http://lib.uliege.be https://matheo.uliege.be Final work : Aeroelastic Calculations on the Front Stage of an Industrial Compressor Auteur : Fasano, Gianmarco Promoteur(s) : Dimitriadis, Grigorios Faculté : Faculté des Sciences appliquées Diplôme : Master en ingénieur civil en aérospatiale, à finalité spécialisée en "turbomachinery aeromechanics (THRUST)" Année académique : 2017-2018 URI/URL : http://hdl.handle.net/2268.2/5499 Avertissement à l'attention des usagers : Tous les documents placés en accès ouvert sur le site le site MatheO sont protégés par le droit d'auteur. Conformément aux principes énoncés par la "Budapest Open Access Initiative"(BOAI, 2002), l'utilisateur du site peut lire, télécharger, copier, transmettre, imprimer, chercher ou faire un lien vers le texte intégral de ces documents, les disséquer pour les indexer, s'en servir de données pour un logiciel, ou s'en servir à toute autre fin légale (ou prévue par la réglementation relative au droit d'auteur). Toute utilisation du document à des fins commerciales est strictement interdite. Par ailleurs, l'utilisateur s'engage à respecter les droits moraux de l'auteur, principalement le droit à l'intégrité de l'oeuvre et le droit de paternité et ce dans toute utilisation que l'utilisateur entreprend. Ainsi, à titre d'exemple, lorsqu'il reproduira un document par extrait ou dans son intégralité, l'utilisateur citera de manière complète les sources telles que mentionnées ci-dessus. Toute utilisation non explicitement autorisée ci-avant (telle que par exemple, la modification du document ou son résumé) nécessite l'autorisation préalable et expresse des auteurs ou de leurs ayants droit.
Transcript
http://lib.uliege.be https://matheo.uliege.be
Final work : Aeroelastic Calculations on the Front Stage of an Industrial Compressor
Auteur : Fasano, Gianmarco
Promoteur(s) : Dimitriadis, Grigorios
Faculté : Faculté des Sciences appliquées
Diplôme : Master en ingénieur civil en aérospatiale, à finalité spécialisée en "turbomachinery
aeromechanics (THRUST)"
Année académique : 2017-2018
URI/URL : http://hdl.handle.net/2268.2/5499
Avertissement à l'attention des usagers :
Tous les documents placés en accès ouvert sur le site le site MatheO sont protégés par le droit d'auteur. Conformément
aux principes énoncés par la "Budapest Open Access Initiative"(BOAI, 2002), l'utilisateur du site peut lire, télécharger,
copier, transmettre, imprimer, chercher ou faire un lien vers le texte intégral de ces documents, les disséquer pour les
indexer, s'en servir de données pour un logiciel, ou s'en servir à toute autre fin légale (ou prévue par la réglementation
relative au droit d'auteur). Toute utilisation du document à des fins commerciales est strictement interdite.
Par ailleurs, l'utilisateur s'engage à respecter les droits moraux de l'auteur, principalement le droit à l'intégrité de l'oeuvre
et le droit de paternité et ce dans toute utilisation que l'utilisateur entreprend. Ainsi, à titre d'exemple, lorsqu'il reproduira
un document par extrait ou dans son intégralité, l'utilisateur citera de manière complète les sources telles que
mentionnées ci-dessus. Toute utilisation non explicitement autorisée ci-avant (telle que par exemple, la modification du
document ou son résumé) nécessite l'autorisation préalable et expresse des auteurs ou de leurs ayants droit.
Aeroelastic Calculations on the FrontStage of an Industrial Compressor
Graduation Studies conducted for obtaining the Master’s degree in Civilengineer in aerospace, specialized in “turbomachinery aeromechanics”
Faculty of Applied SciencesUniversity of Liege
Academic year 2017 - 2018
MSc Thesis / Gianmarco Fasano
Abstract
Nowadays trend, in gas turbine, towards more efficient and powerful enginesis leading to design very slender blades with large aerodynamic loads and tooperate the machines at elevated temperatures and speeds. These featuresmakes the blades more susceptible to aeroelastic problems, such as flutter, aself-excited and self-sustained vibratory instability characterized by energytransfer between the blades and the fluid. This interaction can cause sig-nificant damages to the structure in a short period of time, unless properlydamped. Of fundamental importance is the development of reliable toolscapable of detecting potential flutter problems.
The aim of this thesis work is to validate the AU3D CFD tool, developedat the Imperial College London. Flutter analysis is performed on the frontstage of an industrial gas turbine compressor, designed and manufactured atSiemens Industrial Turbomachinery Ltd. An evaluation of different solvers iscarried with respect to accuracy, calculation efficiency and industrial applica-bility. Being able to simulate the machine working operation using differentsoftware is an important goal for a company. On one side, each solver hasits unique algorithms, so testing a case with different solvers could assess thevalidity of the obtained results in a more reliable way. On the other side,different software implement different methods and different approaches sothat selecting a specific solver for a test case can lead to higher efficiency interms of accuracy and time.
The results obtained through AU3D show a good match with what othersolvers, already used as reference in Siemens AB, predicted. The CPU time,for both steady and unsteady calculations, is suitable for industrial applica-tions. Furthermore, the code is able to manage off design operational pointwithout much effort. In this work, only some of the possible analysis method,offered by AU3D, are investigated; many more are available to perform morespecific simulations.
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Acknowledgement
I would like to express my gratitude to Prof. Nenad Glodic, the director ofthe THRUST programme, for carrying on with passion this amazing Masterprogramme and for his extraordinary guidance during the last two years.
Thanks to Prof. Paul Petrie-Repar and Prof. Jens Fridh for all the sharedknowledge and all the stimulating experiences made during the first year atKTH. Thanks to Prof. Ludovic Noels for leading me through the wonderfulexperience at ULg.
Thanks to all the THRUST students for all the unforgettable adventures wehad in the past two years.
Thanks to my supervisor at Siemens AB, Dr. Maria Mayorca for entrustingme for the opportunity of performing my final project in such an inspiringcompany. I am grateful for the exceptional guidance provided and for thevaluable and interesting discussions.
Thanks to the aeromechanical team at Siemens AB, Dr. Erik Munktell,Qingyuan Zhuang and Nikola Kafedzhiyski for the support and for provid-ing an excellent atmosphere at work. Thanks to Aliaksandr Rahachou forintroducing me to the programming world.
Thanks to Lad Bharat, Caetano Peng, Nigel and Senthil Krishnababu for allthe useful advice and the constant encouragement.
Thanks to Prof. Greg Dimitriadis for academically supervising my thesis.
Finally, I would like to express my very profound gratitude to my parents,Vincenzo and Beatrice, to my sister Giorgia and to all my dear friends for pro-viding me with unfailing support and continuous encouragement throughoutmy years of study.
∆T Temporal Phase Lag between Pitchwise Boundaries
∆t Time Step
δ Logarithmic Decrement
ε Turbulence Eddy Dissipation Rate
ηs Isentropic Efficiency
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γ Specific Heats Ratio
κ Turbulence Model Constant
µt Turbulence Viscosity
ν Kinematic Viscosity
ω Angular Frequency
ρ Density
σ Inter-blade Phase Angle
σε Turbulence Model Constant
Θ Pitch
θ Circumferential Coordinate
ξ, η, ζ Orthogonal Blade Modes
Subscripts
0 Total Condition
1 Cascade Inlet
2 Cascade Outlet
h Temporal Harmonic Index
i Time Step Index
j Passage Index
l Disturbance Index
n Travelling Disturbance Index
t Turbulence
Superscripts
ˆ Complex Value
˜ Time Varying Value
Abbreviations
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3D Three Dimensional
AROMA Aeroelastic Reduced Order Modeling Analyses
BCs Boundary Conditions
BW Backward
CFD Computational Fluid Dynamics
CFL Courant–Friedrichs–Lewy
EO Engine Order
FE Finite Element
FW Forward
HB Harmonic Balance
IGV Inlet Guide Vane
LE Leading Edge
M1 Mode 1
M2 Mode 2
ND Nodal Diameter
NRBC Non-Reflective Boundary Conditions
PS Pressure Side
R1 Rotor of the First Stage
S1 Stator of the First Stage
SGT Siemens Gas Turbine
SPMR Single-passage Multi-row
SS Suction Side
TE Trailing Edge
TW Travelling Wave
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1 Introduction
1.1 Gas Turbine
A Gas Turbine is the combustion engine at the heart of a power plant, itturns fuel into electric current with a high efficiency. It relies on turboma-chinery and combustion theory. In figure 1, the Siemens Gas Turbine 800is represented. Starting from left, moving to the right, the surrounding airenters into the compressor which increase the fluid pressure through variousstages, the compressed fluid is then mixed with fuel and in the combustor themixed fluid is burned at extremely elevated temperature. The produced hotgas passes through the turbine stages, it expands causing the turbine rotorsto spin. The turbine shaft, which rotates at high speed, is connected to therod in a generator which turns a big magnet enclosed in multiple coils madeof copper. While the generator magnet rotates, electrons are forced to moveand their motion through a wire is electricity. A combined-cycle power plantgenerates power very efficiently combining a gas turbine with a steam one.
Figure 1: Siemens Gas Turbine 800, courtesy of Siemens Industrial Turbo-machinery AB
In this work, the front stage of an industrial compressor, featuring transonicflow at the rotor, is investigated with a focus in compressor aerodynamicsand aeroelasticity.
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1.2 Axial Flow Compressor
1.2.1 Introduction
An axial flow compressor is a rotating machine which pressurizes the workingfluid by means of multiple stages. A stage is composed by a row of rotatingblades followed by a row of stator vanes. Each stage features a relative smallpressure ratio, therefore, multi-stages are exploited to achieve a high one,increasing the overall machine efficiency. Moreover, a specified number ofvanes are variable, they can be rotated to change the incidence angle so toallow the compressor to operate at off-design, i.e. different speeds.
1.2.2 Basic Operation
The fluid enters the rotor bladerow, which is spinning at elevated speed,driven by the turbine. The rotor blades feed energy into the fluid whichaccelerates increasing its pressure. Through the stator vanes, the fluid decel-erates to achieve an higher pressure at the stator outlet. Moreover, the fluidis redirected to reach a desired flow angle. Figure 2 illustrates the section ofan axial compressor among with a diagram representing the trend of velocityand pressure through the various bladerows.
Figure 2: Pressure and velocity changes through an axial compressor [1]
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1.2.3 Blade Design
The blade profile has to be designed to achieve the maximum efficiency as-suring at the same time its structural integrity. Rotor blades are, usually,designed considering a pressure gradient along the span to guarantee a uni-form axial velocity profile. The centrifugal forces, due to the high rotationalspeed of the shaft, is counteracted with the presence of a high flow pressureclose to the tip region. To achieve these characteristics, the blade has tobe twisted from root to tip, figure 3. This feature ensure the correct flowincidence angle at each span level. Moreover, at the tip and at the root, theblade profile section is more cambered, ‘end-bend’, to avoid regions of slowfluid in the boundary layer.
Figure 3: A typical rotor blade showing twisted contour [1]
1.2.4 Operating Conditions
A compressor is designed to operate at different mass flow rates and differentspeeds which results in different pressure ratio values. Figure 4 shows a typ-ical compressor map. A speedline is defined keeping constant the rotationalspeed and changing back pressure which leads to different mass flow rates.
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Each speedline has two outer limits that the compressor must not overcome:at low mass flow rates, stall and surge phenomena occurs, while, at highmass flow rates, the flow becomes choked. Stall is caused by a large flow sep-aration at the suction side of the blade profile which occurs when the fluidincidence angle is too high or too low. It can lead to severe blade vibrationand, as consequence, machine failure. Surge occurs when the pressure atthe outlet is large enough with respect to the inlet one, that the rotor is nomore able to accelerate the fluid enough to overcome the pressure difference:the flow is reversed causing the complete breakdown of the airflow throughthe whole compressor. This lead to large flow fluctuations which change thethrust force applied on the rotor causing critical damages to the structure.Finally, the choke condition occurs when the fluid reach a Mach value biggerthan one at the blade throat. In this condition, the mass flow rate can notincrease anymore.
Figure 4: Axial compressor characteristics [2]
1.2.5 Transonic Compressor
A transonic compressor features flow velocity above the speed of sound reach-ing Mach values between 0.8 and 1.2. Usually, the rotor blade tip is char-acterized by a low camber profile and the pressure rise is mostly due to thepresence of a shock wave. The latter represent an abrupt variation of flowvariables, i.e. pressure increase and velocity drop, due to velocities thatreach values above the speed of sound. Different shocks develop around the
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blades and their magnitude and location depend on the compressor operatingconditions. In figure 5 different shock patterns close to the tip of a typicaltransonic compressor are illustrated. Following the compressor speedline,from high mass flow rates to small ones, the hereafter described conditionsare found:
• Choked flow : at low back pressure, two oblique shocks are present, thefirst is attached to the LE, while, the second is a passage shock sinceit goes through the blade-to-blade area. The latter moves upstream asthe back pressure increases.
• At Design point : at the designed back pressure, the passage shockbecomes more weak and, in certain cases, disappears; on the contrary,the oblique shock at LE strengthens moving upstream to develop a bowshock.
• Near stall : at high back pressure, there is the detachment, from the LE,of the bow shock which move upstream potentially going to impingeon the TE of the upstream bladerow (IGV or stator).
Figure 5: Shock patterns near the tip of a typical transonic compressor [3]
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1.3 Aeroelasticity
1.3.1 Introduction
Aereolasticity is the study of the interaction of inertial, structural and aero-dynamic forces. In turbomachinery, it represents the study of the interactionsbetween the blades and the surrounding flow. As a matter of fact, a pertur-bation in the flow is related with an unsteady pressure acting on the bladesurface. Depending on the source of the perturbation and on the charac-teristics of the unsteady pressure, i.e. periodic, different phenomena can bedefined.
Figure 6: Collar’s Aeroelastic Triangle
Two main aeroelastic phenomena are present in turbomachinery, both partof the dynamic aeroelasticity field, figure 6: forced response and flutter. Theformer is a synchronous vibration coming from flow non-uniformities, while,the latter is a self-induced and sustained vibration caused by the aerody-namic coupling between adjacent blades. The excitation of structural modescan be investigated by means of the Campbell diagram, figure 7. It representsthe frequency versus the rotational speed of the shaft. Whenever there is acrossing between the structure natural frequency and the Engine Order linesa possible resonant condition can appear and the structure will vibrate witha relative high amplitude which could lead to failure. The Engine Order ex-citation is a periodic force, i.e. wakes from an upstream bladerow interactingwith the downstream blades. This type of excitation is force response whichcan be easily predicted from such plot. Instead, flutter is not correlated to
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EO, therefore the frequency at which it could appear can not be know atpriori.
Flutter is a self-excited and self-sustained vibration phenomenon. The vibra-tion starts by itself if the fluid-structure coupled system becomes unstable.Unless proper damping is provided, vibration amplitudes rapidly escalate.To better understand the phenomenon, consider a single blade oscillating inits cascade, its motion influences the flow surrounding it and its neighboringblades, this is called aerodynamic coupling. The blade feels this flow per-turbation as an unsteady pressure applied on its surface. The pressure canbe expressed as a force and the latter as work per cycle of oscillation. If thework is such that energy is transferred from the fluid to the structure, theblade oscillation amplitude grows leading to structural damages and finally tomachine failure. Flutter is, currently, a difficult phenomenon to understandcompletely. Many researches are being developed to better understand allits different characteristics. The most affected turbomachine parts are fansand compressor front stages. For the latter, depending on the flow charac-teristics, flutter domains can be defined on the compressor map as shown infigure 8.
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Figure 8: Flutter domains on the compressor map [5]
A description of the various types of flutter follows:
• Subsonic and Transonic Stall Flutter : appears when the compressor isoperated close to the stall line, so at a reduced mass flow and increasedpressure ratio, and the angle attack is sufficient high to cause stall;
• Choke Flutter : appears when the compressor is operated close to thechoke line; the flow velocity is very high and a shock develops, the latteroscillates due to the blade motion leading to flow separation that canexcite some particular structural eigenfrequency;
• Supersonic Stalled Flutter : the flow feature a very strong shock and arelevant boundary layer separation, the latter is referred as the maincause of flutter for this type of flow;
• Supersonic Unstalled Flutter : in this case flutter is not due to theboundary layer separation but from the reflection of the shock, imping-ing on the neighbouring blade;
• Acoustic Flutter : while the blade vibrates, acoustic waves are emittedwhich can be reflected back on the blade exciting a vibration modes.
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An important parameter that influence flutter occurrence in turbomachinesis the reduced frequency, discovered by Theodorsen:
fred =2πfc
V∞=
t
T
where f is the blade oscillation frequency, c is the chord and V∞ is the freestream velocity. It can be interpreted as the time, t, a fluid particle takes totravel the entire blade chord over the oscillation period, T . The fluid particlewill take more time to travel the chord if the flow is unsteady. Therefore,small values represent a quasi-steady flow, while, high values characterizedflows with relevant unsteadiness. The lower the reduced frequency, the higherthe flutter risk. The critical values for turbomachines are between 0.2 and0.5.
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1.3.3 Structural Dynamics
Introduction Flutter phenomenon is strictly related to the rotor vibrationcharacteristics. Therefore, it is important to understand the vibratory modesof blades and disks. There are, potentially, infinite modeshapes and each ofthem co-exist at the same time in the considered object. These are obtainedthrough a structural analysis using finite element methods.
Blade Modes The basic vibratory modes of a blade are the chord-wisebending or flap (F), torsion (T), and edge-wise bending (E). The line thatconnect the points that features zero displacement is referred as inflection lineor node line. The order of each mode is defined by the number of inflectionlines present in a specific mode. Figure 9 displays the three described modes,in the upper part of the picture the displacement contour plot is shown, thered color refers to the largest displacement.
Figure 9: Blade modeshapes
Disk Modes The principle is the same as for blades, the main differenceis that inflection lines are now referred as nodal diameter since they span thewhole diameter of the disk. Figure 10 displays the disk modeshape for a nodaldiameter value of 1 and 3. The sign ‘+’ denotes an outward displacement ofthe plane surface, while ‘-’ an inward one.
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(a) ND = 1 (b) ND = 3
Figure 10: Disk modes
Bladed Disk Modes In turbomachinery, the rotor can be made assem-bling the various blades on the disk or manufacturing directly blades anddisk as a single object, called blisk. Either way, the resulting bladed diskfeatures modeshapes that are more complex with respect to the individualblade and disk ones. Firstly, for bladed disk the ND value is limited by thepossible axisymmetric divisions given by the number of blades, NB:
NDmax =NB
2if NB even
NDmax =NB − 1
2if NB odd
The modeshapes are now referred as mode-families: for each blade modecorrespond a group of disk modes that depends on the ND value. These arealso called Travelling Wave Modes (TWM). While the rotor is rotating,the nodal diameters rotate along the disk creating points of non-deformationat the base of the blades at different times. Therefore, the blade will vibratefollowing its basic modeshape and frequency, but each blade will not vibratein phase between each others defining a travelling wave. Furthermore, theND can rotate forward, following the rotor rotational speed direction, orbackward; thus the travelling wave appears in pair of FW and BW. TheIBPA (inter-phase blade angle) is expressed as:
σforward =2 · π ·ND
NB
σbackward =2 · π · (NB −ND)
NB
The aerodynamic coupling is highly influenced by the relative motion of theblades represented by the phase angle value. Bladed disk modeshapes arecomplex numbers. Figure 11 displays three different modeshapes. For a NDvalue of zero, the mode is said to be “standing” because there is no NDrotating along the disk; in this case, all the blades oscillate in phase.
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Figure 11: Bladed disk modeshapes
Depending on the machine part considered, i.e. fan, turbine or compressor,and on its geometry, the bladed disk can be either disk or blade dominated.The disk natural frequency varies changing ND because it can be divided indifferent sectors which are structural coupled. Usually, for relative stiff disks,the frequency is disk-dominated for low ND values and blade-dominated forhigh ones, as shown in figure 12
Figure 12: Bladed disk frequency correlation
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1.3.4 Flutter Stability Formulation
Introduction The fluid-structure system equilibrium equation can be ex-pressed as:
[M ]X + [C]X + [K]X = Faero(t)
where M is the modal mass matrix, C the modal damping matrix and Kthe modal stiffness matrix. X represents the modal coordinates vector andFaero(t) represents the unsteady aerodynamic force vector. The latter is dueto the flow and the motion of the blade. It comes from the unsteady pressureacting on the blade surface when it vibrates. This force can be expressed interms of aerodynamic damping and stiffness which leads to define the coupledequation of motion:
[(−ω2M + iω(C + Caero) + (K +Kaero]eiωt = 0
In blade cascades the structural terms (M, C and K) are characterized byhigher values than the aerodynamic damping term due to high mass ratio, theblade mass is much bigger than the mass of air that surrounds it. The struc-tural and the aerodynamic part are then decoupled: the structural analysisto calculate eigenfrequencies and eigenmodes can be determined assumingno-flow conditions, i.e. vacuum, while the aerodynamic part can be takeninto account performing a purely unsteady aerodynamic analysis [4].
Unsteady Aerodynamic Analysis Consider a 2D section of a blade,figure 13, and assume the blade motion is harmonic and oscillates in travellingwave mode in its cascade. The motion can be considered a complex vectorcomposed of three orthogonal components:
h = (hξ + hη + hζ)eiωt
The unsteady pressure acting on the blade surface can be expressed as:
p = p · ei(ωt+φ)
where φ represent the phase shift between the pressure and the blade motion.
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Figure 13: System of orthogonal modes [4]
From the pressure, the unsteady force acting on the blade can be recoveredas:
F = −pnThe phase shift between force and motion leads to the aerodynamic damping,which can be either positive or negative. Whenever negative damping ispresent, the fluid is feeding energy to the structure and flutter occurs.
Figure 14: Positive damping, motion lagging force [6]
The unsteady work per cycle can, then, be calculated as:
Wcycle =
∫T
F · hdt =
∫T
F · heiωtdt
The work per oscillation cycle represents the energy transfer between struc-ture and fluid. Normalizing the work per cycle by means of the oscillationamplitude [10], the aerodynamic damping parameter can be calculated as:
Ξ =−Wcycle
πh
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This parameter can be used to visualize and evaluate flutter stability, figure15.
Figure 15: Flutter stability [7]
Whenever the aerodynamic damping parameter is positive, no flutter riskis present: the fluid is damping the blade motion preventing the oscillationfrom growing. In contrast, a negative damping parameter denotes an energytransfer from the fluid to the structure and flutter phenomenon occurs if notenough structural damping is provided to the structure.Another commonly used parameter to judge stability is the LogarithmicDecrement, which describes the relative change in amplitude per cycle, or,in other words, the decay or growth of an oscillation.
δ =Waero
4KEaverage
The average kinetic energy of the vibrating blade is calculated as:
KEaverage = ω2nπ
2q2
where q is the scaling factor used for the modeshape.
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2 Objectives and Approach
2.1 Objectives
The main objective of this project is the validation and application of theAU3D CFD aeromechanical tool on the front stage compressor of an indus-trial gas turbine developed at Siemens Industrial Turbomachinery. To assessthe validity of the obtained results a comparison with other solver is carriedin terms of accuracy, calculation efficiency and industrial applicability.Three different sub-objectives are defined as follows:
1. Mesh generation
2. Steady calculation
3. Flutter phenomenon investigation
2.2 Approach
A summary of the main parts of the work is described as follow.
Mesh Generation The mesh for the first stage and half of the compressoris generated by means of the LEVMAP tool embedded in AU3D. Each bladerow is meshed individually to being assembled afterwards. The mesh genera-tor requires that the blade profile is defined following a specific format, calledJH05. A C# code, programmed in Visual Studio, is developed to convert theblade profile definition format, currently used at Siemens, to the LEVMAPone. Finally, the mesh features, the user wants to assign, are defined in ascript, used as input in LEVMAP.
Steady calculation A single passage steady analysis is performed on thefront stage of the compressor. The whole stage is modeled to allow thesetting of reliable boundary conditions. The results are then compared withones previously obtained through other solvers. The comparison is carriedin terms of blade profile loading and Mach number contour plots.
Flutter phenomenon investigation Aeroelastic calculation on the rotorare performed using the time marching unsteady calculation solver embeddedin AU3D. For this analysis, only the rotor blade row is modelled as singlepassage. The boundary conditions are extracted from the steady calculations.Flutter phenomenon is investigated in term of aerodynamic damping.
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3 CFD Methods for Flutter Prediction
CFD methods able to model unsteady turbomachinery flows have becomean important aspect in the design of gas turbines as well as in aero engines.Unsteady phenomena such as flutter or forced response need to be evaluatedwith a certain degree of accuracy to guarantee the integrity of the machineduring its operation. Many different methodologies to solve the flow un-steadiness have been developed in the past years. This review focuses onthe various prediction methods for flutter, on which are based the differentsolvers described later in this work, figure 16.
Figure 16: CFD methods for flutter prediction
3.1 Frequency Domain
Frequency domain techniques are based on the main assumption that anyflow unsteadiness is small compared to the steady part and is periodic intime. The governing fluid equations of motion and the associated boundaryconditions are then linearised: each variable can be expressed as the combi-nation of a mean value U and a small perturbation U ′ of the mean value:
U(x, t) = U(x) + U ′(x, t)
where x = (x, r, θ) represents the location of the various nodes of which thediscretised CFD domain is composed of. The steady term, U , is calculated
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performing a steady-state analysis, while the unsteady term, U ′, is decom-posed into its Fourier components, Un, which are solved independently fromthe steady flow:
U ′(x, t) =
NH∑n=1
Un(x)ejωnt
Assuming the perturbation harmonic in time, the time derivatives are re-placed by jωn, where ωn is the frequency of the unsteady disturbance. Timedoes not appear explicitly and the problem becomes linear and steady withvariable coefficients depending only on the steady-state solution. Since thesteady solutions is assumed periodic, an IPBA can be defined for each har-monic component so that the full annulus solution can be recovered from asingle passage calculation reducing drastically the required CPU time. Thistime-linearised method is accurate only for small harmonic perturbationsaround a uniform mean-flow. Moreover, each Fourier harmonic is solved in-dependently using a pseudo-time marching method [11], which neglects po-tentially important non-linear interactions between the steady and unsteadyflow and between different frequencies perturbations. Therefore, many stud-ies have been made to develop a non-linear frequency domain method. Inthe following two different non-linear techniques, introduced, respectively, byNing and He [12] and by Hall [13], are described.
3.1.1 Non-linear Harmonic Method
The non-linear harmonic method has been introduced by Ning and He [12],it relies on time-linearised frequency methods but exploits a time-averagedflow, despite of a steady state one, to express unsteady perturbations. Theharmonic linearised solution is, then, affected by the time-averaged flow.Solving the time averaging of the non-linear flow equation results in addi-tional terms, called ‘unsteady stress’ terms which account for the unsteadi-ness non-linear effects on the time-averaged flow [5]. Since the resultingequations are steady, they can be solved using conventional time-marchingCFD techniques in pseudo-time. The main advantage of this method is itsability to capture interactions between the mean flow and the first orderperturbation, which are mainly present in transonic flows with shocks wheretime-linearised methods do not take into account non-linearities effects, pre-dicting abrupt pressure peaks [5]. Nonetheless, non-linearities arising fromthe interaction between different disturbances are still neglected [14].
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3.1.2 Harmonic Balance Technique
The harmonic balance technique has been introduced by Hall [13] for mod-eling unsteady nonlinear flows in turbomachinery. This method has somesimilarities to the non-linear harmonic method introduced by Ning and He[12], but it is more general. The temporal behaviour of the solution is ob-tained from uniformly sampled ‘snapshots’ in time. For an analysis includingn unsteady harmonics, steady state solutions at (2n+1) time levels are com-puted. The solutions are linked through phase-lagged boundary conditionsand a spectral time-derivative operator in the interior domain [5]. From theknowledge of the temporal behaviour in time, a set of coupled partial differ-ential equations for the unknown Fourier coefficients of the flow field can bewritten. A pseudo-time term is introduced into the harmonic balance equa-tions so that the equations can be solved using conventional time-marchingCFD techniques [13].
A big advantage of this method is the possibility to include higher orderharmonics; others non-linear frequency domain methods only capture thenon-linear interaction between the time-averaged flow and the first harmon-ics. However, including higher order harmonics is very expensive in termsof CPU time: each harmonic solution requires two steady state calculations,real and imaginary part, and all harmonic solutions must be computed atthe same time. For the complete formulation of the harmonic balance tech-nique and an application of the method to the front stage transonic rotor ofa modern high-pressure compressor, the reader is referred to reference [13].
3.2 Non-linear Time Domain
3.2.1 Whole Annulus
The most accurate approach to solve unsteady flows is to model the wholeannulus, 360°. In this way, the flow characteristics are fully captured since noassumptions are required, such as periodicity or linearity. A time-marchedsolution can be obtained by means of a defined integration scheme. Thismethod is inefficient in terms of simulation time required to solve the full do-main, but, nowadays super performing computers are able to carry a transientsimulation in sufficient time; as a matter of fact, this method is becomingmore and more present in industrial applications.
3.2.2 Single-passage for Axisymmetric Flows
Nevertheless, to evaluate flutter, especially in preliminary design stage, thewhole annulus approach is computationally expensive and is not suitable.
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Therefore, a big effort has been made to develop methods for which mod-elling a single passage is sufficient to obtain accurate results. In this way,the computational cost is drastically reduced. The methods rely on the as-sumption of time-space periodicity: any unsteadiness must take the form of atravelling wave, only in this way phase-lagged periodic boundary conditionscan be defined. To implement the pitch-wise boundary conditions, the CFDdomain mesh is modified adding ‘dummy’ or ‘shadow’ points which extendinto the domain of the adjacent passage as shown in figure 17.
Figure 17: Single passage domain with shadow point boundaries [5]
Each shadow point corresponds to a master point inside the single passagedomain. These shadow points define the pitch-wise boundary conditionscollecting data about the temporal variation of the master points. The phase-lagged periodicity between master and shadow points is expressed as:
UShadow(x, t) = UMaster(x, t± |∆T |)
where the temporal phase shift, |∆T |, is the ratio σ/|ω|.
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4 AU3D Methodology
The different methodologies, implemented in the AU3D code, to evaluateflutter are described to help the reader understanding, in a more completeway, the results that will be illustrated later in this work.
4.1 Introduction
The AU3D code was developed at the Imperial College London in partnershipwith Rolls Royce plc. The main solver of the AU3D suite, is a finite-volumetime accurate Reynolds-Averaged Navier-Stokes solver for multi-stage, multi-passage viscous and inviscid compressible flows in turbomachines for 3Dsteady and unsteady aerodynamics, aeroelasticity and aeroacoustic analy-ses [15]. A full description of the formulation implemented in the code ispresented in the paper by Sayma et al [16]. Flutter stability evaluation canbe performed using either whole annulus or single passage methods. Thefirst is more time consuming but it is able to solve the flow unsteadinessfor all nodal diameters values in one unique simulation. The single-passagemethod, instead, is faster but each ND value requires one separate simula-tion; this means no modal interaction is modelled: single passage analysisassumes that there is negligible coupling between any two modes. Instead,since the whole annulus flutter analysis time-integrates a number of familiesof modes simultaneously, any subsequent coupling is automatically captured.
4.2 Time-domain Fourier Model
As already stated in section 3.2.2, single-passage methods require a definitionof phase-lagged boundary conditions. In this section the BCs implementedin AU3D are described. The method was developed for the most general casewith multiple travelling and stationary disturbances per row. Here, only theformulation for taking into account unsteady perturbations is explained indetail, since no stationary disturbances are present in the type of flutter anal-ysis carried out in this work. Firstly, temporal and circumferential averagesin bladed passages are explained; from the latter a chorochronic periodicitycan be derived and, finally, steady and unsteady perturbations are described.
4.2.1 Temporal and Circumferential Averages in Blade Passages
Consider a row composed of NB blades and assume a travelling disturbance,characterized by frequency 1/T , and a stationary circumferential variation,characterized by wave length 2π/k. The time-averaged flow is the same in
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every passage, if the stationary disturbance is no present. Anyway, whenevera stationary disturbance exists, the time-averaged flow field changes frompassage to passage and a circumferential average, also called passage-average,needs to be defined. The flow variable must depend on the circumferentialcoordinate inside the passage but the passage-averaged flow field is identicalin every passage. We therefore define the vector x = (x, r, ζ) describingthe local coordinate system inside the passage and the vector xg = (x, r, θ)describing the global coordinate system. The transformation from local toglobal coordinate system is expressed by:
xlocal = xglobal = x
rlocal = rglobal = r
θj = ζ + 2πj − 1
NB
Uj(x, r, ζ, t) = U(x, r, θj, t) describes the flow field in the jth passage. Thisallows the definition of a passage-dependent time-average:
U0(x, r, θj) =1
T
∫ T
0
U(x, r, θj, t)dt
and a passage-averaged or circumferentially averaged flow field:
U(x, r, ζ, t) = U(x, t) =1
NP
NP∑j=1
U(x, r, θj, t)
where NP is the minimum number of passages required to complete an integernumber of wave lengths of the stationary disturbance. This leads to thedefinition of the time-space averaged flow field:
U0(x) =1
NP
1
T
NP∑j=1
∫ T
0
U(x, r, θj, t)dt
In the discrete case with NT equally discretised time steps per period of thetravelling disturbance, i.e. ∆t = T
NTand ti = i∆t:
U0(x) =1
NP
1
NT
NP∑j=1
NT∑i=1
U(x, r, θj, ti)
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4.2.2 Chorochronic Periodicity
The method assumes that the flow variables inside the single passage do-main can be expressed as a superposition of a space-time average, stationarycircumferential variations and unsteady disturbances:
U(x, r, θ, t) = U0(x) + U(x, θ) + U ′(x, θ, t)
also illustrated in figure 18. The space-time average, U0(x), changes de-pending on the chosen location inside the passage and, due to interactionwith the steady flow field, the unsteady fluctuation, U ′(x, θ, t), and the sta-tionary variation, U(x, θ), vary inside the single-passage domain in terms ofamplitude and phase. If these conditions are satisfied, a defined time-space(chorochronic) periodicity can be determined between adjacent passages andthe unsteady and steady variations can be expressed by means of temporaland spatial Fourier components respectively.
Figure 18: Flow field decomposition into time-space average, circumferentialvariation and unsteady perturbation, from [5]
4.2.3 Unsteady Perturbations
The unsteady fluctuations typically take the shape of travelling waves withconstant angular velocity v and angular frequency ω. For self-excited vibra-tion, the frequency and circumferential wave number of the unsteady per-turbation are the vibration frequency and nodal diameter pattern. If more
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than one of these perturbations are present, the unsteady fluctuation can bedescribed as a superposition of NL independent disturbances:
U ′(x, θ, t) =
NL∑l=1
U ′l (x, θ, t)
where U ′l (x, θ, t) is the unsteady part of the lth perturbation. The num-ber of independent disturbances depends on the nature of the flow. Forflows without strong interactions the independent disturbances are the fun-damental perturbations at the source of the problem (e.g. travelling dis-tortions, oscillating blades or blade passing disturbances). For flows withsignificant interactions between the fundamental perturbations, the inducedfrequencies should also be included in the independent disturbances [5]. Ifthe time-averaged flow in every passage is the same, i.e. U(x, θ) = 0, andthe unsteadiness consists of one or more disturbances of the nature describedabove, a time-space periodicity applies which is characterised by the angularfrequency ω and circumferential wave number ω/v, where v is the rotationalspeed of the disturbance. It then becomes possible to decompose each dis-turbance into its temporal Fourier components:
U ′l (x, θ, t) =
NH∑h=1
(alh(x)cos(h
ωlvlθ − hωlt) + blh(x)sin(h
ωlvlθ − hωlt)
)where NH is the number of harmonics. Now, we assume that each indepen-dent disturbance features the same number of harmonics NH , so that, wecan combine the indices of independent disturbances l and their harmonicsh into a disturbance index n so that:
NN = NHNL
ωn = ω(l−1)NH+h = hωl
vn = v(l−1)NH+h = hvl
As a consequence, the unsteady perturbation is expressed as a superpositionof NN travelling disturbances:
U ′l (x, θ, t) =
NN∑n=1
(an(x)cos(
ωnvnθ − ωnt) + bn(x)sin(
ωnvnθ − ωnt)
)The time-average is recovered imposing n = 0 and ω0 = 0. The temporalFourier coefficients an(x) and bn(x) are calculating using a partial substi-tution technique developed by Gerolymos [17] which is able continuously
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updates the Fourier coefficients. The flow variables at time step i on theopposite boundary are redefined according to:
U(x, θ + Θ, ti) = U(x) +
NN∑n=1
(an(x)cos(
ωnvn
(θ ±Θ)− ωnti)+
bn(x)sin(ωnvn
(θ ±Θ)− ωnti))
where Θ = 2π/NB is the pitch. Using absolute phase angles in the Fouriertransform and updating the coefficients guarantees that the conditions on theopposite boundary are redefined with the correct phase lag. This formulationcan be further expanded to take into account stationary disturbance, U(x, θ)and circumferentially distorted travelling disturbance but these excitationsdoes not characterize flutter phenomena, they are mainly related to forceresponse and inlet distortions; therefore, this overview will not go further.The reader is referred to [5] for a complete formulation.
4.2.4 Limitations of the Time-domain Fourier Model
The time-domain Fourier method can be applied only to turbomachineryflow fields characterised by the following properties:
• Temporal periodicity: unsteady perturbations are expressed by meansof phase-lagged BCs, therefore, any unsteadiness must take the formof a travelling wave. As a matter of fact, only spinning modes can berepresented by this method, while, all non-rotating modes are lost.
• Preestablished frequency and wave number spectrum: frequencies andwave numbers of the disturbances must be known a priori. This is truefor flutter phenomena, where initial estimates can be made based onrotational speed and circumferentially wave number content. Neverthe-less, this prevents the modelling of undeterministic unsteady features,i.e. rotating stall.
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5 Mesh Generation
5.1 Blade Profile Definition
In order to use the LEVMAP mesh generator, the blade profile definition hasto be specified following a detailed format called JH05. The latter is quitedifferent from the currently used format at Siemens AB named VDA. Thedifferences between the two formats are described in details below along withthe description of the developed code that converts VDA in JH05 format.
5.1.1 VDA and JH05 Blade Profile Formats
VDA The blade profile is generated starting from several blade-to-bladesections at different span levels. Each section contains at least four curves:LE, SS, TE and PS, which are defined as Bezier curves. In figure 19, anexample of a turbine section is shown. There are five Bezier curves of degreefive. Green lines denote control point arms; yellow points represent internalcontrol points and dark blue line, coordinate profile of the section.
Figure 19: Curves and control point definition in a blade section [8]
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The different sections are then radially connected between each other usingBezier patches which will form the various blade surfaces. In figure 19, redsquares represent boundary control points for the patches. The final bladeprofile is illustrated in figure 20.
Figure 20: Final blade profile, VDA format
As can be noted from the picture, in this case, the blade is subdivided in 4main surfaces: LE, PS, TE, SS; and each of them is further subdivided insix different sub-surfaces, which defines Bezier patches.
JH05 The blade profile is based on the coordinate points of the section.A certain number of cross sections is defined to be equally spaced along thespan. Of each section three main groups of parameters are used. The firstone consist of inlet/outlet metal angles and LE/TE radii. The second is astreamtube, a line that starts from the domain inlet surface, intersects theLE and TE of the section and ends at the domain outlet surface. Thanks tothis feature, the mesh generator will automatically create the single passageblade domain. Each streamtube is described in terms of axial and radialcoordinates. Finally, a fixed number of coordinates (r, θ, x) are defined foreach section. Representation of the described quantities is shown in the belowparagraph, Code Algorithm.
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5.1.2 Format Conversion
To perform the change from VDA to JH05 a Siemens software called CATOis used. “CATO (Common Airfoil Tool) is designed for 2D profiling and eval-uation of turbine and compressor airfoils (blades and vanes) as well as for the3D modification of the composed profiles” [8]. The software is based on theC# programming language which is run through Visual Studio Professionalto realize a graphical user interface (GUI). The program was already ableto read a VDA file with all its different features: Bezier curves and patches.Therefore, a new code is developed to extract from the VDA blade profile allthe data required to generate the JH05 format file.
Code Algorithm The first part gets the streamline coordinates. The VDAfile is read and the LE and TE sub-surfaces are located in space. The co-ordinates of the LE and TE are calculated at the hub and at the tip of theblade. To do so, an interpolation is carried between the VDA surfaces andthe compressor channel geometry, stored in a file containing axial and radialcoordinates. In this way, the blade is cut to follow the channel geometry.
Then, LE and TE points are located for a certain number of cross sectionsand from them the different streamlines are defined in terms of axial andradial coordinates. The inlet and outlet line, that will be used from themesher to generate domain surfaces, can be generated in two different ways:use 1/5 of the axial chord of the given streamline as an offset or use an axialposition that will create a radially constant line at the given x position.
Figure 21 shows the streamlines for the three different vanes of the firststage and half of the gas turbine compressor. There are 21 streamlines(coloured lines), the vertical dashed lines represent the inlet and outlet ofeach blade row, the lines in grey are the LE and TE and the thick black linesdefine the channel geometry.
The second part gets the cross section coordinates trough a simple surfaceinterpolation. The coordinates are ordered so that LE point becomes the firstone and coordinates are added clockwise. The 21 equally spaces cross sectionsare depicted in figure 22.
Finally, the outer points of the LE/TE sub-surface at each span level areutilized to calculate:
• LE and TE circle radii for each streamline as the distance between thetwo points
• Inlet and exit metal angles, drawing the tangent lines at the two points.
The code can be easily run by any CATO user thanks to the newly createdGUI shown in figure 23.
Figure 23: Blade profile definition conversion GUI in CATO
The Mercoo file is the one containing the channel geometry. The user has tospecify the number of blade that will be used from LEVMAP. Futhermore,regardind the inlet/outlet location, if the value is left to be 0.00, 1/5 of theaxial chord of the given streamline as an offset is used; if the value is changed,it will represent the axial position at which a radially constant line will betraced. The code output files are:
• The JH05 blade definition file• Inlet/outlet lines coordinates, that will be used to generated the bound-
ary condition files for the CFD analysis• Streamlines and sections coordinates in a format easily readable from
Matlab to visualize blade profile characteristics.
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Starting from the calculated cross sections and exploiting a Matlab tool-box that creates NURBS surfaces, the blades can be visualize in 3D, figure24.
Figure 24: Final blade profile, JH05 format. 1.5 stage
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5.2 LEVMAP, Mesh Generator
LEVMAP (LEVel MAPping) generates semi-structured meshes for axial andradial flow turbomachinery blades. The program has been developed at theImperial College Vibration UTC [18]. The main feature of LEVMAP is itsability to generate semi-structured meshes using a combination of structuredgrids, in the radial direction, and unstructured ones, in the axial and tan-gential directions.
5.2.1 Methodology
“The basic idea relies on the fact that blade-like structures are not stronglythree-dimensional since the radial variation is usually small. It is thereforepossible to start with a structured and body-fitted two-dimensional O-gridaround a given airfoil section to resolve the boundary layer. This core meshis then extended in an unstructured fashion up to the far-field boundaries,the triangulation being performed using an advancing front technique. Oncethis two-dimensional grid is generated, it is projected to the remaining radialsections via quasi-conformal mapping techniques. When all such radial sec-tions are formed, a three-dimensional prismatic grid is obtained by simplyconnecting the corresponding points of different layers. In this way, hexahe-dral elements are generated in the viscous region and triangular prisms inthe rest of the solution domain.” [18]
5.2.2 Input Files
Three files are needed in order to generate a 3D blade mesh:
• .bla: blade profile geometry, defined using JH05 format, previouslygenerated (see subsection 5.1.2);
• .rad: contains the distribution of radial mesh levels of the final 3D meshas a fraction of the hub-to-tip distance;
• .mco: formatted input file for user specified grid control data. This filealong with its parameters is described in the following paragraph.
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MCO (Mesh COntrol file) By means of this file, several parameters canbe set to characterize the resulting mesh. In this paragraph, only the mainvariables, used for this work, are illustrated. To better visualize the latter amesh example is shown in figure 25.
Figure 25: Blade-to-blade mesh
The first parameters to set are the inlet and outlet angles of the do-main. Their value correspond to the flow angles specified in the boundaryconditions extracted from previously performed simulations. In this way thevelocity components will match the domain geometry. The Far-field den-sity represent the background spacing in the blade domain and its value isthe starting point to define whenever the mesh will be coarse or fine. De-pending on the far-field density the spacing of the region around the bladeis chosen to have a smooth variation in cells dimension. At the LE and TEregions the spacing can be controlled independently and its values has tobe chosen to allow corners to be well defined. Furthermore, to better cap-ture fluid phenomena at sensible locations, source points can be definedfor stretching the mesh in some defined region of the blade domain. In figure26, the source points at LE and TE can be noted.
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(a) LE region(b) TE and wake region
Figure 26: Mesh features
Viscous effects are predominant in two separate regions in a blade passage:the boundary layer and the wake regions. LEVMAP exploits hybrid meshing:the boundary layer region is meshed using an O-type body-fitted mesh basedon quadrilateral elements; instead, away from the blade, triangular elementsare used. The ‘O’ mesh can be modified in terms of thickness, number oflayers and grid stretching. At the wake region, a line source, that goes fromthe TE to the outlet boundary, is defined, Figure 26. This allow to have arefined mesh at the wake region to better capture the fluid characteristics.
5.2.3 Tip Gap Modeling
As can be seen in Figure 27a, the tip grid is generated by triangulating theregion within the blade. “The starting point for this mesh is the discretisedline segments at the pressure and suction side of the blade. The mesh is thusmatched with the blade passage mesh. The combination of the passage meshand the in-blade mesh are then projected in the radial direction in the tipgap to produce the required mesh” [18].
The radial specification is supplied by the user at the end of the file.rad, the same way as the radial specification in the blade passage. The gridspacing and stretching can be set in the MCO file. Figure 27b shows a tipgap mesh featuring a uniform radial distribution.
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(a) Blade-to-blade (b) Radial distribution
Figure 27: Tip gap mesh
The resulting mesh is illustrated in the following picture:
Figure 28: LEVMAP mesh
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6 Steady Calculation
To perform flutter calculations, AU3D requires steady state calculations asinput. Therefore, accurate and reliable steady state results are of significantimportance for flutter calculations. The steady calculation is performed onthe first stage and half of the compressor composed by inlet guide vanes, rotorand stator. Modelling the whole stage allow to perform the analysis applyingmore realistic boundary conditions for the CFD domain. A single passagemodel, with periodic BCs, is exploited to drastically reduce the computationtime.
6.1 Setup
6.1.1 Mesh Generation
The mesh is generated for each blade row, by means of LEVMAP, previouslydescribed in section 5.2. Then a multi-bladerows is obtained assembling theindividual ones. The resulting mesh is represented in figure 29 and 30.
Figure 29: Mesh - 1.5 stage
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Figure 30: Mesh top view
The outlet surfaces of each blade passage domain, are defined as far as pos-sible from the blade TE, such that wake effects can be fully captured.
6.1.2 Turbulence Model
The used turbulence model is the Baldwin-Barth, a one-equation model,based on a simplification of the k− ε turbulence equations. It was conceivedto remove the length scale shortcomings of algebraic models. Attached, sep-arated, and merging shear layer flows are handled by using a robust, implic-itly factored Alternating Direction Implicit (ADI) algorithm to solve a fieldequation for the turbulence Reynolds number. The turbulence field equationis loosely coupled to the mean flow equations. The presence of convectiveterms allows upstream turbulence to influence the flow downstream, resultingin improved treatment for shock-induced separated regions [19]. Sufficientgrid resolution in the body-normal direction is required to compute damp-ing functions for wall-bounded flows: y+ lower than 3.5. A more detaileddescription of the model is presented in appendix A.
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6.1.3 Boundary Conditions
Inlet/Outlet The choice of modeling the entire first stage comes from theboundary conditions application at inlet and outlet surfaces of the domain.Indeed, fixing BCs at the inlet of the IGV and at the outlet of the Statorrepresent the best choice to represent the fluid characteristics in a morerealistic way. Applying BCs directly at the Rotor is, generally, a difficulttask. The BCs used to initialize the simulation are derived from the wholecompressor aero calculations performed by MULTIP. The latter is a Siemensin-house code, developed by Denton and Dawes (1998). It is used for 3Dviscous multistage CFD simulations and it is considered as industry standardsolver for performance calculations.1-D non-reflecting BCs are exploited for this case. The latter are the defaultones in AU3D and are studied to avoid numerical reflections, which candisturb the flow field in the neighbourhood of a blade leading to an incorrectsurface pressure distribution and hence aerodynamic work or losses.The parameters required by the solver are: the three components of the free-stream velocity, static pressure and static temperature. Those variable aredefined by means of a radial profile at constant axial location. The solverreads the values provided, and calculates and applies the total pressure, totaltemperature, and inlet flow angle at the inlet surface; while static pressure,static temperature, and outlet flow angle at the outlet surface.
Inter-blade-row surfaces The inter-blade-row surfaces are treated as mix-ing planes with Riemann invariant boundary conditions. In the mixing planeapproach, each fluid zone is treated as a steady-state problem. Flow-fielddata from adjacent zones are passed as boundary conditions that are spa-tially averaged or “mixed” at the mixing plane interface. This mixing re-moves any unsteadiness that would arise due to circumferential variations inthe passage-to-passage flow field (e.g., wakes, shock waves, separated flow),thus yielding a steady-state result. Despite the simplifications inherent inthe mixing plane model, the resulting solutions can provide reasonable ap-proximations of the time-averaged flow field.
Blade surface and Tip Gap modelling The blade surface is treated asa viscous wall with no slip boundary condition. To model the tip leakagegap, zero rotational speed is imposed at the surface representing the casing.
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6.1.4 Convergence Criterion
To evaluate the convergence of the solution, the continuity equation residualcan be monitored for each time step, as illustrated in figure 31. The residualis not normalized as can be noted from its high values in the first iterations.The solution is assumed to converge when the residual reach a constant valuewith order of magnitude -4.
Figure 31: Convergence status - Steady calculation
The convergence is affected by the Courant–Friedrichs–Lewy (CFL) number.AU3D solver is able to obtain reliable results using high CFL values. Forthis analysis a value of 30 is set.
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6.2 Results
The first main part of this study focuses on the compressor operated atdesign point; however, off-design and different blade loading conditions areinvestigated later. At design point, the compressor features a transonic flowat the rotor, which will highly influence the flow characteristics.
6.2.1 Mesh Sensitivity Study
The effect of the mesh definition on the results is investigated. Four differentmeshes are generated changing the far-field density and, proportionally, thespacing around the blade. Only the rotor mesh is varied since it is the mostsensible bladerow of the compressor. The total number of nodes is shown inthe following table for each Case.
Cases Rotor Stage
Case 1 162614 390444
Case 2 223892 451722
Case 3 250230 478060
Case 4 292105 519935
Table 1: Total number of nodes - Mesh sensitivity study
A comparison between the coarsest and the finest generated meshes isillustrated in figure 32. To compare the dependency of the results on the
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grid refinement, the blade profile loading is plotted for the four differentCases, at three span levels: 24%, 50% and 93%, figure 33. Among withthe four investigated Cases, the results obtained by MULTIP are plotted toget an idea about the behaviour of the different meshes; a more detailedcomparison will be discussed in section 6.2.3.
Looking at the three plots, there is not a significant difference between thefour Cases. The main feature that is captured slightly differently is the shock,especially toward the blade tip, where the velocities are higher, figure 33c.Indeed, comparing Case 1 and Case 4, the latter foresees a shock locationmore toward the trailing edge and with higher strength. Moreover, it can benoticed that Case 1, the coarsest mesh, captures the shock more similarly toMULTIP results than the other three Cases. This behaviour comes directlyfrom the AU3D solver; in fact, it is a single precision solver which guaranteea bigger stability and robustness. Therefore, generally, in this version of
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AU3D, coarsest meshes lead to more reliable results. To better visualize theflow development at the blade-to blade passage, the Mach contour plots areevaluated at 93% span level, location at which the four Cases show the maindifference. Figure 34 shows the different results for the two extreme Cases,1 and 4.
Figure 34: Mach contour plot at 93% span
From the picture, an oblique shock that starts from the suction side of theupper blade and travels to the pressure side of the adjacent blade is welldefined. A bow shock is also present right before the LE. As already showedlooking at the blade profile loading, the finest mesh capture the shock witha higher strength; this feature is evident looking at the effect of the shockon the adjacent blade; in fact, the Mach field at the LE of the PS is higherin Case 4. In Case 4, there is a net distinction between the oblique andthe bow shocks; instead for Case 1, they seems to interact in some way.The computational time is investigated for each Case using same number ofiterations, figure 35. All simulations are run in parallel using 8 partitions.Comparing the coarsest, Case 1, and the finest mesh, Case 4, the latter is30% more time consuming.
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Figure 35: CPU time - Mesh sensitivity analysis
From this discussion, the four different Cases do not show significant differ-ences. Although, the coarsest mesh is able to capture the fluid features moreclosely to results obtained with MULTIP and it is the best one in terms ofcomputational time efficiency.
6.2.2 Compressor Map
As stated before, the applied boundary conditions are directly extractedfrom calculations performed with MULTIP. However, each solver uses differ-ent mathematical schemes and algorithms so that the flow characteristics aredifferently captured; i.e. the losses through a blade row could not coincide.Moreover, the number of stages modelled to run the simulation affect theresults: MULTIP models the whole compressor, while, in this work only thefirst and half stage is investigated. Finally, different solvers employ differ-ent meshes, i.e. structured/unstructured, which also affect the results. Toestimate if the comparison is meaningful, two important parameters can bestudied: the mass flow and the stagnation pressure ratio. The latter es-tablish at which operating condition the compressor is working, and, if theoperating point is the same for both solvers, a relevant comparison can beperformed. Furthermore, the behaviour of the AU3D at off-design points is
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investigated building the Compressor Map. The back pressure of the stageis varied to obtain different mass flow values, resulting, in different pressureratio. The results are then compared with ones obtained using another CFDsoftware, STAR-CCM+, developed by CD-adapco, which is able to performhigh-fidelity simulations [20]. The obtained plot is shown in the followingpicture:
Figure 36: Compressor map
MULTIP comparison The compressor map is plotted for two Cases in-vestigated in the mesh sensitivity study, section 6.2.1. The point at whichMULTIP BC’s are used without any modification is indicated with a crossand MULTIP operating point is shown in red. It can be noted that theAU3D point that seems closest to the MULTIP one is at 99% of total backpressure. A deeper investigation is, then, performed in terms of blade profileloading, comparing the two points at 100% and 99% of back pressure, figure37. The AU3D point closer to the MULTIP one in the compressor map isnot the closer in terms of blade profile loading; at 99% of total back pressurethe shock is captured more toward the TE and with a much higher intensity,especially at 93% span level. This leads to higher Mach number values inthe flow field. Moreover, the mass flow value closest to the MULTIP one is
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obtained with 100% of back pressure. For these reason, this point will beconsidered the reference design point for AU3D from now on.
(a) 50% span
(b) 93% span
Figure 37: Blade profile loading - Off-design study
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AU3D Off-design Behaviour A Compressor Map is, generally, calcu-lated on the whole compressor to evaluate its performance; in this work,even if only 1.5 stage is modelled, the CM is useful to evaluate how AU3Dbehave off-design. Looking at figure 36, it can be noted that using a finermesh, the solution converges for a wider range of off-design conditions. Nev-ertheless, at design point, the operating condition are the same for the twodifferent Cases; therefore a coarser mesh, Case 1, can still be taken as refer-ence one. AU3D is an engineering tool with no strict boundary condition’srequirements, so that off design runs are possible but the accuracy of theresults is not assessed. It is, generally, not used for high fidelity performancecalculation, but, it is mainly used for aerolasticity phenomena investigation.Nevertheless, comparing AU3D results with STAR-CCM+ ones, the curvestrend is the same with a small values off-set. Therefore, even if AU3D is notmeant for this kind of analysis, the results show a good degree of accuracy.
To look closer at the possible surge and choke zones, the outer points onthe CM map are considered and Mach contour plots are generated, figure 38.
Figure 38: Mach contour plot at 93% span - Off-design study
At 79% of the total back pressure, it seems that the choke condition is wellcaptured; the Mach number is relatively high leading to a very strong obliqueshock which is now located very close to the TE of the blade. Moreover, a
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well defined bow shock is present before the LE. At 107,5% of back pressure,the compressor should be close to the surge zones but, looking at the Machcontour plot, no evident separation zones are present which would lead tostall. Capturing surge phenomena is, usually, not an easy task. Many solverparameters should be tuned properly to obtain a converged solution. More-over, when the compressor reaches the surge line, only few stages are actuallyin stall; so, probably, the stage being investigated in this work, is not keen tostall. However, generating a finer mesh could help getting closer to the surgeline. Blade profile loading plots, regarding the possible surge/choke zones,can be found in appendix B.
Compressor Efficiency The isentropic efficiency, of the first and halfstage can be calculated as:
ηs =
(p02p01
)( γ−1γ
)− 1
T02T01− 1
Figure 39 shows the isentropic efficency at different operational points.
Figure 39: Efficiency map
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The Case 2 mesh is exploited to have a bigger mass flow range, so that abigger view of the efficiency trend can be visualized. From the plot, it canbe noted that the solution obtained with AU3D, does not converge for lowermass flow values, contrary to STAR-CCM+. Therefore, it is not possible tovisualize the decrease in efficiency for low mass flow values. Moreover, AU3Dpredicts a slightly lower isentropic efficiency at design point.
6.2.3 Results Validation
Solvers Used as Reference To assess the validity of the results obtainedusing AU3D, a comparison is performed looking at different CFD solvers:
• PHAST: Phase Harmonic Aeroelasticity Solver for Turbomachinery,mainly used for flutter and forced response evaluation;
• MULTIP: 3D viscous multistage performance calculations, developedat Cambridge University, it is used as main reference for compressoraerodynamics evaluation;
• CFX: high-performance CFD software tool, mainly used to calculatecompressor performance and to validate unsteady analysis results.
• Star-CCM+: CFD solver developed and distributed by CD-adapco,able to perform high fidelity computations using a friendly GUI.
In the following table the modelled CFD domain, the type of mesh used andthe exploited turbulence model is listed for each solver.
These characteristics highly influence the calculation results: the model affectthe definitions of the boundary conditions, the type of mesh establish how theflow is discretised and the turbulence model define how the various featuresof a turbulent flow are captured.
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Blade Profile Loading The first comparison is carried out in terms ofblade profile loading, at three different span levels: 24%, 50% and 93% toinvestigate the flow close to the hub, at mid-span and close to the tip.
Hub Region Close to the hub, figure 40, the total loading, the areadescribed by the PS ans SS lines, is similar but there are differences in dis-tribution along the chord. At the PS, AU3D predicts a lower pressure valuewith a more flat trend; while at the SS, it is not able to capture the two pres-sure dips in the first half of the blade chord, but the pressure value in thesecond half of the chord perfectly match the one calculated with the othersolvers.
Mid-span At mid-span, figure 41, there is a good match between thesolvers. Along the SS, at around 55-60% of the chord a steep pressure iscaptured since the flow is transonic and a shock develops. PHAST andMULTIP predict the pressure gradient almost equally, while AU3D, CFXand STAR-CCM+ show some differences that will be more evident at thetip region where the velocities are higher.
Tip Region At the tip, figure 42, looking at the PS, almost all thesolvers show the same behaviour except for PHAST solver which predictshigher loading with similar distribution. Moreover, toward the TE there isthe reflection of the shock coming from the adjacent blade and it is differentfor the various solvers since the shock is differently capture. The latter de-velops at the SS, around 70% of the chord. This region is the most difficultto capture correctly since the shock location and intensity can be affectedby multiple variables: boundary conditions, numerical scheme and turbu-lence model. The latter can be classified into zero-, one- and two-equationand higher-order models, depending on the number of transport-equationused. Theoretically speaking, “the more the number of transport equationsinvolved, the more accurate the prediction is, as less assumptions are used”[21]. PHAST and MULTIP use a zero-equation model such as the Baldwin-Lomax one, AU3D a one-equation model (Baldwin-Barth) and CFX andStar-CCM+ use a two-equations model, the k-epsilon one. As a matterof fact, each solver shows a different behaviour. AU3D, CFX and STAR-CCM+ capture the shock with a bigger magnitude, the pressure gradient ismore steep, with respect to PHAST and MULTIP. Regarding the location,AU3D, CFX and STAR-CCM+ are the one at the outer limits, the first pre-dicts a shock at around 72% while the other two at around 65%; MULTIPand PHAST are in between. However, MULTIP is currently used, at Siemens
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AB, for evaluating the compressor performance in high fidelity, meaning thatthe solver is set up so that the results match with the experimental testingresults in terms of loss. The shock is then captured more softly due to itstransient characteristic, it moves in time. In AU3D a sharp shock is cap-tured using the default solver parameters value. Changing the dissipationschemes or using low order schemes could average out the shock strength.Therefore, the numerical parameters can be changed depending on the scopeof the calculation; in this work, the main goal is the evaluation of the flutterstability and studying this unsteady phenomenon for different steady resultscan show important features of flutter mechanism.
Mach Contour Plot To investigate more in deep the differences betweenthe different solvers, the Mach contour plot is extracted. Figure 43, 44 showthe results obtained in PHAST and AU3D, using the same interval for Machvalues and the same number of levels. The first main difference can be notedin the areas of maximum velocity along the SS: PHAST predicts, for all spanlevels, higher velocities toward the LE, while, AU3D predicts it around halfchord. Moreover, the latter captures a stronger shock as visible from theplot at 94% span. The Mach contour plot with AU3D full range of values ispresent in appendix B, along with the plot obtained trough STAR-CCM+.
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Figure 43: Mach contour plot - PHAST solver [9]
Figure 44: Mach contour plot - AU3D with PHAST range of values
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6.2.4 Different Load Case with Different Inlet Conditions
Motivation Whenever the compressor is operated at a different load caseor with different inlet conditions, the flow characteristics can become moredifficult, i.e. the mass flow can be relatively small. The convergence ofthe solution, for a such difficult case, is evaluated with AU3D to study thesolver behaviour. Moreover, the case that will be described in the followingis the one for which the investigated compressor experience the lowest flutterstability, [9]. This will allow, later in this study, to perform unsteady cal-culation not exclusively at the design point, reaching a bigger view on thesolver possibilities and accuracy.
Load Case Compressors are, generally, unstable at low mass flows. Theytend to stall as the incidence on to the airfoils increases. This can lead toa phenomenon called rotating stall and, if the pressure rise is high enough,surge of the engine. This forces the shutting down of the engine if it occursand can cause machine failure. Variable vanes can be altered while the engineis running, so that flow angles can be controlled decreasing the effects ofincidence. Closing variable vanes in the front stages will unload these stagesthus preventing the compressor from stalling. In this way, the compressor ismore stable and the surge margin is increased. In this work, the fully closedIGV condition is investigated. A new mesh is generated taking into accountthe new blade angles for IGV and S1, figure 45.
Inlet Conditions When a new gas turbine is designed, the engineer has totake into account the various possible locations in which it could be installedand operated. A study of the different inlet conditions, from extreme coldconditions to extreme hot conditions, has to be carried out to scan the com-plete turbine operating range on eventual flutter. In this work, the extremehot condition is investigated.
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Figure 45: Mesh top view - fully closed IGV
Results For such operation condition, the convergence of the steady statecalculation, is not trivial. The flow is characterized by relatively low massflow values which renders difficult the pressure build up in the compressor athigh rotational speed. Indeed, a different approach is used to reach conver-gence in this case: the rotational speed is set to zero for the first simulationrun and it is increased progressively, carrying out several calculations, whichstarts from the previously obtained results. the operating line is followeduntil the design rotational velocity is reached. In this way, the pressure isbuilt up step by step and convergence is achieved. The obtained blade pro-file loading, at 93% span, is shown in figure 46. The results from AU3D arelocated in between the PHAST and MULTIP ones, showing a good accuracy.The blade is less loaded with respect to the open IGV condition and the flowis no more transonic, no shock is present. Such features can also be notedfrom the Mach contour plot shown in figure 47. The differences betweentransonic and subsonic flow should influence the unsteadiness mechanism.For this reason flutter will be scanned at different steady state fields in orderto ensure that different unsteady mechanisms are captured.
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Figure 46: Blade profile loading at 93% span - fully closed IGV - hot condi-tions
Time accurate computations are performed to evaluate the rotor one flut-ter stability. Aerodynamic damping is calculated for the first two structuraleigenmodes, 1F and 1T, for all tested cases. The results are validated us-ing results previously obtained with PHAST [9] and the more recent onesobtained with STAR-CCM+ [20]. All solvers models the CFD domain as asingle-passage.
7.1 Methodology for Flutter Calculation
The methodology used by AU3D, PHAST and STAR-CCM+ to performunsteady calculations is quite different, as listed in table 3. Therefore, it isof particular interest the comparison of flutter stability predicted using thesedifferent solvers which relies on different calculation techniques. A closerexplanation of each solver is carried in this section.
AU3D AU3D performs time accurate calculations, as previously describedin section 4.2. At the initial timestep, a velocity is imposed to the blade,the mesh will start moving and it will reach its maximum displacement.Then, the flow will be perturbed and the blade movement will be dampedor amplified, depending on the flow characteristics and the travelling wavemode. The displacement and the respective unsteady pressure, acting onthe blade surface, are calculated, from which the net force acting on theblade is obtained. Finally, the work per cycle is calculated. The output filecontains the time-histories of the logarithmic decrement per vibration cycle,from which the flutter stability can be evaluated.
PHAST PHAST exploits the non-linear harmonic method described insection 3.1.1. Hereafter, a description of the capability of the solver is ex-tracted from reference [22]. “For unsteady flow calculations using the phase
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harmonic solution method, it is assumed that unsteady flow disturbance canbe decomposed in a real-number harmonic form. These equations are thenexpressed in terms of flow solutions at three phases. These three sets ofequations are simultaneously solved in a similar way to that of the RANSequations with the extra term being treated as a source term. A pseudotime derivative term is added to the three sets of equations, so that theRunge-Kutta method can be used to time-march their solutions to a steadystate. More details about the RANS equations and the solution method canbe found in [23]. The main advantage of this phase solution approach incomparison with the conventional harmonic solution method is that the nonlinearity is automatically retained in the nonlinear convection terms in thediscretised equations. This feature leads to much improved solution conver-gence for harmonic disturbances, particularly for cases with flow separation”[22].
STAR-CCM+ STAR-CCM+ exploits the HB technique described in sec-tion 3.1.2. “The unsteady, transient flow is represented in the frequencydomain as a Fourier series in time. All transport equations for momentum,energy and turbulence are decomposed into the frequency domain on thebasis of the fundamental driving modes, usually a blade-passing frequencyor repeating wake modes. Steady-state equations representing the unsteadysolution at discrete time levels in a single unsteady period are solved to ob-tain the Fourier coefficients. The number of time levels required depends onthe number of modes retained in the problem. The steady-state solution inevery time-level is implicitly coupled at the periodic boundaries by the phys-ical time derivatives. The linear system is then subjected to approximatefactorization to achieve implicit coupling between time levels”[24] .
7.2 AU3D Setup
The initial setup is done starting from the results obtained with the steadystate calculation performed for the first and half stage. AU3D allows theuser to split a multi-row mesh into its component single row meshes. Thisroutine is used to extract the flow results from a mixing plane calculationand use them as a starting point for a single row flutter analysis. Rotor 1 isextracted from the whole stage steady analysis.
7.2.1 Mesh Generation
The mesh is generated adding shadow elements to the original Rotor blademesh. Shadow elements are used to impose unsteady, phase-lagged boundary
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conditions on the periodic surfaces. The resulting mesh is shown in figure48.
Figure 48: Single passage mesh - shadow points
From the picture, the shadow elements can be noted looking at the blade-to-blade mesh at the hub and tip. Indeed, the outer jagged elements are theadded shadow elements.
7.2.2 Boundary Conditions
One of the output files of the steady calculation, contains radial profiles, atthe inter-blade-row surfaces, of many flow parameters. These values are usedto set up the boundary conditions for the inlet and outlet surfaces of the Rotordomain. The parameters used are: the three components of the free-streamvelocity, static pressure, static temperature and a turbulence variable. Thetype of BCs used is the 1-D non-reflecting boundary conditions for axial flowturbomachinery. “NRBC decompose the flow field into a sum of modes ofdifferent frequencies and wave numbers superposed onto a steady flow. For
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small amplitude perturbations, this linearisation is exact. Each mode canthen be treated independently and expressed in terms of pressure, vorticityand entropy waves according to characteristic theory. NRBC then forms theboundary condition by using the contribution from the incoming waves fromoutside the domain and from the outgoing waves from inside the domain. Inthe time-domain, the decomposition of the flow into a sum of modes of differ-ent frequencies and radial and circumferential wave numbers is very complex,therefore, the one-dimensional non-reflecting boundary conditions are usedin two or three dimensions by doing a local analysis normal to the boundaryand ignoring all tangential variations” [5]. Moreover, Riemann invariantsboundary conditions are utilized. The latter is of fundamental importancefor obtaining meaningful results; in fact, using the Riemann invariants BCs,the flow parameters value at the inlet and outlet surface is fixed, the solverdoes not recalculate the flow quantities. In this way, the user is sure that theresults obtained previously in the steady analysis using the whole stage arenow assigned at the rotor 1 bladerow without any modification coming fromthe fact that now only one bladerow is modelled to run the analysis.
7.2.3 Convergence Criterion
Because the flow is changing with time, there is no steady-state as such andconvergence cannot be measured in terms of solution residual or mass flowrates when running a single-passage flutter. Instead it is measured in terms ofaerodynamic work done per vibration cycle. Once the flow reaches a regularperiodic cycle, aerodynamic work done per vibration cycle converges to asteady value, figure 49. The picture refers to mode 1 for a nodal diameterof -6 when the compressor is operated at the design point. This is one ofthe more time consuming calculations; in fact, convergence of the solutiondepends on the ND value: some NDs converge faster than others due tothe difference in the development of the flow in time. Approximately, 33cycles correspond to 10000 iterations and a good convergence is reached for50 cycles; that is why 20000 iterations is chosen as a reference value to runall the different NDs and different cases.
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Figure 49: Convergence status, M1, ND: -6, Design Point
7.2.4 Numerical Parameters
The used numerical parameter to perform the time accurate analysis arelisted in the following table.
The number of timesteps per cycle is chosen referring to the AU3D User’s
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Guide, [15], which recommend at least 200 time steps per cycle of the per-turbation of interest, with second order accuracy in time. The transientcalculation exploits the Newton-Raphson method to solve the nonlinearityof the system. The number of Jacobi and Newton iterations are also setthanks to past experience, [15].
7.3 Structural Modes and Mapping Results
The modeshapes were obtained from Abaqus blade alone calculations, sincethe disk influence can be neglected for the investigated compressor rotor.This means the same modeshape was considered for the different nodal di-ameters, i.e. real modes. Because the structural mesh and the fluid meshdo not have one-to-one nodal correspondence, the modeshapes need to beinterpolated onto the CFD mesh. The displacements were mapped from theFE to CFD mesh by using the AROMA program.The format of the mode file depends on the type of CFD analysis to beperformed. In this case, AU3D computations are performed on a single pas-sage mesh. The initial conditions for each mode can be defined in terms ofmodal velocity, modal displacement and modal acceleration. These initialconditions are usually used to create an excitation in the modes for fluttercomputations. In this work, the modal velocity is used because it leads tofaster predictions [15]. For each mode, the frequency and mechanical damp-ing need to be supplied.The displacements values used as input are the mass normalized ones, ex-tracted directly from Abaqus. The oscillation amplitude is not, manually,scaled: AU3D scales it, automatically, depending on the initial condition im-posed for the mode, i.e. initial velocity. The scaling factor resulting from thecalculation can be calculated while post-processing the results and it is offundamental importance. In fact, when flutter phenomena are investigated,the main focus is on the principle of flutter, when it begins; in this conditionthe relation between displacement and pressure acting on the blade is linearand the displacement is relatively small. A representation of the first twomodes is shown in figure 50. Mode 1 corresponds to the first bending modeand mode 2 corresponds to the first torsional mode. Both modes have theirhighest displacement in the tip at the leading edge.
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Figure 50: R1 modeshapes
7.4 Results
7.4.1 Stability Curves
Design Point The first results are obtained for the compressor operatedat the design point. Figure 51 shows the logarithmic decrement as a functionof the inter-phase blade angle for both mode 1 and mode 2. When plot-ting the Log Dec, particular attention have to be payed to define forwardand backward travelling wave NDs. By definition the TW mode directionis positive when it is in the direction of rotation. AU3D is capable to auto-matically define FW and BW values, setting the natural frequency positivefor FW modes and negative for BW ones and taking into account the fixedrotational speed direction. AU3D and PHAST results show the same trendand the same conclusions regarding flutter phenomenon can be stated: rotor1 operated at design point is stable for all nodal diameters. The least stablezone for both M1 and M2 is in the range of the first forward NDs values. Thisresult is quite surprising since the steady calculation were different betweenthe two solvers: AU3D is capturing the shock with an higher magnitude and
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intensity then PHAST. Therefore, at this operating condition, the unsteadi-ness of the flow should be highly affected by the shock. But, from the resultsit seems that the shock does not affect flutter evaluation. A plot of the lo-cal work on the blade surface is necessary to investigate more the unsteadymechanism. Moreover, if for mode 1 the matching is very close, for mode 2more divergent results can be noted; the former is a first bending mode whichdoes not future complex displacements, if the displacement at a certain spanlevel is considered, it is almost constant along the chord, while, the latter isa torsion mode which features a strongly non uniform displacement. There-fore, since flutter stability is evaluated in terms of averaged value, it is morelikely that for a more uniform displacement distribution align the blade theresults between different solver will match more closely.STAR-CCM+ results are different from the ones PHAST and AU3D pre-dicts. The modelled CFD domain is not anymore only the rotor but thewhole first stage and half is tested. As can be noted from figure 51, for anIBPA value between 0 and 1, STAR-CCM+m predicts the same Log Decvalues, but for an IBPA value slightly less than 0, so in BW mode, there isa deep, which AU3D and PHAST do not capture. This difference may bedue to acoustic waves coming from the IGV and the Stator into the Rotordomain. Moreover, for high IBPA values, both FW and BW, show a lowerblade damping. In reference [20] a complete investigation of the HB results,obtained with STAR-CCM+, is carried out.
Different Load Case with Different Inlet Conditions To study morein deep how accurate the AU3D tool is, the most difficult case is then in-vestigated. This is the fully closed IGV case with hot temperature, whichfeatured the smallest Log Dec value previously obtained with PHAST simu-lations [9]. The steady state calculation for this case has been discussed insection 6.2.4. For this case, the unsteady mechanism is no more related tothe shock since here the flow is subsonic. Figure 52 shows that also for thiscase, the results are very similar. For mode 2 the least stable zone is nowmoved to higher NDs values.
Conclusions Calculation results showed that R1 is stable at all nodal di-ameters for all load cases, therefore no R1 flutter risks is feasible duringturbine operation.
The performance of the different solvers is investigated in terms of CPU timerequired to obtain a converged unsteady solution. Table 5 lists the calculationtime, expressed in hours, for a ND value of zero along with important featuresthat highly affects the computation time, such as the CFD domain modelled,the mesh type, the number of nodes and the specific run type performed: inserial or in parallel and, for the latter, how many cores have been used.
Parameter AU3D PHAST STAR-CCM+
Model R1 R1 IGV + R1 + S1
Mesh Type Semi-structured H-grid Polyhedral
Number of Nodes 169 924 478 720 855 803
Number of Iterations 10 000 30 000 1 450
Number of Cores 8 1 180
CPU Time for ND 0 5.20 hrs 30.33 hrs 0.67 hrs
Table 5: CPU time solvers comparison - Unsteady calculation
STAR-CCM+ model the whole first stage and half, therefore, the number ofnodes is bigger with respect to the other solvers. For this test case, AU3Dreaches full convergences with 10 000 iterations, while PHAST requires 20000 more. For STAR-CCM+ the number of iterations is very small, mainlyfor the implemented HB method and for the polyhedral mesh. PHAST canbe run only in serial, increasing drastically its CPU time. STAR-CCM+results to be the faster solver, using the described setup. Additional CPUtime values extracted from PHAST and AU3D are present in appendix B.
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8 Conclusions
In this work, the front stage of an industrial compressor, featuring transonicflow at the rotor, has been investigated with a focus on its aerodynamicand aeroelastic characteristics. The machine analysis has been performedusing the AU3D CFD aeromechanical tool, an engineering tool capable ofgenerating meshes and perform steady and unsteady calculations. To assessthe validity of the obtained results a comparison with other solver has beencarried in terms of accuracy, calculation efficiency and industrial applicability.
The mesh for the first stage and half of the compressor has been generatedby means of the LEVMAP tool embedded in AU3D. The blade profile formatrequired by the tool has been defined by means of a Siemens software calledCATO. The software is based on the C# programming language which isrun through Visual Studio Professional to realize a graphical user interface(GUI). A new code has been developed for this specific blade profile defini-tion. LEVMAP was able to provide a high quality mesh in very few steps;moreover, few grid data need to be changed when switching to a differentgeometry and the tip gap can be easily modelled.
To perform flutter calculations, AU3D requires steady state calculations asinput. The steady calculation has been performed on the first stage and halfof the compressor composed by inlet guide vanes, rotor and stator. At designconditions, the compressor features a transonic flow at the rotor, which highlyinfluence the flow characteristics. The results have been compared, in termsof blade profile loading and Mach number contour plots, with ones previouslyobtained through other solvers. Results showed a good overall accuracy withsome differences in capturing location and intensity of the shock area. Amesh sensitivity study has been performed revealing that coarsest mesheslead to more reliable results. AU3D proved to be an engineering tool withno strict boundary condition’s requirements. Comparing AU3D results withhigh-fidelity solvers, the results showed a good degree of accuracy in off-design and part-load cases.
Aeroelastic calculation on the rotor have been performed using the timemarching unsteady calculation solver embedded in AU3D to evaluate the ro-tor one flutter stability. Calculation results showed that R1 is stable at allnodal diameters for all load cases, therefore no R1 flutter risks is feasibleduring turbine operation. The results have been validated using results pre-viously obtained with PHAST [9] and the more recent ones obtained withSTAR-CCM+ [20]. Flutter phenomenon has been investigated in terms of
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aerodynamic damping. Results show a good match with PHAST ones andthe CPU time required by AU3D was quite small for a time-accurate solver.
Recommendations for future work The post-processing of the unsteadycalculation can be improved to get a deeper insight on the unsteady mech-anism that govern a specific test case: through AU3D various animationscan be extracted from the simulation to visualize the trend in time of theunsteady pressure acting on the blade; furthermore, there is the possibility toplot the local work on the blade surface. In this work, flutter calculations areperformed using the single-passage method. However, AU3D implements alsothe whole annulus method which is able to calculate the results for all NDsvalues in one unique simulation leading to relatively fast analysis with no pe-riodicity assumption. Moreover, multi-row analysis can be performed to eval-uate how flutter is influenced by the interaction within different bladerows.Finally, many interesting features of AU3D can still be tested: stagger anglescan be easily modified to estimate manufacturing tolerances, possibility toinvestigate mistuning modifying directly the CFD domain mesh, the sparseassembly/SPMR method can be exploited to evaluate forced response andinlet distortions.
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References
[1] Rolls-Royce, “The Jet Engine,” ISBN 0902121 235, Derby, UK, 1996.
[2] V. H. Ashrafi F, Michaud M, “Delay of rotating stall in compressorsusing plasma actuators,” ASME. Turbo Expo: Power for Land, Sea,and Air., vol. 2C: Turbomachinery:V02CT44A009, p. 14, June 2015.
[3] J.D. Denton, “The effects of lean and sweep on transonic fan perfor-mance: A computational study,” Whittle Laboratory, Cambridge Uni-versity Engineering Department, Madingley Road, Cambridge, CB3ODYUK, 2002.
[4] D. Vogt, “Experimental Investigation of Three-Dimensional Mechanismsin Low-Pressure Turbine Flutter,” Doctoral Thesis, KTH, 2005.
[5] S. C. Stapelfeldt, “Advanced methods for multi-row forced response andflutter computations,” Doctor of Philosophy in Mechanical Engineeringof Imperial College London and Diploma of Imperial College London,Department of Mechanical Engineering, 2014.
[6] S. Hammer, “3D FLUTTER ANALYSIS OF SPACE TURBINES,”Master of Science Thesis, KTH, 2012.
[7] “Lecture Notes,” Course in Thermal Turbomachinery, KTH, 2016.
[8] A. Rahachou, “CATO User Manual,” Siemens Industrial Turbomachin-ery, 2012.
[11] R.H. Ni and F.Sisto, “Numerical computation of nonstationary aerody-namics of at plate cascades in compressible flow,” Journal of Engineeringfor Power, 98:165-170, 1976.
[12] W. Ning and L. He, “Computation of unsteady flows around oscillat-ing blades using linear and linear harmonic Euler methods,” Journal ofTurbomachinery, 120(3):508-514, 1998.
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[13] K. C. Hall, J. P. Thomas, and W. S. Clark, “Computation of unsteadynonlinear flows in cascades using a harmonic balance technique,” AIAAJournal, 40(5):879-886, 2002.
[14] G. Saiz, “Turbomachinery Aeroelasticity Using a Time-Linearized MultiBlade-Row Approach,” PhD thesis, University of London, Imperial Col-lege London of Science Technology and Medicine, 2008.
[15] B. Lad, “User Guide to AU3D system release 16.1,” Rolls-Royce plc,DNS 206350, 2016.
[16] A.I. Sayma, M. Vahdati, and M. Imregun, “Multistage whole annulusforced response predictions using an integrated linear analysis technique,Part 1: Numerical model,” Journal of Fluids and Structures, 14(1):87-101, 2000.
[17] G. A. Gerolymos, G. J. Michon, and J. Neubauer, “Analysis and applica-tion of chorochronic periodicity in turbomachinery rotor/stator interac-tion computations,” Journal of Propulsion and Power, 18(6):1139-1152,2002.
[18] M. K.H. Kim, “User Guide to LEVMAP 6.1: A semi-structured meshgenerator for turbomachinery blades,” VUTC-1-02005, Vibration Engi-neering UTC, Imperial College London, 2003.
[19] K. J. Renze, “A comparative study of turbulence models for oversetgrids,” Retrospective Theses and Dissertations, Iowa State University,1992.
[20] J. L. Lacabanne, “Embedded Blade Row Flutter Calculations using aHarmonic Balance Method in an Industrial Compressor Front Stage,”Master of Science Thesis, 2018.
[21] E.Y.K. NG and S.T. TAN, “Comparison of Various Turbulence Modelsin Rotating Machinery Blade-to-Blade Passages,” International Journalof Rotating Machinery, Vol. 6, No. 5, pp. 375-382, 2000.
[22] M.T. Rahmati, D.X. Wang, L. He, “PHAST (Phase Harmonic Aeroelas-ticity Solver for Turbomachinery) User’s Guide (Version 14.03),” 2014.
[23] M. T. Rahmati, L. He, and R. G. Wells, “Interface treatment for har-monic solution in multi-row aeromechanic analysis,” Proceedings of theASME Turbo Expo, 2010.
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[24] F. M. . P. Shankara, “Harmonic Balance Method: A Break From Tra-ditional Simulation of Turbomachinery Flows,” CD-adapco, 2012.
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A Baldwin-Barth Turbulence Model Descrip-
tion
A summary of the B-B turbulence model follows below [19].
The main equation that models the Turbulence Reynolds Number is:
D(νRT )
Dt= (cε2f2 − cε1)
√νRTP + (ν +
νtσε
)∇2(νRT )− 1
σε(∇νt) · ∇(νRT )
The functions used in this field equation are given below.
1
σε= (cε2f2 − cε1)
√cµκ2
The turbulent kinematic viscosity and the turbulent eddy viscosity are de-fined as:
νt = cµ(νRT )D1D2
µt = ρνt
The damping functions are:
D1 = 1− e−y+/A+
D2 = 1− e−y+/A+2
f2(y+) =
cε1cε2
+(
1− cε1cε2
)( 1
κy++D1D2
)[√D1D2+
y+√D1D2
(D2
A+e−y
+/A+
+D1
A+2
e−y+/A+
2
)]Finally, the production term is:
P = νt
(∂ui∂xj
+∂uj∂xi
)∂ui∂xj− 2
3νt
(∂uk∂xk
)2The following constants are used in the equations above:
