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 (see Equation 5.15) Low Reynolds number model damping fuoction (see Table 2.2) y Dimensional vertical coordinate (see Figure 3.1) Low Reynolds number model damping fuoction (see Table 2.2) y+ DimensioDless coordinate (see Equation 2.31) Low Reynolds number model damping fuoction (see Table 2.2) Y DimensioDless vertical coordinate (see Equation 5.15) Apparent focal length ofLOA Dimensional horizontal coordinate (see Figure 3.1) Mass flowrate across control volume face i (see Equation 6.15) ilb component ofthe acceleration due to gravity Generation ofturbulence (see Equation 5.14) Greek Variables Dimensionless generation ofturbulence (see Equation 5.32) Grashof number based on the laminar viscosity (see Equation 2.5) Laminar thermal diffusivity ofthe fluid Grashof number based on the turbulent viscosity (see Equation 5.33) Turbulent thermal diffusivity ofthe fluid (see Equation 5.11) Local convective heat transfer coefficient Coefficient ofthermal expansion ofthe fluid Cavity height Distance from separation height (see Equation 4.2) Flu: across control volume face i (see Equation 6.2) General diffusion coefficient Dimensional turbulent kinetic energy (see Equation 2.17) Kronecker delta (see Equation 2.16) Dimensionless turbulent kinetic energy (see Equation 5.19) Horizontal distance between neighbouring nodes (see Figure 6.2)

ix NOMENCLATURE (Continued) NOMENCLATURE (Continued)

Vertical distance between neighbouring nodes (see Figure 6.2) Turbulent Length of time step T 11JR Turbulent kinetic energy source term Volume ofthe control volume Viscous source term Width of control volume (see Figure 6.2) VIS VP Variable property model source term Heigbt ofcontrol volume(see Figure 6.2) West control volume face (see Figure 6.2) Absolute error in variable t!> (see Equation B.2) West node (see Figure 6.2) Dimensional rate of dissipation ofturbulent kinetic energy w Dimensionless rate ofdissipation ofturbulent kinetic energy (see Equation 5.20) Dimensionless temperature (see Equation 5.17) Supersaipts Half angle between laser beams von Karman"s constant H Refer to the hybrid differencing scheme Wavelength oflaser light Q Refer to the QUICK differencing scheme (see Equations 6.18-6.25) Laminar dynamic viscosity ofthe fluid overbar Tune averaged term Laminar kinematic viscosity of the fluid Fluctuating term Turbulent kinematic viscosity (see Equation 2.27) Dimensionless turbulent viscosity (see Equation 7.7) Pi Fluid density Prandtl-Scbmidt number for dissipation rate (see Table 2.1) Prandtl-Scbmidt number for turbulent kinetic energy (see Table 2.1) Turbulent Prandtl number (see Table 2.1) Shear stress General flow variable

",. Dimensionless stream function (see Equation 7.6) "i' Turbulence intensity (see Equation 2.46)

Subsaipts

b Constant part BUOY Buoyancy source term DIS Dissipation ofturbulent kinetic energy source term East control volume face (see Figure 6.2) E East node (see Figure 6.2) GEN Generation ofturbulent kinetic energy source term L Laminar LIN Linearized term LR Low Reynolds number model source term North control volume face (see Figure 6.2) lib Neighbouring points N North node (see Figure 6.2) P Computational node being solved for QU QUICK source term South control volume face (see Figure 6.2) South node (see Figure 6.2)

xi xii

usr OF FIGURES (Continued) Page 7.21 Verification Study 3-Vertical Velocity Profile 137 LIST OF TIGURES 7.22 Verification Study 3-Temperature Profile on Vertical Centreline 139 Page 7.23 Verification Study 3-Proflles ofHeat Flux along the Vertical Walls 140 7.24 Verification Study 3-Proflles ofTurbulence Parameters 141 2.1 Problem Specification 5 on Horizontal Centreline 2.2 Natural Convection in a Venical Rectangular Cavity 12 7.25 Verification Study 3·Horizontal Profiles of Turbulent Viscosity Ratio 143 2.3 Wall Bounded Turbulence Models 24 7.26 Grid Dependence Study-Vertical Velocity Profile along Horizontal Centreline 147 3.1 The Flow Cell 41 7.27 Grid Dependence Study-Temperature Profile along Horizontal Centreline 148 3.2 The Heated Wall 42 7.28 Grid Dependence Study-Horizontal Velocity Proftle along Vertical Centreline 149 3.3 The Flow Loop 44 7.29 Grid Dependence Study-Temperature Profile along Vertical Centreline 150 3.4 Particle Streak Flow Visualization 46 35 Laser Induced Fluorescence Flow Visualization 47 7.30 Turbulence Model Study-Vertical Velocity Proftle 152 3.6 LDA Components 50 along Horizontal Centreline 7.31 Turbulence Model Study-Horizontal Velocity Profile 154 6.1 Control Volume Orientation 79 along Vertical Centrelina 6.2 Typical Control Volume 81 6.3 Boundary Nodes with Typical Control Volume 88 7.32 Variation ofThermophysical Properties 156 7.33 Variation of Dimensionless Buoyancy Source Term 158 7.1 Observed Flow Regimes 103 7.34 Secondary Model Study-Vertical Velocity along Horizontal Centreline 161 7.2 Case I-Flow VISUalization lOS 7.35 Secondary Model Study-Horizontal Velocity along Venical Centreline 163 7.3 Case 2-Flow Visualization 106 7.4 Case 3-Flow VISUalization 107 7.36 Case I-Low Reynolds Number Turbulence Model Predictions 165 75 Case 4-Flow Visualization 108 7.37 Case 2-Low Reynolds Number Turbulence Model Predictions 166 7.38 Case 3-Low Reynolds Number Turbulence Model Predictions 167 7.6 Case 1-LDA Results 111 7.39 Case 4-Low Reynolds Number Turbulence Model Predictions 168 7.7 Case 2-LDA Results 112 7.8 Case 3-LDA Results 113 7.40 Case I·FLUENT Two Dimensional Predictions 171 7.9 Case 4-LDA Results 114 7.41 Case 2-FLUENT Two Dimensional Predictions 172 7.10 LDA Measurements in Separation Region 118 V~2 Case 3-FLUENT Two Dimensional Predictions 173 7.1I LDA Measurements in Exit Region 119 7.43 Case 4-FLUENT Two Dimensional Predictions 174 7.44 Case I-FLUENT Three Dimensional Predictions 176 7.12 Typical Temperature Traces 121 7.45 Case 2-FLUENT Three Dimensional Predictions 177 7.13 Horizontal Temperature Proftles 123 7.46 Case 3-FLUENT Three Dimensional Predictions 178 7.14 Vertical Temperature Profile 125 7.47 Case 4-FLUENT Three Dimensional Predictions 179

U5 Verification Study l-Streamlines 128 7.48 Case l-Streatnline Comparison 184 7.16 Verification Study l-U Velocity Contours 129 7.49 Case 2-Streamiine Comparison 186 7.17 Verification Study I-V Velocity Contours 130 750 Case 3-Streamiine Comparison 187 7.18 Verification Study I-Temperature Contours 131 751 Case 4-streamline Comparison 189 752 Case 2-Dimensionless Turbulent Kinetic Energy Comparison 193 7.19 Verification Study 2-Laminar Mixed Convection 134 753 Case 2-Dimensionless Turbulent Viscosity Comparison 194 7.20 Verification Study 2-Streamlines 135 754 Case 3-Dimensionless Turbulent Kinetic Energy Comparison 195 755 Case 3-Dimensionless Turbulent Viscosity Comparison 196 756 Case 4-Dimensionless Turbulent Kinetic Energy Comparison 197 757 Case 4-Dimensionless Turbulent VISCOSity Comparison 198

xiii xiv LISf OF FlGURES (Continued) Page 7.58 Case 2-Horizontal Temperature Profile Comparison 201 7.59 Case 3-Horizontal Temperature Prome Comparison 202 LIST OF TABLES 7.60 Case 4-Horizontal Temperature Profile Comparison 203 Page

7.61 Case 2-Nusselt Numbet Profile Comparison 206 2.1 Constants used in the Ie-e Model 21 7.62 Case 3-Nusselt Number Profile Comparison 207 2.2 Typical Low Reynolds Number Models 32 7.63 Case 4-Nusselt Number Profile Comparison 208 4.1 Counter Processor Settings 56 A.l QUICK Differencing 224 4.2 Velocity Measurement Locations in the Plane ofSymmetry 57 4.3 Velocity Measurement Locations in the Separation Region 58 4.4 Velocity Measurement Locations in the Exit Region 59 4.5 Velocity Measurement Locations in the Inlet and Outlet Planes 59 4.6 Velocity Measurement Locations Adjacent to the Heated Wall 60 4.7 Horizontal Temperature Measurement Locations 61

6.1 The Function A[P] for Different Schemes 84 6.2 Summary of the Dimensionless Governing Equations 90 6.3 Summary of Source Term Linearization 94

7.1 Summary ofCases 100 7.2 Summary ofDimensionless Separation Heights from Flow Visualization 110 7.3 Summary of Dimensionless Experimental Separation Locations 120 7.4 Verification Study I-Summary of Important Flow Variables 132 7.5 Summary of Secondary Models 160 7.6 Summary of Dimensionless Separation Locations for the Low Reynolds 169 Number Turbulence Model 7.7 Summary of Dimensionless Separation Locations for FLUENT Predictions 182 7.8 Separation Location Comparison 190 7.9 Average Nusselt Number Comparison 210

B.l Uncertainties in the Basic Measurements 227 B.2 Uncertainties in the Thermophysical Properties 228 B.3 Uncertainties in the Dimensiunless Parameters 229 B.4 Uncertainties in the Measured Quantities 230

1.2 Practictl Applications

CHAPTER 1 The study of :nixed convection in enclosures has many practical applications in

engineering. In many of these applications, the geometry is complex and the flow structures INTRODUCTION typically exhibit turbulence and recirculation which are difficult to model. Although the present

work uses a simple rectangular geometry, it provides a fundamental understanding of the 1.1 Nature oC the Flows and Heat TransCer buoyancyfmertial interaction of the flow that is necessary for the prediction of more complex The mechanism of convective heat transfer is referred to as heat transfer to a fluid in geometries. motion. The two modes ofconvection are free and forced convection and the heat transfer rates Some direct applications include the heating, cooling and ventilation of rooms and are usually estimated by considering one of these modes dominant. buildings, the cooling ofelectronic components, the study ofthermal pollution in water resources, As heat is transferred to or from a fluid, the temper-..rure gradients present give rise to the study offluid motion in storage tanks used in solar heating, the analysis ofheat exchangers, density gradients. In the presence ofa body force, such as gravity, these density gradients bring and many applications in the nuclear industry. about fluid motion. This mode ofheat transfer is called free or natural convection. Since the velocity field is determined by the buoyancy effect of the fluid, the velocity and temperature 1.3 Objectives oC the Present Work fields are strongly coupled. The heat transfer is usually dependent on fluid properties through The present work: is a combination of both an experimental and a numerical study of the Prandtl and Grashof numbers (see section 2.1 for a definition of these parameters). mixed convection in enclosures. While each ofthese studies complement the other. each has its In the case offorced convection, the fluid motion is due primarily to some external force. o~objectives. niis external force may be the result of a fan. pump, the wind, etc. With this type of In the experimental study. the velocity. temperature. and turbulent fields are obtained in convection, usually the velocity field is determined fllSt and the temperature field is determined the enclosure. In addition, overall heat transfer correlations are presented. The main motivation from it. The heat transfer rate is a function ofthe Reynolds and Prandtl numbers. for the experimental study is to provide experimental data which may be used to compare and In many engineering applications the effect ofbuoyancy and forced convection are ofthe verify numerical simuiations. This is necessary as there are a limited number of experimental same order. In this case. the heat transfer is referred to as mixed or combined convection. The studies. present study collSiders this type of convection in rectangular enclosures. The numerical study will aid in the future development ofnumerical codes. Some ofthe

mathematical modelling and some aspects of numerical formuiatiollS are strongly problem

dependent. As there are no extensive numerical studies ofmixed convection in enclosures in the current literature. the present numerical srudy will aid in the testing of the performance ofsome of the models and formulations. CHAPTER 2

LITERATURE SURVEY

2.1 Introduction

Before a summary ofprevious literature is presented. a brief explanation ofterminology

and parameters used in mixed convection problems is presented. In the present study, the

velocity, temperature, and turbulence fields will be obtained both experimentally and numerically

in a rectangular cavity. The specific cavity is shown schematically in Figure 2.1. The enclosure

shown is a rectangular cavity with a horizontal jet entering in the upper left comer. This inlet

jet bas specified velocity and temperature profiles (uJ and TJ respectively). The fluid discharges by way ofa horizontal duet located in the lower left comer ofthe cavity. All solid boundaries

are insulated with the exception ofthe right wall, which is maintained isothermally at Tw. This problem may be specified in terms ofdimensionless parameters. These are the fluid

Prandtl number, the cavity inlet ratio, the cavity aspect ratio, the inlet Reynolds number, and the

Gramof number. The Prandtl number, Pr, is a dimensionless paratneter defined as the ratio of

momentum to thermal diffusion and is expressed as

~ Prz ILLCp. (2.1) K aL where ILL is the dynamic viscosity of the fluid,

<; is the specific heat at constant pressure,

K is the thermal conductivity ofthe fluid,

lit. is the kinematic viscosity, and

aL is the thermal diffusivity ofthe fluid.

The inlet ratio, a, is defined as the ratio of the inlet height to the cavity height and is expressed

as

(2.2)

-----I~~ UJ TJ It must be noted that, in the current study. the outlet height equals the inlet height. The cavity

aspect ratio, A, is the ratio of the height to width of the cavity

A = ~ (2.3) Tw l The inlet Reynolds number, Re, is the ratio of inettial to viscous forces and is defined as The system Grashof number, Gr, is the ratio of buoyant to viscous forces and is expressed as H

(2.5)

In this notation, the subscript i refers to the component ofgravity and the subscript H refers to

the length scale. In the present work, the non-zero component ofgravity is 1;-. The length scale .J------w------1 is the height of the cavity as an important boundary layer forms on the heated wall. Another

cOnvenient paratneter that is often used with mixed convection is the Archimedes (Ar) or D Richardson (Ri) number. In general, this variable is,

(2.6)

This paratneter is defined as the ratio ofbuoyant forces to inertial forces and is derived from the

buoyant term in the dimensionless momenrum equations. This term appears as a source term in

the dimensionless govemiDg equations. Ifthe Grashof and Reynolds numbers do not have the Figure 2.1: Problem Specification same length scale additional dimensionless parameters may be present in this term. In Figure 2.1, the fluid inenia drives the fluid across the top boundary. As it meets the 2.2.1 Previous Numerical Rese:u-ch solid right wall, the fluid turns d<.wn the right boundary, and as it meets the lower boundary, the Oberkampf and Crow [2) considered an open reservoir with a horizontal inler jer at the fluid turns horizontally toward the exit. As the fluid flows adjacent to the right boundary, heat surface on one end ofthe reservoir and a horizontal outler pon at various heights on the opposite is transferred to or from the wall. Ifthe boundary temperature is less than the fluid temperature, end. The study considered the effects of inflow and outflow, heat traIISfer and wind shear at the the fluid traIISfers heat to the wall resulting in a downward buoyancy force. Since the buoyancy surface. In a well mixed reservoir, the fluid circulates top to bottom by way of large vortices, force is acting in !!Ie same direction as !!Ie inenial force, the flow is said to be ·aiding flow'. while in a stratifioo reservoir, the venical motion of the fluid is inhibited by buoyancy. Wmd Tf the temperature ofthe wall is greater than the fluid temperature, heat is traIISferred to the fluid speed also had a significant effect on the temperature and flow field. An aiding wind increased resulting in an upward buoyancy force. Since the buoyancy and inenial forces oppose each other, the speed of the flow while dissipating a considerable amount of heat (by convection and the flow is said to be ·opposing flow·. The aiding or opposing nature of the flow may be evaporation). An opposing wind forced the heated jetbelow the surface and significantly reduced characterized in tenos ofthe Grashofnumber. the heat dissipation to the atmosphere. The researchers also reported a high amount ofcomputer

GrL > 0 for opposing flow time necessary for :he simulations.

Gr R 0 for isothermal flow L (2.7) To study the flow inside solar energy storage tanks, Cabelli (3) used a stream function

GrL < 0 for aiding flow vorticity approach to calculate a laminar flow field. The results indicate that for a Richardson

number greater than about unity, buoyant effects were dominant. 2.2 Mixed Convection in Enclosures Hjenager and Magnussen [4,5) predicted three-dimensional turbulent flow in a room. The study ofmixed convection in internal vertical and horizontal flows has been studied In, this work both the isothermal and buoyant flows were considered. Comparisons with e'nensively in the past. Typically, the geometry used in most of the research includes flow experiments, indicated that for isothermal flows the prediction is adequate. However, inadequate between parallel plates and flow in tubes and duets (both circular and non-circular). While the results were obtained with buoyant flows. Reference (5) states that reliable measurements are re5'Jlts ofthis research is insightful, it is not presented in this literature survey for the sake of needed for the validation ofthe computer codes. brevity. However, Gebhart er al. [I) provides a complere literature survey on this topic and Shoukri and Ahluwalia (6) studied turbulent mixed convection in enclosures. 10 this others. The research ofmixed convection in confined eotclosures is limited. A detailed survey study, an explicit numerical scheme and a two equation, k-e, turbulence model were employed. ofthe current literature follows. The boundary conditions consisted of a square enclosure with two inlets (one at each end) and

an outler port (m the centre) ofthe bottom wall. The fluid was heated volumetrically in a region

in the centre of the cavity. The study indicated that there are strong interactions between the

10 inenial forces and the buoyancy forces and that these interactions had a significant effect on the Kumar and Yuan nO) used a SIMPLE family algorithm to predict laminar mixed temperature field. For the case considered, it was found that for an Archimedes number greater convection. In this study, a venical isothennal jet entered the cavity in the upper left comer. than approximately 0.2, the flow was buoyancy dominated and for Archimedes numbers less than The outlet pon (venical as well) was ill the upper right comer. The entire cavity was maintained this value, the flow was dominated by inenia. The results also indicated that turbulence at a constant temperature. This study concluded that buoyancy effects have been found to be modelling was needed for such flow interactions. The k-c model appeared to give adequate significant on the flow and temperature fields and on the friction factor and heat transfer rate. results. Recently, papers have been written which deal with reviews of current numerical

Cha and laIuria [7,8) studied mixed convection flow for energy extraction. Hot fluid analysis. These include:he papers ofPatankar (11) and Leschziner (12). While these papers are was withdrawn from the top ofthe cavity and cold Water was placed in the caVity at the bottom DOt written specifically for modelling mixed convection, many ofthe principles presented tnay to preserve thermal stratification. The effect of buoyancy was found to be significant for be employed when modelling such flows. Richardson numbers greater thaI: 0.1 and was very strong for Ri greater than about 1. The horizontal spread and the vertical mixing was found to be dependent on the inlet parameters. 2.2.2 Previous EKperimentni Rese:m:h

Oosthuizen and Paul [9) modelled laminar mixed convection using a finite element Neiswanger, Johnson and Carey [13] conducted flow visualization and measured local stream function-vorticity formulation. The problem specification is identical to that of Figure heat transfer for cross flow mixed convection in a rectangular enclosure with restricted inlet and

2.1, with the exception ofthe left wall being maintained isothermally at the same temperature as outlets. Until the publication of this paper, the authors claimed that no detailed experimental the jettemperature. This study found that the buoyancy forces increase the average heat traIISfer studies had been conducted on high Rayleigh number mixed convection near venical walls in fo,r aiding flow and decrease the heat transfer rate in the case of opposing flow. However, for en,c1osures. For Reynolds numbers less than 2000, the flow was laminar. The flow structure opposing flow, the heat transfer is enhanced by buoyancy ifthe Reynolds number is very smaIl. changed considerably at a Reynolds number of5000. Neiswanger er aI. suggest that there is a

The effect ofbuoyancy tnay be neglected if need for more experimental data for different geometries and Prandtl numbers.

JaIuria and Cooper (14) studied negatively buoyant wall flows. 10 this study, a detailed NUJorad > 2.5 (2.8) NUJ"~ investigation ofpenetration, entrainment, and heat transfer characteristics ofbuoyant wall jets is For buoyancy dominated flow, the inertial force tends to make a more uniform heat transfer presented. distribution. For the opposing flow case, the local heat transfer rate is significantly inf1uetJCed by the Reynolds number. 11 12

2.2.3 Natural Convection in Enclosures

A brief review ofnatural convection in enclosures is presented as this has been studied

extensively in lbe past. In addition, natural convection may be considered a limiting case of

mixed convection (completely buoyancy dominated). Thus some aspects of natural convection

may be applicable to mixed convection.

The present survey is limited to rectangular vertical cavities. In lbis configuration, lbe

two vertical walls are differentially heated and lbe horizontal walls are isothermal (see Figure H T H Tc 1 2.2). Natural convection probletns are usually specified in tertns oflbe fluid Prandll number. ~I

lbe system Rayleigh number and lbe cavity aspect ratio. The Prandll number and aspect ratio g,

have been previously defined and the Rayleigh number is defined as the product of the Grashof

and Prandll numbers. The review papers of Catton [lS] and Osttach [16,17] present complete literature reviews of rectangular vertical cavities as well as others of special interest. Osttach J [16] states that vertical cavities contain all ofthe essential physics relevant to all confined natural w I convection probletns.

In 1978, the conference on Numerical Methods in Laminar and Turbulent Flow was held

at.university College, Swansea. During !his conference, a session concerning the comparison Ra = gP(TH -TclH' va: of numerical techniques applied to standard probletns was held. This session concluded that Pr=.!... buoyancy driven flow in a cavity with differentially heated sides was an adequate problem for a a: comparison. For this standard problem, the Rayleigh numbers based on the temperature A. =!!. w difference ofthe vertical walls were set at lQ3, 104, lOS and 1()6. The fluid in the cavity bad a

Prandll number of 0.71 which corresponds to air.

A year later, after a great deal ofdiscussion with various mathematicians and engineers,

it was decided that the twO-dimensional problem of Mallison and de Vabl Davis [18] was an Figure 2.2: Natural Convection in a Vertical Rectangular Cavity adequate test case. Following the discussions, Jones and de Vabl Davis invited coDttibutors to

13 14 subtnit a solution for the flow and thermal fields. The contributors were requested to submit the (see section 2.5). In this section, specific emphasis is placed on the effect of turbulence following information modelling. (a) the average Nusselt number,

(b) the maximum and minimum local Nusselt numbers on the hot wall and their locations, 2.3 Turbulence Modelling

(c) the maximum vertical velocity on the horizontal midplane and its location, Turbulent motion is described as being rotational, intertnittent, highly disordered, and

(d) the maximum horizontal velocity on the vertical midplane and its location, dissipative [23]. In principle, the instantaneous conservation equations apply equally to a

(e) contour plots of the velocity components, and, if available, the stream function, the turbulent motion as to a laminar one. However, at present, this is not a practical route as the

pressure and the temperature. important details ofturbulence are small scale in character. For example, in gaseous flow, the

In 1983. de Vabl Davis and Jones published two papers on this subject [19,20]. The eddies responsible for the decay of turbulence are of lbe order of 0.1 mm [24]. In most first was a summary and assessment of37 solutions and the second was the description ofabench engineering applications, the flow domain is many orders of magnitude larger and hence direct mark solution. The authors claim that the bench mark solution is accurate to within one percent. simulations require large computing times and computer memory.

The motivation for obtaining a highly accurate solution was to have something to compare with A complete description ofdifferent turbulence models is beyond the scope ofthe present for validating computer programs. In 1990, Hortmannn et aI.[21] used a finite volume multigrid project. Thus the objective of this chapter is to provide an understanding of the need of procedure to predict the bench mark solution. They predicted their solution to be accurate to turbulence models and to provide background information necessary for the discussion ofthe wall within 0.01%. bounded models presented in section 2.4. A comprehensive discussion ofturbulence modelling

Hyun and Lee [22] predicted transient natural collvection in asquare cavity. This study ~y be found in references [23.24,25]. concluded that if the Prandll number is greater than unity, oscillations in the solution were evident if 2.3.1 Time-Averaged Governing Equations

(2.9) Fortunately. in most applications, the engineer is only concerned with the time-averaged

The present literature survey for laminar natural convection is restricted to these cases effects of turbulence. Thus. the first step towards the modelling of turbulent flow is the as the bench mark problem may be considered the definitive test case. Several researchers have decomposition of the flow variables into a time-averaged and fluctuating component. Thus, a ~. studied turbulent natural convection using both experimental and numerical techniques. A given flow variable, is decompnsed by detailed discussion ofthis literature is presented after turbulence modelling bas been introduced (2.10) 15 16

The overbar denotes the time-averaged value while the prime denotes the random variation about stress and turbulent heat flux terms. This is a possible solution to the closure problem but the the mean. With the substitution of equation (2.10) into the governing equations, the time- resulting equations for these terms contain higher order terms [25]. For example, the partial averaged governing equations may be derived. For nearly incompressible flow, the resulting two- differential equation for the Reynolds stress term. /iiUJ, contains a third order term, namely, dimensional cartesian equations of motion written in tensor notation are: Uj UJ Uk. Equations for the third order terms contain fourth order terms. In general, closure

Conrinuity schemes are usually stopped at second order terms and the third order terms are approximated

by known variables. (2.11)

Momennun 2.3.2 The Turbulent VIScosity Concept

(2.12) Another approach to the closure ofthe equations is to use a simple turbulence transport model. These models use an eddy or turbulent viscosity concept. Some examples ofthe models Energy include the zero, one and two equation models where the number of equations refers to the

Ei + oiijI = .i- [a jJ.. - "iE'T] (2.13) number ofpartial differential equations in the model. or OXj ib:j L OXj J Boussinesq introduced the first concept of a turbulence model [24]. Analogous to In this form, the buoyancy source term in the momentum equation has been modelled with the Newton's law ofviscosity, Boussinesq approximation. This approximation states that the density ofthe fluid is considered

OUj OUj) constant except in the buoyancy source term. This term is the last term in equation 2.12 and is TL =JLL - +- (2.14) [ OXj OXj reSponsible for the motion ofthe fluid due to density gradients. Without this ap~roximaIion, the Boussinesqsuggested using a turbulent shear stress, TT, equal to the product ofthe mean velocity buoyancy source term is modelled by + isg. The use ofthe Boussinesq equation is discussed in gradient and a turbulent viscosity. Thus Chapter 7. For simplicity, it is usual to drop the overbar in the governing equations on all the terms except the Reynolds stress term, /iiUJ and the turbulent heat flux term, iijT. This TT •- p u/ u! • JLT [~ + ~3 pajJk (2.15) J ib: 5]ib: - j j notation has been adopted in the present work. These equations are not closed as there are more The turbulent viscosity, Jkr, is not a property ofthe fluid but is dependent on the local turbulence. unknowns than equations. The averaged equations of motion differ from their instantaneous Its value may vary from flow to flow and even point to point within a given flow. The first term counterparts, as the former contains the Reynolds stress term and the turbulent heat flux term. on the right hand side ofequation 2.15 accounts for the shear stress due to turbulent motion just The obvious approach to close the set ofequations is to derive exact equations for the Reynolds as equation 2.14 accounts for the shear stress due to molecular motion. The second term on the

17 18 right hand side of equation 2.15 accounts for the normal stresses due to turbulent motion just as history of turbulence. As a result, the model incorrectiy predicts the turbulent viscosity to be the static pressure, PIp. accounts for the normal stresses due to molecular motion. The variabie, zero whenever the velocity gradient is zero [25].

O;j' is the Kronecker delta and is defined as, In 1930, von Karman proposed that the mixing length may be modelled using

Oij = 1 if i=j j (2.16) t m • I: I oU/ox , I (2.20) = 0 if i¢j a'lu/oxj

The symbol, k, denotes the turbulent kinetic energy and is defined as where I: is von Karman's constant (I: = 0.42). Von Karman's proposal is limited at inflection

(2.17) points ofthe velocity profile (i.e. a'lu/ib:J = 0). At this location the mixing length is infinite and cannot be used to compute the fmite shear stress. In a similar manner the turbulent heat flux can be expressed as, It must be noted that equation (2.19) and equation (2.20) are based on erroneous (2.18) physical arguments but can be regarded as definitions for the quantities "T and t", which in where CSor is turbulent Prandtl number [26]. simple flows are easier to determine than /iiUJitseif [27]. In many problems it is difficult to

The concept ofturbulent viscosity provides a basis for turbulence modelling but it itself specify the mixing length and in flow situations more complex than shear layers it may be cannot be considered a model since it can't be determined from calculable quantities. Thus impossible. additional modelling must be used in conjunction with the turbulent viscosity concept. Launder and Spalding [24] presented a review of other zero equation models.

2.3.3 Zero Equation Models 2.3.4 One Equation Models

The fU'St proposed zero equation model for the turbulent viscosity, Jkr ' is the mixing In an attempt to overcome limitations of the zero equation models, turbulence models length hypothesis [24). Prandtl suggested that the turbulent viscosity be described as, have been developed to account for the transport and history ofturbulence. This is accomplished

with the use of a transport equation for an acceptable velocity scale of turbulence. Several JLT • pt~ ~ (2.19) I I models use the square root ofthe turbulent kinetic energy, ,fk , for the dependent variable in where the mixing length, tm' must be prescribed algebraically. The mixing length hypothesis the differential transport equation. An exact equation may be derived from the Navier-stokes has been tested extensively in the past and has many limitations. This model is based on the equation. The following equation for the turbulent kinetic energy is exact with the exception of assumption that the turbulence is in local equilibrium. Thus, the turbulence is dissipated and lhe first term on the right hand side. This term, called the diffusion term, assumes that lhe produced at the same rate. This hypothesis does not account for the convection, diffusion or 19 20

diffusion is proportional to the gradient of Ie. The constant, ak' is :ul empirical diffusion number simple shear flows or boundary layers where it is easy to specify l1'1e length scale, L. However

commonly referred to as the turbulent Schmidt number. in complex flows it is no easier to prescribe than tm was in the mixing length hypothesis [25].

Therefore the trend has been to use two equation models where the length scale is also ak ak a [ VT ] ak ~ au, - + Ui -•- 'VL + - - - Ui Uj -- c (2.21) at ax, ax, ak ax, axj detern:ined from a differential equation. The second term on the right hand side accounts for the production of to.lfbulence. The

dissipation term, e, is modelled on the assumption that in an equilibrium flow, the rate ofenergy 2.3.5 Two :F.quation Models

dissipated by the smallest eddies is equal to the rate ofenergy fed down the chain of eddies from For any improvement over the transport models presented thus far, an additional

the largest to the smallest (23). The dissipation is determined from transport equation for the length scale must be used. This allows the length scale to be influenced

by transport and history processes in a similar manner to the turbulent kinetic energy [25]. (2.22) Several models have been adopted such as the k-w, k-e, and the k-kL models where w represents where CD is an empirical constant and L is a length scale. For the case of buoyant flow, as is the square ofthe vorticity and e the rate ofdissipation. These models have achieved moderate the case ofmixed convection, an additional term is included in the differential equation to account success. The kat model will be discussed in detail as it has been used to predict recirculating for the production of turbulent kinetic energy due to buoyancy. This production term is added flow succes..'"fully (23). to the right side ofequation (2.21) and is (28) The equation for the dissipation of turbulent kinetic energy may be derived from the

(2.23) Navier-Stokes equations. However many modelling assumptions must be made and the result is

'Il!is term is naturally obtained as a result of time-averaging the momentum equation when the ve:Y empirical [25]. The usua1 equation for the rate ofdissipation ofturbulent energy, using the

buoyancy term is included. The turbulent Reynolds stresses are related to the turbulent kinetic assumption of infinite Reynolds number, is

energy through the Prandtl-Kolmogorov relationship, ~ u, ~ • +~] ~] C ~ vT [~ ~ ~ (2.25) + .!.. [[VL a~ + 1 +5] -Cz at ax, ax ax, k iJx aXi iJx k (2.24) i j j where a~ is an empirical diffusion constant commonly referred to as the Prandtl-Schmidt number where C is an empirical constant and L is a length scale which must be prescribed from simple Il for the dissipation rate. Just as a buoyancy term appears in the turbulent kinetic energy equation, empirical functions similar to those used with the mixing length (23). Although the equation for some authors use such a term in the rate ofdissipation equation. This source term is the turbulent kinetic energy accounts for the transport and history of kinetic energy, its range of (2.26) applications is limited by the prescription of the length scale. L. This model works well in

21 22

Other authors argue that there is no physical reasoning for including such a term and neglect it au):! [avl:! (au av]:!] (2.30) in the conservation equation [28,29). Some work has indicated that the inclusion ofthis term is G E vT [2[ -ax +2 -aYJ +l-ayax+- completely insignificant [29,30).

The normally used constants in the k·t equation are presented in Table 2.1 [26,31).

2.4 Wall Bounded Tu:-bulent Flows

Table 2.1: Consl:lnls used in the k-e Model As the wall is approach::d in a turbulent flow field, a further computational difficulty arises. Within a very thin region near the boundary, the effective.transport coefficients change Cl' C1 Cz C3 a~ ak '7 by more than an order of magnitude as viscous effects dominate the turbulence effects (30). In 0.7 0.09 1.44 1.92 1.3 1.00 0.9 this situation the models presented in section 2.3 are not capable ofpredicting the flow in their

present form as they are based on the assumption of infinitely high Reynolds number. As

The complete k-e model with the inclusion ofbuoyancy terms in tw

is, complex flow in a cavity. Therefore this section deals solely with wall bounded models used in

conjunction with a two equation model of turbulence. (2.27) In the wall bounded region, the models must be modified or additional modelling must

be incorporated into the fully turbulent models to account for the effect of viscosity near the

(2.28) walls. In an attempt to model this transitional layer, several methods have bee.1 adopted. These

include wall functions, low Reynolds number modelling, parabolic sublayer, and two layer

approaches.

In each subsection ofthis chapter an overview ofthese wall bounded models is presented

(2.29) subjectively. At the end of each subsection a discussion of the merits and limitations of each

method is presented.

where the mechanical production ofturbulence, G, is