Laminar convection in uniformly heated vertical pipes

An exact solution is presented in this paper for the problem of laminar convective flow under a pressure gradient along a vertical pipe, the walls of which are heated or cooled uniformly; the solution is based on the assumption that velocity and buoyancy profiles far from the pipe entrance do not change with height, and entry-lengt effects are ignored. Two different types of behaviour are found accordingly as the pressure gradient and buoyancy forces act together or in opposition near the centre of the pipe.When an upflow is heated (or a downflow cooled) the velocity near the walls is increased relatively and that near the axis decreased until, for sufficiently large Rayleigh numbers, definite velocity and thermal boundary layers are formed.In the case of cooled upflow (or heated downflow) there is an increase in the velocity across the whole profile for small Rayleigh numbers. As the Rayleigh number is increased the velocity and buoyancy increase, slowly at first and then rapidly, and the solution ‘runs away’ at a Rayleigh number of about 33. For higher Rayleigh numbers, laminar Poiseuille flow of an increasingly complicated profile is theoretically possible, but is unlikely to be found in practise.

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Unsteady laminar convection in uniformly heated vertical pipes

Journal of Fluid Mechanics ◽

10.1017/s0022112073001035 ◽

1973 ◽

Vol 57 (1) ◽

pp. 81-102 ◽

Cited By ~ 4

Author(s):

R. K. Gupta

Keyword(s):

Rayleigh Number ◽

Pressure Gradient ◽

Temperature Fields ◽

Volume Flux ◽

Mathieu Functions ◽

Vertical Pipe ◽

Circular Tubes ◽

Time Required ◽

Laminar Convection ◽

Rayleigh Numbers

In this paper an exact solution is presented for the problem of unsteady laminar convective flow under a pressure gradient along a vertical pipe. We have obtained the solution of the problem on the basis of the assumption that the velocity and buoyancy profiles far from the pipe entrance do not change with the height, and the entry lengths have been ignored. The wall of the pipe is heated or cooled uniformly. We have discussed both the cases, when buoyancy forces act together with the pressure gradient or in opposite direction.In the case when the upflow is heated (or a downflow is cooled) the velocity and thermal boundary layers are formed for sufficiently large Rayleigh numbers. In the second case which has been discussed in detail (when the upflow is cooled or the downflow is heated) we have found the critical value of the Rayleigh number R = R, beyond which the velocity profile and the temperature profile become unsteady and turbulent in all the cases. In the case of the elliptical cylinder R, increases up to 1730 as the ellipticity is increased while in the case of the co-axial pipes this Rayleigh number increases as the gap c between the cylinders is decreased (if c = a/b = 1·2 then R, = 60762, but decreases to 1 when c = 4). Besides this, the time required to reach steady state increases as the Rayleigh number increases in both circular and elliptical pipes; it also increases when the eccentricity is decreased. The cases discussed by Morton (1960 and Dalip Singh (1965 are particular cases of the results derived below.In this investigation we have dealt with the following ducts: (i) circular tubes, (ii) elliptical tubes and (iii) co-axial tubes. The general solutions for both velocity and temperature fields have been found for the case when the pressure gradient is an arbitrary function of time, with an arbitrary heat source also present. Particular cases when both the parameters are absohte constants have been discussed in detail.We have made use of finite transforms very frequently; especially for the case of an elliptical tube, a new transform involving Mathieu functions developed by Gupta (1964 has been used. A few new infinite series have been summed with the help of this transform.Various non-dimensional quantities (for both the cylinders) such as the Nusselt number, volume flux and rate of heat transfer have been found when the pressure gradient and source of heat generation are absolute constants.

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

Volume 7: Fluid Flow, Heat Transfer and Thermal Systems, Parts A and B ◽

10.1115/imece2010-38167 ◽

2010 ◽

Cited By ~ 3

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.

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The Effect of Internal and External Heating on the Free Convective Flow of a Bingham Fluid in a Vertical Porous Channel

Fluids ◽

10.3390/fluids4020095 ◽

2019 ◽

Vol 4 (2) ◽

pp. 95 ◽

Cited By ~ 2

Author(s):

D. Andrew S. Rees ◽

Andrew P. Bassom

Keyword(s):

Convective Flow ◽

Internal Heat ◽

Bingham Fluid ◽

Relative Strength ◽

Porous Channel ◽

Free Convective Flow ◽

Bingham Number ◽

Different Types ◽

Yield Surfaces ◽

Rayleigh Numbers

We study the steady free convective flow of a Bingham fluid in a porous channel where heat is supplied by both differential heating of the sidewalls and by means of a uniform internal heat generation. The detailed temperature profile is governing by an external and an internal Darcy-Rayleigh number. The presence of the Bingham fluid is characterised by means of a body force threshold as given by the Rees-Bingham number. The resulting flow field may then exhibit between two and four yield surfaces depending on the balance of magnitudes of the three nondimensional parameters. Some indication is given of how the locations of the yield surfaces evolve with the relative strength of the Darcy-Rayleigh numbers and the Rees-Bingham number. Finally, parameter space is delimited into those regions within which the different types of flow and stagnation patterns arise.

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Laminar convection of a radiating gas in a vertical channel

Journal of Fluid Mechanics ◽

10.1017/s0022112071000673 ◽

1971 ◽

Vol 46 (3) ◽

pp. 513-520 ◽

Cited By ~ 39

Author(s):

R. Greif ◽

I. S. Habib ◽

J. C. Lin

Keyword(s):

Exact Solution ◽

Natural Convection ◽

Temperature Difference ◽

Convective Flow ◽

Vertical Channel ◽

Heat Fluxes ◽

Temperature Profiles ◽

Dimensionless Velocity ◽

Heated Channel ◽

Laminar Convection

An exact solution is obtained for the problem of fully-developed, radiating, laminar convective flow in a vertical heated channel. The effect of radiation is to decrease the temperature difference between the gas and the wall, thereby reducing the influence of natural convection. Thus, the reduction in velocity occurring in a heated upflow is less for a radiating gas. Graphs are presented for the dimensionless velocity and temperature profiles and for the volume and heat fluxes.

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Thermoconvective instabilities in a porous medium bounded by two concentric horizontal cylinders

Journal of Fluid Mechanics ◽

10.1017/s0022112076000669 ◽

1976 ◽

Vol 76 (2) ◽

pp. 337-362 ◽

Cited By ~ 94

Author(s):

Jean-Paul Caltagirone

Keyword(s):

Porous Medium ◽

Finite Elements ◽

Rayleigh Number ◽

Critical Rayleigh Number ◽

Three Dimensional ◽

Two Dimensional ◽

Dimensional Model ◽

Different Types ◽

Rayleigh Numbers ◽

Steady Convection

The study of natural convection in a saturated porous medium bounded by two concentric, horizontal, isothermal cylinders reveals different types of evolution according to the experimental conditions and the geometrical configuration of the model. At small Rayleigh numbers the state of the system corresponds to a regime of pseudo-conduction. The isotherms are coaxial with the cylinders. At larger Rayleigh numbers a regime of steady two-dimensional convection sets in between the two cylinders. Finally, for Rayleigh numbers above the critical Rayleigh number Ra*c the phenomena become three-dimensional and fluctuating. The appearance of these different regimes depends, moreover, on the geometry considered and, in particular, on two numbers: R, the ratio of the radii of the cylinders, and A, the ratio of the length of the cylinders to the radius of the inner one. In order to approach these experimental observations and to obtain realistic theoretical models, several methods of solving the equations have been used.The perturbation method yields information about the thermal field and the heat transfer between the cylinders under conditions close to the equilibrium state.A numerical two-dimensional model enables us to extend the range of investigation and to represent properly the phenomena when steady convection appreciably modifies the temperature distribution and the velocities within the porous layer.Neither of these models allows account to be taken of the instabilities observed experimentally above a critical Rayleigh number Ra*c. For this reason, a study of stability has been carried out using a Galerkin method based on equations corresponding to an initial state of steady convection. The results obtained show the importance of three-dimensional effects for the onset of fluctuating convection. The critical transition Rayleigh number Ra*c is thus determined in terms of the ratio of the radii R by solving an eigenvalue problem.A numerical three-dimensional model based on the method of finite elements has thus been developed in order to point out the different types of evolution with time. Steady two-dimensional convection and fluctuating three-dimensional convection have been actually found by calculation. The solution of the system of equations by the method of finite elements is briefly described.The experimental and theoretical results are then compared and a general physical interpretation is given.

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Transition from laminar convection to thermal turbulence in a rotating fluid layer

Journal of Fluid Mechanics ◽

10.1017/s0022112069001327 ◽

1969 ◽

Vol 35 (3) ◽

pp. 609-620 ◽

Cited By ~ 209

Author(s):

G. Küppers ◽

D. Lortz

Keyword(s):

Steady State ◽

Convective Flow ◽

Fluid Layer ◽

Stable Steady State ◽

State Equations ◽

Stability Calculation ◽

Thermal Turbulence ◽

Laminar Convection ◽

Rayleigh Numbers ◽

Horizontal Fluid Layer

The convective flow in an infinite horizontal fluid layer rotating rigidly about a normal axis is investigated for the special case of infinite Prandtl number and free boundary conditions. For slightly supercritical Rayleigh numbers the solutions of the non-linear steady-state equations are derived approximately by an amplitude expansion. A stability calculation shows that no stable steady-state convective flow exists if the Taylor number exceeds the critical value 2285.

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Heat transport by parallel-roll convection in a rectangular container

Journal of Fluid Mechanics ◽

10.1017/s0022112087003148 ◽

1987 ◽

Vol 185 ◽

pp. 205-234 ◽

Cited By ~ 9

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.

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Convection in an Enclosure-Source and Sink Located Along a Single Horizontal Boundary

Journal of Heat Transfer ◽

10.1115/1.3450782 ◽

1978 ◽

Vol 100 (2) ◽

pp. 205-211 ◽

Cited By ~ 4

Author(s):

L. A. Clomburg

Keyword(s):

Rayleigh Number ◽

Aspect Ratio ◽

Numerical Data ◽

Uniform Heat Flux ◽

Horizontal Boundary ◽

Order Of Magnitude ◽

High Prandtl Number ◽

Laminar Natural Convection ◽

Rayleigh Numbers ◽

Source And Sink

Laminar natural convection in a two-dimensional enclosure with both source (uniform heat flux density) and sink (temperature specified) located on the top horizontal boundary is investigated numerically. Temperature and velocity profiles are presented for a high Prandtl number fluid for length Rayleigh numbers in the range 107 to 109 for length to depth ratios of 1:1 to 4:1. To generalize the results, an order of magnitude analysis is used to determine the dependence of temperature, velocity, and boundary-layer thickness scales on aspect ratio and Rayleigh number. The numerical data are well correlated using these suggested scales. The analysis shows the Nusselt number and the maximum horizontal velocity to depend on the 1/6 and 1/3 powers of the Rayleigh number, independent of aspect ratio.

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