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Helsinki University of Technology Department of Mechanical Engineering Aeronautical Engineering

Mikko Auvinen

NUMERICAL STUDY OF GAS PATH INGESTION INTO TURBINE DISC CAVITY IN AN ENGINE ENVIRONMENT

Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Technology.

Espoo, Finland, April 4, 2005

Supervisor: Professor Jaakko Hoffren Instructor: Charles Faubert, B. Eng.

HELSINKI UNIVERSITY OF TECHNOLOGY

ABSTRACT OF THE MASTER'S THESIS

Author Title of the thesis Date Department

Mikko Auvinen NUMERICAL STUDY OF GAS PATH INGESTION INTO TURBINE DISC CAVITY IN AN ENGINE ENVIRONMENT April 4, 2005 Department of Mechanical Engineering Number of pages 114

Professorship Kul-34 Aeronautical Engineering Supervisor Instructor Professor Jaakko Hoffren Charles Faubert, B. Eng.

In this work computational fluid dynamics (CFD) is utilized to study gas path ingestion into rotor-stator disc cavities in small axial turbine engines. Excessive ingestion leads to over heating and reduced life of critical engine parts in the turbine and therefore must be controlled. This is achieved by supplying cooling flow into the disc cavity from the engine's internal air system at the expense of performance penalties. It is therefore essential for the designers to identify and understand the mechanisms affecting ingestion. This report introduces a new CFD analysis methodology for ingestion studies, which makes use of an interface separating the gas path from the cavity as a solution monitoring and data extraction plane. Series of numerical computations of full Reynolds-averaged Navier-Stokes equations coupled with an appropriate turbulence model were completed with two engine models to identify factors that affect ingestion, obtain rudimentary correlations and record observations relevant to the performed computational analysis. Both 3D models featured detailed CAD geometry of a periodic turbine blade passage sector and its associated upwind rotor-stator disc cavity. The conducted ingestion analysis yielded results that stand in an agreement with prior ingestion studies, but, in the absence of conclusive empirical data, experimental verification could not be attained. The computational solutions allowed the following factors, which affect ingestion, to be identified: Turbine stage loading, the ratio of purge mass flow rate to disc pumping and the spatial design of the annular rim seal. In addition, the obtained correlations for cavity temperature and, especially, pressure levels as a function of ingestion proved encouraging, suggesting a possibility that experimental correction scheme could be used. As a result of this project, the need for further investigations is recognized.

TEKNILLINEN KORKEAKOULU Tekijä Työn nimi Päivämäärä Osasto Professuuri Valvoja Ohjaaja Mikko Auvinen

DIPLOMITYÖN TIIVISTELMÄ

NUMEERINEN TUTKIMUS KAASUKANAVAVIRTAUKSEN TUNKEUTUMISESTA TURBIININ ROOTTORI- JA STAATTORIKIEKON VÄLISEEN ILMATILAAN 4.4.2005 Sivumäärä 114 Konetekniikan osasto Kul-34 Lentotekniikka Professori Jaakko Hoffren Charles Faubert, B. Eng.

Tässä diplomityössä tarkastellaan aksiaaliturbiinin kaasukanavavirtauksen tunkeutumista roottori- ja staattorikiekon väliseen ilmatilaan laskennallisen virtausmekaniikan (computational fluid dynamics, CFD) menetelmin. Liiallinen kuuman kaasukanavavirtauksen tunkeutuminen kyseiseen ilmatilaan aiheuttaa kriittisten moottorikomponenttien ylikuumenemisen ja alentaa niiden elinikää. Tätä haitallista ilmiötä pyritään rajoittamaan paineistamalla kiekkojen välinen ilmatila jäähdytysilmalla, joka vuodatetaan turbiinimoottorin sisäisestä ilmajärjestelmästä. Tällä toimenpiteellä on kuitenkin alentava vaikutus turbiinin suoritusarvoihin ja siksi suunnittelijoiden on tärkeää tuntea ilmiön syntyyn vaikuttavat mekanismit. Työssä esitellään uusi virtauksen tunkeutumisilmiön tutkimiseen sopiva CFDanalyysimenetelmä, jossa hyödynnetään kaasukanavan ja kiekkojen välisen ilmatilan erottavaa rajapintaa laskennan tarkkailussa ja tarkasteltavan aineiston poiminnassa. Raportissa tarkastellaan kahta periodista CAD-geometriaan perustuvaa 3D-mallia, jotka sisältävät yhden turbiinisiiven virtauskanavan ja siihen liittyvän ylävirran puoleisen roottori- ja staattorikiekon ilmatilan. Virtaussysteemin numeerinen analyysi suoritettiin ratkaisemalla diskretoidut Reynolds-keskiarvotetut Navier-Stokes-yhtälöt kytkettyinä tarkoituksenmukaiseen turbulenssimalliin. Sarja laskentatapauksia analysoitiin virtauksen tunkeutumisilmiöön vaikuttavien tekijöiden ja ongelmaan liittyvien riippuvuussuhteiden määrittämiseksi. Lisäksi työ sisältää laskennallisen analyysin toteuttamiseen liittyviä havaintoja. Saadut tulokset suoritetusta virtausanalyysistä ovat yhdenmukaisia aiempien numeeristen tutkimustulosten kanssa. Vakuuttavan empiirisen materiaalin puuttuessa kokeellista vahvistusta ei kuitenkaan saavutettu. Virtaussysteemien laskennalliset ratkaisut sallivat seuraavien virtauksen tunkeutumiseen vaikuttavien tekijöiden identifioimisen: turbiinivaiheen kuormitus, ilmatilaan vuodatetun massavirran suhde roottorikiekon rajakerroksen keskipakoisvoimien aiheuttamaan radiaalimassavirtaan sekä kaasukanavan ja kiekkojen ilmatilan välisen 'virtauskynnyksen' muotoilu. Lisäksi saavutetut ilmatilassa vallitsevan lämpötilan ja painetason korrelaatiot kuuman virtauksen tunkeutumismäärän suhteen osoittautuivat rohkaiseviksi, avaten mahdollisuuden kokeellisen korjausmenetelmän hyödyntämiseen. Lisätutkimuksen tarpeellisuus Pratt & Whitney Canada:lla tuli ilmeiseksi tämän projektin tulosten myötä.

FOREWORD

This study has been completed for Pratt & Whitney Canada under the supervision of Charles Faubert, Manager of Fluid Systems Group, during the period of June 1st ­ December 17th, 2004, in Longueuil, Quebec, Canada. During the following months the report underwent a careful revision at Helsinki University of Technology with the guidance of Professor Jaakko Hoffren.

First and foremost I want to extend my gratitude to Charles Faubert, the instructor of this work, for allowing me to tackle the most interesting and challenging projects in an exceptional industrial environment. His supportive leadership will always be remembered with appreciation. I would also like to single out Dr. Tom Haslam-Jones for sharing his motivational wisdom, Remo Marini for the help with the geometries and the entire Air/Oil & Acoustics/Installations group for making my stay at PWC so special. Moreover, I want to thank my academic supervisor Professor Jaakko Hoffren ­ also on behalf of Pratt & Whitney Canada ­ for his meticulous and critical contribution in the revision process at Helsinki University of Technology. His input will most certainly be appreciated by all readers.

The completion of this Master's Thesis marks my arrival to my next intermediate check point in life, urging a moment to be given for deeper personal reflection. What such mental exercise lays in front of me never fails to draw a reaction that is both emotional and exciting: My own life has become the most commanding testimony of the power of positive support, inspiration and determination. This recognition is always followed by the acknowledgement that I owe this fact to certain individuals who have influenced my life profoundly. I would, therefore, like to take this opportunity to express my gratitude to the following people:

Wendy Lupien for opening the door to Pomfret School and for all the help and support she has provided. All my teachers and coaches at Pomfret. T.J. and the rest of the McMeniman family for the memorable year in Tewksbury. My most respected mentor in both ice hockey and life, Coach Addesa, for all the personal guidance and, especially, for the opportunity to enter Trinity College. Coach Dunham at Trinity for materializing my dream ­ and providing four years of it. My most inspiring teachers at Trinity, especially, Prof. Mertens, Prof. Palladino, Prof. Peattie, Prof. CruzUribe and Prof. Broadbridge whose influence is still driving me forward in the academia. All my dear friends from Trinity for all those special times `neath the elms.

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Finally, I would also like to thank my parents and siblings for their support over the years and, with a more personal note, I wish to extend my warmest embrace to Heini Lähteenmäki to express my gratitude for the happiness and light she has brought to my life. Espoo, April 4th 2005

Mikko Auvinen

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FOREWORD ................................................................................................................................................ 1 LIST OF SYMBOLS.................................................................................................................................... 5 1 INTRODUCTION ................................................................................................................................ 9 1.1 1.2 1.3 2 MOTIVATION AND OBJECTIVES ....................................................................................................... 9 BACKGROUND ............................................................................................................................... 10 PROJECT DESCRIPTION .................................................................................................................. 12

THEORY............................................................................................................................................. 15 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 ON FLUID FLOW IN ROTATING DISC CAVITY SYSTEMS ................................................................. 15 Basic Equations in Stationary and Rotating Frame of Reference ........................................ 15 Boundary-Layer Equations................................................................................................... 20 Integral Equations ................................................................................................................ 22 Applications .......................................................................................................................... 24 COMPUTATIONAL FLUID DYNAMICS ............................................................................................. 26 Governing Equations ............................................................................................................ 26 Rotational Forces ................................................................................................................. 28 Turbulence Models ............................................................................................................... 29 Numerical Discretisation...................................................................................................... 31 CFX-5 ................................................................................................................................... 34

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ANALYSIS.......................................................................................................................................... 36 3.1 3.1.1 3.1.2 3.1.3 3.2 3.2.1 3.2.2 3.2.3 3.3 3.3.1 3.3.2 3.3.3 3.3.4 MODEL GEOMETRY ....................................................................................................................... 36 ATFI HPT ............................................................................................................................. 36 PW307 HPT2........................................................................................................................ 39 On Cavity Modeling Issues................................................................................................... 42 COMPUTATIONAL MODEL ............................................................................................................. 44 Mesh Generation Procedure................................................................................................. 44 ATFI Unstructured Grid ....................................................................................................... 45 PW307 HPT2 Unstructured Grid ......................................................................................... 47 CFD METHODOLOGY .................................................................................................................... 50 Overall Solution Strategy ..................................................................................................... 50 Boundary Conditions and Computational Settings............................................................... 51 Expressions........................................................................................................................... 60 Solution Monitoring and Convergence................................................................................. 63

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RESULTS AND DISCUSSION ......................................................................................................... 68 4.1 4.2 4.3 4.4 4.5 4.6 GENERAL REMARKS ON RESULTS ................................................................................................. 68 ATFI-INGST ................................................................................................................................ 69 ATFI-WAKE................................................................................................................................ 82 ATFI-FER..................................................................................................................................... 86 PW307-INGST ............................................................................................................................. 88 PW307-PTM12442....................................................................................................................... 95

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CONCLUSIONS................................................................................................................................. 99 RECOMMENDATIONS ................................................................................................................. 101

REFERENCES ......................................................................................................................................... 103 APPENDIX A: MATLAB-PROGRAM FOR 2D INLET PROFILE .................................................. 104 APPENDIX B: MATLAB-PROGRAM FOR WAKE GENERATION............................................... 109 APPENDIX C: ADDITIONAL SOLUTION DATA ............................................................................. 113

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List of Symbols

A C D D* E Ff Gf Hf H I L Ma Q R R* R0 Re S T T* U V Vr V Vz Vr* V* Vz* Vfe Y b cp f h k

Area, representative area in pressure dissipation term Constant Diameter Dimensionless diameter Total energy Flux vector Flux vector Flux vector Total enthalpy Rothalpy Reference length Mach number Heat flow Radius Dimensionless radius Universal gas constant Reynolds number Vector of source terms Temperature Dimensionless Temperature Velocity vector, vector of conservative variables Velocity in relative frame of reference Cylindrical velocity components Dimensionless Cylindrical velocity components (Vx* = Vx / Vtot ) Volume of finite element Trend line function Rotor radius Specific heat under constant pressure Dependent variable Enthalpy Thermal conductivity, turbulent kinetic energy Mass flow rate

& m

m* n p p*

& & Mass flow rate ratio, m / m gaspath

Normal vector Pressure Dimensionless pressure 5

q r rp s t u v w wm x y z Greek Symbols

' ij

Heat flux, heat generation term Cylindrical coordinate Vector from an upwind node to an integration point Axial distance between rotor and stator, entropy Time Cartesian velocity component, relative velocity Cartesian velocity component, relative velocity Cartesian velocity component, relative velocity Molecular weight Cartesian coordinate Cartesian coordinate Cartesian coordinate, cylindrical coordinate

Algebraic difference Viscous dissipation term Summation operator Angular speed of the rotor Angular speed of the rotating frame of reference Angle of attack in relative frame of reference, blend factor for advection scheme Kronecker's delta Boundary layer thickness Cylindrical coordinate, nodal value in CFD Dynamic Viscosity Kinematic viscosity 3.14159 ... Geometric angle Fluid density Shear stress Turbulent frequency in k- model Differential operator

Subscripts

E M c s, stat t rot

Energy Momentum Value at inviscid core Static quantity Total quantity Rotational term 6

Cor cfg 0

Coriolis term Centrifugal term Value at rotor

Superscripts

`

Fluctuating value

7

8

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1.1

INTRODUCTION

Motivation and Objectives

This numerical investigation addresses the problem of hot airflow from an axial gas turbine's annular gas path into the internal rotor-stator disc cavity space. See Figure 1.1 for guidance. The phenomenon is referred to as gas path ingestion and, due to its detrimental effect on the engine, it is a matter of particular concern to the designers and, thus, need to be controlled. The ingestion of hot gas from the mainstream into the disc cavity imposes additional thermal stresses to the rotating turbine disc, which leads to reduced operational life, and causes unwanted heating of the air that is internally supplied for blade cooling. These effects have become increasingly dangerous as the temperature levels in modern gas turbines reach the melting point of most materials necessitating elaborate and critical cooling mechanisms, while, at the same time, the performance and reliability requirements for the engines continue to tighten.

Vane Hot Exhaust

Ingestion

Blade

Gas Path

D

isc

Turbine Disc (Rotor)

Stationary Wall

Ca vit y

Rotating Wall

Cooling Air Flow (Purge Flow)

Figure 1.1 A schematic illustration of a single stage blade passage and disc cavity system in an axial gas turbine engine.

The method to control ingestion is based on pressurizing the cavity by supplying adequate cooling air ­ also called purge flow ­ from the internal air system of the engine and utilizing a seal design at the annular outer rim. But, there are limitations and performance cost associated with 9

these remedies: The cooling purge flow that is vented into the gas path results in depreciation of turbine stage efficiency and the rim seal designs are restricted in reducing the space between the rotating disc and the static wall because adequate margins must be included to allow relative movements of engine parts during operation. Therefore it is increasingly important for the designers to understand the ingestion phenomenon so that unnecessary performance penalties can be avoided. However, the complex nature of the flow problem and the inapplicability of existing analytical tools have ruled out most theoretical efforts in the past to understand ingestion in an engine environment. For that reason, the designers at Pratt & Whitney Canada, to this date, have solely relied upon general correlations ­ that do not enjoy high confidence ­ obtained from experimental rig tests to control ingestion. As a consequence, the subsequent predictions for the necessary level of cavity purge flow have been assigned appropriate safety factors at the expense of performance penalties. And, for that reason, the incentive to acquire better understanding of ingestion and to improve the methods of prediction has become greater than before. This project will represent an initial step in the greater effort to achieve these goals and its primary objectives can be identified as follows:

-

Explore available computational fluid dynamics (CFD) capabilities to conduct ingestion analysis.

-

Establish relevant CFD methodology. Introduce underlying dependencies between physical quantities in the cavity and ingestion.

-

Identify factors that affect ingestion. Provide recommendations for future analysis.

1.2

Background

The earlier theoretical efforts ­ by academia and industry ­ to model gas path ingestion into disc cavities range from analytical integral equation formulations (discussed in the next section) to very recent three-dimensional 360º-turbine section and periodic unsteady 1.5-stage engine CFD models. In 2000 Hills, Chew and Turner [1] published a study in which 4 different CFD models of a simplified single stage turbine section including a disc cavity were analyzed and compared to experimental measurements. The different models had the same cavity and gas path geometries but varying vane-blade configurations and numerical analysis techniques. The cases were: Vane only ­ steady state, blade only ­ steady state, vane and blade ­ steady state with mixing plane,

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vane and blade ­ unsteady with sliding plane 1. The investigation concluded that the best agreement with experimental values was achieved with the time accurate vane-blade model, which simulated the vane-blade interaction while requiring the most computational effort by drastic margin. It was also emphasized that, when acting alone, the pressure asymmetry created by the rotor blades initiated far more ingestion than the vanes upstream and that the disc pumping was only observed to have a secondary effect on ingestion. Although the report provided useful information about critical modeling issues, the gross geometric simplifications and impractical operating conditions degraded the results' applicability to real engine applications.

In 2003 the author [2] completed a limited numerical ingestion study at Pratt & Whitney Canada using a finite element in-house CFD code Ns3D. The project attempted to replicate the experimental results obtained by Wilford and Raymond [3] in 1982-rig test, which utilized a simplified gas path and disc cavity model that, for the main portion of the experimental cases, did not include either vanes or blades. It was discovered that the flow system dictated by disc pumping ­ a phenomenon correlated with disc's Reynolds number, Re (= b2/) ­ lead to an unsteady behavior at the rim seal as the purge mass flow rate was reduced below the radial mass flow due to centrifugal forces in the rotor's boundary layer. This subsequent difference in mass flow rates was compensated by ingestion from the gas path that occurred through a moving periodic pattern of inward and outward flowing regions at the rim seal. The net radial mass flow rate through the annular rim was also observed to fluctuate with time further confirming the unsteady nature of ingestion driven by disc pumping. The theoretically predicted time averaged ingestion levels were in an encouraging agreement with the experimental values providing added confidence in CFD analysis of ingestion. Even though the results yielded little information relevant to real engine cases, the lessons learned proved valuable in the current study.

A consortium of industrial and academic researchers at ASME Turbo Expo 2004 published the most recent study [4] on ingestion in an axial gas turbine. The project involved 4 threedimensional turbine stage models, replicating an experimental 1.5 stage turbine rig, with varying degree of geometric detail and utilized commercially available and industrial in-house CFD codes for the numerical computations. The models can be described as follows: 1) Periodic 22.5-degree sector model of 1.5 stage turbine with upwind and downwind turbine disc cavities. Unsteady, sliding mesh computation. 2) Periodic 22.5-degree sector model of 1.5 stage turbine with upwind cavity only. Unsteady, sliding mesh computation.

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The turbine blades in the models were replaced by rotor pegs ­ small cylindrical columns ­ to generate a similar

pressure asymmetry at the annulus.

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3) 360-degree model of cavity only ­ no blades or vanes. Constant radial boundary condition profiles imposed at gas path inlet and outlet. Unsteady computation. 4) Periodic 22.5-degree sector model of cavity and gas path ­ no blades or vanes. Constant radial boundary condition profiles imposed at gas path inlet and outlet. Unsteady and steady state computations. The primary objective of the project was to investigate which model gave the best compromise between sufficient accuracy and computational effort and to verify the existence of a large scale rotating structure in the upwind cavity detected in the experimental test rig. The study concluded the following: - Sufficiently accurate predictions of ingestion could be reached for the following cases: Front cavity with high cooling air mass flow. (Unsteady, sliding mesh models: 1 and 2.) Back cavity with high and low cooling air mass flow. (Unsteady, sliding mesh models.)

- The solutions did not demonstrate noteworthy changes with different turbulence models. - The calculated results for front cavities and low cooling air mass flow are not satisfactory when the simplified sector model (4) is used. - The 360-degree model captured the large-scale flow structure in the cavity while this information is lost when periodic boundary conditions are imposed in a sector model.

Despite the fact that the results of this study offer further encouragement to continue the CFD effort, it falls short in identifying the best analysis technique that would yield acceptable results for practical computational effort. The gap between the 1.5 stage sliding mesh models and the simplified cavity-only models that were investigated in the study is unnecessarily wide effectively leaving out other potential analysis approaches. Moreover, the cavity geometry and rim seal design of the project models, which were based on an experimental rig, were, once again, in a stark contrast to those found in real engines. So, even though significant headway has been made in this field, the issues of gas path ingestion in a real engine environment and the choice of the most appropriate CFD analysis approach remain open questions. And it is these questions that constitute the underlying motivation for this work.

1.3

Project Description

In this study detailed CFD ingestion and cavity flow analysis is conducted with two different computational models that feature the design definition of the particular engine's cavity and blade passage geometry. When the prior studies utilized a species solution to evaluate ingestion, an alternative method of obtaining information concerning ingestion, convergence and flow interaction between the gas path and the disc cavity will be introduced in this study. It utilizes a fluid-fluid interface at the rim seal through which mass and energy flow properties are monitored 12

as the solution advances. This method allows a physically based real-time monitoring of the critical area of the solution field that yields additional information about the nature of the flow system. The CFD models are identified according to the engine they represent and therefore labeled:

1. ATFI High Pressure Turbine Stage (HPT).

nd 2. PW307 High Pressure Turbine 2 Stage (HPT2).

ATFI is an engine that is still in a development phase and scheduled to have its first engine run in January 2005 while PW307 is a complete production engine that is already in service. Both models entail a periodic sector of a single turbine blade passage coupled with an upstream disc cavity. The decision to utilize models that include only a single turbine blade ­ and no up or downstream vanes ­ is based on the reported observations confirming that the pressure asymmetry generated by the blade is the primary force in the gas path driving ingestion into the upwind cavity. The exclusion of the vanes enables the computational model size and subsequent computational time requirement to be dramatically reduced while imposing a secondary accuracy penalty. Since the results produced by the model are reasoned to be off by a systematic margin, the author postulates that the effect of this accuracy penalty can be countered with a simple correction mechanism that utilizes experimental measurements. In addition, due to the intricate cavity features and the small size scale of the modeled engine components, it is assumed that significant large-scale flow structures mentioned above do not form in the system, justifying the use of a periodic sector model. It is acknowledged that imposing a periodic boundary condition will destroy some natural information of longer wavelengths in the cavity, but their influence is considered negligible when compared to the primary `driving' mechanisms of the cavity flow.

In this report the two models and their results are purposefully discussed separately with limited comparisons for they represent two very different engine environments and have been subjected to dissimilar modeling choices. The decision to consider two essentially different models, instead of modifying the initial ATFI model and conducting more thorough parametric study with it, was deliberate because one of the priorities of this project was to focus on generic problems and modeling issues that arise when different engines and configurations are studied. Based on the accumulated experience and results, a series of recommendations is provided and learned lessons documented for future reference. The beginning of this report is devoted for a brief discussion on the analytical and historical basics of cavity flow and ingestion analysis followed by a concise presentation of the fundamentals of computational fluid dynamics ­ the analysis tool of this investigation. In chapter 3 the geometric and computational grid features of the models are illustrated and the relevant CFD methodology and boundary condition data presented. The 13

following chapter 4 contains the presentation and discussion of all significant results, which are shown in a dimensionless form.

All the CFD analysis and subsequent post processing in this work have been performed with commercially available CFX-5 unstructured finite volume flow simulation software. Both threedimensional model geometries have been created with CATIA and spatially discretised with ICEMCFD 4.2 mesh generation software.

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2.1

THEORY

On Fluid Flow in Rotating Disc Cavity Systems

In this section the fundamentals of rotating disc systems are briefly discussed in effort to introduce the integral boundary layer equations, which constitute the basis for analytical disc cavity analysis. It ought to be mentioned that, regardless of the inapplicability of the presented analytical tools and solution techniques to the considered problem, the following presentation is relevant nonetheless to the current study for it introduces the underlying physical foundation of the analysis and provides some historical reference. The presentation follows the approach outlined in Owen and Rogers' [5] and Schlichting's [6] classic textbooks and adopts their notation in order to preserve consistency, although, this author has determined the content and order of the presentation as well as the level of detail of the discussion.

As is the case for all Newtonian flow systems, the Navier-Stokes and the energy equation constitute the mathematical framework for rotating-disc flow analysis. In their complete form, however, these equations lend little help for there exists only few analytical solutions, but their approximate equivalents do provide essential information about these flow systems. For instance, the boundary-layer approximations, that are often valid near a solid surface, result in parabolic differential equations, which yield to a number of solution schemes that are applicable to a variety of cases. The boundary-layer equations can also be expressed in a partially integrated form that is of particular use due to its simplicity and applicability to turbulent flows. These integral equations represent the historical starting point of rotor-stator analysis and have paved the way for attacking problems related to this project. The following presentation describes the underlying analytical foundation of the problem, clarifies the applied approximations and introduces the basic equations before providing a brief discussion on available applications as specified by Owen and Rogers.

2.1.1

Basic Equations in Stationary and Rotating Frame of Reference

Consider a system as shown in Figure 2.1 with two discs of radius b in cylindrical coordinates (r,

, z). The system may also include a cylindrical shroud (see Figure 2.2) that can be attached to

either the rotor (a), the stator (b), or be shared by both (c), at r = b. The rotation of the rotor disc occurs around the z-axis ­ with the shown direction being positive according to the right hand rule ­ and the origin of the coordinate system taken as the point where the z-axis intersects with the rotor. According to this convention the rotor lies in the plane z = 0 and the stator in the plane z =

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s, where s is the distance between the two discs. The angular speed of the rotor is expressed in radians/sec.

Figure 2.1 Schematic diagram of rotor-stator system in cylindrical coordinates.

Figure 2.2 Different axial shroud configurations in rotor-stator (R ­ S) systems.

In a stationary frame of reference, the velocity of the fluid at a point (r, , z) is taken to be U = (Vr, V, Vz) and its density , temperature T and pressure p. For the purpose of this chapter, the discussion is limited to a steady, axisymmetric case in which,

f =0 t

and

f =0

for all dependent variables f.

Momentum and Continuity Equations in Stationary Frame of Reference

Assuming steady flow and that the dynamic viscosity and kinematic viscosity are constant, the continuity and incompressible Navier-Stokes equations for the considered system in a cylindrical coordinate system become:

Vr Vr Vz + + =0 r r z

(2.1)

Vr V V V 1 p + 2Vr - 2 Vr r + Vz r - =- r r r z r

Vr V r + Vz V z - V VrVz = 2V - r2 r

2

(2.2)

(2.3)

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Vr

V Vz 1 p + 2Vz + Vz z = - z z r

(

)

(2.4)

where

2 =

2 2 1 + 2. + r 2 r r z

(2.5)

In reality, however, most flow cases involve turbulence, which obviates the validity of steady-state equations. For these cases, the Navier-Stokes equations require some additional manipulations to take the effect of small-scale fluctuations in flow parameters into account. This is done through a procedure that entails breaking the velocity and pressure variables up into two parts:

V =V +V

and

p = p + p

(2.6)

where the terms are defined as follows,

V =

2 1 V dt (t 2 - t1 ) t1

t

and

2 1 V dt = 0 (t 2 - t1 ) t1

t

with similar expressions for pressure. Thus, it can be seen that the first term represents a timeaveraged value of the variable over a suitably long time interval while the second expression defines the term, identified with a prime, as a purely fluctuating part whose time average is zero. In addition, it is assumed that the time-averaged values of the fluctuation terms' derivatives are also zero.

By substituting (2.6) into equations (2.1 ­ 2.4), averaging the resulting equations and applying manipulations that are in accordance with the fore-mentioned definitions, the continuity and conservation of momentum equations become:

Vr Vr Vz =0 + + z r r

(2.7)

V V V Vr 1 p =- + 2Vr - 2 Vr r + Vz r - r z r r r 2 V 1 2 rV - - z VrVz + r r r

2

(2.8)

( )

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Vr

V r

+ Vz

V z

-

V VrVz = 2V - 2 r r V Vr 1 VVz + - rV Vr - r r r z

(2.9)

(

)

Vr

V z V 1 p + Vz z = - + 2V z z r z

(

)

(2.10)

-

2 1 rVrV z - V z z r r

(

)

The extra terms on the right-hand side of the equations ­ which include the fluctuation variables ­ introduce stress terms that are in addition to stresses due to mean pressure and viscous forces. They arise purely from the turbulent, unsteady nature of the flow. These terms are labeled Reynolds stresses and they are subject to further approximations and turbulence modeling issues.

Momentum Equations in Rotating Frame of Reference

When a rotating system is analyzed using a cylindrical coordinate system it is often convenient to use a frame of reference rotating with angular velocity of

(which may or may not be equal to

the rotational velocity of the rotor, ) about the z-axis. In such a frame of reference the velocity components are relabeled (u,v,w) and defined as follows:

u = Vr

v = V - r

w = Vz

(2.11)

Substituting (2.11) into the original equations (2.1 ­ 2.4) and noting that the static pressure p remains unaffected by the coordinate change, we obtain

u u w + + =0 r r z

u u u v 2 u 1 p +w - - 2v - 2 r = - + 2u - 2 r z r r r

(2.12)

(2.13)

u u

v v uv v + w - + 2u = 2 v - 2 r z r r w w 1 p +w =- + 2w . r z z

(2.14)

(

)

(2.15)

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The terms 2 u and 2 u that appear in equations (2.13) and (2.14) respectively, represent apparent forces, which arise due to the rotating reference frame and are typically referred to as Coriolis terms. This form of the momentum equations is particularly useful when the nonlinear inertial terms are small compared to the Coriolis terms, which would justify their removal completely. The following equations end up being dramatically simplified, but can only be applied to problems in fields like meteorology and oceanography.

Energy Equation for Laminar and Turbulent Flow

For the purpose of laying down the basic analytical framework and presenting existing approximate solution techniques, it is assumed that the flow is incompressible and the fluid possesses constant thermodynamic properties; viscosity , thermal conductivity k and specific heat under constant pressure cp. Even though temperature changes will not affect any of these properties the temperature rise due to heat addition or viscous-dissipation will result in heat flow from one part of the fluid to another. This heat transfer can happen either through thermal convection or conduction, as described by the energy equation for laminar flow:

Vr

T 1 T k + Vz = 2T + r z c p c p

(2.16)

where the viscous-dissipation term is written

2 2 2 2 Vz V V V Vz Vr Vr Vr + - + + = 2 + + 2 + 2 z r z r z r r r 2 2

(2.17)

In equation (2.16) the terms on the left-hand side correspond to convection of heat while the first term on the right-hand side to conduction. Even though the dissipation term is often neglected in analytical considerations, this is most certainly not the case in practical rotor-stator problems for it represents a significant heat addition into the system.

For turbulent flow cases, the energy equation is modified by the same averaging procedure as described earlier in the context of the momentum equations, resulting in

Vr

1 1 k T T rVrT - VzT + . = 2T - + Vz c p r r z z c p r

(

)

( )

(2.18)

The two additional terms on the right-hand side of the equation introduce a contribution that is caused by the fluctuating part of the turbulent velocity and temperature. There are also additional

19

dissipation terms due to turbulence, but, for the sake of conciseness, they will not be presented here.

2.1.2

Boundary-Layer Equations

The mathematical complexity of the momentum and energy equations that are applied to rotorstator systems can be reduced if the distance between the discs is small compared with their radius, s << b. This being the case, the following assumptions can be made according to Owen and Rogers [5]:

(I)

The velocity component, V z , is very much smaller in magnitude than either of the other two components.

(V

z

<< V ,V z << Vr .)

(II)

The rate of change of any variable f (other than the pressure) in the direction normal to the disc is much greater than its rate of change in radial or in tangential direction.

f f f f z >> , z >> r .

(III) The pressure depends only on distance from the axis of rotation.

p = p(r ).

These approximations can also be applied if the flow between the discs is primarily inviscid, except in thin layers close to the discs ­ that is, in boundary layers. Simplifying the original equations (2.1 ­ 2.4), (2.7 ­ 2.10) and (2.16 ­ 2.18) according to the specified assumptions, by removing the insignificant derivatives, leads to what are known as the boundary layer equations:

Vr Vr Vz + + =0 r r z

(2.19)

V V V 1 p 1 r Vr r + Vz r - =- + r z r r z

2

(2.20)

Vr

V r

+ Vz

V z

+

VrV r

=

1 z

(2.21)

0=-

Vr

p z

(2.22)

1 q 1 T T + Vz = + r z c p z c p

(2.23)

where

20

r = =

q = -k

Vr - Vr Vz z V z - V V z

(2.24)

(2.25)

T + c p T Vz z V Vr + z z

(2.26)

= r

(2.27)

Notice that the notation has been simplified by omitting the bars to indicate average values since these equations are valid for both laminar and turbulent flows, with the only exception that the fluctuating terms in and q are identically zero in laminar cases. Inspecting the resulting equations allows one to conclude that they are considerably simpler than the complete versions and demonstrate some noteworthy changes. First, three conservation of momentum equations have been reduced to two as the equation in z-direction collapses entirely. Second, retaining only derivatives with respect to z (compared with those with respect to r and ) has reduced the viscous and Reynolds stresses to a single shear stress vector , whose components are

(

r

, ) .

Third, the mathematical nature of the problems has changed; the original steady-state NavierStokes equations were of elliptic type 2 with respect to the coordinates while the obtained boundary-layer equations end up being parabolic. The profit of ending up with parabolic equations lies in the simplified solution techniques; while elliptic equations, in order to be solved, require known boundary conditions on all borders encircling the solution domain, parabolic equations require only one set of boundary conditions. More specifically, it suffices to specify the physical boundary conditions at the inlet of the solution field to integrate (in the direction of the characteristics) the rest of the solution across the domain.

As can be seen from equation (2.22), which arises from the specified assumptions, the pressure in a rotor-stator system remains constant throughout the thickness of the boundary layer and we can therefore assign the pressure value of the inviscid core ­ space between the two discs where viscous effects can be neglected ­ to the pressure variable in the expressions, p = pc. Further more, we can use the inviscid flow theory to reason that the radial velocity component in the inviscid core area is approximately zero (Vr,c 0) and obtain the following correlation for pressure

2

The full set of compressible flow equations, on the other hand, constitute a hyperbolic group ­ given that density

depends on pressure. See chapter 0 on Computational Fluid Dynamics.

21

1 pc V ,c - = r r

2

(2.28)

where the subscript c refers to the values at the inviscid core of the rotor-stator system. Due to clarity and conciseness, we will now confine our presentation to a system where the space between the rotor and the stator, s, is much larger than the rotor's boundary layer thickness. In this case the treatment is simplified for the rotor's boundary layer can be treated independent of the stator. Regardless of this exclusion, an elementary analysis of the rotor's boundary layer equations provides illustrative insight about the nature of the flow field in a rotor-stator system.

Using the result (2.28) with the equation of continuity, the equations (2.19 ­ 2.23) can be written for the rotor in a convenient form:

(rVr ) + (rVz ) = 0 r z r r 2 2 2 rVr + (rVrVz ) - V = -V ,c + r z z

r 2 2 2 r VrV + r VzV = z z r

(2.29)

( )

(

(2.30)

)

(

)

(2.31)

(rVrT ) + (rVzT ) = r q + r c p z c p r z

(2.32)

These equations can be integrated over the thickness of the boundary layer to reduce the equations to a set of ordinary, first order differential equations, also known as integral form boundary layer equations. This simplification ­ though being restrictive, for further assumptions are required ­ exposes the rotating disc problem to numerous solution schemes.

2.1.3

Integral Equations

In areas where the boundary layer approximations apply, the layer is assumed to be of thickness

, which, in turn, is assumed to be a function of the radius, = (r). Integrating the corresponding

equations (2.29 ­ 2.32), presented in the previous section, across the boundary layer thickness yields a set of integral form boundary layer equations that provide a further simplified approach to the analysis. Considering a rotor-stator system where s >>, the rotor's boundary conditions used for the integration are:

22

Vr(0) = 0, V(0) = V,0 , Vz(0) = 0, T(0) = T0, (0) = 0 Vr 0, V V,c , Vz Vz,c , T Tc, = 0 when z =

(2.33)

where the subscript 0 indicates values at the surface of the rotor. The resulting integrated equations provide the radial dependence of average values (the average being across the layer) of the dependent variables, but assumptions have to be made on how the dependent variables vary with z. Performing the integration from z = 0 to z = on equations (2.29 ­ 2.31) yield,

(r ) r Vr dz + rVz ,c = 0 r 0 (r ) (r ) r Vr 2 dz - V 2 - V ,c 2 dz = - r r ,0 r 0 0

(2.34)

(

)

(2.35)

(r ) 2 2 r VrV dz + r 2V ,cVz ,c = - r , 0 r 0

(2.36)

Here it should also be noted that the radial mass flow-rate within the boundary layer is given by

& m0 = 2r Vr dz

0

(r )

(2.37)

which allows the equation (2.34) to be expressed as

V z ,c = -

& 1 m0 . 2r r

(2.38)

These results are significant because they underline two central features that are common to rotating disc systems: The radial mass flow rate due to `disc pumping' and the presence of a replenishing flow that arise as a consequence. These parameters are illustrated in Figure 2.3 where the flow field is depicted in the absence of a stator. What happens to the flow field when the stator is introduced can now be deduced readily: Due to conservation of mass, the replenishing flow has to be initially `ingested' radially inward along the stationary wall before it is fed into the rotor's boundary layer. This flow behavior, which is illustrated in Figure 2.4, identifies the concept of ingestion in rotor-stator systems.

The momentum equations (2.35), (2.36) combined with (2.38) and the integral energy equation (not given here) establish the starting point for analytical analysis of rotor-stator systems. At this point it can be justifiably concluded that all the assumptions, which were made in the process

23

leading up to these integral equations, create a set of highly restrictive margins for the applicability of the analysis. For instance, one of the very first assumptions made in section 2.1.1

f =0 t

and

f = 0

already eradicates the connection between the presented theory and the physical framework governing gas path ingestion in rotor-stator systems, for the phenomenon is found to be both unsteady and - dependent. This being the case, the detailed discussion of the ensuing solution techniques and applications is left for Ref. [5] and the following section is only devoted for a brief overview on the topic.

Inge stion

r=b

r=b

& m0

Ambient (Core)

Radial Boundary Layer

& m0

Radial Boundary Layer

Ro to r

Vz ,c

Ro to r

Core

r

Vz ,c

St at or

r

r=0

z

r=0

z

Figure 2.3 Schematic illustration of the radial boundary layer on a rotating disc and the subsequent replenishing flow field.

Figure 2.4 Schematic illustration of the flow system in a rotor-stator system.

2.1.4

Applications

The most noteworthy agreement between the values produced by the integral boundary layer equations and experimental data is reached with laminar and turbulent flow over a single rotating disc. The solution techniques primarily involve defining a new independent variable and assuming that the flow variables have a functional form with respect to the new variable. In cases where the boundary layer becomes turbulent, the velocity is assigned either a 1/n-power-law or logarithmic boundary layer profile, which allow explicit expressions for the shear stress at the surface to be derived. This procedure produces a new set of differential equations ­ in laminar cases called von Kármán equations ­ that are solved by numerical integration. The arithmetic for the turbulent momentum equations becomes more involved, but leads eventually to similar, simultaneous differential equations, which can also be solved numerically. 24

For the rotor-stator problems the picture does not seem as positive. Some success has been accomplished in predicting velocity distributions and disc moment coefficients with the integral equations for cases without superimposed flow, but they are limited to such low

Re = b 2 / values that these cases have no practical significance. For higher Re cases

the assumptions begin to obstruct the description of the proper flow physics to such degree that practically all the significant efforts to tackle these problems to this date have utilized numerical, finite-difference schemes for either elliptic Navier-Stokes or parabolic differential boundary layer equations. Similarly for rotor-stator problems with superimposed flow the applicability is very limited. However, some very interesting results have been produced with the integral equations by applying additional assumptions and ensuring that the continuity requirements are satisfied. According to continuity, when the superimposed mass flow is less than the mass flow `pumped' outward by the boundary layer, ingestion must occur from the outside to satisfy conservation of mass. This radial inflow from the surroundings that comes down along the stator disc does manifest in the obtained results. Nonetheless, the overall accuracy of the solution is rather low and the agreement with similar solutions computed by more sophisticated numerical techniques only fair, at best. It should also be mentioned that adding a geometric feature ­ such as a shroud ­ only complicates the analysis and further invalidates the applicability of the integral equations. Therefore, as mentioned by Owen and Rogers, the hope in reliable predictions of ingestion under realistic conditions falls upon the promise of solving the full, three-dimensional, turbulent NavierStokes equations for rotating systems. With genuine excitement and due admiration to the scientific community, it must be said that considering the current state of computational fluid dynamics, this hope has been realized to large extent. It is thereby most appropriate that the following chapter is devoted for a discussion on the modern numerical methods that have made this study possible.

(

)

25

2.2

Computational Fluid Dynamics

The science of fluid dynamics has traditionally relied upon theoretical and experimental 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 continuous 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, as it has developed, an essential and cost effective analysis and design tool in various industries. For instance, the completion of this study has relied upon one of the most advanced CFD tools available today, allowing this report to function as a case-specific demonstration of ­ perhaps some new ­ CFD capabilities. However, even though the field has come a long way, CFD has not removed the need for experimental validation (yet) 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. All the CFD analysis in this study has been performed with commercially available unstructured, finite-volume solver CFX-5 and with the accompanying monitor and post processing modules. For this reason, while trying to keep the theoretical content of the presentation as general as possible, some aspects are discussed according to the approach specified in CFX-5 Solver Theory [7]. A brief description of CFX-5 flow simulation software is provided at the end of this chapter.

2.2.1

Governing Equations

At its most complete form, computational fluid dynamics relies upon the complete theoretical description of fluid- and thermodynamics to achieve quantitative predictions 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 governing equations ­ often referred to as The Navier-Stokes equations ­ are customarily expressed in a compact differential vector form:

26

U F f G f H f + + + +S =0 t x y z

(2.39)

where U = (, u, v, w, H-p)T is a vector of conservative variables which are to be solved,

S=(0, SMx, SMy, SMz, SE)T is the vector including the source terms and Ff, Gf and Hf represent the

flux vectors in x, y and z directions respectively. The flux vectors are written as

u u 2 + p - xx uv - xy Ff = uw - xz uH - u - v - w + q xx xy xz x

(2.40)

v uv - xy Gf = v 2 + p - yy vw - yz vH - u - v - w + q xy yy yz y w uw - xx Hf = vw - xy w 2 + p - xz wH - u - v - w + q xz yz zz z

(2.41)

(2.42)

Here the notation follows the standard adopted in literature (see List of Symbols) qi representing the heat flux term and the stress tensors

ij given by

ij = -

2 3

u u u k ij + i + j x xk j xi

(2.43)

where ij is the Kronecker's delta. It should be noted that the energy equation is written in terms of total enthalpy, H = E + p/, where E is the total energy. Writing the energy equation in this form necessitates the addition of pressure term in the time derivative of the principal equation. This

27

kind of formulation is typically preferred when the solver utilizes pressure correction scheme to obtain the final solution. The presented five equations have seven unknowns (u, v, w, p, T, , H) which calls for two additional equations for closure. This is achieved by adding the equation of state, which relates density to pressure and temperature, and the constitutive equation that relates enthalpy to pressure and temperature. The equation of state for an ideal gas is generally written in a simple form

=

wm ( p - pref ) R0T

(2.44)

with wm as the molecular weight of the gas and R0 as the universal gas constant. The general constitutive equations, on the other hand, require a two-step procedure: For any general change in conditions from (p1, T1) to (p2, T2), the change in enthalpy and entropy, dh and ds, are first computed at constant pressure and then at constant temperature. For enthalpy the steps are calculated with

h2 - h1 = c p dT +

T1

T2

p2

1 T dp 1 + T p p1

(2.45)

where the second term will cancel out when density obeys the ideal gas law. The total change in entropy, in turn, is given by

s2 - s1 =

T2

T1

T

cp

dT +

p2

1 dp. 2 T p p1

(2.46)

In CFX-5 the integrals and differentials are computed numerically using the given expressions for

and cp to construct property tables. The solver then uses adaptive interpolation to assign

enthalpy and entropy values, which define the state of the fluid.

2.2.2

Rotational Forces

In contrast to the treatment of simplified conservation of momentum equations (2.13) and (2.14), in computational fluid dynamics the apparent forces that arise due to the rotating frame of reference are treated as source terms in equation (2.39) and thereby do not require any modification to the principal terms. The Coriolis force and the centrifugal force are entered into the momentum equation via terms:

S M ,rot = S Cor + S cfg

28

(2.47)

where

S Cor = -2 × Vr , f

S cfg = - × ( × r ).

(2.48)

when the given velocity vector represents the relative frame velocity. The energy equation also requires a modification because total enthalpy no longer functions as a conserved quantity in a rotating frame of reference. Therefore, the total enthalpy, H, must be replaced by rothalpy, I ­ a conserved quantity in a rotating frame ­ defined as

1 1 I = hstat + V 2 - h + 2 r 2 . 2 2

Here V is the magnitude of velocity, h enthalpy and r the radial location.

(2.49)

2.2.3

Turbulence Models

It is generally recognized that the Navier-Stokes equations (2.39 ­ 2.42) 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 of three dimensional and unsteady fluctuations in the flow field indicates that the flow has lost its stability and has become turbulent. Turbulent flow occurs when the inertia forces in the flow field become significant compared to the viscous damping forces, a situation which is signified by high Reynolds Number, Re. This 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:

V 2 / L VL Re = . V / L2

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 the mesh to be 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. 29

Earlier, in section 2.1.1, it was shown how the time averaging of the governing equations leads to additional stress terms, which are made up of the fluctuating velocity components. The same procedure is applied to the full 3D Navier-Stokes equations here and the resulting Reynolds Averaged Navier-Stokes Equations can be written in a compact indicial notation while omitting the bar over the averaged components:

ui =0 + x j t

(2.50)

ui ui p + uj =- + t x j xi x j

u u j 2 u i k ij - uiu j + S Mi + - x j xi 3 xk

(2.51)

H p ui H uih u j ij qi - + + - + + S Ei = 0 t t x j x j xi xi

In the energy equation (2.52) the mean total enthalpy is now given,

(2.52)

1 H = h+ V2 +k 2

where k is turbulent kinetic energy, defined by

(2.53)

1 k = V 2 2

(2.54)

The additional stress terms that arise from the time varying velocity components represent the Reynolds Stress terms that introduce new unknowns into the governing set of equations. Since explicit and universally valid expressions do not exist for the Reynolds Stress terms, the closure can only be achieved by formulating equations, which are based on postulated physical behavior of turbulence. Turbulence modeling has been a topic of rigorous research during the past two decades and, at this moment, the available 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.

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 30

between numerical effort and accuracy. Based on general recommendations reported in literature and particularly in CFX solver documentation [7], a two-equation Shear Stress Transport (SST) k-

based turbulence model, which has produced the most encouraging agreements with empirical

results, has been used in all CFD analysis in this study. Although the role and significance of turbulence models in CFD analysis is recognized, it is not a focus of this project due to time and computational limitations.

2.2.4

Numerical Discretisation

Due to the shortcomings of analytical methods to solve the Navier-Stokes equations, numerical approach must be utilized. This translates into replacing the governing differential equations by algebraic approximations, which may be solved using a numerical method. This method requires the spatial domain to be discretised into finite control volumes using a mesh. The governing equations are integrated over each control volume, such that the relevant quantity is conserved in a discrete sense. The spatial discretisation can be accomplished by multiple means, but the most common ways in three-dimensional CFD are based on either tetrahedral or hexahedral finite volume shapes. A mesh that consists of mainly tetrahedral elements is called unstructured while a structured mesh is comprised of hexahedral elements. The flow simulation software CFX-5 utilizes an unstructured mesh, which is preferred in flow simulations that involve complex geometries and flow structures. A structured mesh, on the other hand, can provide a more efficient mesh and higher accuracy in cases where the grid can be aligned with the flow direction.

The control volume approach requires the flux terms of the governing equations (2.50 ­ 2.52) to be transformed, by the use of Gauss' divergence theorem, into a surface integral form. For the case of momentum equation the resulting formulation is

u i dV fe + u j u i dn j = t V s fe u u j 2 u k - pdn j + i + - ij - u iu j dn j + S M dV fe x s s V fe j xi 3 x k

(2.55)

Here s and Vfe denote surface and volume integrals respectively and dnj are the differential components of the outward normal surface vector in Cartesian coordinates that have the units of an area. In the equation, the flux terms are integrated across the surfaces, while the source and accumulation terms are integrated over the control volume. The integral equation, however, is still in a continuous form and therefore has to be discretised to allow numerical solution. To

31

demonstrate with general terms ­ as stated in CFX-5 Solver Theory [7] ­ how the discrete equations are solved with the use of mesh elements consider Figure 2.5.

Figure 2.5 A schematic illustration of a tetrahedral element's isolated surface (face) segment according to CFX-5 Solver Theory.

The surface fluxes are discretely represented at the integration points, ipn, which are located at the center of each surface segment of a 3D finite volume. The integral momentum equation (2.55) can now be written for the discrete system as:

u i - u i t -1 & ( ) V fe t + mip u i ip = ip

( pn )

ip

i ip

u u j 2 u k - ij - u iu j n j + S M V fe + i + xi 3 x k ip x j ip

(2.56)

Here the discretisation of the time derivative is written as a generic, first order scheme and the discrete mass flow rate through the surface of the finite volume is given by

t -1 & mip = (u j n j )ip

(2.57)

while Vfe represents the total volume of the control volume, t is the time step and nj is the outward positive surface vector. The superscript, t-1, refers to values of the old time level.

32

If the grid system is unstaggered, which means that all the solution variables are stored at each node 3, the solution may produce a checkerboard effect where velocity and pressure values have become decoupled. To avoid this, a dissipative fourth order pressure derivative term is added to the mass conservation expression, which distributes the effect of pressure to the neighboring nodes. The principle of the method can be shown through a one-dimensional example of the mass conservation:

3 4 u x A p =0 + & 4m x 4 i x i

(2.58)

where A stands for a representative area. The expression demonstrates how the magnitude of the pressure dissipation term approaches zero when the element size, x, is reduced. CFX-5 utilizes a method that is based on (2.58) with a number of extensions to improve solver robustness.

The method of discretisation of the advection terms in the governing equations is of particular importance due to numerical effects ­ such as numerical diffusion and numerical dispersion ­ and therefore warrants a brief discussion. In CFX-5 the advection schemes are represented in a following simplified form:

ip = up + rp

where up is the value at an upwind node,

(2.59)

is a discrete approximation of the gradient of

and rp is the vector from the upwind node to the integration point, ip. The variable , which can take values between 0 and 1, determines the order of the advection scheme. A value = 0 leads to a first order upwind difference scheme, while = 1 yields a second order accurate discretisation. The benefit of first order upwind scheme is high level of robustness, but it suffers from severe numerical dissipation, which leads to inadequate solution accuracy. The second order scheme, while providing higher accuracy, is less robust and may introduce numerical dispersion into the solution where there are high gradients. values that are in between 0 and 1 allow the user to specify a specific blend of the two schemes. CFX-5 offers a High Resolution option, which determines the value of locally trying to assign it a value that is as close as possible to 1, while making sure that the solution remains bounded. In this study the High Resolution option has been used in all final computations. Lower values of were necessary,

3

In a staggered grid the pressure and velocity variables have been assigned separate nodal systems such that the

velocity nodes fall in between the pressure nodes. This simplifies the numerical solution technique and couples the pressure and velocity values at adjacent nodes. However, this approach dramatically complicates the spatial discretisation for unstructured grids.

33

however, at the start-up of the simulation when no initial solution was available and the complex flow system was prone to failure.

2.2.5

CFX-5

Commercially available CFX-5 multi-physics flow simulation software combines a computational fluid dynamics solver with pre-processing, solution monitoring and post-processing modules. A mesh generation module is also available, but not used in this project. CFX-5 utilizes a threedimensional, unstructured, finite volume mesh for spatial discretisation and enables multiple domain systems to be solved simultaneously. The pre-processor module is primarily used to import the mesh files and define the analysis type, flow physics, boundary conditions and computational parameters. But, it also allows the user to define new solution variables and expressions, which facilitate the monitoring of project specific quantities (in user specified units) through monitoring points, walls or interfaces. All the information is stored in a single definition (.def) file, which is read by the solver at the start of the run.

CFX-5 uses a coupled solver, which solves the hydrodynamic equations (for u, v, w, p) as a single system and, therefore, does not employ a pressure correction scheme. A fully implicit discretisation of the equations is utilized at any given time step and the linearised system of equations is solved using an Algebraic Multigrid method. In steady state analysis the time step can be assigned either a physical or local time scale; A physical time scale selection ensures that the solution is advanced in real time, but in such a manner that the time step functions only as an acceleration parameter for the solution to reach a steady state answer, while the local time scale option provides the user with the ability to impose different time steps in separate regions. In transient, time dependent, analysis the length of time step and the number of coefficient loop iterations within each time step can be controlled explicitly. The solution monitoring module (Solver Manager) offers a graphical and text based user interface for controlling the run in progress and visualizing the development of computational parameters during a run. By default, the residuals of the governing, and turbulence, equations are displayed, but an entire library of generic monitor parameters and expressions is also available. In addition, the user can monitor expressions that were self-defined in the pre-processor module. The Solver Manager also has a feature ­ which unfortunately could not be utilized at Pratt & Whitney Canada ­ that permits the user to edit certain computational and boundary condition values while the run is in progress.

The CFX-5 post-processor module provides the standard visualization capabilities found in most similar software packages, but contains some additional functionalities that simplify the analysis. 34

Its most noteworthy analysis aid is the function calculator, which grants fast and easy access to either exact or averaged values of solution variables at chosen locations in the domain. This feature was heavily utilized in this study and its usability was tremendously appreciated.

35

3

ANALYSIS

As discussed in chapter 1, this numerical study entails CFD analysis of two computational models ­ ATFI and PW307 HPT2 ­ both of which consist of a single periodic turbine blade passage that is isolated from the multi-stage environment and coupled with an upstream disc cavity. Each model's geometry is based on complete CAD blue prints for the 2D cavity and gas path geometry and the 3D turbine blade solid, although, some specific simplifications have been imposed due to modeling issues. The original ATFI model had been assembled and drawn in CATIA by Marini [8] and later modified to suit this study according to the author's requests. The PW307 CATIA model was drawn by Marini according to the specifications the author provided. The computational grids have been generated with ICEMCFD version 4.2 mesh generation software and exported into CFX-5 for computational analysis. In this chapter a description of the models' geometric features is provided followed by an account on mesh generation and a detailed depiction of the CFD methodology utilized in this work.

3.1

3.1.1

Model Geometry

ATFI HPT

The ATFI computational model has been drafted according to the dimensions of cold crosssection (see Figure 3.1) such that no heat or stress induced displacements or relative movements of engine parts have been taken into account. The model is shown in Figure 3.2 and Figure 3.3. The most noteworthy dimensional changes that occur in operation are due to axial movement of the vane section with respect to the turbine disc and radial growth of the turbine disc and blade assembly. These movements tend to close the gaps between the rotating and stationary walls at the rim seal, effectively improving the sealing efficiency, but their quantitative influence on ingestion is not investigated in this report. A fluid-fluid interface has been placed at the rim seal, shown in Figure 3.4, for the purpose of monitoring the flow interaction between the gas path and the disc cavity.

36

[3]

(10) Vane [1] Blade

Gas Path

[2]

[4]

[6]

[1] Gas Path Inlet (Inlet 1) [2] Gas Path Outlet

[7] (9)

[3] Shroud Leakage (omitted) [4] Vane Platform Leakage (omitted)

Di sc

[5] Cover Plate Seal (Inlet 2)

Ca vit y

[6] Blade Platform Leakage (omitted) (8) Turbine Disc [7] Blade Fir tree Fixing Leakage (omitted) (8) Rotating Cover Plate

[5]

(9) Static Cavity Wall (10) Tip Clearance (omitted)

Figure 3.1 ATFI. An Illustrative outline of the ATFI turbine blade passage and disc cavity crosssection.

The following definition of a dimensionless radius is applied to subsequent figures and discussion for both ATFI and PW307. Rmin: Bottom of the disc cavity. Rmax: Top wall of the blade passage.

R* =

r - Rmin Rmax - Rmin

37

Figure 3.2 ATFI. Coupled blade passage and disc cavity computational domain.

Figure 3.3 ATFI. Overview of the periodic blade passage section.

Figure 3.4 ATFI. A fluid-fluid interface, separating the gas path and disc cavity, is utilized to monitor ingestion and other flow conditions.

38

In order to reduce the overall complexity of the system and establish a framework that is suitable for a rather general ingestion analysis, the model was subjected to a number of simplifications. The following realistic details have been removed from the computational model:

1) Blade tip clearance ­ the model's blade is fixed to a rotating shroud. 2) Shroud leakage (inlet). 3) Vane feather seal leakage (inlet). 4) Blade platform leakage (outlet). 5) Blade fir tree fixing leakage (outlet).

The removal of blade tip clearance is well justified for its dimension is very small compared to the blade height and, due to the accelerating nature of the flow, the resulting local alteration in the flow field will not be communicated in the radial direction so that it will affect the flow near the critical lower part of the gas path near the rim seal. The same reasoning can be applied to the shroud leakage. The exclusion of the additional inlets and outlets in the cavity region, however, cannot be defended without difficulty for it requires engagement in an intuitive balancing act with modeling issues. In the absence of any theoretical-versus-experimental investigations on this topic, the decision will have to be based on a priori evaluation of modeling pros and cons, which, at this point, will be mainly based on engineering judgments. For this reason, a brief monologue is offered in section 3.1.3 to elaborate on this issue and to further discuss the progressive decision making process that became part of this project.

3.1.2

PW307 HPT2

The PW307 HPT2 model was drafted in a similar manner as the ATFI by extracting detailed disc cavity, gas path and blade solid dimensions from corresponding CAD blue prints and creating a periodic sector according to the number of blades in the turbine. See Figure 3.5 for cross-section and Figure 3.6 for the complete computational domain. With this model special care was exercised in effort to stay as loyal as possible to realistic dimensions and physical engine features for, unlike the ATFI, the PW307 is an engine in production and air system data is readily available for result comparison. Consequently, the engine's most critical performance point, PTM12442 COND1 (nominal take-off, NTO), was chosen for the analysis and the model drafted according to dimensions that take into account the relative movements of engine parts under this specific condition.

39

[1] Vane

Blade

[2] Gas Path

[3]

D

[7]

is c

[4]

[6]

C av ity

[1] Gas Path Inlet (Inlet_G) [2] Gas Path Outlet (Outlet_G) [3] Vane Platform Leakage (Inlet_Cav3) [4] Cavity Purge Holes (Inlet_Cav2) [5] Brush Seal Leakage (Inlet_Cav1)

[5]

[6] Fir-tree Fixing Leakage (Inlet_Cav4) [7] Blade Platform Leakage (Outlet_Cav)

Figure 3.5 PW307. An illustrative outline of the turbine blade passage and disc cavity crosssection.

R*= 0.992

Inlet_G Blade

Gas Path

Outlet_G

R*= 0.582 R*= 0.536

Inlet_Cav3 Outlet_Cav

R*= 0.314

Inlet_Cav4 Inlet_Cav2 Disc Cavity Rotating Disc

R*= 0.098

Stationary Wall

Inlet_Cav1

Figure 3.6 PW307. Coupled blade passage and disc cavity computational domain. (G = gas path, Cav = cavity.)

40

With the PW307 model a decision was made to attempt to represent all the existing inlets and outlets in the cavity as realistically as possible in effort to simulate the flow patterns closely. See Figure 3.7, Figure 3.8 and Figure 3.9 for the cavity details. This modeling process, however, still required assumptions and approximations for so little is known about the cavity flow behavior and, consequently, it was decided that a modification of the model that did not include Outlet_Cav and Inlet_Cav4 was to be utilized in this study primarily. At the end, the only physical features omitted from the model were:

1) Blade tip clearance 2) Shroud leakage 3) Blade platform and fir tree fixing leakages (in modified model)

The representative cross-sectional area of each inlet and outlet in the cavity was obtained from an in-house memo [9] and incorporated into the 3D models. The cross-sectional areas of the individual, discrete inlets on the stationary wall had to be scaled because their number was not an integer multiple of the number of blades in the turbine. (See section 3.1.3 for a more detailed description on the scaling.) As seen in Figure 3.6 in yellow, two fluid-fluid interfaces were created in order to achieve closer monitoring of physical parameters in the cavity's rim seal area.

D* =R*=16.5-3

Inlet_Cav1 R*=4.15E-4

= 70°

Figure 3.7 PW307. Close-up of the representative brush seal inlet at the bottom of the disc cavity.

Figure 3.8 PW307. Inlet_Cav2: The purge flow is supplied through an angled duct to provide swirl.

41

Inlet_Cav3 Outlet_Cav

R* = 8.96E-4

Inlet_Cav4

R* = 1.48E-3

Figure 3.9 PW307. Close-up of vane platform leakage (Inlet_Cav3), blade platform leakage (Outlet_Cav) and blade fir tree fixing leakage (Inlet_Cav4).

3.1.3

On Cavity Modeling Issues

The study of the ATFI model constituted the first phase of this project when the primary focus was on establishing proper analysis methodology and exploring the possibilities to perform ingestion and cavity flow analysis in an engine environment. Producing values that could have been verified against experimental values was of secondary importance at that point and, for this reason, certain geometric and boundary condition related complexities were purposefully reduced to avoid ­ which were then ­ unforeseeable difficulties. It was also desirable to fix the configuration such that it principally resembled the test rig ­ which was the object of the author's prior CFD study ­ having individual inlets for the main gas path flow and cavity purge flow and a single outlet for the whole system. This approach was reasoned to provide the next logical step forward in the overall learning process. Still, it was recognized that the flow patterns, especially at the top part of the cavity, would be altered by the removal of the additional inlets and outlets, but, since the model represents an experimental engine that will be subjected to multiple design modifications, any attempt to achieve such high accuracy simulation was rendered to be in vain nonetheless.

42

With valuable lessons learned from the ATFI analysis, a decision was made to take a different approach with the second, PW307, phase of the study and explore the territory that comes with the added complexity. Thereby, some underlying objectives were identified in the beginning: 1) The flow entering the cavity through a circumferential inlet (such as Inlet_Cav1) should possess the momentum determined by the known mass flow rate and area of the inlet. 2) The shape and velocity of the jet-like flow that enters the cavity through a discrete inlet (such as Inlet_Cav2) should remain unchanged if scaling is required. 3) Due to the enormous difficulty in determining the nature of the purge flow ­ or drainage ­ through the small-scale holes on the rotating disc, the effective mass flow addition to ­ or subtraction from ­ the flow near the disc should be accounted for with circumferential strips that disrupt the flow on the disc as little as possible.

To meet these objectives the following steps were taken: For the circumferential inlets and outlets the proper radial gaps were determined according to the representative areas given by Liu and Carpenter [9]. For the discrete inlets information from CAD blue prints and air system data were combined to scale the cross-sectional area according to the total mass flow rate across the sector so that velocity remained constant. For instance, when the ratio of purge-holes (Inlet_Cav2) to turbine blades is 1.65 to 1, the resulting scaling yields for the purge-hole's diameter: Dscaled = 1.285Doriginal. Similar scaling is performed for the vane platform leakage (Inlet_Cav3). The drawback of adding discrete objects onto the stationary wall in such coupled analysis is that their relative positions with respect to the rotating blade do not change. From a physical point of view this is clearly incorrect, but the assumption is made that the resulting interaction between the inlets and the blade is negligible.

43

3.2

Computational Model

The spatial discretisation of the computational domains was performed with ICEMCFD 4.2 unstructured mesh generator software, utilizing tetrahedral elements with added prism layers at the surfaces. This section will provide information concerning the grid generation process and illustrate the resulting computational models used in the analysis.

3.2.1

Mesh Generation Procedure

The general procedure in ICEMCFD can be outlined concisely with the following steps. For more specific instructions, an ICEMCFD manual should be consulted: 1) Open the CATIA tetin-file in ICEMCFD. 2) Extract curves from the geometry. 3) Assign all curves and points to their own families, CURVES and POINTS respectively. 4) Set up surface families and assign each surface in the geometry to a representative family (for example Inlet_Cav1, Outlet_G, CavityWall_Low, CavityWall_Top, Interface, Periodic etc.) Each surface family will be assigned specific boundary conditions and mesh parameters. 5) Insert material points: ORFN point outside and LIVE points inside the active flow domain. Note: If the active domains are separated by an interface, each domain requires its own LIVE point. 6) Set model and surface family mesh size parameters (see 3.2.2 and 3.2.3 for details.) 7) Generate tetra mesh and perform appropriate smoothing. Typically steps 6 and 7 require multiple iterations so that desirable outcome is reached. 8) Add prism layers and apply additional smoothing only to TETRA and TRI elements. 9) Follow the steps (1 ­ 9) outlined by Turgeon [10] in an unpublished in-house memo to create separate active domain regions that are split at the interface(s). 10) Specify the solver (CFX-5) and write the boundary condition file (.fbc) for each domain. Check that all the appropriate families appear in the file. No other action is required at this point. 11) Write the grid file (.grd) in binary form.

44

3.2.2

ATFI Unstructured Grid

Since the primary goal of this study is to simulate the flow behavior within the cavity and capture its interaction with the high temperature gas path, the main focus of the mesh generation was directed at the front end of the gas path and the disc cavity. For these reasons the element size was strictly controlled in these critical regions while some accuracy was sacrificed in areas, such as the space downstream of the blade, to keep the element count at a reasonable level. Since the finding of a satisfactory combination of element size parameters is an iterative process and subject to experience and knowledge level of the software, the final mesh of the ATFI model does not represent the optimal set of choices. Especially the issue of wall treatment proved difficult for restraint had to be exercised with the addition of prism layers, whose function is to capture the boundary layer flow. However, even though no documented mesh independence study was performed, different mesh settings were compared and experimented with and, based on primarily physical solution monitoring, the final mesh was rendered satisfactory for the analysis. Table 1 provides a collective set of information concerning the ATFI-model's final grid. Some details of the computational domain are illustrated in Figure 3.10 - Figure 3.13.

Figure 3.10 ATFI. Surface grid on the stationary cavity wall.

Figure 3.11 ATFI. Surface grid on the rotating disc and turbine blade.

45

Table 1

ATFI. Unstructured mesh information.

General Data: Total Number of Elements Total Number of Nodes ICEMCFD Mesh Parameters: Max Size Edge Criterion Tri Tolerance Representative Element Size Parameters: Rotating Disc Upper Cavity Wall Blade Interface Prism Layer Information: Number of Layers First Element Height (in) Expansion Factor 7 0.0003 ­ 0.0004 1.4 0.05 0.05 0.075 0.05

4

1780678 472722 1.0 0.25 0.05 0.0002

Reference Size (in)

Figure 3.12 ATFI. Unstructured mesh crosssection at gas path. The mesh at the main inlet is refined to capture radial and circumferential boundary condition profiles.

Figure 3.13 ATFI. Unstructured mesh in the cavity and front of the gas path.

4

Shown data is ICEMCFD-specific. In retrospect, it is beneficial to keep the maximum element size smaller and control

the coarsening with surface parameters instead. The particular choice is always case specific, but, in general, a value < 0.15 would be recommended for geometries of this scale.

46

3.2.3

PW307 HPT2 Unstructured Grid

The intricate inlet/outlet details in the PW307 model's cavity region introduced even further meshing challenges compared to the ATFI and, therefore, required a process with higher number of mesh parameter iterations. In order to weight the benefits of including the small-scale details on the rotating disc against the added computational effort, meshing difficulties and potentially insignificant accuracy gain, two versions of the model were generated: Complete version with all the details as shown in 3.1.2 and a modified version that excludes Outlet_Cav and Inlet_Cav4. At the end, the main analysis was performed with the modified model due to the significant mesh improvement opportunities that became available when node consuming inlet/outlet details were omitted. Ultimately, due to the author's accumulated experience with ICEMCFD, the final computational grid's overall quality was judged to be good. In Table 2 the elemental and ICEMCFD-specific mesh information is presented and Figure 3.14 ­ Figure 3.19 provide visualization of the computational grid features of the model.

Figure 3.14 PW307. Surface grid on the stationary cavity wall (modified model.)

Figure 3.15 PW307. Surface grid on rotating disc (modified model.)

47

Table 2

PW307. Unstructured Mesh Information.

General Data: Total Number of Elements Total Number of Nodes Total Number of Elements (modified) Total Number of Nodes (modified) ICEMCFD Mesh Parameters (modified model): Reference Size (in) Max Size Edge Criterion Tri Tolerance Representative Element Size Parameters: Rotating Disc Upper Cavity Wall Blade Lower Interface Inlet_Cav1 Prism Layer Information:

5

2448532 725858 1524809 422331 1.0 0.065 0.15 3e-5 0.03 ­ 0.06 0.035 0.075 0.02 0.00075 6 7.5e-5 ­ 4e-4 1.0 ­ 1.5 6+1 1e-5 ­ 3e-4 1.6

Number of Layers First Element Height Expansion Factor Number of Layers (modified) First Element Height (modified) Expansion Factor (modified)

5

In the modified model the prism layers were created by initially adding two layers of equal size (h = 0.006) and splitting

the inner one into 6 layers with the specified expansion factor. This method is recommended by the author, for the second, larger, layer provides a better transition from the fine prism layers to the disproportionally large tetra elements.

48

Figure 3.16 PW307. Unstructured mesh crosssection at gas path (modified model.)

Figure 3.17 PW307. Mesh in the cavity (modified model.)

Figure 3.18 PW307. Close-up of prism layers (modified model.)

Figure 3.19 PW307. Close-up of Outlet_Cav on the rotating disc (complete model.)

49

3.3

3.3.1

CFD Methodology

Overall Solution Strategy

The three-dimensional, viscous, compressible flow analyses, and the associated result acquisition, were performed with commercially available CFX-5 flow simulation software ­ that is briefly described in section 2.2.5 ­ with the following assumptions:

-

The fluid obeys ideal gas law All boundary conditions are time independent All walls are adiabatic Flow structures in the disc cavity have the same periodicity as the blade passage Vane-blade interaction has a secondary effect on ingestion (this assumption is subjected to limited investigation)

-

The interaction between the discrete inlets and the turbine blade is negligible (PW307)

All computations were executed with a steady state solution scheme specified in CFX-5, which employs a physical time step to advance the solution. This time integration technique allows ­ with a properly specified time step ­ any naturally arising unsteadiness to be captured in the flow field, obviating the need to make any assumptions about the nature of the flow behavior in the system. However, the solution technique does not provide time accurate information about the development of the flow field as the iterations advance the solution because the method corresponds to a time-accurate integration algorithm that utilizes only a single inner iteration within a time step. The time step used in the analysis has been determined according to the solution monitoring of physical quantities while making sure the residuals did not demonstrate numerical difficulties. It is generally recommended that the time scale for such rotating domain systems would be in the range of 0.1/ ­ 1/, where is the rotational velocity of the turbine in rad/s. Yet, the time scale used in this analysis fell below this range due to the ever-present activity in the solution space. On the other hand, due to the dramatic time scale difference between the gas path and the cavity flow fields, a higher time step was specified for the energy equation in effort to convect temperatures faster in the cavity region. It was observed that this did not affect the numerical stability noticeably. For spatial discretisation a High Resolution advection scheme was specified for all equations except for turbulence, which utilizes first order upwind scheme. The viscous work option was activated in the energy equation specification to include all the viscous dissipation terms in the 50

computation. For turbulence modeling, the Shear Stress Transport (SST) k- based model was chosen with an automatic near-wall treatment ­ which utilizes a wall function where y+ > 11 ­ to ensure sufficient boundary layer resolution where the mesh would not be adequately refined. Unaware of the apparent benefits of performing multi-domain analysis in CFX-5 6, the separate domains of the ATFI model were merged in ICEMCFD before being imported into CFX-Pre and subsequently configured as a single domain that was solved in a rotational frame of reference. Merging the domains had no effect on the actual numerical treatment, but importing two domains that share an interface separately would have simplified the interface treatment in CFX-Pre. With PW307 model, due to the presence of discrete inlets on the stationary cavity wall, the upper interface was used to separate the gas path from the disc cavity such that the cavity was solved in a stationary and the gas path in a rotating frame of reference. The frame change at the upper interface was specified according to the final assumption so that the relative positions of the two regions were fixed. A detailed account of the applied boundary conditions and domain settings is provided in the following section.

3.3.2

Boundary Conditions and Computational Settings

The performance point specific gas path boundary conditions were obtained from a multi-stage Nistar 7 CFD analysis of the turbine, which provides circumferentially averaged radial profiles of critical flow parameters at specified cross-sectional planes. Since the given profile information was not directly applicable to CFX-5, a Matlab-program (see Appendix A) was drafted by the author to convert and scale the geometric and physical profile information so that it could be utilized in CFX-5 and manipulated as needed. An additional Matlab-program (see Appendix B) was written to create a 3D inlet profile from a radial profile for the purpose of studying the effect of vane-blade interaction on ingestion, without reverting to computationally intensive and time accurate multi-stage analysis.

The cavity boundary conditions at a given performance point were obtained from corresponding air system data, while the values used in the ingestion study were systematically altered according to the author's discretion. Since there is no prior knowledge of the turbulence level of the incoming flow, a recommended option (medium intensity) is primarily used.

6

The author, in retrospect, strongly recommends the use of multi-domain analysis with CFX-5 whenever the analysis Nistar is an in-house CFD code for structured grids tailored to perform steady state multistage turbine analysis. The

involves interfaces based on the ease with which a multi-domain analysis can be set-up and handled in CFX-Pre.

7

interfaces between adjacent stages are mixing planes where appropriate reference frame change is performed and the flow information is circumferentially averaged.

51

The computational cases discussed in this report are categorized as follows:

1) ATFI-INGST: 7 cases 2) ATFI-WAKE: 3 cases 3) ATFI-FER (First Engine Run): 1 case 4) PW307-INGST: 6 cases 5) PW307-PTM12442 (Specific performance point): 3 cases

See sections 4.2 ­ 4.6 for detailed descriptions.

The following sets of data present the applied boundary conditions for each case in a dimensionless form and provide the most relevant computational parameters used in the numerical analysis.

ATFI Computations:

Definition : Dimensionless Variables and Reference Values for ATFI Inlet Conditions. pt_max = Max value at gas path Inlet1 (INGST, WAKE). pt_min = Min value at disc cavity (FER). Tt_max = Max value at gas path Inlet1 (INGST, WAKE). Tt_min = Min value at Inlet2 (FER).

pt * ATFI =

pt - pt _ min pt _ max - pt _ min

Tt * ATFI =

Tt - Tt _ min Tt _ max - Tt _ min

m * ATFI =

& m & mGasPath

1) ATFI-INGST Because the original performance point boundary conditions led to a highly transonic flow in the gas path, the radial profile data was scaled down and the engine RPM's reduced accordingly to relax the flow conditions and improve numerical behavior.

Domain: Frame of Reference: Engine / Frame RPM: Turbulence model: Physical Time Scale (general): Physical Time Scale (energy): Rotating 18100 (1895 rad/s) k- SST 1.0e-5 sec 3.0e-5 sec

52

Boundary Conditions: Walls and Interface Rotating Walls: Stationary Walls: Interface Type: Inlet1 (Gas Path) Inlet Type: B.C Type: Mass and Momentum: Flow Direction: Energy: Turbulence Option: Outlet (Gas Path) Outlet Type: Inlet2 (Cavity) Inlet Type: B.C Type: Mass and Momentum: Subsonic, Stationary Frame Constant Values 1) Mass Flow Rate: m* ATFI_ = 0.01122 2) Mass Flow Rate: m* ATFI_ = 0.00998 3) Mass Flow Rate: m* ATFI_ = 0.00873 4) Mass Flow Rate: m* ATFI_ = 0.00748 5) Mass Flow Rate: m* ATFI_ = 0.00623 6) Mass Flow Rate: m* ATFI_ = 0.00499 7) Mass Flow Rate: m* ATFI_ = 0.00374 Flow Direction: Energy: Turbulence Option: 2) ATFI-WAKE The 2D boundary condition profiles used in ATFI-INGST were modified into 3D Cartesian profiles conserving the circumferential averages for each radial sector. See the Matlab-program in Appendix B for the procedure. The circumferential variation was produced according to the Nistar solution at the inlet's cross-sectional plane. See Figure 3.20 for the solution and Figure 3.21 for its discrete replication. Three profiles were generated with different vane wake locations while all other boundary conditions were kept unaltered. Dimless. Cyl. Vel. Comp: Vr*= 0, V*= 0.809, Vz*= 0.587 Total Temperature in Stn. Frame: Tt* ATFI = 0.076 Medium (Intensity = 5%) Supersonic, Rotating Frame Subsonic, Stationary Frame 2D radial profile Total Pressure in Stn. Frame, pt(r): pt*ATFI_ave = 0.631 Cyl. Vel. Comp, Vr(r), V(r), Vz(r) Total Temperature in Stn. Frame, Tt(r): Tt* ATFI_ave = 0.923 Medium (Intensity = 5%) No slip No slip, Counter Rotating Fluid ­ Fluid

53

Figure 3.20 Nistar solution of total pressure at the inlet1 plane.

Figure 3.21 A synthetic, discrete, replication of the Nistar solution.

Domain: Same as ATFI-INGST

Boundary Conditions: Walls and Interface: Same as ATFI-INGST Inlet1 (Gas Path) Inlet Type: B.C Type: Mass and Momentum: Flow Direction: Energy: Turbulence Option: Subsonic, Stationary Frame 3D Cartesian profile Total Pressure in Stn. Frame, pt(x,y,z): pt*ATFI_ave = 0.631 Dimless. Cyl. Vel. Comp, Vr* (x,y,z), V* (x,y,z), Vz* (x,y,z): Total Temperature in Stn. Frame, Tt(x,y,z): Tt* ATFI_ave = 0.923 Medium (Intensity = 5%)

Outlet (Gas Path): Same as ATFI-INGST

Inlet2 (Cavity) Inlet Type: B.C Type: Mass and Momentum: Flow Direction: Energy: Turbulence Option: Subsonic, Stationary Frame Constant Values Mass Flow Rate: m* = 0.00623 Dimless. Cyl. Vel. Comp: Vr*= 0, V*= 0.809, Vz*= 0.587 Total Temperature in Stn. Frame: Tt* ATFI = 0.076 Medium (Intensity = 5%)

54

3) ATFI-FER This case was added to investigate whether the cavity temperature values predicted by the simplified model compare with experimental measurements taken at the ATFI's first engine run corresponding to the following specifications.

Domain: Frame of Reference: Engine / Frame RPM: Turbulence model: Physical Time Scale (general): Physical Time Scale (energy): Boundary Conditions: Walls and Interface: Same as ATFI-INGST Inlet1 (Gas Path) Inlet Type: B.C Type: Mass and Momentum: Flow Direction: Energy: Turbulence Option: Subsonic, Stationary Frame 2D radial profile Total Pressure in Stn. Frame, pt(r): pt*ATFI_ave = 0.967 Dimless. Cyl. Vel. Comp, Vr*(r), V*(r), Vz*(r): Total Temperature in Stn. Frame, Tt(r): Tt* ATFI_ave = 0.708 Medium (Intensity = 5%) Rotating 19100 (2000 rad/s) k- SST 1.0e-5 sec 3.0e-5 sec

Outlet (Gas Path): Same as ATFI-INGST

Inlet2 (Cavity) Inlet Type: B.C Type: Mass and Momentum: Flow Direction: Energy: Turbulence Option: Subsonic, Stationary Frame Constant Values Mass Flow Rate: m* = 0.00418 Dimless. Cyl. Vel. Comp: Vr*= 0, V*= 0.809, Vz*= 0.587 Total Temperature in Stn. Frame: Tt* ATFI = 0 Medium (Intensity = 5%)

55

PW307 HPT2 Computations:

Definition : Dimensionless Variables and Reference Values for PW307 Boundary Conditions. pt_max = Max value at gas path inlet (INGST). pt_min = Chosen low value at disc cavity (INGST). ps_max = Chosen high value at gas path (INGST). ps_ref = Reference value (Atmospheric). Tt_max = Max value at gas path Inlet (INGST). Tt_min = Min value at disc cavity (INGST).

pt *PW 307 = ps *PW 307 =

pt - pt _ min pt _ max - pt _ min ps - ps _ ref ps _ max - ps _ ref

Tt *PW 307 =

m *PW 307 =

Tt - Tt _ min Tt _ max - Tt _ min

& m & mGasPath

4) PW307-INGST The ingestion study is conducted by primarily reducing the cavity's main inlet (Inlet_Cav2) mass flow rate systematically. The gas path conditions are unaltered throughout the study. The PTM12442-performance point condition is included as the first case. See Figure 3.6 for the naming convention of the boundary condition headers.

Domain_gaspath: Frame of Reference: Engine / Frame RPM: Turbulence model: Physical Time Scale (general): Physical Time Scale (energy): Domain_cavity: Frame of Reference: Turbulence model: Physical Time Scale (general): Physical Time Scale (energy): Boundary Conditions: Walls_gaspath Rotating Walls: Stationary Walls: Walls_cavity Rotating Walls: Stationary Walls: No slip, Rotating with engine RPM No slip 56 No slip No slip, Counter Rotating Stationary k- SST 1.0e-5 ­ 2.0e-5 sec 2.0e-5 ­ 3.0e-5 sec Rotating 27193 (2848 rad/s) k- SST 1.0e-5 ­ 2.0e-5 sec 2.0e-5 ­ 3.0e-5 sec

Interfaces Upper Interface: Lower Interface: Inlet_G Inlet Type: B.C Type: Mass and Momentum: Flow Direction: Energy: Turbulence Option: Outlet_G Outlet Type: B.C Type: Mass and Momentum: Inlet_Cav1 Inlet Type: B.C Type: Mass and Momentum: Flow Direction: Energy: Turbulence Option: Subsonic, Stationary Frame Constant Values Mass Flow Rate: m*PW307 = 1.05e-3 Dimless. Cyl. Vel. Comp: Vr*= 0, V*= 0.707, Vz*= 0.707 Total Temperature in Stn. Frame: Tt*PW307 = 0 Medium (Intensity = 5%) Subsonic, Stationary Frame Averaged Value over the Whole Outlet Static Pressure: ps*PW307 = 0.553 Subsonic, Stationary Frame 2D radial profile Total Pressure in Stn. Frame, pt(r): pt*PW307_ave = 0.915 Dimless. Cyl. Vel. Comp, Vr*(r), V*(r), Vz*(r): Total Temperature in Stn. Frame, Tt(r): Tt*PW307_ave = 0.904 Medium (Intensity = 5%) Fluid ­ Fluid, Frame Change: Frozen ­ Rotor Fluid ­ Fluid

57

Inlet_Cav2 Inlet Type: B.C Type: Mass and Momentum: Subsonic, Stationary Frame Constant Values 1) Mass Flow Rate: m*PW307 = 4.665e-3 (PTM12442) 2) Mass Flow Rate: m*PW307 = 3.642e-3 3) Mass Flow Rate: m*PW307 = 2.978e-3 4) Mass Flow Rate: m*PW307 = 2.313e-3 5) Mass Flow Rate: m*PW307 = 1.647e-3 6) Mass Flow Rate: m*PW307 = 0.983e-3 7) Mass Flow Rate: m*PW307 = 0.318e-3 Flow Direction: Energy: Turbulence Option: Inlet_Cav3 Inlet Type: B.C Type: Mass and Momentum: Flow Direction: Energy: Turbulence Option: Subsonic, Stationary Frame Constant Values Mass Flow Rate: m*PW307 = 1.290e-3 Dimless. Cyl. Vel. Comp: Vr*= 0, V*= 0.707, Vz*= 0.707 Total Temperature in Stn. Frame: Tt*PW307 = 0.146 Low (Intensity = 1%) Normal to Inlet Total Temperature in Stn. Frame: Tt*PW307 = 0.088 Medium (Intensity = 5%)

5) PW307-PTM12442 Three cases were run at PTM-12442-performance point condition with different configurations: 1) Complete model, 2) modified model with Inlet_Cav3 (Mdf_1) and 3) modified model without Inlet_Cav3 (Mdf_2). The total mass flow ratio for each case was m*PW307_total = 0.0070. The detailed specifications are provided in the following tables.

Domain_gaspath and Domain_cavity: Same as PW307-INGST

Boundary Conditions: Walls_gaspath and Walls_cavity: Same as PW307-INGST Interfaces: Same as PW307-INGST Inlet_G: Same as PW307-INGST Outlet_G: Same as PW307-INGST Inlet_Cav1: Same as PW307-INGST

58

Inlet_Cav2 Inlet Type: B.C Type: Mass and Momentum: Subsonic, Stationary Frame Constant Values 1) Mass Flow Rate: m*PW307 = 4.814e-3 2) Mass Flow Rate: m*PW307 = 4.665e-3 3) Mass Flow Rate: m*PW307 = 5.956e-3 Flow Direction: Energy: Normal to Inlet 1) Total Temperature in Stn. Frame: Tt*PW307 = 0.088 2) Total Temperature in Stn. Frame: Tt*PW307 = 0.117 3) Total Temperature in Stn. Frame: Tt*PW307 = 0.123 Turbulence Option: Inlet_Cav3 Inlet Type: B.C Type: Mass and Momentum: Subsonic, Stationary Frame Constant Values 1) Mass Flow Rate: m*PW307 = 1.290e-3 2) Mass Flow Rate: m*PW307 = 1.290e-3 3) Mass Flow Rate: m*PW307 = 0 Flow Direction: Energy: Turbulence Option: Cylindrical Velocity Components: Vr= 0, V= 0.707, Vz= 0.707 Total Temperature in Stn. Frame: Tt*PW307 = 0.146 Low (Intensity = 1%) Medium (Intensity = 5%)

Inlet_Cav4 (only case 1 ­ Complete model) Inlet Type: B.C Type: Mass and Momentum: Flow Direction: Energy: Turbulence Option: Subsonic, Stationary Frame Constant Values 1) Mass Flow Rate: m*PW307 = 1.250e-3 Cylindrical Velocity Components: Vr= 0, V= 0.969, Vz= 0.245 Total Temperature in Stn. Frame: Tt*PW307 = 0.228 Low (Intensity = 1%)

Outlet_Cav (only case 1 ­ Complete model) 8 Outlet Type: B.C Type: Mass and Momentum: Subsonic, Stationary Frame Constant Values 1) Mass Flow Rate: m*PW307 = 1.399e-3

8

Imposing a specific mass flow rate across the outlet may cause some reversed flow at the outlet plane. CFX-5 treats this

situation by adding walls where reversed flow occurs. This may fix the problem, but if numerical instabilities persist, it is recommended that the outlet is switched to an opening with a static pressure and temperature obtained from the solution.

59

3.3.3

Expressions

In order to extract desired physical quantities with specified units and obtain meaningful information about the state of the solutions, a series of expressions were defined in CFX-5 preprocessing module, CFX-Pre. The expressions are evaluated from nodal values once every computational iteration at geometrically defined borders ­ such as inlets, outlets, walls and interfaces ­ and they can be graphically monitored in CFX-Solver Manager module as the solution progresses. In case the pre-defined variables in CFX-5 do not provide access to the desired physical information, additional variables can be created which thereby become available for the expressions.

These analysis tools ­ in particular the ability to create new variables ­ opened up an opportunity to obtain direct and `real time' information concerning hot gas path ingestion into the turbine disc cavity by the use of an interface plane that separates the gas path from the cavity volume. Such a method stands in contrast to earlier documented means to acquire ingestion measurements; in prior published studies the ingestion measurements were obtained by either solving a species solution, which yielded a percentage of gas path fluid species in the cavity, or directly measuring the velocity fields at the rim. The disadvantage of these methods lies in capturing the nature of the ingestion because they require heavy post-processing as multiple solutions must be captured to reveal the time-dependent behavior at the rim. It is therefore highly beneficial to utilize an interface at the rim seal as a monitoring plane, since it allows the critical information to be read easily from the monitoring data and processed efficiently. This method, however, requires the ability to define a new variable since regular solution variables, when averaged over a monitoring plane, do not yield quantitative access to ingestion because it involves reversed flow at varying locations across the rim.

In order to simplify the problem of extracting mass flow information, the interfaces at the rim were drafted as surfaces normal to the x-coordinate axis. Thus, for the purpose of monitoring ingestion and directional heat flow across the interfaces, two additional variables ­ given in Table 3 ­ were defined for both models. Table 3 Additional variables defined in CFX-Pre. Definition:

Variable Name:

magU uTRo

sqrt(u^2) u*T*density

60

Deriving an expression for Ingestion: Now, having the absolute velocity in the x-direction in our disposal (magU = u ), the methodology ­ devised and implemented for this study ­ to extract direct ingestion measurements at the interface, can be written as follows: 1) Define an expression for mass flow rate purged into the disc cavity:

& & & & m Purge = m Inlet _ Cav1 + m Inlet _ Cav 2 + m Inlet _ Cav 3 ...

2) Define an expression to obtain the mass flow rate across the interface:

& m Interface = ( uA) Interface

Note that positive sign corresponds to outward flow and at steady-state conditions it is required that

& & m Interface = m Purge .

3) With area-averaged variables, define an expression for absolute mass flow rate across the interface:

& m Interface = ( u A)Interface .

When there is no ingestion ­ no flow with an opposite sign ­ the expression equals the mass flow rate across the interface defined above. But, in case of ingestion, the expression includes the mass flow entering into the cavity, which we denote

& & m Inward , and a corresponding extra outward component, say mOutward , since the

ingested fluid cannot be stored in the cavity. Therefore the expression can also be

& & & & written: m Interface = m Interface + m Inward + mOutward .

Now we recognize that at steady-state, due to conservation of mass,

& & m Inward = mOutward , which is also valid at transient conditions when time-averaged

over a sufficient time period. Thus, we can write

& & & m Interface = m Interface + 2m Inward .

4) Define an expression that solves for the flow entering the cavity:

& m Inward =

& & m Interface - m Interface

2

& m Inward & m Purge

5) Define the expression for ingestion as a ratio:

Ingestion =

61

Deriving an expression for net heat flow across the interface: Utilizing the grouped variable uTRo (= uT ), a convenient expression for net heat flow across the monitoring plane can be derived as follows: 1) Define an expression for total heat purged into the cavity using area-averaged specific heat values and mass flow averaged temperature values:

& & & Q purge = c p TInlet _ Cav1 m Inlet _ Cav1 + c p TInlet _ Cav 2 m Inlet _ Cav 2 + c p TInlet _ Cav 3 m Inlet _ Cav 3 ...

2) Using area-averaged uTRo-variable define an expression for heat flow, which takes into account the direction such that ingested heat has a negative sign:

Q flow = c p (uT ) Interface AInterface

3) Define net heat flow across the interface with the expressions defined above:

QnetFlow = Q flow - Q Purge .

Notice that at steady-state QnetFlow = 0 while during transient conditions it fluctuates between positive (heat exhaustion) and negative (heat ingestion) values.

Implementation of expressions in CFX-5: Table 4 presents the relevant expressions used in the analysis according to the notation utilized in CFX-5, but in an abbreviated form. The variables, functions and locators are considered selfexplanatory and, therefore, will not be given any further description here. Notice that the generic solution variables used in expressions must be averaged individually, with an appropriate averaging scheme, over the specified surface area. Even though it is recognized that this introduces inaccuracies ­ since, for example, c p T c p T ­ the associated error is considered to be negligible in the applied cases. As shown before, expressions can be defined in terms of other expressions as well.

62

Table 4

Principal expressions defined in CFX-Pre. Definition: (massFlow()@Inlet_Cav1 + massFlow()@Inlet_Cav2 + massFlow()@Inlet_Cav3 + ... etc.) * Nblades [lb s^-1]; massFlow()@Interface * Nblades [lb s^-1]; areaAve(density)@Interface * areaAve(magU)@Interface * area()@Interface * Nblades [lb s^-1]; (absMassInterface - massInterface) / 2.0 [lb s^-1]; massInward / massPurge; (areaAve(Cp )@InletCav1 * massFlow()@InletCav1 * massFlowAve(T)@InletCav1 + areaAve(Cp)@InletCav2 * massFlow()@InletCav2 * massFlowAve(T)@InletCav2 + ... etc.)*Nblades [BTU s^-1] (areaAve(Cp )@Interface * area()@Interface * areaAve(uTRo)@Interface)*Nblades [BTU s^-1] heatFlow ­ heatPurge [BTU s^-1] massFlow()@Inlet_gaspath *Nblades [lb s^-1] massFlow()@Outlet_gaspath *Nblades [lb s^-1] (massInletG + massPurge ­ massOutletG)/massPurge areaAve(T)@Disc [R] areaAve(T)@CavityWall [R]

Expression Name: massPurge massInterface absMassInterface massInward Ingestion

heatPurge

heatFlow netHeatFlow massInletG massOutletG massBalance discTemp cavWallTemp

3.3.4

Solution Monitoring and Convergence

The numerical and physical state of the solutions was primarily monitored with the following quantities and expressions: · · · · · · RMS residuals of mass and momentum equations massInterface massInletG massOutletG massBalance discTemp (multiple locations with PW307) 63

· · ·

cavWallTemp (multiple locations with PW307) heatFlow and netHeatFlow ingestion

Since the coupled systems never demonstrated proper numerical convergence, the criteria for concluding the computations had to be based on the physical behavior of the system. While the RMS residuals typically settled to a constant level during the first 50 ­ 100 iterations after a case was started with an initial solution file 9, the temperature levels in the cavity and the mass flow rate across the interface(s) continued to develop for approximately 600 ­ 1200 iteration before reaching a steady or consistently fluctuating state. See Figure 3.22 and Figure 3.23 for RMS residual plots. If the system ended up demonstrating steady physical behavior, as is the case with most PW307 runs (m* = 0.00266 being the only exception), the ingestion analysis procedure was straightforward: The ingestion values and averaged cavity surface temperatures could be read directly from the monitoring data. See Figure 3.24 and Figure 3.25 for monitor plots.

Figure 3.22 ATFI. Example of Momentum and Mass RMS residual behavior. The physical system is highly unsteady.

Figure 3.23 PW307. Example of Momentum and Mass RMS residual behavior. The particular system reaches a steady physical state.

9

The start-up of a new model, with no initial solution file available, requires 1000+ iterations with primarily first order

upwinding advection scheme. It is also recommended that the transition to a high resolution, second order discretisation scheme, is done gradually for the flow system is initially very prone to numerical failure. For a multi-domain case the amount of iterations could be reduced by individually initializing each domain with appropriate flow parameters. This option, however, was not used in this study.

64

Figure 3.24 PW307. Monitoring of ingestion. The system clearly settles to a steady state. (m* = 0.00399)

Figure 3.25 PW307. Monitoring of mass flow rate at interface. The case with the lowest purge flow rate (m* = 0.00266) demonstrated unsteadiness.

With the ATFI cases, which demonstrated high level of unsteadiness, the computations were continued sufficient amount of time so that multiple fluctuating periods could be included in the `time-averaging' analysis. This typically required 800 ­ 1200 iterations from a restart. Since the approach to acquire ingestion measurements and monitor the state of the solution at an interface plane was introduced in this study, it required an accompanying data analysis procedure to be developed as well. This derived procedure for extracting meaningful values, particularly for ingestion, involves the following steps: · · The massInterface (shown in Figure 3.26), ingestion, netHeatFlow and disc/cavWallTemp monitor plots are exported as text files from CFX-Solver Manager to a spreadsheet. A `backward averaging' is applied to massInterface values. This means that a running average of the values is computed starting from the last iteration and proceeding toward the first. The values are scanned and iterations where the averaged mass flow across the interface value is equal to the cavity purge flow rate (bw_ave(m*interface) = m*purge) are marked. · Depending on the frequency of the occurrence, 2nd ­ 5th marked iteration point is chosen ­ counting from the end ­ as the starting point and plots of ingestion vs. massInterface and netHeatFlow vs massInterface are drawn. Refer to Figure 3.27 and Figure 3.28. · The massInterface vs. netHeatFlow plot should demonstrate linear relationship (as shown in the picture.) If not, a new starting point that is closer to the last iteration is chosen.

65

· ·

Linear curve fitting trend lines, including the formula, are imposed over the two plots. Referring to the trend line formulas, if YnetHeatFlow(m*purge) 0 the justified time-averaged value for ingestion is obtained from Yingestion(m*purge).

Figure 3.26 ATFI. Example of Mass Flow at Interface vs. iteration monitor plot.

Figure 3.27 ATFI. Example of Net Heat Flow vs. Mass Flow at Interface plot with imposed linear trend line. A linear relationship indicates an energy balance has been reached in the system. 66

Figure 3.28 ATFI. Example of Ingestion vs. Mass Flow at Interface plot and an imposed linear trend line for extracting a physically meaningful time averaged ingestion value.

67

4

4.1

RESULTS AND DISCUSSION

General Remarks on Results

The three-dimensional analysis of a flow system, which includes a high temperature, high Mach number turbine blade passage under favorable pressure gradient coupled with a low temperature, low Mach number and radially diffusive rotor-stator disc cavity, brings about a number of computational challenges. For instance, solving such a system that consists of two very dissimilar flow entities and time scales simultaneously imposes two contrasting computational requirements: First, a small time step to ensure numerical stability and, second, a comparatively long time period to reach physical convergence in the cavity. In acknowledgement of the fact that these contrasting requirements and the potentially unsteady nature of the flow system render complete ingestion analysis impractical, this study addresses the issue by presenting potential analysis and modeling methodologies that rely on the use of engineering judgment to find an acceptable balance of computational effort and solution accuracy. While neither of the models analyzed in this investigation represent the ultimately preferred balance of simplifications and realistic detail, the following discussion on the obtained results will provide information that allows valuable conclusions and recommendations to be stated ­ even in the absence of conclusive experimental verification.

The computational results and visualizations of the solution fields are presented and discussed in this section, which is divided according to the case categories specified in 3.3.2:

1) ATFI-INGST 2) ATFI-WAKE 3) ATFI-FER 4) PW307- INGST 5) PW307-PTM12442

For each category the general solution data is provided in a tabular and dimensionless form in Appendix C, while the physical ­ ingestion and temperature ­ relations are presented alongside the discussion graphically in order to capture the nature of the various dependencies. It ought to be mentioned for the record that due to time and computational limitations, rigorous error analysis has not been conducted in this study and, therefore, an appropriate level of discretion is required in analyzing the results. The flow patterns within the computational domain and the relevant flow field solutions are illustrated with contour and vector plots that are generated with CFX-Post. 68

4.2

ATFI-INGST

The gas path mass and momentum boundary conditions for the ATFI-INGST analysis were scaled down in effort to avoid possible numerical difficulties arising from the presence of a shockwave at the trailing edge of the ATFI turbine blade. The mass flow rate was reduced by approximately 3%, which resulted in a 15% reduction in engine RPMs according to the method devised for the MATLAB-program. The program scales the engine RPM for a new mass flow rate such that the angles in the relative velocity triangle ­ at a specified radial plane ­ remain constant. At start-up, when no initial solution was available, an excessive purge flow rate was specified in order to convect the initially ingested hot gas more rapidly out of the cavity. Furthermore, the gas path outlet was initially specified as subsonic in a stationary frame, with a static pressure boundary condition to speed up the convergence, but switched to supersonic in a rotational frame to eliminate oscillations in the flow field near the exit. The system reached a steady state solution after approximately 2000 initial iterations with varying physical time scales

10 and gradually increasing level of second order advection scheme . However, when the purge

flow rate was reduced to a more realistic level, the solution ­ which during the following 1000 iterations seemed to approach a steady state solution with properly converging residuals ­ eventually began to demonstrate fluctuations in the flow properties at the interface. This onset of oscillating behavior was also marked by a sudden elevation of mass and momentum RMS residuals and, since the onset, the unsteadiness never subdued in the following analysis, although, the severity of the oscillations reduced with increasing purge mass flow rate. The steady state time stepping scheme of CFX-5, which utilizes a physical time step to advance the solution toward convergence, has proven extremely fitting for this kind of analysis because it allows physical unsteadiness in the solution field to manifest, but does not resolve the unsteady behavior in a time accurate fashion ­ which would be useless anyways for the simplified model does not take into account the dominating time dependent vane-blade interaction. Therefore the observed unsteady behavior of the flow system cannot be granted, at this point, any physical validity in an engine environment, but it does provide insight about the nature and interaction of the governing driving forces in the system. Referring to the author's previous ingestion study [2] ­ which demonstrated how, in the absence of turbine blades and vanes, the unsteady nature of

10

In the initial phase of this project, CFX-5.6 Solver and Solution Manager were used which allowed computational

parameters to be edited while the runs were in progress. This functionality was regrettably lost with the introduction of CFX-5.7 due to interface incompatibilities.

69

ingestion into the disc cavity was due to excessive viscous disc pumping 11 ­ the presence of similar fluctuations at the ATFI model's rim seal area suggests a direct analogy. However, the existence of the pressure asymmetry in the gas path created by the turbine blade introduces an additional driving mechanism for ingestion.

Figure 4.1 ATFI (m*=0.00623). Stationary frame velocity vector plot at the rim.

Figure 4.2 ATFI (m*=0.00623). Tangentially projected stationary frame velocity vector plot at the rim.

Considering the blade passage flow in a stationary frame of reference, the solutions depict the following qualitative behavior: As the flow near the bottom of the gas path exits the vane passage with a dominating V component, it passes over the annular rim at a large angle from the axial axis setting up a vortex in the region between the stationary rim and the rotating disc. See Figure 4.1 and Figure 4.2 for illustrative vector plots. The vortex structure at the rim would form even in the absence of the blade, but in its presence the resulting pressure asymmetry in the -direction forces the vortex structure into a periodic pattern where it is either strongly driven into the cavity by the high-pressure region or drawn out to the suction side of the blade. The pressure and Mach number contour plots shown in Figure 4.3 and Figure 4.4 demonstrate the dramatic asymmetry created by the turbine blade while Figure 4.5 and Figure 4.6 provide further visualizations of the ingestion phenomena. Therefore, simply put, if the purge flow out of the cavity cannot repel the local surges toward the cavity, ingestion will occur.

11

For clarity, excessive viscous disc pumping refers to a situation where the radial mass flow caused by the centrifugal

forces in the disc's boundary layer exceeds the internally supplied purge flow rate.

70

Figure 4.3 ATFI. Mach number distribution at radial plane R*= 0.690 in the blade passage. Color scale: Mamin_ref=0.2, Mamax_ref=1.1

Figure 4.4 ATFI. Static pressure distribution at radial plane R*= 0.690 in the blade passage.

Figure 4.5 ATFI (m*=0.00623). Ingestion visualization. Interface color: Velocity u: Redpositive (outward); Blue-negative (inward). Blade color: static pressure.

Figure 4.6 ATFI (m*=0.00623). Ingestion visualization with backward traced streamlines from Interface added. Streamline color: static temperature.

71

Since the ATFI has a highly loaded turbine section ­ which translates into drastic pressure asymmetry at the rim ­ and a comparatively spacious upper rim design, it demonstrates high levels of hot gas ingestion into the top region of the cavity. The unsteady interaction of the excessive disc pumping and the periodic flow structure at the rim only further deteriorates the sealing performance for it allows hot gas to sporadically enter deeper into the upper cavity space. This unsteady activity at the rim seal is demonstrated in a series of momentary snap shots of the flow field in Figure 4.7. While the sealing effectiveness at the region of the interface is shown to be poor, a noticeable temperature discontinuity develops at the bottom part of the rim seal, as seen in Figure 4.8 and Figure 4.9, shifting the actual location of the seal. Nonetheless, leakage does occur at this location due to constant fluid exchange between the adjacent cavity regions; the hot air from the upper cavity space enters the inner cavity at the stationary wall and is convected deeper into the cavity by the downward spiraling part of the local vortex structure. Figure 4.10 and Figure 4.11 further illustrate the penetration of the hot air into the cavity as the purge flow is reduced. The tendency of the cavity flow to transport the ingested hot air radially inward places further emphasis on the importance of a high sealing efficiency at the upper rim.

72

Iteration: ITER Momentary m*_interface = 0.00903 Momentary ingestion = 0.39

Iteration: ITER + 20 Momentary m*_interface = 0.00878 Momentary ingestion = 0.80

Iteration: ITER + 40 Momentary m*_interface = 0.00888 Momentary ingestion = 0.79

Figure 4.7 ATFI. Case 3: m* = 0.00873. Illustration of flow at rim seal area. Color: Temperature with velocity vectors.

73

m* = 0.00873 Figure 4.8 ATFI: Total temperature in stn. frame distribution at the outer cavity. Color Scale: T*max_ref=0.36 (Red); T*min_ref=0.09 (Blue)

m* = 0.00499 Figure 4.9 ATFI:. Total temperature in stn. frame distribution in the upper cavity. Color Scale: T*max_ref=0.36 (Red); T*min_ref=0.09 (Blue)

m* = 0.00873 Figure 4.10 ATFI: Total temperature in stn. fr. distribution at the inner cavity. Color Scale: T*max_ref=0.147 (Red); T*min_ref =0.076 (Blue)

m* = 0.00499 Figure 4.11 ATFI: Total temperature in stn. fr. distribution at the inner cavity. Color Scale: T*max_ref=0.147 (Red); T*min_ref =0.076 (Blue)

The quantitative results of the ingestion study that consists of 7 computational cases with systematically varying purge mass flow rates are presented in effort to establish correlations 74

between physical flow parameters in the cavity and ingestion. In Figure 4.12 the ingestion values measured across the interface are plotted as a function of mass flow rate ratio m* =

& & m purge / mGasPath , although, the most appropriate dimensionless variable for ingestion

correlations remains an open question warranting further parametric study. It should be noted, however, that Wilford and Raymond [3] showed that the correct dimensionless variable for a system dictated solely by disc pumping and cavity purge flow is written as

& m purge /( 2b )

0 Re .8

which relates purge flow rate to disc pumping. Here b is the radius of the disc, viscosity of the fluid and Re the disc's Reynolds number. Unfortunately, due to time limitations, the Re -effect on ingestion could not be pursued vigorously in this study.

The ATFI ingestion analysis yielded an intuitive inverse correlation, which, in a qualitative sense, closely resembles the corresponding results of prior ingestion studies. The anticipated asymptotic behavior with increasing purge mass flow rate and the rapid boost in ingestion that occurs as cooling flow is reduced are successfully captured in the analysis. It should be noted that the numerical ingestion values have an apparent vertical offset that is due to the location of the monitoring interface being in the middle of a vortex structure where some of the ingested flow circles back out into the gas path. It can be seen in Figure 4.5 how the interface is not located at the very edge of the rim. In retrospect the monitoring interface should have been placed slightly deeper in the seal. Nonetheless, this does not impose a problem because the level of the offset can be determined from Figure 4.18 by extrapolating the temperature distribution curves to the level where Tt* = Tt*_purge at every radial location. The corresponding ingestion value gives the offset. Being aware of this phenomenon, the location of the interface in the PW307-model was chosen with more care. See Figure 4.31 for comparison.

75

Figure 4.12 ATFI. Ingestion vs. dimensionless purge mass flow rate. The curve has an apparent vertical offset due to the location of the monitoring interface.

The total temperature and static pressure distributions ­ extracted from radial control planes shown in Figure 4.13 ­ are presented in Figure 4.14 through Figure 4.18 as functions of ingestion and purge mass flow rate. The temperature values at the top most cavity plane (R*=0.581) are presented separately in Figure 4.16 due to the heavily fluctuating conditions and the much higher temperature level of that area. The added error bars indicate the approximate range of the oscillating temperature values. Inspection of the temperature distribution plots in Figure 4.14 and Figure 4.15, in turn, yields information on how the reduction in purge flow rate affects the local flow structures in the cavity, especially at the upper regions where disc-pumping effects are more pronounced. Specifically the cavity region above the bend (R*=0.261) at low purge mass flow levels suffers the most dramatic increase in temperature as the vortices become spatially more confined at the top and bottom regions. This localized vortex flow structure is typical in cavity flows, but the spatial size of the individual vortices, particularly in regions close to cavity inlets, is dependent on the velocity with which the cooling flow is purged. It therefore becomes an important modeling issue if the cavity flow patterns are to be carefully simulated. Even though the cross-sectional area of the ATFI model's purge flow inlet has not been sized to achieve a realistic flow velocity, the winding cavity geometry appears to have a stronger influence on the flow patterns ­ see Figure 4.10 ­ and therefore render the velocity effect less significant in this case. 76

R7* = 0.581, (unsteady) R6* = 0.491

R5* = 0.414 R4* = 0.338

R3* = 0.261 R2* = 0.184 R1* = 0.107

Figure 4.13 ATFI. Data presented below refers to the spatially averaged computational values at the shown radial planes. Since the temperature values at the top plane (R7*) are strongly fluctuating, they are presented separately.

ATFI-INGST Cavity Total Temperature vs. Radius

0.275 0.265 0.255 Dimensionless Total Temperature in Stn Frame, Tt* 0.245 0.235 0.225 0.215 0.205 0.195 0.185 0.175 0.165 0.155 0.145 0.135 0.125 0.115 0.105 0.095 0.085 0.075 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

m* = 0.0112 m* = 0.00998 m* = 0.00873 m* = 0.00748 m* = 0.00623 m* = 0.00499 m* = 0.00374

Dimensionless Radius, R*

Figure 4.14 ATFI. Cavity total temperature in stationary frame vs. radius.

77

ATFI-INGST Cavity Total Temperature vs. Purge Mass Flow rate

0.275 0.265 0.255 0.245 0.235 Dimensionless Total Temperature, Tt* 0.225 0.215 0.205 0.195 0.185 0.175 0.165 0.155 0.145 0.135 0.125 0.115 0.105 0.095 0.085 0.075 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012

R* = 0.4912 R* = 0.4144 R* = 0.3377 R* = 0.2609 R* = 0.1842 R* = 0.1074

dimensionless mass flow rate, m*

Figure 4.15 ATFI. Cavity total temperature in stationary frame vs. purge mass flow rate.

Total Temperature at R* = 0.581 vs. Purge Mass Flow rate

0,6

0,5

R* = 0.581

Dimensionless Total Temperature, Tt*

0,4

0,3

0,2

0,1

0 0 0,002 0,004 0,006 dimensionless mass flow rate, m* 0,008 0,01 0,012

Figure 4.16 ATFI. Total temperature in stationary frame at top cavity plane (R* = 0.581) vs. purge mass flow rate.

78

ATFI-INGST Tt*_Stn, Tt*_Rel vs. Radius

0,255 0,235 0,215

Total Temperature, Tt*_Stn / Tt*_Rel m* = 0.00998 [Rel Frame]

0,195 0,175 0,155 0,135 0,115 0,095 0,075 0

m* = 0.00998 [Stn Frame] m* = 0.00499 [Rel Frame] m* = 0.00499 [Stn Frame]

Tt*_Stn_purge = 0.076 Tt*_Rel_purge = 0.094

0,1

0,2

0,3

Dimensionless Radius, R*

0,4

0,5

0,6

Figure 4.17 ATFI. Comparison of Total Temperature values in stationary and relative (rotating) frame of references. Only two cases are included.

ATFI-INGST Cavity Total Temperature vs. Ingestion Tt*_purge = 0.076

0,275 0,25

R* = 0.107

0,225

Total Temperature in Stn Frame, Tt*

R* = 0.184 R* = 0.261 R* = 0.338

0,2 0,175 0,15 0,125 0,1 0,075 0

R* = 0.414 R* = 0.491

0,5

1

1,5

2

2,5

Ingestion (m*_inward / m*_purge)

Figure 4.18 ATFI. Cavity total temperature in stationary frame vs. time averaged ingestion. The apparent offset in ingestion values can be determined from such plot by extrapolating the temperature curves to the horizontal axis where Tt* = Tt*_purge. Here, the extrapolation produces an approximate value of 0.55 for the offset.

79

The fact that the model does not include the blade fixing and vane platform leakages will influence the actual radial temperature distribution in the cavity by reducing the temperature level in the top region where the vane leakage purges cool flow into the area and increasing the temperature in the lower cavity as the radially pumped mass flow is subsequently reduced. It is very difficult, however, to postulate what kind of effect the blade platform and fir tree fixing leakages, which actually manifest as a group of small suction holes on the disc, have on the cavity flow and sealing efficiency. Since realistic modeling of such details is not an option, the matter necessitates the inclusion of approximations, which are either based on intuitive inspection of the physical flow behavior or theoretical results from a simplified CFD analysis.

With the ATFI model the emphasis was placed on having a `clean' disc and omitting all approximating details, which may disrupt the cavity flow field by imposing nonphysical periodic information with wave lengths and frequencies that ­ based on intuition ­ have no physical basis. Instead, the flow extraction is taken into account by subtracting the corresponding mass flow from the main cavity inlet. Since the ATFI engine has been, and will be, subject to design changes, the focus of the analysis was not placed on obtaining experimental verification through cavity temperature and pressure measurements, but to gain understanding of the nature of ingestion and identify factors that influence it. But, in the light of theoretical evidence that the analyzed cavity is prone to ingest, the presented correlations may provide applicable information about the necessary level of purge flow required in a real engine.

The relationship between the static pressure level in the cavity and ingestion, shown in Figure 4.19, and the radial pressure distribution plots in Figure 4.20, prove particularly informative and encouraging in the light of the observation made by Zierer et al. [4] stating that the pressure gradient through the rim seal is a direct indication of the level of ingestion. And, since the radial pressure distribution in the cavity does not display similar sensitivity to local vortex patterns as temperature, it ends up offering more meaningful and generic information about the cavity's sealing performance. This is further supported by the PW307-PTM12442 results, which suggest that the pressure value at the top cavity does not depend on the presence of the vane leakage significantly.

80

ATFI-INGST Ingestion vs. Ave. Cavity Static Pressure 2.5

2

Ingestion (m*_inward / m*_purge)

1.5

1

0.5

0 0.39

0.4

0.41

0.42

0.43

0.44

0.45

0.46

Dimensionless Ave. Static Pressure, Ps*

Figure 4.19 ATFI. Time averaged ingestion vs. spatially averaged cavity static pressure. Pressure values above R3* = 0.261 were used in the averaging.

ATFI-INGST Cavity Static Pressure vs. Radius

0,46 0,45 0,44 0,43 0,42 0,41 0,4 0,39 0,38 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7

Dimensionless Radius, R*

Dimensionless Static Pressure, Ps*

m* = 0.0112 m* = 0.00873 m* = 0.00623 m* = 0.00374

Figure 4.20 ATFI. Cavity static pressure vs. radius.

81

Because the static pressure vs. ingestion correlation possesses a generic graphical form ­ similar to m* vs. ingestion correlation ­ it provides a valuable starting point in the attempt to utilize the obtained theoretical information in the design process. As a result of physical consideration of the flow system and the nature of the flow interaction between the gas path and the cavity, the following deduction is produced: The interaction between the gas path and cavity remain localized in the rim seal region leaving the main cavity flow relatively unaffected notwithstanding the temperature rise due to ingestion. As a consequence, in a fixed cavity and rim seal configuration, the sealing performance of the cavity can be reasoned to be independent of the gas path conditions within reasonable margins. This, in turn, justifies the following assumption: The static pressure vs. ingestion plot will retain its general form ­ again, within reasonable margins ­ while modifications to the boundary conditions or small changes in the geometric features of the model will essentially cause the correlation curve to shift horizontally. This opens the possibility to device a correction mechanism that utilizes cavity temperature and pressure information that is conveniently obtained from air system measurements to horizontally offset the theoretical curve. The legitimacy of such a correction scheme, however, requires experimental validation that falls beyond the scope of this work.

4.3

ATFI-WAKE

The results from the three cases ­ which were run with fixed boundary conditions varying only the location of the synthetically generated wake at the main inlet ­ are discussed and presented here in comparison to the case with circumferentially averaged radial inlet profile.

It is recognized that in the absence of turbine vanes the CFD models under-predict ingestion (Hills et al. [1]) and, therefore, it is necessary to find an assessment for the extent of the margins, which need to be imposed on the blade-only results. In effort to investigate the effect vane-blade interaction will have on ingestion without reverting to extremely time consuming and computationally demanding multi-domain, sliding plane, and time accurate analysis, three cases with a synthetically generated 3D total pressure profiles at gas path inlet were considered in this study. The profiles assimilate the pressure distribution in the circumferential direction and successfully generate a velocity depression in the main stream that is representative of a wake generated by an upstream vane. The three cases, which represent instantaneous snap-shots of the unsteady vane-blade interaction, are illustrated in Figure 4.21 with Mach number contour plots that show the manifestation of the wake. It is assumed that each of the three solutions represents one third of the vane-blade passage time so that averaging the obtained results yields an initial estimate of the overall effect of the vane-blade interaction. This estimate can be used to shift the original ingestion vs. purge flow rate curve for early considerations when experimental 82

data is not yet available. It is important to state that this method is understood to under-estimate the ingestion increase because the approach does not take into account the actual transient effect. But, it does provide useful information about the effect a wake-like velocity depression has on the flow behavior at the rim seal. All cases presented in Figure 4.21 correspond to m* = 0.00623. The ingestion value obtained by averaging the three wake-cases is based on the assumption that the three cases represent equal `snap shots' of time dependent vane-blade interaction. The three cases confirmed what was intuitively predicted: A wake directed at the pressure side of the blade leads to a considerable increase in ingestion, while the wake effect on the suction side is less significant. The timeaveraged results suggest ~25% increase in ingestion (considering the offset in ingestion values to be approximately 0.55) and a subsequent increase in temperature values in the upper cavity as shown in Figure 4.22. Therefore, in accordance with the expected under-estimating nature of the results, it can be recommended that a horizontal shift providing a ~35% increase in ingestion levels should be imposed on the original plot for initial estimations. Figure 4.23 illustrates the suggested correction.

83

Circumferentially Averaged Ingestion = 1.08

Wake at LE / Suction Side Ingestion = 1.15

Wake at Suction Side Ingestion = 1.07

Wake at Pressure Side Ingestion = 1.45

Figure 4.21 ATFI. Visual illustration of the different cases. Left column: pt-profile imposed at Inlet1.

Right column: Mach number contours at a radial plane in gas path. Inlet1 pt-profile visible at bottom.

Ave Ingestionwake = 1.22

84

ATFI-WAKE Comparison: Total Temperature vs. Radius

0,17 0,16

dimensionless Tt in Stn. Frame, Tt*

0,15 0,14 0,13 0,12 0,11 0,1 0,09 0,08 0,07 0,06 0 0,1 0,2 0,3

dimensionless radius, R*

ATFI-WAKE Averaged ATFI-INGST

0,4

0,5

0,6

Figure 4.22 ATFI. Comparison of temperature distributions in the cavity.

ATFI-INGST, WAKE Time Averaged Ingestion vs. Purge Mass Flow Rate

2,25

2

Corrected Correlation Curve (Representative)

1,75

Ingestion [m*_inward / m*_purge]

1,5

WAKE-Averaged

1,25

1

0,75

0,5

0,25

0 0,002

0,003

0,004

0,005

0,006

0,007

0,008

0,009

0,01

0,011

0,012

dimensionless mass flow rate, m*

Figure 4.23 ATFI. Ingestion vs. purge mass flow rate plot with an added point to illustrate the wake effect. A representative correlation curve is also included to demonstrate an estimate which takes into account the addition to ingestion due to vane-blade interaction.

85

4.4

ATFI-FER

This individual case was run in anticipation of the ATFI's first engine run block test. The applied gas path and cavity boundary conditions were specified according to the engine test's performance point data such that the cavity inlet, Inlet 2, was assigned the net cavity purge flow rate. The attained solution demonstrated high levels of ingestion and unsteady behavior at the rim seal, similar to comparable ATFI-INGST cases, yielding a value shown in Figure 4.24. This result is plotted with other ATFI ingestion data showing the appropriate horizontal shift from the original curve that is imposed in accordance with the fore-mentioned assumption.

Because the engine block test has been postponed to a later date, the pressure and temperature distributions in Figure 4.25 and Figure 4.26 are presented in the absence of experimental data.

ATFI-INGST, WAKE, FER

Time Averaged Ingestion vs. Purge Mass Flow Rate

2.25

2

ATFI-FER

1.75

Ingestion [m*_inward / m*_purge]

1.5

WAKE-Averaged

1.25

1

0.75

0.5

0.25

0 0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.011

0.012

dimensionless mass flow rate, m*

Figure 4.24 ATFI. Ingestion vs. mass flow rate plot with FER result added. It should be noted that the boundary conditions for ATFI-FER differ from INGST and WAKE cases. The green curve illustrates a possible ingestion correlation for the FER configuration.

86

ATFI-FER Cavity Static Pressure vs. Radius

0.61

dimensionless static pressure, ps*

0.605

0.6

0.595

0.59

0.585

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 dimensionless radius, R*

Figure 4.25 ATFI. Cavity static pressure vs. radius.

ATFI-FER Total Temperature vs. Radius

0.35

dimensionless Tt in Stn Frame, Tt*

0.3

0.25 ATFI-FER ATFI-FER R7* 0.2

0.15

0.1

0.05

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

dimensionless radius, R*

Figure 4.26 ATFI. Cavity total temperature vs. radius including the value at R* = 0.581.

87

4.5

PW307-INGST

As the ATFI model was purposefully kept simplified to introduce a degree of generality to the ingestion study, the opposite approach was chosen with the PW307 HPT2 model. Principally, the objective has now shifted to investigating how the feasibility of the CFD analysis is affected by the inclusion of realistic cavity details and whether experimental verification can be attained. This different approach will hinder the validity of any direct comparison between the ATFI and PW307 results, but, since the PW307 model represents an engine that differs dramatically from the ATFI, appropriate comparisons will provide valuable information about effects that arise from the contrasts. Because PW307 is a dimensionally smaller engine than ATFI and its corresponding turbine section has a lower level of loading, the following physical differences arise as a result:

-

The viscous disc pumping effect is reduced as seen from the disc's Reynolds

2 number, Re (= R /), ratio:

2 R ATFI = 1.492 2 RPW 307

-

The cavity is volumetrically smaller. The pressure asymmetry created by the blade is less pronounced.

Considering these aspects together with the fact that the design point purge flow rate of PW307 HPT2 cavity is almost twice that of ATFI's gives a strong implication that the system will be far less sensitive to ingestion than the ATFI. In addition, the rim seal design of PW307 HPT2 is tighter than the ATFI's, which also would only further improve its comparative sealing efficiency.

The results from the PW307 ingestion study do stand in an agreement with the anticipated behavior. The flow behavior at the rim seal is dictated by the same principles as in the ATFI and shown in Figure 4.27 and Figure 4.28, but with exception that the vortex structure does not display similar spatial extent and tendency to surge into the cavity. The Mach number and static pressure contour plots in Figure 4.29 and Figure 4.30 when viewed with the ingestion visualizations in Figure 4.31 and Figure 4.32 clearly illustrate how the reduced loading of the blade reduces the force driving ingestion. A visual comparison with the corresponding ATFI images (Figure 4.5 - Figure 4.4) further emphasizes the influence of turbine stage loading on ingestion.

88

Figure 4.27 PW307. Stationary frame velocity vector plot at the rim.

Figure 4.28 PW307. Tangentially projected stationary frame velocity vector plot at the rim.

Figure 4.29 PW307. Mach number distribution at radial plane R*= 0.610 in the blade passage. Color Scale: Mamin_ref=0.2, Mamax_ref=0.95

Figure 4.30 PW307. Static pressure distribution at radial plane R*= 0.610 in the blade passage.

89

Figure 4.31 PW307. Ingestion visualization. Interface color: Velocity u (red-positive, bluenegative.)

Figure 4.32 PW307. Ingestion visualization with backward traced streamlines from Interface added. Streamline color: Velocity. Blade color: Static Pressure.

PW307-INGST Ingestion vs. Purge Mass Flow Rate 0.4 0.35 0.3 0.25 Ingestion 0.2 0.15 0.1 0.05 0 0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.0055

0.006

0.0065

dimensionless mass flow rate, m*

Figure 4.33 PW307. Ingestion vs. purge mass flow rate.

90

Once an initial solution was available, the different cases required 800 ­ 1600 iterations for the temperature values to settle in the cavity. However, as expected, it was observed that as the cavity purge flow level was reduced higher number of iterations was required to convect the temperature values in the cavity. The analysis yielded a correlation between ingestion and cavity purge flow rate, shown in Figure 4.33, which possesses the same characteristics as the corresponding ATFI-INGST curve, while having significantly reduced range of ingestion values. The series of cases considered in the PW307 ingestion study also revealed significant information in support of the observations made with the ATFI study: While all the PW307 cases with m* > 0.00266 demonstrated steady state behavior, the case with the highest ingestion level (m* = 0.00266) exhibited a break-down of stability and manifested oscillations in mass flow rate and heat flow properties at the rim seal. This observation further strengthens the earlier inferences made in relation to the role of excessive disc pumping in cavity-gas path systems. It is thereby concluded that corresponding cases, which demonstrate significant ingestion, necessitate a CFD analysis scheme that is based on physical time stepping.

m* = 0.00532

m* = 0.00399

m* = 0.00266

Figure 4.34 PW307. Total temperature distribution in the cavity. Color Scale: Tt*min_ref =0.040 (Blue); Tt*max_ref=0.212 (Red).

The images shown in Figure 4.34 provide a global view of the temperature distributions within the cavity at different purge flow rates. The subsequent spatial temperature and pressure distribution data ­ taken at the radial monitoring planes defined in Figure 4.35 ­ are presented in Figure 4.36 through Figure 4.39 including the strong correlation, shown in Figure 4.37, between ingestion and upper cavity pressure level. Referring to the discussion on ATFI-INGST, the existence of such systematic correlation, which relates ingestion with a measurable cavity parameter, provides further encouragement about the possibility of developing a method for revising the theoretically predicted curves according to experimental test data. The correction method would still have to rely on a local temperature measurement to indicate the particular ingestion level and it is this 91

issue that motivated the inclusion of realistic cavity details into the model at the expense of complicating the analysis.

The ingestion study was carried out with the modified model (Mdf_1), which does not include the circumferential inlet and outlet on the disc, by varying the mass flow rate of the main cavity inlet (Inlet_Cav2) while maintaining the mass flow rate of the brush seal (Inlet_Cav1) and the vane platform leakage (Inlet_Cav3) constant. The specific temperature values of each inlet were kept unchanged for all the cases. By altering the mass flow rate at one inlet while keeping others fixed, a different purge flow distribution resulted in the cavity with each case. This caused the flow patterns and the radial temperature distributions within the cavity to vary accordingly. Therefore, the temperature distribution plots in Figure 4.38 display some irregular behavior around and below the Inlet_Cav2 region at R* = 0.300. The regional effect is also seen in the pressure distribution plots in Figure 4.36 and therefore the pressure and temperature values at the top three radial planes are considered most useful.

R7* = 0.538 R6* = 0.459

R5* = 0.378 R4* = 0.300 R3* = 0.220 R2* = 0.140 R1* = 0.062

Figure 4.35 PW307. Data presented in the following figures refers to the area-averaged measurements at the shown radial planes.

92

PW307-INGST Cavity Static Pressure vs. Radius 0.86 m* m* m* m* m* m* = = = = = = 0.00598 0.00532 0.00465 0.00399 0.00332 0.00266

0.85 dimensionless static pressure, ps*

0.84

0.83

0.82

0.81

0.8 0 0.1 0.2 0.3 dimensionless radius, R* 0.4 0.5 0.6

Figure 4.36 PW307. Cavity static pressure vs. radius.

PW307-INGST Ingestion vs. Ave. Upper Cavity Static Pressure 0.45 0.4 0.35 0.3 Ingestion 0.25 0.2 0.15 0.1 0.05 0 0.832

0.834

0.836

0.838

0.84

0.842

0.844

0.846

0.848

0.85

0.852

0.854

dimensionless static pressure, ps*

Figure 4.37 PW307. Ingestion vs. averaged static pressure. The pressure values used in the average were taken above Inlet_Cav2 at R5*, R6* and R7*.

93

PW307-INGST Cavity Total Temperature vs. Radius

0.18 dimensionless Total Temperature in Stn Fr, Tt* 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06

0 0.1 0.2 0.3 0.4 0.5 0.6 m* m* m* m* m* m* = 0.00598 = 0.00532 = 0.00465 = 0.00399 = 0.00332 = 0.00266

dimensionless radius, R*

Figure 4.38 PW307. Cavity total temperature in stationary frame vs. radius.

PW307-INGST Cavity Total Temperature vs. Purge Mass Flow Rate

0.18

0.16 dimensionless Tt in Stn Fr, Tt*

0.14

R* R* R* R* R* R* R*

= 0.538 = 0.459 = 0.378 = 0.300 = 0.220 = 0.140 = 0.062

0.12

0.1

0.08

0.06 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 0.0055 0.006 0.0065

dimensionless purge mass flow rate, m*

Figure 4.39 PW307. Cavity total temperature in stationary frame vs. purge mass flow rate.

94

4.6

PW307-PTM12442

In this section the results obtained from the three variations of a PW307 HPT2 model are presented and compared to experimental measurements documented by Liu and Dunning [11].

This portion of the investigation entails three computational cases, each solved according to the PTM12442 performance point specifications, but with varying degree of cavity details included. The main effort, therefore, focuses on finding a modeling approach for cavity flow and ingestion analysis, which provides the best compromise between accuracy and computational effort. As discussed in 3.1.3, the attempt to include a high degree of geometric detail and all of the inlets and outlets in the cavity imposes a number of challenging modeling issues. Ultimately, it is desirable to reduce the level of detail as much as justifiably possible since the added features do not only increase the node count of the computational grids, but also complicate the model creation and especially the meshing procedures. The complete PW307 model used in this study represents an approach that attempts to eliminate as many approximations as possible at the expense of the excessive modeling and computational effort required, while the modified versions ­ Mdf_1 and Mdf_2 defined in 3.3.2 ­ introduce some desirable simplifications. The subsequent comparison of the results, produced by the three models, with experimental air system test measurements is conducted in anticipation that it will provide evidence that leads to a ruling that goes either in favor or against the inclusion of greater cavity detail. As seen from Figure 4.40, Figure 4.41 and Figure 4.42, the Complete and modified cavity configurations generate unique flow patterns in the cavity, which have a significant impact on the spatial temperature distributions. The solutions clearly illustrate the substantial, and somewhat unexpected, influence of Inlet_Cav4 on the lower cavity temperature level and the flow patterns in the central region. These flow effects are largely due to the modeling decision to consolidate all small fir-tree-fixing holes into an approximately positioned and narrow radial inlet with a representative area specified by Liu and Carpenter in [9]. Therefore, the increased temperature level at the bottom cavity, as suggested by the complete model, does not warrant any physical justification and leads to a recommendation against the inclusion of such inlets on the disc 12.

12

It should be noted that typically the fir-tree-fixing holes function as outlets, draining flow from the disc's boundary layer.

Therefore, if the combined drainage through the disc is clearly significant, a single radial outlet (or opening) ­ similar to Outlet_Cav ­ would be recommended.

95

1) Complete

2) Mdf_1

3) Mdf_2

Figure 4.40 PW307. Total temperature in stn. frame distributions in the cavity. Color Scale: Tt*min_ref =0.04 (Blue); Tt*max_ref=0.212 (Red).

1) Complete

2) Mdf_1

3) Mdf_2

Figure 4.41 1) PW307. Projected and normalized vector plots demonstrating cavity vortex structures.

The comparison of static pressure distributions in Figure 4.43 shows how the different configurations, while demonstrating varying pressure levels at lower radial locations, converge toward the top cavity region where the conditions determine the level of ingestion. Especially the close agreement between the Complete and Mdf_1 model is considered indicative of the fact that the exclusion of inlets or outlets on the disc does not affect ingestion. When considering the role of the vane leakage at the top cavity wall, it is predicted that, while its presence does not seem to alter the pressure level considerably under the particular set of boundary conditions, it may have an increasing effect on ingestion at lower purge flow conditions. Nevertheless, because the blade-only models have been verified to under predict ingestion to begin with, the recommendation concerning the vane leakage inlet becomes two-fold: From the pressure point of view the vane leakage can be rendered unnecessary, but, since the introduction of cool flow into 96

the region will alter the temperature values locally, especially in the presence of ingestion, its inclusion is recommended ­ in an alternative form. (See Recommendations for further detail.)

PW307-PTM12442 Total Temperature vs Radius 0.39

Complete Mod_1 Mod_2

0.34 dimensionless Tt in Stn Frame, Tt*

0.29

Air System Measurement

0.24

0.19

0.14

0.09

0.04 0 0.1 0.2 0.3 dimensionless radius, R* 0.4 0.5 0.6

Figure 4.42 PW307. Comparison of total temperature distributions with three air system measurements included. The dramatic discrepancy between the measured and theoretical values at the top cavity necessitates further investigation.

Due to the gross disagreement at the top cavity in both temperature and pressure values in Figure 4.42 and Figure 4.43, the comparison between the theoretical and the air system measurement values, unfortunately, fails to provide any conclusive validation of the results or indication about possible shortcomings in the analysis. This is because the measured temperature values at the top suggest a situation that cannot be physically justified under the considered conditions. The experimental values indicate that significant ingestion occurs, regardless of the high pressure value in the upper cavity, into the top region, but none of the consequently heated air is convected into the main cavity along the static wall ­ which stands in a stark contrast to all preceding ingestion analysis by the author and other scholars referenced in this report. Further consideration of the apparent discrepancy in temperature values reveals additional contradiction between the boundary conditions specified in [9] and the measured values. The blade fir-tree-fixing leakage is predicted to constitute 18% of the purge flow and have a total temperature of Tt* = 0.23, yet the measured value in the region indicates a temperature 97

value Tt* = 0.07, which falls even below the temperature specified for Inlet_Cav2. Although it is recognized that the experimental measurements as well as theoretical predictions must be considered with appropriate error margins, the disagreement of the presented experimental data with the predicted flow behavior in the cavity is so significant that it necessitates further investigation of both theoretical and experimental aspects of cavity flow and ingestion analysis.

PW307-PTM12442 Cavity Static Pressure vs. Radius 0.9 0.88 dimensionless Static Pressure, ps* 0.86 0.84 0.82 0.8 0.78 0.76 0.74 0.72 0 0.1 0.2 0.3 dimensionless radius, R* 0.4 0.5 0.6 Complete Mod_1 Mod_2 Air System Measurement

Figure 4.43 PW307. Comparison of static pressure distributions in the cavity with two air system measurement values included.

98

5

CONCLUSIONS

The following conclusions are presented as a summary of the essential deductions acquired from the analysis.

While complete ingestion analysis with complete turbine vane-blade models and time-accurate solution schemes provide the best accuracy, the study suggests that utilizing a blade-only model with circumferentially averaged gas path boundary conditions enables essential information to be attained about the cavity­gas path flow system. In addition, the simplified model approach allows fundamental dependencies to be established between selected parameters and ingestion.

The series of computational cases completed in this study with different cavity inlet conditions verified that ingestion has two different driving mechanisms: 1. The circumferential pressure asymmetry at the annular rim due to the presence of turbine blades and vanes. 2. Excessive disc pumping; mass flow imbalance within the cavity, which develops when the radial mass flow caused by the centrifugal forces in the disc's boundary layer surpass the purged mass flow rate.

The analysis also provided further confirmation that excessive disc pumping results in timedependent flow behavior in the cavity and unsteady ingestion. This necessitates the use of a time stepping algorithm that advances the solution in physical time in corresponding computational ingestion analysis. Nevertheless, it should be mentioned that a steady-state solution is possible with cases of high purge mass flow rate as the disc pumping effect diminishes.

According to the solutions, the level of ingestion depends on the following parameters or geometrical features: 1. Turbine stage loading. Higher stage loading translates into a more pronounced circumferential pressure asymmetry at the rim, which at the high pressure side drives the vortical flow into the disc cavity. The most appropriate dimensionless variable for stage loading was not investigated. 2. Ratio of purge mass flow rate to disc pumping. Here the appropriate dimensionless variable, introduced in chapter 4.2, is given:

& m purge /(2b )

0 Re .8

,

99

where the disc's Reynolds number, Re (= b2/), characterizes the level of disc pumping. Due to the limited scope of this work, however, corresponding parametric study could not be conducted for validation. 3. Rim seal design. The geometric design and the gap dimensions especially at the outer rim seal have an effect on the vortex structure, which develops between the stationary annular rim and the rotating turbine blade hub. The appropriate geometric parameters and dimensionless variables were not investigated in this report.

The ingestion analysis of this study yielded encouraging correlations that stand in an agreement with related results published prior to this work. The generic forms of the obtained ingestion correlations with purge mass flow rate and cavity static pressure suggest that an experimental correction scheme may be possible. In such case, the theoretical curves could be shifted to include experimentally attained static pressure and total temperature measurements. This, however, requires further theoretical and experimental investigation because of the difficulty in determining the temperature distribution within the cavity.

100

6

RECOMMENDATIONS

The analysis methodology developed for this study, which utilizes an interface at the rim seal to monitor and extract information on ingestion and other flow parameters, is recommended due to its exceptionally flexible and diverse `real time' solution monitoring capabilities. With timedependent cases the interface monitoring system reduces the post-processing requirement dramatically since most information of interest can be obtained directly from the monitoring data.

Addressing the disc cavity modeling issues, based on the limited computational evidence, placing a discrete inlet to the top part of the stationary cavity wall to model a vane platform inlet (see Inlet_Cav3 in Figure 3.9) introduces more disadvantages than offers benefits. The attempt to closely simulate the impinging flow from the vertical strip-like inlet generates very little desirable new information; yet, it requires detrimental assumptions 13 and significantly increases the grid generation effort. In case it is important to capture the temperature distribution within the cavity, it is recommended that a circumferential inlet (similar to Inlet2 in Figure 3.2) is used to ensure that the correct amount of cooling flow is introduced into the region.

Addressing the issue of modeling blade platform and fir-tree fixing leakages, the following recommendations are provided based on computational and grid generation related pros and cons encountered during the study: 1. If both leakages function as outlets and the suction through them is significant 14 they should be consolidated into one circumferential outlet, such as Outlet_Cav in Figure 3.9. This kind of outlet can cause some computational difficulties and therefore it should be declared as an opening with a specified static pressure. Thus, some iterative restarts are required to reach the desired mass flow rate. 2. If one leakage functions as an outlet and the other as an inlet and their mass flow rates are comparable, neglect both and balance the total purge mass flow by altering the main cavity inlet's value. However, if the temperature of the incoming leakage flow is drastically higher than other inlet flow's in the cavity, consolidate the added energy also to the main inlet. In case the outlet mass flow rate is much higher than the inlet's and considered

13

For example, it must be assumed that the relative position of the discrete inlet and the turbine blade remains fixed ­

even though one is stationary and the other rotating ­ for otherwise a time-accurate solution scheme with a sliding plane between the cavity and gas path must be used.

14

What is significant cannot be quantified with confidence at this point. As a rough approximation based on experience, a

suction of >20% of total purge mass flow rate should not be neglected.

101

significant, exclude the inlet with appropriate mass and energy balancing and include the outlet as above.

102

REFERENCES

[1] N. J. Hills, J. W. Chew and A. B. Turner, Computational and Mathematical Modeling of Turbine Rim Seal Ingestion, Vol. 124, American Society of Mechanical Engineers, 2002. [2] M. Auvinen, Prediction of Gas Path Ingestion in Rotating Turbine Disc Cavity Using CFD, A Project memo for Air and Oil Systems, Pratt & Whitney Canada, 2003. (Unpublished.) [3] D.F. Wilford and N. Raymond, Rim Seal Research Program Experimental Testing, Final Report, Pratt & Whitney Canada, 1982. (Unpublished.) [4] R. J. Zierer, L. DeVito, K. Lindblad, J. Larsson, D. Bohn, J. Funcke, A. Decker, Numerical Simulation of the Unsteady Flow Field in an Axial Gas Turbine Rim Seal Configuration, GT2004-53829, American Society of Mechanical Engineers, 2004. [5] J. M. Owen and R. H. Rogers, Flow and Heat Transfer in Rotating-Disc Systems, Volume 1 ­ Rotor-Stator Systems, John Wiley & Sons Inc., New York, 1989. [6] H. Schlichting, Boundary-Layer Theory, 7th Edition, McGraw-Hill Inc, New York, 1987. [7] CFX Ltd., CFX-5 Solver Theory, Version 5.7, ANSYS Co. / CFX Ltd., UK, 2003 [8] R. Marini, ATFI Blade Passage and Disc Cavity CFD Analysis, CEF No.: S-11126M, Pratt & Whitney Canada, 2004. (Unpublished.) [9] X. Liu, D. Carpenter, CFD Analysis on PW307 HPT cavities, SOW: 573093, Pratt & Whitney Canada, 2003. (Unpublished.) [10] E. Turgeon, Flow Ingestion Methodology for Combined Gas Path and Cavity: PW307 HPT 2nd Vane, Pratt & Whitney Canada, 2002. (Unpublished.) [11] X. Liu, J. Dunning, PW307 9813 Build 1 Air System Measurement Results, CEF No.: S-10671T, Pratt & Whitney Canada, 2004. (Unpublished.)

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Appendix A: Matlab-Program for 2D Inlet Profile

% MATLAB PROGRAM ­ rprof.m % AUTHOR: Mikko Auvinen, pw41140 % PROJECT: Predition of Gas Path Ingestion into Rotating Turbine Disc Cavity % DESCRIPTION: The program reads radial profile data from a Nistar mixing plane table and extracts all necessary % boundary condition information from it. The program also performs physically justified scaling of parameters such % that a specified mass flow rate is achieved. % The computed radial profiles are written into .txt files. % - - - Start of Program - - - - - - - - - - - - % FLUID DATA (SI - units). % Air: Ideal Gas R = 287; gm = 1.4; cp = 1004.5; % CONVERSION FACTORS PsitoPa = 6894.757; RtoK = 0.5555; DensConv = 0.062428; % From kg/m^3 to lbm/ft^3 MtoFt = 3.28084; LbtoKg = 0.453592; % - - LOAD NISTAR INFO - - - - - - - - - - - - - load mixplane.data M = mixplane(:,1); alfa = mixplane(:,2); Pt = mixplane(:,3); Tt = mixplane(:,4); r = mixplane(:,5); phi = mixplane(:,6); clear mixplane.data % Convert Angles From Degrees to Radians alfa = alfa.*(pi/180); phi = phi.*(pi/180); % Length of the arrays N = size(M,1); % - - GEOM SCALING - - - - - - - - - - - - - - - - - scale = 1; if scale == 1 R1 = r(1); R2 = r(N); Rn1 = input(' Inner radius of inlet (in) : ') Rn2 = input(' Outer radius of inlet (in) : ') % Compute linear scaling factors A & B: ( Rn = C1 * r + C2 ) C1 = (Rn2 - Rn1)/(R2 - R1); C2 = (R2*Rn1 - R1*Rn2)/(R2 - R1); % Compute new radius values for i = 1:N r(i) = C1*r(i)+C2; end end

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% - - MODEL DATA - - - - - - - - - - - - - - - - - % Geometric Data Nblades = input(' Number of blades in the turbine section : ') secang = 2*pi/Nblades; % Sector Angle % Rotational Velocity (rad/s) omega = 21389 * (2*pi)/60; % - COMPUTATION OF PARAMETERS ACCORDING TO COMPRESSIBLE FLUID EQNS - % - - - Refer to GAS DYNAMICS text book for eqns - - - - - - - - - - Ar = zeros(N,1); % Area of Segment Ts = zeros(N,1); % Static Temperature (R) Ps = zeros(N,1); % Static Pressure (psi) ro = zeros(N,1); % Density (lbm/ft^3) a = zeros(N,1); % Speed of Sound (ft/s) Cx = zeros(N,1); % Axial Speed (ft/s) Ctheta = zeros(N,1); % Theta Abs Velocity (ft/s) Cr = zeros(N,1); % Radial Velocity (ft/s) % Relative values in rotating frame Vtheta = zeros(N,1); % Theta Rel Velocity (ft/s) Mrel = zeros(N,1); % Relative Mach No. Ptrel= zeros(N,1); % Rel Total Pressure Ttrel= zeros(N,1); % Rel Total Temperature beta = zeros(N,1); % Unit Velocity Vector's Components in Stationary Frame Ctcomp = zeros(N,1); Cxcomp = zeros(N,1); Crcomp = zeros(N,1); % Unit Velocity Vector Components in Relative Frame Vtcomp = zeros(N,1); % Mass Flow Rate mdot = zeros(N,1); % Velocity * Area VA = zeros(N,1); for i = 2:N Ar(i) = pi*((r(i)/12)^2 - (r(i-1)/12)^2)/Nblades; Ts(i) = Tt(i)/(1+(gm-1)/2*M(i)^2); Ps(i) = Pt(i)/(1+(gm-1)/2*M(i)^2)^(gm/(gm-1)); ro(i) = (Ps(i)*PsitoPa/(R*Ts(i)*RtoK))* DensConv; a(i) = (gm*R*Ts(i)*RtoK)^(0.5) * MtoFt; Cx(i) = M(i)*a(i)*cos(alfa(i))*cos(phi(i)); Ctheta(i) = -M(i)*a(i)*sin(alfa(i))*cos(phi(i)); Cr(i) = M(i)*a(i)*sin(phi(i)); Vtheta(i) = (Ctheta(i) + omega*r(i)/12.0); Mrel(i) = (Vtheta(i)^2 + Cx(i)^2 + Cr(i)^2)^0.5/a(i); Ttrel(i) = Ts(i)*(1+(gm-1)/2*Mrel(i)^2); Ptrel(i) = Ps(i)*(1+(gm-1)/2*Mrel(i)^2)^(gm/(gm-1)); beta(i) = atan(Vtheta(i)/Cx(i)); Ctcomp(i) = Ctheta(i)/(M(i)*a(i)); % Angle between Vtheta and Cx

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Cxcomp(i) = Cx(i)/(M(i)*a(i)); Crcomp(i) = Cr(i)/(M(i)*a(i)); Vtcomp(i) = Vtheta(i)/(M(i)*a(i)); mdot(i) = ro(i)*Cx(i)*Ar(i); VA(i) = Vtheta(i)*Ar(i); CxA(i) = Cx(i)*Ar(i); end Mdot = sum(mdot)*Nblades; str1 = ' Mass Flow Rate: mdot = '; str2 = num2str(Mdot); str3 = ' lbm/s '; disp(strcat(str1,str2,str3)); % AVERAGES Arsum = sum(Ar); Vthave = sum(VA)/Arsum; Cxave = sum(CxA)/Arsum; % - - SCALED MASS FLOW RATE, TEMP AND PRESSURE - - - - - - - mscale = 1; if mscale == 1 Mtar = input(' TARGET TOTAL MASS FLOW RATE (lb/s) : '); roA = zeros(N,1); for i=2:N mrat(i) = mdot(i)/(Mdot/Nblades); mdot(i) = mrat(i)*Mtar/Nblades;

% Ratios per segment % New mdots based on ratios

% SOLVE MACH NO. ITERATIVELY FOR EACH SEGMENT if Ar(i) > 0 K1 = 1; K2 = 0; dM = 1e-4; tick = 0; while (abs(K1 - K2) > 1e-3) if K2 > K1 M(i) = M(i) - dM; else M(i) = M(i) + dM; end tick = tick +1; K1 = sqrt(R*Tt(i)*RtoK/gm)*mdot(i)*LbtoKg/(Pt(i)*PsitoPa*(Ar(i)/MtoFt^2)*Cxcomp(i)); K2 = M(i)*(1+(gm-1)*M(i)^2/2)^((gm+1)/(2-2*gm)); end %while disp(strcat(' tick = ',int2str(tick))); % COMPUTE NEW Pt fm = sqrt(gm)*M(i)*(1+(gm-1)*M(i)^2/2)^((gm+1)/(2-2*gm)); Pt(i) = mdot(i)*LbtoKg*sqrt(R*Tt(i)*RtoK)/(fm*(Ar(i)/MtoFt^2)*Cxcomp(i)) * PsitoPa^-1; % NEW STATIC VALUES, DENSITY AND SPEED OF SOUND Ts(i) = Tt(i)/(1+(gm-1)/2*M(i)^2); Ps(i) = Pt(i)/(1+(gm-1)/2*M(i)^2)^(gm/(gm-1)); ro(i) = (Ps(i)*PsitoPa/(R*Ts(i)*RtoK))* DensConv; a(i) = (gm*R*Ts(i)*RtoK)^(0.5) * MtoFt;

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roA(i) = ro(i)*Ar(i); % NEW VELOCITY COMPONENTS Cx(i) = M(i)*a(i)*cos(alfa(i))*cos(phi(i)); Ctheta(i) = -M(i)*a(i)*sin(alfa(i))*cos(phi(i)); Cr(i) = M(i)*a(i)*sin(phi(i)); Vtheta(i) = Cx(i)*tan(beta(i)); % FROM SIMILARITY TRIANGLES (WILL BE RECOMPUTED!) Ctcomp(i) = Ctheta(i)/(M(i)*a(i)); Cxcomp(i) = Cx(i)/(M(i)*a(i)); Crcomp(i) = Cr(i)/(M(i)*a(i)); omega = (Vtheta(i) - Ctheta(i))/(r(i)/12.0)*60/(2*pi); % NEW MASS FLOW RATE mdot(i) = ro(i)*Cx(i)*Ar(i); % NEW RELATIVE VALUES Mrel(i) = (Vtheta(i)^2 + Cx(i)^2 + Cr(i)^2)^0.5/a(i); Ttrel(i) = Ts(i)*(1+(gm-1)/2*Mrel(i)^2); Ptrel(i) = Ps(i)*(1+(gm-1)/2*Mrel(i)^2)^(gm/(gm-1)); end % if Ar > 0 end % for loop roAvg = sum(roA)/sum(Ar)/DensConv % FIX NEW TURBINE ROTATIONAL VELOCITY mid = round(size(M,1)/2); omega = (Vtheta(mid-3) - Ctheta(mid-3))/(r(mid-3)/12.0); disp(' ') disp(strcat(' New Engine Omega = ',num2str(omega*60/(2*pi)))); % COMPUTE VTHETA AGAIN BASED ON FIXED OMEGA for i = 1:N Vtheta(i) = (Ctheta(i) + omega*r(i)/12.0); end % NEW TOTAL MASS FLOW RATE (CHECK!) Mdot = sum(mdot)*Nblades; str1 = ' NEW Mass Flow Rate: mdot = '; str2 = num2str(Mdot); str3 = ' lbm/s '; disp(strcat(str1,str2,str3)); end %if mscale = 1 % - - WRITE DATA TO FILES - - - - - - - - write = 1; if write == 1 fid = fopen('Cxprofile.txt','w'); for j = 2:N fprintf(fid,'%12.8f\t %12.8f\n',r(j),Cx(j)); end fid = fopen('Crprofile.txt','w'); for j = 2:N fprintf(fid,'%12.8f\t %12.8f\n',r(j),Cr(j)); end % FOR CHECKING THE VALUE RANGE

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fid = fopen('Cthetaprofile.txt','w'); for j = 2:N fprintf(fid,'%12.8f\t %12.8f\n',r(j),Ctheta(j)); end fid = fopen('Mprofiles.txt','w'); for j = 2:N fprintf(fid,'%12.8f\t %12.8f\n',M(j),Mrel(j)); end fid = fopen('Pprofiles.txt','w'); for j = 2:N fprintf(fid,'%12.8f\t %12.8f\t %12.8f\t %12.8f\n',r(j),Pt(j),Ptrel(j), Ps(j)); end fid = fopen('Tprofiles.txt','w'); for j = 2:N fprintf(fid,'%12.8f\t %12.8f\t %12.8f\t %12.8f\n',r(j),Tt(j),Ttrel(j),Ts(j)); end fid = fopen('ro_a_profiles.txt','w'); for j = 2:N fprintf(fid,'%12.8f\t %12.8f\n',ro(j),a(j)); end fid = fopen('AbsVelcomp_profiles.txt','w'); for j = 2:N fprintf(fid,'%12.8f\t %12.8f\t %12.8f\t %12.8f\n',r(j),Cxcomp(j),Ctcomp(j),Crcomp(j)); end fclose(fid); end

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Appendix B: Matlab-Program for Wake Generation

% MATLAB PROGRAM ­ wakebc.m ­ for Wake Generation % AUTHOR: Mikko Auvinen, pw41140 % DESCRIPTION: The program reads radial profile data and generates a 3D profile according to specified % circumferential information while conserving the original circumferential averages of each sector. % - - - Start of Program - - - - - - - - - - - - - - - - - - - - % - - Load 1-D array data as written by rprof.m program load Geomprofile.txt Ar1 = Geomprofile(:,1); r1 = Geomprofile(:,2); clear Geomprofile.txt load Tprofiles.txt Tt1 = Tprofiles(:,2); Ttrel1 = Tprofiles(:,3); Ts1 = Tprofiles(:,4); clear Tprofiles.txt load Pprofiles.txt Pt1 = Pprofiles(:,2); Ptrel1 = Pprofiles(:,3); Ps1 = Pprofiles(:,4); clear Pprofiles.txt load AbsVelcomp_profiles.txt Cx1 = AbsVelcomp_profiles(:,2); Ctheta1 = AbsVelcomp_profiles(:,3); Cr1 = AbsVelcomp_profiles(:,4); clear AbsVelcomp_profiles.txt load ro_a_profiles.txt ro1 = ro_a_profiles(:,1); a1 = ro_a_profiles(:,2); clear ro_a_profiles.txt % - - MODEL GEOMETRY AND FLUID DATA - - - - - - - gm = 1.40; % gamma R = 287.0; % %Number of Segments in Theta Direction Nseg = 20; % Number of entries in r-direction Nr = size(Ar1,1); % Total Angle TotAng = 7.2; % DEGREES TotAng = TotAng * pi/180; % RADIANS % Segment Angle (Const.) SegAng = TotAng / Nseg; % - - CREATE 2-D ARRAYS - - - - - - - - - - - % Initialize Ar = zeros(Nr,Nseg); r = zeros(Nr,Nseg); Pt = zeros(Nr,Nseg); Ptrel =zeros(Nr,Nseg);

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Ps = zeros(Nr,Nseg); Tt = zeros(Nr,Nseg); Ttrel =zeros(Nr,Nseg); Ts = zeros(Nr,Nseg); Cx = zeros(Nr,Nseg); Cr = zeros(Nr,Nseg); Ctheta = zeros(Nr,Nseg); ro = zeros(Nr,Nseg); a = zeros(Nr,Nseg); % Theta coord. of value point theta = zeros(Nr,Nseg); % Z, Y coordinates z = zeros(Nr,Nseg); y = zeros(Nr,Nseg); % Copy values, compute theta and modify areas for j = 1:Nseg for i = 1:Nr Ar(i,j) = Ar1(i)/Nseg; r(i,j) = r1(i); Pt(i,j) = Pt1(i); Ptrel(i,j) = Ptrel1(i); Ps(i,j) = Ps1(i); Tt(i,j) = Tt1(i); Ttrel(i,j) = Ttrel1(i); Ts(i,j) = Ts1(i); Cx(i,j) = Cx1(i); Cr(i,j) = Cr1(i); Ctheta(i,j) = Ctheta1(i); ro(i,j) = ro1(i); a(i,j) = a1(i); % Value points are in the middle of the segments theta(i,j) = -TotAng/2 +(j-1)*SegAng + SegAng/2; z(i,j) = r(i,j)*cos(theta(i,j)); y(i,j) = r(i,j)*sin(theta(i,j)); end end % - - SPECIFY THE LOCATION OF WAKE - - - - - - wakeMid = 4; wakeLeft = wakeMid + 1; wakeRight = wakeMid - 1; wakeRight2 = wakeRight - 1; wakeRight3 = wakeRight2 - 1; % - - LOCATION OF HIGH PT AS NISTAR SOLUTION SUGGESTS - PtHigh1 = wakeLeft + 2; PtHigh2 = PtHigh1 + 1; PtHigh3 = PtHigh2 + 1; PtHigh4 = PtHigh3 + 1; PtHigh5 = PtHigh4 + 1; % - - FRACTION OF TOTAL PRESSURE AT WAKE (FROM SOLUTION) - - - fracPt = 0.70; fracPt2 = 0.83; fracPt3 = 0.90;

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% - - FACTOR OF PT INCREASE - - - IncfactPt = 1.085; % - - RESIDUAL FACTOR (SEE BELOW) - - - resfac = (1-fracPt)+3*(1-fracPt2)+(1-fracPt3)+5*(1-IncfactPt); %---------------------------------% - NOTE! THE CHANGES IN TOTAL TEMPERATURE ARE CONSIDERED NEGLIGIBLE %---------------------------------% - - RESIDUALS (SAVED IN ORDER TO PRESERVE TOTAL AVG VALUES) - resPt = zeros(Nr,1); for i = 1:Nr resPt(i) = Pt(i,wakeMid)*resfac; Pt(i,wakeMid) = fracPt * Pt(i,wakeMid); Pt(i,wakeLeft) = fracPt2 * Pt(i,wakeLeft); Pt(i,wakeRight) = fracPt2 * Pt(i,wakeRight); Pt(i,wakeRight2) = fracPt2 * Pt(i,wakeRight2); Pt(i,wakeRight3) = fracPt3 * Pt(i,wakeRight3); Pt(i,PtHigh1) = IncfactPt * Pt(i,PtHigh1); Pt(i,PtHigh2) = IncfactPt * Pt(i,PtHigh2); Pt(i,PtHigh3) = IncfactPt * Pt(i,PtHigh3); Pt(i,PtHigh4) = IncfactPt * Pt(i,PtHigh4); Pt(i,PtHigh5) = IncfactPt * Pt(i,PtHigh5); end % for for j = 1:Nseg for i = 1:Nr if j~= wakeMid | j~= wakeRight | j~= wakeLeft | j~= wakeRight2 | j~= wakeRight3 Pt(i,j) = Pt(i,j) + resPt(i)/(Nseg-5); end % if end end % - - X Location - - - - - - - - - x = input(' X-location : '); fid = fopen('Ptprofile2D.txt','w'); for j = 1:Nseg for i = 1:Nr fprintf(fid,'%12.8f\t %12.8f\t %12.8f\t %12.8f\n',x, y(i,j), z(i,j), Pt(i,j)); end end fid = fopen('Ttprofile2D.txt','w'); for j = 1:Nseg for i = 1:Nr fprintf(fid,'%12.8f\t %12.8f\t %12.8f\t %12.8f\n',x, y(i,j), z(i,j), Tt(i,j)); end end

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fid = fopen('CxVel_Comp2D.txt','w'); for j = 1:Nseg for i = 1:Nr fprintf(fid,'%12.8f\t %12.8f\t %12.8f\t %12.8f\n',x, y(i,j), z(i,j), Cx(i,j)); end end fid = fopen('CthetaVel_Comp2D.txt','w'); for j = 1:Nseg for i = 1:Nr fprintf(fid,'%12.8f\t %12.8f\t %12.8f\t %12.8f\n',x, y(i,j), z(i,j), Ctheta(i,j)); end end fid = fopen('CrVel_Comp2D.txt','w'); for j = 1:Nseg for i = 1:Nr fprintf(fid,'%12.8f\t %12.8f\t %12.8f\t %12.8f\n',x, y(i,j), z(i,j), Cr(i,j)); end end fclose(fid)

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Appendix C: Additional Solution Data

Definition : Dimensionless Variables.

pt * = ps * = Tt * =

p t - p t _ min p t _ max - p t _ min p s - p s _ ref p s _ max - p s _ ref Tt - Tt _ min Tt _ max - Tt _ min

m* = R* =

& m & mGasPath

r - Rmin Rmax - Rmin

Data : ATFI-INGST. General Solution Data: Pressure Values at Inlets / Outlet. (Note, values are not time averaged.) Case: m* 0.0112 0.00998 0.00873 0.00748 0.00623 0.00499 0.00374 pt*_Inlet2 (cavity) 0.0929 0.0853 0.0622 0.0523 0.0369 0.0171 0.0 ps*_Inlet2 (cavity) 0.435 0.440 0.425 0.424 0.415 0.400 0.387 ps*_Inlet1 (gaspath) 0.473 0.467 0.466 0.462 0.460 0.457 0.457 ps*_Outlet (gaspath) 0.198 0.188 0.184 0.182 0.178 0.172 0.168

Data : ATFI-FER. General Solution Data for ATFI-FER: Pressure Values at Inlets / Outlet. (Note, values are not time averaged.) Case: m* 0.00418 pt*_Inlet2 (cavity) 0.189 ps*_Inlet2 (cavity) 0.592 ps*_Inlet1 (gaspath) 0.660 ps*_Outlet (gaspath) 0.299

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Data : PW307-INGST. General Solution Data for PW307-INGST: Pressure Values at Inlets / Outlets. (Note: Dimensionless static and total pressure values have different reference values.) Constant values: Inlet / Oulet ps*_Inlet_G ps*_Outlet_G ps*_Inlet_Cav1 ps*_Inlet_Cav3 pt*_Inlet_Cav1 pt*_Inlet_Cav2 pt*_Inlet_Cav3 Variable values: m* = 0.00598 ps*_Inlet_Cav2 pt*_Inlet_Cav2 0.8882 0.1563 m* = 0.00532 0.8730 0.0096 m* = 0.00465 0.8591 -0.1121 m* = 0.00399 0.8476 -0.2062 m* = 0.00332 0.8360 -0.2760 m* = 0.00266 0.825 -0.320 0.515 0.481 0.985 Total / Static Pressure 0.917 0.554 0.819 1.431

Data : PW307-PTM12442. General Solution Data for PW307-PTM12442 (m*purge = 0.0070): Pressure Values at Inlets / Outlets. (Note: Dimensionless static and total pressure values have different reference values.) Static / Total Pressure ps*_Inlet_G ps*_Outlet_G ps*_Inlet_Cav1 ps*_Inlet_Cav2 ps*_Inlet_Cav3 ps*_Inlet_Cav4 ps*_Outlet_Cav pt*_Inlet_Cav1 pt*_Inlet_Cav2 pt*_Inlet_Cav3 pt*_Inlet_Cav4 0.515 0.481 0.583 0.189 1) PW307_Complete 0.917 0.554 0.819 0.924 0.955 0.847 0.707 0.510 0.432 0.583 2) PW307_Mdf_1 0.907 0.554 0.815 0.917 0.955 0.430 0.899 3) PW307_Mdf_2 0.907 0.554 0.917 0.985 -

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NUMERICAL STUDY OF GAS PATH INGESTION IN COUPLED TURBINE BLADE PASSAGE AND DISC CAVITY SYSTEMS

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NUMERICAL STUDY OF GAS PATH INGESTION IN COUPLED TURBINE BLADE PASSAGE AND DISC CAVITY SYSTEMS