Interblade Phase Angle Research Articles

Computation of Mistuning Effects on Cascade Flutter

A computational method is described for predicting eutter of turbomachinery cascades with mistuned blades. Themethodsolves theunsteadyEuler/Navier‐Stokesequationsformultiple-bladepassageson aparallel computer usingthemessagepassinginterface. Asecond-orderimplicit schemewithdualtime-steppingand multigrid isused. Each individual blade is capable of moving with its own independent frequency and phase angle, thus modeling a cascade with mistuned blades. Flutter predictions are performed through the energy method. Both phase-angle and frequency mistuning are studied. It is found that phase-angle mistuning has little effect on stability, whereas frequencymistuningsignie cantlychanges theaerodynamicdamping.Theimportant effectoffrequencymistuning is to average out the aerodynamic damping of the tuned blade row over the whole range of interbladephase angles (IBPA). If a tuned blade row is stable over most of the IBPA range, the blades can be stabilized for the complete IBPA range through appropriate frequency mistuning. Nomenclature Ch = coefecient of aerodynamic force in h direction CW = aerodynamic work coefe cient c = blade chord length E = total energy per unit mass h = translational blade displacement p = static pressure T = e utter period t = time u, v = e ow velocity components in x and y directions ub, vb = grid velocity components in x and y directions ¯ u, ¯ v = relative velocity components (u i ub, v i vb) a = rotational blade displacement h = stream tube thickness ratio in x direction N = e utter damping coefe cient q = density r = interblade phase angle x = e utter frequency

Single-passage solution of three-dimensional unsteady flows in a transonic fan rotor

Calculations of unsteady turbomachinery flows using conventional time domain methods typically would lead to the use of multiple passages/whole annulus owing to the required direct periodic condition. This makes conventional unsteady solutions extremely time consuming and is one of the major difficulties in applying unsteady calculations in a blading design environment. A single-passage approach to three-dimensional unsteady Navier-Stokes calculations using the Fourier series based shape correction method has been developed. The shape correction method gives a single-passage solution to unsteady flows in blade rows under multiple disturbances with arbitrary interblade phase angles. Therefore the computing time can be dramatically reduced. A case study for a transonic fan rotor subject to long-scale circumferential inlet distortions shows that a factor of 5 speed-up could be achieved. Detailed comparisons in both unsteady and time-averaged flow results between the single-passage and multiple-passage solutions demonstrate the validity and effectiveness of the present single-passage approach. A further calculation of a transonic rotor simultaneously under the influences of both upstream and downstream blade rows also shows the capability of the shape correction method in dealing with multiple disturbances.

Comparative Study of Coupled-Mode Flutter-Analysis Methods for Fan Configurations

A comparative computational study of current e utter-analysis methods, for fan geometry in subsonic e ow, is carried out to evaluate the relative advantage of the methods. The e utter characteristics are computed using the followingthreemethods:timedomain, workpercycle, andtheeigenvalueanalysis.Theeigenvalueanalysisisbased on a pulse method in combination with the ine uence coefe cient method. The aeroelastic analyses are performed using a time-accurate solver based on Euler equations. The aeroelastic characteristics of the fan are obtained by coupling the unsteady Euler equation solution with a normal-mode analysis of elastic blades. For the coupledmodee utteranalyzed,thetime-domainmethod indicated stability orinstability fasterthan theeigenvaluemethod, whereas the work-per-cycle method failed to predict the e utter. The eigenvalue method provided the results in the form of aerodynamic damping and e utter frequency for each interblade phase angle analyzed. However, it was more expensive than the time-domain method, and it was better suited for analysis with a fewer number of modes.

Reduced-Order Modeling of Unsteady Viscous Flow in a Compressor Cascade

A simultaneously coupled viscous-inviscid interaction (VII) analysis is used to model the unsteady viscous separated flow through a subsonic compressor. The inner viscous flow around the airfoil and in the wake is modeled using a finite difference discretization of the boundary-layer equations and a one-equation turbulence transport model. The outer inviscid flow is modeled using a variational finite element discretization of the compressible full potential equation. The viscous and inviscid regions are simultaneously coupled using a injection type boundary condition along the airfoil and wake. The resulting nonlinear unsteady equations are linearized about the nonlinear steady flow to obtain a set of linear equations that describe the unsteady small-disturbance behavior of the viscous flow through the cascade. The discretized small-disturbance VII equations are used to form a generalized, quadratic, non-Hermitian eigenvalue problem that describes the eigenmodes (natural modes) and eigenvalues (natural frequencies) of fluid motion about the cascade. Using a Lanczos algorithm, the eigeninformation is computed efficiently for various steady flow inflow angles and unsteady interblade phase angles

The effects of unsteady interactions on blade flutter

(1) This note has relieved the basic assumption of constant interblade phase angle, and developed a new method for steam turbine flutter prediction. The unsteady interaction of interblade phase angle has a very large effect on the aeroelastic stability of the steam turbine blade. (2) An aeroelastic model has been developed to analyze aeroelastic stability of grouped blades of a steam turbine. The results indicate that the aeroelastic stability of the blades located at the middle of a blade group is better than that of the blades located at its edge.

A Navier–Stokes Analysis of Airfoils in Oscillating Transonic Cascades for the Prediction of Aerodynamic Damping

An unsteady, compressible, two-dimensional, thin shear layer Navier–Stokes solver is modified to predict the motion-dependent unsteady flow around oscillating airfoils in a cascade. A quasi-three-dimensional formulations is used to account for the stream-wise variation of streamtube height. The code uses Ni’s Lax–Wendroff algorithm in the outer region, an implicit ADI method in the inner region, conservative coupling at the interface, and the Baldwin–Lomax turbulence model. The computational mesh consists of an O-grid around each blade plus an unstructured outer grid of quadrilateral or triangular cells. The unstructured computational grid was adapted to the flow to better resolve shocks and wakes. Motion of each airfoil was simulated at each time step by stretching and compressing the mesh within the O-grid. This imposed motion consists of harmonic solid body translation in two directions and rotation, combined with the correct interblade phase angles. The validity of the code is illustrated by comparing its predictions to a number of test cases, including an axially oscillating flat plate in laminar flow, the Aeroelasticity of Turbomachines Symposium Fourth Standard Configuration (a transonic turbine cascade), and the Seventh Standard Configuration (a transonic compressor cascade). The overall comparison between the predictions and the test data is reasonably good. A numerical study on a generic transonic compressor rotor was performed in which the impact of varying the amplitude of the airfoil oscillation on the normalized predicted magnitude and phase of the unsteady pressure around the airfoil was studied. It was observed that for this transonic compressor, the nondimensional aerodynamic damping was influenced by the amplitude of the oscillation.

A REVIEW OF AEROELASTICITY METHODS WITH EMPHASIS ON TURBOMACHINERY APPLICATIONS

This paper is a review of current aeroelasticity methods, with particular emphasis on turbomachinery applications. It also includes a brief survey of the computational fluid dynamics procedures that can be used for the modelling of unsteady flows in such environments. The aeroelasticity phenomena encountered in turbomachinery are introduced from an industrial perspective and the predictive methods are divided into two classes: classical and integrated. The former combine the fluid and structural equations in an uncoupled fashion while the latter attempt to solve the equations together by either exchanging information at each time step or by formulating the aeroelasticity problem in a single unified fluid-structure medium. Further topics, such as the availability of experimental data for model validation, nonlinear structural behaviour, general interblade phase angle, blade mistuning and future directions in aeroelasticity analyses, are also discussed in some detail.

Euler Solutions for Transonic Oscillating Cascade Flows Using Dynamic Triangular Meshes

The modified total-variation-diminishing scheme and an improved dynamic triangular mesh algorithm are presented to investigate the transonic oscillating cascade flows. In a Cartesian coordinate system, the unsteady Euler equations are solved. To validate the accuracy of the present approach, transonic flow around a single NACA 0012 airfoil pitching harmonically about the quarter chord is computed first. The calculated instantaneous pressure coefficient distribution during a cycle of motion compare well with the related numerical and experimental data. To evaluate further the present approach involving nonzero interblade phase angle, the calculations of transonic flow around an oscillating cascade of two unstaggered NACA 0006 blades with interblade phase angle equal to 180 deg are performed. From the instantaneous pressure coefficient distributions and time history of lift coefficient, the present approach, where a simple spatial treatment is utilized on the periodic boundaries, gives satisfactory results. By using this solution procedure, transonic flows around an oscillating cascade of four biconvex blades with different oscillation amplitudes, reduced frequencies, and interblade phase angles are investigated. From the distributions of magnitude and phase angle of the dynamic pressure difference coefficient, the present numerical results show better agreement with the experimental data than those from the linearized theory in most of the cases. For every quarter of one cycle, the pressure contours repeat and proceed one pitch distance in the upward or downward direction for interblade phase angle equal to −90 deg or 90 deg, respectively. The unsteady pressure wave and shock behaviors are observed. From the lift coefficient distributions, it is further confirmed that the oscillation amplitude, interblade phase angle, and reduced frequency all have significant effects on the transonic oscillating cascade flows.

Inviscid analysis of transonic oscillating cascade flows using a dynamic mesh algorithm

A locally implicit total-variation-diminishing scheme and a rigid-deformable dynamic mesh algorithm are formulated on the quadrilateral-triangular meshes. The unsteady Euler equations with moving domain effects are solved in a Cartesian coordinate system. For transonic flows around an oscillating cascade of four biconvex blades with different oscillation amplitudes, reduced frequencies, and interblade phase angles, the calculated distributions of magnitude and phase angle of the first harmonic dynamic pressure difference coefficient agree better with experimental data than those from linearized theory and related numerical results on triangular meshes in most of the cases. Also, the numerical wiggles of instantaneous blade surface pressure coefficient distributions, which appeared on the triangular meshes, are eliminated. From the instantaneous pressure and Mach number contours, the unsteady flow phenomena, such as periodical characteristics, pressure wave and shock behaviors, and time-variations of zones with high Mach number gradient normal to the blade surfaces, are investigated. Furthermore, the lift coefficient distributions indicate that the oscillation .amplitude, reduced frequency, and interblade phase angle all have significant effects on the transonic oscillating cascade flows.

Analysis of Unstarted Supersonic Flutter in Cascade by Semiactuator Disk Theory.

Unstarted supersonic flutter in a cascade is characterized by the existence of detached bow shock waves ahead of leading edges of the blades. The oscillating cascaded blades accompanied by the detached bow shock waves are analyzed based on the semiactuator disk theory. The effect of the detached bow shock waves on the stability of blade oscillation in a bending mode with interblade phase angle is investigated, in consideration of the pressure wave reflected at the shock wave surface and the vorticity wave and entropy wave generated by the oscillation of the shock wave and blades. The results show that the detached bow shock waves greatly destabilize the blade oscillation in bending mode, and the flutter boundary markedly extends to a high-frequency region.

Unsteady Aerodynamic Characteristics of Pitch-Oscillating Transonic Cascade Case with an In-Passage Shock.

A numerical study based on the thin-layer Navier-Stokes equations was carried out on the pitch-oscillating two-dimensional transonic cascade around the midchord. A Baldwin-Lomax turbulence model was used to simulate the effects of turbulence. The unsteady aerodynamic characteristics and the role of shock wave behavior were investigated. The numerical results indicate that the shock wave movement acts as an aerodynamically unstable factor at high reduced frequency and in a range of interblade phase angle, but the pitch-oscillating transonic cascade remains aerodynamically stable in the range of calculations. However it becomes unstable if the pitching axis is located in front of X=0. 37.

Analysis of mistuned cyclic systems using mistune modes

The bladed disk assembly, a prime example of a cyclic system, forms an important component of modern turbomachinery. Vibrations of the blades leads to fatigue damage and limits their service life. Analysis of these systems is complicated by the combination of coupling of blade dynamics with mistuning of the blade properties. The present study of mistuned cyclic systems encompasses a complete model of a bladed disk assembly which includes multiple degrees-of-freedom per blade, aerodynamic coupling, and structural coupling. A systematic analysis of this model allows the determination of the effect of mistune pattern, along with excitation mode, on the forced response amplitudes of the blades. The main motivation of this study is to gain a better understanding of the dynamics of cyclic systems. Through this understanding, better designs of these components will be possible. The concept of mistuning modes is introduced to provide a generalized method of describing the mistuning of bladed disk assemblies. Perturbation analyses reveal which constant interblade phase angle modes will contribute to the response as a result of given mistune modes and excitation modes. Results generated through a computational study demonstrate the range of validity of the perturbation approaches.

Unsteady Aerodynamic Analysis of Subsonic Oscillating Cascade

This paper describes the development of the source-doublet-based potential paneling method for oscillating cascade unsteady aerodynamic load predictions. By using the integral influence coefficient method and by using the interblade phase angles, the unsteady loads on an oscillating cascade can be accurately predicted at a minimum cost. As the grids are placed only on the blade surfaces, the blades are allowed to vibrate without grid deformation problems. Four notable subsonic oscillating cascade test cases that cover most important parameters, e.g., blade geometry, interblade phase angle, flow coefficient, flow speed, frequency, etc., are studied in this paper. The agreement between the present solutions and other numerical/experimental results demonstrates the robustness of the present model. Applicability of the method for realistic compressible flow cascades is also discussed.

Effect of Wind Tunnel Acoustic Modes on Linear Oscillating Cascade Aerodynamics

The aerodynamics of a biconvex airfoil cascade oscillating in torsion is investigated using the unsteady aerodynamic influence coefficient technique. For subsonic flow and reduced frequencies as large as 0.9, airfoil surface unsteady pressures resulting from oscillation of one of the airfoils are measured using flush-mounted high-frequency-response pressure transducers. The influence coefficient data are examined in detail and then used to predict the unsteady aerodynamics of a cascade oscillating at various interblade phase angles. These results are correlated with experimental data obtained in the traveling-wave mode of oscillation and linearized analysis predictions. It is found that the unsteady pressure disturbances created by an oscillating airfoil excite wind tunnel acoustic modes, which have detrimental effects on the experimental results. Acoustic treatment is proposed to rectify this problem.

Numerical Analysis of Two-Dimensional Turbulent Flows through an Oscillating Cascade.

A numerical technique for computation of two-dimensional turbulent flows through an oscillating cascade is presented. To consider the interblade phase angle, a time phase shifted boundary condition is introduced at the periodic boundaries. A finite volume method is used to obtain the spatially discretized governing equations, while the Adams-Bashforth method of second-order accuracy is employed for the time integration. Furthermore, a two-equation model of turbulence is introduced to estimate the turbulence effect. To verify the effectiveness of the present method, computations are carried out for the flow through cascades of flat plate blades (inviscid flow analysis), lens-type blades and compressor blades. The present method gives unsteady periodic flow fields including the periodic aerodynamic force and moment acting on the blade clearly, and the negative damping force is obtained from the computational result of the flat plate cascade flow.

Analysis of Unstarted Supersonic Flutter in Cascade by Semiactuator Disk Theory.

Unstarted supersonic flutter in a cascade is characterized by the existence of a detached bow shock wave ahead of each airfoil. To study the influence of the bow shock wave on this flutter, analysis is performed on the corresponding flow field, with use of the semiactuator disk theory. We investigate oscillation in the bending mode with the interblade phase angle, considering the effect of reflection of the pressure wave and production of a vorticity wave and an entropy wave by the bow shock wave. The results show that the existence of detached bow shock waves greatly destabilizes bending oscillation, and the flutter boundary drastically extends to a high-frequency region.

Aeroelastic tailoring analysis for preliminary design of advanced propellers with composite blades

An analytical study of aeroelastic tailoring has been conducted to determine the flutter characteristics of advanced turbo propellers for preliminary design purposes. The structural dynamic model for composite propeller blades is combined with an unsteady cascade theory with three-dimensional corrections to produce an aeroelastic analysis tool for pretwisted propeller blades with homogeneous anisotropy exposed to a high subsonic flow. The free-vibration analysis of the SR-3 propeller model built of unidirectiona l graphite/epoxy , revealed that the first-bending frequency of the blade can be changed by 53% from the baseline value by changing fiber orientation. Using/?-k modal flutter analysis it was also found that the fiber orientation with a little larger sweep angle than that of the elastic axis can eliminate flutter without any weight penalty, and that the corresponding first natural mode has the least bending-torsion coupling. However, the flutter velocity is sensitive to fiber orientation and interblade-phase angle.

Wind Tunnel Wall Effects in a Linear Oscillating Cascade

Experiments in a linear oscillating cascade reveal that the wind tunnel walls enclosing the airfoils have, in some cases, a detrimental effect on the oscillating cascade aerodynamics. In a subsonic flow field, biconvex airfoils are driven simultaneously in harmonic, torsion-mode oscillations for a range of interblade phase angle values. It is found that the cascade dynamic periodicity—the airfoil-to-airfoil variation in unsteady surface pressure—is good for some values of interblade phase angle but poor for others. Correlation of the unsteady pressure data with oscillating flat plate cascade predictions is generally good for conditions where the periodicity is good and poor where the periodicity is poor. Calculations based upon linearized unsteady aerodynamic theory indicate that pressure waves reflected from the wind tunnel walls are responsible for the cases where there is poor periodicity and poor correlation with the predictions.

Numerical analysis of flow through oscillating cascade sections

The design of turbomachinery blades requires the prevention of flutter for all operating conditions. However, flow field predictions used for aeroelastic analysis are not well understood for all flow regimes. The present research focuses on numerical solutions of the Euler and Navier-Stokes equations using an ADI procedure to model two-dimensional, transonic flow through oscillating cascades. The model prescribes harmonic pitching motions for the blade sections for both zero and nonzero interblade phase angles. The code introduces the use of a deforming grid technique for convenient specification of the perioidic boundary conditions. Approximate nonreflecting boundary conditons were coded for the inlet and exit boundary conditions. Sample unsteady solutions were performed for an oscillating cascade and compared to experimental data. Also, test cases were run for a flat plate cascade to compare with the unsteady, small-perturbation, subsonic analysis. The predictions for oscillating cascades with nonzero interblade phase angle cases, which were near a resonant condition, differ from the experiment and theory. The zero degree interblade phase angle cases, which were near a resonant condition, differ from the experiment and theory. Studies on reflecting versus nonreflecting inlet and exit boundary conditions show that the treatment of the boundary can have a significant effect on the first harmonic, unsteady pressure distribution for certain flow conditions.

Turbomachinery unsteady load predictions with nonuniform inflow

A two-dimensional frequency domain source-doublet based potential paneling formulation is developed to predict steady and unsteady turbomachinery blade loadings. The method employed blade-to-blade periodic boundary condition and interblade phase angles to simulate the complex blade-wake interactions in turbomachinery stages. The method is used in the present paper to simulate the unsteady response on Space Shuttle main engine high-pressure oxidizer turbopump diffuser blades. The amplitude of unsteady pressure and the wave propagation characteristics are discussed by comparing similar studies.