A transcendental approximation for natural convection at small prandtl numbers

1968 ◽  
Vol 46 (5) ◽  
pp. 321-324 ◽  
Author(s):  
D. A. Cygan ◽  
P. D. Richardson
Keyword(s):  
Natural Convection ◽  
Prandtl Numbers

scholarly journals A Multiple-Grid Lattice Boltzmann Method for Natural Convection under Low and High Prandtl Numbers

Fluids ◽  
2021 ◽  
Vol 6 (4) ◽  
pp. 148
Author(s):  
Seyed Amin Nabavizadeh ◽  
Himel Barua ◽  
Mohsen Eshraghi ◽  
Sergio D. Felicelli
Keyword(s):  
Natural Convection ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Large Scale ◽  
Bgk Model ◽  
Flow Equations ◽  
Wide Range ◽  
Prandtl Numbers ◽  
Boltzmann Method ◽  
Benchmark Solutions

A multi-distribution lattice Boltzmann Bhatnagar–Gross–Krook (BGK) model with a multiple-grid lattice Boltzmann (MGLB) model is proposed to efficiently simulate natural convection over a wide range of Prandtl numbers. In this method, different grid sizes and time steps for heat transfer and fluid flow equations are chosen. The model is validated against natural convection in a square cavity, since extensive benchmark solutions are available for that problem. The proposed method can resolve the computational difficulty in simulating problems with very different time scales, in particular, when using extremely low or high Prandtl numbers. The technique can also enhance computational speed and stability while keeping the simplicity of the BGK method. Compared with the conventional lattice Boltzmann method, the simulation time can be reduced up to one-tenth of the time while maintaining the accuracy in an acceptable range. The proposed model can be extended to other lattice Boltzmann collision models and three-dimensional cases, making it a great candidate for large-scale simulations.

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.

Numerical Study of Three-Dimensional Oscillatory Natural Convection at Low Prandtl Number in Rectangular Enclosures

2000 ◽  
Vol 123 (1) ◽  
pp. 77-83 ◽  
Author(s):  
Shunichi Wakitani
Keyword(s):  
Natural Convection ◽  
Prandtl Number ◽  
Numerical Study ◽  
Three Dimensional ◽  
Width Ratio ◽  
Three Dimensional Flow ◽  
Longitudinal Vortices ◽  
Aspect Ratios ◽  
Vertical Walls ◽  
Prandtl Numbers

Numerical investigations are presented for three-dimensional natural convection at low Prandtl numbers (Pr) from 0 to 0.027 in rectangular enclosures with differentially heated vertical walls. Computations are carried out for the enclosures with aspect ratios (length/height) 2 and 4, and width ratios (width/height) ranging from 0.5 to 4.2. Dependence of the onset of oscillation on the Prandtl number, the aspect ratio, and the width ratio is investigated. Furthermore, oscillatory, three-dimensional flow structure is clarified. The structure is characterized by some longitudinal vortices (rolls) as well as cellular pattern.

Natural convection in a corrugated slot

2017 ◽  
Vol 815 ◽  
pp. 537-569 ◽  
Author(s):  
Arman Abtahi ◽  
J. M. Floryan
Keyword(s):  
Natural Convection ◽  
Heat Flow ◽  
Convective Heat ◽  
Phase Difference ◽  
Conductive Heat ◽  
Heat Transfer Augmentation ◽  
Conductive Heat Flow ◽  
Secondary Motion ◽  
Prandtl Numbers

Analysis of natural convection in a horizontal slot formed by two corrugated isothermal plates has been carried out. The analysis is limited to subcritical Rayleigh numbers$Ra$where no secondary motion takes place in the absence of corrugations. The corrugations have a sinusoidal form characterized by the wavenumber, the upper and lower amplitudes and the phase difference. The most intense convection occurs for corrugation wavelengths comparable to the slot height; it increases proportionally to$Ra$and proportionally to the corrugation height. Placement of corrugations on both plates may either significantly increase or decrease the convection depending on the phase difference between the upper and lower corrugations, with the strongest convection found for corrugations being in phase, i.e. a ‘wavy’ slot, and the weakest for corrugations being out of phase, i.e. a ‘converging–diverging’ slot. It is shown that the shear forces would always contribute to the corrugation build-up if erosion was allowed, while the role of pressure forces depends on the location of the corrugations as well as on the corrugation height and wavenumber, and the Rayleigh number. Placing corrugations on both plates results in the formation of a moment which attempts to change the relative position of the plates. There are two limiting positions, i.e. the ‘wavy’ slot and the ‘converging–diverging’ slot, with the latter being unstable. The system would end up in the ‘wavy’ slot configuration if relative movement of the two plates was allowed. The presence of corrugations affects the conductive heat flow and creates a convective heat flow. The conductive heat flow increases with the corrugation height as well as with the corrugation wavenumber; it is largest for short-wavelength corrugations. The convective heat flow is relevant only for wavenumbers of$O(1)$, it increases proportionally to$Ra^{3}$and proportionally to the second power of the corrugation height. Convection is qualitatively similar for all Prandtl numbers$Pr$, with its intensity increasing for smaller$Pr$and with the heat transfer augmentation increasing for larger$Pr$.

A Numerical Study of Laminar Natural Convection in Shallow Cavities

1981 ◽  
Vol 103 (2) ◽  
pp. 226-231 ◽  
Author(s):  
G. S. Shiralkar ◽  
C. L. Tien
Keyword(s):  
Heat Transfer ◽  
Natural Convection ◽  
Prandtl Number ◽  
Liquid Metals ◽  
Numerical Study ◽  
Aspect Ratios ◽  
Shallow Cavities ◽  
Horizontal Cavity ◽  
Side Walls ◽  
Prandtl Numbers

Heat transfer by natural convection in a horizontal cavity with adiabatic horizontal walls and isothermal side walls is investigated numerically for high aspect ratios (width/height). Comparison is made with existing analytical and experimental results. Agreement is generally good at moderate and high Prandtl numbers to which most previous works have been restricted. Improvements of the existing correlation have been proposed in regions of discrepancy. Extension to the low Prandtl number case, including the range of liquid metals, has been made on the basis of an analytical model for high Rayleigh numbers as well as by numerical solution of the full equations. The agreement between the two is found to be very good. A correlation for the heat transfer is proposed for each of the two different cases of high and low Prandtl number.

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