forced response excitation due to stagger angle … · forced response excitation due to stagger...

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Forced Response Excitation due to Stagger Angle Variation in a Multi-Stage Axial Turbine Thomas Hauptmann 1 , Jens Aschenbruck 2 and Joerg R. Seume 1 1 Institute of Turbomachinery and Fluid Dynamics, Leibniz Universit¨ at Hannover Appelstr. 9, 30167 Hannover, GERMANY 2 Formerly at the Institute of Turbomachinery and Fluid Dynamics, Leibniz Universit¨ at Hannover Appelstr. 9, 30167 Hannover, GERMANY ABSTRACT Blade repair is often economically more attractive than the replace- ment of damaged blades by spare parts. Such regenerated turbine blades, however, can introduce non-uniform flow conditions which lead to additional forced response excitation of blades. A forced response excitation due to a typical geometric variation, introduced through current repair methods applied in an upstream stage, is in- vestigated using a fluid-structure interaction (FSI) model previously experimentally validated in a five-stage axial turbine. In this study, geometrical variations are applied to the stator vane of the fourth stage of the five-stage axial turbine. The reference configuration, without variations, is compared with experimental data. The focus of the analysis is the determination of the aerodynamic excitation in a multi-stage setup. For both configurations, with and without vari- ations, the stage loading coefficient of the last turbine stage remains constant. In contrast, the aerodynamic work acting on the last rotor blade increases by a factor of 4 dependent on the operating point. The vibration amplitude of the downstream blade is determined us- ing a unidirectional fluid-structure interaction approach. The impact of the variations on the vibration amplitude decreases by a factor of 10 with increasing number of blade rows between the modified vane row and the analyzed blade row. However, the geometric variations induce vibration amplitudes 4 times higher than the reference case. Based on the methodology used, a linear correlation between the excitation of the blade by the aerodynamic work and the vibration amplitude is shown to exist. NOMENCLATURE c absolute velocity c p pressure coefficient D damping ratio f frequency F force h enthalpy i inner ˙ m mass flow rate n normal vector to blade surface o outer p pressure ˜ p pressure fluctuation P power r radius rpm rotor speed T temperature T period of time u circumferential velocity u mode displacement W aero aerodynamic work α yaw angle γ pitch angle λ stagger angle ψ stage loading coefficient Indices dyn dynamic in turbine inlet local local max maximum out turbine outlet stat static tot total u circumferential 1 stage inlet 2 stage outlet Abbreviations BPF blade passing frequency CFD computational fluid dynamics CRC collaborative research center EO engine order FSI fluid-structure interaction LE leading edge MP measuring plane OP operating point PS pressure side SS suction side TE trailing edge INTRODUCTION The overhaul process of jet engines, also referred to as regenera- tion, represents 8% of the operating costs of an airplane [1] and is therefore of great interest. The main cost factor at the overhaul is caused by the blades, which are responsible for approximately 50% of the cost. This cost is mainly incurred by the replacement of worn blades from the high-pressure turbine, because they are one of the most highly loaded parts and therefore subject to substantial wear. For this reason the collaborative research center (CRC) 871 “Re- generation of Complex Capital Good” aims to develop the scientific basis for the overhaul of jet engines. The main objective is to save as many of the worn components as possible [2]. International Journal of Gas Turbine, Propulsion and Power Systems October 2017, Volume 9, Number 3 Copyright © 2017 Gas Turbine Society of Japan Manuscript Received on January 30, 2017 Review Completed on August 9, 2017 1

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Page 1: Forced Response Excitation due to Stagger Angle … · Forced Response Excitation due to Stagger Angle Variation in a ... tor vanes one stage further upstream of the ... highest impact

DRAFT: International Journal of Gas Turbine, Propulsion and Power Systems

Forced Response Excitation due to Stagger Angle Variation in aMulti-Stage Axial Turbine

Thomas Hauptmann1, Jens Aschenbruck2 and Joerg R. Seume1

1Institute of Turbomachinery and Fluid Dynamics, Leibniz Universitat HannoverAppelstr. 9, 30167 Hannover, GERMANY

2Formerly at the Institute of Turbomachinery and Fluid Dynamics, Leibniz Universitat HannoverAppelstr. 9, 30167 Hannover, GERMANY

ABSTRACTBlade repair is often economically more attractive than the replace-ment of damaged blades by spare parts. Such regenerated turbineblades, however, can introduce non-uniform flow conditions whichlead to additional forced response excitation of blades. A forcedresponse excitation due to a typical geometric variation, introducedthrough current repair methods applied in an upstream stage, is in-vestigated using a fluid-structure interaction (FSI) model previouslyexperimentally validated in a five-stage axial turbine. In this study,geometrical variations are applied to the stator vane of the fourthstage of the five-stage axial turbine. The reference configuration,without variations, is compared with experimental data. The focusof the analysis is the determination of the aerodynamic excitation ina multi-stage setup. For both configurations, with and without vari-ations, the stage loading coefficient of the last turbine stage remainsconstant. In contrast, the aerodynamic work acting on the last rotorblade increases by a factor of 4 dependent on the operating point.The vibration amplitude of the downstream blade is determined us-ing a unidirectional fluid-structure interaction approach. The impactof the variations on the vibration amplitude decreases by a factor of10 with increasing number of blade rows between the modified vanerow and the analyzed blade row. However, the geometric variationsinduce vibration amplitudes 4 times higher than the reference case.Based on the methodology used, a linear correlation between theexcitation of the blade by the aerodynamic work and the vibrationamplitude is shown to exist.

NOMENCLATUREc absolute velocitycp pressure coefficientD damping ratiof frequencyF forceh enthalpyi innerm mass flow rate�n normal vector to blade surfaceo outerp pressurep pressure fluctuationP powerr radiusrpm rotor speedT temperatureT period of timeu circumferential velocity

�u mode displacementWaero aerodynamic workα yaw angleγ pitch angleλ stagger angleψ stage loading coefficient

Indicesdyn dynamicin turbine inletlocal localmax maximumout turbine outletstat statictot totalu circumferential1 stage inlet2 stage outlet

AbbreviationsBPF blade passing frequencyCFD computational fluid dynamicsCRC collaborative research centerEO engine orderFSI fluid-structure interactionLE leading edgeMP measuring planeOP operating pointPS pressure sideSS suction sideT E trailing edge

INTRODUCTIONThe overhaul process of jet engines, also referred to as regenera-tion, represents 8% of the operating costs of an airplane [1] and istherefore of great interest. The main cost factor at the overhaul iscaused by the blades, which are responsible for approximately 50%of the cost. This cost is mainly incurred by the replacement of wornblades from the high-pressure turbine, because they are one of themost highly loaded parts and therefore subject to substantial wear.For this reason the collaborative research center (CRC) 871 “Re-generation of Complex Capital Good” aims to develop the scientificbasis for the overhaul of jet engines. The main objective is to saveas many of the worn components as possible [2].

International Journal of Gas Turbine, Propulsion and Power Systems October 2017, Volume 9, Number 3

Copyright © 2017 Gas Turbine Society of JapanManuscript Received on January 30, 2017 Review Completed on August 9, 2017

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One approach in the CRC 871 is to investigate and reduce the aero-dynamic excitation which occurs due to regeneration-induced vari-ances. This excitation can be introduced by geometrical variationsin the upstream blade rows. Hence, the main objective of the presentstudy is to determine the effect of regeneration-induced variancesin the blading on the excitation of the blade row one stage furtherdownstream. Additionally, the focus is set on estimating the vibra-tion amplitude due to geometric variations using a forced responsemethod.Several numerical studies exist in the literature [3–8] investigatingthe influence of geometric variations on the aerodynamic force ina single stage turbine. All these studies have shown that the influ-ence of several kinds of geometric variation in an upstream statorvane row is significant. These results were verified by experimen-tal investigations in [7,9]. The aerodynamic excitation mechanismsoccuring in a high pressure turbine stage were numerically investi-gated in [10]. It was shown that the potential excitation mechanismin the investigated turbine stage was dominant in comparison withthe wake excitation effect. In [11] geometrical variations on the sta-tor vanes one stage further upstream of the excited blade row wasinvestigated. It was shown that a stagger angle variation has thehighest impact in the vibration analysis by modifying the vane ge-ometry of the fourth stage of a five-stage axial turbine even thoughit is one stage further downstream.To determine the aerodynamic excitation in turbomachinery, a uni-directional fluid-structure interaction (FSI) is used because the re-quired computational effort is lower compared to bidirectional FSI.Additionally, the unidirectional FSI assumes that the vibrations aresmall, and therefore the reaction on the fluid is negligible. This isonly valid for a forced response analysis. For using this approachunsteady CFD calculations are conducted in order to determine theunsteady aerodynamic loads acting on the structure. Subsequently,the aerodynamic loads are applied to a structural model in a finiteelement analysis. This analysis is conducted using damping valuesdetermined from tip-timing measurements.

OUTLINEThe first part of this paper presents the five-stage axial air turbineand the geometric variation implemented into the turbine blading.The geometric variation is typical for those introduced through cur-rent repair methods [12]. The influence of this variation on the aero-dynamics downstream of the varied vane is numerically analyzedusing CFD. The reference configuration is also compared with ex-perimental data. The focus is on the impact of the variation onestage downstream, in order to determine the influence of the varia-tion on the wake behavior, as the wake excitation is one of the mainexcitation sources [10].Finally, forced response analyses are conducted with the varied sta-tor vanes. The analyzed blade row is not adjacent to the varied vanerow. The vibration amplitude is numerically predicted using a uni-directional fluid-structure approach validated experimentally in [9].The results are compared with the reference case, and additionallywith a variation in the vane row adjacent to the excited blade.

TEST FACILITYA multi-stage axial turbine test facility in the Institute of Turboma-chinery and Fluid-Dynamics is used for the numerical and experi-mental investigations in this study. The modular casing of the tur-bine enable the use of several turbine configurations using variousinner casing contours and blade designs. In this study, a five-stageaxial turbine configuration is used to investigate the influence of theregeneration-induced variances on the aerodynamic and aeroelasticperformance (see Fig.1).The single solid rotor consists of 30 axial fir tree grooves, in whichthe rotor blades are mounted. Therefore, the rotor blade count ofthe fifth stage is identical to stages one to four. Additionally, allfive vane rows have the same vane count of 29. The stator vanesare mounted in the inner casing of the turbine. The fifth stage of

MP2_10 MP2_51

MP3_02

Diffuser St. 5 St. 4 St. 1 … 3

MP2_10

Tip-Timing

MP2_52

MP2_52

MP3_02

Tip-Timing

Modified vane row

MP2_41

MP2_41 MP2_51

Fig.1: Five-stage axial turbine

Table 1: Operating points of the five-stage axial turbine

Operating Points OP1 OP2 OP3

Mass flow rate m in kg/s 3.3 4.3 8.5

Rotor speed in n rpm 2000 4000 7500

Pressure ratio pin/pout 1.33 1.58 2.74

Inlet temperature Tin in K 341 364 423

Outlet temperature Tout in K 320 324 330

the axial turbine was specially designed for the investigation of theaeroelastic behavior. Further information, including the design pro-cess are presented in [8]. In contrast to the investigations in [3], [5],and [9] geometric variances were applied on the stator vanes onestage upstream of the excited blade row. Therefore, the stator vanesof the fourth stage were modified with a regeneration-specific geo-metric variation. This variation was implemented in the fourth stageof the turbine on every second stator vane, with reference vanes inbetween. In the present study a stagger angle variation of λ = 1.5deg is implemented in the fourth stage and compared with the refer-ence case where all vanes are identical. The alternating distributionof the geometric variation is typical after the regeneration of turbineblades because the mistuning results in a positive effect of the flutterstability ([13] and [14]).This study is performed for two part-loaded operating points (OP1and OP2) and the design point (OP3) listed in Tab.1. At these oper-ating points only one crossing exist with the eigenfrequencies andthe blade passing frequency at OP1. At OP2 and OP3, no resonancecase exists with the BPF. The first and the second eigenfrequenciescan only be excited by the 8th and the 15th engine order as depictedin the Campbell Diagram (Fig. 2). These engine orders normally donot occur. However, the 15th engine order can appear as a result ofan alternating vane pattern of reference and varied blades. Such adistribution can lead to an excitation of the second eigenfrequencyby the 15th engine order.

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0

500

1000

1500

2000

2500

3000

3500

4000

0 2000 4000 6000 8000

Fre

qu

en

cy i

n H

z

Rotational speed in rpm

EO 29 EO 30

EO 1

2nd: 1841 Hz

1st: 967 Hz

OP1 m=3.3 kg/s π= 1.33

OP2 m=4.3 kg/s π= 1.58

OP3 m=8.5 kg/s π= 2.74

. . .

3rd: 2952 Hz

4th: 3954 Hz

EO 15

1st Mode 2nd Mode 3rd Mode 4th Mode

EO 8

Fig.2: Campbell-Diagram and eigenmodes of rotor blade 5 [9]

EXPERIMENATL SETUPFor the investigation of the excitation and the vibration behavior ofthe fifth rotor blade row, detailed instrumentation is selected to col-lect detailed flow data. In Figure 1, the main measurement planesused for the aerodynamic measurements are indicated. In MP2 10,radial probe traverses using 5-hole vector-probes are conducted tocapture the inlet boundary conditions. The outlet boundary con-ditions are captured by rotatable total pressure and total tempera-ture rake-probes in MP3 02. Five rake-probes are equally spacedin circumferential direction and implemented with combined kiel-head and 5-hole-probe rakes. In addition to these measurements,radial and circumferential 5-hole-probe traverses were conducted inMP2 51 and MP2 52 in front and behind the rotor blade row 5, inorder to capture the aerodynamic flow field exciting the rotor bladerow. The mass flow rate is determined by a Venturi nozzle located9 m upstream of the turbine inlet.The probe traverses are conducted with pneumatic 5-hole-probeswith a probe head diameter of 3 mm (see Fig. 3). The 5-hole probesare calibrated for a range yaw (−24◦ <α < 24◦) and pitch (−30◦ <γ < 30◦) angles and for Mach numbers between 0.1 < Ma < 0.9.The thermocouple is located in a kiel-head above the probe headfor measuring the temperature. From measuring pressure and tem-perature with the 5-hole-probe, velocity, the Mach number, and flowangles can be calculated. These values are then used to determinethe excitation of the rotor blade row. As described in [15], [16], and[17], the total pressure, static pressure, and flow angles can be cal-culated from the measured pressures and the calibration coefficientsof the probe.For the detection of the blade vibration, a commercial optical tip-timing system by AGILIS is used. The vibration amplitudes and theeigenfrequencies are determined by eight optical probes circumfer-entially distributed at the same axial position at the trailing edgeof the fifth rotor blade row. All probes measure the time of arrival(TOA) of all blades on each revolution. The arrival times are thenconverted to deflections, as the rotational velocity and the radius atthe measurement location is known. A detailed description is givenin [9].The first and second eigenmode of the fifth rotor blade have theirhighest mode displacement at the trailing edge. These eigenmodes

10mm

Fig.3: Pneumatic 5-hole probe with probe-head diameter of 3 mm

are of particular interest because they can be excited by the bladepassing frequency and lower engine orders at the relevant operatingpoints. The tip-timing probes are therefore placed at this axial po-sition to determine the highest blade deflection and to get accuratedata for the eigenfrequency.The circumferential probe positions are equal to the positions in[9]. In this case, eight probes are circumferentially distributed us-ing an algorithm by AGILIS to ensure that the eigenfrequencies andengine orders can be captured accurately. This probe location algo-rithm is based on the Campbell-Diagram, and is optimized for themeasurement of synchronous blade vibrations. In this study, thefocus is on the investigation of synchronous vibrations. Therefore,a least-square model fitting (LSMF) analysis is used which is rec-ommended for these kind of vibrations. This method is briefly de-scribed in [18]. The LSMF analysis determines vibration frequency,phase, and deflection of each rotor blade.

AERODYNAMIC ANALYSISIn this section, the impact of the stagger angle variation on the aero-dynamic behavior is investigated. These results are necessary toshow the influence on the aerodynamic excitation of a geometricvane variation on the adjacent blade row and on the blade row onestage downstream. The results one stage downstream are of par-ticular interest, because they indicate the impact of the flow on thevibration behavior of the fifth rotor blade row. A dependence onsuch typical variations on the aerodynamic behavior is shown andcompared to the reference case.

Numerical SetupThe numerical investigations of the aerodynamic behavior wereconducted in detail for the reference case and the stagger angle vari-ation in the fourth stator vane row. The numerical model used in thisstudy is depicted in Fig. 4. The inlet conditions for the CFD simula-tions were determined by steady CFD simulations of the completefive stage air turbine. The velocity direction and the temperaturewere derived from the preceding simulations according to the in-vestigated operating points. Afterwards, they were used as inletboundary condition for stage four with a specified mass flow rate.The outlet boundary condition was set with a specified static pres-sure at diffuser outlet. A medium turbulence intensity of 5% wasused at the inlet of stage 4.The numerical model consists of two passages of the fourth and thefifth stage and one pitch of the diffuser. Instead of 29 stator vanesand 30 rotor blades, the simulations were conducted with 30 vanesand 30 blades. This ensures an equal pitch of the domains for theunsteady simulations. The vane size was not modified. The scalingof 29 vanes to 30 vanes has a negligible effect on the aerodynamics

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sliding mesh

sliding mesh

sliding mesh St. 4 vane

St. 4 blade St. 5 vane

St. 5 blade

flow direction

Fig.4: Numerical model

and turbine performance, as shown in [19].For the current investigation, the reference and modified statorvanes of the fourth stage were modeled as an alternating distri-bution. Rotational periodicity was specified at the circumferentialboundaries of the 24 degree segment. In the steady simulations, therotor-stator interfaces were defined as a frozen rotor interface. Thisinterface is selected to ensure that the wake behavior is propagatedthrough the stages. The interface between the rotor blade row 5 andthe diffuser was set to a mixing plane interface in order to reducecomputational effort.For the unsteady simulations, the frozen rotor interfaces were re-placed by a transient rotor-stator interface (sliding mesh). The nu-merical model was discretized with 14.96 million nodes in total.All domains were meshed solely with hexahedral elements. In par-ticular, the stator and rotor domains have a high mesh resolutionto minimize discretization errors and to ensure the propagation ofthe wake through all stages. This high resolution was necessary toobtain the accurate excitation on the blade surface for the forcedresponse analysis. It is important that the numerical error causedby the mesh resolution is negligible. A mesh study of the numeri-cal domain has been performed already in previous studies of thisproject to confirm this (see [12] and [11]).All simulations were performed with the CFD software ANSYSCFX 15.0 using the SST turbulence model. In the unsteady compu-tations, one half rotation of the rotor was simulated for stabilizationwith 32 time steps per pitch. Subsequently, another half rotationwith 64 time steps per pitch was conducted. These time steps wereused for the determination of the unsteady surface pressures.

Steady Aerodynamic ResultsThis section presents the conducted measurements of the referenceconfiguration and the comparison with the predicted flow fieldby the numerical model. Additionally, the simulated case ofthe stagger angle variation in the fourth stage is analyzed forcomparison. The experimental measurements depicted in thefollowing sections are indicated with error bars. All error bars areindicated with a 95% confidence interval of the measured values.These error bars include the error propagation due to accuracy ofthe measurement instrumentation, repeatability and calibration.In a first step the aerodynamic performance of the fifth stage isevaluated with the stage loading coefficient ψ

0 0.4 0.8 1.2 1.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

stage loading coefficient ψ

Span

hei

ght

Exp Reference OP2CFD Reference OP2

Fig.5: Radial distribution of the stage loading at OP2

∆htot =Pm

=∫ 2

1u ·dcu = u · (cu2− cu1) (1)

ψ =−∆htot

u2 (2)

The index 1 denotes the measuring plane between stator 5 and rotor5 (MP2 51) and the index 2 the plane behind the rotor (MP2 52).By determining the velocity in circumferential direction of theprobe measurements, the stage loading coefficient of the fifth rotorblade can be calculated.Figure 5 shows the absolute value of the stage loading from hubto shroud for the reference configuration at OP2. It shows a goodagreement between the simulations and the experiments over thecomplete channel height. The stage loading coefficient decreasesfrom hub to shroud and reaches its maximum at 10% channelheight.The averaged values of the stage loading coefficient at all operatingpoints are presented in Fig. 6. The stage loading coefficient ψ

at the investigated operating points is almost identical for bothconfigurations. Therefore, the angular momentum of the flow in thecircumferential direction, averaged over the span height, upstreamand downstream of the fifth rotor blade row is almost identical forthe configurations with and without modifications.For the aeroelastic investigation of the fifth rotor blade row, theinflow conditions must be determined in order to extract the wakeexcitation source. For this purpose, the aerodynamic flow field isanalyzed at different axial positions. The influence of the staggerangle variation on the aerodynamics downstream of the variedvane is compared to the reference case, in order to estimate theinfluence of the wake. Beside the potential effect, the wakes ofthe vanes are the main excitation mechanism of the rotor blades.Accordingly, the wake behavior is of main interest in the analysisof the aerodynamic behavior of the flow. In MP2 51, the simulateddata of the reference configuration are compared with experimentaldata by circumferential probe traverses at OP2.First, the pressure distribution of vane row 5 has to be determined

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OP1 OP2 OP30.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

Operating Points

stag

e lo

adin

g co

effic

ient

ψ

CFD ReferenceCFD Multi−Stage Alternating Vanes

Fig.6: Averaged stage loading coefficient for all operating points

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

150% span heigth − OP2

normalized chord length

norm

aliz

ed P

ress

ure

CFDExperiment

Fig.7: Pressure distribution at midspan of vane 5

to make sure that the predicted inflow conditions are in accordancewith the experimental data. Figure 7 shows the comparison ofthe pressure distribution throughout the chord length between theexperimental data and CFD simulations at 50% span height forOP2. The experimental and simulated data are in a good agreementaround the chord length. Thus, the inflow condition to the fifth rotorblade are predicted well by CFD. To analyze the wake behavior,the total pressure distribution is investigated behind stator vane row4 and behind stator vane row 5 at 80% span. The investigationin these planes also indicates the propagation of the vane wakethrough the blade rows. The simulated total pressure distributionis calculated from the time-averaged data of the unsteady CFDsimulations. The relative total pressure ptot /ptot,in downstream ofthe stator vane row 4 and 5 is shown as a function of the normalizedpitch in Fig. 8 to Fig. 11. The stagger angle variation causes ashift by 6.5% of the wake position in the circumferential directiondirectly behind the modified vane row in MP2 41 at OP3 (seeFig.8). Additionally, the alternating stagger angle variation causes a0.44% higher deficit in relative total pressure. The large differencein total pressure in the mid-passage between both configurations is

0,535

0,540

0,545

0,550

0,555

0,560

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

pto

t/p

tot,

in

Pitch

CFD Reference CFD Multi-Stage Alternating Vanes

0.44%

6.5% 0.535

0.540

0.545

0.550

0.555

0.560

pto

t/p

tot,

in

0,535

0,540

0,545

0,550

0,555

0,560

1 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2

pto

t/p

tot,

in

Pitch

CFD Reference CFD Multi-Stage Alternating Vanes

Pitch

0.0 1.2 1.4 1.6 1.8 2.0 0.2 0.4 0.6 0.8 1.0

Fig.8: Circumferential total pressure distribution for operatingpoint 3 (OP3) in MP2 41 at 80% span height

0,742

0,744

0,746

0,748

0,750

0,752

0,754

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

pto

t/p

tot,

in

Pitch

CFD Reference CFD Multi-Stage Alternating Vanes

0.07%

6%

0.754

0.752

0.750

0.748

0.746

0.744

0.742

pto

t/p

tot,

in

Pitch

0,742

0,744

0,746

0,748

0,750

0,752

0,754

1 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2

pto

t/p

tot,

in

Pitch

CFD Reference CFD Multi-Stage Alternating Vanes

0.0 1.2 1.4 1.6 1.8 2.0 0.2 0.4 0.6 0.8 1.0

Fig.9: Circumferential total pressure distribution for operatingpoint 2 (OP2) in MP2 41 at 80% span height

caused by the change of the stagger angle of every second vane. Asalready shown in [12], a stagger angle variation causes a shift of thewake in circumferential direction, and also causes a reduction oftotal pressure. Because of the stagger angle variation more lossesare generated in the vane passage due to an earlier separation.Compared to OP3, the stagger angle variation causes a change ofthe wake deficit in relative total pressure of 0.07% in MP2 41 asshown in Fig. 9. However, the stagger angle variation also results ina shift of the wake position in the circumferential direction by 6%.Therefore, a significant disturbance of the flow field, especiallyin the wake region is detected downstream of the stagger anglevariation at both operating points.Figure 10 illustrates the impact of the stagger angle variation on theflow field one stage downstream of the implemented variation inMP2 51 at OP3. One stage downstream, no remarkable shift of thewake position is detected. Apart from that, the alternating staggerangle variation in vane row 4 causes a reduction in the wake deficitby 0.2%. Total pressure in the passage area is reduced by 0.1%due to the influence of the stagger angle variation. Beside thesedifferences, the total pressure distribution is similar to the referencecase.In comparison with OP3 it can be seen in Fig. 11 that the stagger

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0,440

0,445

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0,465

0,470

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

pto

t/p

tot,

in

Pitch

CFD Reference CFD Multi-Stage Alternating Vanes

0.2%

0.440

0.445

0.450

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0.465

0.470 p

tot/p

tot,

in

Pitch

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0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

pto

t/p

tot,

in

Pitch

CFD Reference CFD Multi-Stage Alternating Vanes

0.0 1.2 1.4 1.6 1.8 2.0 0.2 0.4 0.6 0.8 1.0

Fig.10: Circumferential total pressure distribution for operatingpoint 3 (OP3) in MP2 51 at 80% span height

0,701

0,702

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0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

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PitchExp Reference CFD Reference CFD Multi-Stage Alternating Vanes

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pto

t/p

tot,

in

Pitch

Exp Reference CFD Reference CFD Multi-Stage Alternating Vanes

0.08%

0.1%

0.0 1.2 1.4 1.6 1.8 2.0 0.2 0.4 0.6 0.8 1.0

0.709

0.708

0.707

0.706

0.705

0.704

0.703

pto

t/p

tot,

in

0.702

0.701

Pitch

Fig.11: Circumferential total pressure distribution for operatingpoint 2 (OP2) in MP2 51 at 80% span height

angle variation also causes a negligible shift of the wake positionin the circumferential direction one stage further downstream inMP2 51 at OP2. The variation causes a reduction of the wakedeficit by 0.08%. However, at the same time 0.1% higher totalpressure is detected in the passage area. This is the same differenceas determined at OP3 in the passage area. In conclusion, adisturbance of the flow field compared to the reference case isdetected for both operating points one stage further downstreamas it was detected immediately downstream of the modified statorvane row 4. These changes in the flow field between referencecase and the investigated geometric variation lead to small pressureperturbations on the rotor blade surface. The influence on theexcitation of the rotor blade in blade row 5 is shown in the nextsection.In Fig. 11 experimental data of the measurements of the referencecase is included and indicated with error bars. All error barsindicate a 95% confidence interval of the measured values. In thepassage area, the predicted total pressure distribution is in accor-dance to the experimental data. The wake region is overestimatedby the numerical prediction and shows a difference of 0.4% inthe wake deficit. The reason for errors for an accurate predictionof the measured flow field, is the influence of the potential effect

0,000

0,002

0,004

0,006

0,008

0,010

0,012

0,014

0,016

Pre

ssure

Coeff

icie

nt

|cp|

Ref

Var

TE LE TE-180

-120

-60

0

60

120

180

PS SS

Phase

in °

0.016

0.014

0.012

0.010

0.008

0.006

0.004

0.002

0.000

Pre

ssu

re C

oef

fici

ent

|cp|

TE PS LE SS TE

Fig.12: Unsteady pressure amplitudes along the chord length forthe blade passing frequency (30EO) at OP3 - 80% spanheight

by the pneumatic five-hole probe. In [15] the authors showed, bysimulating the flow field including the five-hole probe, that thewake region is predicted accurately.

Unsteady Aerodynamic ResultsThe vibration of the rotor blades are caused by pressure fluctuationson the blade surface. These pressure fluctuations are influenced bythe change of the flow field, which can be modified by geometricvariations as described before. For this purpose, the unsteadypressure on the rotor blade is analyzed by the pressure coefficient∣∣cp

∣∣= pptot,MP51− pstat,MP51

=p

pdyn,MP51(3)

In this equation p denotes the pressure fluctuation on the blade sur-face and pdyn,MP51 the dynamic pressure in MP2 51. The rotorblades are not instrumented with unsteady pressure sensors. There-fore, the analysis is conducted with simulated data. In Figure 12, theunsteady surface pressure amplitude occurring at the blade passingfrequency (30EO) at OP3 is shown at 80% span height. The pres-sure fluctuation distribution close to the leading edge of the rotorblade differs only slightly on the pressure and suction side. Furtherdownstream on the rotor blade, the pressure fluctuations in the refer-ence case increases compared to the alternating stagger angle vari-ation. In contrast to that, the pressure fluctuations for the case withstagger angle variation occurring at the half blade passing frequency(15EO) are significantly higher around the chord length comparedto the reference (see Fig. 13). In the reference case only the 30EO(BPF) occurs. Thus, the pressure fluctuations of the reference caseare negligible for the 15EO, as shown in Fig. 13. The stagger anglevariation with the alternating vane pattern causes this additional fre-quency of the 15EO. This results in a higher excitation of the rotorblade in most of the chord length at 80% span height.

AERODYNAMIC WORKIn order to examine the excitation of the unsteady surface pressureson the eigenmodes of the rotor blade, the aerodynamic work Waerohas to be determined. The calculation of the aerodynamic work inEq.(4) and Eq.(5) is based on the method of [20] and [21]. Theintegral value of the aerodynamic work can be examined by

Waero =

∫ T0(∫ ro

ri

∮S p ·~n ·~u ds dr

)dt

max(|~u|)(4)

with the unsteady surface pressures p on the rotor blade, the normalblade surface vector ~n, and the local deflection ~u of the blade for aspecific eigenmode as described in [10]. As the pahse shift betweenaerodynamic excitation and vibration behavior of the blade is

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0,000

0,002

0,004

0,006

0,008

0,010

0,012

0,014

0,016

Pre

ssure

Coeff

icie

nt

|cp|

Ref

Var

TE LE TE-180

-120

-60

0

60

120

180

PS SS

Phase

in °

0.016

0.014

0.012

0.010

0.008

0.006

0.004

0.002

0.000

Pre

ssu

re C

oef

fici

ent

|cp|

TE PS LE SS TE

Fig.13: Unsteady pressure amplitudes along the chord length forone half of the blade passing frequency (15EO) at OP3 -80% span height

unknown, the most critical case is assumed. This is the case whenthe aerodynamic work is maximal. The maximum aerodynamicwork is calculated with

|Waero,max|=

∣∣∣∣∣∫ T

0(∫ ro

ri

∮S p ·~n ·~u ds dr

)dt

max(|~u|)

∣∣∣∣∣ . (5)

In Figure 14 the maximum aerodynamic work on the blade isshown for the reference case and the alternating stagger anglevariation implemented in stage 4 at all operating points. Theaerodynamic work is normalized to the aerodynamic work of thereference case in OP1. At OP1, there is a crossing of the bladepassing frequency at the first eigenfrequency. Thus, the maximumaerodynamic work is examined with the first eigenmode caused bythe unsteady surface pressure at the BPF.At OP2 the aerodynamic work is also examined for the firsteigenmode but with the unsteady surface pressure at the 15EO. Atthis operating point, a resonance case with the half blade passingfrequency (15EO) exists. As shown in the Campbell-diagram (seeFig. 2), a crossing between the 15EO and the second eigenmodeexists at OP3. Therefore, the aerodynamic work is determined withthese conditions and compared to the other operating points.Figure 14 shows that the alternating stagger angle variationreduces the excitation of the blade passing frequency at OP1.The maximum aerodynamic work is 22% lower compared to thereference case. At the other operating points an increase of themaximum aerodynamic work for the alternating vanes is detected.The implemented variation in an alternating pattern in vane row 4causes an increase by a factor of 3.5 at OP2, and by a factor of 4.0at OP3 compared to the reference case. This is due to the increasein the unsteady surface pressures around the chord length shown inFig. 13.The implemented stagger angle variation causes an additionalexcitation frequency. These results show a remarkable increase ofthe excitation of the first eigenfrequency at OP2 and of the secondeigenfrequency at OP3. In addition, it also causes a reduction ofthe excitation of the first eigenfrequency at OP1. At OP1, the firsteigenmode is close to resonance with the blade passing frequency.The excitation caused by vane row 4 is then superimposed withthe excitation by vane row 5 and the intensity at the blade passingfrequency weakened by vane row 5. This causes the reduction ofthe maximum aerodynamic work at OP1.The aerodynamic work can also be determined as local value onrotor blade with

Waero,local =

∫ T0 p ·~n ·~u dtmax(|~u|)

(6)

This local aerodynamic work is determined for each element of the

OP1 OP2 OP30.0

0.2

0.4

0.6

0.8

1.0

1.2

Operating Points

norm

aliz

ed a

erod

ynam

ic w

ork

W aero

ReferenceMulti−Stage Alternating Vanes

Fig.14: Maximum aerodynamic work on rotor blade 5 for all oper-ating points, normalized with the aerodynamic work of thereference configuration at OP1

rotor blade with the respective unsteady surface pressure and dis-placement of the eigenmode at the specific surface element. In Fig-ure 15 the local aerodynamic work on the rotor blade surface isshown. The trend of aerodynamic work as a function of the operat-ing point is the same as for the maximum aerodynamic work in Fig.14. At OP1, the excitation of the first eigenfrequency with its bend-ing mode shape is clearly visible. The highest aerodynamic workis located at these positions as the eigenmode has its highest dis-placements. The nodal line of the bending mode is indicated withthe white areas on the pressure side of the rotor blade. Compared toOP1, the aerodynamic work acting on the rotor blade is much lowerat the other operating points. For the reference configuration, almostno work is acting on the blade with the first eigenfrequency excitedby the 15EO at OP2. This is in accordance with Fig. 13 and Fig. 14.In contrast to this, the 15EO excites the first eigenfrequency in thecase of the stagger angle variation. The highest aerodynamic workacting on the rotor blade is located at the tip trailing edge wherethe highest mode displacement occurs. Similar results are shown atOP3. The second eigenfrequency has a low excitation at the 15EOin the reference case. The reference configuration has a negligible15EO content due to the numerical model, which used two passageswith periodic boundary condition. For the stagger angle variation,a significant increase of the local aerodynamic work is determined.In this case, the nodal line of the torsion mode is indicated by thewhite areas from hub to tip. The highest excitation is again locatedat the tip trailing edge.To conclude the aerodynamic analysis, the relative total pressureshows remarkable change of the wake position and wake deficit di-rectly behind the implemented stagger variation. One stage furtherdownstream, the total pressure shows a negligible effect on the wakeposition and minor effect on the wake deficit. Nevertheless, the re-sults in the flow passage and the unsteady pressure fluctuations onthe rotor blade in particular reveal that the alternating vane patternstill has an effect one stage downstream of the implemented vari-ation. The high difference in pressure fluctuations at one half ofthe blade passing frequency shows that the alternating vane patternstill has an effect on the excitation of the rotor blade even if thevariation is not implemented in the upstream vane row immediatelyadjacent. This confirms the results of the analysis of aerodynamicwork. The stagger angle variation in an alternating pattern causesa higher excitation compared to the reference case due to the ad-

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Pressure Side Suction Side O

P1

O

P2

O

P3

Reference Alternating Vanes (Multi-Stage)

low high 0

Aerodynamic Work in J/m

low high 0

Aerodynamic Work in J/m

Pressure Side Suction Side

nodal line nodal line nodal line

nodal line nodal line

Fig.15: Local aerodynamic work on the blade surface of rotor 5

ditional frequency at the 15EO. The next step is to investigate theaeroelastic behavior caused by the excitation sources.

FORCED RESPONSE ANALYSISIn this section the focus is on the influence of the alternating varia-tion of stator vane on the vibration amplitude of the rotor blade onestage downstream. For this purpose, a forced response analysis wasconducted. A unidirectional fluid-structure interaction (FSI) ap-proach was used to calculate the vibration amplitudes. This methodwas previously used and described in [8, 9], and is summarized be-low.

Forced Response MethodThe vibration amplitude of the rotor blades resulting from the aero-dynamic excitation is predicted numerically by performing a forcedresponse analysis. A unidirectional FSI approach is used whichis shown in Fig. 16. In this approach unsteady CFD calculationshave to be performed in order to extract the unsteady blade sur-face pressures. Subsequently the unsteady surface-pressures aremapped from the CFD mesh onto the FEM mesh and transformedinto the frequency domain using a FFT. Afterwards, harmonic anal-yses are conducted seperately for the main occurring excitation fre-quencies. The responses for each excitation frequency are then su-perimposed. For this analysis, the values of the aerodynamic andstructural damping must be estimated. In this study, the values ofthe damping ratio are determined from the tip-timing measurementswith the half power bandwidth method. A detailed description ofthe calculation of the damping ratio is described in [9, 22]. As thedamping ratio is estimated from tip-timing measurements, this ratioincludes the aerodynamic and structural damping of the blade. Theanalysis is performed with a damping ratio of D = 0.0012 for OP1,D = 0.0016 for OP2 and D = 0.00175 for the design point OP3. Thisapproach of the forced response analysis is described in detail in [8]and was developed to reduce the computational effort in comparisonto a bidirectional fluid-structure analysis.

Steady CFDUnsteady

CFD

Fast Fourier

Transform

Pressure

Time

Domain

Initial

CFD

Solution

Averaging

Static

Structual

Analysis

Mean

Pressure

Pressure

Frequency

Domain

Harmonic

Response

Analysis

Pre-

Stress

ResponsesSuperimpose

Responses

Time

Series of

Deflection

Fig.16: Flow chart of the unidirectional FSI approach using theharmonic analysis [9]

Forced Response ResultsIn this section, the results of the unidirectional FSI simulations arepresented. This approach was conducted for the reference casewith identical vanes, and for the case with stagger angle variationin stator vane row 4 in an alternating pattern. The following bladedisplacements are evaluated at the tip trailing edge of rotor blade 5,which is equal to the axial position of the tip-timing probes. Besidethe amplitudes determined by the FSI, the vibration amplitudes ofthe tip-timing measurements for the reference configuration areincluded for comparison.At OP1 high amplitudes occur because of the resonance betweenthe BPF and the first eigenfrequency for the reference configura-tion. The vibration amplitudes of the numerical and experimentalresults are normalized to their respective amplitude of the referenceconfiguration at OP1. Although the absolute value of the dampingratio cannot be determined precisely in simulations, the relativemagnitude of vibration amplitudes can be compared betweensimulations and experiments using this normalization. In thefollowing sections, the vibration amplitudes determined by thetip-timing measurements are a data average of all 30 blades. Ascatter of the vibration amplitude for the reference configuration isshown in [9]. In addition, the vibration amplitudes were determinedonly with specific mode content for better comparison with thestagger angle variation, which has a dominant 15EO content.The alternating vane pattern in the fourth vane row decreases thevibration amplitude to 0.94 (see Fig. 17). This is in accordancewith the reduction of the aerodynamic work at OP1. The decreasedaerodynamic work, which includes the unsteady pressure ampli-tudes and modal displacement, causes a lower excitation of therotor blade. At OP2, the normalized vibration amplitudes increasesfrom less than 0.02 for the reference configuration to 0.062 forthe alternating vane configurations because the alternating staggerangle variation causes an excitation of the first eigenfrequency bythe 15EO. Thus, the aerodynamic work on the rotor blade is alsohigher by a factor of 3 compared to the reference. Similar resultsare shown in Fig. 17 at OP3. The aerodynamic work increases

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OP1 OP2 OP30.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

Operating Points

norm

aliz

ed V

ibra

tion

Am

plitu

de

Reference Num.Reference Exp.Multi−Stage Alternating Vanes Num.

Fig.17: Vibration amplitude of rotor blade 5 at different operat-ing points normalized with the vibration amplitude of thereference configuration at OP1 - error bars indicate a 95%confidence interval of the measurements

from the reference case to the case of stagger angle variation by afactor of 4. The alternating pattern induces the additional excitationfrequency of the 15EO, which in turn causes a greater excitation ofthe second eigenfrequency as shown in Fig. 15. This leads to theincreased vibration amplitude (0.16) at OP3.As already described in [9], some uncertainties exist which cancause differences between measurements and simulations. Oneuncertainty may be the variable contact of the blade in the fir treeat partly loaded operating conditions due to the lower rotationalspeed. Additionally, excitation mechanisms in the turbine canoccur, which cannot be captured accurately by the numericalsimulations (e.g. imbalance in the rotor, excitations caused bythe drive train, excitations caused by supporting rib, clearances,etc.). As the vibration amplitudes are higher in the experimentswhich is not caputured by the numerics it does not result from theaerodynamics. Additionally, [23] and [24] show that the mistuningof the blades has a large impact on the amplitude of the blades. Theexperimental validation in these studies examined that the vibrationamplitude can be significantly higher in the experiments comparedto the numerical amplitudes of a tuned model. These differencesmay cause the differences at OP2.

Comparison to a single stageIn this section, the stagger angle variation of 1.5 deg applied to vanerow 4 is compared with a stagger angle variation of 1.5 deg in vanerow 5, in order to determine the influence on the vibration ampli-tude of the blade row 5. The geometric variation in vane row 5 hasalready been investigated in [9]. This comparison is performed inorder to investigate the influence of the variations at different loca-tions. In Figure 18, the unsteady pressure amplitudes at 80% spanheight for the 15EO at OP3 is shown for the stagger angle variationin the adjacent vane row. The stagger angle variation in vane row 5(green) generates higher unsteady pressure amplitudes on the rotorblade. On the pressure side, the maximum cp is at 0.006. On thesuction side the pressure amplitudes have their maximum (0.025)at the leading edge and a second maximum (0.005) at 70% chordlength, both of which are higher compared to the pressure ampli-tudes caused by the variation in vane row 4.The comparison of the experimental and simulated normalized vi-

0,000

0,005

0,010

0,015

0,020

0,025

0,030

0,035

0,040

Pre

ssure

Coeff

icie

nt

|cp|

Ref

Alternate

Alternate Multi-Stage

TE LE TE-180

-120

-60

0

60

120

180

PS SS

Phase

in °

0.040

0.035

0.030

0.025

0.020

0.015

0.010

0.005

0.000

Pre

ssu

re C

oef

fici

ent

|cp|

PS SS LE TE TE

Fig.18: Comparison of the unsteady pressure amplitudes along thechord length for one half of the blade passing frequency(15EO) at OP3 with the variation in stage 5 - 80% spanheight

OP1 OP2 OP30.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

Operating Points

norm

aliz

ed V

ibra

tion

Am

plitu

de

Reference Num.Reference Exp.Alternating Vanes Num.Alternating Vanes Exp.Multi−Stage Alternating Vanes Num.

Fig.19: Comparison of the vibration amplitudes with the variationin stage 5 - error bars indicate a 95% confidence interval ofthe measurements (see [9])

bration amplitudes for the alternating vane configuration of vanerow 5 was published in [9]. Here, the focus is on the impact of thestagger angle variation in vane row and the difference caused bythe one stage in between. In Figure 19, the normalized vibrationamplitude is presented for all operating points, including the com-parison of the single stage variation. It shows that the impact of ageometrical variation in the adjacent blade row is higher by a factorof 10 compared to a defect one stage upstream for OP2 and OP3.In the adjacent blade row, the excitation sources, the potential ef-fect, and the wake excitation, are not extenuated. As the wakes aremixed out in the flow passage and the potential effect is a localizedeffect, the influence of the variation is much lower after one stagedownstream. Nevertheless, this geometric variation causes highervibration amplitude at OP2 and OP3. At OP1 the excitation of theblade passing frequency is reduced by the stagger angle variationin vane row 5. Thus, a geometric variation one stage between theexcited blade row and the geometric variation induces a higher vi-bration amplitude. This is still lower (0.94) than the reference case.

These results are in accordance with the results shown in Fig. 20.This diagram shows a correlation between the normalized vibrationamplitude and the normalized aerodynamic work. As above, the

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

3.75

4.00

normalized aerodynamic work

no

rmal

ized

vib

rati

on

am

pli

tud

e

OP1 Reference

OP1 Alternating Vanes

OP1 Multi-Stage Alternating Vanes

OP2 Reference

OP2 Alternating Vanes

OP2 Multi-Stage Alternating Vanes

OP3 Reference

OP3 Alternating Vanes

OP3 Multi-Stage Alternating Vanes

Mode 1

Mode 2

Fig.20: Correlation between normalized vibration amplitude andnormalized aerodynamic work

amplitudes are normalized to that of mode 1 in the reference case atOP1. The three cases are presented for all operating points. An al-most linear correlation exists which shows that higher aerodynamicwork causes higher vibration amplitudes. As the first and secondeigenmode have different mode shapes, the linear correlation is alsodifferent. The response to aerodynamic work is different for bothmode shapes. Consequently, the influence of variations one stagedownstream on the vibration amplitude is non-negligible. As thepotential effect is a localized effect between stator vane row 5 androtor blade row 5, the remaining main excitation mechanism due tothe variances in stator vane row 4 is the wake excitation. The poten-tial effect on the excitation at half of the blade passing frequency isof minor importance, because the vanes in row 5 have no variation.As a consequence of the mixing process, the wake excitation dueto variances in an upstream stage has a lower influence comparedto the adjacent vane row. If the variation was applied to the adja-cent vane row both, the wake excitation, as well as excitation bythe potential effect, could have a significant impact on the vibrationamplitude.

CONCLUSIONS AND OUTLOOKThe effect on the aerodynamic excitation caused by an upstreamblade row, which is not adjacent to the excited blade row isinvestigated using a FSI method. This method was previouslyvalidated against experimental data. For this purpose, a typicalregeneration-induced variance is applied to the fourth stage ofthe five-stage axial turbine in an alternating distribution, whichwas previously shown in [11]. It was shown that a variation instagger angle has the highest impact on the aeroelastic behavior ofthe downstream turbine blades. The influence of a stagger anglevariation on the aerodynamic excitation and the vibration amplitudeof the last rotor blade row is analyzed in detail.The prediction of the forced response amplitude using the unidi-rectional FSI approach shows an increase of the amplitudes dueto the geometric variation. These variations lead to an additionalexcitation frequency. Dependent of the operating points, this cancause up to a fourfold increase in aerodynamic work acting onthe analyzed blade row, the additional excitation induces fourfoldgreater vibration amplitudes compared to the reference case. If ageometric variation is applied to the adjacent vane row upstreamof the analyzed blade, the impact on the vibration amplitudeis greater by a factor of 10 compared to the case where thevariation is applied one stage further upstream. At other operating

points the geometric variation can have a positive effect on thevibration amplitude. The additional excitation frequency causesa reduction of the excitation by the blade passing frequency,which can in turn reduce the vibration amplitude up to a factorof 0.25 depending on in which vane row the geometric variationis present. Additionally, a linear correlation between the aero-dynamic work and the vibration amplitude was found. A futurestep will be the experimental validation of the geometric varia-tion vane row 4 in the axial air-turbine for experimental verification.

ACKNOWLEDGMENTSThe authors kindly thank the German Research Foundation (DFG)for the financial support to accomplish the research project C4“Regeneration-induced Variances of Aeroelastic Properties of Tur-bine Blades” within the Collaborative Research Center (CRC) 871.Furthermore, the authors thank ANSYS for providing CFX in anacademic license and the Leibniz Universitat Hannover IT Services(LUIS) for the computational resources provided. Finally, we ac-knowledge the valuable suggestions of the anonymous reviewers.

References[1] Rupp, O., 2001, “Instandhaltungskosten bei zivilen

Strahltriebwerken”, Deutscher Luft- und Raumfahrtkongress2001, Hamburg DGLR-2001-008.

[2] Aschenbruck, J., Adamczuk, R., and Seume, J., 2014, “Re-cent Progress in Turbine Blade and Compressor Blisk Re-generation”, Proceedings of 3rd International Conferenceon Through-life Engineering Services, November 4-5 2014,Cranfield, England.

[3] Vahdati, M., Sayma, A., and Imegrun, M., 2000, “An Inte-grated Nonlinear Approach for Turbomachinery Forced Re-sponse Prediction. Part II: Case Studies”, Journal of Fluidsand Structures, Vol. Vol. 14(1), pp. 103–125.

[4] Breard, C., Green, J., and Imregun, M., 2003, “Low-Engine-Order Excitation Mechanisms in Axial-Flow Turbomachin-ery”, Journal of Propulsion and Power, Vol. Vol. 2003(19),pp. 704–712.

[5] Di Mare, L., Imregun, M., Smith, A., and Elliott, R., 2007, “ANumerical Study of High Pressure Turbine Forced Responsein the Presence of Damaged Nozzle Guide Vanes”, Aeronau-tical Journal, Vol. Vol. 111 / 3177, pp. 751–757.

[6] Meyer, M., Parchem, R., and Davison, P., 2011, “Predic-tion of Turbine Rotor Blade Forcing due to in-service StatorVane Trailing Edge Damage”, Proceedings of ASME TurboExpo, June 6-10 2011, Vancouver, British Columbia, Canada,GT2011-45204.

[7] Petrov, E., Di Mare, L., Hennings, H., and Elliott, R., 2010,“Forced Response of Mistuned Bladed Disks in Gas Flow:A Comparative Study of Predictions and Full-Scale Experi-mental Results”, Journal of Engineering for Gas Turbines andPower, Vol. Vol. 132(5) / 052504.

[8] Aschenbruck, J., Meinzer, C., Pohle, L., Panning-von Scheidt,L., and Seume, J., 2013, “Regeneration-induced Forced Re-sponse in Axial Turbines”, Proceedings of ASME Turbo Expo,June 3-7 2013, San Antonio, Texas, USA, GT2013-95431.

[9] Aschenbruck, J., and Seume, J., 2015, “Experimentally Veri-fied Study of Regeneration-Induced Forced Response in Ax-ial Turbines”, ASME Journal of Turbomachinery, Vol. Vol.137(3) / 031006.

[10] Jocker, M., Hillion, F., Fransson, T., and Wahlen, U., 2002,“Numerical Unsteady Flow Analysis of a Turbine Stage withExtremely Large Blade Loads”, ASME Journal of Turboma-chinery, Vol. Vol. 124(3) / 429.

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[11] Hauptmann, T., Aschenbruck, J., and Seume, J., 2015,“Forced Response Excitation due to Variances in a Multi-Stage Axial Turbine”, International Gas Turbine Congress,November 15-20 2015, Tokyo, Japan.

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[13] Campobasso, M., and Giles, M., 2000, “Analysis of the Effectof Mistuning on Turbomachinery Aeroelasticity”, Proceed-ings of the 9th International Symposium on Unsteady Aero-dynamics, Aeroacoustics and Aeroelasticity of Turbomachines(ISUAAAT 2000), Lyon, Fr.

[14] Zhai, Y., Bladh, R., and Dyverfeldt, G., 2012, “AeroelasticStability Assessment of an Industrial Compressor Blade In-cluding Mistuning Effects”, Journal of Turbomachinery, Vol.Vol. 134(060903).

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[16] Herbst, F., Bluemel, S., Fakiolas, E., and Seume, J., 2011,“Numerical Investigation of the Interaction between Probe,Flow and Blading in an Axial-Turbine”, Proceedings of theInternational Gas Turbine Congress, 13-18 November 2011,Osaka, Japan, IGTC2011-0194.

[17] Rieß, W., and Braun, M., 2003, “Stationares und instationaresVerhalten verschiedener Typen von Stroemungs-Messsondenin instationaerer Stroemung”, DFG Final Report, Ri 375/13-1, Institute of Turbomachinery and Fluid Dynamics, LeibnizUniversitat Hannover, Germany.

[18] Andersson, C., Grasbon, P., and Merchant, S., 2010, “Vi-brations of the LH2 Turbine Rotor during the Vinci EngineTest Tip Timing Measurements and Predictions”, Proceed-ings of the ASME Turbo Expo, June 14-18 2010, Glasgow,UK, GT2010-23413.

[19] Lastiwka, D., Chang, D., and Tavoularis, S., 2013, “Effectsof Rotor Blade Scaling in High-Pressure Turbine UnsteadyLoading”, International Journal of Turbo Jet-Engines, Vol.Vol. 30(1).

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[21] Gezork, T., Mayorca, M., Groth, P., Vogt, D., and Fransson, T.,2014, “Influence of Tip Shroud Cavity Detailing on TurbineBlade Forcing Calculations”, Proceedings of ASME TurboExpo, June 16-20, Dusseldorf, Germany, GT2014-26724.

[22] Jinting, W., Dandan, L., Feng, J., and Z., C., 2013, “Accuracyof the half-power bandwidth method with a third-order correc-tion for estimating damping in multi-DOF systems”, Journalof Earthquake Engineering and Engineering Vibration, Vol.Vol.12(1), pp. 33–38.

[23] Pohle, L., Panning-von Scheidt, L., Aschenbruck, J., Seume,J., and Wallaschek, J., 2014, “Dynamical Behavior of a Mis-tuned Air Turbine: Comparison between Simulations andMeasurements”, Proceedings of ASME Turbo Expo, June 16-20 2014, Duesseldorf, Germany, GT2014-26025.

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