COMPRESSIBILITY EFFECTS ON THE TWO GENERATIONS OF UNSTEADY FLOW TYPES IN AXIAL FLOW COMPRESSORS.pdf

Proceedings of GT2005 ASME Turbo Expo 2005: Power for Land, Sea and Air June 6-9, 2005, Reno-Tahoe, Nevada, USA GT2005-68928 COMPRESSIBILITY EFFECTS ON THE TWO GENERATIONS OF UNSTEADY FLOW TYPES IN AXIAL FLOW COMPRESSORS Xinqian Zheng, Sheng Zhou, Anping Hou, Jinsong Xiong School of Jet Propulsion Beijing University of Aeronautics and Astronautics Beijing, 100083, P. R. China ABSTRACT There occurred unsteady separated flows inside axial flow compressors, which was however not taken into consideration in the present aerodynamic design system. This discrepancy indicates that the potential underlying unsteady separated flows is yet to be explored, hence the present research team proposes the concept of two generations of unsteady flow types, i.e. Unsteady Natural Flow Type (UNFT) and Unsteady Cooperative Flow Type (UCFT). Numerical simulations are carried out in the present paper to study the compressibility effect on the unsteady cooperative flow type in axial flow compressors. The studies show that aerodynamic performances are remarkably enhanced by means of transforming the flow type from UNFT into UCFT by imposing unsteady excitations. In the case of 2D subsonic cascade, performances are greatly improved in a wide range of Ma number (Ma<0.8) and the maximum relative reduction of the loss coefficient reaches 40.2%. In the case of 2D trans-supersonic cascade, positive effects can’t be captured. However, in the case of a 3D trans-supersonic single rotor, the adiabatic efficiency is increased from 87.0% to 90.2%. NOMENCLATURE A b Ca C a ( Y ,t ) velocity CL Cu d d′ f shed Lift coefficient Circumferential component of velocity Design point of UNFT Design point of UCFT Characteristic frequency of vortex shedding fe fe Excitation frequency Relative excitation frequency f e = f e f shed i k Incidence Load/loss Ratio k = L ω Relative variation rate of k δk δk = ( k ex − k un ) k un × 100% Lu Work Lu = U (C u 2 − C u 1 ) Load coefficient L = Lu U 2 Mach number Mass flow rate Static pressure (averaged by mass rate) Total pressure (averaged by mass rate) Instantaneous total pressure Time Relative excitation amplitude Chord length Axial component of velocity Instantaneous axial component of velocity L Ma & m p p∗ p t T ∗ C a min Ca Minimum axial component of velocity Averaged axial velocity component of ∗ ( y ,t ) Total temperature 1 Copyright © 2005 by ASME U Rotating velocity Cartesian coordinate in axial direction （non-dimensioned by b ） x y Cartesian coordinate in circumferential direction （non-dimensioned by b ） Y Cartesian coordinate in circumferential direction （non-dimensioned by τ ） ω δω Loss coefficient ∗ ∗ ω = ( p1∗ − p2 ) ( p1 − p1 ) Relative variation rate of ω δω = (ω ex − ω un ) ω un × 100% τ π η Pitch ∗ ∗ Total pressure ratio π = p 2 p1 Adiabatic efficiency η = (π γ −1 γ − 1) ( α α s1 α s2 α s3 Subscripts T2∗ − 1) T1∗ Attack angle Attack angle at stall of the first generation aeronautic flow type Attack angle at stall of the second generation aeronautic flow type Attack angle at stall of the third generation aeronautic flow type Inlet of the computational domain Outlet of the computational domain The condition of no excitation The condition of excitation 1 2 un ex 1 INTRODUCTION Real flows inside axial flow compressors are a kind of unsteady vorticity flows over multi-bodies with relative movement. Since the stator blades are arranged in a stagger way with respect to the rotor blades, this kind of unsteady flows involves at least two cylindrical coordinate systems fixed respectively at the stator and the rotor and rotating around the same axis. As for turbo-fan engines used in modern airplanes, double axes are used and three cylindrical coordinate systems will be involved. The relative rotating of the rotors with respect to the stators leads to inherent unsteadiness. However, the aerodynamic design and analysis system developed in the past century is based on the basic assumption of steady flow [1,2]. Therefore, the current aerodynamic design and analysis system for axial flow compressors is on the verge of major changes, i.e. abolishing the restraint of steady flow assumption and transforming from steady system into unsteady system. Based on analyzing the contradiction between the real unsteady flows inside axial flow compressors and the current steady design system, the present research team put forth a concept of unsteady flow types, i.e. the Unsteady Nature Flow Type (UNFT) and the Unsteady Cooperative Flow Type (UCFT). Since the relevant governing equations don’t contain time variable, the current aerodynamic design and analysis systems for axial flow compressors couldn’t provide the unsteady time-space structure of the real flow. Hence, the actual flows inside axial flow compressors designed on the basis of steady flow assumption are principally in a chaotic state, and may well be classified as unsteady natural flow type in axial flow compressors. If external unsteady excitations can trigger basic changes in the time-space structure of the unsteady vorticity flow over the blade row under study, and transform it from chaotic to orderly, then the resulting flows will be classified as unsteady cooperative flow type. The unsteady aerodynamic design and analysis system of axial flow compressors should take it as its objective to realize unsteady cooperative flow type [3]. The concept of flow type was proposed by Prandtl to indicate a steady and well-behaved flow pattern. Küchemann pointed out that the aim of aerodynamic design of airplane is to realize certain flow type [4]. Up to now, three generations of aeronautic flow types have been proposed. The first generation of aeronautic flow type is steady attached- vortices flow type. It is characterized by the fact that separated flows should be avoided to the utmost except for a thin vortices sheet separated at the trailing edge, hence it applies only to the case of small angle of attack. The second generation of aeronautic flow type is steady unattached-vortices flow type, in which not all separated flows are considered harmful to the performances. Under proper control conditions, the concentrated vortices formed by the rolling-up of steady separated vortices can produce high lift on the wings, and hence only uncontrollable unsteady separation is considered harmful. Thus the angle of attack can be greatly enhanced and the lift coefficient be significantly increased. However, for further improving aerodynamic performances, the restraints of forbidden unsteady separation in the second generation flow type must be broken through, and this is the aim of the third generation of aeronautic flow type, which is called unsteady unattached-vortices flow type. Look back at the development of airplanes in the last hundred years from the viewpoint of aerodynamics, the first two generations of aeronautic flow types have played important roles. Now people are engaged in the studies on the third generation of aeronautic flow type, which is realized by means of unsteady excitations mainly. Lift coefficient curves corresponding to three generations of aeronautic flow types are shown in Fig. 1. Each transition in the flow type generation brought about an essential enhancement in airplane aerodynamic performances: essential delay of the stall incidence, which was delayed from α s1 to α s 2 and further to α s 3 , resulting in dramatic enhancement of the lift coefficient. The concept of two generations of unsteady flow types for axial flow compressors was proposed by referring to external flows over airplanes. The steady design system can’t take the flow-field time-space structure into consideration, and thus can’t but be satisfied with the resulting unsteady natural flow type, whereas the unsteady design system can improve the flow-field time-space structure by properly organizing the interactions 2 Copyright © 2005 by ASME between separate but interacting unsteady flows, resulting in unsteady cooperative flow type. The characteristics of pressure ratio corresponding to two generations of unsteady flow types are given in Fig. 2. As with the transition of aeronautic flow types, the transition of unsteady flow types aims at improving aerodynamic performances near the stall (large incidence), delaying the occurrence of stall, enlarging the stall margin and moving the design point to a higher load, i.e. from point d corresponding to unsteady natural flow type to point d ′ corresponding to unsteady cooperative flow type, rather than improving the general performances at the design point. αs1 αs2 αs3 α Fig. 1 Sketch of the lift coefficient curves corresponding to three generations of aeronautic flow types Total pressure ratio π d′ d U N FT U CFT Fig. 2 Sketch of the characteristics of pressure ratio corresponding to two generations of unsteady flow type in axial flow compressors Then, how to transform the flow type from UNFT to UCFT inside axial flow compressors? The alternative arrangement of stator blade rows and rotor blade rows form multiple bodies with relative motions. Even under design working conditions, unsteady separated flows are inevitable, and present a unique time-space structure. Unsteady flow over each blade row in axial flow compressors exhibits the interaction between two main unsteady factors. The first factor is the time-space & (kg/s) M ass flo w ra te m structure of the unsteady separated flow over the blade row under study, i.e. a series of vortices with different sizes and frequencies competing and merging continuously, resulting in a chaotic state. The second factor is the excitations forced by surrounding relatively moving bodies and impacting on the unsteady separated flow over the blade row under study, such as the excitations caused by its upstream vorticity wakes. When the frequencies and amplitudes of the excitations forced by surrounding bodies are within a proper range so that the time-space structure of unsteady separated flow over the blade row under study is transformed from chaotic into orderly state, it can be said that the flow type has been transformed from unsteady Natural flow type into unsteady cooperative flow type. As to unsteady separated flows inside axial flow compressors, the influence of the wakes of upstream blade rows on the time-space structure of downstream blade rows is a specific example of the above-mentioned mechanism and it is very practical in applications. It is proposed in the present paper to utilize the upstream wakes as “unsteady excitations” to control the unsteady separated flows over downstream blade rows and generate the unsteady cooperative flow type. For brevity’s sake, the unsteady excitations of upstream wakes will be shortened to “unsteady excitation”, the frequency with which the upstream wake passes the downstream row is called “unsteady excitation frequency” and the defect peak of the upstream wake is called “unsteady excitation amplitude”, when the flow-fields are not influenced by the upstream wakes, the case will be described as ‘unexcited’. The experiments were carried out under subsonic conditions: the free stream Ma number is 0.5 in the planar cascade experiments [5], the free stream Ma number is 0.07 in the stationary annular cascade experiments [6] and the relative Ma number at inlet is 0.2 in the experiments on single-stage low speed axial flow compressor [7]. These low speed experiments have their importance, however, the free stream Ma number of aeronautic axial flow compressors varies in a wide range, in particular the relative Ma number at the entrance of fans has reached 1.7 or more. Thus there emerges a question whether the unsteady cooperative flow type can improve the performances of axial flow compressors at high Ma number? What is the functional relationship between the potential of improving performances and the magnitude of Ma number? Compressibility effect impacts directly the applications of unsteady cooperative flow type to aeronautic engines and is an important fundamental problem that needs study and must be resolved. Therefore, the present paper will, based on previous studies, focus on the compressibility effect on unsteady cooperative flow type in axial flow compressors. 2 SUBSONIC CASCADES 2.1 Numerical Method Two-dimensional orthogonal coordinate system fixed on the cascade is adopted, in which the unsteady compressible Reynolds-averaged N-S equations can be expressed in non-dimensional form as Lift coefficient CL ∂Q ∂E ∂F + + =0 ∂x ∂y ∂t (1) 3 Copyright © 2005 by ASME and a third-order TVD Runge-Kutta time-marching scheme was used for time integration, because time accuracy was critical for this simulation. In order to obtain precise time dependent solutions, in particular to obtain real time-space structures of unsteady separated flows, we must adopt a space discretization format with high precision. The sixth-order compact finite difference scheme [8] is used for viscous fluxes, and the fifth-order accurate generalized compact scheme [9] is adopted for in-viscous fluxes, which is in conformity with “the principle of restraining fluctuations” and “the principle of stability”, so that virtual and unreal numerical fluctuations can be effectively restrained. Typical numerical experiments show that this format can effectively solve Euler Equations and N-S Equations. Besides, B-L model is adopted as turbulence model. H-type grid is adopted, which is generated by solving relevant elliptic differential equation. The grid influences are analyzed to make sure that the computation results are independent of the grid adopted. The influences of different factors are analyzed, such as the forward and backward extensions of the computational domain, the number of grid points and the local densification of grid points. After overall consideration of compromise between computation work and computation precision, the grid adopted in the present paper is as follows. The computational domain covers six chord lengths, one chord length before the cascade and 4 chord lengths behind the cascade; the number of grid points is 321 (axial direction)× 65 (circumferential direction). For improving computation precision, grid points are locally densified near the leading and trailing edges and the cascade walls, in particular within boundary layers, the first level grid points above the wall are located at y+≤10. The problem how to select initial values has also been studied. Commercial software NUMECA is first used to obtain a converged steady solution, which is then taken as the initial flow-field for our unsteady simulation, thus the convergence of the unsteady simulation can be greatly accelerated and computation time be saved. For different initial flow-fields, the unsteady simulations can invariably lead to periodic or quasi-periodic converged solutions, and the influences on the time-averaged performances are also small. The influence of the initial values exhibits chiefly in the converging speed. Data are recorded only when the computation is converged to a periodic or quasi-periodic solution. Because for this kind of massively separated and unsteady flow the initial transient stage is quite random and has a profound effect on the detailed fluctuating history of the flow, it is impractical to reach an exact grid and initial value independence. Overall inaccuracy by grid and initial value is lower than 0.5%. Total pressure of upstream stator will be defective due to the effects of viscous dissipation, so the spatial distribution of total pressure is non-uniform, which leads to inherent flow unsteadies at inlet boundary of rotor under consideration just because the adjacent blade rows are moving relatively. So, in order to represent the effect of upstream wake, total pressure fluctuations is taken as the proper variable to simulate the inherent unsteadiness, which is defined as: ∗ p (∗y ,t ) = p1 {1 + A sin[2π (Y + f e t )]} ∗ in which p1 is the averaged total pressure at entrance; A is the relative peak of wake defect (unsteady excitation amplitude), A ＝0 corresponding to a uniform steady boundary condition, i.e. the flow field is not influenced by upstream wakes; f e is the frequency of the wake passing (unsteady excitation frequency); Y is cartesian coordinate in circumferential direction （non-dimensioned by pitch） , Y ∈ [0,1] and it equals zero at the suction side and equals 1 at the pressure side. Boundary conditions are given at the inlet and outlet boundaries, in which total temperature, total pressure and incidence are given at the inlet, whereas only static pressure was fixed at the outlet while velocity and density can be derived by extrapolation. Non-slip, impermeability and adiabatic boundary conditions are imposed on the profile surface. Periodic conditions are imposed on the boundaries of the extended domains before and behind the cascade. 2.2 Influences of Excitation Frequency, Excitation Amplitude and Incidence There are four parameters the influences of which are most obvious: (1) Relative excitation frequency f e ( f e = f e f shed , f shed is the characteristic frequency of vortex shedding from the trailing edge [10]); (2) Unsteady excitation amplitude A ; (3) Incidence i; (4) Ma number (the compressibility effect). In order to isolate the compressibility effect, the other parameters need be fixed. At first, the influences of the first three parameters are analyzed whereas Ma number is fixed at 0.5. Fig. 3 gives the influence of relative excitation frequency f e . The dashed-line denotes the performance without excitation, which is taken as the baseline (same for all other figures). It can be seen from Fig. 3 that when A ＝15%, the averaged performances of the flow-field have been significantly improved in a wide range of frequency (0.3< f e <2.5). When the frequency of the wake passing equals the frequency of vortex shedding from the trailing edge, i.e. f e ＝1, the loss coefficient ω is reduced by 33% and the work/loss ratio is enhanced by 71%. When f e >2.5, the unsteady excitation effects aren’t obvious and the performances are nearly the same as unexcited ones. Hence, it can be concluded that performances can be improved, but not for any excited frequencies. At A ＝10%, the positive effects of unsteady excitations are reduced for all values of f e . The influences of the unsteady excitation amplitude on the averaged performances are given in Fig. 4, from which it can be concluded that the larger the unsteady excitation amplitude, the better the averaged performances, exhibiting a monotonic relationship. Modern advanced axial flow compressors are designed to move the working point toward a higher load, resulting in higher wake defects, thus the applications of two generations of unsteady flow types to enhance their performances are quite promising. However, whether unsteady excitation can lead to positive effects under all incidences? The range of incidences with positive effects is roughly shown in Fig. 5, from which it can be seen that the unsteady excitation can play a prominent positive role mainly in the cases of large incidence. (2) 4 Copyright © 2005 by ASME Now that positive effects of unsteady excitations are obtained, what about the improvement of the corresponding time-space structure? The comparison of the instantaneous vorticity field between excited and unexcited cases is given in Fig. 6. It can be seen that the unsteady excitations effectively restrain boundary layer separation, reduce the separation zone (blue color) and delay the separation point and weaken the vorticity intensity of trailing edge vortexes (red color). The stream-wise distribution of loss coefficient is shown in Fig. 7. The reduction of the loss coefficient is brought about chiefly by the improvement of the time-space structure of flow-fields in cascade passages. Controlling the unsteady separated flow by means of unsteady excitation, the key is the influence of excitation on the time-space structure of unsteady separated flows. Before forcing excitation, unsteady separated flow is a highly nonlinear multi-frequency system and exhibits characteristics of randomness, broad spectrum and chaos. If the time-averaged performances are significantly enhanced by unsteady excitations, it means the time-space structure of flow-field is essentially improved. In this case, small modes are merged into larger modes and the mode frequencies are reduced in number. This kind of mode-merging process is progressing stepwise; hence the frequency response to the external excitation has a broad spectrum. As a result, the dominant mode of the separated flow is locked to the dominant mode and harmonic modes of unsteady excitations, most of random modes will be smoothed out and the original chaotic state changes towards coherent and more orderly [11]. It can be concluded from the above analyses that reasonable arrangement of the upstream wakes relative to the unsteady separated flows over the downstream blade rows can enhance the averaged aerodynamic performances, effectively restrain boundary layer separation, turn the flow-field from chaotic into orderly and transform the flow from unsteady natural flow type (UNFT) to unsteady cooperative flow type (UCFT). 0.25 0.24 0.23 Loss coefficient ϖ 0.22 0.21 0.20 0.19 0.18 0.17 0.16 0.15 0 2 4 6 8 10 Unexcitation f e =1 12 14 (16 %) Relative excitation amplitude A Fig. 4 Influence of relative excitation amplitude, i =10° Ma =0.5 0.50 0.45 0.40 Loss coefficient ϖ 0.35 0.30 0.25 0.20 0.15 0.10 0 Unexcitation f e=1, A=15% 5 10 15 Incidence i ( ) Fig. 5 Comparison of the curve of loss coefficient vs incidence between excited and unexcited cases, f e =1， Ma =0.5 o 20 25 0.25 0.24 0.23 Loss coefficient ϖ 0.22 0.21 0.20 0.19 0.18 0.17 0.16 0.15 0 1 2 3 4 5 6 Unexcitation A =10% A =15% Relative excitation frequency f e Fig. 3 Influence of relative excitation frequency, i =10° Ma =0.5 (a) Unexcited case 5 Copyright © 2005 by ASME 0 -10 δϖ (%) -20 -30 -40 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (b) Excited case. f e =1， A ＝15% Fig. 6 Instantaneous vorticity field. i =10°, Ma =0.5 0.25 0.20 Ma Fig. 8 Influence of Ma number on the loss coefficient in the case of compressor cascade 100 Leading edge Trailing edge 80 Loss coefficient ϖ 0.15 60 0.10 0.05 0.00 -0.05 -1 0 1 2 3 4 5 Unexcitation A =15% δk (%) 40 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x Fig. 7 Streamwise distribution of loss coefficient, i =10° Ma =0.5 Ma Fig. 9 Influence of Ma number on the work/loss ratio in the case of compressor cascade 2.3 Compressibility Effects Detailed studies have been carried out in a wide range of Ma number at the inlet. The influences of the unsteady excitation frequency, excitation amplitude and incidence corresponding to several Ma numbers are obtained, and the resulting well-regulated variations are qualitatively consistent with the curves shown in Fig. 3~5 [12]. In order to study the compressibility effect, the other parameters are all chosen to be the optimized ones. The incidence and relative excitation frequency are chosen as i ＝ 10° and f e ＝ 1, respectively. The major part of the total pressure defect in the wake is brought about by velocity defect, and the proportion of the dynamic pressure in the total pressure increases with the Ma number. Therefore, the relative excitation amplitude (i.e. the peak of relative total pressure defect) A should be chosen to increase with Ma number. When Ma number increases from 0.2 to 0.5 and 0.8, A is chosen to be 1.5%, 15% and 25%, respectively. Positive effects of the unsteady excitations are shown in Fig. 8 and Fig. 9, from which it can be seen that not only remarkable positive effects can be obtained at low speed (e.g. δω ＝－40.2% at Ma=0.2), but the positive effects are also obvious even at relatively large Ma number: at Ma=0.7, the relative reduction of loss coefficient δω ＝－ 17% and the relative increment of work/loss ratio δk ＝50%. At Ma=0.8, although the positive effects can still be obtained, however they are greatly reduced. Detailed analyses revealed that local supersonic area occurred on the suction side. When Ma reaches 0.9, the local supersonic area expands drastically and nearly no positive effects can be obtained. The impacts of shock waves on the unsteady excitation effects will be analyzed in the next section. 3 SUPERSONIC CASCADES The code used in supersonic cases has been presented in section 1.1. It has high precision and high resolution. In particular, it can properly capture shock waves and thus is specifically suited to supersonic flow simulation. The profile to be simulated is chosen to be ATS-2, the blade tip profile of a 6 Copyright © 2005 by ASME high-pressure ratio single stage fan rotor with designed Ma number of 1.6. The grid points are 321 (axial) × 61 (circumferential). At the entrance boundary, the total temperature, total pressure and circumferential velocity component are given with the velocity obtained by extrapolation, whereas the unsteady excitations simulating the upstream wakes are given by equation (2). Numerical simulations of supersonic flows are carried out for Ma=1.2, 1.3, 1.4, 1.5, 1.6 and 1.7, and qualitatively consistent results are obtained. For brevity’s sake, only the results at Ma=1.6 are presented as a typical case. Unexcited entropy flow-field is displayed in Fig. 10. The boundary layer increases sharply after the shock, and a typical Karman vortex street can be seen downstream of the trailing edge. The unexcited flow-field structure is displayed in detail in Fig. 11, where only the region within the block as shown in Fig. 10 is magnified and shown. It can be seen from Fig. 11 (b) that a normal shock is clearly captured at the position of maximum thickness and two expansion waves are generated at the leading edge and extend separately to the pressure side and the suction side. The expansion wave above the pressure side is intersected with the normal shock in the passage and weakens the latter thereby, whereas the expansion wave above the suction side extends upstream to infinity. Positive effects of unsteady excitations can’t be obtained in the 2D cascade under supersonic condition, although great efforts have been made. The results mentioned in section 1 reveal that under subsonic condition the enhancement of aerodynamic performances is brought about by the improvement of the flow-field time-space structure (Fig. 6). Hence problem arises as what are the effects of unsteady excitations on the flow-field time-space structure under supersonic condition? The excited flow-field structure is shown in Fig. 12. The comparison of Fig. 11 and Fig. 12 clearly shows that the unsteady excitations have little effects on the structure of supersonic flow-field, and have no effects after the shock. The shock is supposed to block all the disturbances caused by the imposed unsteady excitations. Hence the authors take the privilege to speculate that the blocking effects of the shock are the main reason why no positive effects can be obtained in supersonic cases. (a) Entropy (b) Ma number (c) Vorticity Fig. 11 Flow-field in supersonic compressor cascade, unexcited (a) Entropy (b) Ma number (c) Vorticity Fig. 12 Flow-field in supersonic compressor cascade, excited 4 THREE-DIMENSIONAL TRANSONIC ROTORS It has been mentioned previously that within the scope of 2D compressor cascades, no positive effects can be captured in transonic and supersonic cases due to the shock blocking effects. What will be the results in the case of 3D trans-supersonic rotor? 3D trans-supersonic rotors are thus taken as our next object. Governing equations are 3D unsteady compressible Reynolds averaged N-S equations. Sixth-order symmetric difference scheme and sixth-order GVC scheme are adopted for non-viscous terms and time derivative respectively. The mixing length model is applied to treat turbulence. A 3D right-angle coordinate system fixed on the rotor is adopted, in which the non-dimensional N-S equations are as follows: ∂Q ∂ ∂ (FI + FV ) + ∂ (G I + GV ) = S ω (3) + (E I + EV ) + ∂t ∂x ∂y ∂z The rotor under study is a stage of a multi-stage high-pressure compressor. The computational domain covers six chord lengths, one chord length before the rotor and 4 chord Fig. 10 Entropy flow-field in supersonic compressor cascade 7 Copyright © 2005 by ASME lengths behind. AutoGrid in the commercial software NUMECA is used to generate computational grid points, which are 160 (40＋60＋60, axial)×50 (circumferential)×40 (radial), totaling to 320,000 points. Boundary conditions are given at the entrance and exit boundaries, in which total temperature and three velocity components are given at the entrance with the static pressure obtained by extrapolation, whereas only the static pressure at the blade tip is given at the exit with the distribution of static pressure derived by the equation of radial balance. Non-slip, non-penetrating and adiabatic boundary conditions are imposed on the solid surfaces, and periodic conditions are imposed on the boundaries of the extended domains before and behind the rotor. In the present simulation, the effects of the upstream wakes are represented by the defect of the unsteady axial velocity component at the entrance boundary. The defect is of the shape of a half sine wave as follows: 1.32 1.28 Total pressure ratio π 1.24 1.20 1.16 1.12 1.08 20 22 24 26 28 30 Steady Calculation With Numeca Time Averaged Unsteady Calculation C a (Y ,t ) = C a min + C a A sin[π (Y + f e t )] (4) in which C a (Y ,t ) , C a min and C a denote the instantaneous, minimum and averaged axial velocity components, respectively. Please refer to equation (2) for the definitions of other parameters. The axial velocity component is given by equation (4) with unsteady excitation, whereas the axial velocity component is given by averaged axial velocity C a of equation (4) without excitation. It is very important and complicated to calibrate code. For validating the present code’s reliability, the well-known NASA67# is taken as the rotor to carry out the calibration computation. NASA67# is design by NASA Glenn as the first stage of a double stage fan, the pressure ratio is designed to be 1.629 and the relative Ma number at the blade tip is 1.38. In the calibration, the computational results given by commercial software NUMECA are taken as the benchmark, which are compared with the time-averaged results given by the present code with uniform boundary conditions imposed at the entrance. The comparison is shown in Fig. 13, from which it can be concluded that the efficiency of the present unsteady calculation is about 1% lower than that of the steady calculation, and the time-averaged total pressure ratio is intersected with the steady one, however the difference is somewhat of the same order of magnitude as the grid error. 0.92 0.88 0.84 0.80 0.76 0.72 0.68 0.64 20 22 24 26 28 30 Steady Calculation With Numeca Time Averaged Unsteady Calculation (b) Total pressure ratio π Fig. 13 Comparison of the present results with the steady calculations given by NUMECA The total pressure ratio and adiabatic efficiency are given in Fig. 14 for unexcited case as well as excited case, in which relative excitation frequency f e ＝ 1 (the characteristic frequency of vortex shedding from the trailing edge f shed is 24686Hz). It can be seen that the excited performances are remarkably improved in comparison with the unexcited ones, and the positive effects increase with the unsteady excitation amplitude. The maximum increments of aerodynamic performances are as follows. The adiabatic efficiency η increases from 87.0% to 90.2% and the total pressure ratio π increases from 1.324 to 1.34. The larger the unsteady excitation amplitude, the more remarkable the excitation effects, which is qualitatively consistent with the results of 2D subsonic cascade (see fig. 4). Generally, the positive effects are more prominent in the cases of small mass flow rate, which is qualitatively consistent with the results of 2D subsonic cascade (see fig. 5) and the experiment results of low speed axial-flow compressor (see fig. 15) [7]. Thus, unsteady cooperative flow type realized by means of unsteady excitations is able to move the design point toward a higher load (see fig. 2). Why positive effects are not captured in 2D trans-supersonic cascade, whereas they are obtained in 3D trans-supersonic rotor? Since both 2D cascade and 3D rotor are simplified models of real multi-stage axial compressors, different models correspond to physically and mathematically different deterministic problems, leading to great differences between the time-space structure of 2D cascade and that of 3D rotor. In the case of 2D supersonic cascade, the shock penetrates the whole passage and the unsteady excitations are blocked by the shock and can’t influence the separated flow behind the shock. As to the 3D trans-supersonic rotor, although there is also a shock, however it can’t penetrate the whole passage. From the blade root via middle part to blade tip, the flow becomes subsonic, transonic and supersonic consecutively. Hence, the disturbances caused by unsteady excitations can be transmitted to and thus influence the separated flow, at least, part of the separated flow, resulting in enhanced aerodynamic performances. & (kg/s) Mass flow rate m Adiabatic efficiency η (a) Efficiency η & (kg/s) Mass flow rate m 8 Copyright © 2005 by ASME 0.90 Adiabatic efficiency η 0.89 0.88 0.87 0.86 0.85 0.84 24 25 26 27 Unexcitation A =08% A =16% A =24% A =32% from 87.0% to 90.2%, and the larger the unsteady excitation amplitude, the more remarkable the excitation effects. 4) The studies on both 2D subsonic cascade and 3D trans-supersonic rotor show that the realization of unsteady cooperative flow type aims mainly at improving aerodynamic performances near the stall (large incidence), delaying the occurrence of stall. Thus, the design point could be moved to a higher load. ACKNOWLEDGMENTS This work was supported by a grant from the National Natural Science Foundation of China (No. 10477002) and from the Ph.D. Innovative Foundation by Beijing University of Aeronautics and Astronautics. REFERENCES [1] Johnson, I., Bullock, R. eds., 1965, “Aerodynamic Design of Axial Flow Compressors”, NASA SP-36. [2] Adamczyk, J. J., 1999, “Aerodynamic of Multistage Turbomachinery Flow in Support of Aerodynamic Design,” ASME 99-GT-80. [3] Zhou, S., Hou, A. P., Gong, Z. Q., Lu, Y. J., Ge, J. D., 2004, “On the Two Generations of Unsteady Flow Types in Axial Flow Compressors,” ACTA Aeronautics et Astronautics Sinica, 25(6). [4] Küchemann, D., 1978, “The Aerodynamic Design of Aircraft,” Pergamon Press. [5] Hou, A. P., Jiang Z. L., Ling D. J., Zhou, S., 2005, “Experimental Research of Sound Excitation on Flow of Subsonic Compressor Blade Profile in Plan Cascade,” Journal of Beijing University of Aeronautics and Astronautics, 31(1). [6] Qiu, Y. X., Ge, J. D., Lu, Y. J., Zhou, S., and Li, Q. S., 2003, “Research on Sound-vortex Resonance in Enhancing Performance of an Annular Cascade,” ASME GT2003-38022. [7] Li, Z. P., Gong, Z. Q., Liu, Y., and Lu, Y. J., 2004, “The Experiment Research on the Performance of Low Speed Axial-Compressor by External Acoustic Excitation,” ASME GT2004-53183. [8] Lele, S. K., 1992, “Compact Finite Difference Schemes with Spectral-like Resolution,” J. of Computational Physics, 103, pp. 16-42. [9] Shen, M. Y., Niu, X. L., and Zhang, Z. B., 2001, “The Three-Point Fifth-Order Accurate Generalized Compact Scheme and Its Applications,” ACTA Mechanica Sinica, 17(2), pp. 142-150. [10] Zheng, X. Q., Zhou, X. B., and Zhou, S., 2004, “Investigation on a Type of Flow Control to Weaken Unsteady Separated Flows by Unsteady Excitation in Axial Flow Compressors,” ASME Journal of Turbomachinery (to be published). [11] Wu, J. Z., Vakill, A. D., and Wu, J, M., 1991, “Review of the Physics of Enhancing Vortex Lift by Unsteady Excitation,” Progress Aerospace Sci, 28, pp. 73-131. [12] Zheng, X. Q., and Zhou, S., 2004, “Impact of Wake on Downstream Adjacent Rotor in Low-speed Axial Compressor,” Journal of Thermal Science, 13(2), pp. 114-120. 28 & (kg/s) Mass flow rate m (a) Efficiency η 1.34 1.32 1.30 1.28 1.26 1.24 1.22 24 25 26 27 28 Total pressure ratio π Unexcitation A =08% A =16% A =24% A =32% (b) Total pressure ratio π Fig. 14 Comparison of the performances of 3D rotor before and after being excited, f e =1 1.0175 Mass flow rate m & (kg/s) Total pressure ratio π 1.0170 Unexcitation excitation 1.0165 1.0160 1.0155 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 & (kg/s) Mass flow rate m Fig. 15 comparison of the performances of low speed axial flow compressor experiments with and without excitation [7]. 5 CONCLUSIONS 1) Enhancement of cascade performances by unsteady excitations is effective not only at low speed, but also at high subsonic speed (Ma<0.8), which makes the application of unsteady cooperative flow type to axial flow compressors highly promising. 2) When there occurs a local supersonic area near the leading edge of cascade, the positive effects of unsteady excitations are dramatically reduced. They can’t be captured in the case of 2D supersonic cascade because of the shock blocking effect. 3) The studies on 3D trans-supersonic single stage rotor show that unsteady excitations can increase adiabatic efficiency 9 Copyright © 2005 by ASME