Kenneth C. Hall

Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique

A harmonic balance technique for modeling unsteady nonlinear e ows in turbomachinery is presented. The analysis exploits the fact that many unsteady e ows of interest in turbomachinery are periodic in time. Thus, the unsteady e ow conservation variables may be represented by a Fourier series in time with spatially varying coefe cients. This assumption leads to a harmonic balance form of the Euler or Navier ‐Stokes equations, which, in turn, can be solved efe ciently as a steady problem using conventional computational e uid dynamic (CFD) methods, including pseudotime time marching with local time stepping and multigrid acceleration. Thus, the method is computationally efe cient, at least one to two orders of magnitude faster than conventional nonlinear time-domain CFD simulations. Computational results for unsteady, transonic, viscous e ow in the front stage rotor of a high-pressure compressor demonstrate that even strongly nonlinear e ows can be modeled to engineering accuracy with a small number of terms retained in the Fourier series representation of the e ow. Furthermore, in some cases, e uid nonlinearities are found to be important for surprisingly small blade vibrations.

Proper Orthogonal Decomposition Technique for Transonic Unsteady Aerodynamic Flows

A new method for constructing reduced-order models (ROM) of unsteady small-disturbance flows is presented. The reduced-order models are constructed using basis vectors determined from the proper orthogonal decomposition (POD) of an ensemble of small-disturbance frequency-domain solutions. Each of the individual frequency-domain solutions is computed using an efficient time-linearized flow solver. We show that reduced-order models can be constructed using just a handful of POD basis vectors, producing low-order but highly accurate models of the unsteady flow over a wide range of frequencies. We apply the POD/ROM technique to compute the unsteady aerodynamic and aeroelastic behavior of an isolated transonic airfoil and to a two-dimensional cascade of airfoils

Limit cycle oscillation of a fluttering cantilever plate

A general technique for constructing reduced order models of unsteady aerodynamic flows about two-dimensional isolated airfoils, cascades of airfoils, and three-dimensional wings is developed. The starting point is a the domain computational model of the unsteady small disturbance flow. For illustration purposes, we apply the technique to an unsteady incompressible vortex lattice model. The eigenmodes of the system, which may be thought of as aerodynamic states, are computed and subsequently used to construct computationally efficient, reduced order models of the unsteady flowfield. Only a handful of the most dominant eigenmodes are retained in the reduced order model. The effect of the remaining eigenmodes is included approximately using a static correction technique. An important advantage of the present method is that once the eigenmode information has been computed, reduced order models can be constructed for any number of arbitrary modes of airfold motion very inexpensively. Numerical examples are presented that demonstrate the accuracy and computational efficiency of the present method. Finally, we show how the reduced order model may be incorporated into an aeroelastic flutter model

Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter and Limit Cycle Oscillations

By the use of a state-of-the-art computational e uid dynamic (CFD) method to model nonlinear steady and unsteady transonice owsin conjunction with a linearstructural model,an investigationismadeinto how nonlinear aerodynamics can effect the divergence, e utter, and limit-cycle oscillation (LCO) characteristics of a transonic airfoil cone guration. A single-degree-of-freedom (DOF) model is studied for divergence, and one- and two-DOF models are studied for e utter and LCO. A harmonicbalancemethod in conjunction with the CFD solver is used to determine the aerodynamics for e nite amplitude unsteady excitations of a prescribed frequency. A procedure for determining the LCO solution is also presented. For the cone guration investigated, nonlinear aerodynamic effects are found to produce a favorable transonic divergence trend and unstable and stable LCO solutions, respectively, for the one- and two-DOF e utter models. Nomenclature a = nondimensional location of airfoil elastic axis, e=b b, c = semichord and chord, respectively cl, cm = coefe cients of lift and moment about elastic axis, respectively e = location of airfoil elastic axis, measured positive aft of airfoil midchord h, ® = airfoil plunge and pitch degrees of freedom I® = second moment of inertia of airfoil about elastic axis

Calculation of Unsteady Flows in Turbomachinery Using the Linearized Euler Equations

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).

Three-Dimensional Transonic Aeroelasticity Using Proper Orthogonal Decomposition-Based Reduced-Order Models

The proper orthogonal decomposition (POD) based reduced order modeling (ROM) technique for modeling unsteady frequency domain aerodynamics is developed for a large scale computational model of an inviscid flow transonic wing configuration. Using the methodology, it is shown that a computational fluid dynamic (CFD) model with over a three quarters of a million degrees of freedom can be reduced to a system with just a few dozen degrees of freedom, while still retaining the accuracy of the unsteady aerodynamics of the full system representation. Furthermore, POD vectors generated from unsteady flow solution snapshots based on one set of structural mode shapes can be used for different structural mode shapes so long as solution snapshots at the endpoints of the frequency range of interest are included in the overall snapshot ensemble. Thus, the snapshot computation aspect of the method, which is the most computationally expensive part of the procedure, does not have to be fully repeated as different structural configurations are considered.

Modeling Viscous Transonic Limit Cycle Oscillation Behavior Using a Harmonic Balance Approach

Presented is a harmonic-balance computational fluid dynamic approach for modeling limit-cycle oscillation behavior of aeroelastic airfoil configurations in a viscous transonic flow. For the NLR 7301 airfoil configuration studied, accounting for viscous effects is shown to significantly influence computed limit-cycle oscillation trends when compared to an inviscid analysis. A methodology for accounting for changes in mean angle of attack during limit-cycle oscillation is also developed.

Eigenmode Analysis in Unsteady Aerodynamics: Reduced Order Models

In this article, we review the status of reduced order modeling of unsteady aerodynamic systems. Reduced order modeling is a conceptually novel and computationally efficient technique for computing unsteady flow about isolated airfoils, wings, and turbomachinery cascades. Starting with either a time domain or frequency domain computational fluid dynamics (CFD) analysis of unsteady aerodynamic or aeroacoustic flows, a large, sparse eigenvalue problem is solved using the Lanczos algorithm. Then, using just a few of the resulting eigenmodes, a Reduced Order Model of the unsteady flow is constructed. With this model, one can rapidly and accurately predict the unsteady aerodynamic response of the system over a wide range of reduced frequencies. Moreover, the eigenmode information provides important insights into the physics of unsteady flows. Finally, the method is particularly well suited for use in the active control of aeroelastic and aeroacoustic phenomena as well as in standard aeroelastic analysis for flutter or gust response. Numerical results presented include: 1) comparison of the reduced order model to classical unsteady incompressible aerodynamic theory, 2) reduced order calculations of compressible unsteady aerodynamics based on the full potential equation, 3) reduced order calculations of unsteady flow about an isolated airfoil based on the Euler equations, and 4) reduced order calculations of unsteady viscous flows associated with cascade stall flutter, 5) flutter analysis using the Reduced Order Model. This review article includes 25 references.

Reduced-order modelling of unsteady small-disturbance flows using a frequency-domain proper orthogonal decomposition technique

A new method for constructing reduced-order models (ROM) of unsteady small-disturbance flows is presented. The reduced-order models are constructed using basis vectors determined from the proper orthogonal decomposition (POD) of an ensemble of small-disturbance frequencydomain solutions. Each of the individual frequency-domain solutions is computed using an efficient time-linearized flow solver. We show that reduced-order models can be constructed using just a handful of POD basis vectors, producing low-order but highly accurate models of the unsteady flow over a wide range of frequencies. In this paper, we apply the POD/ROM technique to compute the unsteady aerodynamic and aeroelastic behavior of an isolated nansonic airfoil, and to a two-dimensional cascade of airfoils. Nomenclature A = matrix defining homogeneous part of discretized aerodynamic operator A = reduced-order form of A. b = airfoil semi-chord b = vector defining inhomogeneous part of discretized aerodynamic operator Ba, B1 = matrices relating airfoil motion h and h to b c c

A comparison of classical and high dimensional harmonic balance approaches for a Duffing oscillator

The present study focuses on a novel harmonic balance formulation, which is much easier to implement than the standard/classical harmonic balance method for complex nonlinear mathematical models and algorithms. Both harmonic balance approaches are applied to Duffings oscillator to demonstrate the advantages and disadvantages of the two approaches. A fundamental understanding of the difference between these two methods is achieved, and the properties of each method are analyzed in detail.