We study, using direct numerical simulations, the effect of geometrical confinement on heat transport and flow structure in Rayleigh-Bénard convection in fluids with different Prandtl numbers. Our simulations span over two decades of Prandtl number $Pr$, $0.1 \leq Pr \leq 40$, with the Rayleigh number $Ra$ fixed at $10^8$. The width-to-height aspect ratio $Γ$ spans between $0.025$ and $0.25$ while the length-to-height aspect ratio is fixed at one. We first find that for $Pr \geq 0.5$, geometrical confinement can lead to a significant enhancement in heat transport as characterized by the Nusselt number $Nu$. For those cases, $Nu$ is maximal at a certain $Γ= Γ_{opt}$. It is found that $Γ_{opt}$ exhibits a power-law relation with $Pr$ as $Γ_{opt}=0.11Pr^{-0.06}$, and the maximal relative enhancement generally increases with $Pr$ over the explored parameter range. As opposed to the situation of $Pr \geq 0.5$, confinement-induced enhancement in $Nu$ is not realized for smaller values of $Pr$, such as $0.1$ and $0.2$. The $Pr$ dependence of the heat transport enhancement can be understood in its relation to the coverage area of the thermal plumes over the thermal boundary layer (BL) where larger coverage is observed for larger $Pr$ due to a smaller thermal diffusivity. We further show that $Γ_{opt}$ is closely related to the crossing of thermal and momentum BLs, and find that $Nu$ declines sharply when the thickness ratio of the thermal and momentum BLs exceeds a certain value of about one. In addition, through examining the temporally averaged flow fields and 2D mode decomposition, it is found that for smaller $Pr$ the large-scale circulation is robust against the geometrical confinement of the convection cell.
We report experimental measurements of heat transport in rotating Rayleigh-B{é}nard convection in a cylindrical convection cell with aspect ratio $Γ= 1/2$. The fluid was helium gas with Prandtl number Pr = 0.7. The range of control parameters was Rayleigh number $4 \times 10^9 < {\rm Ra} < 4 \times 10^{11}$ and Ekman number $2 \times 10^{-7} < {\rm Ek} < 3 \times 10^{-5}$(corresponding to Taylor number $4 \times 10^9 < {\rm Ta} < 1 \times 10^{14}$ and convective Rossby number $0.07 < {\rm Ro} < 5$). We determine the crossover from weakly rotating turbulent convection to rotation dominated geos…
Results from direct numerical simulation for three-dimensional Rayleigh-Bénard convection in samples of aspect ratio $Γ=0.23$ and $Γ=0.5$ up to Rayleigh number $Ra=2\times10^{12}$ are presented. The broad range of Prandtl numbers $0.5<Pr<10$ is considered. In contrast to some experiments, we do not see any increase in $Nu/Ra^{1/3}$, neither due to $Pr$ number effects, nor due to a constant heat flux boundary condition at the bottom plate instead of constant temperature boundary conditions. Even at these very high $Ra$, both the thermal and kinetic boundary layer thicknesses obey Prandtl-Blasiu…
In agreement with a recent experimental discovery by Xia et. al. (2009), we also find a sloshing mode in experiments on the large-scale circulation (LSC) of turbulent Rayleigh-Benard convection in a cylindrical sample of aspect ratio one. The sloshing mode has the same frequency as the torsional oscillation discovered by Funfschilling and Ahlers (2004). We show that both modes can be described by an extension of a model developed previously [Brown and Ahlers (2008)] which consists of permitting a lateral displacement of the LSC circulation plane away from the vertical center line of the sample…
Experimental and numerical data for the heat transfer as a function of the Rayleigh-, Prandtl-, and Rossby numbers in turbulent rotating Rayleigh-Benard convection are presented. For relatively small $Ra ~ 10^8$ and large Pr modest rotation can enhance the heat transfer by up to 30%. At larger Ra there is less heat-transfer enhancement, and at small Pr = 0.7. there is no heat-transfer enhancement at all. We suggest that the small-Pr behavior is due to the breakdown of the heat-transfer-enhancing Ekman pumping because of larger thermal diffusion.
The shape of velocity and temperature profiles near the horizontal conducting plates in turbulent Rayleigh-Bénard convection are studied numerically and experimentally over the Rayleigh number range $10^8\lesssim Ra\lesssim3\times10^{11}$ and the Prandtl number range $0.7\lesssim Pr\lesssim5.4$. The results show that both the temperature and velocity profiles well agree with the classical Prandtl-Blasius laminar boundary-layer profiles, if they are re-sampled in the respective dynamical reference frames that fluctuate with the instantaneous thermal and velocity boundary-layer thicknesses.
Convective turbulence in a Rayleigh Benard system has shown a marked reluctance to exhibit clear scaling in the energy or entropy spectrum. The recent numerical simulation of Pandey, Verma and Mishra has shown significantly better evidence of scaling in the infinite Prandtl number limit. This prompted us to look at this limit analytically. We find that the inevitable presence of sweeping helps give a very good understanding of the results of Pandey et. al. In the presence of rotation, the Rayleigh number dependence of the Nusselt number shows a strong increase in slope in recent experiments an…
Direct numerical simulations of turbulent convection at high Rayleigh numbers in large aspect ratio cells are challenging because of the prohibitive computational resources required. One can achieve high Rayleigh numbers at affordable costs for low aspect ratios, but the effect of small aspect ratio remains to be understood fully. In this work, we explore integral quantities in convection in a cell with the small aspect ratio of 0.1 by varying both the Rayleigh and Prandtl numbers systematically. We find that the heat transport in this flow is within 10\% of that in cells with large aspect rat…
We experimentally test the effects of tilting a turbulent Rayleigh-B{é}nard convection cell on the dynamics of the large-scale circulation (LSC) orientation $θ_0$. The probability distribution of $θ_0$ is measured, and used to obtain a tilt-induced potential acting on $θ_0$, which is used in a low-dimensional model of diffusion of $θ_0$ in a potential. The form of the potential is sinusoidal in $θ_0$, and linear in tilt angle for small tilt angles, which is explained by a simple geometric model of the vector direction of the mean buoyancy force acting on the LSC. However, the magnitude of the …
We report Particle Image Velocimetry of the Large Scale Circulation and the viscous boundary layer in turbulent thermal convection. We use two parallelepipedic Rayleigh-B{é}nard cells with a top smooth plate. The first one has a rough bottom plate and the second one has a smooth one so we compare the rough-smooth and the smooth-smooth configurations. The dimensions of the cell allow to consider a bi-dimensional mean flow. Lots of previous heat flux measurements have shown a Nusselt--Rayleigh regime transition corresponding to an increase of the heat flux in presence of roughness which is highe…
Results from direct numerical simulations of vertical natural convection at Rayleigh numbers $1.0\times 10^5$-$1.0\times 10^9$ and Prandtl number $0.709$ support a generalised applicability of the Grossmann-Lohse (GL) theory, which was originally developed for horizontal natural (Rayleigh-B{é}nard) convection. In accordance with the GL theory, it is shown that the boundary-layer thicknesses of the velocity and temperature fields in vertical natural convection obey laminar-like Prandtl-Blasius-Pohlhausen scaling. Specifically, the normalised mean boundary-layer thicknesses scale with the $-1/2$…
Bounds for the poloidal and toroidal kinetic energies and the heat transport are computed numerically for rotating convection at infinite Prandtl number with both no slip and stress free boundaries. The constraints invoked in this computation are linear or quadratic in the problem variables and lead to the formulation of a semidefinite program. The bounds behave as a function of Rayleigh number at fixed Taylor number qualitatively in the same way as the quantities being bounded. The bounds are zero for Rayleigh numbers smaller than the critical Rayleigh number for the onset of convection, they…
This work applies a continuous data assimilation scheme---a particular framework for reconciling sparse and potentially noisy observations to a mathematical model---to Rayleigh-Bénard convection at infinite or large Prandtl numbers using only the temperature field as observables. These Prandtl numbers are applicable to the earth's mantle and to gases under high pressure. We rigorously identify conditions that guarantee synchronization between the observed system and the model, then confirm the applicability of these results via numerical simulations. Our numerical experiments show that the ana…
A low-dimensional model of large Prandtl-number ($P$) Rayleigh Bénard convection is constructed using some of the important modes of pseudospectral direct numerical simulations. A detailed bifurcation analysis of the low-dimensional model for $P=6.8$ and aspect ratio of $2\sqrt{2}$ reveals a rich instability and chaos picture: steady rolls, time-periodicity, quasiperiodicity, phase locking, chaos, and crisis. Bifurcation analysis also reveals multiple co-existing attractors, and a window with time-periodicity after chaos. The results of the low-dimensional model matches quite closely with some…
We test the ability of a low-dimensional turbulence model to predict how dynamics of large-scale coherent structures such as convection rolls change in different cell geometries. We performed Rayleigh-Bénard convection experiments in a cubic container, in which there is a single convection roll known as the large-scale circulation (LSC). The model describes the motion of the orientation $θ_0$ of the LSC as diffusion in a potential which is predicted as a function of the shape of the cell from an approximate solutions of the Navier-Stokes equations. The model predicts advected oscillation modes…
We report on experimental determinations of the temperature field in the interior (bulk) of turbulent Rayleigh-Benard convection for a cylindrical sample with aspect ratio (diameter over height) of 0.50, both in the classical and in the ultimate state. The Prandtl number was close to 0.8. We find a "logarithmic layer" in which the temperature varies as A*ln(z/L) + B with the distance z from the bottom plate of the sample. The amplitude A varies with radial position r. In the classical state these results are in good agreement with direct numerical simulations (DNS); in the ultimate state there…
Numerical data for the heat transfer as a function of the Prandtl (Pr) and Rossby (Ro) numbers in turbulent rotating Rayleigh-Benard convection are presented for Rayleigh number Ra = 10^8. When Ro is fixed the heat transfer enhancement with respect to the non-rotating value shows a maximum as function of Pr. This maximum is due to the reduced efficiency of Ekman pumping when Pr becomes too small or too large. When Pr becomes small, i.e. for large thermal diffusivity, the heat that is carried by the vertical vortices spreads out in the middle of the cell, and Ekman pumping thus becomes less eff…
Horizontal Convection (HC) at large Rayleigh and Prandtl numbers, is studied experimentally in a regime up to seven orders of magnitude larger in terms of Rayleigh numbers than previously achieved. To reach Rayleigh up to $10^{17}$, the horizontal density gradient is generated using differential solutal convection by a differential input of salt and fresh water controlled by diffusion in a novel experiment where the zero-net mass flux of water is ensured through permeable membranes. This setup allows for measuring accurately the Nusselt number in solutal convection by carefully controlling the…
We analyze the Prandtl number (Pr) dependence of spectra and fluxes of kinetic energy, as well as the energy injection rates and dissipation rates, of turbulent thermal convection using numerical data. As expected, for a flow with $ \mathrm{Pr} \lessapprox 1 $, the inertial-range kinetic energy flux is constant, and the kinetic energy spectrum is Kolmogorov-like ($ k^{-5/3} $). More importantly, we show that the amplitudes of the kinetic energy fluxes and spectra and those of structure functions increase with the decrease of Pr, thus exhibiting stronger nonlinearity for flows with small Prandt…
We present results of direct numerical simulations on homoclinic gluing and ungluing bifurcations in low-Prandtl-number ($ 0 \leq Pr \leq 0.025 $) Rayleigh-Bénard system rotating slowly and uniformly about a vertical axis. We have performed simulations with \textit{stress-free} top and bottom boundaries for several values of Taylor number ($5 \leq Ta \leq 50$) near the instability onset. We observe a single homoclinic ungluing bifurcation, marked by the spontaneous breaking of a larger limit cycle into two limit cycles with the variation of the reduced Rayleigh number $r$ for smaller values of…
We present a detailed bifurcation structure and associated flow patterns for low-Prandtl number ($P=0.0002, 0.002, 0.005, 0.02$) Rayleigh-Bénard convection near its onset. We use both direct numerical simulations and a 30-mode low-dimensional model for this study. We observe that low-Prandtl number (low-P) convection exhibits similar patterns and chaos as zero-P convection \cite{pal:2009}, namely squares, asymmetric squares, oscillating asymmetric squares, relaxation oscillations, and chaos. At the onset of convection, low-P convective flows have stationary 2D rolls and associated stationary a…
We invoke the concepts of magnetic boundary layer and magnetic Rayleigh number and use the magnetic energy dissipation rates in the bulk and the boundary layers to derive some scaling laws expressing how Nusselt number depends on magnetic Rayleigh number, Prandtl number and magnetic Prandtl number for the simple case of turbulent magnetohydrodynamic Rayleigh-Benard convection in the presence of uniform vertical magnetic field.
We discuss two aspects of turbulent Rayleigh-Bénard convection (RBC) on the basis of high-resolution direct numerical simulations in a unique setting; a closed cylindrical cell of aspect ratio of one. First, we present a comprehensive comparison of statistical quantities such as energy dissipation rates and boundary layer thickness scales. Data are used from three simulation run series at Prandtl numbers $Pr$ that cover two orders of magnitude. In contrast to most previous studies in RBC the focus of the present work is on convective turbulence at very low Prandtl numbers including $Pr=0.021$ …
We compute the spectra and fluxes of the velocity and temperature fields in Rayleigh-Bénard convection in turbulent regime for a wide range of Prandtl numbers using pseudo-spectral simulations on $512^3$ grids. Our spectral and flux results support the Kolmogorov-Obukhov (KO) scaling for zero Prandtl number and low Prandtl number ($P=0.02$) convection. The KO scaling for the velocity field in zero-Prandtl number and low-Prandtl number convection is because of the weak buoyancy in the inertial range (buoyancy is active only at the very low wavenumbers). We also observe that for intermediate Pra…
We consider axisymmetric Boussinesq convection in a shallow cylinder radius, L, and depth, H (<< L), which rotates with angular velocity $Ω$ about its axis of symmetry aligned to the vertical. Constant heat flux boundary conditions, top and bottom, are adopted, for which the onset of instability occurs on a long horizontal length scale provided that $Ω$ is sufficiently small. We investigate the nonlinear development by well-established two-scale asymptotic expansion methods. Comparisons of the results with the direct numerical simulations (DNS) of the primitive governing equations are good at …
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 $Pr\gtrsim 1$ 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 $Ra\lesssim 5\times10^8$ 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 …
The boundary layer structure of the velocity and temperature fields in turbulent Rayleigh-Benard flows in closed cylindrical cells of unit aspect ratio is revisited from a transitional and turbulent viscous boundary layer perspective. When the Rayleigh number is large enough, the dynamics at the bottom and top plates can be separated into an impact region of downwelling plumes, an ejection region of upwelling plumes and an interior region away from the side walls. The latter is dominated by the shear of the large-scale circulation (LSC) roll which fills the whole cell and continuously varies i…
A series of numerical simulations of Rayleigh-B{é}nard convection in a cubic cavity are conducted in order to examine the structure of the thermal boundary layer in case of mixed boundary conditions. The main goal of the study is the physical mechanism which provides increasing of heat flux with spatial frequency of conducting-adiabatic pattern. Different spatial configuration of conducting plates, including the fractal one, are considered for Rayleigh numbers from $\Ray=10^{7}$ to $\Ray=2.0\times 10^{9}$. We have shown that the temperature boundary layer in case of mixed boundary conditions a…
We report the existence of two new limiting turbulent regimes in horizontal convection (HC) using direct numerical simulations at intermediate to low Prandtl numbers. The flow driven by a horizontal gradient along a horizontal surface, perpendicular to the acceleration of gravity is shown to transition to turbulence in the plume and the core, modifying the rate of heat and momentum transport. These transitions set a sequence of scaling laws blending the theoretical arguments from both the Shishkina, Grossmann & Lohse (SGL) theory with the Hughes, Griffiths & Mullarney (HGM) regime. These resul…
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 …
In turbulent wall sheared thermal convection, there are three different flow regimes, depending on the relative relevance of thermal forcing and wall shear. In this paper we report the results of direct numerical simulations of such sheared Rayleigh-Bénard convection, at fixed Rayleigh number $Ra=10^6$, varying the wall Reynolds number in the range $0 \leq Re_w \leq 4000$ and Prandtl number $0.22 \leq Pr \leq 4.6$, extending our prior work by Blass et al. (2020), where $Pr$ was kept constant at unity and the thermal forcing ($Ra$) varied. We cover a wide span of bulk Richardson numbers $0.014 …
We study the structure of the thermal boundary layer (BL) in Rayleigh-Bénard convection for Prandtl number ($Pr$) 0.021 by conducting direct numerical simulations in a two-dimensional square box for Rayleigh numbers ($Ra$) up to $10^9$. The large-scale circulation in the flow divides the horizontal plates into three distinct regions, and we observe that the local thermal BL thicknesses in the plume-ejection region are larger than those in the plume-impact and shear-dominated regions. Moreover, the local BL width decreases as $Ra^{-β(x)}$, with $β(x)$ depending on the position at the plate. We …
Results on the Prandtl-Blasius type kinetic and thermal boundary layer thicknesses in turbulent Rayleigh-Bénard convection in a broad range of Prandtl numbers are presented. By solving the laminar Prandtl-Blasius boundary layer equations, we calculate the ratio of the thermal and kinetic boundary layer thicknesses, which depends on the Prandtl number Pr only. It is approximated as $0.588Pr^{-1/2}$ for $Pr\ll Pr^*$ and as $0.982 Pr^{-1/3}$ for $Pr^*\ll\Pr$, with $Pr^*= 0.046$. Comparison of the Prandtl--Blasius velocity boundary layer thickness with that evaluated in the direct numerical simula…
The Rayleigh (Ra) and Prandtl (Pr) number scaling of the Nusselt number Nu, the Reynolds number Re, the temperature fluctuations, and the kinetic and thermal dissipation rates is studied for (numerical) homogeneous Rayleigh-Benard turbulence, i.e., Rayleigh-Benard turbulence with periodic boundary conditions in all directions and a volume forcing of the temperature field by a mean gradient. This system serves as model system for the bulk of Rayleigh-Benard flow and therefore as model for the so called ``ultimate regime of thermal convection''. With respect to the Ra dependence of Nu and Re we …
We investigate the structures of the near-plate velocity and temperature profiles at different horizontal positions along the conducting bottom (and top) plate of a Rayleigh-Bénard convection cell, using two-dimensional (2D) numerical data obtained at the Rayleigh number Ra=10^8 and the Prandtl number Pr=4.4 of an Oberbeck-Boussinesq flow with constant material parameters. The results show that most of the time, and for both velocity and temperature, the instantaneous profiles scaled by the dynamical frame method [Q. Zhou and K.-Q. Xia, Phys. Rev. Lett. 104, 104301 (2010) agree well with the c…
Three-dimensional miscible Rayleigh--Taylor (RT) turbulence at small Atwood number and at Prandtl number one is investigated by means of high resolution direct numerical simulations of the Boussinesq equations. RT turbulence is a paradigmatic time-dependent turbulent system in which the integral scale grows in time following the evolution of the mixing region. In order to fully characterize the statistical properties of the flow, both temporal and spatial behavior of relevant statistical indicators have been analyzed. Scaling of both global quantities ({\it e.g.}, Rayleigh, Nusselt and Reynold…
Using direct numerical simulations, we study the statistical properties of reversals in two-dimensional Rayleigh-Bénard convection for infinite Prandtl number. We find that the large-scale circulation reverses irregularly, with the waiting time between two consecutive genuine reversals exhibiting a Poisson distribution on long time scales, while the interval between successive crossings on short time scales shows a power law distribution. We observe that the vertical velocities near the sidewall and at the center show different statistical properties. The velocity near the sidewall shows a lon…
Natural convection is of great importance in many engineering applications. This paper presents a smoothed particle hydrodynamics (SPH) method for natural convection. The conservation equations of mass, momentum and energy of fluid are discretized into SPH equations. The body force due to the change of density in a temperature field is considered by the Boussinesq approximation. The numerical method is validated by comparing numerical results with experimental results from literature. The numerical and experimental results reach a good agreement. Then the SPH method is applied to study the nat…
The heat transport in rotating Rayleigh-Bénard convection is considered in the limit of rapid rotation (small Ekman number $E$) and strong thermal forcing (large Rayleigh number $Ra$). The analysis proceeds from a set of asymptotically reduced equations appropriate for rotationally constrained dynamics; the conjectured range of validity for these equations is $Ra \lesssim E^{-8/5}$. A rigorous bound on heat transport of $Nu \le 20.56Ra^3E^4$ is derived in the limit of infinite Prandtl number using the background method. We demonstrate that the exponent in this bound cannot be improved on using…
We present high-resolution numerical investigations of heat transport by two-dimensional (2D) turbulent Rayleigh-Bénard (RB) convection over the Rayleigh number range $10^8 \leqslant Ra\leqslant 10^{10}$ and the Prandtl number range $0.7\leqslant Pr \leqslant10$. We find that there exist strong counter-gradient local heat flux with magnitude much larger than the global Nusselt number $Nu$ of the system. Two mechanisms for generating counter-gradient heat transport are identified: one is due to the bulk dynamics and the other is due to the competitions between the corner-flow rolls and the larg…
A model for three-dimensional Rayleigh-Bénard convection in low-Prandtl-number fluids near onset with rigid horizontal boundaries in the presence of a uniform vertical magnetic field is constructed and analyzed in detail. The kinetic energy $K$, the convective entropy $Φ$ and the convective heat flux ($Nu-1$) show scaling behaviour with $ε= r-1$ near onset of convection, where $r$ is the reduced Rayleigh number. The model is also used to investigate various magneto-convective structures close to the onset. Straight rolls, which appear at the primary instability, become unstable with increase i…
We prove the first rigorous bound on the heat transfer for three-dimensional Rayleigh-Bénard convection of finite-Prandtl-number fluids between free-slip boundaries with an imposed heat flux. Using the auxiliary functional method with a quadratic functional, which is equivalent to the background method, we prove that the Nusselt number $Nu$ is bounded by $Nu \leq 0.5999 R^{1/3}$ uniformly in the Prandtl number, where $R$ is the Rayleigh number based on the imposed heat flux. In terms of the Rayleigh number based on the mean vertical temperature drop, $Ra$, we obtain $Nu \leq 0.4646 Ra^{1/2}$. …
We have developed novel instrumentation for making Lagrangian measurements of temperature in diverse fluid flows. A small neutrally buoyant capsule is equipped with on-board electronics which measure temperature and transmit the data via a wireless radio frequency link to a desktop computer. The device has 80 dB dynamic range, resolving milli-Kelvin changes in temperature with up to 100 ms sampling time. The capabilities of these "smart particles" are demonstrated in turbulent thermal convection in water. We measure temperature variations as the particle is advected by the convective motion, a…
Results from direct numerical simulations for three dimensional Rayleigh-Benard convection in a cylindrical cell of aspect ratio 1/2 and Pr=0.7 are presented. They span five decades of Ra from $2\times 10^6$ to $2 \times10^{11}$. Good numerical resolution with grid spacing $\sim$ Kolmogorov scale turns out to be crucial to accurately calculate the Nusselt number, which is in good agreement with the experimental data by Niemela et al., Nature, 404, 837 (2000). In underresolved simulations the hot (cold) plumes travel further from the bottom (top) plate than in the fully resolved case, because t…
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 …
Using direct numerical simulations of Rayleigh-Bénard convection (RBC), we perform a comparative study of the spectra and fluxes of energy and entropy, and the scaling of large-scale quantities for large and infinite Prandtl numbers in two (2D) and three (3D) dimensions. We observe close similarities between the 2D and 3D RBC, in particular the kinetic energy spectrum $E_u(k) \sim k^{-13/3}$, and the entropy spectrum exhibits a dual branch with a dominant $k^{-2}$ spectrum. We showed that the dominant Fourier modes in the 2D and 3D flows are very close. Consequently, the 3D RBC is quasi two-di…
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…
Transition to hyperchaotic regimes in Rayleigh-Benard convection in a square periodicity cell is studied by three-dimensional numerical simulations. By fixing the Prandtl number at P=0.3 and varying the Rayleigh number as a control parameter, a bifurcation diagram is constructed where a route to hyperchaos involving quasiperiodic regimes with two and three incommensurate frequencies, multistability, chaotic intermittent attractors and a sequence of boundary and interior crises is shown. The three largest Lyapunov exponents exhibit a linear scaling with the Rayleigh number and are positive in t…
Gravity-driven convection of fluid at parameters near its thermodynamic critical point inside a porous layer heated from below (Rayleigh-Darcy convection) is studied. The fluid having the temperature slightly above the critical one is one-component. The hydrodynamic model describing a high compressible fluid phase at variable physical properties inside a solid matrix at a uniform porosity is analyzed. A near-critical fluid is assumed to be the van der Waals gas. In the limit of small variations in the density and thermodynamic coefficients where the Oberbeck-Boussinesq approximation is applica…
We study the stability of steady convection rolls in 2D Rayleigh--Bénard convection with free-slip boundaries and horizontal periodicity over twelve orders of magnitude in the Prandtl number $(10^{-6} \leq Pr \leq 10^6)$ and five orders of magnitude in the Rayleigh number $(8π^4 < Ra \leq 3 \times 10^7)$. The analysis is facilitated by partitioning our modal expansion into so-called even and odd modes. With aspect ratio $Γ= 2$, we observe that zonal modes (with horizontal wavenumber equal to zero) can emerge only once the steady convection roll state consisting of even modes only becomes unsta…
The structure of the boundary layers in turbulent Rayleigh-Benard convection is studied by means of three-dimensional direct numerical simulations. We consider convection in a cylindrical cell at an aspect ratio one for Rayleigh numbers of Ra=3e+9 and 3e+10 at fixed Prandtl number Pr=0.7. Similar to the experimental results in the same setup and for the same Prandtl number, the structure of the laminar boundary layers of the velocity and temperature fields is found to deviate from the prediction of the Prandtl-Blasius-Pohlhausen theory. Deviations decrease when a dynamical rescaling of the dat…
The boundary layer flow in a Rayleigh-Bénard convection cell of rectangular shape has been visualized in this fluid dynamics video. The experiment has been undertaken in air at a Rayleigh number $Ra=1.3\times 10^{10}$ and a Prandtl number $Pr=0.7$. Various sequences captured at selected positions of the heating plate show that the boundary layer is a very transient flow region characterized by coherent structures that permanently evolve. It becomes fully turbulent in the areas where the large-scale circulation impinge or leave the bottom plate.
