Effect of departures from the Oberbeck-Boussinesq approximation on the heat transport of horizontal convecting fluid layers

1980 ◽  
Vol 98 (1) ◽  
pp. 137-148 ◽  
Author(s):  
Guenter Ahlers
Keyword(s):  
Nusselt Number ◽  
Rayleigh Number ◽  
Critical Rayleigh Number ◽  
Boussinesq Approximation ◽  
Initial Slope ◽  
Nusselt Numbers ◽  
Fluid Layers ◽  
Different Temperatures ◽  
Rayleigh Numbers ◽  
Experimental Scatter

Measurements are presented of the Nusselt numbers N and Rayleigh numbers R for shallow layers of 4He gas heated from below. By choosing different temperatures between 2·3 K and 5·1 K and different pressures between 0·07 bar and 1 bar, the extent Q of departures from the Oberbeck-Boussinesq approximation was varied. When R was evaluated at the static temperature at the midplane of the cell, both the critical Rayleigh number Rc and the initial slope N1 of the Nusselt number were found to be independent of Q within experimental scatter. This result agrees with the prediction of Busse (1967). When R was evaluated at the cold end temperature of the cell, both Rc and N1 depended strongly upon Q.

Subcritical convective instability Part 1. Fluid layers

1966 ◽  
Vol 26 (4) ◽  
pp. 753-768 ◽  
Author(s):  
Daniel D. Joseph ◽  
C. C. Shir
Keyword(s):  
Nusselt Number ◽  
Rayleigh Number ◽  
Heat Source ◽  
Linear Theory ◽  
Convective Instability ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Heat Sources ◽  
Fluid Layers ◽  
Rayleigh Numbers

This paper elaborates on the assertion that energy methods provide an always mathematically rigorous and a sometimes physically precise theory of sub-critical convective instability. The general theory, without explicit solutions, is used to deduce that the critical Rayleigh number is a monotonically increasing function of the Nusselt number, that this increase is very slow if the Nusselt number is large, and that a fluid layer heated from below and internally is definitely stable when $RA < \widetilde{RA}(N_s) > 1708/(N_s + 1)$ where Ns is a heat source parameter and $\widetilde{RA}$ is a critical Rayleigh number. This last problem is also solved numerically and the result compared with linear theory. The critical Rayleigh numbers given by energy theory are slightly less than those given by linear theory, this difference increasing from zero with the magnitude of the heat-source intensity. To previous results proving the non-existence of subcritical instabilities in the absence of heat sources is appended this result giving a narrow band of Rayleigh numbers as possibilities for subcritical instabilities.

scholarly journals Effect of Rayleigh Number on Internal Eccentricity in a Heated Horizontal Elliptical Cylinder to its Coaxial Square Enclosure

2020 ◽  
Vol 25 (3) ◽  
pp. 17-29
Author(s):  
Abdelkrim Bouras ◽  
Djedid Taloub ◽  
Zied Driss
Keyword(s):  
Heat Transfer ◽  
Rayleigh Number ◽  
Outer Cylinder ◽  
Boussinesq Approximation ◽  
Square Enclosure ◽  
Nusselt Numbers ◽  
Flow And Heat Transfer ◽  
Commercial Code ◽  
Volume Method ◽  
Rayleigh Numbers

AbstractThis paper deals with numerical investigation of a natural convective flow in a horizontal annular space between a heated square inner cylinder and a cold elliptical outer cylinder with a Newtonian fluid. Uniform temperatures are imposed along walls of the enclosure. The governing equations of the problem were solved numerically by the commercial code Fluent, based on the finite volume method and the Boussinesq approximation. The effects of Geometry Ratio GR and Rayleigh numbers on fluid flow and heat transfer performance are investigated. The Rayleigh number is varied from 103 to 106. Throughout the study the relevant results are presented in terms of isotherms, and streamlines. From the results, we found that the increase in the Geometry Ratio B leads to an increase of the heat transfer coefficient. The heat transfer rate in the annulus is translated in terms of the average Nusselt numbers along the enclosure’s sides. Tecplot 7 program was used to plot the curves which cleared these relations and isotherms and streamlines which illustrate the behavior of air through the channel and its variation with other parameters. The results for the streamlines, isotherms, local and average Nusselt numbers average Nusselt numbers are compared with previous works and show good agreement.

scholarly journals The global characteristics of the three-dimensional thermal convection inside a spherical shell

1997 ◽  
Vol 4 (1) ◽  
pp. 19-27 ◽  
Author(s):  
J. Arkani-Hamed
Keyword(s):  
Boundary Layer ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Spherical Shell ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Horizontal Layer ◽  
Boundary Layer Theory ◽  
Physical Parameters ◽  
Rayleigh Numbers

Abstract. The Rayleigh number-Nusselt number, and the Rayleigh number-thermal boundary layer thickness relationships are determined for the three-dimensional convection in a spherical shell of constant physical parameters. Several models are considered with Rayleigh numbers ranging from 1.1 x 102 to 2.1 x 105 times the critical Rayleigh number. At lower Rayleigh numbers the Nusselt number of the three-dimensional convection is greater than that predicted from the boundary layer theory of a horizontal layer but agrees well with the results of an axisymmetric convection in a spherical shell. At high Rayleigh numbers of about 105 times the critical value, which are the characteristics of the mantle convection in terrestrial planets, the Nusselt number of the three-dimensional convection is in good agreement with that of the boundary layer theory. At even higher Rayleigh numbers, the Nusselt number of the three-dimensional convection becomes less than those obtained from the boundary layer theory. The thicknesses of the thermal boundary layers of the spherical shell are not identical, unlike those of the horizontal layer. The inner thermal boundary is thinner than the outer one, by about 30- 40%. Also, the temperature drop across the inner boundary layer is greater than that across the outer boundary layer.

Relation between Nusselt number and Rayleigh number in turbulent thermal convection

1976 ◽  
Vol 73 (3) ◽  
pp. 445-451 ◽  
Author(s):  
Robert R. Long
Keyword(s):  
Nusselt Number ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Thermal Convection ◽  
Similarity Theory ◽  
Unknown Constant ◽  
Velocity Defect ◽  
Fluid Layers ◽  
Rayleigh Numbers ◽  
Existing Data

A theory is developed for the dependence of the Nusselt number on the Rayleigh number in turbulent thermal convection in horizontal fluid layers. The theory is based on a number of assumptions regarding the behaviour in the molecular boundary layers and on the assumption of a buoyancy-defect law in the interior analogous to the velocity-defect law in flow in pipes and channels. The theory involves an unknown constant exponentsand two unknown functions of the Prandtl number. For eithers= ½ ors= 1/3, corresponding to two different theories of thermal convection, and for a given Prandtl number, constants can be chosen to give excellent agreement with existing data over nearly the whole explored range of Rayleigh numbers in the turbulent case. Unfortunately, comparisons with experiment do not permit a definite choice ofs, but consistency with the chosen form of the buoyancy-defect law seems to suggests= 1/3, corresponding to similarity theory.

scholarly journals NUMERICAL STUDIES OF NATURAL CONVECTION IN A SQUARE CAVITY

2006 ◽  
Vol 5 (1) ◽  
pp. 68
Author(s):  
Viviana Cocco Mariani ◽  
Ivan Moura Belo
Keyword(s):  
Boussinesq Approximation ◽  
Square Cavity ◽  
Flow Equations ◽  
Nusselt Numbers ◽  
Numerical Studies ◽  
Numeric Simulation ◽  
Different Temperatures ◽  
Convection Problem ◽  
Volume Method ◽  
Rayleigh Numbers

In the present work a numeric study of thermal and fluid dynamics behavior of natural air convection in a bi-dimensional square cavity is presented, in a laminar flow. The square cavity has two walls heated with different temperatures and two isolated walls, the Boussinesq approximation is used and a constant Prandtl number. The Finite Volume Method is used for the discretization of flow equations. The staggered load of variables is adopted and Power-Law and SIMPLE models are used. The numeric simulation is made up of several Rayleigh numbers, 104 Ra 106, and the results of average Nusselt numbers are compared to values obtained in the literature. Flow and isotherm lines are presented and analyzed. The numerical results presented here in this work agree with the ones available in the literature and can be used by researchers who work in the convection problem numeric simulation area.

scholarly journals Numerical Analysis of the Improving Thermal Energy Efficiency of Taylor-couette Flow

2021 ◽  
Vol 15 ◽  
pp. 236-247
Author(s):  
Khaoula Ben Abdelmlek ◽  
Fayçal Ben Nejma
Keyword(s):  
Energy Efficiency ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Numerical Study ◽  
Rotation Velocity ◽  
Radius Ratio ◽  
Nusselt Numbers ◽  
Cylindrical Annulus ◽  
The Mean ◽  
Rayleigh Numbers

This paper deals with adimensionnal analysis of natural convection in a horizontal cylindrical annulus. The inner cylinder is isothermally heated and rotates with an angular velocity Ω, however the outer one is kept cold and motionless. The gap between cylinders is defined by an adimensional radius ratio f. The numerical study was carried out using COMSOL Multiphysics. The effects of Rayleigh number ranging from 102 to 106, radius ratio and rotation velocity on the flow pattern and the thermal behavior in the annulus are then elaborated. Particular attention is paid to the effect of different parameters on the local Nusselt numbers on the inner and outer cylinders, the mean Nusselt number and the energy efficiency of the process. Results show that the mean Nusselt number increases with the increase of Rayleigh number. However, it decreases with the increase of the radius ratio f because of the narrowing of the annulus. The results prove also that the heat transfer rate drops with the rise of rotation velocity. Finally, it was found that the energy efficiency achieved its maximum for lower Rayleigh numbers Ra=103, and lower rotation velocities.

Free Convection Heat Transfer across Rectangular-Celled Diathermanous Honeycombs

1980 ◽  
Vol 102 (1) ◽  
pp. 75-80 ◽  
Author(s):  
D. R. Smart ◽  
K. G. T. Hollands ◽  
G. D. Raithby
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Critical Rayleigh Number ◽  
Convection Heat Transfer ◽  
Aspect Ratios ◽  
Cell Dimensions ◽  
Free Convection Heat ◽  
Rayleigh Numbers ◽  
Honeycomb Material

Experimentally obtained Nusselt number-Rayleigh number plots are presented for free convective heat transfer across inclined honeycomb panels filled with air. The honeycomb cells were rectangular in shape with very long cell dimensions across the slope and comparatively short dimensions up the slope. Elevation aspect ratios, AE, investigated were 3, 5 and 10; angles of inclination, θ, measured from the horizontal, were 0, 30, 60, 75 and 90 deg. The effect on the Nusselt number, of the emissivities of the plates bounding the honeycomb, and of the emissivity of honeycomb material, was also investigated. The measurements confirmed that the critical Rayleigh number and the post-critical heat transfer depend on the radiant properties of the honeycomb cells. The critical Rayleigh numbers at θ = 0 were well predicted by the methods of Sun and Edwards. For 0 < θ ≤ 75 deg, the critical Rayleigh numbers and the Nusselt-Rayleigh relations were both found to be essentially the same as their horizontal counterparts provided the Rayleigh number was first scaled by cos θ. For θ = 90 deg, the form of the Nusselt-Rayleigh relation was found to be very different from that for θ ≤ 75 deg and similar to that observed for square-celled honeycombs for θ ≥ 30 deg. The θ = 90 deg data were found to be closely correlated by an equation of the form recently proposed by Bejan and Tien.

Heat transport by parallel-roll convection in a rectangular container

1987 ◽  
Vol 185 ◽  
pp. 205-234 ◽  
Author(s):  
R. W. Walden ◽  
Paul Kolodner ◽  
A. Passner ◽  
C. M. Surko
Keyword(s):  
Rayleigh Number ◽  
Heat Transport ◽  
Critical Rayleigh Number ◽  
Equation Model ◽  
Onset Of Convection ◽  
Infinite Layer ◽  
Lateral Extent ◽  
Prandtl Numbers ◽  
Rayleigh Numbers ◽  
Good Agreement

Heat-transport measurements are reported for thermal convection in a rectangular box of aspect’ ratio 10 x 5. Results are presented for Rayleigh numbers up to 35Rc, Prandtl numbers between 2 and 20, and wavenumbers between 0.6 and 1.0kc, where Rc and kc are the critical Rayleigh number and wavenumber for the onset of convection in a layer of infinite lateral extent. The measurements are in good agreement with a phenomenological model which combines the calculations of Nusselt number, as a function of Rayleigh number and roll wavenumber for two-dimensional convection in an infinite layer, with a nonlinear amplitude-equation model developed to account for sidewell attenuation. The appearance of bimodal convection increases the heat transport above that expected for simple parallel-roll convection.

scholarly journals Numerical Analysis on Natural Convection Heat Transfer in a Single Circular Fin-Tube Heat Exchanger (Part 1): Numerical Method

Entropy ◽  
2020 ◽  
Vol 22 (3) ◽  
pp. 363 ◽  
Author(s):  
Jong Hwi Lee ◽  
Jong-Hyeon Shin ◽  
Se-Myong Chang ◽  
Taegee Min
Keyword(s):  
Natural Convection ◽  
Heat Exchanger ◽  
Rayleigh Number ◽  
Numerical Method ◽  
Stokes Equations ◽  
Convection Heat Transfer ◽  
Boussinesq Approximation ◽  
Tube Heat Exchanger ◽  
Fin Tube ◽  
Rayleigh Numbers

In this research, unsteady three-dimensional incompressible Navier–Stokes equations are solved to simulate experiments with the Boussinesq approximation and validate the proposed numerical model for the design of a circular fin-tube heat exchanger. Unsteady time marching is proposed for a time sweeping analysis of various Rayleigh numbers. The accuracy of the natural convection data of a single horizontal circular tube with the proposed numerical method can be guaranteed when the Rayleigh number based on the tube diameter exceeds 400, which is regarded as the limitation of numerical errors due to instability. Moreover, the effective limit for a circular fin-tube heat exchanger is reached when the Rayleigh number based on the fin gap size ( Ra s ) is equal to or exceeds 100. This is because at low Rayleigh numbers, the air gap between the fins is isolated and rarely affected by natural convection of the outer air, where the fluid provides heat resistance. Thus, the fin acts favorably when Ra s exceeds 100.

Effect of Thermal Instability on Thermally Developing Laminar Channel Flow

1976 ◽  
Vol 98 (1) ◽  
pp. 62-66 ◽  
Author(s):  
Y. Kamotani ◽  
S. Ostrach
Keyword(s):  
Rayleigh Number ◽  
Thermal Instability ◽  
Reynolds Numbers ◽  
Appreciable Change ◽  
Nusselt Numbers ◽  
Laminar Channel Flow ◽  
Predicted Values ◽  
Plate Channel ◽  
Rayleigh Numbers ◽  
Thermal Entrance

Results are reported of an experimental investigation in the thermal entrance region of a horizontal parallel-plate channel when the lower plate is heated and the upper one is cooled. The experiments covered a range of Rayleigh numbers between 103 and 3.1 × 104 and Reynolds numbers between 30 and 1100 using air. Measurements of Nusselt numbers and temperature distributions indicate much higher critical Rayleigh numbers than the theoretically predicted values. Beyond critical Rayleigh numbers second-type vortex rolls are predominant and the local Nusselt number increases gradually with the Rayleigh number. The thermal-entrance length as determined from temperature profiles does not show appreciable change with the Rayleigh number.

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