Proceedings of GPPS Forum 18Global Power and Propulsion Society

Montreal, 7th-9th May 2018www.gpps.global

GPPS-2018-29

AEROELASTIC ASSESSMENT OF A HIGHLY LOADED HIGH PRESSURECOMPRESSOR EXPOSED TO PRESSURE GAIN COMBUSTION DISTURBANCES

Victor Bicalho Civinelli de AlmeidaChair for Aero Engines

Department of Aeronautics and AstronauticsTechnische Universitat BerlinVictor.Bicalho@ilr.tu-berlin.de

Berlin, Germany

Dieter PeitschChair for Aero Engines

Department of Aeronautics and AstronauticsTechnische Universitat BerlinDieter.Peitsch@ilr.tu-berlin.de

Berlin, Germany

ABSTRACTA numerical aeroelastic assessment of a highly loaded

high pressure compressor exposed to flow disturbances ispresented on this paper. The disturbances originate from novel,inherently unsteady, pressure gain combustion processes, suchas pulse detonation, shockless explosion, wave rotor or pistontopping composite cycles. All these arrangements promiseto reduce substantially the specific fuel consumption ofpresent-day aeronautical engines and stationary gas turbines.However, their unsteady behavior must be further investigatedto ensure the thermodynamic efficiency gain is not hinderedby stage performance losses. Furthermore, blade excessivevibration (leading to high cycle fatigue) must be avoided,especially under the additional excitations frequencies fromwaves traveling upstream of the combustor.

Two main numerical analyses are presented, contrastingundisturbed with disturbed operation of a typical industrialcore compressor. The first part of the paper evaluatesperformance parameters for a representative blisk stage withhigh-accuracy 3D unsteady Reynolds-averaged Navier-Stokescomputations. Isentropic efficiency as well as pressure andtemperature unsteady damping are determined for a broadrange of disturbances. The nonlinear harmonic balance methodis used to determine the aerodynamic damping. The secondpart provides the aeroelastic harmonic forced response of therotor blades, with aerodynamic damping and forcing obtainedfrom the unsteady calculations on the first part. The influenceof blade mode shapes, nodal diameters and forcing frequencymatching is also examined.

INTRODUCTIONIncrease in overall efficiency of aeronautical engines and

stationary gas turbines, reducing specific fuel consumption,is being extensively researched. Since traditional engine

configurations are reaching saturation development state, noveltechnological breakthroughs are sought. Among these radicalchanges, the most significant tackle either the core/bypassflows or the combustor losses. To address core flow hurdles,intercooling, Rankine bottoming and recuperation might beviable options. Reheating is also particularly developed forland-based applications. Concerning bypass losses, gearedturbofan and open rotor concepts are receiving considerableattention [1–3].

With respect to the combustion losses, several innovativeapproaches have been proposed. Most of them aim atimproving current constant-pressure (in fact, pressure-loss)combustion with a pressure-increase process. This so-calledpressure gain combustion (PGC) can be accomplished bydifferent means. The leading paths are pulse detonation [4, 5],rotating detonation [6–8], wave rotors [9, 10] and shocklessexplosion [11, 12] combustion. Some other concepts are thepiston topping [13] and the nutating disk [14]. All these setupsare theoretically supported by substantial thermodynamicgains, which for succinctness will not be here discussed.

Despite fairly different construction and engine-integration characteristics, all these PGC processes introduceadditional unsteadiness into the adjacent turbomachinerycomponents, namely, the compressor upstream and the turbinedownstream. This unsteadiness is directly linked with theinherently periodic nature of the combustion mechanisms,and affect the operation of both stationary and rotating parts(traditionally designed by steady state workflows). Theintegration between combustion units and engine (mostlikely through plena interfaces) is key to estimate properboundary conditions for the neighboring components, such asperturbation signatures, operation frequencies and amplitudes.

Although fundamental combustion research on pulsedetonation had been restricted to low frequency processes,massive increase in the number of ignitions per second has

This work is licensed under a Creative Commons Attribution 4.0 International License CC-BY 4.0

been achieved. Kilohertz pulse detonation frequencies wererecently reported by [15]. As for rotating detonation, it alreadytakes place on the few kilohertz range [16, 17]. Similarperiodicity is reported for the other PGC devices. Thisfrequency range is also expected to but pushed up, once PGCtechnology takes off.

With respect to the amplitudes of such disturbances, theyare also strongly linked to the PGC operation (such as filling,ignition and purging phases), but also to the plena interfaces.Limited PGC experimental data with upstream measurementsis available, such as [5, 18]. Some numerical work usedpressure amplitude values at the combustor interfaces of a fewpercent up to approximately 40% [19–21]. This disturbanceamplitude range is already significant from the turbomachinerypoint of view, justifying detailed investigations.

The need for assessing the performance and aeroelasticityeffects of these disturbances has been recognized by thecombustion and turbomachinery academic community [3, 5,17, 22, 23]. However, quantitative information to evaluate theextent of these changes is scarce, especially for state-of-the-art industrial machines. Up to the current date, there has beenno full integration between those PGC cores to commercialengines. Therefore, the current exploratory research shedssome light into the effects of this interfacing (specificallyupstream), by numerical means. Both performance andaeroelasticity assessments will be presented.

The paper is organized as follows. On the case descriptionsection, the simulated high pressure compressor geometry ischaracterized. Next, the numerical methodologies both for thecomputational fluid dynamics (CFD) and computational soliddynamics (CSD) are presented. Subsequently, results are givenfor a broad range of disturbance amplitudes and frequencies,followed by the main outcomes.

CASE DESCRIPTIONA modern industrial high pressure compressor, depicted

on Fig. 1, is here used for the simulations [24, 25]. Practical3D geometries and operation points have been provided byan industrial partner. It is important to say that the thewhole engine has been designed for traditional deflagrationcombustion, without specific considerations for PGC and itseffects on turbomachinery components. Therefore, the presentstudy analyses how this traditional setup would potentiallyreact to unsteady disturbances and depart from the referencedesign.

The simulated domain is the HPC 6th stage, the lastwith a bladed disk (blisk) design. Although it is not thevery last stage right before the combustion chamber, it hasbeen chosen considering that blisk manufacturing becomesincreasingly more common in axial compressor design [26–29]. For this stage, 82 rotor blades are followed downstreamby 108 stator vanes. The operating conditions for theengine have been extracted from the cruise altitude workingline and define a highly stressed performance state, yieldinglarge overall pressure ratio and isentropic efficiency. Dueto confidentiality constraints, the speedline itself and someabsolute design quantities will not be published here, but

FIGURE 1: The E3E high pressure compressor rotor [25].

rather normalized values, enough for full comprehension andcomparative evaluations.

NUMERICAL METHODOLOGYFor brevity purposes, neither the fluid nor the solid domain

state and constitutive equations will be derived here. Rather,focus will be given to the interface treatment of necessaryaeroelastic quantities, such as aerodynamic work or pressureharmonics. The next sections describe respectively the CFD(steady and unsteady) and CSD methodologies used along thiswork. The space and time independence studies for the gridsare likewise presented.

Computational fluid dynamicsDue to the unsteady behavior associated with the new

combustion processes, 3D unsteady viscid fluid dynamicssimulations are employed, both in time and frequencydomain. Time-domain calculations are here used todetermine performance parameters, unsteady damping andblade surface pressure harmonics, to be later used in thestructural simulations. Frequency-domain runs determine theaerodynamic damping.

(i) Steady state: the Navier-Stokes equations areboundary-layer resolved with the finite volumes solver CFX®

[30], with air (ideal gas) as compressible medium. Fourthorder Rhie & Chow smoothing is employed [31], whileReynolds stresses are computed by the two-equation Menter’sSST turbulence model [32]. Since the simulated stage islocated substantially far from the engine inlet, the flow issafely assumed to be fully turbulent, so no transition modelswere shown necessary. Double-precision, properly convergedsteady state solutions (with root-mean-square residuals lowerthan 10-5, and imbalances lower than 5 ·10-6) are used as initialconditions for the unsteady runs.

The 3D geometries were meshed with AutoGrid5®,developed by Numeca [33]. Both root fillet and tip gaphave been modeled. Multi-block structured hexaedra mesheswere employed, with directional boundary-layer refinement onall walls (dimensionless wall distance y+ ≈ 1), keeping the

expansion ratio below 1.2. The topology for both rotor andstator vanes is the O4H. The tip gap was modeled with at least30 cells in the radial direction.

Several meshes were generated in order to perform aproper grid independence study, done separately for rotorand stator. The grids were refined as uniformly as practical,however always keeping the boundary layer on the wallsresolved. Figure 2 displays three chosen meshes for rotor(labeled from coarsest to finest as R-A, R-B and R-C) andstator (labeled as S-A, S-B and S-C), with isentropic efficiencyand mass flow values normalized by the finest grid, as afunction of cell count. The deviation between the finest-and coarsest-grid values is less than 1% for both domains,in the sense that all displayed grids capture the global flowbehavior accurately enough. The Grid Independence Index[34], formally derived based on the Richardson extrapolation[35], has been calculated and showed no oscillatory behavior.For the isentropic efficiency, the index between finest andmedium meshes is GCIBC = 0.24%, with apparent order of1.94.

0 0.5 1 1.5 2

0.992

0.994

0.996

0.998

1

Millions of cells

Ro

tor

adia

bat

ic e

ffic

ien

cy n

orm

aliz

ed

R−A

R−B

R−C

(a) Rotor

0 0.5 1 1.5 2

0.994

0.996

0.998

1

Millions of cells

Sta

tor

mas

s fl

ow

no

rmal

ized

S−A

S−B

S−C

(b) Stator

FIGURE 2: Grid independence study for fluid system, normalized byfinest grid values.

To further establish the grid which would bring accurateresults with the least possible computational cost, local effectshave been assessed. Figure 3 shows the local pressurecoefficient cp distribution on both rotor and stator at 25% span,for the aforementioned grids labeled on Fig. 2. Particularlyclose to the leading edge, the cp trough on the pressure side isbetter modeled by the medium (B) and fine (C) grids, and notso well by the coarse mesh (A). Although all meshes provideacceptable results for the current investigations, the mediumresolution grids for both rotor and stator (R-B and S-B) havebeen elected for further simulations, in order to achieve a goodcompromise between computational efficiency and adequateaccuracy. The grids utilized for the current stage are assembledtogether and shown on Fig. 4. The number of nodes for thestage (if using one passage for rotor and one for stator) isapproximately 1.3 million.

0 0.2 0.4 0.6 0.8 1 Streamwise position

c p

Grid R−CGrid R−BGrid R−A

(a) Rotor

0 0.2 0.4 0.6 0.8 1 Streamwise position

c p

Grid S−CGrid S−BGrid S−A

(b) Stator

FIGURE 3: Pressure coefficient cp distribution at 25% span, forrotor and stator grids (A: coarse; B: medium; C: fine).

As inlet boundary conditions, total pressure andtemperature along with flow direction are implemented, whileon the outlet the static pressure is prescribed. Both on inletand outlet, radial profiles experimentally validated by themanufacturer are utilized. For the steady runs only, mixing-plane interface, with constant total pressure (that is, no loss inthrough the interface), is used to account for the non-unitarypitch ratio.

(ii) Unsteady: two types of time-resolved simulationsare performed, with and without the PGC disturbances.The time discretization scheme utilized is the secondorder backward Euler, with a bounded high resolutiondiscretization for advection and viscous terms. The solutionis deemed periodically converged when the relative differencebetween two consecutive periods for some globally integratedparameter (such as compression ratio or efficiency) or pressureprobe falls below 0.05%. The attainment of quasi-convergencedepends not only on the blade passing frequency (BPF), butalso on the combustion-originated frequency and intensity,which in turn determines the required total run time.

The time resolution must be properly chosen to modelthe necessary phenomena without excessively consumingcomputational resources. For the present case, the main (apriori known) frequencies to be modeled by the time march are

FIGURE 4: Grids R-B and S-B for the simulated stage (rotor onthe front and stator on the back), obtained after gridindependence study.

the rotor BPF and the disturbance frequency fd . A temporalresolution study was performed to determine the minimumtime discretization required, using the number of time steps perrotor passing period (TSRPP) as a parameter. Figure 5 depictstwo points on each (one on each domain) where pressureprobes are located, whose time traces are subsequently shownon Fig. 6 (after periodic convergence is achieved). For bothdomains, 50 TSRPP are enough to accurately capture theunsteady behavior. To enhance resolution for the disturbedruns, a slightly higher value of 55 TSRPP has been chosen.

It is important to highlight that, under unsteadydisturbances (additional to the rotor-stator frequency), therequired time step could become smaller. To assess whether55 TSRPP is enough for the outlet disturbance with the highestfrequency here simulated (3.0 times the BPF), an extra runwith halved time step (i.e., 110 TSRPP) was carried out, foran extra full rotor revolution. In comparison with the 55TSRPP setup, a relative difference of 0.09% and 0.005% forisentropic efficiency and compression ratio, respectively, wasfound. Unsteady damping, as well as modal forcing (bothdefined further), showed similar minimal changes. From that,the current time resolution was deemed enough within thesimulated disturbance range. Higher frequency disturbancescould possibly require further time step refinement.

For the transient calculations, the number of rotor bladeswas reduced from 82 to 81, so that 3 rotor passages and 4 statorpassages yield a unitary pitch ratio. The total number of nodesfor this setup is approximately 4.6 million. This modificationprevents phase errors which would incur when using forexample the profile-scaling method [36]. Time-inclining or-shifted methods, such as [37] or [38], are not physicallyconsistent when disturbance frequencies different from theBPF are present in the system. In summary, the currentunsteady simulation setup does not induce frequency or phaseerrors due to non-unitary pitch ratio. All unsteady disturbances

FIGURE 5: Blade-to-blade cut on 25% span. Points for pressureprobes on rotor (P1) and stator (P2) domains.

0 0.5 1 1.5 2Period fraction t/T

c p

2030405055607080100

(a) Pressure coefficient cp on probe P1 downstream of rotor blade

0 0.5 1 1.5 2 2.5 3Period fraction t/T

c p

2030405055607080100

(b) Pressure coefficient cp on probe P2 upstream of stator vane

FIGURE 6: Pressure probe periodically converged time trace, onrotor and stator domains. Several values for time stepsper rotor passing period (TSRPP) are shown.

are modeled by the time marching scheme without furthersimplifications.

To model the PGC disturbances, time periodic boundaryconditions on the stage outlet have been implemented.To the knowledge of the authors, no effective test rig

or prototype coupling industrial axial compressor andturbine to a pressure gain combustor has been built, ina fully coupled fashion. Ongoing investigations focuson how to connect these components through adequateplena geometries, to minimize unsteadiness and thereforeprevent extra losses. Accordingly, due to the lack ofmore precise plenum information, it is assumed that theunsteadiness reaching the compressor/combustor interfacedoes not vary circumferentially, being here described byharmonic fluctuations, as in Eq.(1).

p(x, t) = p(x) [1+Ad · sin(2π fd t)] (1)

Here, Ad and fd stand respectively for amplitude and frequencyof the imposed harmonic disturbance on the static pressuredistribution p(x). Indeed, since the pressure is on this workprescribed radially, p(x) = p(r) for radius r. This approachassumes that the number of blades and vanes is rather largerthan the number of combustion units downstream, so thatno disturbance wave interaction among combustor units ismodeled.

(iii) Time-spectral: the frequency-domain methodemployed here is the nonlinear harmonic balance [39] (alsoknown as time spectral method [40]). The time and spatialdependent state variables are written as a Fourier series for apredetermined fundamental frequency, whose first harmonicsare iteratively and simultaneously solved with a pseudo timemarch. Along this work, 3 harmonics (producing 7 timelevels) have been used, after parametric studies showed thatmore harmonics did not change the solution substantially.Additionally, 20 pseudo time steps per oscillation period wereenough to achieve the desired computational accuracy at rapidconvergence. The nonlinear harmonic balance is here utilizedto determine the aerodynamic work Waero, as defined on Eq.(2).

Waero =∫ t+T

t

∫`

p v · n d` dt (2)

Here, p is the static pressure, v the wall velocity vector and nthe wall outwards pointing normal versor. The surface integraltakes place along the blade walls `. The aerodynamic workis integrated in time t during one period T . For negativeWaero, energy is being fed from the blade motion to the fluidstream, in a stable fashion; for positive Waero, energy is beingtransfered from the fluid to the structure, hinting instability.This is the same approach to assess flutter stability, basedon the well-established energy method [41], with traveling-wave configuration [42, 43]. Interblade phase angles are heresymbolized by σ . The wall movement is prescribed accordingto the desired rotor blade mode shape. More details will befurther given on the modal analysis section.

With this approach, extensively used on industrial design,two assumptions are implied: constant vibration frequency(for each natural mode) and independence between damping

and forcing (justified, for example, by [44]). It is importantto remark that the main vibration damping contribution for ablisk structure comes from aerodynamic forcing, since frictiondamping is not present, and material damping is negligible.Therefore the need of precisely determining the aerodynamicwork on the rotor blade.

The aerodynamic work is used to compute the dampingratio ξ , as in Eq.(3) (see for example [45–47]).

ξ =−Waero

2π ω2 q2 (3)

Here, ω = 2π f is the angular vibration frequency of theblade (obtained from a prestressed modal analysis), and qis the scaling factor of the natural mode shape for the CFDcomputation (including the modal scaling, in case mode shapesare obtained as mass normalized).

The time-spectral analyses simulate the rotor domain only,with the same inlet boundary conditions as for the stageruns (total pressure, total temperature and flow direction).Regarding the rotor outlet, design static pressure radialprofiles for the corresponding axial position (validated by themanufacturer) are employed.

On this work, the rotor blades (which in general are undermore mechanical stresses than the stator) vibrate on the firstthree natural frequencies (the most energetic, sufficient foraeroelastic structural response [48–50], with a maximum tipdisplacement set to 0.5% of blade height. The normalizationprovided by Eq.(3) makes sure that the damping ratio isindependent of the chosen scaling factor, within the quadraticregime. Parametric simulations have made sure that the chosenvalues for the scaling factor and natural frequencies lie withinthe linear range.

Computational solid dynamics(i) Prestressed static and modal analyses: the current rotor

blisk is modeled as a cyclic symmetric sector, with a typicalmodern industrial titanium alloy. Here, focus will be givento the blade geometry and mesh, sufficing to say that thedisk displacement is fixed at the bore, that is, shaft flexibilityis ignored, as standard with blisk analyses [51, 52]. In theprestressed static analysis, constant rotational velocity and themean static pressure on the blade surface are prescribed asloads. The pressure load comes from CFD-only simulationwithout any external disturbance, that is, a baseline unsteadyrun. The nodal pressure time history on the blade surface isexpanded into a Fourier series, whose first term correspond tothe mean static load (which is used in the prestressed staticanalysis). The following terms determine the engine ordermultiples (these shall be later used in the harmonic responseanalyses).

The finite elements method (FEM) solver ANSYS®

[30] has been used for the structural analyses. Aftersome preliminary studies comparing different finite elementsfamilies, high-order tetrahedra have been chosen in place ofhexaedra, pyramids or wedges. Although hexaedra could

facilitate the mapping between solid and fluid grids, tetrahedraelements have proven to better model the current physics ofthe problem, particularly because regions with high stressgradients are better meshed. Similarly to the fluid domain, agrid independence study was performed for the solid meshes,this time with the natural frequency of the first natural modeshapes as comparative parameter. The grid convergence fortwo modes is displayed on Fig. 7.

0 0.1 0.21

1.0002

1.0004

1.0006

1.0008

Millions of cells

Nat

ura

l fre

qu

ency

no

rmal

ized

Blisk−D

Blisk−C

Blisk−B

Blisk−A

Blisk−E

(a) Mode 2

0 0.1 0.21

1.005

1.01

1.015

1.02

1.025

Millions of cells

Nat

ura

l fre

qu

ency

no

rmal

ized

Blisk−A

Blisk−B

Blisk−C

Blisk−D

Blisk−E

(b) Mode 4

FIGURE 7: Grid independence study for solid system, normalizedby finest grid values.

Even the coarsest of the generated meshes representsremarkably good the displayed modes (very similar behaviortakes place for the first 20 mode shapes). However, the lastbut one finest mesh (labeled as Blisk-D) has been chosen toguarantee a good mapping between fluid and solid systemvariables. The sector has 96322 nodes. This grid, expandedto cover the whole blisk, is shown on Fig.8, contoured withthe second blade natural mode displacement, with 40 nodaldiameters.

For brevity purposes, only a brief remark is madeabout the use of cyclic symmetry. In order to prevent theexpensive modeling of the whole blisk structure, just onesector is meshed. Fourier decomposed boundary conditions areimplemented on the symmetrical faces in the circumferentialdirection, with proper consideration of the total number ofsectors [53, 54].

(ii) Harmonic forced response: subsequently to theprestressed modal analyses, mode-superposition harmonicresponse is computed. The time and frequency domainrelations are displayed on Eqs.(4).

MU(t)+CU(t)+KU(t) = F(t) (4a)

d j(ω)+2ω j ξ j d j(ω)+ω2j d j(ω) = f j(ω) (4b)

FIGURE 8: Grid Blisk-D for the simulated rotor, obtained aftergrid independence study. Whole-circle expandedresult displays the second blade natural mode (torsion)contours.

Here, as in classic FEM, M, C and K are the mass, dampingand stiffness matrices, plus the displacement and forcingvectors U(t) and F(t). Note that, since the forcing vector F(t)is complex when expanded into an harmonic series, real andimaginary aerodynamic forcing components must be inputed.Eq.(4b) shows the nodal coordinate d j(ω) for one generalnatural mode j with corresponding natural frequency ω j (moredetails given by [47, 55]). Both the complex-valued forcing( f j(ω)) from the unsteady time-domain runs and the dampingratio (ξ j from Eq.(3)) from the frequency-domain runs areused, in order to obtain the maximum stresses and strains onthe blisk.

In the mode-superposition harmonic analyses, the first 117natural modes have been used. A phase sweep was done,according to the studied target-frequency. The highest naturalfrequency obtained in the modal analysis setup was always stleast 1.5 times the excitation frequency.

RESULTSComputational fluid dynamicsOn this subsection, the CFD-only results are presented.

Initially, time-domain calculations are assessed, showing howthe disturbances affect performance, as well as the unsteadydamping. Next, the frequency-domain results for aerodynamicdamping are given.

(i) Performance drop: the isentropic efficiency of thesimulated stage was calculated for several disturbancespatterns. Outlet static pressure amplitude and frequencyhave been varied according to current (and foreseen) PGCranges. The amplitude Ad is normalized as a percentage ofthe outlet mean static pressure, while the frequency is writtenas a function of both BPF multiple and Strouhal number St.The last is defined here as St = ( fd cr)/vaxial , where cr is areference rotor chord length and vaxial the mass-flow-averagedinlet axial velocity [19]. Figure 9 depicts the drop in stageisentropic efficiency η as a function of disturbance frequencyand amplitude.

As a first trend, higher disturbance amplitudes Ad lead to

0.31 0.61 1.22 1.83 2.44 3.05 3.66 4.88 6.11 7.33St

0.125 0.25 0.5 0.75 1 1.25 1.5 2 2.5 3

−16

−14

−12

−10

−8

−6

−4

−2

0

BPF multiple

η dr

op [%

]

Disturbanceamplitude A

d

5%10%15%20%

FIGURE 9: Isentropic efficiency η drop as a function of disturbancefrequency fd (function of St and BPF multiple). Resultsfor several disturbance amplitudes (Ad on Eq.(1)).

larger drops in η . This behavior is to some extent expectedonce the flow conditions depart from the steady state design.However, for low disturbance frequencies (such as 0.125 and0.25 times the BPF), η decrease is evidently larger, reachingmore than 15% for Ad = 20%. This performance depletionshould be accounted for, when designing a compressor stagefor a particularly perturbed environment. Although smalldecreases in efficiency might not represent a setback for theuse of unsteady combustion, higher disturbance amplitudes,especially at lower frequencies, can lead to strong flowseparation on the blade surface, being profoundly detrimentalfor stable compressor operation. This was the case forlower disturbance frequencies, such as fd = 0.25 BPF andAd = 20%, where due to the disturbance waves the adversepressure gradient increases on the stator, followed by the rotor,inflating recirculation areas adjacent to the blade (see Fig. 11,depicting the static entropy and surface streamlines aroundthe rotor blade, at the same time instant for the undisturbedand the disturbed cases). These recirculation areas adjacent tothe blade surface are not present on the undisturbed transientinstance. This performance depletion region is also indicatedat the upper left corner on the efficiency drop map on Fig. 10.

(ii) Unsteady damping ε: this quantity is defined onEqs.(5).

∆φ =φmax −φmin

φmean(5a)

ε =∆φ2 −∆φ1

∆φ2(5b)

∆φ represents the periodic variation of any (properly space-averaged) quantity in time, normalized by its mean. φ canbe any state variable, being restricted on this manuscript to(total and static) pressure and temperature. This definitionis similar to the one given by [19]. However, since

0.61 1.22 1.83 2.44 3.05 3.66 4.88 6.11 7.33St

0.25 0.50 0.75 1.00 1.25 1.50 2.00 2.50 3.000.00

0.05

0.10

0.15

0.20

BPF multiple

Pre

ssur

e di

stur

banc

e am

plitu

de A

d

η drop [%]

−14 −12 −10 −8 −6 −4 −2

FIGURE 10: 2D isentropic efficiency η drop map, as function ofdisturbance frequency and amplitude ( fd and Ad onEq.(1)).

(b) Undisturbed (c) Disturbed

FIGURE 11: Static entropy contour and surface streamlines aroundthe rotor blade (cross section at 40% span), forundisturbed and disturbed cases, with fd = 0.25 BPFand Ad = 20%. A substantial increase in entropy andrecirculation zones are present on the disturbed run.

the disturbances now propagate upstream, the order of thestreamwise-station indexes 1 and 2 is reversed. The unsteadydamping computational is carried out for a few periods (thelongest among disturbance and blade passing periods), andonly after the unsteady run has achieved proper periodicconvergence as previously described. The streamwise stations

are set right before and after each blade/vane (circa 10% ofchord).

The essence of the unsteady damping is to determinehow much a state property unsteady variation is reduced(ε > 0) or amplified (ε < 0), as it travels from stations 2 to1. Note that the unsteady damping is completely unrelatedto the aerodynamic damping (presented on the next section).Figure 12 depicts the unsteady damping for total pressure andtemperature, at Ad = 20%, along the whole stage (that is, inthis case station 2 is right aft stator vane and station 1 rightfore rotor blade).

0.25 0.5 0.75 1 1.25 1.5 2 2.5 30.75

0.8

0.85

0.9

0.95

1

BPF multiple

ε stag

e

PressureTemperature

(a) Static quantities

0.25 0.5 0.75 1 1.25 1.5 2 2.5 30.88

0.9

0.92

0.94

0.96

0.98

1

BPF multiple

ε stag

e

Total pressureTotal temperature

(b) Total quantities

FIGURE 12: Unsteady damping εstage, for disturbance amplitudeAd = 20%. Pressure and temperature (static and total)values shown for several frequencies.

All unsteady damping values obtained are positive andclose to 1 (that is, almost complete attenuation). Static andtotal unsteady damping show very similar behavior. However,very low and very high frequencies are less damped than theones close to the BPF, when the disturbance propagates fartherupstream. This is a valuable information for the aeroelasticitydesigner, since the blade structure should avoid (or at leastwithstand) vibrations at the adjacent excitation frequencies.The unsteady damping pinpoints how far the unsteadinesspropagates along the streamwise direction, with high values(say, more than 95%) indicating that these fluctuations areweak enough when reaching the next stage, and therefore do

not necessarily prompt redesign considerations.

(iii) Aerodynamic damping: the aerodynamic work wasused to determine the damping ratio ξ , according to Eqs.(2)and (3). The first three natural blade modes were simulatedwith the nonlinear harmonic balance, along the whole nodaldiameter (conversely, interblade phase angle) range. A sampleresult is shown on Fig. 13, for the second blade naturalmode (torsion) with 40 nodal diameters (σ ≈ 177◦) travelingforward. One can see that the wall power density contoursare similar to the mode displacement contours shown on Fig.8,indicating good FEM-CFD mapping and also that the energyexchange distribution due to blade motion relates, in this case,directly to the mode shape itself.

FIGURE 13: Power density on blade wall, for second blade naturalmode (torsion), 40 nodal diameters, as a forwardtraveling wave. Mesh velocity vectors are alsodisplayed (exaggerated displacement for visualizingpurposes).

Subsequently, the damping ratio was calculated from theaerodynamic work, shown on Fig. 14. Firstly, all simulatedmodes were shown to be stable for all nodal diameters (that is,ξ > 0). Secondly, all mode shapes present a relatively smoothharmonic behavior (mostly on its first order), as predictedby the aerodynamic influence coefficients theory, in travelingwave mode setups [43, 56]. The minimum damping ratio fromthe simulated natural frequencies will be used on the harmonicresponse analyses, as a conservative measure.

Computational solid dynamicsAfter the prestressed modal analysis, the next step is to

identify the frequencies and forcing shapes which excite thestructure. For these loads, the Campbell and interferencediagrams, respectively, are convenient. Due to the exploratorycharacter of this study, several cases have been run, howeverrestricted to the nominal rotational speed of the machine.

Along this manuscript, focus is cast on the disturbancesoriginated from the combustor, although the stator perturbation

(a) Displacement contours of first three blade-dominated natural modes

−40 −30 −20 −10 0 10 20 30 401

2

3

4

5

6

7

8

9

Nodal diameter

ξ no

rmal

ized

Mode 1Mode 2Mode 3

(b) Normalized damping ratio

FIGURE 14: Natural modes displacement contours andcorrespondent normalized damping ratio ξ , as afunction of nodal diameter.

(as seen by the rotor) is also naturally calculated. Onthat account, a sample interference diagram is shown onFig.15, with five circumferentially scattered disturbanceunits downstream of the stage (which could correspond todetonation tubes, rotating-detonation chambers etc.). Theinclination of the rotational speed line is determined notonly by how fast the shaft spins, but also by the number ofdisturbance units. Therefore, for each choice of combustorunits downstream, a new interference diagram is produced.The integral crossings between the speedline and the naturalfrequencies indicate which nodal diameter shapes are expectedto be excited [57, 58].

The simulated engine orders are directly correlated withthe excitation frequency fd . A Fourier decomposition of thetransient pressure signal (from the CFD runs) was performedfor each one of the disturbance frequencies. The samplingrate and interval was regarded so to achieve the properNyquist frequency, taking into account again the longestamong disturbance and blade passing periods. In general,a full wheel rotation produced periodically converged resultsenough to cover several multiples of the Nyquist frequency.It is important to check the transient signal to be used forthe Fourier decomposition specially for the lower disturbancefrequencies.

0 5 10 15 20 25 30 35 40Nodal diameter

Fre

quen

cy

Mode family 1Mode family 2Mode family 3Mode family 4Mode family 5Mode family 6Mode family 7Mode family 8Mode family 9Mode family 10StandingwaveBackward traveling waveForward traveling waveRotational speed: 100 %

FIGURE 15: Sample interference diagram for simulated blisk, withthe first ten mode shape families. An illustrativerotational speed line is indicated, for good visibilitypurposes.

Figure 16 shows the maximum von-Mises stress on therotor blade for the undisturbed and the disturbed case, forfd = 0.125 BPF run with Ad = 20%. The disturbed case showsclearly a stress distribution much higher than the undisturbed,particularly close to the root leading edge, right above the fillet.For this specific disturbance frequency, the first natural mode(bending) is the one most excited.

(b) Undisturbed (c) Disturbed

FIGURE 16: Normalized von-Mises stresses on rotor blade forundisturbed and disturbed case, for fd = 0.125 BPFcase with Ad = 20%.

The maximum von-Mises stress on the rotor blisk isshown on Fig.17, for the disturbed computations. Thenormalization was done in two different ways. On Fig.17(a), all stresses were divided by the stress obtained on the1.0 BPF disturbed run (any other value could be arbitrarilychosen, which would change only the vertical axis scale). Theobserved behavior matches with the drop in efficiency, as wellas the unsteady damping plots (depicted respectively on Figs.9and 12). The stronger disturbance propagation observed onthe low-frequency range excites the rotor blade more thanthe higher frequencies. Therefore, larger displacements and

stresses are expected.With respect to Fig.17(b), the normalization is

accomplished dividing the maximum von-Mises stressesby the corresponding values obtained in the counterpartundisturbed runs. This is done independently for eachexcitation frequency. Low disturbance frequencies, again,promote higher stresses on the blade. The high values areexpected, considering that, for the baseline undisturbed case,the strain induced on the blade due to the specified excitationfrequencies is indeed minimal (except if specifically adjacentto BPF’s or eventual system fluctuations, such as rotatinginstabilities, are present). On the other hand, for the disturbedruns, the Fourier decomposition of the pressure will invariablyyield much higher values close to the exciting frequencies.In spite of elevated relative values, the dynamic stress stillremains under acceptable material limits (at least for thedisturbance amplitudes here considered).

0.31 0.61 1.22 1.83 2.44 3.05 3.66 4.88 6.11 7.33St

0.125 0.25 0.5 0.75 1 1.25 1.5 2 2.5 30

20

40

60

80

100

BPF multiple

Max

imum

von

−M

ises

str

esse

s no

rmal

ized

(a) Normalized by 1.0 BPF multiple

0.31 0.61 1.22 1.83 2.44 3.05 3.66 4.88 6.11 7.33St

0.125 0.25 0.5 0.75 1 1.25 1.5 2 2.5 30

500

1000

1500

2000

BPF multiple

Max

imum

von

−M

ises

str

esse

s no

rmal

ized

(b) Normalized by undisturbed case

FIGURE 17: Maximum von-Mises stresses on rotor blade fordisturbed runs.

CONCLUSIONSNumerical aeroelastic and high-accuracy CFD analyses

were presented on this paper, contrasting undisturbed andpressure gain disturbed operation of a typical industrial highpressure compressor stage. Due to the exploratory natureof this study, a wide range of disturbance frequencies andamplitudes was considered. The principal outcomes are:

(i) From the turbomachinery component performanceviewpoint, decrease in stage isentropic efficiency η of morethan 15% (for pressure disturbance amplitude of Ad = 20%and frequency fd = 0.125 BPF) was observed. Additionally,momentary strengthening in recirculation zones adjacent tothe blade surface (due to the increase in the adverse pressuregradient), also occurred for the lower frequencies. Finally,η drop was shown to be squarely proportional to the relativevariation in outlet pressure, for the simulated range.

(ii) The pressure and temperature unsteady damping (asdefined on Eq.(5)) was higher than 95%, for disturbancefrequencies between 0.75 and 2.5 multiples of the BPF. Loweror higher frequencies are less unsteadily damped, meaningthat the propagating disturbance should be perceptible to agreater extent upstream of the simulated stage. However,for amplitudes up to Ad = 20%, two stages should suffice tocompletely damp the combustion-originated disturbances, inthe current amplitude and frequency range.

(iii) The aerodynamic damping calculations exhibited nounstable nodal diameter for the operating point. The three firstmodes simulated with the nonlinear harmonic balance methoddisplayed a fairly sinusoidal-like smooth behavior along theinterblade phase angle spectrum. The global damping ratio ξ

was calculated from the aerodynamic work to be further usedon the harmonic response analysis.

(iv) After a prestressed modal run, the rotor blisk wasexcited with the unsteady Fourier transformed forces,on mode-superposition harmonic response analyses.The horizontal comparison between different excitationfrequencies followed the same behavior of the efficiency dropand unsteady damping. That is, the upstream propagatingdisturbances excite the rotor blade substantially more onthe low-frequency end. Additionally, the von-Mises stressesobtained on the disturbed runs, anew for the lower frequencies,were several hundred times higher than the correspondingstresses for the undisturbed cases. For excitation frequencieshigher than 1 BPF, this stress magnification is much moremodest. Nonetheless, the stresses in all cases were still belowmaterial limits.

Further studies varying the disturbance pattern (suchas circumferentially rotating waves) as well as multistagedisturbed runs shall be performed in future work.

NOMENCLATUREA disturbance amplitudeC damping matrixc reference chordcp pressure coefficientd complex modal coordinated overdot for time derivative

(d = ∂d/∂ t

)f complex modal forcingfd disturbance frequencyK stiffness matrix` surface integration coordinateM mass matrixn normal versorp pressureq scaling factor for FEM-CFD mappingr radiusSt Strouhal number (St = fd cr/vaxial)

t timeT periodU displacement vectorv velocity vectorx spatial coordinates vectorWaero aerodynamic worky+ dimensionless wall distance

Greekε unsteady dampingη isentropic efficiencyµ dynamic viscosityξ damping ratioρ densityσ interblade phase angleφ generic state variableω angular frequency of vibration

Subscripts1 streamwise station numberingd disturbancej natural mode indexr rotors stator

AbbreviationsBPF blade passing frequencyCFD computational fluid dynamicsCSD computational solid dynamicsFEM finite elements methodGCIBC grid independence index for grids B and CHPC high pressure compressorPGC pressure gain combustionTSRPP time steps per rotor passing period

ACKNOWLEDGMENTThe authors greatly acknowledge the support from the

Berlin International Graduate School in Model and Simulationbased Research (BIMoS), as well as Rolls Royce DeutschlandGmbH & Co KG. for providing the test case data.

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