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On least-order flow representations for aerodynamics and aeroacoustics

Published online by Cambridge University Press:  16 March 2012

Michael Schlegel*
Affiliation:
Institut für Strömungsmechanik und Technische Akustik, Technische Universität Berlin MB1, Straße des 17. Juni 135, D-10623 Berlin, Germany
Bernd R. Noack
Affiliation:
Institut P′, CNRS–Université de Poitiers–ENSMA, UPR 3346, Département Fluides, Thermique, Combustion, CEAT, 43 rue de l’Aérodrome, F-86036 Poitiers CEDEX, France
Peter Jordan
Affiliation:
Institut P′, CNRS–Université de Poitiers–ENSMA, UPR 3346, Département Fluides, Thermique, Combustion, CEAT, 43 rue de l’Aérodrome, F-86036 Poitiers CEDEX, France
Andreas Dillmann
Affiliation:
Institut für Aerodynamik und Strömungstechnik, Deutsches Zentrum für Luft- und Raumfahrt, Bunsenstraße 10, D-37073 Göttingen, Germany
Elmar Gröschel
Affiliation:
Aerodynamisches Institut, Rheinisch-Westfälische Technische Hochschule Aachen, Wüllnerstraße 5a, D-52062 Aachen, Germany ABB Turbo Systems AG, Bruggerstraße 71a, 5400 Baden, Switzerland
Wolfgang Schröder
Affiliation:
Aerodynamisches Institut, Rheinisch-Westfälische Technische Hochschule Aachen, Wüllnerstraße 5a, D-52062 Aachen, Germany
Mingjun Wei
Affiliation:
Mechanical and Aerospace Engineering, New Mexico State University, PO Box 30001/Dept 3450, Las Cruces, NM 88003-8001, USA
Jonathan B. Freund
Affiliation:
Mechanical Science & Engineering, University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, IL 61801, USA
Oliver Lehmann
Affiliation:
Northeastern University, Department of Electrical and Computer Engineering, 440 Dana Research Building, Boston, MA 02115, USA
Gilead Tadmor
Affiliation:
Northeastern University, Department of Electrical and Computer Engineering, 440 Dana Research Building, Boston, MA 02115, USA
*
Email address for correspondence: michael.schlegel@tu-berlin.de

Abstract

We propose a generalization of proper orthogonal decomposition (POD) for optimal flow resolution of linearly related observables. This Galerkin expansion, termed ‘observable inferred decomposition’ (OID), addresses a need in aerodynamic and aeroacoustic applications by identifying the modes contributing most to these observables. Thus, OID constitutes a building block for physical understanding, least-biased conditional sampling, state estimation and control design. From a continuum of OID versions, two variants are tailored for purposes of observer and control design, respectively. Firstly, the most probable flow state consistent with the observable is constructed by a ‘least-residual’ variant. This version constitutes a simple, easily generalizable reconstruction of the most probable hydrodynamic state to preprocess efficient observer design. Secondly, the ‘least-energetic’ variant identifies modes with the largest gain for the observable. This version is a building block for Lyapunov control design. The efficient dimension reduction of OID as compared to POD is demonstrated for several shear flows. In particular, three aerodynamic and aeroacoustic goal functionals are studied: (i) lift and drag fluctuation of a two-dimensional cylinder wake flow; (ii) aeroacoustic density fluctuations measured by a sensor array and emitted from a two-dimensional compressible mixing layer; and (iii) aeroacoustic pressure monitored by a sensor array and emitted from a three-dimensional compressible jet. The most ‘drag-related’, ‘lift-related’ and ‘loud’ structures are distilled and interpreted in terms of known physical processes.

Type
Papers
Copyright
Copyright © Cambridge University Press 2012

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References

1. Afanasiev, K. 2003 Stabilitätsanalyse, niedrigdimensionale Modellierung und optimale Kontrolle der Kreiszylinderumströmung [Stability analysis, low-dimensional modelling, and optimal control of the flow around a circular cylinder]. PhD Thesis, Technische Universität Dresden, Germany.Google Scholar
2. Ben-Israel, A. & Greville, T. N. E. 2003 Generalized Inverses: Theory and Applications, vol. 15, 2nd edn. CMS Books in Mathematics , Springer.Google Scholar
3. Bergmann, M., Cordier, L. & Brancher, J.-P. 2005 Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced order model. Phys. Fluids 17, 121.CrossRefGoogle Scholar
4. Borée, J. 2003 Extended proper orthogonal decomposition: a tool to analyse correlated events in turbulent flows. Exp. Fluids 35, 188192.CrossRefGoogle Scholar
5. Cavalieri, A. V. G., Daviller, G., Comte, P., Jordan, P., Tadmor, G & Gervais, Y. 2011a Using large eddy simulation to explore sound source mechanisms in jets. J. Sound Vib. 330, 40984113.CrossRefGoogle Scholar
6. Cavalieri, A. V. G., Jordan, P., Agarwal, A. & Gervais, Y. 2011b Jittering wavepacket models for subsonic jet noise. J. Sound Vib. 330, 44744492.CrossRefGoogle Scholar
7. Cavalieri, A. V. G., Jordan, P., Gervais, Y. & Colonius, T. 2011 c Axisymmetric superdirectivity in subsonic jets. In 17th AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2011-2743.Google Scholar
8. Cazemier, W., Verstappen, R. W. C. P. & Veldman, A. E. P. 1998 Proper orthogonal decomposition and low-dimensional models for driven cavity flows. Phys. Fluids 7, 16851699.CrossRefGoogle Scholar
9. Coiffet, F., Jordan, P., Delville, J., Gervais, Y. & Ricaud, F. 2006 Coherent structures in subsonic jets: a quasi-irrotational source mechanism? Intl J. Aeroacoust. 5 (1), 6789.CrossRefGoogle Scholar
10. Crighton, D. G. 1975 Basic principles of aerodynamic noise generation. Prog. Aerosp. Sci. 16, 3196.CrossRefGoogle Scholar
11. Crighton, D. G. & Huerre, P. 1990 Shear layer pressure fluctuations and superdirective acoustic sources. J. Fluid Mech. 220, 355368.CrossRefGoogle Scholar
12. Ffowcs Williams, J. E. 1963 The noise from turbulence convected at high speed. Phil. Trans. R. Soc. Lond. A 231, 505514.Google Scholar
13. Franzke, C. & Majda, A. J 2006 Low order stochastic mode reduction for a prototype atmospheric GCM. J. Atmos. Sci. 63 (2), 457479.CrossRefGoogle Scholar
14. Freund, J. 2001 Noise sources in a low Reynolds number turbulent jet at Mach 0.9. J. Fluid Mech. 438, 277305.CrossRefGoogle Scholar
15. Freund, J. & Colonius, T. 2002 POD analysis of sound generation by a turbulent jet. AIAA Paper 2002-0072.Google Scholar
16. Freund, J. & Colonius, T. 2009 Turbulence and sound-field POD analysis of a turbulent jet. Intl J. Aeroacoust. 8 (4), 337354.CrossRefGoogle Scholar
17. Gerhard, J., Pastoor, M., King, R., Noack, B. R., Dillmann, A., Morzyński, M. & Tadmor, G. 2003 Model-based control of vortex shedding using low-dimensional Galerkin models, AIAA Paper 2003-4262.Google Scholar
18. Gröschel, E., Schröder, W., Schlegel, M., Scouten, J., Noack, B. R. & Comte, P. 2007 Reduced-order analysis of turbulent jet flow and its noise source. ESAIM: Proc. 16, 3350.CrossRefGoogle Scholar
19. Gröschel, E., Schröder, W., Renze, P., Meinke, M. & Comte, P. 2008 Noise prediction for a turbulent jet using different hybrid methods. Comput. Fluids 37, 414426.CrossRefGoogle Scholar
20. Guj, G., Carley, C. & Camussi, R. 2003 Acoustic identification of coherent structures in a turbulent jet. J. Sound Vib. 259 (5), 10371065.CrossRefGoogle Scholar
21. Hileman, J. I., Caraballo, E. J., Thurow, B. S. & Samimy, M. 2004 Differences in dynamics of an ideally expanded Mach 1.3 jet during noise generation and relative quiet periods. AIAA Paper 2004-3015.Google Scholar
22. Hileman, J. I., Thurow, B. S., Caraballo, E. J. & Samimy, M. 2005 Large-scale structure evolution and sound emission in high-speed jets: real-time visualization with simultaneous acoustic measurements. J. Fluid Mech. 544, 277307.CrossRefGoogle Scholar
23. Hoarau, C., Borée, J., Laumonier, J. & Gervais, Y. 2006 Analysis of the wall pressure trace downstream of a separated region using extended proper orthogonal decomposition. Phys. Fluids 18, 055107.CrossRefGoogle Scholar
24. Holmes, P., Lumley, J. L. & Berkooz, G. 1998 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.Google Scholar
25. Howe, M. S. 2003 Theory of Vortex Sound. Cambridge University Press.Google Scholar
26. Jordan, P. & Gervais, Y. 2008 Subsonic jet aeroacoustics: associating experiment, modelling and simulation. Exp. Fluids 44, 121.CrossRefGoogle Scholar
27. Jordan, P., Schlegel, M., Stalnov, O., Noack, B. R. & Tinney, C. E. 2007 Identifying noisy and quiet modes in a jet. In 13th AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2007–3602.Google Scholar
28. Jørgensen, B. H., Sørensen, J. N. & Brøns, M. 2003 Low-dimensional modelling of a driven cavity flow with two free parameters. Theor. Comput. Fluid Dyn. 16, 299317.Google Scholar
29. Juvé, D., Sunyach, M. & Comte-Bellot, G. 1980 Intermittency of the noise emission in subsonic cold jets. J. Sound Vib. 71 (3), 319332.CrossRefGoogle Scholar
30. Lall, S., Marsden, J. E. & Glavaški, S. 1999 Empirical model reduction of controlled nonlinear systems. In Proceedings of the 14th IFAC World Congress, vol. F, pp. 473478. International Federation of Automatic Control (IFAC), Laxenburg, Austria.Google Scholar
31. Lall, S., Marsden, J. E. & Glavaški, S. 2002 A subspace iteration approach to balanced truncation for model reduction of nonlinear control systems. Intl J. Robust Nonlinear Control 12, 519535.CrossRefGoogle Scholar
32. Laurendeau, E., Jordan, P., Bonnet, J. P., Delville, J., Parnaudeau, P. & Lamballais, E. 2008 Subsonic jet noise reduction by fluidic control: the interaction region and the global effect. Phys. Fluids 20, 101519.CrossRefGoogle Scholar
33. Lee, H. K. & Ribner, H. S. 1972 Direct correlation of noise and flow of a jet. J. Acoust. Soc. Am. 52 (5), 12801290.CrossRefGoogle Scholar
34. Lighthill, M. J. 1952 On sound generated aerodynamically: I. General theory. Proc. R. Soc. Lond. A 211, 564587.CrossRefGoogle Scholar
35. Luchtenburg, D. M., Günther, B., Noack, B. R., King, R. & Tadmor, G. 2009 A generalized mean-field model of the natural and high-frequency actuated flow around a high-lift configuration. J. Fluid Mech. 623, 283316.CrossRefGoogle Scholar
36. Lugt, H. J. 1996 Introduction to Vortex Theory. Vortex Flow Press.Google Scholar
37. Maurel, S., Borée, J. & Lumley, J. L. 2001 Extended proper orthogonal decomposition: application to jet/vortex interaction. Flow Turbul. Combust. 67, 125136.CrossRefGoogle Scholar
38. Meinke, M., Schröder, W., Krause, E. & Rister, T. R. 2002 A comparison of second- and sixth-order methods for large-eddy simulations. Comput. Fluids 21, 695718.CrossRefGoogle Scholar
39. Morzyński, M. 1987 Numerical solution of Navier–Stokes equations by the finite element method. In Proceedings of SYMKOM 87, Compressor and Turbine Stage Flow Path – Theory and Experiment, Reports of the Institute of Turbomachinery 527, Cieplne Maszyny Przepłwowe 94, pp. 119–128. Technical University of Łódź.Google Scholar
40. Morzyński, M., Stankiewicz, W., Noack, B. R., King, R., Thiele, F. & Tadmor, G. 2007 Continuous mode interpolation for control-oriented models of fluid flow. In Active Flow Control: Papers Contributed to the Conference ‘Active Flow Control 2006’, Berlin, Germany, September 27 to 29, 2006 (ed. R. King), Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 95, pp. 260–278. Springer.Google Scholar
41. Noack, B. R., Afanasiev, K., Morzyński, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335363.CrossRefGoogle Scholar
42. Noack, B. R., Morzyński, M. & Tadmor, G.  (Eds) 2011 Reduced-Order Modelling for Flow Control. CISM Courses and Lectures , vol. 528. Springer.CrossRefGoogle Scholar
43. Noack, B. R., Papas, P. & Monkewitz, P. A. 2005 The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523, 339365.CrossRefGoogle Scholar
44. Noack, B. R., Schlegel, M., Ahlborn, B., Mutschke, G., Morzyński, M., Comte, P. & Tadmor, G. 2008 A finite-time thermodynamics of unsteady fluid flows. J. Non-Equilib. Thermodyn. 33 (2), 103148.Google Scholar
45. Noack, B. R., Schlegel, M., Morzyński, M. & Tadmor, G. 2010 System reduction strategy for Galerkin models of fluid flows. Intl J. Numer. Meth. Fluids 63 (2), 231248.Google Scholar
46. Noack, B. R. & Niven, R. K. 2012 Maximum entropy closure for a Galerkin model of an incompressible periodic wake. J. Fluid Mech. (in press).Google Scholar
47. Panda, J., Seasholtz, R. G. & Elam, K. A. 2005 Investigation of noise sources in high-speed jets via correlation measurements. J. Fluid Mech. 537, 349385.CrossRefGoogle Scholar
48. Pastoor, M., Henning, L., Noack, B. R., King, R. & Tadmor, G. 2008 Feedback shear layer control for bluff body drag reduction. J. Fluid Mech. 608, 161196.CrossRefGoogle Scholar
49. Picard, C. & Delville, J. 2000 Pressure velocity coupling in a subsonic round jet. Intl J. Heat Fluid Flow 21, 359364.CrossRefGoogle Scholar
50. Protas, B. & Wesfreid, J. E. 2003 On the relation between the global modes and the spectra of drag and lift in periodic wake flows. C. R. Méc. 331, 4954.Google Scholar
51. Rempfer, D. & Fasel, H. F. 1994 Dynamics of three-dimensional coherent structures in a flat-plate boundary layer. J. Fluid Mech. 275, 257283.CrossRefGoogle Scholar
52. Rodriguez Alvarez, D., Samanta, A., Cavalieri, A. V. G., Colonius, T. & Jordan, P. 2011 Parabolized stability equation models for predicting large-scale mixing noise of turbulent round jets. In 17th AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2011-2743.Google Scholar
53. Rowley, C. W. 2005 Model reduction for fluids using balanced proper orthogonal decomposition. Intl J. Bifurcation Chaos 15 (3), 9971013.CrossRefGoogle Scholar
54. Rowley, C. W., Colonius, T. & Murray, R. M. 2004 Model reduction for compressible flows using POD and Galerkin projection. Physica D 189 (1–2), 115129.CrossRefGoogle Scholar
55. Samimy, M., Kim, J.-H., Kastner, J., Adamovich, I. & Utkin, Y. 2007 Active control of a Mach 0.9 jet for noise mitigation using plasma actuators. AIAA J. 45 (4), 890901.CrossRefGoogle Scholar
56. Schaffar, M. 1979 Direct measurements of the correlation between axial in-jet velocity fluctuations and far field noise near the axis of a cold jet. J. Sound Vib. 64 (1), 7383.CrossRefGoogle Scholar
57. Schaffar, M. & Hancy, J. P 1982 Investigation of the noise emitting zones of a cold jet via causality correlations. J. Sound Vib. 81 (3), 377391.CrossRefGoogle Scholar
58. Scharton, T. D. & White, P. H. 1972 Simple pressure source model of jet noise. J. Acoust. Soc. Am. 52 (1), 399412.CrossRefGoogle Scholar
59. Schlegel, M., Noack, B. R., Comte, P., Kolomenskiy, D., Schneider, K., Farge, M., Scouten, J., Luchtenburg, D. M. & Tadmor, G. 2009 Reduced-order modelling of turbulent jets for noise control. In Numerical Simulation of Turbulent Flows and Noise Generation (ed. Brun, C., Juvé, D., Manhart, M. & Munz, C.-D. ). Notes on Numerical Fluid Mechanics and Multidisciplinary Design , vol. 104, pp. 327. Springer.CrossRefGoogle Scholar
60. Seiner, J. M. 1974 The distribution of jet source strength intensity by means of a direct correlation technique. PhD Thesis, Pennsylvania State University, University Park, PA.Google Scholar
61. Seiner, J. M. & Reetoff, G. 1974 On the distribution of source coherency in subsonic jets. AIAA Paper 1974-4.Google Scholar
62. Siegel, S. G., Seidel, J., Fagley, C., Luchtenburg, D. M., Cohen, K. & McLaughlin, T. 2008 Low-dimensional modelling of a transient cylinder wake using double proper orthogonal decomposition. J. Fluid Mech. 610, 142.CrossRefGoogle Scholar
63. Sirovich, L. 1987 Turbulence and the dynamics of coherent structures, Part I: Coherent structures. Q. Appl. Math. XLV, 561571.CrossRefGoogle Scholar
64. Tam, C. 1998 Jet noise: since 1952. Theor. Comput. Fluid Dyn. 10, 393405.CrossRefGoogle Scholar
65. Townsend, A. A. 1956 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
66. Tröltzsch, F. 2005 Optimale Steuerung partieller Differentialgleichungen: Theorie, Verfahren und Anwendungen. Vieweg.CrossRefGoogle Scholar
67. Wei, M. 2004 Jet noise control by adjoint-based optimization. PhD Thesis, University of Illinois at Urbana-Champaign, IL.Google Scholar
68. Wei, M. & Freund, J. 2006 A noise-controlled free shear flow. J. Fluid Mech. 546, 123152.CrossRefGoogle Scholar
69. Wei, M. & Rowley, C. W. 2009 Low-dimensional models of a temporally evolving free shear layer. J. Fluid Mech. 618, 113134.CrossRefGoogle Scholar
70. Willcox, K. 2006 Flow sensing and estimation via the gappy proper orthogonal decomposition. Comput. Fluids 35 (2), 208226.CrossRefGoogle Scholar
71. Willcox, K. & Megretski, A. 2005 Fourier series for accurate, stable, reduced-order models in large-scale applications. SIAM J. Sci. Comput. 26 (3), 944962.CrossRefGoogle Scholar
72. Willcox, K. & Peraire, J. 2002 Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40 (11), 23232330.CrossRefGoogle Scholar
73. Wu, J.-Z., Ma, H.-Y. & Zhou, M.-D. 2006 Vorticity and Vortex Dynamics. Springer.CrossRefGoogle Scholar
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