scholarly journals Bounds for rotating Rayleigh–Bénard convection at large Prandtl number

2021 ◽  
Vol 930 ◽  
Author(s):  
A. Tilgner
Keyword(s):  
Momentum Equation ◽  
Total Momentum ◽  
Kinetic Energy Density ◽  
Maximal Velocity ◽  
The Maximum Principle ◽  
Bénard Convection ◽  
Benard Convection ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Bounds are derived for rotating Rayleigh–Bénard convection with free slip boundaries as a function of the Rayleigh, Taylor and Prandtl numbers ${\textit {Ra}}$ , ${\textit {Ta}}$ and ${\textit {Pr}}$ . At infinite ${\textit {Pr}}$ and ${\textit {Ta}} > 130$ , the Nusselt number ${\textit {Nu}}$ obeys ${\textit {Nu}} \leqslant \frac {7}{36} \left ({4}/{{\rm \pi} ^2} \right )^{1/3} {\textit {Ra}} {\textit {Ta}}^{-1/3}$ , whereas the kinetic energy density $E_{kin}$ obeys $E_{kin} \leqslant ({7}/{72 {\rm \pi}}) \left ({4}/{{\rm \pi} } \right )^{1/3} {\textit {Ra}}^2 {\textit {Ta}}^{-2/3}$ in the frame of reference in which the total momentum is zero, and $E_{kin} \leqslant ({1}/{2{\rm \pi} ^2})({{\textit {Ra}}^2}/{{\textit {Ta}}})({\textit {Nu}}-1)$ . These three bounds are derived from the momentum equation and the maximum principle for temperature and are extended to general ${\textit {Pr}}$ . The extension to finite ${\textit {Pr}}$ is based on the fact that the maximal velocity in rotating convection at infinite ${\textit {Pr}}$ is bound by $1.23 {\textit {Ra}} {\textit {Ta}}^{-1/3}$ .

scholarly journals Bulk scaling in wall-bounded and homogeneous vertical natural convection

2018 ◽  
Vol 841 ◽  
pp. 825-850 ◽  
Author(s):  
Chong Shen Ng ◽  
Andrew Ooi ◽  
Detlef Lohse ◽  
Daniel Chung
Keyword(s):  
Nusselt Number ◽  
Natural Convection ◽  
Temperature Gradient ◽  
Power Law ◽  
Exact Relation ◽  
Bénard Convection ◽  
Benard Convection ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Previous numerical studies on homogeneous Rayleigh–Bénard convection, which is Rayleigh–Bénard convection (RBC) without walls, and therefore without boundary layers, have revealed a scaling regime that is consistent with theoretical predictions of bulk-dominated thermal convection. In this so-called asymptotic regime, previous studies have predicted that the Nusselt number ($\mathit{Nu}$) and the Reynolds number ($\mathit{Re}$) vary with the Rayleigh number ($\mathit{Ra}$) according to $\mathit{Nu}\sim \mathit{Ra}^{1/2}$ and $\mathit{Re}\sim \mathit{Ra}^{1/2}$ at small Prandtl numbers ($\mathit{Pr}$). In this study, we consider a flow that is similar to RBC but with the direction of temperature gradient perpendicular to gravity instead of parallel to it; we refer to this configuration as vertical natural convection (VC). Since the direction of the temperature gradient is different in VC, there is no exact relation for the average kinetic dissipation rate, which makes it necessary to explore alternative definitions for $\mathit{Nu}$, $\mathit{Re}$ and $\mathit{Ra}$ and to find physical arguments for closure, rather than making use of the exact relation between $\mathit{Nu}$ and the dissipation rates as in RBC. Once we remove the walls from VC to obtain the homogeneous set-up, we find that the aforementioned $1/2$-power-law scaling is present, similar to the case of homogeneous RBC. When focusing on the bulk, we find that the Nusselt and Reynolds numbers in the bulk of VC too exhibit the $1/2$-power-law scaling. These results suggest that the $1/2$-power-law scaling may even be found at lower Rayleigh numbers if the appropriate quantities in the turbulent bulk flow are employed for the definitions of $\mathit{Ra}$, $\mathit{Re}$ and $\mathit{Nu}$. From a stability perspective, at low- to moderate-$\mathit{Ra}$, we find that the time evolution of the Nusselt number for homogenous vertical natural convection is unsteady, which is consistent with the nature of the elevator modes reported in previous studies on homogeneous RBC.

scholarly journals Vertical natural convection: application of the unifying theory of thermal convection

2015 ◽  
Vol 764 ◽  
pp. 349-361 ◽  
Author(s):  
Chong Shen Ng ◽  
Andrew Ooi ◽  
Detlef Lohse ◽  
Daniel Chung
Keyword(s):  
Boundary Layer ◽  
Natural Convection ◽  
Temperature Fields ◽  
Mean Flow ◽  
Bénard Convection ◽  
Benard Convection ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Gl Theory

AbstractResults from direct numerical simulations of vertical natural convection at Rayleigh numbers $1.0\times 10^{5}$–$1.0\times 10^{9}$ and Prandtl number $0.709$ support a generalised applicability of the Grossmann–Lohse (GL) theory, which was originally developed for horizontal natural (Rayleigh–Bénard) convection. In accordance with the GL theory, it is shown that the boundary-layer thicknesses of the velocity and temperature fields in vertical natural convection obey laminar-like Prandtl–Blasius–Pohlhausen scaling. Specifically, the normalised mean boundary-layer thicknesses scale with the $-1/2$-power of a wind-based Reynolds number, where the ‘wind’ of the GL theory is interpreted as the maximum mean velocity. Away from the walls, the dissipation of the turbulent fluctuations, which can be interpreted as the ‘bulk’ or ‘background’ dissipation of the GL theory, is found to obey the Kolmogorov–Obukhov–Corrsin scaling for fully developed turbulence. In contrast to Rayleigh–Bénard convection, the direction of gravity in vertical natural convection is parallel to the mean flow. The orientation of this flow presents an added challenge because there no longer exists an exact relation that links the normalised global dissipations to the Nusselt, Rayleigh and Prandtl numbers. Nevertheless, we show that the unclosed term, namely the global-averaged buoyancy flux that produces the kinetic energy, also exhibits both laminar and turbulent scaling behaviours, consistent with the GL theory. The present results suggest that, similar to Rayleigh–Bénard convection, a pure power-law relationship between the Nusselt, Rayleigh and Prandtl numbers is not the best description for vertical natural convection and existing empirical relationships should be recalibrated to better reflect the underlying physics.

THE 1:2 MODE INTERACTION IN RAYLEIGH–BÉNARD CONVECTION WITH WEAKLY BROKEN MIDPLANE SYMMETRY

2001 ◽  
Vol 11 (01) ◽  
pp. 27-41 ◽  
Author(s):  
ISABEL MERCADER ◽  
JOANA PRAT ◽  
EDGAR KNOBLOCH
Keyword(s):  
Steady State ◽  
Symmetry Breaking ◽  
Mode Interaction ◽  
Reflection Symmetry ◽  
Steady State Mode ◽  
Bénard Convection ◽  
Benard Convection ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

The effects of weak breaking of the midplane reflection symmetry on the 1:2 steady state mode interaction in Rayleigh–Bénard convection are discussed in a PDE setting. Effects of this type arise from the inclusion of non-Boussinesq terms or due to small differences in the boundary conditions at the top and bottom of the convecting layer. The latter provides the simplest realization, and captures all qualitative effects of such symmetry breaking. The analysis is performed for two Prandtl numbers, σ=10 and σ=0.1, representing behavior typical of large and low Prandtl numbers, respectively.

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