Gas turbines  internal flow systems modeling

Cambridge University Press 978-1-107-17009-4 — Gas Turbines Bijay Sultanian Frontmatter More Information Gas Turbines This long-awaited, physics-first, design-oriented text describes and explains the underlying flow and heat transfer theory of secondary air systems An applications-oriented focus throughout the book provides the reader with robust solution techniques, state-ofthe-art three-dimensional computational fluid dynamics (CFD) methodologies, and examples of compressible flow network modeling It clearly explains elusive concepts of windage, nonisentropic generalized vortex, Ekman boundary layer, rotor disk pumping, and centrifugally driven buoyant convection associated with gas turbine secondary flow systems featuring rotation The book employs physics-based, designoriented methodology to compute windage and swirl distributions in a complex rotor cavity formed by surfaces with arbitrary rotation, counterrotation, and no rotation This text will be a valuable tool for aircraft engine and industrial gas turbine design engineers as well as graduate students enrolled in advanced special topics courses Bijay K Sultanian is founder and managing member of Takaniki Communications, LLC, a provider of web-based and live technical training programs for corporate engineering teams, and an adjunct professor at the University of Central Florida, where he has taught graduate-level courses in turbomachinery and fluid mechanics since 2006 Prior to founding his own company, he worked in and led technical teams at a number of organizations, including Rolls-Royce, GE Aviation, and Siemens Power and Gas He is the author of Fluid Mechanics: An Intermediate Approach (2015) and is a Life Fellow of the American Society of Mechanical Engineers © in this web service Cambridge University Press Tai ngay!!! Ban co the xoa dong chu nay!!! www.cambridge.org Cambridge University Press 978-1-107-17009-4 — Gas Turbines Bijay Sultanian Frontmatter More Information Cambridge Aerospace Series Editors: Wei Shyy and Vigor Yang 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 J M Rolfe and K J Staples (eds.): Flight Simulation P Berlin: The Geostationary Applications Satellite M J T Smith: Aircraft Noise N X Vinh: Flight Mechanics of High-Performance Aircraft W A Mair and D L Birdsall: Aircraft Performance M J Abzug and E E Larrabee: Airplane Stability and Control M J Sidi: Spacecraft Dynamics and Control J D Anderson: A History of Aerodynamics A M Cruise, J A Bowles, C V Goodall, and T J Patrick: Principles of Space Instrument Design G A Khoury (ed.): Airship Technology, Second Edition J P Fielding: Introduction to Aircraft Design J G Leishman: Principles of Helicopter Aerodynamics, Second Edition J Katz and A Plotkin: Low-Speed Aerodynamics, Second Edition M J Abzug and E E Larrabee: Airplane Stability and Control: A History of the Technologies that Made Aviation Possible, Second Edition D H Hodges and G A Pierce: Introduction to Structural Dynamics and Aeroelasticity, Second Edition W Fehse: Automatic Rendezvous and Docking of Spacecraft R D Flack: Fundamentals of Jet Propulsion with Applications E A Baskharone: Principles of Turbomachinery in Air-Breathing Engines D D Knight: Numerical Methods for High-Speed Flows C A Wagner, T Hüttl, and P Sagaut (eds.): Large-Eddy Simulation for Acoustics D D Joseph, T Funada, and J Wang: Potential Flows of Viscous and Viscoelastic Fluids W Shyy, Y Lian, H Liu, J Tang, and D Viieru: Aerodynamics of Low Reynolds Number Flyers J H Saleh: Analyses for Durability and System Design Lifetime B K Donaldson: Analysis of Aircraft Structures, Second Edition C Segal: The Scramjet Engine: Processes and Characteristics J F Doyle: Guided Explorations of the Mechanics of Solids and Structures A K Kundu: Aircraft Design M I Friswell, J E T Penny, S D Garvey, and A W Lees: Dynamics of Rotating Machines B A Conway (ed.): Spacecraft Trajectory Optimization R J Adrian and J Westerweel: Particle Image Velocimetry G A Flandro, H M McMahon, and R L Roach: Basic Aerodynamics H Babinsky and J K Harvey: Shock Wave–Boundary-Layer Interactions C K W Tam: Computational Aeroacoustics: A Wave Number Approach A Filippone: Advanced Aircraft Flight Performance I Chopra and J Sirohi: Smart Structures Theory W Johnson: Rotorcraft Aeromechanics vol W Shyy, H Aono, C K Kang, and H Liu: An Introduction to Flapping Wing Aerodynamics T C Lieuwen and V Yang: Gas Turbine Emissions P Kabamba and A Girard: Fundamentals of Aerospace Navigation and Guidance R M Cummings, W H Mason, S A Morton, and D R McDaniel: Applied Computational Aerodynamic P G Tucker: Advanced Computational Fluid and Aerodynamics Iain D Boyd and Thomas E Schwartzentruber: Nonequilibrium Gas Dynamics and Molecular Simulation Joseph J S Shang and Sergey T Surzhikov: Plasma Dynamics for Aerospace Engineering Bijay K Sultanian: Gas Turbines: Internal Flow Systems Modeling © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-17009-4 — Gas Turbines Bijay Sultanian Frontmatter More Information Gas Turbines Internal Flow Systems Modeling B I J A Y K S U L T A N I A N © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-17009-4 — Gas Turbines Bijay Sultanian Frontmatter More Information University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence www.cambridge.org Information on this title: www.cambridge.org/9781107170094 DOI: 10.1017/9781316755686 © Bijay K Sultanian 2018 This publication is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published 2018 Printed in the United States of America by Sheridan Books, Inc A catalogue record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Sultanian, Bijay K Title: Gas turbines : internal flow systems modeling / Bijay K Sultanian Description: Cambridge, United Kingdon ; New York, NY, USA : Cambridge University Press, 2018 | Series: Cambridge aerospace series | Includes bibliographical references and index Identifiers: LCCN 2018010102 | ISBN 9781107170094 (hardback) Subjects: LCSH: Gas-turbines–Fluid dynamics–Mathematics | Gas flow–Mathematical models | BISAC: TECHNOLOGY & ENGINEERING / Engineering (General) Classification: LCC TJ778 S795 2018 | DDC 621.43/3–dc23 LC record available at https://lccn.loc.gov/2018010102 ISBN 978-1-107-17009-4 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-17009-4 — Gas Turbines Bijay Sultanian Frontmatter More Information To my dearest friend Kailash Tibrewal, whose mantra of “joy in giving” continues to inspire me; my wife, Bimla Sultanian; our daughter, Rachna Sultanian, MD; our son-in-law, Shahin Gharib, MD; our son, Dheeraj (Raj) Sultanian, JD, MBA; our daughter-in-law, Heather Benzmiller Sultanian, JD; and our grandchildren, Aarti Sultanian, Soraya Zara Gharib, and Shayan Ali Gharib, for the privilege of their unconditional love and support! © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-17009-4 — Gas Turbines Bijay Sultanian Frontmatter More Information © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-17009-4 — Gas Turbines Bijay Sultanian Frontmatter More Information Contents Preface Acknowledgments About the Author page xi xv xvii Overview of Gas Turbines for Propulsion and Power Generation 1.0 Introduction 1.1 Primary Flow: Energy Conversion 1.2 Internal Flow System (IFS) 1.3 Physics-Based Modeling 1.4 Robust Design Methodology 1.5 Concluding Remarks Worked Examples Problems References Bibliography Nomenclature 1 15 18 23 23 26 29 30 31 Review of Thermodynamics, Fluid Mechanics, and Heat Transfer 2.0 Introduction 2.1 Thermodynamics 2.2 Fluid Mechanics 2.3 Internal Flow 2.4 Heat Transfer 2.5 Concluding Remarks Worked Examples Problems References Bibliography Nomenclature 34 34 34 46 92 105 119 120 131 136 137 138 1-D Flow and Network Modeling 3.0 Introduction 3.1 1-D Flow Modeling of Components 3.2 Description of a Flow Network: Elements and Junctions 143 143 144 153 vii © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-17009-4 — Gas Turbines Bijay Sultanian Frontmatter More Information viii Table of Contents 3.3 Compressible Flow Network Solution 3.4 Concluding Remarks Worked Examples Problems Project References Bibliography Nomenclature 156 161 162 172 177 177 177 178 Internal Flow around Rotors and Stators 4.0 Introduction 4.1 Rotor Disk 4.2 Cavity 4.3 Windage and Swirl Modeling in a General Cavity 4.4 Compressor Rotor Cavity 4.5 Preswirl System 4.6 Hot Gas Ingestion: Ingress and Egress 4.7 Axial Rotor Thrust 4.8 Concluding Remarks Worked Examples Problems Projects References Bibliography Nomenclature 182 182 182 186 190 200 206 209 218 221 222 225 227 229 230 234 Labyrinth Seals 5.0 Introduction 5.1 Straight-Through and Stepped-Tooth Designs 5.2 Tooth-by-Tooth Modeling 5.3 Concluding Remarks Worked Examples Project References Bibliography Nomenclature 237 237 238 242 248 248 254 255 255 256 Whole 6.0 6.1 6.2 6.3 6.4 6.5 258 258 259 261 268 271 291 Engine Modeling Introduction Multiphysics Modeling of Engine Transients Nonlinear Convection Links Role of Computational Fluid Dynamics (CFD) CFD Methodology Thermomechanical Analysis © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-17009-4 — Gas Turbines Bijay Sultanian Frontmatter More Information Table of Contents ix 6.6 Validation with Engine Test Data 6.7 Concluding Remarks Project References Bibliography Nomenclature 305 306 307 309 311 312 Appendix Appendix Appendix Appendix Appendix 317 322 323 334 A B C D E Review of Necessary Mathematics Equations of Air Thermophysical Properties Transient Heat Transfer in a Rotor Disk Regula Falsi Method Thomas Algorithm for Solving a Tridiagonal System of Linear Algebraic Equations Appendix F Solution of an Overdetermined System of Linear Algebraic Equations Epilogue Current Research Work and Challenges Index © in this web service Cambridge University Press 337 340 347 352 www.cambridge.org Cambridge University Press 978-1-107-17009-4 — Gas Turbines Bijay Sultanian Frontmatter More Information © in this web service Cambridge University Press www.cambridge.org C.3 Analytical Solution 327 whose substitution in Equation C.13 yields Sr dSτ d2 S ¼ Sτ 2r  π Sr Sτ α dτ dr dSτ d2 Sr ¼  π ¼ λ αSτ dτ Sr dr where λ is the separation constant, which leads to two ordinary differential equations: dS ỵ S ẳ d (C.18) d2 Sr ỵ Sr ¼ dr (C.19) where β2 ¼ λ  π such that λ > π : The general solution of Equation C.18 can be written as Sτ ¼ C eα λτ (C.20) Sr ¼ C cos r ị ỵ C sin r Þ (C.21) and that of Equation C.19 as Now we can write the general solution of Equation C.13:    C cos r ị ỵ C sin r ị T^m r; ị ẳ C e   T^m r; ị ẳ e C~ cos r ị ỵ C~ sin r ị (C.22) where C~ ẳ C1 C2 and C~ ¼ C C , which along with the characteristic parameter β need to be evaluated from the given initial and boundary conditions, Equations C.14, C.15, and C.16, yielding the final solution T^m ðr; ị ẳ X nẳ1 e  n !     sin βn r rim cos βn r     βn r rim þ sin βn r rim cos βn r rim (C.23) where   β r cot βn r rim ¼ n rim Bi (C.24) n ẳ ỵ 2n (C.25) hrim r rim km (C.26) Bi ¼ Note that Equation C.24 is transcendental in βn For Bi ¼ 1, the first ten roots of this equation are tabulated here: 010 15:48:28 328 C.4 Transient Heat Transfer in a Rotor Disk n βn r rim 10 0.8603336 3.4256185 6.4372982 9.5293344 12.6452872 15.7712849 18.9024100 22.0364967 25.1724463 28.3096429 Numerical Solution In Section C.2, we performed the energy balance on a disk element and derived Equation C.5, which is a nonlinear second-order partial differential equation to compute transient temperature in the disk resulting from one-dimensional radial heat conduction with simultaneous convection on disk surface Because the analytical solution of Equation C.5 with realistic disk geometry and thermal boundary conditions is not possible, we resort to its numerical solution, the methodology for which is presented in this section For deriving the discretized algebraic equations corresponding to Equation C.5, we again use the control volume approach of energy balance on a disk element without making it differential Toward this approach, the rotor disk is first divided into N finite control volumes, as shown in Figure C.2a We place a node at the center of each control volume To facilitate the application of thermal boundary conditions on the disk bore and rim surfaces, we create two fictitious boundary nodes, which are placed across the disk faces at a distance equal to that of the adjacent interior node, as shown in the gure, resulting in N ỵ total number of nodes In the control volume approach presented by Patankar (1980), the first node and the last node are placed directly on the end surfaces The energy balance on the control volume associated with an interior node i, shown in Figure C.2b, yields  Tmi  Tmi ị Ai1 kmi ỵ k mi1    δmi cmi ¼ Tmi1  Tmi Δτ r i  r i1  Ai2 kmi ỵ kmiỵ1  (C.27)    Tmi  Tmiỵ1 r iỵ1  r i    hi1 Si1 Tmi  Tawi  hi2 Si2 Tmi  Tawi where, except for Tmi ðτ Þ, which is known at the current time, all other quantities pertain to the time ỵ Other quantities in Equation C.27 are dened as follows: 010 15:48:28 C.4 Numerical Solution 329 Ai1  Area of conduction from node i  to node i Ai2  Area of conduction from node i to node i ỵ cm  Temperature-dependent specic heat of disk material hi1  Heat transfer coefficient on Face of node i control volume hi2  Heat transfer coefficient on Face of node i control volume k mi , kmi1 , kmiỵ1  Disk material thermal conductivities evaluates at temperatures Tmi , Tmi1 , Tmiỵ1 , respectively r i  Radius of node i r i1  Radius of node i  r iỵ1  Radius of node i ỵ Si1  Convection surface area on Face of node i control volume Si2  Convection surface area on Face of node i control volume Tmi , Tmi1 , Tmiỵ1  Disk temperature at node i, i  1, and i ỵ 1, respectively Tawi  Adiabatic wall temperature on Face of node i control volume Tawi  Adiabatic wall temperature on Face of node i control volume δmi  Mass of node i control volume τ  Time Δτ  Time step W Figure C.2 (a) Rotor disk control volumes and nodes (b) energy balance on an interior control volume for numerical solution .010 15:48:28 330 Transient Heat Transfer in a Rotor Disk We can write Equation C.27 in the following form: bi Tmi1 ỵ di Tmi ỵ Tmiỵ1 ẳ ci (C.28) where i ẳ 2,3, , N ỵ and  Ai1 kmi þ k mi1   bi ¼  r i  r i1   Ai1 kmi ỵ kmi1 Ai2 k mi ỵ kmiỵ1 mi cmi   ỵ   ỵ hi Si ỵ hi Si ỵ di ¼ 1 2 Δτ r i  r i1 r iỵ1  r i  Ai2 kmi ỵ k miỵ1   ẳ  r iỵ1  r i ci ẳ mi cmi Tmi ị ỵ hi1 Si1 Tawi ỵ hi2 Si2 Tawi Note that the system of equations that result from Equation C.28 is not closed for it has N number of equations and N ỵ unknown temperatures To close this system, we need two more equations from the thermal boundary conditions specified at disk bore and rim surfaces, relating the temperatures at fictitious boundary nodes to those at their next interior nodes Let us now establish equations relating temperatures at bore-side and rim-side fictitious nodes to those at their neighboring interior nodes, which are shown in Figure C.3 Disk bore Three types of thermal boundary conditions can in general be specified at the disk bore wall, allowing us to develop an equation to relate the fictitious node to interior node in the form of Equation C.28 Type 1: Specified surface temperature Twbore Twbore ¼ Tm1 þ Tm2 Tm1 þ Tm2 ¼ 2Twbore (C.29) Comparing Equation C.29 with Equation C.28 for i ¼ yields b1 ¼ 0, d ¼ 1, a1 ¼ 1, and c1 ¼ 2Twbore (C.30) Type 2: Specified heat flux q_ bore
 Tm1  Tm2 ¼ q_ bore r2  r1   r  r q_ bore Tm1  Tm2 ¼ kmbore kmbore 010 15:48:28 (C.31) C.4 Numerical Solution 331 Figure C.3 (a) Type BC on disk bore, (b) Type BC on disk bore, (c) Type BC on disk bore, (d) Type BC on disk rim, (e) Type BC on disk rim, and (f ) Type BC on disk rim Comparing Equation C.31 with Equation C.28 for i ¼ yields in this case   r  r q_ bore b1 ¼ 0, d ¼ 1, a1 ¼ 1, and c1 ¼ kmbore (C.32) Note that, for disk bore cooling, q_ bore is negative, and for heating, it is positive Type 3: Convective boundary condition with specified hbore and Tawbore
k mbore Tm2  Tm1  ẳ hbore
Tm1 ỵ Tm2   Tawbore r2  r1   2kmbore Tm2  Tm1 ẳ hbore Tm1 ỵ Tm2  2Tawbore r2  r1

2k mbore r2  r1 
þ hbore Tm1  2kmbore r2  r1   hbore Tm2 ¼ 2hbore Tawbore (C.33) Comparing Equation C.33 with Equation C.28 for i ¼ yields in this case b1 ẳ 0, d1 ẳ 2kmbore r2  r1 ỵ hbore , a1 ẳ  2kmbore r2  r1 ỵ hbore , and c1 ¼ 2hbore Tawbore (C.34) Disk rim At the disk rim also, in general, we can specify one of the three types of thermal boundary conditions For each type, let us develop an equation to relate the ctitious node N ỵ to interior node N ỵ in the form of Equation C.28 .010 15:48:28 332 Transient Heat Transfer in a Rotor Disk Type 1: Specified surface temperature Twrim Twrim ẳ TmNỵ1 ỵ TmNỵ2 (C.35) TmNỵ1 ỵ TmNỵ2 ẳ 2Twrim Comparing Equation C.35 with Equation C.28 for i ẳ N ỵ yields bNỵ2 ẳ 1, d Nỵ2 ẳ 1, aNỵ2 ẳ 0, and cNỵ2 ¼ 2Twrim Type 2: Specified heat flux q_ rim
T  mNỵ2  TmNỵ1 kmrim ẳ q_ rim r Nỵ2  r Nỵ1   r Nỵ2  r Nỵ1 q_ rim TmNỵ1 ỵ TmNỵ2 ẳ kmrim (C.36) (C.37) Comparing Equation C.37 with Equation C.28 for i ẳ N ỵ yields in this case   r Nỵ2  r Nỵ1 q_ rim (C.38) bNỵ2 ẳ 1, dNỵ2 ẳ 1, aNỵ2 ẳ 0, and cNỵ2 ẳ kmrim Note that, for disk rim cooling, q_ rim is negative, and for heating, it is positive Type 3: Convective boundary condition with specied hrim and Tawrim kmrim

T  mNỵ1  TmNỵ2 r Nỵ2  r Nỵ1 2kmrim r Nỵ2  r Nỵ1
 ẳ hrim
T mNỵ1 ỵ TmNỵ2   Tawrim   TmNỵ1  TmNỵ2 ẳ hrim TmNỵ1 ỵ TmNỵ2  2Tawrim 
 2kmrim  hrim TmNỵ1 ỵ ỵ hrim TmNỵ2 ẳ 2hrim Tawrim r Nỵ2  r Nỵ1 r Nỵ2  r Nỵ1 2k mrim (C.39) Comparing Equation C.39 with Equation C.28 for i ẳ N ỵ yields in this case bNỵ2 ẳ  2kmrim r Nỵ2  r Nỵ1 ỵ hrim , dNỵ2 ẳ 2kmrim r Nỵ2  r Nỵ1 ỵ hrim , aNỵ2 ẳ 0; and cNỵ2 ẳ 2hrim Tawrim (C.40) Now we have all the needed equations for each of the three thermal boundary conditions at disk bore and rim to include two more equations, one for i ¼ and the other for i ẳ N ỵ 2, in the system of equations, which arise from Equation C.28 This, at each time step, results in a closed (number of equations equals number of unknowns) 010 15:48:28 Reference 333 tridiagonal system of equations whose solution by the Thomas algorithm is presented in Appendix E Reference Patankar, S V 1980 Numerical Heat Transfer and Fluid Flow Boca Raton, FL: Taylor & Francis .010 15:48:28 Appendix D Regula Falsi Method D.1 Regula Falsi Method In gas turbine internal flow systems modeling and many other engineering applications, we often need to find the root x of the equation f xị ẳ where f ðxÞ is an analytic function When we are given an equation of the form g1 xị ẳ g2 ðxÞ, where both g1 ðxÞ and g2 ðxÞ are analytic, the equation is called the transcendental equation and may the rewritten in the standard form f xị ẳ g2 xị  g1 xị ẳ The Regula Falsi method, presented in detail in Carnahan, Luther, and Wilkes (1969), is a powerful technique to iteratively solve
f xị ẳ for its root that lies within the limits xL  x  xR such that
 f xL < and f xR > Note that in MS Excel, one can use the “Goal Seek” function to find the root of f x
ị ẳ
 Knowing xL , xR , f xL , and f xR , the Regula Falsi algorithm can be summarized as follows: Evaluate x by the equation xC ¼ xL f R  xR f L fR  fL (D.1) where f L and f R represent values of f ðxÞ evaluated at xL and xR , respectively
 Evaluate f C ¼ f xC If f C  E, where E is an acceptable error in f xị ẳ 0, we have obtained xC as the desired root of the equation, and the iteration can be terminated Otherwise, go to step 3 If f L f C < 0, set xR ¼ xC and f R ¼ f C ; otherwise, set xL ¼ xC and f L ¼ f C Repeat steps from to until the convergence condition in step is satisfied The subroutine REGULA listed at the end of this appendix uses the foregoing algorithm of the Regula Falsi method Example D.1 For the total-pressure mass flow function F^f t ¼ 0:4 and κ ¼ 1:4, use the Regula Falsi method to find the Mach number from the equation sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κ ^ Ff t ẳ M
ỵ1 1 1 1ỵ M Tabulate values of Mach number and function f ðM Þ for the first five iterations and graphically represent the iteration process for the first two iterations 334 011 15:47:50 335 D.2 REGULA Subroutine Solution: First, we cast the given equation in the following form: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κ ^ f M ị ẳ M
ỵ1  Ff t ẳ 1 ỵ 1 M Using F^f t ¼ 0:4 and κ ¼ 1:4 in this equation, and starting with M L ¼ 0:1 and M R ¼ 1, we obtained the following results for the first five iterations of the Regula Falsi algorithm: Iteration ML fL MR fR MC fC 0.10000 0.10000 0.10000 0.10000 0.10000 –0.28239 –0.28239 –0.28239 –0.28239 –0.28239 1.00000 0.54814 0.39651 0.37016 0.36652 0.28473 0.14441 0.02754 0.00386 0.00051 0.54814 0.39651 0.37016 0.36652 0.36603 0.14441 0.02754 0.00386 0.00051 0.00007 The results show that Mach number of 0.36603 obtained in just five iterations is within 0.02 percent of the exact value of 0.36596 Figure D.1 shows how the Regula Falsi method works for the first two iterations k = 1.4 Fˆft= 0.4 0.3 0.2 MR(2) = 0.39651 0.1 f ML(0) = 0.1 0.1 0.2 0.3 0.4 0.5 0.6 –0.1 –0.2 0.7 M 0.8 0.9 1.1 1.2 MR(0) = 1.0 MR(1) = 0.54814 –0.3 –0.4 Figure D.1 Computation of Mach number for a given total-pressure mass flow function using Regula Falsi method (Example D.1) D.2 REGULA Subroutine The call statement for subroutine REGULA is of the form CALL REGULA (N, XL, XR, XC) 011 15:47:50 336 Regula Falsi Method Listing of Subroutine REGULA C C C C C C C C C C C C C C SUBROUTINE REGULA (N, XL, XR, XC) IMPLICIT DOUBLEPRECISION (A-H, O-Z) USING REGULA FALSI METHOD, THIS SUBROUTINE FINDS THE SOLUTION OF A TRANSCENDENTAL EQUATION TOL = 1.0D-5 CALL FUNCTION (XL, FL) CALL FUNCTION (XR, FR) IF (FL*FR GT 0.) THEN PRINT*, “REVISE VALUES OF XL AND XR TO YIELD FL*FR < 0.” RETURN ELSE END IF COMPUTE THE TANSCENDENTAL EQUATION ROOT -DO I = 1, N XC = (XL*FR - XR*FL)/(FR - FL) CALL FUNCTION (XC, FC) CHECK FOR CONVERGENCE -IF (F LT TOL) RETURN KEEP RIGHT OR LEFT SUBINTERVAL -IF (FC*FL LT 0.0) THEN XR = XC FR = FC ELSE XL = XC FL = FC END IF END DO RETRUN END Reference Carnahan, B., H A Luther, and J O Wilkes 1969 Applied Numerical Methods New York: John Wiley & Sons .011 15:47:50 Appendix E Thomas Algorithm for Solving a Tridiagonal System of Linear Algebraic Equations E.1 Thomas Algorithm We briefly present here the Thomas algorithm, which is widely used to solve a tridiagonal system of linear algebraic equations Additional details with its applications are given in Patankar (1980) Let us consider N nodes arranged along the west-east direction numbered 1, 2, 3, , N From a control volume analysis, the nodal values ϕi are governing by the following linear algebraic equation: bi i
1 ỵ di i ỵ iỵ1 ẳ ci (E.1) where i ẳ 1, 2, 3, , N For each value of i, except for i ¼ with b1 ¼ and i ¼ N with aN ¼ 0, Equation E.1 relates ϕi to its before-node value ϕi
1 and its after-node value iỵ1 Furthermore, when write Equation E.1 all values of i, the resulting system of equations assumes the following matrix equation where the nonzero elements of the coefficient matrix fall along three diagonals, hence it is called tridiagonal matrix In this matrix, the coefficient vector d i occupies the main diagonal, vector bi the lower or before diagonal, and vector the upper or after diagonal — the choice of symbols for these vectors reflects this fact 32 d a1 6 b2 d a2 6 b3 d a3 6 6 6 6 bN
1 dN
1 aN
1 bN dN 76 76 76 76 76 76 76 76 76 76 76 76 76 74 ϕ1 c1 ϕ2 7 6 ϕ3 7 7 ¼ 7 7 ϕN
1 c2 7 c3 7 7 7 7 cN
1 ϕN cN (E.2) The first equation in Equation E.2 involves only ϕ1 and ϕ2 , which implies that we can express ϕ1 in terms of ϕ2 Marching forward, while expressing value at the preceding node in terms of the value at the current node, we obtain a relation between the values at 337 012 15:47:36 338 Thomas Algorithm for Solving a Tridiagonal System of Linear Algebraic Equations the current node and the node that follows Recognizing this fact, let us assume the following algebraic equation form for this relation i ẳ i iỵ1 þ βi (E.3) ϕi
1 ¼ αi
1 ϕi þ βi
1 (E.4) which also yields Using Equation E.4 to eliminate ϕi
1 in Equation E.1, we obtain   bi i
1 i ỵ i
1 ỵ d i i ỵ iỵ1 ẳ ci i ẳ
ai c
bi i
1 iỵ1 ỵ i di ỵ bi i
1 di ỵ bi i
1 (E.5) Comparing Equations E.4 and E.5 yields i ẳ
ai di ỵ bi αi
1 (E.6) βi ¼ ci
bi βi
1 di þ bi αi
1 (E.7) For i ¼ with b1¼ 0, Equations E.6 and E.7 yield α1 ¼
a1 c and β1 ¼ , d1 di and for i ¼ N with aN ¼ 0, we obtain αN ¼ Substitution of αN ¼ in Equation E.3 yields ϕN ¼ βN , which, through back substitution in this equation, allows us to unpack all remaining values of ϕN
1 , ϕN
2 , , ϕ1 Thus, we can summarize the Thomas algorithm for solving the tridiagonal system of linear algebraic equations represented by Equation E.1as follows:
a1 c Compute α1 ¼ and β1 ¼ d1 di Use the recurrence Equations E.6 and E.7 to compute αi and βi for i ¼ 2, 3, , N Set ϕN ¼ βN Use Equation E.3 to compute ϕN
1 , ϕN
2 , , ϕ1 through back substitution E.2 THOMAS Subroutine The call statement for subroutine THOMAS is of the form CALL THOMAS (NJ, AA, BB, CC, DD, TM) Listing of Subroutine THOMAS SUBROUTINE THOMAS (N, A, B, C, D, PHI) IMPLICIT DOUBLE PRECISION (A-H, O-Z) 012 15:47:36 Reference C C C C C C C C C C 339 PARAMETER (ND = 100) DOUBLE PRECISION A(ND), B(ND), C(ND), D(ND), PHI(ND) DOUBLE PRECISION ALPHA(ND), BETA(ND) USING THOMAS ALGORITHM, THIS SUBROUTINE SOLVES A TRIDIGONAL SYSTEM OF LINEAR ALGEBRAIC EQUATIONS SMALL = 1.0D-20 ALPHA(1) = -A(1)/D(1) BETA(1) = C(1)/D(1) COMPUTE COEFFICIENTS OF RECURRENCE FORMULA -DO I = 2, N DENOM = D(I) + B(I)*ALPHA(I-1) IF (DENOM LE SMALL) DENOM = SMALL ALPHA(I) = -A(I)/DENOM BETA(I) = (C(I) - B(I)*BETA(I-1))/DENOM END DO PHI(N) = BETA(N) NM1 = N - BACK SUBTITUTION TO OBTAIN REMAINING PHI’S -DO I = NM1, 1, -1 PHI(I) = ALPHA(I)*PHI(I+1) + BETA(I) END DO RETRUN END Reference Patankar, S V 1980 Numerical Heat Transfer and Fluid Flow Boca Raton, FL: Taylor & Francis .012 15:47:36 Appendix F Solution of an Overdetermined System of Linear Algebraic Equations F.1 Introduction In gas turbine internal flow systems modeling and many other engineering applications that require finding a regression equation as a curve-fit to a large number of experimental or numerical data, we often need to solve an overdetermined system of linear algebraic equations In such a system, the number of equations is more than the number of unknowns Of course, a method capable of solving such a system can also provide a direct solution when the system is closed with equal number of equations and unknowns When the coefficient matrix depends on the solution itself, we have a system of nonlinear equations, having no known direct solution method The numerical solution of such a system requires iterative direct solutions of a series of intermediate systems of linear equations, updating the coefficient matrix as needed from iteration to iteration Hopefully, such an iterative solution method leads to the correct converged solution of the system of nonlinear equations on hand F.2 Linear Least-Squares Data Fitting Sultanian (1980) used the robust Householder reflection method for linear least-squares curve fitting The method was first proposed by Golub (1965) In this section, we provide some introductory details to facilitate the use of the computer code CURVE listed at the end
 Suppose we are given m data points θi ; μi , which may be experimental or numerical, where θi are the values of the independent variable like dimensionless temperature, μi the values of the dependent variable like fluid viscosity, and the subscript i ¼ 1,2, , M We not know the exact analytical relation to compute μi for a given θi The following regression or curve-t equation provides approximations to i : ^ị ẳ C1 ị ỵ C 2 ị ỵ    ỵ C N N ị (F.1) where we must have M  N Typically, in a regression problem, M is much greater than N Note that ϕj may be nonlinear functions of θ, however, the equation is linear because the coefficients C j appear linearly Our task is to determine these coefficients in the least-squares sense 340 013 15:47:24 F.3 Computer Program HOME 341 Applying Equation F.1 to each data point, we obtain the following matrix equation: 32 3 C1 μ1 a11 a12 a1N a a a 76 C μ 2N 76 21 22 27 76 7 76 ¼ 54 CN aM1 aM2 aMN μM (F.2) which can be written in a compact form as [A]{X} =
{B}  where [A] is a M  N coefficient matrix with its elements given by aij ¼ ϕj θi The column vector {X}, which is a N  matrix, represents the column vector {C} on the left-hand side of Equation F.2, while the column vector {B}, which is a M  matrix, represents the column vector {μ} on the right-hand side of the equation F.3 Computer Program HOME The computer program listing at the end consists of the main program HOME and subroutines REDUCE and PIVOT The main program is currently designed for curvefitting of data using a polynomial function, internally generating the coefficient matrix [A] Taking data points as input, the program returns the coefficients of the polynomial function as the curve-fit equation One can easily modify the main program for different basis functions for curve-fitting or directly input the elements of the coefficient matrix for solving a closed system of linear algebraic equations Subroutine REDUCE computes various characteristic parameters of Householder reflections and reduces the coefficient matrix into an upper triangular matrix To minimize the round-off errors in numerical computation, at each step, the subroutine PIVOT chooses the column the coefficient matrix with the largest sum of squares to be reduced next Listing of Program HOME C C C C C MAIN PROGRAM - HOME -PROGRAM HOME IMPLICIT DOUBLE PRECISION (A-H, O-Z) PARAMETER (ND = 100) DOUBLE PRECISION A(ND, ND), X(ND), B(ND), THETA(ND) DOUBLE PECISION ICONT(ND), DUMP (ND) COMMON A, B, X, M, N, ICONT, RELERR CHARACTER TITLE *10 CHARACTER INFLNM *80 INPUT FOR INDEPENDENT VARIABLE ARE STORED IN THETA(I) INPUT FOR DEPENDENT VARIABLE ARE STORED IN B(I) 013 15:47:24