The Pennsylvania State University. The Graduate School. Department of Mechanical Engineering STUDY OF GAS TURBI E BLADE CO JUGATE HEAT TRA SFER TO

The Pennsylvania State University The Graduate School Department of Mechanical Engineering STUDY OF GAS TURBI E BLADE CO JUGATE HEAT TRA SFER TO DETERMI E BLADE TEMPERATURES A Thesis in Mechanical Engineering by Mangesh A. Kane 2009 Mangesh Kane Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 2009

The thesis of Mangesh Kane was reviewed and approved* by the following: Savas Yavuzkurt Professor of Mechanical Engineering Thesis Advisor Anil Kulkarni Professor of Mechanical Engineering Karen A. Thole Professor of Mechanical Engineering Head of the Department of Mechanical and Nuclear Engineering *Signatures are on file in the Graduate School. i

Abstract Conjugate heat transfer for turbulent flow over flat plates and turbine blades have been simulated and studied using a commercial computational fluid dynamics code FLUENT. The three two-equation turbulence models, k-ε Standard, k-ε RNG and k-ε Realizable were used. Computational grids were created using a preprocessor GAMBIT. Near wall treatment has been used for each model to resolve the flow near the solid surfaces. Experimental data for flat plate and turbine blade without film cooling was used for code validation. For the baseline case of a flat plate turbulent boundary layer, all models performed relatively similar to each other and results were within 5% and 7% of the data for skin friction coefficient and Stanton numbers, respectively. For the baseline case of a turbine blade, all models performed similar except Standard k-ε model. Results are within 3% and 5% of the data for heat transfer coefficient and surface temperatures, respectively. For above mentioned cases another approach is to use an iterative method. In this approach experimental data is used to derive a boundary condition which facilitates to solve only for conduction within the solid body and still contains the effect of conjugate heat transfer. The results were compared with the experimental data as well for three turbulence models. Results were within 5% of the data for surface temperature. Furthermore Mark-II blade which has internal cooling was simulated and studied using the same code FLUENT. This case was solved using constant wall temperature approach, Conjugate approach and the iterative method. Results for all were compared with 10% and 15% for surface temperature and Stanton number. Realizable and RNG k-ε models produced good results than the standard k-ε model. The purpose of this research is to ii

show the advantages of the conjugate approach which is faster and more accurate to calculate the heat transfer coefficient and temperatures and where it should be used. An iterative CHT approach is developed to show its advantages over conjugate approach. iii

Table of Contents List of Figures...vii List of Tables..ix Nomenclature...x Acknowledgements... xiii Chapter 1... 1 Introduction... 1 1.1 Motivation... 1 Chapter 2... 6 Literature Survey... 6 2.1 Analytical Studies on Conjugate Heat Transfer... 6 2.2 Experimental and Numerical Studies on Conjugate Heat Transfer. 8 2.3 Summary of Literature Review...11 Chapter 3... 13 k-ε Models and Results of Baseline Studies... 13 3.1 Boundary Layer Equations... 13 3.2 k-ε Models... 15 3.2.1 Standard k-ε model... 16 iv

3.2.2 RNG k-ε model. 18 3.2.3 Realizable k-ε model 20 3.3 Validation of baseline studies - Flat plate BL.....21 Chapter 4... 27 Conjugate Method and Iterative Approach... 27 4.1 Boundary Conditions...27 4.2 Iterative Approach...27 4.3 Results and Discussion 29 Chapter 5... 33 Turbine Blade with No Cooling.33 5.1 Flow Facility of Experiment...33 5.2 Numerical Analysis.34 5.2.1 Computational Domain 35 5.2.2 Surface Heat Flux Boundary Conditions.. 36 5.2.3 Results and Discussion.. 37 Chapter 6... 46 Heat Transfer of Turbine Blade With Internal Cooling... 46 6.1 Experiment Facility..... 46 6.2 CFD set up for Simulations... 47 v

6.2.1 Boundary Conditions... 47 6.2.2 Computational Grid..48 6.2.3 Non Conjugate Analysis 48 6.2.4 Conjugate Analysis...49 6.2.5 Iterative Analysis... 50 6.3 Results and Discussion... 51 Chapter 7... 64 Conclusions and Suggestions for Future Studies... 64 7.1 Conclusions... 64 7.2 Suggested Future Studies... 67 Bibliography... 68 Appendix A Experimental Data in Tabulated Form... 72 Appendix B User Defined Functions Program... 78 vi

List of Figures Figure 1.1: Pratt & Whitney s F-119 turbofan engine used to power the F-22 advanced fighter (courtesy of Pratt & Whitney).4 Figure 1.2: Gas turbine power output as a fuction of turbine inlet temperature at different pressure ratios..5 Figure 3.1: Grid for flat plate baseline case. 23 Figure 3.2: Stanton number for a flat plate BL as a function of Reynolds number..... 24 Figure 3.3: Skin Friction Coefficient for flat plate BL as a function of momemtum thickness Reynolds number...25 Figure 4.1: Grid and Boundary conditions for a flat plate with finite thickness... 30 Figure 4.2: Heat Transfer Coefficient Distribution along flat plate...31 Figure 4.3: Calculated and measured temperature distribution along the flat plate...32 Figure 5.1: Schematic of corner test section of stator vane cascade... 39 Figure 5.2: Grid and Boundary conditions for turbine blade case... 40 Figure 5.3: Heat flux along the pressure surface of the blade..41 Figure 5.4: Heat flux along the suction surface of the blade....42 Figure 5.5: Heat transfer coefficient along blade surface calculated using FLUENT. 43 Figure 5.6: Temperature along blade surface...44 Figure 5.7: Stanton number along blade surface... 45 Figure 6.1: Schematic of facility instrumentation for Mark II blade.. 53 Figure 6.2: Mark II vane coordinate system.... 54 Figure 6.3: Mark II vane cooling hole locations.. 55 vii

Figure 6.4: Mark II blade boundary conditions for 2D calculation..... 56 Figure 6.5: 2-D Grid for Mark II blade simulation generated in GAMBIT.....57 Figure 6.6: Heat Transfer Coefficient along the pressure surface 58 Figure 6.7: Heat Transfer Coefficient along the suction surface. 59 Figure 6.8: Calculated Heat Transfer Coefficient along the blade surface.. 60 Figure 6.9: Temperature along the pressure surface of the blade.... 61 Figure 6.10: Temperature along the suction surface of the blade.... 62 Figure 6.11: Simulated Temperature distribution of the blade... 63 viii

List of Tables Table A-1: Skin Friction Coefficient data for Flat Plate.......72 Table A-2: Stanton number data for Flat Plate.....73 Table A-3: Mark-II vane coordinates.........74 Table A-4: Temperature along the surface for Run 42......75 Table A-5: Convective Heat Transfer Coefficient for Cooling hole......76 Table A-6: Heat Transfer Coefficient for the blade surface........77 ix

omenclature Br x = local Brun number c p = specific heat at constant pressure C = true chord length C f = skin friction coefficient C 1ϵ,C 2ϵ = constants for turbulence model C v,c µ = constants for turbulence model shown C p = pressure coefficient G k = generation of turbulent kinetic energy due to shear G b = generation of turbulent kinetic energy due to buoyancy h = heat transfer coefficient, h = q w /(T w -T inlet ) k = turbulent kinetic energy, k = 0.5(u rms 2 + v rms 2 + w rms 2 ); k = thermal conductivity of the fluid Ma = Mach number Nu = Nusselt number, Nu = h d / k f P = Vane pitch Pr = Prandtl number q = convective heat flux w q rad = radiation heat flux x

q cond = conduction heat flux Re = Reynolds number Re θ = Reynolds number based upon momentum thickness s = surface distance along vane measured from flow stagnation point St = Stanton number, t = time T w = wall temperature T inlet = inlet air temperature Tu = turbulence level based on inlet velocity unless specified T = Free stream temperature u, v, w = local mean velocity components u + = streamwise velocity along a streamline in wall coordinates U = mean velocity in incident (X) flow direction U = Free stream mean velocity in incident (X) flow direction u v = Reynolds shear stress u w = Reynolds shear stress W = mean velocity component in spanwise direction y + = normal distance from vane surface in wall coordinates Greek Letters δ = 99% boundary layer thickness δ * = displacement thickness xi

δ ij = Kronecker delta = von Karman s constant and wave number ρ = density θ = momentum thickness νˆ = ratio of viscosities τ = time lag w = wall shear stress, µ = dynamic viscosity µ t = turbulent viscosity µ eff = effective turbulent viscosity xii

Acknowledgements I would like to begin my acknowledgements by thanking my thesis advisor as well as my mentor Dr. Savas Yavuzkurt for his endless support and encouragement that he provided me throughout my Masters program. His constructive criticism of my work always showed room for improvement and needless to say, his contribution to this thesis was invaluable. I will always cherish the stimulating discussion that I had with him during the course of my research. I would also like to extend my appreciation to Dr. Anil Kulkarni for being such wonderful committee member. His suggestions also aided in improving the quality of this thesis. I would like to take this opportunity to thank Dr. Kulkarni for his constant support throughout my stay in graduate school at Penn State. Besides the academicians, I would like to thank a number of people whose support kept me going. I want to profusely thank my best friend and my wife Mugdha, who stood by me during this phase of studies. My in-laws were also a big help during this time. I would also like to thank Jennifer Houser and LaTrisha Hough for their administrational support. Many thanks to my friends Chaitanya, Piyush, Neeraj, Venkatesh, Vickey, Ankit, Kaushik, Bhaskar, Somesh for their good wishes. Last but not the least; I would like to thank my mother, father, and brother and sister in-law. I really don t have enough words to thank them. They taught me the value of hard work and provided me with enormous support throughout my life. With my fondest love and regards, I dedicate this thesis to them. xiii

Chapter 1 Introduction 1.1 Motivation To increase the power output and efficiency of the gas turbine cycle there are many modifications and improvements have taken place over the past 70 years. These gas turbines are widely used in the areas of Aircraft propulsion and power generation. Figure 1.1 shows a modern Pratt & Whitney F-119 turbofan engine that powers the F-22 fighter. This engine produces 35,000 pounds of thrust, costs on the order of $7-8 million, and has turbine rotor inlet temperatures of 1700K. The ultimate goal of the gas turbine industry is to produce a gas turbine with an increased thermal efficiency, higher power-to-weight ratios and reliability. Turbine industry is a very competitive field due to these requirements from customers. The ideal cycle for the simple gas turbine is the Joule cycle which is also called Brayton cycle [1]. Fig 1.2 shows that the power output of a gas turbine engine is a function of the turbine inlet temperature. The gas turbine industry is continuously putting effort on increasing the turbine inlet temperature to increase the turbine efficiency. But increasing inlet temperature has adverse effects on the life and reliability of the components of the gas turbine. Thus it is important to accurately predict the heat load applied to the components near the combustor for these high inlet temperatures. Inaccurate predictions can result in reduced lifetime, poor reliability, and costly redesign of components or cooling schemes. 1

Cooling of turbine vanes and blades is a technique which allows an increases turbine inlet temperature while maintain the material temperature within acceptable limits. The main types of cooling are external cooling and internal cooling. Surface exposed to hot gases are cooled by means of external cooling. The interior of the blades or vanes are cooled by internal cooling. Currently combination of convention cooling, impingement cooling and film cooling is used as the most efficient cooling concept. With the use of cooling, turbine inlet temperatures can be as high as 300 o C above the maximum tolerable temperature of alloy. A precise calculation of the thermal loads is of great importance, as a small increase in the material temperature reduces the life span of the component significantly. Due to uncertainty in the heating loads, designers are often required to overcool the components resulting in additional losses of turbine working fluid. The flow around a vane or blade airfoil is a complex combination of effects of many parameters including turbulence, surface curvature and roughness, favorable and unfavorable pressure gradients, boundary layer transition, and stagnation point flow. This complex flow structure makes the prediction of the heat loads very difficult. The task in front of the researchers is the aero-thermal design of a cooling concept for a gas turbine blade. For modern types of blades which include film cooling makes it more difficult due to introduction of many interdependent parameters. In such cases, problem can be solved through simultaneous modeling of the whole system using commercial software like FLUENT [2]. 2

Modern turbine blades have internal cooling, blade conduction, and convection over the film cooled blade. Simultaneous calculation of conduction in the solid material of the blade and convection in the working and cooling fluid is called conjugate heat transfer. In the analysis of the conjugate problem, the energy conservation equations of conduction and convection are coupled by the condition of heat flux continuity at the solid-fluid interface [3]. Computational fluid dynamics (CFD) is a design tool gaining increased use in the gas turbine industry because it has the potential to unveil the detailed understanding of the effects of different parameters on design. CFD is a predictive tool which is based on the first principles. It has reached considerable maturity during the past decade. CFD analysis allows for the design and optimization of each of the components without costly large-scale experiments where it is not possible to simulate the effects of all the parameters of an entire gas turbine. The objective of this research is to show the importance of conjugate heat transfer (CHT) in calculating the gas turbine blade surface temperatures. The experimental data from different researchers is observed and used to find a boundary condition which can be used in the numerical simulation. CFD simulations for conjugate heat transfer were performed for a comparison with experimental data to determine which model predicts the accurate results. To reduce effort and time for conjugate calculation another approach of iterative method is used to solve only for conduction in the blade material. This method is used to include data from experiments. 3

Figure 1.1 Pratt & Whitney s F-119 turbofan engine used to power the F-22 advanced fighter (courtesy of Pratt & Whitney). 4

Figure 1.2 Gas turbine power output as a function of turbine inlet temperature at different pressure ratios [4]. 5

Chapter 2 Literature Survey Numerous experimental and numerical studies on turbine blade heat transfer have been presented in the open literature by researchers. Since 1937, Gas turbine engines have been operating and cutting edge research is still going on to improve the efficiency of turbines. Researchers presented their work which ranges from examining overall cycle issues to examining particular engine components. In this chapter recent studies on conjugate heat transfer related to flat plate and turbine blades are presented. Section 2.1 will discuss the historical developments. Section 2.2 will discuss the experimental investigations of turbine blade heat transfer followed by computational investigations of turbine blade heat transfer and conjugate heat transfer. 2.1 Analytical studies on Conjugate heat transfer In early studies analytical solution for conjugate heat transfer in compressible flow over flat plate were targeted. Over 100 papers were presented before 1972 on conjugate heat transfer [5] but derived solutions were very complex and that was the main disadvantage as for complex geometries these solutions were not so much of use. Luikov then presented a solution for laminar, incompressible flow over a flat plate in 1974 [3]. He presented the solution for conjugate problem by differential energy equations and by boundary layer equations. His results for above mentioned solutions made a good agreement with the exact solution for local Brun number in the range 0 < 6

Br x < 1.5. He also made a point that for Br > 0.1 the finite thickness plate should be solved by conjugate method. In a similar study by Yu et al. [6], the different solution method was proposed. Authors presented the heat transfer rates and temperature over the entire thermo-fluid field for any Prandtl number. Solutions are presented for flat plate as well flow along wedges and a rotating cone. Authors also presented the agreement of correlation equations of the local Nusselt numbers with numerical data. Authors tabulated the regime of conjugation parameter similar to Brun number for different geometries where problem should be solved as a conjugate heat transfer problem. Yu et al. [7] presented conjugate parameters, dimensionless coordinates and temperatures for flow near vertical and horizontal flat plates. From the correlations presented in the paper, heat transfer rates and local temperatures at the interface can be calculated. Authors concluded that the analytical solutions were in agreement with numerical data which was obtained from Keller s finite difference scheme known as box method [8]. Vynnycky, M. et al. [9] showed in their investigation, the effects of the Reynolds number, Prandtl number and thermal conductivity ratio on the heat transfer properties. They solved the conjugate heat transfer problem for incompressible flow over a flat plate. In this work, authors solved two dimensional conjugate problem whereas Luikov [3] approximated this as one dimensional conduction process in flat plate. Authors concluded that the conjugate method was able to produce good results than the non-conjugate 7

method. They achieved the accuracy of 1to 2% for Nusselt number for low values of thermal conductivity whereas 5% for higher values of thermal conductivity. Mosaad [10] presented the analytical solution for a laminar flow over a flat plate with finite thickness. He used the conjugate approach and combined the energy equation for plate and fluid under the heat flux continuity at the interface. He derived the simplified expressions for Nusselt number, heat flux and temperature and showed that they all are functions of Brun number. He concluded that the critical local Brun number value is 0.15 above which problem need to be solved by conjugate approach. 2.2 Experimental and umerical studies on Conjugate heat transfer Cooling of a gas turbine blade was the most important problem which was targeted by different researchers in the past. The combination of impingement cooling, convection cooling and film cooling is now considered as the most efficient way of cooling the turbine blade. The introduction of cooling air into the interior of the blade through channels is called convection cooling. Also admission of cooling air into the mainstream which creates a film over the blade surface and avoids direct contact of hot gas with the blade is called film cooling. The studies of convection cooling and film cooling were presented by many researchers some of which are summarized below. An enormous number of publications on the topic of conjugate heat transfer can be found in open literature. But most of them are numerical findings; in fact experimental studies of turbine blade heat transfer are very limited where conduction in the blade is considered. Hence researchers created simpler geometries to validate their studies. 8

Rigby and Lepicovsky [11] presented two simple cases to validate the conjugate heat transfer code for turbomachinery applications. The first case was the laminar flow in a thin walled metal pipe. The results for wall temperatures were in excellent agreement with the experimental data. The second case was hot flow on one side of the plate and the cross cooling flow on the other side. The one dimensional heat conduction assumption in the experiment to find out heat transfer coefficients was the main reason for the discrepancies in the Nusselt number comparisons. Takahashi et al., [12] performed a conjugate heat transfer analysis on turbine blade which was cooled by round, smooth channels. For the vane internal surface they used heat flux boundary condition. But the heat flux varied in radial direction. The results of this conjugate calculation could not be compared because of unavailability of experimental data. The study involved the parameters variation to find out the effects on metal temperature distribution. Han et al., [13] performed conjugate heat transfer simulation of a hollow turbine blade with four cooling cavities. They developed their own code for the simulation. Unstructured grid was used to mesh the domain and k-ω turbulence model was used. Heat flux boundary condition was used on cooling holes instead of solving the cooling flow. 3D vane was developed by extruding the 2D vane. This work was also not compared with the experimental data because of non-availability. Hylton et al., [14] presented a report on analytical and experimental evaluation of heat transfer distribution over the turbine blade. In this research they have tested C3X and Mark II blade for heat transfer rates. The experimental conditions ensured the coincident 9

similarity in principal independent aero-thermo parameters, Reynolds number, and turbulence intensity to those with current engines. Thus they tried to avoid the shortcomings of the previous experiments and tried to collect the data over the wide range of operating conditions. Their analytical evaluation was based on the differential method with zeroth order turbulence modeling. The numerical schemes were developed to find the solution for the heat transfer problems involved in internally cooled turbine blade. The convection outside the blade surface and inside the cooling channels and the conduction in the blade material increases the complexity of the problem. Hence the industrial practice was to solve the outside convection first then correct heat transfer coefficient for film cooling and find out the heat transfer coefficients and bulk temperature over the surface and then passes this information as a boundary condition in the finite element simulation of the blade conduction. This process is a conventional approach which requires more time and the accuracy is lost in the decoupling. Bohn et al. [15] used the conjugate approach to overcome this difficulty in their work. The external surface temperature at the midspan was in good agreement with the experimental data provided by Hylton et al., [14]. Bohn et al. [16] presented a 3D simulation of the film cooled turbine blade and showed that for 3D simulation for exact determination of boundary conditions from the experiments is necessary. For 3D calculation, the boundary conditions they used were only the averaged values of pressure, temperature and flow angels in inlet and outlet. This shows that, there is a high degree of freedom in the conjugate approach. Their conjugate studies to a film cooled turbine blade predicted 8% of difference in temperatures for conjugate and decoupled conventional approach. 10

York et al., [17] presented a complete 3D conjugate heat transfer simulation on C3X turbine blade and compared the results with the data of Hylton et al., [14]. They simulated the two cases for different Mach numbers and their results showed a very good agreement with the experimental results. They used the GAMBIT to mesh the grid and used FLUENT to solve the simulation. Realizable k-ε turbulence model was used and boundary conditions were taken from the experimental report. Temperature distribution at different spans was plotted and the study showed that the conjugate heat transfer gives the better accuracy than the conventional approach. 2.3 Summary of Literature review From the above mentioned brief review of literature it can be inferred that the conjugate heat transfer approach has an importance to get accurate turbine blade temperatures. Conjugate heat transfer study needs attention on every detail at the interface of the solid and fluid. After introduction of the film cooling in the blade it becomes more difficult to solve the problem with number of variables like local heat transfer coefficient, bulk temperature, local temperature etc. In the previous studies spanwise averaged values of heat transfer coefficients were used to find out the temperature distribution along the blade surface. To be more accurate local heat transfer coefficient values corrected for variable surface temperatures need to be used and that is the aim of this research. The conjugate approach involves time consuming calculation as the number of grid points increase as both fluid and solid domain have to be taken in consideration. An 11

enormous care need to be taken while meshing the interface as the temperature gradient is high in this region and also turbulence need to be resolved accurately. So to avoid this problem a new approach was introduced where only conduction in the blade material was solved, where experimental data on heat transfer coefficient was used as the boundary condition. The objective of the research is as follows a) To determine the temperature distribution on a flat plate with a finite thickness using a full conjugate approach and a new iterative approach and compare the results with non-conjugate approach. b) To compare conjugate, non-conjugate and iterative approach for an internally cooled gas turbine blade. For flat plate cases different turbulence models (k-ε Standard, k-ε RNG and k-ε Realizable) were used but for gas turbine blade calculations realizable k-ε model was used since other studies have shown that this model worked the best. 12

Chapter 3 k-ε models and Results of Baseline Studies Researchers either use their own codes to solve turbulent boundary layer problems using finite difference techniques or they use commercial available codes. Writing a code for turbulent boundary layer is a laborious work and they don t work for complicated geometries. So it becomes practical to use existent codes wisely for one s own research. The disadvantage of using commercial code is that one does not have full access to the source code. In this study CFD simulations have been performed using the commercially available modeling package FLUENT [2]. This package allows solutions of transport equations for fluid and conduction for solid using a variety of turbulence models including Standard k-ε [18], RNG k-ε [19], and realizable k-ε [20] and many others. 3.1 Boundary Layer Equations The governing equations of velocity and temperature fields within a boundary layer for a steady, two dimensional, incompressible and turbulent flow are continuity, momentum and enthalpy equations which are as follows: Continuity: + =0 (3.1) 13

x-momentum: ρu + ρv = μ ρ u'v' (3.2) Enthalpy: ρu + ρv = μ ρ h 'v' + (3.3) U[ 1 μ ρ u 'v'] In equations (3.1) to (3.3), x-coordinate follows the surface (in the direction of flow), and y-coordinate is in the perpendicular direction to the flow. U, V, u, v are the time averaged mean velocities and fluctuating velocity components in x and y direction respectively. P is the static pressure and is assumed constant throughout the boundary layer field. h is the mean static enthalpy and h is the fluctuating static enthalpy. In deriving the above equations, conventional time averaging is employed. By specifying appropriate initial and boundary conditions, and a turbulence model for u'v' and h 'v', these equations can be solved. 14

3.2 k-ε models The additional term, ρu v for two-dimensional flows, is called the Reynolds stress. In order to be able to solve equation 3.2, a method is required to model the additional term. A transport equation for the Reynolds stresses can be developed, but in doing so, even more unknown variables are introduced in the form of triple correlations. For this reason, different techniques have been developed that model the Reynolds stresses and provide closure for the Reynolds averaged equations. One of the most widely used methods for modeling the Reynolds stress is to use the Bossinesq approach. This approach relates the Reynolds stress to the mean velocity gradients. Using this technique, the Reynolds stress can be determined from equation 3.4 below. ( ) = + + ρ µ u u u i u j 2 ρ µ u k i i j t t δij x j xi 3 xi (3.4) In equation 3.4, µ t is the eddy or turbulent viscosity and can be computed using equation 3.5 below. 2 µ t = ρc k µ ε (3.5) 15

In equation 3.5, k is the turbulent kinetic energy, while ε is the rate of dissipation. Using this method, the assumption is made that µ t is a scalar isotropic quantity, which is not the case for many types of flow. Many models exist which use this type of modeling to close the RANS equations. The standard k-ε model and its two other variants RNG k-ε model and the realizable k-ε models are used widely. 3.2.1 Standard k-ε model In this research FLUENT solver is used as a CFD tool hence the explanation given for turbulence models written below follows that given in FLUENT [2]. In the development of the k-ε model, it was assumed that the flow is fully turbulent and the effects of molecular viscosity are negligible. Standard k-ε model is a semi-empirical model which is based on model transport equation for k and ε that are given below in equations 3.6 and 3.7. ρ Dk Dt µ µ t k = + + G k + G b ρε YM (3.6) xi σ k xi D ρ ε µ µ t ε ε ( ) ρ ε 2 = + C ε G k C εg b C ε Dt xi σε xi + 1 + 3 2 k k (3.7) 16

In these two equations, G k represents the generation of turbulent kinetic energy due to the mean velocity gradients. G b is the generation of turbulent kinetic energy due to buoyancy effects. Y M is the contribution of the fluctuating dilation in compressible turbulence to the overall dissipation rate. C 1ε, C 2ε and C µ are constants. σ k and σ ε are turbulent Prandtl numbers for k and ε respectively. Due to the assumption of isotropy, the standard k-ε model fails for flows with streamwise curvature. In addition to problems associated with streamwise curvature, another potential problem with the standard k-ε model is that the production of turbulent kinetic energy is proportional to the mean rate of strain. The turbulence can have a tendency to be over-predicted in regions of high acceleration or deceleration. The turbulent production term, G k, in equations 3.6 and 3.7 can be written as shown in equation 3.8. G 2 k = µ t S (3.8) For high Reynolds number the µ eff is used instead of µ t. In equation 3.8, S is the modulus of the mean rate of strain tensor and is defined as S 2 S ij S ij (3.9) 17

In equation 3.9, the mean rate-of-strain tensor is given below in equation 3.10. Sij 1 u u i j = + 2 x j x i (3.10) 3.2.2 R G k-ε model The renormalization group (RNG) k-ε turbulence model is similar in form to the standard k-ε model; however, it was developed in a different manner. The RNG k-ε model was derived from the instantaneous Navier-Stokes equations using a mathematical technique called renormalization group methods. Similar equations for k and ε are developed, however, the equations contain additional terms. The RNG k-ε equations for k and ε are defined below in equations 3.11 and 3.12. ρ Dk Dt = x i α kµ eff k + G k + G b ρε YM (3.11) x i ρ ε 2 D ε ε = α εµ eff + C1ε ( G k + C3εG b) C2ε Dt x x ρε R i i k k (3.12) 18

The RNG method also results in a differential equation for the turbulent viscosity given below as 2 ρ k ν d 172 εµ = $. 3 ν$ 1+ Cν dν$ (3.13) where ν $ = µ eff / µ ; C ν =100 Equation 3.13 can be integrated to determine how the effective turbulent transport varies with the effective Reynolds number. In the high Reynolds number limit, equation 3.13 reduces to equation 3.5. The additional term, R, in the differential equation for the dissipation is defined below in equation 3.14. R C µ o = ρη 3 ( 1 η / η ) 2 ε 3 1+ βη k (3.14) In equation 3.14, η Sk / ε andη o = 4. 38, β=0.012. The constant C µ is a derived value and is not a result of a fit to experimental data. In regions of high rates of strain, S, the value of η will exceed η o. When this occurs, the additional term R makes a negative 19

contribution in the equation for the dissipation. The larger dissipation in turn decreases the turbulent kinetic giving lower values for the turbulent viscosity. Thus, it should be expected in regions of high rates of strain that the RNG k-ε model should give lower values for the turbulent kinetic energy production. 3.2.3 Realizable k-ε model The realizable k-ε model used in this study was developed by Shih, et al. (1995). In regions of large mean strain rate, eddy viscosity models can predict regions of negative normal stresses. The realizable k-ε model was developed to insure positive values for the normal stresses as well as insure that the Shwarz inequality is not violated. To satisfy these constraints, a variable form of C µ that is a function of the mean strain rate, turbulent kinetic energy and dissipation rate was employed. Shih, et al. [20] also used a different form of the dissipation equation that was developed from the dynamic equation of the mean-square vorticity fluctuation at large turbulent Reynolds number. The equation for the dissipation rate is given in Equation 3.15. Dε 2 t ρ = C1S C 2 C1 C3 G b Dt x µ+ +ρ ε ρ + ε ε (3.15) j σε x j k+ υε k µ ε ε ε The parameters defined in equation 3.15 are defined in Shih, et al. (1995). In this formulation, the production of dissipation is proportional to the mean rate of strain. The 20

form of the dissipation transport equation used in the realizable k-ε model is thought to provide a better representation of the spectral energy transfer. 3.3 Validation of Baseline Studies - Flat Plate BL: As initial baseline studies, simulations of flow over the flat plate were compared to experimental data by Ames and Moffat [21] and Weighardt and Tillman [29]. Iyer and Yavuzkurt in their research worked on flat plate to find out effect of high free stream turbulence on heat transfer [22]. He used commercial code TEXSTAN to simulate the flow. Since profiles generated by FLUENT code are compared with his data, the initial conditions and boundary conditions are matched with the same. To accurately resolve the region near the plate, the first grid point near the wall was located at y + =1. Near wall treatment [2] has been used. The grid set up for this problem is shown in figure 3.1. The inlet boundary condition was specified at start of the flat plate, at a location where experimental measurements could be used for the inlet boundary conditions. The outflow boundary condition was located at the end of the plate. The velocity U = 0 at the flat plate and U = U in the free stream were applied. The geometry and mesh were created in GAMBIT [23]. The resulting mesh contained 7,200 cells. After simulation y + is calculated and it is found that it is in the range of 0.9 to 1.4 which is suggested in FLUENT user s manual [2] for turbulent flows. Baseline (Tu = 1%) test cases were run with the abovementioned boundary conditions for all three k-ε models with turbulent Prandtl number (Pr t = 0.7) being constant. The results were compared with experimental data and empirical correlations. 21

Results were also compared with the results found by Iyer [22]. Figure 3.2 shows Stanton (St) number versus Reynolds number based on length of heating. Here Standard, RNG and Realizable model predictions compare reasonably well (within about 5%) with data and the correlation of Kays and Crawford [24] given below. St * Pr 0.4 = 0.03 * Re x -0.2 (3.16) Standard k-ε model did not perform well comparatively (within 10% of the data and the correlation). Standard model starts predicting well near trailing edge of the plate. RNG model starts over predicting near trailing edge. Realizable model predicts very well over the entire length within 2%. It can be seen that all three models perform better than models used in TEXSTAN. Figure 3.3 shows the plot of skin friction coefficient versus Reynolds number based on momentum thickness. The figure indicates that Standard and RNG model predicted results within 11% and the prediction matched very well for Realizable k-ε model within 3% with the data and correlation of Kays and Crawford [24]. Equation is as follows. C f / 2 = 0.0125 * Re θ -0.25 (3.17) 22

Top Symmetry Velocity Inlet U = 9 m/s Pressure Outlet T = 400 K Ti = 1% Fig 3.1: Grid for a flat plate baseline case 23

3.0E-03 St 1.0E-03 1.0E+06 Ames and Moffat data(1990) St*Pr^0.4=0.03*Rex^-0.2 (Kays and Crawford, 1987) Texstan_KYC (Iyer, 1998) Texstan_LB (Iyer,1998) Texstan_JL (Iyer, 1998) k-ε-std k-ε-rng k-ε-realizable Re x 6.0E+06 Fig 3.2: Stanton number for a flat plate BL as a function of Reynolds number 24

3.0E-03 Weighardt-Tillman data (1968) Cf/2=0.0125*Reθ^-0.25 (Kays and Crawford, 1987) k-ε-std k-ε-rng k-ε-realizable C f /2 1.0E-03 1E+03 Re θ 1E+04 Fig 3.3: Skin Friction Coefficient for a flat plate BL as a function of momentum thickness Reynolds number. 25

Chapter 4 Conjugate Method and Iterative Approach When one needs the flow and temperature field over a flat plate with a finite thickness, one needs to solve the convection over the plate and conduction in the plate simultaneously. As mentioned before, this is called conjugate method. Analytical solutions for simple cases like horizontal flat plate, vertical flat plate are available in literature. Flow over flat plate with finite thickness is solved in FLUENT using coupling of momentum and heat transfer equations. Boundary conditions used for this simulation are given in the following sections. 4.1 Boundary Conditions Bottom wall of the plate was kept at constant temperature. Two side walls of the plate are insulated. The velocity U = 0 at the plate (no slip condition) and U = U in the free stream were applied. The solid-fluid interface of the plate is coupled. Turbulence intensity and viscosity ratio were specified as 1% and 10% respectively. All three k-ε models (Standard, RNG and Realizable) were used to solve the problem for temperature along the flat plate. 26

To accurately resolve the region near the plate, the first grid point near the wall was located at y + =1 as suggested in FLUENT manual. Boundary layer grid is used to mesh the region near the wall and course grid used to mesh the free stream region. Care was taken to resolve the flow accurately. Resulting mesh contained 7,200 cells in flow domain and plate material is meshed using 600 numbers of cells. Figure 4.1 shows the mesh generated in GAMBIT. 4.2 Iterative approach The above mentioned simulations need number of iterations to attain converged temperature distribution. One has to be careful while meshing the near wall region. Boundary layer has to be resolved properly to obtain accurate results. One also needs enough boundary conditions to set the problem. In both cases temperature at bottom wall is given and side walls are insulated. The only concern was the boundary condition at the top surface of the plate. The experimental data of Ames and Moffat [21] gives Stanton number along the plate length. One can obtain heat transfer coefficient from the Stanton number from the following relation. St = ρ (4.1) Using this data we can plot heat transfer coefficient versus flat plate length. Fig 4.2 shows the h as a function of length along the flat plate. 27

As can be seen in the Figure 4.2 a third degree polynomial curve fitting was done to the data points using Microsoft Excel. In FLUENT, the User Defined Functions (UDF) provides functions written in C programming language directly to the solver. A small program written below using UDF allows us to provide the variable property as a boundary condition given in equation 4.2. Here we used heat transfer coefficient as a function of distance as a boundary condition. h = -0.051x 3 + 0.869x 2-5.805x + 43.37 (4.2) #include "udf.h" DEFINE_PROFILE(heatco_profile,h,i) { real x[2]; face_t f; begin_f_loop(f,h) { F_CENTROID(x,f,h); F_PROFILE(f,h,i)= (-0.051*x[0]*x[0]*x[0] + 0.869 * x[0]*x[0] - 5.805 * x[0] + 43.37); } end_f_loop(f,q) } The documentation about the syntax and the variable names used in UDF can be found in UDF manual [2]. Simulation was carried out for conduction into flat plate material using above mentioned boundary conditions. 28

4.3 Results and Discussion The time and effort required for this simulation is less than the conjugate approach because solving only for conduction is faster than solving for both conduction and convection. Figure 4.3 shows the plot of temperature distribution on top surface of the plate as a function of distance. In the first approach conjugate heat transfer calculations predicted the data well within 3% for Realizable k-ε model, 4 to 5% for RNG k-ε model and 8% for Standard k-ε model. Using iterative technique accuracy of 5% was achieved. In this approach conduction was solved and used UDF to apply a boundary condition to solver. The approximation of third degree polynomial shows some over prediction in results but that can be improved by more accurate curve fitting. 29

Top Symmetry Velocity Inlet U = 9 m/s T = 400 K Ti = 1% Fluid domain Pressure Outlet Interface Plate Bottom wall T=300 K Fig 4.1 Grid and Boundary Conditions for a flat plate with finite thickness 30

40 35 h (W/m 2 K) 30 25 h(x) Curve fit 20 0 2 4 6 8 x (m) Fig 4.1: Heat transfer coefficient distribution along the flat plate 31

T (K) 350 340 330 Ames & Moffat data (1990) Standard k-ε ( Conjugate) RNG k-ε ( Conjugate) Realizable k-ε ( Conjugate) Iterative Calculation 320 310 300 0 2 4 6 8 10 x (m) Fig 4.2 Calculated and measured temperature distribution along the flat plate 32

Chapter 5 Turbine blade with no cooling Radomsky [25] at university of Wisconsin had conducted an experiment on a large-scale, non film-cooled, gas turbine vane. He had examined the heat transfer on the surface of the blade which is used here for comparison. In his experiment he had found the temperature distribution along the surface of the blade. Also he had tabulated all heat transfer data containing heat transfer flux, heat transfer coefficient etc. He compared that data with TEXSTAN code and also FLUENT 5 code. He had analyzed the data for different turbulence level. In this research, his results were compared with the results which were found using FLUENT 6.3. The basic difference here is the approach of the simulation. He had solved the full simulation while we used the boundary condition which was derived from the experimental data. He had done simulations for different turbulence level and only one turbulence level which is turbulence intensity of 10% was used here. Comparison with experimental data and TEXSTAN code simulation is presented. 5.1 Flow facility of experiment [25] Brief summary of experimental facility used in University of Wisconsin before 1997 is mentioned here to show the efforts and time required to put in experiments on 33

turbine blades. To find the detail surface heat transfer and flow field measurements, stator vane test section have been scaled up. All of the experimental measurements, with the exception of the boundary layer measurements, were performed while the wind tunnel facility housing the vane test section was located in the Convective Heat Transfer Laboratory at the University of Wisconsin. In July 1999, this facility was relocated to the Experimental and Computational Convection Laboratory (ExCCL) at Virginia Tech where the boundary layer measurements were performed. Experimental measurements were performed over a range of turbulence levels representative of those exiting current gas turbine combustors. The experimental measurements included surface temperature measurements on the stator vane using thermocouples, and on the vane platform (endwall) using an infra-red (IR) camera. Detailed velocity field measurements were performed at the vane midspan and in the endwall region. The flowfield measurements were made using a two-component laser Doppler velocimeter (LDV) system. In some cases, two orientations of the LDV system were required to obtain all three components of the velocity. Hot-wire measurements were used to document the streamwise energy spectra and turbulent length scales. The schematic of experiment is shown in figure 5.1. 5.2 umerical analysis CFD simulations have been used to determine the required heat transfer data on the vane surface. A series of CFD simulations were performed to compare various turbulence models to determine which are best suited to model the highly turbulent flow 34

in the stator vane passage. The aim here is to model the stator vane boundary layer region and the region above the boundary layer. 5.2.1 Computational Domain The geometry and mesh were created in GAMBIT [23]. A hybrid-meshing scheme, available in GAMBIT was used in which quadrilateral cells were placed near the surface of the vane and tetrahedral cells were used in the freestream region. The boundary conditions used were the same as mentioned in the study of Radomsky and Thole [26]. These are described here briefly. About one chord upstream the incoming velocity is unaffected hence the inlet boundary condition was set there. This facilitates the velocity and turbulence inlet boundary conditions to be constant across the pitch. The outlet boundary location was placed at one and a half chord downstream of the trailing edge of the vane. Periodic boundary conditions were used everywhere except for the inlet, outlet, and vane surface. The boundary condition at the exit was specified as an outflow boundary condition, which shows small flow gradients. A no-slip boundary condition was imposed at the vane surface. The inlet velocity at one chord upstream was set to the uniform velocity of 5.85 m/s which is used by Radomsky in his simulation. Turbulence intensity and length scale were given in the experimental data. 120,000 cells were used for this simulation. The grid and boundary conditions are shown in figure 5.2. For all of the cases, a converged solution was typically achieved after 7500 iterations by monitoring the residuals for continuity, x-momentum, y-momentum, energy, turbulent kinetic energy and its dissipation rate. In FLUENT, the residual is computed 35

from the imbalance in the conservation equation for each of the variables. The residual is then scaled by a factor representative of the flow rate through the domain. For the continuity equation, the scaling is based on the largest residual in the continuity equation. The CFD simulation was considered converged when the all residual dropped below 1x10-6. 5.2.2 Surface Heat Flux Boundary Condition The inlet and outlet boundary conditions are mentioned before. The periodic boundary conditions are used to perform cascade simulation. Surface heat flux along the blade surface was obtained from the experimental data. The plot of heat flux for pressure surface and the suction surface is plotted as a function of distance along the surface. Figure 5.2 and figure 5.3 show the heat flux as a function of blade surface distance for pressure and suction surfaces respectively. Curve fitting is done using a sixth degree polynomial. MS Excel was used for curve fitting. The higher order curve fitting can be done using MATLAB. Heat flux as a function of distance for a pressure surface and suction surface are given as equation 5.3 and 5.4 respectively. q = 36622x 6 + 44251x 5 + 20331x 4 + 45094x 3 + 5295.x 2 + 393.9x + 759.6 (5.1) q = 30380x 6-50071x 5 + 24884x 4-1595.x 3-1425.x 2 + 220.3x + 756.3 (5.2) After getting these equations for Heat flux as a function of distance, the UDF was written for the same. The User Defined Functions are very useful in FLUENT to specify variable boundary conditions. 36

5.2.3 Results and Discussion The CFD simulations have been performed in FLUENT using three turbulence models k-ε Standard [18], k-ε RNG [19], k-ε Realizable [20]. Equation 5.3 is the fundamental heat transfer equation relating the heat flux, heat transfer coefficient and temperature difference for convection. q(x) = h(x) * ( T(x) - T ) (5.3) q(x) was used as a boundary condition and both h(x) and T(x) were unknown. Both heat transfer coefficient and temperature were iteratively solved till there values did not change. Three cases were run one for each turbulence model and then results were plotted for heat transfer coefficient along the blade surface, Stanton number along the blade surface and the temperature distribution along the surface of the blade and compared with the experimental data. Heat transfer coefficient along the blade surface is shown in figure 5.4. From this plot we can see that all three models reasonably match with the experimental data. All these models under predict the results and the Standard k-ε model under predicts the most. Realizable k-ε model matched very well with the experimental data within 5%. RNG model and Standard k-ε model predicted the heat transfer coefficient within 8% and 14% respectively. All three models predicted good results on pressure surface. On suction surface deviation near the leading edge is about 16% and at the trailing edge it is around 20% by Standard k-ε model. Temperature distribution along the blade surface is shown in 37

figure 5.5. From this plot we can see that all three models reasonably match with the experimental data. All these models over predict the results and the Standard k-ε model predicted the results within 15% of experimental data. Realizable k-ε model matched very well with the experimental data within 4%. The ultimate goal is to find out the surface temperature distribution as accurately as possible. Stanton number as a function of the blade surface is shown in figure 5.6. The deviation in the results near the end of suction and pressure surface is because of the approximation of polynomial curve for heat flux. Realizable k-ε model matched very well with the experimental data within 4%. RNG model and Standard k-ε model predicted within 6% and 9% respectively. All three models predicted good results on pressure surface. On suction surface deviation near the trailing edge it is around 16% by Standard k-ε model. The important result here is that the technique of using UDF for boundary condition was worked. This will help to proceed to the case where cooled turbine blade is simulated. Next chapter deals with the case where turbine blade is internally cooled and UDF is used to hook up the boundary condition to the solver and flow simulation will be turned off. Only conduction is solved in the blade material and results are discussed. 38

Active turbulence generator grid Window 17.8b Inlet measurement Y, Z, X,U Plexiglass wall Trip wire Boundary layer bleed 88b 4.6C 16b Flow removal to downstream of test Figure 5.1 Schematic of corner test section containing vane cascade [25]. 39

Velocity Inlet U = 5.25 m/s T = 298 K Periodic Pressure Outlet Periodic Boundary Layer Fig 5.2: Grid and boundary conditions for turbine blade case. 40

765.0 760.0 Heat Flux Curve fit 755.0 750.0 745.0 740.0 q" (w/m 2 ) 735.0-0.500-0.400-0.300-0.200-0.100 0.000 x(m) 730.0 Fig 5.3 Heat flux along the pressure surface of the blade 41

q" (w/m 2 ) 795.0 790.0 785.0 780.0 775.0 770.0 765.0 760.0 755.0 750.0 Heat Flux Curve fit 745.0 0.000 0.200 0.400 0.600 0.800 x (m) Fig 5.4 Heat flux along the suction surface of the blade 42

90 70 Heat Transfer coefficient_experiment Realizable k-ε RNG k-ε Standard k-ε h (W/m 2 K) 50 30 Pressure Surface Suction Surface 10-1.000-0.500 0.000 0.500 1.000 1.500 s/c Fig 5.5 heat transfer coefficient along blade surface calculated using FLUENT 43

320 314 Temperature_Experimental data Realizable k-ε RNG k-ε Standard k-ε T (K) 308 Pressure Surface Suction Surface 302-1.000-0.500 0.000 0.500 1.000 1.500 s/c Fig 5.6 Temperature along blade surface 44

0.020 St 0.015 0.010 St_Experiment Realizable k-ε RNG k-ε Standard k-ε 0.005 Pressure Surface Suction Surface 0.000-1.000-0.500 0.000 0.500 1.000 1.500 s/c Fig 5.7 Stanton number along blade surface 45

Chapter 6 Heat Transfer of Turbine Blade with Internal Cooling 6.1 Experimental Analysis The experimental investigation on Mark-II blade was carried out at DDA Aerothermodynamic Cascade Facility (ACF). A research in high temperature turbine component modeling with internal cooling techniques, aerodynamics was the objective of the experiments. Dynamic similarity in free stream and boundary layer Reynolds number were employed. Detailed information about the test facility and instrumentation is described in Hylton et al., [14]. Figure 6.1 shows the schematic of the facility instrumentation. Figure 6.2 shows the cascade coordinate systems used for Mark-II blade. The vane coordinates are given in table A-3. This blade has 10 internal cooling holes and the configuration of the holes is shown in figure 6.3. Each cooling hole on test vane was supplied with air from a separate line. To establish periodicity, pressure taps were provided at inlet and exit of the cascade. Thermocouples were used to major the surface temperature of the blade. The vane is made of ASTM 310 stainless steel. The thermal conductivity of this material is low which attracted researchers to use this material for the vane. Experimental data on heat transfer coefficients, blade temperatures were obtained for the blade over the range of operating conditions. These conditions include free stream condition, exit Reynolds number, and turbulence intensity at inlet. All tests are carried 46

out at nominal temperature of 811K. For numerical simulations run no 42 was used from the test to select the initial and boundary conditions. 6.2 CFD Set up for Simulations An incompressible, 2D, numerical simulation for Mark-II blade was done using FLUENT code. The simulation is prepared using the preprocessor GAMBIT. The schematic of geometrical configuration is shown in figure 6.4. The unstructured grid is used for the domain and boundary layer grid was used to resolve the near wall region. Also boundary layer grid was used near each cooling hole. The first cell center in the flow region has a dimensionless wall distance of y + = 0.5. This allows the accurate determination of the local heat flux for the coupling of solid and fluid regions. The problem is solved by two approaches, one is constant wall temperature and other is conjugate in which both solid and fluid regions were solved simultaneously. The iterative approach was finally applied to solve conduction in the blade. The two-equation Realizable k-ε model was used to solve the simulation. 6.2.1 Boundary Conditions Periodic boundary conditions were employed to replicate the multiple blade passages in the experiment, and therefore only one vane was included in the domain. Other parameters were directly taken from the experiment. The vane has a true chord of approximately 136 mm. The vane is created in x-y plane using the data points provided in 47

table A-3. The velocity inlet condition of 106 m/s was applied at inlet which was calculated from inlet Mach number 0.19 mentioned in the experiment. The total pressure at the inlet was 334 kpa and the temperature at the inlet was specified as 788K. The static pressure of 325.68 kpa was calculated from these conditions [27]. Turbulence intensity of 6.5% was applied at the inlet. At the outlet static pressure of 167 kpa was applied to maintain the inlet to outlet pressure ratio of 0.5. 6.2.2 Computational Grid The grid was generated using FLUENT s pre-processor GAMBIT. The grid consisted of 111,815 unstructured quadrilateral cells. The number of grid cells in the boundary layer is important to capture the both fluid and thermal boundary layer. The boundary layer consisting of 6 cell rows was also generated to resolve the turbulence near the blade surface. The first row height is 5 x 10-4 m and the growth factor used is 1.120. From this we achieved the boundary layer of 0.0041 m of total depth. The gradual change in the size of cells from fluid zone to solid zone is also important to capture the accurate conjugate heat transfer from the blade surface. Near the cooling holes boundary layer grid was generated to capture the correct heat transfer from cooling air to blade. 6.2.3 on-conjugate Analysis For non-conjugate analysis constant wall temperature at the pressure and suction surface of the blade was applied. This analysis is done to compare the conjugate analysis 48

and its advantages over the constant wall temperature presumption which are discussed further in this chapter. Temperatures over the pressure and suction surface are listed in table A-4. Averaging the temperature over each surface was realistic approach and used as a constant wall temperature over the blade surface. For pressure surface 522 K and for suction surface 539 K were used. The convective heat transfer coefficients for all ten cooling holes were supplied which are taken from experimental values given in table A-5. The simulation was carried out in FLUENT using realizable k-ε turbulence model with enhanced wall treatment. 6.2.4 Conjugate Analysis Conjugate analysis was done using the same boundary conditions mentioned in 6.2.1. Only difference in non-conjugate analysis and conjugate analysis is the interface of the blade and the outer flow domain was coupled. The simulation solves the conduction in the blade and convection above the blade surface simultaneously. In the analysis of the blade temperatures, thermal conductivity was specified as temperature dependent. The results were compared with the non-conjugate simulation. For this case also the realizable k-ε turbulence model with enhanced wall treatment was used in the flow domain. The CFD simulation was considered converged when all residuals in conservation equations dropped below 1x10-6 for both cases. 49

6.2.5 Iterative Analysis In this case, only conduction heat transfer within the blade was considered. The boundary conditions at the blade surfaces and cooling holes were used. From the experimental data on the heat transfer coefficient tabulated in table A-6 over the blade surface was plotted in MATLAB. The curve fitting of 10 th order polynomial is used to fit the curve and equation was obtained for the heat transfer along the pressure and suction surfaces. Figure 6.5 shows the heat transfer distribution along the pressure surface and the curve fitting. The equation 6.1 obtained is given as follows. h = 5.4513e15 * x 10-2.4961e15 * x 9 + 4.1125e14 * x 8-2.1497e13 * x 7-1.7935e12 * x 6 + 3.2103e11 * x 5-1.9097e10 * x 4 + 5.4964e8 * x 3-6.9926e6 * x 2 + 11574 * x + 669.97 (6.1) Figure 6.6 shows the heat transfer coefficient along the suction surface and the curve fitting. The equation 6.2 obtained from curve fitting is as below. h = 6.939e14 * x 10-7.7899e14 * x 9 + 3.44e14 * x 8-8.0649e13 * x 7 + 1.1119e13 * x 6-9.311e11 * x 5 + 4.668e10 * x 4-1.318e9 * x 3 + 1.8729e7 * x 2-1.167e5 * x + 890.71 (6.2) 50

The boundary condition of heat transfer coefficient on the pressure and suction surfaces was supplied to FLUENT using the UDF program which is given in appendix. The value supplied by UDF is assigned to the midpoint of the face cell. Hence fine the grid more is the accuracy of the calculation. 6.3 Results and Discussion Figure 6.7 shows the heat transfer distribution along the blade surface with both non-conjugate and conjugate results. Conjugate results are 13% higher compared to the data. But near the leading edge error is about the 40% and it can be because of acceleration after stagnation point. Relaminarization near the leading edge also contributed to this. The main goal here is to compare the conjugate and non-conjugate simulations hence this deviation is not considered. Near the trailing edge the experimental data is not available. The non-conjugate simulation results are under predicted in the range of 20% accuracy. Near the leading edge the error is around 40% and for the downstream of suction edge the prediction is under predicted by around 25%. The sudden increase in the heat transfer coefficient was found due to sudden change in profile disturbing the boundary layer. For high speed flows the flow along the suction side of the turbine blade has significant influence on the heat transfer profile along that surface. From the results it can be seen that the more realistic results are obtained by conjugate analysis where both conduction and convection are solved simultaneously. Figure 6.8 and figure 6.9 show the temperature distribution along pressure and suction surfaces of the blade respectively. Conjugate results and iterative conjugate method results were compared with the experimental data. The trend was predicted well 51

with the Realizable k-ε model. The average error on the pressure side is much less than the error on the suction surface which is around 14%. Discrepancies up to 10% are generally accepted when simulations are done by k-ε turbulence model. The error near the leading edge is large as 20%. The surface temperatures are over predicted by conjugate simulation. The predicted cross sectional temperature distribution through the blade surface is shown in figure 6.10. The maximum temperature was observed near the trailing edge because of the recirculation occurring at that zone. The boundary layer breaks up there due to abrupt ending of the blade. The lowest temperature was observed in between cooling holes two and three. These holes are farthest from the hot air and also they have highest heat transfer coefficients. In iterative conjugate analysis experimental heat transfer coefficient is used as a boundary condition and hence the results were much more accurate than the conjugate results. The surface temperatures were over predicted near the leading edge by 7% and under predicted over the surface by 4%. Near the trailing edge again the results were over predicted by 5%. The average error with the iterative method is around 5%. 52

Fig 6.1 Schematic of facility instrumentation for Mark II blade [14]. 53

Fig 6.2 Mark II vane coordinate system [14]. 54

Fig 6.3 Mark II vane cooling holes locations [14]. 55

Fluid Zone Solid wall Fig 6.4: Mark II blade boundary conditions for 2D calculation. 56

Fig 6.5: 2-D Grid for Mark II blade simulation generated in GAMBIT 57

Fig 6.6: Heat transfer coefficient along the pressure surface [14].. 58

Fig 6.7: Heat transfer coefficient along the suction surface [14]. 59

1.3 1.1 0.9 Hylton et al (1983) Conjugate- k- ε- Realizable Non-conjugate- k- ε- Realizable h ref = 1135 W/m 2 K h/ h ref 0.7 0.5 0.3 0.1 Pressure Side Suction Side -1-0.5 0 0.5 1 x/l Fig 6.8 Calculated Heat transfer coefficient along the blade surface 60

1.00 0.90 0.80 Hylton et al., 1983 k-ε Realizable Iterative Method T / T ref 0.70 0.60 0.50 0.40 T ref = 811K 0.30 0.00 0.05 0.10 0.15 s/c Fig 6.9 Temperature along the pressure surface of the blade 61

1.0 0.9 0.8 Hylton et al., 1983 k-ε Realizable Iterative Method T / T ref 0.7 0.6 0.5 0.4 T ref = 811K 0.3 0.00 0.05 0.10 0.15 s/c Fig 6.10 Temperature distribution along the suction surface of the blade 62

Fig 6.11: Simulated temperature distribution of the blade 63