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Register a new userPrandtl and Rayleigh number dependence of heat transport in high Rayleigh number thermal convection
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Results from direct numerical simulation for three-dimensional Rayleigh-Benard convection in samples of aspect ratio $Gamma=0.23$ and $Gamma=0.5$ up to Rayleigh number $Ra=2times10^{12}$ are presented. The broad range of Prandtl numbers $0.5<Pr<10$ is considered. In contrast to some experiments, we do not see any increase in $Nu/Ra^{1/3}$, neither due to $Pr$ number effects, nor due to a constant heat flux boundary condition at the bottom plate instead of constant temperature boundary conditions. Even at these very high $Ra$, both the thermal and kinetic boundary layer thicknesses obey Prandtl-Blasius scaling.
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Using direct numerical simulations, we study the statistical properties of reversals in two-dimensional Rayleigh-Benard convection for infinite Prandtl number. We find that the large-scale circulation reverses irregularly, with the waiting time between two consecutive genuine reversals exhibiting a Poisson distribution on long time scales, while the interval between successive crossings on short time scales shows a power law distribution. We observe that the vertical velocities near the sidewall and at the center show different statistical properties. The velocity near the sidewall shows a longer autocorrelation and $1/f^2$ power spectrum for a wide range of frequencies, compared to shorter autocorrelation and a narrower scaling range for the velocity at the center. The probability distribution of the velocity near the sidewall is bimodal, indicating a reversing velocity field. We also find that the dominant Fourier modes capture the dynamics at the sidewall and at the center very well. Moreover, we show a signature of weak intermittency in the fluctuations of velocity near the sidewall by computing temporal structure functions.
We study, using direct numerical simulations, the effect of geometrical confinement on heat transport and flow structure in Rayleigh-Benard convection in fluids with different Prandtl numbers. Our simulations span over two decades of Prandtl number $Pr$, $0.1 leq Pr leq 40$, with the Rayleigh number $Ra$ fixed at $10^8$. The width-to-height aspect ratio $Gamma$ spans between $0.025$ and $0.25$ while the length-to-height aspect ratio is fixed at one. We first find that for $Pr geq 0.5$, geometrical confinement can lead to a significant enhancement in heat transport as characterized by the Nusselt number $Nu$. For those cases, $Nu$ is maximal at a certain $Gamma = Gamma_{opt}$. It is found that $Gamma_{opt}$ exhibits a power-law relation with $Pr$ as $Gamma_{opt}=0.11Pr^{-0.06}$, and the maximal relative enhancement generally increases with $Pr$ over the explored parameter range. As opposed to the situation of $Pr geq 0.5$, confinement-induced enhancement in $Nu$ is not realized for smaller values of $Pr$, such as $0.1$ and $0.2$. The $Pr$ dependence of the heat transport enhancement can be understood in its relation to the coverage area of the thermal plumes over the thermal boundary layer (BL) where larger coverage is observed for larger $Pr$ due to a smaller thermal diffusivity. We further show that $Gamma_{opt}$ is closely related to the crossing of thermal and momentum BLs, and find that $Nu$ declines sharply when the thickness ratio of the thermal and momentum BLs exceeds a certain value of about one. In addition, through examining the temporally averaged flow fields and 2D mode decomposition, it is found that for smaller $Pr$ the large-scale circulation is robust against the geometrical confinement of the convection cell.
Steady flows that optimize heat transport are obtained for two-dimensional Rayleigh-Benard convection with no-slip horizontal walls for a variety of Prandtl numbers $Pr$ and Rayleigh number up to $Rasim 10^9$. Power law scalings of $Nusim Ra^{gamma}$ are observed with $gammaapprox 0.31$, where the Nusselt number $Nu$ is a non-dimensional measure of the vertical heat transport. Any dependence of the scaling exponent on $Pr$ is found to be extremely weak. On the other hand, the presence of two local maxima of $Nu$ with different horizontal wavenumbers at the same $Ra$ leads to the emergence of two different flow structures as candidates for optimizing the heat transport. For $Pr lesssim 7$, optimal transport is achieved at the smaller maximal wavenumber. In these fluids, the optimal structure is a plume of warm rising fluid which spawns left/right horizontal arms near the top of the channel, leading to downdrafts adjacent to the central updraft. For $Pr > 7$ at high-enough Ra, the optimal structure is a single updraft absent significant horizontal structure, and characterized by the larger maximal wavenumber.
Vertical convection is investigated using direct numerical simulations over a wide range of Rayleigh numbers $10^7le Rale10^{14}$ with fixed Prandtl number $Pr=10$, in a two-dimensional convection cell with unit aspect ratio. It is found that the dependence of the mean vertical centre temperature gradient $S$ on $Ra$ shows three different regimes: In regime I ($Ra lesssim 5times10^{10}$), $S$ is almost independent of $Ra$; In the newly identified regime II ($5times10^{10} lesssim Ra lesssim 10^{13}$), $S$ first increases with increasing $Ra$ (regime ${rm{II}}_a$), reaches its maximum and then decreases again (regime ${rm{II}}_b$); In regime III ($Ragtrsim10^{13}$), $S$ again becomes only weakly dependent on $Ra$, being slightly smaller than in regime I. The transitions between diffeereent regimes are discussd. In the three different regimes, significantly different flow organizations are identified: In regime I and regime ${rm{II}}_a$, the location of the maximal horizontal velocity is close to the top and bottom walls; However, in regime ${rm{II}}_b$ and regime III, banded zonal flow structures develop and the maximal horizontal velocity now is in the bulk region. The different flow organizations in the three regimes are also reflected in the scaling exponents in the effective power law scalings $Nusim Ra^beta$ and $Resim Ra^gamma$. In regime I, the fitted scaling exponents ($betaapprox0.26$ and $gammaapprox0.51$) are in excellent agreement with the theoretical predication of $beta=1/4$ and $gamma=1/2$ for laminar VC (Shishkina, {it{Phys. Rev. E.}} 2016, 93, 051102). However, in regimes II and III, $beta$ increases to a value close to 1/3 and $gamma$ decreases to a value close to 4/9. The stronger $Ra$ dependence of $Nu$ is related to the ejection of plumes and larger local heat flux at the walls.
We investigate the dependency of the magnitude of heat transfer in a convection cell as a function of its inclination by means of experiments and simulations. The study is performed with a working fluid of large Prandtl number, $Pr simeq 480$, and at Rayleigh numbers $Ra simeq 10^{8}$ and $Ra simeq 5 times 10^{8}$ in a quasi-two-dimensional rectangular cell with unit aspect ratio. By changing the inclination angle ($beta$) of the convection cell, the character of the flow can be changed from moderately turbulent, for $beta = 0^o$, to laminar and steady at $beta = 90^o$. The global heat transfer is found to be insensitive to the drastic reduction of turbulent intensity, with maximal relative variations of the order of $20%$ at $Ra simeq 10^{8}$ and $10%$ at $Ra simeq 5 times 10^{8}$, while the Reynolds number, based on the global root-mean- square velocity, is strongly affected with a decay of more than $85%$ occurring in the laminar regime. We show that the intensity of the heat flux in the turbulent regime can be only weakly enhanced by establishing a large scale circulation flow by means of small inclinations. On the other hand, in the laminar regime the heat is transported solely by a slow large scale circulation flow which exhibits large correlations between the velocity and temperature fields. For inclination angles close to the transition regime in-between the turbulent-like and laminar state, a quasi-periodic heat-flow bursting phenomenon is observed.
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