Category UNSTEADY AERODYNAMICS, AEROACOUSTICS AND AEROELASTICITY OF TURBOMACHINES

On Fig.4 changes on a stators pitch of total pressure upon mean radius be­fore R and static pressure upon the case behind R, measured in axial gaps at 1-st assembly of the compressor on the design operating mode at Uup = 125m/s, ф = 0,5. Thus, in the researched compressor R blades work in conditions of the disturbed ft>w on an input owing to wakes behind the IGV vanes and on an output owing to potential influence upwards on a ft>w of the S vanes. Dis­placement on a phase of these disturbances by mutual circumferential shift of the IGV and S (changes of parameter v), is the basic source of clocking effect.

Change of the overall parameters in 1-st assembly of the compressor in de­pendence on a mutual circumferential position of stators is presented on Fig.5. Change of efficiency makes 0,8 ^ 1,0% and this change, basically, is con­nected to change of a total pressure ratio. At other assemblies of the compres­sor change of efficiency on v did not exceed 0, 5% and is commensurable with a margin error its definitions during experiment. It gives the basis to consider 1-st assembly of the compressor with identical number vanes in stator rows and with identical axial gaps Д12 = Д23 = 0,15 the optimal on clocking effect on overall characteristics of the compressor.

The purposes of a series of the experiments, executed in work, are the deter­mination of inflience of axial gaps between rows and numbers of IGV vanes on unsteady fhw parameters of the R blades and an experimental estimation of inflrence of unsteadiness on losses in the compressor. Practical result of the analysis is definition of optimization criteria of mutual circumferential position of stators.

Axial gaps Д12 (between IGV and R) and Д23 (between R and S) at re­searches of the clocking effect are chosen, recognizing that according to Refs. [1,2,7,8,9,12], the source of the clocking effect in system of rows IGV-RK – S is change of flow in the R at mutual circumferential shift of the IGV and S. As criteria of an estimation of a unsteadiness level and clocking effect the generalized parameters were used:

(< Гя >t )v – averaged on mutual circumferential position of stators v (0 < v < 1) of RMS deviation on time t of velocity circulation on the R blade

Гя = rR (t);

<< Гя >t>v – RMS deviation < Гя >t from (< Гя >t)v for all mutual circumferential position of stators. Values (< Гя >t)v and << Гя >t>v were calculated under the theory of potential-vortical interaction for cascades on mean radius of the compressor at various combinations of axial gaps Д12 and Д23 in a wide range of their change from 5 up to 60mm (from 0,05 up to 0,62 from the R blade axial projection length).

By results of calculations 2 assembly of the compressor were chosen for research, providing essentially various levels of parameter << Г я >t >v at close values of parameter < Гя >t, i. e. essentially various levels of clocking effect at close values of fbw unsteadiness on the R blades:

1-st compressor assembly – Д12 = Д23 = 15mm (Д12 = Д23 = 0,15), << TR >t>v= 41%, (< TR >t)v = 4,2%; _

3-th compressor assembly – Д12 = 60mm, Д23 = 5mm (Д12 = 0,62, Д23 = 0, 05), « TR >t>v= 15%, (< TR >t)v = 4,3%;

Besides the specified configurations of the compressor, for which experi­ment was carried out at numbers of blades 36-38-36, 2-nd compressor as­sembly was executed at numbers of blades 18-38-36. By a calculated esti­mation reduction of numbers of the IGV vanes in 2 times results in increase of common ft>w unsteadiness on the R blades, however considerably reduces clocking effect.

Flow calculations were carried out on a design operation mode of the com­pressor for 2D-ft>w in system of cascades IGV-R-S on mean radius. From the measured radial distributions of stagnation pressures and temperatures on a path of the compressor, given on Fig.2, it is visible that fbw radial distortion is not great, and flow on the mean section on blade height is representative enough for the characteristic of flow in all compressor.

In the theory of potential-vortical interaction of cascades it is supposed that vortical wakes behind cascades in relative movement are given as universal velocity distribution in automodel area. In details specified theory is described in works [10,11] and used in [9].

At numerical fbw simulation averaged on Reynolds Navier-Stokes equa­tions were used, closed by two-parametrical (q, ш) model of turbulence. Fea­tures of the calculating scheme and construction of the grids are described in [13]. In the given work conditions of periodicity were imposed on the bottom and top borders of the calculating area, containing accordingly 18, 19, 18 air­foils in the IGV, R, S cascades at full simulation of conditions of experiment or 1, 1, 1 airfoils at the simplified flow simulation. In the latter case the gen­eral number of grid units has made 12700 at the minimal size of a cell at an airfoil, equal to 1,5 x 10-3mm. The step of integration on time was equaled T/3800, where period T = hR/u, hR is the R cascade pitch. Calculations were executed by V. G. Krupa.

The basic advantage of numerical simulation in comparison with the model of potential-vortical interaction consists in an opportunity of a direct estima­tion of influence of row unsteady interaction on time-averaged gasdynamical characteristics. From comparison of experimental and calculated circumferen­tial distributions of time-averaged stagnation pressures at the compressor exit for an optimum point (Uup = 125m/s, ф = 0, 5) on Fig.3 it is visible, that the calculated data are close to experimental.

It is necessary to note, that attempts of the simplified calculations by re­placement of the real relation of numbers of blades in the rows 183 19, 18 on simplified 1 1 1 resulted in essential deviations of calculated values from ex­perimental. Therefore all calculations, for which comparisons with experiment are presented, were received for real relation of numbers of blades.

The experimental information on a few structure in the compressor at var­ious circumferential positions of stator rows was obtained as follows. On a constant operation mode of the compressor on the given frequency of the R rotation and the air fbw rate by circumferential shift of IGV relative S the rel­ative mutual circumferential position of stators (parameter v) is changed from 0 (some starting position of stators) up to 1 (shift of IGV relative S on one full pitch of stator vanes Hst) with the pitch Д v = 0,1. In turn at each of 11 values V by simultaneous circumferential shift of IGV and S value of the relative cir­cumferential position of stators (parameter Yst) is changed from 0 up to 1 with the pitch ДУ^ = 0,1. Thus, the received files of measured parameters pro­vided the information on their circumferential distortion and on their change in dependence on mutual stators position.

At measurements by high-response sensors the information on change of fbw parameters on time within 1 second with phase synchronization of mea­surements with the help of the R blade passing frequency (BPF) mark was provided. After amplification of electric signals of sensors and their transfor­mation to physical values with the help of analog-digital converters measured parameters entered in a digital kind to PC. Frequency of interrogation of high – response sensor made 19,2kHz, that at BPF 1,2kHz provided 15 measurements on the period of BPF. It is enough received volume of the experimental infor­mation for obtaining of change of flow parameters as in steady (connected with stators), and rotating (connected with a rotor) systems of references by results of measurements by high-response sensors in the steady system of reference, realized in experiment. The technique of recalculation of the measured pa­rameters for their representation in rotating system of reference is described in [12,14]. For allocation of a periodic signal the indications of sensors were av­eraged on ensemble on 114 periods of BPF (3 revolutions of R). Random fbw pulsations were determined by subtraction of periodic components of pulsa­tions from the measured values. It is necessary to notice, that random pul­sations are caused not only by flow turbulence, but also small deviations in geometry of the R blades and operation mode of the compressor.

Any few parameter n(r, x,y, t,v) (for example, pressure, velocity and etc.) generally is the function, dependent from radial r, axial x and circumferential y coordinates of a considered point, time t and a mutual circumferential position of stators v. For reception of integrated estimations of interactions of rows effects by results of experiment or calculation at fixed values r, x and v were determined the following generalized parameters:

(n)t – averaged on time value П;

П >t – mean root square (RMS) deviation of П from its average on time value (n)t;

((n)t )y, (< П >t)y – averaged on period H (pitch of stators or rotor) values (n)t and < П >t;

П >y, << П >t>y – RMS deviations of values (П) and < П >t from their averaged values ((n)t)y and (< П >t)y on circumferential period H.

Averaged on circumferential period H values ((n)t )y and (< П >t )y char­acterize some stationary fbw, obtained by averaging of real unsteady fbw, and RMS deviations < П >y and << П >t >y determines a level of unsteady in­teractions of mutually moving rows and for the isolated row of identical blades < П >y and << П >t >y are equal to zero. The specified parameters may be determined in the R and S systems of reference. Conformity between the parameters, determined in different systems of reference, is given in [14].

Experimental installation includes the researched compressor, consisting of inlet guide vanes (IGV), rotor (R) and stator (S). The compressor has a cylindri­cal fl)wing path and is designed as a typical stage of the low-speed compressor on the following parameters: external diameter Dc = 1.2m, hub/tip ratio di­ameter d = 0.8, relative maximal airfoil thickness Cmax = 10%, frequency of the R rotation n = 2000rot/min, the R tip speed Uup = 125m/s, air fbw rate G = 30.8kg/s, ft>w coefficient ф = 0.5, loading coefficient ф = 0.32, total pressure ratio nt = 1.055, efficiency pad = 0.90, inlet and outlet ft>w angles a = a3 = 90°, a degree of reaction т = 0.8. Flowing path and measurements are shown on Fig.1.

In initial assembly the compressor has numbers of blades IGV, R and S equal, accordingly, 36, 38, 36. The construction of the compressor allows to change over a wide range axial gaps between blade rows from 5 up to 90mm (from 0.05 up to 0.93 lengths of an axial projection of the R blade) and numbers of vanes in stators. During experiment independent turn of the IGV and S is carried out, that allows to measure circumferential distortion of a flow field and their change owing to mutual shift of the stators. The IGV vanes, having a symmetric airfoil, are located along an axis of the compressor with a zero incidence angle.

At experimental researches the measurements were made, allowing to re­ceive average on time gasdynamical parameters of the compressor, and also structure of unsteady ft>w outside of and inside blade rows R and S. Unsteady values of pressure were measured with the help of high-response sensors ‘En – devco”on the R case in 6 points, on the suction and pressure surfaces S vanes in 12 points and in a ft>w behind the R and S on mean radius (see Fig.1).

Flow velocity was measured with the help of two-component laser anemome­ter (LDA) on mean radius in an axial gap between R and S.

N. M. Savin, V. E. Saren

The Central Institute of Aviation Motors Aviamotornaya st. 2, Moscow, 111116, Russia *

Abstract Results of researches of blade rows unsteady interaction in the three-rows stage of the axial compressor are presented with the purpose of definition the gas – dynamical mechanism of clocking effects in system of rows stator-rotor-stator. Experiments were carried out on the large-dimensional low-speed compressor with measurements of stationary and unsteady fbw parameters. Mutual circum­ferential position of stator rows, axial gaps and numbers of blades in rows are varied. Results of measurements are compared to calculation of few param­eters, received on the half-analytical theory of potential-vortical interaction of the airfoil cascades and by numerical simulation of averaged on Reynolds 2-D unsteady Navier-Stokes equations.

Keywords: Subsonic compressor, clocking effect

1. Introduction

Perfection of modern axial turbomachines demands already at a design stage of the account of unsteady interaction of rotors and stators. As have shown researches of last years, this circumstance is connected not only to a traditional problem of resonant blade vibrations, but also with infhence of unsteadiness on the time-averaged gasdynamical characteristics of turbomachines. The level of periodic pressure pulsations in a flawing path of the axial turbomachine, designed on specified parameters, is determined by aerodynamic loading of the rows, axial gaps between them and the relation of numbers blades (or pitches) in rotors and stators.

*Work is executed at financial support of International scientific and technical centre (ISTC), the grant number 672.2.

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One of displays of the mutual inflience of rows on their fbw is the effect of a mutual circumferential position of stators and (or) rotors (clocking effect). As have shown the first experiments at a transonic compressor stage with IGV [1,2] and on 4-stages turbine [3,4], at typical axial gaps between rows and equal (or multiple) numbers of vanes in stators their mutual circumferential position essentially infhences on a ft>w in rotor [1,2], and also on efficiency of stags [3,4].

According to the further experimental researches on turbines [5,6] change of efficiency at change of a mutual circumferential position of stators, having equal numbers of vanes, achieves 0,5 %. Detailed researches of atypical sub­sonic stage of the high pressure compressor [7,8] have shown, that appropriate change of efficiency achieves 1,5 %. Thus the level of stagnation pressure pul­sation in an absolute ft>w changes twice. The done work changes a little bit, and change of efficiency is provided, mainly, due to change of useful work [7,8].

From theoretical methods of research of effect of mutual circumferential po­sition of stators have received application a half-analytical method of potential – vortical interaction of mutually moving cascades [1,2,9-11] and a method of direct numerical integration of averaged on Reynolds 2-D unsteady Navier – Stokes equations, closed by this or that model of turbulence [3,4-8]. First of the specified methods is based on the most simple quasi-steady model of an incompressible flow and allows to describe qualitative features of investigated flow. This method theoretically predicted influence of a mutual circumferential position of stators on a flow of the located between them rotor before perfor­mance of the first experiments.

Numerical methods are widely used by various authors, however for a task unsteady interaction of rotors and stators demand significant computing re­sources. Besides it is necessary to recognize that for similar tasks there are not clear requirements to construction of calculation grid zones, and also inflience of used model of turbulence on accuracy of description of the unsteady vorti­cal wakes dissipation. By that less, in overwhelming majority of the mentioned above works calculation and experimental data are compared.

Thus, available theoretical and experimental results allow to consider op­timization of a mutual circumferential position of blade rows in an axial tur­bomachine as an effective control mean of unsteady interaction of rotors and stators for decrease of losses and pressure pulsations.

In given paper results of theoretical and the experimental researches are presented, executed on the large-dimensional compressor, which was specially created in CIAM for research of effects of unsteady interaction of blade rows of the axial compressor. The work was executed at financial support of the In­ternational Science and Technology Centre (ISTC), the Grant No.672-98. The first results, received on this compressor, were reported on previous ISUAAAT

Symposium in Lyons [12]. As against the experiments executed earlier at mod­elling stages of the axial compressor [1,2,7,8], on the created compressor the complex research was carried out, including fbw laser anemometry between rows and digital processing of results of measurements of instant values of static pressure on the rotor case and on the stator vanes and also stagnation pressure between rows and behind a compressor. Researches are carried out at various axial gaps between rows and various numbers of blades in rows.

The basic attention in the paper is given to statement of the research prob­lem, the developed measurement technique and the analysis of the results, al­lowing to reply on the questions, connected to use in the practical purposes of effect of a mutual circumferential position of stators in system of compressor rows stator-rotor-stator.

The structure rotor blades model is based on a modal approach of the cou­pled fbid-structure problem (Bathe and Wilson 1976, Rzadkowski 1998). The first step of the modal approach consists of solving the problem of the natu­ral mode shapes and eigenvalues without damping and in a vacuum. Then the displacement of each blade can be written as a linear combination of the first N modes shapes with the modal coefficients depending on time. Taking into account the orthogonality property of the mode shapes the equation of motion reduces to the set of independent differential equations relatively to modal co­efficients of natural modes. The modal forces are calculated for each iteration with the use of the instantaneous pressure field calculated form the fbw code (Gnesin and Rzadkowski 2000).

2. Numerical Results

The numerical calculations presented below were carried out for the stage of the turbine with rotor blades length of 0.765 m. The number of stator blades is equal to 56, the number of rotor blades is equal to 96. The stator to rotor blade number ratio of 56:96 (7:12). All geometrical parameters of the blade are presented in Rzadkowski 1998.

It was assumed that the pressure behind the rotor blades is changing in the circumferential direction (measured by the angle around the axis of rotation of the turbine). For circumferential angle а Є (0, 90°) p2=6000 Pa, а Є (90°, 180°) p2=7500 Pa, а Є(180°, 270°) p2=9000 Pa, а Є (270°, 360°) p2=7500 Pa (see Figures 2 pressure = p2/(pka2),pka2 =9467 Pa). The unsteady forces acting on the ith rotor blades, in axial, tangential and radial directions were found.

The numerical and experimental verification of the numerical code is pre­sented in Rzadkowski and Gnesin 2000.

The numerical calculations have been made using the computational H-grid of 11*24*60 grid points for each stator passage and 11*14*60 grid points for each rotor passage.

One of the important aspects of stator-rotor interaction is the effect of the blade response with taking into account the excitation caused by the flow uni­formity and excitation due to blades oscillations.

The blade vibrations are defined with taking into account the first ten natural modes shapes of rotating blade. The values of natural frequencies and the mechanical damping coefficients hi = 2ші fi, are given in Table 1. The modal damping coefficients were assumed (Rzadkowski 1998): = 0.00075, =

0.00094, ^3 = 0.0011, £з = ^4 = £io.

The unsteady force is the unperiodic function in time. The forces acting on the various blades differ one from another. We are using here the term the unsteady modal force which is equal along the blade length and correspond to the particular mode shape. This is disadvantage of the modal superposition calculations, where modal force averaged along the length of the blades is calculated. After the start regime, there began the coupled vibrations where unsteady forces in the turbine stage are the result of continuous interaction between gas fbw, rotation of the rotor wheel and blades vibration. So, it is impossible to separate the unsteady effects caused by the external excitation and the unsteady effects due to blades vibration.

Figures 3 – 4 shown the unsteady modal forces corresponding to the 1st, 2nd, 4th and 8th modes for the 1st blades. Generally the low frequency excitation is predominant.

Figures 5 a, b present the modal components of the unsteady modal force corresponding to the first mode. The high frequency excitation appeared for 2800 Hz and is equal to 1 % of the steady force Ao=27.5 [N]. The low fre­quency excitation caused by non-uniform pressure distribution is 158 % of Ao for frequency 50 Hz (see Figure 5b).

Figures 6a, b present the modal components of the unsteady modal force corresponding to the second mode. The high frequency excitations appeared for 2800 Hz and is equal to 2% of the steady force Ao=35.5 [N]. The low frequency excitation is 38 % of Ao for frequency 50 Hz (see Figure 6b).

Figures 7a, b present the modal components of the unsteady modal force corresponding to the fourth mode. The high frequency excitations appeared for 2800 Hz and is equal to 2 % of the steady force Ao=27.0 [N]. The low frequency excitation is 78 % of Ao for frequency 50 Hz (see Figure 7b).

Figures 8a, b present the modal components of the unsteady modal force corresponding to the 8th mode. The high frequency excitations appeared for 2800 Hz and is equal to 5 % of the steady force Ao = 6.8 [N]. The low frequency excitation is 600 % of Ao for frequency 50 Hz (see Figure 8b).

It should be noted that only first four modes bring their contribution to the blade motion. The low frequency unsteady forces caused by non-uniform pres­sure distribution are higher in comparison to the high frequency excitations.

The modal coefficients of the 1st blade motion corresponding to the 1st, 2nd, 4th and 8th modes shape have been shown in Figures 9 -12.

The unsteady amplitude of the first mode (see Figure 9) has frequency 73 Hz (99 Hz the natural frequency) and the frequency closes to 100 Hz. The

unsteady amplitude of the second mode (see Figure 10) has frequency 70 Hz and 157 Hz (160 Hz the natural frequency).

The unsteady amplitude of the fourth mode (see Figure 11) has frequencies 77 Hz and 280 Hz (297 Hz the natural frequency). The unsteady amplitude of the 8th mode (see Figure 12) has frequencies 77 Hz (862 Hz the natural frequency).

The spectrum includes mainly the blade oscillation frequencies closed to their natural ones (not multiple to the rotation frequency).

3. Conclusions

A partially – integrated method based on the solution of the coupled aero­dynamic and structure problem is used for calculation of the unsteady 3D fbw through a turbine stage with taking into account the rotor blades oscillations. The paper has investigated the mutual influence of both outer nonuniform dis­tribution of the pressure behind the rotor blade and rotor blades rotation and oscillations. The interblade phase angle of blades oscillations depends not

Figure 6. The amplitude-frequency spectrum for the modal force of the 2nd mode

only on unsteady forces lag but on the blade natural frequencies, as well. The low frequency unsteady forces caused by non-uniform pressure distribution are higher in comparison to the high frequency excitations. It has shown that

References

Bakhle, M. A., Reddy, T. S.R., and Keith T. G. (1992). Time Domain Flutter Analysis of Cascades Using a Full-Potential Solver, AIAA J. vol.30, No 1, p.163.

Bathe K., Wilson E. (1976). Numerical Methods in Finite Element Analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.

Bendiksen O. (1998). Nonlinear blade vibration and flitter in transonic rotors, Proc. of IS – ROMAC – 7, The 7th Intern. Symp. on Transport Phenomena and Dynamics of Rotating Machinery, 22-26 February, Honolulu, Hawaii, USA, 664.

Carstens V., Belz J. (2000). Numerical investigation of nonlinear fliid-structure interaction in vibrating compressor blades, ASME paper 2000-GT-0381, 2000.

Amplitude-frequency spectrum of the blade oscillations by 4th mode

Amplitude-frequency spectrum of the blade oscillations by 8th mode

Chew J. W., Marshall J. G., Vahdati M. and Imregun M. (1998). Part-Speed Flutter Analysis of a Wide-Chord Fan Blade, T. H. Fransson(ed.), Unsteady Aerodynamics and Aeroelasticity of Turbomachines, Kluwer Academic Publishers, Printed in the Netherlands. 707-724.

Gnesin V., Rzadkowski R. and Kolofyazhnaya, L., V. (2000). A coupled flud-structure analysis for 3D flitter in turbomachines, ASME paper 2000-GT-0380.

Gnesin V., and Rzadkowski R. (2000). The theoretical model of 3D flitter in subsonic, transonic and supersonic inviscid fbw, Transactions of the Institute of Fluid-Flow Machinery, No. 106, 45-68.

Gnesin V., Rzadkowski R. and Kolodyazhnaya, L., V. (2000). A coupled fliid-structure analysis for 3D fitter in turbomachines, ASME paper 2000-GT-0380.

Hall, K. C. and Silkowski, P. D. (1997). The Infhence of Neighbouring Blade Rows on the Un­steady Aerodynamic Response of Cascades, ASME Journal of Turbomachinery, 119,85-93.

He L. (1994). Integration of 2D fliid/structure coupled systems for calculation of turbomachin­ery aerodynamic, aeroelastic instabilities, Journal of Computational Fluid Dynamics. 3, 217.

He L. and Ning W. (1998). Nonlinear harmonic analysis of unsteady transonic inviscid and viscous fbws, unsteady aerodynamics and aeroelasticity of turbomachines, Proceedings of the 8th International Symposium held in Stockholm, Sweden, 14-18 September, 183-189.

Moyroud F., Jacquet-Richardet G., and Fransson T. H. (1996). A modal coupling for fliid and structure analysis of turbomachine flitter application to a fan stage, ASME Paper 96-GT – 335, 1-19.

Namba, M. and Ishikawa, A. (1983). Three-dimensional Aerodynamic Characteristics of Os­cillating Supersonic and Transonic ennular Cascades, ASME J. of Engineering for Power 105,138-146.

Rzadkowski R., Gnesin V. (2000). The numerical and experimental verification of the 3D invis­cid code, Transactions of the Institute of Fluid-Flow Machinery, No. 106, 2000, 69-95.

Rzadkowski R., Gnesin V. (2002). 3D Unsteady Forces of the Transonic Flow Through a Tur­bine Stage with Vibrating Blades, ASME Paper GT-2002-300311.

Rzadkowski R. (1998). Dynamics of steam turbine blading. Part two: Bladed discs, Ossolineum, WrocSaw-Warszawa.

An ideal gas flow through the mutually moving stator and rotor blades with periodicity on the whole annulus is described by the unsteady Euler conser­vation equations, which are integrated using the explicit monotonous finite – volume difference scheme of Godunov-Kolgan and moving hybrid H-H grid.

The algorithm proposed allows calculate unsteady forces of the turbine stages with an arbitrary pitch ratio of stator and rotor blades.

The 3D transonic flow of inviscid non-heat conductive gas through an axial turbine stage is considered in the physical domain, including the nozzle cas­cade (NC) and the rotor wheel (RW), rotating with constant angular velocity. In general case both NC and RW have an unequal number of blades of the arbi­trary configuration (see Figures 1). Taking into account the ft>w unperiodicity from blade to blade (in the pitchwise direction) it is convenient to choose the calculated domain including all blades of the NC and RW assembly, the entry region, the axial clearance and the exit region. Each of passages is dicretized using H-type grid for stator domain and hybrid H-H grid for rotor domain (Rzadkowski and Gnesin 2002). Here outer H-grid remains stationary during the calculation, while the inner H-grid is rebuilt in each iteration by a given algorithm, so that the external points of the inner grid remain unmoved, but the internal points (on the blade surface) move according to the blade motion.

It is assumed that the unsteady flow fluctuations are due to both the rotor wheel rotation and to prescribed blade motions, and the flows far upstream

and far downstream from the blade row are at most small perturbations of uniform free streams. So, the boundary conditions formulation is based on one – dimensional theory of characteristics, where the number of physical boundary conditions depends on the number of characteristics entering the computational domain (Gnesin and Rzadkowski 2000).

In the general case, when axial velocity is subsonic, at the inlet boundary initial values for total pressure, total temperature and fbw angles are used, while at the outlet boundary only the static pressure has to be imposed. Non­refecting boundary conditions can be used, i. e., incoming waves (three at inlet, one at the outlet) have to be suppressed, which is accomplished by setting their time derivative to zero.

Romuald Rzadkowski

Institute of Fluid-Flow Machinery, Polish Academy of Sciences 80-952 Gdansk, ul. Fiszera 14, Polish Naval Academy z3@imp. gda. pl

Vitaly Gnesin, Luba Kolodyazhnaya

Department ofAerohydromechanics, Institute for Problems in Machinery

Ukrainian National Academy of Sciences 2/10 Pozharsky st., Kharkov 310046, Ukraine

gnesin@ipmach. kharkov. ua

Abstract A three-dimensional numerical analysis for aerodynamic unsteady forces and flitter parameters of the last stage steam turbine 13K215 rotor blades have been presented. The low frequency excitation was simulated for a 94 rotating blades with 54 nozzles. It was assumed that the pressure behind the rotor blades is changing in the circumferential direction. The flitter parameters of this stage were calculated.

Keywords: unsteady forces, inviscid flow, rotor blades, stator blades

1. Introduction

The classical and partial integration flitter analysis (Bakhle et al. 1992, He 1994; Moyround et al. 1996, Rzadkowski 1998, Rzadkowski and Gnesin 2000, 2001, He and Ning 1998, Bendiksen 1998, Gnesin et al. 2000, 2001, Carstens and Belz 2000) take into consideration only the rotor blades. The stator blades are modelled by the interblade phase angle of the rotor blades as the initial condition.

Hall and Silkowski 1997 can be cited as one of a few papers investigating into the effect of neighbouring blade rows. Namba and Ishikawa 1983 give an analytical study on contra-rotating annular cascades with oscillating blades. For the first time the coupled solution of an aeroelastic problem for turbine

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stage with vibrating blades was presented by Rzadkowski and Gnesin 2002 for uniform distribution of the pressure behind the rotor blade.

In this paper a three-dimensional numerical analysis for aerodynamic un­steady forces of the last stage steam turbine 13K215 rotor blades have been presented for non-uniform pressure distribution behind the rotor blades.

The numerical calculation of the 3D transonic ft>w of an ideal gas through turbomachinery blade rows moving relatively one to another without taking into account the blades oscillations is presented.

An ideal gas ft>w through the mutually moving stator and rotor blades with periodicity on the whole annulus is described by the unsteady Euler conser­vation equations, which are integrated using the explicit monotonous finite – volume difference scheme of Godunov-Kolgan and moving hybrid H-H grid.

It was assumed that the pressure behind the rotor blades is changing in the circumferential direction (measured by the angle around the axis of rotation of the turbine). For circumferential angle ( а Є (0, 90°) p2=6000 Pa, а Є (90°, 180°) p2=7500 Pa, а Є(180°, 270°) p2=9000 Pa, а Є (270°, 360°) p2=7500 Pa. The unsteady forces acting on the ith rotor blades, in axial, tangential and radial directions were found.

The fuel injection in the turbine-combustor results in the modified the tan­gential forces in the turbine, as shown in Table 2. In situ reheat decreased tangential force Fy on the first blade row but increased tangential force on the subsequent rows. Since the tangential force decrease on the first stage is smaller than the increase on the subsequent stages, the power of the turbine – combustor increased for all cases with combustion. Although the variation of the averaged blade force Ftot is rather small, as shown in Table 2 and Fig. 4, the power increase of approximately 5% is significant.

The time variation in of the rotor blade tangential forces, shown in Fig. 5, indicates that the largest amplitudes occur in the last rotor row and the small­est amplitudes occur in the first rotor row. This conclusion is valid for every combustion or no combustion case.

(d) Fourth rotor

Figure 5. Variation of tangential forces on the rotors

A phase shift caused by fuel injection is visible for the first and second rotor blades. The larger unsteadiness within the second rotor makes this phe­nomenon more clearly distinguishable in Fig. 5(b). The patches of burning mixture and the reduced degree of mixedness are the probable causes for this tangential force phase shift in the upstream region.

Figure 6 shows the fast Fourier transform of the tangential forces. They have been nondimensionalized by the average tangential force obtained in the case without fuel injection. The blades of the fourth rotor are excited the most. This excitation occurs at the first blade passing frequency (BPF), which is 1920 Hz. For the rest of the blades, the excitation due to the second BPF is comparable in amplitude to the excitation of the first BPF. Except for first rotor in case 1 and third rotor in case 3, the fuel injection has the effect of increasing the excitation of the first BPF. The largest amplitude increase is 216% and occurs on the third row blades in case 2. The unsteady force, however, is approximately 50% of the maximum amplitude value that occurs on the fourth rotor blade at BPF.

3. Conclusions

The complexity of the transport phenomena in a multi-stage turbine-combustor makes it one of the most challenging numerical simulation problems. The large unsteadiness and straining of the fbw along with the wide range of velocity variation lead to a widely spread of local characteristic time scales for flow and combustion, which strongly impacted the on-going reactions. As a first step in the numerical simulation of the in situ reheat, a two-reaction chemistry mechanism has been considered.

The numerical simulation was used to predict the airfoil temperature vari­ation and the unsteady blade loading in a four-stage turbine-combustor. The in situ reheat decreased the power of the first stage, but increased more the power of the following stages. The power of the turbine increased between 2.8% and 5.1%, depending on the parameters of the fuel injection. The largest excitation of the four-stage turbine-combustor corresponded to the rotor of the fourth stage, with or without combustion. The highest excitation corresponded to the first blade passing frequency, for all cases analyzed.

TION