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.

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.

Finite amplitude cellular convection

1958 ◽  
Vol 4 (3) ◽  
pp. 225-260 ◽  
Author(s):  
W. V. R. Malkus ◽  
G. Veronis
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Heat Transport ◽  
Stability Problem ◽  
Dominant Role ◽  
Linear Equations ◽  
Finite Amplitude ◽  
Set Up ◽  
Rayleigh Numbers ◽  
Linear Stability Problem

When a layer of fluid is heated uniformly from below and cooled from above, a cellular regime of steady convection is set up at values of the Rayleigh number exceeding a critical value. A method is presented here to determine the form and amplitude of this convection. The non-linear equations describing the fields of motion and temperature are expanded in a sequence of inhomogeneous linear equations dependent upon the solutions of the linear stability problem. We find that there are an infinite number of steady-state finite amplitude solutions (having different horizontal plan-forms) which formally satisfy these equations. A criterion for ‘relative stability’ is deduced which selects as the realized solution that one which has the maximum mean-square temperature gradient. Particular conclusions are that for a large Prandtl number the amplitude of the convection is determined primarily by the distortion of the distribution of mean temperature and only secondarily by the self-distortion of the disturbance, and that when the Prandtl number is less than unity self-distortion plays the dominant role in amplitude determination. The initial heat transport due to convection depends linearly on the Rayleigh number; the heat transport at higher Rayleigh numbers departs only slightly from this linear dependence. Square horizontal plan-forms are preferred to hexagonal plan-forms in ordinary fluids with symmetric boundary conditions. The proposed finite amplitude method is applicable to any model of shear flow or convection with a soluble stability problem.

Towards numerical computation of double-diffusive natural convection within an eccentric horizontal cylindrical annulus

2016 ◽  
Vol 26 (5) ◽  
pp. 1346-1364 ◽  
Author(s):  
Chahinez Ghernoug ◽  
Mahfoud Djezzar ◽  
Hassane Naji ◽  
Abdelkarim Bouras
Keyword(s):  
Mass Transfer ◽  
Nusselt Number ◽  
Natural Convection ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Content Type ◽  
Cylindrical Annulus ◽  
Buoyancy Ratio ◽  
Convective Mode ◽  
Double Diffusive

Purpose – The purpose of this paper is to numerically study the double-diffusive natural convection within an eccentric horizontal cylindrical annulus filled with a Newtonian fluid. The annulus walls are maintained at uniform temperatures and concentrations so as to induce aiding thermal and mass buoyancy forces within the fluid. For that, this simulation span a moderate range of thermal Rayleigh number (100RaT100,000), Lewis (0.1Le10), buoyancy ratio (0N5) and Prandtl number (Pr=0.71) to examine their effects on flow motion and heat and mass transfers. Design/methodology/approach – A finite volume method in conjunction with the successive under-relaxation algorithm has been developed to solve the bipolar equations. These are written in dimensionless form in terms of vorticity, stream function, temperature and concentration. Beforehand, the implemented computer code has been validated through already published findings in the literature. The isotherms, streamlines and iso-concentrations are exhibited for various values of Rayleigh and Lewis numbers, and buoyancy ratio. In addition, heat and mass transfer rates in the annulus are translated in terms of Nusslet and Sherwood numbers along the enclosure’s sides. Findings – It is observed that, for the range of parameters considered here, the results show that the average Sherwood number increases with, while the average Nusselt number slightly dips as the Lewis number increases. It is also found that, under the convective mode, the local Nusselt number (or Sherwood) increases with the buoyancy ratio. Likewise, according to Lewis number’s value, the flow pattern is either symmetric and stable or asymmetric and random. Besides that, the heat transfer is transiting from a conductive mode to a convective mode with increasing the thermal Rayleigh number, and the flow structure and the rates of heat and mass transfer are significantly influenced by this parameter. Research limitations/implications – The range of the Rayleigh number considered here covers only the laminar case, with some constant parameters, namely the Prandtl number (Pr = 0.71), and the tilt angle (α=90°). The analysis here is only valid for steady, two-dimensional, laminar and aiding flow within an eccentric horizontal cylindrical annulus. This motivates further investigations involving other relevant parameters as N (opposite flows), Ra, Pr, Le, the eccentricity, the tilt angle, etc. Practical implications – An original framework for handling the double-diffusive natural convection within annuli is available, based on the bipolar equations. In addition, the achievement of this work could help researchers design thermal systems supported by annulus passages. Applications of the results can be of value in various arrangements such as storage of liquefied gases, electronic cable cooling systems, nuclear reactors, underground disposal of nuclear wastes, manifolds of solar energy collectors, etc. Originality/value – Given the geometry concerned, the bipolar coordinates have been used to set the inner and outer walls boundary conditions properly without interpolation. In addition, since studies on double-diffusive natural convection in annuli are lacking, the obtained results may be of interest to handle other configurations (e.g., elliptical-shaped speakers) with other boundary conditions.

Prandtl Number Dependence of Natural Convection in Porous Media

1987 ◽  
Vol 109 (2) ◽  
pp. 371-377 ◽  
Author(s):  
T. Jonsson ◽  
I. Catton
Keyword(s):  
Heat Transfer ◽  
Stainless Steel ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Horizontal Layer ◽  
Silicon Oil ◽  
Bed Height ◽  
Oil Water ◽  
Convection In Porous Media

The effect of Prandtl number of a medium on heat transfer across a horizontal layer was measured. Stainless steel particles of diameters 1.6, 3.2, and 4.8 mm, glass particles of diameters 2.5 and 6.00 mm, and lead particles of diameter 0.95 mm were used with silicon oil, water, and mercury as working fluids. The bed height was varied from 2.5 to 12 cm. Experimental results are presented showing Nusselt number as a function of medium Rayleigh number with the effective Prandtl number, defined as the product of medium Prandtl number and Kozeny–Carmen constant, serving as a parameter. Correlations for Nusselt number are given for effective Prandtl number less than 0.1 and for effective Prandtl number greater than 0.1, which corresponds to an infinite effective Prandtl number. For the steel–water case the wavenumber is shown as a function of medium Rayleigh number.

The effect of distant side-walls on the evolution and stability of finite-amplitude Rayleigh-Bénard convection

1981 ◽  
Vol 378 (1775) ◽  
pp. 539-566 ◽  
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Side Walls ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

A recent study by Cross et al . (1980) has described a class of finite-amplitude phase-winding solutions of the problem of two-dimensional Rayleigh-Bénard convection in a shallow fluid layer of aspect ratio 2 L (≫ 1) confined laterally by rigid side-walls. These solutions arise at Rayleigh numbers R = R 0 + O ( L -1 ) where R 0 is the critical Rayleigh number for the corresponding infinite layer. Nonlinear solutions of constant phase exist for Rayleigh numbers R = R 0 + O ( L -2 ) but of these only the two that bifurcate at the lowest value of R are stable to two-dimensional linearized disturbances in this range (Daniels 1978). In the present paper one set of the class of phase-winding solutions is found to be stable to two-dimensional disturbances. For certain values of the Prandtl number of the fluid and for stress-free horizontal boundaries the results predict that to preserve stability there must be a continual readjustment of the roll pattern as the Rayleigh number is raised, with a corresponding increase in wavelength proportional to R - R 0 . These solutions also exhibit hysteresis as the Rayleigh number is raised and lowered. For other values of the Prandtl number the number of rolls remains unchanged as the Rayleigh number is raised, and the wavelength remains close to its critical value. It is proposed that the complete evolution of the flow pattern from a static state must take place on a number of different time scales of which t = O(( R - R 0 ) -1 ) and t = O(( R - R 0 ) -2 ) are the most significant. When t = O(( R - R 0 ) -1 ) the amplitude of convection rises from zero to its steady-state value, but the final lateral positioning of the rolls is only completed on the much longer time scale t = O(( R - R 0 ) -2 ).

A Numerical Study of Laminar and Turbulent Natural Convective Heat Transfer From an Isothermal Vertical Plate With a Wavy Surface

2010 ◽  
Author(s):  
Patrick H. Oosthuizen
Keyword(s):  
Heat Transfer ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Turbulent Flow ◽  
Convective Heat Transfer ◽  
Transfer Rate ◽  
Surface Waviness ◽  
Dimensionless Amplitude ◽  
Buoyancy Forces ◽  
Rayleigh Numbers

Natural convective heat transfer from a wide isothermal plate which has a wavy surface, i.e., has a surface which periodically rises and falls, has been numerically studied. The main purpose of the study was to examine the effect of the surface waviness on the conditions under which transition from laminar to turbulent flow occurred and to study the effect of the surface waviness on the heat transfer rate. The surface waves, which have a saw-tooth cross-sectional shape, are normal to the direction of flow over the surface and have a relatively small amplitude. The range of Rayleigh numbers considered in the present study extends from values that for a non-wavy plate would be associated with laminar flow to values that would be associated with fully turbulent flow. The flow has been assumed to be steady and fluid properties have been assumed constant except for the density change with temperature that gives rise to the buoyancy forces, this being treated by means of the Boussinesq type approximation. A standard k-epsilon turbulence model with full account being taken of the effects of the buoyancy forces has been used in obtaining the solution. The solution has been obtained using the commercial CFD solver FLUENT. The solution has the following parameters: the Rayleigh number based on the plate height, the Prandtl number, the dimensionless amplitude of the surface waviness, and the dimensionless pitch of the surface waviness. Results have been obtained for a Prandtl number of 0.7 and for a single dimensionless pitch value for Rayleigh numbers between approximately 106 and 1012. The effects of Rayleigh number and dimensionless amplitude on the mean heat transfer rate have been studied. It is convenient in presenting the results to introduce two mean heat transfer rates, one based on the total surface area and the other based on the projected frontal area of the surface.

A Numerical Study of Natural Convection in a Cavity With Two Fins Attached to Its Vertical Walls

2007 ◽  
Author(s):  
G. A. Sheikhzadeh ◽  
M. Pirmohammadi ◽  
M. Ghassemi
Keyword(s):  
Nusselt Number ◽  
Natural Convection ◽  
Rayleigh Number ◽  
Numerical Study ◽  
Convection Heat Transfer ◽  
Square Cavity ◽  
Vertical Walls ◽  
The Mean ◽  
Energy Equations ◽  
Rayleigh Numbers

Numerical study natural convection heat transfer inside a differentially heated square cavity with adiabatic horizontal walls and vertical isothermal walls is investigated. Two perfectly conductive thin fins are attached to the isothermal walls. To solve the governing differential mass, momentum and energy equations a finite volume code based on Pantenkar’s simpler method is developed and utilized. The results are presented in form of streamlines, isotherms as well as Nusselt number for Rayleigh number ranging from 104 up to 107. It is shown that the mean Nusselt number is affected by the position of the fins and length of the fins as well as the Rayleigh number. It is also observed that maximum Nusselt number occurs about the middle of the enclosure where Lf is grater the 0.5. In addition the Nusselt number stays constant and does not varies with width of the cavity (lf) when Lf is equal to 0.5 and Rayleigh number is equal to 104 and 107 as well as when Lf is equal to 0.6 and low Rayleigh numbers.

scholarly journals Effect of Various Tilted Positions of a Thin Fin on Natural Convection of Laminar Viscous Flow in a Square Cavity

2021 ◽  
Vol 39 (5) ◽  
pp. 1634-1642
Author(s):  
Syed Fazuruddin ◽  
Seelam Sreekanth ◽  
G Sankara Sekhar Raju
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Fluid Flow ◽  
Natural Convection ◽  
Rayleigh Number ◽  
Finite Difference ◽  
Average Nusselt Number ◽  
Vertical Position ◽  
Inclination Angles ◽  
Rayleigh Numbers

An exhaustive numerical investigation is carried out to analyze the role of an isothermal heated thin fin on fluid flow and temperature distribution visualization in an enclosure. Natural convection within square enclosures finds remarkable pragmatic applications. In the present study, a finite difference approach is performed on two-dimensional laminar flow inside an enclosure with cold side walls and adiabatic horizontal walls. The fluid flow equations are reconstructed into vorticity - stream function formulation and these equations are employed utilizing the finite-difference strategy with incremental time steps. The parametric study includes a wide scope of Rayleigh number, Ra, and inclination angle ϴ of the thin fin. The effect of different Rayleigh numbers ranging Ra = 104-106 with Pr=0.71 for all the inclination angles from 0°-360° with uniform rotational length of angle 450 of an inclined heated fin on fluid flow and heat transfer have been investigated. The heat transfer rate within the enclosure is measured by means of local and average Nusselt numbers. Regardless of inclination angles of the thin fin, a slight enhancement in the average Nusselt number is observed when Rayleigh number increased for both the cases of the horizontal and vertical position of the thin fin. When the fin has inclined no change in average Nusselt number is noticed for distinct Rayleigh numbers.

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.

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