Chinese Journal of Aeronautics 25 (2012) 143-154 Contents lists available at ScienceDirect Chinese Journal of Aeronautics journal homepage: www.elsevier.com/locate/cja Flow Stability Model for Fan/Compressors with Annular Duct and Novel Casing Treatment LIU Xiaohua, SUN Dakun, SUN Xiaofeng*, WANG Xiaoyu School of Jet Propulsion, Beihang University, Beijing 100191, China Received 25 March 2011; revised 21 April 2011; accepted June 2011 Abstract A three-dimensional compressible flow stability model is presented in this paper, which focuses on stall inception of multi-stage axial flow compressors with a finite large radius annular duct configuration for the first time It is shown that under some assumptions, the stability equation can be obtained yielding from a group of homogeneous equations The stability can be judged by the non-dimensional imaginary part of the resultant complex frequency eigenvalue Further more, based on the analysis of the unsteady phenomenon caused by casing treatment, the function of casing treatment has been modeled by a wall impedance condition which is included in the stability model through the eigenvalues and the corresponding eigenfunctions of the system Finally, some experimental investigation and two numerical evaluation cases are conducted to validate this model and emphasis is placed on numerically studying the sensitivity of the setup of different boundary conditions on the stall inception of axial flow fan/compressors A novel casing treatment which consists of a backchamber and a perforated plate is suggested, and it is noted that the open area ratio of the casing treatment is less than 10, and is far smaller than conventional casing treatment with open area ratio of over 50, which could result in stall margin improvement without obvious efficiency loss of fan/compressors Keywords: compressor; stability; rotating stall; casing treatment; eigenvalue Introduction1 Considerable work was completed in the past tens of years on investigating the phenomenon of rotating stall of fan/compressors Since the classical explanation presented by Emmons, et al [1] in 1955, there have been some developments in the model studies of stall inception Nenni and Ludwig’s work [2] resulted in an analytical expression for the inception condition of two-dimensional incompressible rotating stall, which was later extended to compressible flow in 1979 and a three-dimensional incompressible model without any relevant numerical results reported [3] Stenning [4] also *Corresponding author Tel.: +86-10-82317408 E-mail address: sunxf@buaa.edu.cn Foundation items: National Natural Science Foundation of China (50736007, 50890181); the Innovation Foundation of BUAA for PhD Graduates (300383) 1000-9361/$ - see front matter © 2012 Elsevier Ltd All rights reserved doi: 10.1016/S1000-9361(11)60373-7 studied the rotating stall based on a linearized small perturbation analysis in 1980 It is verified that all these models can predict the instability inception condition with a satisfactory accuracy as long as sufficient loss and performance characteristics of the compressors concerned are given Furthermore, in recent years, more attention is paid to the compressible flow stability of rotating stall in multi-stage compressors In 1996, X F Sun [5] firstly developed a three-dimensional compressible stability model including the effect of the casing treatment, and this work was extended to transonic compressors stability prediction recently [6] However, most existing works are based on the assumption that the circumferential flow passage comprises of two flat plates which model the hub and tip duct wall of compressors It should be noted that this assumption implicates an infinite curvature radius, which is an approximate analysis of the actual annular duct This simplification makes it straightforward to gain the perturbation expression and artificially decompose the radial modes for an eigenvalue So a · 144 · LIU Xiaohua et al / Chinese Journal of Aeronautics 25(2012 ) 143-154 model concerning a finite large radius, which searches for an eigenvalue for a coupled multi-radial mode distribution of perturbation in fan/compressors system, should be acceptable and viewed as an improvement towards the real configuration On the other hand, casing treatment has long been an effective strategy to meet the requirement of stall margin with the increase of blade loading Although there were successful practical applications in the research institutions [7-12] and industrial departments [13-14], a large number of different configurations [15-18] were used in various experiments without enough unambiguous or unified explanation Up to now, most designs of casing treatments have been based on try and error Smith and Cumpsty [19] admitted that the reason for the effectiveness of casing treatment is not really understood It was noted that with the discovery of stall precursor in compressors, the idea of a novel casing treatment was proposed by D K Sun, et al [20-21] to suppress the precursor or delay its evolution in order to enhance stall margin It is obvious that such casing treatment is aimed at affecting stall precursors instead of improving the blade tip flow structure like the conventional casing treatments So, theoretically it is possible to design a kind of advanced casing treatment with low open area ratio to suppress the stall precursor, which can result in the stall margin improvement without the obvious efficiency loss of fan/compressors The present work will introduce a three-dimensional compressible flow stability model of multi-stage compressors with a finite large radius duct, which is obtained by the following steps First, under the assumptions of uniform mean flow and large radius assumption, the perturbation field is described by linearized Euler equations, which can be solved by mode decomposing plus appropriate boundary condition Further the compressor stability can be described as an eigenvalue problem, which is established by using the mode-matching technique and applying the conservation law and conditions reflecting the loss characteristics of fan/compressors at the two sides of rotors and stators, which are modeled as a series of actuator disks Then the eigenvalue equations in matrix form are solved using the winding number integral approach, and the stability can be judged by the non-dimensional imaginary part of the resultant complex frequency eigenvalue Furthermore, based on the analysis of the unsteady phenomenon caused by casing treatments, the function of casing treatments is modeled by a wall impedance condition which is included in the stability model through the eigenvalues and the corresponding eigenfunctions of the system Finally, some experimental investigation and two numerical evaluation cases are conducted to validate this model and emphasis is placed on numerically studying the sensitivity of setup of different boundary conditions on the stall inception of axial flow fan/compressors A novel casing treatment which consists of a backchamber and a perforated plate is suggested, and it is noted that the open area ratio of the casing treatment is less than 10, far No.2 smaller than the conventional casing treatment with open area ratio of over 50, which could result in stall margin improvement without an obvious efficiency loss of fan/compressors Theoretical Model of Stall Inception 2.1 Governing equations and pressure perturbation In the present work, a linear cascade of blades is modeled by the three-dimensional actuator disk The actual compressor configuration is simplified as uniform annular duct with finite large radius, and the radial mean flow is ignored A three-dimensional, compressible, inviscid, non-heat-conductive flow is considered The governing equations for a small disturbance problem are the linearized Euler equations as follows, which reflect the conservation relations for mass, momentum and energy wv c wv c w(r vrc ) wU c wUc wU c U0 U0 T V0 U0 x U0 rwr rwT rwT wt wx wx (1) vc wvrc wv c wv c V0 r U r 2V0 T wt wx r wT r wvT c wv c wv c vc V0 T U T V0 r wt wx r wT r wvxc wv c wv c V0 x U x wt wx r wT wpc wpc wpc V0 U0 r wT wt wx k wpc U wr (2) wpc r wT (3) U0 wpc U wx (4) p0 § wU c wU c wU c à ă V0 U0 ă U â wt r wT wx áạ (5) where U is density, U and V are the velocity components, p is the pressure, k the specific heat ratio, and v the fluctuating velocity The subscript “0” represents the mean flow, while the superscript “Ą” represents perturbation x, r and ș represent axial, radial and circumferential coordinates, respectively A large radius assumption is made that the curvature radius rm of annular duct is not infinite but much greater than the axial perturbation wavelength Ȝx: Ox rm f (6) Given that most previous stability models consider the actual annular duct as rectangular channel, i.e., infinite curvature radius, a model concerning a finite radius should be acceptable and viewed as an improvement Under this condition, all items including V0 become high-level minim in each equation and can be omitted The simplified equations are obtained as follows: wv c wv c w(r vrc ) wU c wU c U0 U0 T U0 x U r wr r wT wt wx wx (7) No.2 LIU Xiaohua et al / Chinese Journal of Aeronautics 25(2012) 143-154 · 145 · wvxc wv c U0 x wt wx wpc U wx (8) wvrc wv c U0 r wt wx wpc U wr (9) w w °°bmn1 wr (J m ( Pmn rh )) bmn wr (N m ( Pmn rh )) (17) ® w w °b (J ( r )) b (N ( r )) P P mn1 m mn t mn m mn t wr wr ¯° wvT c wv c U0 T wt wx wpc r wT (10) where the undetermined coefficients bmn1 and bmn2 cannot be zero at the same time for a nontrivial solution, so wpc wpc U0 wt wx U0 p § wU c wU c à k 0ă U0 U0 ăâ wt wx áạ (11) From Eqs (7)-(11), it can be shown that fluctuating variables related to pressure satisfy the wave equation in the form of (1 Max ) w2 pc w2 pc w2 pc w2 p c wx r wT wr a02 wt 2Max w2 pc wpc a0 wxwt r wr f f ¦ ¦ pmn< mn (r ) ei(mT Zt D mn x ) (13) m f n where m is the circumferential mode number, or ordinal number of harmonic, n the radial mode number, Z the eigenfrequency, D mn the axial wave number, and pmn the wave amplitude Substituting Eq (13) into Eq (12) yields an equation about the radial eigenfunction < mn (r ) : r2 w2< mn (r ) wr f f ¦ ¦ ( pmn j eiD j j f j Z j D j where rh and rt are the radius at hub and tip, respectively So the radial wave number for hard wall can be solved by the following equations: ) j · j (r ) ¸ ei(mT Zt ) Imn (21) ¸ ¹ j mn j f j rh , rt j f iD ( x x ) § mp j e mn Ư Ư U j r ăă Zmn U jD j < mn j (r ) m f n © mn f j iD mn ( x x mpmn e j ) · j (r ) ei(mT Zt ) < mn Ư ¦ (a j )2 pmn j eiD f f m f n j iD pmn e j mn j (22) j Z U 0jD mn U pc ( x, r ,T , t ) (20) j iD mn (x x ) e i Đ pmn ă Ư Ư U j ă Z U jD j Imn j (r ) m f n © mn f U 0j (15) (16) j j j iD mn ( x x ) · i mT Zt pmn D mn e j (r ) ¸ e < mn j j ¸ Z U D mn ¹ vpT c ( x, r , T , t ) (19) iD ( x x ) § p jD j e mn j Ư Ư j ă mn mn j j < mn (r ) ă Z U D mn m f n U0 © f vpxc ( x, r ,T , t ) (14) where Jm(ȝmnr) and Nm(ȝmnr) are Bessel functions of the first and second kind, respectively, bmn1 and bmn2 are two undetermined coefficients, and ȝmn is the radial wave number For hard wall boundary condition on both hub and tip wall, w< mn (r ) wr r j < mn (r ) wherex j is the axial coordinates for an arbitrary reference plane; “+j ” and “ j ” represent the waves traveling downstream and upstream from the plane x j, respectively After substituting Eq (19) into Eqs (8)-(11), the other perturbation corresponding to pressure can be obtained j < mn (r ) bmn1J m ( Pmn r ) bmn N m ( Pmn r ) (18) (xx j ) j j iD mn ( x x pmn e Assume the solution of Eq (14) is j iD mn ( x x ) j pmn e < mn (r ))ei(mT Zt ) vprc ( x, r , T , t ) w< mn (r ) r wr êĐ Z ẵ à ô ằ r m ắ< mn (r ) D Ma đ ă x mn D mn ằ ôơâ a0 ¼ j mn m f n (12) w (N m ( Pmn rh )) wr w (N m ( Pmn rt )) wr Meanwhile, the axial wave number Įmn is solved in the above process The complete expression for pressure perturbation is p c( x, r , T , t ) where Max is the axial Mach number, a0 the sound speed Assume the solution of Eq (12) is p c( x, r , T , t ) w (J m ( Pmn rh )) wr w (J m ( Pmn rt )) wr j j mn ( x x ) j < mn (r ) j j < mn (r ) ei(mT Zt ) ( x x ) where I mn is the deriative of < (23) mn 2.2 Vortex wave Since the vortex wave will not cause the pressure · 146 · LIU Xiaohua et al / Chinese Journal of Aeronautics 25(2012 ) 143-154 variation, the solutions related to vortex mode can be given by the homogeneous form of Eqs (8)-(10) It is noted that vr is composed of two parts: one is the contribution by the pressure wave vpr and the other one is by the vortex wave vvr Since the wave lengths of the pressure wave and the vortex wave are different from each other, vvr can be assumed not to contribute to the left-hand side of Eq (16) For this reason, vvr is concluded to satisfy the following condition on solid wall: vvrc |r rh , rt (24) In the similar way, the corresponding solutions for the homogeneous form of Eqs (8)-(10) are vvxc ( x, r ,T , t ) vvr c ( x, r ,T , t ) vvT c ( x, r ,T , t ) f ¦ f ¦ vvxmnj < vmnj (r )ei(mT Zt )e i Z U 0j < Ivmnj (r ) bmn1 With the above basic solutions, it is shown that pressure perturbation is ¦ ¦ pmn j eiD f p c j ( x, r ,T , t ) f j iD pmn e j mn f f ¦¦ m f n f f ê f Ư ¦ « (a j )2 ( pmn j eiD U c j ( x, r ,T , t ) ¦ ¦ vvTjmn< vmnj (r )ei(mT Zt )e Z U 0j m f n ¬ bmn1J m ( P P U vmnj < vmnj (r )e i Z U 0j ( x x j ) º » ei(mT Zt ) ằ ẳ (27) j vmn r ) ê § p jD j eiDmn j ( x x j ) Ư Ư ô U j ăă mnZ mnU jD j