Non-Oberbeck-Boussinesq (NOB) effects on the flow organization in two-dimensional Rayleigh-Benard turbulence are numerically analyzed. The working fluid is water. We focus on the temperature profiles, the center temperature, the Nusselt number, and on the analysis of the velocity field. Several velocity amplitudes (or Reynolds numbers) and several kinetic profiles are introduced and studied; these together describe the various features of the rather complex flow organization. The results are presented both as functions of the Rayleigh number Ra (with Ra up to 10^8) for fixed temperature difference (Delta) between top and bottom plates and as functions of Delta ("non-Oberbeck-Boussinesqness") for fixed Ra with Delta up to 60 K. All results are consistent with the available experimental NOB data for the center temperature Tc and the Nusselt number ratio Nu_{NOB}/Nu_{OB} (the label OB meaning that the Oberbeck-Boussinesq conditions are valid).
Beyond Ra ~ 10^6 the flow consists of a large diagonal center convection roll and two smaller rolls in the upper and lower corners. In the NOB case the center convection roll is still characterized by only one velocity scale.
The original Oberbeck-Boussinesq (OB) equations which are the coupled two dimensional Navier-Stokes(NS) and heat conduction equations have been investigated by E.N. Lorenz half a century ago with Fourier series and opened the way to the paradigm of chaos. In our former study-Chaos, Solitons and Fractals78, 249 (2015)-we presented fully analytic solutions for the velocity, pressure and temperature fields with the aim of the self-similar Ansatz and gave a possible explanation of the Rayleigh--Bènard convection cells. Now we generalize the Oberbeck-Boussinesq hydrodynamical system, going beyond t…
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 numerically analyze Non-Oberbeck-Boussinesq (NOB) effects in two-dimensional Rayleigh-Benard flow in glycerol, which shows a dramatic change in the viscosity with temperature. The results are presented both as functions of the Rayleigh number (Ra) up to $10^8$ (for fixed temperature difference between the top and bottom plates) and as functions of "non-Oberbeck-Boussinesqness'' or "NOBness'' ($Δ$) up to 50 K (for fixed Ra). For this large NOBness the center temperature $T_c$ is more than 5 K larger than the arithmetic mean temperature $T_m$ between top and bottom plate and only weakly depen…
A thermodynamic argument is proposed in order to discuss the most appropriate form of the local energy balance equation within the Oberbeck-Boussinesq approximation. The study is devoted to establish the correct thermodynamic property to be used in order to express the relationship between the change of internal energy and the temperature change. It is noted that, if the fluid is a perfect gas, this property must be identified with the specific heat at constant volume. If the fluid is a liquid, a definitely reliable approximation identifies this thermodynamic property with the specific heat at…
As shown in earlier work (Ahlers et al., J. Fluid Mech. 569, p.409 (2006)), non-Oberbeck Boussinesq (NOB) corrections to the center temperature in turbulent Rayleigh-Benard convection in water and also in glycerol are governed by the temperature dependences of the kinematic viscosity and the thermal diffusion coefficient. If the working fluid is ethane close to the critical point the origin of non-Oberbeck-Boussinesq corrections is very different, as will be shown in the present paper. Namely, the main origin of NOB corrections then lies in the strong temperature dependence of the isobaric the…
In this article we will present pure two-dimensional analytic solutions for the coupled non-compressible Newtoniain Navier-Stokes --- with Boussinesq approximation --- and the heat conduction equation. The system was investigated from E.N. Lorenz half a century ago with Fourier series and pioneered the way to the paradigm of chaos. We present a novel analysis of the same system where the key idea is the two-dimensional generalization of the well-known self-similar Ansatz of Barenblatt which will be interpreted in a geometrical way. The results, the pressure, temperature and velocity fields are…
The influence of symmetrical non-Oberbeck-Boussinesq (SNOB) effect on statistical and scaling properties of temperature field in turbulent convection is investigated experimentally in $SF_{6}$ in the vicinity of its gas-liquid critical point (CP). The main conclusion of the studies is that besides the strong $Ra$ and $Pr$ dependence of the rms of temperature fluctuations normalized by the temperature difference across the cell, different from the Oberbeck-Boussinesq (OB) case of turbulent convection, all rest of statistical and scaling properties of temperature field discussed in details are t…
We establish stochastic functional integral representations for solutions of Oberbeck-Boussinesq equations in the form of McKean-Vlasov-type mean field equations, which can be used to design numerical schemes for calculating solutions and for implementing Monte-Carlo simulations of Oberbeck-Boussinesq flows. Our approach is based on the duality of conditional laws for a class of diffusion processes associated with solenoidal vector fields, which allows us to obtain a novel integral representation theorem for solutions of some linear parabolic equations in terms of the Green function and the pi…
We present the adaptation to non--free boundary conditions of a pseudospectral method based on the (complex) Fourier transform. The method is applied to the numerical integration of the Oberbeck--Boussinesq equations in a Rayleigh--Bénard cell with no-slip boundary conditions for velocity and Dirichlet boundary conditions for temperature. We show the first results of a 2D numerical simulation of dry air convection at high Rayleigh number ($R\sim10^9$). These results are the basis for the later study, by the same method, of wet convection in a solar still.
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 linear stability threshold of the Rayleigh-Benard configuration is analyzed with compressible effects taken into account. It is assumed that the fluid obeys a Newtonian rheology and Fourier's law of thermal transport with constant, uniform (dynamic) viscosity and thermal conductivity in a uniform gravity field. Top and bottom boundaries are maintained at different constant temperatures and we consider here boundary conditions of zero tangential stress and impermeable walls. Under these conditions, and with the Boussinesq approximation, Rayleigh (1916) first obtained analytically the critic…
The progress in our understanding of several aspects of turbulent Rayleigh-Benard convection is reviewed. The focus is on the question of how the Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the Prandtl number Pr, and on how the thicknesses of the thermal and the kinetic boundary layers scale with Ra and Pr. Non-Oberbeck-Boussinesq effects and the dynamics of the large-scale convection-roll are addressed as well. The review ends with a list of challenges for future research on the turbulent Rayleigh-Benard system.
In an attempt to understand the role of the strong radial dependence of thermal diffusivity on the properties of convection in sun-like stars, we mimic that effect in non-Oberbeck-Boussinesq (NOB) convection in a horizontally-extended rectangular domain (aspect ratio 16), by allowing the thermal diffusivity $κ$ to increase with the temperature (as in the case of stars). Direct numerical simulations (i.e., numerical solutions of the governing equations by resolving up to the smallest scales without requiring any modeling) show that, in comparison with Oberbeck-Boussinesq (OB) simulations (two o…
The Boussinesq equations for Rayleigh-Benard convection are simulated for a cylindrical container with an aspect ratio near 1.5. The transition from an axisymmetric stationary flow to time-dependent flows is studied using nonlinear simulations, linear stability analysis and bifurcation theory. At a Rayleigh number near 25,000, the axisymmetric flow becomes unstable to standing or travelling azimuthal waves. The standing waves are slightly unstable to travelling waves. This scenario is identified as a Hopf bifurcation in a system with O(2) symmetry.
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…
We propose an analytical model to estimate the interface temperature $Θ_Γ$ and the Nusselt number $Nu$ for an evaporating two-layer Rayleigh-Bénard configuration in statistically stationary conditions. The model is based on three assumptions: (i) the Oberbeck-Boussinesq approximation can be applied to the liquid phase, while the gas thermophysical properties are generic functions of thermodynamic pressure, local temperature, and vapour composition, (ii) the Grossmann-Lohse theory for thermal convection can be applied to the liquid and gas layers separately, (iii) the vapour content in the gas …
Penetrative turbulent Rayleigh-Bénard convection which depends on the density maximum of water near $4^\circ\rm{C}$ is studied using two-dimensional (2D) and three-dimensional (3D) direct numerical simulations (DNS). The working fluid is water near $4^\circ\rm{C}$ with Prandtl number $Pr=11.57$. The considered Rayleigh numbers $Ra$ range from $10^7$ to $10^{10}$. The density inversion parameter $θ_m$ varies from 0 to 0.9. It is found that the ratio of the top and bottom thermal boundary-layer thickness ($F_λ=λ_t^θ/λ_b^θ$) increases with increasing $θ_m$, and the relationship between $F_λ$ and …
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…
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…
We have found a multi-scale steady solution of the Boussinesq equations for Rayleigh-Bénard convection in a three-dimensional periodic domain between horizontal plates with a constant temperature difference by using a homotopy from the wall-to-wall optimal transport solution given by Motoki et al. (J. Fluid Mech., vol. 851, 2018, R4). The connected steady solution, which turns out to be a consequence of bifurcation from a thermal conduction state at the Rayleigh number $Ra\sim10^{3}$, is tracked up to $Ra\sim10^{7}$ by using a Newton-Krylov iteration. The exact coherent thermal convection exhi…
Rigorous upper limits on the vertical heat transport in two dimensional Rayleigh-Benard convection between stress-free isothermal boundaries are derived from the Boussinesq approximation of the Navier-Stokes equations. The Nusselt number Nu is bounded in terms of the Rayleigh number Ra according to $Nu \leq 0.2295 Ra^{5/12}$ uniformly in the Prandtl number Pr. This Nusselt number scaling challenges some theoretical arguments regarding the asymptotic high Rayleigh number heat transport by turbulent convection.
We investigate theoretically the nonlinear state of ideal straight rolls in the Rayleigh-Bénard system of a fluid layer heated from below with a porous medium using a Galerkin method. Applying the Oberbeck-Boussinesq approximation, binary mixtures with positive separation ratio are studied and compared to one-component fluids. Our results for the structural properties of roll convection resemble qualitatively the situation in the Rayleigh--Bénard system without porous medium except for the fact that the streamlines of binary mixtures are deformed in the so-called Soret regime. The deformation …
We study shallow moist Rayleigh-Benard convection in the Boussinesq approximation in three-dimensional direct numerical simulations. The thermodynamics of phase changes is approximated by a piecewise linear equation of state close to the phase boundary. The impact of phase changes on the turbulent fluctuations and the transfer of buoyancy through the layer is discussed as a function of the Rayleigh number and the ability to form liquid water. The enhanced buoyancy flux due to phase changes is compared with dry convection reference cases and related to the cloud cover in the convection layer. T…
Steady but generally unstable solutions of the 2D Boussinesq equations are obtained for no-slip boundary conditions and Prandtl number 7. The primary solution that bifurcates from the conduction state at Rayleigh number $Ra \approx 1708$ has been calculated up to $Ra\approx 5. 10^6$ and shows heat flux $Nu \sim 0.143\, Ra^{0.28}$ with a delicate spiral structure in the temperature field. Another solution that maximizes $Nu$ over the horizontal wavenumber has been calculated up to $Ra=10^9$ and its heat flux scales as $Nu \sim 0.115\, Ra^{0.31}$ for $10^7 < Ra \le 10^9$, quite similar to 3D tur…
We study the evolution of a melting front between the solid and liquid phases of a pure incompressible material where fluid motions are driven by unstable temperature gradients. In a plane layer geometry, this can be seen as classical Rayleigh-Bénard convection where the upper solid boundary is allowed to melt due to the heat flux brought by the fluid underneath. This free-boundary problem is studied numerically in two dimensions using a phase-field approach, classically used to study the melting and solidification of alloys, which we dynamically couple with the Navier-Stokes equations in the …
We present measurements of the orientation $θ_0$ and temperature amplitude $δ$ of the large-scale circulation in a cylindrical sample of turbulent Rayleigh-Benard convection (RBC) with aspect ratio $Γ\equiv D/L = 1.00$ ($D$ and $L$ are the diameter and height respectively) and for the Prandtl number $Pr \simeq 0.8$. Results for $θ_0$ revealed a preferred orientation with upflow in the West, consistent with a broken azimuthal invariance due to Earth's Coriolis force [see \cite{BA06b}]. They yielded the azimuthal diffusivity $D_θ$ and a corresponding Reynolds number $Re_θ$ for Rayleigh numbers o…
We apply the weak formalism on the Boussinesq equations, to characterize scaling properties of the mean and the standard deviation of the potential, kinetic and viscous energy flux in very high resolution numerical simulations. The local Bolgiano-Oboukhov length $L_{BO}$ is investigated and it is found that its value may change of an order of magnitude through the domain, in agreement with previous results. We investigate the scale-by-scale averaged terms of the weak equations, which are a generalization of the Karman-Howarth-Monin and Yaglom equations. We have not found the classical Bolgiano…
We revisit the optimal heat transport problem for Rayleigh-Bénard convection in which a rigorous upper bound on the Nusselt number, $Nu$, is sought as a function of the Rayleigh number $Ra$. Concentrating on the 2-dimensional problem with stress-free boundary conditions, we impose the full heat equation as a constraint for the bound using a novel 2-dimensional background approach thereby complementing the `wall-to-wall' approach of Hassanzadeh \etal \,(\emph{J. Fluid Mech.} \textbf{751}, 627-662, 2014). Imposing the same symmetry on the problem, we find correspondence with their result for $Ra…
Recently, the discrete unified gas-kinetic scheme (DUGKS) [Z. L. Guo \emph{et al}., Phys. Rev. E ${\bf 88}$, 033305 (2013)] based on the Boltzmann equation is developed as a new multiscale kinetic method for isothermal flows. In this paper, a thermal and coupled discrete unified gas-kinetic scheme is derived for the Boussinesq flows, where the velocity and temperature fields are described independently. Kinetic boundary conditions for both velocity and temperature fields are also proposed. The proposed model is validated by simulating several canonical test cases, including the porous plate pr…
We investigate the influence of the thermal properties of the boundaries in turbulent Rayleigh-Bénard convection on analytical upper bounds on convective heat transport. We model imperfectly conducting bounding plates in two ways: using idealized mixed thermal boundary conditions of constant Biot number $η$, continuously interpolating between the previously studied fixed temperature ($η= 0$) and fixed flux ($η= \infty$) cases; and by explicitly coupling the evolution equations in the fluid in the Boussinesq approximation through temperature and flux continuity to identical upper and lower cond…
In this paper we investigate two-dimensional (2D) Rayleigh-Bénard convection using direct numerical simulation in Boussinesq fluids of Prandtl number $P = 6.8$ confined between thermally conducting plates. We show through the simulation that in a small range of reduced Rayleigh number $r$ ($770 < r < 890$) the 2D rolls move chaotically in a direction normal to the roll axis. The lateral shift of the rolls may lead to global flow reversal of the convective motion. The chaotic traveling rolls are observed in simulation with {\it free-slip} as well as {\it no-slip} boundary conditions on the velo…
The pattern selection problem in binary-mixture convection in an extended channel with a lateral through-flow is presented. The through-flow breaks left--right parity and changes pattern dynamics dramatically. The problem is studied based on computer simulation of the complete set of hydrodynamic equations (Oberbeck-Boussinesq approximation) in the two-dimensional rectangular channel with aspect ratio $Γ= 12$ and convection-suppressing lateral boundary conditions. A wide variety of new dynamical patterns is obtained, discussed and classified.
In this paper we present numerical simulations of rapidly-rotating Rayleigh-Bénard convection in the Boussinesq approximation with stress-free boundary conditions. At moderately low Rossby number and large Rayleigh number, we show that a large-scale depth-invariant flow is formed, reminiscent of the condensate state observed in two-dimensional flows. We show that the large-scale circulation shares many similarities with the so-called vortex, or slow-mode, of forced rotating turbulence. Our investigations show that at a fixed rotation rate the large-scale vortex is only observed for a finite ra…
We examine the dynamics associated with weakly compressible convection in a spherical shell by running 3D direct numerical simulations using the Boussinesq formalism [1]. Motivated by problems in astrophysics, we assume the existence of a finite adiabatic temperature gradient $\nabla T_{\rm{ad}}$ and use mixed boundary conditions for the temperature with fixed flux at the inner boundary and fixed temperature at the outer boundary. This setup is intrinsically more asymmetric than the more standard case of Rayleigh-Bénard convection in liquids between parallel plates with fixed temperature bound…
We present high resolution numerical simulations of convection in multiphase flows (boiling) using a novel algorithm based on a Lattice Boltzmann method. We first validate the thermodynamical and kinematical properties of the algorithm. Then, we perform a series of 3d numerical simulations at changing the mean properties in the phase diagram and compare convection with and without phase coexistence at $Ra \sim 10^7$. We show that in presence of nucleating bubbles non-Oberbeck Boussinesq effects develops, mean temperature profile becomes asymmetric, heat-transfer and heat-transfer fluctuations …
A low-dimensional model (LDM) for turbulent Rayleigh-Benard convection in a Cartesian cell with square domain, based on the Galerkin projection of the Boussinesq equations onto a finite set of empirical eigenfunctions, is presented. The empirical eigenfunctions are obtained from a joint Proper Orthogonal Decomposition (POD) of the velocity and temperature fields using the Snapshot Method on the basis of a direct numerical simulation (DNS). The resulting LDM is a quadratic inhomogeneous system of coupled ordinary differential equations which we use to describe the long-time temporal evolution o…
The letter considers non-isothermal fluid flows and revises simplifications of basic hydrodynamic equations for such flows arriving eventually to a generalization of the Oberbeck-Boussinesq approximation valid for arbitrary equation of state including both non-ideal gases as well as liquids. The proposed approach is based on a suggested general definition of potential temperature. Special attention is put on the energy conservation principle, and it is shown that the proposed approximation exactly preserves the total mechanical energy by approximate equations of motion. The principal importanc…
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…
Rotating spiral patterns in Rayleigh-Bénard convection are known to induce azimuthal flows, which raises the question of how different neighboring spirals interact with each other in spiral chaos, and the role of hydrodynamics in this regime. Far from the core, we show that spiral rotations lead to an azimuthal body force that is irrotational and of magnitude proportional to the topological index of the spiral and its angular frequency. The force, although irrotational, cannot be included in the pressure field as it would lead to a nonphysical, multivalued pressure. We calculate the asymptotic…
Natural convection in porous media is a fundamental process for the long-term storage of CO2 in deep saline aquifers. Typically, details of mass transfer in porous media are inferred from the numerical solution of the volume-averaged Darcy-Oberbeck-Boussinesq (DOB) equations, even though these equations do not account for the microscopic properties of a porous medium. According to the DOB equations, natural convection in a porous medium is uniquely determined by the Rayleigh number. However, in contrast with experiments, DOB simulations yield a linear scaling of the Sherwood number with the Ra…
Large-scale patterns, which are well-known from the spiral defect chaos regime of thermal convection at Rayleigh numbers $Ra < 10^4$, continue to exist in three-dimensional numerical simulations of turbulent Rayleigh-Bénard convection in extended cylindrical cells with an aspect ratio $Γ=50$ and $Ra>10^5$. They are uncovered when the turbulent fields are averaged in time and turbulent fluctuations are thus removed. We apply the Boussinesq closure to estimate turbulent viscosities and diffusivities, respectively. The resulting turbulent Rayleigh number $Ra_{\ast}$, that describes the convection…
We simulate numerically Boussinesq convection in non-rotating spherical shells for a fluid with a unity Prandtl number and Rayleigh numbers up to $10^9$. In this geometry, curvature and radial variations of the gravitationnal acceleration yield asymmetric boundary layers. A systematic parameter study for various radius ratios (from $η=r_i/r_o=0.2$ to $η=0.95$) and gravity profiles allows us to explore the dependence of the asymmetry on these parameters. We find that the average plume spacing is comparable between the spherical inner and outer bounding surfaces. An estimate of the average plume…
We analyze the reversals of the large scale flow in Rayleigh-Bénard convection both through particle image velocimetry flow visualization and direct numerical simulations (DNS) of the underlying Boussinesq equations in a (quasi) two-dimensional, rectangular geometry of aspect ratio 1. For medium Prandtl number there is a diagonal large scale convection roll and two smaller secondary rolls in the two remaining corners diagonally opposing each other. These corner flow rolls play a crucial role for the large scale wind reversal: They grow in kinetic energy and thus also in size thanks to plume de…
Thermal convection in an inclined layer between two parallel walls kept at different fixed temperatures is studied for fixed Prandtl number Pr=1.07. Depending on the angle of inclination and the imposed temperature difference, the flow exhibits a large variety of self-organized spatio-temporal convection patterns. Close to onset, these patterns have been explained in terms of linear stability analysis of primary and secondary flow states. At larger temperature difference, far beyond onset, experiments and simulations show complex, dynamically evolving patterns that are not described by stabili…
Convection is a key transport phenomenon important in many different areas, from hydrodynamics and ocean circulation to planetary atmospheres or stellar physics. However its microscopic understanding still remains challenging. Here we numerically investigate the onset of convective flow in a compressible (non-Oberbeck-Boussinesq) hard disk fluid under a temperature gradient in a gravitational field. We uncover a surprising two-step transition scenario with two different critical temperatures. When the bottom plate temperature reaches a first threshold, convection kicks in (as shown by a struct…
We consider stability of steady convective flows in a horizontal layer with stress-free boundaries, heated below and rotating about the vertical axis, in the Boussinesq approximation (the Rayleigh-Benard convection). The flows under consideration are convective rolls or square cells, the latter being asymptotically equal to the sum of two orthogonal rolls of the same wave number k. We assume, that the Rayleigh number R is close to the critical one, R_c(k), for the onset of convective flows of this wave number: R=R_c(k)+epsilon^2; the amplitude of the flows is of the order of epsilon. We show t…
