Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 51, 3, pp. 523-531, Warsaw 2013 THE OFF-DESIGN PERFORMANCE PREDICTION OF AXIAL COMPRESSOR BASED ON A 2D APPROACH X.C. ZHu, J.F. Hu, H. Ou-Yang, J. Tian, X.Q. Qiang, Z.H. Du Shanghai JiaoTong University, School of Mechanical Engineering, Shanghai, China e-mail: zhxc@sjtu.edu.cn The two-dimensional compressor flow simulation approach has always been a very valuable tool in compressor preliminary design studies, as well as performance predictions. In this context, a general development of the streamline curvature (SLC)method is elucidated fir- stly. Then a numerical method based on SLC is developed to simulate the internal flow of the compressor according to the development analysis and conclusion. Two certain trans- onic axial compressors are calculated by this 2D method. The speed lines and span-wise aerodynamic parameters are compared with the experiment data in order to demonstrate the method presented in this paper. Key words: two-dimensional, compressor, streamline curvature, deviation, loss model Nomenclature c – blade chord C,V – absolute and relative velocity, respectively CD – drag coefficient D – diffusion factor G – mass flow I – relative stagnation enthalpy S – entropy M – Mach number P,T – static pressure and static temperature, respectively i,k,β,ϕ,φ,θ – incidence, blade, relative flow, deflection, sweep and camber angle (inclu- ding circumferential direction), respectively m,n – direction of meridional and of computation station z,r – axial and radial (including radius) direction, respectively γ – specific heats ratio σ – solidity ρ – density ω – rotational speed Θ – momentum thickness Subscripts: 1, 2 – upstream and downstream of blade row, respectively. 1. Introduction In the last two decades, with the advances of computer resources and computational fluid dyna- mics (CFD), 3D viscous based on unsteady Reynolds-average Navier-Stokes (RANS) has been widely applied to simulation and analysis of the compression systems.Denton andDawes (1999) suggested that “little has changed” on the streamline curvature approach because of the CFD 524 X.C. ZHu et al. development. The computational precision of SLCheavily depends on predictionmodels. Accor- ding to the history of SLC development, incidence and deviation have been changed less than the loss recently. A great deal of effort is used to improve or develop the loss model and shock loss, especially to increase accuracy for a variety of compressor cascades (Aungier, 2003; Boyer, 2003; Pachidis et al., 2006; Swan, 1961). In the present work, the general development of SLC is elucidated at first. According to analysis of it, one numericalmethod is set upwhich considersmain factors of SLC inmaximum. Thedeviation is setupbasedonthe referenceminimumincidenceandconsiders themain impacts at off-design points for a transonic compressor. The total pressure loss consists of profile loss, secondary loss and shock loss. Every component is calculated respectively at the design and off-design points. A certain transonic axial rotor is calculated and analysised by this 2Dmethod at first. Beside that, one stage compressor is also calculated. The speed lines and span-wise aerodynamic parameters are compared with the experimental data. The result validates that the method presented in this paper is correct and applicable. 2. Development of SLC The SLCmethod basically solves the discrete equation of full radial equilibrium equation incor- poratingmanymodels and equations such as deviation and the loss model, state and continuity equations, etc. on computational grids which are constructed in the meridian plane through streamlines and stations (see Fig. 1). The governing equations are derived from the well-known Euler equations ∂Vm ∂n = 1 Vm [∂I ∂n −T ∂S ∂n − (Cθ r −ω )∂(rCθ) ∂n ] +Vmcos(φ−ϕ) ∂φ ∂m − sin(φ−ϕ) 1−M2m Vm [ (1−M2θ) sinφ r − tan(φ−ϕ) ∂φ ∂m + 1 cos(φ−ϕ) ∂φ ∂n ] G= T ∫ H 2πrρVmcos(φ−ϕ) dn (2.1) Fig. 1. The computational grids in the meridian plane The simulation accuracy of SLCheavily depends onpredictionmodels such asminimum loss, incidence angle, deviation angle, total pressure loss and blockage models etc.With the develop- ment of experimental equipments and methods, many model modifications or new correlations have appeared to accommodatemodern compressors. Butmost of them are still developed from low-speed correlations of the 1950s. NASASP36 low-speed “reference condition” correlations presented byLieblein (1957, 1959), Lieblein and Roudebush (1956) in the 1950s have been widely used in the deviation and loss The off-design performance prediction ... 525 prediction (see Fig. 2). Creveling andCarmody (1968), Cetin et al. (1987) recommended aseries ofmodifications for deviation calculation accounting for transonic, 3D effects and off-design con- dition. Swan (1961), Koch and Smith (1976) developed improved profile loss predictionmethods according to Lieblein’s work then. Considering additional losses caused by secondary flow,Koch and Smith (1976), Hearsey (1994), Aungier (2003) etc. developed many empirical and semi- -empiricalmodels andmodifications. Especially, Hearsey’s secondary lossmodel was established on the basis of the profile loss with different distribution along the blade span. Fig. 2. Illustration of the reference condition The shock loss is one indispensable part of the total loss at high Mach number in modern compression systems.Miller’s normal shock loss (Miller etal, 1961)model has been very popular in shock loss calculation because of its simple algorithm. But the shock loss predicted by it is always larger than in fact at a high Mach number. According to passage shock geometries at different operating conditions in supersonic compressor cascades, recent works by König et al. (1994,a,b), Bloch et al. (1999) andBoyer (2003) demonstrated that the efficiency characteristic of transonicmachines is largely determinedby the shock loss. Someshock loss prediction algorithms of arbitrary shape cascades over the entire operating range were then proposed. More recently, Templalexis et al. (2006) and Pachidis et al. (2006) published several zooming research efforts using 2D SLC component models. According to the analysis above, the development of SLC ismainly based on itsmodels. The incidence is divided into two forms to a certain extent. One is the reference incidence, another is the minimum incidence. The deviation is mostly set up according to the reference incidence, and develops more slowly than loss. There are two main methods for loss calculation: a) the total loss at the design condition is calculated firstly, and loss at off-design is then calculated based on it, Mach number and incidence, b) the partial losses, such as profile, shock etc. are calculated firstly at every operating condition. The sumof them is the total loss. For calculation of the loss at the off-design condition, there are two reference criterions, which are based on the reference incidence (low-speedminimum incidence) and theminimum incidence. The recent representations are Swan, Pachidis, Boyer, Aungier’s works. 3. Individual work 3.1. Numerical models 3.1.1. Incidence angle In general, the SLC approach proceeds around one reference condition. Hence, the reference incidence appears and is used as the reference parameter. The other prediction models are established referringto it.Thereference incidence isbasedon low-speed, twodimensional cascade 526 X.C. ZHu et al. data from NASA SP36 data or curves. Recently, Milan has introduced one modification which can be used to revise it. The correlations of flow angle and profile geometry parameters can be seen in Fig. 3. Themain equations are given by i=β1−k1 iref =(Ki)sh(Ki)t(i0)10+nθ+Ci (3.1) where (i0)10 represents the zero-camber, minimum loss incidence for 10% thickness NACA 65 series. The constant (Ki)sh and (Ki)t are thickness and thickness distribution correction factors for particular blade profiles. For example, (Ki)sh is equal to 0.7 for Multiple-circular-Arc type, 1 for NACA 65 series, and 1.1 for C4 series blades. In the present work, (Ki)t is calculated as a function of cascade thickness, n is the slope of the variation in incidence with the camber angle θ and Ci is the revision based onMilan et al. (2009). Fig. 3. The correlations between cascade and aerodynamics parameters 3.1.2. Deviation angle It is generally accepted that blade deviation is a crucial parameter for the accurate off- -design compressor performance simulation. In the relative frame of referencethe deviation angle is defined by β2 = δ+k2 (3.2) where k2 is the outlet blade angle. For a transonic compressor, there are some factors which should be considered in deviation prediction at off-design conditions. According to the previous works, the deviation is calculated as the following formula in the present work. This approach is based on the reference deviation angle and consideration of the effects of various real-flow phenomena as separating individual deviation sources δ= δref + δi+ δVA+ δ3D (3.3) where δref is the reference deviation when the incidence angle equals the reference incidence, which is determined using Carter’s rule with themodification recommended by Cetin, δi is the deviation when the actual incidence angle is different from the reference. It is determined from a four-piece curve using NACA-65 series cascade data as its basis (Creveling and Carmody). δVA is the expression of the deviation due to axial velocity ratio and δ3D is the deviation due to the three dimensional effects. It is calculated based on the NASA SP36 curves according to Boyer’s method. 3.1.3. Total pressure loss model Modern transonic compressors have a complex three-dimensional flow field, and thus the computation of the total loss coefficient remains difficult. Lieblein (1957), Swan (1961), Koch The off-design performance prediction ... 527 and Smith (1976) et al. proposed many correlations for simulation of the loss. According to their works, the loss model of present work is broken down into three categories as the profile, shock, and secondary, referring to Swan, Boyer, Vassilios Pacthe hidis and Aungie’s methods. The following equation identifies the components used in determination of the total pressure loss coefficient ̟= fRe̟pro+̟sec+̟M ̟prof =2fReσ (Θ c )cos2β1 cos3β2 ̟sec = fsecσ (cosβ1 cosβ2 )2 CD ̟M = 1− [ (γ+1)M2 B (γ−1)M2 B +2 ] γ γ−1 [ γ+1 2γM2 B −(γ−1) ] 1 γ−1 1− [ 1+ γ−1 2 M21 ] −k k−1 (3.4) and 0.02¬Re ·10−5 ¬ 0.76 fRe =−8.735294Re ·10 −5+8.66888 0.76