussinesq equations using a series of gradually refined finite volume grids. Horizontal and spanwise cube boundaries are assumed to be either perfectly thermally conducting or perfectly thermally insulated, which results in four different sets of thermal boundary conditions. Critical Grashof numbers are obtained by interpolation of numerically extracted growth/decay rates of oscillations amplitude to zero. Slightly supercritical flow r…
The convective instability in a plane liquid layer with time-dependent temperature profile is investigated by means of a general method suitable for linear stability analysis of an unsteady basic flow. The method is based on a non-normal approach, and predicts the onset of instability, critical wave number and time. The method is applied to transient Rayleigh-Benard-Marangoni convection due to cooling by evaporation. Numerical results as well as theoretical scalings for the critical parameters as function of the Biot number are presented for the limiting cases of purely buoyancy-driven and pur…
We find steady channel flows that are locally optimal for transferring heat from fixed-temperature walls, under the constraint of a fixed rate of viscous dissipation (enstrophy = $Pe^2$), also the power needed to pump the fluid through the channel. We generate the optima with net flux as a continuation parameter, starting from parabolic (Poiseuille) flow, the unique optimum at maximum net flux. Decreasing the flux, we eventually reach optimal flows that concentrate the enstrophy in boundary layers of thickness $\sim Pe^{-2/5}$ at the channel walls, and have a uniform flow with speed $\sim Pe^{…
The present paper numerically analyzes a passive cooling system using enclosures with different geometries filled with thermal conductivity-enhanced phase change material (PCM). A numerical code is developed using an unstructured finite-volume method and an enthalpy-porosity technique to solve for natural convection coupled to a solid-liquid phase change. Five geometries containing the same volume of PCM are compared while cooling the same surface. The unsteady evolution of the melting front and the velocity and temperature fields is detailed. Other indicators of cooling efficiency are monitor…
We present the results of direct numerical simulations of flow patterns in a low-Prandtl-number ($Pr = 0.1$) fluid above the onset of oscillatory convection in a Rayleigh-Bénard system rotating uniformly about a vertical axis. Simulations were carried out in a periodic box with thermally conducting and stress-free top and bottom surfaces. We considered a rectangular box ($L_x \times L_y \times 1$) and a wide range of Taylor numbers ($750 \le Ta \le 5000$) for the purpose. The horizontal aspect ratio $η= L_y/L_x$ of the box was varied from $0.5$ to $10$. The primary instability appeared in the …
The logarithmic law of mean temperature profile has been observed in different regions in Rayleigh-Bénard turbulence. However, how thermal plumes correlate to the log law of temperature and how the velocity profile changes with pressure gradient are not fully understood. Here, we performed three-dimensional simulations of Rayleigh-Bénard turbulence in a slim-box without the front and back walls with aspect ratio, $L:D:H=1:1/6:1$, in the Rayleigh number $Ra=[1\times10^8, 1\times10^{10}]$ for Prandtl number $Pr=0.7$. The velocity profile is successfully quantified by a two-layer function of a st…
We study the topology and the temporal dynamics of turbulent Rayleigh-Benard convection in a liquid metal with a Prandtl number of 0.03 located inside a box with a square base area and an aspect ratio of five. Experiments and numerical simulations are focused on the moderate Rayleigh number range, where a new cellular flow regime has been reported by a previous study (Akashi et al., Phys. Rev. Fluids, vol.4, 2019, 033501). This flow structure shows symmetries with respect to the vertical planes crossing at the center of the container. The dynamic behaviour is dominated by strong three-dimensio…
Direct Numerical Simulations of turbulent convection in a large aspect-ratio box are carried out in the range of Rayleigh number $7 \times 10^4 \le Ra \le 2 \times 10^6$ at Prandtl number Pr=0.71. A strong correlation between the vertical velocity and temperature is observed in the turbulent regime at almost all the length scales. Frequency spectra of all the velocities and temperature show a $-5/3$ law for a wide band of frequencies. The variances of horizontal velocities at different points in the flow yield a single power-law. Probability density functions of velocities and temperature are …
Thermal convection in fluid layers heated from below are usually realized experimentally as well as treated theoretically with fixed boundaries on which conditions for the temperature and the velocity field are prescribed. The thermal and velocity boundary layers attached to the upper and lower boundaries determine to a large extent the properties of turbulent convection at high Rayleigh numbers. Fixed boundaries are often absent in natural realizations of thermal convection. This paper studies the properties of convection driven by a planar heat source below a cooling source of equal size imm…
The phenomenon of convection is found in a wide variety of settings on different scales -- from applications in the cooling technology of laptops to heating water on a stove, and from the movement of ocean currents to describing astrophysical events with the convective zones of stars. Given its importance in these diverse areas, the process of convection has been the focus of many research studies over the past two centuries. However, much less research has been conducted on how the presence of an obstruction in the flow can impact convection. In this work, we find that the presence of an obst…
After the sunset, under calm and clear sky conditions, aerosol laden surface air-layer, cools radiatively to the upper atmosphere. Predominant effect of the radiative cooling on the vertical temperature profile extends to several hundred meters from the surface. This results in the development of a stable, nocturnal inversion layer. However, ground surface, owing to its higher thermal inertia, lags in the cooling process. Due to this about a meter thick air layer just above the ground can be (2-6)deg Celsius cooler than the ground. Thus, at the surface an unstable convective layer is present, …
The feasibility of using a thermomagnetically pumped ferrofluid to enhance the performance of a natural convection cooling loop is investigated. First, a simplified analytical estimate for the thermomagnetic pumping action is derived, and then design rules for optimal solenoid and ferrofluid are presented. The design rules are used to set up a medium-scale (1 m, 10-1000 W) case study, which is modeled using a previously published and validated model (Aursand et al. [1]). The results show that the thermomagnetic driving force is significant compared to the natural convection driving force, and …
Rayleigh-Bénard convection, i.e. the flow of a fluid between two parallel plates that is driven by a temperature gradient, is an idealised setup to study thermal convection. Of special interest are the statistics of the turbulent temperature field, which we are investigating and comparing for three different geometries, namely convection with periodic horizontal boundary conditions in three and two dimensions as well as convection in a cylindrical vessel, in order to work out similarities and differences. To this end, we derive an exact evolution equation for the temperature probability densit…
As they keep cooling and contracting, Solar System giant planets radiate more energy than they receive from the Sun. Applying the first and second principles of thermodynamics, one can determine their cooling rate, luminosity, and temperature at a given age. Measurements of Saturn's infrared intrinsic luminosity, however, reveal that this planet is significantly brighter than predicted for its age. This excess luminosity is usually attributed to the immiscibility of helium in the hydrogen-rich envelope, leading to "rains" of helium-rich droplets. Existing evolution calculations, however, sugge…
The present work investigates the internal flow structure of two condensing droplets of aqueous solution during interaction. The velocity of fluid inside the droplets was investigated by confocal micro-PIV technique. Condensation on the droplets was carried out inside a closed chamber by creating a difference in vapor pressure between the droplet interface and a reservoir fluid surrounding the droplet in the room temperature without any cooling. Condensation on the droplet leads to spatial variation of solute concentration inside the droplet causing buoyancy driven Rayleigh convection. Fluid f…
Using numerical simulations of rapidly rotating Boussinesq convection in a Cartesian box, we study the formation of long-lived, large-scale, depth-invariant coherent structures. These structures, which consist of concentrated cyclones, grow to the horizontal size of the box, with velocities significantly larger than the convective motions. We vary the rotation rate, the thermal driving and the aspect ratio in order to determine the domain of existence of these large-scale vortices (LSV). We find that two conditions are required for their formation. First, the Rayleigh number, a meaure of the t…
In this paper, we extend Grossmann and Lohse's (GL) model [Phys. Rev. Lett. {\bf 86}, 3316 (2001)] for the predictions of Reynolds number (Re) and Nusselt number (Nu) in turbulent Rayleigh-Bénard convection (RBC). Towards this objective, we use functional forms for the prefactors of the dissipation rates in the bulk and the boundary layers. The functional forms arise due to inhibition of nonlinear interactions in the presence of walls and buoyancy compared to free turbulence, along with a deviation of viscous boundary layer profile from Prandtl-Blasius theory. We perform 60 numerical runs on a…
We present the results of direct numerical simulations of Rayleigh-Bénard convection in the presence of a uniform vertical magnetic field near instability onset. We have done simulations in boxes with square as well as rectangular cross-sections in the horizontal plane. We have considered horizontal aspect ratio $η= L_y/L_x =1$ and $2$. The onset of the primary and secondary instabilities are strongly suppressed in the presence of the vertical magnetic field for $η=1$. The Nusselt number $\mathrm{Nu}$ scales with Rayleigh number $\mathrm{Ra}$ close to the primary instability as $[\mathrm{\{Ra …
Convection in the Sun occurs at Rayleigh numbers, $Ra$, as high as $10^{22}$, molecular Prandtl numbers, $Pr$, as low as $10^{-6}$, under conditions that are far from satisfying the Oberbeck-Boussinesq (OB) idealization. The effects of these extreme circumstances on turbulent heat transport are unknown, and no comparable conditions exist on Earth. Our goal is to understand how these effects scale (since we cannot yet replicate the Sun's conditions faithfully). We study thermal convection by using direct numerical simulations, and determine the variation with respect to $Pr$, to values as low a…
This investigation deals with the analysis of stagnation point heat transfer and corresponding flow features of hydromagnetic viscous incompressible fluid over a vertical shrinking sheet. The considered sheet is assumed to be permeable and subject to addition of stagnation point to control the generated vorticity in the boundary layer. The sheet is placed on the right side of the fluid saturated porous medium which is having permeability of specified form. Nonlinear convection waves in the flow field are realized due to the envisaged nonlinear relation between density and temperature. The equa…
Coupled mixed convective and stratified systems are common in natural flows. To study experimentally the associated dynamics, we use a singular property of water: its non-linear equation of state is characterised by a maximum density close to $4^{\circ}$C. By heating the top of a tank at $35^{\circ}$C and cooling the bottom at $0^{\circ}$C, a two-layer configuration spontaneously appears. The convective motion in the bottom layer consists mostly of a large-scale circulation and rising cold plumes. This turbulent flow generates propagating internal gravity waves in the upper stably-stratified l…
Statistical properties of turbulent Rayleigh-Benard convection at low Prandtl numbers (Pr), which are typical for liquid metals such as mercury, gallium or liquid sodium, are investigated in high-resolution three-dimensional spectral element simulations in a closed cylindrical cell with an aspect ratio of one and are compared to previous turbulent convection simulations in air. We compare the scaling of global momentum and heat transfer. The scaling exponents are found to be in agreement with experiments. Mean profiles of the root-mean-square velocity as well as the thermal and kinetic energy …
We present measurements of the large-scale circulation (LSC) of turbulent Rayleigh-Benard convection in water-filled cylindrical samples of heights equal to their diameters. The orientation of the LSC had an irregular time dependence, but revealed a net azimuthal rotation with an average period of about 3 days for Rayleigh numbers R > 10^10. On average there was also a tendency for the LSC to be aligned with upflow to the west and downflow to the east, even after physically rotating the apparatus in the laboratory through various angles. Both of these phenomena could be explained as a result o…
We report on a numerical study of turbulent convection driven by a combination of internal heat sources and sinks. Motivated by a recent experimental realisation (Lepot et al. 2018), we focus on the situation where the cooling is uniform, while the internal heating is localised near the bottom boundary, over approximately one tenth of the domain height. We obtain scaling laws $Nu \sim Ra^γPr^χ$ for the heat transfer as measured by the Nusselt number $Nu$ expressed as a function of the Rayleigh number $Ra$ and the Prandtl number $Pr$. After confirming the experimental value $γ\approx 1/2$ for t…
We present a direct numerical simulation on the dynamics of large-scale circulation (LSC) of turbulent Rayleigh-Bénard convection of air (Pr = 0.7) contained in a cubic cell for Rayleigh number range 2X10^6 < Ra < 10^9 . The strength and orientation of LSC are quantified by the amplitude and phase of the first Fourier mode of the vertical velocity. The plane containing LSC is generally aligned along one of the diagonals of the box accompanied by a four-roll structure in the other. However, an abnormal single-roll state with substructures is noted at the face planes for low Ra. In addition to t…
We summarize the physical equations and analytic solutions of three versions of the box model equations, suitable for the integral formulation of axisymmetric gravity-driven particle currents with constant volume. The first model is based on a simple constant resisting stress, while the second and third models assume flow dilution by particle deposition. The third model is characterized by assuming an interstitial fluid lighter than the ambient fluid. All the calculations are performed on a flat topography. Ambient fluid entrainment and cooling effects are not considered. All particles are ass…
Turbulent and internal wave motions are important for the exchange of momentum, heat and suspended matter in the deep-sea which is generally stably stratified in density. Turbulence-generation models involve shear of vertical current differences that deforms the stratified waters, and convection that is drive by (unstable) buoyancy. Shear-generation is found more general in the well-stratified ocean-interior, while convection is known to occur near the sea-surface, e.g. via nighttime cooling. Far below the surface, the Western-Mediterranean Sea is very weakly stratified and offers opportunity …
We study thermal convection in a rotating fluid in order to better understand the properties of convection zones in rotating stars and planets. We first derive mixing-length theory for rapidly-rotating convection, arriving at the results of Stevenson (1979) via simple physical arguments. The theory predicts the properties of convection as a function of the imposed heat flux and rotation rate, independent of microscopic diffusivities. In particular, it predicts the mean temperature gradient; the rms velocity and temperature fluctuations; and the size of the eddies that dominate heat transport. …
To study properties of magneto-hydrodynamic (MHD) convection and resultant dynamo activities in proto-neutron stars (PNSs), we construct a "PNS in a box" simulation model with solving compressible MHD equation coupled with a nuclear equation of state (EOS) and a simplified leptonic transport. As a demonstration, we apply it to two types of PNS models with different internal structures: fully-convective model and spherical-shell convection model. By varying the spin rate of models, the rotational dependence of convection and dynamo that operate inside the PNS is investigated. We find that, as a…
Buoyancy-induced (Rayleigh-Benard) convection of a fluid between two horizontal plates is a central paradigm for studying the transition to complex spatiotemporal dynamics in sustained nonequilibrium systems. To improve the analysis of experimental data and the quantitative comparison of theory with experiment, we have developed a three-dimensional finite-difference code that can integrate the three-dimensional Boussinesq equations (which govern the evolution of the temperature, velocity, and pressure fields associated with a convecting flow) efficiently in large box-shaped domains with experi…
The elliptical instability is an instability of elliptical streamlines, which can be excited by large-scale tidal flows in rotating fluid bodies, and excites inertial waves if the dimensionless tidal amplitude ($ε$) is sufficiently large. It operates in convection zones but its interactions with turbulent convection have not been studied in this context. We perform an extensive suite of Cartesian hydrodynamical simulations in wide boxes to explore the interactions of the elliptical instability and Rayleigh-Bénard convection. We find that geostrophic vortices generated by the elliptical instabi…
Horizontally extended turbulent convection, termed mesoscale convection in natural systems, remains a challenge to investigate in both experiments and simulations. This is particularly so for very low molecular Prandtl numbers as in stellar convection and in Earth's outer core. The present study reports three-dimensional direct numerical simulations of turbulent Rayleigh-Bénard convection in square boxes of side length $L$ and height $H$ with the aspect ratio $Γ=L/H$ of 25, for Prandtl numbers that span almost 4 orders of magnitude, $10^{-3}\le Pr \le 7$, and Rayleigh numbers $10^5 \le Ra \le …
We investigate the dynamics of a fluid layer subject to an imposed bottom heat flux and a top monotonically-increasing temperature profile driving horizontal convection. We use direct numerical simulations and consider a large range of flux-based Rayleigh numbers $10^6 \leq Ra_F \leq 10^9$ and imposed top horizontal to bottom vertical heat flux ratios $0 \leq Λ\leq 1$. The fluid domain is a closed two-dimensional box with aspect ratio $4\leq Γ\leq 16$ and we consider no-slip boundaries and adiabatic side walls. We demonstrate a regime transition from Rayleigh--Bénard convection (RB) to horizon…
This paper treats topology optimization of natural convection problems. A simplified model is suggested to describe the flow of an incompressible fluid in steady state conditions, similar to Darcy's law for fluid flow in porous media. The equations for the fluid flow are coupled to the thermal convection-diffusion equation through the Boussinesq approximation. The coupled non-linear system of equations is discretized with stabilized finite elements and solved in a parallel framework that allows for the optimization of high resolution three-dimensional problems. A density-based topology optimiz…
One of the most prominent dynamical features of turbulent, rapidly-rotating convection is the formation of large-scale coherent structures, driven by Reynolds stresses resulting from the small-scale convective flows. In spherical geometry, such structures consist of intense zonal flows that are invariant along the rotation axis. In planar geometry, long-lived, depth-invariant structures also form at large scales, but, in the absence of horizontal anisotropy, they consist of vortices that grow to the domain size. In this work, through the introduction of horizontal anisotropy into a numerical m…
We present numerical simulations, using two complementary setups, of rotating Boussinesq thermal convection in a three-dimensional Cartesian geometry with misaligned gravity and rotation vectors. This model represents a small region at a non-polar latitude in the convection zone of a star or planet. We investigate the effects of rotation on the bulk properties of convection at different latitudes, focusing on determining the relation between the heat flux and temperature gradient. We show that our results may be interpreted using rotating mixing length theory (RMLT). The simplest version of RM…
