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A computational and experimental investigation of film cooling effectiveness Zhou, Jian Ming 1994

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A Computational and Experimental Investigation ofFilm Cooling EffectivenessbyJIAN-MING ZHOUB.Sc. (Applied Mechanics), Fudan University, Shanghai, China, 1984M.Sc., The University of British Columbia, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIES(Department of Mechanical Engineering)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1994© Jian-ming Zhou, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of____________________________The University of British ColumbiaVancouver, CanadaDateDE.6 (2188)AbstractFilm cooling is a technique used to protect turbine blades or other surfaces from a hightemperature gas stream. This thesis presents an experimental and computational study offilm cooling effectiveness based on two film cooling models in which coolant is injectedonto a flat plate from a uniform slot (2-D) and a row of discrete holes (3-D). The existingturbulence models and near-wall turbulence treatments are evaluated. The transportequations are solved by the control volume finite difference and multigrid formulation, andthe flow and heat transfer near the injection orifices and the film cooled wall are resolvedby grid refinement. To verify the numerical model, physical experiments based on theheat-mass transfer analogy were carried out. Film cooling effectiveness and flow fieldswere measured using a flame ionization detector and hot-wire anemometry.For the 2-D model, the turbulence is modelled by the multiple-time-scale (M-T-s)turbulence model combined with the low-Re k turbulence model in the viscosity-affectednear-wall region. Comparisons of the film cooling effectiveness and flow fields betweencomputations and experiments for mass flow rate (RM) of 0.2,0.4,0.6 show that the M-TS model provides better agreement than the k-E model especially at high RM. Also, thelow-Re k turbulence model used in the near-wall region allows for grid refinement near thefilm cooled wall, giving better flow and heat transfer predictions downstream of injectionthan the wall function method.For the 3-D model, a non-isotropic k-E turbulence model is used in combinationwith the low-Re k turbulence model as the near-wall treatment. Comparison of thespanwise averaged film cooling effectiveness between computation and experiment showsgood agreement for mass flow ratios of 0.2, 0.4; however, the numerical values areconsistently lower than the measured results for RM = 0.8. Comparison of the meanvelocity and turbulence kinetic energy shows good agreement, especially near theinjection. Further work to extend the M-T-S model to the 3-D model is suggested.Parametric tests of film cooling by single and double-row injection were carriedout computationally to investigate the effects of mass flow rate, injection direction, holespacing and stagger on the film cooling effectiveness. The superior performance of thelateral injection at high mass flow ratio, mainly near the injection orifice, is demonstrated.For the double-row injection, consistently better performance of the arrangement withstagger factor A/d=3 is found for the range of parameters investigated.ilTable of ContentsAbstract iiTable of Contents iiiList of Tables viiList of Figures viiiNomenclature xivAcknowledgment xviChapter 1 Introduction 11.1. Background 11.2. Posing the Film Cooling Models 21.3. Objectives and Scope of the Thesis 5Chapter 2 Literature Survey 112.1. Experimental Studies 112.2. Computational Studies 162.2.1. Numerical Solution Techniques 162.2.2. Turbulence Modelling 212.3. Remarks Arising from Work Reviewed 24Chapter 3 Experimental Investigation 273.1. Heat and Mass Transfer Analogy 273.2. Experimental Facility and Equipment 283.2.1. WindTunnel 283.2.2. Injection System 293.2.3. Traverse Mechanism 293.2.4. Data Acquisition System 303.3. Measurement Techniques 30ifi3.3.1. Measurement of Concentration 313.3.2. Measurement of Fluid Velocity 323.3.3. Determination of 2-D Injection Flow Reattachment 323.3.4. Measurement Uncertainties 333.4. Experimental Measurements 333.4.1. 2-D and 3-D Wind Tunnel Models 333.4.2. Upstream Boundary Layer 343.4.3. Preliminary Tests for the 2-D Model 343.4.4. Preliminary Tests for the 3-D Model 353.4.5. Measurement Procedure 36Chapter 4 Mathematical Formulation 454.1. Turbulence Models 464.1.1. Eddy Viscosity and Diffusivity Concepts 464.1.2. The k- Turbulence Model 474.1.3. Nonisotropic Eddy-Viscosity Relation 494.1.4. The Multiple-Time-Scale Turbulence Model 504.2. Near-Wall Turbulence Treatments 524.2.1. Wall Function Approach 524.2.2. Low-Re k Model with Fine Grid Treatment 54Chapter 5 Computational Procedure 565.1. Finite Volume Formulation 565.2. False Diffusion 605.3. Solution Algorithms 625.3.1. Modified simpler Algorithm 625.3.2. Vanka’s Algorithm 645.4. Multi-Grid Computational Procedure 65Chapter 6 Results I: Two-Dimensional Case 69iv6.1. Computational Domain and Boundary Conditions 696.2. Numerical Grid and Effect of Grid Refinement 716.3. Predictions and Comparison with Experimental Data 746.3.1. Mean Velocity 746.3.2. Turbulent Kinetic Energy 766.3.3. Film Cooling Effectiveness 77Chapter 7 Results II: Three-Dimensional Case 917.1. Predictions and Comparison with Experimental Data 927.1.1. Computational Domain and Grid Arrangement 927.1.2. MeanVelocity 947.1.3. Turbulent Kinetic Energy 947.1.4. Film Cooling Effectiveness 957.2. Parametric Analysis 977.2.1. Computational Domain and Boundary Conditions 977.2.2. Finite Array Effects 997.2.3. Single-Row Injection vs. Double-Row Injection 1007.2.4. Hole Spacing Effect in Single-Row Injection 1017.2.5. Stagger Effect in Double-Row Injection 102Chapter 8 Conclusions and Recommendations 132References 135Appendices 141A. Experimental Measurement Uncertainty Analysis 141A. 1. Effectiveness Measurement 141A.2. Velocity Measurement 142A.3. Mass Flow Ratio 143B. Detailed Flow and Effectiveness Distributions 145B. 1. Spanwise Hole Spacing Effects in Single-Row Film Cooling 145vB.2. Hole Staggering Effects in Double-Row Film Cooling 145C. 2-D Computations with the Algebraic Reynolds Stress Model 163D. 3-D Computations with the Multiple-Time-Scale Model 163viList of TablesTable 3.1: Sampling parameters of the data acquisition system. 30Table 3.2: Summary of the measurement uncertainties. 33Table 3.3: Flux balance of 2-D injectant. 35Table 4.1: k-E turbulence model constants. 48Table 4.2: M-T-S turbulence model constants. 51Table 6.1: Arrangement of four progressively refined grids. 72Table 6.2: Multi-grid parameters in the 2D-MGFD code. 74Table 6.3: Comparison of reattachment lengths. 75Table 7.1: Multi-grid parameters in the 3D-MGFD code. 93Table 7.2: 3-D test parameters. 97Table B. 1: Location and Circulation of Vortices at X/d=3. 146VIIList of FiguresA typical turbine rotor blade cross section with cooling flow. 8Flat plate film cooling models. 9Schematics of 2-D and 3-D film cooling flows. 10Schematic of wind tunnel.Schematic of wind tunnel test section.Schematic of data aquisition system.Schematic of FID system.Calibration of FID system.Calibration of hot-wire probe.Geometric description of 2-D and 3-D models in the windtunnel test section.Measured mean velocity of the upstream boundary layer on thelogarithmic coordinate (X=-120 mm).Measured RMS turbulence intensity of the upstream boundarylayer (X=-120 mm).Measured concentration along the streamwise vertical linesdownstream of the injection holes (X/d=3, d=6.35 mm, 2-Dmodel).Measured surface concentration downstream of the injectionholes (X/d=3, d—12. 7 mm, 3-D model).Measured concentration along the central planes downstreamof the injection hole (X/d=3, d=12. 7mm, 3-D model).Figure 5.1: Relative location of staggered grid and calculated variables.Figure 1.1:Figure 1.2:Figure 1.3:Figure 3.1:Figure 3.2:Figure 3.3:Figure 3.4:Figure 3.5:Figure 3.6:Figure 3.7:Figure 3.8:Figure 3.9:Figure 3.10:Figure 3.11:Figure 3.12:37383839394041424243434459vuiFigure 5.2:Figure 5.3:Figure 5.4:Figure 6.1:Figure 6.2:Figure 6.3:Figure 6.4:Figure 6.5:Figure 6.6:Figure 6.7:Figure 6.8:Figure 6.9:Figure 6.10:Figure 6.11:Figure 6.12:Figure 6.13:Figure 6.14:Figure 6.15:596063797980818283848586878889908081A typical control volume.Finite difference grid.Control volume for the continuity equation.Computational domain for 2-D computations.Typical grid arrangement for 2-D computations (Grid 3).Predicted 2-D vertical velocity distribution at the slot exit(KE&LK, RM=0.4).Mean velocity and turbulent kinetic energy downstream ofinjection (X/d=3) predicted by four progressive refined grids(KE&LK, RM=0.4).Film cooling effectiveness predicted by four progressiverefined grids (KE&uc, RM = 0.4).Estimated false diffusion coeficient distribution on Grid 3(KE&LK, RM=0.4).Typical 2D-MGFD multi-grid iteration convergenceperformance (RM = 0.4).2-D mean velocity distribution (RM = 0.2).2-D mean velocity distribution (RM = 0.4).Predicted vector fields by the 2-D MTS&LK numerical model.2-D turbulent kinetic energy distribution (RM = 0.2).2-D turbulent kinetic energy distribution (RM = 0.4)2-D film cooling effectiveness distribution.2-D concentration distribution on the vertical streamwise plane(RM = 0.2).2-D concentration distribution on the vertical streamwise plane(RM = 0.4).xComputational doanun for 3-D film cooling modelVertical mean velocity at the hole exit predicted by fourprogressively refined grids (KE&LK, RM = 0.4)Mean velocity and turbulent kinetic energy at X/d=3 predictedby four progressively refined grids (KE&LK, RM = 0.4)Film cooling effectiveness predicted by four progressivelyrefined grids (KE&LK, RM = 0.4)Estimated false diffusion coefficient on the vertical streamwiseplane Z=O using Grid 3 (KE&LK, RM = 0.4)Estimated false diffusion coefficient on the vertical cross planeX!d—O.5 using Grid 3 (KE&LK, RM = 0.4).Typical 3D-MGFD iteration convergence performance(RM = 0.4)Mean velocity distributions (3D model, RM = 0.4)Mean velocity distributions (3D model, RM = 0.8)Predicted vector fields on the vertical streamwise planeZ=0(RM =0.4,0.8)Predicted vector fields on the vertical cross planeX/d=3 (RM =0.4,0.8).Turbulent kinetic energy distributions (3D model, RM = 0.4).Turbulent kinetic energy distributions (3D model, RM = 0.8).Film cooling effectiveness (3D model, RM =0.2,0.4,0.8)Film cooling effectiveness distribution on the wall surface(3D model, RM = 0.2).Film cooling effectiveness distribution on the wall surface(3D model, RM = 0.4).Figure 7.1:Figure 7.2:Figure 7.3:Figure 7.4:Figure 7.5:Figure 7.6:Figure 7.7:Figure 7.8:Figure 7.9:Figure 7.10:Figure 7.11:Figure 7.12:Figure 7.13:Figure 7.14:Figure 7.15:Figure 7.16:104105105106106107107108109110110111112113114115xFigure 7.17: Film cooling effectiveness distribution on the wall surface(3D model, RM = 0.8). 116Figure 7.18: Computational doamin for 3D parametric tests. 117Figure 7.19: Predicted mean velocity at the hole exit (RM = 0.4) 118Figure 7.20: Film cooling effectiveness predicted by a) no slot and anassumed uniform injection (without slot) and b) including theslot (with slot) (RM =0.4). 118Figure 7.21: Film cooling effectiveness predicted by linear and uniforminjection flow profiles (S/d=4) 119Figure 7.22: Schematic description of the periodic boundary condition 120Figure 7.23: Predicted surface cooling effectiveness of fmite array andperiodic array of lateral injection (S/d=4, RM = 0.8) 121Figure 7.24: Predicted surface cooling effectiveness of finite array andperiodic array of lateral injection (S/d=4, RM = 1.2) 122Figure 7.25: Predicted lateral averaged effectiveness of finite array andperiodic array of lateral injection (S/d=4, RM = 0.8 and 1.2) 123Figure 7.26: Predicted lateral averaged effectiveness by streamwise andlateral injection (S/d=4, RM = 0.4) 124Figure 7.27: Predicted surface cooling effectiveness by streamwise andlateral injection (S/d=4, RM = 0.4) 125Figure 7.28: Vector fields and concentration distributions on the cross planeX/d=3 predicted by streamwise and lateral injection (S/d=4,RM=0.4) 126Figure 7.29: Predicted lateral averaged effectiveness vs. X/d (single-rowinjection, S/d=4, 5) 127Figure 7.30: Predicted lateral averaged effectiveness vs. relative mass flowratio R (single-row injection, S/d=4, 5) 128xiFigure 7.31: Predicted lateral averaged effectiveness vs. mass flow ratio RM(double-row injection) 129Figure 7.32: Predicted lateral averaged effectiveness vs. X/d (double-rowinjection) 130Figure 7.33: Predicted lateral averaged effectiveness vs. stagger factor Aid(double-row injection) 131Figure B. 1: Predicted surface cooling effetiveness (single-row injection, 147S/d—4)Figure B .2: Predicted surface cooling effetiveness (single-row injection, 148S/d—5)Figure B .3: Predicted vector fields and concentration distributions atX/d=3 149(single-row injection, S/d=4).Figure B .4: Predicted vector fields and concentration distributions atX/d=3 (single-row injection, S/d=5). 150Figure B .5: Predicted surface cooling effetiveness (double-row injection,AJdO,1, RM=0.4). 151Figure B.6: Predicted surface cooling effetiveness (double-row injection,A/d2,3, RM—0.4). 152Figure B.7: Predicted surface cooling effetiveness (double-row injection,AJdO,1, RM=0.8). 153Figure B.8: Predicted surface cooling effetiveness (double-row injection,A/d=2,3, RM=0.8). 154Figure B.9: Predicted surface cooling effetiveness (double-row injection,AJdO,1, RM=1.2). 155xi’Figure B. 10: Predicted surface cooling effetiveness (double-row injection,A./d=2,3, RM=1.2). 156Figure B.ll: Predicted vector fields and concentration distributions atX/d=3 (double-row injection, AJd=O,1, RM = 0.4) 157Figure B.12: Predicted vector fields and concentration distributions atX/d=3 (double-row injection, Aj’d=2,3, RM = 0.4) 158Figure B. 13: Predicted vector fields and concentration distributions atX/d=3 (double-row injection, AJd=O,], RM = 0.8) 159Figure B. 14: Predicted vector fields and concentration distributions atX/d=3 (double-row injection, AJd=2,3, RM = 0.8) 160Figure B. 15: Predicted vector fields and concentration distributions atX/d=3 (double-row injection, A/d=O, 1, RM = 1.2) 161Figure B. 16: Predicted vector fields and concentration distributions atX/d=3 (double-row injection, AJd—2,3, RM = 1.2) 162xmNomenclatureACç,c1,c2CJjf,CplCp Cr3,c1’c2c3EGPRMRRSTU,v,wUtx,Y,zdhkk , k111, lqNear-wall turbulence damping function constants.Stagger factor (see Figure 7.18).Concentration.The k-E turbulence model constants.The M-T-S turbulence model constants.Voltage.Generation rate of turbulent kinetic energy.Mean static pressure.p.U.Mass flow ratio or blowing rate (RM=Relative mass flow ratio (R = RM )SidInjection row spacing (see Figure 7.18).Injection hole spacing (see Figure 7.18).Temperature.Mean velocity components.Friction velocity (U =Cartesian coordinates.Injection hole diameter.Film cooling heat transfer coefficient.Turbulent kinetic energy.Turbulent kinetic energy of large eddies and small eddies.Length scales for turbulent eddy viscosity and dissipation rate.Fluctuating pressure.Heat flux per unit time and area.xivu’ , , w’ Fluctuating velocity components.x, y, z Cartesian coordinates.Dimensionless distance from the solid wall = YU)Lateral and streamwise components of injection angle (see Figure7.18).Boundary layer thickness.E Dissipation rate of turbulent kinetic energy.£ £ Energy transfer rate, dissipation rate of turbulent kinetic energy.TI Film cooling effectiveness.von Karman constant (K—O.41).Dynamic viscosity.Turbulent eddy viscosity.V Kinematic viscosity.p Fluid density.cYk , GE Turbulence model constants.Turbulent Prandtl number.Wall shear stress.Mean and fluctuating values of scalar.Subscriptsaw Adiabatic wall.j Coolant injection.t Turbulent.main streamxvAcknowledgmentI would like to express gratitude to my supervisors, Dr. M. Salcudean and Dr. I.S.Gartshore. Their guidance and encouragement have made this project an enjoyableexperience and their financial assistance has allowed me to participate in this project.I would express my appreciation to professors in the Department of MechanicalEngineering and the Institute of Applied Mathematics, especially to Dr. U. Asher and Dr.P. Hill.Thanks are due to all members of the CFD group in the Department of MechanicalEngineering for their helpful suggestions and discussions. Thanks to Mr. M. Findlay forhis proofreading of this thesis.Finally, I wish to thank my wife, Lu, for her patience and support.xviChapter 1Introduction1.1. BackgroundFor high efficiency operation of gas turbine engines, thermodynamic analysis shows thatthe temperature of the combustion gas at the inlet to the turbine should be as high aspossible. The working temperature has now reached about 1800 K for modern gas turbineengines. As the temperatures rise, the problem of protecting the surfaces from thermaldamage becomes critical since a difference of 15°C in the average blade temperature canmean a factor of two in the blade service life. Reliable operation and prolonged useful lifeof turbine blades require an effective cooling system to maintain the blade temperature andto keep thermal stresses within allowable limits for the material.Film cooling, often used in conjunction with internal convection cooling, is apromising thermal protection method available for the outer surface of blades at the firststages of gas turbine stators and rotors. In the film cooling process, the coolant is injectedinto the boundary layer through rows of holes to generate an insulation film on the bladesurface downstream of the holes. There are many configurations of film cooling. Forexample, near the leading edge of turbine blades, full coverage cooling (or shower-headfilm cooling) is used to protect the critical leading edge region. Figure 1.1 shows a typicalrotor blade cross-section with the cooling air flow.Film cooling design aims at maximizing the thermal protection for the blades withthe smallest amount of coolant injected, resulting in the smallest possible penalty to theengine cycle. The thermal protection by discrete hole film cooling depends crucially onadopting the correct spacing between holes, the right velocity of injection relative to thatof the external stream, and the best injection hole orientation. Clearly, other practical1Chapter]. Introduction 2factors such as manufacturing methods, internal cooling, and structural integrity of theblade also play crucial roles. These are beyond the scope of the present work, however.A large number of experimental studies have been undertaken to provide designdata for the effects of various parameters on the film cooling. Current design is mostlyempirical and relies heavily on the correlations of overall film cooling effects based upon alarge experimental data base. In addition, experimental investigations are usuallyexpensive and time-consuming under realistic conditions. Thus there is an urgent need toreduce the level of empiricism in the design practice and to develop a truly predictivecapability for film cooling design.Increasing efforts have been made to deduce film cooling performance through thecomputational modelling of transport phenomena in the film cooling process. With theadvent of high speed and large capacity computers, numerical simulation has become apromising tool. However, successful application still depends heavily on an understandingof basic transport mechanisms in the film cooling process and the improvement ofmodelling and solution techniques.1.2. Posing the Film Cooling ModelsBasic film cooling research has been carried out to measure and predict the relationshipbetween the wall temperature distribution and heat transfer for a given geometry andmainstream and secondary flows. Unfortunately, due to the presence of many parameterssuch as pressure gradient, Reynolds number, inlet geometry, etc., the prediction of theblade surface temperature contains significant uncertainties. This situation makes itnecessary to study each parameter independently, with the hope of later establishing theirmutual couplings. For simplicity, effects present in operating high temperature enginessuch as blade curvature, variable fluid properties, and fluid compressibility are left out ofthe present study.Chapter]. Introduction 3In the present work, the film cooling process is idealized by considering the flow assteady, and the velocities and temperature variations as sufficiently small so that the fluidproperties can be considered as constant. With this assumption, the velocity field isindependent of the temperature field. By using the superposition method, the heat transferbetween the mainstream and the film cooled wall can be described by the followingequation (see Eckert, 1984 for details):q=h(T—T) (1.1)where q is the heat flux per unit time and area from the mainstream to the film cooledsurface, h is the heat transfer coefficient, T is the wall temperature, and Taw is theadiabatic wall temperature. Taw and h are two important variables for the prediction ofblade surface temperature. The present work studies only the adiabatic wall temperaturewhich can be expressed in a dimensionless form, called the film cooling effectiveness:(1.2)where T is the mainstream temperature and Tc is the coolant temperature. Thesignificance of the film cooling effectiveness is that it can vary considerably and is harderto predict than the heat transfer coefficient.One important feature of film cooling is the highly complex nature of the flow fieldcreated by the coolant jet interacting with a hot cross-stream. Most recent studies of filmcooling effectiveness have been done on flat plates with the objective to identify andunderstand the thermal and aerodynamic behavior of the coolant film. Figure 1.2 showsthe typical 2-D and 3-D flat plate film cooling models. These basic models includecomplex features involved in real turbine blade cooling; therefore, modelling experiencebased on these models could provide useful information for the prediction of film coolingin real situations.Figure 1.3 shows the schematics of the typical structures appearing in 2-D and 3-Dfilm cooling flows. In the 2-D film cooling case, both the external flow and the secondaryChapter]. Introduction 4fluid are uniform across the span. A shear layer separates the injected turbulent jet flowand the free-stream turbulent boundary layer. The injected flow may separate at the rearof the slot and subsequently reattach. Downstream of the slot, the instability due to theshear layer causes the turbulent mixing of the free-stream and the injected film. The filmcooling effectiveness is closely related to the enhanced mixing resulting from the coolantjet separation and reattachment, and also related to the interaction between the coolant jetand the free stream boundary layer. The effects of the mass flow rate and injection angleof the coolant on the cooling effectiveness have been widely studied in the 2-D model.In the 3-D film cooling case, one or several rows of holes are used for the coolantinjection instead of a continuous slot. A complicated flow pattern is found after thecoolant is injected streamwise into the crossing boundary layer through a discrete hole.Due to the mutual deflection of the jet and cross flow, the mainstream moves upwards andalong the sides of the jet. In the wake regions of the jets, the streamwise velocityincreases and the conservation of mass requires fluid to move from the sides towards theplane of symmetry. Two vortices are formed in the kidney-shaped cross section of the jet.Very close to the wall a reverse-flow region forms. Cross-stream fluid enters this regionand travels upstream where it is lifted by the jet fluid and is then carried downstream.Unlike the flow in 2-D situations, the flow does not recirculate and the reverse flow isrestricted to a region very near the wall. The mixing between the external flow and jetflow is enhanced by the vortices along the jet. The discrete hole injection produces a 3-Dspanwise, non-uniform flow and cooling effectiveness. A spanwise averaged film coolingeffectiveness defined as:—S/2 (1.3)where S denotes the hole spacing, is used to evaluate the film cooling performance. Thefilm cooling performance is affected by factors such as the mass flow rate of the coolant,hole arrangement, and injection angle.Chapter]. Introduction 51.3. Objectives and Scope of the ThesisA research collaboration between Pratt and Whitney Canada and the University of BritishColumbia has been developed to investigate the film cooling at the leading edge of turbineblades. Experimental investigations have been performed and a numerical tool formodelling the fthn cooling process has been developed. The numerical tool can be usednot only to provide a detailed database but also to improve our understanding of thethermal and aerodynamic mechanisms involved.Due to the complex flow structures involved, direct application of the existingnumerical methods to film cooling prediction has raised two important problems: 1) Filmcooling flow involves a wide range of length scales. Nevertheless the resolution of theflow and heat transfer near the regions of injection and film cooled wall surface is crucialfor accurate predictions of film cooling effectiveness. To obtain such flow resolution, anefficient numerical method covering a wide range of scales is needed. 2) The adequacy ofexisting turbulence models for complex film cooling flows is not clear. These difficultiesdiscourage further investigations of the heat transfer coefficient h, and the model withvariable fluid properties until there is a clear evaluation of the numerical methods that areused to simulate simpler but related physical phenomena.The main objective of this work is to develop a numerical modelling methodsuitable for the prediction of film cooling effectiveness. This objective is approached byfour steps. The first is to adapt grid refinement to resolve flow and heat transfer near thewall and injection orifice regions, and to obtain numerical solutions with the efficientmulti-grid iteration method. The second is to evaluate the standard k-E turbulence modeland the wall function treatment, and to explore additional turbulence models and near-wallturbulence treatments. The third is to verify the numerical model by conducting physicalmeasurements of film cooling effectiveness and flow fields in wind tunnel experiments.Chapter 1. Introduction 6The fourth and final step is to apply the present numerical tool to study the double rowfilm cooling on a flat plate, and to compare the results obtained with the experimentalwork of Gartshore et al. (1993). The work described in this thesis can be summarized asfollows.Two film cooling models have been investigated, representing film cooling on a flatplate from vertical injection of coolant firstly through a uniform slot and, secondly from arow of discrete square holes. The models have been simplified so that a Cartesiancoordinate system can be used, and numerical errors arising from the analysis of generalcomplex geometries can be avoided. Despite this simplification of the geometries and theassumption that the fluid is incompressible, steady-state, and has constant fluid properties,the numerical models retain the main features of flow and heat transfer which are presentin the real situation of complex film cooling. The 2-D model represents the effect of theseparation and the reattachment of injectant while the 3-D case represents morecomplicated effects, such as the formation of the kidney-shaped vortex along thetrajectory of injectant as well as the complex detaching and reattaching flow downstreamof injection.The performance of standard high-Re k-a turbulence models with the wall functionhas been evaluated for the film cooling geometries described above. In the present work,a low-Re k turbulence model was used in the region very close to the adiabatic wall inorder to resolve the flow and heat transfer near the wall. The multiple-time-scaleturbulence model was used in the 2-D computations to improve the prediction at highmass flow ratios. In the 3-D computations, a non-isotropic k-a model was used toaccount for the anisotropy of turbulence near the wall.The transport equations were solved on refined grids by using the multi-gridmethod with efficient reduction of the numerical errors. The increased accuracy of thecomputations makes it possible to evaluate the turbulence models.Chapter]. Introduction 7Since there is no directly comparable experimental data in the literature for thesesimplified models, physical experiments were carried out to provide detailed flow field andcooling effectiveness for verification of the numerical model. Film cooling effectivenessmeasurements were made using a flame ionization detector, based on the heat-masstransfer analogy, and the mean flow and turbulence were measured using hot-wireanemometry. Comparisons between experiment and computation are presented in thisthesis.The numerical model was applied in parametric studies to show the effects of holespacing, hole stagger, and coolant mass flow rate on the film cooling performance in singleand double-row film cooling. The parameter values are the same as those used in theexperimental work of Gartshore et al. (1993). The structure of vortices behind coolantinjection locations and its effects on the film cooling performance were investigatednumerically.The main contributions of the present study can be summarized as follows.• The present experimental measurements of film cooling effectiveness, meanflow and turbulence in the wind tunnel provide a systematic database forverification of turbulence models.• The present numerical work applies and evaluates the low-Re k model and theuse of a fine grid as a near-wall turbulence treatment to resolve the heattransfer in the region close to the film-cooled wall surface. It also applies themultiple-time-scale turbulence model to allow for non-equilibrium turbulencein the film cooling.• The present work provides increased accuracy in the assessment of turbulencemodels by using the multi-grid method and grid refinement in the computationsreducing discretization errors and improving iterative convergence.• The parametric tests provides some insight into vortex formation downstreamof the coolant injection and the consequent effect on the cooling performance.Chapter 1. Introduction 8Figure 1.1: A typical turbine rotor blade cross section with cooling flow.Chapter 1. Introduction 9MainstreamFigure 1.2: Flat plate film cooling models.Chapter 1. Introduction 10MainstreamBoundary Layer Mixing Shear LayerSeparation RegionInjection Flow(a) 2D film cooling flowMainstreamBcIryL/ Induced VortexJet Detachment xInjection Flow Reattachment(b) 3D film cooling flowFigure 1.3: Schematics of 2-D and 3-D film cooling flows.Chapter 2Literature SurveyFilm cooling has been a subject of research for over forty years. The large body of filmcooling papers in the open literature is divided here into broad categories of experimentand computation. General reviews have been undertaken by Goldstein (1971) for earlywork and Moffat (1986) for more recent developments. Experimental works includestudies which generate data on adiabatic film cooling effectiveness and providemeasurements of flow and heat transfer. Computational works consist of studies of thenumerical solution techniques and turbulence modelling methods. In this chapter,previous experimental and computational papers are reviewed. Since there have been alarge number of computational papers in the areas of turbulent heat transfer other thanfilm cooling, review of those works are made at the same time. Finally, some remarks areaddressed to the motivation of the present work.2.1. Experimental StudiesThere are many papers in the open literature reporting film cooling effectivenessmeasurement data for design use, especially for slot, transpiration, and single holeinjection configurations. Recent work has been done on single- and double-row holeinjection on either flat plate or curved leading edge surfaces. Most of the early discrete-jetexperiments were conducted with the coolant jet-to-crossflow density ratio close to unity.Experiments were performed using either a thermal approach or the heat-mass transferanalogy. The measurements of cooling effectiveness and coolant distribution were mainlymade using thermocouples, visualization on thermal-sensitive material, and foreign-gasdetectors. The flow fields were measured using hot-wire anemometry or laser Doppler11Chapter 2. Literature Survey 12velocimetry. A brief review of the most significant papers related to the present work isprovided in the following discussion.Goldstein et al. (1970), and Goldstein and Eckert (1974) investigated the angledinjection of air through discrete holes into a turbulent boundary layer of air on a flat plateto determine the effect on the film cooling effectiveness. In their experiments, the injectedair had a higher temperature (25°C) than the mainstream, and the effectiveness wasmeasured using the non-dimensional adiabatic wall temperature measured by wallthermocouples. They observed that the film cooling effectiveness downstream of theholes increases as the boundary layer thickness just upstream of the injection location isdecreased. Comparison between secondary air injected by a single hole and a row of holesshowed that data from single hole tests are similar to a row of holes for low blowing rates,but significant differences are observed at higher blowing rates. Also, they found thatlateral injection spreads the protection of the cooling film over a wider area than wheninjection is normal to the flow or inclined downstream only. The interaction of the coolantjet with the mainstream was found to affect the development of cooling effectiveness. Atsurface locations near the injection holes, as mass flow ratio is increased, effectiveness firstrises and then reaches a sustained maximum value. Further downstream, the effectivenesslevels are generally lower but rise continuously as mass flow ratio increases.Bergeles et al. (1976) studied experimentally the near-field character of a circularjet discharged normally to a main stream. In their work, the film cooling effectiveness wasmeasured by adding a tracer of helium (one percent by volume) to the secondary stream.The concentration of helium on the surface of the test plate was obtained by withdrawingsamples of air/helium mixture through static pressure taps and measuring the heliumconcentration with an on-line kathometer. For mass flow ratios between 0.046 and 0.5, aclearly identifiable reverse flow region was detected on the downstream side of the holeusing flow visualization. The velocity distribution in the jet at discharge was found to begreatly affected by the presence of the external stream. Measurements of the local coolingChapter 2. Literature Survey 13effectiveness showed the peak values of effectiveness immediately downstream ofdischarge occur off the centerline, which is consistent with the jet assuming a kidney shapeas it is bent over by the external stream.Foster and Lampard (1980) conducted detailed studies of effectiveness and flowdownstream of a row of 15 holes on a flat plate. A heat-mass transfer analogy experimentwas carried out in which the injected gas was a mixture of Freon and compressed air, andthe effectiveness was measured by katharometer. They studied the effect of thestreamwise injection angle on film cooling and showed that a small injection angleprovides the best cooling effectiveness at low blowing rates while large injection anglesare best at high blowing rates. At high blowing rates, the cross-streamwise distribution ofthe effectiveness downstream of injection is more uniform for 900 injection than 350injection. It was also observed that an increase in the upstream boundary layer thicknessproduces a reduction in the effectiveness due to increased lateral mixing in the near-wallregion. The use of a small spacing-to-diameter ratio gives improved lateral coverage at allblowing rates, and alleviates jet lift-off effects at high blowing rates.Jubran and Brown (1985) measured the cooling effectiveness from two rows ofholes inclined in the streamwise and spanwise directions. In their experiment, cold air wasinjected into the hot main stream and the adiabatic wall temperature was measured usingthermocouples. Cholesteric liquid crystals were applied to the working surface fortemperature-flow visualization as a backup to the thermocouple temperaturemeasurements. They found that an increase in the distance between two rows of holesreduced both local and lateral averaged cooling effectiveness downstream of the secondrow of holes, especially in the region close to the second row of holes at higher mass flowratios. The influence of free-stream turbulence intensity and velocity gradients on filmcooling performance showed that the averaged effectiveness downstream of the secondrow of holes is reduced by increased turbulence intensity for all streamwise positions atChapter 2. Literature Survey 14low blowing rates. The improved cooling of two rows of holes over one row was alsoshown in their study.Honami et al. (1991) carried out an experimental study of film cooling using lateralinjection. In their heat transfer experiment, the surface temperature was visualized bycovering the test surface with a thin sheet of encapsulated temperature-sensitive liquidcrystal. An image processing system based on the temperature and hue of the liquidcrystal was used. A double-wire probe (with a constant temperature hot-wire anemometerand a constant current thermal resistance meter) was used for simultaneousvelocity/temperature correlation field measurements. From their experiments on threetypes of hole arrangements: lateral, streamwise and inlined injection, and on three massflow ratios: 0.5, 0.85, and 1.2, the highest spanwise-averaged film cooling effectivenesswas observed for lateral injection for the same coolant flow per unit span. The lateralinjection produced asymmetric structures with a large scale vortex motion promoted bythe primary stream on one side of the jet, but suppressed on the other side. It was alsofound that this asymmetry increases as the mass flux ratio increased, resulting in low filmcooling effectiveness.Recently, Ligrani et al. (1992) presented a detailed systematic study on thedevelopment and structure of flow downstream of either one row or two staggered rowsof film cooling holes with compound angle orientations. The effectiveness was measuredby thermocouples and upstream boundary layer properties were measured using a fivehole pressure probe with a conical tip. They found that the spanwise-averaged values ofeffectiveness measured downstream (as far as 20 hole-diameters) of two staggered rowsof holes were highest with a blowing ratio of 0.5, and decreased as the blowing ratioincreased above 0.5 because of injection lift-off effects. However, as the boundary layerconvected farther downstream, the effectiveness increased with blowing ratio. It was alsofound that with one row of holes the local effectiveness variations are spanwise periodic,Chapter 2. Literature Survey 15where higher values corresponded to locations where injectant is plentiful near the testsurface.Mehendale and Han (1992) studied the effects of injection hole geometry on theleading edge effectiveness and heat transfer under high mainstream turbulence conditions.It was found that, due to an increase in mainstream turbulence, the effect of secondaryflow turbulence is considerably reduced. The effectiveness was found to decrease withincreasing mainstream turbulence; however, this effect reduces with increasing blowingrate. In addition, it was found that larger spanwise distances for the case of spacing-to-diameter ratio of 4 cause larger spanwise variation as compared to the smaller spacing-to-diameter ratio of 3. The best effectiveness for the case of mass flow ratio RM = 0.8 wasfound with spacing-to-diameter ratio of 4 while the best effectiveness for the case ofRM = 0.4 was found with spacing-to-diameter ratio of 3.Recently, a systematic experimental investigation of film cooling effectiveness nearthe leading edge of a turbine blade has been carried out in the Department of MechanicalEngineering at the University of British Columbia. The measurements of film coolingeffectiveness was made using a flame ionization technique based on the heat-mass transferanalogy. The turbine blade model has a semi-cylindrical leading edge bonded to a flatafter-body. Both air and CO2 were used as the secondary flow. The secondary flow wasinjected in the boundary layer through 4 rows of holes located at ±15° and ±44° about thestagnation line of the leading edge. These holes of diameter d had a 30° spanwiseinclination and a 4d spanwise spacing. Adjacent rows of holes were staggered by 2d. Apaper by Salcudean et al. (1994a) showed that the strong pressure gradient near theleading edge produces a strong non-uniform flow division between the first and the secondrow of holes at low overall mass flow ratios. Best effectiveness were obtained in a verynarrow range of mass flux ratios near 0.4. The effectiveness values deteriorates abruptlywith decreasing mass flow ratios, and substantially with increasing mass flow ratios. Inthe study of the effects of coolant density, it was found that air appears better close to theChapter 2. Literature Survey 16first row of holes and CO2 better at some distance downstream of both rows (Gartshore etal., 1993). Double row cooling with air as coolant showed that the relative stagger of thetwo rows is an important parameter (Salcudean et al., 1994b). Holes in line with eachother in successive rows can provide improvements in spanwise-averaged film coolingeffectiveness of as much as 100% over the common staggered arrangement.2.2. Computational Studies2.2.1. Numerical Solution TechniquesFor many years, researchers have been using parabolic-type solution procedures for theprediction of heat, mass, and momentum transfer in film cooling. The parabolic-typeprocedure cannot be used for the simulation of recirculating flow in a plane parallel to thedirection of the free stream, but it can be used to simulate vertical flows in the cross-stream plane.Bergeles et al. (1976b) used a partially parabolic numerical scheme to predict themean velocity and temperature for laminar flow for a single row of inclined holes and for asurface with multiple row of holes in a staggered array. Their calculation showed that thestrong acceleration reduced the lateral rate of spreading. A counter rotating vortex pair iscreated downstream from the hole which shifts the minimum effectiveness away from themid-plane between holes.Bergeles et al. (1981) used a semi-effiptic procedure and a nonisotropic k-Eturbulence model for the prediction of film cooling from two rows of holes. Comparisonwith measurements obtained for an injection angle of 300 and mass flow ratios in therange of 0.2 to 0.5 showed good agreement in the majority of cases. Discrepancies wereobserved, however, with small boundary layer thicknesses or large injection rates. In theirpaper, they indicated that the cause of the discrepancies was the local equilibriumChapter 2. Literature Survey 17assumption in the turbulence modelling and the inability of the semi-elliptic procedure toproperly simulate the zone of flow recirculation downstream of the hole.Demuren et al. (1986) used a locally-elliptic calculation to investigate the influenceof different parameters on the cooling effectiveness. The predicted temperature fieldsagreed fairly well with available measurements. The film cooling effectiveness was notalways in good quantitative agreement with the data. The agreement was satisfactory forthe mass flow ratios up to 1 for small spacings. For high blowing rates and largerspacings, only the general trends of the measurements were predicted with the calculatedcooling effectiveness lower than observed. It was suggested in their work that a morerefined treatment of the region near injection would be necessary to represent the complexflow there.With the aid of a locally-elliptic calculation, Schonung and Rodi (1987) developeda two-dimensional boundary-layer method for film cooling through discrete-hole injection.Due to the high computational cost of conventional effiptic methods, the elliptic reverse-flow region in the vicinity of the injection holes and the 3-D effects were taken intoaccount by two added ‘injection’ and ‘dispersion’ models. Haas et al. (1991) extended thisboundary-layer method to study the influence of density difference between hotmainstream and cool secondary gas from a row of holes.By using a laterally periodic 3-D parabolic procedure, Sathyamurthy and Patankar(1990) investigated the effect on the film cooling effectiveness of variations in the lateralangle of injection, the spacing between the holes, and the blowing rates. The computedresults were found to be in good agreement with the previous experimental measurements.From their study, it was found that lateral injection can operate at high blowing rates andcan achieve better film coverage than streamwise injection. Increased blowing rates andreduced spacing between the injection holes increase the film cooling effectiveness whenthe jets are injected across the mainstream. It should be noted that due to the limitation ofChapter 2. Literature Survey 18their parabolic procedure, their computations could not properly represent the film coolingnear the injection.White (1981) solved the fuily elliptic transport equations of a single jet injectioninto a cross-flow mainstream. The distortion of the flow within the injection hole wasfound to have a significant effect on the predicted flow field. The separation andreattachment of the non-uniform slot flow suggest that the prediction of coolingeffectiveness is restricted to injection at low blowing rates unless a fully elliptic solver isused.Several elliptic-type Navier-Stokes solvers have been developed in the past years.The “Semi-Implicit Method for Pressure-Linked Equations” (SIMPLE) algorithm (Patankar,1980) and its revised versions have been widely used for numerical simulation ofincompressible flows. The SIMPLE-type procedures employ a segregated solutionapproach in which the variables are solved separately. Based on a strategy different fromthe sequential update philosophy of SIMPLE, Vanka (1986) developed a block-implicitsolution algorithm, in which the pressures and velocities are updated simultaneously butwithout the pressure correction equations. In Vanka’s algorithm, the continuity equationis retained in its primitive form in terms of velocities and the discretized momentum andcontinuity equations are treated as one large set of non-linear algebraic equations to besolved. Rapid convergence was reported with a modest requirement for computer storageand time per iteration for the calculation of laminar square cavity flows, sudden expansionflows, and turbulent flows involving sudden, axisymmetric expansion geometries.Recently, many commercial Computational Fluid Dynamics codes have beendeveloped based on elliptic-type Navier-Stokes solvers. Jubran (1989) used the PHOENICSpackage to predict the film cooling effectiveness and the velocity field from two rows ofholes inclined in the streamwise and spanwise directions. With the k-E turbulence model,they reported successful predictions of cooling effectiveness at the centerlines of holes forlow blowing rates. They also found that the elliptic procedure showed no significantChapter 2. Literature Survey 19improvement over the semi-elliptic procedures and concluded that the main problem inpoor predictions, especially at high blowing rates, is the inadequacies of turbulencemodels. Later, Amer et al. (1992) also used the PHOENICS package for an investigation ofthe performance of k-a and k-o models in a prediction of film cooling from two rows ofholes (see Section 2.2.2).Leylek and Zerkle (1993) carried out a large scale numerical analysis of discrete-jetfilm cooling with a fully-coupled and elliptic computation of flow in plenum, film hole, andcross-stream regions by using the PHOENICS system of codes. The standard k-a model wasemployed with the generalized wall function treatment. Their computations were carriedout for a single row of jets with film-hole length-to-diameter ratios of 1.75 and 3.5,blowing ratios from 0.2 up to 2, coolant-to-cross-flow density ratio of 2, streamwiseinjection angle of 35°, and pitch-to-diameter ratio of 3. Because of the use of the wallfunction, the nodes adjacent to the adiabatic wall surfaces were located at = 50. Thecomputed flow within the film-hole showed that the strength of counter-rotating vorticesand local jetting effects were controlled mainly by the film-hole length-to-diameter ratio,the blowing ratio, and the streamwise injection angle. Comparison with experimental dataon film cooling effectiveness showed that the computation predicted the correct trends foroverall streamwise variation of effectiveness but that the predicted values wereconsistently higher at the blowing ratio of 0.5, and much improved for the blowing ratio of1.0. Comparison of the lateral variation of effectiveness showed that the lateral rate ofspreading of film from the jets was lower in the computation than in the measured dataand that the prediction missed the jet detachment-reattachment behavior.Early work has indicated that in solving the transport equations, the numericalfalse diffusion error, which results mainly from the upwinding difference scheme for theconvective terms, needs to be reduced. Demuren (1985) presented detailed computationsof the steady flow of a row of turbulent jets issuing normally into a nearly uniform crossflow. His use of a three-dimensional QUICK scheme, which employs a higher-orderChapter 2. Literature Survey 20accuracy difference approximation, produced better results than the more widelyemployed hybrid (central/upwind) scheme.Since false diffusion is proportional to the magnitudes of the velocity vector, thegrid mesh sizes, and the angles between the velocity vector and any of the grid lines(Patankar, 1980), grid refinement is one of the techniques which can reduce the error andthus improve the accuracy of the numerical approximations. However, as the number ofdiscrete variables and algebraic discretized equations increases, traditional iterativeprocesses encounter a deterioration of convergence.As a faster iterative technique, the multi-grid method has demonstrated itspotential in the field of computational fluid dynamics. The multi-grid method efficientlyeliminates most of the work related to the repetition of iterations and rapidly solves thealgebraic system of equations with a convergence rate insensitive to the number of gridpoints. As a pioneer in this area, Brandt (1977) introduced a multi-level adaptivetechnique with a nonlinear Fast Approximation Scheme (FAS) and subsequentlydeveloped the Distributive Gauss-Seidel relaxation method as a smoother for solving theNavier-Stokes equations on a staggered grid (Brandt, 1980). Vanka (1986) used themulti-grid method for his Symmetrical Coupled Gauss-Seidel relaxation method forprimitive variable solutions. The TEACH code, which was based on the SIMPLE algorithm,was modified to include the multi-grid formulation by Zhou (1990) for film coolingcomputations. Recently, a multi-grid segmentation numerical code for 3-D turbulent flowwas developed at the University of British Columbia (Nowak, 1991), which permits theentire computational domain to be broken into several segments with different grid sizesand uses the multi-grid method to enhance the iterative convergence of computations overall segments in the domain.A calculation of 3-D turbulent jets in crossflow was done by Demuren (1990)using a multigrid method. His computations obtained fairly rapid convergence using thek-a turbulence model, but computations with a full Reynolds stress turbulence model wereChapter 2. Literature Survey 21not very efficient. His tests of grid independence showed that there were slight differencesbetween results obtained on the two finest grid levels.2.2.2. Turbulence ModellingFor the successful computation of film cooling, turbulence modelling problems have to beaddressed. Recent studies have shown that the selection of the turbulence model is one ofthe serious problems affecting film cooling computations.In previous studies, various k-E models were most commonly used to describe theeffects of turbulence for the prediction of film cooling. The k-E turbulence models makeuse of both the eddy-viscosity concept introduced by Boussinesq (1877) and theKolmogorov-Prandtl expression. The k-E model of Jones and Launder (1972) has beenparticularly popular and has been applied successfully to a wide variety of 2-D flows thatinclude wall boundary flows, recirculating flows, confined flows, shear flows, and jet flows(Launder and Spalding, 1974). In addition to the k-E model there are other two-equationmodels which have been used to model turbulence. One such alternative is the k-o model,where x is another choice of the second dependent variable complementing the equationfor k and is referred to as the rate of dissipation per unit turbulence kinetic energy(w k/e). Although various two-equation models are sometimes believed to differ merelyin mathematical form and not in content (Launder and Spalding, 1974), different resultsmay be obtained from different models due to the fact that the boundary condition for Eon a solid wall may not be identical to that for o.Amer et al. (1992) evaluated two two-equation turbulence models, namely the k-eand k-o models for the prediction of film cooling effectiveness from two rows of holesinclined in the streamwise direction. The comparison between the predicted results andprevious experimental results indicates that the ability of the turbulence models to predictthe experimental results depends heavily on the blowing rate as well as on the downstreamdistance from the injection holes. For some cases the k-E model performs better than theChapter 2. Literature Survey 22k-o model and vice versa for other cases; however, it was concluded that the two-equation turbulence models do not work well for film cooling, especially in the vicinity ofthe holes and at high blowing rates. It should be noted that their computation used acoarse grid arrangement near the jet orifice and wall, so that their solution may dependheavily on the grid used.The failure of the two-equation models with wall function treatment for filmcooling can be attributed to: 1) increased turbulence generation at high mass flow ratiosdue to the fact that the flow near the holes is disturbed and unsteady, and cannot berepresented by the equilibrium turbulence assumption in the model, 2) film jet spreadingcan not be represented by isotropic eddy-viscosity, 3) flow and heat transfer in the near-wall viscosity-affected sublayer need to be resolved.For complex 3-D flows, the k-E models may have to be replaced with higher-orderturbulence models. The transport equations for the turbulence stresses uu and theturbulence scalar flux u.O can be obtained by applying the Reynolds decomposition. Themodelled Reynolds stress equations are, however, extremely difficult to solve for a 3-Dflow. Several attempts have been made to simplify the Reynolds stress transportequations. The simplification of these transport equations results in the algebraic stressmodels which model uu, and u0 by algebraic transport equations at each point in theflow (Launder, 1988). The algebraic stress models have considerable appeal but there isstill less experience with them than with the k-E models (Ferziger, 1987).Bergeles et al. (1978) refmed the k-e model by introducing the algebraic stressmodel. For the computation of discrete hole film cooling the proposed model accounts forthe anisotropic nature of the eddy viscosity and diffusivity. This formulation has beenwidely used in recent film cooling computations by Demuren et al. (1986) and Jubran(1989). However, their work indicated the need to allow for the nonequilibrium ofturbulence in order to obtain satisfactory predictions of film cooling effectiveness at highblowing rates, especially in the region close to the injection.Chapter 2. Literature Survey 23Kim and Chen (1989) developed a multiple-time-scale (M-T-s) turbulence model tosimulate non-equilibrium turbulence. The M-T-S model partitions the turbulent kineticenergy spectrum into the turbulent kinetic energy of large eddies and that of the fine-scaleeddies instead of using a single time scale to describe both the turbulent transport anddissipation of the turbulent kinetic energy. In the M-T-S model the turbulent transport ofmass and momentum is described using the time scale of the large eddies and thedissipation rate is described using the time scale of the small eddies. Therefore, the M-T-Smodel is more able to resolve non-equilibrium turbulence by considering the generation,cascade, and dissipation of the turbulent kinetic energy. The M-T-S model has beensuccessfully applied to several complex flow situations, such as divergent channel flows,wall jet, and backward-facing step flow (Kim and Chen, 1989; and Kim, 1991). Recently,Kim and Benson (1993) have applied the M-T-S model to the flow of a row of jets in aconfined crossflow and reported the inability of the k- model to predict the horseshoevortex located along the circumference of the jet exit.For flow close to a solid wall, a high-Re number turbulence model is no longervalid. A treatment is needed which takes into account the influence of the wall upon thedevelopment of near-wall turbulence. As the most popular approach, wall functions areused to account in an overall fashion for the effective convection, diffusion, sources, andsinks of the flow in the region between the near-wall node and the wall (Launder andSpalding, 1974). In more complex flows, however, difficulties are often encounteredbecause fme grids are necessary to accurately compute near-wall flow characteristics suchas the reattachment length following a flow separation.An alternative is to solve the flow equations elliptically through a fine grid to thewall. Low-Re k-E models for different flows have been studied by some researchers(Jones and Launder, 1972; and Nagano and Tagawa, 1990). One disadvantage of thisapproach is that many grid points (usually more than 30) are required within the viscoussublayer. For economy of nodal points, a two-layer approach was used in the recent workChapter 2. Literature Survey 24done by Yap (1987) and Rodi (1991). In the two-layer approach, the flow far away fromthe wall is simulated with a high-Re turbulence model, such as the k-E model, while theviscosity-affected near-wall region is resolved with a simpler low-Re k equation modelemploying a prescribed length-scale distribution. This near-wall treatment has found awide range of applications in 2-D flow situations and the results have been encouraging.However, very few simulations have been done in 3-D flow situations.2.3. Remarks Arising from Work ReviewedA large experimental database of film cooling effectiveness measured by direct thermalmethods and heat-mass transfer analogy methods is available in the open literature. Sofar, however, no detailed flow measurements, including upstream boundary layerthickness and turbulence level, have been reported together with the measurement ofeffectiveness. Most computational works have been compared with publishedeffectiveness values using an assumed mainstream boundary layer. Therefore, thecomparison between computational and experimental results is not reliable since it isknown that the mainstream boundary layer thickness and turbulence level affect the mixingof the coolant jet with the mainstream and thus the effectiveness. Furthermore, detailedflow information downstream of the coolant injection and particularly close to the coolantholes is important for the verification of turbulence models. Therefore, extensivesystematic measurements of the flow and heat transfer have to be made to achievesatisfactory flow and heat transfer code validations.In real film cooling situations, the coolant is usually injected through circular holes.This geometry requires the numerical model to use a curvilinear body-fitted coordinatesystem rather than a Cartesian coordinate system. Although there are such commercialComputational Fluid Dynamics codes available, their accuracy and efficiency generallycannot provide good flow resolution near the wall. In an effort to reduce the numericalChapter 2. Literature Survey 25errors involved, the Cartesian model is used in the present work, meeting our objective toevaluate various turbulence models.In previous film cooling computations, the evaluations of the turbulence modelsare often based on rather coarse computational grids. In order to separate the numericalerrors from the deficiencies of the turbulence model, it is important to ensure thatcomputations reach grid independence and that numerical errors (e.g., the false diffusion)resulting from the discretization are properly evaluated. For film cooling, an ellipticNavier-Stokes solver which has the capability of handling local grid refinement near thewall and near the injection orifice region appears necessary, since the flow field has verynonuniform length scales. Convergence for a large number of computational grid nodescan be accelerated by applying the multi-grid method.The failure of numerical computation for high mass flow ratios of coolant injectionhas been shown during the present study (Zhou et al., 1993a and 1993b). This failure ispartly due to the fact that at high mass flow ratios, turbulence is neither in equilibrium noris it isotropic. The algebraic Reynolds stress models may be required in 3-D flows.However, even the algebraic forms of these equations are not efficient and are not alwaysnumerically stable. Therefore, a simplified form of the algebraic Reynolds stress modelwhich represents the non-isotropic turbulence is appropriate at this stage of computation.Although successful predictions using two-equation models other than the k-amodel have been reported (e.g., the k-o model of Wilcox, 1993), these models are allbased on the equilibrium eddy-viscosity concept. In order to take some account of thenon-equilibrium turbulence in film cooling flow, the multiple-time-scale turbulence modelcan be introduced into film cooling computations. However, the applicability of themultiple-time-scale turbulence model needs to be evaluated.Grid refinement near the solid wall requires an adequate near-wall turbulencetreatment in order to resolve the large gradients of velocity and temperature in theviscosity-affected sublayer. Most reports of film cooling computation use wall functionsChapter 2. Literature Survey 26as the near-wall treatment for turbulent flow and heat transfer. Two-layer modellingapproaches that use the k-E model and low-Re k model have been successfully applied toboundary layers and separated and attached flows. The two-layer near-wall treatment isnot computationally expensive and does not cause severe iteration convergence problems.This treatment has not been used in previous film cooling computations. Its applicabilityto 3-D film cooling flow needs to be assessed.In view of this survey of published literature on film cooling research, theobjectives of the present work can be summarized as follows:• Carry out measurements of film cooling effectiveness, mean flow, andturbulence in the wind tunnel to provide new experimental data for thevalidation of the present numerical predictions.• Solve the governing equations using the multi-grid method on a refined grid toachieve an accurate prediction of film cooling effectiveness. The gridrefinement is able to reduce the false diffusion and to resolve important flowstructures.• Use the low-Re k model and fine grid as a near-wall turbulence treatment inorder to resolve the flow and heat transfer in the viscosity-affected region closeto the film-cooled wall surface and thus to improve the prediction of filmcooling effectiveness.• Apply the multiple-time-scale turbulence model to the film coolingcomputation in order to allow for non-equilibrium turbulence.• Study the formation of the vortices formed downstream of injection and theireffect on the cooling performance using a highly refined mesh near the coolingorifices and the surface. The present flat plate tests can provide some insightinto film cooling in real situations.Chapter 3Experimental InvestigationIn this chapter, measurements of the film cooling effectiveness, mean flow, and turbulenceare described for 2-D and 3-D film cooling studies. The physical experiments use theheat-mass transfer analogy which replaces the actual heat transfer process. The windtunnel and injection system for the experiments, the measurement techniques, and therelated calibration procedures are presented. The condition of the turbulent boundarylayer and injection flow are investigated. The results of flow field and effectivenessmeasurements are presented in Chapters 6 and 7.3.1. Heat and Mass Transfer AnalogyIn order to obtain local film cooling effectiveness values, direct thermal experiments haveto be performed using a thermally insulated wall. However, it is difficult to maintain anadiabatic wall during experiments especially near the coolant injection orifices. Thisdifficulty can be avoided by employing the heat-mass transfer analogy. In the heat-masstransfer analogy, the small temperature difference between the mainstream and theinjection flow can easily be created by the use of a tracer gas mixed with the injectionflow. The impermeable wall then gives the analogous boundary condition of the adiabaticwall.The mass transfer process is analogous to the heat transfer process if theequivalent dimensionless parameters of the flow are the same in the two cases and if theLewis number is unity (Goldstein, 1971). The Lewis number of the gas mixture is fairlyconstant near unity. Available experimental evidence, as well as theoreticalconsiderations, have pointed to the fact that the turbulent Lewis number, which is definedas the ratio of turbulent heat transfer diffusivity and the turbulent mass transfer diffusivity,27Chapter 3. Experimental Investigation 28has the value unity (Eckert and Drake, 1987). It is found that for heat transfer in theenvironment of turbulent flow, i.e., away from the near-wall region, the turbulentcontribution to the energy and to the heat flux is generally more important than themolecular contribution. Therefore, even though the laminar Lewis number may deviatefrom unity, the mass transfer process still represents the heat transfer process adequately.In the present experiments, the injection gas consists of air with about 0.03 percentpropane so that the density ratio between the mainstream and injection flow is essentiallyunity, thus representing results comparable to low temperature differences. The filmcooling effectiveness can therefore be expressed by the relative concentration of propaneto the plenum concentration(3.1)where C,, is the relative concentration on the wall and C, and C are the relativeconcentration in the mainstream and injection, respectively. In the present experiments,the injected gas contains a single constituent not contained in the mainstream, thus C,0, C = 1 and(3.2)3.2. Experimental Facility and Equipment3.2.1. Wind TunnelThe experiments were performed in a low speed, blower-type, boundary layer wind tunnelwith a test section measuring 406 mm wide, 267 mm high, and 800 mm long (See Figure3.1). The tunnel had a turbulence intensity of less than 0.5 percent at a free streamvelocity of 10 mIs. The side walls and the tunnel floor in the test section were constructedof plexiglass. The top wood roof was adjusted to have zero pressure gradient on the flowwhen no secondary flow is injected. In order to ensure that the boundary layer formed byChapter 3. Experimental Investigation 29the free stream is fully turbulent, a trip wire was used at the inlet of the test section toensure transition. The boundary layer thickness upstream of the holes was about 1.5 cm.3.2.2. Injection SystemThe test section was augmented for the experiments by the addition of a plenum module.The plenum beneath the test section (shown in Figure 3.2) is 908 mm long and 406 mm indiameter. The plenum module facilities controlled the injection of the secondary streaminto the mainstream. The plenum floor was designed to accommodate a wide variety ofslot and orifice geometries. The plenum air was supplied from the building’s main aircompressor which has a rated capacity of 250 scfm at 150 psi. Before reaching theplenum, the compressor air travels a long distance through piping, a condensing filter, twopressure regulators, and a rotameter. The condensing filter removes water and oil fromthe compressed air. The two pressure regulators were used in series to reduce pressurefluctuations in the supply line. The propane injection tap was connected to the pipingupstream of the rotameter. The propane and the compressed air were well mixed by a fanmixer. The flow rate fluctuation was less than ±0.25 scfm in the present experiments. Therotameter was used to measure the mass flow rate of air into the plenum and therefore intothe mainstream.3.2.3. Traverse MechanismAbove the wind tunnel roof are two rails which support the hot wire anemometer traversemechanism. These rails run the full length of the test section. The traverse mechanism iscapable of movement in the longitudinal, lateral, and vertical directions. Good qualitybearings were used in the traverse mechanism to allow accurate positioning (with accuracyof about 0.0 127 mm in these three directions). Movement of the traverse mechanism wasdone by hand with the aid of one dial gauge which is accurate to 0.254 mm. The windChapter 3. Experimental Investigation 30tunnel roof had several slots, allowing two-dimensionality checks of the boundary layer atthe locations of interest.3.2.4. Data Acquisition SystemIn order to obtain and process data, a computer data acquisition system was developedwith a Lab-PC A/D board of National Instruments (see National Instruments Lab-PCmanual, 1991). A schematic of the data acquisition system is shown in Figure 3.3. Theboard has a 12-bit successive approximation analog-to-digital converter with eight analoginputs, and two 12-bit digital-to-analog converters with voltage outputs. The boardconverts the voltage signals from the hot-wire anemometer and the flame ionizationdetector to a personal computer, and delivers analog signals to control the scanning valvefor the sampling of each tube. The data acquisition program written in the C languagecommunicates with the board and also processes the data. The sampling parameters areshown in Table 3.1. The A/D board is accurate to within 0.01%, so the error can beignored.Table 3.1: Sampling parameters of the data acquisition system.Source of the Sampling Sampling SamplingVoltage Signals Frequency Numbers GainHot-Wire Anemometer 3 kHz 25,000 1Fifi 150Hz 2,500 13.3. Measurement TechniquesThe quantities measured in this study include concentration of propane, mean flowvelocity, and turbulence intensity. A general outline of the techniques and equipment usedfollows.Chapter 3. Experimental Investigation 313.3.1. Measurement of ConcentrationThe concentration of propane was measured by the flame ionization detector (FID) whichprovides continuous measurements of fluctuating concentration in turbulent flows. Figure3.4 illustrates the FID system (See Fackrell, 1980 for the details of this instrument). Thesample gas, i.e. the mixture of air and propane, is sucked directly into the FID along ashort length of tubing. The FID consists of a hydrogen-in-air flame burning in an insulatedflame chamber across which a voltage is applied. The introduction of a hydrocarbon gasinto the flame leads to the production of ions and hence a current. An electrometeramplifier is used to convert this small current to a suitable voltage output. The voltageoutput is assumed to be linear with the concentration of sampled gasE—E0=cLC (3.3)where E is the voltage output, E0 is the voltage output when sample gas is absent, C isthe concentration of sample gas, and o is a constant. The linearity of the FID system overthe range of our experiments is shown in Figure 3.5.In the experiments, concentrations were measured with a rake of eleven very finesample tubes (0.3 mm outside diameter) at locations of interest./Gas mixture was sampledthrough these tubes and sent, through a scanning valve, to the FID, which accuratelymeasures mean propane concentration. Since the response was found to be linear withconcentration (at least in the range 1 to 10,000 ppmv propane), only one calibrationconstant and a zero reading need be obtained. Before beginning the test, each tube wascalibrated individually from two known concentrations of propane, i.e. C, and C. Thiscalibration was then recorded in the computer. By sampling alternately from the plenumto check C and then from a tube to find C, before switching to a new sampling tube,errors were minimized to within 2.5%. Random spikes observed on the output signal atthe high sensitivity setting, caused by small particles in the sampled air, were reduced byChapter 3. Experimental Investigation 32electronically low-pass filtering the signal at a low frequency of 300 Hz. The overall errorof cooling effectiveness measurements was estimated to be ±0.05 as shown in Table Measurement of Fluid VelocityThe mean and fluctuating velocities were measured using a DISA constant temperatureanemometer system. The hot-wire probe is standard DISA single wire probe, with a 5 jimdiameter, 1.25 mm length platinum-coated tungsten wire. The hot-wire anemometerbridge was operated at a 1.6 overheat ratio. This ratio is 20 percent lower than the ratiorecommended by DISA; however, the lower ratio allows a longer useful life of the wire.The voltage signal, produced by the anemometer bridge, is passed through a 10 kHz lowpass filter before reaching the AD converter. The 10 kHz frequency was chosen toeliminate high frequency noise without affecting the lower frequency signal components.The hot-wire probe was calibrated, using King’s law with an exponent of 0.45, against aPitot static probe in low turbulence conditions (with the turbulence intensityq/u <0.5%). A typical calibration is shown in Figure 3.6. A lookup table wasobtained based on the calibration and to be used for the measurements. A digital samplingrate of 3 kHz was used for all measurements. The uncertainty in the measurements wasusually within ±0.02 m/s for the mean velocities and within ±0.04 m/s for the fluctuatingvelocities given in Table 3.2. The hot-wire probe was mounted to a dial gauge which wasused to measure the position normal to the plate surface with accuracy of ± 1.3 x 10-2 mm.3.3.3. Determination of 2-D Injection Flow ReattachmentIn two-dimensional tests, the size of the separation bubble downstream of the injectionorifice was estimated using a simple flow visualization technique. Tufts were arranged onthe wall surface and the point of reattachment was determined by the direction change ofthe tufts on the surface. The error involved was estimated at less than ±1/2 slot width.Chapter 3. Experimental Investigation 333.3.4. Measurement UncertaintiesThe measurement uncertainties were estimated by taking into account the flow meters,Pitot tubes, gauges, rulers, and other control parameters used in the experiments. Table3.2 summarizes the uncertainties involved in the measurements.Table 3.2: Summary of the measurement uncertainties.Measured Quantity Estimated UncertaintyU (Hot wire) ±0.02 m/s (for J/u <0.5%)(Hot wire) ±0.04 m/s (for <0.5%)ri (FID) ±0.05x lmm(forl)y 0.254 mm (for velocities)y lnlm(forl)z lmm(forl)3.4. Experimental Measurements3.4.1. 2-D and 3-D Wind Tunnel ModelsThe 2-D model was built to carry out measurements of film cooling effectiveness andvelocity fields. The model and the corresponding coordinate system are shown in Figure3 .7a. The 2-D model consists of a vertical slot made in the plexiglass floor of the testsection. The slot has a width of d=6.35 mm and a height of about Sd.Figure 3.7b shows the 3-D film cooling model and the corresponding coordinatesystem. The 3-D model consists of a row of six square holes made in the plexiglass floorof the test section. Each injection slot has a cross section width of d=12. 7 mm, a heightof about Sd, and a spanwise spacing of S=3d.Chapter 3. Experimental Investigation 343.4.2. Upstream Boundary LayerTests were carried out to investigate the two—dimensionality of the upstream boundarylayer. The mean velocity and turbulence intensity within the boundary layer weremeasured upstream of the injection orifices at three different spanwise locations, as shownin Figures 3.8 and 3.9. The two-dimensionality of the upstream boundary layer is reachedwith maximum deviations of ±0.3 rn/s in the mean velocity and ±0.6 rn/s in the fluctuatingvelocity. Figure 3.11 shows the_mean velocity profile with respect to the logarithmiccoordinate y (y = , U = /‘). A clear logarithmic region is observed showingthat the upstream boundary layer is fully turbulent. The turbulent shear stress t, iscalculated by fitting the velocity profile to the logarithmic law(3.4)where ic is the von Karman constant (ic=0.41), and E is a constant of integration (E=9.O).It is estimated that r,—O.32 kg/ms2. Near the viscosity-affected region, the velocityprofiles do not fit the linear profile as expected. This discrepancy is due to the fact thatmean velocity cannot be measured accurately very close to the wall using the present hot-wire anemometer. Since the upstream boundary layer has a significant influence on theeffectiveness downstream, the measured flow fields were used as the inlet boundaryconditions for the computations (Details will be given in Chapter 6).3.4.3. Prelfrnlnnry Tests for the 2-D ModelPreliminary tests were carried out to investigate the two-dimensionality of the coolantinjected from the slot and conservation of the injected gas. The vertical distributions ofconcentration at three different locations three slot-widths downstream of injection(X/d—3) are shown in Figure 3.11. The two-dimensionality of the injection flow isreached with a maximum deviation in the cooling effectiveness of ±0.08. The amount ofinjected gas mixture at the slot must be the same as that observed downstream of theChapter 3. Experimental Investigation 35injection. Since the propane was well mixed with the air before the injection, theconservation of the injected gas can be examined by integrating the velocity andconcentration profiles along a cross-section at locations downstream of injectionM=JCUdy (3.4)where M is mass flux of injected gas, C is the concentration of the injected gas and U isthe local mean velocity. Table 3.3 shows the flux balances between the injected mixtureand the measured mixture at a location downstream of the injection for two mass flowratios. The error in the flux balances observed are less than 10% which is consistent withthe errors involved in both the velocities and concentration measurements. This checkprovides confidence in the techniques involved and also in the two-dimensionality of theinjection system.Table 3.3: Flux balance of 2-D injectant, (MmesirepMirjeced)/Minjected.I X/d RM=0.2 I RM=0.4 I3 -1.58% I +7.67% I3.4.4. Preliminary Tests for the 3-D ModelThe uniformity of the injection jets from each hole is examined by measuring the surfaceconcentration and the concentration along the center line of each hole downstream ofinjection at X/d=3. These results are shown in Figures 3.12 and 3.13. The variation in themeasured cooling effectiveness from all the holes is within ±0.10. While larger deviationswere found only in the two side holes, i.e. Holes 1 and 6, the uniformity can be consideredadequate. The measurement used here were taken from downstream of Hole 3.Chapter 3. Experimental Investigation 363.4.5. Measurement ProcedureIn the 2-D measurements, the concentration of the injection flow was measured along thestreamwise vertical plane downstream of the injection slot and the measurements werecarried out for three mass flow ratios, RM = 0.2,0.4,0.6 with the main stream velocity,= lOm/s. Both mean velocity and turbulence intensity were measured downstream ofthe injection but only for mass flow ratios of RM = 0.2,0.4. The reattachment lengths ofinjected fluid were observed from the simple flow visualization (see Section 3.3.3). Themeasured effectiveness distribution and flow fields are described in Chapter 6 where theyare compared with computed results.In the 3-D measurements, the concentration of the injection flow was measured onthe wall surface and the measurements were carried out for three mass flow ratios,RM = 0.2,0.4,0.8 with the main stream velocity, U,,. = lOm/s. For mass flow ratios ofRM = 0.4,0.8, both mean velocity and turbulence intensity were measured downstreamalong the central line of the injection holes. Because of the symmetry of the flow field, thevelocity measured along the central line of injection holes by the normal single wirerepresents the local streamwise mean velocity and turbulence intensity. The experimen1almeasurement data are presented in Chapter 7 where they are compared to numericalpredictions.Chapter 3. Experimental Investigations 374-aFigure 3.1: Schematic of wind tunnel.Chapter 3. Experimental InvestigationsWind Tunnel Test SectionFigure 3.2: Schematic of test section.38I Personal ComputerTest SectionFigure 3.3: Schematic of data acquisition systemChapter 3. Experimental Investigations0.35Figure 3.4: Schematic of FID system.39BleedValveHSample in(Air/Propane)4—-,Hydrogen0.4o.s V.5,Cwiti_______0 50 100 150 200 250Propane Concentration [PPM]300 350Figure 3.5: Calibration of FID system.5—4.9 -4.8 -4.7 -4.6 -4.5 -4.4t6Chapter 3. Experimental Investigations 40w1.8 2 2.2 2.4UAO.452.6 2.8 3Figure 3.6: Calibration of the hot wire probe (B in volts, U in m/s).Chapter 3. Experimental Investigations 41(a) 2D ModelTop ViewFront ViewTop ViewFront View—400 Ixz‘ 350635570—265 YI(b)3D ModelHole 1Hole 2400 Hoie3Hole4HoleSHole6z35012.7563—265— xFigure 3.7: Geometric description of the 2-D and 3-D Models.7V101 102Y+Figure 3.8: Measured mean velocity of the upstream boundary layer oncoordinate (X = —120mm).the logarithmicFigure 3.9:(X = —120mm)Measured turbulence intensity of the upstream boundary layerChapter 3. Experimental Investigations 42u+20 -15 -1050A Measurement (ZId=-1 0)Measurement (ZId=0)Measurement (ZId=1 0)u+=Y*U=1/KLn(EY)10°.11 o0.000.0 10.0 20.0 30.0 40.0 50.0Y(mm)Chapter 3. Experimental Investigations 431.000.75A•ZJd-oC——El—— ZId—20.g —E3-- ZId--20&50 9,CC0C.)0.250.00 _.pJ0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0Y/dFigure 3.10: Measured concentration along the streamwise vertical lines downstream ofthe injection slot (X/d 3, d = 6.35mm, 2-D model).0.50.4,.-4::0.3/0.2 4” :::.:•0 i.iole301 HoIe-5—-A—— Hole-S0.0 I....-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5ZfdFigure 3.11: Measured surface concentration downstream of the injection holes on thetranspiated coordinate (X/d = 3, d = 12.7mm, 3-D model).Chapter 3. Experimental Investigations 440.50.4--R--HoIe-1O HoIe-30.3—-----HoIe-4—-a.---HoIe-6LU°•2o-J0.10.0Figure 3.12: Concentr:tion distribution the central planes downstream of the injection hole (X/d = 3, d = 12.7mm, 3-D model).Chapter 4Mathematical FormulationIn the present computations of film cooling effectiveness, the flow and heat transfer isassumed as steady state and incompressible. From the principles of conservation of mass,momentum and energy and by using the Reynolds ensemble-averaging procedure, theequations for the ensemble-averaged properties of turbulent flows and the associated heattransfer in Cartesian tensor co-ordinates are (see Rodi, 1984 for details):Mass conservation equation:=0 (4.1)ax,Momentum conservation equation:a(pukU,) a a (au. aUNi=——+-— (4.2)aXk ax, ax,, aX,, ax, )Scalar transport equation:a(pUk) a(aI (43)aXk aXk aX,,Ic jwhere U (i= 1,2,3) and P represent the mean velocities and static pressure, respectively, 1denotes a scalar variable, and t and y are the fluid dynamic viscosity and the scalarmolecular diffusivity. This system of equations contains unknown variables, the Reynoldsstresses uu, and turbulent scalar flux u4’ . In order to obtain a closed set of equations,some assumptions must be made to relate the Reynolds stresses and the turbulent scalarflux to other existing variables through the procedure called turbulence modelling.In the following sections, two turbulence models used in the present study aredescribed, i.e., the k-E model and the multiple-time-scale model (M-T-s). A simplifiedalgebraic stress model is introduced for the 3-D film cooling flow. Two near-wallturbulence treatments, i.e., the wall function (wF) and the low-Re k model (LK) with anear-wall fine grid are also discussed.45Chapter 4. Mathematical Fonnulation 464.1. Turbulence ModelsThe essence of turbulence modelling is to represent the unknown Reynolds stresses andturbulent scalar flux in terms of known parameters. There are two main categories ofmodelling approaches. One category, called turbulent-viscosity modeffing, is based on thesuggestion of Boussinesq (1877) that Reynolds stresses can be represented in terms ofmean strain-rates (by analogy with laminar Newtonian flows). The second category,called turbulent-stress modelling, is based on the development of differential equationsdescribing the transport of individual stresses (Launder and Spalding, 1972).4.1.1. Eddy Viscosity and Diffusivity ConceptsThe eddy-viscosity concept can be represented by the following equation:2 (au.—puu1 =——kö,+pI __L÷__ (4.4)3 x1 x1jwhere ji is the turbulent eddy viscosity, k is the kinetic energy of the turbulent motionk=--uu (4.5)and ö is Kronecker delta. By direct analogy to turbulent momentum transport, turbulentscalar transport is often assumed to be related to the gradient of the transport quantity—u——I (4.6)whereF is the turbulent diffusivity of the scalar. The Reynolds analogy between scalartransport and momentum transport suggests that r’, is closely related to , i.e.(4.7)where g is the turbulent Prandtl number or the turbulent Schmidt number.Experiments have shown that unlike the turbulent diffusivities for momentum andscalar quantities, o varies only slightly across any flow or from flow to flow (Rodi,1984). Many turbulence models make use of Equation 4.7 with the turbulentChapter 4. Mathematical Formulation 47PrandtllSchmidt number as a constant (e.g., cy = 0.9 for wall bounded flow). The relationgiven by Equation 4.6 has proved useful in many practical calculations.The major drawbacks associated with the standard turbulent-viscosity models aretheir assumptions of isotropic and equilibrium turbulence and that turbulent scalar fluxmust be zero at zero gradient points. An alternative approach is the use of the Reynoldsstress model which calculates the individual Reynolds stresses from their respectivetransport equations that are obtained directly from the instantaneous momentumequations. However, these transport equations contain further unknown, higher-orderstatistical correlations which have to be modelled in terms of known, mean parameters.Previous work has shown that there is still much work to be done to attain a complete andaccurate (or even useful) representation of the correlations.The turbulence model incorporated in the present study is based on the turbulent-viscosity approach. The concept of anisotropic turbulence is introduced through theimplementation of the turbulent eddy-viscosity and eddy-diffusivity plus a simplifiedformulation of the Reynolds stress model. Non-equilibrium turbulence is introducedthrough the multiple-time-scale model, which takes into account the time scales of bothlarge and small eddies.4.1.2. The k-a Turbulence ModelThe two-equation k-a turbulence model uses the eddy-viscosity concept and theKolmogorov-Prandtl expression in which the eddy viscosity can be consideredproportional to a velocity characterizing the fluctuating motion proportional to -/ and toa typical length of this motion,iocpsThl (4.8)where 1 is the mixing length. The quantities k and 1 are related to the dissipation rate ofturbulent kinetic energy, a, by dimensional analysis (Rodi, 1984)Chapter 4. Mathematical Formulation 48k312 (4.9)Combining these two expressions, we obtain(4.10)where C is an empirical constant.For high turbulent Reynolds number flow (Ret =k2/vE> 100), the distribution ofk can be determined by solving the transport equation which can be expressed asaPUik!j pG (4.11)aX, aX1 k aX,where Gk is assumed to be a constant and G is the generation rate of turbulent kineticenergyG=uu,3!LJ (4.12)The equation forE contains complex correlations whose behavior is not well known and isusually presented in the following form:aPUIE=C1pG—C2p— (4.13)Jx1 XGX1J k kwhere Gk C1,C2 are the modelling constants. These constants have been obtained basedon the experimental observations of grid-generated turbulence and near-wail turbulentflows (Launder and Spalding, 1974). The commonly accepted values of these constantsfor incompressible flows are shown in Table 4.1 and are used here. Solutions of the twotransport equations for k and e completely define the turbulent parameters which cansubsequently be used to close the Reynolds-averaged Navier-Stokes equations.Table 4.1: k-E turbulence model constants.C C1 C20.09 1.44 1.92 1.0 1.3Chapter 4. Mathematical Formulation 494.1.3. Nonisotropic Eddy-Viscosity RelationIn film cooling flow with discrete injection, the pressure field created by the injection issufficiently strong so that the Reynolds stress terms in the momentum equation (Equation4.2) have little direct effect on the mean velocity field in the immediate vicinity of theinjection orifice. Downstream of injection, however, it is the turbulent stress field thatcauses the flow field to approach a two-dimensional form. In the film cooling flow, theexistence of nonisotropic turbulence near the adiabatic wall results in the underpredictionof the lateral near wall spreading by computations which use the isotropic k-E turbulencemodel.A nonisotropic eddy-viscosity relation was proposed by Bergeles et al. (1981),based on the algebraic Reynolds stress model of Launder et al. (1975)UUI=_Uk+CS_(GU—_L3UGkk;J (4.14)where C3 is a constant (C3 = 0.27) and the local generation rates of uu, are:(---u.----au.I (4.15)Xk)By assuming that the flow is in local equilibrium and fully developed, the primaryturbulence stresses can be simplified by (not tensor form):au --- au—pu v = — — pu w = t3 — (4.16)ay azwhere y is the direction perpendicular to the wall, and x and z represent the streamwiseand the cross-streamwise directions of the flow, respectively. The turbulent viscosity isrelated to the turbulence intensity (detailed derivation is given by Bergeles et al., 1981)1=0.27kuI i=x,y,z (4.17)By using the measured data of fully developed pipe flow, the following curve fit was usedto represent a linear decay of the turbulence anisotropy from the wall to the outer edge ofthe boundary layer____(t)2=1.0+3.5(1— y<6 (4.18)(vt)2Chapter 4. Mathematical Fonnulation 50_:=.y6 (4.19)(v’)where is the local boundary layer thickness defined in a suitable way.Within the boundary layer, the lateral component of Reynolds stresses and scalarflux can then be written as(au aw (au av—pu’ w’=p+— pu’ Vt = Et+ —) (4.20)—pi (4.21)Pr az PraywhereJ..t=.Ij1+3.5u—yIo)] y<8t,y = y (4.22)yöl_Lt,y=I.ItThis replacement provides increased eddy viscosity and diffusivity in the lateral directionover that in the normal direction in the boundary layer region.4.1.4. The Multiple-Time-Scale Turbulence ModelFor film cooling at high mass flow rates, the turbulence exhibits more nonequilibriumbehaviour: The production of turbulent kinetic energy and the dissipation rate vary widelyin space. Such nonequilibrium turbulence cannot be resolved by the equilibrium eddy-viscosity concept (discussed in Section 4.2.2), which uses a single time scale to describeboth the turbulent transport and the dissipation of the turbulent kinetic energy.The multiple-time-scale (M-T-s) turbulence model developed by Kim and Chen(1989) considers separate time scales for large eddies and for fine-scale, small eddies. Theturbulent kinetic energy spectrum is partitioned into the turbulent kinetic energy of largeeddies k, and that of small eddies k. The turbulent kinetic energy k and k aregoverned by a system modelling equation. Instead of only considering the generation anddissipation of turbulent kinetic energy as in the k-E model, the M-T-S model considers theChapter 4. Mathematical Formulation 51generation of turbulent kinetic energy from large eddies, the energy transfer rate fromlarge eddies to small eddies, and the dissipation of small eddies. The turbulent kineticenergy and the energy transfer rate equations for large eddies are:aU.k a F(+_ak 1———I I I—a- I=pG—pE (4.23)ax, ax Gk ,) ax1jandaPu1E a pG2 pGe-‘ — - I I I -‘ I—p1 + p2 p3 ( )clx, cix, [I cY ) cix,j k krespectively. The turbulent kinetic energy and the dissipation rate equations for smalleddies are:apU1k, a L+]ak=pE—pE, (4.25)ax, ax, Gk ax,andapu,e, af[i+i.ttJat=pE,E—c’,3’- (4.26)ax1 ax1 L a ax1 k, k,respectively, where the and equations were obtained from a physical dimensionalanalysis and the model constants were determined from the assumptions that theturbulence field of a uniformly-sheared flow can approach an asymptotic state in whichG/ becomes a constant and that the ratio of depends on the ratio of G Ie. (detailsare given by Kim and Chen, 1989). The values of the modelling constants are as follows:Table 4.2: M-T-S turbulence model constants.C, C,,, C C3 C C,2 c3 k G0.09 0.21 1.24 1.84 0.29 1.28 1.66 0.75 1.15The influence of nonequiibrium turbulence on turbulence transport is introducedthrough the eddy-viscosity equation, in which(4.27)Chapter 4. Mathematical Formulation 52where C = 0.09 and k = k + k. The use of the energy transfer rate instead of thedissipation rate a indicates that the turbulence transport of mass and momentum isgoverned by the time scale of the energy containing large eddies rather than small eddies.The resulting eddy-viscosity coefficient C (which is defined by j.t1 = Ck I a) is equal towhich is a function of the ratio of the production of turbulent kinetic energy k andits dissipation rate G I a as observed in experiments, instead of a constant as used in thek-a turbulence model. Thus, the development of the mean fluid flow and the turbulencefield is influenced by the spatially-varying turbulent viscosity, and the spatially-varyingturbulent viscosity depends not only on the turbulence intensity but also on the degree ofnonequilibrium turbulence.42. Near-Wall Turbulence TreatmentsThe k-a turbulence model is generally restricted to high Reynolds number conditions,where the effects of laminar viscosity can be neglected. Very close to the wall surface, inthe laminar sublayer region, this assumption is no longer valid. The turbulence modelsshould then incorporate the effects of laminar viscosity. In this thesis, two near-wallturbulence treatments are used: 1) wall function and 2) low-Re k model with near-wallrefined grid. The use of the low-Re k model with fine near-wall grid allows for animprovement in the prediction of the flow and heat transfer near the wall surface, andhence an improved prediction of the film cooling effectiveness. The influence of thesetreatments is examined for both the flow field and heat transfer in film cooling.4.2.1. WaIl Function ApproachSince the velocity gradients are steep near the wall, an accurate representation of flowfields would require substantial numbers of grid points near the surface. For economy ofcomputational cost, one common approach is to avoid the calculation of the flow withinChapter 4. Mathematical Formulation 53the viscous sublayer by using wall functions. Wall functions bridge the laminar sub-layerregion by matching the dependent variables appearing in the turbulence models touniversal values at some point beyond the viscosity-affected region.The standard wall function (Launder and Spalding, 1974) provides the boundaryconditions, such as wall shear stress the mean dissipation rate in the k equation and thedissipation rate, for a solid wall by locating the first computational grid point at a locationsufficiently remote from the wall (say, =30—300) where the flow is fully turbulent. Thewall function method is based on two assumptions: first, the flow in the vicinity of a solidwall behaves locally as a one-dimensional Couette flow; second, the near wall turbulencecharacteristics are the same as those within the fully turbulent region.In the fully turbulent region, the following assumptions were made in deriving thenear-wall equations: 1) the turbulent shear stress is approximately constant and equal tothe wall shear stress, 2) the pressure gradient is negligible, 3) the turbulent effect isdominant (i.e. >> j.t), and 4) the flow is assumed to be in local equilibrium (i.e. theproduction and the dissipation rate of turbulent kinetic energy are locally in balance).Based on this assumption, the wall shear stresses can be determined from=picC’4k”2UIln(Eyj (4.28)The boundary conditions used for the kinetic energy and dissipation equations are:k = C”- (4.29)pk312 (4.30)icyDetailed descriptions can be found in Launder and Spalding (1974).For the M-T-S model, the following boundary conditions can be used based on thelocal equilibrium assumption in the near-wall, fully turbulent region:K2—1 (4.31)k a8C2(C3—Cr1—Cr2)(4.32)Chapter 4. Mathematical Formulation 544.2.2. Low-Re k Model with Fine Grid TreatmentIn film cooling flow, very steep gradients of velocity, turbulent, and scalar quantities existin the near-wall viscosity affected region and their modelling can have a significant impacton the prediction of film cooling effectiveness. The wall function treatment, however,limits the grid refinement near solid walls since it requires that the first grid point belocated in the fully turbulent region. In order to achieve better numerical resolution,proper near-wall turbulence treatment is required.The two-layer modelling approach has been introduced as a practical near-wallturbulence treatment. The approach summarized by Rodi (1991) uses the k-E model awayfrom the wall and resolves the viscosity-affected near-wall layer with a low-Reynoldsnumber k model and a near-wall refined grid. In the model, the use of prescribed lengthscales in the viscosity-affected layer avoids the computation of the E equation in thisregion where the gradient of a is steep and higher numerical resolution is needed. Since arather stiff differential equation for a is omitted from the near-wail computational region,the level of computational difficulty is reduced. Only six or more grid points are requiredin the viscosity-affected layer for the two-layer approach.The calculation procedure can be described as follows:• The turbulence eddy-viscosity p near the wall is obtained using the turbulencelength scale l with the aid of the van Driest damping function.= pCk’2l (4.33)= C3”4[1—3exp(Re/A)] (4.34)where is a constant.• The turbulent kinetic energy is calculated based on a prescribed length scale forthe dissipation rateaUk a1 k312——I —i-— (4.35)aX, ax, k ax,jChapter 4. Mathematical Formulation 55k312 (4.36)£where a damping function similar to that used for the length scale l is assumedas follows:= KC’4[1 — exp(Re1/4)] (4.37)and 4 is a constant. The boundary condition for the k equation on the wall issimply k=O.• The wall shear stress (‘ç) is determined from the velocity at the first nodeadjacent to the wall by assuming that this node is within the viscous sublayer sothat:(4.38)• The matching point for the standard k-E model and the low-Re number kmodel is located at y = 50. Usually, a sufficient number of grid nodes in thesublayer are about six or more.The constants recommended by Rodi (1991) are:= 50.5,4 = 2C314 (4.39)For the M-T-S model, the energy transfer rate and the dissipation rate inside thenear-wall layer are given by Kim and Benson (1993) based on the local equilibriumassumption:k312Ep = (4.40)and the values at the wall for k and k are assumed to be zero.Chapter 5Computational ProcedureIn the present computations, two CFD codes: 2D-MGFD developed by Zhou (1990) and3D-MGFD by Nowak (1991) were used for the 2-D and 3-D models, respectively. 2D-MGFD uses the traditional multi-grid FAS (Full Approximation Scheme) algorithm for theflow equations. In the present work, an improved SIMPLER solution algorithm for the flowequations and the M-T-S turbulence model are introduced into the code. This code wasused as a testing tool for achieving efficient and stable solutions when a new turbulencemodel was introduced and also provided useful information for the improvement of the3D-MGFD code. 3D-MGFD uses a new multi-grid correction scheme and Vanka’s solutionalgorithm for the flow equations. In the present work, the LK near-wall turbulencetreatment and the simplified algebraic Reynolds stress turbulence model were introducedinto the code.In this chapter, the finite volume formulation of the transport equations based onthe hybrid difference scheme is described. A formulation is introduced to determine thenumerical false diffusion resulting from the use of the upwinding scheme. The improvedSIMPLER solution algorithm and the Vanka’s scheme for the flow equations are presented.The multi-grid procedures used in 2D-MGFD and 3D-MGFD codes are also presented.5.1. Finite Volume FormulationThe transport equations described in the previous chapter can be represented by thegeneral transport equation(pU1)=——Ir-’1+s (5.1)56Chapter 5. Computational Procedure 57where 0 can be replaced for different equations (= U for the U-momentum equation forexample), F is a general diffusivity coefficient, and a general source term. Thediscretized system of transport equations is formed on a staggered grid. The calculationdomain is divided into a number of nonoverlapping control volumes and the staggered gridis arranged such that the velocity components are calculated on the faces of the controlvolumes. The locations for the velocity components and scalars (such as pressure,turbulence variables, and concentration/temperature) are shown in Figure 5.1. Due to thestaggered grid arrangement, three different control volumes are required for the threevelocity components, U, V, W.The finite volume form of the general equation is obtained by integrating over thecontrol volume. Figure 5.2 shows a typical control volume for any variable. Using Gauss’divergence theorem, the volume integrals can be transformed into surface integrals for theconvective and diffusive fluxes across the control volume faces:apiJ J IPUO—F——I dydz—J J IPUO—F—I dydzy Zb L Jx J. )‘. L 1x Jr,,, r i , r 1+J J pV — F — dxdz — J J I p VØ — F I dxdzL aYJ X, Zb L )Y Jy yF;r ti r+j J “Ipw0r:LI dxdy_J J “IpWO—F—I dxdyYs L Z J, X,5 L z Jr,,= JJJS,dxdydzwhere F,. . ., F, represent the sum of the convective and diffusive fluxes across the faces e,,b, respectively. For example, 1 Ce + De withCe = 1 Z [p U0] dydz and De = s: _[r’ dydz (5.3)The convective and diffusive fluxes across the control volume boundaries areexpressed in terms of the nodal values of the dependent variable by using finite differenceChapter 5. Computational Procedure 58approximations. The value of the dependent variable has to be determined at the controlvolume faces. The convective and diffusive flux terms (Ce De etc.) can be approximatedusing a power-law difference scheme (Patankar, 1980). The power-law scheme smoothlyblends central differencing when the cell Reynolds number (based on the control volumewidth and the velocity of flow through the control volume face) is low, and upwinddifferencing when the cell Reynolds number is high. For problems where diffusion isdominant, central differencing is most appropriate and is second-order accurate. The useof central differencing leads to numerical instabilities, however, when the cell Reynoldsnumber is larger than 2. Upwind differencing is used to counteract this instability, wherethe value of the dependent variable at the upstream node is assumed to prevail at thecontrol volume face. This leads to an approximation which is unconditionally stable, butonly first order accurate.By using the power-law scheme for the surface integral, the general algebraicequation is obtained for each node P:a,,d?,, = aEØE + aww + aNN + aø + aTØT + aBøB + b,, (5.4)witha,, =a—S (i=E,W,N,S,T,B) (5.5)andae = DeA(IFI)+max(—F,0) and a = (5.6)where A(PI)=max[0(1—0.1Pl)}. Pe=Fe/De, and P=E/D. The variable brepresents the constant part of the linearized source term for each control volume. Thesource term is linearized as follows:JJJ Sdxdydz = SØJ. + S (5.7)where S,, and S are derived using central differencing approximations. The modificationof the discretization near the boundaries is described in detail by Djilali( 1990).ChapterS. Computational Procedure 59Figure 5.1: Relative location of velocity and scalar variables in the staggered grid.X/wFigure 5.2: A typical control volume.TAT zVVw EBChapter 5. Computational Procedure 605.2. False DiffusionThe upwind differencing approximation of convective terms in the conservation equationwhich is employed for stability when the cell Reynolds number is larger than 2 is a first-order approximation. For the conservation equation, the first-order approximation of theconvection term Ce $ $[pUip]dydz results in false diffusion which can bedemonstrated in the following 1-D example as shown Figure 5.3.Figure 5.3: Finite difference grid.I IW P E= [1e +i(_)2[1e+••• (5.8)where Ø is the solution at node P and is the approximate solution at the node e.According to the upwind differencing scheme, = 4,, when the cell Reynolds number ate is greater than 2. The error in the estimate of the convection flux due to thisapproximation isPe’e CD[1e (5.9)This has the form of a flux of by false diffusion with a diffusion coefficient‘FaIse PeUe. In 3-D flow, the false diffusion in directions normal and coincident withthe velocity vector can be expressed by (see Demuren, 1985)U’(Ux+VAy) 510f’ 2(U2+V2)w[u3w+v3wAy+(u2 +V2)2zz]2(U2 V2 W2)(U2 V2)(uzsx + V3zS.y + Wz\z)F = (5.12)2(U2+V2+W2)Chapter 5. Computational Procedure 61where U, 17, W are the velocity components in the Cartesian coordinates and (, i, ) is anew Cartesian coordinate system which is chosen such that the C-axis is tangential to thevelocity vector, and the and i axes are normal to the velocity vector with the -axis inthe x-y plane. It can be shown that the false diffusion in 3-D computations is related to theinclination of the grid line to the mainstream and the gradient across the streamline. Theseexpressions are useful in evaluating the false diffusion, and thus, estimating the numericaluncertainty involved in the computations.Excessive false diffusion in the numerical scheme prohibits accurate representationof turbulent flow and the associated heat transfer. Most turbulence models (such as k-Eand M-T-S models) use the eddy-viscosity concept and the effective viscosity term varieswith different turbulence models. The necessity of separating the numerical errors fromthe turbulence models becomes obvious when it is considered that one of the objectives ofthe present work is to evaluate the turbulence modelling methods for film cooling. In filmcooling computations, the false diffusion coefficients in directions normal to the velocityvector can be significant sources of numerical errors, since convection often dominatesand the gradient of physical variables along the streamline is relatively small.Methods which can be used to reduce the false diffusion in computations include:1) reducing the grid size, 2) altering the grid to follow the streamlines, 3) using a higher-order approximation scheme. A number of upstream-weighted differencing schemes ofhigher-order accuracy and satisfactory numerical stability have been proposed (Raithby,1976 and Sidillcova and Ascher, 1994), but these methods suffer from complexity and lackof boundedness. In general, the higher-order approximation methods developed in thepast tend to produce spurious overshoots and undershoots. Also, general coordinatesystems were used to align the grid with the streamlines. However, the computationalcosts of these approaches are relatively high. In practice, grid refinement has been foundto be an effective approach especially since the resulting large memory from the gridChapter 5. Computational Procedure 62refinement can be solved by faster and larger computers and multi-grid solution techniqueswhich provides enhanced iteration convergence.5.3. Solution AlgorithmsFor the nonlinear Navier-Stokes equations, a solution algorithm is needed to solve thecoupled continuity and momentum equations. A modified SIMPLER algorithm and Vanka’salgorithm are used in the 2D-MGFD and 3D-MGFD codes, respectively and are described inthe following sections.5.3.1. Modified SIMPLER AlgorithmThe SIMPLE pressure-correction scheme of Patankar (1980) is a widely used algorithm forfluid flow and heat transfer. In the SIMPLE algorithm, the momentum and continuityequations are solved in a decoupled manner. The momentum equations are solved basedon an approximate pressure field and a pressure-correction equation is used to correct theflow and pressure field to satisfy the continuity equation. Although SIMPLE and its revisedversion SIMPLER have been found satisfactory in most simple calculations, improvementsare still needed. First, the pressure-correction equation in SIMPLE is not strictly derived;some bold approximations are made. Second, the coefficients of the pressure-correctionequation are very complicated, especially when applied to a general coordinate system.To overcome these difficulties, a new correction scheme is developed in thepresent work based on the projection method proposed by Ascher et al. (1994) for higherindex differential-algebraic equations (DAEs). At each iteration, the velocities on thecontrol volume surface (see Figure 5.4), which are calculated based upon the pressurefield from the previous iteration step, are corrected through a new function N’pU = pU(0l +(), i = e,w (5.13)axChapterS. Computational Procedure 63pv(new) = pv(old), j n,s (5.14)‘\aY )so that the new velocity field satisfies the approximate continuity equation.—pU’]LSy — [pV(new) — PV(new)]&= :v7°— U,°M]— p[v(old — Vold) ]x+ [[p-) —() ]+ [3] — [i] ](5.15)E!PNVnUw Pp UeI VsISFigure 5.4: Control volume for the continuity equation.By using the central difference scheme for the derivative term of N, an algebraic equationfor y can be obtained. In the present computations, velocity components normal to all theboundaries (inlet, solid wall, no flux surface, and exit) are imposed, thus no correction isneeded. Therefore, there is no need to impose boundary conditions for Ni on theboundaries. After obtaining the solution of ‘qi, the velocity field can be updated byEquations 5.13 and 5.14.The correction equation ensures that the flow fields satisfy the constraint of thecontinuity equation after each iteration. The advantage here is that the coefficients of thedifference equation for the correction equation (5.15) only depend on the geometry of theChapterS. Computational Procedure 64grid; not on the flow field. Therefore, a simpler formulation and better stability for thecorrection are obtained.5.3.2. Vanka’s AlgorithmVank&s algorithm solves the flow field by coupling the velocity and pressure. A briefdescription of Vanka’s solution procedure is presented by considering a single controlvolume in two space dimensions as shown in Figure 5.4 (details are given by Vanka,1986):An equation for the variable Ue can be obtained from the momentum equation inthe form of Equation 5.4:aeUe=abUb+be+(PP—PE) (5.16)where Ub are the neighbor values for Ue ab are their coefficients, and Ay, /Xx are theappropriate mesh sizes. Equation (5.16) can be rewritten as:UeBe+(PpPE)Ce (5.17)where Be = abUb + be )/ae and Ce = /ae. Similarly(5.18)= Bfl+(Pp—PN)Cfl (5.19)Equation (5.19) can be put in the form(5.20)where B = B— PNCfl. Similarly(5.21)The continuity equation for the control volume surrounding P, takes the formDeUeDwUw+DnVnDsVs =0 (5.22)with D = pA for each control volume face where p is the fluid density and A the area ofthe face. Substituting Equations 5.18-19 and 5.20-21 into Equation 5.22 givesChapterS. Computational Procedure 65De[Be +(p1._PE)Cel_Dw[Bw +( —P)c] (5.23)+D[B +(i.—PN) Cfl]—DS[BS +(i —P)C]= owhich is the same ascIPW+PP+YPE =6 (5.24)where N’ P are incorporated into 6.For the i-th cell in a row, Equation (5.24) can be written as+ + = 6 for i = 1,2,..., n (5.25)where a., f3.,6 are calculated using the latest values of the flow parameters. Thissystem of equations can be solved with appropriate boundary conditions (details are givenby Salcudean et al., 1992 and Nowak, 1991). Then, the velocity components are foundfrom the explicit formulas given by Equations 5.17-5.21. This ‘line Vanka’ procedure isrepeated for all the lines parallel to the three coordinate directions, usually in the ‘zebra’fashion: odd-numbered rows of cells go first, followed by the even-numbered rows.In contrast with the decoupled solution technique used in the simple algorithm, theVanka scheme solves the momentum and continuity equations with an implicit pressure-velocity coupling and therefore eliminates the need for the pressure-correction equation.The velocities and pressures are simultaneously updated and iterations are made to removethe nonlinearities. It was found that with the Vanka scheme calculations of complexturbulent recirculating and reacting flows have been made in computational times a factorof ten smaller than those required by SIMPLE (Vanka, 1986).5.4. Multi-Grid Computational ProcedureThe iteration method used to solve the discretized nonlinear equations faces slowconvergence as an increased number of grid points is used to achieve high flow resolution.The slow convergence of the solution algorithm is generally due to the persistence of lowfrequency errors that are not effectively removed on a grid which is small relative to theChapter 5. Computational Procedure 66wave lengths of the errors. In order to remove these low frequency components, amultigrid solution algorithm is used. In this method, relaxation techniques are applied ona hierarchy of grids, so that error components corresponding to a wide range offrequencies are effectively removed (Brandt, 1977).The basic full multi-grid FAS procedure used in 2D-MGFD can be described by atwo-grid system: a coarse grid çH and a fine grid (detailed description and theperformance of 2D-MGFD are given by Zhou, 1990). The fine grid is obtained by dividingthe cells of the coarse grid along each direction by two. Therefore, for a three dimensionaldomain, the fine grid has eight times the number of cells of the coarse grid. Assuming thediscretized system of flow equations on a given grid 2” has a formL’Q” = Fh (5.26)the multi-grid procedure is given by• Step 1. Coarse grid pre-iteration, LHQH = F”:The solution q” is obtained by performing NH,1 iteration sweeps on 2”. It isusually not necessary to obtain a ‘fully converged’ solution on the coarse grid,as this is used to provide an initial guess for the fine grid computations.• Step 2. Prolongation from i)” to 2h, IZ:The solution q” obtained on cI” is prolongated to 2”. The prolongation isdone by interpolating the solution to the new grid points which lie between thecoarse grid points.• Step 3. Fine-grid iteration, LhQ = F”:The solution q” is obtained by performing Nh, i relaxation sweeps on i2” withthe prologated solution. This relaxation process removes the high frequencycomponents of the error from the solution on 2h. Nh 1 is not a large numbersince the relaxation on can reduce the high frequency components of theerror quickly. The residuals r” = F” — Lhqh are calculated on 2”. TheChapter 5. Computational Procedure 67residuals r” contain the low frequency component of the error in the solution.If the solution is fully converged, these residuals are close to zero.• Step 4. Restriction from 12” to 12”, ia”:The residuals are restricted to the coarse grid. The restriction process is alsoan interpolation, such that the residuals are represented on the coarse grid. Atthe same time, the solution q” is also restricted to the coarse grid by the samerestriction process.• Step 5. Coarse-grid correction, LHQH = Ijr” + LF(Jqh):The solution q” to the correction equations is obtained by performing N112iteration sweeps on 12”. This solution is not expensive to obtain because it isobtained on the coarse grid. The multigrid correction is obtained bysubtracting the solution to the correction problem from the restricted solution= qH — J,Hqh• Step 6. Prolongation of the correction v” to 12”, If,:The multigrid correction vH is prolongated to 12”, and then used to correct thefme grid solution qew) = + qh Since the multigrid correction specificallytargets the low frequency errors, the residuals are reduced effectively.• Step 7. Fine-grid post iteration L”Q” = F”:With the corrected solution q(new)’ Nh,2 iteration sweeps is performed on12hThe multigrid cycle from Step 3 is then repeated.The multigrid procedure in 3D-MGFD uses a different approach in the coarse-gridcorrection (Step 5): LHQH = Ir’ + L”(qg), where q is the coarse-grid solution(Nowak, 1991). The coarse-grid solution qg instead of the restricted solution JIqhi usedon the right hand side of the correction equation simplifies the programming. It isimportant to note that for some problems (although not in the present computations) inwhich the solutions on 12” and 12” have significant different characteristics, such a multi-grid correction scheme may not improve iteration convergence. This is because unlike theChapterS. Computational Procedure 68restricted solution JHqh the coarse-grid solution q does not contain flow information onthe fme grid.In the present computations, the convergence criterion which measures the degreeto which a computed solution satisfies the finite difference equations (Equation 5.4) on thefine grid 2h is based on the normalized absolute residual errors of the equations beingsolved. These residuals are defined as follows(a—s) aflbøflb_s,1E =where, for the mass equation, 1 is the total inflow of mass; for the momentum equation,is the total inflow of momentum; and for the scalar equations, 1 is the product oftotal volumetric inflow and the inlet scalar quantity. The solution is regarded asconverged when these normalized absolute residuals become less than a prescribed smallvalue. In this work, the value of E i05 was considered to be acceptable for the flowequations and scalar equations while E io for the turbulence equations. A reduction ofthese values by a factor of 10 did not result in any appreciable change in the computed filmcooling effectiveness and flow fields.Chapter 6Results I: Two-Dimensional CaseThis chapter presents both the experimental and computational results for the 2-D filmcooling model. Two alternative turbulence models, the multiple-time-scale k-e model(M’rs) and the standard k-E model (KE), were used combined with two near-walltreatments, the low-Re k model (KE&LK in short) and the standard wall function (KE&wFin short). Comparison between experiments and computations are described bydistributions of 1) film cooling effectiveness, 2) mean flow velocity and turbulent kineticenergy, and 3) coolant distribution on the vertical streamwise plane downstream ofinjection. These quantities indicate the shear in the mean flow, turbulent mixing of themainstream and coolant, and also film cooling performance. Although the 2-Dcomputation is not our primary objective, it is necessary to investigate the proposedturbulence modelling methods in the 2-D case since most of the turbulence models weredeveloped for 2-D flows. Further improvement towards more general turbulencemodelling is suggested as a result of the present work.In the following sections, the computational domain and boundary conditiontreatments are described. The grid independence of the present computation and the falsediffusion involved in the numerical discretization are discussed.6.1. Computational Domain and Boundary ConditionsThe computational domain for the 2-D model is shown in Figure 6.1. The treatment ofboundary conditions on each side can be described as follows:• Mainstream:The upstream boundary was located at a distance X/d=1O from the injectionorifice, where the experimental values of mean velocity and turbulence intensity were69Chapter 6. Results I: Two-Dimensional Case 70measured. The mean velocity and turbulent kinetic energy k profiles were tuned tomatch the measurements closely. It has been found that the downstream effectivenessvalues are sensitive to the upstream conditions. Proper comparisons with theexperimental results must include these well-defined inlet conditions.Within the boundary layer, the streamwise mean velocity distribution was obtainedfrom the logarithmic law (Equation 3.4) in the fully turbulent region and the linear lawUy+ (6.1)in the laminar sublayer region. The turbulent kinetic energy was linearly interpolatedfrom the measured fluctuating velocity based on the isotropic assumptionk=u2 (6.2)The other mean velocity components were assumed asV=0; W=0 (6.3)The inlet condition for the dissipation rate of was prescribed based on theformulations described by Yap (1988):e=Ck2Il2 (6.3)i dy)where 1= min(icy, 0. 09), y is the distance from the wall and the von Karman constantK = 0.4.In the mainstream flow at the inlet, the following expressions were usedU=U; V=0; W=0 (6.4)k=1.5(iUj2 (6.5)= Ck3”2I L (6.6)where i = I U, is the mainstream turbulence intensity which is 0.5% asmeasured in the experiment. The length scale was given as the height of inlet domain,L=30d.Chapter 6. Results I: Two-Dimensional Case 71In the calculation with M-T-S, an arbitrary ratio of k I k =4 was used as an inletboundary condition. A change of this ratio from] to 20 did not have a noticable effecton the turbulence field downstream.• Injection:Uniform injection velocity, turbulent kinetic energy and energy dissipation wereimposed at the inlet of the injection slot.k = l.5i2V, = Id (6.7)where V is the mean injection flow velocity, the injection jet turbulence intensity= (v I v) is specified as 5%, and d is the slot width.• Outlet Condition:The zero gradient condition was applied at the outlet boundary, which is locateddownstream at X/d=40. It was sufficiently far downstream to ensure that the flow inthe upstream region was not affected by downstream conditions.• Axis of Symmetry:The top boundary was treated as a no-flux boundary, where a zero gradient acrossthe boundary was imposed on all variables. No effect was found for any furtherextension.• Adiabatic Wall:At the bottom wall, zero normal and tangential velocities as well as zero heat fluxwere imposed. Two alternate treatments of the turbulence were used near the wall:the wall function and the low-Re k model.6.2. Numerical Grid and Effect of Grid RefinementPreliminary runs were made to determine the effect of grid refinement and to monitor thenumerical false diffusion involved in the computations. Four progressively refined gridsChapter 6. Results I: Two-Dimensional Case 72were used for computations at RM =0.4. The computations were carried out using acombination of the standard k-E model with the low-Re k model (KE&LK).To refine the grid near the slot and the solid walls, a uniform grid was used in theX-direction across the slot and exponential expanding grids were used to avoidunnecessary use of fine grids in the region away from the injection. The grid expansionratios used in the X-direction upstream and downstream of the slot were 1.12 and 1.06respectively. The grid expansion ratio used in the y-direction above the adiabatic wall was1.06. The grid expansion ratio used in the y-direction under the wall in the slot was 1.2.Table 6.1 shows the four progressively refined grids with the grid size within the slot andthat of the first node next to the wall, as well as the overall number of grid points. Atypical numerical grid (Grid 3)is shown in Figure 6.2.Table 6.1: Arrangement of four progressively refined grids.Grid 1 Grid 2 Grid 3 Grid 4Ax within the slot d/4 d18 d112 d116Ay next to the wall d116 d132 d148 d/64Grid Points 58x66 74x78 90x86 106x94Figure 6.3 shows the predicted mean velocity distributions at the exit of the slotusing the four progressively refined grids. The nonuniformity of the injected slot flow isclearly presented, which shows that the injection jet is compressed by the upstreamboundary layer and is concentrated near the downstream exit. The finer grid tends to givesharper variation and higher maximum velocity values. Very little difference is foundbetween Grids 3 and 4.Figure 6.4 gives the predicted mean velocity and turbulent kinetic energy at thelocation X/d=3 downstream of the slot. Identical mean velocity distributions are achievedChapter 6. Results I: Two-Dimensional Case 73in the cases of Grid 3 and Grid 4, and the overall discrepancy among the four grids isrelatively small. For the turbulent kinetic energy, the finer grid tends to give higher values.This indicates that the coarse grid generally ‘smears out’ the turbulence generation, thusunderpredicting the resulting turbulent kinetic energy. No further difference is foundbeyond Grid 3. Figure 6.5 gives the film cooling effectiveness distributions by the fourgrids. Again, the grid independence of effectiveness is achieved by Grid 3.Figure 6.6 shows the distribution of the false diffusion coefficient by Grid 3, whichis estimated by Equation 5.10. The coefficients have values greater than unity near theslot orifices. From this preliminary test, it is shown that Grid 3 is a reasonable grid forlater computations.For the computations with wall function (wu) treatment, the first node next to thewall was located at = —--, which corresponds to = 20—40 for the presentcomputations in the range RM = 0.2 —0.8. The use of the coarse grid near the wallresulted in high false diffusion, especially near the injection location.A three-level multi-grid iteration was used in the computations. The iterationparameters used in the 2D-MGFD code are listed in Table 6.2. Typical convergenceperformance for the computations at RM = 0.4 with all the turbulence model options isshown in Figure 6.7. Numerical instability was observed in the computations using the MT-S model, so 5 iteration sweeps were added for the turbulence equations in eachsmoothing cycle. Computations using the k-E model and WF treatment (KE&wF) have thebest iteration performance. The multi-grid convergence rate deteriorated for computationsusing the fme grid low-Re k model (Lx) since more nodal points were used than in thecomputations using the WF treatment. These results apply in general for 2-D for variousRM. For the present nonlinear system of governing equations, reasonable iterationconvergence was obtained although the convergence rate was sensitive to the number ofgrid points.Chapter 6. Results I: Two-Dimensional Case 74Table 6.2: Mu1ti-rid parameters in the 2D-MGFD code.Number of Under-relaxationDescription Smoothings FactorSolution on the coarsest grid 10 0.8Correction on coarse grid 5 0.8Pre-Solution on the finest grid 5 0.8Post-Solution on the finest grid 5 0.8Multi-grid steps for solution on the finer grid 30 --40 N/A6.3. Predictions and Comparison with Experimental DataComputations were carried out using the standard k-E model and the M-T-S model withtwo near-wall turbulence treatments: the standard wall function (wF) and the low-Re kmodel with refined grid (LK). The computed film cooling effectiveness and velocityprofiles are compared with the experiments.6.3.1. Mean VelocityFigures 6.8 and 6.9 show a comparison between the computed mean velocity distributionsand the measured data at mass flow ratios of RM = 0.2 and 0.4, respectively. The meanvelocity is normalized by the mainstream velocity, which is Urn = 10 m/ s in the presentcase.Generally, all the options provide fairly good predictions of the mean velocity forthe upstream boundary layer flow (X/d=-3). After the injection, the coolant separates anda recirculation zone is formed immediately downstream of the slot orifice. Typicalpredicted vector fields using the M-T-S model with the Lx treatment are shown in Figure6.10. The computed reattachment lengths are compared with those observed as shown inChapter 6. Results I: Two-Dimensional Case 75Table 6.3. Recognizing the uncertainties involved in the experimental observations, thiscomparison gives a rough indication of the accuracy of the prediction of the wall shearstress. The computations using the WF treatment did not find a separation bubble forRM = 0.2 since a coarse grid had to be used near the wall while the refined-gridcomputations using the LK treatment gave satisfactory agreement. By examining thereattachment length, it is found that the WF computations tend to over predict thereattachment length at RM = 0.4,0.6. Apparently, this is because the coarse grid cannotgive a good representation of the separation bubble.Table 6.3: Comparison of reattachment lengths.RM=O.2 [ RJb=O.4 R=O.6Experiment 0.5 - 1.0 2.0- 3.0 4.5- 5.5MTS with LK 1.56 2.56 4.81MTS with WF < grid cell size 5.98 9.01k-Ewithuc 1.69 2.81 5.3k- with WF < grid cell size 10.9 16.0After the injection at X/d=3, all the options underestimate the velocity gradientnear the wall, although the Tic treatment gives relatively larger velocity gradients than theWF treatment. However, better agreement is found further downstream. At X/d—1O theprediction using the uc treatment recovers and agrees reasonably well with experiments,but the prediction using the WF treatment is not good. At X/d=20 fair agreement is foundin all options and the best agreement is found using the M-T-S model with the tictreatment.Chapter 6. Results I: Two-Dimensional Case 766.3.2. Turbulent Kinetic EnergyFigures 6.11 and 6.12 compare the computed turbulent kinetic energy with theexperimental measurements at RM = 0.2,0.4 respectively. The turbulent kinetic energy kin the experiment is calculated from the measured turbulence intensity g(u’ )2•considering the nonisotropic turbulence near the wall, an expression for k (k = 1. 1(u’ )2) isused in the present work, which best fits Equation 4.18. The turbulent kinetic energy isnormalized by the mainstream velocity Urn = lOm/ s.Reasonable agreement with experiments is found for all treatments upstream of theslot at X/d=-3. Downstream of the slot, both turbulence models with the wi treatmentshow consistent underprediction of turbulent kinetic energy and this underestimation issignificant for RM = 0.4. With the LK treatment, the turbulent kinetic energy predicted byboth turbulence models agree reasonably well with measured values in magnitude.However, they fail to predict the high peak of turbulent kinetic energy near the wall. Thisfailure could be caused by inadequate treatment of the turbulence in the separation bubbleas well as unsteady phenomena which are not captured by the steady-state model. Furtherdownstream at X/d=1O, agreement between the LK treatment and observations is excellentwhereas the WF treatment severely underestimates the turbulent kinetic energy. Within theboundary layer, the M-T-S model generally gives higher turbulence than the k-E model. AtX/d=20, reasonable agreement is obtained with all options except the k-E model with thew1 treatment. The best agreement is found using the M-T-S model with LK treatment.Turbulent kinetic energy is underpredicted by k-E/LK. The improvement with the M-T-Smodel is significant at the higher mass flow rate RM = 0.4, which indicates the necessity ofthe non-equilibrium turbulence assumption.Chapter 6. Results I: Two-Dimensional Case 776.3.3. Film Cooling EffectivenessFigure 6.13 shows a comparison of the measured and computed film cooling effectivenessat RM = 0.2,0.4,0.6. Detailed concentration distributions on the streamwise verticalplane at RM = 0.2,0.4 are also shown in Figures 6.14 and 6.15.At the low mass flow ratio RM = 0.2, the WF treatment underpredicts theeffectiveness just after the injection because of the poor resolution and the false diffusion,as demonstrated by the fact that the effectiveness never reaches unity. Downstream ofinjection, both turbulence models with the LK treatment give good agreement with theexperiments. In the recovery region, the best agreement is obtained by M-T-S/LK.However, both turbulence models with the WF treatment consistently overpredict theeffectiveness because of the poor representation of the separation region and theunderpredicted turbulence. Hence the mixing is overpredicted due to the coarse grid.As the mass flow ratio increases to RM = 0.4, M-T-S/LK gives the best overallagreement with the measured effectiveness; however, the agreement deterioratesdownstream and only the rate of change of effectiveness with downstream distance ispredicted well in the recovery region. As the mass flow rate is further increased toRM = 0.6, the slope of effectiveness shows that M-T-S/LK predicts the measured valuesvery well in the recovery region. However, it is clear that the mixing near the slot isunderpredicted because of the inadequate representation of the turbulence in theseparation region. The representation of the turbulence length scale near the slot, theunsteadiness of reattachment, and the existence of secondary flow are poorly captured bythe present turbulence models. It is expected that the agreement between computationsand experiments will deteriorate further with increasing mass flow ratio.An attempt was made in the 2D-MGFD code to use the algebraic Reynolds stressmodel of Launder et al. (1975) together with the k-e model. However, the direct iterativemethod used in the present work for solving the algebraic equations for the ReynoldsChapter 6. Results I: Two-Dimensional Case 78stresses failed to provide a converged solution (see Appendix D for details). Theperformance of the algebraic Reynolds stress model needs to be investigated.It is interesting to note that the WF treatment predicts a sudden drop ineffectiveness near the slot, which is observed at RM = 0.4 and which is more pronouncedat RM = 0.6. This is probably due to the overprediction of the separation bubble and jetpenetration resulting from the underpredicted turbulence mixing with the coarse gridarrangement, which can be shown clearly by the concentration distributions in Figures6.14 and 6.15.Chapter 6. Results I: Two-Dimensional CaseFigure 6.1: Computational domain for 2-D computations.-5 0 5 10 15XIdgrid arrangement79No FluxAdiabatic WallOutlet30dInlet1OdZxInjection40d2015YId1050-5-10 20 25 30 35 40Figure 6.2: Typical for 2-D computations (Grid 3).Chapter 6. Results I: Two-Dimensional Case 800.75 - A Grid 1Grid2--V-- Grid3V/U • Grid 4m0.50--::: I-0.50 -0.25 0.00 0.25 0.50XIdFigure 6.3: Predicted 2-D vertical velocity distribution at the slot exit (KE&LK,RM = 0.4).3..3.53.0 Grid 1 3.0Grid2 Grid 12.5 Grid 3 2.5 Grid 2Grid4 / - Grid32.0 / 2.0 Grid 4YId / Y/d1.5 1.51.0 1.0 -05 05 N0 C 0 C0.00 0.25 0.50 0.75 1.00 0.0 1.0 2.0 3.0 4.0 5.0 6.0U/Urn kIUm2Xl 02Figure 6.4: Mean velocity and turbulence kinetic energy downstream of injection(X/d = 3) predicted by four progressive refined grids (KE&LK, RM = 0.4, 2-D model).Chapter 6. Results I: Two-Dimensional Case 811.00/Gridi075- Grid2U) Grid3U) ----- Grid4-> 0.50 -Ui0.250.00 ‘ I I I0 )(/d1° 15 20Figure 6.5: Film cooling effectiveness predicted by four progressive refined grids(KE&T Ti D - — n A ‘) Th -L-.1\::///////,/////////YY1d15 --0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5XIdFigure 6.6: Estimated false diffusion coefficient for Grid 3 (KE&LK, RM = 0.4, theestimated values are indicated on the contours, 2-D model).Chapter 6. Results I: Two-Dimensional Case 82KE&LK (RM=O.4) KE&WF (RM=O4)10--------- ReU 10 ---- ReU102 -------- ReV 102 ---i---- ReV------- ReM ---7-- ReM101 ----4- ReT 101 -------- ReT----•---- ReK ReKLU 100 ----i--- ReD LU 100 4 ReD10.1 10.11 O 1 02::1o io1O6 1O6o 0Z io- Z io1 08 I I I 10.8_____________________________________________10 20 30 10 20 30Multi-Grid Iteration Multi-Grid IterationMT(RM_O.4) MTS&WF(RM=O.4) I10-- ReU 10 -- ReU -1 2------- ReV 102 -------- ReV -0ReM k. ReMO 101 ---- ReT 0 101 4 ----+---- ReT -ReKp 1 ReKPLU 100 •a ReDp LU 10 . < ------- ReDp -z i- - Z io -1O8 I .1 .1. 108 .1 .1.10 20 30 10 20 30Multi-Grid Iteration Multi-Grid IterationFigure 6.7: Typical 2D-MGFD multi-grid iteration convergence performance (RM 0.4).Chapter 6. Results I: Two-Dimensional Case 83°tñu175 1.00Figure 6.8: 2-D mean velocity distribution (RM = 0.2).Chapter 6. Results I: Two-Dimensional Case 84. ExperimentMTS&WF3.0 MTS&LK .-KE&WF---—-——KE & 1KJ3j00 0.25 0.50 0.75U/Urn0 50 0.75 0 50 0.75 1.00•UIUrn li/UrnFigure 6.9: 2-D mean velocity distribution (RM = 0.4).Chapter 6. Results I: Two-Dimensional CaseYIdYIdYldMTS&LK (RM=O.2)-0.5-1.0MTS&LK (RM=O.4)-0.5-1.0 -0.5 0.0MTS&LK (RM=O.6)-0.5-1.02.01 . . OmIs1 OmIslOm/s857777/ii! ti//I‘.1.1.1 t i.t.t .1 I,.., I.,,, I.,,. I,,., I,,,, I,,,, I,,,-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0XId0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0XIdI If II it--0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0XIdFigure 6.10: Predicted Vector fields by the 2-D MTS/LK model (RM = 0.2,0.4,0.6).Chapter 6. Results I: Two-Dimensional Case0.00.0-I.0.c0.00.06.0 0.0—— 0.06.0 0.0863.0YId2.01.0• ExperimentMTS&WFMTS & LKKE&WFKE&LKXId=-33.0YId2.01.0ExperimentMTS&WFMTS & LKKE&WFtXId=3‘1.5 30 4.5kIUmX1 021.5 30 4.5 6.0kIUmX1 023.0Y/d2.01.0• ExperimentMTS&WFMTS & LKKE&WF-. KE&LKX/d=10• \•\‘S.5”“UI.13.0Y/d2.01.0ExperimentMTS&WFMTS & LK! KE&WFKE&LKIXId=20•1.5 30 4.5kIUmX1 021.5 30 4.5 6.0kIUmX1 02Figure 6.11: 2-D turbulent kinetic energy distributions (RM = 0.2).0.00.0 1.5 3 4.5kIUm x102Chapter 6. Results I: Two-Dimensional Case 87a3.0YId2.01.0ExperimentMTS & WFMTS & LKKE&WFKE & LKXId=3. ExperimentMTS&WFMTS & LKKE&WF--- KE&LK6.0III6.0ExperimentMTS&WFMTS&LKKE&WF— KE&LKX/d=1 3 4.5 3 4.5kfUmXlO2 k!UmX1O26.0Figure 6.12: 2-D turbulent kinetic energy distributions (RM 0.4).Chapter 6. Results I: Two-Dimensional Case 88RM=O.2• Experiment- MTS&WFMTS & kKE&WF-‘-•-•-•-•-•- KE&LK0 5 10 15 20 25RM=O.41.000.750.500.,U)C)C13UIMTS&WFMTS & LKKE&WF-•-•-•-•-•-•- KE&LK0 5 10 15 XId 20 25 30RM=O.G0.750.500.250.00• ExperimentMTS&WFMTS & LKKE&WF-•-•-•-•-•-•- KE&LK0 5 10 15 XId 20 25 30Figure 6.13: 2-D film cooling effectiveness (RM = 0.2, 0.4, 0.6).C-I3L.bbbbb0.P 000001-<p-F.)C.).obbbb-f-I Cl) pa I 0CD I. CD 0 CD 0 I-i ox0 CD CD C, I-i CD CD CDCD CDCD——— Im pa I II 0r©mF.)OI1M01.JMCfl1MCJ14xxmO1%JMC)1.JMcli.)M014CoCD 01 0 CD 0 0 0 CD CD Cl) CD Cl) CD CD-I Cl) 2° r 9’ 001CC000-MClC000P o0obb10-‘bbbom 2° I 9’ 00MC00000-I Im CD CDCDx 0.M011MO1%lr0014MO1%MO1JIOOI1lam-IccChapter 7Results II: Three-Dimensional CaseThis chapter presents the measured and computed results of film cooling effectiveness andflow field for the simplified geometry of vertical injection with square holes. Thissimplified geometry produces a flow with many of the characteristics of actual filmcooling; however, no direct application of numerical results to actual turbines is implied.The present numerical model uses the simplified nonisotropic k-E model with the near-walltreatment of the low-Re k model (KE&LK). The improvement of the present model overthe original 3D-MGFD code, which uses the standard k-E model and the standard wallfunction (KE&wF), is shown by comparing the computed and measured coolingeffectiveness, mean velocity, and turbulent kinetic energy. The computational domain andboundary condition treatments are described. The grid independence of the presentcomputation are discussed.The present numerical model was applied to investigate the effect of differentparameters, namely mass flow ratio, hole spacing, and hole stagger on film coolingeffectiveness with double-row injection. In the present computations, the results forinclined injection cases were obtained using a prescribed velocity imposed at the slot exitwhich is inclined uniformly in the slot direction. It is believed that realistic trends arerepresented by these parametric studies, which are intended primarily as illustrations of theinsight one can obtain through numerical computations. The observations are strictly validonly for the range of parameters investigated.91Chapter 7. Results II: Three-Dimensional Case 927.1. Predictions and Comparison with Experimental Data7.1.1. Computational Domain and Grid ArrangementFigure 7.1 shows the computational domain and the boundary conditions for the present3-D computations. The problem shown consists of an infinite number of holes in thespanwise direction. Because of symmetry on the vertical streamwise planes betweenholes, the computational domain is reduced to the region between two neighboringsymmetric planes and the symmetry conditions are imposed on these two planes. Theboundary condition treatments for inlet, outlet, top wall, and the adiabatic wall are thesame as those discussed in the 2-D case in Section 6.2.The grid independence of the numerical model is determined using fourprogressively refined grids. These grids were selected based on the grid independencetests in the 2-D case discussed in Section 6.3. Grids 1 through 4 introduced in the 2-Dcase were used in the X- and Y-directions and uniform square grid cells are used over theinjection hole and in the Z-direction (see Figure 7.1 for detailed description of thecoordinate system). Computations using the LK treatment were carried out for the fourgrids at RM = 0.4.Figure 7.2 shows the predicted vertical mean velocity distribution along thestreamwise center line at the exit of the injection hole. It is shown that a refined grid givesincreased peak velocity and therefore an increased penetration of injected coolant. Theprofiles predicted by Grid 3 and Grid 4 are nearly identical. Figure 7.3 shows thepredicted mean velocity and turbulent kinetic energy downstream of the injection at X/d=3on the plane Z/d=O. The mean velocities predicted by Grid 3 and Grid 4 reach goodagreement with each other, although the turbulent kinetic energy predicted by Grid 4 isslightly higher than that predicted by Grid 3. However, such slight differences do notaffect the predicted values of effectiveness using Grid 3 or 4 as shown in Figure 7.4.The cross-streamwise false diffusion resulting from the use of Grid 3 wascomputed based on Equations 5.10-5.12. It was found that the projected component ofChapter 7. Results II: Three-Dimensional Case 93the false diffusion coefficient on the X-Y plane has relatively higher values than theprojected components on other planes. Figures 7.5 and 7.6 show the false diffusioncoefficient distributions on the X-Y plane at Z=O and the Y-Z plane at X/d=O.5. Overall,coefficients greater than unity are generally located near the injection exit. Since it wasfound that further refinement gives identical results in terms of film cooling effectiveness,Grid 3 was chosen for later computationsA two-level multi-grid iteration was used in the computations. The iterationparameters used in the 3D-MGFD code is listed in Table 7.1. Typical convergenceperformance for both near-wall treatments is shown in Figure 7.7. The multi-gridconvergence rate deteriorates for computations using the LK treatment with refined grid.This is due partly to two factors: 1) In the 3D-MGFD code, on the coarse grid correctionlevel the restriction of the finer grid solution was not used. Instead the original convergedcoarse grid solution was used. This formulation prohibits the transfer of information onthe finer grid to the coarse grid, and 2) A refined grid was used near the solid wall. Forthis nonlinear system, the convergence rate is sensitive to the number of grid pointsdeteriorating with an increasing number of grid points.Table 7.1: Multi-grid parameters in the 3D-MGFD code.Number of Under-relaxationDescription Smoothing FactorSolution on the coarser grid 500 0.6Correction on the coarser grid 10 0.6Pre-Solution on the finer grid 5 0.6Post-Solution on the finer grid 5 0.6Multi-grid steps for solution on the finer grid 30 40 N/AChapter 7. Results II: Three-Dimensional Case 947.1.2. Mean VelocityThe comparison of the experimental and computational mean velocity profiles is carriedout both upstream and downstream of the injection holes. The mean velocities weremeasured along the center line of the injection hole. Although the symmetry of thevelocity field should be observed, the uncertainties involved in the measured values due tothe injection flow condition and the positioning accuracy of the measuring equipmentshould be recognized.Figures 7.8 and 7.9 show the mean velocity distribution at RM = 0.4,0.8. Thevelocities are normalized by the free stream velocity U, 10. Good agreement isobserved by both treatments upstream of the injection at X/d—-5.Downstream of the injection, the flow detaches and reattaches and a pair ofkidney-shaped vortices are formed along the jet. Figures 7.10 and 7.11 show the predictedvector fields on the vertical streamwise plane (Z=O) and on the cross section (Y-Z) atX/d=3.The comparison of the mean velocities downstream of injection indicates that goodagreement between computation and experiment is achieved by the LK treatment atRM = 0.4 in regions near the injection at XId=3 and farther downstream at XId=10, 20.However, the agreement deteriorates in regions farther downstream at the higher massflow ratio, RM = 0.8. The WF treatment generally cannot predict the mean flow gradientand the wall shear stress is not properly calculated by the coarse grid.7.1.3. Turbulent Kinetic EnergyFigures 7.12 and 7.13 compare the measured and computed turbulent kinetic energy atRM = 0.4,0.8. The measured turbulent kinetic energy k is calculated from the measuredturbulence intensity g(u?)2, as described in Section 6.4.2 (i.e. k = 1.1(u’)2.) It should beChapter 7. Results II: Three-Dimensional Case 95noted here that the measurement of the turbulence intensity along the centerline containsuncertainties about position and that there is an assumption of symmetry. Fartherdownstream these uncertainties should be reduced as spanwise gradients decrease.Both computations give good agreement with experiments upstream of theinjection X/d=-5. Downstream, but near the injection at X/d=3, the LK treatment givesgood agreement at RM = 0.4,0.8 although slightly overpredicting the peak value. The WFtreatment underestimates the turbulent kinetic energy and the position of the peak value islower than measured.Farther downstream at X/d=1O and X/d—20, good agreement is achieved by bothtreatments. The LK treatment gives a higher peak value of the turbulent kinetic energythan the measured one while the WF treatment gives lower values. The underprediction ofk using the w treatment was found near the wall (Y/d<1), while good agreement wasachieved using the ix treatment. At the higher mass flow ratio RM = 0.8, theunderprediction using the WF treatment becomes larger.7.1.4. Film Cooling EffectivenessFigures 7.14 shows a comparison of the computed and measured spanwise-averagedeffectiveness as well as the effectiveness along the center line downstream of an injectionhole at RM = 0.2,0.4,0.8. Detailed surface cooling effectiveness distributions are shownin Figures 7.15-7.17. In the experiments, measurements of cooling effectiveness werecarried out in half of the domain due to the symmetry situation.At mass flow ratio RM 0.2, the LK treatment gives good averaged and center lineeffectiveness everywhere downstream of the injection. The wi treatment severelyoverpredicts the center line effectiveness. Good prediction of averaged effectiveness isobtained using the WF treatment except in the region near the injection where theeffectiveness is underpredicted due to the false diffusion on the coarse grid.Chapter 7. Results II: Three-Dimensional Case 96As mass flow ratio increases to RM = 0.4,0.8, the agreement between averagedeffectiveness calculated using the LK treatment and the experimental measurements nearthe injection hole deteriorates, although good agreement is still obtained in the recoveryregion. The underprediction of the averaged effectiveness is probably caused byinadequate modeling of the complex flow immediately downstream of the injection. Fromthe surface cooling effectiveness distribution, it is found that the LK treatment overpredictsthe penetration of the jet due to the fact that the increased turbulent mixing resulting fromnonequilibrium turbulence and swirling flow at high mass flow ratio is not well representedby the present model. The computations generally under-predict the spanwise spreadingof the jet, despite the use of the nonisotropic turbulent eddy-viscosity.Conversely, as the mass flow ratio increases to RM = 0.4,0.8 the WF treatmentconsistently gives lower averaged effectiveness than the measured values even fartherdownstream while the center line effectiveness appears to be improved at RM = 0.8. Dueto the lack of near-wall flow resolution, the penetration of the jet is overpredicted and thespanwise spreading of the jet is severely underpredicted.The present comparison shows that the LK treatment improves the prediction ofeffectiveness. However, it also suggests that the k-E turbulence model cannot correctlydescribe the turbulence stresses and scalar fluxes when the shear flow between the mainstream and jets is high and the associated streamwise vorticity is strong. Based on the 2-D computations, it appears that the turbulence modelling of the complex flow at high massflow ratio can be improved by using the M-T-S turbulence model which takes into accountthe increased non-equilibrium turbulence.In the present work, the M-T-S turbulence model was added to the 3D-MGFD code.However, the code could not provide a converged solution due to the fact that in the codeall variables are solved simultaneously in a coupled nature (see Appendix D). An attemptshould be made to decouple the turbulence equations from the mean flow in order tostablize the turbulence equations.Chapter 7. Results II: Three-Dimensional Case 97Also, it appears that the stronger vortices downstream of injection at higher massflow ratios require a solution of the full Reynolds stress equation in order to take intoaccount the nonisotropic turbulence resulting from the swirling flow in the vortices.7.2. Parametric Analysis7.2.1. Computational Domain and Boundary ConditionsComputations have been carried out for single and double staggered cooling orifices. Thecomputational domain is shown in Figure 7.18. The parameters used in the tests werechosen as suggested in the work of Gartshore et a!. (1991) (See Table 7.2). For theinclined jets, the coolant orifice was not square at the exit surface and this was taken intoaccount. The arrangement of the computational grid was the same as in the previoussection. For the two-row injection, a uniformly refined grid was used between the rows toensure that the interaction between the jets is well represented.Table 7.2: 3-D parametric tests.Parameter ValueMass Flow Ratio RM O.4,O.8,1.2Injection Angle to the Mainstream o = 00, 900Injection Angle to the Surface 13 = 300Hole Spacing S/d= 4, 5Row Spacing RId=3Stagger factor A/d=O, 1, 2, 3Non-uniformity of the injection flow at the slot exit has been observed in thecomputed results for the vertical cooling orifices. Figure 7.19 shows the predicted meanChapter 7. Results II: Three-Dimensional Case 98velocity at the exit of an injection hole. The mean injection flow velocity is 4 mIs. It isfound that at the exit surface, the vertical velocity reaches a peak value of about 7 mIswhile the streamwise velocity has a more uniform value of about 2 mIs over the wholesurface. The influence of this nonuniformity on the cooling effectiveness is studied bycarrying out a computation with no slot and an assumed uniform injection at the injectionlocation. Figure 7.20 shows the predicted film cooling effectiveness by computationswith and without a slot. The comparison indicates that the averaged effectiveness has lessthan ±0.05 difference near the injection and no appreciable difference farther downstream.However, the center line effectiveness has ±0.10 difference near the injection and less than±0.02 difference farther downstream. This suggests that computations within the injectionoriface might be required in order to study the detailed distribution of effectiveness nearthe injection.Since the present code is limited to a Cartesian coordinate system, specialtreatment is needed for computations with an inclined jet. Nevertheless, some indicationof the effects of the non-uniform flow can be determined by assuming a flow profile basedon the observations in the vertical cooling orifice computations. This issue of theuniformity of the flow at the slot exit needs to be explored further using codes able tocorrectly represent non-Cartesian systems.In the present computations, the jet flow at the exit is assumed to follow theinjection direction. Therefore, the velocity components of the jet can be expressed asU = V2. cos I sin x, V = VT sin 3, W = 17?. cos cos o (7.1)where 14 denotes the total velocity at each point on the exit surface. The total velocitycan be distributed uniformly: 14 (x, z) = VM, where VM is the jet mean velocity. However,in order to take into account the non-uniformity of the jet, a simple linear distribution for14 is assumed on the exit surface. For example, for the lateral injection,(2x Y2z ‘\14(x,z)= VM1—+111 —+11 (7.2)x Az IChapter 7. Results II: Three-Dimensional Case 99where d and d represent the hole width in X and Z directions.Figure 7.21 shows the predicted cooling effectiveness values resulting from theuniform and linear jets. The linear jet has a higher peak velocity at the edge of the exit,thus higher lateral injection momentum, which prevents the coolant from detaching fromthe surface. The resulting cooling effectiveness near the injection orifice with the linearprofile is higher than with the uniform profile. However, the difference is reduced at thehigher mass flow ratio RM =0.8. No significant effects have been found far downstreamof the injection.7.2.2. Finite Array EffectsFor the film cooling with lateral injection, it is assumed that there are an infinite number ofholes in the lateral direction. Because of periodicity in that direction, the computationaldomain is restricted to one period as shown in Figure 7.18. In the present work, theperiodic boundary condition is imposed at the level of the discretized equations by addingextra computational cells outside of the computational domain and imposing periodicvalues. Such treatment is introduced in the 3D-MGFD code and the procedure can bedescribed as follows: Considering a row of grid cells, which includes two surfaces wherethe periodic boundary conditions need to be imposed (See Figure 7.22). The periodiccondition is imposed before each smoothing by lettingUINk = U2, U1 = Uk_l (73)5i,Nk = S2 5i1 5i,Nk—1With this updated flow field, the velocity components WNk, T4’,2 on the surfaces arecalculated in the same way as those in the interior nodes. The mass fluxes through bothsurfaces are monitored to determine the convergence of this iterative procedure. Lessthan 1% imbalance in the mass fluxes on the surfaces was found once the iterations in theentire computational domain have converged to specified criteria.Chapter 7. Results II: Three-Dimensional Case 100The computations with the periodic array represents film cooling with a largenumber of holes in a row. It is different from the case of a finite array (e.g. 5 to 6 holes),which is often used in experimental studies. This difference can be demonstrated bycomparing the film cooling effectiveness distributions predicted by a finite array (5 holes)and the periodic array as shown in Figures 7.23 and 7.24. For the finite arraycomputations, two edge-walls (ZJd=±16) were treated as solid walls.At the lower mass flow ratio RM = 0.8, the surface effectiveness near the injection(X/d<15) downstream of the middle hole is close to that observed in the periodic array.However, at the higher mass flow ratio RM = 1.2, the effectiveness distributiondownstream of each hole of the finite array varies significantly from one hole to another.The difference can also be shown by comparing the averaged effectiveness of the periodicarray and middle section of the finite array (see Figure 7.25). This effect increases withincreasing RM, presumably because the coolant layer becomes thicker and the disturbanceat the edge of the finite array becomes larger and more significant.7.2.3. Single-Row Injection vs. Double-Row InjectionFigure 7.26 shows the spanwise-averaged cooling effectiveness for streamwise and lateralinjection for a single-row of holes and for lateral injection for a double-row of holes. Thetotal mass flow ratio was chosen to be equal for single and double row injection and itsvalue, RM, is defined by the mass flow ratio for single row cases. Lateral injectionperforms much better than streamwise injection near the coolant orifice but loses itsadvantage farther downstream. The best cooling performance is given by the double-rowlateral injection. The superior performance of the lateral injection is attributed to: 1) thehigher spanwise spreading of the laterally injected coolant which forms more uniformspanwise effectiveness distributions downstream of the injection, and 2) the singledominant vortex formed downstream of each hole which interacts with the neighboringfluid to push cold fluid towards the surface near Z=0. Comparison of the single- andChapter 7. Results II: Three-Dimensional Case 10]double-row lateral injections shows a lower penetration for the double row but also astrong interaction of vortices which tends to push colder fluids toward the surface.Figure 7.27 illustrates the film cooling effectiveness distribution on the wall. Thesuperior coverage of the lateral injection arrangement is obvious. This is further illustratedin Figure 7.28 which shows the concentration in planes perpendicular to the cooledsurface at a distance of X/d—3 from the coolant orifice. As expected, the spanwisespreading of the coolant distribution is much higher for lateral than for streamwiseinjection.The flow distribution in the film cooling process can be shown with vector velocityfields in a cross flow plane at X/d=3 (Figure 7.28). For streamwise single row injection,two symmetric vortices are formed which lift the coolant jet near Z=0 and entrain the hotfluid towards the surface and therefore deteriorate the cooling effectiveness. For thelateral injection a single dominant vortex is formed and its interaction with the neighboringfluid tends to push the cold fluid towards the surface.7.2.4. Hole Spacing Effect in Single-Row InjectionThe effect of spanwise hole spacing on the cooling effectiveness is studied for lateralinjection with hole spacings of S/d=4 and 5. In the present tests, a mass flow ratio perunit span R3=-is introduced which is proportional to the mean flow from the coolantholes divided by flow across the holes in an area one d high and S width, is introduced.The tests are carried out at three mass flow ratios, R = 0.1,0.2,0.3. Figure 7.29 showsthe averaged cooling effectiveness (i) of lateral injection. There exists a small region justafter the injection where i decreases with an increase of R. As the spacing increasesfrom S/d=4 to 5, the drop of r with R becomes larger and this drop can be found fartherdownstream (up to d=58). Far downstream of the injection, r increases with R.Chapter 7. Results II: Three-Dimensional Case 102Overall, wider spacing gives consistently lower rj in the present testing range, which isshown in Figure 7.30.The detailed surface cooling effectiveness distribution, the vector field and theconcentration contours in a cross plane at X/d—3 of a single-row lateral injection forX/d=4 and 5 are shown in Appendix B.1. It is clear that small spacing is preferred foroverall cooling performance (although in real design, there exist structural and machiningproblems if the holes are too close together). Also, low mass flow ratio is preferred forbetter performance near the injection region. The film cooling process can be exploredfurther by investigating the flow and coolant distribution farther downstream. For thelateral injection, a dominant vortex is formed for each injection orifice. This vortex liftsthe cool fluid away from the surface which is then pushed back towards the surface by theneighboring vortex. Such interaction is enhanced as R increases. It is found that thevortex interaction has a negative effect on the overall r in the region near injection wherethe fluid has higher temperature gradients near the wall surface than in the fluid fartherdownstream. Farther downstream, TI increases with R since the vortex interaction isweaker. For S/d=5, the interaction between the vortices loses more cold fluid into themainstream than for S/d=4, due to the larger spacing. This phenomenon is more severefor higher mass flow ratios.7.2.5. Stagger Effect in Double-Row InjectionThe cooling performance of double row lateral injection is studied on four staggered-holearrangements, A/d—0, 1, 2, 3, where A indicates the shift between the two rows of holesthat have hole spacing S/d=4. (A=0 represents in line holes and A=0 is the same as A =4.)The total mass flow ratio, RM, was chosen to be equal to the mass flow ratio for singlerow cases.Figures 7.31 and 7.32 show the spanwise-averaged cooling effectiveness fordouble row lateral injection with four staggered-hole arrangements. The arrangement ofChapter 7. Results II: Three-Dimensional Case 103A/d=3 shows superior performance over the other arrangements for RM = 0.4,0.8,1.2while A/d=1 shows reduced overall performance. The arrangement of A/d=0 (in line)gives promising performance especially, near the injection region as RM increases.Figure 7.33 show the predicted effectiveness vs. the stagger factor Aid resultingfrom the four hole arrangements. The lowest value of effectiveness can be foundconsistently between A/d=1 and 2 in the present range of parameters. The bestperformance can be found consistently between A/d=3 and 4. The fully staggeredarrangement is commonly used in the real design because of the needs for structuralintegrity and efficient internal cooling. These tests suggest that it is possible toconsistently achieve better performance, even with the fully staggered arrangement, bychanging the injection direction slightly streamwise. Further investigation needs to becarried out using a generalized coordinate system.The detailed surface effectiveness distributions are shown in Appendix B.2. Theflow and coolant distribution downstream is studied in order to gain insight into thedouble-row film cooling process. The vector fields and concentration contours in a crossflow plane at Xid=3 are given in Appendix B.2 with discussions.Chapter 7. Results II: Three-Dimensional Case 104ii’No FluxInlet Outlet______25d6 Adiabatic Wall11±Injection10d d 40dSymmetric Boundary:::tb0DomarnFigure 7.1: Computational domain for 3-D film cooling model.Chapter 7. Results II: Three-Dimensional CaseFigure 7.2: Vertical mean velocity at the hole exit predicted by four progressively refinedgrids (KE&LK, RM = 0.4).orFigure 7.3: Mean velocity and turbulence kinetic energy at X/d = 3 predicted by fourprogressively refined grids (KE&LK, RM = 0.4).1050.75V/Urn0.500.250.00-0.50 -0.25 0.00 0.25XId0.502.52.0Y/d1.51.00.5Grid-iGrid-2Grid-3Grid-4XId=30.00 0.25 0.50 0.75 1.00U/Urn2.0 3.0kl(Um)2xl 02Chapter 7. Results II: Three-Dimensional Case0.75000)C0.50LU0.25-0.000.0 2.5X1d5°Grid 1-•-•-•-•-•-•- Grid2- Grid3Grid 47.5 10.0106Figure 7.4: Film cooling(KE&LK, RM = 0.4).effectiveness predicted by four progressively refined grids-1.0 0.0 1.0 2.0 3.0XIdFigure 7.5: Estimated false diffusion coefficient on the vertical streamwise plane Z 0using Grid 3 (KE&LK, RM = 0.4).‘I’-I’I’V3.02.0YId1.0-0.0/11-__5).510.50.5/<1\Chapter 7. Results II: Three-Dimensional Case 107Figure 7.6: Estimated false diffusion coefficient on the vertical cross plane X = 0.5 usingGrid 3 (KE&LK, RM = 0.4).XId=O .520 ‘1_—1.5::;05:o12:5KE&LK (RM=O.4) KE&WF (RM=O.4)10110°10.11 021 o1 o1 -°10.61 oI-2UICu•0U)a)a)NCu0zEl ReUA ReV4 ReWReM•- ReKI . . . . I . . . . I . . . .10110°10i0io.3. -4a) ‘UNio.51 061 oE’---•-- ReU -A ReVV-•--•-- ReW -e--•-•-- ReM•---- ReK-. ReD -%p•I..1l-14%6----- ReT -66 -0 10 20 30 40 0 10 20 30 40Multi-Grid Iteration Multi-Grid IterationFigure 7.7: Typical 3D-MGFD iteration convergence performance (RM = 0.4).Chapter 7. Results II: Three-Dimensional CaseU/Urn2.52.0Y/d1.51.00.5u.c0.00 0.25 0.50 0.75U/Urn1.001082.52.0Y/d1.5ExperimentKE & LKKE&WFXId=-51ExperimentKE&LK- KE&WFUXId=30.25 0.50 0.75 0.25 0.50 0.75U/Urn U/UrnFigure 7.8: Mean velocity distribution (3-D model, RM = 0.4).Chapter 7. Results II: Three-Dimensional Case2.52.0Y/d1.51.00.5• ExperimentKE & LKKE&WFa0.25 0.50 0.75U/UrnU/Urn109X/d=-5D.C0.00 1.00U/Urn0.00 0.25 0.50 0.75 1.00 0.25 0.50U/UrnFigure 7.9: Mean velocity distribution (3-D model, RM = 0.8).IIIICD•CDCfloc.e_____________CD—ci..‘...I-—cia,L-I..,--------©0••EEEEER°ào________,—.injJX\k,7.—_—/——CDI—__CD—--——————II______Cj)_____-.—-c--CD______.p -CDCDin-.-=-----------—I—CD———____________________CoinCDCDCDCDI-i—-infl‘\\\\\\\\\\\\EEEEEEEEEEEEEj,cz__________________________________________________— 0I.arm Qo I II 000 in0•0 01F\_‘_\,\\Chapter 7. Results II: Three-Dimensional Case0.0.0 1.0k/U 2x10111. ExperimentKE & LKKE&WF2.52.0YId1.51.00.5. ExperimentKE & LKKE&WFXId=32.52.0YId XId=-52.0 3.0 4.0k/Um2X1 022.5 - Experiment -KE & LKKE&WF2.OL20X/d=10kIUm2Xl024.02.52.0Y/d1.51.00.5. ExperimentKE & LK. KE&WFX/d=20.0 1.0k/LJ2x1O4.0Figure 7.12: Turbulent kinetic energy distributions (3-D model, RM 0.4).Chapter 7. Results II: Three-Dimensional Case0...0.0 1.0I(/U2x10112a Experiment -KE & LKKE&WFa ExperimentKE & LKKE&WF2.52.0YId1.51.00.5nw’XId=-52.52.0Y/d1.51.00.5k/U 2x104.0 4.0a ExperimentKE & LKKE&WFa ExperimentKE & LKKE&WF2.52.0Y/d1.51.00.5nl’a••a!aI:.(:.‘UX/d=1O2.52.0Y/d1.51.00.5‘UXId=20‘0.0 1.0k/U 2x101.0k/U 2x1 024.0 4.0Figure 7.13: Turbulent kinetic energy distributions (3-D model, RM = 0.8).Chapter 7. Results II: Three-Dimensional Case 113RM=O.21.000.75 4 . Expt (Center-line)(l • Expt (Averaged)KE&WF (Center-line)\ ....... KE&WF (Averaged)> 0 50 • \ ••. KE&LI( (Center-line)0 5 10XId 1520 25RM=O.41.00 40.75 Expt (Center-line)• Expt (Averaged)KE&WF (Center-line)• \\.. KE&WF (Averaged)W_______________________0 5 10 XId 15 20 2RM=O.81.000.75 Expt (Center-line)0 • Expt(Averaged)KE&WF (Center-line)KE&WF (Averaged)> 0 50 KE&LK (Center-line)KE&LK (Averaged)LIJ ‘, \..0.250.00 I I0 5 10XId 15 20 25Figure 7.14: Film cooling effectiveness (3-D model, RM = 0.2, 0.4, 0.8).Chapter 7. Results II: Three-Dimensional Case 114C C In C 10 It) It) CF..CCd F-CCd F..CCd CCdCC”I0CCdC tI)C..InCd)DC.) C C C C C C C C C C C C C CII. IU C.) < 0) F. tO F) Cd —It) U) C IL) C IL) C It)F.. II) 1.1 F.. C Cd P. C Cd I•. It) CdF)P..I1)0)FdID CdP.CCCdCCl C C C C C C C C C C 0 C C C C-Jq U? U?.009.;. ,;.U? U?— — 0 0 0 — —>-U? U? U?—— 0 09.;• .;Figure 7.15: Film Cooling effectiveness distribution on the wall surface (RM = 0.2).Chapter 7. Results II: Three-Dimensional Case 115U’ U U’ U’ U) U) U) U)I.-U)c.l t-tflCU l’-U)C1 PU)CCOF’.rIflU’NCD COt...1flU’C1O0 0 0 0 0 0 0 0 0 0 C C C C C0II4-CC)E1a)a.‘4uJcuLILUinc in q in q iq—— 0 0 0 — —L CL C L—— 0 0 0 — —Vu 0 C ‘DC U)— — 0 00Figure 7.16: Film Cooling effectiveness distribution on the wall surface (RM = 0.4).Chapter 7. Results II: Three-Dimensional CaseIfl W ID ID ID II II ID1-IDC’I 1-IDC4 1-IDC1 r-IDUID1ID IDIDID0 0 0 0 0 0 0 e 0 0 0 0 0 0 0116If Lq o 0.;. .;.c q Lq— — 0 0 0>-0 Ifl 0 U 0 tO—— 0 0 0 — —>-Figure 7.17: Film Cooling effectiveness distribution on the wall surface (RM = 0.8).Chapter 7. Results II: Three-Dimensional Case 117Periodic BoundariesMainstreamFigure 7.18: Computational domain for 3-D parametric tests.Chapter 7. Results II: Three-Dimensional CaseVertical Mean Velocity (RM=O.4) V (mis)________________________________________F 7.63E 7.12D 6.61C 6.10B 5.60A 5.099 4.588 4.077 3.566 3.055 2.544 2.033 1.532 1.021 0.51118Figure 7.19: Predicted Mean velocity at the hole exit (KE&LK, RM = 0.4).1.000.75COCl)G)0.5013Ui0.250.000 5 10XId 15 20 25Figure 7.20: Film Cooling effectiveness predicted by a) No slot and an assumed uniforminjection (without slot) and b) Including the slot (with slot) (KE&LK, RM = 0.4).Vector Field (U-W) at Hole Exit4 mIs0.250.00ZId-0.25/ / — — — — — — — — — // — — — —.. — —.. — —.. — /\ ,.— — — —. —. —. -.-- -.-- —\ -.. —.. —.. — —505\r0.50 -0.25 0.00 0.25 0.50XId-0.50 -0.25 0.00 0.25 0.50X/dRM=O.4Expt (Center-line)• Expt (Averaged)Without Slot (CerWithout Slot (Averaged)With Slot (Center-line)With Slot (Averaged)Chapter 7. Results II: Three-Dimensional Case 119Figure 7.21:profiles (S/dFilm cooling effectiveness predicted by linear and uniform injection flow=4).I.... IRM=O.4Linear Profile/1UnormPfdeormProfule1.00C’)00.7500.5000)•t;0.500.250.005 10 15 200XIdRM=O8Linear Profile -:Unorm Profile\—0 5 10 15 20XIdChapter 7. Results II: Three-Dimensional Case 120W(i4k+1)U(i,Nk) — S(i,Nk) .U(i+1,Nk)(i,Nk)U(i,Nk-1) — S(Nk-1) U(i+1,Nk-1)W i,Nk-iI Computational DomainWi,3) IU(i,2)_ S(2) ...U(i+1,2)W,2)U(i,1)_ S(1) U(i+1,1)W i,1)Figure 7.22: Schematic description of the periodic boundary condition.Chapter 7. Results II: Three-Dimensional Case 121Finite Array (RM=O.8)15 LeveIc10 C 0.752Zido ZZEEt :: :::—150 5 10 15 20 25XIdPeriodic Array (RM=O.8)5ZIdO- -I-50 5 10 15 20 25XIdFigure 7.23: Predicted surface effectiveness of finite array and periodic array of lateralinjection (S/d = 4, RM = 0.8).Chapter 7. Results II: Three-Dimensional Case 122Finite Array (RM=l .2)15 - 1Ei--__—-. Level ç1—15o 5 10 15 20 25XIdPeriodic Array (RM=l .2)5ZId05 4-50 5 10 15 20 25XIdFigure 7.24: Predicted surface effectiveness of finite array and periodic array of lateralinjection (S/d = 4, RM = 1.2).Chapter 7. Results II: Three-Dimensional Case1.000.750.001.000.750.00123CoCOa)Ca)0.50a)0)0.250 5 10 15 20 25XIdCOCOa)Ct0.50ta)0.250 5 15 20 25Figure 7.25: Predicted lateral averaged cooling effectiveness of finite array and periodicarray of lateral injection (RM = 0.8 and 1.2, S/d = 4).Chapter 7. Results II: Three-Dimensional Case 1241.000gj 1-Row Streamwise Injectionc 0.75 1-RowLateralinjection2-Row Lateral Injectionta 1:O.50fjHoJo::.Figure 7.26: Predicted averaged cooling effectiveness by streamwise and lateral injection(RM = 0.4, S/d = 4)..<-‘MOCnO)-JCD>WOQm11‘MtO31O’4WWOOflflaaaaaaaaaappoCCpPaaaaaaaaaaaaaaaPacCaaaaaaaaaaaaPaba’wwwb——abaCfl-JWU1”JC,3ah3mo1-.J(oC)MCfl4COO)M(flC.)CMmfl-4Wr.3U14MC314NC$1JP3011MCJIMCJ1’-4U14h3V1lMC31.4P3Cn4M1JC.J(UiUiUiUiUiUiUiUiUiUiUiiUUiUiiiiUiUiUiUiViUiViUiboaaabba-‘Fi)2.aabaaaaIICtxCD I I. 0-I1-RowStreamwiseInjection1-Rowlateral Injection2-RowLateralInjection4.0IIIiiiiittiii112rrid’s3.0lIltIlIlItIIttiiiittIlitIIillIlIIIIll/Il/Iiiii/t/II\IIIIII//iii//ii\\\\\\\\\IYIdII!I///n,/,,,,.,,\\\\\\\\\20/h//,‘,-4.0iiI11111111II111111IIiiIliiiIIt3.0IIII111111IIIIItttIi-iCDCo2•—CDe+ CDCI-I‘-I—.CDCDCl)0—S0 CDIIF-I•3 0II0 Cl) i-I CD C, ‘-‘ CDYId 2.011111Ittt11111lItI1111111tI11111Itti11111liiitiltItilttilttItIltII111111it II ii Ii StI‘I‘IviI4.0ttIItlltiItIttiIIiIItltIltttlttIll2IllIs3.0ItIt1111I111111111IItIittIIIItillY/d 2.0ttitlllttIlIIIttiiiItIiIilIIlIlItIltIltitttiittttiIttlIIII//tltltIiiiiiti\\ILI!I/!i,,iiillIII/I\\\\I////iiiIIFtliii\I,,!ii,,,,itt1.0jk_________°92.0-1.00.0ZId1.02.0Level1-RowStreamwiseInjectionF0.7000.651D0.6014.0C0.552B0.503A0.45490.40480.3553.070.30660.25650.207Yld40.15830.1092.020.05910.0101.0IIlilthullIiihlIItiiiiuuttttItIIt/IllllllIIIIiIIIIhutIIII/liiih,IuuS/1/IllII,I /SSttIFI//____——I/..——-..-,———1-RowLateralInjection-2.0-1.0°°ZId1.02-RowLateralInjectionlI2mIsCD Cl) Ci) 11 CD CDLevelcF0.7000.651D0.601C0.552B0.503A0.45490.40480.35570.30660.25650.20740.15830.10920.05910.010LevelcF0.7000.6510.6010.5524.00.6030.4540.4040.3550.3063.00.2560.2070.1580.1090.0590.0101.0-2.0-1.00.0ZId1.02.0°°ZId1.0-1.00.0Z/d1.0Chapter 7. Results II: Three-Dimensional CaseSId=41270 5 10 15XIdSId=5R=0.1R=0.2R=0.3200 5 10 15 201.00U)U)R=0.1F0.001.00C’)C’) 7.29: Predicted averaged cooling effectiveness vs. X/d for S/d = 4 and 5.Chapter 7. Results II: Three-Dimensional Case 128X/d=10.70060 --- SId=4C ---A-- SId=50.500.400.300.2000.100.00 I I I0.05 0.10 0.15 0.20 0.25 0.30 0.35Relative Mass Flow Ratio (Re)X/d=30.70---U - SId=4o 0.60c A. SId=50.500.400.300.20.:0.100.00 I I0.05 0.10 0.15 0.20 0.25 0.30 0.35Relative Mass Flow Ratio (Re)X/d=1O0.70---E--- SId=4o 0.60.A.. SId=50.500.400.30a). I I I0.05 0.10 0.15 0.20 0.25 0.30 0.35Relative Mass Flow Ratio (Re)Figure 7.30: Predicted averaged cooling effectiveness vs. Rs (S/d = 4 and 5).Chapter 7. Results II: Three-Dimensional Case 129X/d= 10.70CoCo0 0.60C0_.__.u—-.0.50 GLUa)0.308)----E--- A!d=00.20---A--- A! ---G--- Nd=30.00 I I I0.40 0.60 0.80 1.00 1.20Mass Flow Ratio (RM)XId=30.70---E--- AId=00.60 - A!d=1-.-.•-.-.-A/d=2-.-.--.-.- AId=30.50040 -----;:-0.30o :::_..__0.200.100.00 I I I0.40 0.60 0.80 1.00 1.20Mass Flow Ratio (RM)X/d=1 00.70Co Nd=00.60 - --A--- AId=1--.-•-.-.-Nd=2-.-.--.-.- A/d=30.500.400.30020•_ --A- - A0.100.00 I I0.40 0.60 0.80 1.00 1.20Mass Flow Ratio (EM)Figure 7.31: Predicted averaged cooling effectiveness vs. mass flow rate RM.Chapter 7. Results II: Three-Dimensional Case 130RM=O.41.00000.75 AId=01.00 .0 5XId10 15 20RM=l.21.000XId10 15 20Figure 7.32: Predicted averaged cooling effectiveness vs. X/d.Chapter 7. Results II: Three-Dimensional Case 131XId=10.70--—B--’- R =0400.60 M0 --—A——- R =0.88) MC‘\ •‘“ R=l.2 -,0.50 ‘ - ---_, - - ---——.-.__.___._._—.--—:-.-0.40 ‘“‘- .7 ///t /0••/0.30 -.00.20(U.2i0.100.00 I00 1.0 2.0 3.0 4.0Staggering Factor (Aid)X/d=30.70 .0 0.600 ---A---0C0.500.400.30 “‘-.\I—0.202.23010_J0.00 I I0.0 1.0 2.0 3.0 4.0Staggering Factor (Aid)XId= 100.70— R =0400.60 Ma, —--A——- RM=O.8C ‘‘‘‘ n =1.2o.so0.400C,o 0.30 L _.-—‘‘ ‘‘‘ -0.00 I I0.0 1.0 2.0 3.0 4.0Staggering Factor (Aid)Figure 7.33: Predicted averaged cooling effectiveness vs. stagger factor A/El.Chapter 8Conclusions and RecommendationsExperimental and computational work have been carried out for two-dimensional andthree-dimensional film cooling. The film cooling effectiveness, mean flow, and turbulencewere measured in the wind tunnel based on the heat-mass transfer analogy using the flameionization detector and hot-wire anemometry. The numerical models are assessed byusing the new experimental data. In the present computations, the flow and the associatedheat transfer are resolved using grid refinement. The converged refined-grid solutions areobtained efficiently by using the multi-grid method. The numerical model uses improvedturbulence models in order to take into account the non-equilibrium of the turbulence,viscosity affected near-wall turbulence, and the nonisotropic turbulence, none of which areconsidered in the traditional modeffing approach of the k-E model with a wall function.The present investigation of the simplified 2-D and 3-D film cooling models has illustratedthe deficiency of traditional turbulence modeffing approaches and has presented acomputational method suitable for real geometries in film cooling.In the 2-D computation, the M-T-S model combined with the LK treatment showsthe best agreement with the experiments. As the mass flow rate increases, theimprovement of the present model over the traditional approach using the k-E model witha wall function is significant. It is necessary to use the LK treatment with a near-wallrefined grid in order to predict accurately the flow and heat transfer, thus the effectiveness.However, it is found that within the separation bubble the Lx treatment cannot predict theincreased mixing due to the separation. The disagreement between the measured andcomputed effectiveness suggests that the extra mixing created by the unsteadiness in theactual flow near reattachment is not well represented with the present modelling.132Chapter 8. Conclusions and Recommendations 133Comparisons between experiments and computations of the 3-D case show goodagreement at low mass flow ratios. However the agreement deteriorates at higher massflow ratios due to deficiencies in the turbulence modeling. There is a need for animproved definition of the Reynolds stresses in order to reflect the increased turbulencegeneration and subsequent increased diffusion observed in the experimental results. It isimportant to note that for three-dimensional complex flows, there is a need for animproved definition of the Reynolds stresses, which can not be represented by theisotropic assumption in the k-E models. Consideration should also be given to theunsteady nature of the separation and reattachment of the flow. The algebraic turbulentstress model and the multi-time scale model may improve the modeling of turbulence in 3-D film cooling flow.Parametric studies have been carried out in order to understand the flowphenomena in the film cooling process. Film cooling through single and double rows ofholes with streamwise and spanwise injection have been presented. The superiorperformance of lateral injection, mainly near the coolant orifices has been illustrated anddiscussed. It was shown that lower jet penetration and a favorable interaction of vorticesproduced by the jets issuing from the two rows of holes are responsible for the superiorperformance of the lateral injection near the holes. For double-row injection, consistentlybetter performance of the arrangement with staggering factor A/d=3 for two rows of holesis found for the range of parameters investigated. Such behavior is observed inexperimental work on a more realistic turbine model (Gartshore et al., 1993).Some recommendations are made as follows:1) Experimental data in the near-wall and injection exit regions are needed toguide the turbulence modelling of these regions. Non-intrusive measurementsusing laser Doppler velocimetry (LDV) and particle image velocimetry (PIV)are suggested in order to obtain detailed data of mean velocity and turbulentChapter 8. Conclusions and Recommendations 134shear stresses in the 3-D flow field. Measurements of wall shear stress andlocal pressure distribution on the wall surface are also suggested.2) Computations of more complex geometries, such as injection through circularorifices and curved surfaces which are present near the leading edge of realturbine blades, should be carried out using the present numerical model withthe Navier-Stokes solver on a curvilinear coordinate system.3) The multiple-time scale model should be introduced into the 3-D simulationsto improve the prediction of effectiveness at high mass flow ratios where thenonequilibrium turbulence is significant. The algebraic Reynolds stress modelor the full Reynolds stress model should also be used to handle the turbulentmixing and transport in the flow with swirling flow, pressure gradients, andstrong streamline curvature. Also, an appropriate near-wall turbulencetreatment to represent the increased mixing occurring in the separation bubbleshould be further investigated.4) The present results show that the vertical jet film cooling arrangementconstitutes a severe test for discretization schemes as well as turbulencemodels and near-wall turbulence treatments. In both cases, inadequaciesappear to be magnified as a result of the high streamline curvature and largegradients in the flow field. These features, as well as the simple uniform flowupstream boundary condition, suggest the adoption of this flow configurationas a benchmark test for numerical methods and turbulence models.References[1] A.A. Amer and B.A. Jubran and M.A. Hamdan (1992), “Comparison of differenttwo-equation turbulence models for prediction of film cooling from two rows ofholes,” Numerical Heat Transfer, Part A, Vol. 21, pp. 143-162.[2] U.M. Ascher, H. Chin, and S. Reich (1994), “ Stabilization of DAEs and invariantmanifolds”, Numer. 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Ramsey (1970), “Film coolingfollowing injection through inclined circular tubes”, Israel Journal of Technology,Vol. 8, pp. 145-154.[21] R.J. Goldstein and E.R.G.Eckert (1974), “Effects of hole geometry and density onthree-dimensional film cooling”, mt. J. Heat Transfer, Vol. 107, pp. 595-607.[22] R.J. Goldstein (1971), “Film cooling,” Advances in Heat Transfer, pp. 32 1-379.References 137[231 R.J. Goldstein, Y. Kornblum, and E.R.G. Eckert (1982), “Film cooling effectivenesson a turbine blade”, Israel Journal of Technology, Vol. 20, pp. 193-200.[24] W. Haas, W. Rodi, and B. Schonung (1991), “The influence of density differencebetween hot and coolant gas on film cooling by a row of holes: predictions andexperiments”, ASME 91-GT-255.[25] S. Honami, T. Shizawa, A. Uchiyama, and M. Yamamoto (1991),” An experimentalstudy of film cooling in the lateral injection”, 91-Yokohama-IGTC-30.[261 W.P. Jones and B.E. Launder (1972), “The prediction of laminarization with a two-equation model of turbulence”, mt. J. Heat Mass Transfer, Vol. 15, pp. 301-314.[27] B.A. Jubran and A. Brown (1985), “Film cooling from two rows of holes inclined inthe streamwise and spanwise directions”, J. Engr. Gas Turbine Power, Vol. 107, pp.85-91.[28] B.A. Jubran (1989), “Correlation and prediction of film cooling from two rows ofholes”, J. Turbomachinery, Vol. 111, pp. 502-509.[29] B.E. Launder and D.B. Spalding (1974), “The numerical computation of turbulenceflow”, Comp. Meths. Appl. Mech. Engng., Vol. 3, pp. 263-289.[30] B.E. Launder, G.J. Reece, and W. Rodi (1975), “Progress in the development of aReynolds-stress turbulence closure,” I. Fluid Mech., vol. 68, pp. 566-570.[31] S.-W. Kim and C.-P. Chen (1989), “A multiple-time-scale turbulence model based onvariable partitioning of the turbulent kinetic energy spectrum”, Numerical HeatTransfer, PartB, Vol. l6,pp. 193-211.[32] S.-W. Kim (1991), “Calculation of divergent channel flows with a multiple-time-scaleturbulence model”, AL4A Journal, Vol. 29, No. 4, pp. 547-554.[33] S.-W. Kim and T.J. Benson (1993), “Fluid flow of a row of jets in crossflow- anumerical study”, AJAA Journal, Vol. 31, No. 5, pp. 806-811.[34] J.H. Leylek and R.D. Zerkle (1993), “Discrete-jet film cooling: A comparison ofcomputational results with experiments”, ASME 93-GT-207.References 138[35] P.M. Ligrani, S. Ciriello, and D.T. Bishop (1992), “Heat transfer, adiabaticeffectiveness, and injectant distributions downstream of a single row and twostaggered rows of compound angle film cooling holeS’, J. of Turbomachinery, Vol.114, pp. 687-700.[36] S. McCormick (1989), “Multilevel adaptive methods for partial differentialequations”, SIAM.[37] A.B. Mehendale and J.C. Han (1992), “Influence of high mainstream turbulence onleading edge film cooling heat transfer: effect of film hole spacing”, J. of Heat MassTransfer, Vol. 135, pp. 2593-2604.[38] R.J. Moffat (1986), “Turbine blade cooling,” Heat Transfer and Fluid Flow inRotating Machinery, pp. 1-26.[39] Y. Nagano and M. Tagawa (1990), “An improved k-c model for boundary layerflows”, Journal Fluids Engineering, Vol. 113, pp. 33-39.[40] P. Nowak (1991), “3-D segmented multi-grid computing code”, Technical Report,Department of Mechanical Engineering, UBC, Canada.[41] S.V. Patankar, A.K. Rastogi, and J.H. Whitelaw (1973), “The effectiveness of three-dimensional film cooling slots - II. predictions,” mt. J. Heat Mass Transfer, Vol. 16,pp. 1665-1681.[42] S.V. Patankar, D.K. Basu, and S.A. Alpay (1977), “Prediction of the three-dimensional velocity field of a deflected turbulent jet,” Journal of Fluid Engineering,pp. 758-762.[43] S.V. Patankar (1980), Numerical Heat Transfer and Fluid Flow, Hemisphere.[44] W. Rodi (1984), Turbulence Models and Their Application to Hydraulics - A Stateof the Art Review.[45] W. Rodi (1991), “Experience with two-layer models combining the k-c with a one-equation model near the wall,” AIAA-91-0216.References 139[46] M. Salcudean, Z. Abdullah, and P. Nowak (1992), “Mathematical modeling ofrecovery furnaces,” 1992 International Chemical Recovery Conference, Seattle,WA.[47] M. Salcudean, I.S. Gartshore, K. Zhang, and I. McLean (1994a), “An experimentalstudy of film cooling effectiveness near the leading edge of a turbine blade,” ASMEJournal of Turbomachinery, January.[48] M. Salcudean, LS. Gartshore, K. Zhang, and Y. Barnea (1994b), “Leading edge filmcooling of a turbine blade model through single and double row injection: Effects ofcoolant density”, to be presented at ASME IGTCE, June 1994, The Hague, Holland.[49] P. Sathyamurthy and S.V. Patankar (1990), “Prediction of film cooling with lateralinjection”, Heat Transfer in Turbulent Flow, pp. 6 1-70.[50] B. Schonung and W. Rodi (1987), “Prediction of film cooling by a row of holes witha two-dimensional boundary-layer procedure,” ASME Journal of Turbomachinery,Vol. 9, pp. 579-587.[51] D. Sidilkover and U.M. Ascher (1994), “A multigrid solver for the steady stateNavier-Stokes equations using the pressure-Poisson formulation,” Technical Report94-3, Department of Computer Science, the University of British Columbia, Canada.[52] S.P. Vanka (1986), “Block-implicit multigrid solution of Navier-Stokes equations inprimitive variables”, Journal of Computational Physics, Vol. 65, pp. 138-158.[53] A.J. White (1980), “The prediction of the Flow and Heat Transfer in the Vicinity of aJet in Crossflow”, ASME 80-WAIHT-26.[54] D.C. Wilcox (1993), Turbulence Modelling for CFD, DCW Industries, Inc., LaCanada, California.[55] C. Yap (1987), Turbulent Heat and Momentum Transfer in Recirculating andImpinging Flows, Ph.D. thesis, University of Manchester, England.[56] J.M. Zhou (1990), “A multi-grid computation of film cooling flow,” M.Sc. Thesis,University of British Columbia, Canada.References 140[57] J.M. Zhou and M. Salcudean (1990), “A multi-grid local mesh-refinement methodfor recirculating flows”, Computational Fluid Dynamics, M. Rahman ed.,[58] J.M. Thou, M. Salcudean, and I.S. Gartshore (1993a), “A numerical computation offilm cooling effectiveness,” Near-Wall Turbulent Flows, R.M.C. So, C.G. Spezialeand B.E. Launder eds., Elsevier, pp. 377-386.[59] J.M. Zhou, M. Salcudean, and I.S. Gartshore (1993b), “Prediction of film cooling bydiscrete-hole injection,” ASME 93-GT-75.[601 J.M. Zhou, M. Salcudean, and I.S. Gartshore (1994), “Application of the multiple-time scale turbulence modeling to the prediction of film cooling effectiveness,” paperaccepted by the 10th International Conference of Heat Transfer, August 1994,Brighton, England.AppendicesA. Experimental Measurement Uncertainty AnalysisThe experimental data used in the present study were obtained using FID, hot-wireanemometry, Pitot tubes, and flow meters. Uncertainties are expected in the measureddata due to changes in the process over the time interval required to make themeasurements, as well as errors introduced from the instrumentation system (Daily et al.,1984). The uncertainty analysis describes the error which may be present in the measureddata.A.1. Effectiveness MeasurementThe error in effectiveness measurement is mainly associated with the FID. In the presentmeasurements, the output of the FID voltage, E, and the concentration of propane in air,C, has a linear relationshipC=c(E—E0) (A.1)where is a calibration constant and E0 is the voltage output of pure air. From thecalibration (see Figure 3.5), the standard deviation of propane concentration S’ can beexpressed as:It(c-c)2=N(A.2)where N is the number of calibration points and Creg is the concentration from the linearregression. The standard deviation was found as 3.2 ppm.During the experiment, the propane concentration in the injection chamber, C,may change due to the instability involved in the supply of compressed air and propane.By adjusting two flowmeters of compressed air and propane, the relative fluctuation in141Appendix 142C, eC, was controlled to within ±2.5%. Since the propane concentration in the pure airCC,. =0, the uncertainty in the effectiveness r (see Equation 3.1) can be calculated by onlyconsidering S and eC(A,3)(A.4)In the present work, C is about 150 ppm and r ranges from 0 to 1, Therefore, theuncertainty in effectiveness is less than ±3.29%.A.2. Velocity MeasurementThe error in velocity measurement is mainly associated with the hot-wire anemometry. Inthe present experiments, the output of the bridge voltage, E, and the fluid velocity, U, arerelated through the following equation at the calibration condition:E2=A+BUN (A.5)where A and B are calibration constants and N=O.45. During calibration, the hot-wireprobe was calibrated against a Pitot tube manometer. The velocity was obtained from theequationpU2=p1g (A.6)where PAl is the density of alcohol and h is the vertical height of the alcohol column in themanometer. A scale of 1:10 inclination of the column was chosen in calibration. Thereading from the column gave an accuracy of ±0.5 mm. Converting this accuracy to thevertical height scale gives the error from reading öh=±0.05 mm. By differentiatingEquation A.6, the error in air velocity from the reading on the column scale ise=öh=pAlghi61g (A.7)pUAppendix 143At the room temperature, say 20°C, p=l.l64kg/rn3 and PAl =806.6 kg/rn3, this error canbe simplified to= 0.34 (A.8)The error is inversely proportional to air velocity. The velocity was calibrated over arange of 3.5-1 1.0 mIs, and the error with respect to these two limits is in a range from0.03 1 m/s to 0.097 m/s.Similar to Equation A.2, the standard deviation of air velocity regression (seeFigure 3.6) is 0.O7mIs. Changes of room temperature during experiments were less than2°C, and thus can be neglected. The velocity range in the measurements is from 2.0 mIsto 10.0 m/s, therefore the uncertaity_in the velocity is ±6%.The turbulence intensity .J(u )2 is calculated through_u)g( )2 = =‘Nwhere U is mean velocity of all sample velocities u (i = 1,. . ., N) and N is the number ofsamples. The uncertaity in the intensity can be similarly calculated as ±6% in theenvironment with turbulence intensity less than 6%.A.3. Mass Flow RatioIn the experiments, the mainstream velocity was measured using the Pitot tube manometerand the injection flow was measured by the flowmeter. The mainstream velocity was 10mis in the experiments, therefore, using Equation A.8 the uncertainty in the mainstreamvelocity is 0.034 mIs. The uncertainty in the injection velocity from the flowmeter’sreading error is ±4.7x103 rn/s. The uncertainty in the mass flow ratio, RM, can beobtained by the error propagation formulaAppendix 144—j RM--— (A.1O)In the present experiment, RM varies from 0.2 to 0.8, therefore the uncertainty in RM is=±0.41% (A.11)RMAppendix 145B. Detailed Flow and Effectiveness DistributionsB.1. Spanwise Hole Spacing Effects in Single-Row Film CoolingFigure B.1 and B.2 show the detailed surface cooling effectiveness distribution. Figure B.3and B.4 show the vector field and concentration contour in a cross flow plane at X/d=3 ofa single-row lateral injection for X/d=4 and 5.B.2. Hole Staggering Effects in Double-Row Film CoolingThe detailed surface cooling effectiveness distributions are shown in Figures B.5-B. 10.Figures B. 11-B. 16 show the vector fields and concentration contours in a cross-flow planeat X/d=3 for A/d—0, 1, 2, 3. It is found that two vortices from both front (X/d=O) andback (X/d=-3) rows merge forming a single vortex, except at A/d=2 (fully staggered)where two vortices are still visible. However, as RM increases to 1.2, two vortices mergebefore X/d=3.For different hole arrangements, the interaction between the vortices from frontand back rows results in different lateral momentum, which can be observed by theposition of the vortex. The lateral momentum in the direction of the injection pushes thecooler fluid back to the surface. Table B. 1 shows the location of the vortex centermeasured from the front hole center and the circulation of positive vorticity F over thearea of the cross section (-2<Z’d<2, 0< Y/d<4) at X/d=3, where F= if — --ds. It3Z Y)Areais found that AId=3 gives consistently higher lateral movement than other arrangements.There is no clear evidence on the effect of the circulation on the effectiveness.Appendix 146Table B. 1: Location and circulation of vortices downstream of injection (X/d=3).XId=3RM Aid Location (ZId,YId) Circulation (m2Is)0.4 0 (1.1, 0.7) 4.46x1031 (0.8, 0.6) 4.69x1032 (0.8,0.6)&(0.9,0.9) 3.39x103 (1.9, 0.9) 4.22x1030.8 0 (0.9, 0.8) 1.34x1021 (1.35, 0.8) 1.21x102 (1.3,0.8)&(1.5,1.3) 8.69x1033 (2.4, 1.0) 1.02x1021.2 0 (3.0, 1.15) 1.88x1021 (1.95, 1.1) 2.07x102 (1.7, 1.15) 1.20x103 (3.1, 1.2) 1.63x102Appendix C. Predicted Flow Fields and Effectiveness in Parametric Tests 147LevelFEDCBA98765432cw0.93750.8750.81 250.750.68750.6250.56250.50.43750.3750.31250.250.1 8750.1250.062500 — 2:5 5:0 X!d7510•°2.01.0YId0.0-1.0-°8p.V0.0 2.5 5.0 XId 7.5 10.0M12V..,/4//)1a.m....0.0 2.5 5.0 XId ‘.5 10.0Figure C.1: Predicted surface cooling effectiveness (1-Row injection, S/d = 4).Appendix C. Predicted Flow Fields and Effectiveness in Parametric TestsFigure C.2: Predicted surface cooling effectiveness (1-Row injection, S/d = 5).148RM=O.4-.,i-->-,4-_________-____ - -1—-—LevelFEDCBA98765432cw0.93750.8750.81 250.750.68750.6250.56250.50.43750.3750.31250.250.1 8750.1250.06252.01.0YId0.0-1.0-• 0.0 2.5 5.0 75X/d 10.0 12.5RM=O.8‘///“4/ 2010.0 2.5 5.0 7•5d 10.0 12.5RM=l.2cG/m(OM 2.5 5.0 7•5X!d 10.0 12.5Appendix C. Predicted Flow Fields and Effectiveness in Parametric TestsLevelF 0.700- 1 0.651D 0.601C 0.552B 0.503A 0.4549 0.4049 0.3507 0.3066 0.2565 0.2074 0.1583 0.1092 0.059I 0.010Level vF 0.700- 7 0.6510 0.601C 5.552B 0.003A 0.4549 0.4048 0.3557 0.3066 0.2565 0.2074 0.1583 5.1092 0.0591 0.015LevelF 0.7000.65l0 0.601C 0.552B 0.503A 0.4549 0.4049 0.3557 0.3066 0.2565 0.2074 0.1553 0.1092 5.0591 0.010149Figure C.3: Predicted vector fields and concentration distributions at X/d 3 (1-Rowinjection, S/cl = 4).RM=O.4 RM=O.4lm/s -4.03.0YId2.011111111?I//I(li/i4.03.0Y/d2.01.0_..0 -1.0RM=O.B1.0 [BIB2.0Y/d2.0III’’,llllttlItIIII ll1lIII),i\\\\ \ I HI!,!!,, ,,,,,..\\\ \ I I It/I/i,,‘\\\ \ il/il,,,:2.o -1.0 0.0 Z/d1.0RM=O.84.03.0Y/d2.0I-2.01.0-1.0 0.0 ZId1.0 2.0 -24.0.0 -1.0 0.0 Z/d1.0RM=l.2I huB2.!! IIIIIIIIIIHI1/!I/,,, / li/Ii’\ \ \ \\\III II Hill/i,,, / /l/Ils.\\\\\\\\\i Intl/i/u1,,,,,,1/I////,’,’,..YId-‘.‘. ‘Y/d2.01 .0:2o 1.0 0.0 Z/d 1.0 2.0Appendix C. Predicted Flow Fields and Effectiveness in Parametric Tests 150Figure C.4: Predicted vector fields and concentration distributions at X/d = 3 (1-Rowinjection, S/d = 5).RM=O.44.03.0YId2.01 mIsRM=O.4hullbill!, ,.,.. \\\iilI,!,,,Ill//f ___.....‘.‘.\\\\I I_z:j14.03.0Yld2.01.00.0-2.0 -1.0 ZId1.0RM=O.S2.04.03.0LevelF 0.700—0.651V0.6010.552B 0.503A 0.4540.404V 8 0.3557 0.366025602070.158VV 3 0.1082 0.0591 0.010LevelF 0.7000.6510.6010.5520.5030.4540.4540.3550.3060.2560.2070.1580.1050.0590.0 101 mIs-2.0 -1.0 0.0 ZId1.0RM=O.B2.0I I I I II lii II I I I I l’ I IV\IIIII I III,,,,,,II I Ill/Ill,\\\\ I I Il/iii-’ . . . - -\\\\H 111//u,,,/ II /hi1./ I1//-’0.0 Z/d 1.0RM=l.24.03.01 mIsRM=l.2Y/d\\\\\\\\lTIIIIIIl//l//T177l;7uI’N\\\\\\ ti////////,,,,_— ///Ill .\IV<LevelF 0.7005.6514.0 0.6010.552B 0.503A 0.4040.4043.0 8 0.3557 0.306o:7Y!d ,—.. 0.158°-2.0 -1.0 0.0 ZId1.0 2.0-2.0 -1.0 0.0 ZId1.0 2.0Appendix C. Predicted Flow Fields and Effectiveness in Parametric Tests 151IL) IL) U) LI) U) Lb U) U)F- LI) C%1 r— II) (‘1 F- U) C’1 F- U) C4C)F-.U)C’1U) CF..Ifl4(D€.? d d o o o a o a a a a a d ea),.\c1_______o c c 0 0 o o o 0‘- C1 C4 ,-O - Cc -dFigure C.5: Predicted surface film cooling effectiveness (A/d = 0, 1, RM = 0.4).Appendix C. Predicted Flow Fields and Effectiveness in Parametric Tests 152U) U) U) U) U) U) U) U)U) C’1 F U) C1 r- U) C’l 1- U) Clc,-U)U)C’lU) - )CCOd o o o o o o o o o d o oa)U)\-- U) XCILF d :C’)IIJ___________o o q c q c q 0ci -0d - ci -o -I IFigure C.6: Predicted surface film cooling effectiveness (A/d = 2, 3, RM = 0.4).Appendix C. Predicted Flow Fields and Effectiveness in Parametric Tests 153U) U) U) U) U) U) U) U)r- U) Cl 1- U) C.I F- U) C’l F U) (‘1O)F-..-U)C’lCD C,U)C’4(Od o o o o o o o o o o o o d ouj C.) m i. D U) () C’1 —C,,\\ \\ cD\\\__\*riNd uu \0I.J... Io c 0 0 0 0 0 0 q c-i -d -— I I — I I>- >-Figure C.7: Predicted surface film cooling effectiveness (A/d = 0, 1, RM = 0.8).Appendix C. Predicted Flow Fields and Effectiveness in Parametric Tests 154U) U) U) U) U) U) U) U)1- U) C. I. U) CN 1- U) C’l 1% U) CNC )lDe o o o o o o o o o d d o o0U,Nc\ \ \U) \ x\X.\ %LII___D -I.. .... ..C C C 0 0 C 0 C 0 Cc.i ‘i-ed - s c.i— I I - I>-Figure C.8: Predicted surface film cooling effectiveness (A/d = 2, 3, RM = 0.8).Appendix C. Predicted Flow Fields and Effectiveness in Parametric Tests 155U) U) U) U) U) U) U) U)F- U) C%1 r- U) C’l F- U) C.1 1- U) C’lc r.. — U) C1 CD C) P.. — U) CD U)CDCDCDU)U)..Qd c a a a a a a a a d a aUi C) I 0) CD P.. CD U) C) CJ —..,.,.,,‘c)NOxI X____________________o c o 0 0 0 o oc -O i- C4 C4 -O -Figure C.9: Predicted surface film cooling effectiveness (AId = 0, 1, RM 1.2).I-..>1 p C, CD C CD C C)) ii! IA/d=22.01.0YId 0.0-1.0- 0.0-1.0-2.0oq CD p I. C I-i CD C, Ct Ct I-i C, CD S C, C C CD IAId=3LevelF0.9375E0.87500.8125C0.75B0.6875A0.62590.562580.570.437560.37550.312540.2530.187520.12510.0625I. C;’O)Appendix C. Predicted Flow Fields and Effectiveness in Parametric TestsIllI!llJ!I/!’’It Ill/I/I//I,,,,,,I II hi/i/u ‘.‘—.-_ - - . . __-.‘s\\\ \ \ 1/ /1/,, .... - \ \I,,‘,_-_.._ /1—’I I / /4ltt lit I\ \ \\\\I lii IISSSS/ / ,,,,,_ \\\\\.\\h IIIlu, --. ___\\\\\\\\h IF/ / — —— — ‘ \ \ J:1.57C0.7000.6510.6010.5520.5030.4540.4040.3550.3060.2560.2070.1580.1090.0590.010Level cF 0.7000.6510 0.6010.552O 0.503P. 0.4540.4045 0.3557 0.3065 0.2565 0.2074 0.1583 0.1092 0.0591 0.010Figure C.11: Predicted vector field and(AId = 0, 1, RM = 0.4).concentration distribution at X/d = 3AId=O1 mIs4.03.0YId2.0LevelF4.03.0Yld2.0AId=ODCBA98765432-1.0 0.0ZId1.0 2.01.0(1AId=14.03.0lm/sIIIAId=1it I 1111111 III I 111111?Ill / I//i,,,,/ “‘Y/d2.04.03.0Y/d2.01.0.-2.0 -1.0 0.0Z/d1.0 2.0-2.0 -1.u 1.0Appendix C. Predicted Flow Fields and Effectiveness in Parametric TestsI I liii II II II I I II I30.IIIIIHIIIIIIIIIIIIIIIIIIIIIIIIIII I Ill I I II lIlt I I I I I I I IlIlt IIIIIII1IIIIIIIIIII1IIIIIIII I IIIll lit till, / / I III I I I I I I I Iii It It ti/Il / / / ii I,SISI\\ \ \III IItII//u,,,,,,,,, \1I titI//// —---S.”I ‘,,,____ -\\2.0158Level cF 0.7000.651D 0.601C 0.552B 0.503A 0.4549 0.4048 0.3557 0.3066 0.2565 0.2074 0.1583 0.1092 0.0591 0.010Level cF 0.7000.651D 0.601C 0.5525 0.503A 0.4549 0.4048 0.3557 0.3066 0.2565 0.2074 0.1583 0.1092 0.0591 0.010AId=21 mIsAId=24.0II IIIIIIIIIIIIIIIIIIIIIIIIIIIIII II30[IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIY/d2.01.0II IIIIII1IIIIIIIIIIIII IIIIIIIIIIIIIIIIIII I I I IIIsI I I 1111 II I I /1/ -.--I I //,__ ._—‘II,,, --S S —IIIIIIIIIIIIIIII it111111,,, IIIlI/, II,iII IIIII//,s,,I/iIf,‘I.,;4.03.0Y/d2.01.0-0.0-2.0°°ZId1.0AId=34.0:2.o-1.0 0.0ZId1.0A/d=31 mIs2.0Y/d2. a1.0Figure C.12: Predicted vector field and(A/d = 2,3, RM = 0.4).:2.o-1.0 0.0Z/d 1.0 2.0concentration distribution at X/d 3Appendix C. Predicted Flow Fields and Effectiveness in Parametric Tests 159Level cF 0.7000.651D 0.601C 0.552B 0.503A 0.4549 0.4048 0.3557 0.3066 0.2565 0.2074 0.1583 0.1092 0.0591 0.010AId=O1 mIsAId=O4.03.0Y/d2.01.0I 1111 II III 111111 III I Il//i ‘SI\\IIIItIt!!!/, S\\\\1lHlI////\\\\\t I1IIi,,,-__. //————-—--.\\1 1/11/1/7 II°°ZId1.0A/d= 1-0.0-—1.040 1.0-o.c-1.0 0.ZIdA/d= 14.03.0Y!d2.01.0—>i misIIIIIII2.0Level cF 0.700Y/dIIIIIISI,, IIIIIIIIIIIII,),,,, SS\SI.I II,,,’.IIIt -_-._‘_‘_‘.\ \ I111/i, __. S/ 1J//z_-_-__ \10.651D 0.601C 0.552B 0.503A 0.4549 0.4048 0.3557 0.3066 0.2565 0.2074 0.1583 0.1092 0.0591 0.010°ZId1.0-D.c-2.0Figure C.13: Predicted vector field and(A/d = 0, 1, RM = 0.8).-1.0°°Z/d1.0 2.0concentration distribution at X/d = 3Appendix C. Predicted Flow Fields and Effectiveness in Parametric TestsIt tttIIIItIIIIttIIIItItIIIIIIIII ItIt ttIIIIItIlIIItItIIIIIIIIIIIIIt IIII tIIItIItIttIIIItItIIIIIIIIIIIiI I It I I I I lilt I I I I III III11IIIIIIIIIIIIIIIIIIIIIIIII ItII IIIIIIIIIIIIIIIIIIIIIIIIIIIIII III,,\ i1I!lf//i — ——_.__\\\.I,,..,-,1jS--—...‘Level cF 0.7000.651D 0.601C 0.552B 0.503A 0.4549 0.4048 0.3557 0.3066 0.2565 0.2074 0.1583 0.1092 0.0591 0.010Figure C.14: Predicted vector field and(Aid = 2,3, RM = 0.8).concentration distribution at X/d 3Nd=24.03.0YId2.0160Level cNd=2 F 0.7000.651ii.i,S D 0.601A0 C 0.552B 0.503A 0.4549 0.4048 0.3553.0 7 0.3066 0.2565 0.207YId 4 0.1583 0.1092.0 2 0.0591 0.0100•2O-1.0 0.0ZId2.0AId=31 mIs4.0-1.0 0.0ZId1.0 2.0AId=34.03.0Y/d2.01.0till I I till I I I ilIlIttIii\itiilllIlI 1111111, i,,,,,,S S S \\\ I III I I I I 1/1/! / /\\\\III Ill!?,,,, ,,.....\\\\\1 Ill/ui,, 2.0\\\\-‘ U/,///”7_ ‘‘.‘\\l I //‘3.0Y/d-0.0-2.01.0°°ZId 1.011’-1.0 0.0Z/d1.0 2.0Appendix C. Predicted Flow Fields and Effectiveness in Parametric Teststill, Ill,, It,,,,,,\\\\i it IIllI,,,’,,,.,....-- -—---.--‘ \\\ \ ‘ I / I / / /It Iii,\ \ \ I I / / / ,,‘—__///‘1/L161Level CF 0.7005 0.651D 0.601C 0.552B 0.503A 0.4549 0.4048 0.3557 0.3066 0.2565 0.2074 0.1583 0.1092 0.0591 0.010Level cA/d= 1 F 0.7000.651D 0.601C 0.552B 0.503A 0.4549 0.4048 0.3557 0.3066 0.2565 02074 0.1563 0.1092 0.0591 0.010AId=Oliii1 mIsAId=O4.03.0YId2.04.03.0Yld2.01.0—i:2.O -1.0°°Z/d1.0 2.0-0.c-2.0 -1.0°°Z/d1.0 2.0AId=1fm/s4.03.0YId2.01.0A AlII I\ l’lllIIliiII IIII,,,,,tt,,I,,,I,It\ ‘I tttI Ill/I /\ \ \IItII/// /‘-, \ \\IIt!I/i,N\\\ tti/i,4.03.0YId2.01.0-2.0 -1.0°°Z/d1.0Figure C.15: Predicted vector field and(A/d=O,1, RM 1.2).:2.0 1.0 0.0 Z/d1.0 2.0concentration distribution at X/d = 3Appendix C. Predicted Flow Fields and Effectiveness in Parametric TestsIt I ItlItlIttI I 1111111 ItlItIttIt11111111111111 IltIll I tittIttill III llIIlIItIIIttIIItIlIIItIIIII III3.0 I I I I r I I HI I I I I I I IIII IIIIIIIIII 11111111 ItItIltIll IllII IIIIIIIIIIIIIIIIIIIIIIIIIIIII/t I IIIIII IltItIlIflIlt IIItIIIt III\\\/lIt I I1, - ..\\\‘/H1 I ii‘‘-‘‘.-..... _\\\\It I i,_\\\U4//__.zz I—, I I\162Level cF 0.700— 0.651O 0.601C 0.552B 0.503A 0.4549 0.4048 0.3557 0.3066 0.2565 0.2074 0.1583 0.1092 0.0591 0.010Level cF 0.7000.651D 0.601C 0.5528 0.503A 0.4549 0A048 0.3557 0.3065 0.2565 0.2070.1583 0.1090.0591 0.010AId=24.0AId=2Iti misY/d2.014.03.0Y/d2.01.0a--2.0 -1.0 0.0ZId1.0AId=31 mIs4.03.0-tIlIltIll IIIIt till I/IYld2.0-:20-1.0 0.0ZId 1.0 2.0A/d=34.03.0Y/d__..‘.\\ \ I / lilt t / /,I III i ii,‘ ,.t / ///\\\ \ I / ///_/1’- I‘2.01.0_n a°°ZId 1.0Figure C.16: Predicted vector field and(A/d = 2,3, RM = 1.2).1.0 0.0ZId 1.0 2.0concentration distribution at X/d = 3Appendix 163C. 2-D Computations with the Algebraic Reynolds Stress ModelThe algebraic Reynolds stress model of Launder et al. (1975) was investigated in thepresent 2-D computations. An attempt was made in the 2D-MGFD code to use the k-equations (Equations 4.11 and 4.13) together with the algebraic expression for theReynolds stresses (Equation 4.14) to solve for uu, k, and E. During each smoothingcycle, each component of uu was calculated iteratively based on the mean flow field, k,and e. However, no converged solution was obtained. This failure to converge is due tothe fact that the algebraic equation of Equation 4.14 cannot be solved by the directiterative method. Since the algebraic Reynolds stress model is derived empirically and allthe stress components are strongly coupled through Equation 4.14, an adequate method tosolve this system of equations needs to be devised.U. 3-fl Computations with the Multiple-Time Scale ModelThe multiple-time scale model of Kim (1989) was explored within the present 3-Dcomputations. The M-T-S model was implemented into the 3D-MGFD code but noconverged solution was obtained even on a coarse grid. Based on the experience of 2-Dcomputations as stated in Section 6.2, extra iterations for the M-T-S model equations areneeded in order to obtain stable iterative convergence. However, in the 3D-MGFD code allthe variables are solved simultaneously in a coupled nature. In order to stablize theturbulence equations, an attempt should be made to decouple the turbulence equationsfrom the mean flow. Due to the limitation of time, further implementation of the 3D-MGFD code was not made.


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