Effect of vertical heat transfer on thermocapillary convection in an open shallow rectangular cavity

In order to understand the effect of the vertical heat transfer on thermocapillary convection characteristics in a differentially heated open shallow rectangular cavity, a series of two- and three-dimensional numerical simulations were carried out by means of the finite volume method. The cavity was filled with the 1cSt silicone oil (Prandtl number Pr = 13.9) and the aspect ratio ranged from 12 to 30. Results show that thermocapillary convection is stable at a small Marangoni number. With the increase of the heat flux on the bottom surface, thermocapillary convection transits to the asymmetrical bi-cellular pattern with the opposite rotation direction. The roll near the hot wall shrinks as the Marangoni number increases. At a large Marangoni number, numerical simulations predict two types of the oscillatory thermocapillary flow. One is the hydrothermal wave, which is dominant only in a thin cavity. The other appears in a deeper cavity and is characterized by oscillating multi-cellular flow. The critical Marangoni number for the onset of the oscillatory flow increases first and then decreases with the increase of the vertical heat flux. The three-dimensional numerical simulation can predict the propagating direction of the hydrothermal wave. The velocity and temperature fields obtained by three-dimensional simulation in the meridian plane are very close to those obtained by two-dimensional simulation.     

Numerical simulation of convective heat transfer to a radial free surface jet impinging on a hot solid

The convective heat transfer between a circular free surface impinging jet and a solid surface has been studied numerically. The thin liquid film formed on the surface has been assumed to be in non-turbulent free surface flow. The effects of surface tension, viscosity, gravity and heat transfer between the film flow and the solid surface have been taken into account. The flow structure on a non-heated surface has been investigated first. Next, the steady-state flow structure in the liquid film as well as the heat transfer has been examined. The predicted results have been compared with experimental data for the purpose of validating the analysis. The hydrodynamics of the liquid film and the heat transfer processes have been investigated numerically to understand the physics of the phenomena. Received on 5 October 1998     

Natural convection and conduction heat transfer in open shallow cavities with bounding walls

A numerical study has been carried out on inclined open shallow cavities, which are formed by a wall and horizontal fins. Constant heat flux is applied on the surface of the wall inside the cavity while its other surface was kept isothermal. The wall and the fins are conductive. Conjugate heat transfer by natural convection and conduction is studied by numerically solving equations of mass, momentum and energy. Streamlines and isotherms are produced, heat and mass transfer is calculated. A parametric study is carried out using following parameters: Rayleigh number from 10⁶ to 10¹², conductivity ratio from 1 to 60, open cavity aspect ratio from 1 to 0.125, dimensionless end wall thickness from 0.05 to 0.20, horizontal walls from 0.01 to 0.15 and inclination of the end wall from 90° to 45°. It is found that the volume flow rate and Nusselt number are a decreasing function of the cavity aspect ratio, horizontal fin thickness and conductivity ratio. They are an increasing function of end wall thickness and inclination angle, except in the latter case optima exist at high Rayleigh numbers.     

Natural convection in an inclined enclosure containing internal energy sources and cooled from below

Natural convection in an inclined enclosure from below and containing internally heated fluid has been investigated using a finite difference calculation procedure. Results have been obtained for Rayleigh number values up to 10⁶ and for inclination angles of 30 and 60°. For internal Rayleigh numbers that are much larger than the external Rayleigh number, the flow rises in the interior and moves down both the hot and cold walls. On the other hand, if the external Rayleigh number has a larger magnitude, the flow moves upwards along the hot surface and downwards along the cold surface. For the latter situation, the inner core is multicellular in nature at large external Rayleigh numbers. The average heat flux ratio along the cold surface (convective heat flux/corresponding conduction heat flux) increases with increasing external Rayleigh number and decreasing internal ratio is non-monotonic in nature. The heat flux ratio along both surfaces is observed to be strongly dependent on the inclination angle at high external Rayleigh numbers. A maximum in the local heat flux along the cold surface is obtained in the vicinity of x/L = 1 where hot fluid, either from the interior or directly from the opposite hot wall, meets the surface. Along the hot wall, a maximum in the heat flux ra flo     

Numerical simulation of laminar natural convection in a laterally heated vertical cylindrical enclosure: application to crystal growth

Laminar natural convection has been studied in a laterally heated vertical cylindrical enclosure with a free insulated surface and a centrally located constant temperature wall at the top. These conditions are a simplification of the conditions existing in a Czochralski crystal pulling system. The laminar, axisymmetric flow of a Newtonian, constant physical properties fluid under Boussinesq’s approximation has been considered. Governing equations in primitive variable form are solved numerically by control volume method. SIMPLE algorithm due to Patankar has been used for the numerical simulation. The effects of the constant wall heat flux boundary condition at the side wall have been investigated whereas the bottom wall is considered to be insulated. Streamlines and isotherms are presented for various Rayleigh numbers and Prandtl numbers. Heat flux vectors through the melt are plotted for selected cases. The axial velocity and temperature variations at different horizontal sections of the crucible have been presented graphically to explain the transport processes inside the crucible. It has been observed that in case of low Pr and high Ra, flow separation occurs at the vertical wall of the crucible which leads to an oscillatory flow as Ra increases. The investigation has been extended to the oscillatory regime of flow in the zone of supercritical Rayleigh numbers and some unsteady results are also presented. Finally a heat transfer correlation has been developed for steady-state case.     

Buoyancy and surface-tension induced instabilities of fluid in open and closed vertical cylindrical containers

Rayleigh and Marangoni types of instability of completely and partially confined fluid columns are considered. The appropriate differential systems are reformulated in a form that fully accounts for the presence of the surrounding solid and free surfaces as well as for the surface-tension variations in the latter. The present work is carried out under the assumption that the free surface, if present, is constrained to stay horizontal. It is shown that if this assumption and the principle of exchange of stabilities hold the differential systems for both open and closed containers, govern no less than four dependent variables. By adopting the Jeffreys-Goldstein method, series solutions for circular cylindrical geometries are obtained. Special attention is paid to the problem of termwise differentiation of infinite series. Despite truncation, results are found to be physically plausible and in agreement with existing analyses. An increase in the conductivity of the vertical surfaces tends to diminish the temperature variations inside the container and has, therefore, a stabilizing effect. An increase in the conductivity of either the medium above the exposed surface or the solid walls retards both Marangoni and Rayleigh types of instability. For very slender containers, the effect of surface tension is negligible and the critical values of Rayleigh number obtained here are close to those found by Yih. The approximation adopted restricts the results of this work to cases in which the container's height to radius ratio is of the order unity or larger. Most of the work summed up in this paper was carried out at the Mechanics Department of the State University of New York in Stony Brook, U.S.A.     

A numerical study of the longitudinal thermoconvective rolls in a mixed convection flow in a horizontal channel with a free surface

This paper presents a numerical study of three-dimensional laminar mixed convection within a liquid flowing on a horizontal channel heated uniformly from below. The upper surface is free and assumed to be flat. The coupled Navier–Stokes and energy equations are solved numerically by the finite volume method taking into account the thermocapillary effects (Marangoni effect). When the strength of the buoyancy, thermocapillary effects and forced convective currents are comparable (Ri ⋍ O(1) and Bd = Ra/Ma ⋍ O(1)), the results show that the development of instabilities in the form of steady longitudinal convective rolls is similar to those encountered in the Poiseuille–Rayleigh–Bénard flow. The number and spatial distribution of these rolls along the channel depend on the flow conditions. The objective of this work is to study the influence of parameters, such as the Reynolds, Rayleigh and Biot numbers, on the flow patterns and heat transfer characteristics. The effects of variations in the surface tension with temperature gradients (Marangoni effect) are also considered.     

Onset of Benard-Marangoni convection in a rotating liquid layer with nonuniform volumetric energy sources

The criteria for the onset of natural convection in a rotating liquid layer with nonuniform volumetric energy sources from absorbed thermal radiation are determined via linear stability analysis. The linearized perturbation equations are solved by using a numerical technique to obtain the eigenvalues that governs the onset of convection in a microgravity environment. The stability criteria are obtained in terms of the Marangoni number as function of the optical thickness. The influences of the Rayleigh number, Taylor number, Bond number, Crispation number, and Biot number on convection are examined in detail. These parameters provide a relationship between the critical Marangoni number and the Coriolis force, the buoyancy force, the interfacial tension, and the heat transport mechanisms.     

Natural convection in an inclined rectangular porous cavity with uniform heat flux from the side

The problem of natural convection in an inclined rectangular porous layer enclosure is studied numerically. The enclosure is heated from one side and cooled from the other by a constant heat flux while the two other walls are insulated. The effect of aspect ratio, inclination angle and Rayleigh number on heat transfer is studied. It is found that the enclosure orientation has a considerable effect on the heat transfer. The negative orientation sharply inhibits the convection and consequently the heat transfer and a positive orientation maximizes the energy transfer. The maximum temperature within the porous medium can be considerably higher than that induced by pure conduction when the cavity is negatively oriented. The peak of the average Nusselt number depends on the Rayleigh number and the aspect ratio. The heat transfer between the two thermally active boundaries is sensitive to the effect of aspect ratio. For an enclosure at high or low aspect ratio, the convection is considerably decreased and the heat transfer depends mainly on conduction.     

Experimentelle Untersuchung des Wärmeüberganges durch freie Konvektion an waagerechten Rippenrohren in einem zylindrischen Behälter

Experimental measurements of heat transfer by free convection in a cylindrical waterfilled enclosure from a horizontal finned tube of small fin height is reported. The measured heat transfers are presented in form of correlation equationsNu=f(Ra). The Rayleigh numbers have been alternatively defined with the annulus height, core diameter and an equivalent diameter defined for the finned tube. The correlation equations have been formulated for heat transfer (i) in an infinite medium and (ii) in an annulus. Results show that for the experimental annulus gap/enclosure diameter ratio of 24.66 and 40.74, the heat transfer analysis based on infinite medium assumption shows a somewhat better correlation although the influence of the enclosure cannot be completely neglected. Defining the Rayleigh number with an equivalent diameter makes it possible to compare the experimental measurements with those from a smooth tube.     

等热流密度加热垂直下降液膜的研究

利用等热流密度加热条件下降膜流动的三维模型方程进行线性稳定性分析和数值模拟。线性稳定性分析表明，模型方程在小到中等Reynolds数下都适用，并且流向不稳定性增长率随着Reynolds数和Marangoni数增加而增加，展向不稳定性增长率则随着Marangoni数增加而增加，随着Reynolds数增加而减小，流向和展向对扰动波数都存在一个不稳定区间。三维数值模拟表明，在等热流密度加热条件下，液膜在随机扰动的情况下最终会形成带孤立波的三维溪流状结构，液膜与气体的换热也因溪流状结构的出现而加强；在随机扰动的基础上引入占优势地位的展向最不稳定扰动会使得换热增强，液膜会提前破裂；在随机扰动的基础上引入占优势地位的流向最不稳定扰动时，液膜的换热会增强，但不会提前破裂；在随机扰动的基础上同时引入占优势地位的流向和展向最不稳定扰动时，换热会加强且液膜会提前破裂。     

Heat transfer and supersonic flow structure in surface cavities

The characteristics of the flow and heat transfer in two- and three-dimensional open cavities on plane and cylindrical surfaces in a supersonic stream in the presence of a turbulent boundary layer have been investigated experimentally. The effects of the Mach number, boundary layer thickness, the shape of the cavity, and its angle of inclination to the free-stream direction on the flow parameters in the mixing layer above the cavity and the heat flux and pressure distribution on the surface of the cavity and its bottom are descirbed. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 74–80, July–August, 1998.     

Axisymmetric thermocapillary convection in a slowly rotating annular cylindrical container

In a slowly rotating annular cylindrical container the free liquid surface (liquid-gas interface) is subjected to a temperature gradient in radial direction. The temperature dependent surface tension creates a shear stress on the interface which is transmitting a thermocapillary convection in the bulk of the liquid. For constant temperature T ₁ of the inner and T ₂ of the outer wall a steady Marangoni convection takes place, exhibiting a double vortex ring of equal directional flow. For time-oscillatory temperatures of the walls a time-dependent thermocapillary convection appears, which will create on the free liquid surface various wave patterns. They shall, depending on the forcing frequency of the temperature, exhibit resonance peaks. The velocity distribution and the response magnitude inside the container has been determined. Received on 3 September 1997     

Transient numerical simulation of buoyancy driven flow adjacent to an elliptic tube

In this paper the problem of laminar, transient, two-dimensional free convective heat transfer from the surface of a horizontal elliptic tube is considered. The tube, whose surface is suddenly subjected to uniform heat flux, is placed in a quiescent Boussinesq Newtonian fluid with its major axis horizontal. The details of both flow and thermal fields are obtained by solving the full governing Navier–Stokes and energy equations. These equations, expressed in terms of stream function, vorticity and temperature, are numerically solved using an implicit spectral finite difference procedure. The parameters involved are the modified Rayleigh number, Prandtl number and axis-ratio. The investigation covers a Rayleigh number range up to 10⁷. The minor–major axis ratio of elliptic cylinder ranges between 0.05 and 0.998 and Prandtl number ranges between 0.1 and 10. The effects of these parameters on the surface temperature distribution and heat transfer coefficients are determined and the different aspects of the results are discussed for some selected cases.     

波纹竖板层流降膜蒸发传热的分析

对高粘度液体在等温正弦形波纹壁面上的自由降落与蒸发建立了摄动分析模型。得到了流动的分析解和蒸发传热的数值解。考察了壁面波纹的波幅和波数、液膜表面张力及贝克利数对流动与传热的影响,结果表明,加大波纹的波幅、适当选择波数、减小贝克利数可增强传热,而表面张力对蒸发传热的影响较小。     

Natural convection heat transfer along vertical rectangular ducts

Experimental investigations have been reported on steady state natural convection from the outer surface of vertical rectangular and square ducts in air. Seven ducts have been used; three of them have a rectangular cross section and the rest have square cross section. The ducts are heated using internal constant heat flux heating elements. The temperatures along the vertical surface and the peripheral directions of the duct wall are measured. Axial (perimeter averaged) heat transfer coefficients along the side of each duct are obtained for laminar and transition to turbulent regimes of natural convection heat transfer. Axial (perimeter averaged) Nusselt numbers are evaluated and correlated using the modified Rayleigh numbers for laminar and transition regime using the vertical axial distance as a characteristic length. Critical values of the modified Rayleigh numbers are obtained for transition to turbulent. Furthermore, total overall averaged Nusselt numbers are correlated with the modified Rayleigh numbers and the area ratio for the laminar regimes. The local axial (perimeter averaged) heat transfer coefficients are observed to decrease in the laminar region and increase in the transition region. Laminar regimes are obtained at the lower half of the ducts and its chance to appear decreases as the heat flux increases.     

Buoyant Marangoni convection of nanofluids in square cavity

The buoyant Marangoni convection heat transfer in a differentially heated cavity is numerically studied. The cavity is filled with water-Ag, water-Cu, water-Al₂O₃, and water-TiO₂ nanofluids. The governing equations are based on the equations involving the stream function, vorticity, and temperature. The dimensionless forms of the governing equations are solved by the finite difference (FD) scheme consisting of the alternating direction implicit (ADI) method and the tri-diagonal matrix algorithm (TDMA). It is found that the increase in the nanoparticle concentration leads to the decrease in the flow rates in the secondary cells when the convective thermocapillary and the buoyancy force have similar strength. A critical Marangoni number exists, below which increasing the Marangoni number decreases the average Nusselt number, and above which increasing the Marangoni number increases the average Nusselt number. The nanoparticles play a crucial role in the critical Marangoni number.     

Numerical estimation of mixed convection heat transfer in vertical helically coiled tube heat exchangers

A numerical investigation of the mixed convection heat transfer from vertical helically coiled tubes in a cylindrical shell at various Reynolds and Rayleigh numbers, various coil‐to‐tube diameter ratios and non‐dimensional coil pitches was carried out. The particular difference in this study compared with other similar studies is the boundary conditions for the helical coil. Most studies focus on constant wall temperature or constant heat flux, whereas in this study it was a fluid‐to‐fluid heat exchanger. The purpose of this article is to assess the influence of the tube diameter, coil pitch and shell‐side mass flow rate on shell‐side heat transfer coefficient of the heat exchanger. Different characteristic lengths were used in the Nusselt number calculations to determine which length best fits the data and finally it has been shown that the normalized length of the shell‐side of the heat exchanger reasonably demonstrates the desired relation. Copyright © 2010 John Wiley & Sons, Ltd.     

Control of Thermo- and Solutocapillary Flows in FZ Crystal Growth by High-Frequency Vibrations

The paper deals with the numerical investigation of the possibilities to control convective flows in the liquid bridge in zero gravity conditions applying axial vibrations. The surface tension is assumed to be dependent both on the temperature and on the solute concentration. The free surface deformations and the curvature of the phase change surfaces are neglected but pulsational deformations of the free surface are accounted for. The first part of the paper concerns axisymmetric steady flows. The calculations show that the evolution of convective flow with the variation of thermal Marangoni number at a fixed value of the solutal Marangoni number is accompanied by the hysteresis phenomenon, which is related to the existence of two stable steady regimes in a certain parameter range. One of these regimes is thermocapillary dominated, it corresponds to the two-vortex flow, and the other is solutocapillary dominated, it corresponds to the single-vortex flow. Under vibrations, the range of the Marangoni numbers where hysteresis is observed becomes narrower and is shifted to the area of larger values. The second part of the paper concerns the stability of axisymmetric thermo-and solutocapillary flows and the transition to three-dimensional regimes. Significant mutual influence of flows generated by each process on the stability of the other is discovered. Stability maps in the parametric plane for the thermal Marangoni number, the solutal Marangoni number, are obtained for different values of vibration parameters. It is shown, that vibrations exert a stabilizing effect, increasing critical Marangoni numbers for all modes of instability. However, this effect is different for different modes and at high vibration intensity destabilization is possible. Consequently, vibrations can modify the scenario of the transition to the three-dimensional mode.     

Thermo-Bioconvection in a Square Porous Cavity Filled by Oxytactic Microorganisms

This paper studies the thermo-bioconvection in a square porous cavity filled by oxytactic microorganisms. The Darcy model with Boussinesq approximation has been used to solve the flow and heat and mass transfer in the porous region. The governing equations formulated in terms of the dimensionless stream function, temperature and concentration have been solved using the finite difference method. Comparison with results from the open literature of the mean Nusselt number for a square cavity filled with a regular porous medium is made. It is shown that the results are in very good agreement. The main objective was to investigate the influence of the traditional Rayleigh number Ra = 10, 100, bioconvection Rayleigh number Rb = 10, 100, Lewis number Le = 1, 10, and Péclet number Pe = 0.1, 1 on the fluid flow and heat and mass transfer. Comprehensive analysis of an effect of these key parameters on the Nusselt and Sherwood numbers at the vertical walls has been conducted.