Journal articles: 'Convection de Rayleigh- Bénard et Bénard-Marangoni'

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Ostilla-Mónico, R. "Mixed thermal conditions in convection: how do continents affect the mantle’s circulation?" Journal of Fluid Mechanics 822 (June 1, 2017): 1–4. http://dx.doi.org/10.1017/jfm.2017.247.

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Natural convection is omnipresent on Earth. A basic and well-studied model for it is Rayleigh–Bénard convection, the fluid flow in a layer heated from below and cooled from above. Most explorations of Rayleigh–Bénard convection focus on spatially uniform, perfectly conducting thermal boundary conditions, but many important geophysical phenomena are characterized by boundary conditions which are a mixture of conducting and adiabatic materials. For example, the differences in thermal conductivity between continental and oceanic lithospheres are believed to play an important role in plate tectonics. To study this, Wang et al. (J. Fluid Mech., vol. 817, 2017, R1), measure the effect of mixed adiabatic–conducting boundary conditions on turbulent Rayleigh–Bénard convection, finding experimental proof that even if the total heat transfer is primarily affected by the adiabatic fraction, the arrangement of adiabatic and conducting plates is crucial in determining the large-scale flow dynamics.

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SCHEEL, J. D., P. L. MUTYABA, and T. KIMMEL. "Patterns in rotating Rayleigh–Bénard convection at high rotation rates." Journal of Fluid Mechanics 659 (June 30, 2010): 24–42. http://dx.doi.org/10.1017/s0022112010002399.

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We present the results from numerical and theoretical investigations of rotating Rayleigh–Bénard convection for relatively large dimensionless rotation rates, 170 < Ω < 274, and a Prandtl number of 6.4. Unexpected square patterns were found experimentally by Bajaj et al. (Phys. Rev. Lett., vol. 81, 1998, p. 806) in this parameter regime and near threshold for instability in the bulk. These square patterns have not yet been understood theoretically. Sánchez-Álvarez et al. (Phys. Rev. E, vol. 72, 2005, p. 036307) have found square patterns in numerical simulations for similar parameters when only the Coriolis force is included. We performed detailed numerical studies of rotating Rayleigh–Bénard convection for the same parameters as the experiments and simulations. To better understand these patterns, we compared the effects of the Coriolis force as well as the centrifugal force. We also computed the coefficients of the amplitude equation describing one-, two- and three-mode bulk solutions to rotating Rayleigh–Bénard convection. We find that squares are unstable, but we do find stable limit cycles consisting of three coupled oscillating amplitudes, which can superficially resemble squares, since one of the three amplitudes is rather small.

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Yang, Yantao, Roberto Verzicco, and Detlef Lohse. "Two-scalar turbulent Rayleigh–Bénard convection: numerical simulations and unifying theory." Journal of Fluid Mechanics 848 (June 8, 2018): 648–59. http://dx.doi.org/10.1017/jfm.2018.378.

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We conduct direct numerical simulations for turbulent Rayleigh–Bénard (RB) convection, driven simultaneously by two scalar components (say, temperature and concentration) with different molecular diffusivities, and measure the respective fluxes and the Reynolds number. To account for the results, we generalize the Grossmann–Lohse theory for traditional RB convection (Grossmann & Lohse, J. Fluid Mech., vol. 407, 2000, pp. 27–56; Phys. Rev. Lett., vol. 86 (15), 2001, pp. 3316–3319; Stevens et al., J. Fluid Mech., vol. 730, 2013, pp. 295–308) to this two-scalar turbulent convection. Our numerical results suggest that the generalized theory can successfully capture the overall trends for the fluxes of two scalars and the Reynolds number without introducing any new free parameters. In fact, for most of the parameter space explored here, the theory can even predict the absolute values of the fluxes and the Reynolds number with good accuracy. The current study extends the generality of the Grossmann–Lohse theory in the area of buoyancy-driven convection flows.

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SHISHKINA, OLGA, and ANDRÉ THESS. "Mean temperature profiles in turbulent Rayleigh–Bénard convection of water." Journal of Fluid Mechanics 633 (August 25, 2009): 449–60. http://dx.doi.org/10.1017/s0022112009990528.

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We report an investigation of temperature profiles in turbulent Rayleigh–Bénard convection of water based on direct numerical simulations (DNS) for a cylindrical cell with unit aspect ratio for the same Prandtl number Pr and similar Rayleigh numbers Ra as used in recent high-precision measurements by Funfschilling et al. (J. Fluid Mech., vol. 536, 2005, p. 145). The Nusselt numbers Nu computed for Pr = 4.38 and Ra = 108, 3 × 108, 5 × 108, 8 × 108 and 109 are found to be in excellent agreement with the experimental data corrected for finite thermal conductivity of the walls. Based on this successful validation of the numerical approach, the DNS data are used to extract vertical profiles of the mean temperature. We find that near the heating and cooling plates the non-dimensional temperature profiles Θ(y) (where y is the non-dimensional vertical coordinate), obey neither a logarithmic nor a power law. Moreover, we demonstrate that the Prandtl–Blasius boundary layer theory cannot predict the shape of the temperature profile with an error less than 7.9% within the thermal boundary layers (TBLs). We further show that the profiles can be approximated by a universal stretched exponential of the form Θ(y) ≈ 1 − exp(−y − 0.5y2) with an absolute error less than 1.1% within the TBLs and 5.5% in the whole Rayleigh cell. Finally, we provide more accurate analytical approximations of the profiles involving higher order polynomials in the approximation.

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STEVENS, RICHARD J. A. M., ROBERTO VERZICCO, and DETLEF LOHSE. "Radial boundary layer structure and Nusselt number in Rayleigh–Bénard convection." Journal of Fluid Mechanics 643 (January 15, 2010): 495–507. http://dx.doi.org/10.1017/s0022112009992461.

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Results from direct numerical simulation (DNS) for three-dimensional Rayleigh–Bénard convection in a cylindrical cell of aspect ratio 1/2 and Prandtl number Pr=0.7 are presented. They span five decades of Rayleigh number Ra from 2 × 106 to 2 × 1011. The results are in good agreement with the experimental data of Niemela et al. (Nature, vol. 404, 2000, p. 837). Previous DNS results from Amati et al. (Phys. Fluids, vol. 17, 2005, paper no. 121701) showed a heat transfer that was up to 30% higher than the experimental values. The simulations presented in this paper are performed with a much higher resolution to properly resolve the plume dynamics. We find that in under-resolved simulations the hot (cold) plumes travel further from the bottom (top) plate than in the better-resolved ones, because of insufficient thermal dissipation mainly close to the sidewall (where the grid cells are largest), and therefore the Nusselt number in under-resolved simulations is overestimated. Furthermore, we compare the best resolved thermal boundary layer profile with the Prandtl–Blasius profile. We find that the boundary layer profile is closer to the Prandtl–Blasius profile at the cylinder axis than close to the sidewall, because of rising plumes close to the sidewall.

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CIONI, S., S. CILIBERTO, and J. SOMMERIA. "Strongly turbulent Rayleigh–Bénard convection in mercury: comparison with results at moderate Prandtl number." Journal of Fluid Mechanics 335 (March 25, 1997): 111–40. http://dx.doi.org/10.1017/s0022112096004491.

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An experimental study of Rayleigh–Bénard convection in the strongly turbulent regime is presented. We report results obtained at low Prandtl number (in mercury, Pr = 0.025), covering a range of Rayleigh numbers 5 × 106 < Ra < 5 × 109, and compare them with results at Pr∼1. The convective chamber consists of a cylindrical cell of aspect ratio 1.Heat flux measurements indicate a regime with Nusselt number increasing as Ra0.26, close to the 2/7 power observed at Pr∼1, but with a smaller prefactor, which contradicts recent theoretical predictions. A transition to a new turbulent regime is suggested for Ra ≃ 2 × 109, with significant increase of the Nusselt number. The formation of a large convective cell in the bulk is revealed by its thermal signature on the bottom and top plates. One frequency of the temperature oscillation is related to the velocity of this convective cell. We then obtain the typical temperature and velocity in the bulk versus the Rayleigh number, and compare them with similar results known for Pr∼1.We review two recent theoretical models, namely the mixing zone model of Castaing et al. (1989), and a model of the turbulent boundary layer by Shraiman & Siggia (1990). We discuss how these models fail at low Prandtl number, and propose modifications for this case. Specific scaling laws for fluids at low Prandtl number are then obtained, providing an interpretation of our experimental results in mercury, as well as extrapolations for other liquid metals.

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Kunnen, R. P. J., H. J. H. Clercx, and G. J. F. van Heijst. "The structure of sidewall boundary layers in confined rotating Rayleigh–Bénard convection." Journal of Fluid Mechanics 727 (June 27, 2013): 509–32. http://dx.doi.org/10.1017/jfm.2013.285.

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AbstractTurbulent rotating convection is usually studied in a cylindrical geometry, as this is its most convenient experimental realization. In our previous work (Kunnen et al., J. Fluid Mech., vol. 688, 2011, pp. 422–442) we studied turbulent rotating convection in a cylinder with the emphasis on the boundary layers. A secondary circulation with a convoluted spatial structure has been observed in mean velocity plots. Here we present a linear boundary-layer analysis of this flow, which leads to a model of the circulation. The model consists of two independent parts: an internal recirculation within the sidewall boundary layer, and a bulk-driven domain-filling circulation. Both contributions exhibit the typical structure of the Stewartson boundary layer near the sidewall: a sandwich structure of two boundary layers of typical thicknesses ${E}^{1/ 4} $ and ${E}^{1/ 3} $, where $E$ is the Ekman number. Although the structure of the bulk-driven circulation may change considerably depending on the Ekman number, the boundary-layer recirculation is present at all Ekman numbers in the range $0. 72\times 1{0}^{- 5} \leq E\leq 5. 76\times 1{0}^{- 5} $ considered here.

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Huang, Shi-Di, and Ke-Qing Xia. "Effects of geometric confinement in quasi-2-D turbulent Rayleigh–Bénard convection." Journal of Fluid Mechanics 794 (April 6, 2016): 639–54. http://dx.doi.org/10.1017/jfm.2016.181.

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We report an experimental study of confinement effects in quasi-2-D turbulent Rayleigh–Bénard convection. The experiments were conducted in five rectangular cells with their height $H$ and length $L$ being the same and fixed, while the width $W$ was different for each cell to produce lateral aspect ratios (${\it\Gamma}=W/H$) of 0.6, 0.3, 0.2, 0.15 and 0.1. Direct flow field measurements reveal that the large-scale flow slows down as ${\it\Gamma}$ decreases and there are more plumes travelling through the bulk region. Moreover, the reversal frequency of the large-scale flow is found to increase drastically in smaller ${\it\Gamma}$ cells, by more than 1000-fold for the highest value of Rayleigh number reached in the experiment. The reversal frequency can be well described by a stochastic model developed by Ni et al. (J. Fluid Mech., vol. 778, 2015, R5) and the probability density functions (PDF) of the time interval between successive reversals are found to follow Poisson statistics as in the 3-D system. It is further observed that the bulk temperature fluctuation increases significantly and its PDF changes from exponential to Gaussian as ${\it\Gamma}$ decreases. The influences of geometric confinement on the global heat transport are also investigated. The measured Nu–Ra relationship suggests that, as the lateral aspect ratio decreases, the relative weight of the boundary layer contribution in the global heat transport increases compared to that from the bulk. These results demonstrate that in the quasi-2-D geometry, geometric confinement has strong effects on both the global and local properties in turbulent convective flows, which are very different from the previous findings in 3-D and true 2-D systems.

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Weiss, Stephan, Xiaozhou He, Guenter Ahlers, Eberhard Bodenschatz, and Olga Shishkina. "Bulk temperature and heat transport in turbulent Rayleigh–Bénard convection of fluids with temperature-dependent properties." Journal of Fluid Mechanics 851 (July 20, 2018): 374–90. http://dx.doi.org/10.1017/jfm.2018.507.

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We critically analyse the different ways to evaluate the dependence of the Nusselt number ($\mathit{Nu}$) on the Rayleigh number ($\mathit{Ra}$) in measurements of the heat transport in turbulent Rayleigh–Bénard convection under general non-Oberbeck–Boussinesq conditions and show the sensitivity of this dependence to the choice of the reference temperature at which the fluid properties are evaluated. For the case when the fluid properties depend significantly on the temperature and any pressure dependence is insignificant we propose a method to estimate the centre temperature. The theoretical predictions show very good agreement with the Göttingen measurements by He et al. (New J. Phys., vol. 14, 2012, 063030). We further show too the values of the normalized heat transport $\mathit{Nu}/\mathit{Ra}^{1/3}$ are independent of whether they are evaluated in the whole convection cell or in the lower or upper part of the cell if the correct reference temperatures are used.

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Wei, Ping, and Guenter Ahlers. "On the nature of fluctuations in turbulent Rayleigh–Bénard convection at large Prandtl numbers." Journal of Fluid Mechanics 802 (August 3, 2016): 203–44. http://dx.doi.org/10.1017/jfm.2016.444.

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We report experimental results for the power spectra, variance, skewness and kurtosis of temperature fluctuations in turbulent Rayleigh–Bénard convection (RBC) of a fluid with Prandtl number $Pr=12.3$ in cylindrical samples with aspect ratios $\unicode[STIX]{x1D6E4}$ (diameter $D$ over height $L$) of 0.50 and 1.00. The measurements were primarily for the radial positions $\unicode[STIX]{x1D709}=1-r/(D/2)=1.00$ and $\unicode[STIX]{x1D709}=0.063$. In both cases, data were obtained at several vertical locations $z/L$. For all locations, there is a frequency range of about a decade over which the spectra can be described well by the power law $P(f)\sim f^{-\unicode[STIX]{x1D6FC}}$. For all $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D6E4}$, the $\unicode[STIX]{x1D6FC}$ value is less than one near the top and bottom plates and increases as $z/L$ or $1-z/L$ increase from 0.01 to 0.5. This differs from the finding for$Pr=0.8$ (He et al., Phys. Rev. Lett., vol. 112, 2014, 174501) and the expectation for the downstream velocity of turbulent wall-bounded shear flow (Rosenberg, J. Fluid Mech., vol. 731, 2013, pp. 46–63), where $\unicode[STIX]{x1D6FC}=1$ is found or expected in an inner layer ($0.01\lesssim z/L\lesssim 0.1$) near the wall but in the bulk. The variance is described better by a power law $\unicode[STIX]{x1D70E}^{2}\sim (z/L)^{-\unicode[STIX]{x1D701}}$ than by the logarithmic dependence found or expected for $Pr=0.8$ and for turbulent shear flow. For both $\unicode[STIX]{x1D6E4}$, we found that, independent of Rayleigh number, $\unicode[STIX]{x1D701}\simeq 2/3$ near the sidewall ($\unicode[STIX]{x1D709}=0.063$), where plumes primarily rise or fall and the large-scale circulation (LSC) dynamics is most influential. This result agrees with a model due to Priestley (Turbulent Transfer in the Lower Atmosphere, 1959, University of Chicago Press) for convection over a horizontal heated surface. However, we found $\unicode[STIX]{x1D701}\simeq 1$ along the sample centreline ($\unicode[STIX]{x1D709}=1.00$), where there are relatively few plumes moving vertically and the LSC dynamics is expected to be less important; that result is consistent with one of two possible interpretations by Adrian (Intl J. Heat Mass Transfer, vol. 39, 1996, pp. 2303–2310) of a model due to Libchaber et al. (J. Fluid Mech., vol. 204, 1989, pp. 1–30). We discuss the composite nature of fluctuations in turbulent RBC, with contributions from intrinsic background fluctuations, plumes, the stochastic dynamics of the LSC, and the sloshing and torsional mode of the LSC. None of the models advanced so far explicitly consider all of these contributions.

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Wan, Zhen-Hua, Ping Wei, Roberto Verzicco, Detlef Lohse, Guenter Ahlers, and Richard J. A. M. Stevens. "Effect of sidewall on heat transfer and flow structure in Rayleigh–Bénard convection." Journal of Fluid Mechanics 881 (October 24, 2019): 218–43. http://dx.doi.org/10.1017/jfm.2019.770.

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In Rayleigh–Bénard convection experiments, the thermal coupling between the sidewall and fluid is unavoidable. As a result, the thermal properties of the sidewall can influence the flow structure that develops. To get a better understanding of the influence of the sidewall, we performed a one-to-one comparison between experiments and direct numerical simulations (DNS) in aspect ratio (diameter over height) $\unicode[STIX]{x1D6E4}=1.00$ samples. We focus on the global heat transport, i.e. the Nusselt number $Nu$, and the local vertical temperature gradients near the horizontal mid-plane on the cylinder axis and close to the sidewall. The data cover the range $10^{5}\lesssim Ra\lesssim 10^{10}$ where $Ra$ is the Rayleigh number. The $Nu$ number obtained from experimental measurements and DNS, in which we use an adiabatic sidewall, agree well. The experiments are performed with several gases, which have widely varying thermal conductivities, but all have a Prandtl number $Pr\approx 0.7$. For $Ra\gtrsim 10^{7}$, both experiments and DNS reveal a stabilizing (positive) temperature gradient at the cylinder axis. This phenomenon was known for high $Pr$, but had not been observed for small $Pr\approx 0.7$ before. The experiments reveal that the temperature gradient decreases with decreasing $Ra$ and eventually becomes destabilizing (negative). The decrease appears at a higher $Ra$ when the sidewall admittance, which measures how easily the heat transfers from the fluid to the wall, is smaller. However, the simulations with an adiabatic sidewall do not reproduce the destabilizing temperature gradient at the cylinder axis in the low $Ra$ number regime. Instead, these simulations show that the temperature gradient increases with decreasing $Ra$. We find that the simulations can reproduce the experimental findings on the temperature gradient at the cylinder axis qualitatively when we consider the physical properties of the sidewall and the thermal shields. However, the temperature gradients obtained from experiments and simulations do not agree quantitatively. The reason is that it is incredibly complicated to reproduce all experimental details accurately due to which it is impossible to reproduce all experimental measurement details. The simulations show, in agreement with the models of Ahlers (Phys. Rev. E, vol. 63 (1), 2000, 015303) and Roche et al. (Eur. Phys. J. B, vol. 24 (3), 2001, pp. 405–408), that the sidewall can act as an extra heat conductor, which absorbs heat from the fluid near the bottom plate and releases it into the fluid near the top plate. The importance of this effect decreases with increasing $Ra$. A crucial finding of the simulations is that the thermal coupling between the sidewall and fluid can strongly influence the flow structure, which can result in significant changes in heat transport. Since this effect goes beyond a simple short circuit of the heat transfer through the sidewall, it is impossible to correct experimental measurements for this effect. Therefore, careful design of experimental set-ups is required to minimize the thermal interaction between the fluid and sidewall.

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Castillo-Castellanos, Andrés, Anne Sergent, Bérengère Podvin, and Maurice Rossi. "Cessation and reversals of large-scale structures in square Rayleigh–Bénard cells." Journal of Fluid Mechanics 877 (September 2, 2019): 922–54. http://dx.doi.org/10.1017/jfm.2019.598.

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We consider direct numerical simulations of turbulent Rayleigh–Bénard convection inside two-dimensional square cells. For Rayleigh numbers $Ra=10^{6}$ to $Ra=5\times 10^{8}$ and Prandtl numbers $Pr=3$ and $Pr=4.3$, two types of flow regimes are observed intermittently: consecutive flow reversals (CR), and extended cessations (EC). For each regime, we combine proper orthogonal decomposition (POD) and statistical tools on long-term data to characterise the dynamics of large-scale structures. For the CR regime, centrosymmetric modes are dominant and display a coherent dynamics, while non-centrosymmetric modes fluctuate randomly. For the EC regime, all POD modes follow Poissonian statistics and a non-centrosymmetric mode is dominant. To explore further the differences between the CR and EC regimes, an analysis based on a cluster partition of the POD phase space is proposed. This data-driven approach confirms the successive mechanisms of the generic reversal cycle in CR as proposed in Castillo-Castellanos et al. (J. Fluid Mech., vol. 808, 2016, pp. 614–640). However, these mechanisms may take one of multiple paths in the POD phase space. Inside the EC regime, this approach reveals the presence of two types of coherent time sequences (weak reversals and actual cessations) and more rarely intense plume crossings. Finally, we analyse within a range of Rayleigh numbers up to turbulent flow, the relation between dynamical regimes and the POD energetic contents as well as the residence time in each cluster.

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BROWN, ERIC, and GUENTER AHLERS. "The origin of oscillations of the large-scale circulation of turbulent Rayleigh–Bénard convection." Journal of Fluid Mechanics 638 (October 1, 2009): 383–400. http://dx.doi.org/10.1017/s0022112009991224.

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In agreement with a recent experimental discovery by Xi et al. (Phys. Rev. Lett., vol. 102, 2009, paper no. 044503), we also find a sloshing mode in experiments on the large-scale circulation (LSC) of turbulent Rayleigh–Bénard convection in a cylindrical sample of aspect ratio one. The sloshing mode has the same frequency as the torsional oscillation discovered by Funfschilling & Ahlers (Phys. Rev. Lett., vol. 92, 2004, paper no. 1945022004). We show that both modes can be described by an extension of a model developed previously Brown & Ahlers (Phys. Fluids, vol. 20, 2008, pp. 105105-1–105105-15; Phys. Fluids, vol. 20, 2008, pp. 075101-1–075101-16). The extension consists of permitting a lateral displacement of the LSC circulation plane away from the vertical centreline of the sample as well as a variation of the displacement with height (such displacements had been excluded in the original model). Pressure gradients produced by the sidewall of the container on average centre the plane of the LSC so that it prefers to reach its longest diameter. If the LSC is displaced away from this diameter, the walls provide a restoring force. Turbulent fluctuations drive the LSC away from the central alignment, and combined with the restoring force they lead to oscillations. These oscillations are advected along with the LSC. This model yields the correct wavenumber and phase of the oscillations, as well as estimates of the frequency, amplitude and probability distributions of the displacements.

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Weiss, Stephan, and Guenter Ahlers. "Heat transport by turbulent rotating Rayleigh–Bénard convection and its dependence on the aspect ratio." Journal of Fluid Mechanics 684 (September 2, 2011): 407–26. http://dx.doi.org/10.1017/jfm.2011.309.

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AbstractWe report on the influence of rotation about a vertical axis on heat transport by turbulent Rayleigh–Bénard convection in a cylindrical vessel with an aspect ratio $\Gamma \equiv D/ L= 0. 50$ ($D$ is the diameter and $L$ the height of the sample) and compare the results with those for larger $\Gamma $. The working fluid was water at ${T}_{m} = 4{0\hspace{0.167em} }^{\ensuremath{\circ} } \mathrm{C} $ where the Prandtl number $\mathit{Pr}$ is 4.38. For rotation rates $\Omega \lesssim 1~\mathrm{rad} ~{\mathrm{s} }^{\ensuremath{-} 1} $, corresponding to inverse Rossby numbers $1/ \mathit{Ro}$ between zero and twenty, we measured the Nusselt number $\mathit{Nu}$ for six Rayleigh numbers $\mathit{Ra}$ in the range $2. 2\ensuremath{\times} 1{0}^{9} \lesssim \mathit{Ra}\lesssim 7. 2\ensuremath{\times} 1{0}^{10} $. For small rotation rates and at constant $\mathit{Ra}$, the reduced Nusselt number ${\mathit{Nu}}_{red} \equiv \mathit{Nu}(1/ \mathit{Ro})/ \mathit{Nu}(0)$ initially increased slightly with increasing $1/ \mathit{Ro}$, but at $1/ \mathit{Ro}= 1/ {\mathit{Ro}}_{0} \simeq 0. 5$ it suddenly became constant or decreased slightly depending on $\mathit{Ra}$. At $1/ {\mathit{Ro}}_{c} \approx 0. 85$ a second sharp transition occurred in ${\mathit{Nu}}_{red} $ to a state where ${\mathit{Nu}}_{red} $ increased with increasing $1/ \mathit{Ro}$. We know from direct numerical simulation that the transition at $1/ {\mathit{Ro}}_{c} $ corresponds to the onset of Ekman vortex formation reported before for $\Gamma = 1$ at $1/ {\mathit{Ro}}_{c} \simeq 0. 4$ and for $\Gamma = 2$ at $1/ {\mathit{Ro}}_{c} = 0. 18$ (Weiss et al., Phys. Rev. Lett., vol. 105, 2010, 224501). The $\Gamma $-dependence of $1/ {\mathit{Ro}}_{c} $ can be explained as a finite-size effect that can be described phenomenologically by a Ginzburg–Landau model; this model is discussed in detail in the present paper. We do not know the origin of the transition at $1/ {\mathit{Ro}}_{0} $. Above $1/ {\mathit{Ro}}_{c} $, ${\mathit{Nu}}_{red} $ increased with increasing $\Gamma $ up to ${\ensuremath{\sim} }1/ \mathit{Ro}= 3$. We discuss the $\Gamma $-dependence of ${\mathit{Nu}}_{red} $ in this range in terms of the average Ekman vortex density as predicted by the model. At even larger $1/ \mathit{Ro}\gtrsim 3$ there is a decrease of ${\mathit{Nu}}_{red} $ that can be attributed to two possible effects. First, the Ekman pumping might become less efficient when the Ekman layer is significantly smaller than the thermal boundary layer, and second, for rather large $1/ \mathit{Ro}$, the Taylor–Proudman effect in combination with boundary conditions suppresses fluid flow in the vertical direction.

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Wei, Ping, and Guenter Ahlers. "Logarithmic temperature profiles in the bulk of turbulent Rayleigh–Bénard convection for a Prandtl number of 12.3." Journal of Fluid Mechanics 758 (October 14, 2014): 809–30. http://dx.doi.org/10.1017/jfm.2014.560.

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AbstractWe report measurements of logarithmic temperature profiles $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\varTheta (z,r) = A(r)\times \ln (z/L) + B(r)$ in the bulk of turbulent Rayleigh–Bénard convection (here $\varTheta $ is a scaled and time-averaged local temperature in the fluid, $ z$ is the vertical and $r$ the radial position, and $L$ is the sample height). Two samples had aspect ratios $\varGamma \equiv D/L = 1.00$ and 0.50 (where $D=190\ \mathrm{mm}$ is the diameter). The fluid was a fluorocarbon with a Prandtl number of $\mathit{Pr} = 12.3$. The measurements covered the Rayleigh-number range $2\times 10^{10} \lesssim \mathit{Ra} \lesssim 2\times 10^{11}$ for $\varGamma = 1.00$ and $3\times 10^{11} \lesssim \mathit{Ra} \lesssim 2\times 10^{12}$ for $\varGamma = 0.50$. In contradistinction to what had been found for $\varGamma = 0.50$ and $\mathit{Pr} = 0.78$ by Ahlers et al. (Phys. Rev. Lett., vol. 109, 2012, art. 114501; J. Fluid Mech., 2014, in press), the measurements revealed no $\mathit{Ra}$ dependence of the amplitude $A(r)$ of the logarithmic term. Within the experimental resolution, the amplitude was also found to be independent of $\varGamma $. It varied with $r$ in a manner consistent with the function $A(\xi ) = A_1/\sqrt{2\xi - \xi ^2}$, where $\xi \equiv (R-r)/R$ with $R=D/2$ and $A_1 \simeq 0.0016$. The results for $A(r)$ are smaller than those obtained from experiments and direct numerical simulations (Ahlers et al., Phys. Rev. Lett., vol. 109, 2012, art. 114501) at similar values of $\mathit{Ra}$ for $\mathit{Pr} = 0.7$ and $\varGamma = \frac{1}{2}$ by a factor that depended slightly upon $\mathit{Ra}$ but was close to $2$.

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Jiang, Hechuan, Xiaojue Zhu, Varghese Mathai, Xianjun Yang, Roberto Verzicco, Detlef Lohse, and Chao Sun. "Convective heat transfer along ratchet surfaces in vertical natural convection." Journal of Fluid Mechanics 873 (June 28, 2019): 1055–71. http://dx.doi.org/10.1017/jfm.2019.446.

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We report on a combined experimental and numerical study of convective heat transfer along ratchet surfaces in vertical natural convection (VC). Due to the asymmetry of the convection system caused by the asymmetric ratchet-like wall roughness, two distinct states exist, with markedly different orientations of the large-scale circulation roll (LSCR) and different heat transport efficiencies. Statistical analysis shows that the heat transport efficiency depends on the strength of the LSCR. When a large-scale wind flows along the ratchets in the direction of their smaller slopes, the convection roll is stronger and the heat transport is larger than the case in which the large-scale wind is directed towards the steeper slope side of the ratchets. Further analysis of the time-averaged temperature profiles indicates that the stronger LSCR in the former case triggers the formation of a secondary vortex inside the roughness cavity, which promotes fluid mixing and results in a higher heat transport efficiency. Remarkably, this result differs from classical Rayleigh–Bénard convection (RBC) with asymmetric ratchets (Jiang et al., Phys. Rev. Lett., vol. 120, 2018, 044501), wherein the heat transfer is stronger when the large-scale wind faces the steeper side of the ratchets. We reveal that the reason for the reversed trend for VC as compared to RBC is that the flow is less turbulent in VC at the same $Ra$. Thus, in VC the heat transport is driven primarily by the coherent LSCR, while in RBC the ejected thermal plumes aided by gravity are the essential carrier of heat. The present work provides opportunities for control of heat transport in engineering and geophysical flows.

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Hewitt, Duncan R., Jerome A. Neufeld, and John R. Lister. "Convective shutdown in a porous medium at high Rayleigh number." Journal of Fluid Mechanics 719 (February 19, 2013): 551–86. http://dx.doi.org/10.1017/jfm.2013.23.

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AbstractConvection in a closed domain driven by a dense buoyancy source along the upper boundary soon starts to wane owing to the increase of the average interior density. In this paper, theoretical and numerical models are developed of the subsequent long period of shutdown of convection in a two-dimensional porous medium at high Rayleigh number $\mathit{Ra}$. The aims of this paper are twofold. Firstly, the relationship between this slowly evolving ‘one-sided’ shutdown system and the statistically steady ‘two-sided’ Rayleigh–Bénard (RB) cell is investigated. Numerical measurements of the Nusselt number $\mathit{Nu}$ from an RB cell (Hewitt et al., Phys. Rev. Lett., vol. 108, 2012, 224503) are very well described by the simple parametrization $\mathit{Nu}= 2. 75+ 0. 0069\mathit{Ra}$. This parametrization is used in theoretical box models of the one-sided shutdown system and found to give excellent agreement with high-resolution numerical simulations of this system. The dynamical structure of shutdown can also be accurately predicted by measurements from an RB cell. Results are presented for a general power-law equation of state. Secondly, these ideas are extended to model more complex physical systems, which comprise two fluid layers with an equation of state such that the solution that forms at the (moving) interface is more dense than either layer. The two fluids are either immiscible or miscible. Theoretical box models compare well with numerical simulations in the case of a flat interface between the fluids. Experimental results from a Hele-Shaw cell and numerical simulations both show that interfacial deformation can dramatically enhance the convective flux. The applicability of these results to the convective dissolution of geologically sequestered ${\mathrm{CO} }_{2} $ in a saline aquifer is discussed.

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Sun, Z. F., and K. T. Yu. "Rayleigh–Bénard–Marangoni Cellular Convection." Chemical Engineering Research and Design 84, no. 3 (March 2006): 185–91. http://dx.doi.org/10.1205/cherd.05057.

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Ahlers, Guenter, Eberhard Bodenschatz, and Xiaozhou He. "Logarithmic temperature profiles of turbulent Rayleigh–Bénard convection in the classical and ultimate state for a Prandtl number of 0.8." Journal of Fluid Mechanics 758 (October 9, 2014): 436–67. http://dx.doi.org/10.1017/jfm.2014.543.

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AbstractWe report on experimental determinations of the temperature field in the interior (bulk) of turbulent Rayleigh–Bénard convection for a cylindrical sample with an aspect ratio (diameter $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}D$ over height $L$) equal to 0.50, in both the classical and the ultimate state. The measurements are for Rayleigh numbers $\mathit{Ra}$ from $6\times 10^{11}$ to $10^{13}$ in the classical and $7\times 10^{14}$ to $1.1\times 10^{15}$ (our maximum accessible $\mathit{Ra}$) in the ultimate state. The Prandtl number was close to 0.8. Although to lowest order the bulk is often assumed to be isothermal in the time average, we found a ‘logarithmic layer’ (as reported briefly by Ahlers et al., Phys. Rev. Lett., vol. 109, 2012, 114501) in which the reduced temperature $\varTheta = [\langle T(z) \rangle - T_m]/\Delta T$ (with $T_m$ the mean temperature, $\Delta T$ the applied temperature difference and $\langle {\cdots } \rangle $ a time average) varies as $A \ln (z/L) + B$ or $A^{\prime } \ln (1-z/L) + B^{\prime }$ with the distance $z$ from the bottom plate of the sample. In the classical state, the amplitudes $-A$ and $A^{\prime }$ are equal within our resolution, while in the ultimate state there is a small difference, with $-A/A^{\prime } \simeq 0.95$. For the classical state, the width of the log layer is approximately $0.1L$, the same near the top and the bottom plate as expected for a system with reflection symmetry about its horizontal midplane. For the ultimate state, the log-layer width is larger, extending through most of the sample, and slightly asymmetric about the midplane. Both amplitudes $A$ and $A^{\prime }$ vary with radial position $r$, and this variation can be described well by $A = A_0 [(R - r)/R]^{-0.65}$, where $R$ is the radius of the sample. In the classical state, these results are in good agreement with direct numerical simulations (DNS) for $\mathit{Ra} = 2\times 10^{12}$; in the ultimate state there are as yet no DNS. The amplitudes $-A$ and $A^{\prime }$ varied as ${\mathit{Ra}}^{-\eta }$, with $\eta \simeq 0.12$ in the classical and $\eta \simeq 0.18$ in the ultimate state. A close analogy between the temperature field in the classical state and the ‘law of the wall’ for the time-averaged downstream velocity in shear flow is discussed. A two-sublayer mean-field model of the temperature profile in the classical state was analysed and yielded a logarithmic $z$ dependence of $\varTheta $. The $\mathit{Ra}$ dependence of the amplitude $A$ given by the model corresponds to an exponent $\eta _{th} = 0.106$, in good agreement with the experiment. In the ultimate state the experimental result $\eta \simeq 0.18$ differs from the prediction $\eta _{th} \simeq 0.043$ by Grossmann & Lohse (Phys. Fluids, vol. 24, 2012, 125103).

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20

Trouette, Benoît, Eric Chénier, Frédéric Doumenc, Claudine Delcarte, and Béatrice Guerrier. "Transient Rayleigh-Bénard-Marangoni solutal convection." Physics of Fluids 24, no. 7 (July 2012): 074108. http://dx.doi.org/10.1063/1.4733439.

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21

OR, A. C., and R. E. KELLY. "Feedback control of weakly nonlinear Rayleigh–Bénard–Marangoni convection." Journal of Fluid Mechanics 440 (August 10, 2001): 27–47. http://dx.doi.org/10.1017/s0022112001004670.

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We study the effect of proportional feedback control on the onset and development of finite-wavelength Rayleigh–Bénard–Marangoni (RBM) convection using weakly nonlinear theory as applied to Nield's model, which includes both thermocapillarity and buoyancy but ignores deformation of the free surface. A two-layer model configuration is used, which has a purely conducting gas layer on top of the liquid. In the feedback control analysis, a control action in the form of temperature or heat flux is considered. Both measurement and control action are assumed to be continuous in space and time. Besides demonstrating that stabilization of the basic state can be achieved on a linear basis, the results also indicate that a wide range of weakly nonlinear flow properties can also be altered by the linear and nonlinear control processes used here. These include changing the nature of hexagonal convection and the amount of subcritical hysteresis associated with subcritical bifurcation.

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22

Azmi, H. M., and R. Idris. "Effects of Controller and Nonuniform Temperature Profile on the Onset of Rayleigh-Bénard-Marangoni Electroconvection in a Micropolar Fluid." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/571437.

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Linear stability analysis is performed to study the effects of nonuniform basic temperature gradients on the onset of Rayleigh-Bénard-Marangoni electroconvection in a dielectric Eringen’s micropolar fluid by using the Galerkin technique. In the case of Rayleigh-Bénard-Marangoni convection, the eigenvalues are obtained for an upper free/adiabatic and lower rigid/isothermal boundaries. The influence of various parameters has been analysed. Three nonuniform basic temperature profiles are considered and their comparative influence on onset of convection is discussed. Different values of feedback control and electric number are added up to examine whether their presence will enhance or delay the onset of electroconvection.

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23

Touazi, O., E. Chénier, F. Doumenc, and B. Guerrier. "Simulation of transient Rayleigh–Bénard–Marangoni convection induced by evaporation." International Journal of Heat and Mass Transfer 53, no. 4 (January 2010): 656–64. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2009.10.029.

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24

Senin, Nor Halawati, Nor Fadzillah Mohd Mokhtar, and Mohamad Hasan Abdul Sathar. "Thermocapillary Convection in a Deformable Ferrofluid Layer." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 76, no. 2 (October 23, 2020): 144–53. http://dx.doi.org/10.37934/arfmts.76.2.144153.

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A linear stability evaluation is conducted to explore the effect on the onset of Marangoni-Bénard convection in a ferrofluid layer system. The system is heated from below with treatment of both the lower and upper boundaries to completely insulate the temperature disturbance. The eigenvalue problem is solved by using regular perturbation technique to obtain the critical number of Marangoni and also the critical number of Rayleigh. It is observed that the increase in the value Crispation, the magnetic number of Rayleigh and also the magnetic number will destabilize the system while the increasing number of Bonds will delay the convection.

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25

Chen, Jun, Chao-Qun Shen, He Wang, and Cheng-Bin Zhang. "Rayleigh-Bénard-Marangoni convection characteristics during mass transfer between liquid layers." Acta Physica Sinica 68, no. 7 (2019): 074701. http://dx.doi.org/10.7498/aps.68.20181295.

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26

Hashim, Ishak. "On competition between modes at the onset of Bénard-Marangoni convection in a layer of fluid." ANZIAM Journal 43, no. 3 (January 2002): 387–95. http://dx.doi.org/10.1017/s144618110001258x.

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AbstractIn this paper we use classical linear stability theory to analyse the onset of steady and oscillatory Bénard-Marangoni convection in a horizontal layer of fluid in the more physically-relevant case when both the non-dimensional Rayleigh and Marangoni numbers are linearly dependent. We present examples of situations in which there is competition between modes at the onset of convection when the layer is heated from below.

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27

Sun, Z. F. "Onset of Rayleigh–Bénard–Marangoni convection with time-dependent nonlinear concentration profiles." Chemical Engineering Science 68, no. 1 (January 2012): 579–94. http://dx.doi.org/10.1016/j.ces.2011.10.023.

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28

DOUMENC, F., T. BOECK, B. GUERRIER, and M. ROSSI. "Transient Rayleigh–Bénard–Marangoni convection due to evaporation: a linear non-normal stability analysis." Journal of Fluid Mechanics 648 (April 7, 2010): 521–39. http://dx.doi.org/10.1017/s0022112009993417.

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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 wavenumber and time. The method is applied to transient Rayleigh–Bénard–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 purely surface-tension-driven convection. Critical parameters from calculations are in good agreement with those from experiments on drying polymer solutions, where the surface cooling is induced by solvent evaporation.

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29

Kiran, R. V., and Attluri Kalyani. "Gradient on Rayleigh–Bénard – Marangoni – Magnetoconvection in a Micropolar Fluid with Maxwell – Cattaneo Law." Mapana - Journal of Sciences 14, no. 3 (July 22, 2015): 1–22. http://dx.doi.org/10.12723/mjs.34.1.

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The effect of non-uniform temperature gradient on the onset of Rayleigh-Bénard-Marangoni- Magneto-convection in a Micropolar fluid with Maxwell-Cattaneo law is studied using the Galerkin technique. The eigen value is obtained for rigid-free velocity boundary combination with isothermal and adiabatic condition on the spin-vanishing boundaries. A linear stability analysis is performed. The influence of various parameters on the onset of convection has been analyzed. One linear and five non-linear temperature profiles are considered and their comparative influence on onset is discussed. The classical approach predicts an infinite speed for the propagation of heat. The present non-classical theory involves a wave type heat transport (Second Sound) and does not suffer from the physically unacceptable drawback of infinite heat propagation speed.

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30

Köllner, Thomas, and Thomas Boeck. "Numerical simulation of solutal Rayleigh-Bénard-Marangoni convection in a layered two-phase system." PAMM 14, no. 1 (December 2014): 643–44. http://dx.doi.org/10.1002/pamm.201410306.

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31

Xu, B., X. Ai, and B. Q. Li. "Stabilities of combined radiation and Rayleigh–Bénard–Marangoni convection in an open vertical cylinder." International Journal of Heat and Mass Transfer 50, no. 15-16 (July 2007): 3035–46. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2007.01.007.

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32

Chen, Jun, Jixiang Wang, Zilong Deng, Xiangdong Liu, and Yongping Chen. "Experimental study on Rayleigh-Bénard-Marangoni convection characteristics in a droplet during mass transfer." International Journal of Heat and Mass Transfer 172 (June 2021): 121214. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2021.121214.

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33

Lyubimov, D. V., T. P. Lyubimova, N. I. Lobov, and J. I. D. Alexander. "Rayleigh–Bénard–Marangoni convection in a weakly non-Boussinesq fluid layer with a deformable surface." Physics of Fluids 30, no. 2 (February 2018): 024103. http://dx.doi.org/10.1063/1.5007117.

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34

Es Sakhy, Rachid, Kamal El Omari, Yves Le Guer, and Serge Blancher. "Rayleigh–Bénard–Marangoni convection in an open cylindrical container heated by a non-uniform flux." International Journal of Thermal Sciences 86 (December 2014): 198–209. http://dx.doi.org/10.1016/j.ijthermalsci.2014.06.036.

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35

Sun, Z. F., and M. Fahmy. "Onset of Rayleigh−Bénard−Marangoni Convection in Gas−Liquid Mass Transfer with Two-Phase Flow: Theory." Industrial & Engineering Chemistry Research 45, no. 9 (April 2006): 3293–302. http://dx.doi.org/10.1021/ie051185r.

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Hadji, Layachi. "Nonlinear analysis of the coupling between interface deflection and hexagonal patterns in Rayleigh-Bénard-Marangoni convection." Physical Review E 53, no. 6 (June 1, 1996): 5982–92. http://dx.doi.org/10.1103/physreve.53.5982.

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37

Baudey-Laubier, Louis-Henri, Benoît Trouette, and Eric Chénier. "Sensitivity of lateral heat transfer on the convection onset in a transient Rayleigh-Bénard-Marangoni flow." International Journal of Thermal Sciences 130 (August 2018): 353–66. http://dx.doi.org/10.1016/j.ijthermalsci.2018.04.014.

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38

Li, You-Rong, Lu Jia, Li Zhang, Yu-Peng Hu, and Jia-Jia Yu. "Direct numerical simulation of Rayleigh-Bénard-Marangoni convection of cold water near its density maximum in a cylindrical pool." International Journal of Thermal Sciences 124 (February 2018): 327–37. http://dx.doi.org/10.1016/j.ijthermalsci.2017.10.034.

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39

Plaksina, Yu Yu, A. V. Pushtaev, N. A. Vinnichenko, and A. V. Uvarov. "The Effects of Small Contaminants on the Formation of Structures during Rayleigh–Bénard–Marangoni Convection in a Planar Liquid Layer." Moscow University Physics Bulletin 73, no. 5 (September 2018): 513–19. http://dx.doi.org/10.3103/s0027134918050156.

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40

Samoilova, Anna E., and Alexander Nepomnyashchy. "Feedback control of Marangoni convection in a thin film heated from below." Journal of Fluid Mechanics 876 (August 1, 2019): 573–90. http://dx.doi.org/10.1017/jfm.2019.578.

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We use linear proportional control for the suppression of the Marangoni instability in a thin film heated from below. Our keen interest is focused on the recently revealed oscillatory mode caused by a coupling of two long-wave monotonic instabilities, the Pearson and deformational ones. Shklyaev et al. (Phys. Rev. E, vol. 85, 2012, 016328) showed that the oscillatory mode is critical in the case of a substrate of very low conductivity. To stabilize the no-motion state of the film, we apply two linear feedback control strategies based on the heat flux variation at the substrate. Strategy (I) uses the interfacial deflection from the mean position as the criterion of instability onset. Within strategy (II) the variable that describes the instability is the deviation of the measured temperatures from the desired, conductive values. We perform two types of calculations. The first one is the linear stability analysis of the nonlinear amplitude equations that are derived within the lubrication approximation. The second one is the linear stability analysis that is carried out within the Bénard–Marangoni problem for arbitrary wavelengths. Comparison of different control strategies reveals feedback control by the deviation of the free surface temperature as the most effective way to suppress the Marangoni instability.

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41

Sun, Z. F. "Onset of Rayleigh−Bénard−Marangoni Convection in Gas−Liquid Mass Transfer with Two-Phase Flow: Comparison of Measured Results with Theoretical Results." Industrial & Engineering Chemistry Research 45, no. 18 (August 2006): 6325–29. http://dx.doi.org/10.1021/ie060178f.

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42

Anonymous. "Erratum: Nonlinear analysis of the coupling between interface deflection and hexagonal patterns in Rayleigh-Bénard-Marangoni convection [Phys. Rev. E 53, 5982 (1996)]." Physical Review E 55, no. 3 (March 1, 1997): 3793. http://dx.doi.org/10.1103/physreve.55.3793.2.

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43

Karimfazli, I., I. A. Frigaard, and A. Wachs. "Thermal plumes in viscoplastic fluids: flow onset and development." Journal of Fluid Mechanics 787 (December 18, 2015): 474–507. http://dx.doi.org/10.1017/jfm.2015.639.

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The purely conductive state in configurations such as the Rayleigh–Bénard one is linearly stable for yield stress fluids at all Rayleigh numbers, $Ra$. However, on changing to localized heater configurations the static background state exists only if the yield stress is sufficiently large. Otherwise, thermal plumes may be induced in a stationary viscoplastic fluid layer, as illustrated in the recent experimental study of Davaille et al. (J. Non-Newtonian Fluid Mech., vol. 193, 2013, 144–153). Here, we study an analogous problem both analytically and computationally, from the perspective of an ideal yield stress fluid (Bingham fluid) that is initially stationary in a locally heated rectangular tank. We show that for a non-zero yield stress the onset of flow waits for a start time $t_{s}$ that increases with the dimensionless ratio of yield stress to buoyancy stress, denoted $B$. We provide a precise mathematical definition of $t_{s}$ and approximately evaluate this for different values of $B$, using both computational and semianalytical methods. For sufficiently large $B\geqslant B_{cr}$, the fluid is unable to yield. For the flow studied, $B_{cr}\approx 0.00307$. The critical value $B_{cr}$ and the start time $t_{s}$, for $B<B_{cr}$, are wholly independent of $Ra$ and $Pr$. For $B<B_{cr}$, yielding starts at $t=t_{s}$. The flow develops into either a weakly or a strongly convective flow. In the former case the passage to a steady state is relatively smooth and monotone, resulting eventually in a steady convective plume above the heater, rising and impinging on the upper wall, then recirculating steadily around the tank. With strongly convecting flows, for progressively larger $Ra$ we observe an increasing number of distinct plume heads and a tendency for plumes to develop as short-lived pulses. Over a certain range of $(Ra,B)$ the flow becomes temporarily frozen between two consecutive pulses. Such characteristics are distinctly reminiscent of the experimental work of Davaille et al. (J. Non-Newtonian Fluid Mech., vol. 193, 2013, 144–153). The yield stress plays a multifaceted role here as it affects plume temperature, size and velocity through different mechanisms. On the one hand, increasing $B$ tends to increase the maximum temperature of the plume heads. On the other hand, for larger $B\rightarrow B_{cr}$, the plume never starts.

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Chong, Kai Leong, and Ke-Qing Xia. "Exploring the severely confined regime in Rayleigh–Bénard convection." Journal of Fluid Mechanics 805 (September 23, 2016). http://dx.doi.org/10.1017/jfm.2016.578.

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We study the effect of severe geometrical confinement in Rayleigh–Bénard convection with a wide range of width-to-height aspect ratio $\unicode[STIX]{x1D6E4}$, $1/128\leqslant \unicode[STIX]{x1D6E4}\leqslant 1$, and Rayleigh number $Ra$, $3\times 10^{4}\leqslant Ra\leqslant 1\times 10^{11}$, at a fixed Prandtl number of $Pr=4.38$ by means of direct numerical simulations in Cartesian geometry with no-slip walls. For convection under geometrical confinement (decreasing $\unicode[STIX]{x1D6E4}$ from 1), three regimes can be recognized (Chong et al., Phys. Rev. Lett., vol. 115, 2015, 264503) based on the global and local properties in terms of heat transport, plume morphology and flow structures. These are Regime I: classical boundary-layer-controlled regime; Regime II: plume-controlled regime; and Regime III: severely confined regime. The study reveals that the transition into Regime III leads to totally different heat and momentum transport scalings and flow topology from the classical regime. The convective heat transfer scaling, in terms of the Nusselt number $Nu$, exhibits the scaling $Nu-1\sim Ra^{0.61}$ over three decades of $Ra$ at $\unicode[STIX]{x1D6E4}=1/128$, which contrasts sharply with the classical scaling $Nu-1\sim Ra^{0.31}$ found at $\unicode[STIX]{x1D6E4}=1$. The flow in Regime III is found to be dominated by finger-like, long-lived plume columns, again in sharp contrast with the mushroom-like, fragmented thermal plumes typically observed in the classical regime. Moreover, we identify a Rayleigh number for regime transition, $Ra^{\ast }=(29.37/\unicode[STIX]{x1D6E4})^{3.23}$, such that the scaling transition in $Nu$ and $Re$ can be clearly demonstrated when plotted against $Ra/Ra^{\ast }$.

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45

Krug, Dominik, Xiaojue Zhu, Daniel Chung, Ivan Marusic, Roberto Verzicco, and Detlef Lohse. "Transition to ultimate Rayleigh–Bénard turbulence revealed through extended self-similarity scaling analysis of the temperature structure functions." Journal of Fluid Mechanics 851 (July 30, 2018). http://dx.doi.org/10.1017/jfm.2018.561.

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In turbulent Rayleigh–Bénard (RB) convection, a transition to the so-called ultimate regime, in which the boundary layers (BL) are of turbulent type, has been postulated. Indeed, at very large Rayleigh number $Ra\approx 10^{13}{-}10^{14}$ a transition in the scaling of the global Nusselt number $Nu$ (the dimensionless heat transfer) and the Reynolds number with $Ra$ has been observed in experiments and very recently in direct numerical simulations (DNS) of two-dimensional (2D) RB convection. In this paper, we analyse the local scaling properties of the lateral temperature structure functions in the BLs of this simulation of 2D RB convection, employing extended self-similarity (ESS) (i.e., plotting the structure functions against each other, rather than only against the scale) in the spirit of the attached-eddy hypothesis, as we have recently introduced for velocity structure functions in wall turbulence (Krug et al., J. Fluid Mech., vol. 830, 2017, pp. 797–819). We find no ESS scaling at $Ra$ below the transition and in the near-wall region. However, beyond the transition and for large enough wall distance $z^{+}>100$, we find clear ESS behaviour, as expected for a scalar in a turbulent boundary layer. In striking correspondence to the $Nu$ scaling, the ESS scaling region is negligible at $Ra=10^{11}$ and well developed at $Ra=10^{14}$, thus providing strong evidence that the observed transition in the global Nusselt number at $Ra\approx 10^{13}$ indeed is the transition from a laminar type BL to a turbulent type BL. Our results further show that the relative slopes for scalar structure functions in the ESS scaling regime are the same as for their velocity counterparts, extending their previously established universality. The findings are confirmed by comparing to scalar structure functions in three-dimensional turbulent channel flow.

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Bouillaut, Vincent, Simon Lepot, Sébastien Aumaître, and Basile Gallet. "Transition to the ultimate regime in a radiatively driven convection experiment." Journal of Fluid Mechanics 861 (January 4, 2019). http://dx.doi.org/10.1017/jfm.2018.972.

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We report on the transition between two regimes of heat transport in a radiatively driven convection experiment, where a fluid gets heated up within a tunable heating length $\ell$ in the vicinity of the bottom of the tank. The first regime is similar to that observed in standard Rayleigh–Bénard experiments, the Nusselt number $Nu$ being related to the Rayleigh number $Ra$ through the power law $Nu\sim Ra^{1/3}$. The second regime corresponds to the ‘ultimate’ or mixing-length scaling regime of thermal convection, where $Nu$ varies as the square root of $Ra$. Evidence for these two scaling regimes has been reported in Lepot et al. (Proc. Natl Acad. Sci. USA, vol. 115, 2018, pp. 8937–8941), and we now study in detail how the system transitions from one to the other. We propose a simple model describing radiatively driven convection in the mixing-length regime. It leads to the scaling relation $Nu\sim (\ell /H)Pr^{1/2}Ra^{1/2}$, where $H$ is the height of the cell and $Pr$ is the Prandtl number, thereby allowing us to deduce the values of $Ra$ and $Nu$ at which the system transitions from one regime to the other. These predictions are confirmed by the experimental data gathered at various $Ra$ and $\ell$. We conclude by showing that boundary layer corrections can persistently modify the Prandtl number dependence of $Nu$ at large $Ra$, for $Pr\gtrsim 1$.

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He, Xiaozhou, Eberhard Bodenschatz, and Guenter Ahlers. "Azimuthal diffusion of the large-scale-circulation plane, and absence of significant non-Boussinesq effects, in turbulent convection near the ultimate-state transition." Journal of Fluid Mechanics 791 (February 17, 2016). http://dx.doi.org/10.1017/jfm.2016.56.

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We present measurements of the orientation ${\it\theta}_{0}$ and temperature amplitude ${\it\delta}$ of the large-scale circulation in a cylindrical sample of turbulent Rayleigh–Bénard convection (RBC) with aspect ratio ${\it\Gamma}\equiv D/L=1.00$ ($D$ and $L$ are the diameter and height respectively) and for the Prandtl number $Pr\simeq 0.8$. The results for ${\it\theta}_{0}$ revealed a preferred orientation with up-flow in the west, consistent with a broken azimuthal invariance due to the Earth’s Coriolis force (see Brown & Ahlers (Phys. Fluids, vol. 18, 2006, 125108)). They yielded the azimuthal diffusivity $D_{{\it\theta}}$ and a corresponding Reynolds number $Re_{{\it\theta}}$ for Rayleigh numbers over the range $2\times 10^{12}\lesssim Ra\lesssim 1.5\times 10^{14}$. In the classical state ($Ra\lesssim 2\times 10^{13}$) the results were consistent with the measurements by Brown & Ahlers (J. Fluid Mech., vol. 568, 2006, pp. 351–386) for $Ra\lesssim 10^{11}$ and $Pr=4.38$, which gave $Re_{{\it\theta}}\propto Ra^{0.28}$, and with the Prandtl-number dependence $Re_{{\it\theta}}\propto Pr^{-1.2}$ as found previously also for the velocity-fluctuation Reynolds number $Re_{V}$ (He et al., New J. Phys., vol. 17, 2015, 063028). At larger $Ra$ the data for $Re_{{\it\theta}}(Ra)$ revealed a transition to a new state, known as the ‘ultimate’ state, which was first seen in the Nusselt number $Nu(Ra)$ and in $Re_{V}(Ra)$ at $Ra_{1}^{\ast }\simeq 2\times 10^{13}$ and $Ra_{2}^{\ast }\simeq 8\times 10^{13}$. In the ultimate state we found $Re_{{\it\theta}}\propto Ra^{0.40\pm 0.03}$. Recently, Skrbek & Urban (J. Fluid Mech., vol. 785, 2015, pp. 270–282) claimed that non-Oberbeck–Boussinesq effects on the Nusselt and Reynolds numbers of turbulent RBC may have been interpreted erroneously as a transition to a new state. We demonstrate that their reasoning is incorrect and that the transition observed in the Göttingen experiments and discussed in the present paper is indeed to a new state of RBC referred to as ‘ultimate’.

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Fantuzzi, Giovanni, and Andrew Wynn. "Exact energy stability of Bénard–Marangoni convection at infinite Prandtl number." Journal of Fluid Mechanics 822 (June 1, 2017). http://dx.doi.org/10.1017/jfm.2017.323.

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Using the energy method we investigate the stability of pure conduction in Pearson’s model for Bénard–Marangoni convection in a layer of fluid at infinite Prandtl number. Upon extending the space of admissible perturbations to the conductive state, we find an exact solution to the energy stability variational problem for a range of thermal boundary conditions describing perfectly conducting, imperfectly conducting, and insulating boundaries. Our analysis extends and improves previous results, and shows that with the energy method global stability can be proven up to the linear instability threshold only when the top and bottom boundaries of the fluid layer are insulating. Contrary to the well-known Rayleigh–Bénard convection set-up, therefore, energy stability theory does not exclude the possibility of subcritical instabilities against finite-amplitude perturbations.

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Liu, Q. S., B. H. Zhou, and Z. M. Tang. "Oscillatory instability of Rayleigh-Marangoni-Bénard convection in two-layer liquid system." Journal of Non-Equilibrium Thermodynamics 30, no. 3 (January 19, 2005). http://dx.doi.org/10.1515/jnetdy.2005.022.

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Fahmy, Muthasim, and Zhifa Sun. "TRANSIENT RAYLEIGH-BÉNARD-MARANGONI CONVECTION ENHANCED GAS-LIQUID SOLUTE TRANSFER IN THIN LAYERS." Frontiers in Heat and Mass Transfer 2, no. 4 (January 19, 2012). http://dx.doi.org/10.5098/hmt.v2.4.3003.

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Journal articles on the topic 'Convection de Rayleigh- Bénard et Bénard-Marangoni'

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