A method for calculating unsteady flows in cascades is presented. The model, which is based on the linearized unsteady Euler equations, accounts for blade loading shock motion, wake motion, and blade geometry. The mean flow through the cascade is determined by solving the full nonlinear Euler equations. Assuming the unsteadiness in the flow is small, then the Euler equations are linearized about the mean flow to obtain a set of linear variable coefficient equations which describe the small amplitude, harmonic motion of the flow. These equations are discretized on a computational grid via a finite volume operator and solved directly subject to an appropriate set of linearized boundary conditions. The steady flow, which is calculated prior to the unsteady flow, is found via a Newton iteration procedure. An important feature of the analysis is the use of shock fitting to model steady and unsteady shocks. Use of the Euler equations with the unsteady Rankine-Hugoniot shock jump conditions correctly models the generation of steady and unsteady entropy and vorticity at shocks. In particular, the low frequency shock displacement is correctly predicted. Results of this method are presented for a variety of test cases. Predicted unsteady transonic flows in channels are compared to full nonlinear Euler solutions obtained using time-accurate, time-marching methods. The agreement between the two methods is excellent for small to moderate levels of flow unsteadiness. The method is also used to predict unsteady flows in cascades due to blade motion (flutter problem) and incoming disturbances (gust response problem).

Calculation of unsteady flows in turbomachinery using the linearized Euler equations

The problems associated with the prohibitive number of possible system modes for a fluttering compressor or turbine blade row are eliminated by the development that comprises the present report. The existence and uniqueness of extremely simple system flutter modes are proved for blade rows consisting of identical blades equally spaced about a common rotor. These simple system modes, if properly interpreted, have the effect of reducing by a factor of n the number of degrees of freedom necessary to analyze an w-bladed configuration. Stated differently, the system of n blades may be considered, with no loss of generality whatsoever, in terms of a single "equivalent blade." The proof holds under any type of flow and any and all types of interblade coupling, so long as a linear analysis is permissible. Moreover, since it is the flutter-inception point that is of interest in predicting critical velocity or rotational speed, it may well be that the conclusions developed apply even to the onset of stall flutter. Practical application of the method to stall-flutter calculations would, of course, require the availability of aerodynamic stallflutter coefficients. The development is carried out first under the assumption of infinite rotor inertia or, in other words, constant rotor velocity. This restriction is then relaxed, and the treatment is expanded to permit torsional oscillations of the rotor itself. I t is proved that under certain conditions the assumption of infinite rotor inertia introduces no error whatsoever.

System Mode Shapes in the Flutter of Compressor Blade Rows

This paper presents an investigation of the effects of blade mistiming on the aeroelastic stability and response of a cascade in incompressible flow. The aerodynamic, inertial, and structural coupling between the bending and torsional motions of each blade and the aerodynamic coupling between the blades are included in the formulation. A digital computer program was developed to conduct parametric studies. Results indicate that the mistuning has a beneficial effect on the coupled bending-torsion and uncoupled torsion flutter. The effect of mistuning on forced response, however, may be either beneficial or adverse, depending on the engine order of the forcing function. Additionally, the results illustrate that it may be feasible to utilize mistuning as a passive control to increase flutter speed while maintaining forced response at an acceptable level. [A ] [Ar] {AD } { ADr } a b c [D], [Ds [E] E(s,r) [G],[G S

Flutter and Response of a Mistuned Cascade in Incompressible Flow

Propagation of sound waves through a blade row: II. Analysis based on the acceleration potential method

A survey of vibration localization phenomena in the context of structural dynamics and vibrations is presented The review covers the more common and relevant cases where mode localization and vibration confinement are likely to occur in engineering structures Examples considered include periodic or nearly periodic multi-span beams and multi-bay trusses, large space structures, space antennas, and almost periodic (ap) structures with circular symmetry, eg, bladed disks in turbomachines Both analytical and numerical methods for analyzing and predicting localization in finite and infinite systems are discussed In this paper, we show how the problem of mode localization and vibration confinement can be formulated as a problem in the theory of stability of differential equations with ap coefficients Using stability theory, new definitions of mode localization can be established for both linear and nonlinear structures The possibility of stabilizing certain nonconservative fluid-structure systems using structural disorder is demonstrated, and stability theorems are given for aeroelastic systems governed by normal operators We also illustrate how the results from localization theory and the associated stability theory can be applied to the vibration control problem, by triggering vibration confinement by active or passive means

Localization phenomena in structural dynamics

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Theoretical investigation of subsonic oscillatory blade-row aerodynamics

An Investigation of the Flutter Characteristics of Compressor and Turbine Blade Systems

SUMMARY A method is presented for calculating the aerodynamic lifts and moments experienced by a cascade or two-dimensional approximation to a compressor or turbine blade row in subsonic flow under harmonic oscil- lation. The most significant features of the method stem from the utilization of Fourier transforms of blade pressure-jump functions. This permits expres- sion in closed form of the kernel function appearing in the integral equa- tion relating upwash to the pressure-jump transform. phenomena, first discovered in connection with the subsonic wall inter- ference problem, are shown to occur in the case of cascade oscillation even in the presence of stagger and arbitrary interblade phase-lag angles. The resonance points are shown to be related to the poles of the kernel function for the Fourier transform formulation of the problem. techniques are developed for the direct solution of the transform rather than of the pressure jump proper, and it is shown that lift and moment niay be easily expressed in terms of the pressure-jump transform without requiring any inversion. The method is applied to computation of a zero- stagger cascade in antiphase motion (corresponding to the tunnel-wall interference problem) in order to permit comparison of results with those obtained by solution of the more conventional integral equation based upon the Hankel function series type of kernel. obtained in all cases. Arbitrary stagger and interblade phase-lag angles are permitted. So-called resonance Numerical Excellent agreement is INTRODUCTION The aerodynamic forces and moments experienced by compressor blades undergoing oscillatory motion or in the presence of certain classes of oscillatory inflow are of interest to the aircraft-jet-engine'industry for several reasons. First, while the occurrence of classical flutter in compressor blades

Frank Lane

Papers

System Mode Shapes in the Flutter of Compressor Blade Rows

Theoretical investigation of subsonic oscillatory blade-row aerodynamics

An Investigation of the Flutter Characteristics of Compressor and Turbine Blade Systems

National advisory committee