th
6 Australasian Congress on Applied Mechanics, Mechanics, ACAM 6 12-15 December 2010, Perth, Australia
Fluid-structure interaction study of gas turbine blade vibrations vibrations *
Gareth L. Forbes , Osama N. Alshroof and Robert B. Randall The School of Mechanical and Manufacturing Engineering, University of New South Wales, Australia *
Corresponding author. Email:
[email protected] g.forbes10 @gmail.com
Abstract: A recent research program has identified the possibility of using the analysis of casing wall pressures in the direct measurement of gas turbine rotor blade vibration amplitudes. Currently the dominant method of non-contact measurement of gas turbine blade vibrations employs the use of a number of proximity probes located around the engine periphery measuring the blade tip (arrival) time (BTT). Despite the increasing ability of this method there still exist some limitations, viz: the requirement of a large number of sensors for each engine stage, sensitivity to sensor location, difficulties in dealing with multiple excitation frequencies and sensors being located in the gas path. Analytical modelling of the casing wall pressures and reconstruction of rotor b lade vibration amplitudes from the analysis of these simulated pressure signals has shown significant improvement over current non-contact rotor blade vibration measurement limitations by requiring only a limited number of sensors and providing robust rotor blade vibration amplitude estimates in the presence of simulated measurement noise. However, this modelling was conducted with some fundamental assumptions about the casing wall pressures being made. One of these assumptions presumed that during blade motion the pressure profile around the rotor blades follows the blade’s motion while it oscillates around its equilibrium position. This assumption is investigated in this paper through the numerical modelling of the fully coupled two-way rotor blade motion and fluid pressure interaction. Keywords: blade vibration, casing wall pressure, fluid-structure interaction, gas turbine.
1 Introduct Introduct ion The greatest cause of failures in gas turbines, reported to be up to 42% of total gas turbine failures, come from blade faults [1]. Blade vibration is unavoidable and inherent in the operation of any gas turbine and without proper design for the excitation forces present, blade vibrations can be a cause of blade degradation and lead to failure. It is paramount that blade vibration can be measured and blade fatigue be estimated. Gas turbine blade vibration measurement is motivated by the desire to acquire either the blade’s forced vibration magnitude and frequency, or to estimate the modal parameters of the blade. The impetus for this information is generally driven, respectively, by the need for knowledge of High Cycle Fatigue estimates for blade life, or the use of blade modal parameter values for condition monitoring of the blades. Measurement of blade vibration can be achieved by directly attaching strain gauges to the blade surface, however the attachment of sensors to all blades within the engine is never desirable, and is certainly not practical outside of the design stage. Such are the difficulties of direct measurement of blade vibration, non-contact blade vibration measurement has been sought, with BTT methods showing the most promise and receiving research attention since the 1970’s. Despite the promise of BTT methods they are still not without limitations or shortcomings four decades after the ir initial use. In a recent research project it was proposed that blade vibration would have an effect on the casing wall pressure and casing vibration, and thus measurement of these parameters could be used for monitoring of blade vibration parameters [2]. It was shown that direct demodulation of simulated internal casing pressure signals could yield the measurement of rotor blade vibration amplitudes [3], and that observation of casing vibration signals could lead to the ability to estimate rotor blade natural frequencies [4]. The analysis performed in both of the previous studies was based on the assumption that the pressure profile around the rotor blade followed the blade’s motion as it oscillated about its equilibrium position and that the pressure at the rotor blade tip was not appreciably changed from that which is imposed on the turbine casing. The first of these assumptions is investigated further in this paper using both finite element and computational fluid dynamic modelling of the fluid structure interaction of the operating conditions inside a gas turbine whilst the rotor blades are vibrating.
Computational fluid structure interaction studies of gas turbine flow and blade motion are usually confined to investigations into blade flutter. Numerous computational and experimental studies into blade flutter arising from the flow and blade structural motion in gas turbines have been made, such as the investigation by Nowinski and Panovsky [5] where computational and experimental results of the influence of adjacent blade pressure under forced flutter conditions (i.e. the blades were externally forced to vibrate at their natural frequency) was investigated. This type of fluid-structure interaction only considers the one-way interaction of the structural motion on the fluid. In fact it is still uncommon to find many realistic two-way fluid structure computational studies for turbine blade-flow interaction. Studies of this nature are generally undertaken with in-house software and investigations implementing commercial multi-physics computational solvers are even more rare. A computationally efficient harmonic balance computational analysis, using in-house software, of the one-way fluid-structure interaction with turbine flow and rotor blade motion was investigated by Hall and Ekici [6]. Once again the motion within this study was enforced on the computational fluid domain. The effect that enforced blade motion due to the stator passing, i.e. forced at a harmonic of shaft speed, was shown to have little effect on the time averaged pressure over the rotor blade surface. As the pressure was time averaged, this does not necessarily mean that the pressure on the rotor blade surface does not change with time, and thus must follow the blade motion, but the small changes in the rotor blade pressure could tentatively lead to this conclusion. Another previous experimental and computational investigation has proven successful in using gas turbine pressure signals for determining changes in rotor blade geometry, e.g. blade twist faults [7][8]. Identification of pressure signal fault signatures for different geometric changes and relation of these back to a given fault from the measured pressure, was undertaken in these studies. This work did not however consider any blade vibration in the computational modelling. A two dimensional two-way coupled finite element/computational fluid dynamic model of a rotor blade undergoing forced stator passing vibration is investigated in this paper. Before the results of this numerical study are discussed the forces driving the rotor blade motion in a gas turbine will first be investigated.
2 Rotor blade vibration As has already been stated, vibration of gas turbine blades is an inherent attribute of their use, and measurement of the characteristics of this blade vibration is not new. Rotor vibration is driven by two groups of forces, those being from stochastic processes in the fluid flow and from the periodic traversing of high and low pressure regions from the preceding stator blade row. Shown in Figure 1 is a general schematic of a gas turbine where the stator and rotor blade stage arrangement can be seen. As the turbine engine rotates, rotor blade stages rotate around past stator blade stages such that the blades “cut” or pass through the trailing wakes of the stator blades. This wake passing is illustrated in Figure 2 with the plot of the velocity contours for three rotor and stator blade passages for three different angular offsets during rotation. The forces on the rotor blades from this wake crossing are periodic with stator blade passing frequency, sp f , the oscillating rotor blade surface pressure at this stator passing frequency is shown in Figure 8 in the results of this study. This is a typical result which has been found in many experimental investigations, with a characteristic type result for rotor blade lift coefficients versus angle of rotation is shown in Figure 3, taken from ref. [9].
Figure 1: Schematic of gas turbine, displayed is one rotor and stator blade stage without the external casing.
Figure 2: Longitudinal velocity contours plotted for three rotor-stator blade passages at three different angular offsets. Note the passing of the rotor blades through the trailing stator blade wakes
Figure 3: Unsteady aerodynamic lift coefficients calculated from measurements by pressure probes located on rotor blades, for different blade rows and for different engine design points as shown. [9]
th
The driving force on the r rotor blade may then be described mathematically as a sum of periodic forces at multiples of shaft speed, being: [4]
f t r
r
A cos i i
i 0
2 s r 1 b
spf
t r
(1)
s r 1 round 2 b
(2)
where Ai are the Fourier coefficients, ‘s’ is the number of stator blades and ‘b’ the number of rotor blades and r is phase offset between when each rotor blade aligns with a stator blade. If the forces acting on the rotor blades are harmonic with stator pass frequency, then the motion of the rotor blades will also be periodic with the same frequency as the driving forces. It has been shown that if the pressure profile around the rotor blades follows the motion of the oscillating blade then processing of this measured pressure signal could lead to a means of measuring blade vibration information [2]. This interaction between the blade motion and the forces driving the motion is complex and although the pressure profile around the moving blade will change with time, it is thought that it will in general have the same profile and be steady with time as compared to the mean pressure. This hypothesis that the rotor blade pressure profile follows the rotor blade motion will now be investigated through numerical FEA and CFD modelling of a gas turbine with coupled fluid and blade vibration interaction.
3 Computational Models This study employs the computational modelling of the rotor blade fluid structure interaction and examines how the rotor blade vibrations affect the pressure profile around the rotor blade. Within this section the computational models and modelling procedure will be explained. A fluid structure interaction model was developed using the FEA and CFD multi-physics environment available in ANSYS 12.1 Workbench. The time transient model developed is coupled by first solving the fluid domain until convergence is reached for the given boundary conditions at a given time step in ANSYS CFX. The forces from the fluid domain at the fluid structure boundary, being the rotor blade surface, are then passed to the ANSYS structural solver and the resulting displacement is calculated. These resulting displacements are then passed back to the CFD solver, with this process continuing until convergence between the boundary loads and displacements is achieved. The solver then moves onto the next time step and the process is then repeated. 200 time steps per stator passing period was also used in the model resulting in an RMS Courant number less than 1.
3.1 FEA The finite element model was constructed to represent rotor blade vibration limited to a single equivalent blade mode by attaching an essentially stiff aerofoil to a soft spring and limiting the motion to a single direction as shown in Figure 4(a). Properties of the finite element model can be found in Table 1. Table 1: Structural domain properties Property Rotor blade natural frequency (bounce) Stator passing frequency,
spf
Equivalent spring stiffness Maximum blade displacement
Value 13360 Hz 5093 Hz 5000 N/m 0.2 mm
(b)
(a)
(c)
Figure 4: Computational mesh for the rotor blade structural domain, zoom around rotor blade tip, and coupled fluid-structure domain respectively. The rotor blade vibration is restricted in the direction shown in (a).
3.2 CFD A pseudo two dimensional computational fluid dynamic model of a single rotor and stator blade passage was constructed using ANSYS CFX. The domain is not a conventional two dimensional model in the fact that a thin slice of a helical section of the fluid is modelled with a finite width in order to satisfy the commercial CFD code boundary conditions. The fluid domain was constructed from 5 2.25x10 elements, with the mesh around the rotor blade leading edge shown in Figure 4(b). The current number of elements is not sufficient to resolve some of the flow and will need to be increased in further analysis. Boundary conditions for the fluid domain are shown in Figure 5 and listed in Table 2. The aerofoil profile is a NACA4506 with a chord length of 50mm. The flow angle of incidence over each blade was made as close to 0 degrees as possible to give the most favourable flow conditions. The fluid domain was also limited to a single stator and rotor blade passage as shown in Figure 5 to reduce the computational domain, meaning that the underlying rotor to stator blade stage numbers would be of an equal number. A more realistic ratio of stator to rotor stage blade numbers would be of a non-integer number requiring the fluid domain to include as many passages as the lowest common denominator of the rotor and stator blade numbers. A first order backward Euler differencing scheme was used for the transient term with a SST turbulence model employed for the advection terms in the Navier-Stokes equation. The solution was -6 assumed to have converged when the residuals of all equations had been reduced to 10 at each time step.
Rotational periodicity
Rotational periodicity
Fluid-structure boundary
Inlet
Rotational periodicity
Outlet
Figure 5: Fluid domain schematic with boundary conditions shown
Table 2: Fluid domain boundary conditions Boundary
Condition Vu = 89 m/s Vv = 122 m/s Average static pressure = 0 Free slip wall Free slip wall Stationary no slip wall Rotating no slip wall, 100 rad/s
Inlet Outlet Casing surface Hub surface Stator blade Rotor blade
4 Results/Discussion Results from two computational models, one without any fluid-structure coupling i.e. the rotor blade rotates but is not permitted to vibrate, and a second where the complete two way fluid-structure interaction is present are investigated and reported below. The pressure in the fluid domain is interrogated at various locations, as shown in Figure 6, being across the rotor blade surface, the variation of the pressure with time at four specific locations and the changes in pressure across a vertical plane.
(4)
Vertical Plane 1
(3) (2)
(1)
Figure 6: Pressure measurement locations on rotor blade. Four pressure tapping locations and vertical plane.
2
x 10
4
1 0 ) −1 a P ( e r −2 u s s e r P−3
−4 −5 −6 0
Suction side uncoupled Pressure side uncoupled Suction side coupled Pressure side coupled 0.2
0.4 0.6 0.8 Normalised distance from TE ’x/c’
1
Figure 7: Surface pressure on rotor blade for both the coupled and uncoupled case averaged over two stator passing periods Shown in Figure 7 is the time averaged pressure over the rotor blade surface for both the coupled and uncoupled case, averaged over two stator passing periods after reaching a quasi-steady solution. The difference in pressure can be seen to be minimal and that the same pressure distribution shape is observed for when blade vibration is both present and absent. This small change in the time averaged surface pressure on the rotor blade with and without blade motion was also observed by Hall and Ekici [6]. Although the time averaged pressure shows little change between the coupled and uncoupled case, it does not describe how the pressure varies with time, and whether this variation is different between the two cases, i.e. does the pressure profile around the rotor blade follow the blade’s motion? To examine the change in pressure with time four pressure tappings will be observed over two stator passing periods, the normalised absolute pressure at these four pressure tapping locations for the uncoupled case is shown in Figure 8(a). It is observed in this figure, that the pressure on the rotor blade is periodic with stator passing frequency with a variation of approximately less than 10% over the period for the four locations. This variation with pressure is what causes the oscillating forces on the blade and thus drives blade motion. The same corresponding pressure tapping locations for the coupled case are shown in Figure 8(b). Again the periodic pressure fluctuation pattern can be observed as in the uncoupled case for all locations. The pressure locations 1 and 4 are significantly less smooth in the coupled case when compared with the uncoupled case. From observation of the velocity contours of the coupled case the rotor blade trailing edge suffers from some wake separation which is most likely the cause of the less comparable results for location 1. Increasing of the mesh density should also provide a smoother pressure profile around the leading edge which can been seen to affect the pressure distribution on the pressure side of the leading edge in Figure 7, which would also affect the stability of the pressure measurements at the tapping location 4. The displacement of the rotor blade over two periods is shown in Figure 9(b). It is clear that the displacement is periodic with stator passing frequency. It may be noted however that the minimum displacement is not the same for both periods shown. From observations of previous periods, this difference is reducing for each cycle, and should be removed if the computations are continued over further cycles. The time averaged pressure over two cycles along the vertical plane, c.f. Figure 6, is also shown in Figure 9(a), the maximum and minimum pressures for this two cycle period are shown in addition, with only minor variations in this pressure distribution with time.
Finally a comparison of the change in pressure over time for both cases is shown in Figure 10, although there are differences in the magnitude of the pressure for the two cases, the general trend of the pressure takes the same form. 1.02 1
1
(a)
(b) 0.95
e r u s 0.98 s e r p e t u 0.96 l o s b a d 0.94 e s i l a m r o 0.92 n
e r u s s e r p e 0.9 t u l o s b a d e 0.85 s i l a m r o n
location 1 location 2 location 3 location 4
0.9 0.88 0
0.5
1 stator passing period
location 1 location 2 location 3 location 4
0.8
1.5
2
0.75 0
0.5
1 stator passing period
1.5
2
Figure 8: Instantaneous pressure at the 4 pressure tapping locations on the rotor blade surface. (a) uncoupled case. (b) coupled case. 1
−0.21
e 0.8 c a f r 0.6 u s l i o 0.4 f o r e a
time ave max min
(a)
(b)
−0.215 ) m m ( t −0.22 n e m e c a l p −0.225 s i d e d a l b −0.23 r o t o r
0.2
m o r f 0 e c n a −0.2 t s i d d −0.4 e s i l a −0.6 m r o n
−0.235
−0.8 −1 −15000
−10000
−5000
0
−0.24
0
pressure (Pa)
0.5
1 stator passing period
1.5
2
Figure 9: (a) Pressure distribution along vertical plane 1, c.f. Figure 6, for the coupled case. Plotted are the max, min and time averaged values over two stator passing periods. (b) Rotor blade displacement for the coupled case over two stator passing periods. −2.75
x 10
4
uncoupled coupled −2.8
−2.85 ) a P −2.9 ( e r u s s e −2.95 r p
−3
−3.05
−3.1
0
0.5
1 stator passing period
1.5
2
Figure 10: Comparison for the coupled and uncoupled case of pressure tapping at location 3 on the rotor blade over two stator passing periods
5 Conclusion s Reported in this paper are the results of a computational study into the changes in the pressure around a gas turbine rotor blade whist it is vibrating. A two-way coupled finite element and computational fluid dynamic model was constructed using the commercial ANSYS Workbench software. Results for this fully coupled model were compared against the pressure distribution results for a CFD model of the same system without blade vibration being present. It was observed that the time averaged pressure along the rotor blade surface does not vary significantly between the uncoupled and coupled case, with both models displaying the same pressure distribution profile. Examination of changes in the pressure at four locations across the blade surface with time, showed that both the uncoupled and coupled models produced similar pressure changes which are harmonic with the stator passing frequency, driving blade motion at this frequency. The blade motion amplitude in two-way coupled model was reasonably small at 0.4% of the interblade spacing distance, and as such the change in pressure between the cases would be assumed to be negligible. It can be tentatively concluded from these results that the rotor blade pressure profile follows the blade motion. Before this conclusion can be made with more certainty, longer computation time is required to obtain complete convergence to the quasi-steady solution at the same time as increasing the CFD mesh density so that all flow effects can be resolved and the effects of the larger blade motion amplitudes need to be investigated.
Ac kn ow led gement s Grateful acknowledgment is made for the financial assistance given by the Australian Defence Science and Technology Organisation, through the Centre of Expertise in Helicopter Structures and Diagnostics at UNSW.
References 1 2 3
4 5 6 7
8 9
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