Function Of Rayleigh Number Research Articles

Experimental Investigation of Free Convection in a Vertical Rod Bundle—A General Correlation for Nusselt Numbers

Free convection in two vertical, enclosed rod bundles has been experimentally investigated for a wide range of Rayleigh numbers. A uniform power dissipation per unit length is supplied to each rod, and the enclosing outer cylinder is maintained at constant temperature. Nusselt numbers for each rod, as well as an overall value for each bundle, have been obtained as a function of Rayleigh number. Comparison of the results for air and water as the working fluid indicate that, for a fixed Rayleigh number, an increase in the Prandtl number produces a reduction in the Nusselt number. This is contrary to what has been reported for vertical cavities and is attributed to curvature effects. Furthermore, the data reveal the interesting fact that it is quite possible for the individual rods in the bundle to exchange energy with the working fluid via different but coexisting regimes at a given power dissipation. Also, as the Rayleigh number is increased, the rods each tend to assume nearly the same heat transfer coefficient. Finally, a correlation for the overall convective Nusselt number is developed in terms of Rayleigh number and geometric parameters.

Wave number selection in Rayleigh-Bénard convection: A numerical study

A phenomenological amplitude equation introduced earlier to calculate steady states of convection in a finite container, is studied numerically as a function of space and time. The effect of sidewalls on wave number selection is demonstrated by observing the response of the system to the sudden imposition of boundary conditions. The wave vectors of stable steady states are studied as a function of Rayleigh number near threshold, and it is shown that more than one state can be stable for fixed Rayleigh number.

Dynamics of defects in Rayleigh-Bénard convection

The behavior of an extra roll extending into an otherwise regular convection pattern is studied as a function of Rayleigh number, Prandtl number, $P$, and wavelength, by means of a fully resolved numerical simulation of the Boussinesq equations with free-slip boundary conditions. For reduced Rayleigh numbers of order one or less and $P\ensuremath{\gtrsim}40$, numerical simulations of the lowest-order amplitude equations reproduce the Boussinesq results semiquantitatively. In particular, we find that when this class of defects is stable, they move with constant velocity $v$, parallel to the roll axis and give rise to a slow modulation of the roll pattern of the form $f(x,y\ensuremath{-}vt)$. Both $f$ and $v$ have been calculated analytically within a linearized theory. The envelope function $f$ depends in an essential way on $v$ such that the limit $v\ensuremath{\rightarrow}0$ cannot be sensibly taken.

Combined Natural Convection and Radiation in a Volumetrically Heated Fluid Layer

The effect of radiation in combination with turbulent natural convection on the rates of heat transfer in volumetrically heated fluid layers characterized by high temperatures has been considered in this study. It is demonstrated that even at high Rayleigh numbers the radiation mode is as effective as the turbulent natural convection mode in removing the heat from the upper surface of the molten pools with adiabatic lower boundary. As a result of this improved heat transfer, it is shown that considerably thicker molten pools with internal heat generation can be supported without boiling inception. The total Nusselt number at a moderate but fixed value of conduction-radiation parameter, can be represented as a function of Rayleigh number in a simple power-law form. As a consequence of this relationship it is shown that maximum nonboiling pool thicknesses vary approximately inversely as the 0.9 power of internal heat generation rate. A comparison between exact analysis using the integral formulation of radiation flux and Rosseland approximation shows that the latter approximation bears out very adequately for optically thick pools with conduction-radiation parameter ≳ 0.4 inspite of the fact that individual components of Nusselt number due to radiation and convection, respectively, are grossly in error. These errors in component heat fluxes are compensating due to the total heat balance constraint. However, the comparison between Rosseland approximation and exact formulation gets poorer as the value of conduction-radiation parameter decreases. This increase in error is principally incurred due to the error in estimating wall temperature differences.

Thermal convection with spatially periodic boundary conditions: resonant wavelength excitation

Thermal convection in a fluid contained between two rigid walls with different mean temperatures is considered when either spatially periodic temperatures are prescribed at the walls or surface corrugations exist. The amplitudes of the spatial non-uniformities are assumed to be small, and the wavelength is set equal to the critical wavelength for the onset of Rayleigh-Bénard convection. For values of the mean Rayleigh number below the classical critical value, the mean Nusselt number and the mean flow are found as functions of Rayleigh number, Prandtl number, and modulation amplitude. For values of the Rayleigh number close to the classical critical value, the effects of the non-uniformities are greatly amplified, and the amplitude of convection is then governed by a cubic equation. This equation yields three supercritical states, but only the state linked to a subcritical state is found to be stable.

Natural Convection In Compound Parabolic Concentrators—A Finite-Element Solution

Natural convection heat transfer coefficients between absorber surfaces and cover plates for vertically oriented two-dimensional compound parabolic concentrators (CPC) are evaluated using finite-element techniques. Values of the critical Rayleigh number for different concentrations (2 < C < 10) with 1/3, 2/3, and full CPC heights are determined. Generalized charts for estimating the average absorber plate Nusselt number as a function of Rayleigh number and concentration for both full and truncated CPC cavities are given. The results are useful for evaluating the covective loss coefficients from such collectors.

3次元ベナー対流の数値的な研究

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.

Nonlinear penetrative convection

Convection in water above ice penetrates into the stably stratified region above the density maximum at 4 °C. Two-dimensional penetrative convection in a Boussinesq fluid confined between free boundaries has been studied in a series of numerical experiments. These included cases with a constant temperature at both boundaries as well as cases with a fixed average flux at the lower boundary. Steady convection occurs at Rayleigh numbers below the critical value predicted by linear theory. At high Rayleigh numbers, resonant coupling between convection and gravitational modes in the stable layer excites finite amplitude oscillations. The problem can be described by a simplified model which allows for distortion of the mean temperature profile and balances the convected and conducted flux. This model explains the finite amplitude instability and predicts the Nusselt number as a function of Rayleigh number. These predictions are in excellent agreement with the computed results.

An experimental study of natural convection and wall effect in liquid metals contained in vertical cylinders

Experimental results are presented giving Nusselt Number as a function of Rayleigh Number for liquid mercury and lead when submitted to an upward temperature gradient in a cylindrical vessel. The wall effects have been evaluated by using containers with different height to diameter ratios. For values of Ra ranging from 2.104 to 2.107, Nu may be determined by the relationship: (Nu±12%)=0.078 (0.68)L/D Ra1/3 the region in the neighborhood of Ra=104 is characterized by a wide dispersion of data points which has been attributed to the transition from laminar to turbulent regime. The extension of these results to a wide variety of metallurgical problems is discussed.