Mixed Convection inside a cavity resulting from thermal buoyancy force under local modulation via rotating flat plate has been investigated. The present model consists of a square cavity with the left and right vertical walls fixed at constant high and low temperatures respectively while the top and bottom walls are supposed to be adiabatic. Two clockwise rotating flat plates, having negligible thickness in comparison to their lengths, acting as flow modulators have been placed vertically along the centerline of the cavity. The moving boundary problem due to plate motion in this study has been solved by implementing \textit{Arbitrary Lagrangian Eulerian (ALE)} finite element formulation with triangular discretization scheme.Simulations are conducted for air ($Pr=0.71$) at different Rayleigh numbers ($10^2 \leq Ra \leq 10^6$). Rotational Reynolds number based on plate dynamic condition has been considered to be constant at 430. Numerical results identify critical Rayleigh number $Ra_{cr}=0.41\times10^6$ beyond which two smaller flow modulators are more effective than a single larger modulator. Thermal oscillating frequency was observed to be insensitive to Rayleigh number for the case of double modulators.
Thermal convection in a horizontal layer with uniform internal heating and stress-free constant-temperature boundaries is analyzed numerically. The work is motivated by the questions arising in development of liquid metal batteries, in which convection is induced by the Joule heating of electrolyte. It is demonstrated that three-dimensional convection cells exist at subcritical Rayleigh numbers.
I consider a dielectric fluid heated from above and subjected to an electric potential difference between its top and bottom. I show that for a suitably chosen electric potential difference, the layer of fluid can become unstable. For the case of a strongly polar fluid like pure water, an electric potential difference of a few hundreds of volts can trigger convection. Although the analysis in this paper cannot explain the phenomenon described in [Gross & Porter(1966)], it could be because of unavailability of accurate physical parameters of the fluid used in the experiment.
We offer a unifying theory for turbulent purely internally heated convection, generalizing the unifying theories of Grossmann and Lohse (2000, 2001) for Rayleigh--Bénard turbulence and of Shishkina, Grossmann and Lohse (2016) for turbulent horizontal convection, which are both based on the splitting of the kinetic and thermal dissipation rates in respective boundary and bulk contributions. We obtain the mean temperature of the system and the Reynolds number (which are the response parameters) as function of the control parameters, namely the internal thermal driving strength (called, when nond…
Oscillatory instability of buoyancy convection in a laterally heated cube with perfectly thermally conducting horizontal boundaries is studied. The effect of the spanwise boundaries on the oscillatory instability onset is studied. The problem is treated by Krylov-subspace-iteration based Newton and Arnoldi methods. The Krylov basis vectors are calculated by a novel approach that involves the SIMPLE iteration and a projection onto a space of functions satisfying all linearized and homogeneous boundary conditions. The finite volume grid is gradually refined from 1003 to 2563 finite volumes. A se…
Convection of an internally heated fluid, confined between top and bottom plates of equal temperature, is studied by direct numerical simulation in two and three dimensions. The unstably stratified upper region drives convection that penetrates into the stably stratified lower region. The fraction of produced heat escaping across the bottom plate, which is one half without convection, initially decreases as convection strengthens. Entering the turbulent regime, this decrease reverses in two dimensions but continues monotonically in three dimensions. The mean fluid temperature, which grows prop…
Motivated by the mushy zones of sea ice, volcanoes, and icy moons of the outer solar system, we perform a theoretical and numerical study of boundary-layer convection along a vertical heated wall in a bounded ideal mushy region. The mush is comprised of a porous and reactive binary alloy with a mixture of saline liquid in a solid matrix, and is studied in the near-eutectic approximation. Here we demonstrate the existence of four regions and study their behavior asymptotically. Starting from the bottom of the wall, the four regions are (i) an isotropic corner region; (ii) a buoyancy dominated v…
The mechanisms governing the transition to turbulence in natural convection boundary layers along strongly heated vertical walls remain neither very clear nor well understood, because of the lack of experiments and the difficulties of physical modelling. Our measurements bring experimental data focusing on this transition in quiescent air along radiating and conducting plates in the whole range of 2000 to 8000 W/m\^2 heating rate. The analysis of the time series obtained by sliding window cross-correlation thermoanemometry leads us to point out coherent turbulent structures on short heights th…
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…
A water-filled differentially heated rotating annulus with initially prepared stable vertical salinity profiles is studied in the laboratory. Based on two-dimensional horizontal particle image velocimetry (PIV) data, and infrared camera visualizations, we describe the appearance and the characteristics of the baroclinic instability in this original configuration. First, we show that when the salinity profile is linear and confined between two non stratified layers at top and bottom, only two separate shallow fluid layers can be destabilized. These unstable layers appear nearby the top and the …
In the above paper the author treats the boundary layer flow along a vertical flat plate, immersed in a Darcy Brinkman Forchheimer porous medium. The porosity and the permeability of the porous medium are variable across the boundary layer. In addition a magnetic field with constant strength is applied normal to the plate. The fluid temperature at the plate is constant and different from that of the ambient fluid. This temperature difference creates a buoyancy force and the flow is characterized as mixed convection. The partial differential equations of the boundary layer flow are transformed …
The flow of fluid confined between a heated rotating cylinder and a cooled stationary cylinder is a canonical experiment for the study of heat transfer in engineering. The theoretical treatment of this system is greatly simplified if the cylinders are assumed to be of infinite length or periodic in the axial direction, in which cases heat transfer occurs only through conduction as in a solid. We here investigate numerically heat transfer and the onset of turbulence in such flows by using both periodic and no-slip boundary conditions in the axial direction. We obtain a simple linear criterion t…
A sequence of three steady - oscillatory transitions of buoyancy convection of air in a laterally heated cube with perfectly thermally insulated horizontal and spanwise boundaries is studied. The problem is treated by Newton and Arnoldi methods based on Krylov subspace iteration. The finite volume grid is gradually refined from 100^3 to 256^3 finite volumes. It is shown that the primary instability is characterized by two competing eigenmodes, whose temporal development results in two different oscillatory states that differ by their symmetries. Bifurcations due to both modes are subcritical. …
A series of laboratory experiments has been carried out in a thermally driven rotating annulus to study the onset of baroclinic instability, using horizontal and uniformly sloping bottom topographies. Different wave flow regimes have been identified and their phase boundaries -- expressed in terms of appropriate non-dimensional parameters -- have been compared to the recent numerical results of \citet{thomas_slope}. In the flat bottom case, the numerically predicted alignment of the boundary between the axisymmetric and the regular wave flow regime was found to be consistent with the experimen…
Heat transport in turbulent thermal convection increases with the thermal forcing, but in almost all studies the rate of this increase is slower than it would be if transport became independent of the molecular diffusivities -- the heat transport scaling is slower than the mixing-length (or `ultimate') scaling. In configurations driven by either thermal boundary conditions or internal heating, thermal boundary layers instead lead to a boundary-limited (or `classical') scaling. With net-zero internal heating and cooling in different regions, mixing-length scaling can occur because heat need not…
Inertia-gravity waves (IGWs) play an essential role in the terrestrial atmospheric dynamics as they can lead to energy and momentum flux when propagating upwards. An open question is to which extent IGWs contribute to the total energy and to flattening of the energy spectrum observed at the mesoscale. In this work, we present an experimental investigation of the energy distribution between the large-scale balanced flow and the small-scale imbalanced flow. Weakly nonlinear IGWs emitted from baroclinic jets are observed in the differentially heated rotating annulus experiment. Similar to the atm…
Recent planetary data and geophysical modelling suggest that hydrothermal activity is ongoing under the ice crust of Enceladus, one of Saturn's moons. According to these models, hydrothermal flow in the porous, rocky core of the satellite is driven by tidal deformation that induces dissipation and volumetric internal heating. Despite the effort in the modelling of Enceladus' interior, systematic understanding---and even basic scaling laws---of internally-heated porous convection and hydrothermal activity are still lacking. In this article, using an idealised model of an internally-heated porou…
From Sheared Annular Centrifugal Rayleigh-Bénard Convection to Radially Heated Taylor-Couette Flow: Exploring the Impact of Buoyancy and Shear on Heat Transfer and Flow Structure
This work reviews basic features of both Rayleigh-Bénard (RB) convection and internally heated (IH) convection, along with findings on IH convection from laboratory experiments and numerical simulations. In the first chapter, six canonical models of convection are described: three configurations of IH convection driven by constant and uniform volumetric heating, and three configurations of RB convection driven by the boundary conditions. The IH models are distinguished by differing pairs of thermal boundary conditions: top and bottom boundaries of equal temperature, an insulating bottom with h…
An experimental and numerical smoothed particle hydrodynamics (SPH) analysis was performed for the convective flow arising from a horizontal, thin cylindrical heat source enclosed in a glycerin-filled, slender enclosure at low Rayleigh numbers ($1.18\leq {\rm Ra}\leq 242$). Both the experiments and the SPH calculations were performed for positive ($0.1\leqΔT\leq 10$ K) and negative ($-10\leqΔT\leq -0.1$ K) temperature differences between the source and the surrounding fluid. In all cases a pair of steady, counter-rotating vortices is formed, accompanied by a plume of vertically ascending flow …
We use direct numerical simulation (DNS) to explore the effect of tilt on two-dimensional turbulent thermal convection on a half-soap bubble that is heated at its equator.In the DNS, the bubble is tilted by an angle $δ\in[0^{\circ},90^{\circ}]$, the Rayleigh number is varied between $Ra\in[3\times10^6, 3\times10^9]$, and the Prandlt number is fixed at $Pr=7$.The DNS reveals two qualitatively different flow regimes: the dynamic plume regime (DPR) and the stable plume regime (SPR).In the DPR, small dynamic plumes constantly emerge from random locations on the equator and dissipate on the bubble.…
Two widely used sub-grid scale models, the standard and the dynamic Smagorinsky models were tested in a simulation of the flow in a thermally driven 3D cavity at Rayleigh number of Ra=1E9. The main focus of the research is the response of the subgrid scale models to the coarse mesh and their ability to predict the flow in a thermally driven 3D cavity with the mesh resolution below generally accepted LES standards. The research is motivated by investigating a feasible modeling strategy for the particulate flow in the wall-bounded side-wall heated cavity. As URANS and hybrid RANS-LES models fail…
Convection over a wavy heated bottom wall in the air flow has been studied in experiments with the Rayleigh number $\sim 10^8$. It is shown that the mean temperature gradient in the flow core inside a large-scale circulation is directed upward, that corresponds to the stably stratified flow. In the experiments with a wavy heated bottom wall, we detect large-scale standing internal gravity waves excited in the regions with the stably stratified flow. The wavelength and the period of these waves are much larger than the turbulent spatial and time scales, respectively. In particular, the frequenc…
The two-dimensional flow of viscous incompressible liquid in a square cavity with a free boundary and differentially heated vertical sides is considered in the present work. The influence of gravitational and thermocapillary convection on temperature and velocity fields is studied in large range of dimensionless parameters and similarity criteria using equations in a Boussinesq approximation. Limiting cases of dimensionless parameters are analyzed numerically.
We consider convection in an internally heated layer of fluid that is bounded below by a perfect insulator and above by a poor conductor. The poorly conducting boundary is modelled by a fixed heat flux. Using solely analytical methods, we find linear and energy stability thresholds for the static state, and we construct a lower bound on the mean temperature that applies to all flows. The linear stability analysis yields a Rayleigh number above which the static state is linearly unstable ($R_L$), and the energy analysis yields a Rayleigh number below which it is globally stable ($R_E$). For var…
We study the conductive and convective states of phase-change of pure water in a rectangular container where two opposite walls are kept respectively at temperatures below and above the freezing point and all the other boundaries are thermally insulating. The global ice content at the equilibrium and the corresponding shape of the ice-water interface are examined, extending the available experimental measurements and numerical simulations. We first address the effect of the initial condition, either fully liquid or fully frozen, on the system evolution. Secondly, we explore the influence of th…
We present a numerical study of double-diffusive convection characterized by a stratification unstable to thermal convection while at the same time a mean molecular weight (or solute concentration) difference between top and bottom counteracts this instability. Convective zones can form in this case either by the stratification being locally unstable to the combined action of both temperature and solute gradients or by another process, the oscillatory double-diffusive convective instability, which is triggered by the faster molecular diffusivity of heat in comparison with that one of the solut…
The evolution of three-dimensional, cellular convective flows in a plane horizontal layer of a Boussinesq fluid heated from below is well studied. Here we review results from the investigation of this system as well as a number of related and novel numerical findings. We present theoretical results for pattern formation in Rayleigh-Bénard convection with solving NS equation under the Boussinesq approximation . System of equations were reduced to 2-dimension form to simplify the analysis.The evolution of the flow agree with the idea of the flow achieving an optimal form.
The dynamics of a small Prandtl number binary mixture in a laterally heated cavity is studied numerically. By combining temporal integration, steady state solving and linear stability analysis of the full PDE equations, we have been able to locate and characterize a codimension-three degenerate Takens-Bogdanov point whose unfolding describes the dynamics of the system for a certain range of Rayleigh numbers and separation ratios near S=-1.
We present numerical simulations of three-dimensional thermal convective flows in a cubic cell at high Rayleigh number using thermal lattice Boltzmann (LB) method. The thermal LB model is based on double distribution function approach, which consists of a D3Q19 model for the Navier-Stokes equations to simulate fluid flows and a D3Q7 model for the convection-diffusion equation to simulate heat transfer. Relaxation parameters are adjusted to achieve the isotropy of the fourth-order error term in the thermal LB model. Two types of thermal convective flows are considered: one is laminar thermal co…
New bounds are proven on the mean vertical convective heat transport, $\overline{\langle wT \rangle}$, for uniform internally heated (IH) convection in the limit of infinite Prandtl number. For fluid in a horizontally-periodic layer between isothermal boundaries, we show that $\overline{\langle wT \rangle} \leq \frac12 - c R^{-2}$, where $R$ is a nondimensional `flux' Rayleigh number quantifying the strength of internal heating and $c = 216$. Then, $\overline{\langle wT \rangle} = 0$ corresponds to vertical heat transport by conduction alone, while $\overline{\langle wT \rangle} > 0$ represent…
The paper is devoted to the study of the formation of stratification in an incompressible fluid due to convective laminar flows in horizontal layers heated from the side. Medium and intensive modes of stationary laminar thermal, concentrational and thermo-concentrational (in particular thermohaline) convection are considered, in which nonlinear flow features are manifested, which can radically change the flow structure and the characteristics of heat and mass transfer. The solutions of the problems of laminar problems of convection show the features of the formation of layered structures, vert…
Fully resolved simulations are used to quantify the effects of heat transfer in the presence of buoyancy on the drag of a spatially fixed heated spherical particle at low Reynolds numbers ($Re$) in the range $10^{-3} \le Re \le 10$ in a variable property fluid. The amount of heat addition from the sphere encompasses both, the heating regime where the Boussinesq approximation holds and the regime where it breaks down. The particle is assumed to have a low Biot number which means that the particle is uniformly at the same temperature and has no internal temperature gradients. Scaling buoyancy wi…
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…
The present work focuses on the study of mixed convection of a purely viscous shear-thinning fluid in a horizontal annular eccentric duct. The inner and outer cylinders are heated with constant and uniform heat flux densities. The objective of this work is to study the effect of the variation of eccentricity, rheological behavior of the fluid as well as the thermodependency of the rheological parameters on the reorganization of the flow and thermal stratification caused by the buoyancy forces. At the entrance of the heating zone, the dynamic regime is assumed to be established and the temperat…
This paper numerically investigates the physical mechanism of flow instability and heat transfer of natural convection in a cavity with thin fin(s). The left and the right walls of the cavity are differentially heated. The cavity is given an initial temperature, and the thin fin(s) is fixed on the hot wall in order to control the heat transfer. The finite volume method and the SIMPLE algorithm are used to simulate the flow. Distributions of the temperature, the pressure, the velocity and the total pressure are obtained. Then, the energy gradient theory is employed to study the physical mechani…
This paper experimentally investigates the convection in a fast rotating Tangent Cylinder (TC), for Ekman numbers down to $E=3.36\times10^{-6}$, in a configuration relevant to the liquid core of the Earth. In the apparatus, the TC results from the Proudman-Taylor constraint incurred by rotating a hemispherical fluid vessel heated in its centre by a protruding heating element of cylindrical shape. The resulting convection that develops above the heater, i.e within the TC, is shown to set in for critical Rayleigh numbers and wavenumbers respectively scaling as $Ra_c\sim E^{4/3}$ and $a_c\sim E^{…
Mixed convection above a horizontal disk rotating in a semi-infinite fluid is examined when the disk is heated so that its temperature varies quadratically with distance away from its centre. Steady similarity solutions are presented for a range of values of the two dimensionless parameters, a Prandtl number and a Grashof number. The results are corroborated by asymptotic analyses undertaken in the limits of small and large Prandtl number. For finite Prandtl number, the existence of multiple solutions at fixed Grashof number is revealed. The similarity structure for steady solutions fails beyo…
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 simplest model to couple the heat conduction and Navier-Stokes equations together is the Oberbeck-Boussinesq(OB)system which were investigated by E.N. Lorenz and opened the paradigm of chaos. In our former studies - Chaos, Solitons and Fractals 78, 249 (2015), ibid, 103, 336 (2017) - we derived analytic solutions for the velocity, pressure and temperature fields. Additionally, we gave a possible explanation of the Rayleigh-Bènard convection cells with the help of the self-similar Ansatz. Now we generalize the OB hydrodynamical system, including a viscous source term in the heat conduction …
We obtain an analytical bound on the mean vertical convective heat flux $\langle w T \rangle$ between two parallel boundaries driven by uniform internal heating. We consider two configurations, one with both boundaries held at the same constant temperature, and the other one with a top boundary held at constant temperature and a perfectly insulating bottom boundary. For the first configuration, Arslan et al. (J. Fluid Mech. 919:A15, 2021) recently provided numerical evidence that Rayleigh-number-dependent corrections to the only known rigorous bound $\langle w T \rangle \leq 1/2$ may be provab…
We present in this article the work of Henri Bénard (1874-1939), French physicist who began the systematic experimental study of two hydrodynamic systems: the thermal convection of fluids heated from below (the Rayleigh-Bénard convection and the Bénard-Marangoni convection) and the periodical vortex shedding behind a bluff body in a flow (the Bénard-Kármán vortex street). Across his scientific biography, we review the interplay between experiments and theory in these two major subjects of fluid mechanics.
Turbulence in the flow of fluid through a pipe can be suppressed by buoyancy forces. As the suppression of turbulence leads to severe heat transfer deterioration, this is an important and undesirable phenomenon in both heating and cooling applications. Vertical flow is often considered, as the axial buoyancy force can help drive the flow. With heating measured by the buoyancy parameter $C$, our DNS show that shear-driven turbulence may either be completely laminarised or transitions to a relatively quiescent convection-driven state. Buoyancy forces cause a flattening of the base flow profile, …
Coherent large-scale circulations of turbulent thermal convection in air have been studied experimentally in a rectangular box heated from below and cooled from above using Particle Image Velocimetry. The hysteresis phenomenon in turbulent convection was found by varying the temperature difference between the bottom and the top walls of the chamber (the Rayleigh number was changed within the range of $10^7 - 10^8$). The hysteresis loop comprises the one-cell and two-cells flow patterns while the aspect ratio is kept constant ($A=2 - 2.23$). We found that the change of the sign of the degree of…
