R LIBRARY WWWMW!” Michigan State 006439222 A University This is to certify that the dissertation entitled EXPERIMENTAL STUDY OF LOCAL NATURAL CONVECTION HEAT TRANSFER IN INCLINED AND ROTATING ENCLOSURES presented by Fakhri Hamady has been accepted towards fulfillment , ofthe requirements for i ' Ph.D. clegmeinMechanical Engineering Date May 15, 1987 0-12771 MSU is an Affirmative Action/Equal Opportunity Institution RETURNING MATERIALS: IVIESI.J Piace in book drop to LJBRARJES remove this checkout from -_ your record. FINES will be charged if book is returned after the date stamped beiow. OCT 3 1991 27 5 WSWOFLOCALWCONVBCIIORHEAT msmmmcunnmnommcmcmsunzs By Pakhri Handy A nIssmnnon Suhitted to Hichigan State University in partial fulfil-eat of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Kechmical Engineering 1987 WWOPIDCALNATURALCONVECTIONHEAT msmmmmmmmcmcmsum By Pakhri Hanady The local and mean natural convection heat transfer characteristics have been studied experimentally in an air-filled differentially heated enclosure with cross-sectional aspect ratio one. In the investigation, a Mach-Zehnder interferometer was employed to reveal the entire temperature field which enables the measurement of the local and mean Nusselt numbers at the hot and cold surfaces. The first part of this investigation was a study of the inclination effect on the flow and heat transfer behaviors. The measurements of local and mean Nusselt numbers are obtained at various inclination angles, ranging between 0 deg. (heated from above) and 180 deg. (Benard convection, heated from below), for Rayleigh numbers between 10‘ and 106 . The measured heat flux at the hot and cold boundaries showed a strong dependence on the angle of inclination and Rayleigh number. In addition, new results and details are made available concerning the local heat transfer distribution as a function of the inclination angle and Rayleigh number. In the second part, the study is extended to include the effect of combined heating and rotation on the thermal and hydrodynamic boundary layers. The enclosure is rotated about its longitudinal horizontal axis. The experimental results showed how the centrifugal and Coriolis forces arising from rotation have remarkably influenced the local heat transfer behavior when compared with the non-rotating results. Local heat fluxes are obtained as a function of Taylor 4 4 5 (Ta < 10 ) and Rayleigh (10 < Ra < 3x10 ) numbers at different angular positions of the enclosure. Correlations are established for the non-rotating and rotating conditions as a function of Rayleigh and Taylor numbers . Photographs of the flow pattern and isotherms at different inclination angles and rotational speeds are shown, in order to give a greater understanding of the flow and heat transfer behaviors. D-ICATION To my beloved parents and brothers ii assi: ...gin C00pe Ratio .Rgin WW I would like to record my indebtedness and thank to my advisors, Dr. J. R. Lloyd, and Dr. K. T. Yang for their valuable assistance and advice throughout the completion of my research studies at Michigan State University; to my committee members, Dr. J. Beck, Dr. A. Atreya, Dr. R. Bartholomew, and Dr. D. Yen for their constructive comments, and assistance; to the machinists in the Department of Mechanical Engineering, Mr. L. Eisele, and R. Rose, for their assistance and cooperation in building our Theme-Optics Heat Transfer laboratory. I also like to express my gratitude for the support of the National Science Foundation NSF (Grant 713828) and for use of the MSU Engineering Case Center computing facilities. TABLE OF CONTENTS LIST OF FIGURES ..................................... LIST OF TABLES ..................................... NOMENCLATURE ..................................... CHAPTER I. INTRODUCTION ..................................... 1.1 Problem Statement ........................... 1.2 Literature Survey ........................... 1.2.1 Thermal Convection in Vertical Enclosures ........................... 1.2.2 Thermal Convection in Inclined Enclosures ........................... 1.2.3 Thermal Convection in Rotating Fluids ............................... 2.1 Governing Equations ......................... 2.2 Governing Equations in a Rotating Enclosure ................................... 3. EXPERIMENTAL FACILITIES AND TECHNIQUES ........... 3.1 Mach-Zehnder Interferometer Facilities ....... 3.1.1 Optical Plates ....................... 3.1.2 Light Source ......................... 3.1.3 Photographic Equipment ............... 3.2 Experimental Apparatus ....................... .2.1 Moving Frame ......................... .2.2 Rotating Frame ....................... .2.3 Test Section ......................... .2.4 Thermocouple Wiring .................. .2.5 Temperature Controlled Section ....... .2 6 Driving Motor ........................ UMUUUU 3.3 Experimental Procedure ....................... iv 35 45 48 SO 50 52 52 54 57 59 62 66 68 3.3.1 Test Section Assembly and Leveling ... 68 3.3.2 Insulation ........................... 71 3.3.3 Thermocouples ........................ 73 3.3.4 Inclination Angle and Rotational Speed ................................ 73 3.3.5 Temperature Measurements ............. 74 3.3.6 Photography .......................... 74 4. RESULTS AND DISCUSSION ........................... 75 4.1 Natural Convection in an Inclined Rectangular Enclosure ...................... 75 4.1.1 Mean Nusselt Number Results .......... 76 4.1.2 Local Nusselt Number Results ......... 82 4.2 Natural Convection in a Heated Rotating Enclosure .................................. 99 4.2.1 Heat Transfer Results in a Heated Rotating Enclosure at Angular position 90 deg. ..................... 100 4.2.2 Heat Transfer Results in a Heated Rotating Enclosure at Angular position 180 deg. .................... 130 4.2.3 Heat Transfer Results in a Heated Rotating Enclosure at Angular position 0 deg. ...................... 136 5. SUMMARY AND CONCLUSIONS .......................... 144 APPENDICES ......................................... 148 1 PHYSICAL PROPERTIES .............................. 148 2 INTERFEROGRAM ANALYSIS ........................... 149 3 HEAT FLUX MEASUREMENT ............................ 154 4 ERROR ANALYSIS ................................... 160 5 COMPUTER PROGRAMS AND SAMPLE CALCULATION ......... 162 REFERENCES .......................................... 197 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 10 ll 12 l3 14 15 l6 l7 18 LIST OF FIGURES Geometric configurations of the enclosure ............................ Rotation referred to a Cartesian frame of reference, O-XYZ ................ Mach-Zehnder Interferomter ............... Interferometer and experimental setup .... Camera setup ............................. Moving and rotating assembly ............. Side view of the test section in the rotating frame ....................... Slipring-brush assembly .................. Rotating section ......................... Dimensions of the test section and the gage block ...................... Schematic diagram of slipring-brush assembly ................................ Rotating union assembly ................. Schematic side view of the rotating union .......................... DC motor with variable speed control .... Top view of the end region .............. Comparison of mean Nusselt numbe results, for ¢ - 90 deg ................ Effect of inclination angle on mean Nusselt number at Ax-l.0 ........... Mean Nusselt number at various inclination for air at Ax-l.0 ........... vi page 39 46 49 51 53 55 56 58 6O 63 -64 65 67 72 77 79 81 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Effect of inclination angle on the and heat transfer, at ¢ - 0 deg . Effect of inclination angle on the and heat transfer, at ¢ - 30 deg Effect of inclination angle on the and heat transfer, at d - 60 deg Effect of inclination angle on the and heat transfer,at é - 90 deg . flow flow Local Nusselt number distribution along the hot and cold walls, for air at 5 Ax-1.0, Ra- 1.1x10 and d - 105 deg .................. Effect of inclination angle on the flow and heat transfer, at d - 120 deg ...... Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-l.0, Ra- 1.1x10 and d - 135 deg .................. Effect of inclination angle on the and heat transfer, at ¢ - 150 deg flow Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra- 1.1x10 and ¢ - 165 deg .................. Effect of inclination angle on the and heat transfer, at d - 180 deg Comparison of local Nusselt number distribution for air, at Ax-1.0, and d - 90 deg .................. Comparison of local Nusselt number distribution for air, at Ax-1.0, and d - 90 deg .................. flow Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-l.0, Ra- 1.2x10 rotational rate— + 6.0 rpm 83 85 86 88 89 9O 91 93 94 95 97 98 (vertical configuration) ................. 102 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra- 1.2x10 rotational rate- + 8.5 rpm, (vertical configuration) ................. 103 vii Figure Figure Figure Figure Figure Figure Figure Figure Figure 33 34 35 36 37 38 39 40 41 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra- 1.2x10 rotational rate— + 10.2 rpm, (vertical configuration) ................. 105 Local Nusselt number distribution along the hot and cold walls, for air at 5 Ax-l.0, Ra- 1.2x10 rotational rate— + 12.5 rpm, (vertical configuration) ................. 106 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-l.0, Ra- 1.2x10 rotational rate— + 15.0 rpm, (vertical configuration) ................. 107 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra- 1.2x10 rotational rate- + 17.5 rpm, (vertical configuration) ................. 108 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra- 2.0x10 rotational rate- + 8.5 rpm, (vertical configuration) ................. 110 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra- 2.0x10 rotational rate- + 12.2 rpm, (vertical configuration) ................. 111 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra- 2.0x10 rotational rate- + 15.1 rpm, (vertical configuration) ................. 112 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-l.0, Ra- 2.0x10 rotational rate- + 17.5 rpm, (vertical configuration) ................ 113 Local Nusselt number distribution along the hot and cold walls, for air at 5 Ax-1.0, Ra- 3.0x10 rotational rate- + 6.1 rpm, (vertical configuration) ................. 114 viii Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 42 43 44 45 46 47 48 49 50 51 Local Nusselt number distribution along the hot and cold walls, for air at 5 Ax-1.0, Ra- 3.0x10 rotational rate- + 8.5 rpm, (vertical configuration) .................. 115 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-l.0, Ra- 3.0x10 rotational rate- + 12.2 rpm, (vertical configuration) .................. 116 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra- 3.0x10 rotational rate- + 15.1 rpm, (vertical configuration) .................. 117 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra- 3.0x10 rotational rate- + 17.5 rpm, (vertical configuration) .................. 118 Local Nusselt number distribution along the hot and cold walls, for air 4 at Ax-l.0, Ra- 7.4x10 rotational rate- + 8.5 rpm, (vertical configuration) .................. 120 Local Nusselt number distribution along the hot and cold walls, for air 4 at Ax-l.0, Ra- 7.4x10 rotational rate— + 12.2 rpm, (vertical configuration) .................. 121 Local Nusselt number distribution along the hot and cold walls, for air at Ax-l.0, Ra- 7.4x10‘ rotational rate- + 15.1 rpm, (vertical configuration) .................. 122 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-l.0, Ra- 7.4x10 rotational rate- + 17.5 rpm, (vertical configuration) .................. 123 Effect of rotation on mean Nusselt number, for air at Ax-l.0, (vertical configuration) .................. 124 Effect of rotation on mean Nusselt number, for air at Ax-1.0, ix Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 52 53 54 55 S6 S7 58 59 61 (vertical configuration) .................. 126 Mean Nusselt number as a function of Taylor and Rayleigh numbers ............ 127 Local Nusselt number distribution the hot and cold walls, for air 5 at Ax-1.0, Ra- 1.2x10 rotational rate- - 8.5 rpm, along (vertical configuration) .................. 128 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-l.0, Ra- 1.2x10 rotational rate- - 17.5 rpm, (vertical configuration) .................. 129 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra- 1.2x10 rotational rate— + 8.5 rpm, (heated from below) ....................... 132 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra— 1.2x10 rotational rate- + 12.2 rpm, (heated from below) ....................... 133 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra- 3.0x10 rotational rate— + 8.5 rpm, (heated from below) ....................... 134 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra- 3.0x10 rotational rate- + 12.2 rpm, (heated from below) ....................... 135 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra- 1.2x10 rotational rate- + 8.5 rpm, (heated from above) ....................... 137 Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra- 1.2x10 rotational rate- + 12.2 rpm, (heated from above) ....................... 138 Local Nusselt number distribution along X Figure Figure Figure Figure Figure Figure Figure 62 63 64 65 66 67 68 the hot and cold walls, for air 5 at Ax-1.0, Ra- 3.0x10 rotational rate- + 8.5 rpm, (heated from above) ....................... Local Nusselt number distribution along the hot and cold walls, for air 5 at Ax-1.0, Ra- 3.0x10 rotational rate- + 12.2 rpm, (heated from above) ....................... Effect of rotation on mean Nusselt number at various angular positions ....... Interference fringe patterns, 5 at Ra- 3.0x10 ............................. 5 Fringe shift evaluation, at Ra - 1.1x10 ¢ - 90 deg ................................ Estimation of the temperature gradient 5 at the cold wall, for Ra - 1.1x10 f - 0.134, and ¢ - 90 deg ................. 5 Fringe shift distribution for Ra - 1.1x10 141 143 151 156 g - 0.134, and 5 - 90 deg ................. 157 5 Temperature profile, for Ra - 1.1x10 g - 0.134, a - 90 deg ..................... 158 xi LIST OF‘TABLES Table 1 Sample calculation of the fringe shift Table 2 Table 3 5 and the temperature, at Ra - 1.1x10 and height,(y/H) - 0.140 ................... 159 Sample calculation of the fringe shift and the temperature distribution, from the interference fringe patterns in Figure 65 ........ , ....................... 179 Sample calculation of local Nusselt number obtained from the temperature gradients, in Table 2. xii Nu Nu Pr Hi» W4 Ru Aspect ratio, (height/width) Longitudinal aspect ratio, (length/width) Specific heat at constant pressure, J/kg.K Enclosure width, m 2 Gravitational acceleration, m/sec 5 Gladstone-Dale constant, m /kg 5 2 Grashof number - gfiH (TH - TC)/u Enclosure height (H - D), m Convective heat transfer coefficient, W/m2.K Thermal conductivity, W/m.R Length of the enclosure, m A0 Local Nusselt number - kC A£ Mean Nusselt number 2 Pressure, N/m (Pa) Dynamic pressure, Pa Perturbation pressure Prandtl number - v/a Position vector, m Non-dimensional position vector - f/H Universal gas constant, J/kg.K xiii Ra Ta x,y,z <4 Rayleigh number - gflH3(TH-TC)/au 2 4 Rotational Rayleigh number - an H (TH-TC)/av Rotational Reynolds number - 0H2/v Time, sec Temperature, C 2 4 2 Taylor number - 40 H /u Coordinates in the x, y, and z-directions Velocity vector, m/sec Non-dimensional velocity vector - vH/a Greek.Letters 2 Thermal diffusivity, m /sec Volumetric thermal expansion coefficient, l/K Non- dimensional distance along the y-direction - y/H Non-dimensional temperature - (T - TC)/(TH - TC) Dynamic vicosity, kg/m.sec 2 Kinematic viscosity, m /sec Non-dimensional distance along the x-direction - x/H 5 Density, kg/m 2 Non-dimensional time c at/H Inclination angle between the cold surface and the horizontal direction, deg. Dissipation function Angular speed, rev/sec xiv subscripts a Air C Cold surface H Hot surface r Rotation 3 Surface XV CBAPTERI 1 . 1 Problem Statement Natural convection in enclosures continues to be the subject of extensive research effort. The convection phenomena studied are usually induced by gravitational body forces. In a fluid where the body force is normal to the direction of the density gradient, flow is produced no matter how small the gradient may be, since no hydrostatic pressure distribution can balance the consequent variation of the buoyancy forces produced. For a fluid with the density gradient parallel and opposite to the gravitational acceleration, the flow can only be observed when the density gradient exceeds a critical value. The critical value depends upon the conditions at the top and bottom surfaces, which may be either free or constrained by rigid conducting surfaces. In enclosures inclined relative to the gravitational field, distinctly three-dimensional flows occur which exhibit patterns closely dependent on the angle of enclosure inclination. A comprehensive review of enclosure heat transfer research up to 1978, has been presented by Ostrach [1] and Catton [2]. If a differentially heated enclosure undergoes rotation, centrifugal and Coriolis force effects on the fluid will interact with the gravitational force in a manner which results in a complex three- dimensional flow and temperature fields. The flow and temperature fields are directly coupled in the momentum equations and this I consequently effects the surface to fluid energy transfer. Cases like that occur in rotating fluid machinery, where the centrifugal acceleration, which is proportional to the square of the angular velocity, may be very large. In geophysical applications involving atmospheric circulation and oceanic rotational motion, the flows are driven primarily by the Coriolis forces and the centrifugal effect can be neglected. Analysis of flow and heat transfer associated with rotating systems is quite complex due to the simultaneous influence of centrifugal and Coriolis forces. As a consequence, a large amount of analytical and experimental study has been and continues to be devoted for the development of a general treatment of the influence of rotation on the flow and heat transfer in simple geometries. However, studies of natural convection in rotating systems, concerning the effects of centrifugal and Coriolis forces are very few, and experimental data are almost non-existent. In this respect, it is further required that special techniques and experimental methods have to be devised to analyze the process with a view to provide information on the heat transfer rate. The objective and exclusive purpose of the present research study is to present a systematic study to reveal the motion due to the interaction between gravitational field and the rotationally induced body forces in an air-filled differentially heated rectangular cavity, or slot rotating about its longitudinal horizontal axis. As would be anticipated, the influence of the centrifugal and Coriolis forces arising from the rotation will affect significantly the flow and heat transfer characteristics of natural convection. Emphasis will be centered on the local heat transfer measurements and visualizations to integrate the qualitative visual observations with the quantitative measurements of heat transfer. This class of rotating flow geometry, may lead to a new area of experimental research, to supplement the available information of natural convection in cavities, in addition to numerous future applications in spacelab, and design development in cooling rotating equipments. The first part of this investigation will be devoted to the study of thermal convection in a differentially heated, air-filled enclosure driven by the gravitational field only. The effect of inclining the boundaries on the thermal convection for various Rayleigh numbers is to be studied. Visualization of the flow patterns will be employed to give greater insight into the physical phenomena, and to enable greater understanding of the flow and the local heat transfer profiles. This will provide a firm basis of comparison to sort out the effects of rotation. The second part of the study will focus on the effects of rotation on the flow and heat transfer in enclosures. Local heat transfer measurements with the flow visualization will be performed for various Taylor numbers, to show the significant amount of distortion of the hydrodynamic and thermal boundary layers. The Rayleigh number will be gradually increased to reveal the interaction between centrifugal and gravitational buoyancy on the flow and heat transfer. The data will be correlated based upon the Rayleigh and Taylor numbers. The Mach-Zehnder interferometer will be used to explore how the local heat flux distribution is influenced by the Rayleigh number and the Taylor number (the square of the ratio of Coriolis force to viscous frictional force), to examine both the effect of the Coriolis force on the thermal convection of fluid at low rotational speeds, and the effect of the buoyant interaction at higher rotational speeds, where the centrifugal buoyancy will become more important relative to the Co he fo be Coriolis acceleration. Local values of Nusselt number (dimensionless heat flux) along with the average values will be measured and plotted for different values of Rayleigh and Taylor numbers, and at angular positions of 90 deg. (vertical configuration), 180 deg. (heated from below), and 0 deg. (heated from above). fe. in 0f sol t8: dim. l . 2 Literature Survey The general purpose of this review is to present the main features of natural convection in enclosures or cavities, as presented in the technical literature. The main concern will be devoted entirely to aspects of theoretical and experimental investigations of heat transfer and flow pattern in vertical, inclined, and rotating fluid systems . 1.2.1 Thermal Convection in Vertical Enclosures The first part of this review is therefore to draw together some of the recent information about the physical processes in vertical enclosures, which are composed of two vertical plates differentially heated in the horizontal direction, where all the other surfaces are either perfectly conducting or insulated. G.l(. Batchelor [3] , in 1954 initiated the first analytical study of free convection in enclosed gas layers, and through an approximate solution he concluded that the heat transfer is a function of the temperature difference between the walls, the Prandtl number of the enclosed fluid, and the aspect ratio of the enclosure. Poots [4] numerically solved the equations for a very limited range of Grashof number and substantiated Batchelor's prediction. His solution is applicable only for Gr <2000 and only for air. In 1956, Carlson [5] conducted an experimental study using optical techniques. His results showed that the mean temperature of the cell is a function of the height increasing with the vertical dimension. The precise variation is not known, and it is expedient to consi: the b Ha:n-. condi recen into ‘ revie cannot tendi1 the IN be de1 and t1 analy, aspect EQOmE ar~ more °n th. QAErgn consider the mean temperature as a constant, given by the average of the hot and cold wall temperatures. Eckert and Carlson [6] , used the Mach-Zehnder interferometer to investigate the temperature and flow conditions in an air layer enclosed between two vertical plates. Local heat transfer coefficient were derived from the local temperature gradients normal to the plate surfaces. They concluded that various flow regimes exist depending on the value of Rayleigh number and the enclosure geometry. That is, below a certain Grashof number and above a certain aspect ratio heat is transferred primarily by conduction in the core of the layer. Convection contributes only in the corner regions. On the other hand, for large Grashof number, and below a certain aspect ratio, boundary layers exist along the surfaces of the enclosure. Excellent review papers on internal flows have appeared recently, including enough information to allow one to gain insight into the natural convection phenomena. Ostrach [1,7] emphazised in his review works that, after all the research activities, the flow pattern cannot be predicted in advance from the given geometry and boundary conditions. Because in confined natural convection the core region and the boundary layer near the walls are closely coupled, neither one can be determined from the boundary conditions. Hence, the boundary layer and the core region coupling makes it very difficult to obtain an analytical solution to internal problems. Catton [2] considered in his review enclosures of different aspect ratios and at various inclination angles, besides the honeycomb geometries that can be used efficiently in solar collectors. His article contributed greatly to the physical understanding of the theoretical and experimental work in this field by providing insights on the coupling between the conservation equations for momentum and energy, and by obtaining engineering correlations for heat transfer. subjec recer uses: {8} i rectaz probl his s: level throng 129,10 inter; thenm regi01 relat {tape appea reSemi setox turbug The intensifying desire to present all the aspects of this subject contributed in an expansion of knowledge in fields considered recently, and still attracts the attention and interest of many investigators from many conventional fields of fluid mechanics. Gill [8] in his work on the boundary layer region for convection in a rectangular cavity, attempted to find an analytical solution to the problem mentioned earlier by Batchelor. In this regard, comparison of his solution with Elder's experimental results showed a satisfying level of agreement, except near the horizontal boundaries. Elder through his experimental works on natural convection in a vertical slot [9,10] , explained the features of the boundary layer flows and the flow interaction between the thermal boundary layers. The influence of the thermal conductivity will accordingly become important in the near-wall regions. In other words a boundary layer flow might be expected in a relatively thin region close to the walls which results in a vertical temperature gradient. He also claimed that the secondary flow appearance in the interior region at Rayleigh numbers greater than 105 resembles a ”cats-eye" pattern. In addition to this, he studied in a second report the transition process from laminar to unsteady to turbulent flow. Mynett and Duxbury [11] investigated natural convection in a vertical enclosure for Rayleigh numbers within the range, 103 ¢ > 100 deg. (heated from below), the instability growth appeared as longitudinal rolls with axes oriented along the upslope. The circulation modes generated indicate a relative dependence on the Rayleigh numbers. For 100 deg.> d > 80 deg. instabilities are characterized by transverse waves oriented across the slope. However, for 80 deg.> :5 >5 deg. and large Rayleigh number, longitudinal instabilities are the overwhelming modes where one expected to examine stable convection for cavities heated from above. Based on experimental observations instability occurred when a critical value of Rayleigh number is exceeded, and more apparent by increasing Rayleigh number. It is, also important to note that, besides the qualitative study, Hart provided quantitative analysis for some of the experimental results. But in spite the influence of the type of flow regime on heat transfer, no measurements have been made. Hollands and Konicek [66,67] attempted to study the various flow regimes in terms of the range of values of the critical Rayleigh numbers closely related to the stability of the horizontal, vertical, and inclined air layers. Their results for the measured critical Rayleigh numbers are in close agreement with the work reported by Unny [68] , and through their experimental analysis the principal modes of flow were discussed. For the horizontal mode (Benard problem), the instability is associated with what may be called a "top-heavy" situation. Heat is transferred by conduction across the fluid layer, where the Rayleigh number is less than the critical (Rac< 1708) value, while for the inclined layer two types of instabilities should be considered, the static top-heavy, and the gravitational buoyancy associated with the vertical slot. The relative magnitude of influence of ca which a tra an in large COB; rec: mm: bOuJ rm "3] 17 of each depends on the angle of inclination, and at a critical angle, which was determined by Hart to be 162 deg. and by Unny to be 168 deg. , a transition from one mode into another, can be seen. The instability in an inclined layer manifests itself in longitudinal rolls [61,65] for large Prandtl number. Clever [69] considered the effects of the angles of inclination on the transition process, and pointed out that, a visual study of the hydrodynamic instabilities near the vertical can yield a definite conclusion on the preferred mode for larger Prandtl numbers, since their experiments were performed only on air and water as the contained fluid. On this basis of analysis, Nusselt number results were expressed as a function of Rayleigh number, and the angle of inclination where the flow is still in longitudinal rolls. Recent developments, however, pointed to the need of more comprehensive treatment of the subject. Ayyaswamy and Catton [70] analytically considered fluid flow in a differentially heated, inclined rectangular cavity, and developed a correlation for the average Nusselt number in terms of the average Nusselt number for the vertical case and the angle of inclination. The result for a given aspect ratio and boundary layer regime predicted a satisfying level of agreement with their numerical solutions. In addition, their correlation is an adequate guide for the applied engineer to establish the first insight into the measurements of the heat transfer rates in many cases. Catton, Ayyaswamy and Clever [28] have applied the Galerkin method for the solution of natural convection flow of a large Prandtl number in a differentially heated inclined rectangular slot. The method has been demonstrated, in the case of Boussinesq approximations, for Rayleigh numbers up to 2x106 and aspect ratio range between 0.1 and 20. For various angles of inclination both temperature and velocity profiles 18 were obtained to reveal the heat transport and flow structure dependence on the aspect ratio, angle of inclination and Rayleigh numbers effects. In this respect, they suggested that, the mean Nusselt number should be expressed in terms of these parameters and the aspect ratio effect can not be ignored as illustrated in Dropkin and Somerscales [62] correlations. Comparisons of their results with De Vahl Davis [21,60] numerical analysis showed good agreement for an aspect ratio of 5 but did not agree for the square cavity. In their discussion the essential role of inclined boundaries has been considered together with the effect of increasing Rayleigh numbers on the heat transport and the flow configuration. In addition, they analysed the case for inclination angles when the hot plate is on top, and predicted that a large amount of heat is associated with the convective mechanisms. Further, decreasing the aspect ratio induces a transverse fluid motion which enhances the convective heat transport, and the presence of the end walls inhibits the longitudinal mode. In a series of papers Ozoe, et al. [71-74] carried out a number of experimental and numerical studies to determine the nature of the fluid flow, temperature distributions and the corresponding heat transfer rates under vertical, horizontal, and inclined conditions with various thermal boundary conditions, and aspect ratios of the enclosures. Their studies were thoroughly conducted and provided valuable information concerning flow patterns, velocities, and temperature distribution. The flow patterns which the authors observed were of the following nature. In a vertical square channel (angle of 90 deg.) the preferred mode was a single longitudinal two-dimensional roll-cell dominating the flow. This pattern remained until the angle of inclination increased up to about 170 deg. where the flow modification was observed. For large angles between 170 deg. and 180 19 deg. (horizontal), a complex flow pattern was found. Near 179 deg. of inclination or more a gradual rearrangement of the convective flow pattern was seen and a series of two- dimensional roll cells aligned with their axes horizontal and perpendicular to the long dimension of the channel appeared. The heat transfer measurements follow quite well what would be anticipated based on the observed flow patterns. From both the numerical and experimental results, which compared well with [70], the heat transfer rate was found to reach a maximum at about 130 deg. of inclination where the longitudinal mode prevailed and a minimum at about 170 deg. from the horizontal where the transition from the longitudinal roll cell to a series of two dimensional roll-cells has occurred. Additional studies were conducted for aspect ratios of l,2,3,4.2,8.4 and 15.5 and Rayleigh number from 3x103 to 105 to reveal what effect they may have on the heat transfer rates and flow patterns. A consequence of this study is that the critical angle of inclination related to the transition in the mode of circulation appeared to be strongly dependent on the aspect ratio and weakly dependent on the Rayleigh number. For aspect ratios of 3 and 4 the transition from a longitudinal roll-cell to a series of oblique roll-cells with their axes directed upslope occurred at small angles of inclination. Furthermore, the heat transfer rate decreased with a slight decrease in the angle of inclination from the horizontal which is caused by the greater drag of the oblique cells and by the significant reduction of circulation. When the angle was further decreased a minimum value was reached, followed by a maximum value at an inclination of about 130 deg. from the horizontal. In further papers Ozoe, et a1. [75-78] have generalized the work to include various heat transfer problems. For instance, in [75,76] along with three-dimensional numerical results, a 20 visual study presented the flow pattern in three orthogonal planes which agreed reasonably with numerical results for various inclination angles. Also, the average Nusselt number measurements showed significant dependence on the flow patterns, and results from both numerical and experimental works compared very well despite the differences in Rayleigh, Prandtl number and the finite length of the box. I In view of the very wide applications, natural convection was considered in doubly inclined rectangular boxes, and inclined rectangular boxes with the lower surface half heated and half insulated. The same authors showed in [77] that, the heat transfer rate is influenced by both the angles of inclination and rotation in case of two different aspect ratios and hence, the flow mechanism will be composed of one or two roll-cells each in the form of closed helix pairs. However, there was some uncertainty in the numerical results due to the extrapolation from finite grid size, and in the experimental values because of the heat losses involved in the long duration of the experiments. In [78] the roll-cells remarkably appeared on the heated segment rather than on the whole surface and apparently the result is, a decrease in the absolute heat transfer compared to that of a uniform temperature on the lower surface. Arnold, Catton, and Edwards [79] conducted an extensive experimental study to examine the influence of the angle of inclination and the aspect ratio on the heat transfer rate. At a given Rayleigh number their results showed an increase in Nusselt number as the angle of inclination increased from 0 deg. (heated from above), until a local maximum was attained at an angle of 90 deg. (vertical). However, a local minimum is always appeared for angles between 90 and 180 deg. , and smaller than the value at either 90 or 180 deg. for all the aspect 21 ratios. Further, it was noted that for smaller aspect ratio the local minimum becomes more pronounced and closer to the 180 deg. (horizontal). This occurrence of a local minimum predicts that the unicell and the Benard convection do- not superpose. Although, their results agreed reasonably with those of Ozoe, et a1. , the location of their minimum seems to be closer to the vertical. Values from their scaling law are valid only for inclination angles from 180 to 90 deg. Buchberg, et a1. [80] presented a review of the subject, in which a set of empirical correlations were obtained. These correlations have been used to calculate the Nusselt number which is of great importance in the application of solar collectors, where the free convection constitutes the main mode of the heat loss. Another interesting experimental analysis across inclined air layers was developed by Hollands, et a1. [81] . The Rayleigh number considered is from subcritical to 105 and angle of inclination from 180 deg. up to 110 deg.. Subsequent results, showed considerable deviation from the Nusselt values obtained from the horizontal layer when Ra is replaced by Raocos¢ , especially in the range l708< Raocosd <10‘ ,and 150 deg. < d < 120 deg. However, under these conditions their correlation predicted excellent agreement with the experimental data points, and fits closely all data as reported. Quite recently, local heat transfer measurements in inclined air-filled enclosures have been made by Linthorst, Shinkel, and Hoogendoorn [82-84] to further elaborate on essential features of the physical phenomena, at the side wall region. For the determination of the local heat transfer rates a holographic interferometer was employed. Results of the local Nusselt number measurements obtained on the basis of the local wall temperature gradients, provided valuable information on the physics of the heat transfer near the walls. At he be ef b0 31; an Eh em as: la in: 22 angle of inclination of 160 deg., local heat transfer rate measurements showed several maximum and minimum along the hot plate, while for 100 deg. the local heat transfer rate was maximum at the lower region and minimum at the upper region. In general, their predictions of the flow regime were consistent with other investigators works [65,71,72] . Interferograms for angles of inclination ranging from 180 up to 150 deg. and from 150 to 90 deg. were similar to those of the horizontal and vertical orientation.respectively. However, for the range between 150 and 130 deg. transition from one mode into another has occurred. More importantly, average heat transfer rate values were maximum at the horizontal when plotted for various orientations, and in good agreement with those of De Graaf and Van der Held for Ra-l.4x10‘. However, for Ra > 105 results compared favorably with those of Jannet and Mozeas [85] . A similar experimental study using interferometric techniques was reported by Randal, Mitchell and El-Wakil [86] , along with the local heat transfer measurements. Grashof number, inclination angle and aspect ratio effects on both the local and average Nusselt number have been considered. As a result it was shown that is no substantial effect of the aspect ratio on the average heat transfer in the laminar boundary layer regime. Correlations for both the local and average Nusselt numbers are expressed in terms of Grashof number and the angle of inclination, and indicated fair agreement with the results of other authors [80,81]. In further study [87] , the same authors extented their work to include enclosures of moderate aspect ratio, in which emphasis is made on the convection heat transfer dependence on the aspect ratios less than 4. It has also shown that, for the same aspect ratio adding spacers directed upward between the plates yield an increase in heat transfer rate and a decrease when oriented downward. 23 Some of the very recent contribution to the basic aspects of natural convection in inclined enclosures, was the introduction of, variable properties in the numerical methods and the establishment of the limits of validity of Boussinesq approximations. Zhong, Lloyd and Yang [88], and Yang, Yang, and Lloyd [89] have applied in their numerical analysis to a study of the effects of variable thermal properties, a very general law of dependence of thermal conductivity, viscosity and heat capacity on temperature. Throughout their study the various types of flow patterns and heat transfer were analyzed for a complete set of inclination angles. Conductive heat transfer was found to dominate the flow motion for inclination angles ranging from 0 deg. (heated from above) up to 45 deg. For larger angles of inclination convection was remarkably the prevailing heat transfer mode because of the apparent increase of gravitational effects along the differentially heated walls. Transition from unicell to three-dimensional oblique rolls have been seen at critical angle between 150 deg. and 180 deg. (horizontal), with a local minimum in heat transfer rate. Moreover, their correlation for the Nusselt number based on the full variable properties compared favorably with the results of [28,79] . Although papers in this field are still appearing [90-97] it would seem that, much is yet to be learned about the effects of inclined boundaries and their important functions in heat transfer. It is apparent from the preceeding sections that a substantial amount of theoretical and experimental work has been conducted to develop the important aspects of heat transfer in enclosures. However, local Nusselt number distribution has not received such detailed attention, especially for the heated from below configuration (Benard convection), which is not reported to the best of my knowledge in the literature. 24 1.2.3 Thermal Convection in Rotating Fluids Studies of the effect of the various physical factors on the motions produced in a rotating fluid have been conducted with great concern over the last two decades. Most of the theoretical and experimental works dealt mainly with slow rotational speed, and it was assumed that the fluid was in a solid body rotation. A monograph of the theory of rotating fluids by Greenspan [98] contains most of what is necessary for a basic understanding of the theory to treat both large-scale atmospheric convection and small-scale flows commonly produced in laboratory experiments. Further, the general theory of contained rotating fluid motions has been presented at length in [99,100] for viscous incompressible fluids with the development of new theorems and results. A second, somewhat similar monograph on heat transfer and fluid flow in rotating coolant channels was prepared by Morris [101] . Interest was centered on design requirement for cooling rotating components, beside an inclusive review of current research in cylindrical and rectangular ducts which are constrained to rotate about an axis parallel to the duct (parallel-mode rotation), or perpendicular to the duct (orthogonal-mode) which is important in cooling turbine rotor blades. The first treatment of heat transfer in a tube rotating about a perpendicular axis to cool turbine blades, was studied by Schmidt [102] . He suggested flowing cooling fluid through narrow radial passes in a blade opened to a reservoir of circulating cool fluid in the hub. The density gradients caused by the hot wall will interact with the centrifugal acceleration to force the cold denser fluid in the hub to 25 flow radially outward, whereas the hot and less dense fluid near the walls flows inward toward the axis of rotation. Mori and Nakayama [103], analyzed theoretically the fully developed laminar flow, and temperature field in a pipe rapidly rotating around a perpendicular axis using an integral method. Assuming the validity of a velocity and a thermal boundary layers analysis, it was shown that the resistance coefficient and the Nusselt number increase remarkably due to a secondary flow driven by the Coriolis force. In this event, the authors reported that the effect of the secondary flow predominates over almost the whole cross section of the pipe (core region), except near the pipe wall, where the viscosity and heat conduction effects are confined (boundary layer). The Coriolis force, due to the velocity component in the direction of pipe axis is normal to the main flow, while the Coriolis force due to the secondary flow and angular velocity is in the direction of the pipe axis. This secondary Coriolis force yields a characteristic influence on the flow, and distorts both the velocity and temperature fields from their patterns under stationary conditions. In a second report, Mori, et al. [104] extended their work to include both an experimental and a theoretical analysis of the turbulent regime with fully developed velocity and temperature fields. Applying a boundary layer technique, it was found that the increase in Nusselt number and flow resistance for turbulent flow, when compared to those without a secondary flow, are less than the corresponding increase for laminar region. Thus, the influence of a secondary flow produced by the Coriolis force is not as significant in turbulent flow as in laminar flow. The increase in mean Nusselt number is more than 10 percent, and a little variation of the local Nusselt number in the circumferential direction was shown to exist. Moreover, experimental 26 results of heat and mass transfer obtained by using the naphthalene- sublimation technique in laminar, and turbulent regions are in good agreement with the theoretical solution. Cannon and Keys [105] examined heat transfer to a fluid flowing through a pipe rotating about its horizontally aligned axis. That can be applied to cooling power transmission rotating shafts. Their experimental results revealed that rotation stablized laminar flow and delayed transition to turbulence to higher Reynolds numbers. However, in other cases transition was characterized by periodic bursts of turbulence due to local collapse of the rotating laminar structure, and hence heat transfer was found to be a mixture of laminar conduction and turbulent mixing of the bursts. Flow and heat transfer in a circular tube rotating about an axis parallel to its axis, have been considered recently in many theoretical and experimental works. Also the effect of rotation on the hydrodynamic and thermal characteristics was treated in detail according to its significant practical importance in the design of cooling systems for electrical machines. Morris [106] , and Davis and Morris [107] studied such a model to examine the influence of rotation on laminar convection when the fluid flows through a vertical tube rotating about a parallel axis subjected to a uniform heat flux. Solutions obtained for the velocity and temperature distributions by using a series expansion are valid only for low rates of heating. Their results indicated clearly that rotation induces a secondary free convection flow in a plane perpendicular to the axis, and caused a distortion in both the velocity and temperature profiles which modify the resistance to the flow and heat transfer rate. Secondary flows augment the flow resistance and heat transfer coefficient. Mori and Nakayama [108] employed an analytical procedure 27 to analyze the effect of the secondary flow on convective heat transfer to laminar flow in a straight pipe rotating about a parallel axis. Their analysis of fully developed flow under uniform heat flux showed that, the secondary flow is driven by centrifugal buoyancy in almost the whole region of the cross section, and distorts the axial flow and the temperature distributions. Apparently, this is similar to the gravity distortion in a horizontal heated pipe. However, the results indicated that the effect of Coriolis force is to rotate the plane of symmetry to a plane passing through the pipe axis and the axis of rotation. Consequently, this will reduce the secondary flow which reduces both the pressure drop and the heat transfer rate. Further, this phenomenon becomes more evident in a pipe with a small radius of rotation. and constant circumferential velocity, due to the increase of the Coriolis effect. Humphreys, Morris, and Barrow [109] experimentaly studied the characteristics of the local and mean heat transfer for air in turbulent flow through the entrance region of a tube rotating about a parallel axis. The authors pointed out that the development of the secondary flow components in a plane perpendicular to the main flow direction caused under the influence of a body force, an asymmetry in the thermal boundary condition, or an inherent characteristic of the flow. In any real situation, these components may occur simultaneously. However, in an attempt to obtain a clear understanding of these individual secondary flow effects, their treatment of this problem illustrated which one of the effects in dominant. Also their results showed a significant change in heat transfer over the range of Reynolds numbers and rotational rates studied. For higher rotational speeds, it was observed that the combined effect of entry swirl and 28 centrifugal buoyancy dominated the heat transfer, not the Reynolds number. Measurement of heat and mass transfer coeffecients were considered by Sakamoto and Fukui [110] for air and oil flowing through a rotating tube about a parallel axis. Results showed that the significant increase in heat and mass transfer over the range of rotational speed, 420-2700 rpm, are closely related to the increase of rotational number and Graetz number. Studies of flow and heat transfer in square-sectioned tubes rotating about a parallel axis with laminar or turbulent flow are few and very limited. Recently Morris [101], analyzed theoretically this case for laminar and turbulent heated flow. However, limited data are available to determine the flow resistance and heat transfer. Neti, et a1. [111,112] presented a finite difference solution of laminar heat transfer for air in a rotating rectangular duct of aspect ratio two. Their results showed the effect of the secondary flow caused by the Coriolis force and density gradient on heat transfer and pressure drop characteristics. Moreover, solutions for Nusselt number and friction factor were obtained in both the inlet and fully developed regions for different Grashof and Reynolds numbers. Study of flow and heat transfer due to rotating disks has been considered with great concern, in view of its relevant importance in cooling design and performance. Recently, Luk, Millsaps, and Pohlhausen [113] developed an exact solution for the combined free and forced convection in flows adjacent to a uniformly heated rotating disk. Solutions of velocity and temperature fields were obtained by numerically solving the system of governing equations. Riley [114] considered situations in which rotating fluid is bounded by plane surfaces having the same angular velocity. Initially 29 both fluid and plane are in a solid body rotation. However, changes in the boundary temperature disturbed the rotation of the fluid, since changes in temperature yields density changes which will modify the effect of pressure gradient, and thus a radial flow will be developed. Analysis was confined to fluids of small viscosity such that the resulting radial flow disturbances are effectively considered to be in a thin Ekman layer near the surface. By using a boundary layer approach, Edwards [115] obtained a solution for the problem of heat transfer at the boundary of a finite disk rotating with the same angular velocity as the fluid. Furthermore, Kreith, Dougham and Kozlowski [116] , conducted an experimental study of mass and heat transfer from an enclosed rotating disk. Correlations for heat and mass transfer were developed for turbulent flow, as well as the visual study of the flow pattern. Axisymmetric convection in a fluid contained between two finite infinite horizontal disks rotating about a vertical axis with the same angular velocity was presented by Duncan [117] . In his investigation inertial accelerations were neglected in comparison with Coriolis accelerations and viscous effects are confined to Elanan layers at the disks. Conditions required for either conduction or convection to predominate. The corresponding structures of the velocity and temperature fields were also discussed. Centrifugally driven thermal convection in a vertical rotating cylinder heated from above, have recently been considered by Homsy and Hudson [118] . By applying boundary layer methods, solutions for both conducting and insulated side walls boundary conditions, were obtained on top, bottom, and in the inviscid core of the cylinder, where the axial flow was strongly influenced by the horizontal Ekman layers. Critical parameters governing the solutions in all boundary conditions 30 were found to be the aspect ratio, the Prandtl number, the Ekman number and the thermal Rossby number for the flow. Moreover, gravity is seen to have at most only a local effect on the flow near the side walls, whereas the heat transfer was considerably increased by rotation. In a further study [119], the same authors extended their analysis to reveal the effect of the side wall and heat losses on the Nusselt numbers which was determined for the top and bottom surfaces of the cylinder, where constant heat flux boundary conditions have been applied instead of the isothermal conditions considered in [118] . Abell and Hudson [120] examined this problem experimentally in an effort to provide more physical insight into the mathematical analysis and the experimental data. Distinction between rotating and non-rotating convection due to the various important forces involved was presented in terms of the centrifugal acceleration, which is a strong function of the radial position, and the Coriolis acceleration that may play a Eignificant role in transfering heat as a consequence of the induced secondary flow. Nusselt number correlations were derived for low and high visosity oil in terms of thermal Rossby and Ekman numbers which reflect the influence of temperature difference and rotational rate in increasing the heat transfer rate. The stability of a horizontal layer of fluid heated from below subject to the gravity field and the Coriolis force resulting from a rotation about a vertical axis normal to the surface was examined by Chandrasekhar [121] . It was shown that the effect of the Coriolis force is to inhibit the onset of convection; and that the extent of the inhibition depends on the value of Taylor number. Critical Rayleigh number for the onset of convection as a function of Taylor number was determined for three types of boundary conditions: (a) both boundary 31 surfaces free, (b) both boundary surfaces rigid, and (c) one boundary surface free and the other rigid. Veronis [122,123] has conducted a numerical study of Benard convection in a rotating fluid confined between free boundaries over a certain range of Taylor, and Rayleigh numbers. In his analysis it was indicated that for Prandtl number greater than J2 the velocity and temperature fields were dominated by the rotational constraint even for moderate values (103) of Taylor number. That is, the horizontal temperature gradients is largely balanced by the vertical shear of the velocity component normal to the temperature gradient. On the other hand, for Prandtl number less than J2 , the possibility of a finite- amplitude instability at subcritical Rayleigh number exhibited a different structure of steady velocity and temperature fields from that of a fluid with large Prandtl number. Nonetheless the heat flux, once convection is established, was closely dependent on both the Rayleigh and Taylor numbers. Recently, an extensive study of Benard convection with and without rotation was presented by Rossby [124] . Stability of fluids was considered in detail relative to parameters describing its state such as Rayleigh, Taylor, and Prandtl numbers. Measurements performed on two Prandtl numbers corresponding to water and mercury exhibited markedly different behaviors. The results for water showed excellent agreement with the predictions of the linear stability theory. At Taylor number greater than 5x104, the presence of a subcritical instability appeared at larger Taylor numbers. Thus, the difference between the measured critical Rayleigh number and that predicted by Chandrasekhar's linear stability theory [121] is as much as 30 percent. Further, water exhibited a maximum heat flux at large Taylor which is an increasing function of Rayleigh number, and that was attributed to ‘Ekman-layer 32 like' modification of the viscous boundary layer. In contrast, mercury responded quite differently from water, and consequently the heat flux was found to be a decreasing function of Taylor number. This was evident, because oscillatory convection was the preferred mode which is inefficient at transporting heat, and no steady flow was observed. Nusselt number results were also plotted as a function of Rayleigh number for three values of Taylor number. Catton [125] has employed the Landau method to analyze the effect of rotation on natural convection in horizontal liquid layers. Convective heat transfer was measured over a wide range of Rayleigh and Taylor numbers. Results compared quite well with Rossby's experimental results up to a Taylor number 10!5 . However, the theoretical predictions were found to deviate at large Rayleigh numbers and this deviation was believed to be due to centrifugal effects in thinning the boundary layers. Effects of centrifugal convection on the onset of gravitational instabilities of a bounded rotating fluid heated from below was treated by Homsy and Hudson [126]. In their treatment, dependence of the dimensionless temperature, velocity field, and thus the Nusselt number on Prandtl number, Ekman number, aspect ratio, inverse Froude number, and thermal Rossby number was emphazised. Furthermore the authors pointed out that the centrifugal effects can account for the distortion of the vertical temperature gradient, and therefore, for the initial increase of the Nusselt number which was observed in Rossby's experiment. In addition, the effect of the centrifugal circulation away from the cylinder wall was found to increase the critical Rayleigh number at which gravitational instability may exist. For large Rayleigh numbers in excess of 1010 , Hunter and Riahi [127] conducted a theoretical study of nonlinear convection in a 33 horizontal layer of fluid rotating about a vertical axis. From their solutions, obtained by using the boundary layer method, it was noted that the influence of rotation should not always be to delay the onset of convection, because the heat transfer was increased in the intermediate range of rotational parameter. In addition, maximum heat transfer was attained when the thermal and Ekman boundary layers coincide. Using the Galerkin method, Clever and Busse [128] showed the existence of a steady subcritical finite amplitude two-dimensional solutions in a horizontal fluid layer heated from below for Prandtl numbers less than one. This was established by superimposing arbitrary three-dimensional cellular convection. Transition from two dimensional roll-like convection to three-dimensional cellular convection was also considered in detail in their analysis. In this event, results indicated that beyond a certain Rayleigh number for a large range of rotational rates, Coriolis forces enhance the Nusselt number, but there is no evidence that heat transport in a rotating layer will exceed the heat transport in a non-rotating layer at a given value of the Rayleigh in the case of large Prandtl number. A recent theoretical and experimental study of thermal cellular convection in rotating rectangular boxes by Buhler and Oertel [129] , was centered on the effects of variable Prandtl numbers. Linear stability theory with Boussinesq approximations was employed in the analysis and a differential interferometer was used to study the stability behavior and the configuration of the three-dimensional convection flow. The computations showed that the roll-cells change their orientation with increasing Taylor number, and concluded that the centrifugal forces dominate in high Prandtl fluids, while the Coriolis force dominates in low Prandtl number fluids. 34 In closing this review, it is fitting to mention the vital role played by the analytical, numerical, and experimental studies in describing the fundamental. aspects and demonstrating the influence of rotation on the convective heat transfer in ducts of arbitrary cross- section which are constrained to rotate in either a parallel or orthogonal modes. In addition, natural convection was only considered in enclosures differentially heated in the vertical direction (Benard convection, or heated from above), and rotated about a vertical axis passing through the center point of the enclosure. Unfortunately, all theoretical and experimental investigations to date have not included the influence of Coriolis acceleration and buoyant interaction between centrifugal and gravitational buoyancy on the flow and heat transfer in a differentially heated enclosure rotating about its longitudinal horizontal axis. Therefore the purpose of this experimental study is to investigate the effect of the additional controlling parameters, like the Coriolis acceleration and centrifugal buoyancy on the flow and heat transfer mechanisms relative to the non-rotating situation. The way in which rotation affects the variations of local and mean Nusselt number will be presented when the enclosure is at angular positions of 90 deg. (vertical configuration), 180 deg. (heated from below), and 0 deg. (heated from above). On the whole, the articles presented in this review are by no means the only important ones; but the intent was to emphazise on the essential part of the basic source materials related to the subject treated in this investigation. 2.1 1 consi of s: enclo uncle: Surfa angle basis abow incli enClc horiz CODVe c°ntii m2 MATHEMATICAL FORMUIATI. 2.1 Governing Equations In the present work, the problem of natural convection is considered in a square cross-section rectangular air-filled enclosure of side H and horizontal length L, as is shown in figure 1. The enclosure is differentially heated from the sides, which are maintained under an isothermal temperature conditions, whereas all the other surfaces are insulated. The inclination angle, 4, is defined as the angle between the horizontal plane and the cold surface. On this basis, inclination angles less than 90 deg. represent heated from above, and greater than 90 deg. heated from below. In addition, inclinations of 90 deg. and 180 deg. describe respectively a vertical enclosure (differentially heated in the horizontal direction), and a horizontal enclosure (heated from below). The governing conservation equations for laminar natural convection of Newtonian fluids with the gravitational body force included in the momentum equations are written as follows: Continuity Equation %§+V.(p3) - o (2-1) 35 36 3) Three-dimensional enclosure, rotating about the z-axis .9 I l I Y? I | \ I ' . T 4 -§ -9 v-v + (wxr) (2.12) [g] "(10% + 23 x 3 (2.13) inertial rotational 39 SKAJ k. 1'./ '——¢7—-'§‘{--- -- __.___.___’.'_\__ \ ' \ >4 \ \ \. \ Figure 2 Rotation referred to a Cartesian frame of reference, O-XYZ. an: em 4O Substitution of equation (2.12) into equation (2.13) gives, [BIZ] -[Dt (;r+;X-£)] +3(vr+3xf) inertial rotational [1):] ' [Dc + 200 x V: + w(w x r) (2.14) inertial rotational Note that the two terms from equation (2.14) involving the angular velocity of the rotating frame are used to determine the acceleration with respect to the inertial frame oxyz. The term 3 x (:1 x it) represents the centrifugal acceleration, whereas 2; x 3r represents the Coriolis acceleration. It is also desirable to indicate that, 3 x (3 x f) - - 02:1 , f1 - xi + yj (2.15) and 2-5 -5 -5 V (v + w x r) - V .vr (2.16) V(v +23xr)-v v r r Under these conditions the momentum equations (2.2) yield, using equations (2.14), (2.15) and (2.16), -5 2-5 2-5 l -5 -5 p Dt + 2pw x vr - p0 r1 - - Vp + pu V vr + 3 puV(V.vr) + pg (2.17) U! 41 Applying the generalized Boussinesq approximations [118,130] in the equations of motion, which are based on the assumption of constant physical properties, except the density when multiplied.by the centrifugal force or gravity, and negligible viscous dissipation. The equation of state of the fluid is, p - Pcll - fi(T-Tc)] (2.18) and the continuity equation is v.3 - 0 (2.19) these can be substituted in equation (2.17) to give, Dv Vp __z - 2-5 - - -_l . -> -5 - - 2-5 - - -5 m: w v ”c 25 x vr 5(1 TC) 0 r1 5(2 TC)g (2.20) where, 2 2 p1 - p + pcsy - pch' (0 r1 )] (2.21) and p1 is defined as the perturbation pressure. In this respect, equations (2.3), (2.19), (2.20) can be written in nondimensional forms. As follows 55¢ l 2 L114- '0 [he Fc be 73 42 r _ a? (2.22) n Continuity Equation v.Vr - 0 (2.23) Energy Equation 21 2 Dr - v o (2.24) Momentum Equations 0v L—I-Ver--VP pr Dr 2 ° (Ta) 0‘ X Vr) - Raro R + Ra 0 j (2.25) For the onset of steady convection the following set of equations can be deduced from (2.23), (2.24), and (2.25) v.3: - 0 (2.26) Vr.vo - v o (2.27) it re Ef the C0: 43 l— - 2 -9 - - - 1/2 -5 -> - -+ g» Pr (Vt-V V )Vr sz (Ta) (k x Vr) Raro R + Ra 0 3 (2.28) The important dimensionless parameters entering the equations of motion are Pr- 9K (Prandtl number) sfin3l O O O O i>| | #0 «~03: OHUflfl—Jz :00! OH aha—Duh a: how}: so. ...2 . a: 1: u 1 .—«-9.« q d u q “ddqqd q 1 Id —-qdqd - 1 q " on ..X e. ..s a. ..+ a: ..0 on. ..4 2: ..n ..N ..000. oaocc xuos uceeeum ..n 4 owl... -n o I]. .. \ anal @ A . .603 392 .4. s\ so: .uom Shoes... ..m ..m o. qmnu 3 I assnn U93“ 2:: 'n 82 Figure 18 illustrates the general trend in which the inclination angle affects the mean Nusselt number when plotted against the Rayleigh number. Moreover, it suggests a significant improvement in heat transfer with increasing the angle of inclination. 4.1.2 Local Nusselt Number Results In this sub-section the analysis of the experimental data will be centered on the influence of the inclination angle on local heat transfer rate and flow behaviors. In addition, a series of interferograms will be presented in this connection along with a smoke flow visualization to facilitate understanding of the fundamental aspects of the convective structure. Local Nusselt numbers will be presented along the hot and cold surfaces as a function of the non- dimensional enclosure height with the angle of inclination as a parameter varying between 0 and 180 deg. Figure 19c.shows the profiles of Nusselt numbers in terms of the non-dimensional height at 0 deg. (heated from above). It is clearly noted that the local values of the heat transfer rate are almost equal to one as a consequence of pure conduction along the differentially heated walls except near the ends, which is an evidence of a convective flow existence. The flow pattern picture, Figure 19a, indicates this convective nature near the corners, however the flow is extremely slow because of the gravitational stable condition. It is for this reason mainly that the results of mean Nusselt number are different from unity, as reported in previous numerical and theoretical calculations which utilize Boussinesq approximation (linear variation of the density) and adiabatic boundary condition. The later is a basic element required to eliminate the end walls effect. Moreover, the interference 83 Hot w‘all Hot wall Cold wall _ Cold wall a) 1’10"! pattern in xy-plane b) Interference fringe at Ra- 3.0x 10 pattern, at Ra= 1.1x 105 8 L U Athotsfll + Al told Iall Local Nusselt number, Nu .b 9 “fimsgameafiefiluwg o 1 1 l 1 L I I l l 0.0 0.2 0.4 0.6 0.8 1.0 Y/H c) Local Nusselt Number distribution along the hot and cold walls, at Ax=1.0 and Ra= 1.1x 105 F' ure 19 Effect of inclination angle on the 19 flow and heat transfer, at ¢ = 0 deg. 84 fringe pattern of the isotherms, Figure 19b, depicts a stable stratified condition, but the isotherms bending near the end walls indicates the existence of some convective motion. Another aspect to consider from the isotherms picture is the fact that they represent the thermal boundary layer configuration in the immediate vicinity of the differentially heated surfaces. Thus a qualitative understanding of the local heat transfer distribution may be obtained which in turn is related to the inverse of the thermal boundary layer thickness. For inclination angles of 30 and 60 deg. , the existence of the convective flow near the heated walls is noticeable as is shown in Figures 20a and 21a. At 30 deg. , Figure 20a shows that the core region is almost quiescent and the flow is confined to the boundary layer region. At 60 deg. , as shown in Figure 21a, the driving potential of the convective flow is extended to include not only the boundary layer, but also the core region. This fact is clear from the appearance of two symmetrical vortical tubes that reside near the centers of the differentially heated walls. Actually no matter how slow the flow is inside these vortical tubes, thermal energy is still be transported via convection in the core region. Accordingly the interference patterns of the isotherms, Figures 20b and 21b, will make the above analysis more understandable due to the observed development of the thermal boundary layer regions near the differentially heated walls. Also, the thinning of the thermal boundary layers seems to be strongly dependent on the angle of inclination, which in turn yields a consequent increase of the convective flow . In accordance with this qualitative agreement between the flow and temperature fields, the experimental data of the local Nusselt numbers, Figures 19c, 20c, and 21c, demonstrate quantitatively the effect of inclination angle on the heat transfer and the transition process from the conduction dominated gravitationally stable condition 85 a) Flow pattern in xy-plane at Ra- 3.0x 105 Local Nusselt number, Nu c) Figure b) Interference fringe pattern, at Ra= L1X105 8 _ u A! hetvsll + M told sell 5 .— 4 _. -G a + + o a m + + ‘3 a ... + 2 ' ° e + a + + + Cl + + + + + m a a m~ m m l | l l l l l l l 0.0 0.2 0.4 0.5 0.8 Y/H Local Nusselt Number distribution along the hot and cold walls, at Ax-l.0 and Ras L1x105 20 Effect of inclination angle on the flow and heat transfer, at (P - 30 deg. 86 ' a) Flow patter? in xy-piane at Ra-3nx10 b) Interference fringe pattern, at Raa 1.1x 105 8 g . D M to! sell a + M cold well La .8 s — 5 . m + .1.” a + o 4 + + n F' + + VI :1 G a + D . + a i + + m o 2 J" a U D 3 _ + a a El 0 I l l 0.0 0.2 0.4 0.5 0.8 l Y/H c) Local Nusselt Number distribution along the hot and cold walls, at Ax-l.0 and Ra-14x105 Figure 21 Effect of inclination angle flow and heat transfer, at on the - ¢ = 60 deg. ... a 87 at 0 deg. (heated from above) to a convective thermally stratified condition at 60 deg. When the enclosure is in a vertical configuration, gravitational buoyancy becomes very apparent in a thin layer of fluid near the wall region, and buoyancy-driven convective flow will be the preferred mode for transporting energy. The effect of gravity on the flow can be noted in Figure 22a. The first development in this event is the repositioning of the vortical tubes, near the upper and opposite lower edge of the hot and cold walls. This consequent skewness in the vortical structure is responsible for the change in direction of the velocity field in the core region. It also tends to minimize the heat transfer from the heated wall by thickening the thermal boundary layer as seen in Figure 22b. Based on these factors, the experimental determination of the heat transfer rate, Figure 22c, showed maxima near the lower and opposite upper corners of the hot and cold walls respectively and minima on the other facing edges. These findings are in good .agreement with both the flow and thermal boundary layer configurations. A further increase of the inclination angle beyond the vertical, will permit the heated from below configuration to commence its influence on the flow pattern and energy transport process. Figure 24a at 120 deg. shows the flow structure as a single roll-cell oriented with the enclosure longitudinal axis (z-axis). A growing secondary flow near the upper and opposite lower corners of the enclosure also appears. Under these circumstances the corresponding developments in the thermal boundary layers are illustrated in Figure 24b. This in turn will provide the necessary information for the local heat transfer distribution, and demonstrates clearly the shifting of the maximum value of local heat transfer toward the center of the differentially heated walls. This is related to the flow development in the Cold wall Hot wall Cold wall Hot wall .———————————. a) Flow pattern in xy-plane b) Interference fringe at Ra- 3flxlo pattern, at Ra: L1x105 8 :I z _ a M hot "ll 1: + M cold well 0 E 5‘ m a I: _ D D G + + U B a ... + + + ".3 a w :I + a z - + a H 4' m 8 2 4' + m m 3 + + '3 m 0 L l . 1 . 1 . l . 0.0 0.2 0.4 0.6 0.8 1.0 Y/H c) Local Nusselt Number distribution along the hot and cold walls, at Ax-1.0 and Raa le105 Figure 22 Effect of inclination angle on the flow and heat transfer,at (,0 - 90 deg. Y.) Lu 89 .000 won I Av use. m3 x: sum .o.alx4 as has Lou .maams oaou one no: on» anode cowusnuuumwn panes: naemmsz Hmuoa nu assume . E» . . . . . F m o m 0 ¢ o N o o o 4 _ _ . _ . _ . o a a + + B + + l N U +. I mu B .+ +. nu n nn 1. +. +. nu .1 ¢ N + + + + a a n. .... I... B I 3 n— U u B I mm .J In: ...: .< +. .. N In: «3. z n. n a) Local Nusselt number, Nu c) 90 Flow patter? in xy-plane, at Ra= 3.0x 10 Interference fringe pattern, at Ra- 1.1x 105 _ C! M hot "II + M cold "I! El E! El . a a a + m + a + + + + —a o + e + 1+ + a + .. + + a _ + B + + m a a l . 4 l I l l . .4 0.8 0 8 10 0 0 0 2 0 Y/H Local Nusselt Number distribution along the hot and cold walls, at Ax-l.0 and Ra=14x105 Figure 24 Effect of inclination angle on the flow and heat transfer, at ¢ - 120 deg. .moo m3 .6 use. m2 x3 aux .o.aux< am has L0u .maaas taou use so: 91 on» meoHe consonauummo Loses: uaemmsz Houoa mm vacuum . I; . . .. . . ._ no oo .‘o as oo . . s . _ . o n. a + B JN n. .+ I .u + a ++ ++ 14 +. nu nu l. ++++BB U 1 .u [a 2:28: a 2:12: 0N 'Jaqmnu atassnn {9301 92 hydrodynamic boundary layer near the wall region. Figures 23, 24c, and 25 at 105, 120, and 135 deg. of the local heat transfer experimental data can be used to notice the shifting in the location of the maximum value of local Nusselt number distribution along the walls. At inclination angle of 150 deg., Figure 26a, the longitudinal roll-cell is well defined and nearly in a circular form. In addition, the secondary flow which is developed at the upper and opposite lower corners of the enclosure is almost stagnant, which in turn may lead to a reduction in the convective heat transfer. The interferogram for this case, Figure 26b, shows the corresponding effect of the circulating flow on the temperature field. The thermal boundary layer clearly demonstrates the shifting of the location of the maximum value of local Nusselt number from the near ends toward the center of the heated walls, which is proportional to the inverse of the boundary layer thickness. For further insight to this process, the experimental data of local heat transfer plotted in Figures 26c and 27 at inclination angles of 150 and 165 deg. reflect these important features of the thermal boundary layer and the flow pattern. Accordingly, it is worth pointing out that the location of the local maximum value can not exceed the midheight of the isothermal walls. This is due to the buoyancy-induced secondary flow in the outer region near the upper and lower corners of the enclosure. The onset of cellular convection at 180 deg. (Benard convection) is shown in Figure 28a, which presents a picture of the flow pattern in the yz-plane, where the flow is aligned in a series of roll cells with their axes normal to the isothermal walls. Furthermore, the subsequent unstable temperature stratification in the vertical direction is illustrated by the interference fringe pattern Figure 28b. It is apparent, from the isotherm structure that the temperature 93 a) Flow pattern in xy-plane, b) Interference fringe at Ra-3ox105 pattern, at Ra=11x105 8 a z o AlhotIaH ~ + M cold "ll 8 a — E E. I: ' a '3 ‘3 . a + + + u E El + '3 4 — + El + a a + a + a -a + a + z + a g 2 J- + a u + a a o I I I I I l I I l 0.0 0.2 0.4 0.6 0.8 I. Y/H 0 c) Local Nusselt Number distribution along the hot and cold walls, at Ax=1.0 and Ra=14x105 Figure 26 Effect of inclination angle on the flow and heat transfer, at ¢- 150 deg. 94 .36 mg I A» 0cm. .o.AIx< um awe ecu mo_Xas «mm .mgges oaou can be: ecu mcon cofiusnduunmc Lanes: uaommsz Hmuoa hw ousmmm . z; . . . . . _ no no as we so 1 — 1, J 5 d d — u c + nu +. 4. nu 4. nuI l. .u n. L. l. + n. I? + + a a n. .. nu ++ n. n. nu [a 2:23: . ......s .< nu 'Jeqmnu itassnu {e301 95 Cold wall Hot wall a) Flow pattern in yz-plane, at Ra= 3.0x 105 Cold wall b) Interference fringe pattern, at Ra=1.1x105 Hot wall 8 5 . a A! hot wall 2 + M cold wall I: s — 0 n E! El £1 a ' ¢ + s m a a + + Q g + _ + I3 a 4. El '5 4 $1 -+ m + + VII 3 in "E g + H 2 — 8 o L u-l o I I I I I I I I I 0.0 0.2 0.4 0.5 0.8 1.0 Y/H c) Local Nusselt Number distribution along the hot and cold walls, at Ax=1.0 and Ra=14x105 Figure 28 Effect of inclination angle on the flow and heat transfer, at (P a 180 deg. 96 distributions along the insulated walls are almost linear, while the core region remains nearly at a uniform temperature. Close to the isothermal walls, as is shown in Figure 28b, the thermal boundary layers indicate the existence of two maxima of local Nusselt number which are symmetric about a vertical plane passing through the center of the enclosure. The measurements of local Nusselt numbers displayed in Figure 28c, are in good qualitative agreement with the picture which emerged from the thermal boundary layer structure. So far in this sub-section the investigations have been entirely focussed on the influence of inclination angle on some hydrodynamic and thermal aspects of the flow, besides the measurement of local Nusselt number. Let us now consider the experimental data from these analyses in relation to the available experimental and numerical data, in order to outline some levels of agreement. A comparison of the present experimental data for vertical enclosure with the experimental data of Bajorek, and Lloyd [39] and numerical predictions of Catton, Ayyaswamy, and Clever [28] is given in Figure 29. Agreement with the results of Bajorek, and Lloyd [39] is very good. Their experiments are performed under similar boundary conditions. Although the trend of the numerical predictions of [28] are similar to the experimental data, the quantitative difference is large near the ends of the walls. The calculations in [28] are presented for adiabatic and perfectly conducting boundary conditions. Thus, it is interesting to note from these results near the ends of the walls, the difference in the meaning between insulated and adiabatic boundary conditions. Figure 30 presents another comparison of local Nusselt number ‘variations from various experimental and numerical results. In general terms the experimental data of [39] are in excellent agreement with the 97 .men cm I A» use 5.752 be ..So ecu x; consensus Len—Es: Somme: anus uo confinemeou mm 95m: o e F m e o o e o ¢ 0 o N e c c a o q — q - u — a — d o o o r/ h ’14 ~ a Au 4 j / .I n.~ Iafi a. I . ... § 1 n / N M. I a .. s e a . ‘ I. I . , e . H II I O D W. D [In] \\ I “a. I a a [fl/III \ ‘ ”a... no. Xodumm .3 .33 .uom III. a 4. 4]!!! \ I u . D 4 I.” \ B I. .25....3 .8 .32 small /!..I\ 4 I mé m a 4 B we Xmé and .nmna .umm q a a m 4m.— .. :1 I N no— x3 Iom . xuoa accused _U I n ad— 98 600 cm I 6 :2 o.— o.o o.o v.o one .o.au~< um .umm Lam c.o 33 dual... 3: Jamal sexism .33 com a name GHOU no. x: Isa . sue) accused U n°~ In“ as x3 In: comusnmuumwc Lanes: uawmmsz saved mo comfiuemsou on eusmwm «.0 fl o.o nu 'Jaqmnu atessnu {sooq 9.:— 99 present results. However, the numerical predictions of [19,60] , after assuming similar local Nusselt number variations along the hot wall, are in a quantitative disagreement with the experimental data and even with each other, this could partially accounted for the different grid- sizes and the calculations schemes used in their numerical analysis. 4.2 Natural Convection in a Heated Rotating Enclosure In this section the simultaneous influence of the Coriolis, centrifugal and gravitational forces on the natural convection heat transfer will be presented for the case where the air-filled differentially heated rectangular enclosure is constrained to rotate about its longitudinal horizontal axis (z-axis). The influence of rotation on the behavioural pattern of natural convection, can best be described by referring to equation (2.25) in Chapter 2 which introduces the Taylor and rotational Rayleigh numbers which emerge from the Coriolis and centrifugal terms respectively in the momentum equations. In this instance, the rotational Rayleigh number Rar is similar to the Rayleigh number Ra encountered in the gravitational buoyancy but, the gravitational acceleration is replaced by the centrifugal acceleration. In a rotating flow the Taylor number Ta describes the importance of Coriolis effect. Measurements of local and mean Nusselt numbers are presented as a function of Tayor (Ta < 10‘), and Rayleigh number (104 < Ra < 3x105) at angular positions of 90 deg. (vertical enclosure), 180 deg. (heated from below), and 0 deg. (heated from above) for the rotating enclosure. 100 4.2.1 Heat Transfer Results in a Heated Rotating Enclosure at Angular Position of 90 Deg. (Vertical Configuration) In view of the comments made earlier, the influence of rotation on the heat transfer will be discussed based on the local and mean Nusselt number results. In this event, the effect of the Coriolis- induced flows will be characterized by the Taylor number, Ta, and the centrifugal buoyancy induced flows by the rotational Rayleigh number, Rar , which tends to reduce the Coriolis acceleration effect at high rotational rate. At low rotation rate the Coriolis effect will be very important due to the induced cross-secondary flow near the differentially heated walls which causes the fluid to recirculate toward the core region. At this point, the centrifugal buoyancy effect is less significant than the Coriolis effect and results in a slight distortion of the temperature distributions. Figure 31 compares the variations of the local Nusselt number along the hot and cold surfaces under rotating and non-rotating conditions. The reduction in local heat transfer at a Taylor number of 1.12x103(-6.l rev/min), is very significant compared to the non-rotating results as shown in the figure, although the same qualitative local heat transfer distribution is maintained at this level of rotation. The decrease in the heat transfer is related to the increase of the thermal boundary layer thickness near the wall region. This is largely generated by the effect of the influence of the Coriolis force relative to the centrifugal buoyancy. This effect promotes the mixing and thereby reduces the temperature difference between the heated walls. 101 The complex interaction of the Coriolis and centrifugal effects on the flow and heat transfer behaviors starts to develop at a Taylor number of 2.27x103(-8.5 rev/min). Figure 32 shows the importance of the Coriolis and the centrifugal forces. It is apparent from the local heat transfer variations the amount of distortion caused by the interaction between the centrifugal buoyancy flow which tends to improve the heat transfer and the Coriolis effect which is still influencing the temperature and the flow fields. At Taylor numbers of 3.28x103(-10.2 rev/min), and 4.69x103(-12.5 rev/min) , Figures 33 and 34 reveal the effect of centrifugal buoyancy through observation of the enhancement of the local heat transfer with increases in the rotational rate. It is interesting to note in Figure 34 the movement of the local maximum of Nusselt number to the upper end of the hot wall from the lower end for the non-rotating condition. This in turn could be a result of an interaction between the centrifugal buoyancy, which has the tendency to force the cold denser fluid to move towards the outer region of the enclosure, and the gravitational buoyancy, which in turn maintains the boundary layer flow in a relatively thin region near the heated walls. Hence the hydrodynamic and thermal boundary layers near the walls will grow and the movement of local Nusselt number maximum was observed along the hot and cold surfaces. Figures 35 and 36 show how the local Nusselt at Ra-l.2x105number responded to increases in rotational speeds as characterized by Taylor numbers of 7.10x103(-15.l rev/min), and 9.66:10 (-l7.5 rev/min), respectively. The important feature to observe is the increase in local heat transfer as a result of increasing the rotational speed. However, 102 e .Acouuousouucoo acouuuo>v .aau c.m + Iouou accomuouoL m2 Xe... Ioz 6.7.2 as 3o Lam 5:9: 300 one so: was means coqusauuuuun Lanes: uaoemsz Houoa an ousmum o.o . I; . . . . o. a o a o ... o N o q d d u d u — q I .yI .u .+ In mm. + + + +1 I me n. I -r .+l_.+ n. mg n. I? T IT . D a .L .T . . s s I _I _l l_ _I a: co -T ......g... a ... .21.... .. - convenes cud: couuouOL snore“: o “N '1anqu atassnu '[RDO'I 103 .acouuousmfiucou amouuue>v .EQL m.o + Iuaou decouueuoL .auaos pace use no: ma_x~s Iom .o.HIx< so has saw ecu occao cowusnmuumwc bones: saunas: nouoa an ousomm . :\> . . . . . o — m c m o w o N o o o u _ q — — . _ - D If... .u .+ 4. I B + + l N n. + + B + I? 1. .+ .u nu nu n. I II a B m I_ n” a? ¢ m _- .l N ._ .. a s I _I .u I. mm .1 w m. on on a. a _"_usuuog “« U ..o: so; .< I I u gonna?“ :33 convenes £59.33 M 104 the general character of the local distribution of Nusselt number remains almost the same. This in fact implies an effective increase in the relative strength of the buoyant interaction which tends to improve the local heat transfer. On the basis of the above discussion it may be proposed that at low rotational rates the reduction in local heat transfer is attributed mainly to the Coriolis-induced secondary flow in the xy cross-plane. Increases in the rotational rates will then produce a relatively thin boundary layer near the wall region which tends to improve the heat transfer as compared to the results at low rotational speeds. So far, the discussion has focussed on the influence of the controlling parameters on the heat transfer results, and the buoyant interaction between centrifugal and gravitational buoyancy at a single gravitational Rayleigh number 1.2x105. An assessment of the effect of gravitational buoyancy on the local variations of Nusselt number will reveal more details on the inlfuence of Coriolis acceleration and centrifugal buoyancy. In this respect, Figures 37-40, demonstrate that the general trend of the local heat transfer distribution at gravitational Rayleigh number of 2.0x105, at various rotational speeds is similar to the local distributions of Nusselt number at Ra-l.2x1015 as presented earlier for various rotational speeds. However, a related improvement in heat transfer is evident in connection with the increase of the operating temperature difference between the walls, which resulted in increasing the buoyant interaction effect on the flow and temperature fields. 105 .Acouuousumucoo Hmumuuo>v .aau m.o~ + nouns docowueuoL . waxes Inc 6.7.2 be to Lou .23: 33 one no: sea mcodo couponuuumdo Lanes: saunas: Houoa mm ousmmm nu 'Iaqmnu itassnn teaoq I; o._ w.o o.o ....o N.o o.o - — q — A q — — u o a + + + I B B + .+. + B B B D N + m. s . a + + + + + .. a a a + + a a + a a .I v 1 o I: :3 L< + I 23...... 3. D m 106 .AcowuousmuucOU Houmuuo>v .eau m.~a + Iouou decomuouou o o nu 'qumnu :Iassnn Ieooq . ae_xas Io: .o.HIx< no one ecu .mHHos oaou oco uos ecu moods coauonwuumwo Lanes: uaummoz Hoooq em sesame .I\> . . . . . _ no mo two No C . _ _ A I _ .4 .+ .1 +8 ++ U 08+ .+ B B I + B "U .+ II B + + GB + __oa u_ou .< .+ I __oa no: u< U 107 .Acofiuousmwucoo Houwuuo>v .emu o.ma + Imuou decomuouou no_x&a Iom .o.HIx« an own new .mdamz oaou Oco so: an» anode cowusnauumwo Lanes: uaemnsz Houos mm ousofim ~ o.o ¢.o N.o :3. 33 I :2. .2. 1 I... U «a _ _ 4-0 B -+ o nu 'qumnu itassnn teooq 108 ..couuousm.ucou ~oo.uuo>. .emu m.s~ + Isuou .oc0.bouoL . oq_XQ_.Iom .o.AIx< on bus Lag .ednos unou one bag or» acoHo comuan.uum.v honest uaommsz Houon an ousmum . I; . . . . . . o. m o m o tn. 0 N o o o q — . d . — q A _ O n s s n. +. .+ _I mm + n. + +I a a .. n. I B n. + I D + 4. _I l_ _U .FI ¢ 0 + u + s 1. n. I. _I I. I. .I I .+ I? a a + + B B .L o n. ...; ...ou .< ... I ...; .3. .< D :3. .3. .< I couuouoL as“: :ouumuoL booms“: m nu 'Jaqmnu atassnu Isooq 109 At a higher Rayleigh number, 3.0x105, obtained by increasing the temperature difference between the two walls of the enclosure, Figures 41-45 suggest a relative improvement in _the heat transfer with increasing Rayleigh number. However, the general trend of the local distribution of Nusselt number was in a good qualitative agreement with the results of local Nusselt number at Ra-l.2x105 for various rotational speeds. In this case the centrifugal buoyancy effect on the local heat flux distribution has occurred at a rotational speed of 12.2 rev/min instead of 8.5 rev/min as described earlier, due to the combined effects of the Coriolis and centrifugal acceleration on the flow. This would imply that in the case of hot air an increase in the tempreature difference between the walls can enhance the Coriolis effect on the flow and decrease the heat transfer. Figures 46-49, illustrate the local distribution of Nusselt number at Rayleigh number 7.4x10‘. It is interesting to note in these figures that the buoyant interaction commences its influence on the flow and heat transfer at rotational speed 6.1.rev/min, compared with the local Nusselt number results presented in Figures 41-45. Hence, by decreasing the imposed temperature difference between the walls, the Coriolis effect on the local heat transfer variation can be made less significant. Thus far in this analysis the influence of rotation on the local Nusselt number variation showed a significant response to different ~values of the temperature difference between the walls. This was largely a consequence of the buoyant interaction. In this accord, the influence of rotation on mean Nusselt number can be explicitly connected to the above analysis. Figure 50, illustrates the results of 110 ..comuousmfiucou Houuuuo>v .Eau m.o + nouns accomuouOu . as x34. In: 6482 us .So Lou ego: 300 can no: on» use: comusnmuuumn Lanes: Seems: Hooo... 5m 0.5m: . I; . . . . . o— m o m 0 ¢ 0 N o o o a _ a _ . q q _ a O m. + + I N n. B + + + E n. n. W n. W + + ... + I + + a a m. + . . ...... + + a s1 a. m n. m. B I a u .I m m :2231 + - . __....__2 u m 111 ..comuousmmucoo soouuuo>v .eau ~.- + nouns decomuouOL . mc_XQ« Iom .o.HIx¢ be u.o saw .maaos naou one no: on» mcouo co.u:n.uum.c cones: buomnsz Aeoos mm ouso.m . I; . . . o— m o o o e o N... o.o . _ I _ . q . _ . o a + N +. .J a + + + a a + m n. a + - + + + + + I B B .+ B I. ¢ N G n. m. n. m . u I a m :32: i + .. .n ...; ...... .< n. n .Acofiumuammucoo anuwuuo>v .Eau ~.m~ + uuumu HmcofiuouOu . m2 x3 lam 5.752 3 u: ..Ou 6:03 300 can yo: an» ocean cowuanfluumwo hone:c uaonmaz douoa mm ausmwu 112 . I; . . . . . o— w o m 0 ¢ o N o o o q _ d J— . A q — d O ...—1 II. B + N + + + + m + a a 1 1 4. .T nu Hu 1- m U B D B I D E + I V W B B + + a a a a + + .. m u i m ... :3. 33 2 + 1 m __o; “a; u< D 113 .Acouuousmfiucou Hauwauo>1 .Bnu m.ha + loamy HucowumuOu . n2 x313. 6.7.2 an in you 5:2. Sou can go: an» ocean cofiuanwuummo umnsac uaommaz anuoq ow ouaowm . I; . . . . . o — m o m o V o N o o o . _ q _ . _ . — q C +i N nu 1. 1. 1. n. n. fig 1. n_ .T nu .T .U I1 ¢ 1. _U B B + + .. mg n. 1. 1. 1. .1 a H. n. :0; v.00 u< 1. 1 I... go.— .< n. an ‘Jaqmnu atassnn 19301 114 § ..coduuuamwucoo Huufluuo>1 .aau H.@ + lousy Hacowuquu mc_xad lam .o.Hux< um nun new .maama oaou can go: on» macaw :owuanuuummv hunsac uaoumzz H0004 av ouaofih m.o ....o «.0 o.o :2. 33 2 :3. .... 2 1T _ q 1 o l ‘9 I no @— nu 'Jaqmnu atassnu 13301 115 .Acomuousmwucou amuuuuu>1 .aQu m.» + nouou aucofiuouOu . ma x...» no: 6.752 3 a: new $32, oaou can ya: onu ocean comusnuuuawo gonad: uaommaz Huuoa ~¢ ousmmm I; o 1 m o m o ¢ o N o o o 1 — . fl q — 1 1 - O a a + + 9? II. N a + a + + .u nu .+ 1 a + + a m l 1. 1. 1. .+ .u 1 1. 1. .+ .+ nu nu nu n. B I o n. _u :2. 3.: 2 + .. __ou 1.:. .< nu - an ‘Jaqmnu :Iassnu 19301 .Acoflunuamflucou Hmumuuo>v .aQu «.ma + noun. docowuquu . no. x...» Inc 6.7.2 an .30 .3... .325 300 0:0 ac... any anode :oduanuuuamv hogan: adummaz Houoa no ousmum 116 _._\> o1 m.o 0.0 ¢.o «.0 o.o . A . — . J 1 d d D .4 IT n. 1. I. N a + + + + B + n. n. 1. + + + l n. n. a B + m a. nu ". + + l w N n1 5 1. a 1. nu ha 1 "H a n. n. W .1 m ... :3. 3.: I + - m. .3313. ~< n. 117 ..cowuounmmucou Huufiuuu>v .Enu a.mH + noun. HucomunuOH . max...” .3. 5.72 an .2. .8 .2.2. Boo can 3.. on» macaw couaanmuumwu hoses: uaommaz doooq v¢ «gamma . I; . . . .. o F m o m o w o N o 0.0 . _ . _ . _ . 1 . o L + l N + n. 1. +. +. .+ nu ma + a + 1 m B D -1- D n: -T n: 1. .1 w “u . -T n -T mm s + a a a a a 1 n. D m. u .. m m ... Z: :3 1< + .. N 2:10; 3 U n 118 .Acowuuusuuucou Haufluuo>1 .aau m.na + noun. Hmco1uou0u . m2 Xe.» IS. $4.52 an a: new 5:2» ":00 9:. ac: osu ocean cowu=n1uuufic hangs: uaumnaz Houoa mv ousuwh . I; . . . . . o— no no to we so fi ~ ~ ._ 1 fl 4 _ d O L l N n. + + + + + B 1. .1. B B B 1 + + + a ma m L ... 1. B 1. B a + B - a a + + n. a l o I: 33:. + 1 29.12.: D m nu ‘Jaqmnu atassnu 19301 119 mean Nusselt number obtained with all rotational speeds at various gravitational Rayleigh numbers. This figure suggests that the Coriolis acceleration becomes very important at low values of rotational speed. At Taylor number of 2.1x103(-8.5 rev/min), it is observed that the mean Nusselt number attains its minimum value which indicates quantitatively the interaction between Coriolis acceleration and buoyant forces. However, with a further increase of the rotational speed the mean Nusselt number gradually increases due to the prevailing effect of the buoyant interaction on the flow and temperature fields. Figure 51a, shows the influence of rotation on mean heat transfer when plotted against the gravitational Rayleigh numberu Ift is evident that a gradual increase of mean Nusselt number occurs due to increases in the rotational speeds. Figure 51b suggests a similar improvement in heat transfer when plotted as a function of the rotational rate, with the gravitational Rayleigh number as a parameter. This is a clear indication that, the buoyant interaction enhances the mean Nusselt number values. Furthermore, The experimental data displayed in Figures 51a and 51b, are correlated by the following equation, Nu - 0.068 Rao'123 Tao'288 (4.2) this equation predicts the experimental data within i 4 % over a 4 5 3 Rayleigh number range of 10 - 3x10 , and Taylor numbers between 10 - 4 10 . Figure 52 shows a comparison between the experimental data and the values from equation (4.2). 120 ..cowuouamwmcoo Houfiuuo>0 .emu m.0 + nouns HocowuouOu «.3 x: new .0452 on .So ecu 63o: vaoo can no... 0 an» ocean comuanwuummc names: uaowmzz «0004 we ouzmwh I; 0.— 0.0 0.0 #0 «.0 0.0 . _ _ . _ . an 1. .+ 1. .u n. .+.1 B + .+. n. B .+. B a + B + n. + 1 n. n. m + + IL J I: :3 2 + 1 2:12. I U 0 an 'Jaqmnu :Iassnn {coca 121 .1comuouommucoo Hoowuuo>1 .anu «.NH + noumu Hocowumuou . vq_xxh mom .0.Hux2 on u1o sou .mdaoa vaoo can so: an» anode co1usn1huufic noneac uaomuaz Hoooq he ouomwm :\> 0.1 0.0 0.0 e0 «.0 0.0 q #1 d W. 11 — . — a O nu 1. .+ nu n_ 14 N 1. .u 1. n. mm B + + n. a + 1 + + nu .+ nu 1. a + + n. n. a + + 1 ¢ 0 n. J .1 0 ...; Eon 1< 1.. 1 ..o-1o31< n. nu 'Jaqmnu :Iassnn IUDOfl 122 .Aco1uouam1ucou Hoowuuo>v .anu H.m~ + loam» Huco1umu0u v3 X: Ina .0.Hux2 an in ecu s 0.0 ....0 «.0 .maama 0100 can go: as» ocean :01u5n1uun10 tones: uqomuaz Hobo; 00 ououwm 0.0 :3. 23 1 +. :3. .3. 2 a fi 1 _ ..L 0 ON ‘Jaqmnu atassnu Inooq 123 .Aco1uouammucou Hoo1uuo>1 .anu m. «a + noun» Hocowuouou v S x: 131:0 .752 um .53 .53 .31..) goo 0cm yon on» ocean coau=n1uummu cones: uaommsz Houoa we ouomdm . I; . . . . . 01 00 00 ¢0 «0 00 . — q 1— — _ O + + + N + a 1 B a n. n. + + + 1 B 1. 1. n. 1.. B l t B n. n. + 1 l 0 2: :3 2 + 1 2:12. 2 U nu 'Jaqmnu atassnn {coca 124 .Acowumuomwucoo Hoowuuo>1 .0.Hux< um 51m ecu .uonasc uaommaz come no cofiumuOu no uuouum 0m ouammm .. 0000— 0000 0000 000* 000« 0 _ . _ . _ _ . o .1 « 4 m 1 a 4m + 0 mm. m. m .2 1. .. + o + 1+ .1 0 ...—x31... ..+ 1 “Ex: I... ..0 sexual... ..4 l o .2551... ..U 0— .aqmnu grassnu use" 211': 125 The way in which rotation affects the local and mean Nusselt numbers has been presented for the condition where the enclosure is rotated in the direction of the flow along the hot wall (aiding configuration). In contrast, for an enclosure rotating in the opposite direction to the flow along the hot wall (opposing configuration), it was found that the effect of rotation tends to decrease the heat transfer relative to the stationary condition with increases of the rotational speeds. Figures 53 and 54, clearly illustrate the reduction in the local heat transfer with increasing rotation rate. This may be accounted for by the complex interaction between the Coriolis acceleration and the centrifugal buoyancy, which apparently tend to reduce the the heat transfer within the rotational speed limit (17.5 rev/min) considered in this study. 126 10 I3 2 ‘ 0.. THII4JI ; A.. turns." a 0.. [F’.’...l I: . O 0 :1 0 A ‘ 3 a Q a '3 N m :3 2 I: G O 2 1 l n 1 1 1 n 1 ll 1 n 1 l n n n n 10‘ 105 105 In a) Mean Nusselt number versus Rayleigh number 10 F1 9 2 8 . 0.. Int]! an 1.. 1... "-1.1: 5+: on 5 0.. 1mm: as g 5 +.. Ila-2.92 as c t 1. A :1 41 + i a . 1 8 3 n 3 m :5 z c 2' a o 2 1 n n n j 1 1 1 ll 1 1 n I n 1 1 n 103 10‘ 105 TI b) Mean Nusselt number versus Taylor number Figure 51 Effect of rotation on mean Nusselt number, for air at Ax-l.0, (vertical configuration). 127 (h x <0 .muonsoc :mwoaaom nan madame no cofiuucsu o no human: uuommsz coo: «m ouamfim so. me. .o. q u d q q 1~.¢. - —u-q-1 - u —q-quq q u .om A ODQN‘DLD fi' 0— tassnn ueau nu 'Jaqmnu : 128 .Asomuouomuucoo Houmuuo>1 .aau m.0 1 loans Honouuouou e2 x...— no: .0452 an be you .932. 300 can so: ecu ocean cowuanauunwu homes: udonnsz ”noon mm ouau1m I; . . . . . 0 0 m 0 ... 0 « 0 0 0 . _ a . 1 _ a a 1 a + n. + nu B l 1.. n. B + 1 + + a + + + + + a a 1 1. n1 :3. 23 2 + 1 :3. .3. 2 n. 0 nu 'Jaqmnu :Iassnn 19301 129 .Acowuouamuucou Hmuwuuo>v .aau m.na n Imuau Hmcomumuou . me x3 12. $4de an .30 ha 5:03 300 one was o._ w.o . a :2. 33 2 :2. .2. 2 +. nu 0:... mega cog—52.33 .385: udommsz :00; 3 0.33m N.o o.o o nu 'Jaqumu :Iassnn 19301 130 4.2.2 Heat Transfer Results in a Heated Rotating Enclosure at Angular Position of 180 Deg. (Heated from Below) The present sub-section examines the effect of rotation on the onset of cellular motions in a rotating heated from below air-filled enclosure (Benard convection). This should lead to better insight into the hydrodynamic and thermal boundary layer developments under the influence of the controlling parameters, namely, the Taylor, rotational and gravitational Rayleigh numbers. The onset of convection in a non-rotating enclosure was described previously in Figures 28a,b. Unstable temperature stratification in the vertical direction, a flow of a series of roll- cells with their axes oriented normally to the isothermal walls were observed. Accordingly, the inhibiting effect of rotation on the onset of the thermal instability will be analyzed relative to these aspects. The measurements of local Nusselt number will demonstrate the corresponding variations of the thermal boundary layer thickness, and the extent of inhibition on the onset of cellular convection. The way in which rotation affects the local distribution of Nusselt number is demonstrated in Figure 55 for Ta- 2.278x103(-8.5 rev/min), Rat-1.21x102, and Ra-l.2X106. It is evident from this figure that rotation produces a marked effect on the local heat transfer distribution as compared to the stationary experimental data. Two explanations may be argued for the characterization of rotational effect. First, it is reasonable to assume that the combined effect of Coriolis force and buoyant interaction has a strong influence on the cellular motions (unstable condition), which in turn tends to inhibit the thermal instability. Secondly, for the current operating 131 conditions, it is found that transition from a cellular motion to a longitudinal roll-cell has occurred, since in this case buoyant interaction between centrifugal and gravitational buoyancy dominates the flow development and consequently the thermal boundary layer. Moreover, the local distribution of Nusselt number reflects clearly the subsequent thinning in the thermal boundary layer near one end of the isothermal wall and a relative thickening near the other. This is significantly different from the thermal boundary layer encountered in the stationary situation of the unstable fluid. This aspect is mainly a result of the longitudinal flow pattern generated in the vicinity of the wall region, which indicates the significant effect of rotation on the flow, and hence delays the onset of thermal instability to at least a higher operating temperature difference, and slightly improves the heat transfer. Both these concepts offer areas where further development can be extended in the future to the case of higher rotational speeds and various Prandtl numbers. Figure 56 shows the distribution of local heat transfer at a rotational speed 12.2 rev/min. It is interesting to note that the distribution of local Nusselt number is very similar to the results presented in Figure 55. Hence it suggests no significant changes in the hydrodynamic and thermal boundary layers have taken place. A small increase in heat transfer has occurred. The local Nusselt number distributions for rotational rates of 8.5 and 12.2 rev/min, and for Rayleigh number 3.0x10‘5 are illustrated in Figures 57 and 58. It is apparent that the distribution of local Nusselt number is almost as before, however the improvement in heat transfer with increases in rotational speed and imposed temperature difference is detectable but not very marked. 132 ..30uon Ecuu vouoocv .Eau m.o + nouou accomueuou . m3 XS tom 6.712 no in ecu .naaoa unou one ac: or» ocean couosaduunun eonEsc uaonnaz Aouoq mm ousmdb I; . . . . . no no we we co 4 q — q — q — - fl - o l N + + B + n. n. I I B + +- B I B l ... I + I I nu I I I _- l. . $u L "U _. 4. H# - I + I+ n. n. I m B + m + _ _os ...ou ”_< i. 4. 1 23.2.20 a a a + + 222;: .- couuouOu so“: COmufluOh 8:088m3 nN ‘Jaqmnu atassnn 12301 133 ..aoHon scum cosmos. .aau ~.~H + nouns Hocowunuou . mq_XQ—umm .o.Hux4 on two new .maaoe taco can no: man macaw coaosnuuumwc nonssc uaonmaz Hoooq om ousmwm . I; . . . o— m o m o ta o N.o .- d 1 A a — a — G L l. 4. nu + n. B1 .+ n. B .. 1+ 1.. n. +. :1 nu n. .+ 1 D + n. + l B B 1? B 1? 1+ 1 a + 1.. __ua v_oo ~< .+ ..o3 “a: ~< U 1 o— nu 'zaqmnu :Iassnu 19:01 134 Jason so: cannon: .89. m.» + no»: Hocowumuou o2 x3" mom $452 an .Sm ecu 53o: 300 can so: ecu ocean cofiusnwuumfiu amass: uaonmaz Hoooq hm ousmmm z\> o.— m.o o.o ¢.o N.o o.o 1 — W fl 4 — d — u o + J N + a m 1 n. + a + .1 a. m. nu l. n + a + 1 m S E D i m u + + a + u + n. n. + 1 m. n. m a a a + + + l m 1: In: :3 .< + w. .2: “3. .< D 1 o— .A3o~on scum cosmos. .aau ~.~H + nouns HocomuouOu nq—xqa uom .c.als< on has new .nHHo3 odou Uco no: on» mcoHo couusnmuunmo noses: uaoumaz douoq om ousmum 135 . I; . . . o— m o m o ¢ 0 N.c 9.0 u — d _ d I— d — d o L N + + n_ i. nu mm 1 + m. n. .1 w + a + a a + nu IL 0 ...—I .+ B + .. a + B .u an i. l. 1L m :2. :3 1 + a a a + + ..eo “as ~< U A c— an 'Jaqmnu atassnn 19:01 136 6.2.3 Heat Transfer Results in a Heated Rotating Enclosure at Angular Position of 0 deg. (Heated from Above) The flow and temperature fields of natural convection in a non- rotating enclosure heated from above are usually characterized by stable stratification, except near the corners under certain boundary conditions as mentioned earlier. In addition local and mean Nusselt values are approximately equal to one for all imposed temperature differences, as might be expected any disturbance in the flow pattern may cause an increase in the heat transport. Therefore since rotation is capable of distorting the flow pattern, in particular at a large impo'sed temperature difference as a consequence of the gravitational and centrifugal buoyant interaction, one should expect to observe significant effects on heat transfer as a result of rotation. At a rotational rate of 8.5 rev/min and Rayleigh number of 1.2x105, Figures 59 and 60 demonstrate the local variation in heat transfer together with the experimental data in a non-rotating enclosure. Here, the distortion in heat transfer profiles are quite evident. However, the increase in average heat transfer is not significant. In contrast, with increasing the imposed temperature difference between the walls of the test section, the buoyant interactions are clearly in evidence as suggested by a marked enhancement in heat transfer as shown in Figures 61 and 62. Also a transition to a longitudinal roll-cell has occurred near the wall region, which contributes substantially to the thinning of the thermal boundary layer, and produces an increase in the heat transfer of approximately 40 % . 137 .Ao>ono souu nouoocv .aau m.c + nouns Hocoauouou nexus sum 6.7.2 on .2: ecu .32.: 300 one no: or» ocean couusnuuuuun tones: amouasz Houoq an ousmuu :\> o._ m6 m5 Tc N.o o.o J _ _ _ a . o I I I I I I 1 EIBIB IaI I + a i .t 1r + I a IL + + n. a w a a + N + + + + + + a a B n. n. a I e l m ..eI v_.:. .< 1? 1 ..uI no; .< D __ua do; .< I couuouOu so“: cofiuouOu escrow: nu 'Jaqmnu :Iassnn 12:01 138 I .Ao>ono souu cosmos. .aau N.NH + nouou accomuouou u¢_x«4 Ina .o.aux< no umo new .naaoz taco use so: or» usage :ofiusnfiuunmn noses: uaonnsz Houoa on canvas 5» . . . m o m 0 ¢ 0 N.o o.o - _ .- — . — q _ q C + + 1 n_ mg .+ 1? nu n. + + + a n. B 01 m. m 4 ... m a m ... m :2231 + .u :22..: n. 1 n 139 ..o>ono souu cosmos. .aau m.o + nouns accomuouOu n°.xqa lam .o.aux< no nun ecu .maama taco 0cm no: ocu.ucoHo coqusnwuunflc hoses: uaommsz Hmuoa Ho ousmmm .1). . . . . _ no we we No so u — q — q _ 1— u o ++++ + IN Damn. +++ 3n. + IT EBB 1 ++ BB ++++ an. 11¢ BBB—u L . lo . _oy ...ou ..< 1+ 1 ..o; so; .< D nu ‘Jaqmnu atassnn teooq .Ao>ono scum wouoocv .aau N.NH + noun» AccomumuOu as Xe.” lam .o.AIx< um 13o new .muamas UHOU «Em yon on» mcoHo cofiusnwuumwu amass: uuommsz Hmuou Nm ousmwm 140 2: . . . . . no mo ¢o No co - — q _ q d . — a c + + +1.. man. ++ 1N n. +. nu n. .+ 1? +1 1 B + was +++++ Emu 1* n. BBBB1 lo :2232+ 1 ..93 “o: .< D nu ‘Jaqmnu atassnn teaoq 141 Figure 63 reveals the manner in which rotation influences the mean Nusselt number result at angular positions of O, 90, and 180 deg. relative to the stationary results. It is noticed that there is a pronounced improvement in heat transfer at 0 deg. , a reduction at 90 deg. , and a small increase at 180 deg. Figure 64 shows pictures of the interference fringe patterns which in turn illustrate the thermal boundary layer configuration in the vicinity of the wall region in non-rotating and rotating enclosures at various angular positions. These interferograms provide a comprehensive qualitative description of the general trend of the local heat transfer variation along the differentially heated walls of the test section. For instance, at 0 deg. (heated from above) the developed thermal boundary layer in the rotating enclosure, Figure 64b, gives a clear indication of the convective flow existence which is produced by the combined effect of heating and rotation. At angular position 90 deg. the recirculation of the thermally induced secondary flow from the upper and lower corners of the rotating enclosure toward the core region contributed to thickening the thermal boundary layer as shown in Figure 64d, and a consequent decrease in the heat transfer. At 180 deg. (heated from below) the thinning of thermal boundary layer near one end of the rotating enclosure Figure 64f, is a result of the transition from a cellular motion to a longitudinal flow pattern. 142 10 5 a) Ra- 1.2x 10 T"... lg 0% Ted." [+3 13 0 I 3 " I: .’./'/ u /.’// "J ’0’ / I I. / a a/ 3 2 I: I I z 1 111111.1111111 10 b) Ra- 3.0x105 mu.- 1 —1A Tet-1.17 {+1 --¢ fad." [+3 /.¢' /1-/ /./. / Mean Nusselt number, FIT: llllLlllJlALlllllL 0 50 100 150 200 Anglo Figure 63 Effect of rotation on mean Nusselt number at various angular positions. 143 Without rotation With rotation a) ¢=0 deg., 0 rpm b) d>=0 deg.,+17.7 rpm C) ¢=90 deg., 0 rpm d) ¢=90 deg.,+15.5 rpm e) ¢= 180 deg., 0 rpm f) ¢> =180 deg-1+1? 5 rpm Figure 64 Interference fringe patterns at Ra: 3.0x 105 m 5 SW AND CONCIHSIONS In this experimental study local and mean natural convection heat transfer characteristics have been considered in an air-filled differentially heated enclosure with cross-sectional aspect ratio one. The Hach-Zehnder interferometer was employed to reveal the entire temperature field which enabled the measurement of local and mean Nusselt numbers along the heated surfaces, while a laser sheet-smoke visual study of the flow patterns provided a greater understanding of the hydrodynamic and thermal boundary layers interaction. The effects of the various physical factors on the heat transfer behaviors are analyzed in inclined and rotating enclosures which are inclined or rotated about their longitudinal horizontal axes. The first part has shown the importance of including the effect of inclination angle in treating problems of natural convection in an inclined enclosure. The following features are worthy of note for Rayleigh numbers ranging from lO‘to 10. and inclination angles between 0 and 180 deg. 1. For d s 30 deg. (heated from above), it was shown that the conduction dominates the heat transfer and that the flow is very slow. However, at 30 deg. the existence of the thermal and hydrodynamic boundary layers were observed which' in turn contributed to an improvement in the heat transfer. 144 145 2 . For 30 < o s 90 deg. buoyancy-driven flow commenced its influence on the flow . At 60 deg. the thermally induced secondary flow in the form of two vortical tubes is generated in the core region near the center of the heated walls at Rale 5. Skewness in the vortical structure at 90 deg. implied a relatively thin boundary layer near the lower and opposite upper corners of the hot and cold surfaces, which in fact exhibited a substantial increase in the heat transfer. 3. For 90 < o s 120, a transition of the thermally induced secondary flow from the core region at 90 deg. to the upper and opposite lower corners of the enclosure has occurred at 120 deg. This in fact contributed to improving the heat transfer at an angle between 110 -120 deg. 4. For 120 < 4’ s 180 deg. , it was interesting to note that the thermally induced secondary flow occurred near the upper and opposite lower corners of the enclosure. In contrast, the core region was characterized by a longitudinal roll-cell before a transition to three- dimensional flow approximately between 150-160 deg. and then to cellular motions at 180 dog. The local distribution of Nusselt number reflected the manner in which the thermal boundary layer responded to the flow development. 5. The influence of inclination on mean Nusselt number was characterized by a local maximum value between 110-120 deg. and a local minimum between 150-160 deg. , as a consequence of the disappearance of the secondary flow from the core region and the translation in flow pattern respectively. On the whole, the average Nusselt number results were in excellent agreement with existing numerical and experimental data. In the second part the influence of combined gravitationally driven and rotationally driven flows on the thermal and hydrodynamic 146 boundary layers was discussed in accordance with the Taylor, the rotational and the gravitational Rayleigh numbers as the governing parameters. In view of the experimental results, it may concluded that: 6. For the angular position of 90 deg. (vertical enclosure), the Coriolis and buoyant interaction effect on the local distribution of Nusselt number illustrated clearly the subsequent development of the thermal boundary layer and showed a minimum in the heat transfer near 8.5 rev/min. This reduction may be argued in connection with the recirculated secondary induced flow radially toward the core region which in turn generated a complex flow pattern. However, a relative improvement in heat transfer was observed at higher rotational speed as a result of the buoyant interaction dominating effect. 7. For angular position of 180 deg. (heated from below), it was found that rotation indeed inhibits the onset of the cellular convection, and a longitudinal flow pattern was generated. Also, a small increase in heat transfer was produced. 8. For angular position of 0 deg. (heated from above) an increase in heat transfer was shown with increases in the rotational speed. However, it was more markedly influenced by the imposed temperature difference. In this event, the flow pattern was characterized by a longitudinal roll-cell. Finally, progress has been made in the quantitative description of heat transfer in the inclined and rotating enclosures, but much research remains to be done. For instance, study the effect of rotation on the flow and heat transfer at higher rotational speeds over a wide range of Rayleigh number values, and for different Prandtl numbers remains to be done. Also, investigation of the possible delay of transition to turbulent flow as a result of the stabilizing influence 147 of rotation on the flow field. In addition, it is very important to consider the direction of rotation in rotating enclosures. APPENDIX.1 PHYSICAL1PROPERIIES Interferometry is an optical technique where the index of refraction distribution throughout a transparent medium of fluid can be obtained directly. The density field can then be determined by the Lorentz-Lorenz relation, 2 (£131 1 (n +2) p - C (A.l.l) where C is a characteristic constant of the fluid independent of the temperature. Since n-l.000276 for air, equation (A.l.l) can be rearranged to, c - “1:11 (11.1.2) values for (n-l) were taken from the American Institute of Physics Handbook [147] as follows: Air at A - 5461 A, (n-l) - 0.29371110‘“ Using the perfect gas relationship, to measure the air-density P - pRT (A.l.3) The Gladstone-Dale constant was found to be: 3 c - 2.270141110’“ “- a kg The physical properties of air, u, k, Pr, were taken from Ref. [148]. 148 APPENDIX 2 INTERFEROGAH ANALYSIS 2.1 Relation between Fringe Shift and Telperature (mange The difference in the optical paths of the two light beams results in a phase shift of one beam with respect to the other. The equation for th fringe shift along the light beam can be expressed as, L e - i— I [n(x,y,z) -'n 1112. (A.2.1) 0 O 0 Where (none) is the change of the index of refraction at some point in the test region relative to the reference region, "0 is the wavelenght of light in the reference beam (S461A). In interferometric analysis the variation in the index of refraction along the light beam, n(z), is usually very small, thus equation (A.2.1) integrates to, e - ‘1" [n(x,y) - no] (A.2.2) 0 Then the Gladstone-Dale equation (A.l.2) and the perfect-gas relationship (A.l.3) can be introduced into equation (A.2.2), fig [fie—f '1'] (11.2.3) 6 - which gives the relationship between fringe shift and temperture change. In the present study the reference temperature is equal to the cold wall temperature which maintained close to the room temperature. Then , 149 150 e_£112[.l_l] (11.2.4) XOR Tc T In this equation all the terms are known except 6 and T. Since 6 can be measured directly from the interferogram, only T remains to be found, then equation (A.2.4) can be solved for the temperature, TCGLP T (A.2.5) -GI.P-TeAR Co 2.2 Evaluation of the Fringe Shift A traveling microscope made by Gaertner Scientific Corporation of 0.0001 X-Y resolution was used to analyse the interferograms. The interferogram was placed on the micropositioning stage and then aligned such as the hot and cold walls are parallel to the Y-crosshair of the microscope. The sides of the interferogram were measured precisely by using the XY Vernier Hicrometers, in order to determine the scaling factor which is equal to the ratio of the enclosure height to the interferogram height. The photo was then taped to the stage and a glass cover palced on it. The Interfergram height was divided in equisteps of 0.0511, to determine the local heat transfer coefficients along the hot and cold surfaces. For instance at a given height y on the wall with respect to origin (xo,yo) , the fringe shift was determined at 30 points between the hot and cold walls. However, the points were more concentrated near the isothermal walls. At least five points were taken in 1.0 mm range to determine the temperature gradient at that location. Figure 65 of the interferogram demonstrates the procedure to estimate the fringe shift along y starting from the cold wall where the fringe shift is zero. Fractions of a fringe shift are measured by the 151 ? rho -l ‘ 'l xco -—>I . . interference fringe patterns .: 7* x 1‘ I >1 Y2 H fi/ '3 .. 3 ... o no ~ 3 H 1 o u U o ‘ a: 41 1‘ I ya Y yb 3— ‘ V ’11 I ’ . (xmyo) x-axis Figure 65 Fringe shift evaluation, at Ra .11 1.1x105 ¢ - 90 deg. 152 relation (ya - y)/(ya - yb) as indicated in the figure. A computer program called FRNG was used to calculate and plot the fringe shift as a function of the non-dimensional distance along the x-axis, (x/H). A second program called TEMP is used to measure and plot the fringe shift correction due the end effects and the non-dimensional temperature versus (x/H). The program was used also to estimate the non-dimensional temperature gradients along the hot and cold surfaces, by using a least square fit to the experimental data. In addition, the standard deviation was estimated on each measurement and local Nusselt numbers were plotted as function of the non-dimensional height of the test section. A listing of the two programs and a sample calculation are given in Appendix 5. 2. 3 End Effect Correction The fringe shift at the hot wall can be determined from, - 9L3 [ 1— - 1- ] (A.2.6) o XOR TC TH 6 However, The existence of the optical flats at the ends of the test section will in turn contribute to an additional fringe shift, because the measuring beam did not reach immediately the ambient temperture. The fringe shift correction due to the end effect is given in [149] as, l. in A. - A0 KAT n or dx (A.2.7) Since the thermal conductivity of glass is large compared to air, then a linear horizontal temperature distribution in the glass will be a good assumption. Therfore, a linear fringe shift correction can be applied, 6 - em - 16(1- §)[(5H - .C) - e0] (A.2.8) 153 where e represents the corrected fringe shift, emthe measured fringe shift from the interferogram, eois measured from equation (A.l.6), and ‘C’ ‘H are the fringe shifts at the cold and hot walls determined from the plots of ‘11: versus (x/H). Detailed analysis of the end effect correction is given in Ref . [40,149]. APPENDIX 3 The picture in Figure 65 illustrates the interference fringe patterns of the refractive index distribution, which in turn allows a subsequent calculation of the entire temperature field, and hence the local temperature gradient at the wall. The coefficient of heat transferis defined as . _ Q h AT (A.3.l) where q is the heat transferred per unit area and AT is the imposed temperature difference between the walls. In a thin layer of air in the immediated vicinity of the isothermal walls, heat transfer will take place largely by conduction, q - 4143:) (A.3.2) s where ks is the thermal conductivityt of air at the temperature of the heated surface, (aT/ax)s the temperature gradient within the thin layer of air, and x the distance normal to the isothermal surfaces. From equation (A.3.l) and (A.3.2) the non-dimensional local heat transfer coefficient, the local Nusselt number hi! Nu - k 51 8(T-TC)/(TH-Tc) becomes, Nu“) - [kc 3(X/H) ]s _“a M or, Nu“) - ( ) (A.3.3) kc A5 154 155 in equation (A.3.3) kH, k are evaluated at the hot and cold walls C temperature,0 is the non-dimensional temperature and 5 is the non- dimensional distance normal to the heated wall. (AO/Aé) can be determined from the temperature profile near the isothermal walls as demonstrated in Figure 66. In addition, Figures 67 and 68 represent the corresponding fringe shift and the temperature profile at {-0.1340. Experimental data is given in Table l. The mean Nusselt number is determined by numerically integating the local values over the entire wall height, 1 R - EH Imam; (A.3.4) C 0 The Simpson's rule Ref. [150] was used to evaluate the integral. 156 .000 cm a $ Uco .vnaé ... v . mg x: I or he .39.. Boo on» no used—Conn ensueuonaou 23 no couuoemuou mm 0.5m: .=\uv ease denouncoauoucoz no.0 mo.o oo.o Nc.o co.o . _ . _ . co... Nc.c vo.c C 11 cc 9 11 00.: - o..o amused“: teuoysuamrp-uon 157 to . .63 cm a $93 $36 .. w as x: n or ..Om eouusnauunao 32m 005.:— 3 95m: 2qu 3.3 Hocoaocosmoncoz so; a... a... +... a... as .W 1 _ 1 1 . a . _ . wife 3 1 MN mm 1 mm wm b a 4 J: we Mm - . . 1 W... . . We a... 1 me am 1. “a u I .. W. m... c—W « mc— a :m 1 w: u 2.“. 1 mfim n—uk ma: 1. u. ... n— um— 158 .93 8 1 e :2... 1 u a3 x: ... or ..Ou .oZuoun ousuouoasoa mm 0.53m 2:: e28 decomncoauoucoz o._ ad ed to N... c.o - q u q a O . 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Own—.0 uZ\> .oom~.o u .:\a av u use neaxaa a or no .comosnfiuumwo ououmuodsou on» one uuwzm oucwuu on» no cofiuoasuHou ounsow H oaaoa APPENDIX 4 m ANALYSIS To ensure confidence in the experimental results, it is desirable to monitor the accuracy of the experiments by checking that all the measurements are adequately within an acceptable error bound. The types of error involved in this study are optical and measurement errors 4.1 Optical Errors The optical errors are caused by refraction and and effects in the interferometric system. The refraction errors encountered in the measurements are due to the gradient of the refractive index in the direction normal to the beam of light. Consequently, the tempreature distribution or the temperature gradient measured from the interferogram can involve significant errors. Now, assuming the temperature variation in the direction of light is very small compared to the temperature difference between the walls, then the gradient of the index of refraction dn/dx, can be considered as a constant, and the temperature difference between the refracted and unrefracted beams according to Ref. [151] is 2 2 AT - 431-37 (g) (11.4.1) 8n°RT 160‘ 161 In the present study the maximum temperature difference between the walls was 27.42 K, and A0/Aé-ll.86. This in turn yields a temperature gradient of AT/Ax-5687.32 K/m and the refraction error from equation (A.4.l) is T-0.026 K. However, as described earlier in Appendix 2 the fringe shifts were corrected so that the total fringe shift would agree with the fringe shift related to the wall temperature. This in fact, will account for the refraction error. 0n the other hand, correction for the end effect was given in Appendix 2. 4 . 2 Heasurenent Errors Errors in measurements are mainly encountered in the determination of the center of the fringe from the micrometer readings. This error is estimated to be within the range of 1 1/20 of a fringe or approximately i 0.0012 cm in the micrometer readings. This resulted in a fringe error of less than 0.01 %. Another important source of error is occurred in estimating the temperature gradient at the isothermal walls. For this reason mainly the estimated standard deviation were measured on the local values of the temperature gradients as will be illustrated in the sample calculation in Appendix 5. 0n the whole, the error in estimating the temperature gradient along the isothermal walls are within i 3 % and the Total possible error is in the order of 6 % . APPENDIX 5 mmmmww 162 163 ---------------- PROGRAM FRNG.--------------------- THIS PROGRAM IS USED TO EVALUATE THE FRINGE SHIFT FROM THE DATA MEASURED BY THE MICROMETER. --------------- VARIABLE IDENTIFICATION. ----------- B COEFFICIENT OF CUBICAL EXPANSION. C GLADSTONE-DALE CONSTANT. DW TEST SECTION WIDTH. EPS FRINGE SHIFT. G GRAVITATIONAL ACCELERATION. GR GRASHOF NUMBER. PRANDTL NUMBER. RA GAS CONSTANT. RAY RAYLEIGH NUMBER. SF SCALING FACTOR. TAUY DIMENSIONAL LENGTH NORMAL TO THE HOT PLATE. TAUX DIMENSIONAL LENGTH ALONG THE PLATE. TC COLD WALL TEMPERATURE. TH HOT WALL TEMPERATURE. V KINEMATIC VISCOSITY. X COORDINATE IN DIRECTION ALONG THE PLATE. Y COORDINATE NORMAL TO THE HOT PLATE. -------------- ENTRY AND STORAGE BLOCK. -------—----- nnnnnnnnnnnnnnnnnnnnnnnnnnnnn 'U m REAL TAUPT(40),EPSPT(40) CHARACTER *80 TITLE REAL*8 GRX,GRW,RAY C DATA IN/10/,IO/11/,INI/12/ C C .................................................... C ---------------- PROCESS BLOCK. -------------------- C CALL INITT(480) CALL OPENTK('GFNG',IERROR) OPEN(10,FILE='RA3F8') OPEN(11,FILE-'OFILE') OPEN(12,FILE-'INPUT') READ(IN,16) TITLE 16 FORMAT(A80) C C PHYSICAL CONSTANT REQUIRED FOR THE CALCULATION C or GRASHOF NUMBER. 0000 0000 nnnn GOOD nnnnn 164 G-9.8 RA-286.87081 C82.27014E-04 PHYSICAL FACTORS DERIVED FROM THE KNOWN TEMP. OF THE HOT AND COLD WALLS. READ(IN,*) JJ,KK READ(IN,*)TH,TC READ(IN,*)B,V,DW,PR WRITE(INI,180) TH,TC INSERTING THE SCALE FACTOR AND THE LENGTH OF THE HOT PLATE BY READING THE TWO EDGES. READ(IN,*) SF,Xl,X2 CALCULATION OF THE FRINGE SHIFT DEPENDING ON THE GIVEN DATA. DO 50 J=l,JJ NPT=0 WRITE(IO,160) INPUT OF THE MICROMETER READINGS CORRESPONDING TO THE FRINGE PATTERNS, AND THE LOCATIONS OF THE HOT AND COLD PLATES. READ(IN,*) XO,YHO,YCO NON-DIMENSIONALIZING THE HEIGHT, AND CALCULATION OF THE LOCAL GRASHOF NUMBER. TAUx=(xo-x1)/(x2-x1) x=2.54*SF*(xo-x1) xn-o.01*x GRx=(G*B*(TH-TC)*(XM**3))/(v**2) GRW=(G*B*(TH-TC)*(DW**3))/(V**2) RAY-GRW*PR WRITE(IO,*) WRITE(IO,*) WRITE(IO,*) WRITE(IO,17) TITLE WRITE(IO,60) TAUX WRITE(INI,65) TAUX WRITE(IO,70) GRx 000 165 WRITE(INI,75) GRX WRITE(IO,85)GRW WRITE(INI,95)GRW WRITE(IO,96)RAY WRITE(INI,105)RAY WRITE(IO,80) X0 WRITE(IO,90) X1 WRITE(IO,100) X2 WRITE(IO,110) YHO WRITE(IO,120) YCO FRINGE SHIFT CALCULATIONS. 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I> x .000 000.500I ogaalLOQEOa ...: «or .000 00lo.0c< co.vlc..uc. u< .. 0150.0 uI\> .oou oouo.uc< co.u-c..uc. u< .. 0150.0 uzx> ...c.u:ou. m oHnme 196 Table 3 Sample calculation of local the gusselt number from the temperature gradients, m Table 2. Local Nuaaalt number aa a function of the dimonaionlaaa distanca along tha hot and the cold wall for Rayleigh number RAI LIX 105 and incllnation angla-9O dog. V/H NU NU E.S.D. E.S.D. Hot Wall Cold Hall Hot Hall Cold wall 0.0195 3.888 2.093 0.085 0.091 0.0787 4.847 1.398 0.024 0.027 0.1340 5.144 1.572 0.087 0.080 0.1913 5.448 2.057 0.075 0.027 0.2488 5.422 2.830 0.052 0.055 0.3058 5.313 3.108 0.078 0.024 0.3831 5.111 3.442 0.079 0.040 0.4204 4.731 3.808 0.050 0.040 0.4777 4.817 4.184 0.135 0.087 0.5349 4.304 4.309 0.085 0.033 0.5922 3.892 4.727 0.049 0.051 0.8495 3.389 5.135 0.038 0.138 0.7088 2.880 4.898 0.088 0.048 0.7840 2.557 5.133 0.044 0.058 0.8213 1.838 4.858 0.057 0.089 0.8788 1.830 4.791 0.050 0.098 0.9359 1.507 4.041 0.052 0.088 0.9748 2.000 3.304 0.080 0.147 loan Nuaaalt numbar at the not wall no I 3.759 loan Nuaaalt numbar at tho cold wall NU 8 3.479 10 11 12 13 Ostrach, 8., "Natural Convection in Enclosures," Advances in Heat Transfer, Vol. 8, pp. 161-227, 1972. Catton, I., "Natural Convection in Enclosures,” Proceedings of the Sixth International Heat Transfer Conference, Vol. 6, pp.13- 31, 1978. 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