Application of Conjugate Heat Transfer (Cht) Methodology for Computation of Heat Transfer on a Turbine Blade

APPLICATION OF CONJUGATE HEAT TRANSFER (CHT) METHODOLOGY FOR COMPUTATION OF HEAT TRANSFER ON A TURBINE BLADE

A Thesis

Presented in Partial Fulﬁllment of the Requirements for the Degree Master of Science in the

Graduate School of The Ohio State University

Jatin Gupta, B.Tech.

*****

The Ohio State University

2009

Master’s Examination Committee: Approved by

Prof. Seppo Korpela, Adviser Dr. Ali A. Ameri, Co-adviser Adviser Prof. Sandip Mazumder Graduate Program in Mechanical Engineering c Copyright by

Jatin Gupta

2009 ABSTRACT

The conventional thermal design method of cooled turbine blade consists of independent decoupled analysis of the blade external ﬂow, the blade internal ﬂow and the analysis of heat conduction in the blade itself. To make such a calculation, proper interface conditions need to be applied.

In the absence of film cooling, a proper treatment would require computation of heat transfer coefficient by computing heat flux for an assumed wall temperature.

This yields an invariant h to the wall and free stream temperature that is consistent with the coupled calculation. When film-cooling is involved the decoupled method does not yield a heat transfer coefficient that is invariant and a coupled analysis becomes necessary. Fast computers have made possible this analysis of coupling the conduction to both the internal and external flow. Such a single coupled numerical simulation is known as conjugate heat transfer simulation.

The conjugate heat transfer methodology has been employed to predict the flow and thermal properties, including the metal temperature, of a particular NASA turbine vane. The three-dimensional turbine vane is subjected to hot mainstream flow and is cooled with air flowing radially through ten cooling channels. The passage flow and heat transfer, internal coolant flow and heat transfer, and conduction within the metal are solved simultaneously using conjugate heat transfer methodology. Unlike

ii the decoupled approach, the conjugate simulation does not require thermal boundary

conditions on the turbine walls.

It is believed that turbulence has a large inﬂuence on heat transfer and hence in the

present study, a modified k-ω model by Wilcox [26] was used for the external flow and it was compared to Wilcox’s standard k-ω model [25] to see if the new model provided any improvement in predicting the heat transfer on the blade surface. Results here have shown that, for the present geometry, the standard model [25] is still much better in predicting temperature and hence the heat transfer over the blade surface than the modified model [26].

A thermal barrier coating (TBC) “system” is used to thermally protect turbine engine blades and vanes from the hot gases in gas turbine engines. The overall results from this study, including turbulence modeling and the use of TBC, are encouraging and indicate that conjugate heat transfer simulation with proper turbulence closure is a viable tool in turbine heat transfer analysis and cooling design.

iii This is dedicated to all those who supported me in this eﬀort. . .

iv Acknowledgements

Current work is the outcome of the rewarding research in my graduate study and

I would like to express my gratitude to my mentors, friends, and family who helped me with their advice and support during these years. I am truly and extremely grateful to my mentor, Dr. Ali A. Ameri, without whom the success of this study would be impossible. His consistent presence in the times of need and his guidance and advice have helped me through the obstacles, improved the quality of my work, and trained me to be a successful researcher. I was privileged to have a continuous learning experience during my work under his supervision.

I am extremely grateful to my advisor Prof. Seppo Korpela for his valuable support throughout the year. He provided an excellent research environment and great freedom to carry out research. I’ll always remember the intellectual discussions

I had with him from time to time and his great foresight in world oil issues. Despite his position, his priorities have always been to guide his students and travel an extra mile if that means giving practical experience to his students.

I would also want to thank Dr. David L. Rigby, at NASA Glenn, who has been a great inspiration giving valuable comments, especially with setting up the test case and generating the required ﬁles for diﬀerent cases. I must acknowledge his knowledge and experience in this area which has saved me many times from getting at a problem.

Finally, I would like to express my sincere gratitude to all those people who have been constantly around me, have supported me and helped me when I needed them.

To them, I dedicate this work.

v TABLE OF CONTENTS

Page

Abstract ...... ii

Dedication ...... iv

List of Tables ...... viii

List of Figures ...... ix

Chapters:

1. Introduction ...... 1

1.1 Gas Turbines ...... 1 1.2 Increasing eﬃciency through cooling ...... 3 1.3 Flow phenomena inside turbine blades ...... 6 1.4 Background ...... 6 1.5 Thermal Barrier Coating ...... 10 1.6 Project Description ...... 13

2. Fluid Motion and Heat Transfer ...... 15

2.1 Governing Equations ...... 15 2.1.1 Continuity ...... 15 2.1.2 Momentum ...... 16 2.1.3 Energy ...... 16 2.2 Computational Fluid Dynamics ...... 18 2.2.1 Governing Equations ...... 19 2.2.2 Turbulence Models ...... 20 2.3 Mathematical formulation ...... 26 2.3.1 Modeling for h and bulk temperature boundary conditions . 26 2.3.2 TBC Modeling ...... 27

vi 3. Code Description ...... 29

3.1 CHT Calculation using Glenn-HT ...... 30

4. Analysis ...... 31

4.1 Model Geometry ...... 31 4.2 Numerical Methodology ...... 33

5. Results and Discussion ...... 38

6. Summary and Conclusions ...... 48

7. Future Work ...... 50

Bibliography ...... 52

vii LIST OF TABLES

Table Page

4.1 Geometry Parameters for C3X vane ...... 33

4.2 Cooling Channel’s Property ...... 35

4.3 Hot gas path ﬂow details ...... 35

viii LIST OF FIGURES

Figure Page

1.1 RM12, Gas Turbine ...... 2

1.2 Turbine Entry Temperature,(Copyright Rolls Royce plc) ...... 4

1.3 A schematic illustration of a modern thermal barrier coating system consisting of a thermally insulating thermal barrier coating, a thermally grown oxide (TGO) and an aluminum rich bond coat. The temperature gradient during engine operation is overlaid [20]...... 11

2.1 Boundary conditions across solid-ﬂuid interface ...... 27

2.2 Boundary conditions across thermal barrier coating ...... 28

4.1 C3X Vane with Cooling Holes ...... 32

4.2 C3X Grid ...... 34

5.1 Experimental and predicted data of P/P0 ...... 39

5.2 Contours of Mach number at midspan ...... 40

5.3 Experimental and predicted data for Twall/T0 under diﬀerent b.c. . . . 42

5.4 Experimental and predicted data for Twall/T0 under diﬀerent turbulence models ...... 43

5.5 Experimental and predicted data for normalised heat transfer coeﬃ- cient under diﬀerent turbulence models ...... 44

5.6 Static Temperature at 50% of blade’s height ...... 45

ix 5.7 Wall temperature on the pressure side and the suction side of C3X blade 46

5.8 Eﬀectiveness of TBC ...... 47

5.9 Temperature variation across TBC ...... 47

x CHAPTER 1

INTRODUCTION

1.1 Gas Turbines

Gas turbine engine is one of the excellent means of producing either thrust or power. Its main advantages are: exceptional reliability, high thrust-to-weight ratio, and relative freedom from vibration. The work from a gas-turbine engine may be provided either as torque in a shaft rotating at a certain speed, or as thrust in a jet of certain velocity. A gas-turbine consists of the following main parts: an inlet, a compressor, a combustor, a turbine and an exhaust, as shown in the cutaway in

Figure 1.1. The inlet section may involve ﬁlters, valves and other arrangements to ensure a high quality of the ﬂow.

The pressure of the air is increased in the compressor, usually consisting of several stages. Depending on the design of the engine the compressor may be of radial type in which the air enters axially but exits radially, or the more common, an axial compressor may be used, in which the ﬂow enters and leaves axially. The rotation of the compressor blades increases the velocity of the air which then undergoes a diﬀusion process in which its dynamic pressure (velocity) is converted to static pressure.

The compressor and turbine, and the shaft connecting them, is called a spool in which the output power of the turbine exceeds the power requirement needed to drive

1 Figure 1.1: RM12, Gas Turbine

the compressor. The compressed air enters a combustor in which fuel is burned and the temperature of the combustion products leaving the combustor is a key variable in the design of the engine. The design and operation of these burners are vital to yield low emissions in a high eﬃciency engine.

The highly energetic gas from the combustor is expanded through a turbine, which drives the compressor in the front of the engine. After the turbine the gas still contains a signiﬁcant amount of energy which can be extracted in various forms. In aircrafts the surplus energy is transformed into a high velocity jet in the nozzle which is the driving force that propels the vehicle through the air. The jet velocity and hence thrust could be further increased, through re-heating the gas in an afterburner. This is common in high performance aircraft, especially for military applications. For stationary, power generating gas turbines, the extra energy is converted into shaft-power in a power turbine [1].

2 1.2 Increasing eﬃciency through cooling

Increased environmental awareness and rising fuel costs has been forcing designers

to strive towards enhanced eﬃciencies for all propulsion systems.

For gas turbine applications, especially in aircraft, not only the speciﬁc fuel con-

sumption (SFC) is of importance, but also the speciﬁc work output. The former is

equivalent to the inverse of the eﬃciency while the latter is a measure of the compact-

ness of the power plant, i.e. the eﬀectiveness. Gas turbines operate on a standard

Brayton cycle, for which the eﬃciency and speciﬁc work output are given as [1]

1 w T3 1 (γ−1)/γ η = 1 − (γ−1)/γ = (γ−1)/γ + 1 − r (1.1) r cpT1 T1 r

where r is the compression ratio, T1 is the inlet temperature and T3 is the turbine

entry temperature (TET). The optimum pressure ratio, when ideal gas approximation

is used, is such that the eﬃciency is given by

sT η = 1 − 1 (1.2) T3

Increasing T3 yields a direct improvement in the efficiency, η. The performance of practical cycles are however lower, owing to pressure and mass flow losses, friction, components efficiency, non-ideal fluids etc. When these losses are taken into account, the efficiency of the simple gas turbine cycle becomes dependent, not only on the temperature ratio, but also the compressor pressure ratio. Thus the gas turbine industries are trying to reach both higher turbine temperatures as well as increased pressure ratio, to improve the efficiency and effectiveness of future engines.

3 Even though exotic materials are used for the most thermally stressed environ- ments, these have in the past, been unable to withstand the demanded TET without deterioration in the harsh environment. High strength material, such as nickel and cobalt based super alloys, (e.g. Inco 738 and Rene 220), will all weaken from increased temperatures, and since the loads in a rotating turbine are extremely high, the structure fails if no counter-measures are taken. Cooling technology is another alternative and even more eﬀective way to protect the blades from the damage high gas temperatures cause. This is done by introduction of relatively cool gas from the compressor in well selected places in the turbine. This practice was already in use during the Second World War and it extends the engine endurance. Figure 1.2 shows the increases in TET through the introduction of diﬀerent cooling techniques.

Figure 1.2: Turbine Entry Temperature,(Copyright Rolls Royce plc)

4 The highest temperatures loads are found at the exit of the combustor, and in the first turbine stage. Comprehensive cooling systems are thus needed for the inlet guide vanes (IGV). These vanes employ both external cooling (film cooling), and internal cooling (convection- and impingement-cooling). The vanes are perforated by a number of small holes, through which compressed air is ejected. If correctly designed, these holes will supply a cool protective air-film covering the vanes.

This technique is called film cooling. The same type of cooling is applied to the platforms, along the tip (shroud) and root (hub) of the vane. Internally, ribs positioned orthogonally to the flow, are introduced in the ducts, which makes the flow repeatedly separate and re-attach with an increase in turbulence level and a con- sequently enhancement of the heat transfer. These rib-roughened serpentine ducts completely fill out the inside of a vane. Finally the cooling air can be guided verti- cally towards some specifically hot regions for effective cooling. The latter is called impingement cooling.

It should be noted that the available pressure difference between the internal cooling air and the external main gas flow, is severely limited. Hence there is a restriction to the number of turns, ribs and other pressure reducing features that can be employed within a given passage. There is also a construction limit to how complex the interior could be made, while still maintaining productivity and being cost effective. Contemporary gas turbines may use as much as 15% of the total mass

flow for cooling air. Even though there is significant advantage to increasing TET, the use of cooling air for achieving this has some drawbacks: 1) the addition of cool air into the main stream reduces the work output from the turbines; 2) protective films along the vanes complicates the aero-thermal design of the blades, as the momentum

5 and blockage effect introduced by the cooling air changes gas angles dependent on engine loads; 3) cooling air does not participate in the energy enriching process in the combustor and hence the effective mass flow is reduced.

1.3 Flow phenomena inside turbine blades

The turbine in a gas turbine engine consists of one to several stages, each stage having both a stationary and a rotating set of blades. The stationary row is positioned upstream of the rotating row to guide the ﬂow from an axial to a tangential-axial direction in order to drive the turbine. To distinguish between the two rows, the stationary row of blades is called a set of “vanes” and the rotating counterparts are referred to as “blades”. The turbine blades, similar to turbine vanes, need to be cooled using pressurized air from the compressor. The lower external gas temperature however reduces the necessary complexity of the cooling system as well as the amount of air needed, compared to the IGV discussed above. The blades are, however, stressed by the centrifugal forces caused by the high rotational speed of the disk on which they are ﬁxed. They also experience thermal stresses from high temperature gradients.

Rotational eﬀects also induce Coriolis forces in the ﬂow, which leads to complicated

ﬂow structure within the cooling passages, making the design and construction of turbine blades one of the most challenging and expensive industrial enterprises today.

1.4 Background

It is well known that the steady increase in turbine inlet temperature over the last several decades has signiﬁcantly enhanced the eﬃciency and performance of gas turbine engines. The tendency toward higher turbine inlet temperature is continuing

6 and it continues to pose considerable challenges for the analysis and design of turbine cooling schemes (Lakshminarayana [2], Han et al. [3]). Computational ﬂuid dynamics

(CFD) techniques based on Reynolds-averaged Navier-Stokes (RANS) equations have matured in recent years and are routinely used to predict the pressure loadings of vanes and blades, but their applications to predicting turbine heat transfer have achieved only limited success [2, 3, 4].

For non-film cooled airfoils, deviation of actual heat transfer predictions from true to indicated levels can most probably be attributed to one or more of the following analytical deficiencies: 1) Uncertainties regarding the surface location at which transition is initiated as well as the surface extent of the transition zone. 2) Uncertainties regarding the influence of free stream turbulence on local heat transfer rates in the laminar region as well as on initiated and extent of the transition region. 3) Limited understanding on how the airfoil surface curvature influences the turbulence production/dissipation and boundary layer stability [5].

Dunn [4] concludes that heat transfer predictions for gas turbines are very diffi- cult because the boundary conditions are not well known. The rate of heat transfer, along with the effect of Mach number, Reynolds number, inlet flow angle, and the free stream turbulence all have a major influence on temperature and heat flux distributions in blades [6]. The difficulty is also often attributed to the deficiency of current turbulence closures [2, 3, 4]. Dunn [4] notes that the ability of most RANS

CFD codes to predict surface-pressure distribution is signiﬁcantly better than the corresponding heat transfer distributions. Several studies have been conducted on turbine nozzle guide vanes and rotor blades to obtain the heat transfer coeﬃcient

7 numerically. In general, numerically predicted results are within 10% of the experimental data. Many experimental research studies assume that the sudden changes in heat transfer coeﬃcient are a result of transition from laminar to turbulent ﬂow.

Rahman [7] suggests that sudden changes in the heat transfer coeﬃcients on the suction side of a turbine blade can, in many cases, also be caused by localized shocks that disturb the boundary layer.

Such uncertainties has led turbine aerodynamics and heat transfer to be treated separately or in a decoupled manner though they are coupled closely in reality. Typ- ically the external and internal heat transfer coeﬃcients (HTC) on a cooled turbine blade are obtained from boundary layer codes or empirically derived correlations, using surface pressure distributions from 3-D RANS analyses. This information is then passed as boundary conditions to ﬁnite element codes to obtain the metal temperature. The decoupled approaches require multiple iterations and some accuracy is lost in decoupling.

Recently there have been increased research eﬀorts in applying the conjugate heat transfer (CHT) methodology to solve turbine heat transfer problems. The studies by

Bohn et al. [8] and other researchers [9]- [18] are noteworthy. The CHT methodology allows a simultaneous and coupled solution of aerodynamics and heat transfer in the external hot gas and the internal cooling passages and conduction within the solid metal, hence eliminating the need for multiple and decoupled solutions. Unlike a decoupled approach in which external and internal heat transfer coeﬃcients need to be assigned on the walls, the conjugate heat transfer method requires ﬂow boundary conditions only at the inlet and exit of the gas and coolant passages.

8 Recent CHT computations have offered promising results and have been applied to realistic vanes and blades with complex film cooling and serpentine rib-roughened passages (e.g., Bohn et al. [11], Heidmann et al. [12], and Kusterer et al. [19]. However, the predictive accuracy of metal temperature by the CHT method is still limited by the accuracy of turbulence closures, as observed in York and Leylek [13] and Facchini et al. [10], among others. In their CHT investigations, Facchini et al. [14] have shown that heat transfer coefficient profiles obtained without taking into account transition severely overestimate experimental data especially near the leading edge.

The results in [13, 14] using two equation eddy-viscosity models showed that the error of metal temperature prediction was within or around 10% for a relative simple cooled blade (the same C3X blade used here). This is equivalent to an error of 150K for a moderate turbine inlet temperature of 1500K, which exceeds the desirable limit of the industry design practice. These and other CHT studies indicate that there is a strong need for improved turbulence models. This work, as well, attempts to compare the results obtained by modiﬁed k-ω turbulence model byWilcox [26] to that of Wilcox’s standard k-ω model [25] as applied to the present geometry.

There are two key modifications in the new model, namely, addition of a cross- diffusion term and a built-in stress-limiter modification that makes the eddy viscosity a function of k, ω and the ratio of turbulence-energy production to turbulence-energy dissipation.

Speziale [28] was the ﬁrst to suggest the addition of cross diﬀusion to the ω equation as a remedy for the original k-ω model’s sensitivity to the freestream value of

ω. Wilcox [26] has shown that the overall eﬀect of introducing the cross-diﬀusion term is reduction in the net production of k. While Wilcox and others, e.g., Menter [29],

9 Speziale [28], Kok [30] and Hellsten [31] have succeeded in using cross-diffusion to eliminate boundary-condition sensitivity, it has come at the expense of the ability to make reasonable predictions for free shear flows. Thus models incorporating cross diffusion term are limited in applicability to wall-bounded flows. In the present work, addition of cross-diffusion term is expected to limit the value of stress near the leading edge region of the blade.

Coakley [32] introduced the stress-limiter modification. Huang [33] has shown in detail that by limiting the magnitude of the eddy viscosity when turbulence-energy production exceeds turbulence-energy dissipation, this modification yields larger sepa- ration bubbles, which greatly improve incompressible- and transonic-flow predictions.

For a transonic airfoil, it has been veriﬁed by Kandula and Wilcox [34], that it improves predictive accuracy of the standard k-ω model without cross diﬀusion and blending functions. In [26], Wilcox has demonstrated that blending functions add little advantage and counter the elegance and simplicity of the k-ω model.

1.5 Thermal Barrier Coating

Thermal barrier coating (TBC) is an essential requirement of a modern gas turbine engine. These coatings provide thermal insulation to the turbine blades of about

∆T ∼ 150oC. This reduction of temperature helps in prolonging the life of the metal alloy substrate and can protect it from other types of environmental dam- ages and thermo-mechanical fatigue. By insulating the turbine blades, TBC helps in enhancing the engine eﬃciency by allowing the components to run at higher gas temperatures [21].

10 Figure 1.3: A schematic illustration of a modern thermal barrier coating system consisting of a thermally insulating thermal barrier coating, a thermally grown oxide (TGO) and an aluminum rich bond coat. The temperature gradient during engine operation is overlaid [20].

First, a thermally protective TBC layer with a low thermal conductivity is required to maximize the thermal drop across the thickness of the coating. Zirconia has emerged as a preferred material, stabilized into its cubic/tetragonal forms by the addition of yttria in solid solution (called as yttria stabilized zirconia or YSZ). This material has low thermal conductivity (∼ 1W/mK) and relatively high (compared to many other ceramics) thermal expansion coeﬃcient, The low thermal conductivity of bulk YSZ results from the low intrinsic thermal conductivity of zirconia (reported to

11 be between 2.5 - 4.0 W/mK depending on the phase, porosity and temperature) and phonon scattering defects introduced by the addition of yttria [20].

However, zirconia is rather impervious to oxygen. Oxygen can diﬀuse through it rapidly and oxidize the MCrAl (where M is Ni or Co) on the bond coat forming

α−Al2O3. This layer is called thermally grown oxide (TGO) and grows to a thickness

of about 5-10 µm. The growth of this oxide layer has important implication for the

durability of the TBC. As the TGO grows, it is subjected to thermo-mechanical

stress. With the depletion of Al, other metal oxides can be formed which occupy

larger volume than Al2O3. The growth of these metal-oxides can create stress leading to crack formation. Cracks can also initiate at other defects. Due to thermal cycling, these cracks can grow and eventually lead to the failure of the TBC.

The bond coat is made of Ni - or Co - based CrAlY alloy and with or without

Pt addition. This layer is the primary source of Al for the TGO as Al diﬀuses to the

surface as well as to the substrate. These coatings produce a very eﬀective α − Al2O3 oxide barrier and therefore protect the underlying substrate.

The life time of the engine critically depends on the life time of the thermal barrier coating. There are a few reasons for the failures of the TBC [23]. The Al concentration in the bond coat reduces because of diﬀusion of Al to the base of the TBC where it is oxidized. Al also diﬀuses into the super-alloy substrate. The depletion of Al creates de-lamination between TBC and the thermally grown oxide layer.

Al diffuses about three times faster than Ni. However, when Al depletes suffi- ciently, then Ni diffuses through the TGO and it oxidizes. The formation of nickel oxide generates additional stress in the TGO and leads to de-lamination between

TBC and TGO.

12 De-lamination zones can also occur due to imperfections in the oxide layer. There is additional stress at the locations of imperfections. Due to external stress there can be wrinkles in the shape of the TGO. The wrinkles can give rise to de-lamination.

Any damage caused by impact of a foreign object can create an imperfection or a hot-area in the oxide layer which then leads to the de-lamination. Additional stress is generated by thermal expansion misﬁts between diﬀerent materials. The cracks that appear then grow with thermal cycles and eventually the TBC spalls from the substrate [21].

1.6 Project Description

A conjugate numerical methodology was employed to predict the metal temperature of a three-dimensional gas turbine vane. Validation of the methodology was accomplished through the comparision of the predicted aerodynamic loading curves and the midspan temperature distribution on the vane external surface with data from a linear NASA-C3X cascade experiment in the literature [5].

The C3X transonic turbine blade of Hylton et al. [5] was selected as the test case, as this case has detailed measurement of external and internal convection and the metal temperature. Code GlennHT, which was originally developed at NASA Glenn, was used to perform computations. The code had to be modified to perform conjugate heat transfer computations, for the cooling channels, from bulk temperature and heat transfer coefficient conditions, as in the case of C3X vane experiments, h and bulk temperature values are provided in literature [5]. After initial success with these changes, a modified k-ω turbulence model by Wilcox [26] was incorporated and the results were compared to Wilcox’s standard k-ω model [25].

13 According to Wilcox, this modiﬁed model will reduce the excessive and non- physical production of turbulent kinetic energy characteristic of the standard k-ω model in the areas of high irrotational strain. This strain condition occurs at the stagnation point on the leading edge of a turbine airfoil and downstream of the leading edge on the suction side.

Also, the eﬀectiveness of thermal barrier coating in providing thermal insulation to the turbine blades has been investigated.

14 CHAPTER 2

FLUID MOTION AND HEAT TRANSFER

The flow of air through gas turbines used for aviation and industrial purposes is typically subsonic. Also the flow of coolant air within coolant passages of turbine blades is subsonic but compressible. Viscous and diffusion effects in the flow are responsible for heat transfer, various secondary flows, and losses. Thus, to simulate the flows in gas turbine engines compressible Navier-Stokes equations are used. The equations governing fluid motion, and an overview of turbulence and heat transfer models with literature references are given below.

2.1 Governing Equations

The equations that govern ﬂuid motion and heat transfer are the continuity, momentum and energy equations. These equations were independently constructed by

Navier (1827) and Stokes (1845) and are referred to as the Navier-Stokes equations.

These can be formulated in either a conservative form, or in the non-conservative form.

2.1.1 Continuity

The continuity equation is a statement of the conservation of mass. Its conservative form is:

15 ∂ρ + ∇ · (ρu) = 0 (2.1) ∂t 2.1.2 Momentum

The momentum equation describes a force-balance, which - from the Newton’s second law - states that mass times acceleration is equal to sum of imposed forces.

Thus the momentum equation takes the following form:

∂(ρu) + ∇ · (ρuu) = −∇p + ∇ · τ (2.2) ∂t 2.1.3 Energy

The ﬁrst law of thermodynamics states that the exchange of energy for a system is the result of applied work and heat transfer through that region. In its most complete formulation the energy equations is given as [35]:

∂(ρe) + ∇ · (ρu(e + p/ρ)) = −∇ · q + ∇ · (τu) (2.3) ∂t where ρ is the density, u = (u, v, w)T is the velocity, p is the thermodynamic (static) pressure, τ is the deviatoric viscous stress tensor, e is the total energy per unit mass,

and q is the heat ﬂux due to diﬀusion.

The viscous stress tensor, total energy and heat ﬂux are given as:

2 1 τ = − µ(∇.u)I + µ(∇u + (∇u)T) (2.4) 3 2

16 where I is the 3x3 identity matrix,

1 e = u + (u.u) (2.5) b 2

and

q = −k∇T (2.6) where µ is the molecular (dynamic) viscosity, k is the thermal conductivity, ub is the internal energy of the gas, T is the absolute temperature and u and v are three component column vectors.

The gas is treated as ideal gas, i.e., dub = cvdT , where cv is the constant volume

speciﬁc heat. In addition, the gas is assumed to be calorically perfect so that cv is

constant, which is reasonable for ﬂows in turbomachinery. Thus the total energy can

be written as

1 e = c T + (u.u) (2.7) v 2

where the reference internal energy has taken to be zero at a temperature of absolute

zero.

The dynamic viscosity, µ, of the gas is assumed to vary with temperature as [37]

µ T 0.7 = (2.8) µ0 T0

17 where µ0 is the reference viscosity at temperature T0. Thermal conductivity of the gas is then given as

c µ k = p (2.9) P r where cp is the constant-pressure speciﬁc heat and Pr is the Prandtl number.

2.2 Computational Fluid Dynamics

The science of ﬂuid dynamics has traditionally relied upon theoretical and exper-

imental eﬀorts. The theoretical aspect of the science focuses on the construction and

solution of the governing equations for diﬀerent ﬂow categories and studies the various

approximations to those equations. Experimental ﬂuid dynamics, on the other hand,

provides validation to theoretical results and delineates the limits of applicability of

various approximations. With the onset of modern computers in the 1950’s a new,

complementing, field of fluid dynamics began to emerge to offer an alternative way

to analyze fluid flow. This field is known as Computational Fluid Dynamics (CFD),

which principally focuses on solving the Navier-Stokes equations in a discrete form

numerically using a grid of discrete points distributed throughout a computational

domain.

Over the decades CFD has become a signiﬁcant ﬁeld of research in academia and

it has developed to an essential and cost eﬀective analysis and design tool in various

industries. However, even though the ﬁeld has come a long way, CFD has not yet

removed the need for experimental validation and remains a subject of continuous

research. In this chapter the fundamental principles of computational ﬂuid dynamics

are discussed and the most essential solution techniques introduced.

18 2.2.1 Governing Equations

At its most complete form, computational ﬂuid dynamics relies upon the complete

theoretical description of ﬂuid- and thermo-dynamics to achieve quantitative predic-

tions of arbitrary ﬂow systems. The principal set of equations that are numerically

solved consists of three dimensional, unsteady, Reynolds Averaged conservation of

mass, momentum and energy equations, with appropriate turbulence modeling terms

coupled with thermodynamic state and constitutive equations. The set of govern-

ing equations referred to as The Navier-Stokes equations are discretized using ﬁnite

volume approach and can be expressed in a compact diﬀerential vector form as

∂U + ∇.(F − T ) = S (2.10) ∂t

where U = (ρ, ρu, ρe, ρk, ρω)T and F and T are a convective ﬂux tensor and a diﬀusive

ﬂux tensor, respectively, deﬁned as follows:

 ρ(u − Ω × r)T   ρ(u − Ω × r)u + pI     T  F =  ρ(e + p/ρ)(u − Ω × r)  (2.11)  T   ρk(u − Ω × r)  ρω(u − Ω × r)T and

 0   τ     T  T =  q + (τu)  (2.12)  ∗ T   (µ + σ µt)(∇k)  T (µ + σ1µt)(∇ω)

19 The source term, S, contains terms arising from the use of non-inertial reference

frame as well as production and dissipation of turbulent quantities, i.e.,

 0   ρ(Ω × u)      S =  0  (2.13)  ∗ ∗   P − D  P1 − D1 Here Ω is a rotation vector describing the rotation of rotating reference frame

X0 relative to the inertial reference frame X. P ∗ and D∗, respectively, represent

production and destruction of turbulent kinetic energy, and P1 and D1, respectively,

represent the production and destruction of ω. The production and dissipation terms

are different for standard and modified turbulence models and are specified in the

next section. The closure for heat equation was obtained by using turbulent Prandtl

number.

With this notation, the convective ﬂux per unit area in direction of a vector a is

given by

a F = F (2.14) a ||a||

and a diﬀusive ﬂux is similarly obtained.

2.2.2 Turbulence Models

It is generally recognized that the Navier-Stokes equations, Eqs. 2.10 - 2.13, fully

describe the ﬂow physics of Newtonian ﬂuids, including the unsteady and randomly

ﬂuctuating behavior that is observed in most ﬂow systems around us. The presence

20 of three dimensional and unsteady fluctuations in the flow field indicates that the

ﬂow has lost its stability and has become turbulent. The familiar dimensionless ﬂow

parameter that relates inertia forces to viscous forces is typically the primary indicator

in specifying the nature of the ﬂow. The Reynolds Number is deﬁned as follows:

ρV 2/L ρV L Re = = µV/L2 µ

Perhaps surprisingly, numerical solutions of the full 3D Navier-Stokes equations succeed in simulating the break down of stability as disturbances begin to amplify in the flow field and the onset of fluctuations in a form of circulating flow structures called eddies, which span a wide range of length and time scales. But, this can only be achieved if the element size is small enough to capture the fluctuations even at the lowest end of the length scale spectrum. For any practical flow problem this would require a mesh so fine that only the most modern super-computers or clusters could handle solving it, taking the running cost and time requirement for even the simplest analysis beyond all reasonable limits. Therefore, an alternate approach of simulating turbulence has been developed, which involves solving the flow equations in a time averaged form and taking turbulence into account through additional, modeled, terms.

Turbulence models (in CFD) can be grouped to three main categories based on the underlying theoretical hypothesis: Eddy Viscosity Turbulence Models, Reynolds

Stress Turbulence Models and Large Eddy Simulation. Numerous volumes have been published on these topics, which, generally speaking, entail high level of expertise and therefore it is seen appropriate that a thorough coverage of the topic is left for textbooks devoted on turbulence.

21 Out of the three categories Eddy Viscosity models are most commonly used in

industry and, at this moment, the two-equation models of this category appear to

provide the best compromise between numerical eﬀort and accuracy. Based on general

recommendations reported in literature a two-equation (κ, ω) turbulence model, which

has produced the most encouraging agreements with empirical results, has been used

in all CFD analysis in this study. Details of the two turbulence models used here are

presented below.

Standard k-ω Model Constitutive Relations

The model uses the following equations to compute the molecular stress tensor,

tij and the speciﬁc Reynolds-stress tensor, τij

2 t = 2µS¯ , ρτ = 2µ S¯ − ρkδ (2.15) ij ij ij T ij 3 ij

¯ 1 ∂uk Sij = Sij − δij (2.16) 3 ∂xk

ρk µ = , (2.17) t ω 1 ∂ui ∂uj The quantity Sij = + is the conventional mean strain-rate tensor, δij 2 ∂xj ∂xi is the Kronecker delta, and ω is the speciﬁc dissipation rate.

Turbulence Model Equations

The turbulent kinetic energy is calculated as

" ! # ∂ ∂ui ∗ ∂ ∗ ρk ∂k (ρk) = τij − β ρkω + µ + σ (2.18) ∂t ∂xj ∂xj ω ∂xj

22 and the dissipation is calculated as

" ! # ∂ γ1 ∂ui 2 ∂ ρk ∂ω (ρω) = τij − β1ρω + µ + σ1 (2.19) ∂t υt ∂xj ∂xj ω ∂xj

∗ ∗ where β , β1, σ , σ1 and γ1 are turbulence model closure co-eﬃcients

Closure Coeﬃcients

The various closure coeﬃcients appearing in the model are given as

∗ ∗ β = 0.09, β1 = 0.075, σ = 0.5, σ1 = 0.5, (2.20)

q ∗ 2 ∗ γ1 = (β1/β ) − σ1k / β , k = 0.41, P rT = 0.9 (2.21)

To enable the simulation of transition, the eddy viscosity is reﬁned as

ρk µ = α∗ (2.22) t ω and the model coeﬃcients are replaced by

∗ σ = 0.5, σ1 = 0.5 (2.23)

∗ β = 0.09Fβ, β1 = 0.075 (2.24)

5 Fα ∗ α = , α = Fµ (2.25) 9 Fµ

where Fβ, Fα and Fµ are given by following expressions

4 5/18 + (ReT /Rβ) Fβ = 4 ,Rβ = 8 (2.26) 1 + (ReT /Rβ) 23 α0 + ReT /Rω Fα = ,Rω = 2.7, α0 = β/3 (2.27) 1 + ReT /Rω

∗ α0 + ReT /Rk ∗ Fµ = ,Rk = 6, α0 = 0.025 (2.28) 1 + ReT /Rk

ρk Re = (2.29) T µω

where ReT is the turbulence Reynolds number.

Modiﬁed k-ω Model Constitutive Relations

The expressions to compute the molecular stress tensor, tij and the speciﬁc Reynolds- stress tensor, τij are given as

2 t = 2µS¯ , ρτ = 2µ S¯ − ρkδ (2.30) ij ij ij T ij 3 ij

¯ 1 ∂uk Sij = Sij − δij (2.31) 3 ∂xk

 v  u ¯ ¯ ρk u2SijSij µ = , ω˜ = ω, C t  (2.32) t ω˜  lim β∗ 

7 C = (2.33) lim 8

∗ Where Clim is the stress-limiter strength and β is a turbulence model closure coeﬃcient.

24 Turbulence Model Equations

The evolution of turbulent kinetic energy is determined by

" ! # ∂ ∂ ∂ui ∗ ∂ ∗ ρk ∂k (ρk) + (ρujk) = ρτij − β ρkω + µ + σ (2.34) ∂t ∂xj ∂xj ∂xj ω ∂xj

and the turbulent dissipation is governed by

" ! # ∂ ∂ ω ∂ui 2 ρ ∂k ∂ω ∂ ρk ∂ω (ρω) + (ρujω) = α ρτij − βρω + σd + µ + σ ∂t ∂xj k ∂xj ω ∂xj ∂xj ∂xj ω˜ ∂xj (2.35)

where α, β, σ and σd are turbulence model closure co-eﬃcients

Closure Coeﬃcients

The various closure coeﬃcients appearing in the modiﬁed k − ω model are given

13 9 1 3 8 α = , β∗ = , σ = , σ∗ = , P r = (2.36) 25 100 2 5 T 9

∂k ∂ω ∂k ∂ω 1 σd = { 0 , ≤ 0, σdo, > 0, σdo = (2.37) ∂xj ∂xj ∂xj ∂xj 8

1 + 85χω β = β0fβ, β0 = 0.0708, fβ = (2.38) 1 + 100χω

Ω Ω Sˆ 1 ∂u ij jk ki ˆ m χω = ∗ 3 , Ski = Ski − δki (2.39) (β ω) 2 ∂xm 1 ∂ui ∂uj where Ωij = − is the conventional rotation tensor. 2 ∂xj ∂xi One key modiﬁcation to the standard k − ω model is the addition of a cross diﬀusion term. The term proporational to σd in Equation 2.35 is referred to as cross

25 diffusion. It depends on gradient of both k and ω. This modification significantly

reduces the original model’s sensitivity to ﬁnite freestream boundary conditions on

turbulence parameters.

Second key modiﬁcation to the standard k − ω model occurs in the expression for the eddy viscosity υT . In the new model, the eddy viscosity is a function of k

to ω and, eﬀectively, the ratio of turbulence-energy production to turbulence-energy

dissipation. Wilcox [26] suggests that this modiﬁcation greatly improves the models

predictions for supersonic and hypersonic separated ﬂows.

2.3 Mathematical formulation

2.3.1 Modeling for h and bulk temperature boundary conditions

To compute the wall temperature, for the cooling channels, from the specified values of h and bulk temperature, conservation of heat flux across the boundary has been employed at the solid-fluid interface as shown in Fig. 2.1

The complete formulation is presented below

t + t t = g i (2.40) w 2

q = h(tw − tbulk) (2.41)

dT ti − tg q = −ks = ks (2.42) dy 2.dn

where tw is the temperature at the wall, tg is the current temperature at the ghost cell, ti is the gas temperature at the interior cell and tbulk is the bulk temperature of

26 Figure 2.1: Boundary conditions across solid-ﬂuid interface

the ﬂuid. Here dn is assumed to be equal to dg. Equating the expressions above and solving for wall temperature, we get the following expression,

  t + ks.ti bulk h.dn tw =   (2.43) 1 + ks h.dn

2.3.2 TBC Modeling

A thermal barrier coating (TBC) “system” is applied on the vane surface to ther-

mally protect turbine engine blade and vanes from the hot gases is the gas turbine

engine. This surface is deﬁned as conjugate and hence the conjugate treatment of the

surface is described below. Since the thickness of TBC layer is very small (∼ 10−5),

a one dimensional heat ﬂow assumption is used.

Conserving the heat ﬂux from the solid side, across the TBC layer, to the ﬂuid

side, we get the following equation(s),

27 Figure 2.2: Boundary conditions across thermal barrier coating

∂T k ! ∂T −ks |s = (Ts − Tf ) = kf |f (2.44) ∂n l tbc ∂n

! Ti − Tw,s k Tg − Tw,f −ks |s = (Tw,f − Tw,s) = kf |f (2.45) dn l tbc dg These equations when solved for TBC wall temperatures on the solid and the ﬂuid side give the following expressions,

ks dg ks l tg + + ti kf dn dn k tbc Tw,s = (2.46) 1 + ks dg + ks l kf dn dn k tbc

h i ks l ks dg tg 1 + + ti dn k tbc kf dn Tw,f = (2.47) 1 + ks l + ks dg dn k tbc kf dn where Ts is the solid temperature at a distance dn away from the surface, Tf is the ﬂuid temperature at a distance dg away from the surface, Tw,s is the thermal barrier coating temperature on the solid side and Tw,f is the thermal barrier coating temperature on the ﬂuid side.

28 CHAPTER 3

CODE DESCRIPTION

The Glenn-HT code is a general purpose three-dimensional flow solver designed for computation of convective heat transfer of flows in complicated geometries. The code solves the full compressible Navier-Stokes equations using a multi-step Runge-Kutta- based multi grid method. The finite volume method is used with central differencing, and artificial dissipation is employed. The overall accuracy of the code is second order. The present version of the code employs the k, ω-turbulence model developed by Wilcox [25, 27], with modifications by Menter [29] and implemented by Chima [36].

Accurate heat transfer predictions are possible with the code because the model integrates to the wall and no wall functions are used. Rather, the computational grid is generated to be suﬃciently ﬁne near walls to produce a y+ value of less than

1.0 at the ﬁrst grid point away from the wall. Laminar viscosity is determined from temperature using a 0.7 power law [37]. Speciﬁc heats are assumed to be constant.

A full description of the code and its recent applications to turbine heat transfer can be found in [39].

29 3.1 CHT Calculation using Glenn-HT

The traditional method for analyzing the heat transfer on a turbine blade, or other convective surface, is to first obtain a fluid-side convection solution assuming either isothermal or constant heat flux conditions at the blade surface. This effectively decouples the fluid solution from the thermal conduction inside the blade material.

For a one-temperature problem (i.e. one without film cooling), the external flow solution is used to compute a heat transfer coefficient distribution on the surface.

For a two-temperature problem (such as one with ﬁlm cooling), a second external

flow solution is obtained using a different wall thermal boundary condition. The two solutions are then used to compute the heat transfer coefficient and film effectiveness distributions.

The current Glenn-HT code allows the user to specify any distribution of wall temperature or heat flux or heat transfer coefficient and bulk temperature over the convective surface. Once the specification is fixed, it is not affected by conduction in the solid. The conjugate method allows for a coupled heat transfer solution between the solid and fluid, and thereby more accurate heat transfer predictions. The energy equation for solids is a degenerate form of the energy equation for fluids (obtained by setting the velocity to zero). Accounting for conjugate heat transfer is particularly important where there are large thermal gradients on the surface or within the solid, such as might be found near film cooling holes or near other complicated geometrical features.

30 CHAPTER 4

ANALYSIS

4.1 Model Geometry

The experimental data for validation of the conjugate simulation of flow and heat conduction need to provide clear boundary conditions and cooling air flow rates of high accuracy. The C3X transonic turbine guide vane of Hylton et al. [5] was selected as the test case, for this work provides detailed measurement of the external and internal convection and the metal surface temperature. The experimental facility consisted of a linear cascade of three C3X turbine vanes. The center vane was cooled by air flowing through ten round flow passages from the hub to the shroud.

In their studies the heat transfer measurements were obtained by measuring the internal and external boundary conditions of the test piece at thermal equilibrium and solving the steady-state heat conduction equation for the internal temperature

field of the test piece [5]. The heat transfer coefficient distributions were directly obtained from the normal temperature gradient at the surface. The temperature profiles on the external boundary were measured using approximately eighty 0.5 mm- diameter sheathed CA thermocouples, installed in grooves on the exterior surface of the test vane. Average heat transfer coefficients and coolant temperatures for each of 10 radial cooling holes provided the internal boundary conditions for the finite

31 element solution. The heat transfer coefficient for each cooling hole was calculated from the hole diameter, the measured flow rate, and the coolant temperature, with a correction applied for thermal entry length. The uncertainty in the heat transfer coefficient measurements varied from 6.8% at the leading edge to 23.5% at the trailing edge. The uncertainties increase significantly beyond mid-chord due to the decrease in airfoil thickness.

The C3X vane has constant cross section with no twist and it has been made by extruding the two-dimesional model through 69.96 mm in the spanwise (which is the z-) direction. The cooling holes are approximately centered on the camber line of the vane, except for the two holes near the leading edge. The cross section of this vane is shown in Figure 4.1 with the the placement and size of the cooling holes marked by white circles. Detailed geometric parameters are listed in Table 4.1.

Figure 4.1: C3X Vane with Cooling Holes

Glenn-HT code uses structured multi-block grids and thus a topology was ﬁrst generated which deﬁned the eight corners of each block and how they are connected.

32 Stagger angle (deg) 59.89 Air exit angle (deg) 72.38 Throat (mm) 32.9 Vane spacing (mm) 117.7 Vane height (mm) 76.2 Suction surface arc (mm) 177.8 Pressure surface arc (mm) 137.2 True chord (mm) 144.9 Axial chord (mm) 78.2

Table 4.1: Geometry Parameters for C3X vane

Once this topology was generated, it was a relatively straightforward process to produce a grid. The topology needed to be generated by hand, which turned out to be be very time-consuming for a complex geometry like that of C3X. Among the features that were desirable for viscous grids is that the grids are clustered along the solid boundaries, and that orthogonality of the grid is maintained as much as possible. A grid generator, GridProTM [40], was used with the topology to generate the ﬁnal grid.

4.2 Numerical Methodology

The computational domain included one vane in the middle of the ﬂow ﬁeld, with

periodic boundary conditions employed to simulate cascade ﬂow. The computational

domain inlet is located one chord length upstream of the leading edge, where the

turbulence level was measured in the experiments. The outlet is located one chord

length downstream of the trailing edge. Meshes are created for the hot gas path and

the solid vane. Grid independence studies were performed to ensure that the solution

was independent of the grid size.

33 Figure 4.2: C3X Grid

The ﬂow was assumed to be fully developed at the hole inlets at the hub of the vane, as there were long tubes feeding the channels in the experiment of Hylton et al [5]. The coolant exited the top of the vane to atmospheric pressure.

The heat transfer coeﬃcients for the internal ﬂow in the cooling holes were obtained from the correlation:

0.5 0.8 NuD = Cr(0.022P r ReD ) (4.1)

34 as given in Hylton [5]. The constant Cr in Eq. 4.1 accounts the thermal entrance effects for the cooling channels [5]. The cooling channel geometries and the coolant conditions are listed in Table 4.2. The hot gas flow path details are specified in

Table 4.3. The values for Tc and ReD were speciﬁed from Run 144 in the study by

Hylton et al. [5]). The vane material is ASTM type 310 stainless steel, which has the

Cooling Diameter Cr in Tc ReD Coolant Flow Rate Channel (mm) Eq. 4.1 (K) (x 10e-4) (kg/sec) 1 6.30 1.118 365.31 19.909 0.211E-01 2 6.30 1.118 365.29 21.133 0.224E-01 3 6.30 1.118 351.09 21.115 0.217E-01 4 6.30 1.118 355.05 21.499 0.223E-01 5 6.30 1.118 344.21 22.327 0.226E-01 6 6.30 1.118 383.91 21.074 0.231E-01 7 6.30 1.118 354.71 21.542 0.223E-01 8 3.10 1.056 380.76 13.865 0.748E-01 9 3.10 1.056 425.30 8.034 0.469E-01 10 1.98 1.025 467.00 6.237 0.246E-01

Table 4.2: Cooling Channel’s Property

−6 −6 Run No. Pt∞(psi) Tt∞ (K) Ma1 Ma2 Re1 × 10 Re2 × 10 144 59.18 815 0.16 0.90 0.63 2.43

Table 4.3: Hot gas path ﬂow details

constant density of 7900 kg/m3 and the speciﬁc heat of 586.15 J/kgK. The thermal

conductivity (of stainless steel) k is speciﬁed (from York [41]) as

35 k = 0.020176 T + 6.811 (4.2) in units of W/m K with T given in ◦C.

A multiblock numerical grid was used in the present work. Multiblock grids are a collection of topologically hexahedral blocks, each of which contains a regular array of grid points. Essentially, multiblock grids are globally unstructured-and hence have the ability to handle complicated geometries -but locally structured and ”body-ﬁtted”

- and thus have the ability to accurately and eﬃciently handle viscous layers on non- slip surfaces. Thus, the use of multiblock grid gives the highest quality grid in all regions with the fewest number of cells.

A two-dimesional grid was ﬁrst created in GridProTM [40]. Because the airfoil has a constant cross-section, the grid was stacked in the spanwise direction to create the full domain. The cells in the near-wall layers were stretched away from the surfaces, and all wall-adjacent grid points were located at a y+ equal to or less then unity to fully resolve the viscous sublayers. In total, there were 440 blocks - about 0.4 million cells for the entire domain, with about 75% of the total for the hot gas path and remaining for the solid blade.

The present simulations employed two variants of the popular, two-equation k-

ω model for comparision purposes. The first model was the “standard” k-ω model originally proposed by Wilcox [25]. The second model is a modifcation of revised k-ω model given by Wilcox [26] (here referred as “modified” k-ω). The revisions to the standard k-ω model include an addition of a cross-diffusion term and also a built-in stress-limiter modification. For sake of computational simplicity, this revised model has been modified by using stress limiter term both in k- and ω- equations.

36 Wilcox suggests that this modiﬁed model will reduce the excessive and non- physical production of turbulent kinetic energy characteristic of the standard k-ω model in the areas of high irrotational strain. This strain condition occurs at the stagnation point on the leading edge of a turbine airfoil and downstream of the leading edge on the suction side.

The present simulations were run using the code Glenn-HT. The code was run till the solution seemed to have converged and no appreciable difference in the output values was seen. The output files were then post-processed and the solution was visualized using TecplotTMand FieldViewTM. In the fluid zones, the steady, time- averaged Navier-Stokes equations were solved, and the pressure velocity coupling was achieved with a pressure-correction algorithm. In the solid zone, only the Fourier equation for heat diffusion was solved.

All equations were discretized with second-order accuracy. At the fluid-solid interfaces, an energy balance was satisfied at each iteration, such that the heat flux at the wall on the fluid side was equal in magnitude and opposite in sign to the heat flux on the solid side. The temperature of the boundary was itself adjusted during each iteration to meet this condition. Thus, all fluid-solid interfaces were fully coupled and required no user specified boundary values. To initialize the solution, the values for pressure and temperature in the passage were specified as the inlet total pressure and inlet total temperature respectively.

37 CHAPTER 5

RESULTS AND DISCUSSION

The results of the simulation are presented and analyzed in this section. For the mainstream conditions mentioned in Table 4.2, the flow field is examined first, followed by the heat transfer. All experimental data presented here is from run 144 from the work of Hylton et al. [5].

Three diﬀerent cases were considered under diﬀerent boundary conditions, namely constant cooling hole (CCH) temperature boundary condition using standard k-ω

model (SKO), specified heat transfer coefficient and bulk temperature of the fluid at

the cooling holes using SKO model and speciﬁed heat transfer coeﬃcient and bulk

temperature of the ﬂuid at the cooling holes using modiﬁed k-ω model (MKO). Where

simulations with diﬀerent boundary conditions/turbulence models are discussed, ap-

propriate acronyms (CCH, SKO or MKO) are attached to label the curve.

The predicted aerodynamic loading in the form of pressure distribution at the

vane midspan is given in Figure 5.1, along with experimental data. The pressure

distribution does not change with the change in turbulence model as the eﬀect is

completely inviscid. Hence, the loading curves predicted by all the the cases were

essentially identical and only SKO result is plotted.

38 Figure 5.1: Experimental and predicted data of P/P0

On the suction side (SS), the gas ﬂow accelerates rapidly from the stagnation point toward the throat, reaching the maximum speed (corresponding to the minimal value of P/P0 = 0.51) around X/Cx = 0.40. The ﬂow then decelerates until the location around X/Cx = 0.65, before resuming a mild acceleration toward the trailing edge

(TE). The ﬂow is under a favorable pressure gradient on the entire pressure side

(PS). The pressure stays almost constant near P0 from the LE to about X/Cx = 0.5 and then falls oﬀ with further distance toward the TE. The predictions exhibit close

39 agreement with the data of Hylton et al. [5], validating the aerodynamic portion of

the model.

The contours of Mach number on the midspan (at 50 % of the blade’s height) are

shown in Figure 5.2. Due to the shape of the airfoil, the ﬂow sees a strong acceleration

Figure 5.2: Contours of Mach number at midspan

along the suction side near the leading edge. The maximum Mach number in the vane passage is about 0.93, and it occurs at a position just oﬀ the SS about 20% of the distance from the LE to TE. Mach number remains quite low near the pressure side until the aft quarter of the vane, when the ﬂow accelerates to the TE where Mach number reaches about 0.9.

40 The standard k-ω model results in a high production of turbulence kinetic energy

at the stagnation point. This is not physically accurate, as the strain at the leading

edge is irrotational in nature and therefore should not result in production of k [42].

A good turbulence model must be able to eliminate this spurious production of k at the leading edge and in the vane passage. In the code, this is implemented by modi- fying the production terms and writing them in terms of the magnitude of vorticity.

Wilcox, thus, suggested a modiﬁed k-ω model [26] which incorporates a stress limiter modiﬁcation which limits the magnitude of eddy viscosity when turbulence-energy production exceeds turbulence-energy dissipation.

Wilcox has shown in [26] that this model greatly improves the model’s predictions for supersonic and hypersonic separated flows, however, in our present case, flow is mostly transonic and it can be seen that this modified model has shown little success in limiting the value of turbulent kinetic energy for our present geometry. Wilcox, however, has also suggested in [26] that discrepancies can be further reduced by increasing the strength of the limiter in specific cases (most notably for transonic

ﬂows).

With the aerodynamics examined, the heat transfer will be investigated next. The distribution of dimensionless temperature (Twall/T0) on the vane external surface at the midspan is plotted for the CCH, SKO and MKO simulations in Figure 5.3 and

Figure. 5.4, along with the experimental data from Hylton et al. [5].

Recall that the intent of the experiment (and therefore the simulations which modeled the experimental conditions) was to achieve uniform temperature distribution on the external wall, and in this case the average was approximately Twall/Tref = 0.75.

41 Figure 5.3: Experimental and predicted data for Twall/T0 under diﬀerent b.c.

The SKO case results show reasonable agreement with the experimental data but, since both pressure and suction side exhibit typical transitional behaviour, proﬁles obtained without explicitly taking transition into account overestimate experimental data especially near the leading edge [38]. The wall temperature plotted in Figures 5.3 and 5.4 is strongly dependent on the local heat transfer coeﬃcient, and, to a lesser extent, the local freestream conditions. It should be noted that for the Reynolds number in the present case, the boundary layer is expected to start laminar at the

42 Figure 5.4: Experimental and predicted data for Twall/T0 under diﬀerent turbulence models

LE and transition to a turbulent boundary layer at some point downstream. The present turbulence models force a turbulent boundary layer on all walls and therefore do not predict transition [42].

Normalised heat transfer coeﬃcient has been plotted in Figure 5.5 for the two turbulence models. Modiﬁed turbulence model is seen to have some advantage here, compared to standard turbulence model even though both the models fail to give a reasonable prediction for heat transfer near the leading edge on the suction side.

43 Figure 5.5: Experimental and predicted data for normalised heat transfer coeﬃcient under diﬀerent turbulence models

The temperature distribution within the metal vane is seen in Figure 5.6, contours

of Ts on the cross-section plane at 50% span. The patterns of temperature are fairly consistent at diﬀerent spanwise locations and hence the plot is only shown for the midspan cross-section. The metal temperatures are much closer to the mainstream temperature than the coolant temperature in all planes. The maximum metal temperature occurs at the thin trailing edge, and the largest temperature gradients in the chordwise direction are also in this region.

44 Figure 5.6: Static Temperature at 50% of blade’s height

Several notable observations may be made. First the vane outside wall on the SS is very near the freestream temperature, indicating a very small thermal resistance and hence a large heat transfer coefficient. The temperature difference across the pressure-side boundary layer is slightly larger than on the SS due to a greater thermal resistance. Secondly, the temperature difference in the solid is small, indicating a small thermal resistance. Lastly, the temperature difference between the inner walls of the channel and the channel freestream is very large, signifying a much greater thermal resistance or a far smaller heat transfer coefficient, than the external convection.

The SS convection and the conduction in the solid have similar resistances, while the resistance of the PS convection is slightly higher. By far the greatest thermal resistance is due to the internal convection. Lower metal temperatures could be

45 Figure 5.7: Wall temperature on the pressure side and the suction side of C3X blade

achieved by reducing this resistance by increasing the heat transfer coeﬃcient in the channels.

Eﬀectiveness of implementing thermal barrier coating (TBC) in lowering metal temperature is studied in Figures 5.8 and 5.9. It can be observed that use of TBC has helped in lowering the overall temperature of the blade, as in the absence of

TBC the average temperature of the blade was close to 0.75, while with the use of

TBC, average temperature of the blade comes close to 0.70. Figure 5.9 shows the temperature variation across a plane crossing through the blade surface. The jump in the temperature proﬁle, as observed here, can be attributed due to large temperature diﬀerence across a very thin TBC layer on blade surface.

46 Figure 5.8: Eﬀectiveness of TBC

Figure 5.9: Temperature variation across TBC

47 CHAPTER 6

SUMMARY AND CONCLUSIONS

A numerical methodology for conjugate heat transfer simulations has been applied. The present study was designed to highlight the promise of this relatively new tool in gas turbine heat transfer design, wherein the individual heat transfer problems (external/internal convection, conduction in the metal) are coupled in a single simulation. The beneﬁts of the conjugate approach are inherently better accuracy and reduced turn-around time as compared to the common practice of decoupled simulations.

The present methodology was tested with three-dimensional simulations of a stainless steel turbine vane at near-engine conditions, at subsonic passage ﬂows. The vane was cooled by air ﬂowing radially through ten smooth-walled channels. Aerodynamic loading curves was predicted in a good manner. The predicted midspan temperature distribution on the vane external surface was in reasonable agreement with experimental data.

The modiﬁed k-ω model did not perform as well, and showed an oﬀset in predicted temperature as compared to the standard k-ω model and the experimental data. The overprediction of the external temperature with the models close to the leading edge

48 is attributed to their spurious production of turbulence in regions of large irrotational strain and the resulting artificially large heat transfer coefficient, which is also observed in heat transfer coefficient plots. There is room for improvement in the turbulence modeling aspect of this methodology, such as the implementation of a model that can accurately predict boundary layer transition.

With the methodology validated, the computational results were fully analyzed.

The highest metal temperature and largest temperature gradients were located at the trailing edge of the vane where the metal is very thin. The metal temperature everywhere was much closer to the temperature in the passage freestream than the coolant temperature. This was due to the fact that the thermal resistance due to internal convection was much greater than the resistance of the external convection or the conduction within the metal.

A thermal barrier coating (TBC) model was also implemented on the blade surface which was found to be eﬀective in lowering the overall temperature of the blade.

Discontinuity was observed in the temperature plot across the TBC layer, which demonstrated a signiﬁcant drop in temperature across the thin TBC layer.

49 CHAPTER 7

FUTURE WORK

This thesis has demonstrated the use of conjugate heat transfer methodology as a eﬀective tool in gas turbine design and analysis. This approach represents a substantial improvement over the traditional method, where internal and external heat transfer are treated separately, and hence, some accuracy was lost in decoupling.

However, more research need to be done before a clear understanding of the inﬂuence of various parameters, in the prediction of temperature proﬁle, and hence the heat transfer on turbine blade can be fully understood.

It was seen that the modified k-omega model was not very successful in predicting the temperature on blade surface and hence it represents an opportunity for improvement. Wilcox [26] suggests that the magnitude of stress limiter may need to be increased for transonic flows (present case) to give better predictions. Thus, more experiments are needed to test the influence of stress limiter on predictions.

The greatest thermal resistance in the blade was found to be due to the internal convection. Thus, lowering the resistance of internal convection by increasing the heat transfer coeﬃcient in the cooling channels would result in a sizeable reduction in metal temperature throughout the vane. This could be accomplished by changing

50 the channel’s shape (increasing the ratio of wet perimeter to crossing surface), and/or placing ribs on the internal channel walls to increase turbulence.

The implementation of thermal barrier coating (TBC) needs further investigation, such as, the optimum thickness of the TBC layer and the suitable thermal conductivity. Also, to perform these computations in a shorter time, a two-dimesional analysis could also be carried out as only slight variation of temperature proﬁle, and hence the heat transfer, in three dimensions was observed.

51 BIBLIOGRAPHY

[1] Cohen, H., Rogers, G.F.C., Saravanamuttoo, H. I. H., Gas Turbine Theory, Longman Publications, 1996.

[2] Lakshminarayana, B.,Fluid Dynamics and Heat Transfer of Turbomachinery, John Wiley and Sons Inc., New York, 1996.

[3] Han, J. C., Dutta, S., and Ekkad, S. V., Gas Turbine Heat Transfer and Cooling Technology, Taylor and Francis.

[4] Dunn, M. G.,Convective Heat Transfer and Aerodynamics in Axial Flow Tur- bines, ASME Journal of Turbomachinery, 123, pp. 637-686, 2001.

[5] Hylton, L. D., Mihelc, M. S., Turner, E. R., Nealy, D. A., York, R. E., Analyt- ical and Experimental Evaluation of the Heat Transfer Distribution Over the Surfaces of Turbine Vanes, NASA CR-168015 DDA EDR 11209, 1983.

[6] Consigny, H., and Richards, B.E., Short Duration Measurements of Heat- Transfer Rate to a as Turbine Rotor Blade,” ASME Journal of Engineering Power, 104, pp. 542-551, 1982.

[7] Rahman F., Visser J.A., Morris R.M., Capturing Sudden Increase in Heat Transfer on the Suction Side of aTurbine Blade Using a Navier-Stokes Solver, ASME Journal of Turbomachinery, 127, pp. 552-556, 2005.

[8] Bohn, D. E., Bonhoﬀ, B., and Schonenborn, H., Combined Aerodynamic and Thermal Analysis of a Turbine Nozzle Guide Vane, IGTC Paper 95-108, 1995.

[9] Han, Z. X., Dennis, B., H., and Dulikravich, G. S.,Simultaneous Prediction of External Flowﬁeld and Temperature in Internally-Cooled 3-D Gas Turbine Blade Material,ASME paper GT2000-253, 2000.

[10] Rigby, D. L., and Lepicovsky, J., Conjugate Heat Transfer Analysis of Internally-Cooled Conﬁgurations, ASME paper GT2001-0405, 2001.

52 [11] Bohn, D. E., Ren, J., and Kusterer, K., Conjugate Heat Transfer Analysis for Film Cooling Conﬁgurations with Diﬀerent Hole Geometries, ASME paper GT2003-38369, 2003.

[12] Heidmann, J. D., Kassab, A. J., Divo, E. A., Rodriguez, F., and Steinthorsson, E., 2003, Conjugate Heat Transfer Eﬀects on Realistic Film-Cooled Turbine Vane, ASME paper GT2003-38553.

[13] York, D. W., and Leylek, J. H., 2003, Three-Dimensional Conjugate Heat Trans- fer Simulation of an Internally-Cooled Gas Turbine Vane, ASME paper GT2003- 38551.

[14] Facchini, B., Magi, A., and Greco, A., S., 2004, Conjugate Heat Transfer of a Radially Cooled Gas Turbine Vane, ASME paper GT2004-54213.

[15] Arnone, A., Liou, M.-S., and Povinelli, L. A., Multigrid Calculation of Three- Dimensional Viscous Cascade Flows, AIAA Paper 91-3238, 1991.

[16] Ameri, A. A. and Arnone, A., 1992, Navier-Stokes Turbine Heat Transfer Pre- dictions Using Two-Equation Turbulence Closures, AIAA Paper 92-3067 and NASA TM 105817, 1992.

[17] Steinthorsson, E., Liou, M.-S., and Povinelli, L. A., Development of an Explicit Multiblock/Multigrid Flow Solver for Viscous Flows in Complex Geometries, AIAA Paper 93-2380 and NASA TM 106356, 1993.

[18] Kassab, A. J., Divo, E., Heidmann, J. D., Steinthorsson, E., and Rodriguez, F., BEM/FVM Conjugate Heat Transfer Analysis of a Three-Dimensional Film Cooled Turbine Blade, International Journal of Numerical Methods for Heat and Fluid Flow, Vol. 13, No. 5, pp. 581-610, 2003.

[19] Kusterer, K., Hagerdorn, T., Bohn, D., Sugimoto, T., Tanaka, R., Improvement of a Film-Colled Blade by Application of Conjugate Calculation Technique, ASME GT2005-68555.

[20] Hass, D.D., Directed Vapor Deposition of Thermal Barrier Coatings, Ph.D. Dissertation, University of Virginia, 2000.

[21] Evans, A.G., Mumm, D.R., Hutchinson, J.W., Meier, G.H., Pettit, F.S., Progress in Materials Science, Vol. 46, pp. 505-553, 2001.

[22] Evans, A.G., Mumm, D.R., Hutchinson, J.W., Meier, G.H., Pettit, F.S., Mech- anisms Controlling the Durability of Thermal Barrier Coatings, Progress in Materials Science, 2000.

53 [23] Agarwal, P., Patnaik, A. R., Dhar, P., Patnaik, P.C., Physics-Based Model of Thermal Barrier Coating.

[24] Terada, Y., Ohkubo, K., Mohri, T., Thermal Conductivity in Nickel Solid So- lutions, Journal of Applied Physics, Vol. 81, No. 5, 1997.

[25] Wilcox, D.C., Turbulence Modeling for CFD, Second Edition, DCW Industries, Inc., La Canada, CA, 1998.

[26] Wilcox, D.C., Formulation of the κ−ω Turbulence Model Revisited, 45th AIAA Aerospace Sciences Meeting and Exhibit, 2007.

[27] Wilcox, D.C., Simulation of Transition with a Two-Equation Turbulence Model, AIAA Journal, Vol. 32, No. 2, pp. 247-255, 1994.

[28] Speziale, C.G., Abid, R., Anderson, E.C., A Critical Evaluation of Two- Equation Turbulence Models for Near wall Turbulence, AIAA Paper 90-1481, Seattle, WA, 1990.

[29] Menter F.R., Improved Two-Equation κ − ω Turbulence Models for Aerody- namic Flows, NASA TM-103975, 1992.

[30] Kok, J.C., Resolving the Dependence on Feestream Values for the κ − ω Tur- bulence Model, AIAA Journal, Vol. 38, No. 7, pp. 1292-1295, 2000.

[31] Hellsten, A., New Advanced κ−ω Turbulence Model for High-Lift Aerodynam- ics, AIAA Journal, Vol. 43, No. 9, pp. 1857-1869, 2005.

[32] Coakley, T.J., Turbulence Modeling Methods for the Compressible Navier- Stokes Equations, AIAA Paper 83-1693, 1983.

[33] Huang, P.G., Physics and Computations of Flows with Adverse Pressure Gra- dients, Modeling Complex Turbulent Flows, Kluwer, pp. 245-258, 1999.

[34] Kandula, M., Wilcox, D.C., An examination of κ − ω Turbulence Model for Boundary Layers, Free Shear Layers and Separated Flows, AIAA Paper 95- 2317, 1995.

[35] Panton, R.L., Incompressible ﬂow. John Wiley and Sons, Inc, 1995.

[36] Chima, R. V., A k-ω Turbulence Model for Quasi-Three-Dimensional Turbo machinery Flows, NASA TM 107051, 1996.

[37] Schlichting, H., Boundary Layer Theory, 7th ed., McGraw-Hill, New York, pp. 312-313, 1979.

54 [38] Ameri,A.A., Arnone, A., Transition Modeling Eﬀects on Turbine Rotor Heat Transfer, ASME Journal of Turbomachinery, 118, No. 2, pp. 307-313, Apr. 1996.

[39] Garg, V. K., Heat Transfer in Gas Turbines at NASA GRC, International Jour- nal of Heat and Fluid Flow, Vol. 23, pp. 109-136, 2002.

[40] Program Development Corporation, GridProTM /az3000 Users Guide and Ref- erence Manual, White Plains, NY, 1997.

[41] York, D. W., and Leylek, J. H., Three-Dimensional Conjugate Heat Trans- fer Simulation of an Internally-Cooled Gas Turbine Vane, ASME Paper No. GT2003-38551, 2003.

[42] Canelli, C., Sacchetti, M. Traverso, S., Numerical 3-D Conjugate Flow and Heat Transfer Investigation of a Convection-Cooled Gas Turbine Vane, 59 Congresso Nazionale, 2006.