Influence of upstream stator on rotor flutter stability in ... - SAGE Journals

25

Influence of upstream stator on rotor flutter stability in a low pressure steam turbine stage X Q Huang1, L He1
, and D L Bell2 1 School of Engineering, University of Durham, Durham, UK 2 ALSTOM Power Ltd, Rugby, Warwickshire, UK The manuscript was received on 18 November 2004 and was accepted after revision for publication on 23 September 2005. DOI: 10.1243/095765005X69170

Abstract: Conventional blade flutter prediction is normally based on an isolated blade row model, however, little is known about the influence of adjacent blade rows. In this article, an investigation is presented into the influence of the upstream stator row on the aero-elastic stability of rotor blades in the last stage of a low pressure (LP) steam turbine. The influence of the upstream blade row is computed directly by a time-marching, unsteady, Navier – Stokes flow solver in a stator – rotor coupled computational domain. The three-dimensional flutter solution is obtained, with adequate mesh resolution, in a single passage domain through application of the Fourier-transform based Shape-Correction method. The capability of this single-passage method is examined through comparison with predictions obtained from a complete annulus model, and the results demonstrate a good level of accuracy, while achieving a speed up factor of 25. The present work shows that the upstream stator blade row can significantly change the aero-elastic behaviour of an LP steam turbine rotor. Caution is, therefore, advised when using an isolated blade row model for blade flutter prediction. The results presented also indicate that the intra-row interaction is of a strong three-dimensional nature. Keywords: turbine flutter, intra-row interaction, single-passage method, non-linear timemarching

1

INTRODUCTION

Blade flutter is of considerable concern in modern turbomachinery development. This is particularly the case as a result of the trend for longer, lighter, and more powerful and efficient designs. The demand for better understanding and predictive capability of turbomachinery blade flutter has boosted relevant research and development activities, especially in the prediction of unsteady flows around vibrating blades, during the last decade. At present, there are two principal types of unsteady aerodynamic methods: frequency domain (time-linearized) and time domain (non-linear time-marching) methods. Investigations by Holmes and Chuang [1], Hall and Lorence [2], Holmes and Lorence [3], Marshall and Giles [4], and Montgomery


Corresponding author: School of Engineering, University of

Durham, South Road, Durham DH1 3LE, UK.

JPE138 # IMechE 2006

and Verdon [5] are notable examples of the former case. Recently, the frequency domain approach has been extended to include some elements of the non-linear effects, for instance the non-linear harmonic method of Ning and He [6] and He and Ning [7], and the harmonic balance technique of Hall et al. [8]. For the non-linear time-marching technique, there has also been considerable progress, such as the work by Giles and Haimes [9], He and Denton [10], Gerolymos and Vallet [11], Isomura and Giles [12], Ji and Liu [13], Gru¨ber and Carstens [14], Vahdati et al. [15], and Sayma et al. [16]. Currently, most computational models of vibrating blades, including the aforementioned, assume an isolated blade row in a truncated domain. Turbomachines, however, normally consist of multiple blade rows (stators and rotors). The existence of these adjacent blade rows leads to some potentially important intra-row interaction phenomena such as acoustic wave reflection, which an isolated blade row

Proc. IMechE Vol. 220 Part A: J. Power and Energy

26

X Q Huang, L He, and D L Bell

model is unable to capture. The relevant questions/ issues here are as follows. 1. Are the intra-row interaction effects significant in blade aero-elastic stability? 2. If important, how can they be predicted effectively? Until now, only a few studies have examined the intra-row interaction effects on the aero-elastic behaviour of turbomachines. Buffum [17], Hall and Silkowski [18], and Silkowski and Hall [19] used a ‘coupled model’ to obtain the behaviour of the multistage machine from the individual cascades’ reflection and transmission coefficients, which were predicted by a time-linearized solver. Their results suggest that the aerodynamic damping of an isolated cascade of vibrating airfoils can differ markedly from the aerodynamic damping when the cascade is surrounded by neighbouring blade-rows and that the intra-row spacing significantly affects the aerodynamic damping. Li and He [20] carried out a systematic examination of rotor –stator gap effects on the aerodynamic damping of a vibrating rotor row embedded in a compressor stage by using a three-dimensional time-domain, single-passage, Navier – Stokes solver. The predicted non-monotonic relationship between the rotor blade aerodynamic damping and gap distance suggests the existence of an optimum gap regarding rotor flutter stability and/or forced response stress level. It has also been shown that the intra-row interaction effect on the rotor aerodynamic damping may be altered by changing the number of stator blade numbers (NBs). All the aforementioned intra-row interaction studies have been performed on compressors and no work has yet been reported for turbines. In the work presented, the influence of the upstream, fixed, nozzle blades on the self-excited aero-elastic behaviour of rotor blades in the last stage of a low pressure (LP) steam turbine is investigated. It should be recognized at this point that time-marching computations of the complete annulus with multiple blade rows is extremely expensive for this type of configuration and not suitable for parametric studies. An efficient single-passage method is, therefore, adopted in this work.

2 2.1

UNSTEADY FLOW MODELLING

diameters (NDs) X (m) ¼ Am0 sin(v0 t þ ms0 )

(1)

where m ¼ 1, 2, . . . , NBR is the index of rotor blades, Am0 is the amplitude of the vibration, and s0 is the inter blade phase angle (IBPA) of the blade motion and has an expression of s0 ¼ 2pND=NBR . In this kind of stage configurations, there are two fundamental periodic sources of unsteadiness. One is the relative motion between stator and rotor blades and the other is the rotor blade vibration. Each flow disturbance, induced by either the relative motion of neighbouring row blades or the rotor vibration, is assumed to have a distinctive periodicity in both time and space (in the circumferential direction). On the basis of assumption of the temporal periodicity, any flow variable U can be decomposed into a time-average part U0 and a number of unsteady disturbances ui identifiable by their temporal periodicities. Each disturbance can be approximated by a set of Fourier series in time

U(x, y, r, t) ¼ U0 (x, y, r) þ

Npt X

ui (x, y, r, t)

(2)

i¼1

ui (x, y, r, t) ¼

Nfou X

(Ani sin(nvi t) þ Bni cos(nvi t))

(3)

n¼1

where Npt is the number of disturbances; vi is the angular frequency of the ith unsteady disturbance; Ani and Bni are the corresponding Fourier coefficients; Nfou is the order of the Fourier series; x, y, and r are axial, circumferential, and radial coordinates, respectively, and t the physical time. In addition, on the basis of the spatial (circumferential) periodicity assumptions, the unsteady disturbances satisfy the phase-shifted periodic condition ui (x, y þ G, r, t) ¼

Nfou X

(Ani sin(n(vi t þ si ))

n¼1

þ Bni cos(n(vi t þ si )))

(4)

where G is the circumferential spacing between two neighbouring rotor blades and si the IBPA of the ith disturbance. Equations (3) and (4) are the principal formulations of the present Shape-Correction scheme, applied to a single-passage domain [21], and are utilized to deal with periodic boundary conditions and the stator– rotor interface.

Basic phase-shift formulations

The unsteady flows to be dealt with are assumed to be periodic. Consider a turbine stage of which the rotor with NBR blades vibrates periodically with frequency v0 in a travelling wave mode of nodal Proc. IMechE Vol. 220 Part A: J. Power and Energy

2.2

Intra-row interaction

Conventionally, the intra-row interaction refers to the wake/blade row interaction and potential JPE138 # IMechE 2006

Influence of upstream stator on rotor flutter stability

interaction, both of which are due to the relative motion of stator and rotor rows. Here, in particular, the intra-row interaction involves a totally different aspect: an acoustic wave (pressure wave) propagation phenomenon in a stator/rotor stage environment. Once the rotor blades are excited mechanically or aerodynamically, the rotor blade vibration will give rise to unsteady flow disturbance, which will radiate away from the rotor in flow characteristics. These characteristics correspond to two pressure waves, one vorticity wave, and one entropy wave. If the flow is subsonic, one pressure wave is running upstream and the other travelling downstream. The entropy and vorticity waves are always convected downstream by the main flow. The corresponding axial wave numbers [22] are

a1,2 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vx (v0 þ bVu ) + c (v0 þ bVu )2  (a2  Vx2 )b2

a3,4 ¼ 

a2  Vx2

bVu þ v0 Vx

experienced by the rotor blade will have a form of prefl ¼ a  porig (t þ DT ) ¼ a  Amp sin (v0 t þ w þ Dw) (7) where a is the reflection coefficient and Dw is the phase difference between the original disturbance and the reflected disturbance, which accounts for the time difference DT for the pressure wave to travel from the rotor to the stator and then back to the rotor Dw ¼

(8)

DT ¼

DX DX 2DX þ  a  Vx a þ Vx a

(9)

Denoted by K, the reduced frequency based on the blade chord length (C) and the speed of sound

Here, b ¼ s0 =G is the circumferential wave number, Vx and Vu are the averaged flow velocities in axial and circumferential directions, respectively, and a is the speed of sound. In an isolated row model, the unsteady flow waves are often allowed to be resolved in a long domain or pass through the inlet/outlet boundary of a truncated domain transparently by adopting non-reflective far-field boundary conditions. In practice, due to the existence of the neighbouring blade rows, the wave propagations will be disturbed. What we are interested in here, is the upstreamrunning pressure wave, which will go through the rotor–stator intra row zone (with a gap distance of DX ), impinge on stator blades, be reflected back, and result in additional unsteady loading on the rotor blades. As discussed earlier, the flow disturbance induced by the rotor vibration is periodic in time and can be expressed by a set of Fourier series, i.e. the fundamental harmonic plus higher harmonics. Furthermore, for a blade undergoing a harmonic structural vibration, only the fundamental harmonic component of unsteady pressure disturbance contributes to the aerodynamic damping (worksum). The fundamental harmonic of the unsteady pressure originated from the rotor vibration is in the form (6)

where Amp is the original pressure disturbance amplitude and w is the phase between the pressure disturbance and the vibration of the reference rotor blade. Then, the reflected pressure disturbance JPE138 # IMechE 2006

DT  2p T

where T is the vibration period. Roughly, the time DT may be approximated by

(5)

porig (t) ¼ Amp sin(v0 t þ w)

27

K ¼

v0 C a

(10)

The rotor vibration period can be expressed as T¼

2pC Ka

(11)

and we have Dw ¼

  DT DX  2p ¼ 2K  T C

(12)

Then, the worksum, an aero-elastic stability related parameter, can be evaluated WORKSUM ¼ Worig þ Wrefl ¼ W sin(w) þ W  a sin(w þ Dw) (13) where W is a factor combining the vibration amplitude, the original pressure disturbance amplitude, and the rotor blade surface area. From equation (13), the following observations can be made. 1. Depending on the reflection magnitude and phasing, the reflected pressure disturbance might contribute more to the total worksum than the original pressure disturbance. That means that the aero-elastic stability of the rotor in a stage can be completely different from that of an isolated rotor. 2. The contribution from reflected pressure disturbance can be positive or negative depending on the intra-row gap distance. In other words, the presence of the stator can have stabilizing or Proc. IMechE Vol. 220 Part A: J. Power and Energy

28

X Q Huang, L He, and D L Bell

destabilizing effects on the rotor at different intrarow gaps. 3. The worksum contributed by the reflected pressure disturbance has a sinusoidal (or sinusoidlike) trend as the intra-row gap changes. This non-monotonic variation suggests a possible optimum gap distances regarding the self-excited aero-elastic stability of the rotor. 2.3

Numerical method

The unsteady Navier – Stokes equations, which represent the conservation of mass, momentum, and energy, are written in an integral form in absolute cylindrical coordinates (x, u, r) as following @ @t

ððð

þð U dV þ

dV

½(F  Uumg )nx dA

~ þ (G  Uvmg )nu þ (H  Uwmg )nr   dA ððð þð ~ ¼ Si dV þ ½Tx nx þ Tu nu þ Tr nr   dA dV

dA

N

Ani ¼

pi vi X (U  Ri ) sin(nvi t)Dt p 1

N

(14) where U, F, G, and H are the standard conservative variable and inviscid flux vectors. The extra inviscid flux terms Uumg , Uvmg , and Uwmg derive from the contribution of moving grid due to the blade vibration and rotation, where umg , vmg , and wmg are meshing moving velocities. Si is the inviscid source term to account for the centrifugal effect and Tx , Tu , Tr the full viscous stress terms. (nx , nu , nr ) is a unit vector in the outgoing normal direction of the cell surface. The governing equations are discretized in space using the cell-centred finite volume scheme, together with the blend second- and fourth-order artificial dissipation [23]. Temporal integration of the discretized equations is carried out using the second-order four-step Runge – Kutta approach. The convergence is accelerated by a time-consistent multi-grid technique [24]. At the inlet, stagnation parameters and flow angles are specified. At the exit, back pressure is specified. A one-dimensional non-reflecting boundary procedure of reference [25] is adopted to prevent artificial reflections of outgoing waves at both inlet and exit. On blade and end-wall surfaces, a logarithmic law is applied to determine the surface shear stress and the tangential velocity is left to slip. In the present single-passage computation, the phase-shifted periodic boundary conditions and the stator –rotor interface interaction are implemented by employing the Shape-Correction scheme. During the computation, the Fourier coefficients, which are used to approximate the flow variables (equation (3)), are stored and updated at the periodic boundaries and Proc. IMechE Vol. 220 Part A: J. Power and Energy

interfaces. At every time step, the flow variables at periodic boundaries are corrected using the stored Fourier coefficients and the current solution. At the stator– rotor interface, instantaneous flow variables on both sides of the interface boundary over the whole annulus are first reconstructed according to the circumferential periodicities of all disturbances (equation (4)). Then, a second-order interpolation and correction method [26] enables local information to be transferred instantaneously across the interface and the intra-row interaction is directly modelled. It is noteworthy that acceleration of updating of Fourier coefficients can be achieved by two measures. One is the partial-substitution technique, which enables us to update the Fourier coefficients once in one period of its corresponding disturbance rather than one beating period, i.e. the Fourier coefficients are evaluated through the following formula

pi vi X Bni ¼ (U  Ri ) cos(nvi t)Dt p 1

(15) PNpt PNfou where Ri ¼ j=1 n¼1 ½Anj sin(nvj t) þ Bnj cos(nvj t) is the contribution of all disturbances except that from the ith perturbation and Npi the number of time steps in one period of the ith disturbance. The other measure is to carry out several timewise integrations of Fourier coefficients starting at different moments in one period of the corresponding disturbance. 2.4

Aerodynamic damping calculation

In a flutter prediction under the influence of neighbouring blade rows, multiple disturbances are introduced into the unsteady flow. Although all disturbances contribute to the unsteady lift, on average, only the disturbance at the fundamental vibration frequency does aerodynamic work (‘Modal Work’) on the sinusoidally vibrating blades. There are two schemes to evaluate the worksum, hence, the aerodynamic damping, in a time-marching flow solver. 1. Utilize the original pressure and vibration history to compute the worksum in one beating period, which is the minimal common multiple of all disturbance periods. 2. Decompose the original pressure and use its component at the fundamental vibration frequency to calculate the worksum in one vibration period. The former one is straightforward and can be easily implemented. However, in an engineering application with arbitrary blade vibration frequencies, a JPE138 # IMechE 2006

Influence of upstream stator on rotor flutter stability

time-dependent computation might take a long time to reach a beating period. Therefore, the latter procedure is pursued. In this procedure, the Fourier transform plays a key role again. From the timemarching solver, we store the converged blade surface pressure history for one stored period (the longest period of all disturbances). The remaining concern is how to integrate the Fourier coefficients using the data of a stored period rather than a beating period. Here, we adopt the partialsubstitution technique, which has been successfully implemented in the present single-passage flow solver. Once the pressure component of the fundamental vibration frequency is obtained, the worksum can be evaluated by following a routine integration. This damping calculation is entirely a postprocessing procedure and does not represent a significant overhead in terms of CPU time. 3

RESULTS

A hypothetical LP steam turbine stage configuration is used for the present work, in which the rotor has a slender tip section and is fitted with a tip shroud. The stage consists of 66 stator blades and 68 rotor blades. For the rotor flutter analysis, the first flap mode is considered. The vibrating mode shape and frequency for the rotor blades are obtained from a standard FE analysis. For convenience, the mode shape is approximated as a solid body vibration with three components (i.e. torsion, axial flap, and tangential flap), which are specified along the blade span. In this section, results are first presented for a parametric investigation on a two-dimensional geometry, extracted at the tip of the three-dimensional configuration. The results for the complete threedimensional stage configuration are then presented. 3.1

29

model is that it is not possible to vary the intra-row spacing in the three-dimensional case without completely changing the steady flow aerodynamics. This is because of the high flare angle on the outer casing, shown in Fig. 5, which is characteristic of an LP steam turbine cylinder. A computational mesh for the simplified twodimensional model is shown in Fig. 1, at an intrarow gap of 0.4 rotor chord lengths. In this mesh, 125  31 grid points are used for the rotor domain and 108  31 for the stator. As the intra-row gap increases, more axial mesh nodes were inserted in the stator domain, such that the spatial mesh density remains unchanged. Computations were performed for both forward and backward travelling waves at four different NDs in the first flap mode. The frequency and modeshape for the backward travelling wave in each ND pattern is provided in Table 1. As shown in this table, the component of tangential flap in this highly staggered tip section is very small at all NDs. It is interesting, however, that the amplitude of torsion in this so-called ‘first flap’ mode is of comparable magnitude to the amplitude of axial flap (see the ratio AmX/(AmTO  C) in Table 1). In fact, as the ND increases, the amplitude of torsion increases, such that it becomes the most dominant component of vibration above 8 NDs. It is also important to note that the phase angle between the torsional and axial flap components of vibration is non-zero, and approximately 958 for all backward travelling waves, and 2958 for the forward travelling counterparts. These trends in the torsional component of vibration, relative to the axial flap, are associated with the tip shroud. The mode is essentially an axial flap, however, as the blades flap,

Two-dimensional case

As previously discussed, the influence of the upstream stator is closely related to the intra-row gap distance. In this section, the intra-row gap effects are examined systematically, using a contrived two-dimensional computational domain. This was composed of the tip section of the stator and rotor blade, located on a constant radius and in a stream tube with constant thickness. Owing to the different radius of the stator and rotor tip sections in the three-dimensional model, the stator blade was restaggered in the two-dimensional model, to produce appropriate flow conditions at inlet to the rotor (incidence and Mach number). Although it is much less expensive to perform parametric investigations using a two-dimensional model, the main reason to adopt this simplified JPE138 # IMechE 2006

Fig. 1

Stator –rotor computational dimensional tip section)

mesh

(two-

Proc. IMechE Vol. 220 Part A: J. Power and Energy

30

X Q Huang, L He, and D L Bell

Table 1 Natural frequency and mode-shape (two-dimensional tip section) Amplitudes (torsion/components of flap)

Phase angles (difference between torsion and flap)

ND

Frequency (cycles/rev)

AmX/(AmTO  C)

AmX/AmY

fX 2 fTO (8)

fX 2 fY (8)

5 8 11 20

2.440 2.465 2.580 4.860

1.996 1.229 0.861 0.373

14.289 36.735 28.966 27.264

97.13 95.82 95.57 97.01

2103.631 298.056 89.871 97.721

Frequency, amplitude, and phase angle for backward travelling waves. Forward travelling waves can be obtained by changing the sign on all phase angles, including IBPA.

torsion is introduced at an appropriate phase and amplitude to enable the tip shroud to remain interlocking. In contrast to the vibration of free standing turbine blades in first flap, the presence of the tip shroud causes the frequency of the natural modes of vibration to increase with ND, because of the trends in mode shape with ND already described. This is reflected by the increase in natural frequency with ND shown in Table 1. 3.1.1

Validation of the single passage solver

It is impossible to fully validate the present solver directly against experimental data, due to the absence of published unsteady loading measurements on oscillating blades embedded in a multirow configuration. In order to verify the validity of the solver, an appropriate combination of component validation and code-to-code comparisons has, therefore, been performed. The capability of the multi-row, multi-passage, baseline solver to predict the aerodynamic response to blade-row interaction and blade oscillation has been verified

Fig. 2

against experimental data in references [24] and [27], respectively. Furthermore, the single-passage methodology has been validated in an isolated row against a semi-analytical solution of Namba [28]. In the present article, a comparison is made between a single-passage and multiple-passage solution. This comparative study is expected to serve a validation purpose for the single-passage method. Figure 2 shows the normalized spectra of the unsteady rotor axial forces for the 20 ND backward travelling wave mode at an intra-row gap of 40 per cent chord. This mode is chosen simply because it has a higher vibration frequency and requires less computational time. When compared with the multiple-passage solution, the single-passage method underpredicts the aerodynamic damping by 2.4 per cent (Fig. 2 indicates 4.3 per cent difference for the vibration induced force component). This discrepancy in the damping calculation is considered acceptable, and demonstrates that the present ShapeCorrection based, single-passage, method is capable of dealing with multiple disturbances in a stage configuration. A speedup of 13 times is also achieved by

Spectra of unsteady axial forces on the rotor (vvib , vibration frequency; vs , stator blade passing frequency)

Proc. IMechE Vol. 220 Part A: J. Power and Energy

JPE138 # IMechE 2006

Influence of upstream stator on rotor flutter stability

the single-passage solution over the multiple-passage solution (33 passages for the stator and 34 passages for the rotor). If the whole annulus domain were used for the multiple-passage solution, the speedup delivered by the single passage method would be doubled to a factor of 25. Furthermore, by examining the spectra of unsteady rotor axial forces, two points are demonstrated. First, the unsteadiness is dominated by its fundamental harmonic. Second, the non-linear interaction between the fundamental disturbances is negligible in the present case. These indicate that a set of Fourier series with an order of as low as one should be sufficient.

3.1.2

Overview of 2D predictions

The results of the two-dimensional predictions are shown in Figs 3 and 4, and summarized in Tables 2

Fig. 3

JPE138 # IMechE 2006

31

and 3. Before the influence of the intra-row gap effect is considered, there are some general trends in the results that are worthy of comment and explanation. First, all predictions for forward travelling waves indicate a stable aero-elastic condition, whereas for backward travelling waves an unstable condition is predicted (in the absence of material damping). These contrasting results are associated with the change in phase angle between torsion and axial flap in the forward travelling wave compared with the backward counterpart. In the forward travelling wave, the phase difference between torsions and axial flap is around 2958, whereas for the backward travelling wave it is þ958. On consideration, these observations suggest that it is the cross coupling of the unsteady flow induced by one component of vibration doing work to the other component of vibration, or vice-versa, by virtue of the phase difference between each component of vibration.

Forward travelling wave modes

Proc. IMechE Vol. 220 Part A: J. Power and Energy

32

X Q Huang, L He, and D L Bell

Fig. 4

Backward travelling wave modes

It is also interesting to note from Tables 2 and 3, that the absolute magnitude of logarithmic decrement (log-dec) (isolated row prediction) tends to increase with ND up to at least the 11 ND pattern. This is associated with the increasing magnitude of the torsional component of vibration through these cases (Table 1). At 20 NDs, the absolute magnitude of log-dec then falls for both forward and backward Table 2 Summary for the forward travelling wave modes

travelling waves. There are two reasons for this. First, the frequency of this ND is considerably higher than the others (Table 1), so the strain energy in the vibration is much higher and, therefore, the log-dec is reduced. Second, the increased frequency results in an increase in reduced frequency, which tends to have a stabilizing effect.

Table 3 Summary for the backward travelling wave modes

Stage prediction (log-dec)

ND

Isolated row prediction log-dec (1022)

Maximum (1022)

Minimum (1022)

Variation range/isolated row (%)

5 8 11 20

2.907 3.571 5.778 2.379

4.814 4.463 3.648 4.044

2.861 1.807 2.140 1.361

67.18 74.38 26.10 112.78

Proc. IMechE Vol. 220 Part A: J. Power and Energy

Stage prediction (log-dec)

ND

Isolated row prediction log-dec (1022)

Maximum (1022)

Minimum (1022)

Variation range/isolated row (%)

5 8 11 20

20.351 20.946 22.021 20.273

20.222 21.029 21.868 20.192

20.505 22.129 23.725 20.586

80.63 116.28 91.89 144.32

JPE138 # IMechE 2006

Influence of upstream stator on rotor flutter stability

33

Fig. 5 Computational mesh for three-dimensional LP steam turbine stage

3.1.3

Intra-row gap effects (2D)

To study the gap effects, for each ND, the stage predictions are presented over a range of intra-row gaps (from 0.40 to 1.41 rotor chord lengths) and then compared with the isolated rotor prediction. The variation of intra-row gap considered here represents a practical range for the stator – rotor gap in the last stage of an LP steam turbine. Figure 3 shows the predicted log-dec of both the stage and the isolated rotor for forward travelling wave modes (all being stable with positive log-dec). Table 2 shows a summary of the results for these forward travelling waves. These results confirm those observations from equation (13). The reason for the lack of sinusoid variations for 5, 8, and 11 ND modes might be that their wavelength is much longer than the range in intra-row gap considered here, i.e. one rotor chord. On the contrary, the wavelength for the 20 ND mode is relatively short and the log-dec variation shows a variation more like to a sinusoidal curve. Figure 4 presents the results for the backward travelling wave modes and Table 3 is the corresponding summary (all unstable). For the 5 ND backward travelling wave mode, the absolute log-dec variation over the concerned gap range is relatively small compared with its forward travelling counterpart. This does not mean that the influence of upstream fixed blades is small for the 5 ND backward travelling wave mode. The absolute variation is more associated with the original disturbance amplitude, while the relative change just reflects the influence of upstream fixed blades. From Table 3, it is shown that the relative change for the 5 ND backward travelling wave mode is still very big. From an engineering point of view, only the least stable mode is of interest in the aero-elastic analysis. Here the 11 ND

JPE138 # IMechE 2006

backward travelling wave mode is the least stable one and both the absolute and relative changes due to the gap distance variation are very considerable. Note that, the presence of upstream fixed blades can significantly change the aerodynamic damping of the least stable mode in both an absolute and relative sense. For the 20 ND backward travelling wave mode, the influence of the upstream stator blades diminishes exponentially as the intra-row gap increases, while its forward travelling wave counterpart does not. This may be because the acoustic pressure waves tend to be cut-on for the forward mode and cut-off for the backward one. Actually, all forward travelling wave modes are cut-on and all backward travelling wave modes are cut-off in the present study (the cut-off waves have complex axial wave numbers, which are obtained by equation (5), while the real axial wave numbers correspond to the cut-on waves). Cut-on waves will propagate unattenuated, while the cut-off waves’ amplitude will decay exponentially as a function the propagation distance and the wavelength. Even for cut-off modes, this attenuation effect will not obviously show up if the concerned gap variation is much shorter than the wavelength. This is the case for the 5, 8, and 11 ND backward travelling wave modes. Overall, the change of aerodynamic damping value by more 100 per cent with the intrarow gap is of considerable engineering significance. 3.2

Three-dimensional stage

Given the significant dependence of flutter stability on the intra-row gap in the two-dimensional configuration, it is naturally of interest to see if one can identify a similar influence in a realistic three-dimensional configuration. The full three-dimensional stage was, Proc. IMechE Vol. 220 Part A: J. Power and Energy

34

X Q Huang, L He, and D L Bell

Table 4 Three-dimensional predictions of logarithmic decrement Vibration mode

Stage prediction (%)

Isolated rotor prediction (%)

11 ND (backward)

20.913

21.318

therefore, analysed, using the computational mesh shown in Fig. 5. In this case, 117  31  41 mesh nodes are used for the stator domain and 125  31  41 nodes for the rotor. In order to identify the influence of the intra-row gap for this threedimensional configuration, baseline computations were obtained for the rotor in isolation, for comparison of aero-elastic stability, i.e. log-dec. The computations were first performed on the isolated rotor in order to identify the most unstable aero-elastic condition, which was found to be the backward travelling wave at 11 NDs. This mode was, therefore, selected for additional modelling in the complete stator– rotor coupled stage configuration. Table 4 shows the predicted stability in this mode of vibration, in terms of log-dec, for both the stage and isolated rotor computations. The results show that the acoustic reflection from the upstream stator blade row leads to a 30 per cent change in the log-dec, which is clearly significant. In contrast to the two-dimensional predictions of the backward travelling wave at 11 ND (Fig. 4(c)), where the rotor is predicted to be less stable in the stage configuration, in the three-dimensional case, the presence of the stator blade row makes the rotor considerably more stable. These opposing trends from the two-dimensional and threedimensional predictions, therefore, suggest that the pressure wave propagation is of a strong threedimensional nature. 4

CONCLUDING REMARKS

In the present study, a Fourier-transform based Shape-Correction, single-passage, Navier – Stokes, flow solver has been validated and applied to rotor blade flutter analysis in a complete LP steam turbine stage. The capability of this single-passage solver has been demonstrated though comparison with results from a solution in a conventional multiple-passage domain. A speedup by a factor of 25 has been achieved by using this single-passage method over the complete annulus solution. On the basis of the present results, some conclusions regarding the influence of upstream stator blades can be drawn. 1. The presence of upstream fixed blades can significantly change the self-excited aero-elastic Proc. IMechE Vol. 220 Part A: J. Power and Energy

stability of turbine rotor blades, and therefore, isolated rotor predictions can be misleading. 2. Given that a significant variation in log-dec is achieved by altering the intra-row gap, it may be possible to optimize the stator– rotor gap to passively control turbine blade flutter.

ACKNOWLEDGEMENTS This work was partially sponsored by ALSTOM Power Ltd. The technical discussions with Dr P. Walker are much appreciated.

REFERENCES 1 Holmes, D. G. and Chuang, H. A. 2D linearised harmonic euler flow analysis for flutter and forced response. Unsteady aerodynamics, aeroacoustics and aeroelasticity of turbomachines and propellers (Ed. H. M. Atassi), 1993, pp. 213 –230 (Springer-Verlag). 2 Hall, K. C. and Lorence, C. B. Calculation of threedimensional unsteady flows in turbomachinery using the linearized harmonic euler equations. J. Turbomach., 1993, 115, 800 – 809. 3 Holmes, D. G. and Lorence, C. B. Three-dimensional linearized Navier– stokes calculations for flutter and forced response. Proceedings of the 8th International Symposium on Unsteady aerodynamics, aeroacoustics and aeroelasticity of turbomachines (Ed. T. H. Fransson), pp. 211 –224 (Kluwer Academic Publishers, Stockholm, Sweden), 14– 18 September 1997. 4 Marshall, J. G. and Giles, M. B. Some applications of a time-linearized Euler method to flutter and forced response in turbomachinery. Unsteady Aerodynamics and Aeroelasticity of Turbomachines (Ed. T. H. Fransson), 1997 (Kluwer Academic Publishers). 5 Montgomery, M. D. and Verdon, J. M. A 3D linearized unsteady Euler analysis for turbomachinery blade rows, Part 1: aerodynamic and numerical formulations, Part 2: unsteady aerodynamic response predictions. Proceeding of the 8th International Symposium on Unsteady Aerodynamic and Aeroelasticity of Turbomachines, Stockholm, Sweden, 14– 18 September 1997. 6 Ning, W. and He, L. Computation of unsteady flows around oscillating blades using linear and non-linear harmonic euler methods. J. Turbomach., 1998, 120(3), 508 – 514. 7 He, L. and Ning, W. Efficient approach for analysis of unsteady viscous flows in turbomachinery. AIAA J., 1998, 36(11), 2005 – 2012. 8 Hall, K. C., Thomas, J. P., and Clark, W. S. Computation of unsteady non-linear flows in cascades using a harmonic balance technique. 9th International Symposium on Unsteady aerodynamics, aeroacoustics and aeroelasticity of turbomachines, Lyon, France, 2000. 9 Giles, M. B. and Haimes, R. Validation of a numerical method for unsteady flow calculations. J. Turbomach., 1993, 115, 110 – 117.

JPE138 # IMechE 2006

Influence of upstream stator on rotor flutter stability

10 He, L. and Denton, J. D. Three-dimensional timemarching inviscid and viscous solutions for unsteady flows around vibrating blades. J. Turbomach., 1994, 116, 469 – 475. 11 Gerolymos, G. A. and Vallet, I. Validation of threedimensional Euler methods for vibrating cascade aerodynamics. J. Turbomach., 1996, 118, 771 – 782. 12 Isomura, K. and Giles, M. B. A numerical study of flutter in a transonic fan. J. Turbomach., 1998, 120, 500 – 507. 13 Ji, S. and Liu, F. Flutter computation of turbomachinery cascades using a parallel unsteady Navier– Stokes code. AIAA J., 1999, 37(3), 320 – 327. ¨ber, B. and Carstens, V. The impact of viscous 14 Gru effects on the aerodynamic damping of vibrating transoni compressor blades-a numerical study. J. Turbomach., 2001, 123, 409 –417. 15 Vahdati, M., Breard, C., Sayma, A. I., and Imregun, M. A computational study of intake duct effects on fan flutter stablity. AIAA J., 2002, 40(3), 408 – 418. 16 Sayma, A. I., Vahdati, M., Green, J. S., and Imregun, M. Whole-assembly flutter analysis of a low pressure turbine blade. Aeronaut. J., 1998, 102(1018), 459 – 463. 17 Buffum, D. H. Blade row interaction effects on flutter and forced response. J. Propul. Power, 1995, 11(2), 205 – 212. 18 Hall, K. C. and Silkowski, P. D. The influence of neighboring blade rows on the unsteady aerodynamic response of cascades. J. Turbomach., 1997, 119, 85 – 93. 19 Silkowski, P. D. and Hall, K. C. A coupled mode analysis of unsteady multistage flows in turbomachinery. J. Turbomach., 1998, 120, 410 –421. 20 Li, H. D. and He, L. Blade aerodynamic damping variation with rotor –stator gap – a computational study using single-passage approach. ASME paper GT200338199, 2003. 21 He, L. Method of simulating unsteady turbomachinery flows with multiple perturbations. AIAA J., 1992, 30(11), 2730 – 2735. 22 Smith, S. N. Discrete frequency sound generation in axial flow turbomachines. Reports and memoranda, No. 3709, Aeronautical Research council, London, 1972.

JPE138 # IMechE 2006

35

23 Jameson, A., Schimidt, W., and Turkel, E. Numerical solutions of the Euler equations by finite volume method using Runge –Kutta time-stepping scheme. AIAA paper 81-1259, 1981. 24 He, L. Three-dimensional unsteady Navier– Stokes analysis of stator– rotor interactions in axial-flow turbines. Proc. Instn Mech. Engrs, Part A: J. Power and Energy, 2000, 214, 13 – 22. 25 Giles, M. B. Nonreflecting boundary conditions for Euler equation calculations. AIAA J., 1990, 28(12), 2050 – 2058. 26 He, L. Computational study of rotating stall inception in axial-flow compressors. AIAA J. Propul. Power, 1997, 13(1), 31 – 38. 27 Bell, D. and He, L. Three-dimensional unsteady flow for an oscillating turbine blade and the influence of tip leakage. J. Turbomach., 2000, 122, 93 – 101. 28 Li, H. D. and He, L. Single-passage analysis of unsteady flows around vibrating blades of a transonic fan under inlet distortion. J. Turbomach., 2002, 124, 285 – 292.

APPENDIX Notation a Am C K

speed of sound vibration amplitude blade chord reduced frequency, K ¼ vC/a

b v f s

circumferential wave number angular frequency (rad/s) phase angle inter blade phase angle (8)

Subscripts X Y TO

axial (flap) component of vibration tangential component of vibration torsional component of vibration

Proc. IMechE Vol. 220 Part A: J. Power and Energy