Turbulent convection in a horizontal layer of water

Chu, T. Y.; Goldstein, R. J.

doi:10.1017/s0022112073000091

Cited by 216 publications

(75 citation statements)

References 19 publications

(22 reference statements)

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“…However, in other experimental studies on convection in water, namely, [35] and [36], somewhat different laws Nu ∼ Ra 0.293 and ∼ Ra 0.278 were obtained.…”

Section: Moderate and Large Supercriticalitiesmentioning

confidence: 81%

Numerical simulation of two-dimensional convection: Role of boundary conditions

Палымский¹

2007

Fluid Dyn

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Abstract-The problem of two-dimensional, periodic in the horizontal coordinate, convection of an incompressible fluid heated from below between two horizontal planes is considered. The problem is solved in two formulations: with (stress-)free and hard (no-slip) boundary conditions on the horizontal planes. It is shown that at small supercriticalities the two-dimensional convection calculation leads to more correct results with hard than with free boundary conditions. It is established that the difference between the free and hard conditions is most strongly manifested in the pulsations of the vertical velocity component, whereas the dependence of the Nusselt number and the pulsations of the horizontal velocity component on the boundary conditions is more weakly expressed. DOI: 10.1134/S0015462807040059Keywords: convection, incompressible fluid, supercriticality, Nusselt number, Rayleigh number, turbulence.The convection problem has been solved in various formulations by many authors . We will reproduce the results obtained earlier, with special attention to experimental and theoretical investigations of convective flows at high supercriticality r = Ra/Ra cr , where Ra and Ra cr are the Rayleigh number and its critical value [1,2].Two formulations of the problem of convection in an infinite horizontal layer can be distinguished: with (stress-)free and hard (no-slip) horizontal boundaries. As a rule, the solution is assumed to be periodic in the horizontal directions. The two formulations often lead to solutions that differ only quantitatively, not qualitatively [1]. This fact and the relative simplicity of the solution of the problem of convection with free boundary conditions explain the popularity of this formulation.The idea of using only a few degrees of freedom on the basis of the Galerkin method was developed in [3][4][5][6] in numerical investigations and in the linear theory of stability [2,7]. This approach may be correct for solving problems of linear stability theory, but at high supercriticality taking an insufficient number of degrees of freedom into account in the calculations may lead to false chaotic solutions due to poor representation of the dissipative part of the spectrum [8].In [8][9][10] convection was calculated on the basis of a three-dimensional model with free and in [11-13] with hard boundary conditions. Using supercomputers made possible the direct numerical simulation of turbulent convection in air [9, 10, 13], but, unfortunately, these papers contain neither a spectral analysis of the numerical methods used nor a detailed comparison with the experimental data.Even nowadays, the complete calculation of three-dimensional unsteady flow at high supercriticality is a very difficult problem requiring enormous computational resources.In order drastically to reduce the computational resources used, fluid convection can be considered in a model two-dimensional formulation. In [14,15], it was shown that using the two-dimensional approximation is justified for convection generated by heating fr...

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“…However, in other experimental studies on convection in water, namely, [35] and [36], somewhat different laws Nu ∼ Ra 0.293 and ∼ Ra 0.278 were obtained.…”

Section: Moderate and Large Supercriticalitiesmentioning

confidence: 81%

Numerical simulation of two-dimensional convection: Role of boundary conditions

Палымский¹

2007

Fluid Dyn

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“…In his experiment the reversed gradient did not appear in the case of R= 3.81 Rc. Chu and Goldstein (1973) also found the reversal gradient. Our result is not inconsistent with the experimental result.…”

Section: Averaged Valuesmentioning

confidence: 82%

A Numerical Study of Three-Dimensional Benard Convection

Kitade¹

1974

Journal of the Meteorological Society of Japan

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A numerical study of three-dimensional Benard convection is carried out to determine the amplitude of convection as a function of Rayleigh number, Prandtl number and horizontal plan forms of convection cell. We treat three types of plan forms, namely, square, rectangle and roll, all of which have the same horizontal wavenumber. The calculations are carried out for air and water in the range of R*8Rc, where R is prescribed Rayleigh number and Rc is the critical Rayleigh number.The upper and lower boundaries are assumed to be free-slip and conducting. The variables such as velocity, temperature and pressure, are expanded in a series consisting of the eigenfunctions of the linear stability problem and the system is truncated to take into account only a limited number of terms. The amplitudes of the eigen-functions are evaluated by numerical integration of resulting non-linear equations.The main results are summerized as follows.(1) In all cases considered, the system achievs a steady state, a single mode being dominant.(2) The non-dimensional amplitudes of convection increase except perturbation temperature as R increases.(3) The velocity pattern depends on Prandtl number more markedly than the temperature pattern does in the three-dimensional convection. On the other hand, in the two-dimensional convection, the dependences of velocity and temperature patterns on Prandtl number are very small.(4) The dependence of vertical heat flux on horizontal plan forms appears more markedly for air than for water. In the case of air Nusselt number for the two-dimensional convection is larger than that for three-dimensional convection.(5) The reversed gradient of horizontally averaged temperature is found in the middle layer at high Rayleigh number. The gradient is larger for the two-dimensional convection than for the three-dimensional convection.(6) The energy budget of convection indicates that the most characteristic difference between threedimensional and two-dimensional convections is found in the conversion rate of kinetic energy through inertial term, that is, the conversion rate for two-dimensional convection is much smaller than that for three-dimensional convection.

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“…The KAERI's results on the relationship between the Nusselt number and Rayleigh number in the molten metal pool region was compared to other correlations, which were developed through experiments on the Rayleigh-Benard problem concerning natural convection in a horizontal fluid layer heated from below and cooled from above. Experimental results have been reported by Globe and Dropkin [29], Rossby [30], Threlfall [31], Heslot et al [32], and Chu and Goldstein [33], among others. The mercury data of Globe and Dropkin and of Rossby represent the only available heat transfer data for liquid metal, but these studies were performed without crust formation.…”

Section: Natural Convection With Crust Formationmentioning

confidence: 84%