CIEPLNE MASZYNY PRZEPLYWOWE No. 115 TURBOMACHINERY 1999

Transcription

1 CIEPLNE MASZYNY PRZEPLYWOWE No. 115 TURBOMACHINERY 1999 Sergey V.YERSHOV and Andrey V.RUSANOV Institute of Mechanical Engineering Problems of NAS of Ukraine NUMERICAL METHOD AND CODE FlowER FOR CALCULATION OF 3D VISCOUS UNSTEADY FLOWS WITHIN AXIAL AND RADIAL TURBOMACHINES The numerical method of the 3D compressible unsteady viscous flow through axial and radial turbomachine is developed. The approach is based on numerical integration of the Reynolds-averaged Navier-Stokes equations with implicit highresolution ENO scheme. To demonstrate the power of the method the following results are presented: 3D flows through straight test cascades; 3D viscous flows in turbine and compressor (axial and radial) stages; unsteady wake propagation through blade-to-blade passages of downstream row; unsteady blade cooling effects. 1. INTRODUCTION At present numerical methods of calculation of 3D viscous flows within turbomachinery cascades are widely used with aim of design (Dawes, 1997). In the first place due to application of these methods the leading manufacturers reduce the advanced technology of three-dimensional blade shaping to practice. Therefore the further increase of flow machine efficiency is largely defined by the power of numerical methods to simulate real viscous flows through blade cascades. Thus it is especially important to develop such approaches that are applicable to different turbomachine types and take into account main flow peculiarities acting on efficiency and reliability. Present paper represents the brief description of numerical technique realised as code FlowER (Yershov and Rusanov, 1996) that is intended for numerical calculations of 3D viscous steady and unsteady flows through multistage axial and radial turbomachines. The new features of code are described. The numerical results are presented. 2. NUMERICAL TECHNIQUE 2.1. Governing equations The flow through multi stage turbomachine is described by Reynolds-averaged Navier-Stokes equations written for local curvilinear co-ordinate system ( ξ, η, ζ ) : Q J E F G E J H v Fv Gv = + + +, (1) t ξ η ζ ξ η ζ

2 Sergey V.Yershov and Andrey V.Rusanov where Q, E, F, G are the vectors of conservative variables and inviscid fluxes; H is the source term vector; E v, F v, G v are the vector of viscous terms; J is the Jakobian of coordinate transformation. The turbulent effects are simulated with the algebraic turbulence model developed by Baldwin and Lomax (1978). The model modified for calculation of 3D cascade flows by Yershov and Rusanov (1997) Boundary conditions At the permeable inlet boundary of a computational domain the total pressure, total temperature and flow angle distributions are specified. At the exit permeable boundary the static pressure distribution is imposed. The temperature or the heat flux as well as the non-slip condition are enforced on the solid walls. The wall pressure can be defined from the relations obtained from governing equations (1) (Yershov and Rusanov, 1998). The generalised time-space periodicity condition is used for calculation of unsteady flows. For steady flow calculations the circumferentially averaged flow parameters (with the conservation of mass, momentum and energy) at the exit of current blade row are fixed as the inlet conditions for downstream one. In the contrary the circumferentially averaged flow parameters at the inlet of current row are specified as the outlet conditions for upstream one. To ensure the well-posedness of inter-row mixing conditions they should be realised in the terms of the 1D non-reflecting conditions Difference scheme The equations (1) are numerically solved with implicit second/third order accurate ENO scheme developed by Yershov (199). The scheme presents the development of well-known Godunov's scheme (Godunov et al., 1976) and ensures the accurate and reliable solutions for both low subsonic speed and Mach number greater than 2. Enhanced scheme stability and the second-order solution at least for the smooth region are enforced by use of ENO approximation of Harten and Osher (1987). Numerical efficiency of the scheme is achieved owing to simplified multi grid algorithm and local time stepping (for steady calculations only). The numerical approach is described in detail by Rusanov and Yershov (1997), Yershov and Rusanov (1998), Yershov et al. (1998). 3. NUMERICAL RESULTS The code FlowER has been subjected to the detailed testing. It includes: the checking of its consistence with different operation systems; the solution of 1D, 2D and 3D model problems and comparison with the exact and numerical solutions as well as experimental data of other authors; the calculations of the flow through turbomachine cascades. The results of this numerical tests are represented in detail in previous papers (Yershov and Rusanov, 1998, Yershov et al., 1998). One can learn from the following papers (Petiel'chits et al., 1977, Gardzilewicz et al., 1997, Lampart et al., 1999, Lampart et al., 1999) about applications of package FlowER fulfilled by different firms. In present paper the emphasis is mainly put on the new features of code FlowER. Particularly

3 Code FlowER for calculation of 3D viscous flow in axial and radial turbomachines we consider the computation of 3D viscous flow within centrifugal compressor and through turbine with wheelspace leakages and extractions. Also we present the results of calculation of unsteady flows in turbine stages D flow through straight test cascades The test calculation has been performed for the 3D viscous flow through the straight cascade investigated by Langston et al. (1977). The calculated pressure contours at the endwalls are displayed in Figure 1 (left). The corresponding experimental endwall flow (Langston et al., 1977) is given in Figure 1 (right). As it can be seen the numerical results are in a good quantitative and qualitative agreement with experimental data. The vortices and secondary flows locations obtained numerically (Yershov and Rusanov, 1978) correspond to Fig. 1. Endwall flow for Langston cascade the physical ideas of the 3D viscous cascade flow D flow in axial turbine stage The calculations have been carried out for 3D viscous flow in high-pressure axial turbine stage. The geometric and gasdynamic data of the stage are presented in Table 1 (stage 1). The stage scheme is given in Figure 2. The computational grid consists of about cells for each blade row. Table 1. The characteristics of the turbine stages under consideration Parameter Stage 1 Stage 2 Stage 3 Stator Rotor Stator Rotor Stator Rotor Midspan blade chord c, m Aspect ratio h/c Pitch chord ratio s/c Midspan inlet angle α1 ( β1 ), deg Midspan exit angle α 2 ( β2 ), deg Hub radius r h, m Blade number n Inlet total pressure, Pa Inlet total temperature, K Exit static pressure, Pa Rotational speed, s Stator surface temperature, K adiabatic adiabatic 700 The two types of flow are considered: the first one is that without leakages through endwall sealings and wheel-spaces (i.e. with constant mass flow rate) and the second one is that with such leakages. The distribution of mass flow rate along stage axe is

4 Sergey V.Yershov and Andrey V.Rusanov displayed in Figure 3. The calculated values are in error that not exceed 0.5 %. Thanks to leakages the mass flow rate through blade-to-blade channels decreases whereas it increases slightly in axial spaces. The entropy contours for midpitch section of flowpath are presented in Figure. The gas jets are clearly seen to enter the flowfield through the endwall orifices (see Figure 2 also). The jets propagate deeply into flow core and can influence cascade flow, particularly secondary flows. Fig. 2. Turbine stage with leakage through endwall sealings and wheel spaces Fig.3. The distribution of mass flow rate along stage flowpath length flow without leakages; flow with leakages Fig.. The entropy contours for flow with leakages So for the flow with leakages the jet inflowing near root before rotor blade leading edges causes the thickening of hub boundary layer. In this connection the strengthening of radial flow along rotor suction side occurs and root passage vortex grows. In the same time the peripheral secondary flows in rotor are significantly diminished owing to the flow extraction near shroud in the axial spacing (see Figure 2).

5 Code FlowER for calculation of 3D viscous flow in axial and radial turbomachines D viscous flows within centrifugal compressor stage The viscous flow through three-row centrifugal compressor is considered. The geometric and gasdynamic data for the computation are given in Table 2. The compressor comprises centrifugal wheel, radial vaned diffuser and converse radial vaned diffuser (Figure 5). The computational grid consists of about cells. To obtain in detail the flow pattern for centrifugal rotor the additional computation with grid consisting of about cells has been performed. The velocity vectors for mid-pitch meridianal section of flowpath are displayed in Figure 5. As it can be seen from the picture the flow pattern is complicated enough. The Table 2. The characteristics of the centrifugal compressor stage under consideration Parameter Rotor Stator I Stator II Leading edge radius, m Leading edge blade height, m ,0532 Trailing edge radius, m Trailing edge blade height, m , Blade number Inlet total pressure, Pa Inlet total temperature, K 288 Exit static pressure, Pa Rotational speed, s Mass flow rate, kg/s 27.8 Axial Mach number Fig. 5. The velocity vectors at the midpitch section of three-row centrifugal compressor

6 Sergey V.Yershov and Andrey V.Rusanov generation of separations and vortices is observed in the flow. It is possible to suppose that the geometry of hub and tip endwalls is far from optimal near channel bend where the massively separation origins. This separation influences flow far downstream. The flow in converse vaned diffuser leaves off being uniform and the stagnation zones appear around main flow jet. It is evident that such flow structure is characterized by high enegy losses. The authors have suggest the modification of channel shape that permits the increase of comppressor efficiency. The results of the modification and corresponding numerical investigations will be published in the nearest future. It is necessary to note that the secondary flows of centrifugal wheels are more complicated than those of axial one. In the first place it is evidenced by generation of passage vortices not only at tip and case endwalls but also at pressure and suction sides of blades. It is evoked by existence of both circumferential pressure gradient and spanning one in the rotor blade-to-blade passage. The entropy contours for different cross-flow sections of wheel vane channel are presented in Figure 6. One can estimate the location of vortices by the position of local maxima of entropy. The analysis of flow pattern permits us to descry the following main vortices: the branch of tip horseshoe vortex v1, the pressure side passage vortex v2, the tip passage vortex v3, the root passage vortex v, the suction side passage vortex v5. The other branches of root and tip horseshoe vortices do not seen in Figure 6 since they are merged in the passage vortices as well as through their small intensity and coarse picture scale. a b c d Fig. 6. Entropy function contours for cross-flow sections of rotor (per cents of chord): a 9%; b 3%; c 59%; d 86%.

7 Code FlowER for calculation of 3D viscous flow in axial and radial turbomachines The results obtained are in good agreement with basic physical ideas of 3D viscous flows through centrifugal turbomachines. 3.. Unsteady wake propagation; The geometric and gasdynamical data of the high-pressure transonic turbine stage are presented in Table (stage 2). The computational grid includes about 120,000 cells for both rows. The solution has converged to the periodically unsteady one after 8 temporal periods of rotor flow. The unsteady stage performances are known to be influenced sufficiently by the interaction between stator wakes and rotor blades. This interaction manifests itself in the fact that the wakes are split by the rotor blades and the wake fragments generated are transported by main flow through the vane channel to the stage exit. Therefore the accurate simulation of the wake transport processes is one of the serious aspects of the method validation. for the midspan rotor cascade section at four time instants during flow period. It is clearly seen that the main effects of wake transport through the rotor cascade including the distortion and the dispersion are simulated physically correctly by suggested method. Figure 7 shows the contours of the entropy function p ρ γ (a) (b) (c) (d) Fig. 7 - The time history of wake transport through rotor cascade (a) t = 1 T ; (b) t = 1 T 2 ; (c) t = 3 T ; (d) t = T 3.5. Blade cooling effects The geometric and gasdynamical data of the medium-pressure subsonic turbine stage tested by Levina et al. (1979) are presented in Table (stage 3). The computational grid includes about 123,000 cells for both rows. The solution has converged to the periodically unsteady one after seven temporal periods of stage flow. One of the interesting and scanty investigated phenomena of the real flow in a stage with cooled stator blades is temperature segregation, i.e. the overcooling of the rotor suction side and the deficient cooling of the pressure one. Evidently such effect in principle can not be simulated without regard to unsteadiness, viscosity and heat

8 Sergey V.Yershov and Andrey V.Rusanov transfer. The temperature contours near the root section of the rotor cascade are presented in Figure 8 for four instants of time during flow period. The low temperature zones corresponding to the positions of split wake segments are clearly seen to transport from pressure rotor surface to the suction one. This process is caused by the deviation of the relative flow in the wakes and it is strengthened by the secondary flows of the same direction of the transport. Therefore at the root the low temperature gas clings to the rotor suction side and moves spanwise along blade. In a result near the root the low temperature zone may be formed. Thus Figure 9 (left) shows the temperature distributions along rotor blade chord at the 5% span section for four instants on the flow period. During the flow period the maximum difference in temperature between the pressure side and suction one reaches 80 K. For comparison the numerically obtained steady chordwise temperature distribution is displayed in Figure 9 (right). In this case the difference in blade surface temperature is underestimated and equals approximately 30 K. (a) (b) (c) (d) Fig. 8 - The time history of temperature wake transport through rotor cascade at the root section. Contour step is 10 K. Isoline No. 17 corresponds to 1070 K. (a) t = 1 T ; (b) t = 1 T 2 ; (c) t = 3 T ; (d) t = T steady flow unsteady flow Fig. 9 - The chordwise temperature distributions at the root section; suction side; pressure side; (a) t = 1 T ; (b) t = 1 T 2 ; (c) t = 3 T ; (d) t = T

9 Code FlowER for calculation of 3D viscous flow in axial and radial turbomachines. CONCLUSIONS The numerical method and application package named FlowER are developed for the calculation of 3D viscous flows within multi stage turbomachines. The testing of the code demonstrates its reliability and feasibility. The code FlowER is widely used now by a number of Ukrainian, Russian and Polish firms for design and modernisation of turbine and compressor stage. It has proved to be a convenient and friendly tool for numerical analysis of flow and for increase of turbomachine efficiency. 5. ACKNOWLEDGMENTS The authors wish to thank Dr.-Ing. V.N.Dovzhenko and Dr.-Ing. A.Gardzilewicz for their interest in this work and for many useful discussions. REFERENCES Baldwin, B.S., and Lomax, M., 1978, "Thin Layer Approximation And Algebraic Model For Separated Turbulent Flows", AIAA Paper No. 0257, 8 p. Dawes, W.N., 1997, "Current & Future Developments In Turbomachinery CFD", Proceedings, 2nd European Conference on turbomachinery, fluid dynamics and thermodynamics, Antwerpen, Belgium, 62 p. Gardzilewicz, A, Yershov, S., Rusanov, A., Lampart, P., Keitlinski, K., and Elszkowski, J., 1997, "Increasing The Efficiency Of Cylindrical Stages Of Impulse Turbines With The Help Of 3D Flow Computations", Proceedings, Modelling and Design in Fluid-Flow Machinery, J. Badur et al. eds., Wydawnictwo IMP PAN, pp Godunov, S.K., Zabrodin, A.V., Ivanov, M.Y., Krayko, A.N., and Prokopov, G.P., 1976, "Numerical Solution Of Multi Dimensional Problems Of Gasdynamics", Moscow, Main editorial office of physical and mathematical literature of "Nauka" Press, 00 p. (in Russian). Harten, A., and Osher, S., 1987, "Uniformly High-Order Accurate Nonoscillatory Schemes", SIAM Journal of Numerical Analysis, Vol. 2, No. 2, pp Lampart, P., and Gardzilewicz, A., 1999, "The Effect Of Endwall Shaping On The Characteristics Of Last Stages Of Steam Turbine", Proceedings, th International Symposium on Experimental and Computational Aerothermodynamics of Internal Flows, Dresden, to be appeared. Lampart, P., Gardzilewicz, A., Rusanov, A., and Yershov, S., 1999 "Radial Lean And Compound Lean of Stator Blades As Means Of Improving Flow Characteristics Of HP Turbine Stages", Proceedings, th International Symposium on Experimental and Computational Aerothermodynamics of Internal Flows, Dresden, to be appeared. Langston, L.S., Nice, M.L., and Hopper, R.M., 1977, "Three-Dimensional Flow Within A Turbine Blade Cascade", Transactions of ASME, Journal of Engineering Power, Vol. 99, pp Levina, M.E., Frolov, B.I., and Grebnev, V.K., 1979, "Turbine Stages Non-Susceptible To The Change Of Radial Gap", Power Machinery, No. 29, pp (in Russian).

10 Sergey V.Yershov and Andrey V.Rusanov Petiel'chits, V.V., Sharowskiy, M.A., Yershov, S.V., and Rusanov, A.V., 1997, "Numerical And Experimental Investigations Of Stage Of Axial Compressor GPA , Designed By Law Of Variant Head Distribution Along Height", Proceedings, The improvement of turboinstallation by methods of mathematical and physical modelling, Kharkov, Institute of Mechanical Engineering Problems, pp (in Russian). Rusanov, A.V., and Yershov, S.V.,1997, "Numerical Method For Calculation Of 3d Viscous Turbomachine Flow Taking Into Account Stator/Rotor Unsteady Interaction", Proceedings, th Colloquium on Process Simulation, A. Jokilaakso ed., June 1997, Espoo, Finland, pp Yershov, S.V., 199, "The Quasi-Monotone ENO Scheme Of Increased Accuracy For The Integrating Euler And Navier-Stokes Equations", Mathematical Simulation, Vol.6, No. 11, pp (in Russian). Yershov, S.V., and Rusanov, A.V., 1996, "The Application Program Package FlowER For The Calculation Of 3D Viscous Flows Through Multi Stage Turbomachines. Certificate Of State Registration Of Copyright", Ukrainian state agency of copyright and related rights, ΠΑ N 77, (in Ukrainian). Yershov, S.V., and Rusanov, A.V., 1997, "Modification Of Algebraic Turbulence Model Used In Code FlowER ", Modelowanie turbulencji w zastosowaniach technicznych. Zeszyty Naukowe IMPPAN 86/18/97, Gdansk, pp Yershov, S.V., and Rusanov, A.V., 1998, "A Numerical Simulation Of Turbulent Separated Flows Within 3D Cascades Using Implicit Godunov's Type ENO Scheme", Journal of Mechanical Engineering, Vol.1, No. 1, pp (in Russian). Yershov, S., Rusanov, A., Gardzilewicz, A., Lampart, P., and Swirydczuk, J., 1998, "Numerical Simulation Of 3D Flow In Axial Turbomachines", TASK Quarterly, Vol. 2, No. 2, pp