Effects of aerodynamic mistuning


Nowadays, the gas turbine design tends toward lighter, more powerful and at the same time more reliable engines. Also the blades tend to be lighter and longer which lead to a more aggressive conditions. The conditions aforementioned lead to a high risk of flutter generally with catastrophic consequences. The flutter phenomenon is a self-excited and self-sustained vibration of blades that occurs when the blade is absorbing energy from the flow, the amplitude increase and leads to blade failure. Therefore, the development of new and more efficiency techniques in flutter analysis is a challenge for the engine design. There are several investigations about how an oscillating low pressure turbine cascade affects to the aerodamping, however from the aerodynamic point of view of the most of them assumes to be symmetric. In real life due to the manufacturing tolerances and assembly inaccuracies the structure is not perfectly symmetric. This phenomenon is called mistuning. The present work investigates the effects of aerodynamic mistuning on aerodamping in an oscillating low pressure turbine cascade. The investigation is conducted numerically using a commercial CFD solver, ANSYS CFX. The traveling wave mode approach is used for getting the aeroelastic response phenomenon, oscillating all the blades with same frequency, amplitude and a constant interblade phase angle between two consecutive blades.. Thereby, the effects of the axial, circumferential and torsional modes are studied, as well as the influence of the different reduced frequencies and different Mach numbers. Aerodynamic mistuning affects positive and negative to the aerodamping.



Aeroelasticity is the study of mutual interaction among inertial, elastic and aerodynamic forces acting on structural members immersed in fluid flow. Collar (1946) classified the problems in aeroelasticity by introducing the concept of aeroelastic triangle. The inertial, elastic and aerodynamic forces are placed at the vortices of the aeroelastic triangle. Different phenomenons are placed at respective places in the triangle according to the nature of interaction involved. For example consider flutter which involves the interaction among elastic, inertial and aerodynamic forces so it is placed inside the triangle. Similarly Buffeting and galloping are also placed inside the triangle. Divergence, control effectiveness and aeroelastic effect on static stability SSA are placed outside of the triangle on left side as they involve the interaction between elastic and aerodynamic forces.

Aeroelastic problems were not prominent until the early stages of World War II. But afterward as the airplane speeds increased and the designed structures had not sufficient stiffness, caused the engineers to face major aeroelastic problems. In modern aircraft designs the aeroelastic phenomenon has profoundly effect. The most important aeroelastic phenomenon to take care of while designing high speed aircraft is flutter. The high speed planes are subject to many kinds of flutter phenomenon that can cause destruction. Buffeting, dynamic loads problems, load distribution and divergence are some of the aeroelastic phenomenon taking care off while designing modern airplanes.

Aeroelasticity Phenomenon in Turbomachines:

In turbomachinery, inertial, elastic and aerodynamic forces act on the several cascades of blades causing blade vibrations that can cause fatigue and in worse case can be source of failure of the machine. In turbomachine, especially in the first few stages of gas turbine, gas temperature is so high that thermal stresses along with the other three forces are essential to consider. Such problems are called aerothermoelastic problem. In the present thesis, the low pressure turbine cascade is worked on and the problem is pure aeroelastic one. In turbomachinery fluid and structural domains interact. The equations describing the interaction are nonlinear that cannot be solved analytically, so, computational techniques are used to solve the equations.

Aeroelasticity methods devoted for turbomachine were started to be developed in 1950s. Aeroelasticity methods are classified into two types called classical methods and integrated methods. In classical methods fluids structure interaction is ignored. The problem is uncoupled and the linear equations for both the fluid and structure are solved separately. In this method the structural response to the fluid is ignored. In the second method called integrated method fluid motion is not uncoupled from structural response, structural motion is modified by the fluid and also structural motion modifies the fluid flow. Integrated methods are further divided into partial integrated and fully integrated methods. In partial integrated methods fluid and structural equation are solved separately but the result of fluid equations act as boundary condition to the solution of equations for structures motion and it holds vice versa. This exchange of information between structure and fluid motion equations is exchanged at every time step. In fully integrated methods the equations describing fluid motion and also the equations for structure motion are solved together with the same integrator. (Marshall and Imregun, 1996).

In turbomachine, large number of blades with complex shapes interacting with each other, boundary layer thickness interaction, shroud interfaces, and flow separation make the aeroelastic problem more complicated. This gives various kinds of aeroelastic phenomenon pertinent to turbomachine. To name few, static aeroelasticity, flutter, forced response and acoustic resonance are most interested aeroelasticity phenomenon for turbomachine designers.

The centrifugal and aerodynamic forces acting on turbomachine blades moving at high speed causes the blades to untwist and elongate. The blades not just untwist along a fixed axis but also the section profile is changed. This study of blades running shapes is named as static aeroelasticity. When the blades are designed their cold shape is manufactured such that on design running condition they would attain the required shape otherwise turbomachine efficiency will be compromised. For The turbomachine running at off design condition, dynamic aeroelastic effects are measured.

Flutter is the most serious phenomenon, encountered in turbomachinery, if initiated then mostly cannot be stopped and can lead to turbomachine blade failure. The causes of this instability are the unsteady aerodynamic forces which are induced by blades vibrations no matter the follow is uniform or not (Srinivasan, 1997). During each cycle of vibrations energy may be added to the blades which can increase stresses to such level that blade failure can happen. The upstream and downstream blades interact with each other through potential and wake effects creating defects in flow. The blades passing through these flow defects experience unsteady loads that cause vibrations in blades called forced vibrations.

Flutter in Turbomachinery:

In turbomachinery, traditionally the flutter problem has been associated with compressor and fan blades. But the developments in low pressure turbines, where major driving factors were highly loaded, less weight and cost, lead to the reduction in thickness of blades and increase in blades aspect ratio. The blades with less thickness and high aspect ratio have lower stiffness. The blades at later stage of low pressure turbine having less stiffness are potentially vulnerable to the flutter problem (Corral et al., 2006). The present work is related to the flutter problem in cascade of low pressure turbine blades. Fig. 2 below shows the LPT blades....

At first the flutter phenomenon was found in wings of airplanes near the stall conditions and early flutter classical methods were based on wings flutter behaviour. But the blades in turbomachinery having more curved shapes, changing in cross section geometries along cord length make their shapes more complex than air plane wings. Also in turbomachinery a large number of blades are structurally and aerodynamically coupled. The complex shapes, centrifugal forces acting on individual blades, the cascade geometries and the coupling between turbomachinery blades make flutter behaviour on turbomachinery blades much different than that on airplane wing.

For flutter in wings an empirical relation, called as mass ratio, has been introduced.

But in case of turbomachines the mass ratio is higher and instead another correlation, found by Meldahl, is used. Meldahl (1946) found that in turbomachinery flutter occurs above certain flow velocities, and he introduced the correlation called reduced frequency, relating flow velocity, blade chord and oscillation frequency as below.

Where f is the oscillation frequency, c the blade chord length and u is the velocity of air relative to blade. The correlation gives the ratio between time taken by fluid particle to travel a chord to the time required to complete one cycle of vibration of blade.

small values of k means that time of flight for fluid particle is comparatively lower than blade's oscillation period. In other words the flow can be settled to changed conditions because of blade vibration. For a certain oscillation frequency, while increasing flow velocity, critical value of reduced frequency will be approached such that below which flutter can occur. Srinivasan in 1997 reported that flutter has been observed in first mode for reduced frequencies less than 0.4 and for the modes with predominance of first torsion mode, reduced frequency found to be in between 0.4 and 0.7.

In Wings having lower mass ratio, flutter occurs as a result of coupling in modes. While in turbomachine blades with higher mass ratio flutter occurs in single mode as the aerodynamic forces acting on blades are much less than stiffness and inertia forces that they cannot cause modal coupling. But in modern turbomachinery flutter phenomenon can be result of coupling between different modes as the blades are designed to be thinner and highly loaded for improved efficiency.

AS mentioned above, in turbomachinery blades are coupled aerodynamically that affect flutter behaviour. Research work is being done to explore the coupling phenomenon. Vogt (2005) found that the coupling is much influenced by the relative motion between two adjacent blades. In 1988 Crawley work showed that in case of tuned blade rows, blades vibrate with same mode, frequency and amplitude in a row are not just structurally coupled but also aerodynamically coupled. In aerodynamic coupling, blades have more coupling affect on the adjacent blades as compared to the one farther away. This existence of aerodynamic coupling between row of tuned blades makes the blades vibrate with same mode, frequency and amplitude but with a certain interblade phase angle. This type of motion reported by Crawley (1988) was named as travelling wave mode (TWM).

The interblade phase angle s of the TWM is defined as:

wherein N denotes the number of blades and l, being the nodal diameter of the disk. For each nodal diameter pattern a pair of travelling waves is induced. For forward travelling wave interblade phase angle is expressed by the above expression. For the backwards travelling wave, existed in opposite direction to forward travelling wave, the phase lag called as interblade phase angle can be expressed as below in eq....

For last couple of decades research work has been done to explore the different parameters affecting turbomachinery flutter. Sadeghi and Liu (2001) computed the effect of phase angle and frequency mistuning on cascade flutter. It was found that Interblade phase angle mistuning has small effect on the cascade flutter, whereas frequency mistuning has a significant effect on the damping coefficient of the blades in cascade. Frequency mistuning averages out the interblade phase angle over all the IBPA range, because of frequency difference between the two adjacent blades, so, if the tuned blade row is stable for most of the IBPA range then the frequency mistuning can make the blade row stable over the whole IBPA range. Nowinski and Panovsky (1998) reported that the stability is more dependent on mode shape while reduced frequency and steady loading have comparatively lesser effect. (4) Panovsky and Kielb (2000) concluded that mode shape is the more controlling parameter rather than reduced frequency in designing for flutter. Vogt (2005) identified that axial bending have more influence on the flutter than that of circumferential bending. If there exist significant structural coupling between the blades it will cause complex mode shapes to exist and making flutter mechanism more complex. Nowinski and Panovski (1998) elaborated that for torsional bending the change in axis location highly affects stability.

Susceptibility of flutter can be reduced by incorporating some design aspects. Stiffeness of the structure play positive role in this regard, by increasing stiffness, flutter susceptibility can be lowered. One of the most effective approach in this regard is the use of shrouded blades where shrouds increase stiffness of the structure. Nowinski and Panovsky (1998) performed experiments to investigate effect of mistuning on flutter behavior. Their study showed that mistuning has positive impact on the flutter susceptibility. Srinivasan (1997) reviewed the mistuning problem. Mistuning can have a positive impact on flutter, many researchers have proposed models to find the mistuning existed in disk blade system and how to exploit it to avoid flutter. Feiner and Griffin (2004) developed the method to determine mistuning, it can be applied to isolated families of mode. They also proposed another method that can determine the mistuning completely from experimental data. Martel et al. reported a method that can be used to get the information about how to mistune the system intentionally to increase the stability of aerodynamic unstable rotor. Present work is focuses on the aerodynamic mistuning only. Aerodynamic damping can provide positive as well as negative damping, leading to stability or instability depending on the condition.

Flutter is the most serious phenomena in turbomachine concerning blades failure. During flutter the resulting vibration is non integral of the rotational speed of machine dictating that flutter is independent of the rotor speed rather depending on the flow velocity and on incidence. Campbell diagram shown below in figure show the lines for various engine orders and the eigen frequencies of the system for different nodal diameters and rotational speed of the rotor.

Resonance conditions are observed at the points of coincidence of frequency-speed characteristics and engine order line. After the coincidence, at depart of frequency-speed characteristics from engine order line, there is high potential for aeroelsatic instability to occur.

The turbomachines have high mass ratio that cause structural terms to be larger than the aerodynamic damping terms. Aerodynamic terms can be decoupled from structural terms leading to simplification of the aeroelastic problem. Impact of structural terms upon stability is calculated by assuming vacuum and on the other hand contribution of aerodynamic term to the stability is got from purely unsteady aerodynamic analysis.

The pure unsteady aerodynamic analysis for flutter stability is treated here. A two dimensional section of a blade, part of a blade row oscillating with travelling wave mode, is considered. Blade profile assumed not to deform during vibration.

Blade is vibrating in axial, circumferential and torsional bending mode. The torsional bending mode, blade rotates around the axis orthogonal to the axial and circumferential directions. In the present work, single mode vibrations has been analysed at one time, no combined mode vibrations. The harmonic motion of the blade can be described by a complex vector.

Verdon (1987) showed that for small perturbations the unsteady pressure due to harmonic motion can be given by Eq. 1-11 represents steady mean press at a certain location, t represent the time varying perturbation part and denoting the complex pressure perturbation amplitude of the harmonic oscillation. Harmonically oscillating pressure consequently will give rise to unsteady force also of harmonic nature.

It is most convenient way to use pressure coefficients for evaluation of flutter. The unsteady pressure coefficient has been got by normalizing the amplitude of unsteady pressure response by the oscillation amplitude and reference dynamic head taken at upstream of blade the row. The unsteady pressure coefficient can be given as Eq. 1-12 Where the reference dynamic head given as pdyn,ref = ?p01- ps1.The same way as for unsteady pressure the aerodynamic force, can be normalized and given as Eq. 1-12 Where F, unsteady aerodynamic force, is calculated from the integration of the unsteady pressure along the blade profile and is given as Eq. 1-12 The unsteady normalized 'f' force can be written in the orthogonal components form given by Eq. 1-13 The above two force components ....are per unit span, as the analysis is done on the two dimensional section of a blade.

The blade vibration causes an unsteady pressure distribution on the blade profile. To know that whether the unsteady pressure distribution around blade profile will lead to the stable or unstable behaviour of flow, work done by fluid on blade per cycle of blade vibration is calculated. In case of positive work, energy being added to blade from fluid, the flow has destabilizing behaviour. The work per oscillation is given by Eq. 1-14 is the representation of blade motion. Integrating above equation yields Eq. 1-15 In above equation only the imaginary terms of purturbation force and moments enters and provide negative contribution that if response lag behind the excitation gives rise to staibilizing behaviour of flow.

To characterize the aeroelastic stability, normalized stability parameter, reported by Verdon (1987), is used. The stability parameter, negative work performed per cycle normalized by the oscillation amplitude, can be expressed as Eq. 1-15 The stability parameter above if positive then the flow has stabilizing behaviour.

Flutter Analysis Approaches

Flutter analysis for turbomachine blades can be approached by three methods names as influencing coefficient, travelling wave method and pulse response method.

Travelling wave method

In this method all the blades vibrate with same amplitude, frequency and a specified interblade phase angle. The travelling wave method requires performing the calculations for each interblade phase angle is done separately. This computational effort can be reduced by using influencing method. Fig . shows the blades vibrating in travelling wave modes.

Influencing Coefficient method

This method, based on principle of linear superposition, is only valid for linear problems. Principle of linear superposition holds for the oscillations with small amplitude (Hanamura et al. (1980 ) and Crawley (1988)). At the onset of flutter when excitations have small amplitudes influencing coefficient method can be used. The principle has bee validated by many researchers including Nowinski and Panovsky (unknwon) demonstrated the validation in torsional bending of LPT blades while Vogt (2005) demontrated the validation during the work on LPT flutter.

AS compared to TWM, Influencing coefficient method has advantage of less computational effort as there is no need to solve the problem separately for different interblade phase angles. In INFC method, at a time, only one blade in the cascade vibrates while the others remain stationary. By applying periodicity only the relative positions of the vibrating 'm' blade and the reference blade are important [12]. All the influencing coefficient are calculated simultaneously. The blade next to vibrating blade gives first harmonic and the next +- 2 gives second harmonic.

The principle of superposition dictates that total unsteady response on a blade can be obtained by linearly superimposing the response of every blade lagged by a specified phase angle. It can be expressed mathematically as in equation below.

Eq. 1-15 Where is the complex pressure coefficient at point with coordinates of (x,y,z) and lying on the blade m, for the case of blades vibrating in traveling wave mode. is the complex pressure coefficient of blade n, acting on the point (x,y,z) lying on blade m. The above expression is the relation between traveling wave mode and influencing coefficient mode. s is the interblade phase angle.

Pulse response method

For the analysis of flutter by traveling wave mode and Influencing coefficient methods, the coefficients has to be calculated separately for different oscillation frequencies. The pulse response method can be used to reduce this effort, the coefficients for several frequencies are calculated simultaneously from the same transient response. This method has been evolved from indicial approach. If need more information then consult paper [12].

The unsteady pressure coefficients expressed as complex value elaborate well the dependence of response on interblade phase angle s and the oscillation of the blade. The excitation motion, of real nature, cause the pressure applied on blade surface to be changed that corresponds with blade load. Fig below visualize the excitation motion, response and the interblade phase angle.

Only on the imaginary part of the response force/pressure effects stability and is used to calculate the work. In case of negative Interblade phase angle, work per cycle becomes negative and making the stability parameter positive. This shows the stable condition where response is lagging the excitation, so, energy is being delivered from structure to fluid.

The blades oscillations are harmonic in nature, so in turn the expected response on the blade is also assumed to be harmonic in nature. The immediate adjacent blades gives the first harmonic variation in interblade phase angle, while The blades farther away the reference blade give proportionally higher order harmonic variation e.g. 2 blades give 2nd harmonic and the 3 give 3rd harmonic etc. The blades farther away from the reference blade have very small effect, and even negligible in some cases, as compared to the adjacent blades. It has been concluded by many researchers that the contribution of the blades following after the 2 blades can be considered to be negligible. The influence of the adjacent blades is of the same order as the reference blade has on itself because of its own oscillation. The single blade would not flutter but the coupling existence between the blades makes the can make the entire setup to be aerodynamically unstable. The diagram shown below, called S-curve, visualize the stability of the setup for different possible interblade phase angles.

In the above diagram positive interblade phase angles show the forward traveling wave while the negative interblade phase angles are for backward traveling wave. The behavior of the S-curve, stability parameter, is different for the forward and backward traveling wave. This feature can be attributed to the blades curvy shape, pressure side and suction side.

Rigid body Motion:

To get transient results, turbine blades have to be oscillated. In the present work the blades has been oscillated harmonically to study the aerodynamic damping. To oscillate blades rigid body motion theory is applied. The theory of rigid body motion assumes zero deformation in body which results in approximation. In mostly cases it can be applied with good accuracy, including the present case, blades vibrates harmonically with very small amplitude. In the present work, blades are vibrated in single mode at a time. Axial, circumferential and torsional bending modes are applied on the blade to get desired transient simulation results.

In rigid body motion theory, the motion is splited into translation motion of the pivotal point (or centre of mass) and the rotational motion of the body about the pivotal point. It gives advantage of expressing motion of the body as a whole rather than describing motion for all of the particles comprising the body. The expressions of rigid body motion depend on the reference of frame chose. In present study following three frames of reference with their distinct coordinate system are used.

Global coordinate system (GS)

Local body surface coordinate system is used to locate any point on the surface of the blade. It has two coordinates in the arcwise and spanwise direction. The arcwise coordinates start from the stagnation point, at nominal flow, and follow the blade surface till trailing edge both on the pressure and suction side. On pressure side the arcwise coordinates are positive while on the suction side arwise coordinates are negative. The local blade surface coordinate system helps to illustrate well the behaviour of complex pressure coefficient distribution on the surface of the blade. In arcwise direction blade coordinates are normalized by the total local arce length. In figure 2-1 normalized arcwise blade profile with spanwise coordinates of 0.5, mid span, is shown.

Axial and Circumferential bending:

In present work, with all blades vibrating, pivotal points are defined for each of the blades. The blades are rotated around their respective pivotal point. Using rigid body motion principal, blades pivotal points are taken to the origin of the global coordinate system and then after applying rotation around the pivotal point the pivotal point is taken back to the original position. As the blade rotates with a certain angle before taking the pivotal point back, so depending on the amplitude and the number of time steps per period, blades surface attain new desired position at the end of the defined rigid body motion.

The motion can be explained well by the following diagram showing four positions of the blade while defining rigid body motion. First the pivota point is rotated around the principle axis of rotation of machine with angle of 'alpha'. In this case the pricinpal axis of rotation of machine is Z-axis. Then translate the pivotal point in the XZ plane to take it to origin of global axis.

General expression for motion of an arbitrary point on the surface of blade can be written as

In equation above xorig are the global x-coordinates of point at blade surface. The rotation is performed by the product of xorig and Rx,aT to get the x-axiz of both the local and global axis be consistent. Translation is done by subtracting xpiv. Again the blade is rotated around y axis by angle n. Angle n is equal to 90 or zero depending on the axial and circumferential bending respectively. Then to perform the oscillation blade is rotated by an angle depending on phi and the number of time steps per period. Again the blade is rotated with negative eta angle around y axis. Then it is translated with displacement of Xpiv and rotated with alpha abgle to get back the pivotal point to its original position. The Axial and Circumferential bending can be visualized well in the figure 2- below.

Torsional bending:

In torsional bending the blade pivotal point is also taken to the origin of global axis the same way as in axial and circumferential bending case above. But here the difference is that in this case harmonic oscillated motion is applied by rotating the blade with angle depending on the phi, oscillation frequency and the step number. Then the blade pivotal point is taken back to its original place. General equation for rotational bending can be expressed as below.

Aerodynamic mistuning: aerodamping, aerodynamic coupling.

Numerical investigation

Several decades ago all the investigation were conducted analytically or experimentally, however nowadays due to the state- of-the-art the investigations can be conducted in computers using CFD methods. ANSYS CFX is a numerical tool which allows running this kind of simulations; however the results should be checked with experimental dates.

The Department of Energy at the Royal, Institute of Technology (KTH) , Sweden, conducted several investigations as " Assessment of a commercial CFD Package for Use in Turbomachinery Aeroelasticity". Walther (2005) used a single blade oscillating in TWM at zero degree IBPA and compared the results to test data obtained from a test facility run at the Heat and Power Department of the Royal Institute of Technology (Vogt, 2005). Pasta (2006) conducted simulations in the INFC domain using a cascade with seven blades. Pleus (2007) researched about using a pulse response method for determining aerodynamic damping properties. Bartelt (2008) studied the aeroelastic properties of combined modes in the same setup.

Numerical Methods in Flutter Predictions

The numerical investigations are usually conducted using standard FEM,CFD and various reduced models which reduce the computation time and provide accurate results. The present investigation is conducted via CFD which is the simulation of fluid engineers system using modeling and numerical methods. Nowadays new techniques have been developed for CFD models, fluid-structure coupling and moving grids. The next investigations predict the aeroelastic behavior as the flutter phenomenon. Some of the numerical investigations during the last decades are the following ones: Lane and Friedman (1958) and Smith (1972) conducted investigations using methods based on the balance of energy between the blades and the fluid. Verdon (1993) used the linearized Euler method. Bakhle (1997), Gerolymos (1994) and He (1989) used the Euler method. Clark and Hall (2000) used the linearized vicous method. Giles and Haimes(1991), Siden (1991), He and Denton (1994) and Marshall (2000) researched about the viscous methods. There are also some investigations about the coupled aeroelastic analysis in turbomachinery. William (1991) researched a way to get eigenvalues using a linear panel method. Gerolymos (1992) and Srivastava and Reddy (1999) used an inviscid aerodynamic analysis for solving the coupled aeroelastic equations. Breard (2000) conducted investigation about the force response of the fan blades using the Navier-Stokes equations. Many models have been developed in recent years. These methods are used for turbomachinery blade rows and use a coupled and time-accurate fluid-structure interaction. The flow is computed by an Euler or Navier-Stokes equations and the motion of the blades uses structural dynamics. A three dimensional aeroelasticity analysis in turbomachiney blades was researched by Vaddati and Imregun (1996) using the Navier-Stokes solver for the structure and a modal model for the structure. The forced response was response was investigated by Sayma (2000) and Vahdati (2000). Flutter simulations were conducted by Carstens (2003), using a structured grid Navier-Stokes solver with structural model. Soi and Alonso (2002) used a fully coupled fluid-structure interaction. Liu (2001) developed a multiblock deformation grid for moving the grids and a multigrid volume algorithm for the Euler Navier-Stokes equations. This method was used for flutter predictions. Furthermore, Sadeghi (2005) adapted the last method for turbomachinery blade vibrations. Nowadays, several investigations are being conducted in order to create more efficiency algorithms. Numerical simulations have a very important role, however all the solution must be validated by experimental test. Vogt (2005) explain the setting for the experimental test and compare the numerical results with the data from the test.

Numerical methods in Flutter prediction:

Computational aeroelasticity has progressed rapidly with the availability of super fast machines and the advent of advanced numerical tools. There are several efficient numerical methods introduced by different researchers to study aeroelastic phenomenon in turbomachinery. The methods have their fidelity with respect to the application. Flutter analysis in turbomachinery has been reported using Euler methods (51Laumert,et al.,60 Lu and Chen) and Navier Stokes equations solutions (55Weber and Platzer, 57Clark and Hall, 65Hhn and Heinig). Reduced models having very few degree of freedom has been introduced to reduce the computaional time(56 B.I. Epureanu and 67Epureanu, et al.). Coupled aeroelastic analysis has been reported by many researchers including 54Gnesin and Rzadkowski(2001),59Carstens, et al.(2002).MOYROUD, at al. (unknown 52) has developed a 3D interfacing methodology between fluid and structure domain for loose coupling methods. In flutter analysis flow unsteadiness is studied by analyzing the blades vibrations and to incorporate the blade's vibratory motion in solvers, mesh moving techniques have been presented by 64Leger, et al.(1999) and 66Srivastava, et al.(2003). Most of the methods developed incorporate approximations of inviscousness or two dimensional geometry. These two dimensional aeroelastic analyses do not capture the effects in third dimension that is radial direction for Cascade flutter problem. Some researchers including (58SADEGHI and Liu(2005),61Ji and Liu (1999),65Hhn and Heinig (2000)), have introduced the models solving Navier stokes equations in three dimensions.

Non linear Navier stokes equations can not be solved analytically, but numerical techniques has been developed for the solution of complex equations with computers. There are many methods introduced by researchers to study aeroelastic phenomenon in turbomachines. The methods have their own fidelity with respect to the application. Some of the numerical investigations performed by different researchers are: Bjrn Laumert ()

Please be aware that the free essay that you were just reading was not written by us. This essay, and all of the others available to view on the website, were provided to us by students in exchange for services that we offer. This relationship helps our students to get an even better deal while also contributing to the biggest free essay resource in the UK!