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Low-dimensional modelling of a transient cylinder wake using double proper orthogonal decomposition

Published online by Cambridge University Press:  08 August 2008

STEFAN G. SIEGEL ,
JÜRGEN SEIDEL ,
CASEY FAGLEY ,
D. M. LUCHTENBURG ,
KELLY COHEN  and
THOMAS MCLAUGHLIN
STEFAN G. SIEGEL
Affiliation:
Department of Aeronautics, US Air Force Academy, HQ USAFA/DFAN, Colorado Springs, CO 80840, USA
JÜRGEN SEIDEL
Affiliation:
Department of Aeronautics, US Air Force Academy, HQ USAFA/DFAN, Colorado Springs, CO 80840, USA
CASEY FAGLEY
Affiliation:
Department of Aeronautics, US Air Force Academy, HQ USAFA/DFAN, Colorado Springs, CO 80840, USA
D. M. LUCHTENBURG
Affiliation:
Institute of Fluid Dynamics and Technical Acoustics, Berlin University of Technology MB1, Straße des 17 Juni 135, D-10623 Berlin, Germany
KELLY COHEN
Affiliation:
Department of Aeronautics, US Air Force Academy, HQ USAFA/DFAN, Colorado Springs, CO 80840, USA
THOMAS MCLAUGHLIN
Affiliation:
Department of Aeronautics, US Air Force Academy, HQ USAFA/DFAN, Colorado Springs, CO 80840, USA
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Abstract

For the systematic development of feedback flow controllers, a numerical model that captures the dynamic behaviour of the flow field to be controlled is required. This poses a particular challenge for flow fields where the dynamic behaviour is nonlinear, and the governing equations cannot easily be solved in closed form. This has led to many versions of low-dimensional modelling techniques, which we extend in this work to represent better the impact of actuation on the flow. For the benchmark problem of a circular cylinder wake in the laminar regime, we introduce a novel extension to the proper orthogonal decomposition (POD) procedure that facilitates mode construction from transient data sets. We demonstrate the performance of this new decomposition by applying it to a data set from the development of the limit cycle oscillation of a circular cylinder wake simulation as well as an ensemble of transient forced simulation results. The modes obtained from this decomposition, which we refer to as the double POD (DPOD) method, correctly track the changes of the spatial modes both during the evolution of the limit cycle and when forcing is applied by transverse translation of the cylinder. The mode amplitudes, which are obtained by projecting the original data sets onto the truncated DPOD modes, can be used to construct a dynamic mathematical model of the wake that accurately predicts the wake flow dynamics within the lock-in region at low forcing amplitudes. This low-dimensional model, derived using nonlinear artificial neural network based system identification methods, is robust and accurate and can be used to simulate the dynamic behaviour of the wake flow. We demonstrate this ability not just for unforced and open-loop forced data, but also for a feedback-controlled simulation that leads to a 90% reduction in lift fluctuations. This indicates the possibility of constructing accurate dynamic low-dimensional models for feedback control by using unforced and transient forced data only.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 610 , 10 September 2008 , pp. 1 - 42
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Afanasiev, K. & Hinze, M. 2001 Adaptive Control of a Wake Flow Using Proper Orthogonal Decomposition. Shape Optimization and Optimal Design. Lecture Notes in Pure and Applied Mathematics, vol. 216. Springer.Google Scholar
Albarede, P. & Monkewitz, P. A. 1992 A model for the formation of oblique shedding patterns and chevrons in cylinder wakes. Phys. Fluids A 4, 744–756.CrossRefGoogle Scholar
Bergmann, M. & Cordier, L. 2006 Control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models. Preprint.Google Scholar
Bergmann, M., Cordier, L. & Brancher, J. P. 2005 Optimal rotary control of the cylinder wake using POD reduced order model. Phys. Fluids 17 (9), 357–392.CrossRefGoogle Scholar
Chomaz, J. M., Huerre, P. & Redekopp, L. G. 1988 Bifurcations to local and global modes in spatially developing flows. Phys. Rev. Lett. 60 (1), 25–28.CrossRefGoogle ScholarPubMed
Cohen, K., Siegel, S., McLaughlin, T. & Gillies, E. 2003 Feedback control of a cylinder wake low-dimensional model. AIAA J. 41, 1389–1391.CrossRefGoogle Scholar
Cohen, K., Siegel, S., Seidel, J. & McLaughlin, T. 2006 System identification of a low dimensional model of a cylinder wake. AIAA Paper 2006-1411.CrossRefGoogle Scholar
Cybenko, G. V. 1989 Approximation by superpositions of a sigmoidal function. Maths Control Signals Syst. 2, 303–314.CrossRefGoogle Scholar
Deane, A. E., Kevrekidis, I. G., Karniadakis, G. E. & Orszag, S. A. 1991 Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders. Phys. Fluids 3 (10), 2337–2354.CrossRefGoogle Scholar
Efe, M. Ö., Debiasi, M., Özbay, H. & Samimy, M. 2004 Modeling of subsonic cavity flows by neural networks. In Intl Conf. Mechatronics (ICM), Istanbul, Turkey, 560–565.Google Scholar
Efe, M. Ö., Debiasi, M., Yan, P., Özbay, H. & Samimy, M. 2005 Control of subsonic cavity flows by neural networks – analytical models and experimental validation. AIAA Paper 2005-0294.CrossRefGoogle Scholar
Fahl, M. 2000 Trust-region methods for flow control based on reduced order modeling. PhD thesis, Trier university.Google Scholar
Faller, W. E. & Schreck, S. J. 1997 Unsteady fluid mechanics applications of neural networks. J. Aircraft 34, 48–55.CrossRefGoogle Scholar
Faller, W. E., Schreck, S. J. & Luttges, M. W. 1995 Neural network prediction and control of three–dimensional unsteady separated flowfields. J. Aircraft 32, 1213–1220.CrossRefGoogle Scholar
Fan, X. 1995 Laminar flow control models with neural networks. PhD thesis, Ohio State University, Columbus, Ohio.Google Scholar
Fan, X., Hofmann, L. & Herbert, T. 1993 Active flow control with neural networks. AIAA Paper 1993-3273.CrossRefGoogle Scholar
Gad-el-Hak, M. 2000 Flow Control: Passive, Active, and Reactive Flow Management, pp. 352–357. Cambridge University Press.CrossRefGoogle Scholar
Galletti, B., Bruneau, C. H., Zannetti, L. & Iollo, A. 2004 Low-order modelling of laminar flow regimes past a confined square cylinder. J. Fluid Mech. 503, 161–170.CrossRefGoogle Scholar
Gerhard, J., Pastoor, M., King, R., Noack, B. R., Dillmann, A., Morzynski, M. & Tadmor, G. 2003 Model based control of vortex shedding using low-dimensional galerkin models. AIAA CP 2003-4261.CrossRefGoogle Scholar
Gillies, E. A. 1995 Low-dimensional characterization and control of non-linear wake flows. PhD dissertation, Faculty of Engineering, University of Glasgow, UK.Google Scholar
Gillies, E. A. 1998 Low-dimensional control of the circular cylinder wake. J. Fluid Mech. 371, 157–178.CrossRefGoogle Scholar
Gillies, E. A. 2000 Multiple sensor control of vortex shedding. AIAA Paper 2000-1933.CrossRefGoogle Scholar
Glezer, A., Kadioglu, Z. & Pearlstein, A. J. 1989 Development of an extended proper orthogonal decomposition and its application to a time periodically forced plane mixing layer. Phys. Fluids A 1 (8).CrossRefGoogle Scholar
Gottlieb, J. J. & Groth, C. P. T. 1988 Assessment of Riemann solvers for unsteady one-dimensional inviscid flows of perfect gases. J. Comput. Phys. 78 (2), 437–458.CrossRefGoogle Scholar
Graham, W. R., Peraire, J. & Tang, K. Y. 1999 a Optimal control of vortex shedding using low-dimensional models. Part I: Open-loop model development. Intl J. Numer. Meth. Engng 44, 945–972.3.0.CO;2-F>CrossRefGoogle Scholar
Graham, W. R., Peraire, J. & Tang, K. Y. 1999 b Optimal control of vortex shedding using low-dimensional models. Part II: Model based control. Intl J. Numer. Meth. Engng 44, 973–990.3.0.CO;2-F>CrossRefGoogle Scholar
Haykin, S. 1999 Neural Networks – A Comprehensive Foundation, 2nd edn. Prentice-Hall.Google Scholar
Hočevar, M., Širok, B. & Grabec, I. 2004 Experimental turbulent field modeling by visualization and neural networks. Trans. ASME 126, 316–322.Google Scholar
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.CrossRefGoogle Scholar
Jørgensen, B. H., Sørensen, J. N. & Brøns, M. 2003 Low-dimensional modeling of a driven cavity flow with two free parameters. Theoret. Comput. Fluid Mech. 16 (4), 299–317.Google Scholar
von Kármán, T. 1954 Aerodynamics: Selected Topics in Light of their Historic Development. Cornell University Press.Google Scholar
von Kármán, T. 1911 Über den Mechanismus des Widerstands, den ein bewegter Körper in einer Flüssigkeit erfährt. Nachr. Ges. Wiss. Göttingen, Math. Phys. Kl., 509–517.Google Scholar
Khibnik, A. I., Narayanan, S., Jacobson, C. A. & Lust, K. 2000 Analysis of low dimensional dynamics of flow separation. Proc. ERCOFTAC/EUROMECH Colloquium 383 Aussois, France, 1998. Vieweg.Google Scholar
Koopmann, G. 1967 The vortex wakes of vibrating cylinders at low reynolds numbers. J. Fluid Mech. 28, 501–512.CrossRefGoogle Scholar
Lee, C., Kim, J., Babcock, D. & Goodman, R. 1997 Application of neural networks to turbulence control for drag reduction. Phys. Fluids 9 (6), 1740–1747.CrossRefGoogle Scholar
Ljung, L. 1999 System Identification – Theory for the User, 2nd edn. Prentice-Hall.Google Scholar
Luchtenburg, M., Tadmor, G., Lehmann, O., Noack, B. R., King, R. & Morzynski, M. 2006 Tuned POD Galerkin models for transient feedback regulation of the cylinder wake. AIAA Paper 2006-1407.CrossRefGoogle Scholar
Ma, X. & Karniadakis, G. 2002 A low-dimensional model for simulating three-dimensional cylinder flow. J. Fluid Mech. 458, 181–190.CrossRefGoogle Scholar
Min, C. & Choi, H. 1999 Suboptimal feedback control of vortex shedding at low Reynolds numbers. J. Fluid Mech. 401, 123–156.CrossRefGoogle Scholar
Morzynski, M., Stankiewicz, W., Noack, B. R., Thiele, F. & Tadmor, G. 2006 Generalized mean-field model for flow control using continuous mode interpolation. AIAA Paper 2006-3488.CrossRefGoogle Scholar
Narayanan, S., Khibnik, A. I., Jacobson, C. A., Keverekedis, Y., Rico-Martinez, R. & Lust, K. 1999 Low-dimensional models for active control of flow separation. Proc. 1999 IEEE Intl Conf. on Control Applications, Kohala Coast-Island of Hawaii, Hawaii, USA, pp. 1151–1156.Google Scholar
Nelles, O. 2001 Nonlinear System Identification. Springer.CrossRefGoogle Scholar
Noack, B. R., Ohle, F. & Eckelmann, H. 1991 On cell formation in vortex streets. J. Fluid Mech. 227, 293–308.CrossRefGoogle Scholar
Noack, B. R., Afanasiev, K., Morzynski, M. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.CrossRefGoogle Scholar
Noack, B. R., Tadmor, G. & Morzynski, M. 2004 a Low-dimensional models for feedback flow control. Part I: Empirical Galerkin models. AIAA Paper 2004-2408.CrossRefGoogle Scholar
Noack, B. R., Tadmor, G. & Morzynski, M. 2004 b Actuation models and dissipative control in empirical Galerkin models of fluid flows. American Control Conf. Boston, MA, USA, Paper FrP15.6.Google Scholar
Nørgaard, M., Ravn, O., Poulsen, N. K. & Hansen, L. K. 2000 Neural Networks for Modeling and Control of Dynamic Systems. Springer.CrossRefGoogle Scholar
Oertel, H. Jr 1990 Wakes behind blunt bodies. Annu. Rev. Fluid Mech. 22, 539–564.CrossRefGoogle Scholar
Panton, R. L. 1996 Incompressible Flow, 2nd edn. John Wiley.Google Scholar
Papangelou, A. 1992 Vortex shedding from slender cones at low Reynolds numbers. J. Fluid Mech. 242, 299–321 (and Corrigendum 248, 1993, 684).CrossRefGoogle Scholar
Park, D. S., Ladd, D. M. & Hendricks, E. W. 1993 Feedback control of a global mode in spatially developing flows. Phys. Lett. A 182–244.CrossRefGoogle Scholar
Pindera, M. Z. 2002 Adaptive flow control using simple artificial neural networks. AIAA Paper 2002-0990.CrossRefGoogle Scholar
Ravindran, S. 2000 Reduced-order adaptive controllers for fluid flows using POD. J. Sci. Comput. 15, 457–478.CrossRefGoogle Scholar
Rempfer, D. 2000 On low-dimensional Galerkin models for fluid flow. Theoret. Comput. Fluid Mech. 14 (2), 75–88.CrossRefGoogle Scholar
Roussopoulos, K. 1993 Feedback control of vortex shedding at low Reynolds numbers. J. Fluid Mech. 248, 267–296.CrossRefGoogle Scholar
Roussopoulos, K. & Monkewitz, P. A. 1996 Nonlinear modeling of vortex shedding control in cylinder wakes. Physica D 97, 264–73.Google Scholar
Sahan, R. A., Koc-Sahan, N., Albin, D. C. & Liakopoulos, A. 1997 Artificial neural network based modeling and intelligent control of transitional flows. Proc. 1997 IEEE Intl Conf. on Control Applications, Hartford, CT, pp. 359–364.Google Scholar
Seidel, J., Siegel, S., Cohen, K., Becker, V. & McLaughlin, T. 2006 Simulations of three dimensional feedback control of a circular cylinder wake. AIAA Paper 2006-1404.CrossRefGoogle Scholar
Siegel, S., Cohen, K. & McLaughlin, T. 2003 Feedback control of a circular cylinder wake in experiment and simulation (Invited). AIAA Paper 2003-3569.CrossRefGoogle Scholar
Siegel, S., Cohen, K. & McLaughlin, T. 2004 Feedback control of a circular cylinder wake in a water tunnel experiment. AIAA Paper 2004-0580.CrossRefGoogle Scholar
Siegel, S. G., Cohen, K., Seidel, J. & McLaughlin, T. 2005 Short time proper orthogonal decomposition for state estimation of transient flow fields. AIAA Paper 2005-0296.CrossRefGoogle Scholar
Siegel, S., Cohen, K. & McLaughlin, T. 2006 Numerical simulations of a feedback controlled circular cylinder wake. AIAA J. 44 (6), 1266–1276.CrossRefGoogle Scholar
Siegel, S., Aradag, S., Seidel, J., Cohen, K. & McLaughlin, T. 2007 Low dimensional POD based estimation of a 3D turbulent separated flow. AIAA Paper 2007-0112.CrossRefGoogle Scholar
Sirisup, S., Karniadakis, G. E., Xiu, D. B. & Kevrekidis, I. G., 2005 Equation-free/Galerkin-free POD-assisted computation of incompressible flows. J. Computat. Phys. 207 (2), 568–587.CrossRefGoogle Scholar
Sirovich, L., 1987 Turbulence and the dynamics of coherent structures. Part I: Coherent structures. Q. Appl. Maths 45 (3), 561–590.CrossRefGoogle Scholar
Strang, W. Z., Tomaro, R. F. & Grismer, M. J. 1999 The defining methods of Cobalt60: a parallel, implicit, unstructured Euler/Navier–Stokes flow solver. AIAA Paper 99-0786.Google Scholar
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech. 4, 1–21.CrossRefGoogle Scholar
Tadmor, G., Noack, B. R., Morzynski, M. & Siegel, S. 2004 Low-dimensional models for feedback flow control. Part II: Observer and controller design. AIAA Paper 2004-2409.CrossRefGoogle Scholar
Taylor, J. A. & Glauser, M. N. 2004 Towards practical flow sensing and control via POD and LSE based low-dimensional tools. Trans. ASME I: J. Fluids Engng 16 (3), 337–345.Google Scholar
Williamson, C. H. K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477–539.CrossRefGoogle Scholar
Zielinska, B. J. A. & Wesfreid, J. E. 1995 On the spatial structure of global modes in wake flow. Phys. Fluids 7 (6), 1418–1424.CrossRefGoogle Scholar