Natural convection in inclined partitioned enclosures

The problem of steady, laminar, natural convective flow of a viscous fluid in an inclined enclosure with partitions is considered. Transverse gradient of temperature is applied on the two opposing regular walls of the inclined enclosure while the other walls are maintained adiabatic. The problem is formulated in terms of the vorticity-stream function procedure. A numerical solution based on the finite volume method is obtained. Representative results illustrating the effects of the enclosure inclination angle and the degree of irregularity on the contour maps of the streamlines and temperature are reported and discussed. In addition, results for the average Nusselt number at the heated wall of the enclosure and the difference of extreme stream-function values are presented and discussed for various Rayleigh numbers, inclination angles and dimensionless partition heights.     

Conjugate conduction-natural convection in an enclosure with time-periodic sidewall temperature and inclination

The unsteady conjugate conduction-natural convection in enclosure is of great theoretical significance and is widely encountered in engineering applications in the areas of fluid dynamics and heat transfer. However, there are relatively few efforts to investigate the unsteady flow physics and heat transfer characteristics in the inclined enclosure of finite thickness walls. In the present work, this problem is numerically investigated by a high accuracy multidomain temporal-spatial pseudospectral method. The enclosure is filled with Boussinesq fluid and is bounded by four finite thickness and conductive walls; one of the vertical sidewall is exposed to time-periodic temperature environment while the opposite sidewall holds constant temperature; the top and bottom walls are assumed to be adiabatic. Particular efforts are focused on the effects of three types of influential factors: the wall thermophysical properties, the time-periodic temperature patterns and the inclination, and the time-periodic flow patterns and heat transfer characteristics are presented. Numerical results reveal that within the present parameter range, the heat transfer rate increases almost linearly with the thermal conductivity ratio and thermal diffusivity ratio but decreases with the inclination angle. Moreover, the heat transfer could be enhanced or weakened by selecting different temperature pulsating period in the case of finite thickness wall, while it is always enhanced if the walls are zero thickness. The back heat transfer and heat transfer resonance phenomena are observed, and their relationships with the time-periodic flow patterns and temperature distributions are analyzed. The findings are helpful to the understandings of the fluid flow and heat transfer mechanisms in the related enclosure configurations, and may be of engineering use in thermal design improvement.     

An experimental study of the effect of wall temperature nonuniformity on natural convection in an enclosure heated from the side

An experimental study of natural convection in a parallelepipedal enclosure induced by a single vertical wall is described. The upper half of this wall was warm and the lower half cold. The other enclosure walls were insulated. The temperature and flow measurements were performed in the high Rayleigh number regime (10¹⁰<Ra<5×10¹⁰) by using water as the working fluid. The Rayleigh number was based on the enclosure height and the temperature difference between the warm and the cold part of the driving wall. The flow field featured two flat cells, one filled with warm fluid along the top horizontal wall, and the other filled with cold fluid along the bottom horizontal wall. Each of these cells was surrounded by an additional cell as tall as half the enclosure height. The above flow structure prohibited extensive thermal contact between warm and cold fluid, thus limiting the role of convection on the heat transfer process in the cavity. The findings of this study differ significantly from the findings of previous studies based on the ‘classical’ enclosure model possessing two isothermal vertical walls, the one warm and the other cold, and support the view that the use of ‘more realistic‘ temperature boundary conditions in enclosure natural convection needs careful examination.     

Coherent motion of turbulent thermal plume in stably stratified fluid

The purpose of this study is to clarify the existence of an ordered and large scale coherent motion in a turbulent plane thermal plume in a thermally-stable stratified fluid inside a comparatively large enclosure. First, the upper part of the thermal plume was carefully observed by a flow visualization. Secondly, a wave form of plume temperature variation was measured. Thirdly, a spectrum analysis was carried out on time series data of the thermal plume. Finally, physical characteristics were investigated on vortices in the thermal plume based on results of the wave form and the spectrum analysis of the plume temperature. As a result, the main conclusions are obtained as follows. (1) An existence of vortices near the upper part of the thermal plume was firstly found by careful flow visualization. (2) From the wave form of temperature variation and the spectrum analysis of the thermal plume, it was clarified that the vortices are generated in the transition state and are transported to the turbulent state. (3) The vortices are ordered and they behave as a large scale coherent motion in the turbulent thermal plume.     

Vanishing Viscosity Method for Transonic Flow

A vanishing viscosity method is formulated for two-dimensional transonic steady irrotational compressible fluid flows with adiabatic constant . This formulation allows a family of invariant regions in the phase plane for the corresponding viscous problem, which implies an upper bound uniformly away from cavitation for the viscous approximate velocity fields. Mathematical entropy pairs are constructed through the Loewner–Morawetz relation via entropy generators governed by a generalized Tricomi equation of mixed elliptic–hyperbolic type, and the corresponding entropy dissipation measures are analyzed so that the viscous approximate solutions satisfy the compensated compactness framework. Then the method of compensated compactness is applied to show that a sequence of solutions to the artificial viscous problem, staying uniformly away from stagnation with uniformly bounded velocity angles, converges to an entropy solution of the inviscid transonic flow problem. Dedicated to Constantine M. Dafermos on the Occasion of His 65th Birthday     

Work optimization in continuous and discrete systems with complex fluids

We extend to the realm of complex fluids and finite durations the classical problem of extremum work delivered from (or consumed by) the nonequilibrium system composed of a complex fluid, a perfect thermal machine and the environment or an infinite reservoir. The fluid constitutes a valuable resource of a finite flow or amount “a finite resource”, and work production (consumption) takes place sequentially, in stages of “endoreversible” thermal machines. At each stage, heat and mass transfer takes place in boundary layers which play the role of resistances in the system model. For the fluid at flow, total specific work is extremized at constraints which take into account dynamics of heat and mass transport and rate of work generation. Finite rate limits are obtained for the work production and consumption, which provide stronger bounds that those predicted by classical thermodynamics. Optimal work functions, which incorporate an inevitable minimum of the entropy production are found as functions of end states, duration and (in discrete processes) number of stages. Formal analogies between the entropy production expressions for work-assisted and conventional mass transfer operations help formulate optimization models.     

Numerical simulation of natural convective heat transfer and fluid flow around a heated cylinder inside an enclosure

 A collocated, non-orthogonal grid based finite volume technique has been applied for investigating the two dimensional natural convective flow and heat transfer around a heated cylinder kept in a square enclosure. The effects of different enclosure wall thermal boundary conditions, fluid Prandtl number and the ratio between enclosure and cylinder dimensions (aspect ratio) upon the flow and thermal features, have been systematically studied. It is observed that the patterns of recirculatory flow and thermal stratification in the fluid are significantly modified, if any of these parameters is varied. The overall heat transfer rates are also affected due to the changes in the flow and temperature patterns. The study presents useful observations regarding the variation of local Nusselt number along each wall, for the different cases considered. Received on 2 August 2000 / Published online: 29 November 2001     

Convection of a heat-generating fluid in a rotating horizontal cylinder

Thermal convection of a fluid in a horizontal cylinder rotating about its own axis with uniformly volume-distributed internal heat sources is experimentally investigated. The enclosure boundary temperature was kept constant. The threshold of the excitation of convective flows and their structure are studied as functions of the heat-release intensity and the rotation velocity. The experiments are performed with water and water-glycerin solutions. It is shown that rapidly rotating fluid is in a stable quasiequilibrium state, namely, the temperature distribution is axisymmetric and has a maximum at the center of the enclosure. It is found that with decrease in the rotation velocity a convective flow arises thresholdwise, in the form of vortex cells periodically arranged along the axis. The thermal convection in the rotating enclosure is shown to be determined by the effects of two different mechanisms. One of these is due to the centrifugal force of inertia and plays the stabilizing role, while the other, thermovibrational mechanism is connected with nonisothermal fluid oscillations under the action of gravity in the enclosure-fitted reference frame and is responsible for the occurrence of mean thermal convection. The boundaries of the convection generation are plotted in the plane of the governing dimensionless parameters and the heat transfer in the supercritical region is studied.     

Natural Convection in a Square Porous Cavity with Sinusoidal Temperature Distributions on Both Side Walls Filled with a Nanofluid: Buongiorno’s Mathematical Model

Natural convection in a two-dimensional, square porous cavity filled with a nanofluid and with sinusoidal temperature distributions on both side walls and adiabatic conditions on the upper and lower walls is numerically investigated. The flow is assumed to be slow so that advective and Forchheimer quadratic terms are ignored in the momentum equation. The applied sinusoidal temperature is symmetric with respect to the midplane of the enclosure. Numerical calculations are produced for Rayleigh numbers in the range of 10– \(10^{4}\) in comparison with other authors. The present models, in the form of an in-house computational fluid dynamics code, have been validated successfully against the reported results from the open literature. It is found that the results are in very good agreement. Results are presented in the form of streamlines, isotherm contours, and distributions of the average Nusselt number.     

Electrohydrodynamic peristaltic flow of a dielectric Oldroydian viscoelastic fluid in a flexible channel with heat transfer

This paper discusses the effects of a vertical a.c. electric field and heat transfer on a peristaltic flow of an incompressible dielectric viscoelastic fluid in a symmetric flexible channel. The mathematical modeling includes interactions among the electric field, flow field, and temperature. The perturbation solution of the modeled problem is derived by considering a small wave number. The influence of pertinent parameters is demonstrated and discussed. The numerical results show that the possibility of flow reversal increases near the lower bound of the channel and decreases near the upper bound of the channel as the electrical Rayleigh number, the Reynolds number, and the Weissenberg number increase, whereas the opposite effect is observed as the temperature parameter and the Weissenberg number increase. It is observed that the size of the trapped bolus decreases at the upper bound of the channel and increases at the lower bound of the channel with increasing electrical Rayleigh number, whereas the opposite effect is observed as the temperature parameter increases. The results also show that the trapped bolus in the case of an Oldroydian fluid is smaller than that for a Newtonian fluid.     

An analytical study of nonlinear double-diffusive convection in a porous medium under temperature/gravity modulation

The article deals with nonlinear thermal instability problem of double-diffusive convection in a porous medium subjected to temperature/gravity modulation. Three types of imposed time-periodic boundary temperature (ITBT) are considered. The effect of imposed time-periodic gravity modulation (ITGM) is also studied in this problem. In the case of ITBT, the temperature gradient between the walls of the fluid layer consists of a steady part and a time-dependent periodic part. The temperature of both walls is modulated in this case. In the problem involving ITGM, the gravity field has two parts: a constant part and an externally imposed time-periodic part. Using power series expansion in terms of the amplitude of modulation, which is assumed to be small, the problem has been studied using the Ginzburg–Landau amplitude equation. The individual effects of temperature and gravity modulation on heat and mass transports have been investigated in terms of Nusselt number and Sherwood number, respectively. Further the effects of various parameters on heat and mass transports have been analyzed and depicted graphically.     

Forced convection — axial conduction between parallel walls with unequal heat fluxes

Laminar flow heat transfer is computed for a situation in which a fluid moves along a parallel plate channel with unequal wall heat fluxes (one wall is insulated). The fluid enters the heating section through an upstream region which is perfectly insulated. This situation serves to describe an upper bound for the commonly encountered case of double pipe heat exchangers with identical thermal conditions. A control volume approach has been employed for the numerical work enabling a fast calculation for the thermally developing regime in the parallel plate channel. The merits of the adopted procedure are assessed by comparison with other results available in the literature for the one-region and for the two-region problem.     

Effect of variable thermal conductivity models on the combined convection heat transfer in a square enclosure filled with a water–alumina nanofluid

In this numerical study, the effects of variable thermal conductivity models on the combined convection heat transfer in a two-dimensional lid-driven square enclosure are investigated. The fluid in the square enclosure is a water-based nanofluid containing alumina nanoparticles. The top and bottom horizontal walls are insulated, while the vertical walls are kept at different constant temperatures. Five different thermal conductivity models are used to evaluate the effects of various parameters, such as the nanofluid bulk temperature, nanoparticle size, nanoparticle volume fraction, Brownian motion, interfacial layer thickness, etc. The governing stream–vorticity equations are solved by using a second-order central finite difference scheme coupled with the conservation of mass and energy. It is found that higher heat transfer is predicted when the effects of the nanoparticle size and bulk temperature of the nanofluid are taken into account.     

Lane-Emden equations of second kind modelling thermal explosion in infinite cylinder and sphere

We study a modified version of the Lane-Emden equation of the second kind modelling a thermal explosion in an infinite cylinder and a sphere. We first show that the solution to the relevant boundary value problem is bounded and that the solutions are monotone decreasing. The upper bound, the value of the solution at zero, can be approximated analytically in terms of the physical parameters. We obtain solutions to the boundary value problem, using both the Taylor series （which work well for weak nonlinearity） and the b-expansion method （valid for strong nonlinearity）. From here, we are able to deduce the qualitative behavior of the solution profiles with a change in any one of the physical parameters.     

Conjugate Natural Convection in a Porous Enclosure with Non-Uniform Heat Generation

Effects of a conductive wall on natural convection in a square porous enclosure having internal heating at a rate proportional to a power of temperature difference is studied numerically in this article. The horizontal heating is considered, where the vertical walls heated isothermally at different temperatures while the horizontal walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and finite difference method is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (0 ???Ra ???1000), the internal heating and the local exponent parameters (0 ????? ???5), (1 ????? ???3), the wall to porous thermal conductivity ratio (0.44 ???Kr ???9.9) and the ratio of wall thickness to its width (0.02 ???D ???0.5). The results are presented to show the effect of these parameters on the fluid flow and heat transfer characteristics. It is found a strong internal heating can generate significant maximum fluid temperature more than the conductive solid wall. Increasing value thermal conductivity ratio and/or decreasing the thickness of solid wall can increase the maximum fluid temperature. It is also found that at very low Rayleigh number, the heat transfer across the porous enclosure remain stable for any values of the thermal conductivity ratio.     

Computational and experimental studies of convective fluid motion and heat transfer in inclined non-rectangular enclosures

Computational and experimental studies of the fluid motion and heat transfer characteristics of an incompressible fluid contained in a non-rectangular inclined enclosure are described in this paper. The enclosure has two 45° inclined side walls one of which was heated and the other cooled. The remaining two sides of the enclosure are parallel and insulated. The enclosure was rotated about the long axis in steps of 30° through 360°. Experiments were performed to study the effects of Rayleigh number, aspect ratios and orientation of the enclosure. The computational method uses a mesh transformation technique coupled with the introduction of ‘false transient’ parameters for the steady state solution of the problem. The experimental method uses smoke for flow visualization studies. With aspect ratios of 3 and 6, the results indicate that the heat transfer and fluid motion within the enclosure is a strong function of both the Rayleigh number and the cavity orientation angle. A minimum and a maximum mean Nusselt number occurred as the angle of inclination was increased from 0 to 360°. A transition in the mode of circulation occurred at the angle corresponding to the minimum or maximum rate of heat transfer. Stream lines and isotherms are presented for the most representative cases     

Transient heat flux measurement of natural convection in an inclined enclosure with time-periodically-varying wall temperature

The transient natural convection in an inclined enclosure filled with water is studied experimentally for the time-periodically-varying wall temperature on one side wall and constant average temperature on the opposing side wall. This system has no temperature difference between the opposing two side walls in time-averaged sense. The temperatures of two opposing walls and the heat flux across the enclosure are measured by a heat flux meter. Based on the experimental results, the effects of time-periodically-varying wall temperature and inclined angles of the enclosure on heat transfer characteristics are studied. The experimental results show that, with the upper wall temperature oscillating, the heat flux across the enclosure is also periodically varied with time, and the net heat flux is from the lower wall to the upper wall. Numerical computations are also conducted and numerical results are qualitatively assured by the experimental measurements.     

Natural convection of SiO_2-water nanofluid in square cavity with thermal square column

A square with a thermal square column is a simple but nontrivial research prototype for nanofluid research. However, until now, the effects of the temperature of the square column on the heat and mass transfer of nanofluids have not been revealed comprehensively, especially on entropy generation. To deepen insight into this important field, the natural convection of the SiO_2-water nanofluid in a square cavity with a square thermal column is studied numerically in this study. The effects of the thermal column temperature(T = 0.0, 0.5, 1.0, 1.5), the Rayleigh number(ranging from 10~3 to 10~6),and the volume fraction of the nanoparticle(varying from 0.01 to 0.04) on the fluid flow,heat transfer, and entropy generation are investigated, respectively. It is found that, no matter at a low or high Rayleigh number, the volume fraction of the nanoparticle shows no considerable effects on the flow field and temperature field for all the temperatures of the thermal column. With an increase in the volume fraction, the mean Nusselt number increases slightly. At the same time, it is found that, with an increase in the temperature of the thermal column, the average Nusselt number gradually decreases at all values of the Rayleigh number. Meanwhile, it is found that, at a high Rayleigh number, the heat transfer mechanism is the main parameter affecting the increase in the total entropy generation rather than the volume fraction. In addition, no matter at a high or low Rayleigh number, when T = 0.5, the total entropy generation is the minimum.     

Effect of Conduction in Bottom Wall on Darcy–Bénard Convection in a Porous Enclosure

Conjugate natural convection-conduction heat transfer in a square porous enclosure with a finite-wall thickness is studied numerically in this article. The bottom wall is heated and the upper wall is cooled while the verticals walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and the COMSOL Multiphysics software is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (100 ≤ Ra ≤ 1000), the wall to porous thermal conductivity ratio (0.44 ≤ K _r ≤ 9.90) and the ratio of wall thickness to its height (0.02 ≤ D ≤ 0.4). The results are presented to show the effect of these parameters on the heat transfer and fluid flow characteristics. It is found that the number of contrarotative cells and the strength circulation of each cell can be controlled by the thickness of the bottom wall, the thermal conductivity ratio and the Rayleigh number. It is also observed that increasing either the Rayleigh number or the thermal conductivity ratio or both, and decreasing the thickness of the bounded wall can increase the average Nusselt number for the porous enclosure.     

Thermomechanical vibration analysis of a functionally graded shell with flowing fluid

This paper reports the results of an investigation into the vibration of functionally graded cylindrical shells with flowing fluid, embedded in an elastic medium, under mechanical and thermal loads. By considering rotary inertia, the first-order shear deformation theory (FSDT) and the fluid velocity potential, the dynamic equation of functionally graded cylindrical shells with flowing fluid is derived. Here, heat conduction equation along the thickness of the shell is applied to determine the temperature distribution and material properties are assumed to be graded distribution along the thickness direction according to a power-law in terms of the volume fractions of the constituents. The equations of eigenvalue problem are obtained by using a modal expansion method. In numerical examples, effects of material composition, thermal loading, static axial loading, flow velocity, medium stiffness and shell geometry parameters on the free vibration characteristics are described. The new features in this paper are helpful for the application and the design of functionally graded cylindrical shells containing fluid flow.