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Register a new userBounds on heat transfer for Benard-Marangoni convection at infinite Prandtl number
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The vertical heat transfer in Benard-Marangoni convection of a fluid layer with infinite Prandtl number is studied by means of upper bounds on the Nusselt number $Nu$ as a function of the Marangoni number $Ma$. Using the background method for the temperature field, it has recently been proven by Hagstrom & Doering that $ Nuleq 0.838,Ma^{2/7}$. In this work we extend previous background method analysis to include balance parameters and derive a variational principle for the bound on $Nu$, expressed in terms of a scaled background field, that yields a better bound than Hagstrom & Doerings formulation at a given $Ma$. Using a piecewise-linear, monotonically decreasing profile we then show that $Nu leq 0.803,Ma^{2/7}$, lowering the previous prefactor by 4.2%. However, we also demonstrate that optimisation of the balance parameters does not affect the asymptotic scaling of the optimal bound achievable with Hagstrom & Doerings original formulation. We subsequently utilise convex optimisation to optimise the bound on $Nu$ over all admissible background fields, as well as over two smaller families of profiles constrained by monotonicity and convexity. The results show that $Nu leq O(Ma^{2/7}(ln Ma)^{-1/2})$ when the background field has a non-monotonic boundary layer near the surface, while a power-law bound with exponent 2/7 is optimal within the class of monotonic background fields. Further analysis of our upper-bounding principle reveals the role of non-monotonicity, and how it may be exploited in a rigorous mathematical argument.
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We prove a new rigorous upper bound on the vertical heat transport for Benard-Marangoni convection of a two- or three-dimensional fluid layer with infinite Prandtl number. Precisely, for Marangoni number $Ma gg 1$ the Nusselt number $Nu$ is bounded asymptotically by $Nu lesssim Ma^{2/7}(ln Ma)^{-1/7}$. Key to our proof are a background temperature field with a hyperbolic profile near the fluids surface, and new estimates for the coupling between temperature and vertical velocity.
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 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.
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.
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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