Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T14:58:22.455Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  04 July 2016

A. L. Gaitonde  and
D. P. Jones
A. L. Gaitonde
Affiliation:
Department of Aerospace Engineering, University of Bristol, UK
D. P. Jones
Affiliation:
Department of Aerospace Engineering, University of Bristol, UK
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
The Aeronautical Journal , Volume 106 , Issue 1063 , September 2002 , pp. 483 - 492
Copyright
Copyright © Royal Aeronautical Society 2002 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Venkatakrishnan, V. and Jameson, A. Computation of unsteady transonic flows by the solution of the Euler equations, AIAA J, 1998, 26, pp 974–981.Google Scholar
2. Brenneis, A. and Erbele, A. Evaluation of an usteady implicit Euler code against two and three-dimensional standard configurations, Paper 10 in AGARD CP-507, 1991.Google Scholar
3. Guruswamy, G.P. Unsteady aerodynamic and aeroelastic calculations for wings using Euler equations, AIAA J, 1990, 28, pp 461–469.Google Scholar
4. Jameson, A.J. Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings, AIAA Paper 91-1596, 1991.Google Scholar
5. Gaitonde, A.L. A dual time method for the solution of the unsteady Euler equations, Aeronaut J, 1994, 98, (978).Google Scholar
6. AGARD manual on aeroelasticity in axial-flow turbomachines, papers 2 and 3, AGARDograph, 1, (298), 1987.Google Scholar
7. Verdon, J.M. and Casper, J.R. Linearised unsteady aerodynamic analysis for transonic flows, J Fluid Mechanics, 1984, 149, pp 403–429.Google Scholar
8. Whitehead, D.S. The calculation of steady and unsteady transonic flow in cascades, Technical Report CUED/A-Turbo/TR 118, Cambridge Univ Eng Dept, UK, 1982.Google Scholar
9. Kreiselmaier, E. and Laschka, B. Small disturbance Euler equations: efficient and accurate tool for unsteady load prediction, J Aircr, 2000, 37, (5), pp 770–778.Google Scholar
10. Hall, K.C. and Crawley, E.F. Calculation of Unsteady Flows in Turbo-machinery using the linearised Euler equations, AIAA J, 1989, 27, (6).Google Scholar
11. Hall, K.C. A Linearised Euler Analysis of Unsteady Flows In Turbo-machinery, ScD Thesis, MIT, 1987.Google Scholar
12. Lindquist, D.R. Computation of Unsteady Transonic Flowfields using Shock Capturing and the Linear Perturbation Euler Equations, PhD Dissertation, Dept of Aeronautics and Astronautics, MIT., 1991.Google Scholar
13. Lindquist, D.R. and Giles, M. B. Validity of linearised unsteady Euler equations with shock capturing, AIAA J, 1994, 32, (1).Google Scholar
14. Kahl, G. and Klose, A. Time linearised Euler calculations for unsteadyquasi-3D cascade flows, Sixth International Symposium on Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turboma chines and Propellors (Univ of Notre Dame, 1991), Springer-Verlag, 1993.Google Scholar
15. Holmes, D.G. and Chuang, H.A. Two-dimensional linearised harmonic Euler flow analysis for flutter and forced response, Sixth International Symposium on Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines and Propellors (Univ of Notre Dame, 1991), Springer-Verlag, 1993.Google Scholar
16. Silva, W.A. Discrete-Time Linear and Nonlinear Aerodynamic Impulse Responses for Efficient CFD Analyses, 1997, PhD Thesis.Google Scholar
17. Hall, K.C. Thomas, J.P. and Dowell, E.H. Proper orthogonal decomposition technique for transonic unsteady aerodynamic flows, AIAA J, 2000, 38, (10), pp 1853–1862.Google Scholar
18. Silva, W.A. and Raveh, D.E. Development of unsteady aerodynamic-state-space models from CFD-based pulse responses, AIAA Paper 2001-1213,2001.Google Scholar
19. Jameson, A. Schmidt, W. and Turkel, E. Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time stepping schemes, AIAA Paper 81-1259, 1981.Google Scholar
20. Wylie, C.R. and Barrett, L.C. Advanced Engineering Mathematics, Fifth Ed, 1985, McGraw-Hill.Google Scholar
21. Doetsch, G. Guide to the Application of the Laplace and Z-Trans-forms. Second Ed, 1971, Van Nostrand Reinhold Company.Google Scholar
22. McGillem, C.D. and Cooper, G.R. Continuous and Discrete Signal and System Analysis, 1984, CBS Publishing, Japan.Google Scholar
23. Gaitonde, A.L. and Jones, D.P. Solutions of the 2D linearised unsteady Euler equations on moving meshes, submitted to IMechE Aerospace Engineering. Google Scholar
24. Obayashi, S. Freestream capturing for moving coordinates in three dimensions, AIAA J, 1992, 30, pp 1125–1128.Google Scholar
25. Thomas, P.D. and Lombard, C.K. Geometric conservation law and its application to flow computations on moving grids, AIAA J, 1979, 17, pp 1030–1037.Google Scholar
26. Ning, W. and He, L. Computation of unsteady flows around oscillating blades using linear and nonlinear harmonic Euler method, ASME J of Turbomachinery, 1998, 120, (3).Google Scholar
27. He, L. and Ning, W. An efficient approach for analysis of unsteady viscous flows in turbomachinery, AIAA J, 1998, 36, (11).Google Scholar
28. Juang, J.N. and Pappa, R.S. An Eigensystem realisation algorithm for modal parameter identification and model reduction, J Guidance, (1985), 8, (5), pp 620–627.Google Scholar
29. AGARD, Compendium of unsteady aerodynamic measurements, AGARD, 1982, Report No 702.Google Scholar
30. Giles, M.B. Analysis of the accuracy of shock-capturing in the steady quasi-ID Euler equations, CFD J, 1996, 5, (2).Google Scholar