https://ntrs.nasa.gov/api/citations/20010018414/downloads/20010018414.pdf

Reduced-order modeling: new approaches for computational physics

A new method for constructing reduced-order models (ROM) of unsteady small-disturbance e ows 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 frequencydomain solutions is computed using an efe cient time-linearized e ow 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 e ow 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.

/pdf/proper-orthogonal-decomposition-technique-for-transonic-1qcwd6gv8a.pdf

Proper Orthogonal Decomposition Technique for Transonic Unsteady Aerodynamic Flows

A computer code, LTRAN2, has been constructed that efficiently computes low-frequency unsteady transonic flows about airfoils in motion. The code solves the two-dimensional, nonlinear, low-frequency, smalldisturbance equation by an alternating-direction implicit (ADI) algorithm. The underlying theory of the governing equation and the construction of the solution algorithm are discussed. As a check on the code, solutions computed for the linear low-frequency small-disturbance equation are compared with known answers from linear theory. Then nonlinear results are presented, and Tijdeman's experimental observations of shock wave motions resulting from airfoil flap oscillations are qualitatively reproduced computationally. These computations compare favorably with those of Magnus and Yoshihara and are obtained in substantially less computer time.

Implicit Finite-Difference Computations of Unsteady Transonic Flows about Airfoils

A general technique for constructing reduced order models of unsteady aerodynamic flows about twodimensional isolated airfoils, cascades of airfoils, and three-dimensional wings is developed. The starting point is a time 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 airfoil 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.

Eigenanalysis of unsteady flow about airfoils, cascades, and wings

Transonic aeroelastic solutions based upon the transonic small perturbation potential equation were studied. Time-marching transient solutions of plunging and pitching airfoils were analyzed using a complex exponential modal identification technique, and seven alternative integration techniques for the structural equations were evaluated. The HYTRAN2 code was used to determine transonic flutter boundaries versus Mach number and angle-of-attack for NACA 64A010 and MBB A-3 airfoils. In the code, a monotone differencing method, which eliminates leading edge expansion shocks, is used to solve the potential equation. When the effect of static pitching moment upon the angle-of-attack is included, the MBB A-3 airfoil can have multiple flutter speeds at a given Mach number.

/pdf/time-marching-transonic-flutter-solutions-including-angle-of-2d7gr7ddeb.pdf

Time-marching transonic flutter solutions including angle-of-attack effects

Analytical and empirical studies of a finite difference method for the solution of the transonic flow about an harmonically oscillating wing are presented along with a discussion of the development of a pilot program for three-dimensional flow. In addition, some two- and three-dimensional examples are presented.

On the computation of the transonic perturbation flow field around two- and three-dimensional oscillating wings

A finite difference method for solving the unsteady transonic flow about harmonically oscillating wings is investigated. The procedure is based on separating the velocity potential into steady and unsteady parts and linearizing the resulting unsteady differential equation for small disturbances. The differential equation for the unsteady velocity potential is linear with spatially varying coefficients and with the time variable eliminated by assuming harmonic motion. A study is presented of the shock motion associated with an oscillating airfoil and its representation by the harmonic procedure. The effects of the shock motion and the resulting pressure pulse are shown to be included in the harmonic pressure distributions and the corresponding generalized forces. Analytical and experimental pressure distributions for the NACA 64A010 airfoil are compared for Mach numbers of 0.75, 0.80 and 0.842. A typical section, two-degree-of-freedom flutter analysis of a NACA 64A010 airfoil is performed. The results show a sharp transonic bucket in one case and abrupt changes in instability modes.

/pdf/a-harmonic-analysis-method-for-unsteady-transonic-flow-and-51fdmfajzf.pdf

A harmonic analysis method for unsteady transonic flow and its application to the flutter of airfoils

Analytical and empirical studies of a finite difference method for the solution of the transonic flow about harmonically oscillating wings and airfoils are presented. The procedure is based on separating the velocity potential into steady and unsteady parts and linearizing the resulting unsteady equations for small disturbances. The steady velocity potential is obtained first from the well-known nonlinear equation for steady transonic flow. The unsteady velocity potential is then obtained from a linear differential equation in complex form with spatially varying coefficients. Since sinusoidal motion is assumed, the unsteady equation is independent of time. An out-of-core direct solution procedure was developed and applied to two-dimensional sections. Results are presented for a section of vanishing thickness in subsonic flow and an NACA 64A006 airfoil in supersonic flow. Good correlation is obtained in the first case at values of Mach number and reduced frequency of direct interest in flutter analyses. Reasonable results are obtained in the second case. Comparisons of two-dimensional finite difference solutions with exact analytic solutions indicate that the accuracy of the difference solution is dependent on the boundary conditions used on the outer boundaries. Homogeneous boundary conditions on the mesh edges that yield complex eigenvalues give the most accurate finite difference solutions. The plane outgoing wave boundary conditions meet these requirements.

Further investigation of a finite difference procedure for analyzing the transonic flow about harmonically oscillating airfoils and wings

A finite difference method for analyzing the unsteady transonic flow about harmonically oscillating wings is discussed The procedure is based on separating the velocity potential into steady and unsteady parts and linearizing the resulting partial differential equations for small disturbances Initial solutions were obtained using relaxation procedures, but the solution range proved to be limited in terms of Mach number and reduced frequency Recent two-dimensional results are presented which have been obtained with direct solution procedures in which the difference equations are solved 'all at once' and these provide reasonable correlation for practical values of Mach number and reduced frequency

Analysis of transonic flow about harmonically oscillating airfoils and wings

A finite difference method for solving the unsteady flow about harmonically oscillating wings is investigated. The procedure is based on separating the velocity potential into steady and unsteady parts and linearing the resulting unsteady differential equation for small disturbances. Solutions are obtained using relaxation procedures. The means for improving the solution stability characteristics of the relaxation process are explored. A direct procedure is formulated which permits obtaining solutions for combinations of Mach number and reduced frequency for which the relaxation process has proved unstable. The pressure distribution for an aspect ratio 5 rectangular wing oscillating in pitch is presented.

Application of a finite difference method to the analysis of transonic flow over oscillating airfoils and wings

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