The use of pulse responses and system reduction for 2D unsteady flows using the Euler equations

Abstract This paper describes the application of the theory of pulse responses to a linearised Euler scheme on a moving grid. The impulse responses of such a linear system are memory functions or temporal representations of the manner in which, and the time over which a perturbation remains active in the response of the system. The exact response of the linear system to an arbitrary input can then be predicted via convolution. The linearised Euler scheme is derived from a full nonlinear Euler method based on Jameson’s cell-centred scheme which has been modified to make it time-accurate and has the necessary terms to account for grid motion. The discrete form of the full Euler equations are linearised about the corresponding nonlinear steady mean flow assuming the unsteadiness in the flow and grid are small. The resulting set of differential equations for the flow perturbations are solved at each time step using a dual-time scheme. These equations can be easily solved to obtain pulse responses, which capture the entire frequency content of the linear system. These pulse responses then provide a simple and efficient method for calculating the system response to many different inputs via convolution. A further use of the pulse responses is their potential to allow a reduced-order linear model to be constructed, which preserves some of the accuracyof the original system. Preliminary work carried out in this area is presented. Results of calculations for heave, pitch and ramp test cases are presented.

[1]  Edward F. Crawley,et al.  A linearized Euler analysis of unsteady flows in turbomachinery , 1987 .

[2]  Jeffrey P. Thomas,et al.  Proper Orthogonal Decomposition Technique for Transonic Unsteady Aerodynamic Flows , 2000 .

[3]  A. Jameson,et al.  Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes , 1981 .

[4]  A. Jameson.  Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings , 1991 .

[5]  E Raveh Daniella,et al.  Development of Unsteady Aerodynamic State-Space Models From CFD-Based Pulse Responses , 2001 .

[6]  W. Ning,et al.  Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines , 1998 .

[7]  Michael B. Giles,et al.  Validity of Linearized Unsteady Euler Equations with Shock Capturing , 1994 .

[8]  Gerhard Kahl,et al.  Time Linearized Euler Calculations for Unsteady Quasi-3D Cascade Flows , 1993 .

[9]  G. Doetsch.  Guide to the applications of the Laplace and Z-transforms , 1971 .

[10]  Michael B. Giles,et al.  Analysis of the accuracy of shock-capturing in the steady quasi-1D Euler equations , 1995 .

[11]  W. Silva,et al.  Discrete-time linear and nonlinear aerodynamic impulse responses for efficient cfd analyses , 1997 .

[12]  Antony Jameson,et al.  Computation of unsteady transonic flows by the solution of Euler equations , 1988 .

[13]  A. Gaitonde,et al.  A dual-time method for the solution of the unsteady Euler equations , 1994, The Aeronautical Journal (1968).

[14]  Edward F. Crawley,et al.  Calculation of unsteady flows in turbomachinery using the linearized Euler equations , 1989 .

[15]  Dana Rae Lindquist.  Computation of unsteady transonic flowfields using shock capturing and the linear perturbation Euler equations , 1992 .

[16]  G. Guruswamy.  Unsteady aerodynamic and aeroelastic calculations for wings using Euler equations , 1990 .

[17]  Jer-Nan Juang,et al.  An eigensystem realization algorithm for modal parameter identification and model reduction. [control systems design for large space structures] , 1985 .

[18]  Boris Laschka,et al.  Small Disturbance Euler Equations: Efficient and Accurate Tool for Unsteady Load Prediction , 2000 .

[19]  Shigeru Obayashi,et al.  Freestream capturing for moving coordinates in three dimensions , 1992 .

[20]  George R. Cooper,et al.  Continuous and discrete signal and system analysis , 1984 .

[21]  Joseph M. Verdon,et al.  A linearized unsteady aerodynamic analysis for transonic cascades , 1984, Journal of Fluid Mechanics.

[22]  P. Thomas,et al.  Geometric Conservation Law and Its Application to Flow Computations on Moving Grids , 1979 .

[23]  Li He,et al.  Computation of unsteady flows around oscillating blades using linear and nonlinear harmonic Euler methods , 1997 .