An unsteady simulation of a two-stage, cooled, high pressure turbine cascade is achieved by applying the harmonic balance method, a mixed time domain and frequency domain computational fluid dynamic technique for efficiently solving periodic unsteady flows. A comparison of computed temperature and pressure profile predictions generated using the harmonic balance method and a conventional steady mixing plane analysis is presented. The predicted temperature and pressure profiles are also compared to experimental data at the stage exit plane. The harmonic balance solver is able to efficiently model unsteady flows caused by wake interaction and secondary flow effects due to cooling flows. It is demonstrated that modeling the unsteady effects is critical to the accurate prediction of time-averaged flow field quantities, particularly for cooled machines.

An efficient three-dimensional Euler analysis of unsteady flows in turbomachinery is presented. The unsteady flow is modelled as the sum of a steady or mean flow field plus a harmonically varying small perturbation flow. The linearized Euler equations, which describe the small perturbation unsteady flow, are found to be linear, variable coefficient differential equations whose coefficients depend on the mean flow. A pseudo-time time-marching finite-volume Lax-Wendroff scheme is used to discretize and solve the linearized equations for the unknown perturbation flow quantities. Local time stepping and multiple-grid acceleration techniques are used to speed convergence. For unsteady flow problems involving blade motion, a harmonically deforming computational grid which conforms to the motion of the vibrating blades is used to eliminate large error-producing extrapolation terms that would otherwise appear in the airfoil surface boundary conditions and in the evaluation of the unsteady surface pressure. Results are presented for both linear and annular cascade geometries, and for the latter, both rotating and nonrotating blade rows.

A computational method for efficiently predicting unsteady transonic flows in two- and three-dimensional cascades is presented. The unsteady flow is modelled using a linearized Euler analysis whereby the unsteady flow field is decomposed into a nonlinear mean flow plus a linear harmonically varying unsteady flow. The equations that govern the perturbation flow, the linearized Euler equations, are linear variable coefficient equations. For transonic flows containing shocks, shock capturing is used to model the shock impulse (the unsteady load due to the harmonic motion of the shock). A conservative Lax-Wendroff scheme is used to obtain a set of linearized finite volume equations that describe the harmonic small disturbance behavior of the flow. Conditions under which such a discretization will correctly predict the shock impulse are investigated. Computational results are presented that demonstrate the accuracy and efficiency of the present method as well as the essential role of unsteady shock impulse loads on the flutter stability of fans.

A novel technique for computing unsteady flows about turbomachinery cascades is presented. Starting with a frequency domain CFD description of unsteady aerodynamic flows, we form a large, sparse, generalized, non-Hermitian eigenvalue problem which describes the natural modes and frequencies of fluid motion about the cascade. We compute the dominant left and right eigenmodes and corresponding eigenfrequencies using a Lanczos algorithm. Then, using just a few of the resulting eigenmodes, we construct a reduced order model of the unsteady flow field. With this model, one can rapidly and accurately predict the unsteady aerodynamic loads acting on the cascade over a wide range of reduced frequencies and arbitrary modes of vibration. Moreover, the eigenmode information provides insights into the physics of unsteady flows. Finally we note that the form of the reduced order model is well suited for use in active control of aeroelastic and aeroacoustic phenomena.

In this paper, we present a computational fluid dynamic model of the unsteady flow associated with the onset of stall flutter in turbomachinery cascades. The unsteady flow is modeled using the laminar Navier-Stokes equations. We assume that the unsteadiness in the flow is a small harmonic disturbance about the mean or steady flow. Therefore, the unsteady flow is governed by a small-disturbance form of the Navier-Stokes equations. These linear variable coefficient equations are discretized on a deforming computational grid and solved efficiently using a multiple-grid Lax-Wendroff scheme. A number of numerical examples are presented which demonstrate the destabilizing influence of viscosity on the aeroelastic stability of airfoils in cascade, especially for torsional modes of blade vibration.

In this paper, we present an analysis of the unsteady aerodynamic response of cascades due to incident gusts (the forced response problem) or blade vibration (the flutter problem) when the cascade is part of a multistage fan, compressor, or turbine. Most current unsteady aerodynamic models assume the cascade to be isolated in an infinitely long duct. This assumption, however, neglects the potentially important influence of neighboring blade rows. We present an elegant and computationally efficient method to model these neighboring blade row effects. In the present method, we model the unsteady aerodynamic response due to so-called spinning modes (pressure and vorticity waves), with each mode corresponding to a different circumferential wave number and frequency. Then, for each mode, we compute the reflection and transmission coefficients for each blade row. These coefficients can be obtained from any of the currently available unsteady linearized aerodynamic models of isolated cascades. A set of linear equations is then constructed that couples together the various spinning modes, and the linear equations are solved via LU decomposition. Numerical results are presented for both the gust response and blade vibration problems. To validate our model, we compare our results to other analytical models, and to a multistage vortex lattice model. We show that the effect of neighboring blade rows on the aerodynamic damping of vibrating cascades is significant, but nevertheless can be modeled with a small number of modes.

A computational method is presented for predicting the unsteady aerodynamic response of a vibrating blade row which is part of a multistage turbomachine. Most current unsteady aerodynamic theories model a single blade row isolated in an infinitely long duct. This assumption neglects the potentially important influence of neighboring blade rows. The present ‘coupled mode’ analysis is an elegant and computationally efficient method for modelling neighboring blade row effects. Using this approach, the coupling between blade rows is modelled using a subset of the so-called spinning modes, i.e. pressure, vorticity, and entropy waves which propagate between the blade rows. The blade rows themselves are represented by reflection and transmission coefficients. These coefficients describe how spinning modes interact with, and are scattered by, a given blade row. The coefficients can be calculated using any standard isolated blade row model; here we use a linearized full potential flow model together with rapid distortion theory to account for incident vortical gusts. The isolated blade row reflection and transmission coefficients, inter-row coupling relationships, and appropriate boundary conditions are all assembled into a small sparse linear system of equations which describes the unsteady multistage flow. A number of numerical examples are presented to validate the method and to demonstrate the profound influence of neighboring blade rows on the aerodynamic damping of a cascade of vibrating airfoils.

A computational method for accurately and efficiently predicting unsteady viscous flow through two-dimensional cascades is presented. The method is intended to predict the onset of the aeroelastic phenomenon of stall flutter. In stall flutter, viscous effects significantly impact the aeroelastic stability of a cascade. In the present effort, the unsteady flow is modeled using a time-linearized Navier-Stokes analysis. Thus, the unsteady flow field is decomposed into a nonlinear spatially varying mean flow plus a small-perturbation harmonically varying unsteady flow. The resulting equations that govern the perturbation flow are linear, variable coefficient partial differential equations. These equations are discretized on a deforming, multi-block, computational mesh and solved using a finite-volume Lax-Wendroff integration scheme. Numerical modelling issues relevant to the development of the unsteady aerodynamic analysis, including turbulence modelling, are discussed. Results from the present method are compared to experimental stall flutter data, and to a nonlinear time-domain Navier-Stoke analysis. The results presented demonstrate the ability of the present time-linearized analysis to model accurately the unsteady aerodynamics associated with turbomachinery stall flutter.

An unsteady inviscid flow through a cascade of oscillating airfoils is investigated. An inviscid nonlinear subsonic and transonic model is used to compute the steady flow solution. Then a small amplitude motion of the airfoils about their steady flow configuration is considered. The unsteady flow is linearized about the nonlinear steady response based on the observation that in many practical cases the unsteadiness in the flow has a substantially smaller magnitude than the steady component. Several reduced order modal models are constructed in the frequency domain using the proper orthogonal decomposition technique. The dependency of the required number of aerodynamic modes in a reduced order model on the far-field upstream Mach number is investigated. It is shown that the transonic reduced order models require a larger number of modes than the subsonic models for a similar geometry, range of reduced frequencies and interblade phase angles. The increased number of modes may be due to the increased Mach number per se , or the presence of the strong spatial gradients in the region of the shock. These two possible causes are investigated. Also, the geometry of the cascade is shown to influence strongly the shape of the aerodynamic modes, but only weakly the required dimension of the reduced order models.

Bicycle stability has been of interest to dynamicists and athletes since before J. W. Whipple described the canonical model for bicycle motion in 1899. Since then, the subject has fascinated many who sought to find a simple way to describe the essence of stability for a hands free bicycle at a prescribed forward speed. Caster and gyroscopic effects have been shown to be helpful, but not necessary for there to exist a stable range of forward speeds. This work focuses on showing how using the eigenvalues of the linearized equations for roll and steer (with and without a steering torque) can illuminate the stabilizing and destabilizing effects of changing bicycle geometry and rider position. Of particular interest is the mathematical demonstration of the decreased stability a cyclist on a time trial bike experiences when in the aerodynamic position, as opposed to riding with hands on the brake hoods or bull horns.

This paper demonstrates the potential of using a multi-degree-of-freedom, traditional van der Pol oscillator to model non-synchronous vibration (NSV) in turbomachinery. It is shown that the two main characteristics of NSV are captured by the reduced-order, van der Pol model. First, a stable limit cycle oscillation (LCO) is maintained for various conditions. Second, the lock-in phenomenon typical of NSV is captured for various fluid-structure frequency ratios. This research identifies values and significance of the coupling parameters used in the van der Pol model. These coefficients are chosen to model confirmed instances of experimental NSV, and to develop a preliminary design tool that engineers can use to better design turbomachinery for NSV. Specifically, coefficient tuning from experimental instances of NSV are considered to identify the unknown coupling coefficients in the van der Pol model. The goal of future research will be to identify values and significance of the coupling parameters used in the van der Pol model, to match these coefficients with confirmed instances of experimental NSV, and to develop a preliminary design tool that engineers can use to better design turbomachinery for NSV. Proper orthogonal decomposition (POD) CFD techniques and coefficient tuning from experimental instances of NSV have been considered to identify the unknown coupling coefficients in the van der Pol model. The finalization of this preliminary-design research will be completed in future research.

The harmonic balance method is a mixed time domain and frequency domain approach for efficiently solving periodic unsteady flows. The implementation described in this paper is designed to efficiently handle the multiple frequencies that arise within a multistage turbomachine due to differing blade counts in each blade row. We present two alternative algorithms that can be used to determine which unique set of frequencies to consider in each blade row. The first, an all blade row algorithm, retains the complete set of frequencies produced by a given blade row’s interaction with all other blade rows. The second, a nearest neighbor algorithm, retains only the dominant frequencies in a given blade row that arise from direct interaction with the adjacent rows. A comparison of results from a multiple blade row simulation based on these two approaches is presented. We will demonstrate that unsteady blade row interactions are accurately captured with the reduced frequency set of the nearest neighbor algorithm, and at a lower computational cost compared to the all blade row algorithm. An unsteady simulation of a two-stage, cooled, high pressure turbine cascade is achieved using the present harmonic balance method and the nearest neighbor algorithm. The unsteady results obtained are compared to steady simulation results to demonstrate the value of performing an unsteady analysis. Considering an unsteady flow through a single blade row turbine blade passage, it is further shown that unsteady effects are important even if the objective is to obtain accurate time-averaged integrated values, such as efficiency.

This paper demonstrates the potential of using a multi-degree-of-freedom, traditional van der Pol oscillator to model Non-Synchronous Vibration (NSV) in turbomachinery. It is shown that the two main characteristics of NSV are captured by the reduced-order, van der Pol model. First, a stable limit cycle oscillation (LCO) is maintained for various conditions. Second, the lock-in phenomenon typical of NSV is captured for various fluid-structure frequency ratios. The results also show the maximum amplitude of the LCO occurs at an off-resonant condition, i.e., when the natural shedding frequency of the aerodynamic instability is not coincident with the natural modal frequency of the structure. This conclusion is especially relevant in preliminary design in industry because it suggests that design engineers cannot treat NSV as a normal Campbell-diagram crossing as they would for preliminary design for forced response; it is possible that by redesigning the blade, the response amplitude of the blade may actually be higher. The goal of future research will be to identify values and significance of the coupling parameters used in the van der Pol model, to match these coefficients with confirmed instances of experimental NSV, and to develop a preliminary design tool that engineers can use to better design turbomachinery for NSV. Proper Orthogonal Decomposition (POD) CFD techniques and coefficient tuning from experimental instances of NSV have been considered to identify the unknown coupling coefficients in the van der Pol model. Both the modeling of experimental NSV and preliminary design development will occur in future research.

The harmonic balance method implemented within STAR-CCM+ is a mixed frequency/time domain computational fluid dynamic technique, which enables the efficient calculation of time-periodic flows. The unsteady solution is stored at a small number of fixed time levels over one temporal period of the unsteady flow in a single blade passage in each blade row; thus the solution is periodic by construction. The individual time levels are coupled to one another through a spectral operator representing the time derivative term in the Navier-Stokes equation, and at the boundaries of the computational domain through the application of periodic and nonreflecting boundary conditions. The blade rows are connected to one another via a small number of fluid dynamic spinning modes characterized by nodal diameter and frequency. This periodic solution is driven to the correct solution using conventional (steady) CFD acceleration techniques, and thus is computationally efficient. Upon convergence, the time level solutions are Fourier transformed to obtain spatially varying Fourier coefficients of the flow variables. We find that a small number of time levels (or, equivalently, Fourier coefficients) are adequate to model even strongly nonlinear flows. Consequently, the method provides an unsteady solution at a computational cost significantly lower than traditional unsteady time marching methods. The implementation of this nonlinear harmonic balance method within STAR-CCM+ allows for the simulation of multiple blade rows. This capability is demonstrated and validated using a 1.5 stage cold flow axial turbine developed by the University of Aachen. Results produced using the harmonic balance method are compared to conventional time domain simulations using STAR-CCM+, and are also compared to published experimental data. It is shown that the harmonic balance method is able to accurately model the unsteady flow structures at a computational cost significantly lower than unsteady time domain simulation.

A nonlinear harmonic balance method for the simulation of turbomachinery flows is presented. The method is based on representing an unsteady, time periodic flow by a Fourier series in time and then solving a set of mathematically steady-state equations to obtain the Fourier coefficients. The steady-state solutions are stored at discrete time levels distributed throughout one period of unsteadiness and are coupled via the physical time derivative and at periodic boundaries. Implicit coupling between time levels is achieved in a computationally efficient manner through approximate factorization of the linear system that results from the discretized equations. Unsteady, rotor-stator interactions are performed to validate the implementation. Results based on the harmonic balance method are compared against those obtained using a full unsteady, time-accurate calculation using moving meshes. The implicitly coupled nonlinear harmonic balance method is shown to produce a solution of reasonable accuracy compared to the full unsteady approach but with significantly less computational cost.

This paper presents the results of a new enforced motion method using harmonic balance computational fluid dynamics (CFD) analysis to design for NSV. Currently, most researchers employ a time domain CFD technique to directly find the frequency of the underlying flow instability which can take significant computational time. NSV is said to occur when the frequency of the instability coincides with a blade mode frequency. The enforced motion design method uses blade motion to attempt to force the fluid frequency to lock-on to the blade vibration frequency at a specified amplitude. For a fixed critical amplitude and blade mode frequency, a range of interblade phase angles (IBPAs) is investigated to determine the aerodynamic damping. A negative value of damping at any IBPA deems the design unacceptable. Furthermore, a procedure for blade re-design (frequency changing) is presented. At the least stable IBPA, the damping is determined for a range of blade frequencies and amplitudes to determine the Limit Cycle Oscillation (LCO) amplitude. A better design is then at the blade frequency that minimizes the blade vibration amplitude. Therefore, these preliminary results indicate that it is advantageous to include blade motion in NSV design approaches. Most significantly, it gives designers a quick and efficient method to assess a design for NSV.

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 blades 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 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.