Robert E. Kielb

Harmonic Balance Analysis of Blade Row Interactions in a Transonic Compressor

In this paper we apply the harmonic balance technique to analyze an inlet guide vane and rotor interaction problem, and compare the computed flow solutions to existing experimental data. The computed results, which compare well with the experimental data, demonstrate that the technique can accurately and efficiently model strongly nonlinear periodic flows, including shock/vane interaction and unsteady shock motion. Using the harmonic balance approach, each blade row is modeled using a computational grid spanning just a single blade passage regardless of the actual blade counts. For each blade row, several subtime level solutions that span a single time period are stored. These subtime level solutions are related to each other through the time derivative term in the Euler (or Navier―Stokes) equations, which is approximated by a pseudo-spectral operator, by complex periodicity conditions along the periodic boundary of each blade rows computational domain, and by the interface boundary conditions between the vane and rotor. Casting the governing equations in harmonic balance form removes the explicit dependence on time. Mathematically, the equations to be solved are similar in form to the steady Euler (or Navier―Stokes) equations with an additional source term proportional to the fundamental frequency of the unsteadiness. Thus, conventional steady-state computational fluid dynamics techniques, including local time stepping and multigrid acceleration, are used to accelerate convergence, resulting in a very efficient unsteady flow solver.

Blade Excitation by Aerodynamic Instabilities: A Compressor Blade Study

In this paper, we investigate non-synchronous vibrations (NSV) in turbomachinery, an aeromechanic phenomenon in which rotor blades are driven by a fluid dynamic instability. Unlike flutter, a self-excited vibration in which vibrating rotor blades and the resulting unsteady aerodynamic forces are mutually reinforcing, NSV is primarily a fluid dynamic instability that can cause large amplitude vibrations if the natural frequency of the instability is near the natural frequency of the rotor blade. In this paper, we present both experimental and computational data. Experimental data was obtained from a full size compressor rig where the instrumentation consisted of blade-mounted strain gages and case-mounted unsteady pressure transducers. The computational simulation used a three-dimensional Reynolds averaged Navier-Stokes (RANS) time accurate flow solver. The computational results suggest that the primary flow features of NSV are a coupled suction side vortex shedding and a tip flow instability. The simulation predicts a fluid dynamic instability frequency that is in reasonable agreement with the experimentally measured value.Copyright

Vibration and flutter of mistuned bladed-disk assemblies

An analytical model for investigating vibration and flutter of mistuned bladed disk assemblies is presented. This model accounts for elastic, inertial and aerodynamic coupling between bending and torsional motions of each individual blade, elastic and inertial couplings between the blades and the disk, and aerodynamic coupling among the blades. The disk was modeled as a circular plate with constant thickness and each blade was represented by a twisted, slender, straight, nonuniform, elastic beam with a symmetric cross section. The elastic axis, inertia axis, and the tension axis were taken to be noncoincident and the structural warping of the section was explicitly considered. The blade aerodynamic loading in the subsonic and supersonic flow regimes was obtained from two-dimensional unsteady, cascade theories. All the possible standing wave modes of the disk and traveling wave modes of the blades were included. The equations of motion were derived by using the energy method in conjunction with the assumed mode shapes for the disk and the blades. Continuities of displacement and slope at the blade-disk junction were maintained. The equations were solved to investigate the effects of blade-disk coupling and blade frequency mistuning on vibration and flutter. Results showed that the flexibility of practical disks such as those used for current generation turbufans did not have a significant influence on either the tuned or mistuned flutter characteristics. However, the disk flexibility may have a strong influence on some of the system frequencies and on forced response.

Modeling Cylinder Flow Vortex Shedding with Enforced Motion Using a Harmonic Balance Approach

In recent years, new aeromechanical problems have been encountered in turbomachinery. In particular, non-synchronous vibrations (NSV) in blades have been observed by engine companies and occur as a result of ∞ow instabilities. As a flrst step towards better understanding the NSV in turbine engine conflgurations, the two-dimensional shedding ∞ow about a circular cylinder is investigated in this study. The governing nonlinear, unsteady Navier-Stokes equations are solved using a novel harmonic balance method. This method requires one to two orders of magnitude less computational time than conventional time-marching computational ∞uid dynamic (CFD) techniques. In this paper, results are presented for a stationary cylinder in cross ∞ow and a cylinder with enforced motion in the low Reynolds number regime (47 < Re < 180). A unique phase error method is used to determine the shedding frequency and oscillatory lift for the stationary cylinder case. A relationship between Reynolds number and Strouhal number is determined and compared with existing computational and experimental data. The lock-in efiect for the prescribed motion case is observed, and results show that cylinder motion does not signiflcantly afiect the unsteady lift for cylinder oscillation amplitudes of 10% or less of the cylinder’s diameter and the lift actually decreases for higher oscillatory amplitudes. This is signiflcant because it implies that it may not be necessary to couple the NSV aerodynamic solution with blade motion for some applications, which would require much less computation time than a fully coupled aerodynamic/structural solution. In all cases, the results agreed well with existing experimental and computational data.

Aerodynamic Asymmetry Analysis of Unsteady Flows in Turbomachinery

A nonlinear harmonic balance technique for the analysis of aerodynamic asymmetry of unsteady flows in turbomachinery is presented. The present method uses a mixed time-domain/frequency-domain approach that allows one to compute the unsteady aerodynamic response of turbomachinery blades to self-excited vibrations. Traditionally, researchers have investigated the unsteady response of a blade row with the assumption that all the blades in the row are identical. With this assumption the entire wheel can be modeled using complex periodic boundary conditions and a computational grid spanning a single blade passage. In this study, the steady/unsteady aerodynamic asymmetry is modeled using multiple passages. Specifically, the method has been applied to aerodynamically asymmetric flutter problems for a rotor with a symmetry group of two. The effect of geometric asymmetries on the unsteady aerodynamic response of a blade row is illustrated. For the cases investigated in this paper, the change in the diagonal terms (blade on itself) dominated the change in stability. Very little mode coupling effect caused by the off-diagonal terms was found.Copyright

Computational Models for Nonlinear Aeroelasticity

Two distinctly different reduced-order models are formulated for fully nonlinear aeroelastic systems. The first is based on Proper Orthogonal Decomposition (POD). The velocity field is decomposed into a finite number of orthonormal modes,effecting order reduction by transforming from physical space to a low-dimensional eigenspace. The second model is based on the method of Harmonic Balancing (HB). It retains the same physical dimensions of a high-order CFD model but transforms from the time domain to the frequency domain,requiring a single solution for each harmonic frequency included in the model. The number of harmonic frequencies is much smaller than the number of time steps required in a time-accurate simulation. Comparisons are made between POD and HB model output and experimental data for a set of canonical problems involving viscous effects,flow separation,and fully nonlinear aeroelastic behavior: flow past a stationary cylinder,a cylinder with forced oscillations,and a self-excited,plunging cylinder.

Flutter of low pressure turbine blades with cyclic symmetric modes: A preliminary design method

A current preliminary design method for flutter of low pressure turbine blades and vanes only requires knowledge of the reduced frequency and mode shape (real). However many low pressure turbine (LPT) blade designs include a tip shroud that mechanically connects the blades together in a structure exhibiting cyclic symmetry. A proper vibration analysis produces a frequency and complex mode shape that represents two real modes phase shifted by 90 deg. This paper describes an extension to the current design method to consider these complex mode shapes. As in the current method, baseline unsteady aerodynamic analyses must be performed for the three fundamental motions, two translations and a rotation. Unlike the current method work matrices must be saved for a range of reduced frequencies and interblade phase angles. These work matrices are used to generate the total work for the complex mode shape. Since it still only requires knowledge of the reduced frequency and mode shape (complex), this new method is still very quick and easy to use. Theory and an example application are presented.

Flutter of a Fan Blade in Supersonic Axial Flow

An application of a simple aeroelastic model to an advanced supersonic axial flow fan is presented. Lanes cascade theory is used to determine the unsteady aerodynamic loads. Parametric studies are performed to determine the effects of mode coupling, Mach number, damping, pitching axis location, solidity, stagger angle, and mistuning. The results show that supersonic axial flow fan and compressor blades are susceptible to a strong torsional mode flutter having critical reduced velocities which can be less than one.

The Effects of Aerodynamic Asymmetric Perturbations on Forced Response of Bladed Disks

Most of the existing mistuning research assumes that the aerodynamic forces on each of the blades are identical except for an interblade phase angle shift. In reality, blades also undergo asymmetric steady and unsteady aerodynamic forces due to manufacturing variations, blending, mis-staggered, or in-service wear or damage, which cause aerodynamically asymmetric systems. This paper presents the results of sensitivity studies on forced response due to aerodynamic asymmetry perturbations. The focus is only on the asymmetries due to blade motions. Hence, no asymmetric forcing functions are considered. Aerodynamic coupling due to blade motions in the equation of motion is represented using the single family of modes approach. The unsteady aerodynamic forces are computed using computational fluid dynamics (CFD) methods assuming aerodynamic symmetry. Then, the aerodynamic asymmetry is applied by perturbing the influence coefficient matrix in the physical coordinates such that the matrix is no longer circulant. Therefore, the resulting aerodynamic modal forces in the traveling wave coordinates become a full matrix. These aerodynamic perturbations influence both stiffness and damping while traditional frequency mistuning analysis only perturbs the stiffness. It was found that maximum blade amplitudes are significantly influenced by the perturbation of the imaginary part (damping) of unsteady aerodynamic modal forces. This is contrary to blade frequency mistuning where the stiffness perturbation dominates.

A New Solution Method for Unsteady Flows Around Oscillating Bluff Bodies

This paper briefly summarizes a body of work that describes a combination of methods that have been found useful in greatly increasing the speed of dynamic simulation of complex dynamical systems of very high dimensions. These were initially developed with fluid-structure interaction phenomena in mind for streamlined bodies that are elastically deforming in a flowing fluid. However they have also been applied to bluff body oscillations, the primary subject of the present paper, as well as to the dynamics of biological molecules. In each of these areas of interest, the traditional time marching simulations of spatially discretized models of the fluid, elastic structure, or atoms comprising a molecule simply take too long for most research purposes, not to mention design and optimization studies. Hence the common challenge is to reduce the cost and time of computation. The methods described here have been developed to achieve this goal. For a classical and recent summary of the literature on bluff body dynamics of fluid-structure interaction, please see references [1,2]. For a recent summary of the nonlinear dynamics of fluid-structure interaction (aeroelasticity) for streamlined bodies, please see references [3,4,5].