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Journal ArticleDOI

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

01 Jan 2002-AIAA Journal (American Institute of Aeronautics and Astronautics (AIAA))-Vol. 40, Iss: 5, pp 879-886
TL;DR: In this paper, a harmonic balance technique for modeling unsteady nonlinear e ows in turbomachinery is presented, which exploits the fact that many unstaidy e ow variables are periodic in time.
Abstract: 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.

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Citations
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Journal ArticleDOI
TL;DR: In this paper, the authors review the development of new reduced-order modeling techniques and discuss their applicability to various problems in computational physics, including aerodynamic and aeroelastic behaviors of two-dimensional and three-dimensional geometries.

732 citations

Journal ArticleDOI
TL;DR: In this article, the authors present a review of the physical models for a fluid undergoing time-dependent motes and their applications in many fields of engineering, such as aeronautic and structural engineering.
Abstract: ▪ Abstract The interaction of a flexible structure with a flowing fluid in which it is submersed or by which it is surrounded gives rise to a rich variety of physical phenomena with applications in many fields of engineering, for example, the stability and response of aircraft wings, the flow of blood through arteries, the response of bridges and tall buildings to winds, the vibration of turbine and compressor blades, and the oscillation of heat exchangers. To understand these phenomena we need to model both the structure and the fluid. However, in keeping with the overall theme of this volume, the emphasis here is on the fluid models. Also, the applications are largely drawn from aerospace engineering although the methods and fundamental physical phenomena have much wider applications. In the present article, we emphasize recent developments and future challenges. To provide a context for these, the article begins with a description of the various physical models for a fluid undergoing time-dependent mot...

556 citations

Journal ArticleDOI
TL;DR: In this article, the authors investigated the effect of nonlinear aerodynamic effects on the divergence, utter, and limit-cycle oscillation (LCO) characteristics of a transonic airfoil cone guration.
Abstract: 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

282 citations

Book ChapterDOI
01 Jan 2014
TL;DR: In this paper, a review of model reduction techniques for fluid dynamics systems is presented, with a focus on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations.
Abstract: This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities — which are mainly related either to nonlinear convection terms and/or some geometric variability — that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and — in the unsteady case — long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

213 citations


Cites methods from "Computation of Unsteady Nonlinear F..."

  • ...To solve this problem, the Harmonic Balance (HB) method used for the study of harmonic ODEs was adapted for the efficient solution of time-periodic flows in [57, 89, 90]....

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  • ...This problem is solved either using the alternating frequency/time domain method, as was done in [57, 90], or by the asymptotic numerical method, as was done in [34]....

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  • ...So far the HB method has been applied mainly to industrial problems, such as the design and simulation of turbomachinery [57] and problems in aeroelasticity [120]....

    [...]

Journal ArticleDOI
TL;DR: The von Karman lecture as mentioned in this paper discusses the state of the art in nonlinear aeroelasticity and unsteady aerodynamics and identifies several of the most significant nonlinearities arising in a structure or in an aerodynamic flow field.
Abstract: In this von Karman lecture, a subject is addressed whose foundations were significantly influenced by the work of Theodore von Karman. A classic paper by von Karman and Sears first considered the determination of aerodynamic forces on an airfoil undergoing general time-dependent motion. Also, early in his career, von Karman investigated fundamental issues in structural mechanics and derived the celebrated von Karman plate equations for determining the large (nonlinear) deflections of an elastic plate under a distributed force. Finally, he authored a widely cited paper on the importance of nonlinearities for engineers and engineering. In this lecture, these themes are recalled and the current state of the art in nonlinear aeroelasticity and unsteady aerodynamics is discussed. Several of the most significant nonlinearities arising in a structure or in an aerodynamic flow field are identified. Recent and relevant theoretical and experimental studies are reviewed and future developments are projected that are expected to have a significant impact on our ability to understand and beneficially use nonlinear dynamic aeroelastic behavior

188 citations

References
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Proceedings ArticleDOI
06 Jan 1992

8,784 citations

Journal ArticleDOI
TL;DR: In this paper, a unified theory for the construction of steady-state and unsteady nonreflecting boundary conditions for the Euler equations is presented, which allows calculatios to be performed on truncated domains without the generation of spurious nonphysical reflections at the far-field boundaries.
Abstract: We present a unified theory for the construction of steady-state and unsteady nonreflecting boundary conditions for the Euler equations. These allow calculatios to be performed on truncated domains without the generation of spurious nonphysical reflections at the far-field boundaries.

800 citations

Journal ArticleDOI
TL;DR: In this paper, the authors review the development of new reduced-order modeling techniques and discuss their applicability to various problems in computational physics, including aerodynamic and aeroelastic behaviors of two-dimensional and three-dimensional geometries.

732 citations

Journal ArticleDOI
TL;DR: In this article, an effective method of interior grid control is presented based on a modified elliptic system containing free parameters for a simply connected region, the free parameters are computed from the Dirichlet boundary values.
Abstract: The generation of computational grids suitable for obtaining accurate numerical solutions to the three-dimensional Navier-Stokes equations is the subject of intensive research. For a wide class of nozzle configurations, a three-dimensional grid can be constructed by a sequence of two-dimensional grids in successive cross-sectional planes. The present paper is concerned with numerical generation of two-dimensional grids. An effective method of interior grid control is presented based on a modified elliptic system containing free parameters. For a simply connected region, the free parameters are computed from the Dirichlet boundary values. The resulting interior grid point distribution is controlled entirely by a priori selection of the grid point distribution along the boundaries of the section.

358 citations

Journal ArticleDOI
TL;DR: In this paper, a method for calculating unsteady flows in cascades is presented, which is based on the linearized Euler equations, accounts for blade loading shock motion, wake motion, and blade geometry.
Abstract: 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).

222 citations