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An Experimental and Numerical Investigation of Mixed Convection in Rectangular Enclosures

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AN EXPERIMENTAL AND NUMERICAL INVESTIGATION OF MIXED CONVECTION IN RECTANGULARENCLOSURES By Gregory P. Numberg, B.Eng., M.Eng. AN INVESTIGATION OF MIXED CONVECTION IN ENCLOSURES A Thesis Submitted to the School ofGraduate Snldies in Partial Fulfilment of the Requiremen<: for the Degree Doctor of Philosophy McMaster Universi~ o Copyright by Gregory P. Numberg, September 1994 DocrOR OF PHILOSOPHY(1994) McMASTER UNIVERSITY (Mechanical Engineering) Hamilton, Ontario An Experimental and Numerical Investigation of Mixed Convection in Rectangular Enclosures AUTHOR: Gregory P. Numberg, B.Eng. (McMaster University) M.Eng.(McMaster University) SUPERVISORS: Dr. M. Shoukri Dr.P. Wood TO JAN NUMBER OF PAGES: xvi,232 prediction of the intermediate Archimed:lS number cases as the predicted flow is dominated by buoyancy while the experiments show more of a balance. Sources for this discrepancy are ABSTRACT discussed. The cooling ofa hot surface by fluid motion has many applications in engineering. For these mixed convective problems, the forced component offluid motion may be in the direction ofthe buoyancy vector or it may oppose the buoyancy force. In the present study, the opposing mode is used to study the interaction of inertia and buoyancy forces in a fluid. Both numerical and experimental techniques are used to study :.'te flow in a re~gular cavity of aspect ratio 2. The inlet Reynolds number is varied between 800 and 1300 and the Grasbof number based on the height ofthe enclosure is varied between 0 and 2.4 X 1010• The cases considered correspond to Archimedes numbers ofapproximately 0, I. 10 and 20. The flow field is observed qualitatively using laser induced fluorescence and a detailed flow field is generated using a laser doppler anemometer. Temperature profiles are fow:d using fine wire thermocouple probes. These detailed measurements may provide a data base for the verification ofcomputer Pl"'grams used to predict mixed convection as there are no such detailed chita bases in the present literature. Numerical modelling is based on the SIMPLER algorithm with QUICK differencing. The observed flow field indicated that some regions in the cavity were turbulent while other regions were laminar. This observation suggests the necessi:y of a low Reynolds number turbulence model. In this study, two forms ofthe low Reynolds number k-c model are used. In addition. the commercial computational fluid dynamics program, FLUENT, is used to predict the flow. Comparison ofthe experimental and computational results suggest that for isothermal and buoyancy dominated flow cases the computational modelling is adequate. Difficulties arise in the iii iv ACKNOWLEDGEMENTS TABLE OF CONTENTS P:lge ABSTRACf iii The author wishes to express his appreciation to the following individuals and ACKNOW"...EDGEMENTS organizations that have contributed to the completion ofthis study: TABLE OF CONTENTS vi NOMENCLATURE ix Dr. M. Shoukri and Dr. P. Wood for their supervision. guidance and fmancial assistance UST OF FlGllRES xiii throughout the course ofthis research program. UST OF TABLES xvi • Mr. Joe Verhaeghe and Mr. Ron Lodewyks of the Department of Mechanical Engineering INTRODUCTION 1.1 Nature ofFlows and Heat Transfer technical staff for their assistance in the instrumentation and construction cf the test loop. 1.2 Practical Applications 1.3 Objectives ofthe Present Work • The Ontario Graduate Scholarship Program and the Department ofMechanical ofEngineering UTERATURE SURVEY 4 for their financial support. 2.1 Introduction 4 2.2 Mixed Convection in Enclosures 7 • Fellow graduate students, with whom I shared a ple:.sant, friendly and cooperative working 2.2.1 Previous Numerical Research 8 2.2.2 Previous E::oerimental Research 10 atmosphere. 2.2.3 Natural Con~ection in E."lclosures 11 2.3 Turbulence Modelling 14 • My family, especially my parents. for their continued support and encouragement. 2.3.1 llD1e Averaged Governing Equations 14 2.3.2 The Turbulent Viscosity Concept 16 • Finally, to Jan and Kelly whose love, continued understanding, support and care provided the 2.3.3 Zero Equation Models 17 2.3.4 One Equation Models 18 . incentive to complete this research. 2.3.5 Two Equation Models 20 2.4 Wall Bounded Turbulent Flows 22 2.4.1 Wall Functions 23 2.4.2 Low Reynolds Number Modelling 29 2.4.3 Two Layer Models 33 2.4.4 Discussion ofNear Wall Turbulence Models 34 for Separated Flows 2.5 Turbulence Modelling in Buoyancy Driven Flows 35 EXPERIMENTAL APPARATUS 39 3.1 Introduction 39 3.2 Description ofTest Cell and Loop 40 3.2.1 The Test Cavity 40 3.2.2 The Test Loop 43 3.3 Flow VISUalization 43 vi TABLE OF CONTENTS (Continued) TABLE OF COI\'TENTS (Continued) Page Page 3.4 Velocity Measurement 48 7.3 Verification ofthe Computer Code 126 3.4.1 Theory of Laser Doppler Anemomerry 48 7.3.1 Verification Study I-Laminar Natural Convection 126 3.4.2 Laser Doppler Apparatus 49 7.3.2 Verification Study 2-Laminar Mixed Convection 133 3.4.3 Counter Processor 51 7.3.3 Verification Study 3-Turbulent Natural Convection 136 3.5 Temperature Measurement 52 7.4 Preliminary Testing of the Code for Mixed Convective Flows 144 7.4.1 In~roduction 144 EXPEUMENTALPROCEDURE 54 7.4.2 Grid Independence Study 145 4.1 Introduction 54 7.4.3 Turbulence Model Study 151 4.2 Flow Visualization 55 7.4.4 Secondary Model Study 155 4.3 Velocity Measurements 55 7.5 Numerical Predictions of Turbulent Mixed 164 4.4 Temperature Measurements 60 Convective Flows in Cavities 7.5.1 Low Reynolds Number Turbulence Model Predictions 164 MATHEMATICAL FORMULATION 62 7.5.2 FLUENT Predictions 169 5.1 Introduction 62 7.6 Comparison of Experimental and Numerical Results 182 5.2 Governing Equations 62 7.6.1 Flow Field and Separation Location 183 5.2.1 Dimensional Governing Equations 62 7.6.2 Temperature and Heat Transfer 200 5.2.2 Dimensionless Variables 65 .... 5.2.3 Dimensionless Governing Equations 66 CONCLUSIONS AND RECOMMENDATIONS 212 5.3 Boundary Conditions 70 8.1 Contributions 212 5.3.1 Laminar Flow 70 8.2 Conclusions 213 5.3.2 Turbulent Flow 71 8.3 Recommendations 215 5.4 Calculation ofThermo!,hysical Properties 74 5.5 Variation ofTurbulent Prandtl Number 75 REFERENCES 216 NUMERICAL FORMULATION 77 APPENDICES 6.1 Introduction 77 A Derivation of QUICK Grid Coefficients for a Non-Uniformly Spaced Grid 223 6.2 The Control Volume Method 78 B Uncertainty Analysis 226 6.3 Discretization Equation 80 B.l Introduction 226 6.4 QUICK Differencing 84 B.2 Thermophysical Properties 227 6.5 Boundary Conditions 86 B.3 DimensioDless Parameters 228 6.6 Linearization of Source Terms 89 B.4 Measured Quantities 229 6.7 Variable Property Model 95 C Experimental Data 231 6.8 Pressure-Velocity Coupling 95 6.9 Summary of Steps in a Computational Cycle 97 6.10 Solution of Algebraic Equations 98 6.11 Description of FLUENT Software 99 RESULTS AND DISCUSSION 100 7.1 Introduction 100 7.2 Experimental Results 101 7.2.1 Flow Visualization 102 7.2.2 Velocity Measurements 110 7.2.3 Temperature Measurements 120 vii viii NOMENCLATURE (Continued) ~OMENCLATURE K Thermal conductivity of the fluid 1m Length of LDA measurement volume t m Mixing length (see Equation 2.20) L Turbulence length scale Inlet ratio of the cavity (see Equation 2.2) ll1 Nu Nusselt number (see Equation 7.7) i discretization coefficient (see Equations 6.6-6.12) P Fluid pressure Cavity aspect ratio (see Equation 2.3) tll P DimensioDless pressure Area of the i face ofthe control volume Pej Peelet number at control volume face i (see Equation 6.15) Archimedes number of the system (see Equation 2.6) Pr Fluid Prandtl number (see Equation 2.1) Boundary condition descriptor (see Figure 6.3) Ra Rayleigh number (see Figure 2.2) Boundary condition descriptor (see Figure 6.3) Re Inlet Reynolds number (see Equation 2.4) Boundary condition descriptor (see Figure 6.3) ~ Turbulent Reynolds number (see Equation 2.49) Empirical turbulence model constant Rer Reynolds number based on the turbulent viscosity (see Equation 5.28) Specific heat of the fluid at constant pressure Ri Richardson number ofthe system (see Equation 2.6) Empirical turbulence model constant (see Table 2.1) S9 Source term for variable q, (see Table 6.2) Empirical turbulence model constant (see Table 2.1) Sc Constant part ofthe source term (see Table 6.3) Empirical turbulence model constant (see Table 2.1) Sj Sign ofthe velocity at control volume face i (see Equation 6.26) Empirical tu.1lulence model constant (see Table 2.1) Sp Variable part ofthe source term (see Table 6.3) Fringe spacing (see Equation 3.3) t Tune DiaMeter ofLOA measurement volume (see Equation 4.1) TJ Temperature ofthe inlet jet Inlet height TR Reference temperature (see Equation 5.5) Low Reynolds number model extra term (see Table 2.2) Tw Temperature ofthe heated wall Dimensionless low Reynolds number model extra term (see Equation 5.46) u Velocity component in the x coordinate Conduet3nce across control volume face i (see Equation 6.14) U DimensioDless velocity (see Equation 5.16) Diameter of laser beam uJ Average volumetric inlet velocity Hydraulic diameter u + DimensioDless velocity (see Equation 2.32) Low Reynolds number model extra term (see Table 2.2) v Velocity component in the y coordinate Dimensionless low Reynolds number model extra term (see Equation 5.48) V DimensioDless velocity (see Equation 5.16) Wall roughness parameter w Velocity component in the z coordinate Doppler frequency (see Equation 3.4) x . Dimensional horizontal coordinate (see Figure 3.1) Low Reynolds number model damping function (see Table 2.2) X DimensioDless horizontal coordinate
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