Turbomachinery Flow Physicsand Dynamic Performance

Meinhard T. Schobeiri

Turbomachinery Flow Physicsand Dynamic Performance

Second and Enhanced Edition

With 433 Figures

ABC

AuthorProf. Dr.-Ing. Meinhard T. SchobeiriDepartment of Mechanical EngineeringTexas A&M UniversityUSA

ISBN 978-3-642-24674-6 e-ISBN 978-3-642-24675-3DOI 10.1007/978-3-642-24675-3Springer Heidelberg New York Dordrecht London

Library of Congress Control Number: 2012935425

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Preface to the Second Edition

The first edition of this book appeared in 2005 and was concentrated on turbo-machinery flow physics, design, off-design and nonlinear dynamics performance. AsI mentioned in the preface to the first edition, in dealing with three-dimensionalturbulent flows through a turbomachinery component, for the reasons I outlined, Irefrained from presenting a multitude of existing turbulence models. In the meantime,only a few of those models have survived exhibiting the standard working toolsimplemented in almost all computational tools dealing with turbomachinery flowsimulations. Experienced turbomachinery aerodynamicist knowledgeable of theunderlying physics of the models in existing commercial codes is able to appropri-ately interpret the results. Without this knowledge, however, the user of commercialcodes may tend to uncritically accept computational results as infallible. This issuehas been addressed in the current second edition.

While the unifying character of the book structure is maintained, all chapters haveundergone a rigorous update and enhancement. Accounting for the need of theturbomachinery community in general and my students in particular, I added threechapters, Chapters 19, 20 and 21, that deal with laminar turbulent transition, turbul-ence and boundary layer. In these chapters, I tried to provide the reader with thephysics underlying the computational tools of these subject areas. Since the turbul-ence and transition models are still far from being complete, I strongly recommendthe readers to stay updated on these issues. With this edition, the readers have a solidsource of information for designing state-of-the art turbomachinery components andsystems at hand.

The current edition has four parts. While Parts I through III contain substantialrevisions and enhancements, Part IV, which contains Chapter 19, 20 and 21, isdedicated to computational aspects of turbomachinery design. In each of these newchapters, several case studies are presented that help the reader better understand thephysics of computational tools.

Here, as in the first edition, typing numerous equations, errors may occur. I triedhard to eliminate typing, spelling and other errors, but I have no doubt that someremain to be found by readers. In this case, I sincerely appreciate the reader notifyingme of any mistakes found; the electronic address is given below. I also welcome anycomments or suggestions regarding the improvement of future editions of the book.

My students and numerous colleagues around the world have encouraged me towrite this current edition. For their encouragement and suggestions, I would like toexpress my sincere thanks.

College Station, May 2011 Meinhard T. Schobeiritschobeiri@mengr.tamu.edu

Preface to the First Edition

Turbomachinery performance, aerodynamics, and heat transfer have been subjects ofcontinuous changes for the last seven decades. Stodola was the fist to make asuccessful effort to integrate the treatment of almost all components of steam and gasturbines in a single volume monumental work. Almost half a century later, Traupelpresented the third and last edition of his work to the turbomachinery community. Inhis two-volume work, Traupel covered almost all relevant aspects of turbomachineryaerodynamics, performance, structure, rotordynamics, blade vibration, and heattransfer with the information available up to the mid-seventies. In the meantime, theintroduction of high speed computers and sophisticated computational methods hassignificantly contributed to an exponential growth of information covering almost allaspects of turbomachinery design. This situation has lead to a growing tendency intechnical specialization causing the number of engineers with the state-of-the-artcombined knowledge of major turbomachinery aspects such as, aerodynamics, heattransfer, combustion, rotordynamics, and structure, to diminish. Another (not minor)factor in the context of specialization is the use of “black boxes” in engineering ingeneral and in turbomachinery in particular. During my 25 years of turbomachineryR&D experience, I have been encountering engineers who can use commercial codesfor calculating the complex turbomachinery flow field without knowing theunderlying physics of the code they use. This circumstance constituted one of thefactors in defining the framework of this book, which aims at providing the studentsand practicing turbomachinery design engineers with a solid background inturbomachinery flow physics and performance. As a consequence, it does not coverall aspects of turbomachinery design. It is primarily concerned with the fundamentalturbomachinery flow physics and its application to different turbomachinerycomponents and systems. I have tried to provide the turbomachinery community,including students and practicing engineers, with a lasting foundation on which theycan build their specialized structures.

I started working on this book when I left Brown Boveri, Gas Turbine Division inSwitzerland (1987) and joined Texas A&M University. A part of my lecture notes ingraduate level turbomachinery constitutes the basics of this book, which consists ofthree parts. Part I encompassing Chapters 1 to 6 deals with turbomachinery flowphysics. Explaining the physics of a highly three-dimensional flow through a turbine ora compressor stage requires adequate tools. Tensor analysis is a powerful tool to dealwith this type of flow. Almost all graduate level fluid mechanics and computationalfluid dynamics (CFD) courses at Texas A&M University, and most likely at otheruniversities, make use of tensor analysis. For the reader, who might not be familiar withthe tensor analysis, I tailored the quintessential of tensor analysis in Appendixes A andB to the need of the reader who is willing to spend a few hours on this topic.

PrefaceVIII

Chapters 2 and 3 deal with the Kinematics and Differential Balances in turbo-machinery in three-dimensional form. In dealing with three-dimensional turbulentflows through a turbomachinery component, I refrained from presenting a multitudeof existing turbulence models for three reasons. Firstly, the models are changingcontinuously; the one which is thought today to be the ultimo ratio may be obsoletetomorrow. Secondly, there is a vast amount of papers and publications devoted toturbulence modeling and numerical methods in turbomachinery. In my view,summarizing these models and methods without getting into details is not appropriatefor a text. Thirdly, the emergence of the direct numerical simulation (DNS) as theultimate solution method, though not very practical at the time being, will in the nearfuture undoubtedly eliminate the problems that are inherently associated withReynolds averaged Navier Stokes equations (RANS). Considering this situation, Ifocused my effort on briefly presenting the RANS-equations and introducing theintermittency function. The resulting equations containing the product of theintermittency and the Reynolds stress tensor clearly illustrate how the accuracy ofthese two models affect the numerical results. As a logical followup, Chapter 4extensively discusses the integral balances in turbomachinery. Chapters 5 and 6 dealwith a unified treatment of energy transfer, stage characteristics, and cascade andstage aerodynamics. Contrary to the traditional approach that treats turbine andcompressor stages of axial or radial configuration differently, I have tried to treatthese components using the same set of equations.

Part II of this book containing Chapters 7 through 11 starts with the treatment ofcascade and stage efficiency and loss determination from a physically plausible pointof view. I refrained from presenting recipe-types of empirical formulas that have nophysical foundation. Chapters 10 and 11 deal with simple designs of compressors andturbine blades and calculations of incidence and deviation. Radial equilibrium isdiscussed in Chapter 11, which concludes Part II.

Part III of the book is entirely dedicated to dynamic performance of turbo-machinery components and systems. Particular attention is paid to gas turbinecomponents, their individual modeling, and integration into the gas turbine system.It includes Chapter 12 to 18. Chapter 12 introduces the non-linear dynamic simulationof turbomachinery systems and its theoretical background. Starting from a set ofgeneral four dimensional partial differential equations in temporal-spatial domain, two-dimensional equation sets are derived that constitute the basis for componentmodeling. The following Chapter 13 deals with generic modeling of turbomachinerycomponents and systems followed by Chapters 14, 15, 16, and 17 in which individualcomponents ranging from the inlet nozzle to the compressor, combustion chamber,turbine, and exhaust diffuser are modeled. In modeling compressor and turbinecomponents, non-linear adiabatic and diabatic expansion and compression calcula-tion methods are presented. Chapter 18 treats gas turbine engines, design and dynamicperformance. Three representative case studies conclude this chapter. In preparingPart III, I tried to be as concrete as possible by providing examples for each individualcomponent.

In typing several thousand equations, errors may occur. I tried hard to eliminatetyping, spelling and other errors, but I have no doubt that some remain to be foundby readers. In this case, I sincerely appreciate the reader notifying me of any mistakes

Preface IX

found; the electronic address is given below. I also welcome any comments or sug-gestions regarding the improvement of future editions of the book.

My sincere thanks are due to many fine individuals and institutions. First andforemost, I would like to thank the faculty of the Technische Universität Darmstadt,from whom I received my entire engineering education. I finalized major chapters ofthe manuscript during my sabbatical in Germany where I received the Alexander vonHumboldt Prize. I am indebted to the Alexander von Humboldt Foundation for thisPrize and the material support for my research sabbatical in Germany. My thanks areextended to Prof. Bernd Stoffel, Prof. Ditmar Hennecke, and Dipl. Ing. BerndMatyschok for providing me with a very congenial working environment. I trulyenjoyed interacting with these fine individuals. NASA Glenn Research Centersponsored the development of the nonlinear dynamic code GETRAN which I used tosimulate cases in Part III. I wish to extend my thanks to Mr. Carl Lorenzo, Chief ofControl Division, Dr. D. Paxson, and the administration of the NASA Glenn ResearchCenter. I also thank Dr. Richard Hearsey for providing me with a three-dimensionalcompressor blade design example that he obtained from his streamline curvatureprogram. I also would like to extend my thanks to Dr. Arthur Wennerstom forproviding me with the updated theory on the streamline curvature method.

I am also indebted to the TAMU administration for partially supporting mysabbatical that helped me in finalizing the book. Special thanks are due to Mrs.Mahalia Nix, who helped me in cross-referencing the equations and figures andrendered other editorial assistance.

Last but not least, my special thanks go to my family, Susan and Wilfried for theirsupport throughout this endeavor.

College Station, Texas, September 2003 Meinhard T. Schobeiritschobeiri@mengr.tamu.edu

Contents

I Turbomachinery Flow Physics

1 Introduction, Turbomachinery, Applications, Types . . . . . . . 3

1.1 Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Application of Turbomachines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Power Generation, Steam Turbines . . . . . . . . . . . . . . . . . . . . . . 91.3.2 Power Generation, Gas Turbines . . . . . . . . . . . . . . . . . . . . . . . . 91.3.3 Aircraft Gas Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.4 Diesel Engine Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Classification of Turbomachines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4.1 Compressor Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4.2 Turbine Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Working Principle of a Turbomachine . . . . . . . . . . . . . . . . . . . . . . . . 14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Kinematics of Turbomachinery Fluid Motion . . . . . . . . . . . 15

2.1 Material and Spatial Description of the Flow Field . . . . . . . . . . . . . . 152.1.1 Material Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 Jacobian Transformation Function and Its Material Derivative 172.1.3 Spatial Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Translation, Deformation, Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Reynolds Transport Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Differential Balances in Turbomachinery . . . . . . . . . . . . . . . . 29

3.1 Mass Flow Balance in Stationary Frame of Reference . . . . . . . . . . . . 293.1.1 Incompressibility Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Differential Momentum Balance in Stationary Frame of Reference . 323.2.1 Relationship between Stress Tensor and Deformation Tensor 343.2.2 Navier-Stokes Equation of Motion . . . . . . . . . . . . . . . . . . . . . . 363.2.3 Special Case: Euler Equation of Motion . . . . . . . . . . . . . . . . . 38

3.3 Some Discussions on Navier-Stokes Equations . . . . . . . . . . . . . . . . . 413.4 Energy Balance in Stationary Frame of Reference . . . . . . . . . . . . . . . 42

3.4.1 Mechanical Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4.2 Thermal Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

ContentsXII

3.4.3 Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4.4 Entropy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.5 Differential Balances in Rotating Frame of Reference . . . . . . . . . . . . 503.5.1 Velocity and Acceleration in Rotating Frame . . . . . . . . . . . . . 503.5.2 Continuity Equation in Rotating Frame of Reference . . . . . . . 523.5.3 Equation of Motion in Rotating Frame of Reference . . . . . . . . 533.5.4 Energy Equation in Rotating Frame of Reference . . . . . . . . . . 55

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Integral Balances in Turbomachinery . . . . . . . . . . . . . . . . . . . . . 59

4.1 Mass Flow Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Balance of Linear Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3 Balance of Moment of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4 Balance of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4.1 Energy Balance Special Case 1: Steady Flow . . . . . . . . . . . . . 774.4.2 Energy Balance Special Case 2: Steady Flow,

Constant Mass Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.5 Application of Energy Balance to Turbomachinery Components . . . 78

4.5.1 Application: Accelerated, Decelerated Flows . . . . . . . . . . . . . 794.5.2 Application: Combustion Chamber . . . . . . . . . . . . . . . . . . . . . 804.5.3 Application: Turbine, Compressor . . . . . . . . . . . . . . . . . . . . . 81

4.5.3.1 Uncooled Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.5.3.2 Cooled Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.5.3.3 Uncooled Compressor . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.6 Irreversibility and Total Pressure Losses . . . . . . . . . . . . . . . . . . . . . . 844.6.1 Application of Second Law to Turbomachinery Components . . 85

4.7 Flow at High Subsonic and Transonic Mach Numbers . . . . . . . . . . . 874.7.1 Density Changes with Mach Number, Critical State . . . . . . . . . 884.7.2 Effect of Cross-Section Change on Mach Number . . . . . . . . . 934.7.3 Compressible Flow through Channels with

Constant Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.7.4 The Normal Shock Wave Relations . . . . . . . . . . . . . . . . . . . . 1094.7.5 The Oblique Shock Wave Relations . . . . . . . . . . . . . . . . . . . . 1154.7.6 The Detached Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.7.7 Prandtl-Meyer Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5 Theory of Turbomachinery Stages . . . . . . . . . . . . . . . . . . . . . . . 123

5.1 Energy Transfer in Turbomachinery Stages . . . . . . . . . . . . . . . . . . . 1235.2 Energy Transfer in Relative Systems . . . . . . . . . . . . . . . . . . . . . . . . 1245.3 General Treatment of Turbine and Compressor Stages . . . . . . . . . . 1255.4 Dimensionless Stage Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.5 Relation between Degree of Reaction and Blade Height . . . . . . . . 1315.6 Effect of Degree of Reaction on the Stage Configuration . . . . . . . . 134

Contents XIII

5.7 Effect of Stage Load Coefficient on Stage Power . . . . . . . . . . . . . . 1365.8 Unified Description of a Turbomachinery Stage . . . . . . . . . . . . . . . 137

5.8.1 Unified Description of Stage with Constant Mean Diameter . 1375.8.2 Generalized Dimensionless Stage Parameters . . . . . . . . . . . . 138

5.9 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.9.1 Case 1, Constant Mean Diameter . . . . . . . . . . . . . . . . . . . . . . 1415.9.2 Case 2, Constant Mean Diameter and

Meridional Velocity Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.10 Increase of Stage Load Coefficient, Discussion . . . . . . . . . . . . . . . . 140References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6 Turbine and Compressor Cascade Flow Forces . . . . . . . . . . 145

6.1 Blade Force in an Inviscid Flow Field . . . . . . . . . . . . . . . . . . . . . . . 1456.2 Blade Forces in a Viscous Flow Field . . . . . . . . . . . . . . . . . . . . . . . 1506.3 The Effect of Solidity on Blade Profile Losses . . . . . . . . . . . . . . . . 1566.4 Relationship Between Profile Loss Coefficient and Drag . . . . . . . . 1566.5 Optimum Solidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.5.1 Optimum Solidity, by Pfeil . . . . . . . . . . . . . . . . . . . . . . . . . . . 1586.5.2 Optimum Solidity, by Zweifel . . . . . . . . . . . . . . . . . . . . . . . . 158

6.6 Generalized Lift-Solidity Coefficient . . . . . . . . . . . . . . . . . . . . . . . . 1626.6.1 Lift-Solidity Coefficient for Turbine Stator . . . . . . . . . . . . . . 1646.6.2 Turbine Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

II Turbomachinery Losses, Efficiencies, Blades

7 Losses in Turbine and Compressor Cascades . . . . . . . . . . . . 175

7.1 Turbine Profile Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1767.2 Viscous Flow in Compressor Cascade . . . . . . . . . . . . . . . . . . . . . . . 178

7.2.1 Calculation of Viscous Flows . . . . . . . . . . . . . . . . . . . . . . . . . 1787.2.2. Boundary Layer Thicknesses . . . . . . . . . . . . . . . . . . . . . . . . . 1797.2.3 Boundary Layer Integral Equation . . . . . . . . . . . . . . . . . . . . . 1817.2.4 Application of Boundary Layer Theory to Compressor Blades 1827.2.5 Effect of Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . 1867.2.6 Stage Profile Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

7.3 Trailing Edge Thickness Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1867.4 Losses Due to Secondary Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

7.4.1 Vortex Induced Velocity Field, Law of Bio-Savart . . . . . . . . 1947.4.2 Calculation of Tip Clearance Secondary Flow Losses . . . . . . 1977.4.3 Calculation of Endwall Secondary Flow Losses . . . . . . . . . . 200

7.5 Flow Losses in Shrouded Blades . . . . . . . . . . . . . . . . . . . . . . . . . . . 2047.5.1 Losses Due to Leakage Flow in Shrouds . . . . . . . . . . . . . . . . 204

7.6 Exit Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

ContentsXIV

7.7 Trailing Edge Ejection Mixing Losses of Gas Turbine Blades . . . . 2127.7.1 Calculation of Mixing Losses . . . . . . . . . . . . . . . . . . . . . . . . 2127.7.2 Trailing Edge Ejection Mixing Losses . . . . . . . . . . . . . . . . . . 2177.7.3 Effect of Ejection Velocity Ratio on Mixing Loss . . . . . . . . . 2177.7.4 Optimum Mixing Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

7.8 Stage Total Loss Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2197.9 Diffusers, Configurations, Pressure Recovery, Losses . . . . . . . . . . . 220

7.9.1 Diffuser Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2217.9.2 Diffuser Pressure Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . 2227.9.3 Design of Short Diffusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2257.9.4 Some Guidelines for Designing High Efficiency Diffusers . . 228

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

8 Efficiency of Multi-stage Turbomachines . . . . . . . . . . . . . . . . 231

8.1 Polytropic Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2318.2 Isentropic Turbine Efficiency, Recovery Factor . . . . . . . . . . . . . . . 2348.3 Compressor Efficiency, Reheat Factor . . . . . . . . . . . . . . . . . . . . . . . 2378.4 Polytropic versus Isentropic Efficiency . . . . . . . . . . . . . . . . . . . . . . 239References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

9 Incidence and Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

9.1 Cascade with Low Flow Deflection . . . . . . . . . . . . . . . . . . . . . . . . . 2419.1.1 Conformal Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2419.1.2 Flow Through an Infinitely Thin Circular Arc Cascade . . . . . 2509.1.3 Thickness Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2569.1.4 Optimum Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2569.1.5 Effect of Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

9.2 Deviation for High Flow Deflection . . . . . . . . . . . . . . . . . . . . . . . . . 2599.2.1 Calculation of Exit Flow Angle . . . . . . . . . . . . . . . . . . . . . . . 261

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

10 Simple Blade Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

10.1 Conformal Transformation, Basics . . . . . . . . . . . . . . . . . . . . . . . . . . 26510.1.1 Joukowsky Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 26710.1.2 Circle-Flat Plate Transformation . . . . . . . . . . . . . . . . . . . . . . 26710.1.3 Circle-Ellipse Transformation . . . . . . . . . . . . . . . . . . . . . . . . 26810.1.4 Circle-Symmetric Airfoil Transformation . . . . . . . . . . . . . . . 26910.1.5 Circle-Cambered Airfoil Transformation . . . . . . . . . . . . . . . . 271

10.2 Compressor Blade Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27210.2.1 Low Subsonic Compressor Blade Design . . . . . . . . . . . . . . . 27310.2.2 Compressors Blades for High Subsonic Mach Number . . . . 27910.2.3 Transonic, Supersonic Compressor Blades . . . . . . . . . . . . . . 280

10.3 Turbine Blade Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28110.3.1 Graphic Design of Camberline . . . . . . . . . . . . . . . . . . . . . . . . 282

Contents XV

10.3.2 Camberline Coordinates Using Bèzier Curve . . . . . . . . . . . . . 28310.3.3 Alternative Calculation Method . . . . . . . . . . . . . . . . . . . . . . . 285

10.4 Assessment of Blades Aerodynamic Quality . . . . . . . . . . . . . . . . . . 287References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

11 Radial Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

11.1 Derivation of Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . . 29211.2 Application of Streamline Curvature Method . . . . . . . . . . . . . . . . . 300

11.2.1 Step-by-step solution procedure . . . . . . . . . . . . . . . . . . . . . . . 30211.2 Compressor Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30611.3 Turbine Example, Compound Lean Design . . . . . . . . . . . . . . . . . . . 309

11.3.1 Blade Lean Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31011.3.2 Calculation of Compound Lean Angle Distribution . . . . . . . . 31111.3.3 Example: Three-Stage Turbine Design . . . . . . . . . . . . . . . . . 313

11.4 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31611.4.1 Free Vortex Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31611.4.2 Forced vortex flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31711.4.3 Flow with constant flow angle . . . . . . . . . . . . . . . . . . . . . . . . 318

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

III Turbomachinery Dynamic Performance

12 Dynamic Simulation of Turbomachinery Components . . . 323

12.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32412.2 Preparation for Numerical Treatment . . . . . . . . . . . . . . . . . . . . . . . . 33012.3 One-Dimensional Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 331

12.3.1 Time Dependent Equation of Continuity . . . . . . . . . . . . . . . . 33112.3.2 Time Dependent Equation of Motion . . . . . . . . . . . . . . . . . . . 33312.3.3 Time Dependent Equation of Total Energy . . . . . . . . . . . . . . 334

12.4 Numerical Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

13 Generic Modeling of Turbomachinery Components . . . . . 341

13.1 Generic Component, Modular Configuration . . . . . . . . . . . . . . . . . . 34213.1.1 Plenum as Coupling Module . . . . . . . . . . . . . . . . . . . . . . . . . 34313.1.2 Group 1: Modules: Inlet, Exhaust, Pipe . . . . . . . . . . . . . . . . . 34513.1.3 Group 2: Recuperators, Combustion Chambers, Afterburners 34613.1.4 Group 3: Adiabatic Compressor and Turbine Components . . 34813.1.5 Group 4: Diabatic Turbine and Compressor Components . . . 35013.1.6 Group 5: Control System, Valves, Shaft, Sensors . . . . . . . . . 352

13.2 System Configuration, Nonlinear Dynamic Simulation . . . . . . . . . 352References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

ContentsXVI

14 Modeling of Inlet, Exhaust, and Pipe Systems . . . . . . . . . . . . 357

14.1 Unified Modular Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35714.2 Physical and Mathematical Modeling of Modules . . . . . . . . . . . . . . 35714.3 Example: Dynamic behavior of a Shock Tube . . . . . . . . . . . . . . . . 360

14.3.1 Shock Tube Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . 361References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

15 Modeling of Recuperators, Combustors, Afterburners . . . 367

15.1 Modeling Recuperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36815.1.1 Recuperator Hot Side Transients . . . . . . . . . . . . . . . . . . . . . . 36915.1.2 Recuperator Cold Side Transients . . . . . . . . . . . . . . . . . . . . . 36915.1.3 Coupling Condition Hot, Cold Side . . . . . . . . . . . . . . . . . . . . 37015.1.4 Recuperator Heat Transfer Coefficient . . . . . . . . . . . . . . . . . . 371

15.2 Modeling Combustion Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . 37215.2.1 Mass Flow Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37315.2.2 Temperature Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37415.2.3 Combustion Chamber Heat Transfer . . . . . . . . . . . . . . . . . . . 376

15.3 Example: Startup and Shutdown of a Combustion Chamber . . . . . . 37815.4 Modeling of Afterburners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

16 Modeling of Compressor Component, Design, Off-Design 383

16.1 Compressor Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38416.1.1 Profile Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38516.1.2 Diffusion Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38716.1.3 Generalized Maximum Velocity Ratio for Cascade, Stage . . 39116.1.4 Compressibility Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39316.1.5 Shock Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39716.1.6 Correlations for Boundary Layer Momentum Thickness . . . . 40616.1.7 Influence of Different Parameters on Profile Losses . . . . . . . 407

16.1.7.1 Mach Number Effect . . . . . . . . . . . . . . . . . . . . . . . . 40716.1.7.2 Reynolds Number Effect . . . . . . . . . . . . . . . . . . . . . 408

16.2 Compressor Design and Off-Design Performance . . . . . . . . . . . . . . 40916.2.1 Stage-by-Stage and Row-by-Row Compression Process . . . 409

16.2.1.1 Stage-by-Stage Calculation of Compression Process 40916.2.1.2 Row-by-Row Adiabatic Compression . . . . . . . . . . . 41116.2.1.3 Off-Design Efficiency Calculation . . . . . . . . . . . . . 415

16.3 Generation of Steady State Performance Map . . . . . . . . . . . . . . . . . 41816.3.1 Inception of Rotating Stall . . . . . . . . . . . . . . . . . . . . . . . . . . . 42016.3.2 Degeneration of Rotating Stall into Surge . . . . . . . . . . . . . . . 422

16.4 Compressor Modeling Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42316.4.1 Module Level 1: Using Performance Maps . . . . . . . . . . . . . . 424

16.4.1.1 Quasi-dynamic Modeling Using Performance Maps 42616.4.1.2 Simulation Example: . . . . . . . . . . . . . . . . . . . . . . . . 427

Contents XVII

16.4.2 Module Level 2: Row-by-Row Adiabatic Compression . . . . 42916.4.2.1 Active Surge Prevention by Adjusting the

Stator Blades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43016.4.2.2 Simulation Example: Surge and Its Prevention . . . . 431

16.4.3 Module Level 3: Row-by-Row Diabatic Compression . . . . . 43616.4.3.1 Description of Diabatic Compressor Module . . . . . 43716.4.3.2 Heat Transfer Closure Equations: . . . . . . . . . . . . . . 439

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

17 Turbine Aerodynamic Design, Performance . . . . . . . . . . . . . 445

17.1 Stage-by-Stage and Row-by-Row Design . . . . . . . . . . . . . . . . . . . . . 44717.1.1 Stage-by-Stage Calculation of Expansion Process . . . . . . . . 44817.1.2 Row-by-Row Adiabatic Expansion . . . . . . . . . . . . . . . . . . . . 44917.1.3 Off-Design Efficiency Calculation . . . . . . . . . . . . . . . . . . . . . 45417.1.4 Behavior under Extreme Low Mass Flows . . . . . . . . . . . . . . 45617.1.5 Example: Steady Design and Off-Design Behavior . . . . . . . 459

17.2 Off-Design Calculation Using Global Turbine Characteristics . . . . 46117.3 Modeling of Turbine Module for Dynamic Performance Simulation 462

17.3.1 Module Level 1: Using Turbine Performance Characteristics 46317.3.2 Module Level 2: Row-by-Row Expansion Calculation . . . . . 46417.3.3 Module Level 3: Row-by-Row Diabatic Expansion . . . . . . . . 465

17.3.3.1 Description of Diabatic Turbine Module, First Method: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

17.3.3.2 Description of Module, Second Method . . . . . . . . . 46917.3.3.3 Heat Transfer Closure Equations . . . . . . . . . . . . . . . 471

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

18 Gas Turbine Engines, Design and Dynamic Performance 473

18.1 Gas Turbine Steady Design Operation, Process . . . . . . . . . . . . . . . 47518.1.1 Gas Turbine Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47718.1.2 Improvement of Gas Turbine Thermal Efficiency . . . . . . . . . 483

18.2 Non-Linear Gas Turbine Dynamic Simulation . . . . . . . . . . . . . . . . . 48518.2.1 State of Dynamic Simulation, Background . . . . . . . . . . . . . . 486

18.3 Engine Components, Modular Concept, Module Identification . . . . 48718.4 Levels of Gas Turbine Engine Simulations, Cross Coupling . . . . . . 49318.5 Non-Linear Dynamic Simulation Case Studies . . . . . . . . . . . . . . . . 494

18.5.1 Case Study 1: Compressed Air Energy Storage Gas Turbine . 49518.5.1.1 Simulation of Emergency Shutdown . . . . . . . . . . . . 497

18.5.2 Case Study 2: Power Generation Gas Turbine Engine . . . . . . 49918.5.3 Case Study 3: Simulation of a Multi-Spool Gas Turbine . . . 504

18.6 A Byproduct of Dynamic Simulation: Detailed Calculation . . . . . . 50718.7 Summary Part III, Further Development . . . . . . . . . . . . . . . . . . . . . 510References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

ContentsXVIII

IV Turbomachinery CFD-Essentials

19 Basic Physics of Laminar-Turbulent Transition . . . . . . . . . . 515

19.1 Transition Basics: Stability of Laminar Flow . . . . . . . . . . . . . . . . . 51519.2 Laminar-Turbulent Transition, Fundamentals . . . . . . . . . . . . . . . . . 51519.3 Physics of an Intermittent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

19.3.1 Intermittent Behavior of Statistically Steady Flows . . . . . . . 51919.3.2 Turbulent/Non-turbulent Decisions . . . . . . . . . . . . . . . . . . . . 52019.3.3 Intermittency Modeling for Flat Plate Boundary Layer . . . . 524

19.4 Physics of Unsteady Boundary Layer Transition . . . . . . . . . . . . . . . 52519.4.1 Experimental Simulation of the Unsteady Boundary Layer . 52719.4.2 Ensemble Averaging High Frequency Data . . . . . . . . . . . . . 53019.4.3 Intermittency Modeling for Periodic Unsteady Flow . . . . . . . 533

19.5 Implementation of Intermittency into Navier Stokes Equations . . . . 53619.5.1 Reynolds-Averaged Navier-Stokes Equations (RANS) . . . . . 53619.5.2 Conditioning RANS for Intermittency Implementation . . . . . 540

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

20 Turbulent Flow and Modeling in Turbomachinery . . . . . . . 545

20.1 Fundamentals of Turbulent Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 54520.1.1 Type of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54720.1.2 Correlations, Length and Time Scales . . . . . . . . . . . . . . . . . . 54820.1.3 Spectral Representation of Turbulent Flows . . . . . . . . . . . . . 55520.1.4 Spectral Tensor, Energy Spectral Function . . . . . . . . . . . . . . 558

20.2 Averaging Fundamental Equations of Turbulent Flow . . . . . . . . . . 56020.2.1 Averaging Conservation Equations . . . . . . . . . . . . . . . . . . . . 561

20.2.1.1Averaging the Continuity Equation . . . . . . . . . . . . . 56120.2.1.2 Averaging the Navier-Stokes Equation . . . . . . . . . . 56120.2.1.3 Averaging the Mechanical Energy Equation . . . . . . 56220.2.1.4 Averaging the Thermal Energy Equation . . . . . . . . 56320.2.1.5 Averaging the Total Enthalpy Equation . . . . . . . . . 56520.2.1.6 Quantities Resulting from Averaging to Be Modeled 568

20.2.2 Equation of Turbulence Kinetic Energy . . . . . . . . . . . . . . . . . 57020.2.3 Equation of Dissipation of Kinetic Energy . . . . . . . . . . . . . . . 576

20.3 Turbulence Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57720.3.1 Algebraic Model: Prandtl Mixing Length Hypothesis . . . . . . 57820.3.2 Algebraic Model: Cebeci-Smith Model . . . . . . . . . . . . . . . . . 58420.3.3 Baldwin-Lomax Algebraic Model . . . . . . . . . . . . . . . . . . . . . 58520.3.4 One- Equation Model by Prandtl . . . . . . . . . . . . . . . . . . . . . . 58620.3.5 Two-Equation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

20.3.5.1 Two-Equation k- Model . . . . . . . . . . . . . . . . . . . . . 58720.3.5.2 Two-Equation k- -Model . . . . . . . . . . . . . . . . . . . . 58920.3.5.3 Two-Equation k- -SST-Model . . . . . . . . . . . . . . . . 590

20.4 Grid Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592

Contents XIX

20.5 Examples of Two-Equation Models . . . . . . . . . . . . . . . . . . . . . . . . . 59420.5.1 Internal Flow, Sudden Expansion . . . . . . . . . . . . . . . . . . . . . . 59420.5.2 Internal Flow, Turbine Cascade . . . . . . . . . . . . . . . . . . . . . . . 59520.5.3 External Flow, Lift-Drag Polar Diagram . . . . . . . . . . . . . . . . 59620.5.4 Case Study: Flow Simulation in a Rotating Turbine . . . . . . . 59720.5.5 Results, Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60520.5.6 Rotating Turbine RANS, URANS-Shortcomings, Discussion 614

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615

21 Introduction into Boundary Layer Theory . . . . . . . . . . . . . . . 619

21.1 Boundary Layer Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 62021.2 Exact Solutions of Laminar Boundary Layer Equations . . . . . . . . . 623

21.2.1 Zero Pressure Gradient Boundary Layer . . . . . . . . . . . . . . . 62421.2.2 Non-zero Pressure Gradient Boundary Layer . . . . . . . . . . . . . 62621.2.3 Polhausen Approximate Solution . . . . . . . . . . . . . . . . . . . . . . 629

21.3 Boundary Layer Theory, Integral Method . . . . . . . . . . . . . . . . . . . . 63121.3.1 Boundary Layer Thicknesses . . . . . . . . . . . . . . . . . . . . . . . . . 63121.3.2 Boundary Layer Integral Equation . . . . . . . . . . . . . . . . . . . . . 633

21.4 Turbulent Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63721.4.1 Universal Wall Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 64021.4.2 Velocity Defect Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

21.5 Boundary Layer, Differential Treatment . . . . . . . . . . . . . . . . . . . . . 64821.5.1 Solution of Boundary Layer Equations . . . . . . . . . . . . . . . . . 653

21.6 Measurement of Boundary Flow, Basic Techniques . . . . . . . . . . . . 65321.6.1 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

21.6.1.1 HWA Operation Modes, Calibration . . . . . . . . . . . . 65421.6.1.2 HWA Averaging, Sampling Data . . . . . . . . . . . . . . 655

21.7 Examples: Calculations, Experiments . . . . . . . . . . . . . . . . . . . . . . . 65721.7.1 Steady State Velocity Calculations . . . . . . . . . . . . . . . . . . . . 657

21.7.1.1 Experimental Verification . . . . . . . . . . . . . . . . . . . . 65921.7.1.2 Heat Transfer Calculation, Experiment . . . . . . . . . . 660

21.7.2 Periodic Unsteady Inlet Flow Condition . . . . . . . . . . . . . . . . 66121.7.2.1 Experimental Verification . . . . . . . . . . . . . . . . . . . . 66421.7.2.2 Heat Transfer Calculation, Experiment . . . . . . . . . . 666

21.7.3 Application of - Model to Boundary Layer . . . . . . . . . . . . 66721.8 Parameters Affecting Boundary Layer . . . . . . . . . . . . . . . . . . . . . . 667

21.8.1 Parameter Variations, General Remarks . . . . . . . . . . . . . . . . 66821.8.2 Effect of Periodic Unsteady Flow . . . . . . . . . . . . . . . . . . . . . 672

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680

A Vector and Tensor Analysis in Turbomachinery . . . . . . . . . 685

A. 1 Tensors in Three-Dimensional Euclidean Space . . . . . . . . . . . . . . . 685A.1.1 Index Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686

A.2 Vector Operations: Scalar, Vector and Tensor Products . . . . . . . . . 687

ContentsXX

A.2.1 Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687A.2.2 Vector or Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688A.2.3 Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688

A.3 Contraction of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689A.4 Differential Operators in Fluid Mechanics . . . . . . . . . . . . . . . . . . . . 690

A.4.1 Substantial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690A.4.2 Differential Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691

A.5 Operator Applied to Different Functions . . . . . . . . . . . . . . . . . . . 693A.5.1 Scalar Product of and V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694A.5.2 Vector Product × V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695A.5.3 Tensor Product of and V . . . . . . . . . . . . . . . . . . . . . . . . . . . 695A.5.4 Scalar Product of and a Second Order Tensor . . . . . . . . . . 696A.5.5 Eigenvalue and Eigenvector of a Second Order Tensor . . . . . 700

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702

B Tensors in Orthogonal Curvilinear Coordinate Systems . . 703

B.1 Change of Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703B.2 Co- and Contravariant Base Vectors, Metric Coefficients . . . . . . . . . 703B.3 Physical Components of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 706B.4 Derivatives of the Base Vectors, Christoffel Symbols . . . . . . . . . . . 707B.5 Spatial Derivatives in Curvilinear Coordinate System . . . . . . . . . . . 708

B.5.1 Application of to Zeroth Order Tensor Functions . . . . . . . . 708B.5.2 Application of to First and Second Order Tensor Functions 709

B.6 Application Example 1: Inviscid Incompressible Flow Motion . . . . . 710B.6.1 Equation of Motion in Curvilinear Coordinate Systems . . . . . 711B.6.2 Special Case: Cylindrical Coordinate System . . . . . . . . . . . . . 712B.6.3 Base Vectors, Metric Coefficients . . . . . . . . . . . . . . . . . . . . . 712B.6.4 Christoffel Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713B.6.5 Introduction of Physical Components . . . . . . . . . . . . . . . . . . . 714

B.7. Application Example 2: Viscous Flow Motion . . . . . . . . . . . . . . . . . 715B.7.1 Equation of Motion in Curvilinear Coordinate Systems . . . . . 715B.7.2 Special Case: Cylindrical Coordinate System . . . . . . . . . . . . . 716

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

Nomenclature

A acceleration, force vectorAc cold side blade surface areaAh hot side blade surface areab trailing edge thickness c blade chord lengthc complex eigenfunction, c = cr + icic speed of soundcax blade axial chord lengthCD cascade drag coefficientCf friction coefficientCi constantsCL blade lift coefficientCL

* camberline lift coefficientcp, cv specific heat capacitiesd projected trailing edge thickness d = b/sin 2D diffusion factorD dimensionless trailing edge thicknessD material or substantial differential operator D deformation tensor De equivalent diffusion factorDh hydraulic diameterDm modified diffusion factore specific total energyei orthonormal unit vectorE source (+), sink (-) strengthE total energyE(k) energy spectrumE Planck’s spectral emissive powerf slot thickness/trailing edge thickness ratio f = s/bfC reheat factor for multistage compressors fS sampling frequencyfT recovery factor for multistage turbines f recovery factor for turbines with infinite number of stages F auxiliary functionF force vectorg blade geometry function

NomenclatureXXII

gi, gi co-, contravariant base vectors in orthogonal coordinate systemgij, gij co-, contravariant metric coefficientsG blade geometry parameterG circulation functionGi auxiliary functionsGi transformation vectorh heighth specific static enthalpyH specific total enthalpyH immersion ratioH12 form factor, H12 = 1/ 2H13 form factor, H32 = 3/ 2i incidence angleI intermittency functionI moment of inertiaI1, I2, I3 principle invariants of deformation tensor J Jacobian transformation functionk thermal conductivityk wave number vectorK specific kinetic energyl, m coordinates introduced for radial equilibriumlm specific shaft workL stage powerLij(x,t) turbulence length scalem massm mass flowmc cooling mass flowM Mach numberM vector of moment of momentumMa axial vector of moment of momentum n number of stationsn polytropic exponentn normal unit vector Nu Nusselt numberp, P static, total pressureP total pressure P = p + V2/2

deterministic pressure fluctuationp+ dimensionless pressure gradient

random pressure fluctuationPr Prandtl numberPre effective Prandtl numberPrt turbulent Prandtl numberq specific thermal energy (heat per unit of mass)qc cold side specific heat rejected from stageqh hot side specific heat transferred to the stage

Nomenclature XXIII

q/ specific heat transferred to stator bladesq// specific heat transferred to rotor bladesq heat flux vectorQ thermal energy (heat)Q thermal energy flow (heat flow)Qc

thermal energy flow (heat flow), blade cold sideQh

thermal energy flow (heat flow), blade hot sideQ/ thermal energy flow (heat flow) to/from stator bladesQ// thermal energy flow (heat flow) to/from rotor bladesr degree of reactionri radius of stage stream surface r radius vector R radius in conformal transformation R density ratio, R = 3/ 2R radiationR radius of the mean flow path Re Reynolds numberRw thermal resistanceR(x, t, r, ) second order correlation tensors specific entropys slot thicknesss blade spacingSi cross section at station iSt Stanton numberStr Strouhal numbert timet thicknesst tangential unit vectorT static temperatureT tangential force componentT stress tensor, T = eiej ijTo stagnation or total temperatureTc static temperature on blade cold side Th static temperature on blade hot sideTW blade material temperatureTr trace of second order tensoru specific internal energyu velocity u wall friction velocityu+ dimensionless wall velocity, u+= u/u

time averaged wake velocity defect

U, V, W rotational, absolute, relative velocity vectorsv specific volumeV volume

NomenclatureXXIV

mean velocity vector

random velocity fluctuating vector

co- and contravariant component of a velocity vector

ensemble averaged velocity vectorVmax maximum velocity on suction surfaceW mechanical energy

mechanical energy flow (power)

shaft power

X state vectory+ dimensionless wall distance, y+= u y/xi coordinatesZ stage profile loss

Greek Symbols

heat transfer coefficient, absolute and relative flow anglesst stagnation angle in conformal transformation

blade stagger angle intermittency factorspecific circulation functionshock angle

min, max, minimum, maximum intermittencycirculation strength circulation vector

ijk Christoffel symbol

deviation angleij Kronecker delta

Kronecker tensor, = eiej ij

1, 2, 3, boundary layer displacement, momentum, energy thickness1, 2 dimensionless displacement, momentum thickness

loss coefficient ratioconvergence toleranceturbulence dissipation

h eddy diffusivitym eddy viscosityijk permutation symbol/ dimensionless parameter for stator blades heat transfer // dimensionless parameter for rotor blades heat transfer

Kolmogorov’s length scaletotal pressure loss coefficientefficiency

Nomenclature XXV

velocity ratiosegment angleblade flow deflection angle shock expansion angletemperature ratio

ij(k1, t) one-dimensional spectral functionisentropic exponentratio of specific heatsstage load coefficientwave lengthload functionmass flow ratioabsolute viscosity

, , velocity ratioskinematic viscosity

m straight cascade stagger angledistance ratio = x/bpressure ratio

stress tensor, = eiejdensity

ij(x,t,r. ) dimensionless correlation coefficientcascade solidity = c/stemperature ratio

o, W wall sear stressKolmogorov’s velocity scalestage flow coefficientdissipation function

, potential, stream function(k.t) spectral tensor

complex functionisentropic stage load coefficientstream functionangular velocityRotation tensor

Subscripts and Superscripts

a, t axial, tangentialc compressiblec cold sideC compressor C, S, R cascade, stator, rotorex exitF flameF fuel

NomenclatureXXVI

Fi filmG combustion gash hot sidein inletmax maximumP, S blade pressure, suction surfaces isentropics shockt turbulentw wallS time averaged

random fluctuationdeterministic fluctuation

* dimensionless+ wall functions/, // stator, rotor