Turbomachines: Aeroelasticity, Aeroacoustics, and Unsteady Aerodynamics

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Edited by V. Skibin V. Saren N. Savin S. Frolov

TORUS PRESS Mos ow 2006

V. A. Skibin

P. I. Baranov Central Institute of Aviation Motors Mos ow 111116, Russia V. E. Saren

P. I. Baranov Central Institute of Aviation Motors Mos ow 111116, Russia N. M. Savin

P. I. Baranov Central Institute of Aviation Motors Mos ow 111116, Russia S. M. Frolov

N. N. Semenov Institute of Chemi al Physi s Russian A ademy of S ien es Mos ow 119991, Russia

Š 39.56 ’ 86 “„Š 62-135:629.7.015;681.883.068

Turboma hines:

Aeroelasti ity,

Aeroa ousti s,

and

Unsteady

/ [Edited by V. A. Skibin, V. E. Saren, N. M. Savin, S. M. Frolov℄. | Mos ow: TORUS PRESS Ltd., 2006. | 472 p. Tabl. 25, ill. 259.

Aerodynami s

ISBN 5-94588-041-8 The book in ludes 34 revised and edited arti les on aeroelasti ity of blades, gas-dynami s of vibrating bladerows, rotor{stator intera tion, aeroa ousti s, and unsteady aerodynami s of turboma hines. The book is published in onne tion with the 11th International Symposium on Unsteady Aerodynami s, Aeroa ousti s and Aeroelasti ity of Turboma hines (ISUAAAT-2006) held in Mos ow, Russia, September 4{8, 2006. The Symposium was organized by P. I. Baranov Central Institute of Aviation Motors. The volume is addressed to resear h s ientists dealing with unsteady pro esses in turboma hines and power plant designers and engineers.

Š 39.56

ISBN 5-94588-041-8

Managing Editor Te hni al Editor

L. Kokushkina T. Torzhkova

Printed in Russian Federation



TORUS PRESS Ltd., 2006

Art Editor Cover Design

M. Sedakova P. Sedakov

Foreword The y li pro ess of energy transformation in turboma hines is a

ompanied with ow os illations in the passage. These os illations indu e me hani al vibrations of stru tural elements, radiation of noise, and

ontribute to gasdynami energy losses. Therefore, su h issues as reliability, durability, eÆ ien y, and e ology of turboma hinery are dire tly determined by the level of our knowledge in unsteady gas-dynami s and aeroelasti ity. It is therefore worth onsidering the a

omplishments in this eld of s ien e and te hnology, as well as the development of ee tive means implementing s ienti results into pra ti al devi es, as the most promising roadway towards further improvements in the performan e of turboma hines. International symposia like ISUAAAT are known to fo us on parti ular issues with their thorough analysis in luding fundamental understanding of relevant problems and elaborating advan ed approa hes to put the new knowledge into pra ti al appli ations. The tradition to publish the ontributions to the symposium in a bound volume is ertainly valuable for the international ommunity of resear h engineers and designers of turboma hinery.

O. N. Favorskii

A ademi ian Russian A ademy of S ien es

v

Prefa e This volume ontinues the tradition of publishing the ontributions submitted to and presented at international symposia on unsteady aerodynami s, aeroa ousti s, and aeroelasti ity in turboma hines (ISUAAAT). The rst symposium in the series was organized by Prof. R. Legendre in Paris in September 1976, i.e., 30 years ago. Subsequent symposia were held in Losanne, Switzerland (1980), Cambridge, England (1984), Aa hen, Germany (1987), Beijin, People's Republi of China (1989), Notre Dane, USA (1991), Fukuoka, Japan (1994), Sto kholm, Sweden (1997), Lion, Fran e (2000), and Darem, USA (2003). As is seen, the tradition of the ISUAAAT meetings every three years is settled. The ISUAAAT-2006, whi h formed the basis for this volume, was held in Mos ow, Russia, September 4{8, 2006, and organized by P. I. Baranov Central Institute of Aviation Motors. The periodi organization of the symposia indi ates the persistent interest to its topi al s ope. Despite the issues addressing unsteady

ow phenomena and aeroelasti ity are dis ussed at many other onferen es related to turboma hinery, the ISUAAAT meetings sustain their signi an e for the ommunity. On the one hand, this is explained primarily by growing requirements to advan ed turboma hines in terms of durability, eÆ ien y, and noise. On the other hand, the variety and

omplexity of unsteady phenomena in turboma hines pose a number of questions related to fundamental hydro- and aerodynami s. The latter is aused, rst of all, by high frequen ies of relevant pro esses and by intera tion of neighboring stator and rotor bladerows. It is ommonly understood that one of the most ompli ated issues to be resolved at the design stage is the a

ount for aeroelasti vibrations of blades in the

ow paths of turboma hines. The main spe i feature of ISUAAAT meetings is that they are fo used on the dis ussion of multiple issues relevant to unsteady phenomena in turboma hines. The book in ludes 34 arti les, whi h are grouped in six hapters a

ording to the ISUAAAT-2006 topi s. When omposing the hapters, the editors intended to sele t the arti les tting a proper topi in vi

Prefa e the general s ope. In parti ular, the rst two hapters deal with aeroelasti ity although blade vibrations in the ow are onsidered solely in Chapter 1. This hapter ontains ontributions on utter and some novel omputational approa hes related to it. Chapter 2 ombines the arti les dedi ated to unsteady aerodynami loading of blades os illating in the preset modes with prede ned frequen ies. The arti les onsidering various aspe ts of rotor{stator intera tion are in luded in Chapter 3. Chapter 4 ontains the arti les aimed at studies of various sour es and propagation modes of a ousti disturban es in the turboma hine passages. The papers on omputational methods for unsteady ows in turboma hines are grouped in Chapter 5, and Chapter 6 deals with physi al ee ts a

ompanying unsteady ows. In general, the ontents of the book re e t the state-of-the-art in the theoreti al and experimental studies of unsteady ows in turboma hines. This an be useful for evaluating the predi ting apability of omputer odes proposed for pra ti al al ulations. The volume was published before the opening of the ISUAAAT2006. We thank all authors for preparing their papers and spending their time and eorts with the editors on improving the text, gures, and s ope of their ontributions to t with the overall goals of the book. On our profound belief, these eorts are justi ed by timely publishing of the Symposium pro eeding. On behalf of the ISUAAAT-2006 Organizing Committee, we thank the members of the International S ienti Committee of ISUAAAT, H. M. Atassi (USA), T. P. Crisval (Fran e), P. Ferrand (Fran e), T. H. Fransson (Sweden), K. C. Hall (USA), M. Imregun (U.K.), R. E. Kielb (USA), M. Namba (Japan), T. Nagashima (Japan), J. M. Verdon (USA), and D. S. Whitehead (U.K.), for their de ision to have the Symposium-2006 in Mos ow and for ontinuous support during Symposium preparations. We would like to all late Professor G. Yu. Stepanov (1922{2005) to our memory, who rendered assistan e to the Organizing Committee at the very beginning of Symposium preparations. Professor G. Yu. Stepanov was a parti ipant of the First ISUAAAT Symposium and edited Russian translation of the Symposium Pro eedings .  Stepanov,

G. Yu., ed.

1979.

Unsteady ows in turboma hines.

Mos ow:

Mir

Publ.

vii

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

It is also appropriate to a knowledge with sin ere thanks A ademi ian G. G. Chernyi, A ademi ian O. N. Favorskii, A ademi ian V. M. Titov, and Professor V. B. Kurzin for their help in preparing the Symposium. We thank the sponsoring agen ies, International S ien e and Te hnology Center (ISTC), Russian Foundation for Basi Resear h (RFBR), NPO \Saturn," V. V. Chernyshev Mos ow Ma hine-Building Plant, \Silovye Mashiny," Ltd., OKBM \Soyus" and UK \Permsky Mashinostroitelnyi Komplex," In ., for their nan ial support, without whi h organization of the symposium like ISUAAAT would not be possible. Spe ial thanks are due to the members of the Symposium working group, V. Glotov, L. Zhemuranova, S. Pen'kov, N. Saren, S. Smirnov, D. Kovalev, M. Nyukhtikov, T. Semenova, L. Buldymenko, and A. Kutina. This volume is the out ome of hard work of several persons, and we highly appre iate their valuable ontribution. In parti ular, we a knowledge the assistan e given at various stages by Ms. Olga Frolova. We thank the sta of TORUS PRESS Publishers for their ex ellent servi e in produ ing this volume. We do hope that this volume will serve as a useful addition to the literature on unsteady phenomena in turboma hines. V. A. Skibin V. E . Saren N. M. Savin S. M. Frolov

viii

Contents Se tion 1: Aeroelasti Analysis of Bladerows

1

An Overview of Computational Turboma hinery Aeroelasti ity M. Imregun and M. Vahdati . . . . . . . . . . . . . . . . Aplli ation of a New Mathemati al Tool \One-Dimensional Spe tral Portraits of Matri es" to the Problem of Aeroelasti Vibrations of Turbine-Blade Cas ades

S. K. Godunov, V.B. Kurzin, V.G. Bunkov, and M. Sadkane . . . . . . . . . . . . . . . . . . .

3

. . . . Numeri al Simulation of Aeroelasti Behavior of Isolated Fan Bladerow

9

. . . .

24

R. Kielb, K. Hall, T. Miyakozawa, and E. Hong . . . . Aeroelasti Vibrations of Axial Turboma hine Bladerow V.E. Saren . . . . . . . . . . . . . . . . . . . . . . . . . Frequen y Model of Vibration for Turboma hine Diagnosti s A. Mironovs . . . . . . . . . . . . . . . . . . . . . . . . .

37

Yu. N. Shmotin, R. Yu. Starkov, P.V. Chupin, V.I. Gnesin, and L.V. Kolodyazhnaya . . . . . .

Mistuning Pattern Ee ts on Probabilisti Flutter and For ed Response

Se tion 2: Aerodynami Damping of Bladerow Vibrations Experimental and Numeri al Study of Unsteady Aerodynami s in an Os illating Low-Pressure Turbine Cas ade of Annular Se tor Shape

D. M. Vogt, H.E. Martensson, and T.H. Fransson

49 61 73

. . .

75

F. Poli, E. Gambini, A. Arnone, and C. S hipani . . . .

87

A Three-Dimensional Time-Linearized Method for Turboma hinery Blade Flutter Analysis

ix

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Vis ous Flutter Analysis of a Three-Dimensional Compressor Blade P. Petrie-Repar, A. M Ghee, and P. Ja obs . . . . . . . 98 Three-Dimensional Vis ous Flutter of Rotor Bladerow ~ adkowski, and L. Kolodyazhnaya . . . . 103 V. Gnesin, R.Rz Some Cal ulations of Unsteady Aerodynami Chara teristi s of Cas ades with Os illating Blades V. B. Kurzin and A. S. Tolstukha . . . . . . . . . . . . . 115 Experimental Study and Numeri al Simulation of Flutter Generation in Compressor Blade Cas ade V. A. Tsymbalyuk, A. P. Zinkovskii, P. Eret, and J. Linhart . . . . . . . . . . . . . . . . .

. . . . . . 128

Se tion 3: Ee ts of Rotor{Stator Intera tion

141

Unsteady Aerodynami For e on Os illating Blades Under Intera tion of Three Bladerows M. Namba, R. Nishino, and H. Nakagawa . . . . . . . . 143 Analysis of Stator Re e tion Ee t on Rotor Flutter Chara teristi s H.-D. Li and L. He . . . . . . . . . . . . . . . . . . . . . 155 Cal ulation of Turboma hine Blade For ed Os illations Caused by Vanes on the Basis of Analysis of Nonstationary Aerodynami Intera tion of Cas ades in a Stage B. F. Shorr, A. A. Osipov, G. V. Mel'nikova, and V. G. Aleksandrov . . . . . . . . . . . .

. . . . . . In uen e of Rotor Loading on the Vortex{Blade Intera tion in a High-Pressure Turbine P. Gaetani and G. Persi o . . . . . . . . . . . . . . . . Gasdynami Ee ts of Tangential Bowing of Stator Vanes in a Subsoni Stage of Axial Compressor V. E. Saren and N. M. Savin . . . . . . . . . . . . . . . Rotor{Stator Intera tions in a One and a Half Transoni Turbine Stage G. Paniagua, G. Persi o, and N. Billiard . . . . . . . x

. 169 . 180 . 201 . 215

Contents Constru tive Methods of De reasing Dynami Stresses in Rotor Blades of Turboma hines V. V. Nitusov and V. G. Gribin . . . . . . . . . . . . . . 227 Se tion 4: Flow Path Aeroa ousti s

235

Aerodynami and A ousti Response of an Annular Cas ade to Turbulen e H. M. Atassi and I. V. Vinogradov . . . . . . . . . . . . 237 Improved Hybrid Method of Predi ting Fan Tone Noise M. Namba, R. Nishino, and S. Ohgi . . . . . . . . . . . 257 Numeri al Method for Cal ulating Three-Dimensional Fan Tonal Noise Due to Rotor{Stator Intera tion M. Nyukhtikov and A. Rossikhin . . . . . . . . . . . . . 268 Mathemati al Simulation of Sho k-Wave Stru tures Arising Ahead of Fan Plane Cas ades and Wheels N. L. Efremov, A. N. Kraiko, K. S. P'yankov,

. . . . . . . . . . . 281 Two-Dimensional Numeri al Simulation of Rotor{Stator Intera tion and A ousti Wave Generation V. Aleksandrov and A. Osipov . . . . . . . . . . . . . . . 295 N. I. Tillyayeva, and Ye. A. Yakovlev

Se tion 5: Unsteady Flows in Turboma hines

305

Evaluation of Unsteady Ee ts in a Multistage Axial Compressor Using a Pre onditioned GMRES Solver M. Stridh and L.-E. Eriksson . . . . . . . . . . . . . . . 307 Numeri al Contribution to Analysis of Surge In eption and Development in Axial Compressors N. Tauveron, P. Ferrand, F. Leboeuf, N. Gourdain,

. . . . . . . . . . . . . . . . . . . . . . 340 Unsteady Flow Computation in Hydroturbines Using Euler Equations and S. Burguburu

S. G. Cherny, D. V. Chirkov, V. N. Lapin, S. V. Sharov, V. A. Skorospelov, and I. M. Pylev

. . . . . . . . . . . . 356 xi

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Investigation of Unsteady Flow in the Tip Clearan e of Axial-Compressor Stage Rotor N. N. Kovsher and K. S. Fede hkin . . . . . . . . . . . . 370 Ee t of Flow Unsteadiness on the Performan e of Airfoil Cas ades: Theoreti al Evaluation V. B. Kurzin and V. A. Yudin . . . . . . . . . . . . . . . 376 Se tion 6: Unsteady Flow Phenomena in Turboma hines

391

Forty Years of Exploring Unsteady Flow Phenomena in Centrifugal Compressors R. A. Izmailov . . . . . . . . . . . . . . . . . . . . . . . . Stability of a Low-Speed Centrifugal Compressor with Casing Treatments A. S. Hassan . . . . . . . . . . . . . . . . . . . . . . . . Propagating Sho k Waves in a Narrow Tube from the Viewpoint of Ultra Mi ro Wave Rotor Design K. Okamoto, T. Nagashima, and K. Yamagu hi . . . . . Self-Exited Os illations in Swirling-Jet Euxes D. G. Akhmetov, V. V. Nikulin, and V. M. Petrov . . . . Laser Doppler Diagnosti of Flow in Draft Tube Behind Hydroturbine Runner

393 406 421 434

V. Meledin, Yu. Anikin, G. Bakakin, V. Glavniy, S. Dvoinishnikov, D. Kulikov, I. Naumov, V. Okulov, V. Pavlov, V. Rakhmanov, O. Sadbakov, S. Ilyin, N. Mostovskiy, and I. Pylev

Author Index

xii

. . . . . . . . . . . . . . . . 446 458

F. Poli et al.

SECTION 1

AEROELASTIC ANALYSIS OF BLADEROWS

Aeroelasti Analysis of Bladerows

AN OVERVIEW OF COMPUTATIONAL TURBOMACHINERY AEROELASTICITY M. Imregun and M. Vahdati

Imperial College Me hani al Engineering Department Vibration UTC Exhibition Road London SW7 2BX, United Kingdom

Introdu tion

This paper is an attempt to overview some of the re ent turboma hinery aeroelasti ity methods as this resear h area has seen a very rapid rate of progress in the last 10 years or so. Indeed, it is now possible to ouple Navier{Stokes representations of the unsteady ow with three-dimensional (3D) nite element representations of the stru ture to undertake multipassage, multirow al ulations for both turbine and ompressor appli ations. Previously intra table phenomena su h as ore- ompressor rotating stall and surge an now be simulated on

ommodity PC lusters. There are attempts to use Large-Eddy Simulation (LES) based omputational uid dynami s (CFD) methods for turboma hinery ombustion studies. The paper is organised around a number of ase studies, all sharing the large-s ale numeri al modeling philosophy where all relevant engineering features are in luded in the numeri al al ulations. A further feature of the paper is the simulation of one-o ases that are of great design interest. Su h ases in lude fan blade bird strike and ompressor vane blo kage. The phenomenon of ompressor aeroelasti ity is dis ussed in some detail with several illustrative examples. The importan e of \ exible" boundary onditions is dis ussed in some detail and a new strategy, based on the use of downstream variable-area nozzles, has been proposed. In terms of ompressor aeroelasti ity, of parti ular interest are M. Imregun and M. Vahdati

3

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

the phenomena of rotating stall and surge, for whi h predi tive models are beginning to emerge. Indeed, resear h has shown the possibility of blade utter during a surge event, highlighting the diÆ ulties of designing optimally-spa ed bladerows. Current 3D methods to predi t blade passing and low engine-order for ed response phenomena are surveyed next, again with some illustrative examples. Su h te hniques usually involve multipassage, multirow models and they also in lude features su h as blade exibility and blade root fri tion dampers. The results indi ate that it is now possible to predi t the for ed vibration levels with reasonable a

ura y. Although the area of low engine-order ex itation is still poorly understood, results indi ate that the investigative tools are now in pla e to study the ee t of various ow nonuniformities arising from uneven stator blade spa ing, ow exit angle, blo ked burners, et . Finally, the omputational aspe ts of 3D aeroelasti ity methods are dis ussed brie y with a view to des ribe urrent status and future requirements. Case Study 1. Compressor For ed Response with Blo ked Vane

It is well know that a loss of symmetry will give rise to low engineorder ex itation harmoni s. A typi al ase is the blade throat width variation due to manufa turing toleran es. Also, in the ase of variable inlet guide vanes, it may not be possible to ontrol all stator blades to the same level of a

ura y, thus reating an angle variation along the ir umferen e. Still worse, under extreme ir umstan es, e.g., due to a malfun tion of the ontrol me hanism, a blade might blo k a stator passage, thus reating signi ant ex itation for both upstream and downstream bladerows. Let us now attempt to assess the additional ex itation due to su h a situation. The analysis was ondu ted for a gradual blo kage by rotating one of the stator blades by 10Æ , 20Æ , and 25Æ , an approa h that allows to obtain the rotor response levels as a fun tion of the blo kage angle. The steady-state Ma h number ontours at the stator exit for the 25Æ blo kage ase are plotted in Fig. 1. It is seen that the ee ts of the blo kage on the ow are on ned to the blo ked passage and to its immediate neighbors. The whirl angle variations arising from su h a ow 4

M. Imregun and M. Vahdati

Aeroelasti Analysis of Bladerows

Figure 1

Figure 2

Steady-state Ma h number ontours at S2 exit

The Fourier transform of the whirl angle at S2 exit for all three

blo kage ases:

1

Æ

| 10 ,

2

Æ

| 20 , and

3

Æ

| 25 ;

4

| refers to datum

will ause low engine-order harmoni s whi h will ex ite the vibration modes of the rotor bladerow. The Fourier transform of the whirl angle at stator exit is plotted in Fig. 2 for all three blo kage ases. Although the blo kage has a relatively small ee t on the main 32EO harmoni , M. Imregun and M. Vahdati

5

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 3 Three-bladerow model for unsteady ow analysis the amplitude of the low engine-order harmoni s in reases sharply as the amount of blo kage in reases. The unsteady ow analysis was performed with a three-bladerow model shown in Fig. 3. Case Study 2. Use of Atmospheri Boundary Conditions for Axial-Flow Core-Compressor Steady-State Flow Simulations

The performan e of an axial ow ompressor, either a fan assembly or a ore ompressor, is often summarised in the form of a pressure rise vs. mass ow hara teristi urve, representing nominally steady and axisymmetri ow operation. At a given shaft speed, the operating zone is bounded by the blades hoking at high mass ow/low pressure, or the blades stalling at low mass ow/high pressure. Sin e the avoidan e of stall is a major design onsideration, a onsiderable amount of resear h eort has been devoted to understanding the physi al me hanisms that give rise to stall. So far, due to modeling diÆ ulties, mu h of the stall resear h has been experimental and, in spite of dramati advan es in hardware and software, the simulation 6

M. Imregun and M. Vahdati

Aeroelasti Analysis of Bladerows

of turboma hinery ows near stall is still fraught with diÆ ulties. A major hurdle is the spe i ation of appropriate inlet and outlet boundary onditions that must be imposed from the outset. However, when the ow onditions are uniform at the boundaries, the ow is stable at lower working lines but numeri al diÆ ulties o

ur at higher working lines. It is well known that rigid boundary onditions, based on imposing given exit pressure distributions, are not suitable for studies near stall. For instan e, in the ase of rotating stall, the downstream exit pressure pro les are neither known nor onstant in time. Similarly, at high working lines, the ow be omes genuinely. More a

urate boundary onditions an be imposed by introdu ing a downstream variable nozzle, thus allowing the pressure behind the fan to adjust automati ally while the pressure behind the nozzle is xed. Su h an approa h makes the omputational domain \less sti" and provides a powerful natural boundary ondition for stall studies. Moreover, sin e the aim is to simulate, as mu h as possible, engine and rig tests, nozzle area hanges an be used to move to any point on the

ompressor hara teristi . Two sets of al ulations, with two dierent boundary ondition strategies, were performed along the 70 per ent speed hara teristi of a large aeroengine fan. The domain for Strategy 1 ex ludes the intake and the variable nozzle of Fig. 2 and may be viewed as a standard 3D single-passage al ulation for this type of blade. The omputational Table 1 Comparison of Strategies 1 and 2 Inlet boundary

ondition Outlet boundary ondition Point ontrol Cal ulation type

Strategy 1 Without nozzle or intake Corre ted atmospheri total pressure 2 ow angles Total temperature Radially- onstant stati pressure Change ba k pressure Steady-state

M. Imregun and M. Vahdati

Strategy 2 With nozzle and intake Atmospheri total pressure Atmospheri total temperature Axial ow Atmospheri stati pressure Change nozzle area Steady-state Time-a

urate at stall

7

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 4

Computational domains for Strategies 1 and 2

domain for Strategy 2 in ludes the upstream and downstream extensions, namely, a symmetri intake with a spinner and a variable nozzle. As listed in Table 1, for Strategy 1, orre ted atmospheri pressure and temperature are imposed at the inlet to

ompensate for the intake losses. More signi antly, a radially onstant stati pressure, whose Figure 5 Compressor hara ter- values determine the a tual isti at 70 per ent speed predi tions point on the hara teristi , is using stati pressure boundary ondiused at the exit (Figs. 4 and 5). tions (1 ); variable nozzle boundary onOn the other hand, for Stratditions (2 ); measured data (3 ), and egy 2, atmospheri total presstall hysteresis loop (4 ) sure, and temperature are imposed at the intake inlet, while atmospheri stati pressure is imposed at the nozzle exit. A riti al dieren e between the two strategies is that both the inlet and outlet boundary onditions remain the same for all the points on the hara teristi for the latter. 8

M. Imregun and M. Vahdati

Aeroelasti Analysis of Bladerows APLLICATION OF A NEW MATHEMATICAL TOOL \ONE-DIMENSIONAL SPECTRAL PORTRAITS OF MATRICES" TO THE PROBLEM OF AEROELASTIC VIBRATIONS OF TURBINE-BLADE CASCADES S. K. Godunov, V. B. Kurzin, V. G. Bunkov, and M. Sadkane

M. A. Lavrentyev Institute of Hydrodynami s Siberian Bran h of the Russian A ademy of S ien es Lavrentyev Ave. 15 Novosibirsk 630090, Russia

Introdu tion

Design of integral stru tures is always based on the omputations simulating operation of these stru tures under ertain onditions. Su h

onditions should be des ribed by the limitations providing operational eÆ ien y and safety and avoiding stru tural failure. Admissible errors should be given for the numeri al values of the limiting parameters, whi h guarantee reliability of re ommendations based on the al ulations. In the aeroelasti ity theory, the omputational pro edures dealing with spe tral analyses of matri es are of great importan e. It is well known that the stability riterion is redu ed to the following statement: all eigenvalues of de nite matri es obtained during modeling lie stri tly in the left part of the omplex plane. A natural question arises: What a

ura y is required for omputing these eigenvalues? The examples indi ating that there is no answer to this question are presented below. The formulation of the question should be modi ed a

ording to the lassi al Lyapunov theory. Based on this theory and on its modern generalizations, an algorithm using the spe tral di hotomy S. K. Godunov et al.

9

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

riteria and one-dimensional (1D) spe tral portraits of matri es illustrating spe trum bration is suggested.

Analysis Naturally, the omputational pro edures used should meet the following requirement: their results must satisfy the guaranteed a

ura y estimates. This requirement an be met if standard algorithms in luded into publi -domain software used in engineering omputations are based on the following natural postulate Postulate: Only those numeri al fun tions f (A) of N N or N M matri es an be al ulated, whi h satisfy the following inequality:

k f (A)

f (B ) k ! k A

Bk

Here ! = !(k A k; f (A)) is a known fun tion independent of N and M (matrix dimensions); k A k and k A B k are the matrix norms. Example of admissible fun tions: j (A) are the singular values of matrix A:

A = QDP 

Q Q = IN

19 > CC> > > 0 C > C = C .. .


C N C CC> .. > . M C > >








A> ; 0 0 0 {z }

0 BBM 0 BB 0 M 1 B D=B BB





BB 0 0  |




0

P  P = IM .. . .. .

M 0 of Lyapunov's matrix equation HA + A H + C = 0 to exist for all positive de nite C = C  > 0. Lyapunov's fun tion (Hx; x) diminishes with in reasing t on the solutions of x_ = Ax; hen e

p

k x(t) k k H kk H 1 k k x(0) k p one ompute the oeÆ ient k H kk H 1 k

(the ratio How an of the maximal and minimal axes of Lyapunov's ellipsoid where the traje tory x(t) lies)? This oeÆ ient depends not only on matrix A but also on the hoi e of the right part of C in Lyapunov's matrix equation. Therefore, the stability analysis should in lude not only the solvability of the equation HA + A H + C = 0, but also introdu e some parti ular C providing an a

eptable value of k H kk H 1 k. Based on the arguments mentioned [1℄, it was suggested to nd H from the equation HA + A H + 2 k A k I = 0 and to take as the

hara teristi of the stability quality the value of  =k H k that ensures the validity of the following estimate:

p

k x(t) k  pe tkAk= k x(0) k= pe

t=

 = kAk

k x(0) k

Here  = (A) is the solution of the extremal problem

8 R1 k x(t) k dt >< (A) = sup R1 x > : exp( 2t k A k) k x(0) k 2

0

(0)

2

0

9 >= > dt ;

The parameter  = = k A k is the hara teristi time of solution de ay. The inequality j(A + B ) (A) < 133(A) k B k = k A k (it holds 2 if k B k = k A k< 10  ) shows that  = (A) is stable with respe t to perturbations of the matrix onsidered in terms of the postulate formulated above. It turned out [2, 3℄ that H = H (A) an be represented as a matrix integral

Z+1 k Ak H (A) = [A + i!I ℄ 

S. K. Godunov et al.

1

1

[A

i!I ℄

1

d! 13

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

whi h makes sense not only for Gourwitz matrixes with the spe trum lo ated stri tly in the left half-plane. For the onvergen e of this integral, only the absen e of eigenvalues at the imaginary axis is ne essary. The quantity  =k H k an be treated as a riterion of spe tral di hotomy with respe t to the imaginary axis, a riterion estimating the distan e of j (A) from this axis, regardless of the number of eigenvalues in the left half-plane and those in the right half-plane. The graph of the dependen e of the di hotomy riterion (A aI) on a illustrates spe trum bration by straight lines Re() = a parallel to the imaginary axis. The examples illustrating the use of su h graphs (1D spe tral portraits) in some simple problems of aerodynami s are given below. The rst example illustrating the use of the di hotomy riterion is a simple atter model proposed by TsAGI [4℄. In [4℄, the plate-airfoil is onsidered as a system with four degrees of freedom. When ignoring the aerodynami ee ts, the vibrations of the plate are des ribed by the following equations: dx = Gy dt dy =x dt 0 37:7

B

G=B 

0

169

0 899

1 CC A

1792

Aerodynami ee ts are modeled by adding new elements depending on the ow velo ity v to the oeÆ ients of the system. The system a quires the following form: dx = vDx dt dy =x dt 14

(G + v2 F )y

S. K. Godunov et al.

Aeroelasti Analysis of Bladerows

Figure 1 Fibration of spe trum with lines parallel to the imaginary axis

for dierent velo ities: (a ) v = 395 m/s and (b ) v = 411 m/s. ( ) Spe tral zones

G = 0:73
10

01 B 2B 

0 0 B0:12
10 F =B  0 0

0

3

1

0 1

1

1 CC A

0:197
10 0 0:176
10 0:154
10

2 3 3

0 0:419
10 0 0

2

0 0:171
10 0 0

1 3C CA

Fibration of the spe trum of this system with lines parallel to the imaginary axis for dierent velo ities v is shown in Figs. 1a and 1b by the solid urve 1 ; the dashed urve 2 shows  = = k A k. S. K. Godunov et al.

15

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

It is onvenient to superimpose the spe tral zones in one gure with Re() as the abs issa axis and the velo ity v as the ordinate. The shaded area in Fig. 1 (internal area bounded by the p dot-anddashed urve) is the domain pof Re() values su h that lg   3:95; the external area refers to lg   3:75. The middle area bounded by the white urve is the domain where  = = k A k  3:75. The graphs presented allow one to evaluate the riti al utter velo ity obtained from the omputed ratio jRe(j )j= k A k and from the proposed riterion (A). The admissible error for (A) should be hosen by analyzing the a

ura y of modeling the phenomenon by dierential equations and validated against experimental data. Subspa es orresponding to lusters of eigenvalues (proje tions onto them or their bases) are omputed simultaneously with the spe tral portraits. This allows one to indi ate the ell-diagonal anoni al form of the matrix examined and, by omputing the similar transform matrix, nd its ondition number. The anoni al form of matrix A (v = 411) is

0 3 67 BB 1 17 BB 0 B 0 =B BB 0 BB 0  0 :

:

Q

1

AQ

0

101 1:32 0 0 0 0 0 0

0 0 0 0 430 393 476 433 0:785 0:711 7:75 7:01 0 0 0 0

0 0 26:4 29:2 179 1770 0 0

0 0 2:68 2:96 18:8 176 0 0

0 0 0 0 0 0 0:673 1:21

0 0 0 0 0 0 97:7 1:67

1 C C C C C C C C C C A

k Q kk Q 1 k = 267:0132 As an example for test omputations of spe tral portraits, let us

onsider oupled bending-torsion vibrations of blades in the as ades of turboma hinery in a gas ow. The system of dierential equations that des ribes small vibrations of su h a as ade has the form [5℄ mn hn + Sn an + Knh hn = Ln + Fn Sn hn + Jn an + Kna an = Mn (n = 1; 2; : : : ; N )

16

S. K. Godunov et al.

Aeroelasti Analysis of Bladerows where hn and an are the generalized oordinates of blade deformation owing to bending-torsion vibrations, mn and Jn are the generalized masses and moments of inertia of the blades, Knh and Kna are the

oeÆ ients of generalized bending and torsion rigidity, Sn are the oef ients of generalized oupling of bending and torsion vibrations, N is the number of blades in the as ade, Fn are the elasti oupling for es of blades with ea h other, and Ln and Mn are the generalized aerodynami for es and moments a ting on the nth blade. In the theory of

as ades in an unsteady ow [6℄, the latter quantities an be presented as Ln = q

X N

r

Mn = q

=1

X N

r

hr + lr00 n;h b

lr0 m0

r

=1

h_ r + lr0 n;h !b

hr + m00r n;h b

n;a

h_ r + m0r n;h !b

ar + lr00

n;a

a_ r n;a !

a + m00 r

r

!

a_ r n;a !

!

where q is the free-stream dynami pressure,  and b are the surfa e area and hord of the blade, lr n and mr n are the aerodynami oeÆ ients of the blade ee t, whi h are fun tions of the Strouhal number k = !b=V , V is the free-stream velo ity, ! = Im , and  is the root of the

orresponding hara teristi equation of this system. Introdu e the notation 2 = !nh

kn h ; mn v=

2 = !na 2 !na 2 ; !nh

kn a ; Jn

"n =



n =

q ; mn b!2

Sn ; mn b hn =

2n =

Jn mn b

hn b

and note that the estimate "n 1 is valid be ause the unsteady aerodynami for es a ting on the blade are mu h smaller than elasti for es and for es of inertia. With this notation, the system takes the form:  + a + K h = " !2 L + h n n n nh n n n  +  (a + v!2 a ) = " !2 M

n h n n n n n nh n (n = 1; 2; : : : ; N ) S. K. Godunov et al.

1

mn

Fn

17

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

where

X N

Ln =

r

=1

X N

Mn =

r

=1

h lr0 n;h r + lr00 b m0

r

h_ r + l0 n;h !b r

hr + m00r n;h b

!

a_ a + lr00 n;a r n;a r !

h_ r + m0r n;h !b

n;a

a + m00 r

r

a_ r n;a !

!

Note that the matrix orresponding to the left-hand side of the system is a Hamiltonian, and the matrix orresponding to the right-hand side

an be onsidered as a perturbing omponent. As the initial parameters required for solving the system, one an use the values of !nh , n , n , v, "n , k, and aerodynami oeÆ ients whose values are summarized in [7℄ as fun tions of as ade geometry and ow parameters (within the framework of the ideal uid model). Figures 2 to 6 show the 1D spe tral portraits of matri es in a system that des ribes vibrations of a as ade of thin blades at = 0 ( as ade density  = 1:5; eje tion angle = 30Æ ; exure of the mid-line of the blade normalized to its hord f = 0:025; Strouhal number k = 0:5; and number of blades in the period N = 10) in a ow of an ideal in ompressible uid. The symbol  is the riterion of di hotomy of the spe trum of matri es by on entri ir les with the enter at the origin

Figure 2

Di hotony riteria



(a ) and

Xa

(b ) for the as ade with blades

possessing identi al inertial and elasti parametrs: and

18

" = 0:01

=

0:3,

v

= 2,



= 1,

S. K. Godunov et al.

Aeroelasti Analysis of Bladerows

Figure 3

Di hotony parameters



(a ) and

blades have the following hara teristi s:

Figure 4

Xn (b ) for the as ade whose v = 1,  = 1, and " = 0:01

= 0,

Di hotony quality for almost the same as ade as that in Fig. 3

but with additional allowan e for elasti oupling of blades with ea h other

of the omplex plane of eigenvalues, depending on the ir le radius, and the symbol Xa denotes the quality of di hotomy by lines parallel to the imaginary axis from the oordinate a of their interse tion with the real axis. The values of R and a for whi h  and Xa are almost in nite determine the absolute values and real parts of eigenvalues, more exa tly, those intervals that are onsidered as reliable on the basis of the omputations performed. Figure 2 shows the di hotomy riteria  (Fig. 2a ) and Xa (Fig. 2b ) for the abovementioned as ade with blades possessing identi al inertial and elasti parameters equal to S. K. Godunov et al.

19

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 5

Di hotony quality of the matrix with the parameters shown in

Fig. 2

=

0:3; v = 2;

 = 1;

" = 0:01

Note, for these parameters of the as ade, the Hamiltonian omponent of the matrix has essentially dierent eigenvalues (Fig. 2a ); therefore, the real values of the total matrix that des ribes as ade vibrations with allowan e for aerodynami intera tion (Fig. 2b ) an be fairly a

urately determined by the perturbation method. Figure 3 shows the di hotomy parameters  and Xn for the as ade whose blades have the following hara teristi s:

= 0;

v = 1;

 = 1;

" = 0:01

In this ase, the absolute values of matrix eigenvalues almost oin ide with ea h other (Fig. 3a ); hen e, the perturbation method annot be 20

S. K. Godunov et al.

Aeroelasti Analysis of Bladerows

Figure 6 In uen e of a small perturbation of the Hamiltonian omponent of the matrix on matrix stability

used to determine the real parts of the eigenvalues. The urve of the di hotomy quality

Xa

hara terizes the positions of these values with

guaranteed a

ura y (Fig. 3b ). For this ombination of blade parameters, several eigenvalues of the matrix are lo ated in the right half-plane, i.e., the orresponding matrix is unstable. Figure 4 shows the di hotomy quality for almost the same as ade as that in Fig. 3 but with additional allowan e for elasti oupling of blades with ea h other. The di hotomy parameter

= 0:2!n2



for the orresponding matrix with

is plotted in Fig. 4a, whi h shows that this matrix onsists

of Jordan ells of dimension 2 and 4. A

ording to available knowledge, su h a matrix should be more sensitive to perturbations, whi h is eviden ed by its di hotomy quality S. K. Godunov et al.

Xa

(Fig. 4b ).

21

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

One advantage of the spe tral portraits of matri es is the possibility of redu ing the analysis of stability of high-order matri es in some ases to the analysis of stability of their submatri es of lower order. The

riterion of existen e of this possibility is the presen e of lusters | numerous eigenvalues lo ated lose to ea h other, whi h are separated by a signi ant distan e in the omplex plane if the di hotomy quality is good. As an example, onsider a matrix whose di hotomy parameters are shown in Fig. 2. It follows from the di hotomy quality of this matrix  with allowan e for elasti oupling Fn (Fig. 5a ) that the riterion indi ated above is satis ed in the ase onsidered. A omparison of the di hotomy quality Xa for the total matrix (Fig. 5b ) with similar dependen es for the orresponding submatri es (Figs. 5 and 5d ) supports this statement. The in uen e of a small perturbation of the Hamiltonian omponent of the matrix on stability of the latter is illustrated in Fig. 6. As an example, onsider the matrix whose spe tral portrait is shown in Fig. 3. Its perturbing omponent des ribes the a tion of blade oupling for es of the form Fn = ( 1)n (hn+1 + hn 1 2hn) Figure 6a shows the spe trum di hotomy by radial ir les. Figure 6b illustrates the portrait of the same spe trum by di hotomy by straight lines parallel to the imaginary axis; some part of the spe trum is seen to lie in the right half-plane. After detuning, the entire spe trum is lo ated in the left half-plane, whi h is illustrated in Fig. 6 . As the detuning parameter in reases, the spe trum is shifted to the left (Fig. 6d ). This example illustrates the known fa t of the in uen e of small geometri inhomogeneity of as ades on stability of their vibrations.

Con luding Remarks The methods of stability analysis des ribed in this paper are implemented with the use of simple iterative algorithms proposed and des ribed in [1, 8, 9℄. The algorithms solve Lyapunov's matrix equations and their generalizations to the ase of spe trum di hotomy. As far as the present authors are aware, these generalizations appeared in the 22

S. K. Godunov et al.

Aeroelasti Analysis of Bladerows

book [10℄. Unfortunately, we did not understand its ontent. The latter retarded justi ation, whi h re eived less attention than onstru ting of omputational s hemes.

A knowledgments The authors are grateful to A. A. Saitgalin for illustrative omputations. This work was supported by the Integration proje t No. 5 of SB RAS.

Referen es 1. Bulgakov, A. Ya. 1980. Ee tively omputed quality parameter of stability of a system of linear dierential equations with onstant oeÆ ients. Sib. Mat. Zh. 21(3):32{41. 2. Bulgakov, A. Ya., and S. K. Godunov. 1988. Cir ular di hotomy of a matrix spe trum. Sib. Mat. Zh. 29(5):59{70. 3. Godunov, S. K. 2002. Le tures on advan ed aspe ts of linear algebra. Novosibirsk: Nau hnaya Kniga. 4. Bun'kov, V. G., and V. A. Mosunov. 1988. Appli ation of Lyapunov's a tion integral for estimating stability of a linear system. U h. Zap. TsAGI 19(2). 5. Bendiksen, O., and P. Friedmann. 1980. Coupled bending-torsion utter in as ade. AIAA J. 18(2):194{201. 6. Gorelov, D. N., V. B. Kurzin, and V. E. Saren. 1971. Aerodynami s of

as ades in an unsteady ow. Novosibirsk: Nauka. 7. Gorelov, D. N., V. B. Kurzin, and V. E. Saren. 1974. Atlas of unsteady aerodynami as ades. Novosibirsk: Nauka. 8. Malyshev, A. N. 1990. Guaranteed a

ura y in spe tral problems of linear algebra. Colle ted Papers of the Institute of Mathemati s (SB RAS) 17:19{104. 9. Godunov, S. K., and M. Sadkane. 2003. Some new algorithms for the spe tral di hotomy methods. Linear Algebra. 10. Daletskii, Yu. A., and M. G. Krein. 1970. Stability of solutions of dierential equations in the Bana h spa e. Mos ow: Nauka, Fizmatgiz.

S. K. Godunov et al.

23

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

NUMERICAL SIMULATION OF AEROELASTIC BEHAVIOR OF ISOLATED FAN BLADEROW 





Yu. N. Shmotin , R. Yu. Starkov , P. V. Chupin , V. I.

y Gnesin ,

and L. V.

y Kolodyazhnaya

 JSC

\NPO "SATURN" Rybinsk, Russia y IPMa h National A ademy of S ien es Kharkov, Ukraine

Introdu tion

The tenden y of in reasing the eÆ ien y of gas-turbine engines implies the ne essity of designing wide- hord unshrouded fan blades of omplex spatial aerodynami shape. However, with su h blades, the risk of aeroelasti instability (e.g., utter) development in reases. For reliable and safe operation of gas-turbine engines with su h fans, it is ne essary to be able to a

urately predi t the aeroelasti behavior of fan blades at the design stage. The onventional approa h to the utter al ulations of the bladed disks is based on the frequen y analysis method assuming that blade motions are des ribed by harmoni time-dependent fun tions with a

onstant phase shift between the adja ent blades, whereas the natural modes and frequen ies of the \rotor{ ow" system remain similar to those in va uum. This method ignores the \feedba k" ee t of the gas

ow on blade os illations. This paper presents the solution methodology of the unsteady multidimensional aerodynami problem oupled with the elasti blade os illation problem in the gas ow. Also presented are the results of

al ulations for aeroelasti hara teristi s of a fan bladerow (BR) of a modern bypass gas-turbine engine. 24

Yu. N. Shmotin et al.

Aeroelasti Analysis of Bladerows

Aeroelasti Model On the one hand, when onsidering the gas ow BR with os illating blades, one has to take into a

ount that the gasdynami parameters depend on the blade velo ity and position determined by a spe i ed os illation law (kinemati ally for ed os illations). On the other hand, when onsidering the problem of for ed os illations, the blade velo ity and positions depend on the aerodynami loads for ing the blades. The prin ipal drawba k of solving these two problems independently is the ignoran e of interferen e between blade motions and aerodynami loads, i.e., aerodamping hara teristi s of the \BR { gas ow" system driven by energy ex hange between the main (averaged) ow and os illating blades. The energy ex hange an manifest itself by energy \swapping" from the gas ow to the moving blade (self-ex ited os illations or utter) or by dissipation of the os illating blade energy in gas

ow (aerodamping), and therefore, it appears to be the most important hara teristi s of aeroelasti stability (instability) of the \BR { gas

ow" system. The adequate a

ounting of energy ex hange an be attained by solving a oupled aerodynami ow { aeroelasti os illations problem, based on simultaneous integration, using the time-mar hing s heme, of the omplete system of equations of unsteady aerodynami s and aeroelasti blade os illations. The three-dimensional transoni ow of invis id gas with zero thermal ondu tivity through the axial turboma hine is onsidered in the physi al domain, in luding a fan BR, rotating at a onstant angular velo ity. The ow is des ribed by the omplete system of unsteady Euler equations presented in the integral form [1℄:  t

where

Z

fd



+

2 3  66v1 7 7 f =6 ; 64v2 7 7 5 v3 E

Yu. N. Shmotin et al.

I

~~ F n d



~ F

+

Z

Hd



=0

(1)

2 3 ~ v 6 v1~ v + Æ1i p7 6 7 6 = 6v2~v + Æ2i p77 ; 4 5

+ Æ3i p ( + p)~v

v3~ v E

25

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

2 3 0 66 a 1 2!v2 77 H=6 64a 2 + 2! 1 775 ; 0

(

e

Æji

e

= 10;; jj ==6 ii

0

is the volume of the nite-volume omputational ell with moving boundaries ; ~n is the unit exterior normal; p and  are the pressure and density, respe tively; ! is the angular rotation velo ity; v1, v2, and v3 are the omponents of the velo ity ve tor ~ v ; a 1 and a 2 are the proje tions of translational a

eleration; E = (" + (v12 + v22 + v32 r2 !2 )=2) is the total energy per unit volume; " is the internal energy per unit mass; and r is the distan e to the rotation axis. The system of Eqs. (1) is supplemented with the ideal gas equation of state: 1 p "= k 1  where k is the spe i heat ratio. Equations (1) are integrated in the omputational domain, in luding the omplete ir umferen e, i.e., the number of interblade hannels is equal to the number of blades. The number of interblade hannels, N , and interblade phase angle Æ are related to ea h other as N Æ = 2j (where j is the integer number). The omputational grid is divided into z segments, either of whi h in ludes one blade and possesses the ir umferential length equal to a rotor pit h. Ea h segment is dis retized using the hybrid H{H grid for rotor hannels. The external H-grid remains xed, while the internal Hgrid is re onstru ted at every iteration in a

ordan e with the position of the os illating blade. The dis rete form of Eqs. (1), obtained for an arbitrary spatially deforming grid, is [2℄: 1 hf +1 2 +1 2 +1 2 +1 2 +1 2 +1 2
t i h (f w ) +1 f +1 2 +1 2 +1 2 +1 2 +1 2 +1 2 + i + (f w ) (f w ) +1 + (f w ) (f w ) +1 + (f w ) e

i

= ;j

i

= ;k

= ;j

n i

26

=

= ;k

i

=

n j

= ;j

i

= ;k

= ;j

e

=

= ;k

n j

n i

=

n k

n k

Yu. N. Shmotin et al.

Aeroelasti Analysis of Bladerows

+ [(F1 ) +1 (F1) + (F2 ) +1 (F2 ) + (F3 ) +1 (F3 ) ℄ + H +1 2 +1 2 +1 2 +1 2 +1 2 +1 2 = 0 i

i

j

i

j

= ;j

= ;k

=

k

i

= ;j

k

= ;k

=

where subs ripts and supers ripts stand for \new" and \old" ells; f = f; ~v; E g is the symboli ve tor of unknown variables; F1, F2 , and F3 are the values of variables in the enters of ell fa es;  and w are the fa e area and normal velo ity in the fa e enter, respe tively. The gasdynami parameters on the fa es are determined from the solution of Riemann problem with the use of iterative pro ess [3℄. It is assumed that nonstationary ee ts in BR are ex ited by fan rotation in the nonuniform ow as well as by blade os illations under the in uen e of nonstationary aerodynami loads, whereas the ow at in nity ahead of and behind the BR is uniform with small perturbations, spreading from the omputational domain both upstream and downstream. Therefore, the boundary onditions at the open boundaries are based on the one-dimensional hara teristi s theory. In general, when the axial velo ity is subsoni , the omplete set of boundary onditions an be written as: n

T0 = T0 (x; y) ; p0 = p0(x; y) ;  = (x; y)   2a = 0

= (x; y) ; d v3  1

at the BR inlet, and p = p(x; y); dp a2 dp = 0 ; dv1




!2 r 2!v2 dt = 0   2 a dv2 + 2!v1 dt = 0 ; d v3 + =0  1

at the BR outlet. Here, T0 and p0 are the total temperature and total pressure in the laboratory frame of referen e;  and are the ow angles in tangential and meridian se tions ahead of BR; and p is the stati pressure behind the BR. The dynami model of the os illating blade in linear formulation is des ribed by the matrix equation: [M ℄fu(x; t)g + [C ℄fu_ (x; t)g + [K ℄fu(x; t)g = [F ℄ Yu. N. Shmotin et al.

(2) 27

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

where [M ℄, [C℄, and [K℄ are mass, damping, and stiness matri es, respe tively; fu(x; t)g is the blade displa ement; and [F ℄ is the ve tor of unsteady aerodynami loads. Using the modal approa h: fu(x; t)g = [U (x)℄fq(t)g =

N X i=1

fUi (x)gqi(t)

where Ui (x) is the blade displa ement ve tor in the ith mode, and qi (t) is the modal oeÆ ient. Based on the ondition of orthogonality of natural modes, Eq. (2) an be transformed to the system of independent dierential equations with respe t to the modal oeÆ ients of natural modes: qi(t) + 2hi q_i(t) + !i2 qi(t) = i (t)

(3)

Here, hi is the damping oeÆ ient of the ith mode; !i is the natural frequen y of the ith mode; i is the modal for e orresponding to the displa ement within the ith mode, al ulated at every iteration based on the instantaneous pressure distribution on the blade surfa e RR 

pU i nÆ d

i = RRR v

2

U i dv

where p is the pressure on the blade surfa e. The modal for e i an be interpreted as a generalized for e, a ting on the blade in the ourse of displa ement within the ith mode and related to the unit mass. Having de ned the modal oeÆ ients qi from the system of dierential Eqs. (3), the blade displa ement and velo ity an be obtained as u(x; t) = u(x; _ t) = 28

X i

X i

Ui (x)qi(t) Ui (x)q_i(t) Yu. N. Shmotin et al.

Aeroelasti Analysis of Bladerows

Numeri al Modeling of Aeroelasti Behavior The numeri al simulation of aeroelasti hara teristi s was performed for a fan, onsisting of 24 blades (Fig. 1) for the regime with n = 0:7 at the operation line. The rst ve natural modes and frequen ies of blade os illation are shown in Table 1 (all natural frequen ies are related to the rst natural frequen y). The al ulations were performed using 10  25  58 (radial  tangential  axial)

omputational grid for ea h interblade hannel. At the rst stage, the aerodynami al ulations for a rotating fan BR with a preset blade os illation law were made. All blades were assumed to exhibit harmoni os illations a

ording to ea h of the natural modes following the same law with a onstant interblade phase angle (IBPA) Æ : Figure 1 Fan bladerow qij = qi0 sin [2it + (j 1)Æ ℄ where qij is the modal oeÆ ient; i is the natural mode number; j is the blade number; qi0 is the os illation amplitude of the ith mode; and i is the natural frequen y. The aerodynami al ulation was ontinued until the ow with a periodi unsteadiness with the frequen y equal to the blade os illation frequen y was established. The onvergen e was ontrolled by omparing the unsteady pressure oeÆ ient along the blade pro le, al ulated for two time instants separated by the os illation period. The error in the unsteady pressure oeÆ ient did not ex eed 0.1%. The al ulations were performed for harmoni os illations of blades with IBPA = 0Æ , 180Æ , 90Æ , and 45Æ with regard for the intera tion between rst ve modes. Yu. N. Shmotin et al.

29

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Table 1

Natural modes and frequen ies (normalized)

Aeroelasti stability of the \air ow{BR" system without regard for me hani al damping was determined by the aerodynami damping

oeÆ ient D:

D= W =

Z1=Zl 0

(F v + M!) dt dl

0

where W is the work oeÆ ient, F is the aerodynami for e ve tor, M is the aerodynami moment; l is the pro le perimeter; v is the translational velo ity ve tor; and ! is the angular velo ity ve tor. When the dire tions of for e (moment) oin ide with the pro le displa ement (rotation), work oeÆ ient W is positive (W > 0; D < 0). In this ase, the main ow energy is transferred to the os illating blade. If the dire tions of for e (moment) are inverse to blade displa ement (rotation), the work oeÆ ient W is negative (W < 0; D > 0). In this

ase, the energy of the os illating blade is transferred to the main ow. The hara ter of energy ex hange between the air ow and the os illating blade along the blade length with dierent IBPA values is shown in Fig. 2a, representing variation of the aerodamping oeÆ ient along 30

Yu. N. Shmotin et al.

Aeroelasti Analysis of Bladerows

Figure 2 Variation of aerodamping oeÆ ient along the blade height for dierent IBPA (1{5 modes) (a ) and the ee t of IBPA on the aerodamping

oeÆ ient averaged over blade height (b )

the blade height. The IBPA values are presented by numbers in Fig. 2a. The in rease of the aerodamping oeÆ ient (stability enhan ement) towards the blade periphery part is typi al. Aerodamping is determined by the phase shift for unsteady pressure and blade motion, whi h, in turn, is de ned by the phase shift for the adja ent blades. Figure 2b shows the ee t of IBPA on the aerodamping oeÆ ient for the preset os illation laws. As is seen from the graph, the aerodamping oeÆ ient urve (averaged over the blade height) exhibits the

hara teristi sinusoidal form. The maximal and minimal aerodamping o

urs at Æ = 180Æ and Æ = +90Æ , respe tively. Yu. N. Shmotin et al.

31

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

The sign of the aerodamping oeÆ ient, al ulated for the ase of preset harmoni blade os illations, an be used only for preliminary

on lusions on the in ipien e riterion of self-ex ited os illations. The nal estimate of aeroelasti behavior of a BR an be obtained only based on the oupled solution of aerodynami and aeroelasti problems given by Eqs. (1) and (3), respe tively. In this ase, blade response will depend not only on \the kinemati history," but also on the mass ow rate, blade mass, and blade natural frequen ies. Figure 3 shows the time histories of the modal oeÆ ients orresponding to ea h of ve modes at IBPA = +90Æ . The time interval 0  t  0:02534 s orresponds to harmoni os illations. Starting from t = 0:02534 s, the harmoni os illations are terminated and further blade motions o

ur under the ee t of aerodynami , aeroelasti , inertial for es with due regard for the intera tion of all ve modes. As is seen from the graphs, the 2nd to 5th modes de ay very qui kly, whereas blade os illations in the 1st mode de ay mu h slower. Similar al ulations were performed for transient pro esses with IBPA 45Æ , 90Æ , and 180Æ. All natural modes were also shown to de ay due to aerodamping. The maximal and minimal de rements of os illation damping were dete ted for the antiphase os illations (IBPA = 180Æ ) and for IBPA = +90Æ , respe tively. Figure 4 shows the peripheral blade se tion displa ements in tangential (hy ) and axial (hz ) dire tions, and rotation with respe t to the se tion gravity enter (') in the oupled os illations with IBPA = 90Æ . As is evident from Fig. 4, harmoni os illations (0  t  0:02534 s) in lude all natural modes. The maximum deviation amplitudes from the stati de e tion are equal to 1.5 mm in the ir umferential dire tion, 2.0 mm in the axial dire tion, and 1:2Æ in angulation. The 1st mode has the smallest bending os illation de rement for damping of harmoni vibrations (t  0:02534 s), while the 1st and the 5th modes | for torsional os illations. By the time instant t  0:15204 s, the peripheral se tion position is approa hing the stati deformation position: the ir umferential bending opposite to the rotation dire tion is equal to 1.5 mm, the axial displa ement opposite to the air ow dire tion is equal to 0:55 mm, and the torsional displa ement with respe t to the

enter of gravity is equal to 0:2Æ. Blade os illations ause unsteady aerodynami loads a ting on the blades. Figure 5 shows the unsteady aerodynami for es, a ting on the 32

Yu. N. Shmotin et al.

Aeroelasti Analysis of Bladerows

Time histories of the modal oeÆ ients of natural modes (IBPA = 90Æ ): ( ) 1st natural mode; ( ) 2nd; ( ) 3rd; ( ) 4th; and ( ) 5th natural mode Figure 3 a

Yu. N. Shmotin et al.

b

d

e

33

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure

4

Cir umferential (a ) and axial (b ) displa ements, and rota-

tion with respe t to the enter of gravity ( ) in the peripheral blade se tion

Æ

(IBPA = 90 , 1{5 modes)

peripheral blade se tion in the ir umferential (Fy ) and axial (Fz ) dire tions, as well as the aerodynami moment (M ) with respe t to the enter of gravity for IBPA = 90Æ. As is seen from the graphs, the ir umferential for e u tuations at a given level of harmoni os illation rea h the value of 0:8 kN/m (18% of the averaged value), the axial for e u tuations rea h the value of 1:0 kN/m (17% of the averaged value), and the aerodynami moment u tuations are about 0:11 kN
m/m (46% of the averaged value). After termination of harmoni os illations, the unsteady aerodynami for es de ay and approa h the ( onstant) stati loads values. 34

Yu. N. Shmotin et al.

Aeroelasti Analysis of Bladerows

Figure 5

Tangential (a ) and axial (b ) for es and aerodynami moment ( )

Æ

in the peripheral blade se tion (IBPA = 90 , 1{5 modes)

Con luding Remarks The numeri al simulation of the aeroelasti fan BR hara teristi s using the mathemati al model of nonstationary aerodynami s oupled with the dynami s of aeroelasti os illations showed the following results: 1. The positive damping oeÆ ient was found to exist for the ase of harmoni os illations of blades with the preset law, for ea h of the natural modes with the intera tion between the modes taken into a

ount, i.e., the os illating blade energy is transferred to the main

ow. Yu. N. Shmotin et al.

35

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

2. The aerodamping of all natural modes in the whole range of interblade phase angles was found to take pla e in oupled os illations of the \rotor { gas ow" system.

Referen es 1. Gnesin, V. I., and L. V. Kolodyazhnaya. 1999. Numeri al modelling of aeroelasti behavior for os illating turboma hines bladerow in 3D transoni ow of ideal gas. Probl. Ma hinery 1(2). 2. Gnesin, V., R. Rzadkowski, and L. Kolodyazhnaya. 2000. A oupled uidstru ture analysis for 3D utter in turboma hines. ASME 2000-GT-380. International Gas Turbine and Aeroengine Congress. Muni h, Germany. 3. Gnesin, V. I. 1999. A numeri al study of 3D utter in turboma hines using

uid{stru ture oupled method. Eng. Me h. 6(4/5):253{67.

36

Yu. N. Shmotin et al.

Aeroelasti Analysis of Bladerows

MISTUNING PATTERN EFFECTS ON PROBABILISTIC FLUTTER AND FORCED RESPONSE R. Kielb, K. Hall, T. Miyakozawa, and E. Hong

Duke University Box 90300, Durham, NC 27708-0300, USA

Introdu tion

This paper presents the results of a probabilisti utter and for ed response study of a mistuned bladed disk using a high delity model in luding both stru tural and aerodynami oupling. The ase study shows that the stability and resonant response of the eet an be signi antly ae ted by the standard deviation of blade frequen ies and the pattern in whi h they are arranged in the wheel. Methods for understanding and identifying the bene ial and detrimental patterns are presented. In addition, the results from a limited study of the perturbation of pure traveling wave for ing fun tions are dis ussed. It is found that perturbations of the for ing fun tion result in relatively small hanges in the maximum blade response. The airfoils in blisk and bladed disk assemblies are oupled stru turally (through the disk, blisk, and/or shrouds) and oupled aerodynami ally. Although it is well known that both oupling me hanisms

an play a signi ant role in determining stability and maximum airfoil resonant response, the vast majority of resear h eorts have on entrated on mistuned for ed response with models only a

ounting for the stru tural oupling. Examples of papers using high delity models ontaining both aerodynami and stru tural oupling to study mistuned

utter are Refs. [1, 2℄. Papers that have spe i ally investigated the ee t of mistuning pattern on utter suppression are Refs. [3{5℄. R. Kielb et al.

37

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Theory

A high delity model, ontaining both stru tural and aerodynami oupling, is used to study the probabilisti utter and for ed response problem. The stru tural model is a redu ed order (ROM) stru tural

oupling model based on the output of nite element y li symmetry analysis. The aerodynami oupling model is based on unsteady omputational uid dynami s (CFD) methods to generate unsteady pressures, whi h are then used to determine the modal for es. The modal utter equations of motion for a mistuned bladed disk or blisk an be written as  0  +
K
!2 (I +
M ) fY g = fF mg + fW g Here, 0 is a diagonal matrix ontaining the squares of the tuned system mode frequen ies due to stru tural oupling only,
K and
M are the perturbationsmin the modal stiness and mass matri es due to frequen y mistuning, F is the modal for e ve tor due to blade motion, and W is the for e ve tor due to an external ex itation. The solutions to the for ed response and utter problems an be written as hh  0

h i

h

ii

i 1

 + Ab [Am℄ Abm !2 [I ℄ fW g hh  h i ii i h 0 + Ab [Am℄ Abm !2 [I ℄ fY g = f0g

fY g = h i

where Ab is the mistuned stiness matrix, [Am ℄ is the tuned aerodyi h nami matrix, and Abm is the mistuned aerodynami matrix. In this work, the mistuned stiness matrix is obtained using the single family of modes (FMM) approa h similar to that of Feiner and GriÆn [6℄ as des ribed in [2℄. The aerodynami for es due to blade motion are onsidered to be tuned in the present work. However, results are shown where the external for ing ve tor, fW g, is mistuned. That is, it is not a pure traveling wave. It is only ne essary to obtain the tuned system mode frequen ies (and stru tural damping), the individual mistuned blade frequen ies, the tuned unsteady aerodynami for es (due to blade motion) as a fun tion of traveling wave index (interblade phase angle), and the unsteady 38

R. Kielb et al.

Aeroelasti Analysis of Bladerows aerodynami for es due to external ex itation. This is the same information, required for a tuned utter or for ed response analysis. With this information, the mistuned aeroelasti stability and for ed response

an be qui kly determined. For example, the utter stabilities for a 1000 ase Monte Carlo simulation are al ulated in a few se onds on a PC.

Probabilisti Flutter A probabilisti utter analysis was presented in [7℄ and the relevant results are repeated herein. A bladed disk, onsisting of 35 blades, and representative of a modern front ompressor stage, is onsidered. The rst bending family of modes, whi h is isolated from the other families, is investigated. For the utter analysis, the tuned blade-alone frequen y is 410 Hz. This results in a redu ed frequen y (tip se tion, based on semi hord) less than 0.2, whi h generally results in utter. The tuned system mode (bladed disk) frequen ies as a fun tion of nodal diameter are su h that the frequen y spread from lowest to highest frequen y is only 3.2%. This is onsidered to be a relatively sti disk. For the tuned rotor, it was found that the 5 nodal diameter forward traveling wave had the least stability ( 0:15% riti al damping ratio). It should also be noted that the 5 nodal diameter ba kward traveling mode is nearly the most stable mode with a damping level of approximately 4%. This level of damping is generally onsidered to be asso iated with a ase of relatively strong aerodynami oupling. It was then assumed that a eet of 1000 engines was assembled with blades from a population of blades with a normal distribution and with various standard deviations. A Monte Carlo simulation was performed with the results shown in Fig. 1. In this gure, the umulative probability is plotted of the damping of the least stable mode. Although not shown, an in rease in population standard deviation in reases the overall stability of the eet. For the 1000 samples with a population standard deviation of 1.0%, the sample standard deviations varied from approximately 0.5 to 1.4. As expe ted, there is a dependen e of stability on sample standard deviation. But there is still a signi ant variation in stability for nearly the same value of sample standard deviation. For example, for standard deviations between 0.95 and 1.05, the aerodyR. Kielb et al.

39

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Comparison of umulative damping probabilities from frequen y and pattern alone mistuning: 1 |  = 0:95% and 2 |  = 1:0% Figure 1

nami damping varies from approximately 0:02% to 0.13%. Thus, the arrangement pattern must also have a signi ant ee t. One of the 1000 blade sets with a standard deviation of 0.95% was

hosen to study the pattern ee t. This was a

omplished by randomly varying the pattern with this same set of 35 blades. Again, a 1000 engine simulation was ondu ted. Figure 1 also shows the umulative probability for this ase. As an be seen, there is less variation than the pervious ase where the sample standard deviation also varies. This

urve appears to have three regions. That is, there are a small number of patterns that result in very low damping, a small number of patterns that result in relatively high damping, and the vast majority of patterns that have aerodynami damping in the narrow range of 0.025% to 0.05%. Thus, given a set of blades, this Monte Carlo simulation method

an be used to identify a high damping arrangement. A Fourier evaluation was made for ea h pattern. That is, the pattern of physi al blade frequen ies was represented as a sum of pure wave patterns. Note that for the worst ases (low damping), there are pa kets of blades that have very little variation in frequen y. This suggests that arranging the blades su h that blades of like frequen y are neighbors may result in low damping. The frequen y patterns of the four worst and four best patterns were evaluated for Fourier ontent. 40

R. Kielb et al.

Aeroelasti Analysis of Bladerows

Figure 2 Comparison of Fourier ontent of 4 worst and 4 best blade arrangements: 1 | best patterns and 2 | worst patterns The results are shown in Fig. 2 where the relative amplitude is plotted vs. the number of ir umferential waves. That is, a wave pattern of 1 represents sinusoidal variation of frequen y vs. blade number with one wave around the rotor. The relative amplitude is the average of the four worst (lowest damping) and four best (highest damping) patterns. As

an be seen, the two worst patterns have relatively high values of 1 and 4 wave ontent, and relatively low values of 11 and 17 wave ontent. Additional mistuned utter analyses were performed, but with pure wave frequen y patterns. The \strength" of the patterns was varied by hanging the variation of amplitude of the wave expressed as the dieren e of the maximum and minimum frequen ies normalized by the average frequen y. The results of this study are shown in Fig. 3 where the damping of the least stable mode is plotted vs. the amplitude of the frequen y pattern for 1, 4, 11, 14, 17, and 17.5 waves. The latter being the ase known as near alternate mistuning. That is, the blades alternate in a high/low arrangement ex ept for blades 0 and 34 (both having high frequen ies). Note that this generally has the same trend as that shown in Fig. 2. The wave patterns of 1 and 4 produ e a modest R. Kielb et al.

41

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 3 Ee t of pure wave mistuning on damping of least stable mode: 1 | Near alternate (17.5%) wave pattern, 2 | 1 wave pattern, 3 | 4 wave pattern, 4 | 11 wave pattern, 5 | 14 wave pattern, and 6 | 17 wave pattern

suppression ee t on utter, and the wave patterns of 11 and 14 show a stronger suppression ee t. The near alternate wave pattern shows the strongest ee t. Note that the 17 wave pattern also has a weak suppression ee t. It is spe ulated that this is due to the fa t that this pattern results in a range of onse utive blades where the blade frequen ies do not vary signi antly.

Probabilisti For ed Response The pre eding se tion demonstrates the ability of frequen y mistuning to suppress utter. However, it is well know that frequen y mistuning is detrimental to for ed response. In this se tion, a limited study of the for ed response behavior, of the system onsidered above, due to a pure 5 nodal diameter ba kward traveling wave is onsidered. Two dierent mistuning patterns are onsidered. First, the \best" pattern (most stable) from the utter Monte Carlo simulation is used. As an be seen in Fig. 1, this pattern results in a 42

R. Kielb et al.

Aeroelasti Analysis of Bladerows

Frequen y response of best and near alternate mistuning patterns: | tuned, 2 | best futter suppression pattern, and 3 | near alternate (1.85%) Figure 4

1

system damping of approximately 0.09% in the least stable mode (predominately, a 5 nodal diameter forward traveling wave). However, as previously mentioned, the for ing fun tion traveling wave (5 nodal diameter ba kward traveling wave) is the nearly the most stable with an aerodynami damping of approximately 4%. The response as a fun tion of ex itation frequen y is shown in Fig. 4. The tuned response (pure traveling wave) has a very broad peak that is expe ted when the damping level is relatively high. However, the \best" utter suppression pattern produ es mu h higher response with individual blade peaks showing mu h less ee tive damping. Examination of the individual blade responses at the peak ex itation frequen y shows highly lo alized modes. That is, the pure traveling wave response is destroyed. Note only the lassi al mode lo alization behavior is o

urring, but this behavior is resulting in a loss of aerodynami damping. Also shown in Fig. 4 is the response with a near alternate mistuning pattern. The level of this pattern (1.85%) was hosen to result in nearly the same utter stability as the best random pattern (see Fig. 3). Note that the response with this pattern is still signi antly higher than the tuned response, but is less than half that of the best random pattern. This behavior is most likely due to the fa t that there is less mode R. Kielb et al.

43

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

lo alization, whi h, in turn, does not ause as mu h degradation in the aerodynami mistuning. Thus, before mistuning is onsidered as a utter suppression method, a areful study of the ee t on for ed response must be made. For ing Fun tion Mistuning It is generally assumed that the for ing fun tion on turboma hinery blades is a pure traveling wave. However, small deviations in blade geometry due to manufa turing or eld usage, su h as erosion and blending, an result in ea h blade being aerodynami ally dierent. One ee t is that the blade for e an be perturbed in both amplitude and phase. Consider a for ing fun tion onsisting of a pure traveling wave. The for ing ve tor in traveling wave form (j is the traveling wave index) an be written as 809 >> >> >< 0 >= W = wj > 1 > >> >> :0; The amplitude of this for ing fun tion is wj . The same for ing fun tion in the xed blade oordinate system (where k is the blade number and N is the number of blades) is written as f

g











8p 9 >< 0 >= P = wj > p1 > ; pk = e :pN 1;

f

g

i2jk=N




In this work two perturbations of this pure traveling wave are onsidered. First, only the for e on blade 0 is perturbed: p 0 = 1 +
p 44

R. Kielb et al.

Aeroelasti Analysis of Bladerows

With this perturbation, it an be shown that the for ing ve tor in the traveling wave form be omes 8 0 9 819 >> >> < 0 =
>< 1 >= (1) f g = >1 +
> + >:


>; >: >; 1 0 The rst term represents a pure traveling wave with perturbed amplitude. The se ond term an be thought of as white noise ontaining all traveling waves at the same amplitude. An extreme ase of this for ing fun tion where blade zero has no for e (
= 1 0) is now

onsidered. From Eq. (1), it is seen that for the ase of = 35, the perturbation in the pure traveling wave is approximately 3%. For this for ing fun tion study, the bladed disk onsidered is the same as that above ex ept that the blade alone frequen y is 977 Hz and the disk is very sti with a frequen y spread (over all nodal diameters of 0.03%). The unsteady aerodynami s were al ulated for this system with the result that the tuned system is stable. Figure 5 shows the resonant response of all blades vs. ex itation frequen y due to the 5 nodal diameter ba kward traveling wave ( 0 = 0). The response is normalized to that of a pure traveling wave. As an be seen, only blades W

wj

p

wj

p

N

N

p

:

N

p

Figure 5 R. Kielb et al.

Frequen y response of rotor with Blade 0, For e = 0.0

45

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 6 wave:

b0

1

Frequen y response of rotor with randomly perturbed traveling

| frequen y mistuned, FF tuned;

= 0; and

1

2

| frequen y mistuned, FF,

| frequen y and FF tuned

1, 0, and 1 have resonant peaks at amplitudes signi antly dierent from that of the pure traveling wave. Blade 1 has an amplitude approximately 14% higher than that of the tuned ase due to a perfe t traveling wave. This ase was repeated but with the blades frequen y mistuned (1 standard deviation = 1:0%). The results, given in Fig. 6, show the response for the ases of only frequen y mistuned and both frequen y and for ing fun tion (FF) mistuning. Only the envelope of maximum responses is shown. For referen e, the response to the tuned rotor due to a pure traveling wave is also shown. The perturbation in the for ing fun tion has only a slight ee t on the envelope of maximum response. The ampli ation ee t seen in the tuned ase has essentially disappeared. The se ond type of for ing fun tion mistuning onsidered is where the for es on all blades are perturbed. This was a

omplished by randomly perturbing both the real and imaginary parts of for ing ve tor (nominally, a pure 5 nodal diameter ba kward traveling wave) on ea h blade with standard deviations of 5.0%. The resulting amplitudes of the blade for es were in the range of 0.92 to 1.09, and the phase perturbations were in the range of 5Æ to +5Æ . A Fourier analysis of this 46

R. Kielb et al.

Aeroelasti Analysis of Bladerows for ing fun tion showed the expe ted 5 nodal diameter ba kward traveling wave with small ontributions from other nodal diameter waves. The amplitude of the next largest wave was less than 2% of that of the 5 nodal diameter. The response of the frequen y mistuned rotor (standard deviation = 1:0%) are shown in Fig. 6 for the ases with and without the for ing fun tion mistuning. As in the previous study, this for ing fun tion perturbation has little ee t on the envelope of blade responses.

Con luding Remarks The results of probabilisti utter and for ed response studies on mistuned bladed disks using a high delity model in luding both stru tural and aerodynami oupling have been presented. The method used does not require any additional information than that required of a tuned

utter analysis, with the ex eption of the mistuned blade frequen ies. The ase study shows that the stability of the eet an be signi antly ae ted by the standard deviation of blade frequen ies and the pattern in whi h they are arranged in the wheel. A method for identifying the bene ial patterns was presented. The omputational method used herein an be used to help determine the required utter margin, random mistuning level ne essary to stabilize a eet of engines, and robustness of intentional mistuning for utter suppression. Also, mistuned utter analyses an be used during development testing by determining the relative stability of the test engine, and pi king a test blade set (or rearranging blades) to minimize stability. Similarly, the stability of produ tion engines an be maximized. Sin e the on lusions herein are based on a single ase study, it is ne essary to ondu t additional numeri al studies. The for ed response studies showed that before intentional mistuning is used to suppress utter, a areful study of the ee t on for ed response must be made. From the limited study of for ing fun tion mistuning, it was found that it is equivalent to adding low level noise of other traveling waves. When frequen y mistuning ee ts are in luded, they dominate those due to the for ing fun tion perturbations. However, it is possible that for ing fun tion perturbation may ause measurable in reases in the response for very sti disks and are nearly frequen y tuned. R. Kielb et al.

47

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

A knowledgments The authors a knowledge NASA Glenn Resear h Center for their nan ial support.

Referen es 1. Seinturier, E., C. Dupont, M. Berthillier, and M. Dumas. 2000. A new method to predi t utter in presen e of stru tural mistuning | Appli ation to a wide hord fan stage. Symposium on Unsteady Aerodynami s, Aeroa ousti s and Aeroelasti ity of Turboma hines Pro eedings . Lyon, Fran e. 739{48. 2. Kielb, R. E., D. M. Feiner, J. H. GriÆn, and T. Miyakozawa. 2004. Flutter of mistuned bladed disks and blisks with aerodynami and FMM stru tural

oupling. ASME GT-2004-54315. Vienna, Austria. 3. Crawley, E. F., and K. C. Hall. 1985. Optimization and me hanisms of mistuning in as ades. J. Engineering Gas Turbines Power 108:418{26. 4. Shapiro, B. 1998. Symmetry approa h to extension of utter boundaries via mistuning. J. Propulsion Power 14(3):354{66. 5. Martel, C., R. Corral, and J. M. Llorens. 2006. Stability in rease of aerodynami ally unstable rotors using intentional mistuning. ASME GT-200690407. Bar elona, Spain. 6. Feiner, D. M., and J. H. GriÆn. 2002. A fundamental model of mistuning for a single family of modes. ASME J. Turboma hinery 124(4):597{605. 7. Kielb, R. E., K. C. Hall, E. Hong, and S. S. Pai. 2006. Probabilisti utter analysis of a mistuned bladed. ASME GT-2006-90847. Bar elona, Spain.

48

R. Kielb et al.

Aeroelasti Analysis of Bladerows

AEROELASTIC VIBRATIONS OF AXIAL TURBOMACHINE BLADEROW V. E. Saren

P. I. Baranov Central Institute of Aviation Motors (CIAM) Aviamotornaya Str. 2 Mos ow 111116, Russia

Introdu tion

One of important tasks in the analysis of aeroelasti vibrations of turbama hine bladerows is the estimation of stability of blade self-ex iting vibrations at dierent operation onditions. The main diÆ ulty of su h an estimation is adequate a

ounting for aerodynami intera tion of vibrating blades in a wide range of ow onditions around the blades and for various vibration modes. The paper presents the results obtained in the ourse of development and appli ation of omputer odes aimed at the margin of rotor blade autoos illation at the map of hara teristi s for the axial ompressor stage of gas-turbine engine. Natural modes and frequen ies of blade vibrations in the absen e of aerodynami intera tion are al ulated for ea h point of ompressor hara teristi s taking entrifugal for es and stationary aerodynami load into a

ount [1℄. The te hnologi al s atter in natural frequen ies of elasti vibrations of blades in the row is in luded into the hara teristi equation for the aeroelasti system. The matrix of aerodynami onne tivity of blades is al ulated based on the solution of two-dimensional gas-dynami equations linearized with respe t to small blade displa ements [2{5℄. To analyze the aeroelasti stability of a bladerow, the stability quality parameter of the orresponding linear system is used [6, 7℄. This method was brie y reported in [8℄. In this paper, the general formulation is des ribed in more details, and the results obtained after algorithm implementation are presented. V. E. Saren

49

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

1

Margin of Autoos illations for Rotor Bladerow at the Plane of Compressor Stage Chara teristi s

The margin of autoos illations for rotor blades of an axial ompressor stage is al ulated at a given rotation frequen y, mass ow rate, and total pressure ratio. Assume that the time-averaged relative ow velo ity in the rotor is known. Then one an al ulate the natural modes and frequen ies of elasti vibrations of rotor blades orresponding to the given operation onditions of the ompressor stage. Contrary to

entrifugal for es, the aerodynami for es are known to weakly distort at least three rst natural vibration modes of blades rigidly xed in the disk. Based on this, the unsteady aerodynami for es aused by small vibrations of blades in the given mode may be onsidered as linearly depending on blade displa ements and not distorting their natural modes. However, as the aerodynami for es are non onservative, the natural frequen y of vibration in the ow is a omplex number ontrary to elasti system vibrations. Therefore, small vibrations of blades in a given mode an damp or not depending on the nature of aerodynami intera tion between vibrating blades. The point at the map of ompressor stage hara teristi s determining the stage operation mode will be treated as lo ated in the stability domain, if natural vibrations of rotor blades in the ow are damping for ea h mode onsidered. The equation of small vibrations of blades in the ow in a given mode takes the form [8℄:

M

 2 uk + Quk t2

XA

Z 1 l=0

l k ul = 0 (k = 0; 1; : : : ; Z

1)

(1)

where M and Q are the generalized mass and stiness of a blade, respe p tively; uk = uk ei!t (i = 1) is the generalized oordinate of kth blade in the row onsisting of Z blades; ! is the vibration frequen y; and t is the time. The Al k omponent of the y li matrix of system (1) with A r = AZ r (r = 0; 1; : : : ; Z 1) is the generalized aerodynami for e a ting on the kth blade, when the amplitude of the lth blade vibration is equal to unity. If elasti properties of all blades in the row are identi al, the nonzero solution of system (1) exists only if its determinant is zero, i.e., 50

V. E. Saren

Aeroelasti Analysis of Bladerows

A(q)

q2

q2



I = 0

(2)

R where q = !b=v; q = !0 b=v; and R = (1=2)(f b3 =M ). Here, I is the unit matrix of the Z th order; b is the hara teristi linear dimension of the blades; V is the hara teristi ow velo ity; f is the ow density; and !0 is the natural frequen y of a blade in the absen e of aerodynami

intera tion between vibrations of blades. It should be noted that Eq. (2) is trans endental, as the omponents of A matrix depend in general on all powers of redu ed frequen y q. For identi al blades, system (1) an be transformed to the diagonal form using the unitary matrix: 

p1

H=

Z

e

i2kl=Z



(H
H  = I )

where H  is the matrix onjugate to H . In this ase, hara teristi Eq. (2) takes the form:

qk2 + q2 Ck (qk ) = 0 (k = 0; 1; : : : ; Z 1) R where ‘ = H  AH is the diagonal matrix with omponents Ck =

Z X1 r=0

Ar ei2rk=Z

(k = 0; 1; : : : ; Z

(3)

1)

In line with the de nition of aerodynami in uen e oeÆ ients

Al k , the value of Ck is the generalized aerodynami for e a ting on

any given blade, hosen as the initial one, at syn hronous vibrations of all blades in the row with unit amplitude and phase shift of  = 2k=Z between vibrations of neighboring blades. Here, k = 0; 1; : : : ; Z 1 determines the distribution of blade displa ements for either time instant, i.e., spe i es the mode of aeroelasti vibrations of bladerow with every blade vibrating in a given natural mode. In their turn, aerodynami in uen e oeÆ ients Al k an be al ulated using the inversion formulae:

Al V. E. Saren

k

=

1

Z

ZX1 r=0

Cr e

i2r(l k)=Z

; k; l = 0; 1; : : : ; Z

1 51

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

The mass riterion for typi al turboma hines is relatively small (about 10 {10 ). This makes it possible to sele t a unique eigenvalue

orresponding to the th vibration mode of the bladerow among the roots of trans endental Eqs. (3) at a spe i ed value of = 0 1 1. This eigenvalue is lose to the redu ed frequen y of elasti vibrations of blades. As a matter of fa t, if k = k( k) is assumed to be the analyti fun tion in the vi inity of point k = , the eigenvalue an be presented in the form of the power series: R

2

3

k

k

;

;::: ;Z

q

C

C

q

q

qk

1

X

= +

qkl R

q

l=1

q

(4)

l

Then, substitution of Eqs. (4) to Eqs. (3) results in the following relations: 1 k = 2 k( ) q 1

qk 2

q

C

q



= 2 k+ = 2k 2 k + qk 1 q





Ck

q 1

(5)



q=q

q

1 k k q q 4 q q where = 0 1 1. At small dieren es in natural frequen ies of elasti vibrations of bladerows, the system (1), in addition to y li matrix , ontains a diagonal matrix
with omponents: Z
k= k =1 ( =0 1 1) k qk 3

k

q 2 q

;



q 1



Ck



q

2

 C

=

q

q

2



2

=

q 1

;::: ;Z

A

q



G

q

2

q

X1

2

R

q

;

Z

q

k

;

;::: ;Z

k=0

where k is the redu ed frequen y of elasti vibrations of the th blade in the row. Applying transformation to
matrix results in onversion of
to y li matrix
=
with the omponents: 1 Z
r i r l k =Z ( = 0 1 1) l k= q

k

H

G

D

X1

d

Z

q e

H

2

(

G

GH

)

k; l

;

;::: ;Z

r=0

Thus, at small dieren es in natural frequen y of blades in the row, the determination of k values in the series (4) redu es to the

al ulation of eigenvalues of matrix = (1 2 )[
( )℄. q 1

B

52

= q

D

C q

V. E. Saren

Aeroelasti Analysis of Bladerows

It is evident that relations (5) for qk2, qk3 : : : remain formally the same, but the free term in expansion (4) in addition to q will ontain the power series with respe t to " = max jqk qj. k From the aforesaid, one an see that the stability analysis of small vibrations of bladerows in a given mode with small dieren es in natural frequen ies is redu ed to al ulation of eigenvalues of matrix B or equivalent matrix D = (1=2q)[
G A(q)℄. For a suÆ iently small R, in a

ordan e with the assumed form of the solution, the onditions Im qk1 > 0

(k = 0; 1; : : : ; Z 1)

(6)

ensure asymptoti damping of initial disturban e of blades in the row in one of natural modes. Therefore, the ondition Im qk1  0 an be treated as the ne essary ondition for autoos illations of the bladerow in a given mode. In the al ulations for real bladerows, the stability riterion (6) proved to be \sensitive" to small variations of omponents of matri es B or D. In parti ular, the eigenvalues of formally equivalent matri es B and D determined by numeri al simulation an dier signi antly, espe ially, for the rst bending mode of blade vibrations. In [8℄, the riterion (6) was substituted by the riterion of stability quality (B). The method of al ulating the (B) riterion with a preset a

ura y was developed in [6, 7℄. The riterion , by de nition, provides the upper estimate for the level of bladerow vibrations at some \initial" blade vibrations. If  = 1, the bladerow in the ow is absolutely unstable. But if  is nite but suÆ iently large, this means that, in a formally stable bladerow, blade vibrations in a given mode an arise with dynami stresses ex eeding a permissible level (usually (5{10)
107 Pa) at \ba kground" dynami stresses  (2{3)
107 Pa. In this

ase, the bladerow is to be onsidered as \virtually unstable." Thus, with the suggested method of estimating the domain of unstable modes on the map of hara teristi s, the problem is redu ed to determining the values of  = , indi ating that the bladerow be omes \virtually unstable." V. E. Saren

53

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

2

Examples

2.1 Autoos illations of rod as ades in the ow

In [8℄, the bending vibrations of the as ade of antilever- xed rods in the in ompressible ow were onsidered as an example of analyti al solution of the problem. The rods were assumed to have thi kness
,

hord b, and span L. Here, some omplementary data are presented to illustrate the method under onsideration. The vibrations of the rods were assumed to o

ur along the normal to the hord. The inertia for es of rod motion along the span were negle ted. In this ase, the mode of rod displa ement was determined by the fun tion:  z   z  K1 (n)K4 n un(z ) = K2 (n )K3 n L L where z is the oordinate measured along the span towards the free end and K1 to K4 are the Krylov fun tions. The values of 1 = 1:875, 2 = 4:694, : : : , n = (2k 1)=2 (n = 3; 4; : : : ) determine natural frequen ies of bending vibrations of the rods without regard for their intera tion with the ow: !0n = n 2

s

EJ m b
L4

Here, m is the rod material density, E is the modulus of elasti ity, and J is the moment of inertia of rod ross se tion. Figure 1 Aerodynami model of Let 0 and be the angles dense as ade of rods formed by the approa hing ow velo ity ve tor V and rod hord with the normal to the as ade front (Fig. 1). If one assumes that as ade solidity  = b=h  1, where h is the as ade pit h, and the ow velo ity omponent along z-axis is negligible, then the spe i aerodynami load a ting on vibrating rods an be determined analyti ally [5℄. A

ording to Eqs. (1){(3), for investigating the stability of small bending 54

V. E. Saren

Aeroelasti Analysis of Bladerows

vibrations of a rod as ades in the ow, it is suÆ ient to onsider rod displa ements in the form: unk (z; t) = uun(z )ei!t eik (n = 1; 2; : : : ; k = 0; 1; 2; : : : ) where n is the number of rod elasti -vibration mode, k is the rod number in the as ade, and  is the parameter determining the mode of as ade vibrations. The dimensionless oeÆ ient of aerodynami load is given by [5℄: F q 2ie i (tg os 0 sin 0)( ) h 2 C (q; ) = 1 F3 q + F4 q F2q + ei ( ) i 0 + 2ei os (  ) + q(F5 q + F6 ) (7)

os where

i

F1 =

2 i 2 os ( ) + 2i( ) + 2ie   sin( 0)

os i i F2 = 2ie ( ) + i 1 + 2e os 0

os

F3 =

2ei os ( ) 2

os 0 + 4ei sin( 0) F4 = 2iei ( ) + 2i

os

os 0 2 F5 = os ( )2 + e i os ( ) 3 sin( 0) ( ) 2 sin( 0) F6 = 2

os2

os

Equation (3) in this ase takes the form: " # 1 1  qn 2 = 1 C (qn; ); q q2 R n

n

1 f Z Rn = 2 m
0

1

un (z ) dz

(8)

where
=
=b; z = z=L. V. E. Saren

55

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 2 Dependen e of q on 0 for bending vibrations of a as ade of rods: q < q is the domain of autoos illations; q > q is the stability domain

Figure 3 In uen e of mass ri-

terion on the frequen y of aeroelasti vibrations of a as ade of rods: 1 | R1 = 10 1 ; 2 | 10 2 ; and 3 | R1 = 10 3

Formulae (7) and (8) make it possible to address some issues of aeroelasti vibrations. In parti ular, if the value of q = q is determined as the maximal positive root of the equation Im C (q; ) = 0 with parameter  varying in the range ( 1; 1), one an readily estimate ee t of aerodynami loading on bending vibrations in the ow. As an example, Fig. 2 shows the al ulated urve q = q (0 ), separating the stability and instability domains of bending vibrations of a rod as ade. The al ulations show that the autoos illations under onsideration are possible only at suÆ iently large values of the angle of atta k, 0 , at 1   < 0. At the notations adopted, this means that the disturban e wave at autoos illations propagates along the front of the as ade towards the rods with lower number (see Fig. 1). This nding an be used as an indi ator for dete ting bending autoos illations. The autoos illation frequen y depends on Rn. Figure 3 shows the dependen e of !=!01 ratio on the angle of atta k, 0 , at  = 0:1 and R1 = 10 1 , 10 2, and 10 3 . One an see that in this ase, the autoos illation frequen y an be onsidered to be lose to the elasti vibration frequen y !01 only at R1  10 3. It should be also noted that fun tion C = C (q; ) an have singular points in the (q ; ) plane (see Eq. (7)), whi h should be taken into a

ount when using iterative methods. 56

V. E. Saren

Aeroelasti Analysis of Bladerows

2.2

Stability quality of bladerows

Figure 4 shows the dependen e of  on the redu ed frequen y q for the uniform bladerow omposed of rigidly xed rods in the disk with the

onstant untwisted pro le along the span. The relative ow at the row inlet had the angle of atta k . Blade vibrations in the rst bending mode were onsidered. Plotted along the x-axis in Fig. 4 is the value of q, whi h were varied by hanging the relative velo ity of the in oming

ow. Curves 1 {3 in Fig. 4 are plotted in the vi inities of riti al values of q obtained from the ondition Im C (q) = 0 at dierent angles of atta k  = 0:035; 0.335; and  = 0:35. One an see in Fig. 4 that  in reases drasti ally at q q . The dashed region shows the se tion of initial rise of  parameter in the interval (8; 20). It is interesting that the mentioned \jump" o

urs at a suÆ iently large distan e from the riti al value q . This allows one to believe that the stability quality of the aeroelasti system under

onsideration is getting worse when the ow parameters approa h the values relevant to unstable regimes. Thus, if one assumes that the

riti al value of  is equal to 20, the regimes with the redu ed frequen y dierent signi antly from the nominal riti al value have to be treated as \virtually unstable."

!

Figure

quen y

2

|

q



4 Stability quality parameter as a fun tion of the redu ed fre = 0 01 ( = 0 035 rad); in the vi inity of riti al values: 1 |

q = 0:06 ( = 0:335 rad); and 3

V. E. Saren

|

q :  : q = 0:07 ( = 0:35 rad)

57

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Table 1 Results of al ulations of  for the rotor of axial ompressor stage at a pressure ratio of 1.7 n N G1 G2 G3

0.6 0.7 0.8 0.9 1.0

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

q



q



q



0.679 5 0.681 6 0.681 6 1.777 4 1.783 4 1.783 4 2.550 2 2.559 2 2.559 2 0.646 6 0.646 6 0.645 6 1.626 6 1.627 6 1.626 6 2.206 3 2.208 5 2.205 7 0.599 7 0.595 7 0.595 7 1.463 8 1.458 9 1.455 9 1.879 5 1.873 5 1.861 5 0.557 9 0.553 10 0.551 13 1.328 8 1.320 8 1.315 9 1.624 8 1.614 44 1.608 1 0.495 19 0.492 21 0.491 22 1.154 8 1.147 9 1.145 9 1.354 7 1.346 8 1.344 8

As was shown in [8℄, the introdu tion of small dieren es in natural frequen ies of blades in the row under onsideration results in a signi ant redu tion of  parameter, and onsequently, in the \improvement" of the aeroelasti system stability quality. Table 1 shows the results of al ulations of  for the rotor of an axial ompressor stage with the designed pressure ratio of 1.7. This example is remarkable as the al ulations were performed using the design data, i.e., prior to stage testing. The al ulations were made for redu ed rotation frequen ies n = 0:6, 0.7, 0.8, 0.9, and 1.0. The variations in the mass ow rate orresponded to the onditions of open throttle (G1 ), onditions of maximal eÆ ien y (G2 ), and to the minimal value whi h ould be attained with the design data (G3). Three rst modes N = 1, 2, and 3 of natural vibrations of blades, whi h were assumed to be rigidly xed in the disk, were investigated. One an see from Table 1 that only the regime orresponding to n = 0:9, G = G3, where autoos illations in the third torsional mode ould be expe ted, is nominally unstable ( = 1) among the ompressor-stage 58

V. E. Saren

Aeroelasti Analysis of Bladerows

Experimental margin of bending autoos illations in the rotor of the axial- ompressor stage at the ow oeÆ ient C 1a { the rotor blade tip speed U or . Symbols
orrespond to the points obtained by al ulations

Figure 5

operation modes onsidered. However, a signi ant \worsening" in the stability quality in the rst mode is observed in the left bran h of the

hara teristi orresponding to = 0 9, as well as at all mass ow rates with the nominal rotation frequen y of = 1. In the rst test runs, it was found that al ulations of hara teristi s underestimated the values of mass ow rates. Nevertheless, at a redu ed rotation frequen y of = 0 9 at small throttling, autoos illations arose in the bending mode N = 1, and the tests were terminated. Before the next tests [8℄, the rotor stage was assembled with alternating the blades with higher and lower natural bending frequen ies within the te hnologi al s atter ( 5%). This made it possible to in rease the range of mass ow rates used, although the left bran hes of the

hara teristi s were ut at = 0 7  0 9 be ause of dete ting bending autoos illations. At elevated rotation frequen ies, the autoos illations in the bending mode were dete ted at = 1 05. The al ulation of the stability quality parameter for the unstable operation onditions n

:

n

n

n

:

:

:

n

V. E. Saren

:

59

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

mentioned, performed with regard for the dieren es in rotor blade frequen ies, gave the value of  = 10  12. Thus, the example presented above indi ates the possibility of using  as a generalized stability riterion of aeroelasti vibrations of bladerow at dierent operation regimes. More omplete experimental data on the limits of bladerow autoos illations were reported in [9℄ for the rotor of axial- ompressor stage designed for the following parameters: a total pressure ratio of 1.51, peripheral velo ity of u = 396 m/s, and ow oeÆ ient of C1a = 0:51. Figure 5 shows the experimental data on the margin of bending vibrations in the (‘1a; u)-plane. The autoos illation regimes al ulated using the pro edure des ribed above at  = 12 are shown by symbols
in Fig. 5.

Referen es

1. Ushakov, A. I., V. A. Fateev, and M. A. Melnikov. 1987. Cal ulation of stress-deformed state and natural vibrations of omplex-shape blades. Aeroelasti ity of turboma hine blades. Mos ow: CIAM. 1221:113{25. 2. Gorelov, D. N., V. B. Kurzin, and V. E. Saren. 1971. Cas ade aerodynami in unsteady ow . Novosibirsk: Nauka. 3. Fayzullin, R. T. 1985. Cal ulation of subsoni ow of ideal gas through the as ade of vibrating airfoils by nite-element method. Aeroelasti ity of turboma hine blades. Mos ow: CIAM. 1127:230{34. 4. Butenko, K. K. 1985. Cal ulation of unsteady aerodynami loads on the

as ade of vibrating airfoils in subsoni or supersoni ows of ideal gas. Aeroelasti ity of turboma hine blades. Mos ow: CIAM. 1127:226{30. 5. Saren, V. E. 1990. Asymptoti theory of high-solidity as ade of unsteady in ompressible uid. In: Unsteady aerodynami s and aeroelasti ity of turboma hines and propellers. 5th International Symposium Pro eedings. International A ademi Publ. Pergamon Press. 178{96. 6. Bulgakov, A. Ja. 1980. EÆ iently al ulated parameter of stability quality for linear dierential equations with onstant oeÆ ients. Siberian Mathemati al J. 21(3):32{41. 7. Bulgakov, A. Ja., and S. K. Godunov. 1981. Numeri al determination of one of stability quality riteria for the linear dierential equations with

onstant oeÆ ients. Novosibirsk: IM SO AN SSSR. Preprint. 8. Saren, V. E. 1995. To the al ulation of bladerow utter. Applied Me hani s Engineering Physi s 38(5):85{92. 9. Zablotski, I. E., Yu. A. Korostelev, and R. A. Shipov. 1977. Nonintrunsive measurements of turboma hine blade vibrations. Mos ow: Mashinostroenie. 60

V. E. Saren

Aeroelasti Analysis of Bladerows

FREQUENCY MODEL OF VIBRATION FOR TURBOMACHINE DIAGNOSTICS A. Mironovs

SIA \D un D entrs" Balta Str. 27, Riga LV-1055, Latvia

Various aspe ts of aerodynami intera tion of blades and vanes are stated in many works, for instan e, on aeroelasti ity [1℄ and vibration diagnosti s [2, 3℄. The pulse model of blades{vanes intera tion des ribed the vibration stru ture of a rotor and stator. Intera tion between wakes of upstream rotating blades and vanes generates pulse series of aerodynami for es. Dissimilarity of blades, rotating with a rotor, reates ir umferential distortion of wakes that modulates the amplitude and phase of aerodynami for e pulses. For es ause vibration of asing details that are measured by an a

elerometer. Above model with some development [4℄ des ribed harmonious and ombinational narrow-band omponents of a spe trum. The subsequent improvement of the model [5℄ has given an opportunity to onne t the random ex itation reated by blade vorti es with random side-frequen y band omponents of the vibration spe trum next to blades rotation frequen y. Until vibration diagnosti s was limited by a low-frequen y band of the spe trum (few kilohertz maximum), su h an approa h was suÆ ient. However, new hallenges made it ne essary to develop a new model of vibration spe trum taking into a

ount the real nature of blades{vanes intera tion.

Simplest Model of Blades{Vanes Intera tion Asymmetri lo ation of a

elerometer

The simplest model of stage (Fig. 1) ontains an ideal guide vane (GV) and a rotor with single blade. Guide vane has four uniform vanes (zv = 4) that are tted by its tip edge. All features of vanes are even. The A. Mironovs

61

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

a

elerometer is lo ated on the

asing next to the vane No. 1. The wake of passing blade generates variable aerodynami for es on vanes. Within one rotor revolution, ea h vane re eives one pulse of aerodynami for es. To rea h the a

elerometer mounted outside GV, for e pulses of vanes need various time be ause of dierent length of signal paths. The delay between the instant of pulse genFigure 1 Model of stage ontaining eration on the ith vane and the four guide vanes and the rotor with one instant of its arrival to the a blade

elerometer de nes the shift i of the orresponding pulse in series. The 1st vane has minimal delay min be ause it is lo ated next to the a

elerometer. The 3rd vane lo ated opposite to the a

elerometer has maximal delay. Therefore, pulses generated by dierent vanes

ome to the a

elerometer with dierent delay that is a fun tion of vane lo ation relative to the a

elerometer. Be ause of above, the instantaneous frequen y of pulses (registered by the a

elerometer) be omes variable in time that leads to a frequen y (angle) modulation of pulse series. The arrier frequen y is the frequen y of aerodynami for e pulses, fp = zv fr , generated on vanes and transmitted to the a

elerometer. The rotational speed fr is then the frequen y of a modulation fun tion. The instantaneous frequen y in ase of angle modulation depends on the properties of the modulating fun tion 1 1 f (t) = = ins T0 + (i i 1) where T0 = t0i t0i 1 is the time period between pulses appearan e on vanes, t0i;i 1 are the instants of ith and (i 1)th pulses appearan e on vanes, and i;i 1 is the time delay of pulses arrival on a

elerometer from the ith and (i 1)th vanes. To illustrate the angle modulation, the model spe trum of aerodynami for e pulse series was al ulated. The model omprised 32 ideal 62

A. Mironovs

Aeroelasti Analysis of Bladerows

Figure 2

Pulse delay fun tion of vane number

vanes intera ting with a single-blade rotor rotated at 300 s 1. Guide vanes were rigidly fastened to the outer shroud ring. The distan e between a

elerometer and the 1st vane determined the minimum delay 0 of its pulse arrival. Figure 2 shows the pulse delay as a fun tion of the vane number. The maximal u tuation (frequen y deviation
f ) of the instantaneous frequen y depends on the vanes number and the length of vibration path for ea h vane. The oeÆ ient of modulation depends on relation between deviation and modulating frequen y 1
f = f fmin = zv = fr fr 1=zv + fr
max

In ase when the pulse series exhibits angular modulationby harmonious fun tion, the Bessel fun tions are most appropriate for spe trum analysis. However, the fun tion in Fig. 2 is not harmonious, so its spe trum analysis must be performed based on pulse series al ulation. The main mission is to al ulate the length of ea h vane vibration path. The

al ulation must take into a

ount the urvature of outer asing surfa e and the sound speed in dierent materials of stator details. A trial al ulation was performed based on the single-blade model and real data for the stage of the turboshaft engine. Figure 3 shows the spe trum of al ulated pulse series. There are three main attributes of this spe trum: ( ) the asymmetry of the modulation spe trum in lower and higher sides of the arrier frequen y; ( ) the amplitude of the arrier omponent is less than most of modulated omponents; and ( ) modulation bands appear not only nearby i

ii

iii

A. Mironovs

63

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 3 Spe trum of al ulated pulse series the arrier omponent but also nearby multiple frequen ies ( arriers); modulation bands are also multiple extending. In addition to angle modulation, there is the amplitude modulation of pulse series on the a

elerometer aused by dierent damping. The damping of pulses depends mainly on the path length so the most remote vanes have the most damped signal. The modulation of pulse series manifests itself as side spe trum omponents dislo ated by fr relative to the arrier frequen y zv fr and also as fr frequen y near to zero. The magnitude of amplitude modulation depends of stru tural features in luding an assembly diameter, vane size, and materials of details. The width of the modulation spe trum is limited by fr frequen y only, so it is less than that of angle modulation. Pulses are also subje t to deformation on its way to the a

elerometer. The pulses oming to the a

elerometer by dierent paths parti ipate in the reation of a nal pulse shape. So, the shape and duration of pulses vary depending on the vane number. The duration of the initial pulse determines its frequen y s ale that is hundreds kilohertz, so it is out of the frequen y band onsidered. In this way, the asymmetri lo ation of the a

elerometer auses distortion of the vibration spe trum nearby the arrier frequen y in omparison to the spe trum of aerodynami for es of vanes. The width of the modulation spe trum depends on the depth of angle modulation and on the order number of arrier frequen y. The rotor rotation frequen y plays the role of the modulating frequen y, so the modulation spe tral omponents have step fr . A tually, the relative width of the modulation spe trum is determined by a maximal delay of a signal from the most remote vane. 64

A. Mironovs

Aeroelasti Analysis of Bladerows

Ee t of Blades/Vanes Dissimilarity The model of a tual intera tion between blades and vanes must a

ount for the a tual number of rotor blades and nonuniformity of blades and vanes assemblies. A

ounting for an a tual number of blades results in the a tual frequen y of pulse series zr zv fr re eived by the a

elerometer. For most of jet engines, this frequen y is 0.1: : : 1 MHz and more. Vanes dissimilarity Variation of aerodynami and elasti properties of dierent vanes determines their ir umferential dissimilarity. In ase of rigid vanes, their dissimilarity is mainly based on random variation of aerodynami properties in luding vane angle, hord, thi kness, et . Dieren e between vanes leads to variation of aerodynami for es even if blades are uniform. The diagram in Fig. 4 illustrates how two identi al wakes with time period Tl ex ite three dierent Figure 4 Ex itation of for es by inpulses of aerodynami for es F1, tera tion of two identi al wakes with three dierent vanes F2 , and F3 on three neighbouring vanes. Amplitudes, phases, and shapes of for es are dierent in spite of the same ex itation. Be ause of su h intera tion, random variation of vane geometry modulates the pulse series of aerodynami for es. For long and thin vanes, the nonuniformity of vane elasti properties must also be taken into a

ount. Under the in uen e of ripple aerodynami for es, a middle se tion of a vane deviates from its stati position. After impa t ompletion, the middle se tion strives to initial position under in uen e of elasti for es that means some os illation with one of natural frequen ies. The ex iting frequen y (zr zv fr ) is one order higher than basi natural frequen ies of vanes. Therefore, vanes mainly u tuA. Mironovs

65

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

ate. However, natural os illations modulate the os illation amplitudes. In this way, random variation of vane elasti properties stipulates the variety of parameters of vane u tuations. Both aerodynami and elasti for es stipulate stress in a Figure 5 Sket hy spe trum of vane pla e of vane fastening to the properties variation shroud ring. Stress waves pass through stator details and ae t the a

elerometer that generates vibration signal. Thus, random variation of vane properties modulates pulse series in the asing even in

ase of uniform blades. It is important that hara teristi s of random variation depend solely on the stru ture and do not depend on the rotor speed. A sket hy spe trum of su h vibration (random modulation) is shown in Fig. 5. The bandwidth fb is very large be ause of a huge value of arrier frequen y. Vane properties, a

ording to dierent evaluations, may vary from the nominal meaning by few tenths to few per ent. For instan e, variation of vane spa ing by 0.1% auses instantaneous frequen y deviation by 1: : : 10 kHz. Su h a bandwidth is mu h larger than the modulating frequen y (rotor speed) of any engine type. For better illustration of random fa tor and a

elerometer asymmetry, some sket hy spe tra are presented in Fig. 6. Figure 6a shows the ase when the a

elerometer is mounted in the enter of uniform GV (assembly), and Fig. 6b relates to a tual blades/vanes intera tion when the arrier frequen y f = zr zv fr is modulated by random vibration. This modulation transforms harmoni spe tral omponents to band omponents whi h width is the fun tion of vane hara teristi variation
f = F (fb). Asymmetri lo ation of the a

elerometer stipulates the appearan e of modulation zones next to arrier (Fig. 6 ). The following equation des ribes the random modulation zones in high-frequen y band: (1) 
f = (f0 fmin ) = zr zv fà  1=(z z f 1) +
 r v r max 66

A. Mironovs

Aeroelasti Analysis of Bladerows

Figure 6

Sket hy spe trum of vibration pulse series depending on a

elerometer position and GV properties: (a ) enter positioned a

elerometer

and ideal vanes; (b ) enter positioned a

elerometer and a tual vane properties; and ( ) a

elerometer on a asing and a tual vanes (random modulation)

where intera tion with zr blades is onsidered. Here, the value of 1=(zr zv fr ) is about 2: : : 3 orders less than
max; so, disregarding small ina

ura y, Eq. (2) an be simpli ed to 
f = zr zv fr 
1 max The frequen y band of vibration measurement is limited by 20: : : 25 kHz. Usually, su h high-frequen y omponents zr zv fr are not measured. However, the a

elerometer rea ts to the envelope of highA. Mironovs

67

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

frequen y signal a ting as a me hani al dete tor be ause energy of highfrequen y vibration of thin-walled asing is not suÆ ient to shake up a more massive bra ket with the a

elerometer. Therefore, the modulation omponents appear in the spe trum. The lower border of the modulation spe trum is limited by zero while the higher border is limited by the frequen y al ulated as 
f = 
1 max Thus, uniform blades{vanes intera tion generates vibration on the

asing- xed a

elerometer. The spe trum of this vibration ontains random band omponents limited by 0: : :
f that do not depend on the rotation speed. Ea h turbine/ ompressor stage generates its own

omponents depending on the stru tural properties. The appearan e of su h omponents aused by asymmetry of a

elerometer lo ation is onsidered as the e ho ee t, the delay of whi h is aused by the dieren e between vibration paths of vanes. For the rst time, this ee t is de ned as dynami or, more exa tly, aerodynami part of frequen y response fun tion (FRF) of a turboma hine stator. It means that vanes transmit any aerodynami ex itation to the asing- xed a

elerometer with a ertain distortion. A

omplished by the stati part of FRF (me hani al resonan es of stator details), the dynami part does not vary on e the stator stru ture remains the same. lm

lm

Blades dissimilarity A tual blade assembly has ir umferential dissimilarity of aerodynami and elasti properties. Variation of blade angles, hords, thi knesses, surfa e quality, et . auses geometri al dissimilarity of blades whi h stru ture hanges with ea h new start of a ma hine. The aerodynami dissimilarity is losely related to the geometri al one and hanges the wakes onve ted downstream. The wakes of blades have pulse nature; therefore, for their analysis, the same approa h as for pulse series on vanes is used. In ase of rigid blades and laminar ow, the ir umferential distortion repeats in ea h revolution. Su h distortion is onsidered as periodi al. In the frequen y domain, su h a distortion looks as the modulation of the arrier frequen y by harmoni s f (Fig. 7  ). r

68

A. Mironovs

Aeroelasti Analysis of Bladerows

Figure 7 Spe trum of asing vibration ex ited by ir umferential air distortion in ase of: (a ) amplitude modulation (rigid blades and laminar

ow); (b ) amplitude and frequen y modulation ( onsole mounting of elasti blades and laminar ow); and ( ) amplitude and frequen y modulation of a tual unsteady ow The onsole way of blade mounting into the dis permits natural os illations of blades. The dieren e in blade elasti properties leads to variation of natural os illation frequen ies of the blades around some frequen y fn . The variation of natural frequen ies determines the bandwidth of the arrier omponent. The os illating blades generate wakes with dierent amplitude, phase, et . Amplitude modulation in frequen y domain is presented by band

omponents displa ed by fn to both sides of the arrier frequen y. Angle modulation reates zones on both sides of the arrier frequen y and its width fan an be both larger and smaller than fn and depends on the os illation amplitude and frequen y. Summary spe trum in ludes both kinds of modulation (Fig. 7b ). There exist few A. Mironovs

69

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

natural frequen ies as well as few lo al extremums of angle modulation in the frequen y domain. Wakes dissimilarity together with the a

ompanying turbulen e exhibits random nature in omparison with the deterministi periodi al one mentioned above. Turbulen e leads to variation of wake parameters and, as a result, to angle modulation of wake series by random ex itation. In the spe trum, the turbulen e modulation zone is not displa ed from the arrier frequen y be ause there is no pronoun ed maximum in turbulen e spe trum. Turbulen e seemingly \smears" the arrier omponent of spe trum and transforms it from the harmoni to band-like omponent with the bandwidth depending on turbulen e intensity. In this way, periodi al and random ingredients of the ir umferential ow distortion provide the modulation of wake series resulting in the spe trum presented in Fig. 7 . Finally, variable ex itation ae ting guide vanes is a modulated pulse series having both harmoni and random ingredients. Wide-band

omponents and some harmoni s form the spe trum of su h a series. Maximum of ffkni ; fani g determines the width of spe trum, but natural frequen ies or angle modulation features provide the extremums. Model of a tual intera tion Ex itation of GV assembly leads to formation of twi e-modulated pulse series. By other words, e ho ee t transposes initial ir umferential ow distortion (Fig. 7 ) des ribed by fun tion Fiffkni ; fani g to the shape as shown in Fig. 8.

Figure 8 Spe trum of a tual GV vibration with e ho ee t (a asing- xed a

elerometer) 70

A. Mironovs

Aeroelasti Analysis of Bladerows

Boundary frequen ies of the a

elerometer vibration an be al ulated for a jth (GV) stage as: 
f = F 
 1 max j

i

j

Sour es of Casing Vibration Casing vibration

The above-mentioned e ho ee t is based on signal treatment as the pulse series in the ultrasound band. However, there exists another approa h. Based on the prin iple of superposition, one an onsider ea h GV as an independent vibration sour e. The resultant vibration spe trum of a signal measured by a asing- xed a

elerometer is the sum of all vane spe tra. By this approa h, the signal of separate vane orresponds to the ex itation spe trum. The vibration model within the audio band (until 20: : : 25 kHz) be omes simpler be ause blade (and multiple) frequen ies be ome arrier frequen ies. Caused by wake modulation, the low-frequen y band of vibration spe trum in ludes both rotor harmoni s provoked by ir umferential dissimilarity and randomband omponents generated by natural frequen ies (Fig. 9). However,

ontrary to the e ho ee t, su h band omponents are related to blade frequen y and depend on rotation speed. As a matter of fa t, both approa hes to asing vibration presentation are useful; so, one an onsider that GV transmits ex itation to the a

elerometer both as independent sour es and as the assembly of vanes providing the e ho ee t.

Figure 9 Complete vibration spe trum on a asing ontaining harmoni and wideband omponents followed by e ho ee t

A. Mironovs

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Rotor vibration Rotor bearings transmit rotor vibration to a asing. The main sour es of rotor vibration are inertial for es and aerodynami for es generated by intera tion between ir umferential air distortion and rotating assembly of dissimilar blades. Rotor vibration passes through bearings that are a ting as low-pass lter and damp high-frequen y vibration. As a result, the asing- xed a

elerometer re eives predominantly a low-frequen y vibration from the rotor.

Con luding Remarks Thus, the vibration spe trum of the asing- xed a

elerometer in the low-frequen y band is formed by the rotor and vanes generated vibration. In the wide frequen y band (up to 20: : : 25 kHz), it is formed, mainly, due to vane vibration. The measurements of asing vibration in the wide frequen y band and the use of the e ho ee t permit to separate the hanges in ma hine ondition from those indi ating variation of stru tural properties and also to verify the sour e of vibration

hanges (rotor or stator).

Referen es 1. Samoilovi h, G. S. 1975. Ex itation of blades vibrations in turbo ma hines . Mos ow: Mashinistroenie. 2. Karasev, V. A., V. A. Maksimov, and M. K. Sidorenko. 1978. Vibration diagnosti s of gas-turbine engines . Mos ow: Mashinistroenie. 3. Kiselev, J. V. 1980. Spe i features of vibration, ex ited by blade assemblies of GTE. In: Problems of applied me hani s in aviation. Kuibyshev. 2:178{91. 4. Mironov, A. G., and S. M. Doroshko. 1986. Appli ation of vibration spe trum hara teristi s for diagnosti s of the du t ow in aviation GTE. Izvestiya VUZov. Aviation Te hni s 2:45{49. 5. Mironov, A. G. 1991. Study of diagnosti features of random vibration in the du t ow of turboma hines. In: The provision of aviation engine reliability. Kiev: KNIGA.

72

A. Mironovs

SECTION 2

AERODYNAMIC DAMPING OF BLADEROW VIBRATIONS

Aerodynami Damping of Bladerow Vibrations EXPERIMENTAL AND NUMERICAL STUDY OF UNSTEADY AERODYNAMICS IN AN OSCILLATING LOW-PRESSURE TURBINE CASCADE OF ANNULAR SECTOR SHAPE

D. M. Vogt, H. E. Martenssony, and T. H. Fransson  Royal

Institute of Te hnology Department of Energy Te hnology Sto kholm S-10044, Sweden yVolvo Aero Department of Aerothermodynami s Trollhattan S-461 81, Sweden The unsteady aerodynami s during ontrolled blade os illation in an annular se tor as ade with low-pressure turbine (LPT) blades has been studied experimentally and numeri ally. Following the in uen e oeÆ ient approa h, a as ade of seven blades has been employed with the middle blade made os illating in ontrolled modes. One of the novelties of the presented study is the ombination of three-dimensional (3D) ow and 3D blade os illation due to in reasing bending amplitude from hub to tip as present for low-order stru tural modes. On the numeri al side, a linear Euler utter predi tion tool has been used at dierent degree of detailing. The results indi ate that the invis id model is apable of

apturing the main features of the unsteady aerodynami s during blade os illation and that it an be used to support design work. Introdu tion

The designing and maintaining of utter-free gas turbine engines remains one of the paramount hallenges for engine manufa turers. Flutter denotes a self-ex ited and self-sustained instability phenomenon that might lead to stru tural failure in a short period of time unless properly damped. To predi t eventual o

urren es of utter during engine opD. M. Vogt et al.

75

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

eration, it is ne essary to a

urately predi t the aeroelasti stability at

ertain operating onditions and take into a

ount stru tural properties of the setup. In a ommon approa h, the aerodynami and stru tural parts are solved separately and nally ombined; in the present paper only the aerodynami part is onsidered. A number of unsteady aerodynami predi tion models have appeared over the last de ades; simple analyti al and potential models have been presented in [1, 2℄ featuring good performan e for quasiisentropi ow (i.e., weak sho ks). This limitation is overruled by the dis rete Euler method although it does not a

ount for linearized shear stress perturbations, whi h are present in boundary layers or around separated ow regions. To resolve time-unsteady ow, the method an be solved in either a time-mar hing manner or as a harmoni solution. The rst approa h [3, 4℄ solves the equation at indi ated time steps and a hieves onvergen e upon a ertain number of os illation periods. The linear harmoni approa h assumes small perturbations of ow variables around a steady mean value that ould be obtained from a steady nonlinear ow analysis [5, 6℄. Validation test ases for unsteady aerodynami models vary widely in geometri al setup, degree of omplexity, and instrumentation. Most fa ilities feature ontrolled blade os illation allowing for a

urate and distin t setting of os illation parameters. Following the in uen e oeÆ ient approa h, one blade only is os illated whereas the unsteady pressure is measured on several blades. Bol s et al. [7℄ have presented one of few on lusive studies on this approa h employing an annular

as ade with one and all blades os illating. For the sake of simpli ity, os illating as ade tests were performed in two-dimensional (2D) setups (i.e., linear as ades) [8, 9℄. Nevertheless, the annular shape of turboma hines indu es radial gradients that an only be obtained in annular test setups. In the present work, an annular test fa ility has been employed for studying the unsteady aerodynami s in a as ade during ontrolled blade os illation. One of the novelties of the present study is a radially varying lo al bending amplitude of the blade in ombination with radial

ow gradients from the mean ow eld. On the numeri al side, a linear Euler utter predi tion tool has been used on 2D as well as two 3D models; one of the 3D models featured tip learan e su h as to in rease further the degree of detailing. 76

D. M. Vogt et al.

Aerodynami Damping of Bladerow Vibrations

In uen e CoeÆ ient Approa h Commonly, utter in turboma hine bladerows is des ribed by the traveling wave mode approa h assuming that all the blades are os illating in the same mode and at the same amplitude and frequen y [10℄. Nowinski and Panovsky [11℄ have shown that this assumption represents the least stable ondition and therefore tends to be over onservative. Considering a bladerow of N blades, the traveling wave mode response ontains

ontributions from all the blades, whi h superimpose linearly at a ertain interblade phase angle su h as

^m; pA;twm (x; y; z ) =

X

N=2 n= N=2

^n;m pA;i (x; y; z ) e

in

(1)

In the above equation, ^m; pA;twm (x; y; z ) is the omplex pressure oeÆ ient at point (x; y; z ), a ting on blade m, with the as ade os illating in the traveling wave mode and at interblade phase angle , and ^n;m pA;i (x; y; z ) is the omplex pressure oeÆ ient of the vibrating blade n, a ting on the nonvibrating referen e blade m at point (x; y; z ). Blade indi es are herein as ending in dire tion of the su tion side and des ending in dire tion of the pressure side, respe tively. The oeÆ ients on the lefthand side of Eq. (1) are des ribing the traveling wave mode domain whereas the ones on the right-hand side are des ribing the in uen e

oeÆ ient domain.

Experimental Setup Figure 1 depi ts pro le se tions of the LPT pro le used in the study at three spanwise heights as well as the test se tion. The pro le features a real hord of 50 mm at mid span and an aspe t ratio (span/ hord) of 1.94 at a radius ratio of 1.25. The blade is assembled in an annular

as ade with ylindri al hub and asing ontours at a pit h/ hord ratio of 0.68 at mid span and a tip learan e of 1% span. Nominal in ow to the as ade was at 26Æ yielding 87Æ in turning. The redu ed frequen y has been varied during the tests by ontrolling the blade os illation frequen y. The redu ed frequen y is based on full D. M. Vogt et al.

77

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Test obje t ( ) and test se tion ( ): 1 | 10%, 2 | 50% 3 | 90%, and 4 | enter of torsion

Figure 1

a

b

hord here. A more omplete des ription of the test ase is presented in [12℄.

Numeri al Method The linearized Euler method employed herein is the small-perturbation harmoni approximation of a nonlinear Euler method. The method is implemented in the 3D solver VOLSOL [13℄, whi h is a stru tured multiblo k ow solver based on nite volumes. While the nonlinear Euler method of VOLSOL uses deforming grids at nite amplitude, the linearized method uses an analyti ally evaluated in nitesimal mesh deformation as a part of the ux al ulation to take are of the mesh movement. The underlying nonlinear Euler model is a high order nite volume te hnique allowing for moving meshes by onsidering a nite volume extending also in time. In order to derive the harmoni linear Euler method, a harmoni linearization in the time dire tion is onstru ted around a steady-state solution, whi h satis es the nonlinear Euler equations on a omputational domain with nonmoving boundaries. The ode features 2D and 3D apabilities, whereof the 2D part has been assessed in [14℄ on the STCF4 ase with good results. The 3D part has been assessed in [15, 16℄ with good results as well. In [15℄, the 78

D. M. Vogt et al.

Aerodynami Damping of Bladerow Vibrations

ee ts of mesh quality, sensitivity to numeri al approximation as well as the o

urren e of spurious instabilities have been addressed.

Results Figure 2 depi ts omparisons of the omputed unsteady pressure obtained from the 2D and 3D models without tip learan e with the test data at midspan for three middle blades. The data are plotted su h that ar = 0 is lo ated at the blade leading edge whereas the negative

Figure 2

Unsteady blade surfa e pressure at midspan; axial bending:

test data,

| 3D, and

2

D. M. Vogt et al.

3

1

|

| 2D

79

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

and positive bran hes are spanning the su tion and pressure sides, respe tively. The top part of the respe tive graph depi ts the pressure amplitude while the pressure phase is in luded in the bottom part. It is re ognized that the dominant part of unsteady pressure response is present on the os illating blade (index 0) as well as the blade surfa es of the nonos illating neighboring blades fa ing the os illating blade (i.e., pressure side of blade +1 and su tion side of blade 1). Both models perform well by apturing the essential hara ter of the unsteady ow; espe ially in regions of onsiderable pressure amplitude, it is important to orre tly apture the pressure phase in order to be able to a

urately predi t aeroelasti stability. Whereas the 2D model

Figure 3 bending:

80

1

Unsteady blade surfa e pressure at midspan; ir umferential

| test data,

2

| 3D, and

3

| 2D

D. M. Vogt et al.

Aerodynami Damping of Bladerow Vibrations

Figure 4 bending:

1

Ar wise stability ontribution at midspan on blade

| test data,

2

| 3D with tip learan e,

3

| 3D, and

1; axial

4

| 2D

predi ts the level of pressure perturbation more a

urately around the peak response (ar = 0:11), the 3D model yields a qualitatively more a

urate though overpredi ted result. The response at ir umferential bending mode is shown in Fig. 3. Again, it an be re ognized that both models apture the overall hara ter of the unsteady ow. On the os illating blade, the 3D model lies within measurement a

ura y whereas the 2D model slightly overpredi ts the response on the pressure side lose to the trailing edge. On blade 1, the 2D model predi ts a broader peak on the su tion side, whi h is not supported from the test data. The ee t of model detailing on predi tion of resolved aeroelasti stability ontribution is addressed in Figs. 4{6. Here, the proje tion of the pressure onto the tested modes is analyzed on blade 1. Using this D. M. Vogt et al.

81

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 5

Ar wise stability ontribution at midspan on blade

ferential bending:

4

1

| test data,

2

| 3D with tip learan e,

3

1; ir um| 3D, and

| 2D

approa h, information on the lo al aeroelasti stability, ontribution of a spe i blade is obtained dire tly. At axial bending mode depi ted in Fig. 4, a distin t improvement is obvious when moving from the 2D model over the 3D model without tip learan e to the 3D model with tip

learan e. This improvement is visible on both the dire t in uen e (top graph) as well as the oupling in uen es (middle and bottom graphs) and underlines the dire t gain in a

ura y obtained with the in reased degree of detailing. At ir umferential bending mode shown in Fig. 5, the in rease in predi tion a

ura y is again obvious, espe ially for the two oupling in uen es. Note that for orre t predi tion of the mode shape sensitivity, 82

D. M. Vogt et al.

Aerodynami Damping of Bladerow Vibrations

Figure 6

1

Ar wise stability ontribution at midspan on blade

| test data,

2

| 3D with tip learan e,

3

| 3D, and

4

1; torsion:

| 2D

not only the dire t but also the oupling in uen es must be predi ted

orre tly. To give the omplete pi ture, the aeroelasti stability ontribution at torsion mode is shoen in Fig. 6. Similar to the two other modes, lear improvement is visible for the dire t and the oupling oeÆ ients. In addition, it is striking to note that the 2D model performs better for this mode than the 3D model without tip learan e.

Con luding Remarks A ombined experimental and numeri al study on the unsteady aerodynami s in an os illating as ade has been presented. The annular se tor as ade of LPT blades was used as the test obje t. Following the D. M. Vogt et al.

83

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

in uen e oeÆ ient approa h, one of the blades was made os illating in ontrolled modes (two bending and one torsion mode). The ombination of three-dimensionally varying bending amplitude and prevalent 3D ow features represents one of the novelties of the present study and was used as a hallenging and representative test ase. The linear Euler method with moving meshes was employed as a predi tion tool. Results were obtained using a 2D model, 3D model without tip learan e, and 3D model with tip learan e. Comparisons of the resolved unsteady blade-surfa e pressure revealed superiority of the more omplex model; although the 3D models were not always able to apture the exa t level of pressure perturbation, it has been re ognized that the overall hara teristi s of the unsteady

ow during ontrolled blade os illation ould be predi ted more a

urately. From proje tions of pressure data onto the respe tive mode shapes, lo ally resolved aeroelasti -stability data were obtained. Correlations of numeri al results and test data revealed a lear improvement when moving from the 2D model to the 3D model with tip learan e. Considering the limited though still satisfa tory predi tion a

ura y of the 2D model, it is on luded that the use of su h models is justi ed for supporting preliminary design work, espe ially when assessing a large number of geometries. The gain in predi tion a

ura y has been demonstrated on the present test ase when moving over a 3D model without tip learan e to a 3D model with tip learan e. For obtaining a

urate results, it is therefore on luded that the use of more omplex models is inevitable.

A knowledgments The present study has been promoted within the Swedish Gas Turbine Center (GTC) and the EU-funded proje t DAIGTS ( ontra t number ENK5-CT2000-00065). The authors would like to a knowledge this nan ial support as well as the support from the Royal Institute of Te hnology, Sweden.

Referen es 1. Verdon, J. M., and J. R. Caspar. 1982. Development of linear unsteady aerodynami s for nite-de e tions as ades. AIAA J . 20(9).

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Aerodynami Damping of Bladerow Vibrations 2. Whitehead, D. S. 1987. Classi al two-dimensional methods. AGARD Manual on Aeroelasti ity in Axial Flow Turboma hines. AGARD. I:31{3-30. 3. Whit eld, D. L., T. W. Swaord, and R. A. Mula . 1987. Threedimensional unsteady Euler solutions for propfans and ounter-rotating propfans in transoni ows. AIAA Paper No. 87-1197. 4. Giles, M. B. 1988. Cal ulation of unsteady wake/rotor intera tion. AIAA J. Propulsion Power . 4(4)356{62. 5. Kahl, G., and A. Klose. 1991. Time linearized Euler al ulations for unsteady quasi-3D as ade ows. 6th International Symposium on Unsteady Aerodynami s, Aero ousti s and Aeroelasti ity of Turboma hines and

Propellers Pro eedings . 6. Hall, K. C. 1999. Linearized unsteady aerodynami s. Le ture series program on aeroelasti ity in axial ow turboma hines. Durham, USA: Duke University. 7. Bol s, A., T. H. Fransson, and D. S hla i. 1989. Aerodynami superposition prin iple in vibrating turbine as ades. AGARD Conferen e Unsteady Aerodynami Phenomena in Turboma hines Pro eedings . Luxembourg: 468. 5.1{5.20. 8. Carta, F. O. 1982. Unsteady gapwise periodi ity of os illating as aded airfoils. ASME J. Engineering Power 105(3):1983. 9. Buum, D. H., and S. Fleeter. 1991. Wind tunnel wall ee ts in a linear os illating as ade. ASME J. Turboma hinery 115(1):147{56. 10. Crawley, E. F. 1988. Aeroelasti formulation for tuned and mistuned rotors. AGARD manual on aeroelasti ity in axial- ow turboma hines. Vol. 2. Stru tural dynami s and aeroelasti ity. Ch. 19. AGARD-AG-298. 11. Nowinski, M., and J. Panovsky. 2000. Flutter me hanisms in low pressure turbine blades. J. Engineering Gas Turbine Power 122:82{88. 12. Vogt, D. M., and T. H. Fransson. 2006. Experimental investigation of mode shape sensitivity of an os illating LPT as ade at design and odesign onditions. ASME Paper GT2006-91196. 13. Lindstrom D., and H. M artensson. 2001. A method for utter al ulations based on the linearised ompressible Euler equations. IFASD Pro eedings . 64. 14. T herny heva, O. T., S. Regard, F. Moyroud, and T. H. Fransson. 2000. Sensitivity analysis of blade mode shape on utter of two-dimensional turbine blade se tions. ASME Paper 2000-GT-0379. 15. M artensson, T. H., D. M. Vogt, and T. H. Fransson. 2005. Assessment of a 3D linear utter predi tion tool using se tor as ade test data. ASME Paper GT2005-68453.

D. M. Vogt et al.

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

16. Vogt, D. M., H. E. M artensson, and T. H. Fransson. 2005. Validation of a three-dimensional utter predi tion tool. NATO Symposium on Evaluation, Control and Prevention of High Cy le Fatigue in Gas Turbine Engines for Land, Sea and Air Vehi les Pro eedings

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. Granada, Spain.

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Aerodynami Damping of Bladerow Vibrations A THREE-DIMENSIONAL TIME-LINEARIZED METHOD FOR TURBOMACHINERY BLADE FLUTTER ANALYSIS







F. Poli , E. Gambini , A. Arnone , and C. S hipani

 \Sergio

1

y

Ste

o" Department of Energy Engineering University of Floren e Via S. Marta 3 Floren e 50139, Italy y Avio Group | R&D Via 1Æ Maggio 99 Rivalta di Torino 10040, Italy

Introdu tion

Nowadays, engine weight redu tion is a major on ern for aeroengine designers: they need to redu e the impa t of in reasing fuel pri e on operation osts on one side and the environmental impa t on the other, by lowering fuel onsumption and emissions. The goal of engine weight redu tion is often a hieved by de reasing the number of me hani al parts and by adopting thin and highly loaded blades. This approa h, while helping to redu e engine life ost, in reases signi antly the relevan e of aerodynami ally indu ed vibration phenomena ( utter and for ed response), whi h an result in blade high- y le fatigue failures. Predi ting and avoiding uid{blade intera tion indu ed vibrations has thus be ome a primary obje tive in aeroengine design. During the ongoing resear h eort on Computational Aeroelasti ity (CA) at the \Sergio Ste

o" Department of Energy Engineering (University of Floren e), an aeroelasti solver has been developed, in ollaboration with Avio Group. This solver, named Lars (time-Linearized Aeroelasti Response Solver), was designed to work together with the Traf steady/unsteady aerodynami solver [1, 2℄. A rst Lars variant F. Poli et al.

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

solves quasi-three-dimensional aeroelasti equations and is also the basis for a utter s reening pro edure for real and omplex modes [3℄; a fully three-dimensional (3D) variant has also been developed. The aim of this paper is to present this 3D method for turboma hinery blade utter analysis. The approa h adopted is un oupled and time-linearized (see [4℄ for an overview of omputational aeroelasti ity methods).

2 2.1

Numeri al Method Linearized un oupled method

The time-linearized un oupled method for utter analysis [5{7℄ works as follows. Blade vibrations are derived from modal analysis results and assigned to the aeroelasti al ulation as input data. These os illations are supposed to be time-sinusoidal and small (ideally in nitesimal); moreover, they are iso-frequen y (the whole bladerow vibrates at one frequen y) and in traveling wave form (there is a onstant phase delay between ea h pair of adja ent blades: the InterBlade Phase Angle, or IBPA for short). A harmoni deformation is de ned for the uid mesh by perturbing the undeformed steady grid: w0

= w + 1 is set at Mn < 1 (where Mn is the Ma h number determined by the V - omponent normal to the

as ade front. The parameters in the right layer are also set arbitrarily (ex eption: the values of pressure p+ and total enthalpies determining the ow are identi al in both layers). When using the dis ontinuity de ay s hemes, the setting of all parameters in the adja ent layers does not overde ne the problem, be ause as a result of dis ontinuity de ays at and + , only the information permitted by rigorous statement of a boundary value problem for 2D nonstationary Euler equations is transmitted. At the lower and upper boundaries of the omputational domain (at  = 0 and  = T = N ), the periodi ity onditions are satis ed with the help of similar layers. The SWS evolution in the regions outside the domain

al ulated by \Chimera" ode is determined by fast ANA algorithm. Figure 6 shows the al ulated (by \Chimera") pressure eld in the ow through the nonideal as ade of blades [14℄ with rounded leading edges with the urvature radius r = 0:1. For the sele ted grids, the SWS

ompletely disappear at a erFigure 6 Pressure eld al ulated tain distan e from the as ade. by \Chimera" ode for a nonideal as- Near the forward front, a solu ade. The insert in the upper left orner tion is suÆ iently pre ise. It alshows the ow in the neighborhood of lows one to use the distributions one of the blades of parameters at n = 0:5{1.0, where the sho k intensities are already insigni ant, for determining the as ade dire ting ee t and for al ulation of SWS using ANA. Then, the spe tra and total intensities of noise radiation determined by fast Fourier transform, showed that for nonideal as ades, the nonlinear SWS damping ampli es with in reasing the bluntness radius. Thus, for as ades with 24 blades with 290

N. L. Efremov et al.

Flow Path Aeroa ousti s

identi al dispersion of installation angles and n = 10, the total intensity of SWS noise at r = 0:1 is 2.3 dB lower than at r = 0. Cal ulation of Sho k-Wave Stru ture Ahead of Ideal Wheel

At high rotation speed of fan wheel, the ow near the ir umferential parts of its blades an be supersoni . For 3D- ow al ulation of a wheel with identi al blades and identi al installation angles, the omputational odes integrating Euler equations in Cartesian oordinates rotating together with the wheel were developed. In the ourse of integration of the invis id 3D- ow in the oordinates, rotating together with the wheel, the steady and periodi in a ir umferential dire tion with period 2=N solution is obtained. The numeri al method, whi h extends to three spatial dimensions the method applied for al ulation of ows in as ades, was implemented both in expli it and impli it versions. The expli it version permits integration with a lo al time step, while the impli it method allows one to redu e the omputational time by a fa tor of 3{4. Figure 6 demonstrates the apabilities of the odes developed. It shows Ma h number distribution at blades, hub, and outer bypasses for rh =rt = 0:3 (rh and rt are the radii of the hub and blade tip) when axial

Figure 7 Predi ted Ma h number distribution at partially supersoni ow through a wheel N. L. Efremov et al.

Figure 8 Ma h number distribu-

tion in the \tip" ylindri al se tion of a wheel shown in Fig. 7

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

subsoni Ma h number Mn = 0:7 and full Ma h number at the periphery M = 1:14. The SWS are well visible at outer bypasses. Their resolution is in omparably better than in the al ulations performed in [20, 21℄. The lo ation of sho ks and other features in Fig. 7 are illustrated by its plane analog (Fig. 8), whi h orresponds to an ideal as ade obtained as the \tip" ylindri al se tion of the wheel in Fig. 7. In this example, a good resolution and a

eptable CPU time are provided by grid adaptation des ribed above and the possibility of al ulation of only one \blade passage," as in the ase of an ideal as ade.

Con luding Remarks The presented examples showed that in the approximation of ideal gas, the methods and algorithms developed allow one to al ulate with a good a

ura y the a ousti performan es of SWS, propagating upstream from the as ades and wheels in the supersoni ow with subsoni axial velo ity omponent. When the SWS initial intensity is determined by the ow in the small neighborhood of blade leading edges, an invis id approximation is suÆ ient. For a wheel, this is valid for blade tip sites in the supersoni ow. Ahead of other parts of blades, SWS do not form. However, the ow ahead of them, whi h in uen es the evolution of \tip" SWS, is ae ted by vis osity in fan blade passages. Estimation of and a

ount for these ee ts are the tasks for further investigations.

A knowledgments This work is arried out with support of the Russian Foundation for Basi Resear h (proje ts 05-01-00846 and 05-01-08054-o p) and the State program of support of the leading s ienti s hools of Russian Federation (proje ts SS-2124-2003.1 and SS-9902.2006.1).

Referen es 1. Taganov, G. I. 1951. Spe i features of ows in axial supersoni ompressors. Tr. TsAGI. 2. Stepanov, G. Yu. 1962. Hydrodynami s of turbine as ades . Mos ow: Fizmatlit.

292

N. L. Efremov et al.

Flow Path Aeroa ousti s 3. Grodzovskii, G. L., A. A. Nikolskii, G. P. Swishov, and G. I. Taganov. 1967. Supersoni gas ows with perforated boundaries. Mos ow: Mashinostroenie. 4. Lawa ze k, O. K. 1972. Cal ulation of the ow properties up and downstream of and within a supersoni turbine as ade. ASME Paper No. GT47. 5. Li htfuss, H. J., and H. Starken. 1974. Supersoni as ade ow. Progress Aerospa e S ien e 15:37{150. 6. Bogod, A. B., A. N. Kraiko, and Ye. Ya. Chernyak. 1979. Investigation of plane as ade streamlining by supersoni ideal gas ow with subsoni \normal" omponent in regimes with atta hed sho ks. Izvestiya Akad. Nauk USSR, Mekhanika Zhidkosti y Gaza 4:108{13. 7. Fink, M. R. 1971. Sho k wave behavior in transoni ompressor noise generation. ASME Paper No. GT-7. 8. York, R. E., and H. S. Woodard. 1976. Supersoni ompressor as ades analysis of the entran e region ow eld ontaining deta hed sho k waves. Trans. ASME. Ser. A. J. Eng. Power 98(2):247{54. 9. Kraiko, A. N., D. E. Pudovikov, and N. I. Tillyayeva. 1995. Design of as ade with minimal drag in supersoni ow with subsoni velo ity omponent normal to as ade front. Izvestiya Ross. Akad. Nauk, Mekhanika Zhidkosti Gaza 1:137{46. 10. Morfey, C. L., and M. J. Fisher. 1970. Sho k wave radiation from a supersoni du ted rotor. Aeronauti al J. Royal Aeronauti al So iety 74(715):579{85. 11. Kraiko, A. N., V. A. Shironosov, and Ye. Ya. Shironosova. 1984. To stationary ideal gas ow in plane as ade. Applied Me hani s Te hni al Physi s 6:35{43. 12. Brailko, I. A., A. N. Kraiko, K. S. Pyankov, and N. I. Tillyayeva. 2003. Numeri al and theoreti al investigation of airdynami and a ousti performan es of supersoni fan as ade with subsoni axial velo ity omponent. Aeromekh. Gaz. Dyn. 4:9{22. 13. Hawkings, D. 1971. Multiple tone generation by transoni ompressors. J. Sound Vibration 17(2):241{50. 14. Kurosaka, M. 1971. A note on multiple pure tone noise. J. Sound Vibration 19(4):453{62. 15. Shim, I. B., J. W. Kim, and D. J. Lee. 2003. Numeri al study of N -wave propagation using optimized ompa t nite dieren e s hemes. AIAA J. 41(2):316{19. 16. Goldsteyn, A. W. 1974. Supersoni fan blading. Patent USA. No. 3820918. 28.06.1974. (Priority 21.01.1972.) N. L. Efremov et al.

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

17. Alexandrov, V. G., A. N. Kraiko, S. Yu. Krasheninnikov, V. Ye. Makarov, V. I. Mileshin, A. A. Osipov, V. A. Skibin, and V. I. Solonin. 2003. Rotor blade wheel of turbojet engine axial ompressor (fan). Patent Russia No. 2213272. 27.09.2003. (Priority 02.04.2002.) 18. Petersson, N. A. 1999. An algorithm for assembling overlapping grid systems.

SIAM J. S i. Comput.

20(6):1995{2022.

19. L ohner, R., D. Sharov, H. Luo, and R. Ramamurti. 2001. Overlapping unstru tured grids. AIAA Paper No. 01-0439. 20. Gliebe, P., R. Mani, H. Shin, B. Mit hell, G. Ashford, S. Salamah, and S. Connell. 2000. Aeroa ousti predi tion odes. NASA/CR-2000-210244. 21. Prasad, A. 2003. Evolution of upstream propagating sho k waves from a transoni ompressor rotor. 125(1):133{40.

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Transa tion ASME. J. Turboma hinery.

N. L. Efremov et al.

Flow Path Aeroa ousti s TWO-DIMENSIONAL NUMERICAL SIMULATION OF ROTOR{STATOR INTERACTION AND ACOUSTIC WAVE GENERATION

V. Aleksandrov and A. Osipov

P. I. Baranov Central Institute of Aviation Motors (CIAM) Aviamotornaya str. 2, Mos ow 111116, Russia

Computational s heme and omputer ode were developed for dire t numeri al simulation (DNS) of unsteady aerodynami intera tion of two plane pro le as ades and tone noise generated by this intera tion with regard for periodi vorti al wakes shedding from trailing edges of the front rotor as ade in vis ous ow. To over ome diÆ ulties related to DNS of unsteady wake formation in pulsating turbulent ow in a turboma hinery stage, an approximate simulation pro edure was developed applying arti ial initiation of shear layers in the al ulated ow eld, whi h (being time-averaged) satis ed the known semiempiri al relations for steady self-similar wakes. The initial velo ity pro le in the wake was introdu ed at some se tion behind the front rotor as ade, and its subsequent evolution during propagation through the stator as ade was des ribed by general equations of two-dimensional (2D) unsteady ow. Numeri al integration of unsteady ow equations was performed by expli it Godunov{Kolgan{Rodionov nite-dieren e s heme of se ondorder a

ura y in time and spa e. The model developed in orporated main qualitative and quantitative features of noise generation me hanism under onsideration and provided e onomi , onvenient and apparently more reliable up-to-date simulation instrument as ompared with DNS. Using the generalized spatial-temporal periodi ity ondition for ow parameters in the stage, the problem was redu ed to ow al ulation in one interblade hannel of ea h as ade for arbitrary numbers of blades in rotor and stator. The method allowed the stage a ousti hara teristi s to be determined as a result of harmoni analysis of V. Aleksandrov and A. Osipov

295

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

unsteady ow parameters at inlet and exit boundaries of the omputational region. Introdu tion

During previous 10{15 years, main resear h eorts in the development of omputational methods for unsteady aerodynami s of aviation turboma hines were on entrated on DNS of transient ow in a turboma hinery stage. Considerable su

ess has been rea hed today in this dire tion due to sweeping progress in omputer hardware. Analysis of relevant a

omplishments indi ates that the reliability of DNS an be ome a

eptable for pra ti al appli ations in the nearest future. However, at present, the a

ura y of su h al ulations is not suÆ ient, and onsiderable resear h eorts and years of rush work are still required. The basi diÆ ulties of numeri al simulation of turboma hinery tone noise onsist in ne essity of a

urate al ulation of omplex vorti al stru tures, whi h form in vis ous turbulent ow downstream the blade row. In the ow through a plane as ade, these vorti al stru tures are presented mainly by wakes shedding from trailing edges. Numeri al simulation of unsteady vis ous turbulent wakes, whi h an provide the a

eptable data for al ulating a ousti hara teristi s of blade row intera tion, has been a too omplex problem yet. For solving this problem, extraordinary grid re nement is required in thin boundary layers on pro le surfa es and in the wakes, so that the time-integration step be omes too small. Besides, at present, there are no reliable methods of turbulent vis osity des ription in transient as ade ows. In view of these diÆ ulties, it seems reasonable to use some simpli ations for approximate simulation of tone noise generation, e.g., to apply a ombined approa h based on the approximate model for time-averaged omponent of vis ous wakes behind the front as ade and DNS for unsteady as ade intera tion, in luding wake impa t on the ba k as ade. As for the approximate wake model, one an use known semiempiri al relationships for steady turbulent as ade wakes or numeri al results for steady vis ous ow through a single as ade. Within su h an approa h, all basi features of sound generation in a plane stage are onsidered in nonlinear manner. Disadvantages of this approa h are related to approximate modeling of unsteady omponents 296

V. Aleksandrov and A. Osipov

Flow Path Aeroa ousti s of wakes be ause unsteady wake formation during shedding of vis ous boundary layers is not resolved. The bene ts of this approa h onsist in a possibility to redu e onsiderably the total number of grid ells and to in rease an integration time step be ause thin boundary layers are not onsidered. The latter means that the approa h does not deal with the problem of wake formation. Instead, shear layers determined on the basis of onservation integrals and generalization of large volume of empiri al data are used. These layers are embedded arti ially into the unsteady ow and

ontain main qualitative and quantitative features of the noise generation me hanism under onsideration. The initial velo ity pro le in the wake is introdu ed at some se tion behind the front rotor as ade, and its subsequent evolution during propagation through the stator as ade is des ribed by general equations of 2D unsteady ow. The model developed provides e onomi , onvenient and apparently more reliable up-to-date simulation instrument as ompared with DNS.

Wake Model and Cal ulation Method For the wake model, approximate semiempiri al relationships obtained in [1℄ are used. The pro le of relative gas velo ity of a self-similar turbulent wake an be written in the form w0

p

w

=

 s

os2

 s

(1)

where s = 1:52  and  = h os . Here, w0 is the velo ity perturbation relative to the ow ore velo ity w;  is the oordinate along the wake axis, =s is the dimensionless oordinate normal to the wake axis, s is the wake width, h is the as ade gap,  is the loss fa tor, and  is the angle between the ore velo ity ve tor and x-axis, normal to the

as ade front. To determine  , an approximate empiri al dependen e  (D) an be used, where D = (wmax w2)=w1 is the diuser fa tor, wmax is the maximal gas velo ity on the as ade pro le, and w1 and w2 are the gas velo ities in front of the as ade and behind it. The value of D an be obtained from steady ow al ulation in a as ade. Equation (1) is used within Godunov{Kolgan{Rodionov nitedieren e s heme for modeling wake initiation in ow eld in any se V. Aleksandrov and A. Osipov

297

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 1 Comparison between al ulated and self-similar wake pro les tion between two grid ell layers, so that the wake appears in ell layer downstream of a given se tion. The subsequent wake evolution during propagation through the stator as ade is des ribed by Euler equations of 2D transient ideal gas ow. Vis ous stress terms are added to the

ow equations for onsidering wake dissipation during its propagation in the ow eld. The value of ee tive vis osity oeÆ ient was taken

onstant in the ow eld and was hosen to t the dependen e of wake width on oordinate  of Eq. (1). Figure 1 ompares the pro les of predi ted steady wake (solid line) and self-similar wake a

ording to Eq. (1) (dashed line) at dierent x along the wake axis. Figure 2 shows the stage on guration with two plane as ades, moving relative to one another at rates W1;2 (W1 = W2 ). Unsteady

ow in this system is time-periodi with periods T1 = h2 =W2 and T2 = h1 =W1 in the referen e frames y1 of the rst (front) and y2 of the se ond (ba k) as ades with gaps h1 and h2, respe tively. Besides, at any xed x- oordinate, the ow satis es the generalized spatial-temporal periodi ity onditions in the form: p(y1 ; t) p(y2 ; t)

298

=p =p



y1



y2

+ h1 ; t +

h1



W2

+ h2 ); t +

h2



(2)

W1

V. Aleksandrov and A. Osipov

Flow Path Aeroa ousti s

With Eqs. (2), the problem is redu ed to ow al ulation in one interblade hannel of ea h as ade for the arbitrary number of blades in rotor and stator. Computational realization of su h a pro edure implies storage of ow parameter at boundaries of the omputational domain during one temporal period of the

ow. The nonre e tion onditions at inlet and exit boundaries of the omputational domain are ensured for all perturbations by using spe ial buer zones with a oarse grid. In the al ulations of as ade intera tions, the wakes were initiated at some small downstream distan e from the trailing edges of the front as ade. Further wake propagation is unsteady due to a ousti waves and free vor- Figure 2 Stage on guration ti es shedding from the trailing edges due to unsteady aerodynami loads on the pro les. For demonstrating the omputational pro edure with unsteady wake behavior, a model problem was solved numeri ally for transient intera tion of a periodi wake set with a ousti waves. Numeri al results were obtained for the wake set propagating in the ow with Ma h number Mx = My = 0:5 and with oblique a ousti waves of dimensionless pressure amplitude P = 0:001, propagating downstream from left to right. Figure 3 shows the instantaneous elds of y omponent of velo ity, pressure, and entropy for this ase. It is seen that the pressure eld is rather omplex and ontains dierent re e ted omponents. Small entropy nonuniformities in the ow eld result from approximation errors in the numeri al s heme, whi h grow in the zones with intense shear in vorti al stru tures. Thus, the entropy eld hara terizes the lo al vorti ity in the velo ity eld. Vorti al stru tures in Fig. 3 demonstrate the growth of instability waves in the wakes. V. Aleksandrov and A. Osipov

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 3 Instantaneous elds of unsteady y- omponent of velo ity, pressure, and entropy in the periodi wake set The data presented in Fig. 3 show that the wake system, initiated arti ially in the al ulated ow, gives no nonphysi al defa ements and re e ts adequately main physi al ow features in a shear layer.

Results of Cal ulations This se tion presents the numeri al results illustrating the generation of tone sound in the stage. The a ousti hara teristi s of the stage are determined as a result of harmoni analysis of unsteady ow eld parameters at inlet and exit boundaries of the omputational domain. A

ording to Eqs. (2), the unsteady pressure in the referen e frames of the rst and se ond as ades an be presented as follows:

300

p1

=

p2

=

X1 X11

j= j=

1

e i!1j t e i!2j t

X1 1 1 X

n=

n=

1

p1jn(x)e1 1jn y1 p2jn(x)ei 2jn y2

V. Aleksandrov and A. Osipov

Flow Path Aeroa ousti s

Figure 4 Fragment of omputa- Figure 5 Predi ted instantaneous

tional grid

eld of Mx in the stage

n

j 1jn  2 + h1 h2 W !1j  2j 2 ; h2 y1 = y2 + W2 t ;



;

n

j + 2jn  2 h2 h1 W !2j  2j 1 h1 W2 = W1



If the nonre e tion ondition is satis ed at the inlet and exit boundaries and pressure perturbations are small at these boundaries, then p1;2jn determines the a ousti eld amplitudes of given frequen y harmoni (j) and y-mode (n). The numeri al pro edure outlined above was applied for al ulating unsteady aerodynami intera tion of two plane as ades. The rst

as ade onsisted of NACA 4404 pro les with hord length and gap h1 equal to 1.0, and the se ond as ade onsisted of NACA 6404 pro les with hord length equal to 1.0 and gap h2 = 0:75. The Ma h number of in oming x-axial uniform ow in front of the stage in the referen e frame of the se ond stator as ade was equal to 0.414, and the Ma h number of relative motion of the rst rotor as ade and stator as ade was equal to 0.45. Figure 4 shows a fragment of the omputational grid V. Aleksandrov and A. Osipov

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Table 1 Frequen y-mode onstitution of a ousti eld n

8 7 6 5 4 3 2 1

1 2 0 0 0 0 0 0 0 0 0 0 0 0.00070 0 0.00015 0.00085 0.00016

3 0 0 0 0 0.00035 0.00026 0.00006 0

j

4 0 0 0.00031 0.00014 0.00007 0.00005 0.00003 0

5 0.00022 0.00004 0.00003 0.00001 0.00002 0.00001 0 0

6 7 0.00004 0.00004 0.00002 0.00001 0.00002 0 0 0 0 0.00001 0.00001 0 0 0 0 0

for a small vi inity of trailing edge of the front as ade and leading edge of the ba k as ade. The grid onsisted of 81 000 ells in this ase. Figure 5 shows the instantaneous al ulated eld of Ma h number Mx for the x- omponent of gas velo ity in this stage. In this pi ture, one

an learly see the stru ture of wakes shedding from the rst as ade and their deformation as result of intera tion with the se ond as ade. Table 1 presents the amplitudes of dierent omponents p1jn of a ousti eld in front of the stage in the referen e frame of the rst

Figure 6 Frequen y-mode sound stru ture at inlet and outlet se tions 302

V. Aleksandrov and A. Osipov

Flow Path Aeroa ousti s

Figure 7

Sound frequen y spe trum:

two as ades without wakes, and (a ) Inlet and (b ) outlet

3

1

| two as ades with wakes,

2

|

| one as ade with in oming wakes.

as ade. All a ousti omponents in Table 1 are propagating (satisfying

ut-on ondition). The transposed table presents the orresponding frequen y-mode onstitution of a ousti eld in the stator referen e frame. It is seen that the generated sound ontains rather intense high harmoni s of rotor blade passing frequen y. Figure 6 shows the frequen y-mode stru ture of generated sound pressure in the form of 2D diagram for two se tions at the stage inlet and exit. The frequen y spe trum data are presented for three ases: two as ades with wakes, two as ades without wakes, and a single ba k

as ade with in oming wakes (Fig. 7).

Con luding Remarks 1. The al ulation method and omputational ode were developed for numeri al simulation of tone sound generated by unsteady aerodynami intera tion of two plane-pro le as ades moving relative to ea h other with regard for the impa t of periodi vorti al wakes shedding from trailing edges of the front rotor as ade in vis ous

ow. The al ulation method is based on the dire t numeri al integration of unsteady ideal-gas ow equations using Godunov{Kolgan{ Rodionov nite-dieren e s heme and the approximate model of vis ous wakes. V. Aleksandrov and A. Osipov

303

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

2. For approximate modeling of wakes, a spe ial numeri al pro edure was developed to initiate shear layers in the ow eld in a

ordan e with semiempiri al relationships for self-similar vis ous turbulent wakes in the as ade ow. 3. Analysis of al ulated unsteady aerodynami hara teristi s of the

ow with arti ially initiated wakes showed that the developed simulation method provided adequate des ription of real wake ows. 4. Numeri al results, obtained using this method, demonstrated its high

apabilities for numeri al simulation of tone sound generation in the plane model of a turboma hinery stage.

A knowledgments This study was supported by the Russian Foundation for Basi Resear h (proje ts 05-01-00846 and 05-01-08054).

Referen es 1. Samoylovi h, G. S. 1962.

Ex itation of turboma hinery blade os illations .

Mos ow: Fismatgiz.

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V. Aleksandrov and A. Osipov

SECTION 5

UNSTEADY FLOWS IN TURBOMACHINES

Unsteady Flows in Turboma hines

EVALUATION OF UNSTEADY EFFECTS IN A MULTISTAGE AXIAL COMPRESSOR USING A PRECONDITIONED GMRES SOLVER M. Stridh and L.-E. Eriksson

Division of Fluid Dynami s Department of Applied Me hani s Chalmers University of Te hnology Goteborg SE-412 96, Sweden The deterministi stress (DS) terms governing medium-s ale deterministi unsteady ee ts in turboma hinery ows an be determined to some degree of a

ura y by solving the linearized Navier{Stokes (LNS) equations. This is usually done by a traditional time stepping pro edure, whi h unfortunately often fails to rea h onvergen e when large separations o

ur in the averaged ow eld. This work presents an alternative solution pro edure for the LNS equations, using a pre onditioned Generalized Minimal RESidual (GMRES) algorithm. As far as the authors are aware, this is one of the rst appli ations where a robust GMRES algorithm has been used to solve the LNS equations. Using this te hnique for a 3.5-stage ompressor, it be ame possible to determine and evaluate the unsteady ee ts aused by the DS terms on the steady ow eld in a multistage environment. Several operating points for the 3.5-stage ompressor have been omputed along a speed-line and the omputational results were ompared with the results obtained by onventional Reynolds averaged Navier{Stokes (RANS) approa h as well as with experimental data. The rst 1.5 stages of the fan were also solved with an unsteady RANS (U-RANS) method, and omparison with RANS and RANS + DS was performed, showing improved averaged properties in the latter ase. Also, a simpli ed test ase was examined, onsisting of two periodi blo ks, separated by a mixing plane (MP). It was found that the use of one-dimensional (1D) absorbing type of boundary ondition at the M. Stridh and L.-E. Eriksson

307

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

MP interfa es had an in uen e on the mixing in the entire upstream domain, i.e., some of the mixing is performed prior to the MP interfa e due to the boundary ondition. The idea with using the DS terms is to

ompensate for the instant mixing at the MP interfa e. This means that if a signi ant part of the mixing pro ess has already been performed upstream of this interfa e, when the DS terms are evaluated mu h of their ee t is lost. This is an important on lusion from a rather simple test ase. An alternative, pressure type, of boundary onditions (b. .) at the MP is suggested, whi h shows promising behavior over MP.

1

Introdu tion

The ow eld in a transoni ompressor is ompressible, threedimensional (3D), and highly unsteady. Due to high Reynolds numbers involved, the omputational ost for a dire t numeri al simulation (DNS) or large-eddy simulation (LES) would be prohibitive and these methods are therefore not appli able for engineering appli ations. Even if ensemble averaging is used for modeling turbulent u tuations, and the U-RANS method with full rotor{stator intera tion is applied, the

omputational ost in a multistage ompressor will be still very high. At present, one of the most widely used omputational methods for analysis of turboma hinery ows is the 3D RANS MP model in whi h tangentially averaged ow properties are transferred between bladerows [1℄. The main dieren e between full unsteady simulation and the MP solution is the la k of all the medium-s ale unsteady effe ts due to bladerow intera tions in the latter ase. When writing the average passage- ow equations, for the MP ase, the so- alled DS terms appear missing [2℄. In addition, the average passage formulation introdu es a losure problem, as both the turbulent and DS terms need to be modeled. Over the years, several losure models for the DS terms have been proposed. All of them have given improved omputational results when

ompared with measurements and unsteady omputations. Most of these models in lude only one part of the total DS term, namely, that whi h an be dedu ed from the steady ow eld of the neighboring bladerows via a ir umferential averaging pro edure (spatial part). As pointed out by Dano et al. [3℄ and Baralon [4℄, the spatial, spatialtemporal and purely temporal parts of the DS are of the same order 308

M. Stridh and L.-E. Eriksson

Unsteady Flows in Turboma hines of magnitude and hen e are equally important for a transoni ow eld. Furthermore, Bardoux and Leboeuf [5℄ showed that by in luding only the spatial orre tion of the DS term, a less a

urate result was obtained

ompared to that without any orrelation at all. Hen e, an approa h for in luding all parts of the DS term is important. In this work, the DS terms are predi ted via a linearized harmoni approa h, whi h was rst ombined with the average passage equations by Giles [6℄. In brevity, the methodology for this approa h is to solve rst for the steady ow in the entire multistage ompressor in the onventional manner without the ee t of DS terms. From this solution, the deterministi perturbations due to bladerow intera tion an be derived as the dieren e between the a tual ow state and the ir umferentially average ow state at the inlet or outlet of ea h blade passage. These perturbations are then onsidered separately using an LNS solver, from whi h the DS terms an be omputed. The underlying assumption in this approa h is that the u tuations are suÆ iently small and hen e are governed to a suÆ ient degree by the LNS equations. This linearized method was rst developed for two-dimensional (2D) potential ows by Casper and Verdon [7℄ for aeroelasti al ulations. Later, Hall and Loren e [8℄ and Ning and He [9℄ developed 3D linearized Euler solvers for vibrating blades. He and Ning [10℄ developed an LNS solver where a frozen eddy vis osity was assumed and this method was later used for simulating rotor{stator intera tion by Chen et al. [11℄, who showed that the DS terms an be a

urately predi ted by this method. This result has been repeated by the present authors and the ee ts of DS terms on the average ow have been studied as well [12, 13℄. The LNS equations in frequen y domain are traditionally solved by a time stepping pro edure by adding a pseudo-time variable. Unfortunately, when using this solution method, onvergen e problems often appear, espe ially in ases with large separations (shear) in the average

ow eld. An alternative solution pro edure for the LNS equations, using a pre onditioned GMRES algorithm has therefore been developed and as far as the authors are aware, this is one of the rst appli ations where a robust GMRES algorithm has been used to solve the LNS equations [14℄. By using this te hnique for a 3.5-stage ompressor, it be ame possible to determine and evaluate the unsteady ee ts on the steady owM. Stridh and L.-E. Eriksson

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

eld in a multistage environment. Several operating points for the 3.5-stage ompressor have been omputed along a speed-line and the

omputational results were ompared with the results obtained by the

onventional RANS approa h as well as with experimental data. Typi al design parameters, su h as, ow angles, losses, et . were evaluated.

2

Average Solver

The average passage equations [2℄ are solved using the G3d ow solvers originally written by Eriksson [15℄. For turbulen e losure, the standard k{ model with wall fun tions is used. In order to over ome some of the weaknesses of this turbulen e model, a realizability limiter and the Kato{Launder x is implemented [16℄. The numeri al method is based on an expli it time-mar hing (threestep Runge{Kutta with lo al time step a

eleration), ell- entered nite-volume approa h using a multiblo k nonorthogonal boundary tted stru tured grid. For further details about the average ow solver, refer to [15, 17, 18℄. At the outlet boundary, either a onstant pressure ondition or a so- alled throttle ondition is used. The throttle onditioned makes it possible to rea h a higher pressure ratio before running into numeri al stall. It works as if an imaginary throttle is pla ed after the ompressor whi h an adjust the mass ow rate through the system. So, for example, if the wanted stati pressure at the ompressor outlet leads to separation that ee ts the mass ow rate, it will temporarily lower the pressure at the outlet boundary. When the mass ow rate in reases again, the pressure at the boundary is also in reased. In this way, the solution is not for ed to stall. A subsoni inlet ondition is used with spe i ed values of h0, p0 ,

ow angles, k, and . Between rotating and stationary bladerows, an MP interfa e is used in the RANS omputations and a sliding grid interfa e in the U-RANS.

3 Linearized Solver From the steady-state solution, the deterministi perturbations Q0 an be approximated as the dieren e between the a tual ow state and the 310

M. Stridh and L.-E. Eriksson

Unsteady Flows in Turboma hines

tangentially average ow state at the outlet of ea h blade passage. This perturbation is then assumed to vary harmoni ally in time as: 1 X Q0 = Q^ nei!n t n= 1

!n = Nbu n
; n = 1; 2; 3; : : : ;

1

where Q^ n is the desired omplex variable eld, !n is the harmoni frequen y de ned by the number of blades Nbu in the upstream bladerow, and
is the relative angular velo ity between adja ent bladerows. The LNS equation in frequen y domain valid for the harmoni frequen y of nth order an be written as:    i!n Q^ n + (A0 Q^ n ) + (B0 Q^ n) + (C0Q^ n) = 0 x

y

z

where A0, B0, and C0 are the ux Ja obian matri es: A0 =

 E  Q

0

;

B0 =

 F  Q

0

;

C0 =

 G  Q

0

The subs ript \0" refers to the referen e solution Q0 obtained by the average ow solver. The dis retized version of the above equation, without and with added pseudo-time, an be written in a ompa t form: Ax = b ; x_ + Ax = b (1) where ve tor x ontains all the degrees of freedom (DOF) of the problem, ve tor b is due to the driving terms in the boundary onditions (inhomogeneous terms), and matrix A represents all of the spatial differen ing terms of the linearized ow solver. As far as the authors are aware, this is one of the rst appli ations of a really robust pre onditioned GMRES algorithm to solve the LNS equation. GMRES by itself [19℄ is a gradient method where the residual forms an orthogonal base spanning the Krylov subspa e (the base ve tors an be obtained via the Arnoldi method). The residual rm to the approximate solution xm is minimized in the l2 norm. M. Stridh and L.-E. Eriksson

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Sin e the spe trum of matrix is extremely ri h, it is required to

ompute a great number of basis ve tors to resolve the ne essary part of the Krylov subspa e needed for rea hing onvergen e via the GMRES method alone. For this reason, one needs to apply some intelligent pre onditioning pro edure for the linear set of Eqs. (1) to transform the spe trum of matrix in su h a way that GMRES works eÆ iently. The pre onditioning involves solving two dierent problem sets: ( ) the homogeneous ase of Eqs. (1) ( (0) 6= 0, = 0) using N pseudotime steps
t and ( ) the inhomogeneous ase of Eqs. (1) ( (0) = 0, 6= 0) using N pseudo-time steps
t. With this pro edure, one an obtain a new linear set of equations: ( {z e T A}) = (| e T{zA ) 1 } | A

A

i

x

b

ii

x

b

x

I

C

I

A

b

d

where e T A is approximated via the 3-stage Runge{Kutta (RK3) time stepping method and T = N
t. Note that max(
t) is determined here by the stability onstraint of the RK3 time stepping method, and is thus xed for a given grid, referen e solution, and spatial dis retization s heme. The main reason for the ee tiveness of GMRES for matrix is due to the spe trum transformation, whi h makes a large part of the spe trum ollapsing towards one point in the imaginary plane. The ee tiveness of the pre onditioning and GMRES algorithm were dis ussed in more detail in a previous paper by Stridh and Eriksson [14℄. C

4 4.1

Results Convergen e

Both the traditional time stepping and a more ee tive pre onditioned GMRES te hnique were used for a 3.5-stage ompressor, where large separation o

urs in the average ow eld. The time stepping pro edure failed to rea h a onverged solution (due to the existen e of several exponentially growing eigenmodes), while the GMRES with the abovementioned pre onditioning was able to rea h a onverged solution, and thus resolved the desired unsteady ee ts in the stator bladerow. The 312

M. Stridh and L.-E. Eriksson

Unsteady Flows in Turboma hines

Figure 1 Residual history using the time-stepping (a ) and GMRES algorithm (b ) for the last stator stage of a 3.5-stage ompressor

residual plotted against the omputational eort measured in the number of pseudo-time steps n performed in both methods is shown in Fig. 1. For GMRES, ea h dot represents a restart of the algorithm. 4.2

Comparison with U-RANS

In addition to the 3.5-stage fan, time a

urate (U-RANS) omputations were also performed for the rst 1.5 stages in an o-design operating point. Figure 2 shows the integrated entropy levels along the ompressor axis. Curves 1, 2, and 3 orrespond to U-RANS, RANS + DS, and

onventional RANS results, respe tively. Figure 2b shows the exploded view of the last rotor{stator interfa e. It is seen that when in luding the DS terms, the nal entropy level in reases and approa hes the nal level predi ted by the U-RANS method. However, a large part of the in rease seems to arise almost instantly at the MP interfa e and the remaining dieren e then stays fairly onstant. The reason for this unphysi al behavior is analyzed in more detail in Se tion 5. 4.3

Compressor map

Figure 3 shows the omputed and measured ompressor hara teristi s for the whole 3.5-stage ompressor. Figure 3a presents the s aled ef ien y and Fig. 3b shows the s aled pressure ratio (the s aling of the M. Stridh and L.-E. Eriksson

313

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 2 (a ) Integrated entropy along the axis of rst 1.5 stage of the

ompressor: U-RANS (1 ), RANS + DS (2 ), and RANS terms (3 ); and (b ) exploded view of the results for the last rotor{stator interfa e

Figure 3 Chara teristi s of a 3.5-stage ompressor: (a ) eÆ ien y (s aled)

and (b ) pressure ratio (s aled). 1 | RANS, 2 | RANS (throttel), 3 | RANS + DS terms, 4 | RANS + DS terms (throttel), and 5 | experiment

314

M. Stridh and L.-E. Eriksson

Unsteady Flows in Turboma hines eÆ ien y and pressure ratio is due to on dentiality of the ompressor performan e). Cir les 1 and 2 represent onventional RANS solutions using pressure and throttle b. ., respe tively. Triangles 3 and 4 orrespond to the solutions ae ted by the DS terms, 3 | using the pressure b. . and 4 | for the throttle b. . Curve 5 orresponds to the measurements performed by Volvo Aero Corp. One an see from this gure that the mass ow rate through the ompressor in reases with DS terms in luded at a given pressure ratio. At the same time, the DS terms de rease the eÆ ien y of the ompressor. One an also see that the ee ts dis ussed in rease with the pressure ratio, and that by in luding the DS terms, a more realisti numeri al predi tion loser to the stall margin is probably obtained. (More omputed points are soon to be in luded, waiting for onvergen e.) In later Se tions, further detailed omparisons between the onventional RANS and RANS + DS results are performed. These omparisons are made at a high pressure-ratio operating ondition, orresponding to the ir le 1 and triangles 3 put in Fig. 3a inside the bla k ir le. Note that inside the ir le, there are two triangles 3, orresponding to the ee ts of DS terms omputed using one or two harmoni frequen ies, respe tively. The larger ee ts were found in the latter ase. 4.4

Integrated entropy

Figure 4 shows the integrated entropy levels along the ompressor axis. Lines 1 and 2 orrespond to the onventional RANS and RANS + DS results, respe tively. Figures 4b to 4d show the exploded views of the rotor{stator MP interfa es. Also, Figs. 4e and 4f show the exploded views of the last rotor and stator passages. With the DS terms, the entropy levels rst in rease at the left-hand side of the MP and then an approximately equally large entropy jump over the interfa es is observed in the predi tions by both methods, resulting in a higher entropy level at the right-hand side of the MP. Thus, both methods give unphysi al instantaneous entropy jumps over the MP. The nal entropy level is higher in the RANS + DS solution, i.e., the DS terms in rease the overall losses in the multistage ompressor at o-design operating onditions. The omparison of entropy growth inside the passage of the last rotor and stator se tions (Figs. 4e and 4f ) indi ates that there is apM. Stridh and L.-E. Eriksson

315

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 4 Integrated entropy at dierent axial positions of the entire 3.5stage ompressor; onventional RANS and RANS + DS results orrespond to 1 and 2, respe tively 316

M. Stridh and L.-E. Eriksson

Unsteady Flows in Turboma hines proximately the same entropy rise in the both passages, a

ording to RANS and RANS + DS. 4.5

Flow pattern ahead of and behind the last stator passage

Figures 5 and 6 show the tangentially averaged ow patterns ahead of and behind the last stator passage as a fun tion of radius (span fra tion), respe tively. The axisymmetri properties plotted in the gures are as follows: radial velo ity V , tangential velo ity W , total pressure P0 , stati pressure PS , Ma h number M, and, nally, ow angle. When omparing the two methods, some dieren es in axisymmetri in ow/out ow parameters an be seen. In the range of span fra tion from 0.1 to 0.6, the in ow azimuthal velo ity W , stati pressure PS , and Ma h number M dier by about 1% at most. At the outlet, the maximum dieren e between the two methods is slightly higher than at the inlet, and appears around span fra tion of 0.2. The radial and azimuthal velo ities dier within about 1.5% at this span fra tion, while the ow angles at this point dier within 4Æ. 4.6

Comparison of ontours

In this se tion, ontour levels at three blade-to-blade sli es, lose to tip, midspan and lose to hub are ompared. These sli es are shown in Fig. 7a. The results are also ompared in the sli es shown in Figs. 7b and 7 and taken at onstant axial positions: Fig. 7b shows the sli es at onstant axial position in the rotor passage and Fig. 7 shows the sli es at onstant axial position in the stator passage. However, the rst omparison is made for the entropy levels in a meridionial plane between blades: Fig. 8a orresponds to RANS; Fig. 8b shows the omparison, and Fig. 8 orresponds to RANS + DS. The bla k and grey lines orrespond to the onventional RANS and to RANS + DS methods, respe tively. This de nition is used in all subsequent gures where the two methods are ompared with ea h other. The entropy levels in Fig. 8 show that entropy in reases with radius and throughout the stages. With the DS terms in luded, entropy in reases everywhere ex ept for the separation zone in the last passage. M. Stridh and L.-E. Eriksson

317

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 5 Tangentially averaged ow patterns at in ow to the last stator passage as a fun tion of span fra tion (0 = hub, 1 = shroud): 1 | RANS and

2 | RANS + DS results

318

M. Stridh and L.-E. Eriksson

Unsteady Flows in Turboma hines

Figure 6 Tangentially averaged ow patterns at out ow to the last stator

passage as a fun tion of span fra tion (0 = hub, 1 = shroud): 1 | RANS and 2 | RANS + DS results M. Stridh and L.-E. Eriksson

319

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 7 Computational on guration with sli es where the omputational

results will be ompared: (a ) three blade-to-blade sli es, lose to tip, midspan, and lose to hub; (b ) onstant x-sli es in last rotor passage; and ( ) onstant x-sli es in last stator passage

In the separation zone, the ee t of the DS terms seems to be opposite, i.e., they redu e the maximal entropy level. Last rotor passage Figure 9 shows the ontours of entropy (Fig. 9a ) and Ma h number (Fig. 9b ) in three blade-to-blade sli es orresponding to hub, midspan, and tip of the rotor passage. Ea h sub gure shows three passages, 320

M. Stridh and L.-E. Eriksson

Unsteady Flows in Turboma hines

Figure 8 Comparison of entropy ontours in the through- ow between blades, predi ted by onventional RANS (bla k) and RANS + DS (grey)

orresponding to onventional RANS (top), omparison ( enter), and RANS + DS (bottom). Dieren es in entropy between the two solutions are evident. With the DS terms in luded, entropy is seen to in rease at the midspan and

lose to the tip, but in the near-hub region, an entropy de reases. For the Ma h number, the ee ts are smaller but still visible in the sli e

lose to the hub. The regions with high Ma h number extend when the M. Stridh and L.-E. Eriksson

321

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 9

Comparison of entropy (a ) and Ma h number (b ) ontours in

blade-to-blade sli e of the last rotor passage

322

M. Stridh and L.-E. Eriksson

Unsteady Flows in Turboma hines

Figure 10 passage: (a )

Figure 11

Entropy ontours at onstant axial positions in the last rotor x

= 0:28 and (b )

x

= 0:3

Contours of kineti energy of the DS

kdet

in the blade-to-blade

sli es in the last rotor passage

M. Stridh and L.-E. Eriksson

323

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 12

Contours of kineti energy of the DS

in the last rotor passage: (a )

x

= 0:28 and (b )

x

kdet

at onstant

x-sli es

= 0:3

DS terms are in luded, i.e., the mass ow rate in the near-hub region in reases. Figure 10 ompares entropy distributions predi ted by the two methods at onstant axial positions in the rotor passage. As indi ated by the integrated entropy levels in Fig. 4, the in rease due to DS terms is almost onstant throughout the rotor passage. Figures 11 and 12 show the levels of the deterministi kineti energy, de ned as kdet =

^

u00k;du00k;d

2 in the orresponding blade-to-blade and axial sli es in the rotor passage. The level is highest at the passage inlet lose to the hub, but de reases rapidly downstream. Last stator passage Similar to the last rotor passage, the ee ts of DS terms were studied for the last stator passage. Figure 13 shows the Ma h number (Fig. 13a ) and entropy (Fig. 13b ) ontours for three blade-to-blade passages. Close to the hub, the regions with high entropy and high Ma h numbers 324

M. Stridh and L.-E. Eriksson

Unsteady Flows in Turboma hines

Figure 13

Entropy (a ) and Ma h number (b ) ontours in blade-to-blade

sli e of the last stator passage

M. Stridh and L.-E. Eriksson

325

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 14

Entropy (a ) and Ma h number (b ) ontours at four axial

positions in the last stator passage

326

M. Stridh and L.-E. Eriksson

Unsteady Flows in Turboma hines be ome smaller while for the midspan and tip sli es, the regions with high Ma h numbers still shrink but the regions with high entropy grow. This is onsistent with what is shown in Fig. 14, where entropy and Ma h number distributions are presented for four axial positions. Close to the hub, one an see a large bubble of high entropy due to ow separation, whi h de reases in intensity when ae ted by the DS terms. The latter is indi ated both by the entropy and Ma h number ontours in this region. Outside the separation zone, the entropy in reases in all axial positions. The Ma h number ontours indi ate that at the inlet of the passage (top), the regions with low parameter levels are shrinking but further downstream, the opposite ee t is observed for a large portion of the passage, i.e., the regions with low Ma h number are slightly extended. In the separation zone, the Ma h number is somewhat lower in the ase with DS terms in luded. Figures 15 and 16 show the orresponding blade-to-blade and axial sli es for the deterministi energy kdet . For the sake of omparison, Fig. 15b presents the levels of turbulent kineti energy k. Ea h sub gure in Fig. 15b also ompares the levels of k between the two methods. The RANS + DS method predi ts de reased k levels lose to the hub and slightly in reased levels at midspan and lose to tip. Figure 16 shows how kdet is developed in axial se tions throughout the stator passage. Worth noting are high levels of kdet at the inlet ( lose to the hub) between blades (around the separation zone) and behind the blade (in the wake region). The stress terms omputed by the GMRES method are shown in blade-to-blade sli es at midspan in Fig. 17. It is the spatial gradients of these terms that in uen e the average ow parameters.

5

Test Case

For studing the ow behavior at MP interfa es, further evaluations were performed on a simpli ed test ase onsisting of two blo ks: the rst blo k is stationary and the se ond blo k rotates at 100 rad/s. At the inlet, a ti ious wake varying only in axial velo ity was spe i ed. In the wake region, the in ow onditions were spe i ed so that one obtained approximately 40 m/s in the wake, and about 107 m/s outside the wake. M. Stridh and L.-E. Eriksson

327

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 15

Contours of deterministi DS

kdet

(a ) and turbulent

k

(b )

kineti energies

328

M. Stridh and L.-E. Eriksson

Unsteady Flows in Turboma hines

Figure 16 Contours of DS kdet kineti energy at onstant x sli es in the last stator passage Density was spe i ed to be onstant through the wake. The turbulen e model and vis ous diusion were made ina tive; hen e, only the Euler part was onsidered. The wake overed 20% of the inlet area. Pressure was set onstant at p = 105 Pa throughout the entire domain for the ideal ase of zero wake mixing. The two blo ks were separated by an MP interfa e, where two types of b. . were used. The rst type is the so- alled absorbing b. ., whi h absorbs the waves traveling with normal in iden e to the boundary. M. Stridh and L.-E. Eriksson

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 17

Contours of DS terms in the blade-to-blade sli e at midspan of

the last stator passage

The se ond type is a ombined subsoni inlet/outlet ondition (so- alled pressure b. .). In the b. . of pressure type at the MP, stati pressure was averaged from the downstream domain and was used as the outlet ondition for the upstream domain. All other variables were extrapolated from the upstream domain. In the upstream domain, the following parameters

330

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Figure 18

Test- ase setup with a wake only in axial velo ity; the downstream blo k rotates and the upstream blo k is stationary: (a ) pressure type b. . and (b ) absorbing type b. .

were azimuthally averaged and used as inlet onditions for the downstream domain: h0, p0, Vrad , Wtang , k, and . The axial velo ity was extrapolated from the downstream domain. The b. . of absorbing type at the MP was based on the propagation dire tion of the hara teristi variables at the boundary. Due to the sign of the hara teristi speeds, the information was taken from the interior or exterior of the ow domain onsidered. This type of MP has shown to be numeri ally robust and has, therefore, been used in all previous omputations. For further details about this type of b. . refer to [20℄. Figure 18 shows the test- ase setup and axial velo ity ontours from the steady-state solution using the pressure type b. . (Fig. 18a ) and the absorbing type b. . (Fig. 18b ) (both gures have the same legend). Ideally, the wake should go straight through the upstream blo k without any disturban e at the MP interfa e as it does using the pressure type b. . With the absorbing type b. ., the wake is learly disturbed as a result of the MP. This happens due to the fa t that the hara teristi variables onne ted with the wake do not propagate orthogonally to the MP boundary and therefore partly re e t and reate a disturban e at the MP. M. Stridh and L.-E. Eriksson

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℄

Figure 19 Deterministi stress term ud ud : (a ) using pressure type MP 00

00

and (b ) using absorbing type MP. Note the dierent levels in the s ale

The Fourier omponents used as the inlet onditions for the linearized solver, will be ae ted by the type of MP onditions used in the averaged ow solver. This ee t is learly seen when the deterministi

℄

stress terms are omputed with ea h method. Figure 19a shows u00d u00d predi ted by the averaged ow using the pressure type b. . at the MP, and Fig. 19b shows the orresponding stress term obtained using the absorbing type b. . at the MP. Note that the s ales for the two sub gures are not the same: the dieren e is a fa tor of two, i.e., the highest level in Fig. 19a is two times higher than in Fig. 19b. The stress terms obtained with the pressure type b. . at the MP are therefore almost two and a half times larger than for the absorbing type.

℄

Note also that in an ideal ase, the stress term u00d u00d would be

onstant throughout the downstream blo k. As Fig. 19 shows, the linearized solver used does not satisfy this requirement. Instead the stress term rapidly de reases in intensity as one approa hes the outlet. This ee t is partly due to numeri al diusion of wave perturbations and partly to the use of the 1D absorbing type b. . at the downstream boundary. Figure 20 shows how entropy and pressure vary with axial positions for a number of dierent omputational setups. For this simple test ase, the entropy and pressures throughout the two blo ks should ideally be onstant at all axial positions. When 332

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Unsteady Flows in Turboma hines

Figure 20 Integrated entropy (a ) and stati pressure (b ) at axial posi-

tions throughout the test ase, omparing dierent types of b. . at the MP: 1 | RANS (with pressure type), 2 | RANS (with absorbing type), 3 | RANS + DS (pressure type (DS are exa t stress terms for the ideal ase)), 4 | RANS + DS (pressure type (DS set to a onstant value obtained from LNS at inlet)), 5 | RANS + DS (absorbing type (DS set to a onstant value obtained from LNS at inlet)), 6 | RANS + DS (absorbing type (DS omputed further upstream in the rst blo k)), 7 | RANS + DS (absorbing (DS from LNS solution)), and 8 | RANS + DS (pressure type (DS from LNS solution))

using the onventional MP interfa e, an instantaneous entropy jump is seen over the MP for both pressure type (1 ) and absorbing (2 ) b. . at the MP. For the pressure, the pressure type b. . at the MP gives an instantaneous jump while the absorbing type b. . at the MP gradually in reases the pressure throughout the rst blo k, i.e., there is some mixing of the wake in the rst blo k ahead of the MP. When in luding the ee ts of the stress terms omputed with the linearized solver, it be ame possible to de rease the instantaneous entropy jump and pressure over the MP, for both types of b. . (7 = absorbing type, 8 = pressure type). It has been found that for this test

℄

ase, u00d u00d mostly governs the pressure jump while, the entropy seems M. Stridh and L.-E. Eriksson

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^ ℄

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

to be sensitive to the relation between h000;du00d and u00d u00d terms. In the

ase of pressure type b. . at the MP, entropy a tually de reases over the MP, and the pressure jump de reases more in this ase, as well. It should be noted that the entropy seems to be very sensitive to the

omputed stress terms. In the ase where entropy jump de reases over the interfa e, the ratio between the two stress terms in reases, i.e., the

ombination of the two stress terms is slightly overpredi ted. The pressure growth in the rst blo k due to the use of absorbing type b. . at the MP de reases with the DS terms in luded but still exists. The rapid growth of pressure lose to the outlet is due to the rapid de rease of the stress terms in this area whi h is onne ted to the absorbing b. . used in the linearized solver. If one instead uses the stress terms obtained at the inlet of the downstream blo k at all axial positions (5 = absorbing, 4 = pressure), the right onstant pressure is obtained in the se ond blo k for both types of b. . at the MP interfa e. For the absorbing type, there is still a pressure rise in the rst blo k, i.e., there is still some mixing of the wake due to the b. . used. The entropy jump for the pressure type b. . at the MP a tually in reases meaning that the mass-averaged entropy is lowered in the se ond blo k as a result of in reasing stress terms. In the absorbing ase, the opposite result is obtained, i.e., entropy in reases slightly due to higher stress terms. For this simple test ase, one an readily ompute the exa t stress levels and spe ify them in the downstream domain. This has been done for the MP using the pressure ondition orresponding to almost

onstant entropy and pressure throughout the test ase as expe ted (3 ). The absorbing type of b. . has shown to be very sensitive to the lo ation where the stress terms are evaluated in the upstream domain. If, for example, they are evaluated further upstream, an overpredi tion of the stress terms results in an entropy in rease over the interfa e (6 )

ompared to the onventional MP without stress terms. The balan e between the two nonzero stress terms seems to determine the dire tion of the jump. The disturban e at the MP when using the absorbing type of b. . also reates stress terms in all other dire tions

℄

(e.g., u00d vd00 ) whi h should not exist for the ideal ase. The idea with the stress terms is to ompensate for the instantaneous mixing of upstream wakes at the MP interfa es. If some of this 334

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Unsteady Flows in Turboma hines

Figure 21 Mass-averaged integrated entropy at dierent axial positions in

the 1.5-stage fan, for dierent types of b. . at the MP interfa es: 1 | RANS (with pressure type), 2 | RANS (with absorbing type), 3 | RANS + DS (pressure type (DS set to a onstant value obtained from LNS at inlet), 4 | U-RANS, and 5 | RANS + DS (absorbing type (DS set to a onstant value obtained from LNS at inlet)

mixing o

urs prior to the interfa e due to the b. . used, some of the idea with the stress terms is lost. The values of the omputed stress terms are ae ted by the re e tions at the interfa e, whi h an lead to less a urate DS terms. To further evaluate the ee t of b. . used at the MP, the 1.5-stage fan was updated using the pressure type of b. . at the MP for the same o-design operating point as prevuosly presented in Fig. 6. A new RANS solution using the pressure type b. . at the MP was rst obtained and then the DS terms were determined to further update the RANS solution using the pressure type b. . at the MP. Figure 21a shows the rotor passage and Fig. 21b shows the last stator passage and the se ond MP interfa e. The mass-averaged entropy in reases in the rotor passage due to the use of pressure type b. . at the MP. This is not seen in the U-RANS omputations. The instantaneous jump of entropy over the MP has vanished with the use of pressure type b. . at the MP. When the DS terms were in luded, the jump in reased but not as M. Stridh and L.-E. Eriksson

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mu h as for the absorbing type b. . at the MP. The nal entropy level in reases with the use of pressure type b. . The ee ts of b. . at the MP is of the same order as the DS terms, so some further evaluations in this area are highly re ommended.

6

Con luding Remarks

The ee ts of DS terms are quite ompli ated and dier depending on the lo ation in the ompressor. In general, the DS terms seem to in rease the total losses and the mass ow rate (at onstant pressure ratio) through the ompressor, whi h is onsistent with time-averaged U-RANS results. Lo ally, the DS terms an work in the opposite manner, i.e., de rease losses in a region; this is observed in the separation zone lose to the hub of the last stator. Outside this separation zone, the DS terms tend to in rease losses and make the wakes wider, i.e., in rease blo kage, but this is not observed in general: instead, the mass ow rate through the ompressor in reases due to the DS terms. The in rease in the mass ow rate is probably obtained be ause of the de rease of the re ir ulation zone lose to the hub leading to in reased

ow turning in this region of the stator. Sin e the ee ts of the DS terms in rease with pressure ratio, a more a

urate numeri al predi tion should be obtain as the ompressor is approa hing the stall line. However, the ee ts obtained for the 3.5stage transoni fan are limited. When ompressor hara teristi s su h as eÆ ien y and pressure ratio were ompared to measurements, no signi ant improvements were observed with the DS terms in luded. One possible explanation for this nding may be that two ountera ting phenomena are aptured. The rst is the so- alled wake-re overy phenomenon, whi h states that a large part of the kineti energy in the upstream wakes is re overed in the downstream passage, by stret hing the wake, resulting in an eÆ ien y in rease. The se ond is the fa t that the DS terms enhan e mixing within a passage and thereby in rease losses as well. Sin e both these phenomena were aptured, the observed net ee t of DS terms appeared to be small. An alternative explanation is that the ow in this transoni ompressor was dominated by invis id ompressible ee ts, i.e., hoking, sho ks, et . It should also be noted that the al ulated eÆ ien y of the ompressor was very sen336

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Unsteady Flows in Turboma hines

sitive to tip learan e and this may well explain a large part of the dis repan y in the predi ted eÆ ien y. The total (temporal and spatial parts) DS terms an su

essfully be determined by solving the LNS equations either via time-stepping or by pre onditioned GMRES algorithm. As shown, the DS terms have large variations in spa e, whi h would not be fully des ribed by existing DS models based on ir umferential averaging of overlapping meshes. To evaluate the observed in rease in the entropy jump at the MP when in luding the stress terms, a simpli ed test ase was examined. It was then found that the use of 1D absorbing type b. . at the MP interfa es had an in uen e on the mixing in the entire upstream domain, i.e., some of the mixing o

urred prior to the MP interfa e due to the b. . The idea with using the DS terms was to ompensate for the instantaneous mixing at the MP interfa e, whi h means that if a signifi ant part of the mixing pro ess has already been performed upstream of this interfa e, then mu h of the ee t of the DS terms is lost. This is a drawba k in all omputations performed in the 1.5- and 3.5-stage fans and should be learly noted. A suggested modi ed b. . at the MP, a so- alled pressure-type b. ., has been tested and has shown promising behavior at the MP interfa e, both for the simpli ed test ase and for the 1.5-stage fan. Further evaluation of this pressure-type b. . is needed but it seems to improve the overall methodology (RANS, LNS, and RANS + DS). The 1D absorbing b. . in the LNS solver should be used with some

are sin e the same kind of nonphysi al wake mixing may o

ur for the perturbational solutions. For eample, if the perturbational wake is not ompletely dissipated during its transport through the bladerow and is exiting more or less orthogonally to the outlet boundary, the resulting nonideal b. . will ae t the solution and thus the omputed stress terms in this region as well. In this ase, a more sophisti ated absorbing b. . (of 2D type) would be more suitable, sin e this type of b. . is able to identify and absorb entropy and vorti ity waves with general orientations.

Referen es 1. Denton, J. D., and U. K. Singh. 1979. Time mar hing methods for turboma hinery ow al ulations. VKI-LEC-SER-1979-7. M. Stridh and L.-E. Eriksson

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2. Adam zyk, J. J. 1984. Model equation for simulating ows in multistage turboma hinery. NASA Te hni al Memorandum 86869 Lewis Resear h Center Cleveland, Ohio. 3. Dano, C., F. Bardoux, F. Leboeuf, and C. Toussaint. 1999. Chara terization of deterministi orrelations for turbine stage. Part 1: Time-average

ow analysis; Part 2: Unstedy ow analysis. J. Turboma hinery 215:687{ 96. 4. Baralon, S. 2000. On multistage analysis of transoni ompressors: From axisymmetri through ow time-mar hing to unsteady three-dimensional methods. PhD. Thesis. Chalmers University of Te hnology. 5. Bardoux, F., and F. Leboeuf. 2001. Impa t of deterministi orrelations on the steady ow eld. Pro . IMe hE, Part A 215:687{96. 6. Giles, M. B. 1992. An approa h for multi-stage al ulations in orporating unstediness. ASME Paper 1992-GT-282. 7. Casper, J. R., and J. M. Verdon. 1982. Development of a linear unsteady aerodynami analysis for nite-de e tion subsoni as ade. AIAA J. 20(9):1259{67. 8. Hall, K. C., and C. B. Loren e. 1993. Cal ulation of three-dimensional unsteady ows in turboma hinery using the linearized harmoni Euler equations. J. Turboma hinery 115(4):800{9. 9. Ning, W., and L. He. 1998. Computation of unsteady ows around os illating blades using linear and non-linear harmoni Euler methods. J. Turboma hinery 120(3):508{14. 10. He, L., and W. Ning. 1998. EÆ ient approa h for analysis of unsteady vis ous ows in turboma hines. AIAA J. 36(11):2005{12. 11. Chen, T., P. Vasanthakumar, and L. He. 2000. Analysis of unsteady blade row intera tion using nonlinear harmoni approa h. ASME TURBOEXPO. Muni h, Germany. 2000-GT-431. 12. Stridh, M., and L. E. Eriksson. 2005. Evaluation of modeled deterministi stress terms and thier ee ts on a 3rd transoni ompressor. ISABE-1100. 13. Stridh, M., and L. E. Eriksson. 2005. Modeling unsteady ow ee ts in a 3rd transoni ompressor. ASME TURBOEXPO. GT2005-68149. 14. Stridh, M., and L. E. Eriksson. 2006. Solving harmoni linear problems in unsteady turboma hinery ows using a pre onditioned GMRES solver. ECOMAS. 15. Eriksson, L. E. 1995. Developmnet and validation of highly modular ow solver version in G2DFLOW and G3DFLOW series for ompressible vis ous rea ting ow. Te hni al Report 9970-1162, 9970-1160, VAC. 338

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Unsteady Flows in Turboma hines 16. Wil ox, D. C. 1998. Turbulen e modeling for CFD. DCW Industries, In . La Canada, CA. 2nd ed. 17. Eriksson, L. E. 1999. Numeri al simulation of ompressible ows. Contribution to CeCOST ourse, VAC/CTH. 18. Eriksson, L. E. 2005. Compressible CFD. Le ture Notes. Division of Fluid Dynami s, Department of Applied Me hani s, Chalmers University of Te hnology. Gothenburg, Sweden. 19. Saad, Y., and M. H. S hultz. 1986. GMRES: A generalized minimal residual algorithm for solving nonsymmetri system. SIAM J. S i. Stat. Comput. 7(3):856{69. 20. Billson, M. 2004. Computational te hniques for turbulent generated noise. PhD. Thesis. Chlamers University of Te hnology, Sweden.

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NUMERICAL CONTRIBUTION TO ANALYSIS OF SURGE INCEPTION AND DEVELOPMENT IN AXIAL COMPRESSORS 

y

y

z

N. Tauveron , P. Ferrand , F. Leboeuf , N. Gourdain , and

S.

z Burguburu

CEA, 17 rue des Martyrs Grenoble 38100, Fran e y LMFA, UMR CNRS 5509, E ole Centrale de Lyon 36 avenue de Collongue E ully 69134, Fran e z ONERA, Applied Aerodynami s Dpt. Ch^atillon 92320, Fran e 

Nomen lature a f L P P S T U V






a ousti velo ity frequen y du t length pressure pressure dieren e se tion period rotation velo ity at midspan volume spe i heat ratio gas density

Subs ripts

p plenum max maximal value

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Unsteady Flows in Turboma hines min +

minimal value negative part of a surge y le positive part of a surge y le

Introdu tion The estimation of ompressor performan e in transient operations is of high importan e for designers. The goal of this paper is to show and ompare dierent des riptions of ompressor surge o

urren e and development in axial ompressors. The rst model is simple: numerous physi al hypotheses are made to build a linear model, whi h an be solved analyti ally. Even if this model is the sum of rough approximations, some qualitative and quantitative parameters of surge o

urren e an be dedu ed and ompared to experimental data [1℄. The se ond model is more traditional: nonlinear and based on performan e maps and an be onsidered as an extension of Greitzer's model [2℄. The third approa h is more original: it onsists in the solution of one-dimensional (1D) axisymmetri Navier{Stokes equations on an axial grid at the s ale of the row with mass, axial momentum, ir umferential momentum, and total enthalpy balan es written in an appropriate frame. The forth approa h primarily on erns the modeling of rotating stall phenomenon with multidimensional (full three-dimensional (3D) or quasi-3D) ow solver. The main advantage of this kind of simulation is taking multidimensional ee ts (for example, tip leakage ow) into a

ount and des ribing the real dynami s of rotating stall phenomenon, whi h is often onsidered as the rst instability en ountered before surge development [1℄. Last se tion is devoted to the omparison between the dierent models.

1 Con guration The dynami behavior of the ompressor is tested in a on guration in whi h the ompressor is onne ted with a plenum and a throttle as shown in Fig. 1. When the opening of the throttle is redu ed, the pressure ratio over the ompressor rises. With the de reasing ow, the \surge-line" is passed and os illatory behavior arises. Greitzer [2℄ N. Tauveron et al.

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 1 Surge test on guration states that the shape and frequen y of ow os illations depend on parameter B , de ned as

r

Vp B = 2Ua SL 2

Simple Zero-Dimensional Model of Deep Surge

In this se tion, the bases of a simple zero-dimensional (0D) model are presented. This model is based on the following strong simpli ations, whi h are oherent with experimental observations by Day [1℄:

{ The dynami s of the phenomenon is quasi-steady; { Pressure and velo ities tend to approximately follow ompressor

hara teristi map in the stable domains; { The phenomenon is essentially driven by ollapse and re overy dynami s; { The time dependent mass onservation equation is used in the model; and { Numerous simpli ations on temperature variations are used and a soni ondition at the throttle is assumed. 342

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Unsteady Flows in Turboma hines Deep surge o

urren e

This model provides a \theoreti al" and simple method to estimate deep surge o

urren e. This o

urren e is dire tly related to the riti al value of B parameter, given as follows: s 0:25 B rit  2 ) + 2
P =(Umax )
P =(Umin This riti al value depends on ompressor design and on its hysteresis as well. These points have been advan ed by Greitzer [2℄, M Caughan [3℄, and Day [1℄ from experimental and numeri al investigations. On the on guration des ribed by Day [1℄, the agreement between 0D model (B rit = 0:34) and experimental results (B rit = 0:39) is satisfa tory. Deep surge development

Consider the os illations of the operating point of the ma hine in the pressure {mass ow diagram during surge. The os illation y le is separated in positive and negative parts (denoted by a supers ript). The positive part orresponds to the period with a positive+axial velo ity. Analyti al formulae for the estimation of the positive T and the negative T periods are derived. As a onsequen e, some quantitative information on deep surge behavior is obtained. Frequen y, proportion of positive period to negative period

As stated by Day [1℄, the total period is the sum of T + and T . The total period an therefore be estimated. The frequen y is then proportional to 1=B2 (or 1=Vp). Figure 2 shows the experimental frequen y measured by Day [1℄. The line orresponds to the regression line. The slope is equal to 10. The simpli ed model gives the same behavior with a value of 9.1 for the slope. The ratio T + =T an also be evaluated. It strongly depends on the throttle setting. If the throttle is set not far from the design setting, this value is lose to 1, and as a onsequen e, T + >+ T , but for throttle setting far from the design value, one an have T < T . N. Tauveron et al.

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Figure 2 tal,

2

Frequen y of deep surge in fun tion of 1=Vp [1℄:

1

| experimen-

| regression line: 9:97954572480941x

3 Axial Turboma hine Modeling with an A tuator Disk Approa h A traditional way to des ribe surge o

urren e and performan e is to use a uid dynami model using an a tuator disk approa h for the entire ompressor (see, for example, [2℄ for an in ompressible model [4℄ in luding energy equation). Alternative models were also developed to simulate long transients for ma hines in a more pre ise fashion than ma hine hara teristi maps use. Davis and O'Brien [5℄ used stage

hara teristi maps instead of global ma hine maps. The interest is that more lo al aspe ts an be des ribed with this approa h, su h as lo al heat ex hange. As this method has been extensively des ribed and used in the s ienti and industrial ommunities, it is not worth to des ribe it in a more pre ise way. These methods provide quantitative information on surge o

urren e and behavior. Rotating stall, lassi al surge, and deep surge an be distinguished.

4 Axial Turboma hine Modeling with a 1D Axisymmetri Approa h A row-by-row des ription is apable of modeling more lo al ee ts. S hobeiri and Abouelkheir [6℄ developed an approa h based on velo ity triangles, turning angle, and loss orrelations. This paper follows 344

N. Tauveron et al.

Unsteady Flows in Turboma hines a similar approa h based on the deviation angle and loss orrelations for ea h bladerow at midspan, but here two time dependent momentum equations are solved. Model

The des ription that has been developed is based on a 1D axisymmetri approa h. The approa h onsists in the solution of 1D axisymmetri time dependent Navier{Stokes equations on an axial grid: mass, axial momentum, ir umferential momentum, and total enthalpy balan es are written in an appropriate frame (absolute or relative). A midspan velo ity triangle diagram is omputed before and after ea h bladerow. Perfe t gas law is also assumed. The assumption of quasisteady response of the blades to upstream disturban es is made. As a onsequen e, orrelation oeÆ ients are derived from steady orrelations. When the negative ow o

urs, spe i orrelations are used. The assumption of quasi-steady behavior is almost justi ed in numerous transient phenomena: the transit time of the working uid through a row in the ma hine is far below delays in numerous transient phenomena if one onsiders variations of pressure, temperature, and the speed of dierent omponents of the system. Surge transients (rotating-stall like behavior, lassi al and deep surges)

At a low value of B parameter, a physi al regime, whi h hara teristi s are approximately steady (averaged mass ow and delivery pressure (Fig. 3a )), is reprodu ed by the model. The rotating stall regime shows the same ma ros opi tenden ies on e the stall pattern is fully developed. It is alled here as \rotating-stall like behavior" as rotating stall is an unstable and nonaxisymmetri phenomenon that annot be orre tly modeled be ause of the assumptions of ir umferentially uniform ow. At a higher value of B parameter, one observes the lassi al surge development and the surge y le (Fig. 3b ). In the same graph, the steady hara teristi s map is presented. One an see that the y les tend to follow the hara teristi s map in the stable part. Complementary simulations have shown that the phenomenon has a lower frequen y when B is greater. N. Tauveron et al.

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Figure 3 Simulation of (a ) rotating-stall like instability and (b ) surge phenomenon (surge y les): pressure oeÆ ient as a fun tion of ow oeÆ ient

Figure 4 Flow oeÆ ient as a fun tion of time for two dierent throttle settings: 1 | 46% and 2 | 86%

Deep surge o

urs if B is greater or if the throttle is losed further than in the lassi al surge ases. Figure 4 presents the dynami model results: relaxed os illations of mass ow and pressure are developed with the mass owrate be oming negative and the frequen y being very small. One an see that the y les are large and they tend to follow the

hara teristi s map in the stable parts, at positive and negative mass

ows (Fig. 5). 346

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Figure 5 Surge y les: pressure oeÆ ient as a fun tion of ow oeÆ ient Lo al aspe ts In [7℄, a very simpli ed way to estimate the performan e of a row in an unstable regime was reported. This pro edure an provide a very simpli ed riterion to test if a parti ular row is in an unstable regime. As a onsequen e, this model allows one to determine whi h row is involved in the unstable regime, and whi h is the \ rst." The model has been tested on two dierent on gurations of a 3-stage ompressor [8℄. A

ording to Table 1, the 1D model reprodu es orre tly the differen es in the instability o

urren e for the two on gurations, whi h is a system ee t (as only last stages stagger angles were modi ed), and also identi es orre tly whi h row is the \ rst" to go in an unstable regime, whi h is a lo al aspe t. This example illustrates the

apability of the model to treat both system and lo al aspe ts of the phenomenon. Table 1 Evaluation of the rst unstable row: omparison between simulation and experimental data [8℄

Dieren e in Mismat hed stagger instability limits angle on last stages, between both rst unstable row

on gurations Simulation Rotor of the 1st stage Rotor of the 1st stage 8% Measurement Rotor of the 1st stage Rotor of the 1st stage 6:5% Standard stagger angle, rst unstable row

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5

Multidimentional Simulations of Rotating Stall

The present ontribution fo uses on the simulation of the rotating stall phenomenon with a quasi-3D and full 3D ow solvers on the whole annulus of a single-stage subsoni ompressor des ribed extensively in [9℄. Even if it was not possible to represent multistage ompressor on guration and surge development during a long time, improvement of the knowledge about instability at low mass ow rate was provided. The main advantage of full 3D simulation is a possibility to take into a

ount 3D ee ts and tip leakage ow. The simulation showed that dierent me hanisms were involved during the stall in eption and development pro esses. The rst one was linked to intera tion between in ow and tip leakage ow and the se ond was onne ted with the growth of a modal wave. The third me hanism dealt with a more lo al ee t on blade boundary layers and manifested itself in a parti ular on guration. At a high value of B parameter, some rotating stall phenomena transitioned to surge.

Model In this paper, only the full 3D (elements for the quasi-3D solver an be found in [10℄) ow solver is des ribed. The simulations performed with the elsA ode developed by ONERA [11℄ are based on the solution of the Reynolds Averaged Navier{Stokes equations using a nite volume method. The governing equations are integrated in time by a 4-step Runge{Kutta s heme and the spa e dis retization is made by a

entered Jameson s heme. For this al ulation, the system of governing equations is losed with the one-equation Spalart{Allmaras turbulen e model, oupled with a wall-law approa h (y+  20). The mesh is generated as follows (Fig. 6): one passage is divided into H-grids and one O-grid around the blade. The tip learan e is taken into a

ount with a H-mesh of 11 points in the radial dire tion, with nonmat hing points at the interfa e with the O-grid of the blade (Fig. 7). The mesh extends up to 3 hords upstream of the rotor blades and 1.5 hords downstream of the stator blades. In order to suppress the

lassi al periodi ity ondition, the whole stage is simulated. The mesh is dupli ated in order to simulate the ow in all ompressor passages (30 348

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Unsteady Flows in Turboma hines

Figure 6 Partial view of the 3D mesh

Figure 7 View of the tip leakage mesh (top of the blade)

rotor hannels and 40 stator hannels). This method leads to a mesh of 31 million of points. Rotating stall phenomenology: Multidimensional simulations The numeri al simulations (listed in Table 2) show that dierent types of rotating stall exist in the axial ompressor studied. The rst type is alled the tip-leakage rotating stall whi h is a lo al instability lo ated near the shroud and near the rotor leading edge. It is the result of intera tion between the tip leakage ow, vortex separation, whi h appears at midspan on the su tion side of the rotor blades, and the in ow. This tip-leakage rotating stall is hara terized by ten

ells (the number of ells is probably xed by a rotor/stator intera tion mode) whi h move at 75% of the rotation speed of the moving bladerow and therefore at a relatively high frequen y (in simulation 1: 787 Hz). The ells extend on a part of the rotor span, so this type of rotating Table 2 Dierent multidimensional simulations Simulation 1a 1b 2 3

N. Tauveron et al.

B

0.23 0.25 0.47 0.51

Simulation type Full 3D Quasi-3D Quasi-3D Quasi-3D 349

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 8

Stati pressure signal history (rotor/stator interfa e)

stall is alled part-span stall (PS). This phenomenon is a lo al instability that ae ts global performan e of the ompressor ( ow oeÆ ient and pressure rise are strongly redu ed). The se ond type is alled the modal rotating stall, whi h is a global instability. This type of rotating stall has an ee t on the entire system of ompression and is indu ed by the development of a long length-s ale disturban e. Contrary to the tip-leakage rotating stall, the modal rotating stall is hara terized by a lower frequen y (173 Hz in simulation 1). This frequen y orresponds to 3 ells, whi h move at 55% of the rotor rotation speed. These ells extend on the whole rotor span. Therefore, this type of rotating stall is alled full-span stall (FS). These two instabilities annot oexist for a long time: when the se ond instability appears (modal rotating stall), the rst one disappears (Fig. 8). When the size of the plenum is in reased (simulation 2) the modal rotating stall is a transient phenomenon and leads to a surge behavior with massive and abrupt extension of negative mass

ow (Fig. 9). The third type is alled pro le rotating stall and o

urs at lower value of mass ow, when the ow in iden e angle (relative to the blade) is large. When a riti al value of the ow in iden e is obtained, the stall phenomenon is established (the pro le boundary layer on the blades is not atta hed any more). In the parti ular on gurations studied (a 350

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Unsteady Flows in Turboma hines

Figure 9 Entropy eld after 13 rotor revolutions (i.e., 0.12 s) single rotor: simulation 3), this type of rotating stall transitions to a surge like behavior.

6

Cross-Comparisons Between the Models

Surge in eption Table 3 presents ross- omparison between theoreti al and experimental data for a multistage environment. They show a good agreement

Table 3 Evaluation of riti al B parameter with dierent methods: with 3 models and experimental data [2℄ for multistage on guration Experiment

Simple 0D model

A tuator disk

1D axisymmetri approa h 0.7{0.75

B rit 0.8 0.725 0.6{0.7 [2℄  For the 0D simple model, the value of
P =(Umin 2 ) was taken equal to 0.3 following [12℄ N. Tauveron et al.

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s Table 4 Evaluation of riti al B parameter with dierent methods: the simplest method and the most sophisti ated one for a single-stage on guration des ribed in [10℄ B rit

Simple 0D model 0.65

 For the 0D simple model, the value of
P

general orrelation by Day et al. [13℄

(

3D simulations  0:47

2 = Umin

) was estimated with the

provided the performan e peak is well known. Table 4 shows a good agreement for a single-stage on guration as well. Surge development Examine rst the frequen y parameter. Figure 10 shows the simulated frequen y ( rosses) with the 1D model as a fun tion of 1=B 2 . The line

orresponds to the regression line. The slope is equal to 2.11. The simple 0D model has given a value of 1.96 for the slope, whi h agrees well with the 1D simulation. The other ee t studied is the ee t of dierent throttle settings on the positive-to-negative period proportion. The simple 0D model gives the following values for the T + =T ratio: 1 and 2.7 for 46% and 86%

Figure 10 Simulated frequen y of deep surge as a fun tion of 1=B 2 obtained by 1D model for the multistage on guration [2℄ 352

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Unsteady Flows in Turboma hines

Table 5 Evaluation of frequen y of axial waves with dierent methods: the simplest method and the most sophisti ated one for a single-stage on guration des ribed in [10℄ f , Hz

Simple 0D model 26

3D simulations 35

of the throttle settings, respe tively. These values agree well with the results of simulations based on the 1D model (Fig. 4). Table 5 presents the results of similar omparisons for a single-stage on guration. Summary As shown in the previous tables, a good agreement between the dierent methods for surge in eption and surge development was obtained. It

an be on luded that the assumptions adopted in the simpli ed model are satisfa tory. Table 6 presents a summary on the hara teristi s and advantages of the methods presented.

7 Con luding Remarks The paper shows that dierent methods exist for modeling surge in eption and development from very simple and based on purely physi al

onsiderations to more omplete and based on numeri al simulation. Of

ourse, ea h method is adapted to the pre ision required, the availability of data and the time that an be devoted to the study. In this sense, ea h method is adapted to ea h phase of ma hine design. At present, multidimensional simulations are useful, but they annot adequately des ribe long transients in multistage ma hines. Moreover, their apability of simulating low positive and negative mass ows is also limited by turbulen e model pre ision in stall regimes. The instability was shown to be generally related to global onditions. As a onsequen e, a good ompromise is to develop oupling strategies between dierent methods. The other on lusion is that in dierent situations simple models are useful to understand the main physi al phenomena at stake and help determining the pre ise onditions of interest for the time

onsuming multidimensional simulations. N. Tauveron et al.

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Table 6 Comparison of dierent methods Rotating stall in eption

Rotating stall dynami s Surge in eption

Surge dynami s Physi al time CPU time Multistage ompressor

Simple A tuator 1D axisym- Quasi-3D Full 3D 0D disk metri (Case 1b) (Case 1a) Limited to axial perturbations Complete if environment fully des ribed Possibility of Tip leakage a ir umferen ow tial des ripdes ribed tion with MGm Impossible Limited to axial develop- Complete Complete ment but approx- ells Possibility imate des ription with MGm Limited to axial perturbations { Des ription of transition Possibility { Identi a- between stall and surge with MGm tion of the { Limitation of des ription rst unstable of the environment row { No need of

orre tion of inertia Some pa- Complete axial des ription { Limited due to CPU time rameters Possibility { Limited due to turbuwith MGm len e model validation on deta hed boundary layers Periodi Full transient 0.6 s 0.15 s regime Immediate Few minutes Many min420 h 3800 h utes No diÆ ulty Possible Totality of DiÆ ult due { No need of the hannels to RAM

orre tion of not respe ted requirement inertia (31 millions { Possibility of of points for stage-by-stage 1 stage) des ription Few ele- Performan e Geometry at Complete geometry ments of map midspan performan e map Impossible Limited to axial Possible (sen- Possible (valsitive study) idation)

Data needed for ompressor Identi ation of pre ursors MGm is the Moore{Greitzer type model [14℄

354

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Referen es 1. Day, I. J. 1994. Axial ompressor performan e during surge. J. Propulsion Power 10:329{36. 2. Greitzer, E. M. 1976. Surge and stall in axial ow ompressors. ASME J. Engineering Power 98:190{217. 3. M Caughan, F. E. 1989. Numeri al results for axial ow ompressor instability. ASME J. Turboma hinery 111:434{41. 4. El-Mitwally, E. S., M. Abo-Rayan, N. H. Mostafa, and A. H. Hassanien. 1996. Modeling te hniques for predi ting ompressor performan e during surge and rotating stall. ASME Fluids Engineering Conferen e Pro eedings. 3. 5. Davis, M. W., and W. F. O'Brien. 1991. Stage by stage post stall ompression system modeling te hnique. J. Propulsion Power 7:997{1005. 6. S hobeiri, T., and M. Abouelkheir. 1992. Row by row o-design performan e al ulation method for turbines. J. Propulsion Power 8:823{28. 7. Tauveron, N., P. Ferrand, and F. Leboeuf. 2006. Simulation of surge in eption and performan e of axial multistage ompressor. ASME Paper GT2006-90163. 8. Longley, J. P., and T. P. Hynes. 1990. Stability of ow through multistage axial ompressors. ASME J. Turboma hinery 112:126{32. 9. Mi hon, G. J., H. Miton, and N. Ouayahya. 2005. Experimental study of the unsteady ows and turbulen e stru ture in an axial ompressor from design to rotating stall onditions. 6th European Conferen e on Turboma hinery. Lille, Fran e. Paper 015-03/39. 10. Gourdain, N. 2005. Simulation numerique des phenomenes de de ollement tournant dans les ompresseur axiaux. Ph.D. Thesis. E ole Centrale de Lyon, Fran e. 11. Cambier, L., and M. Gazaix. 2002. elsA: An eÆ ient obje t-oriented solution to CFD omplexity. 40th AIAA Aerospa e S ien e Meeting and Exhibition. Reno. 12. Ko, S. G., and E. M. Greitzer. 1986. Axisymmetri ally stalled ow performan e in multistage axial ompressors. ASME J. Turboma hinery 108:216{23. 13. Day, I. J., E. M. Greitzer, and N. A. Cumpsty. 1978. Predi tion of ompressor performan e in rotating stall. ASME J. Engineering Power 100:1{ 14. 14. Moore, F. K., and E. M. Greitzer. 1985. A theory of post-stall transient in multi-stage axial ompression system. NASA Report CR-3878. N. Tauveron et al.

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UNSTEADY FLOW COMPUTATION IN HYDROTURBINES USING EULER EQUATIONS 







S. G. Cherny , D. V. Chirkov , V. N. Lapin , S. V. Sharov , V. A.

y Skorospelov ,

and I. M.

z Pylev

 Institute

of Computational Te hnologies Siberian Bran h of the Russian A ademy of S ien es Novosibirsk, Russia y Sobolev Institute of Mathemati s Siberian Bran h of the Russian A ademy of S ien es Novosibirsk, Russia z JSC \Leningradsky Metalli hesky Zavod" St. Petersburg, Russia The paper deals with numeri al investigation of unsteady threedimensional (3D) ow in a real hydrauli turbine. Unsteady ow phenomena are predi ted based on the Euler equations using the omputational uid dynami s (CFD) ode CADRUN developed by the authors. Numeri al results demonstrate that major ow features are aptured by invis id equations.

1

Introdu tion

Fluid ow in hydrauli turbines is always unsteady. There exist different types of unsteadiness in the ow. The rst type is externally for ed unsteadiness, aused by the rotor{stator intera tion. The se ond is free unsteadiness, whi h manifests itself, for example, as vortex shedding at the blade trailing edge. Another example of free unsteadiness is a free vortex rope movement in a draft tube at part-load operation point. Flow turbulen e relates to the third type of unsteadiness. Steady state simulations in hydroturbines are the ommon pra ti e nowadays. However, vorti es and omplex ow stru tures annot be predi ted a

urately by the steady state simulation. Therefore, development of ef ient tools (methods) for simulating unsteady phenomena in real hydro356

S. G. Cherny et al.

Unsteady Flows in Turboma hines turbines is an a tual up-to-date problem. There are three main spe i features of unsteady ows in hydroturbines; namely, omplex 3D geometry, high Reynolds numbers (106{107) and ow turbulen e, arising in the draft tube. It is believed that simulation of unsteady phenomena in hydrauli turbines requires sophisti ated turbulen e modeling, whi h involves large amount of omputational eorts. However, the present authors suppose that major unsteady ow phenomena an be a

urately predi ted in frames of Euler equations. First of all, this applies to simulation of stator{rotor intera tion in the Fran is turbine. In the present work, this problem is addressed by a throughout ow

omputation of the entire turbine passage using in ompressible Euler equations. Analyzed are the pressure os illations, obtained in al ulations and measurements. The Euler model was also applied to the omputation of vortex rope pre ession in a draft tube. Numeri al simulation in this problem formulation provides a good agreement with experiment in terms of vortex rope shape and pre ession frequen y. Vortex formation an be

aused by ir umferential nonuniformity of upstream ow eld and numeri al vis osity, aused by nite-dieren e s heme dissipation. Thus, the largest vortex stru tures, aused by passage geometry, are well des ribed by the numeri al model of Euler equations. However, predi tion of ne-s ale vortex ee ts, aused by vis ous for es still needs an adequate turbulen e model.

2 2.1

Numeri al Model Governing equations

Fluid ow in stati elements of hydrauli turbine (spiral ase, wi ket gate, and draft tube) is des ribed in the absolute referen e frame, while the rotating Cartesian referen e frame x1, x2, and x3 is used for the runner. It is assumed that the runner rotates with angular speed ! around Ox3 axis. The Euler equations for the relative in ompressible motion an be written as follows uj =0 (1)

xj

 r2 !2     (u u ) + u + = f ; i = 1; 2; 3 p+ t x x 2 i

S. G. Cherny et al.

j

i

j

i

i

(2) 357

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

where (f1 ; f2 ; f3) = (2!u2 ; 2!u1; 0). Fluid motion in stati elements is des ribed by Eqs. (2) with ! = 0. 2.2

Computational domain and boundary onditions for Platanovryssi Fran is turbine

The omputational domain onsisted of a spiral ase with 18 stay vanes, wi ket gate with 20 guide vanes, runner with 16 blades, and a one diuser of the draft tube (Fig. 1). Following the idea of domain de omposition, the entire domain was divided into 60 blo ks. At the inlet of the spiral ase, uniform velo ity distribution was spe i ed orresponding to a preset dis harge Q. In the outlet se tion of the draft tube one, the swirling ow stru ture was ompatible with the so- alled pressure radial-equilibrium ondition

p 2u = r r

(3)

Pressure distribution in the outlet se tion was obtained by integration of Eq. (3) along the radius from the draft tube wall (where onstant pressure pout is spe i ed) to the rotation axis.

Figure 1 Computational domain for unsteady ow simulation in the entire turbine passage

358

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Unsteady Flows in Turboma hines Internal boundaries, appeared as a result of domain de omposition, are treated by ex hanging uid ow parameters between neighboring blo ks during iterative solution of the problem.

2.3 Initial ow- eld and frozen rotor approa h As a matter of fa t, the problems under onsideration are periodi al nonstationary problems, and their solutions are independent of the initial data. However, the pro ess of attaining the periodi al regime an be very slow for rough initial ow elds. The initial ow eld, whi h

ould be lose enough to the orresponding unsteady ow eld, ould be found by steady-state al ulations using a frozen rotor approa h. In this ase, the position of the runner is assumed to be xed relative to guide vanes and steady-state solution is sought in the passage. Note that \freezing" the runner does not mean the absen e of its rotation in the model. The angle speed ! is still present in Eqs. (2) for runner

ow al ulations. The frozen rotor statement slightly diers from wellknown periodi blade hannel approa h (or stage averaging al ulation). In the latter, the ow parameters are ir umferentially averaged at the interfa e between the wi ket gate and the runner, while in frozen rotor

ase, steady-state al ulations are arried out in all wi ket gate and runner blade hannels without ir umferential averaging. The resulting quasi-nonstationary ow eld des ribes rather well the stator{rotor intera tion, and at the same time does not require huge amount of CPU time.

2.4 Arti ial ompressibility and nite volume methods Numeri al method for the solution of Eqs. (1) and (2) is based on introdu ing arti ial ompressibility relation into the model by adding a pseudotime derivative of pressure to the ontinuity Eq. (1). Also, a pseudotime derivative of velo ity is added to the momentum Eqs. (2). This approa h an be used to ompute both steady-state and timedependent ow problems. Time a

ura y is obtained in the numeri al solution by subiterating the equations in pseudotime at ea h physi al time step. The equations are dis retized within the frame of impli it nite volume method using a third-order ux-dieren e splitting te hnique for the onve tive and pressure terms. The time derivatives in the S. G. Cherny et al.

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

momentum Eqs. (2) are dieren ed using the se ond-order, three-level, ba kward-dieren e formulae. The details of the method are presented in [1{4℄. 3

Operating Conditions for Cal ulations

Flow al ulations were arried out for a redu ed Platanovryssi turbine with runner diameter D1 = 1 m, head H = 1 m, and gravitational a

eleration g = gD1 =H = 0:188. The eÆ ien y  = 93:15%, taken from the turbine eÆ ien y hill hart, orresponds to the rotation speed n1 = 73:5 rpm, giving the blade passage frequen y of 19.6 Hz and runner rotation frequen y fR = 1:225 Hz. Figure 2 shows the eÆ ien y Figure 2 EÆ ien y as a fun tion of  for a wide range of dis harge Q variation, and is taken from 1 and guide vane opening 0 at 1 = 73 5 rpm: experiment (solid line) and the1 ef ien y hill hart for 1 =

omputations (signs) 73:5 rpm. The vortex rope in Fran is turbine draft tube typi ally appears at part-load operating onditions, with guide vane opening A0 lower than 65 %. In the present work, three interesting operating points have been onsidered: part-load regime (Q1 = 0:5975, A0 = 51:4%, and  = 87:5%), highest eÆ ien y regime (Q1 = 0:8746, A0 = 75%, and  = 93:15%), and regime of nominal power (Q1 = 1:004, A0 = 95%, and  = 91%). These operating points are indi ated in Fig. 2. 0

0

0

0

0

Q

A

0

n

0

0

:

0

0

0

4

Results and Analysis

Ea h of 60 omputational blo ks was overed with stru tured mesh with a total amount of approximately one million ells for the full turbine. The hosen time step
t = 0:0085 s was equal to the time needed for 360

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Unsteady Flows in Turboma hines

the runner to rotate at an angle
 = 3:75Æ. Thus, 6 time steps were neededÆ for the runner to sweep one blade hannel, and 96 steps to rotate at 360 . 4.1

Operating point 1

First of all, three steady-state frozen rotor omputations of the entire turbine passage were arried out for three relative positions of stator and rotor, one of whi h is shown in Fig. 3. Figure 4 shows ow patterns in the interfa e region at z = onst for these al ulations. Figure 5 demonstrates pressure variation along the ir umferential interfa e line, marked in bold in Fig. 4. Unsteady omputation of this regime was performed in the domain in luding only the runner and draft tube one. The inlet velo ity distribution and initial data for this ase were taken from one of frozen-rotor

omputations des ribed above. Figure 6 shows the instantaneous ow stru tures behind the runner obtained from the omputation and experiment. The predi ted vortex rope was visualized by the iso-pressure surfa e. In the experiment, the visualization of vortex rope was performed by inje ting air into the

ow and drawing the pla e of bubble a

umulation. A pronoun ed vortex rope pre ession was observed both in omputations and in the experiment.

Figure

3

Relative position of stay vanes, wi ket gate, and runner blades

for operating point 1

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 4

Quasi-nonstationary rotor{stator intera tion: radial velo ity

ontours

Figure 5

Variation of ir umferential pressure distribution in the ourse of marked blade passage (see Fig. 4): 1 | blade tip is at point 1; 2 | blade tip is at point 2; and

362

3

| blade tip is at point 3

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Unsteady Flows in Turboma hines

Figure 6

Draft tube vortex rope in a part-load regime: (a ) omputation

and (b ) experiment

Figure 7

Pressure pulsations (a ) and their spe trum (b ) at point 1: solid line | omputation; dashed and dotted lines | experiment

Pressure distribution in a horizontal ross se tion of draft tube one at dierent time instants indi ated a repeated behavior of the vortex

ore with a period of 2.652 s (0.38 Hz). The ratio of pre ession frequen y to runner frequen y was 0:31. Dierent experimental data gave the range from 0.2 to 0.5 for this ratio. Figure 7 presents pressure u tuations at point 1, shown in Fig. 6, and its Fourier transform, obtained in experiment and in omputations. S. G. Cherny et al.

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 8

Pressure pulsations and spe trum at point 2

Horizontal axis in Fig. 7b is the ratio of pressure pulsation frequen y f to runner rotation frequen y fR . Verti al axis is the ratio of pulsation amplitude to the net head. Pressure u tuation measurements were obtained in two points situated on the wall in one horizontal se tion, but 90Æ distant from ea h other. As an be seen from Fig. 7, the frequen ies in these points oin ide, but the amplitudes dier signi antly. A high peak on the frequen y{amplitude graph orresponds to the vortex pre ession frequen y. A small peak orresponding to the blade passing frequen y (equal to 16fR ) is also present in the graph. Evidently, vortex rope pulsations dominate at point 1. Frequen y of the high peak is slightly shifted to the right due to the fa t that wi ket gate opening in the omputation was a tually 5% higher than in the experiment for operating point 1. However, it is worth noting that the well pronoun ed sequen e of de reasing peaks observed in the experiment is remarkably repeated in the omputation. In order to investigate the upstream in uen e of vortex rope pre ession, pressure u tuations in stati points 2 and 3 (see Fig. 6), were also analyzed. The results are plotted in Figs. 8 and 9. Two main peaks in ea h Fourier transform graph indi ate main frequen ies and amplitudes of u tuations. It an be seen that pressure u tuations in ea h point ontain a harmoni , orresponding to vortex rope pre ession. Therefore, vortex pre ession ae ts the ow eld in the runner and even upstream the runner. However, this in uen e de reases far from 364

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Unsteady Flows in Turboma hines

Figure

9

Pressure pulsations and spe trum at point 3

the vortex lo ation. At the same time, the amplitude of a harmoni , having a blade passing frequen y (16fR ) in reases, rea hing maximum at point 3 before the inlet edges of the blades. In the present study, in ontrast to [5, 6℄, nonuniformity of the ow eld in the inlet se tion of the one diuser was taken into a

ount, and the omputations were arried out in the frame of Euler equations. Thus, physi al vis osity was not present in the model. However, numeri al simulation in this statement also gave a pre essing vortex rope, whi h was in good agreement with experiments in terms of vortex shape and pre ession frequen y. One an on lude that vortex formation an be aused by upstream ow eld nonuniformity and arti ial vis osity,

aused by numeri al s heme dissipation. 4.2

Operating point 2

In this regime, vortex rope pre ession was not observed. Computed velo ity distributions in se tion z = 1:31 below the runner were in good agreement with the experimental data (Fig. 10). 4.3

Operating point 3

In this ase, axially symmetri rotating ow with negligible pre ession has been observed in the omputation, as well as in the experiment (Fig. 11). Figure 12 ompares the absolute velo ity distribution along S. G. Cherny et al.

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 10 Cir umferential u and axial z velo ity distribution behind the runner ( = 1 31) for highest eÆ ien y operating point (point 2): squares | measurements; dashed line | omputation, left bank radius; solid line |

omputation, tail water radius

z

:

Draft tube vortex rope in nominal power regime: ( ) omputation and ( ) experiment

Figure 11

a

b

the radius behind the runner, obtained experimentally and numeri ally. The results of nonstationary al ulation of the entire turbine passage and periodi omputation of only one runner-blade hannel are shown by solid and dashed urves, respe tively. Clearly, full turbine omputations provide more a

urate results.

366

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Unsteady Flows in Turboma hines

Figure 12

Comparison of ir umferential

ponents at nominal power regime:

u

and axial

z

velo ity om-

symbols | experiment; dashed | one

runner hannel omputation; solid | entire turbine omputation

4.4 Cal ulated eÆ ien y One of the most important hara teristi s of turbine operation is the eÆ ien y. There is no doubt that a

urate predi tion of turbine eÆ ien y requires the use of an advan ed turbulen e model and a very ne mesh, sin e the exa t solution of Euler equations should give 100 per ent eÆ ien y in any operating regime. However, dis retization of Euler equations introdu es s heme dissipation. It appeared that s heme dissipation in the model in uen es the integral ow hara teristi s in the same way, as physi al ow vis osity does. Symbols in Fig. 2 show the values of eÆ ien y omputed using the Euler model on the basi mesh (Æ) on the mesh, whi h is by a fa tor of 1.5 ner (). The eÆ ien y was de ned as =

M! gQH

where M is the runner torque and H is the head: H = E1 S. G. Cherny et al.

E2

(4) 367

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

In Eq. (4), the hydrauli energies E1;2 are E1;2 =

1

Q

Z

S1;2

p g

g0 juj z+ g 2g

2

!

(u
ds)

where S1 is the inlet se tion of the spiral ase and S2 is the outlet se tion of the one diuser. It an be seen from Fig. 2 that the al ulated eÆ ien y depends on the mesh size; however, the tenden y of eÆ ien y

hange with dis harge variation is aptured rather well.

5

Con luding Remarks

The paper presents the results of invis id analysis of the 3D unsteady

ow in Platanovryssy Fran is turbine. The CFD ode CADRUN developed by the authors was employed to ompute the ow eld on the stru tured mesh with approximately one million ells. To validate the present approa h, 3D ows with rotor{stator intera tion and draft tube vortex rope were simulated and then ompared with the experiments. Good agreement with the experiments was obtained. It was shown that the dis rete invis id model ould a

urately predi t major ow features and qualitative behavior of turbine eÆ ien y with variation of the operating regime.

A knowledgments This work was supported by the Russian Foundation for Basi Resear h (proje t No. 04-01-00246).

Referen es 1. Cherny, S. G., Y. A. Gryazin, S. V. Sharov, and P. A. Shashkin. 1996. An eÆ ient LU-TVD nite volume method for 3-D invis id and vis ous in ompressible ow problems. 3rd ECCOMAS Computational Fluid Dynami s Conferen e Pro eedings . Paris: John Wiley & Sons. 90{96. 2. Gryazin, Y. A., S. G. Cherny, S. V. Sharov, and P. A. Shashkin. 1997. On one method for numeri al solution of 3D hydrodynami ow problems. Dokl. Akad. Nauk 353(4)::478{83.

368

S. G. Cherny et al.

Unsteady Flows in Turboma hines 3. Kovenya, V. M., S. G. Cherny, S. V. Sharov, V. B. Karamyshev, and A. S. Lebedev. 2001. On some approa hes to solve CFD problems. Comput. Fluids 30:903{16. 4. Cherny, S. G., S. V. Sharov, V. A. Skorospelov, and P. A. Turuk. 2003. Methods for three-dimensional ows omputation in hydrauli turbines. Russ. J. Numer. Anal. Math. Modelling 18(2):87{104. 5. Rupre ht, A., T. Helmri h, T. As henbrenner, and T. S herer. 2000. Simulation of vortex rope in a draft tube. 20th IAHR Symposium on Hydrauli Ma hinery and Systems Pro eedings . Charlotte. 6. Lapin, V. N., S. G. Cherny, V. A. Skorospelov, and P. A. Turuk. 2004. Problems of ow simulation in turboma hines. International Conferen e on Computational and Informational Te hnologies for Resear h, Engineering and Edu ation Pro eedings

S. G. Cherny et al.

. Almaty. 3:57{66.

369

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

INVESTIGATION OF UNSTEADY FLOW IN THE TIP CLEARANCE OF AXIAL-COMPRESSOR STAGE ROTOR N. N. Kovsher and K. S. Fede hkin

Zhukovsky Air For e Engineering A ademy Planetnaya Str. 3, Mos ow 125190, Russia

Introdu tion

High spe i parameters of modern axial ompressors are determined primarily by the aerodynami quality of its blading whi h depends on the level of losses in the stage blade rowns. As was shown in numerous experimental and numeri al studies, major stage losses are on entrated in the ir umferential and near-tip regions of the blade assembly. A simulation model allowing one to al ulate unsteady ow parameters both in the invis id ow ore [1℄ and in the quasi-steady shear-wall boundary layer has been developed and used for investigating the unsteady ow in the radial learan e of a rotor [2℄. Model

The invis id ow is assumed to be a potential unsteady ow of the ideal in ompressible liquid, governed by the Euler equations:  x + x d (t; r) dt

div (t; r) 

 y  + z =0 y z

1



rP (t; r) = F (t; r)

To al ulate the invis id ow, the method of dis rete vorti es is used. The vis ous ow simulation is applied to quasi-steady onditions and 370

N. N. Kovsher and K. S. Fede hkin

Unsteady Flows in Turboma hines is based on the equations of the three-dimensional (3D) boundary layer theory:

 x1

 pg  h1

u +

 x2

 pg  h2

w +

 p ( gv) = 0 x3



u u g h w u u g h 1 + +v + uu 12 12 1 + 1 h1 x1 h2 x2 h1 ( x"3  g 2 h x#1 x2 ) h h1 g12 1 2g12 2 + uw h1 h2 1 + g h1h2 x2 x1   h g12 h g12 h2 + ww 1 h2 2 g x2 x1 h2 x2 h h2 r h g r h1h22 p h1 g12 !2 r 1 2 + !2 r 1 12 + g dx1 g dx2 g dx1  g 1  u =  1  x3 x3



g12 x1



p dx2 u v 0

 0



h u w w w w h g12 g12 h1 + +v + uu 2 h1 1 h1 x1 h2 x2 x3 g x1 x2 h1 x1 ( "  # )  g12 2 h2 h1 1 h h 1+ 2g12 + uw g 1 2 h1h2 x1 x2   g g h 1 g12 h2 + ww 12 12 2 + g h2 x2 h2 x2 x1 h2 h r h g r h21h2 p h2 g12 p !2 r 1 2 + !2 r 2 12 + g dx2 g dx1 g dx2  ug dx1  1   2 w v =  x3 x3 0

0

Thus, the ow is modeled by the system of dierential equations, whi h is losed by the equations of algebrai Cebesi{Smith turbulen e model:

u v = "1 0

0



u x3



N. N. Kovsher and K. S. Fede hkin

;

w v = "2 0

0



w x3



371

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 1 Model of ow stru ture in the tip learan e 2

"i = L

" 2  2# u w y

+

y

tr

An important feature of the ow in the radial learan e is the tip vortex, whi h ae ts greatly the entire stage ow. When developing the vortex model, it was assumed that inertial separation should take pla e on the blade tip (Fig. 1). This separation o

urs in the form of individual losed vortex frames, onne ted downstream with the lo al speed towards the vane blading.

Results The mathemati al model allows the tip learan e ows to be adequately resolved and to qualitatively des ribe the a

ompanying phenomena (Fig. 2). The estimation of the in uen e of the tip learan e size on the ow in the rotor ir umferential area at dierent operation modes has been investigated in [3℄. The in uen e of the blade rown geometry | the angle of in iden e ( 1 ) (Fig. 3), mid-line amber () (Fig. 4), and solidity (b=t) (Fig. 5) | on the ow in the rotor radial learan e has been estimated as well. The omputational results demonstrated the signi ant ee t of the mid-line amber on the ow in the tip learan e. 372

N. N. Kovsher and K. S. Fede hkin

Unsteady Flows in Turboma hines

Figure 2 Flow stru ture in the tip learan e

Figure 3 In uen e of angle of in iden e on the ow in the rotor tip

learan e: (a ) = 29Æ ; (b ) 39Æ ; and ( ) = 49Æ

N. N. Kovsher and K. S. Fede hkin

373

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 4 In uen e of mid-line amber on the ow in the rotor tip learan e: (a )  = 25Æ ; (b ) 40Æ ; ( ) 55Æ ; and (d )  = 70Æ

Figure 5 In uen e of solidity on the ow in the rotor tip learan e: (a ) b=t = 0:85; (b ) 1.19; and ( ) b=t = 1:56 374

N. N. Kovsher and K. S. Fede hkin

Unsteady Flows in Turboma hines Thus, the mathemati al model developed is apable of providing qualitative and quantitative information on the pro esses o

uring in the tip learan e of an axial- ompressor stage rotor.

Referen es 1. Belo erkovsky, S. 1988. Mathemati al modeling of plane-parallel stalled

ow near bodies . Mos ow: Nauka. 2. Cebe i, T. 1975. Cal ulation of three-dimensional boundary layers. Threedimensional ows in Cartesian oordinates. AIAA J. 13(8):1056{64. 3. Lakshminarayana, B. 1969. Methods of predi ting the tip learan e ee ts in axial ow turboma hinery. ASME Paper No. 16.

N. N. Kovsher and K. S. Fede hkin

375

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

EFFECT OF FLOW UNSTEADINESS ON THE PERFORMANCE OF AIRFOIL CASCADES: THEORETICAL EVALUATION

V. B. Kurzin and V. A. Yudin

M. A. Lavrentyev Institute of Hydrodynami s Siberian Bran h of the Russian A ademy of S ien es Lavrentyev Ave. 15 Novosibirsk 630090, Russia

Introdu tion

Among the fa tors redu ing the turboma hine performan e, the most important one is the energy loss for the produ tion of vorti es arising in the liquid ow as it intera ts with wheel blades. The vorti es generated in the boundary layers at blade surfa es give rise to vorti al wakes behind the blades. These vorti es de ne so- alled pro le losses. The

ow vorti ity resulting from spanwise nonuniformity of the ir ulation in the liquid ow brings about se ondary total-pressure losses on the wheel. In addition, vorti al wakes behind blades are generated owing to temporal variation of the ir ulation about the blades. For periodi nonstationary pro esses indu ed by blade vibrations or ir umferential nonuniformity of the approa hing ow, in luding hydrodynami rotor{ stator intera tion, the energy losses an be analyzed onsidering their period-averaged values. The problem of performan e redu tion aused by periodi pulsations of the ow in turbine as ades and some relevant experimental data were dis ussed in [1℄. For a ompressor stage, this problem, related with the formation of vorti al wakes in the ow past mutually moving as ades, was experimentally examined in [2, 3℄. In the present study, using the linear model of two-dimensional ow, an expression for the mean kineti energy of ow pulsations generated by vorti al wakes 376

V. B. Kurzin and V. A. Yudin

Unsteady Flows in Turboma hines in the periodi ally pulsating ow approa hing an airfoil as ade is derived. Then, a dimensionless quantity that des ribes the redu tion in the as ade performan e aused by energy losses for the produ tion of vorti al wakes is introdu ed. By way of example, this quantity is al ulated for a periodi ally pulsating ow with pulsations resulting from the hydrodynami intera tion of airfoil as ades simulating turboma hinery stages. A omparison of the predi ted values with experimental data is then given.

1 Statement of the Problem Consider an unsteady ow of an ideal in ompressible liquid past an airfoil as ade. The approa hing ow is assumed to be pulsating, the pulsations being weak and periodi , so that

V (x0 ; y0) = V 1 + Vb(x0 ; y0) ; V 1 = onst jV j  jV 1 j ; Vb (x0 ; y0) = Vb(x0 ; y0 + H ) where (x0; y0 ) is some Cartesian referen e frame with respe t to whi h the as ade moves with a onstant velo ity u. The fun tion Vb (x0 ; y0) is expanded in a Fourier series with respe t to the oordinate y0 in the frame (x; y), atta hed to the blade as ade, x = x0 and y = y0 + ut; then, one obtains:

Vb (x0; y0 ) =

X1 V r =1

x

br ( 0 ) exp

Vbr = Vbr1 + jVbr2

j

2r (y

H

ut) (1)

The latter expression de nes the pulsational perturbation introdu ed into the approa hing ow by the as ade. A

ording to Eqs. (1), in the linear approximation, the determination of nonstationary hara teristi s of the airfoil as ade in a given ow velo ity eld redu es to nding a superposition of solutions to problems equivalent to the problem about

as ades syn hronously vibrating at frequen ies !r = !r, ! = 2u=H with identi al amplitudes and with some phase shift r = 2r=H between neighboring as ades. Here, n = 1; 2; : : : ; N ; N = H=h; and h is the pit h in the as ade. V. B. Kurzin and V. A. Yudin

377

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 1 Unsteady ow past an airfoil as ade

Under the assumption that no separation o

urs in the ow, the pulsating omponent of the disturbed ow is potential with the owvelo ity potential equal to '

=

X1 r ( r=1

' x; y

)exp(j!rt)

(2)

where 'r (x; y) = 'r1(x; y) + j'r2(x; y) are the amplitudes of the owvelo ity omponents for the individual terms in the Fourier series (1). These values an be found using pro edures developed in the theory of as ades in unsteady ows [4, 5℄. In the linear approximation, the vorti al wakes atta hed to the airfoils an be modeled with ow-velo ity

onta t dis ontinuity lines Lm that stret h along main- ow streamlines (Fig. 1). Within the framework of the model of ideal in ompressible liquid and under the assumption of no separation, the Cau hy{Lagrange integral is onstant everywhere in the region outside the as ades and the vorti al wakes, ' t

2

+ V2 + p = onst

(3)

where V = V0 +v, p = p0 +~p, and V0, p0 and v =
'; p~ are, respe tively, the mean and time-dependent omponents of the ow velo ity and stati 378

V. B. Kurzin and V. A. Yudin

Unsteady Flows in Turboma hines pressure. It should be noted here that far upstream of the as ade, the pulsating omponent of the ow in the frame (x0 ; y0) is zero, and far downstream of the as ade, the vorti al wakes shedding from the airfoils give rise to ow pulsations in the region outside the wakes. Denote the ow quantities far upstream of the as ade with the subs ript \1", and the ow quantities at a small distan e downstream of the as ade, where the free vorti es atta hed to the airfoils an be assumed fully developed at the onta t dis ontinuity line Lm (the experimental data of [6℄ show that the free vorti es indeed arise near the trailing edges of airfoils and these vorti es bear lose resemblan e to

onta t dis ontinuity lines) with the subs ript \2." Using integral (3), one an write the relation



V 2 '1 V 2 p2 + 2 = p1 + 1 + 2 2 t For the mean values, this relation yields:



V 2 p1 + 1 2

 

V 2 = p2 + 2 2

'2 t





(4)

Equation (4) shows that the average total pressure far downstream of the as ade and the average total pressure right behind the as ade, in se tion 2, are identi al and, therefore,



V 2 p2 + 2 2

 



V 2 hv2 i = p20 + 20 + 2 2

(5)

where

RT

hv i = 2

0

(r')2 dt

T

=

1

1X (r'r
r'r ) ; 2 r=1

T=

2 !

(6)

The rst term in Eq. (5) de nes the total pressure in the mean ow behind the as ade. The se ond term results from the perturbation introdu ed into the ow by unsteady vorti al wakes. This term de nes the portion of the kineti energy of the liquid ow onsumed on the produ tion of the wakes. The term hv2 i=2 is the total-pressure omponent V. B. Kurzin and V. A. Yudin

379

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

that redu es the potential energy due to the ompressor pressure and, in this way, de reases the ompressor performan e. In nonideal liquids, owing to vis ous fri tion, the perturbation introdu ed into the ow by the vorti al wakes will de ay with distan e from the as ade front, and the kineti energy onsumed on the produ tion of the wakes will dissipate. Assuming now that, rst, the ee t due vis ous-fri tion for es is signi ant only in the region o

upied by the unsteady vorti al wakes and, se ond, the wakes themselves fully de ay in the downstream dire tion, one obtains from Eq. (5) that the total pressure far downstream in the liquid ow is equal to the total pres2 =2). Then, the term hv2 i=2 de nes sure in the mean ow, (p20 + V20 the total-pressure losses due to ow unsteadiness. Averaging this term over y and normalizing it by the dynami head in front of the as ade, V 21 =2, one obtains the dimensionless parameter:

1 P

hr'r
r'r i

 = r=1

2V 21

des ribing the redu tion in the as ade performan e aused by the energy loss on the produ tion of the unsteady vorti al wakes. To determine , one has to nd the nonstationary omponent of the liquid ow generated by the vorti al wakes.

2

Velo ity Field Indu ed by Unsteady Vorti al Wakes

To des ribe the dis ontinuous liquid ow in se tion 2 behind the as ade, the natural frame A v with  -axis dire ted along the onta t dis ontinuity line L0 and v-axis normal to this line is introdu ed (see Fig. 1). Sin e in the linear approximation the shedding parti les move with the main- ow velo ity V20, then, a

ording to Eq. (2), the intensities of the vorti al sheets Lm an be represented as

m (; t) =

1 X





mr exp j!r t r=1

mr = mr1 + j mr2 = onst

380

 V20



V. B. Kurzin and V. A. Yudin

Unsteady Flows in Turboma hines For ea h of the time harmoni s, the amplitude fun tions of the wake intensities an be found in the form [4, 5℄:

mr ( ) = 0r exp

 !r  exp ( jmr ) j V

(7) 20 A

ording to Eq. (7), within the framework of the model of ideal in ompressible liquid, the amplitude fun tion of the rth harmoni of the

ow-velo ity potential in the band between vortex lines Lm and Lm+1

an be represented as

     r!  r!v r!v

mr exp + dmr exp 'r (; v) = exp j V V V

(8) 2 20 Be ause fun tion 'r is dis ontinuous a ross the vorti al sheets, the

onstants mr and dmr for dierent bands between vortex lines dier in value. To nd these onstants, one an use the following relations: { ondition of generalized periodi ity: 'r (x; t) = exp ( jmr ) ' (x; y + mh) (9)

20

{ ontinuity of the normal omponent of the ow velo ity on the line Lm :

 ' 

r =0

v

(10)

{ expression for the wake intensities:

mr ( ) =

 ' 

r ; (; v) 2 Lm



(11)

In view of Eq. (9), the y-averaged values of terms hr'r
r'r i in Eq. (6) are onstant within ea h band between neighboring vortex lines Lm . Taking into a

ount the relationship between the oordinates in the frames 0xy and Av, and using Eqs. (9){(11), Eq. (8) an be rewritten as

'r =



j 0r exp(kr v) exp ( jr ) 2 kr exp(r ) exp(jr )  exp( kr v) exp ( jkr  ) (12) + exp( r ) exp(jr )

V. B. Kurzin and V. A. Yudin

381

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

where r = r kr sin , r = kr os , kr = !rh=V2,  = =h, and v = v=h. The integral in Eq. (6) an now be written in the form:

' 'r r ' r r'r
r'r = ' + r   v v and, using Eq. (12), one obtains:

hr'r
r'r i = 4k

20r (1 exp( 2r )) 2 exp( r ) os r + exp( 2r ) r os (1

(13)

3 Stator{Rotor Intera tions In a system of mutually moving intera ting as ades (stators and rotors), these as ades are streamlined by a periodi ally pulsating ow. Consider rst the ase of two mutually moving as ades (Fig. 2) and

al ulate the intensity of vorti al wakes behind the as ades based on the semiempiri al theory of potential{vorti al intera tion [7, 8℄. Within the framework of this theory, the liquid is assumed ideal and in ompressible, and the ow velo ity V 1 far upstream of the as ades is assumed uniform. Cas ade 2, lo ated downstream, moves with respe t to as ade 1 with a onstant velo ity u along the y-axis. The airfoils in both as ades are assumed smooth and having angle-shaped trailing edges. The perturbation introdu ed into the ow by the as ades is assumed weak, and the vorti al wakes behind the airfoils are assumed to interse t the streamlines of the main ow through the as ades. The evolution of the wakes behind as ade 1 as these wakes pass through

as ade 2 is negle ted. Then, the omplex velo ity of the liquid ow at a point z at ea h time t an be represented as

V (z; t) = v(z; t) + J (z ) + J1(z; t) + J2 (z; t) where v(z; t) is a fun tion analyti al with respe t to z everywhere outside the as ades. This fun tion, whi h is to be found, de nes the potential intera tion between the as ades. The fun tion J (z ) des ribes the perturbation of the ow in the unsteady vorti al wakes behind as ade 1 in the absen e of as ade 2, this perturbation being fully de ned 382

V. B. Kurzin and V. A. Yudin

Unsteady Flows in Turboma hines

Figure 2

as ades

Two mutually moving

Figure 3

Steady wakes produ ed behind as ade 1

by the empiri al fa tor  of pro le losses. The form of the fun tion J (z ) was found in systemati wind-tunnel tests performed with various

as ades [1, 9℄; this form depends on the fa tor  (Fig. 3). The fun tion J (z; t) is the omplex- onjugate velo ity indu ed by the onta t dis ontinuity lines Lm behind the th as ade that model the unsteady vorti al wakes behind their airfoils: J (z; t) =

1 H

Z1 X

N 1

0

m=0

m (; t)

1

d

exp(2( ( )

z + imh )=H )

Here,  2 L0 ,  is the ar oordinate on the line L0 , H = N1 h1 = N2 h2 is the total period of the as ades, N1 and N2 are the total numbers of airfoils in the as ades, and h1 and h2 are the as ade pit hes. The parameters m de ne the intensities of the unsteady wakes and

an be found from the ondition that the stati pressure is ontinuous a ross the wake: V. B. Kurzin and V. A. Yudin

383

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

m (; t) = t = t1 + T ( ) ;

1  m V0 ( ) t t=t1 T ( ) =

Z

0

(14)

d V0 ( )

where V0 ( ) is the relative velo ity of the steady ow at the point with the ar oordinate  measured from the trailing edge of the mth airfoil R in the th as ade, and m (t) = Vm (s; t) ds is the ir ulation of the relative velo ity on this airfoil. The relative velo ity Vm (s:t) of the liquid ow on the airfoils an be represented as Vm (s; t) =

1

n XX n=0 r=0

(unr (s) os r(!t + m  )

+ vnr (s) os r(!t + m  )) exp



2n




H

(15)

where ! = 2u=H ,  = ( 1) 2=N , and
is the axial learan e between the as ades. The solution of the orresponding boundaryvalue problem for the analyti al fun tion v(z; t) obtained by means of the theory of fun tions of omplex variable with the use of the Cau hy formula for periodi fun tions nally yields some system of re urrent relations for the expansion oeÆ ients of the ow velo ity in series (15): K1r (U1nr ) = 1nr (U2pq ; p 2 (1; n

1); q 2 (1; p)) K2r (U2nr ) = 2nr (U1pq ; p 2 (1; n); q 2 (1; p))

(16)

where Unr = unr + jvnr ( = 1; 2; i 6= j ). The integral operator Kr in system (16) is the se ond-kind Fredholm operator, and the right-hand sides nr are ompletely de ned by the as ade geometry, by the ow

onditions, and by the set oeÆ ient  for as ade 1. The ir ulations m (t) on the airfoils are also given by Eq. (15), and their harmoni s are given by the equalities  (r;
) = 1(r;
) + j 2 (r;
) =

384

Z

1

X

n=r

Uhr (s) exp



2n
H



ds

V. B. Kurzin and V. A. Yudin

Unsteady Flows in Turboma hines

Then, the relation between these harmoni s with 20r in Eqs. (13) is given, in view of Eqs. (14), by the equality:

20r

= 20r =

r2 !2 2 (r;
) ; V2

V

= lim !1V0 ( ) ;  = 1; 2

(17)

Next, onsider the intera tion in a stator{rotor{stator (or rotor{stator{ rotor) system ontaining three as ades. For the sake of de niteness, it is assumed (Fig. 4) that as ade 2 moves in the negative dire tion of the y-axis with velo ity u. Here again, the velo ity V 1 far upstream of the as ade is assumed to be uniform. Under the assumption that

as ades 1 and 3 intera t weakly with ea h other, the time-dependent perturbation of the ow on the airfoils of as ade 2 an be represented as a superposition of two perturbations introdu ed into the ow by the two intera ting (stator{rotor and rotor{stator) pairs. The result of the summation depends on the mutual ir umferential position of the stators, i.e., as ades 1 and 3, de ned, for instan e, by the parameter  2 [0; 1℄ that shows that as ade 3 is shifted with respe t to as ade 1 by distan e h3 in the dire tion in whi h as ade 2 moves (Fig. 4). Then, a

ording to Eq. (15), the relative velo ity V2 (s; t) of the liquid

ow at the rst airfoil of as ade 2 is given by 1 X V2r (s;
12;
23;  ) exp ( jr!t) V2 (s; t) = r=0

V2r (s;
12;
23;  ) = Ur12(s;
12) + Ur23(s;
23) exp Ur12 (s;
12) = Ur23 (s;
23) =

1 X

n=r

1 X

n=r

 2n
 12 12 U (s) exp 2nr

U223 nr (s) exp



H

2n
23

 2r  j N3



H

where H = h1 N1 = h2N2 = h3N3 is the total period of all as ades 1 to 3, and
12 and
23 are the axial learan es between the as ade pairs 1{2 and 2{3, respe tively. The oeÆ ients U212nr and U223nr in Eq. (15) are given by the solutions of the problems in whi h the intera tion within the as ade pairs 1{2 and 2{3 is onsidered. V. B. Kurzin and V. A. Yudin

385

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 4 Stator{rotor{stator system

Subsequently, the ir ulation harmoni s on as ade 2 an be al ulated by the formulae 2(r;
12;
23;  )

=

21(r;
12;
23;  ) + j 22 (r;
12;
23;  )

=

Z

V2r (s;
12;
23;  ) ds

and the values of 20r in Eq. (17) are given by

20r

4

=

r2!2 22 (r;
12;
23;  ) V22

Results of Cal ulations

The method des ribed above was embodied in a PC ode. For two or three mutually moving as ades, the ode al ulates both nonstationary aerodynami hara teristi s of the airfoils and the value of  for ea h of the as ades. Note that the al ulating time depends weakly on the total number of airfoils in the total period of all the as ades and on the axial learan e between the as ades; for all omputational runs made in the present study, this time never ex eeded a minute. Figure 5 shows the results of al ulations and the experimental data of [10℄. Plotted along the y-axis in Fig. 5 (right olumn) are the 386

V. B. Kurzin and V. A. Yudin

Unsteady Flows in Turboma hines

Figure 5

Level of ex iting for es 2y and the parameter of total-pressure losses  vs. axial learan e, N2 = 10, 2 = 1:33: (a ) N1 = 9, 1 = 0:71, and  = 0:021; and (b ) N1 = 3, 1 = 0:64, and  = 0:04

al ulated values of  on as ade 2. Plotted in Fig. 5 (left olumn) are the al ulated and experimental values of the ex iting for e 2y = (max Y2 (t) min Y2 (t))=Y20, t 2 [0; T2℄, T2 = 2h1=u, on the pro les of

as ade 2, where Y2 (t) is the ir umferential omponent of the for e on the airfoil and Y20 is its mean omponent. Close inspe tion of the urves shows that the parameter  for either as ade varies in proportion to the level of ex iting for es on its airfoils. This result is not surprising sin e the level of ex iting for es in reases due to the in reasing intensity V. B. Kurzin and V. A. Yudin

387

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 6

Nonmonotoni behavior of 2y and  with the axial learan e

of the perturbation introdu ed into the ow by the as ade. The latter, in turn, in reases the intensity of the unsteady vorti al wakes shedding from the airfoils of the as ade; as a result, an in rease in  is observed. Note that the typi al value of  is 0.5%{1%; this value sharply rises as the axial learan es between the as ades be ome small. An important feature in the behavior of the ex iting for e in a system of intera ting as ades is the experimentally observed dependen e of 2y on the axial learan e at large angular oordinates of as ade 1. In [11℄, this feature was explained by superposition of potential and vorti al (from steady vorti al wakes J ) ow perturbations. A omparison between urves in Fig. 6 shows that the parameter  also displays a nonmonotoni behavior in the same range of axial learan es. Figure 7 ompares the present al ulations with the experimental data of [3℄ obtained for a stator{rotor{stator system. The experiment in [3℄ was aimed at studying the ee t due to stator ir umferential position ( lo king). Curve 1 in Fig. 7 shows the experimental values of the parameter
P () h
P i 100%  (v) = h
P i vs. the parameter , where
P is the dieren e between the mean stagnation pressure and the ir umferentially averaged stagnation pressure, both pressures being measured on the mean radius behind as ade 3 and as ade 2 (this dieren e is due to the dissipation of free vorti es 388

V. B. Kurzin and V. A. Yudin

Unsteady Flows in Turboma hines

Figure 7 Ee t of stator lo king on the total-pressure losses: 1 | present

al ulations and 2 | experiment of [3℄, N1 = 18, N2 = 19, N3 = 18, 1 = 0:016, 2 = 0:02, u=V 1 = 1:8

atta hed to as ade 2, and hP i is the value of
P averaged over . Curve 2 is the predi ted dependen e for the parameter ( ) hi 100%  (v ) = h i

First, the satisfa tory agreement between the urves substantiates the on lusion, drawn experimentally in [3℄ that the total-pressure losses are primarily onditioned by the intensity of free vorti es produ ed behind a rotor. Se ond, this agreement proves that the proposed timeeÆ ient al ulation method, although rude, an be useful for preliminary estimations of losses aused by ow unsteadiness in a system of

as ades. Besides, it should be noted that the positions of the maxima and minima in the urves are almost oin ident. It is therefore believed that the proposed al ulation method is apable of adequately predi ting the ee t due to mutual position of stators (or rotors), allowing redu tion in the total pressure losses in airfoil as ades. V. B. Kurzin and V. A. Yudin

389

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

A knowledgments This

work

was

supported

by

the

Siberian

Bran h

of

the

Rus-

sian A ademy of S ien es under Interdis iplinary Integration Proje t No. 117.

Referen es 1. Samoilovi h, G. S. 1975. Turboma hine blades vibration. Mos ow: Mashinostroenie. 2. Saren, V. E. 1995. Relative position of two rows of axial turboma hine: Ee t on aerodynami s in a row pla ed between them. In: Unsteady aerodynami s and aeroelastisity of turboma hines. Amsterdam: Elsevier. 421{25. 3. Savin, N. M., and V. E. Saren. 2000. Hydrodynami intera tion of the blade rows in the stator{rotor{stator system of an axial turboma hine. Fluid Dynami s 35(3):432{41. 4. Samoilovi h, G. S. 1969. Unsteady ow and aeroelasti vibration of turboma hine blades . Mos ow: Nauka. 5. Gorelov, D. N., V. B. Kurzin, and V. E. Saren. 1971. Aerodynami s of airfoil as ades in unsteady ow . Novosibirsk: Nauka. 6. Saren, V. E., and S. A. Smirnov. 2003. Unsteady vorti al wakes behind mutually moving rows of axial turboma hine. Thermophysi s Aerome hani s 10(2):175{87. 7. Yudin, V. A. 1981. Cal ulation of hydrodynami intera tion of pro le

as ades with ee t of wing wake. Trudy CIAM 953:52{66. 8. Yudin, V. A. 2001. Cal ulation of hydrodynami intera tion of pro le as ades with ee t of unsteady wing wake diusion. J. Applied Me hani s Te hni al Physi s 42(5):61{69. 9. Stepanov, G. Yu. 1962. Hydrodynami s of turboma hinery as ades . Mos ow: Fizmatgiz. 10. Ada hi, T., K. Fukusado, N. Takanashi, and Y. Nakamoto. 1974. Study of the interferen e between moving and stationary blade rows in axial

ow blower. Bull. JSME 17(109):904{11. 11. Saren, V. E., and V. A. Yudin. 1984. In uen e of axial learan e on hydrodynami intera tion of pro le as ades. In: Aeroelasti ity of turboma hine. USSR A ad. S i., Institute of Hydrodynami s. 33{42.

390

V. B. Kurzin and V. A. Yudin

SECTION 6

UNSTEADY FLOW PHENOMENA IN TURBOMACHINES

Unsteady Flow Phenomena in Turboma hines

FORTY YEARS OF EXPLORING UNSTEADY FLOW PHENOMENA IN CENTRIFUGAL COMPRESSORS R. A. Izmailov

St. Petersburg State Polyte hni University (SpSPU) Polyte hni heskaya Str. 29, St. Petersburg, Russia

Introdu tion Centrifugal ompressors are ne essary parts of (small) gas turbine engines, natural gas pipeline installations, pro ess equipment, and industrial station for air. For a long period of time, they were onsidered to be free of any pulsations (ex ept for well known surge). Numerous failures during the adaptation period of natural gas pipeline ompressors (not only in Russia) initiated detailed investigations of unsteady

ow phenomena, espe ially in rotating impellers. This paper presents a short des ription of physi al and methodologi al aspe ts of the problem in its histori al development.

General S ope The issue was rst addressed by the present author in the early 1960s. There was no experien e in the measurements of unsteady ow phenomena and there was only some general understanding of these phenomena. Initially, the results obtained from axial turboma hine investigations were used for explaining the physi al pi ture of (possible) phenomena. For the sake of simpli ity, aeroelasti and transient pro esses were ex luded from on ideration. The investigations were on entrated on the remaining pro esses, namely, (i ) deterministi pro esses aused by ir umferential nonuniformities (both in rotating and stationary frames); (ii ) rotating stall; and (iii ) sto hasti phenomena. The frequen y in the experiments ranged from 0 to 40 kHz (at present, the upper limit R. A. Izmailov

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

is extended to 100 kHz). As a basi tool for measurements, miniature strain-gage (stati ) pressure transdu ers, hot-wire anemometers, and sometimes, total pressure transdu ers were hosen. From the very beginning, only semi ondu tor pi k-ups (domesti made) were used. To transfer signals from rotating impellers, high-te h mer ury slip-ring te hniques were applied [1℄. Before measurements, the measuring hannels were alibrated at stati and dynami onditions. Initially, analogue methods were used, but in the 1970s, a sophisti ated data a quisition system was developed based on multi hannel statisti al analyzers in nu lear physi s resear h. Certainly, the issue of data analysis was also ru ial. The experimental units (5 setups suited for measurements in rotating impellers) onsisted of impellers and several types of diusers (vaneless, vaned, and of semivaned type): sometimes, a full stage with inlet hamber and outlet volute was used. All existing types of impellers (with 2 = 21Æ : : : 90Æ ) and vaned diusers (with z = 7 : : : 24), as well as vaneless diusers of varying widths were tested. The maximum level of peripheral velo ities was 293 m/s for measurements in rotating impellers, while in industrial experiments this level was up to 640 m/s (in this ase, the unsteady pressures were measured only in diusers).

Results The hara teristi s of ompressor stages were divided into three domains: (i ) from stonewall to the onset of rotating stall; (ii ) rotating stall; and (iii ) surge [1℄. For a stage with a very short vaneless diuser and an outlet s roll (typi al design for pumps and old-fashioned natural gas pipeline ompressors), there exist only two domains, and the rotating stall domain is absent. Figure 1 presents the results of measurements of unsteady total and stati pressure os illations (at U2 = 250 m/s) at maximal, optimal, and low mass ow rates (a, b, and , respe tively) and for surge onditions (d ). Pressure re ords A to F orrespond to the stati pressures in the impeller; and G, F , and I orrespond to total pressures in the outlet volute. The unsteadiness of the ow in the impeller is learly seen. The level of os illations is low only in the design point (b), while at high (a) and low ( ) mass ow rates, the level of os illations is very high (up to 0:75U22). The prin ipal period of os il394

R. A. Izmailov

Unsteady Flow Phenomena in Turboma hines

Figure 1 Measurements of unsteady total and stati pressure u tuations

(U2 = 250 m/s) for maximal (a ), optimal (b ), and low ( ) mass ow rates and for surge onditions (d )

lations is equal to the rotation period modulated by 5{6 multiples. At the impeller inlet (F ), the level of os illations is of the same order as at the outlet. The origin of these os illations is aused by ir umferential

ow nonuniformities in the volute. The multiples arise due to aerodynami resonan e in impeller hannels and will be dis ussed later. In the volute (H , I ), the rotating wakes an be learly seen, whi h propagate towards the pipe outlet (G). During surge (d), the os illations modulate a low-frequen y sinusoidal pulsation. In reasing of the peripheral velo ity leads to enhan ement of haoti turbulent u tuations, but the main spe i features remain the same. The results presented above are typi al for stages of dierent types for the mass ow rates ranging from maximal values up to the values

orresponding to the in epien e of rotating stall. The main reasons for os illations in this domain are the ir umferential nonuniformities and R. A. Izmailov

395

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

turbulen e. As a rule, the transformation oeÆ ient for the level of these os illations is of order 1 (in subsoni ow). This suggests a simple rule for al ulating the unsteady pressure, whi h auses the os illating stresses in the impeller: to measure the ir umferential distribution of stati pressure on the walls of diuser losely behind the impeller (as lose as possible). The well-known pioneering arti le by Dean and Senoo [2℄ stimulated the present author to address the \jet and wakes" problem. All types of stages were onsidered. The results of measurements of unsteady velo ities in vaneless diusers (phase lo ked) are shown in Fig. 2 (for the maximal mass ow rate). As is seen, these

u tuations propagate up to a diameter of 1:25D2 (in the vaneless diffuser), the ow is three-dimensional (3D), and the de ay law resembles the exponential urve (Fig. 3). For a vaned diuser, the problem of propagating wakes as well as the problem of propagating nonuniformities reated by the diuser inside the impeller are more ompli ated. During experimental investigations of unsteady os illations in the impeller and diuser, the present author has dis overed the phenomenon of aerodynami (a ousti ) resonan e. In ontrast with the well-known a ousti resonan e in axial turboma hines, this phenomenon exists at absolutely rigid blades and has aerodynami grounds. Similar ee ts were dis overed in hydrauli turboma hines and reported by Den-Gartog [3℄. The in oming pressure nonuniformities at the inlet of diuser or impeller hannels propagate towards the outlet and re e t from the hannel open end. If the propagation time \in and out" is equal to the period of nonuniformities, the aerodynami resonan e omes into ee t. In this ase, the nonuniformities are well pronoun ed inside the hannel. Examples of resonant hara teristi s of impeller and vaned diuser are presented in Figs. 4 and 5. In the diuser, the \jets and wakes" propagate downstream the main ow. In the impeller, they propagate upstream the ow. This phenomenon is dangerous for impellers and diusers and is responsible for high level of dynami stresses leading to failures of dis s, blades, and bolts (it was on rmed by industrial experiments on a blast furna e ompressor). Despite the physi s of the ow is very ompli ated, the prin ipal frequen y an be al ulated using Rayleigh's formula for open or losed tubes [4℄. This phenomenon is also responsible for high level of radiated sound and sometimes an result in high-level vibrations of inlet and outlet piping of turboma396

R. A. Izmailov

Unsteady Flow Phenomena in Turboma hines

Figure 2 Unsteady velo ities in vaneless diusers: (a ) z=b3 = 0:12; (b ) 0.5; and ( ) z=b3 = 0:88 R. A. Izmailov

397

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 3 Flu tuations of unsteady velo ities in a vaneless diuser, u2 = 100 m/s 398

R. A. Izmailov

Unsteady Flow Phenomena in Turboma hines

Figure 4 Resonant hara teristi s of impeller

hines, in parti ular, natural gas installations or high-pressure ompressors. As was mentioned above, the se ond domain on the hara teristi of the \impeller + diuser" stage is rotating stall. This phenomenon was dis overed by N. E. Zhukovski during experiments with a fan [5℄. Before the Se ond World War, this phenomenon was des ribed by GerR. A. Izmailov

399

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 5 Resonant hara teristi s of vaned diuser 400

R. A. Izmailov

Unsteady Flow Phenomena in Turboma hines man s ientists Fis her and Thoma [6℄ and Gruenagel [7℄ who studied

entrifugal pumps. The boom of investigations of the rotating stall (in axial ompressors) was indu ed by the work of Emmons et al. [8℄. The investigations of the present author were mostly in uen ed by the experimental and theoreti al works of Jansen [9℄. However, Jansen performed his outstanding experiments not with a full stage of entrifugal

ompressor. He studied a ow in a vaneless diuser with a spe ial ow generator (rotating s reens) simulating the outlet onditions in the impeller. This phenomenon was investigated by the present author in the stages with vaneless, vaned, and ribbed diusers by measuring unsteady stati pressures both in the impellers and diusers. In some experiments, unsteady velo ities in a vaneless diuser were measured with the aid of hot wires. The \frozen" velo ity eld and stati pressure in the diuser are presented in Fig. 6 for several radii and widths. The 3D hara ter of the ow pattern, with the strong reverse- ow zone near the diuser walls is learly evident. The level of velo ity u tuations in reases downstream the main ow (Fig. 3, '2 = 0:057). Due to slow rotation of this nonuniform ow pattern (a typi al value of the rotation speed is less then 0:1!rot), the frequen y of the os illations in the impeller is greater than in the absolute frame and depends on the number of the zones. A detailed analysis of the results for stages with dierent diusers and two-dimensional impellers showed that 3D ow separation on the diuser walls is responsible for the onset of rotating stall. The values of the inlet ow angle leading to rotating stall a

ording to the experiments of the present author and theoreti al al ulations based on the boundary layer theory oin ided with Jansen's results (2  12Æ { 16Æ from the radial dire tion). To the present author's opinion, the Emmons{Pearson{Grant model (large angle of atta k at the bladerow inlet) is valid for 3D impellers. Careful measurements in the narrow parametri domain pre eding the onset of rotating stall made it possible for the present author to dis over the \pre ursor stall" phenomenon. This ee t was dete ted by measuring the os illating pressure inside the impeller operating together with a vaned diuser. Some nonrotating stall zones (5{7) existed in the vaned diuser and the number of these zones hanged in an irregular manner. The ow in the diuser was also irregular. With the appli ation of spe ial methods, this phenomenon was dete ted in a stage with R. A. Izmailov

401

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 6 \Frozen" velo ity and stati pressure elds in a diuser a vaneless diuser as well. The dis overy of this spe i feature of the

ow in entrifugal ompressors manifesting itself in a narrow range of operating modes opened a new door for predi ting the onset of surge. Traditionally, antisurge systems utilize some steady parameters (mass ow rate, pressure, rotation speed, et .) to predi t the proximity 402

R. A. Izmailov

Unsteady Flow Phenomena in Turboma hines

to the surge line. In industry, some oset of this surge line (about 10%) is usually introdu ed to restri t the operational range of ompressors. Moreover, the measurement errors of these parameters are quite large (ex eed 5%), as the ow in the onditions near the surge line is very unsteady. As a result, the measurement error and un ertainties of predi tion tools in rease. The present author has proposed a new method of dete ting the onset of surge based on his detailed experimental investigations of ow onditions in this parametri zone. The underlying prin iple is very simple. A

ording to the experiments, with entrifugal ompressors of all types, the onset of rotating stall o

urs earlier than the onset of surge. This implies the straightforward method: to dete t rotating stall [10℄. Sometimes (for high-eÆ ien y ompressors), the onditions of rotating stall and surge in epien e are very lose (on the hara teristi s). In this ase, one should dete t the onset of the

Figure 7

Short-time orrelation analysis. Numbers

1

to

6

stand for Probe 1

to Probe 6

R. A. Izmailov

403

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

pre ursor stall. The latter deals with the phenomenon of sound/noise level in reasing in very harsh onditions. The problem of dete ting the pre ursor stall was resolved with the aid of short-time orrelation analysis (Fig. 7). Some promising results were obtained using the Hilbert transform. At present, the singular spe trum analysis [11℄ is applied for this purpose whi h is implemented into the state-of-the-art software system \Caterpillar." Very powerful wavelet methods are also used.

Con luding Remarks During the period of time under onsideration, a powerful method of experimental investigations of unsteady ow phenomena of all types in entrifugal ompressors was developed and systemati ally applied. With the aid of a spe ial data a quisition system, typi al stages of entrifugal ompressors were tested with measurements of unsteady pressures in rotating impellers. The physi al pi ture of omplex pro esses involved was revealed and the approa hes for estimating the level of dangerous loads leading to high dynami stresses in impellers were put forward and validated. These investigations resulted in a new approa h to surge dete tion.

A knowledgments Friendly support of Prof. K. P. Seleznev, who showed the best way to su

ess in the omplex and unsteady world is greatfully a knowledged. Many thanks to Prof. G. S. Samoilovi h for fruitful dis ussion, Prof. G. A. Raer for resear h funding and the members of the author's resear h group and postgraduates at LPI (now SpSPU) for their eorts in numerous experimental tests.

Referen es 1. Seleznev, K. P., and R. A. Izmailov. 1984. Instationaere Vorgaenge in Radalverdi htern. Wiss. Zeits hr. Der TU Dresden. J. 33, Heft 4:265{ 68. 2. Dean, R. C., and Y. Senoo. 1960. Rotating wakes in vaneless diuser. Trans. ASME Ser. D. 5:563{74. 3. Den-Gartog, J. P. 1940. The theory of vibrations. Cambridge. 404

R. A. Izmailov

Unsteady Flow Phenomena in Turboma hines 4. Rayleigh, J. W. 1945. The theory of sound. New York: Dover Publi ations. 5. Zhukovski, N. E. 1937. Eddy theory of impellers: Eddy theory of entrifugal ompressor . Vol. 6. Mos ow{Leningrad: ONTI NKTP. 6. Fis her, K., and D. Thoma. 1932. Investigation of the ow onditions in a entrifugal pump. Trans. ASME 54. 7. Gruenagel, E. 1936. Pulsierende Forderung bei Pumpenradern ueber dem Unfang Ing. Ar hiv, Band VII. 8. Emmons, H. W., C. R. Pearson, and H. P. Grant. 1955. Compressor surge and stall propagation. Trans. ASME 77(4):455{69. 9. Jansen, W. 1960. Quasi-unsteady ow in a radial vaneless diuser. MIT, Gas Turbine Lab. Rep. No. 58. 10. Izmailov, R. A., Y. D. Akulshin, and T. E. Krutikov. 2004. Pre-surge diagnosti system for entrifugal ompressors. Turbines Compressors 3{4 (28{29):15{22. 11. Golyandina, N., V. Nekrutkin, and A. Zhigljavsky. 2001. Analysis of time series stru tures: SSA and related te hniques. Bo a Raton: Chapman & Hall/CRC.

R. A. Izmailov

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

STABILITY OF A LOW-SPEED CENTRIFUGAL COMPRESSOR WITH CASING TREATMENTS

A. S. Hassan

Me hani al Engineering Department Fa ulty of Engineering, Assiut University Assiut 71516, Egypt In the present work, a single stage entrifugal ompressor of an a tual air raft turbo harger engine with dierent volute on gurations and asing treatments was investigated. Three dierent ategories of

asing treatments were tested. First, three dierent semi ir ular volute

on gurations with dierent depths were tested and the results were

ompared with the original volute. Se ond, the ompressor with dierent asing treatments through the vaneless region was tested in luding

ir umferential groove, protrude, and ombination of groove and protrude. Third, the ompressor with dierent radial grooves through its front asing mat hing with the diuser passages and the vaneless regions was tested. The time variations of wall stati pressure were observed using ouple of pressure transdu ers with high frequen y response in the vaneless region at dierent ompressor operating onditions. Stall and surge were dete ted by analyzing both of the u tuations of pressure signals and the power spe trum density (PSD) whi h were dedu ed by using the Fast Fourier Transformation analysis (FFT). The number and speed of stall ells relative to the impeller speed were investigated.

Nomen lature b2 H h P Q




406

width at impeller exit groove or protrude height ratio, h=r2 groove or protrude height pressure dieren e volumetri ow rate A. S. Hassan

Unsteady Flow Phenomena in Turboma hines

radius ratio, R = r=r2 radial distan e impeller outer radius groove or protrude depth ratio, t=b2 period of rotating stall ells
time dieren e between signals groove or protrude depth impeller tip speed impeller angular speed propagation speed of stall ells angular gap between sensors, number of stall ells density  ow oeÆ ient,  = Q=(2b2r2U ) pressure oeÆ ient, = 2
P=(U 2) R r r2 T Ti Ti t U !0 !s p  

Subs ripts g p

1

groove protrude

Introdu tion

Compressors with wide operating ranges and high eÆ ien ies are required in order to save energy and to keep the operating osts low. As

ow rate redu es, ompressors have a limited operation range, due to o

urren e of self-ex iting phenomena that result in ma hine fra ture, like rotating stall and surge. Hen e, there has been an extensive sear h for a low-order model apable of des ribing the essential of the dynami s in order to gain a parametri understanding of how to avoid these instabilities [1℄. Resear hes on erning the predi tion and ontrol of rotating stall have been investigated [2, 3℄. Casing treatments have been popular sin e long time, as reported in [4℄. The idea of extending the

ompressor operating range using the asing treatment te hniques has been experimentally studied in [5℄. Re ently, shallow grooves mounted on a asing wall or diuser wall parallel to the pressure gradient alled J-grooves treatment were proposed in [6{9℄. In the present work, different volute on gurations and ir umferential groove, protrude, and A. S. Hassan

407

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

ombination of the both through the asing in the vaneless region were tested. Dierent radial grooves through the ompressor front asing were tested as well.

2

Experimental Apparatus and Pro edure

The general view of the experimental test fa ility is shown in Fig. 1. The test ompressor is onstru ted from radial blade impeller, paraboli vanes diuser, and volute asing. The impeller was run at onstant speed of 3500 rpm. The ompressor drew air at atmospheri onditions and dis harged it into a large tank followed by an ori e owmeter and

ontrol valve for measuring and ontrolling the ow rate. In addition to the pressure tapes that were lo ated through the ompressor system for se uring the ompressor hara teristi s (Fig. 2), the time variation of stati pressure was measured at three points in the vaneless region using pressure sensors with high frequen y response. A dire t urrent ampli er re eived the output signals from the pressure transdu ers and provided a 16-bit analog to-digital onverter board that was supported with PC-SCOPE software for simultaneously sampling pressure signals for 1 s at a rate of 1 kHz. The PC-SCOPE software turned the omputer to os illos ope and saved the pressure waveforms in ASCII le. Subsequently, the data in the le were pro essed using the FFT to estimate the PSD by Wel h's averaged, modi ed periodogram method for dis rete-time signal ve tor.

Figure 1 408

General view of experimental test fa ility A. S. Hassan

Unsteady Flow Phenomena in Turboma hines

Figure 2

Lo ations of pressure taps on the ompressor asing. Dimensions

are in millimeters

In the present work, the ompressor was tested rst without any modi ation and then with dierent asing treatment, namely, (a) three dierent semi ir ular volute on gurations with dierent volute depths, (b) three dierent ir umferential groove depths of Tg = tg =b2 = 0:05, 0.12, and 0.2 at onstant Hg = hg =r2 = 0:2, ( ) three dierent ir umferential groove heights Hg = 0:065, 0.13, and 0.2 at onstant Tg = 0:2, (d) dierent ir umferential protrude widths and heights, (e) a ombination of ir umferential groove and protrude at Hg = 0:13 and 0.06 and Hp = 0:13 and 0.06 at Tg = 0:2 and Tp = 0:14. In addition, several radial grooves in the ompressor asing were manufa tured of Tg = 0:04, 0.08, 0.12, and 0.16 with dierent width ratios, Wg , of 0.019, 0.039, 0.058, and 0.077. The numbers of radial grooves, N , of 7, 10, and 14 were hanged in ea h ompressor front asing. A. S. Hassan

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3 3.1

Results and Dis ussion Rotating stall and surge in vaneless region

The number of stall ells,  = 2
T =( T ), and the propagation speed of these ells, ! =!0 = 2=(!0T ); an be estimated using two pressure tra es that were re orded simultaneously at same radius with 90Æ apart peripherally from ea h. Here, T and
T are the period of rotating stall ells and the time dieren e between signals,  is the angular gap between the oupled sensors, and !0 is the angular velo ity of impeller. At ow ondition  = 0:072, the amplitude of the pressure u tuations rea hed about 30% of the pressure oeÆ ient with a phase shift as shown in Fig. 3. i

s

i

p

i

i

i

p

Flu tuation of pressure oeÆ ient and PSD at  = 0 072: (a ) rst pressure sensor and (b ) se ond pressure sensor

Figure 3

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:

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Unsteady Flow Phenomena in Turboma hines

The dedu ed PSD from the rst pressure sensor shows that there is one predominant frequen y of 17 Hz. This indi ates that the ow suers from rotating stall in this ow ondition. While, the dedu ed PSD from the se ond pressure sensor shows there is same predominant frequen y of 17 Hz with other frequen ies of about 4, 32, and 37 Hz. This means that at this operating ondition, there is ompli ated three-dimensional

ow with rotating stall ells at the higher frequen ies, 17, 32, and 37 Hz, and at the same time surge at frequen y of 4 Hz. A

ording to the above-mentioned equations, the number of stall ells is three and the propagation speed of the stall ells is 26% of !0 . At ow rate of  = 0:068, the amplitude of the pressure u tuation ex eeded 30% of the pressure oeÆ ient and low frequen y of 5 Hz at maximum PSD was

Figure 4 Time variation of pressure oeÆ ient and PSD at  = 0:068:

(a ) rst pressure sensor and (b ) se ond pressure sensor

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observed. In addition, no phase shift was observed between the two pressure waves re orded at the vaneless region 90Æ apart peripherally from ea h other as shown in Fig. 4. Thus, the ompressor is run in surge at this ow ondition with one-dimensional u tuation in ow rate. 3.2 Time variation of pressure oeÆ ient and PSD

The pressure waves at four ow onditions are plotted in Fig. 5. At the

ompressor operating point,  = 0 305, of maximum ow rate, Fig. 5a shows very small value of the amplitude of the pressure u tuations with high frequen y and very small PSD whi h hara terizes the steady state ompressor operation. At the operating points from the maximum ow rate  = 0 305 to just before  = 0 128, the ompressor shows relatively simple u tuations. The amplitude of u tuations of pressure oeÆ ient in reases with the de rease of the ow rate. At the ompressor operating point,  = 0 128 (Fig. 5b ), the amplitude of pressure u tuations in reases at relatively low frequen y, 30 Hz, due to the initiation of rotating stall in the vaneless region. At the ompressor operating point  = 0 121 (Fig. 5 ), the ow instability o

urs with two main predominant frequen ies of 15 and 32 Hz. Sin e, the amplitude of pressure u tuations rea hed about 20% of ompressor maximum pressure oeÆ ient with frequen ies of 15 and 32 Hz, whi h lie in the range of stall frequen ies. At this operating point, also surge frequen ies of 4 to 10 Hz are observed. That is, the ompressor run in presen e of rotating stall with triggering of surge. At the operating point  = 0 108, the amplitude of pressure u tuations ex eeded 40% of ompressor maximum pressure oeÆ ient with predominant surge frequen y of 4 Hz, as shown in Fig. 5d. :

:

:

:

:

:

3.3 Compressor performan e with dierent volute modi ations

Figure 6 shows the ee t of test results of three dierent semi ir ular volute asing with dierent volute radius ratio of = 1 7 (original), = 1 = 1 67, = 2 = 1 64, and = 3 = 1 6 on ompressor performan e. The limit of ow stability due to stall initiation is denoted by empty rhomb on ea h hara teristi urve in these gures. It is lear R

R

412

R

:

R

R

:

R

R

:

:

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Unsteady Flow Phenomena in Turboma hines

Figure 5 Time variation of pressure oeÆ ient and PSD: (a )  = 0:305,

(b ) 0.128, ( ) 0.121, and (d )  = 0 108 :

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Figure 6 Ee t of volute depth on ompressor performan e: 1 | without modi ation, 2 | R1 3 | R2 , 4 | R3 , 5 | surge trigger, and 6 | stall initiation that the volute asing at radius ratio R2 gives the maximum stable operating range and pressure rise oeÆ ient.

3.4 Ee t of asing treatments on pressure u tuations Figure 7 shows sample pressure waves in ase of ompressor with asing treatment (Tg = 0:12 and Hg = 0:2) ompared with the original one. This gure shows sele ted operating onditions overing the presen e of stall and surge for the ompressor with and without asing treatment. Figure 7a shows large amplitudes of pressure u tuations in ranges of stall and surge at dierent ow onditions, while small amplitudes of pressure u tuations, were observed as in Fig. 7b. That is the asing treatment suppresses stall and surge.

3.5 Compressor performan e with dierent ir umferential grooves Figure 8 shows the ee t of dierent ir umferential groove depths and protrude thi kness at onstant height of Hg = Hp = 0:2 on the ompressor performan e. The ompressor with groove of Tg = 0:2, Fig. 8a, gives noti eable in rease in the stall margin, but de rease in ompressor 414

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Unsteady Flow Phenomena in Turboma hines

Figure 7 Suppression of stall and surge by ir umferential groove asing

treatment: (a ) original ompressor and (b ) modi ed ompressor (Hg = 0:2 and Tg = 0:12)

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Figure 8 Ee t of (a ) groove depth (Hg = 0:2: 1 | original; 2 | Hg = 0:05; 3 | 0.12; 4 | Hg = 0:20; and 5 | stall initiation) and (b ) protrude thi kness (Hp = 0:2: 1 | original; 2 | Hp = 0:04; 3 | 0.08; 4 | 0.14; and 5 | stall initiation) on ompressor performan e

Figure 9 Ee t of (a ) groove (Tg = 0:2: 1 | original, 2 | Hg = 0:065, 3 | 0.13, 4 | Hg = 0:20, and 5 | stall initiation) and (b ) protrude heights (Tp = 0:14: 1 | original, 2 | Hp = 0:13, 3 | Hp = 0:20, and 4 | stall initiation) on ompressor performan e

pressure oeÆ ient at low ow rate. At protrude thi kness Tp = 0:14, Fig. 8b, the ompressor gives the highest pressure oeÆ ient at low ow rates while it gives low-pressure oeÆ ient at high ow rates. In reasing the protrude height gives an in rease in the ompressor pressure oef ient at low ow rates. Protrude of Tp = 0:04 and Hp = 0:2 leads 416

A. S. Hassan

Unsteady Flow Phenomena in Turboma hines

to the highest pressure oeÆ ient at the high ow rates and relatively higher pressure oeÆ ient at low ow rates as well as in rease in stable operation range ompared to the original ompressor. Figure 9 shows the ee t of the radial height of the ir umferential groove and protrude on the ompressor performan e at onstant ir umferential groove depth and protrude thi kness. In Fig. 9a, it is shown that in reasing of the ir umferential groove height de reases the ompressor pressure oeÆ ient at low ow rates while in reases the pressure

oeÆ ient at the higher ow rates. Figure 9b indi ates that in reasing of the protrude height in reases the ompressor pressure oeÆ ient at low ow rates. Whereas, the stall margin in reases by in reasing the groove or protrude height, whi h an be aused by the same reason as that mentioned above.

3.6 Compressor performan e with dierent radial grooves Figure 10 shows the ee t of radial groove widths and depths on the

ompressor performan e. It is shown that the radial groove of Tg = 0:08 with the various width ratios de reases the ow oeÆ ient of stall, and in reases the pressure oeÆ ient at stall initiation a hieving an enhan ement in stability and eÆ ien y of the ompressor. In reasing radial

Figure 10

Ee t of radial groove width on the ompressor performan e = 7: (a ) Tg = 0:08: 1 | original, 2 | Wg = 0:019, 3 | 0.039, 4 | 0.058, 5 | Wg = 0:077, and 6 | stall initiation; and (b ) Wg = 0:058: 1 | original, 2 | Tp = 0:04, 3 | 0.08, 4 | 0.12, 5 | Tg = 0:16, and 6 | stall initiation

N

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 11 Ee t of radial groove

widths on ompressor performan e: 1 | Tg = 0:04; 2 | 0.08; 3 | 0.12; and 4 | Tg = 0:16

Figure 12 Ee t of radial groove

numbers on ompressor performan e at Tg = 0:12 and Wg = 0:058: 1 | original; 2 | N = 7; 3 | 10; 4 | N = 14; and 5 | stall initiation

grooves width results in more enhan ements in the ompressor stability. The grooves of g = 0 077 at g = 0 08 lead to an enhan ement in stall stability of about 41.5%. The radial grooves depth gives small enhan ement in ompressor stability. It is lear that an enhan ement of about 41.4% in the ompressor stability ould be a hieved at g = 0 058 and g = 0 04. The ee t of radial groove width g on the ow oeÆ ient orresponding to the stall initiation at various g and = 7 is shown in Fig. 11. The gure shows that at onstant grooves depth, the ow oeÆ ient orresponding to the stall initiation drops down to a minimum value and then in reases with the grooves width. The bottom values depend on the depth and width of the grooves. This means that the enhan ement in the ompressor stability varies with the grooves width to major values depending on the values of the grooves depth. The ee t of radial grooves number on the performan e and stability of the ompressor is shown in Fig. 12 at g = 0 12 and g = 0 058. Changing the number of grooves exerts a small ee t on the enhan ement in the ompressor stability and the pressure oeÆ ient at the stall initiation. Seven grooves of g = 0 058 at g = 0 12 lead to an enhan ement in the stall at stall initiation of about 45.5%. W

:

T

:

W

T

:

:

W

T

N

N

T

:

W

:

W

418

:

T

:

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Unsteady Flow Phenomena in Turboma hines

4

Con luding Remarks

In the present work, ee t of asing treatments on the performan e of a low speed entrifugal ompressor was investigated. Three dierent semi ir ular volute on gurations with dierent depths were tested and the results were ompared with the original volute. Also, dierent asing treatments in the vaneless region; ir umferential groove, protrude and the ombination of the groove and protrude at dierent depths, thi knesses and heights were tested. In addition, dierent radial grooves through the ompressor front asing mat hing the diuser passages and the vaneless regions were tested. The experimental results show that the volute asing at radius ratio R2 gives the maximum stable operating range as well as pressure rise

oeÆ ient. Compressor with groove asing treatment (Hg = 0:2 and Tg = 0:2) gives improvement in stall margin of about 55% and 39% in surge margin but this modi ation de reases the pressure oeÆ ient at low ow rates. Compressor with protrude asing treatment gives improvement about 19% in stall margin, about 26% in surge margin, and about 13%{14% in pressure oeÆ ient. The dombination of groove and protrude (Tg = 0:2, Tp = 0:14, Hg = 0:06, and Hp = 0:13) leads to improvements of about 28% in stall margin, 22% in surge margin, and 4% in pressure oeÆ ients. Radial grooves result in an enhan ement in the ow oeÆ ient at stall initiation rea hing 45.5%.

Referen es 1. Tommy, J. G., W. Frank, J. Bram, and E. Olav. 2003. Modeling for surge

ontrol of entrifugal ompressor and omparison with the experiments. http://aldebaran.elo.utfsm. l/datasheet/ d 00/pdf/ d 001619. 2. Hayami, H., A. S. Hassan, E. Hiraishi, and H. Hasegawa. 1995. Experimental investigation on stall and surge in entrifugal blower. In: Unsteady aerodynami and aeroelasti ity of turboma hines . Eds. Y. Tanida and M. Namba. Amsterdam{Tokyo: Elsevier. 727{36. 3. Hayami, H., and S. Fukuu hi. 1999. Pressure u tuation in pro ess to stall in a transoni entrifugal ompressor. ASME/JSME FEDSSM Pro eedings. Fluid Engineering Division, San Fran is o, California. 4. Prin e, D. C., D. C. Wisher, Jr., and D. E. Hilvers. 1975. A study of asing treatment stall margin improvement phenomena. ASME Paper No. 75-GT60. A. S. Hassan

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5. Amann, C. A., G. E. Nordenson, and G. D. Skellenger. 1975. Casing modi ation for in reasing the surge margin of a entrifugal ompressor in an automotive turbine engine. ASME J. Eng. Power 97:329{36. 6. Takata, H., and Y. Tsukuda. 1977. Stall margin improvement by asing treatment | its me hanism and ee tiveness. ASME J. Eng. Power 99:121{33. 7. Moat, R. J. 1988. Des ribing the un ertainties in experimental results. Int. J. Experimental Heat Transfer. Thermodynami s and Fluid Me hani s

1(1):3{17. 8. Sankar, L. S., K. Juni hi, M. Jun, and K. Takaya. 2000. Passive ontrol of rotating stall in a parallel-wall vaned diuser by J-grooves. Trans. ASME, J. Fluid Engineering 122:90{96. 9. Juni hi, K., L. S. Sankar, M. Jun, and K. Takaya. 2001. Passive ontrol of rotating stall in a parallel-wall vaneless diuser by radial grooves. Trans. ASME, J. Fluid Engineering 123:507{15.

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PROPAGATING SHOCK WAVES IN A NARROW TUBE FROM THE VIEWPOINT OF ULTRA MICRO WAVE ROTOR DESIGN K. Okamoto, T. Nagashima, and K. Yamagu hi

Department of Aeronauti s and Astronauti s University of Tokyo Hongo Bunkyo-ku 7-3-1 Tokyo 113-8656, Japan

Wave rotor topping y le is one of the andidates that improve the performan e of small gas turbines in luding Mi ro-Ele tro-Me hani al System (MEMS) gas turbines. In miniaturization of wave rotors, the ee t of wall fri tion on the propagating pressure wave must be dis ussed arefully. In this study, sho k waves propagating in a small tube of 3 mm square ross se tion were observed with pressure measurement and visualization by S hlieren method. Two dierent tube lengths (42 and 168 mm) were employed to on rm the sho k waves learly visible in both ases, on luding that the dissipation ee t of pressure waves was not so signi ant in the present ases. Also, the pressure time tra es at the stagnation were dis ussed in relation to the dependen e on nondimensional parameters.

1

Introdu tion

The resear h on miniature gas turbines [1℄ in luding MEMS gas turbines [2℄ is be oming popular with in reasing demands in distributed ele tri generators, mobile ele tri sour es, propulsion systems of mi ro air vehi les, et . One of serious problems in su h small gas turbines is poor y le eÆ ien y derived from the size ee ts, su h as large heat and vis ous losses. A wave rotor has been suggested as a devi e that has a potential to improve the gas turbine performan e drastiK. Okamoto et al.

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Figure 1

Wave rotor

ally [3, 4℄. A wave rotor onsists of du ts (ports) and a rotor with many straight tubes ( ells), as shown in Fig. 1. The ompression and expansion of gas and air are arried out with the unsteady propagation of sho k and expansion waves in the ells; therefore, the less wall fri tion loss an be expe ted ompared to the onventional turboma hines with steady inner ow. In addition, a wave rotor an be operated with slower rotor speeds ompared to turboma hines, whi h will be another advantage in miniaturization. A wave rotor is used as a topping y le when it is applied to a gas turbine (Fig. 2). A wave rotor has better heat resistan e be ause of its self ooling feature; therefore, higher maximum temperature and pressure ratio an be a hieved without hanging the

ompressor and turbine Figure 2 Wave rotor topped gas tur(Fig. 3). A

ording to the y- bine 422

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Figure 3

2

Wave rotor topping y le:

| baseline engine. pressure gain

Figure 4

T5

1

| wave rotor topped engine and

is the turbine inlet temperature;

P5

P5base

=

Wave rotor topped ultra mi ro gas turbine (imagination)

le analysis [5℄, the bene t of wave rotor topping will be larger, when the baseline gas turbine is smaller. The authors have suggested an ultra mi ro wave rotor for an ultra mi ro gas turbine of less than 100-watt output (Fig. 4) [6℄. The most important point in the wave rotor design is to open and to lose the ports to the ells at exa t timings a

ording to the pressure wave arrivals at K. Okamoto et al.

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the ell ends. In addition, dissipation of pressure waves is an important issue to be treated arefully in the miniaturization. Therefore, experimental observation of sho k waves in a small ell is signi ant to obtain better design pro edure of ultra mi ro wave rotors. For this purpose, a new test rig was built in the present study, and S hlieren method was adopted to visualize the sho k waves propagating in the ell. Also, two test se tions of dierent lengths were applied and

ompared to investigate the wall fri tion ee t.

2 2.1

Experiment Test fa ility for visualization

Figure 5 shows the test rig built for the present study. In this test rig, the harging and dis harging ports are rotating and a ell is xed in order to make it suitable for visualization of sho k waves and pressure measurement. This arrangement is ompletely opposite to the real wave rotors; therefore, the ee t of rotation su h as entrifugal for e was not taken into a

ount, although it did not have large in uen e on the pressure wave generation and propagation a

ording to the threedimensional numeri al simulation [7℄. In this experiment, high-pressure air ompressed by a s rew ompressor was used as the driver gas. The pressure was kept as 0.25 MPa

Figure

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5

Test fa ility for visualization K. Okamoto et al.

Unsteady Flow Phenomena in Turboma hines

Figure 6

Test se tion. Dimensions are in millimeters

and the temperature was room temperature during this experiment. First, the high-pressure air is harged in the buer tank, and then ows into the ell through the harging port. After generating sho k waves, the air ows out through the dis harging port in the radial dire tion. The rotor has two pairs of harging and dis harging ports so that one rotation orresponds to two y les. The rotor is driven with an ele tri motor and the rotor speed was varied. As shown in Fig. 5, this test rig has the ports of only one side of the ell, and the other end of the ell is always losed, whi h is dierent from a real wave rotor. This on guration was urrently introdu ed for simpli ity of the test rig, and the generating pro ess of sho k waves for air ompression does not dier mu h from the real inner ow dynami s. Two test se tions with dierent lengths (42 and 168 mm) were prepared for this experiment (Fig. 6). The ross se tion was 3 mm square in both test se tions that were made of rystal glass. As mentioned above, one end of the ell was losed and a pressure transdu er was dire tly mounted at the ell end for high-speed pressure measurement at the stagnation. (In Fig. 6, the pressure transdu er is mounted only on the shorter test se tion.) As for the visualization, S hlieren method was adopted with highspeed Charge-Coupled Devi e (CCD) amera. The path of the light beam was set horizontally and the sho k waves were visualized in the K. Okamoto et al.

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radial dire tion. Depending on the trigger pulse from the rotor, the CCD amera aptured a photograph as a ti le in exa t timing, and the shutter speed was set to 500 ns with image intensi er for high sensitivity. As for the high-speed pressure measurement at the ell end, the data were obtained with a high-speed os illos ope and the whole data a quisition system assured the frequen y up to 100 kHz. 2.2

Design parameters

A

ording to the previous studies [8, 9℄, there are three dominant fa tors for the dis ussion of wave rotor performan e loss, and nondimensional parameters were suggested, orresponding to ea h of the fa tors as follows. Gradual Passage Opening: 

=

Passage Opening Time = Wave Travel Time

Wall Fri tion: F

=

Leakage: G

=



W ell r!

.  L a

L Dh

2Æ H ell

Here, W ell is the ell width; r is the mean rotor radius; ! is the angular speed of the rotor; a is the speed of sound; L is the ell length; Dh is the hydrauli diameter of the ell; Æ is the axial learan e gap between the xed end wall and the rotor, and H ell is the ell height. The \gradual passage opening ee t" means the ee t of gradual opening and losing of the ports to the ells, and this has a large in uen e on the sho k wave generating pro ess. The value of ea h parameter shows the amplitude of ea h loss fa tor. That is, the larger the values of these parameters, the larger is the performan e loss. As easily found from the de nition of parameter  , the longer ell allows the slower rotor speed with keeping the same  value, although the value of parameter F for wall fri tion in reases. Therefore, the balan e of these two parameters is very important to a hieve slower rotor speeds in the ultra mi ro wave rotor. 426

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Table 1 Comparison of wave rotor on gurations Cell Cell Length Mean width height Clearan e Rotor L 2Æ speed, L, radius   H W ell , H ell , Æ , mm D

ell h rpm r, m m m m 0.23, 0.15 0.00635, 0.0102 0.127{ 1850{ 0.08{ 20{58 0.025{ 0.46 0.0127 0.381 7400 0.35 0.075

NASA (3-port) NASA 0.152 0.0815 0.00875 0.022 & Allison Kent eld 0.28 0.102 0.0168 0.0559 General 0.3 0.058 0.01 0.0231 Ele tri Comprex R 0.0932 0.048 0.009 0.01 (ABB) Visualization 0.042, test rig 0.06 0.003 0.003 (this work) 0.168

0.13 0.18

16,800 0.194 5500

0.35

12.1 0.0118 10.5 0.006

0.64

19,000 0.0982

21.5 0.0554

0.15

14,000 0.467

9.8

0.1

3600{ 0.18{ 10,800 1.1 14, 56 0.067

0.03

 Wall thi kness between the ells in luded.  Cal ulated with atmospheri value.  D is the hydrauli diameter.

h

Table 2 Operating rotor speeds and  values in the experiment No. S1 S2 S3 S4 S5

L = 42 mm Rotor Speed [rpm℄ 3600 5400 7200 9000 10800



1.10 0.73 0.55 0.44 0.37

No. L1 L2 L3 L4 L5

L = 168 mm Rotor Speed [rpm℄ 1800 2700 3600 4500 5400



0.55 0.37 0.27 0.22 0.18

Table 1 shows the omparison of these parameters with other wave rotors [8{11℄. As is seen here, the present design of the longer ell brings the larger fri tion loss, so that its ee t is easy to appear in the

ow dynami s, while the parameter value of the shorter ell is almost

onventional. The value of  was varied by hanging the rotor speed in this experiment, and the details are shown in Table 2. As mentioned above, the same value of  an be a hieved with the longer test se tion and the slower rotor speed (for example, see ases `S5' and `L2'). K. Okamoto et al.

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3 3.1

Results S hlieren pi tures

Figure 7 shows the S hlieren pi tures of ase \S3" in Table 2. The length of this test se tion is 42 mm, but 3 mm of it at the left end was not visualized due to xing to the endwall. In this gure, the ports are rotating at the left end and moving downward. As is seen in this gure, a sho k wave, generated by opening of the harging port, begins to appear at 60 s. The density gradient in this pressure wave is not so steep at the beginning, owing to the gradual passage opening ee t. Then, the pressure wave is gradually strengthened during its propagation. The sho k wave is re e ted at the right end around 90 s, and the re e ted sho k wave propagates ba k against the in ow and rea hes the left end of the ell. In this pro ess, both sho k waves were visualized learly and they did not disappear. Figure 8 shows the S hlieren pi tures of ase \L3." As is seen in this gure, the sho k waves were visible even in this longer ell. In par-

Figure

428

7

L = 42 mm, 7200 rpm)

S hlieren pi tures (

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Unsteady Flow Phenomena in Turboma hines

Figure 8

S hlieren pi tures (

L = 168 mm, 3600 rpm)

ti ular, the re e ted sho k wave seems to be sharp and normal without skew, in spite of propagating against the in ow; therefore, the boundary layer of the in ow does not mu h ae t the sho k wave propagation even in this ase. As is seen in both gures, the sho k waves seem to propagate with almost onstant speeds; therefore, those propagating velo ity an be

al ulated roughly with these photographs. In both test se tions, the average propagating velo ity of primary and re e ted sho k waves was about 400 and 300 m/s, respe tively. The velo ity of the primary sho k wave was faster than the sound speed, whi h means that it was really a sho k wave. Furthermore, the velo ities of both sho k waves did K. Okamoto et al.

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not dier in both test se tion. Therefore, it an be on luded that the propagation of sho k waves was not seriously ae ted by the wall fri tion ee t, even in the longer test se tion. 3.2

Pressure measurement at the stagnation

Figure 9 shows the time history of pressure at the stagnation in the shorter ell. On the one hand, when the rotor speed was 3600 and 5400 rpm, the rate of in rease in pressure was mu h less than the other results of faster rotor speeds. On the other hand, the rate of pressure in rease did not dier when the rotor speed was faster than 7200 rpm. Therefore, it an be on luded that the pressure wave be ame a sho k wave when the rotor speed was faster than 7200 rpm. The orresponding  parameter value was 0.55. In the ase of the longer ell, the similar trend appeared, when e the limiting rotor speed for sho k wave development seemed about 3600 rpm (Fig. 10). The orresponding parameter value was 0.27. It must be noti ed that the pressure wave an be a sho k wave even with this long

ell, as the rotor speed in reases large enough. Therefore, the pressure wave pro ess an be a hieved in the parameter range of the present design, wherein the limit  parameter values may dier depending on

L = 42 mm) 1 | 3600 rpm, 2 |

Figure 9 Pressure history at stagnation (

5400,

430

3

| 7200,

4

| 9000, and

5

| 10,800 rpm

K. Okamoto et al.

Unsteady Flow Phenomena in Turboma hines

Figure 10 Pressure history at stagnation (L = 168 mm) 1 | 1800 rpm,

2 | 2700, 3 | 3600, 4 | 4500, and 5 | 5400 rpm

Figure 11 Comparison for dierent ell lengths: and 2 | long, 2700 rpm

1 | short, 108000 rpm,

the value of L=Dh . It must be noti ed that the rotor speed an be redu ed from 7200 to 3600 rpm that will be a signi ant advantage for ultra mi ro wave rotors. To ompare the results of short and long ells, normalized time is introdu ed, whi h means the time divided by L=a (a is the sound speed). K. Okamoto et al.

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

If the uid is invis id and the s ale ee t an be negle ted, both plots have to be identi al. Figure 11 shows the results of ases `S5' and `L2' in Table 2. As is seen here, the both results are almost identi al; therefore, the similarity is kept between these two results, whi h means the spe ial treatment for miniaturization is not ne essary for the dis ussion of inner ow dynami s in the present range of experiments for design parameters.

4

Con luding Remarks

To obtain the basi omprehension for designing ultra mi ro wave rotors, a new test rig was built to investigate the sho k waves propagating in a small ell in the present study. A

ording to the S hlieren pi tures, the sho k waves ould propagate without dissipation even in an extremely long ell, and the adverse ee t by the wall fri tion was not observed in the sho k wave propagation. Also, in the pressure measurement at the stagnation, it was on rmed that the sho k wave was generated at a ertain rotor speed, even with the longer ell. The limiting rotor speed for sho k wave generation was slower in the longer ell than in the shorter ell; therefore, it an be on luded, with respe t to the sho k wave propagation, that the rotor speed an be redu ed, advantageously for ultra mi ro wave rotors with extreme rotation, by extending the ell length, while keeping the present range of design parameters.

Referen es 1. Nagashima, T., S. Teramoto, K. Yamagu hi, et al. 2005. Lessons learnt from ultra-mi ro gas turbine development at University of Tokyo. Von Karman Institute for Fluid Dynami s Le ture Series on Mi ro Gas Turbines. 2. Epstein, A. H., S. D. Senturia, O. Al-Midani, et al. 1997. Mi ro-heat engines, gas turbines, and ro ket engines. AIAA Paper No. 97-1773. 3. Wilson, J., and D. E. Paxson. 1993. Jet engine performan e enhan ement through use of a wave-rotor topping y le. NASA Te hni al Memorandum 4486. 4. Wel h, G. E., S. M. Jones, and D. E. Paxson. 1997. Wave-rotor-enhan ed gas turbine engines. J. Engineering Gas Turbines Power 119(2):469{77.

432

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Unsteady Flow Phenomena in Turboma hines 5. Fatsis, A., and Y. Ribaud. 1999. Thermodynami analysis of gas turbines topped with wave rotors. Aerospa e S ien e Te hnology 5:293{99. 6. Okamoto, K., T. Nagashima, and K. Yamagu hi. 2005. Design and performan e of a mi ro wave rotor. ISABE Paper No. 2005-1270. 7. Larosiliere, L. M. 1995. Wave rotor harging pro ess: Ee ts of gradual opening and rotation. J. Propulsion Power 11(1):178{84. 8. Paxson, D. E. 1995. Comparison between numeri ally modeled and experimentally measured wave-rotor loss me hanisms. J. Propulsion Power 11(5):908{14. 9. Wilson, J. 1998. An experimental determination of losses in a three-port wave rotor. J. Engineering Gas Turbine Power 120:833{42. 10. Gyarmathy, G. 1983. How does the Comprex R pressure-wave super harger work? SAE Te hni al Paper 830234. 11. Snyder, P. H., and R. E. Fish. 1996. Assessment of a wave rotor topped demonstrator gas turbine engine on ept. ASME Paper No. 96-GT-41.

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SELF-EXITED OSCILLATIONS IN SWIRLING-JET EFFLUXES





y

D. G. Akhmetov , V. V. Nikulin , and V. M. Petrov

 M. A.

Lavrentyev Institute of Hydrodynami s Siberian Bran h of the Russian A ademy of S ien es Lavrentyev Prosp. 15 Novosibirsk 630090, Russia y Institute of Thermophysi s Siberian Bran h of the Russian A ademy of S ien es Lavrentyev Prosp. 1 Novosibirsk 630090, Russia

1

Introdu tion

Gas or uid ow behind a turbine-wheel is often swirled. As a onsequen e, one of the essential sour es of ow disturban es in turbomashines whi h, in parti ular, lead to a ousti radiation (e.g., in air) are spe i self-exited os illations developing in swirling ows. It is known that in euxes of submerged swirling jets into open spa e, os illation phenomena often arise. These phenomena manifest themselves as strong periodi pulsations of pressure and velo ity. The results of the rst [1℄ and later [2{7℄ systemati investigations in this eld of resear h are presented in the literature. However, although the above ee ts have been known for a long time, the sour es of these periodi pulsations, as well as their generation me hanism, have remained poorly understood so far. Su h a status is explained by the ompli ated three-dimensional and unsteady hara ter of the ow. The most widely known hypotheses on the generation me hanism are the pre ession of the swirl ow

enter near the outlet ori e of a vessel [2℄, or the pre ession of the entire ore about the hamber symmetry axis [5℄, or the rotation of the vortex ore that has been twisted into a spiral after its es ape from the

hamber [4℄. It was also assumed that a reversed ow from open spa e into the hamber is on entrated near the symmetry axis, where a free 434

D. G. Akhmetov et al.

Unsteady Flow Phenomena in Turboma hines stagnation point an appear due to the ollision of ows oming from inside and outside the hamber [3℄. This paper reports the results of systemati experimental investigations of self-exited os illations developing at the outlet of swirling devi e in euxes of submerged swirling jet into open spa e in air and in water. The main governing parameters of the pro ess were determined and a dependen e relating the os illation frequen y with the governing parameters over a wide range of parameter variation was established experimentally. A new generation me hanism for the os illations was found. It was established that, in the ase of eux of a submerged swirling jet through a nozzle from a vortex hamber, the vortex ore that oin ided with its axis inside the hamber sharply deviated in the nozzle away from the symmetry axis towards the nozzle wall. As a result, a jet bend resembling the end of ho key sti k was formed. The bent part of the vortex ore rotated around the symmetry axis of the

hamber with a onstant angular velo ity, produ ing periodi pulsations of the ow parameters. When the devi e operated in air, the jet eux was a

ompanied with strong a ousti radiation.

2 Air Experiments In this se tion, the results for the dependen e of the a ousti os illation frequen y on the parameters of the ow and swirling devi e are presented. Experimentally, a swirling jet was produ ed as a result of air out ow from a vortex hamber into open spa e. Su h a pro edure imitated typi al ows arising in various swirling systems. The vortex hamber was a hollow ylinder with one end plugged by a piston and the other end equipped with a nozzle [6℄. The pro le of the nozzle had the shape of a onvergent 10-millimeter long onfuser that further smoothly transitioned to a 10-millimeter long ylindri al part. The hamber and the nozzle are shown in Figs. 1 to 3. Air was delivered into the vortex hamber tangentially through six identi al slit hannels made on the hamber surfa e near the plugged end. The entire design had axial symmetry. The slit width was 2 mm, and the slit length ould be varied from 1 to 15 mm by means of axial displa ement of the piston. The diameter of the vortex hamber was xed and equal to 28 mm. The hamber length

ould be redu ed stepwise by 46 mm by removing a ylindri al insert of D. G. Akhmetov et al.

435

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 1 Air bubbles visualizing the vortex axis. The photographs orrespond to dierent exposures: (a ) with the ash lamp and (b ) with a long (2 s) exposure time

the same diameter. When piston was displa ed, the length of the hamber with the insert ould be varied from 71 to 85 mm. The experiments were performed with outlet nozzles of three diameters, namely, 8, 14, and 20 mm, or without a nozzle (the outlet diameter was equal to the

hamber diameter in this ase). Pressure u tuations were measured by a mi rophone mounted at a distan e of 1 m from the nozzle exit. It follows from the des ription of the setup and the operating prin iple of the vortex hamber that the basi parameters determining the swirling-jet ow are: the radii of the vortex hamber and the nozzle R and r, respe tively, the hamber length L, the total ross-se tion area of the slits , the air velo ity at the hamber inlet slits, V , al ulated using the Bernoulli equation, V = (2
P=)0 5 (here,
P is the pressure dierential between the value at the hamber inlet and on the inner ylindri al surfa e of the hamber at a distan e of 5 mm from the slit end toward the nozzle, and  is the air density in the experiment

onditions), and the kinemati vis osity of air  . Sin e the experiments were performed at subsoni air supply velo ities (V  70 m/s), the :

436

D. G. Akhmetov et al.

Unsteady Flow Phenomena in Turboma hines

Figure 2 Vortex photograph ol- Figure 3 Flow at the nozzle exit

ored by an ink delivered at the en- se tion. The visualization is made by ter of the vortex- hamber bottom an ink delivered to the symmetry axis. boundary The photograph was taken with the illumination by a ash lamp

ompressibility of the medium ould be ignored. The parameter to be determined was the frequen y f of the pressure u tuations indu ed by swirling jet. If the parameters R and V were hoosen as the basi parameters with independent dimensions, then in a

ordan e with the dimensionality theory, it an be anti ipated that all the nondimensional hara teristi s of the swirling jet, in luding the nondimensional frequen y or Strouhal number Sh = fR=V , will be the fun tions of four dimensionless riteria: r=R, L=R, the geometri swirl parameter s = rR=, and the Reynolds number Re = V R= . Hen e, a fun tional dependen e Sh = (r=R; L=R; s; Re) must hold. Figure 4 shows the typi al spe tra of pressure u tuations for two values of V (Re = 2:8 104 and 5:3 104) at xed r=R = 1=2, L=R = 6, and s = 1:9. It follows from Fig. 4 that ea h spe trum exhibits one


D. G. Akhmetov et al.




437

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 5 Dependen e of the ostion spe tra for = 1 2, = 6, illation frequen y on the inlet ve= 1 9: 1 | Re = 2 8 104 and 2 | lo ity for dierent nozzle radii and = 5 5,  2 = 0 42: 1 | Re = 5 3 104 = 2 7, 2 | 1/2, 3 | 5/7, and 4 | =1

Figure 4 Typi al pressure u tuar=R

s

:

=

:

:

L=R




L=R




r=R

:

=R

:

=

r=R

pronoun ed peak orresponding to the os illation frequen y and this frequen y in reases with . Figure 5 shows the dependen e ( ) for dierent at xed = 5 5 and  2 = 0 42. The straight lines represent the linear regression plotted through experimental points. For other values of the parameters, the shape of the ( ) dependen e is qualitatively similar. Experiments in short (without insert) and long (with insert) vortex hambers showed that variation of the hamber length had only an insigni ant impa t on the measured frequen y. It follows from Fig. 5 that the os illation frequen y in reases almost linear with the inlet velo ity. Despite this fa t, for the given hamber geometry, the Strouhal number is not onstant and in reases slightly with Re. A similar dependen e of Sh on Re was noted in [8℄. As follows from Fig. 5, the in onstan y of Sh is attributable to the fa t that the linear-regression graph plotted using the experimental points rosses the abs issa axis at a ertain point 0 0 rather than departs from the

oordinate origin. A

ordingly, the frequen y an be represented in the form: V

r

L=R

f V

:

=R

:

f V

V

>

f

438

D. G. Akhmetov et al.

Unsteady Flow Phenomena in Turboma hines f = b(V

V0 )

(1)

where b is a onstant depending only on the geometri parameters of the hamber. It follows from Eq. (1) that, in ontrast to Sh, the parameter b, whi h is the slope of the graph f (V ), and the orresponding dimensionless parameter bR are independent of Re. Therefore, this parameter is more onvenient for nondimensional analysis. To hara terize the os illation frequen y f , a new dimensionless parameter whi h, in view of Eq. (1), is independent of Re is worth to be introdu ed: Y =

2br2

(2)

R

This parameter has the following physi al meaning. Sin e V0 is small as ompared with V (see Fig. 5), Y  2fr2 =(V R). If one assumes that the ir ulation is onserved up to the nozzle exit radius (be ause the Reynolds number is large) then Y is approximately the ratio of the os illation frequen y to the uid rotation frequen y at the nozzle radius. In Fig. 6a, the points representing the dependen e of Y on s for various r=R and L=R are plotted. The graph also in ludes the points from Fig. 1 of [2℄. Clearly, the points are grouped around a ertain des ending urve with a s atter less than 20%. However, the points will be grouped even more losely if instead of Y one introdu es the parameter Y1 = Y

 R 0 25 :

r

(3)

This is illustrated by Fig. 6b whi h shows Y1 vs. s and two points from [2℄. The solid urve is the fun tion g1(s) = 2=s0:45 approximating the experimental data. In a

ordan e with Eq. (1), to nd the dependen e f (V ), it is ne essary to know the value of V0. Figure 7 shows the experimental data representing the dependen e of Re0 = V0 R= on s. The solid urve is the fun tion g0 = 1300s0:5. The larger s atter of the values of Re0 as ompared with Y and Y1 is probably onne ted with the greater sensitivity of the value of V0 to errors in measuring the frequen y and velo ity. Thus, a variation of the extreme frequen y value in Fig. 5 by only 5% results in a 15 per ent hange in V0 and less than 5 per ent D. G. Akhmetov et al.

439

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 6 parameter

Dependen e of the parameter

s: 1

|

r=R

= 2=7,

2

| 1/2,

3

Y

(a ) and

| 5/7,

4

|

Y1 (b ) on r=R = 1,

the swirl

and 5 | 0:45

data from Fig. 1 of [2℄. The ontinuous urve shows the fun tion 2=s

hange in the values of Y and Y1 . Thus, using Eqs. (1) to ( 3) and approximating Y1 by the solid urve in Fig. 6b, one an obtain the uni ed empiri al relationship between Sh, Re, s, and r=R for the experimental ranges of the parameters 1 < s < 30, 0:58
104 < Re < 5:8
104 (the

hamber inlet velo ity was varied from 6.2 to 62 m/s): 1 Sh = 

 1 75 R r

:

s

0:45



1

Re0 Re



(4)

For estimating Re0 , one an use the values given by the solid urve in Fig. 7. As follows from Eq. (4), Figs. 6b, and 7, the greatest error arises when al ulating Re0 and this error de reases with Re. 440

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Unsteady Flow Phenomena in Turboma hines

Figure 7

Relation between the Re orresponding to zero frequen y and

the swirl parameter

5 | 0:5 1300s and

s: 1

|

r=R

=

= 2 7,

2

| 1/2,

3

| 5/7,

4

|

r=R

= 1,

data from Fig. 1 of [2℄. The ontinuous urve shows the fun tion

As follows from Fig. 6a variations of parameter

are not large: Y mentioned above, it follows that the os illation frequen y is losely allied to the

uid rotation frequen y at the nozzle exit radius. This nding must be

onne ted with the generation me hanism of the os illations.

Y

3

Y

 1  0 5. From this and from the physi al meaning of :

Water Experiments

For visual observation of the swirl ow and for understanding the generation me hanism of os illations, experiments in water were arried out. The experimental setup was the same as des ribed in previous se tion. The vortex hamber was installed verti ally, with the nozzle dire ted upward. The uid exited from this hamber into a vessel having the shape of a re tangular parallelepiped (180  180  300 mm in size). From the vessel, the uid exited through drains in its upper part. The dimensions of the vessel were suÆ iently large so that the uid in the regions outside the jet was virtually not involved in motion. In the experiments, the unsteady motion of the vortex ore and the qualitative stru ture of the ow near the nozzle output were studied. The mass

ow rate of the liquid was determined. Based on the liquid mass ow D. G. Akhmetov et al.

441

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

rate and the input-slit areas, the input ow velo ity was al ulated. In the basi series of the experiments, the lengths of the hamber and the slits were 77 and 6.7 mm, respe tively. The output diameter of the nozzle was 14 mm. The position of the vortex ore was visualized using small air bubbles. The bubbles were introdu ed into the hamber through either a small hole 0.6 mm in diameter at the losed-end enter or via tangential inputs along with the uid. This method is based on the assumption that, due to the uid rotation, the pressure near the vortex- ore axis is lowered. If the pressure is low enough, then the bubbles are olle ted near the vortex axis, thereby visualizing its position. In the basi series of the experiments, the following phenomena were observed. At Re < 6
103, bubbles left the ore rather rapidly. At Re  6
103, a single bubble is trapped near the outlet, where this bubble then rotates around the symmetry axis. A similar pattern was also observed in [1℄. The number of trapped bubbles in reased with the Reynolds number, and they aligned in a single row. At Re > 1:4
104, the bubbles merge, forming a ontinuous near-axial avity. Figure 1 shows photographs of bubbles obtained in the same onditions with Re = 7:5
103: Figs. 1a and 1b orrespond to exposures made with a ash lamp (exposure time  1{2 ms) and with a long exposure (exposure time  2 s), respe tively. Figure 2 presents a photograph of the vortex that was obtained with the ash lamp upon introdu ing a

olorant through the hole at the enter of the losed hamber end. It an be seen from Fig. 1 that the vortex ore undergoes a sharp bend inside the nozzle, whereas its shape upstream and downstream the nozzle is virtually a straight line. It is also seen that, immediately downstream the nozzle, the ore is absent, sin e the bubbles leave this region rather rapidly and oat upward. This means that a sharp pressure jump takes pla e in this region. This jump serves as a barrier to the upward oat of the bubbles. The de ay of the vortex immediately beyond the nozzle is also seen in Fig. 2. Figure 1b also shows that, after

urving ( ore bend), a part of the ore rotates around the symmetry axis to form a oni al gure visible in the photograph. Before bending, the ore is re tilinear and virtually oin ides with the rotation axis. Using a strobos ope, it was found that the bend rotates with a onstant angular velo ity. Under the onsidered onditions, the rotation frequen y was 17 Hz. In this ase, ore pre ession was not observed 442

D. G. Akhmetov et al.

Unsteady Flow Phenomena in Turboma hines inside the hamber, and the bubbles only slightly drifted along the rotation axis (Fig. 1b ). Using a hydrophone mounted in the nozzle exit se tion (immediately near the nozzle outlet), the pressure pulsations were dete ted, and the frequen y at whi h the pulsation amplitude attained maximum was determined. This frequen y oin ided with the

ore-bend rotation frequen y. It was also found that this frequen y did not vary either in the presen e or in the absen e of bubbles. Thus, the bubbles did not noti eably ae t the ow. With an in rease in the Reynolds number (when a

ontinuous avity formed along the vortex axis), the pattern remained similar to that shown in Fig. 1. The avity whi h was re tilinear inside the hamber was sharply deviated in the nozzle, and the bend rotated with a

onstant angular velo ity. To explain the ow features observed, it is ne essary to model its kinemati stru ture, whi h qualitatively diers from Figure 8 S hemati pattern of the those proposed previously [3, 5℄. streamlines in the plane of the vortex In the development of su h a axis in the rotating frame of referen e model, it was taken into a

ount that the bend rotates with a

onstant angular velo ity, and that the ollision of the jet issuing from the hamber with the ounter ow from the open spa e o

urs. The existen e of the ounter ow was known earlier [3℄ and was on rmed in the experiments des ribed above. As a result, the instantaneous lo al pattern of streamlines before vortex de ay in the frame of referen e lying in the vortex-axis plane and rotating along with the bend an be presented qualitatively as a s hemati of Fig. 8. The symmetry axis is shown by the OO line and a is the stagnation point. Note that this stru ture is not ompletely at rest but exhibits small os illations in both axial and radial dire tions, whi h was on rmed by observations of bubble motion. In addition, in the laboratory frame of referen e, this pattern rotates around the OO axis. As follows from Fig. 8, the vortex D. G. Akhmetov et al.

443

Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

ore and major part of the uid issuing from the hamber along its axis are deviated in one dire tion. To verify the ow stru ture presented above, the following experiment was performed. A thin tube 0.6 mm in diameter was pla ed along the ontinuation of the hamber axis at a distan e of 4 mm from the nozzle exit. Through this tube, a olorant was slowly introdu ed. Figure 3 shows a photograph showing the olorant out ow from the tube. The photograph was obtained with illumination by a ash lamp. It is seen that the olored line is deviated from the hamber axis in one dire tion. In the laboratory frame of referen e, the line rotates around this axis. Note that, sin e the pre ession of the vortex ore before the appearan e of the bend was not observed in the present experiments, this implies that the vortex- ore axis prior to the bend was lose to or

oin ided with the OO line. Otherwise, pre ession would be observed. Experiments performed with nozzles 8 and 20 mm in diameter yielded results qualitatively similar to those presented above. As it was mentioned in the previous se tion, the os illation frequen y was losely allied to the uid rotation frequen y at the nozzle radius. Based on the above results, this fa t may be explained by the fa t that the generation of the os illations is aused by a bend of the vortex ore forming and rotating in the nozzle.

4

Con luding Remarks

The frequen y of the periodi pressure os illations developing in the swirling jet issuing from a vortex hamber was investigated systemati ally. The frequen y in reased linearly with the air velo ity at the

hamber inlet; however, the Strouhal number based on the inlet velo ity and the hamber radius was not onstant. The variation of the Strouhal number is a

ounted for by the linear regression graph representing the dependen e of the frequen y on the velo ity did not depart from the

oordinate origin. The proximity of the uid rotation frequen y at the nozzle radius and os illation frequen y was pointed out. An empiri al equation for al ulating os illation frequen y was proposed. It has been shown that the generation of the os illations is aused by a bend of the vortex ore, resembling a ho key sti k, forming in the nozzle and rotating around the symmetry axis of the hamber. This 444

D. G. Akhmetov et al.

Unsteady Flow Phenomena in Turboma hines

me hanism explains the observed proximity of the uid rotation frequen y at the nozzle radius and os illations frequen y. Note that the symmetry violation observed took pla e even at the axial symmetry of the hamber; i.e., it is a property of the rotating ow itself. The properties of swirling ows established in this work may be responsible for strong disturban es arising in nozzle blo k of turbomashines in some onditions.

A knowledgments This work was supported in part by the NATO grant SfP 981461.

Referen es 1. Vonnegut, B. A. 1954. A vortex whistle. J. A oust. So . Amer. 26(1):18{20. 2. Chanaud, R. C. 1963. Experiments on erning the vortex whistle. J. A oust. So . Amer. 35(7):953{60. 3. Chanaud, R. C. 1965. Observations of os illatory motion in ertain swirling

ows. J. Fluid Me hani s 21(1):111{27. 4. Kiyasbeili, A. Sh., and M. E. Perel'stein. 1974. Vortex uid owmeters. Mos ow: Mashinostroenie. 5. Knysh, Yu. A., and S. V. Luka hev. 1977. Experimental study of a vortex sound generator. A oust. J. 23(5):776{82. 6. Akhmetov, D. G., V. V. Nikulin, and V. M. Petrov. 2004. Experimental study of self-os illations developing in a swirling-jet ow. Fluid Dynami s 39(3):406{13. 7. Akhmetov, D. G., and V. V. Nikulin. 2004. Me hanism of generating selfexited os illations in swirling-jet euxes. Doklady Physi s 49(12):743{46. 8. Cassidy, J. J., and H. T. Falvey. 1970. Observations of unsteady ow arising after vortex breakdown. J. Fluid Me hani s 41(4):727{36.

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LASER DOPPLER DIAGNOSTIC OF FLOW IN DRAFT TUBE BEHIND HYDROTURBINE RUNNER 

   S. Dvoinishnikov , D. Kulikov ,   V. Pavlov , V. Rakhmanov , O. y y N. Mostovskiy , and I. Pylev



   I. Naumov , V. Okulov ,  y Sadbakov , S. Ilyin ,

V. Meledin , Yu. Anikin , G. Bakakin , V. Glavniy ,

 Institute

of Thermophysi s Siberian Bran h of the Russian A ademy of S ien es Lavrentyev Str. 1 Novosibirsk 630090, Russia y Laboratory of Water Turbines JSC Leningradsky Metalli hesky Zavod (LMZ) Sverdlovskaya Nab. 18 St.-Petersburg 195009, Russia Introdu tion

Experimental investigation of the vortex ow behind the vane wheel rotor of the water turbine is one of the most ompli ated problems of applied uid dynami s. For this purpose, using Laser Doppler Semi ondu tor Anemometry (LDSA) is very attra tive. Unsteady ow in the turbine draft tube downstream of the rotor wheel has a omplex stru ture whi h is signi ant for turbine operation. Despite many years of study and a qualitatively lear pattern of the ow, the problem of high-a

ura y diagnosti s for this kind of ows has not been solved yet. Dierent visualization te hniques and measurements show that the ow behind the rotor wheel (for the optimal mode of the hydroturbine universal model) is lose to the axial ow, but for most of operating regimes this is a swirling ow. The swirling ow auses rarefa tion in the enter and produ es a

avitation avity | a rope lled with air bubbles. This rope has usually the shape of a rotating heli al spiral. The pre ession motion of the rope is the sour e of pressure pulsations in the ow and on the walls of the draft tube. Sometimes, those pulsations rea h the spiral hamber and 446

V. Meledin et al.

Unsteady Flow Phenomena in Turboma hines aggravate the vibration of the omponents. Usually, the ow stru ture in the draft tube has, besides the vortex rope, an array of vorti es shedding from the edges of wheel blades. Those vorti es merge with the entral vortex and en han e its intensity. For several operation modes, interblade vorti es ome also into ee t. They are lo alized at the draft tube walls and have smaller intensity than the entral vortex, but still ontribute to the nonstationary ow pattern. This qualitative pi ture of the ow in the draft tube is absolutely insuÆ ient for elaborating parti ular tools aimed at damping the resultant perturbations. It is also insuÆ ient for developing simulation methods for turbine ow analysis. The latter relates mainly to the orre t statement of boundary onditions at the outlet of the rotor wheel and a

urate hoi e for the ow model in the interblade hannels. Moreover, a

urate des ription of the vortex system behind the rotor wheel is a way for estimating the losses arising at o-optimal operation modes of the turbine. Obviously, most of intrunsive methods do not work here be ause of high sensitivity of swirling ow to external perturbations. The oarse Lagrangian methods of ow diagnosti s like Parti le Image Velo imetry (PIV) and Parti le Tra king Velo imetry (PTV) fail to provide a

urate measurements [1, 2℄: these te hniques exhibit 10 per ent error for standard plane-parallel ows, and the error rises up to 20% in the ase of os illating swirling ow, mainly be ause of Lagrangian randomization of parti le-tra ers [3℄. For the type of experiments onsidered herein, the appli ation of tra e diagnosti methods be omes even worse due to obvious three-phase nature of the ow. For example, this is valid for the ore of the vortex rope. In this ow region, tra ers will intera t with air bubbles, and this intera tion will disturb the Lagrangian pi ture of tra er motion. In view of the aforesaid, it is ne essary to use the nonintrunsive methods with high a

ura y for perfoming measurements behind the rotor wheel. Su h methods must be suitable for diagnosti s of omplex three-phase ows. Two-Dimensional Laser Doppler Semi ondu tor Anemometer

Laser Doppler anemometry diagnosti tool of the vortex ow behind the rotor wheel of water turbine was assembled from a high-a

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dimensional (2D) LDSA LAD05-LMZ, designed for spe i onditions of a large-s ale avitation-study test-ben h for hydroturbine model in the fa ilities of LMZ [4, 5℄. A new on ept for this measurement omplex was applied. It allowed working without powerful and expensive gas lasers (traditionally used with the LDSA equipment produ ed by DANTEC, TSI, et .). Instead, a moderate-power semi ondu tor lasers were used. The LDSA equipped with a red-emission laser and a spe ial signal-pro essing te hnique allows to perform the measurements with a turbid water and with air bubbles lo alized in the vortex rope at te hni al onditions of the LMZ test-ben h. The stru ture of 2D LDSA for the diagnosti s of vortex ow behind the rotor wheel of the water turbine is presented in Fig. 1. The LDSA operates as follows. The ray of the semi ondu tor laser 1, after passing through ompounding opti al units, arrives at opti al{a ousti modulators 2 and 3, routing the ultrasoni waves to axes X and Y , respe tively. In the modulators operating at Bragg dira tion, three light beams, dira ted in zero, X -minus and Y -minus rst orders are shaped. The split beams pass sequentially through rotary prisms 4 and 12, the diele tri mirror with shaped over 11 and a lens 9 and are routed to the test ow eld whi h velo ity has to be measured. Being rossed in the ow, the laser beams form an interferen e eld with a known periodi stru ture. Its image in a diused light is shaped by opti al units on a photosensitive surfa e of a photodete tor 13. The size of the image is limited by the eld diaphragm determining spatial ltering in the opti al hannel. When a s attering parti le interse ts the probe opti al eld, a radio pulse of photoele tri signal appears on the photodete tor exit. Its frequen y is known to be a linear fun tion of Doppler shift of frequen y, and its duration is equal to transmitting time of the diusing lter through the interferen e eld. Ampli ers- ommutators 15 and 16 in lude modulators 2 and 3, and are onne ted to a photodete tor 13, quadrature demodulator 18, and Doppler signal prepro essor 17 after arrival of N radio pulses on its entry. With de reasing N , the sampling rate of the information for ea h velo ity omponent in reases, rea hing maximum at N = 2. Swit hing of opti al hannels o

urs in instants when signals of gating are absent, and no more information is a

umulated by FPGA prepro essor. 448

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Unsteady Flow Phenomena in Turboma hines

Figure 1 Stru ture of 2D LDSA for diagnosti s of vortex ow behind the rotor wheel of the water turbine

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LAD05 LMZ LDSA signal prepro essor is intended for omplex demodulation of a Doppler radio signal. Prepro essing unit realizes a method of temporary sele tion and ontrols sele tion of an opti al measuring hannel. In a ombination with the PC signal pro essor, it provides self-a ting adaptive swit hing of opti al measuring hannels and measuring of two orthogonal omponents of the velo ity ve tor. As light emitter, a serial semi ondu tor laser operating in visible band in a single mode with linear polarization of output opti al radiation is used. Opti al{a ousti ommutators are fabri ated from opti al{ a ousti modulators. In the LDSA, the opti al{a ousti ommutatorampli ers have the following parameters: standing-wave ratio = 1:8, ee tiveness = 81%, transmission oeÆ ient = 96%, gamma = 1000 : 1, pulse rise time of laser light at diameter of in ident beam of 0.2 mm = 50 ns.

Software Software of the measuring omplex in ludes tools for diagnosti s and tuning of the equipment, ontrol of the equipment during experiment and visualization of the measurments. The software allows to determine a Doppler frequen y from the light s attering parti les in the ow, dire tion of the velo ity ve tors of the registered light s attering parti les and to arry out statisti al data pro essing. The software has a modular stru ture. Ea h fun tional group is sele ted as a separate program omponent. Program omponents an be updated without relinking the entire ode. The spe ial attention is paid to the onstru tion of the omponents responsible for data pro essing. As the probability of data type modi ation is rather high ( aused by modi ations of the measuring omplex), the omponents have been designed to allow easy introdu tion of exible modi ations. The software makes it possible to use the hannel apa ity on an optimal way: it minimizes the transmission of inee tive information and the number of ommands for prepro essor a tivity.

Te hni al Data The key parameters of the 2D anemometer mat h to the best world analogs (DANTEC, TSI) at essentially smaller overall dimensions and 450

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Table 1 Basi te hni al data of LDSA Spe i ation Gamut of measured velo ities, m/s Stati error of Doppler signal spe tral peak measuring, % Measuring error of average velo ity, % Fo al distan e of output lens, over the range (F ), m Size of the probing opti al eld, no more than, mm Gauge LDSA length, mm Pit h of transition of sounding LDSA eld, mm Input of LDSA unit, W Weight of LDSA unit, kg Resour e of a tivity, h

Value

0 005 0 1 0 5 :

:::

30

:

:

0.25 : : : 1.0 ?0:05  1 for F = 0:5 > 500 < 0:2 80 < 33 up to 40 000

smaller osts (Table 1). For the rst time the new 2D laser anemometer anamorphi s heme was developed that allowed one to apply modern high-power semi ondu tor lasers with a low degree of spatial and temporal oheren e. The latter improved essentially the performan e of the anemometer. For the rst time, the hardware system of signal pro essing was developed on the basis of parallel programmed logi al operations \Field

Figure 2 Exterior of LDSA (a ) and the anemometer installed at a large-

s ale avitational ben h for measuring vortex ow behind the rotor wheel of the water turbine (b ) V. Meledin et al.

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Programmable Gate Array" (FPGA), and the new pro edure of adaptive sele tion of a Doppler frequen y was realized. The laser anemometer has an integral stru ture. The optoele troni unit ontains all units of opti al and ele troni subsystems and is onne ted with an exterior standard PC through a standard network hannel. Due to implementation of TCP proto ol, the PC an be lo ated in an arbitrary pla e (for example, in the other ity), and physi al experiments an be ondu ted remotely (Fig. 2).

Diagnosti of Vortex Flow Behind the Rotor Wheel The measurement fa ility LDSA LAD05-LMZ was applied to the study of ow stru ture behind the rotor wheel of the Fran is turbine for the

ase of transition from the optimal operation mode to a boost operation mode. The tests were performed at a large avitation-study setup for hydroturbine model with a radial rotor wheel equipped with 14 blades (RO 140{46). The average pro les of the ir umferential and axial velo ity omponents in the draft tube were measured along the radius in the tube ross se tion. This measurement site was hosen at the distan e of 240 mm from the lower rim of the rotor wheel. p In test series A, the redu ed rotation frequen y (n11 = nD1 = H , where D1 = 0:46 m is the nominal diameter of the rotor wheel) was in the range from 64.5 to 65 r.p.m. (Table 2 and Figs. 3 and 4, Case A), and in the tests of series B, the redu ed frequen y was xed in the range from 70 to 71.5 r.p.m. (see Table 2 and Fig. 4, Case B) at a given head of H = 20 m and the same avitation number. The turboma hine operation modes were varied through in rementing the opening in the guide apparatus: its hara teristi is the diameter of the maximal ir le imbedded into the interblade spa e of the apparatus. The operation modes were tested with the guide openings of a = 24, 28, 30, and 36 mm. Note that the optimal mode for the two values of the rotor wheel rotation rate orresponds to the openings from 24 to 28 (for the frequen ies of series A) and from 25 to 31 (for the frequen ies of series B). As for the for ed regimes, this parameter is a = 30 and 36 (for Case A) and a = 36 (for Case B). The pro les of measured axial velo ity omponent are plotted in Fig. 4 , while the ir umferential velo ity omponent is plotted in 452

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Unsteady Flow Phenomena in Turboma hines Table 2

Testing of LDSA measuring pro edure

Flow rate from Flow rate from Dieren e , Opening 0 , LDSA measurements,

hara teristi s, r.p.m. mm
11 , % 3 3 11 , m /s 11 , m /s Series A 64.5 24 0.611 0.646 5.4 65 28 0.729 0.729 0.0 64.5 30 0.744 0.768 3.1 65 36 0.849 0.865 1.8 Series B 70 28 0.685 0.721 5.0 71.5 30 0.784 0.76 32 70 36 0.829 0.86 3.6 n11

a

Q

Q

Q

:

Figure 3 Velo ity pro les of the vortex ow behind the rotor wheel of Frensis water turbine for two versions of blade surfa es (1 ) and (2 ). Parameter is the stepover distan e of measuring points from the illuminator interior surfa e. The symmetry axis of the dra tube passes through the = 258 mm value. z

z

Fig. 4b. Note that transformation of the ow stru ture during transition from the optimal mode to the boost mode is similar for both frequen y ranges of Cases A and B. The optimal modes at a = 24 mm (for Case A) and a = 28 (for Case B) exhibit the most uniform axial ow in the draft tube (dashed V. Meledin et al.

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Turboma hines: Aeroelasti ity, Aeroa ousti s, and Unsteady Aerodynami s

Figure 4 Radial LDSA pro les of axial (a ) and tangential (b ) velo ities in a draft tube behind the rotor wheel with the rotation frequen y ranging from 64.5 to 65 r.p.m. (Case A) and from 70 to 71.5 r.p.m. (Case B). Case A: a = 24 mm (dashed line), 28 (dash-dotted), 30 (dotted), and 36 (solid); Case B: a = 28 mm (dashed line), 30 (dash-dotted), and 36 (dotted) lines in Figs. 4a and 4b ). In this ase, the ow swirling is minimal. There is a weak vortex with rotation opposite to ow swirling in the spiral hamber, whi h is explained by de eleration of the ow downstream of the rotor wheel rotating with a onstant angular velo ity. 454

V. Meledin et al.

Unsteady Flow Phenomena in Turboma hines Nevertheless, the ow at the draft tube walls still retains some of ow rotation following the swirl in the spiral hamber. With in reasing the opening of the guide apparatus to a = 28 mm for Case A and to a = 30 for Case B (dash-dotted lines in Fig. 4), a distin t negative axial vortex is formed with a typi al distribution of ir umferential velo ity. Inside the ore (estimated by the maximal level of ir umferential velo ity), a trail-like pro le of axial velo ity is evident whi h is typi al of a left-hand vortex [6, 7℄. Exa tly this kind of vortex indu es intense ounter ow along the axis [8℄. Nearby the draft tube wall, the velo ity pro le remains lose to the equilibrium one. Further opening of the guide apparatus to a = 30 mm for Case A and to a = 36 for Case B (dotted lines in Fig. 4) results in intensi ation of the axial vortex and ounter ow. At last, the mode with a = 36 mm for Case A (solid lines in Fig. 4) demonstrates a ertain attenuation of the ounter ow, and the region o

upied by vortex (estimated by the maximal value of the ir umferential velo ity) in reases signi antly. This an be explained by the formation of a distin t heli al region in the entral heli al stru ture and smearing of averaged pro les (see Fig. 4b, Case B). Table 2 presents the results of testing the measurement te hnique by

omparing the water ow rate al ulated based on the pro les of axial velo ity (see Figs. 3 and 4) and the data taken from the universal hara teristi s of the hydroturbine model Ž 140{46 for the orresponding regime parameters. p The omparison of normalized ow rates Q11 = Q=(D12 H ) demonstrates high a

ura y of the LDSA measuring te hnique when applied to the three-phase swirling ows behind the rotor wheel of the water turbine.

Con luding Remarks Laser Doppler semi ondu tor anemometer LAD05 LMZ was developed and tested in the rotating turbid three-phase ow behind the rotor wheel of the water turbine with depth ex eeding 500 mm. The anamorphi opti al s heme of the 2D laser anemometer allowed the use of modern high-power semi ondu tor lasers with low degree of spatial and temporal oheren e and improved essentially the performan e of the V. Meledin et al.

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anemometer. For the rst time, the built-in system of signal pro essing based on parallel programmed logi al operations FPGA was developed and the new pro edure of adaptive sele tion of Doppler frequen y was realized. Thus, the FPGA te hnology was applied for the rst time for high-a

ura y LDSA measurements of velo ity pro les in a two-phase swirling ow behind the rotor wheel of the model Fransis hydroturbine, in the lower part of the draft tube one. The total error in the ow rate estimated based on the measured velo ity pro les or taken from the universal hara teristi s of the turbine does not ex eed 5%. The velo ity pro les in the draft tube were ompared for dierent operation modes: the dieren e in swirling ow stru ture was observed between optimal and for ed regimes of the Fran is turbine operation.

A knowledgments The authors are grateful to LMZ top managers and experts, I. Kuznetsov and A. Malyshev, for their attention to the work and help in arranging full-s ale experiments. The authors also express their thanks to A. Sharhov for useful dis ussions and parti ipation in assembling the LDSA LAD05 LMZ. This work was partly supported by the Russian Foundation for Basi Resear h (grant 04-01-00124) and INTAS (youth grant 03-55571).

Referen es 1. Durrani, T. S., and C. A. Greated. 1977. Laser systems in ow measurement. New York: Plenum Press. 2. Adrian, R. 1991. Parti le imaging te hniques for experimental uid me hani s. Annual Review Fluid Me hani s 23:261{304. 3. Kozlov, V. V. 2003. On sto hasti plane-parallel ows of ideal uids. In: Fundamental and applied problems of the vorti ity theory. Ed. A. Borisov. Izhevsk: Institute of Computer Te hnologies. 303{7. 4. Meledin, V., I. Naumov, Yu. Anikin, O. Sadbakov, et al. 2004. Laser Doppler measuring system for diagnosti of gas{liquid ows LAD05 LMZ. Des ription and the Manual. 5 123 00 00 00. Novosibirsk: Institute of Thermophysi s of the Russian A ademy of S ien e, Siberian Bran h.

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Unsteady Flow Phenomena in Turboma hines 5. Sadbakov, O., V. Okulov, I. Naumov, Yu. Anikin, V. Meledin, N. Mostovskiy, and S. Ilyin. 2004. Laser Doppler diagnosti s of ow stru ture behind a rotor wheel in hydroturbine in optimal and for ed regimes. Thermophysi s Aerome hani s 11(4):561{66. 6. Okulov, V. 1996. Transitions from right- to left-hand heli al symmetry during vortex breakdown. Te h. Phys. Lett. 22(11):47{54. 7. Alekseenko, S., P. Kujbin, and V. Okulov. 2003. Introdu tion to the theory of on entrated vorti es. Novosibirsk: Nauka. 8. Okulov, V., Z. Sorensen, and L. Voigt. 2002. Alternation of right- and lefthanded rotational stru tures at magni ation of urling ow intensity in

ylindri al aw with rotating end fa es. Te h. Phys. Lett. 28(2):55{58.

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Author Index Akhmetov D.G., 434 Aleksandrov V., 169, 295 Anikin Yu., 446 Arnone A., 87 Atassi H. M., 237 Bakakin G., 446 Billiard N., 215 Bunkov V. G., 9 Burguburu S., 340 Cherny S.G., 356 Chirkov D.V., 356 Chupin P.V., 24 Dvoinishnikov S., 446 Efremov N. L., 281 Eret P., 128 Eriksson L.-E., 307 Favorskii O.N., v Fede hkin K.S., 370 Ferrand P., 340 Fransson T. H., 75 Frolov S.M., vi Gaetani P., 180 Gambini E., 87 Glavniy V., 446 Gnesin V. I., 24, 103 Godunov S. K., 9 458

Gourdain N., 340 Gribin V. G., 227 Hall K., 37 Hassan A.S., 406 He L., 155 Hong E., 37 Ilyin S., 446 Imregun M., 3 Izmailov R.A., 393 Ja obs P., 98 Kielb R., 37 Kolodyazhnaya L.V., 24, 103 Kovsher N. N., 370 Kraiko A.N., 281 Kulikov D., 446 Kurzin V.B., 9, 115, 376 Lapin V.N., 356 Leboeuf F., 340 Li H.-D., 155 Linhar J., 128 Martensson H. E., 75 M Ghee A., 98 Mel'nikova G.V., 169 Meledin V., 446 Mironovs A., 61 Miyakozawa T., 37 Mostovskiy N., 446

Author Index Nagashima T., 421 Nakagawa H., 143 Namba M., 143, 257 Naumov I., 446 Nikulin V. V., 434 Nishino R., 143, 257 Nitusov V. V., 227 Nyukhtikov M., 268 Ohgi S., 257 Okamoto K., 421 Okulov V., 446 Osipov A. A., 169, 295 Paniagua G., 215 Pavlov V., 446 Persi o G., 180, 215 Petrie-Repar P., 98 Petrov V. M., 434 Poli F., 87 Pyankov K. S., 281 Pylev I. M., 356, 446 Rakhmanov V., 446 Rossikhin A., 268 Rz adkowski R., 103

Sadbakov O., 446 Sadkane M., 9 Saren V. E., vi, 49, 201 Savin N. M., vi, 201 S hipani C., 87 Sharov S. V., 356 Shmotin Yu. N., 24 Shorr B. F., 169 Skibin V. A., vi Skorospelov V. A., 356 Starkov R. Yu., 24 Stridh M., 307 Tauveron N., 340 Tillyayeva N. I., 281 Tolstukha A. S., 115 Tsymbalyuk V. A., 128 Vahdati M., 3 Vinogradov I. V., 237 Vogt D. M., 75 Yakovlev Ye. A., 281 Yamagu hi K., 421 Yudin V. A., 376 Zinkovskii A. P., 128

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