Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced-order model

Physics of Fluids

Volumn 17, Issue 9, 2005, Pages 1-21

  Bergmann, Michel ^a   Cordier, Laurent ^a   Brancher, Jean Pierre ^a  

^a NANCY UNIVERSITÉ   (France)

Author keywords

[No Author keywords available]

Indexed keywords

CYLINDERS (SHAPES); DRAG; INCOMPRESSIBLE FLOW; LAGRANGE MULTIPLIERS; MATHEMATICAL MODELS; NAVIER STOKES EQUATIONS; OPTIMAL CONTROL SYSTEMS; OPTIMIZATION; VISCOUS FLOW; WAKES;

CYLINDER WAKE; DRAG OPTIMIZATION; PROPER ORTHOGONAL DECOMPOSITION; REDUCED ORDER MODEL;

FLOW CONTROL;

WAKE;

EID: 27144538190     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2033624     Document Type: Article

References (59)

1. L. Prandtl, "The magnus effect and windpowered ships," Naturwiss. 13, 93 (1925). NATWAY 0028-1042
2. M. Gad-el-Hak, The MEMS Handbook (CRC, Boca Raton, Florida, 2002).
3. "One percent saving in world consumption of jet fuel is worth about 1.25 million dollars a day of direct operating costs (in 2002)," quoted from Ref..
4. M. D. Gunzburger, Flow Control (Springer, New York, 1995).
5. M. Gad-el-Hak, Flow Control: Passive, Active and Reactive Flow Management (Cambridge University Press, London, United Kingdom, 2000).
6. T. R. Bewley, "Flow control: New challenges for a new Renaissance," Prog. Aerosp. Sci. 37, 21 (2001). PAESD6 0376-0421 10.1016/S0376-0421(00)00016-6
7. J. A. Atwell, "Proper orthogonal decomposition for reduced-order control of partial differential equations," Ph.D. thesis, Virginia Tech, 2000.
8. A. J. Booker, J. E. Dennis, Jr., P. D. Frank, D. B. Serafini, V. Torczon, and M. W. Trosset, "A rigorous framework for optimization of expensive functions by surrogates," Struct. Optim. 17, 1 (1999). SOPTEQ 0934-4373 10.1007/s001580050031
9. A. C. Antoulas, D. C. Sorensen, and S. Gugercin, "A survey of model reduction methods for large-scale systems," Contemp. Math. 280, 193 (2001). CTMAEH 0271-4132
10. R. D. Prabhu, S. S. Collis, and Y. Chang, "The influence of control on proper orthogonal decomposition of wall-bounded turbulent flows," Phys. Fluids 13, 520 (2001). PHFLE6 1070-6631 10.1063/ 1.1333038
11. K. Ito and S. S. Ravindran, "A reduced-order method for simulation and control of fluid flows," J. Comput. Phys. 143, 403 (1998). JCTPAH 0021-9991 10.1006/jcph.1998.5943
12. J. Burkardt, Q. Du, M. D. Gunzburger, and H.-C. Lee, "Reduced order modeling of complex systems," Biennal Conferences on Numerical Analysis, 24-27 June 2002, University of Dundee, Scotland, UK.
13. M. Couplet, C. Basdevant, and P. Sagaut, "Calibrated reduced-order POD-Galerkin system for fluid flow modeling," J. Comput. Phys. 207, 192 (2005). JCTPAH 0021-9991
14. X. Ma and G. E. Karniadakis, "A low-dimensional model for simulating three-dimensional cylinder flow," J. Fluid Mech. 458, 181 (2002). JFLSA7 0022-1120 10.1017/S0022112002007991
15. B. R. Noack, K. Afanasiev, M. Morzýski, G. Tadmor, and F. Thiele, "A hierarchy of low-dimensional models for the transient and post-transient cylinder wake," J. Fluid Mech. 497, 335 (2003). JFLSA70022-112010.1017/S0022112003006694
16. B. Jorgensen, J. N. Sørensen, and M. Brøns, "Low-dimensional modeling of a driven cavity flow with two free parameters," Theor. Comput. Fluid Dyn. 16, 299 (2003). TCFDEP 0935-4964
17. C. H. K. Williamson, "Vortex dynamics in the cylinder wake," Annu. Rev. Fluid Mech. 28, 477 (1996). ARVFA3 0066-4189 10.1146/ annurev.fluid.28.1.477
18. F. Abergel and R. Temam, "On some control problems in fluid mechanics," Theor. Comput. Fluid Dyn. 1, 303 (1990). TCFDEP 0935-4964 10.1007/BF00271794
19. T. R. Bewley, "The emerging roles of model-based control theory in fluid mechanics," in Advances in Turbulence IX, Proceedings of the Ninth European Turbulence Conference, edited by I. P. Castro, P. E. Hancock, and T. G. Thomas (CIMNE, Barcelona, 2002).
20. J.-W. He, R. Glowinski, R. Metcalfe, A. Nordlander, and J. Périaux, "Active control and drag optimization for flow past a circular cylinder. Part 1. Oscillatory cylinder rotation," J. Comput. Phys. 163, 83 (2000). JCTPAH 0021-9991 10.1006/jcph.2000.6556
21. C. Homescu, I. M. Navon, and Z. Li, "Suppression of vortex shedding for flow around a circular cylinder using optimal control," Int. J. Numer. Methods Fluids 38, 43 (2002). IJNFDW 0271-2091 10.1002/fld.203
22. B. Protas and A. Styczek, "Optimal rotary control of the cylinder wake in the laminar regime," Phys. Fluids 14, 2073 (2002). PHFLE6 1070-6631 10.1063/1.1476671
23. An exception is the seminal work of Bewley (Ref.), where the optimal control theory is used to determine controls that reduce the drag of a turbulent flow in a three-dimensional plane channel simulated at Re = 180.
24. M. D. Gunzburger, "Adjoint equation-based methods for control problems in incompressible, viscous flows," Flow, Turbul. Combust. 65, 249 (2000). FTCOF9 1386-6184
25. N. Alexandrov, J. E. Dennis, Jr., R. M. Lewis, and V. Torczon, "A trust region framework for managing the use of approximation models in optimization," NASA CR-201745 ICASE Report No. 97-50, October 1997, p. 19.
26. P. T. Tokumaru and P. E. Dimotakis, "Rotary oscillatory control of a cylinder wake," J. Fluid Mech. 224, 77 (1991). JFLSA7 0022-1120
27. T be the mean temporal drag coefficient estimated over a finite horizon T, the relative mean drag reduction is defined as
28. B. Protas and J. E. Wesfreid, "Drag force in the open-loop control of the cylinder wake in the laminar regime," Phys. Fluids 14, 810 (2002). PHFLE6 1070-6631 10.1063/1.1432695
29. B. R. Noack and H. Eckelmann, "A global stability analysis of the steady and periodic cylinder wake," J. Fluid Mech. 270, 297 (1994). JFLSA7 0022-1120
30. D. Barkley and R. D. Henderson, "Three-dimensional Floquet stability analysis of the wake of a circular cylinder," J. Fluid Mech. 322, 215 (1996). JFLSA7 0022-1120
31. J. L. Lumley, "The structure of inhomogenous turbulent flows," in Atmospheric Turbulence and Radio Wave Propagation, edited by A. M. Yaglom and V. I. Tatarsky (Nauka, Moscow, 1967), pp. 166-178.
32. L. Cordier and M. Bergmann, "Proper orthogonal decomposition: An overview, Von Kármán Institute for Fluid Dynamics," Lecture Series 2002-04 on Post-Processing of Experimental and Numerical Data, 22-26 April 2002.
33. W. R. Graham, J. Peraire, and K. T. Tang, "Optimal control of vortex shedding using low order models. Part 1. Open-loop model development," Int. J. Numer. Methods Eng. 44, 945 (1999a). IJNMBH 0029-5981 10.1002/ (SICI)1097-0207(19990310)44:7<945::AID-NME537>3.0.CO;2-F
34. W. R. Graham, J. Peraire, and K. T. Tang, "Optimal control of vortex shedding using low order models. Part 2: Model-based control," Int. J. Numer. Methods Eng. 44, 973 (1999b). IJNMBH 0029-5981 10.1002/ (SICI)1097-0207(19990310)44:7<973::AID-NME538>3.3.CO;2-6
35. M. Bergmann, "Optimisation aérodynamique par réduction de modèle POD et contrôle optimal: Application au sillage laminaire d'un cylindre circulaire," Ph.D. thesis, Institut National Polytechnique de Lorraine, Nancy, France, 2004.
36. In the following, all variables are assumed to be nondimensionalized with respect to the cylinder diameter D and the oncoming flow U.
37. L. Cordier and M. Bergmann, "Two typical applications of POD: Coherent structures eduction and reduced order modelling, Von Kármán Institute for Fluid Dynamics," Lecture Series 2002-04 Post-Processing of Experimental and Numerical Data, 22-26 April 2002.
38. B. Protas and J. E. Wesfreid, "On the relation between the global modes and the spectra of drag and lift in periodic wake flows," C. R. Mec. 331, 49 (2003). CRMOC9 1631-0721
39. L. Sirovich, "Turbulence and the dynamics of coherent structures. Part 1. Coherent structures," Q. Appl. Math. 45, 561 (1987). QAMAAY 0033-569X
40. A. E. Deane, I. G. Kevrekidis, G. E. Karniadakis, and S. A. Orszag, "Low-dimensional models for complex geometry flows: Application to grooved channels and circular cylinders," Phys. Fluids A 10, 2337 (1991). PFADEB 0899-8213
41. G. S. Karamanos and G. E. Karniadakis, "A spectral vanishing viscosity method for large eddy simulations," J. Comput. Phys. 162, 22 (2000). JCTPAH 0021-9991
42. S. Sirisup and G. E. Karniadakis, "A spectral viscosity method for correcting the long-term behavior of POD model," J. Comput. Phys. 194, 92 (2004). JCTPAH 0021-9991
43. T. R. Bewley and S. Liu, "Optimal and robust control and estimation of linear paths to transition," J. Fluid Mech. 365, 305 (1998). JFLSA7 0022-1120 10.1017/S0022112098001281
44. M. D. Gunzburger, "Introduction into mathematical aspects of flow control and optimization, Von Kármán Institute for Fluid Dynamics," Lecture Series 1997-05 on Inverse Design and Optimization Methods, April 21-25 1997.
45. This was not true for the original optimization problem (25) involving J since the arguments α and γ were constrained to satisfy N(α, γ(t))=0.
46. T. R. Bewley, P. Moin, and R. Temam, "DNS-based predictive control of turbulence: An optimal benchmark for feedback algorithms," J. Fluid Mech. 447, 179 (2001). JFLSA7 0022-1120
47. J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research (Springer, Berlin, 1999).
48. The value of J is 11.85 at the beginning of the iterative process and 3.70 when convergence is obtained. To be used as a reference value, J is approximately equal to 10 for an uncontrolled flow.
49. S. J. Baek and H. J. Sung, "Numerical simulation of the flow behind a rotary oscillating circular cylinder," Phys. Fluids 10, 869 (1998). PHFLE6 1070-6631 10.1063/1.869610
50. A. R. Conn, N. I. M. Gould, and P. L. Toint, Trust-Region Methods (SIAM, Philadelphia, 2000).
51. Contrary to the Navier-Stokes equations, the adjoint and optimality equations are linear in the adjoint variables, and hence, the numerical costs necessary to solve these equations are usually lower than for the Navier-Stokes equations. However, since the adjoint equations are defined in reverse time, the adjoint fields may grow exponentially when they are calculated, requiring the use of regularization methods as recently discussed in Ref. 56. In this study, no exponential growth was noticed with the adjoint equations (27).
52. M. Fahl, "Trust-region methods for flow control based on reduced-order modeling," Ph.D. thesis, Trier University, 2000.
53. The trust-region mechanism gives a measure of how well the approximation model is predicting improvement in the high-fidelity model, and thus, suggests criteria for automatically changing or improving the reduced model when poor or incorrect prediction of improvement is obtained. Fahl presents in Ref. 52 an algorithm that implements the combination of PODbased reduced-order modeling and trust-region methods, the trust-region proper orthogonal decomposition (TRPOD).
54. B. R. Noack, P. Papas, and P. A. Monkewitz, "The need for a pressure-term representation in empirical galerkin models of incompressible shear-flows," J. Fluid Mech. 523, 339 (2005). JFLSA7 0022-1120
55. S. S. Collis, R. D. Joslin, A. Seifert, and V. Theofilis, "Issues in active flow control: Theory, control, simulation and experiment," Prog. Aerosp. Sci. 40, 237 (2004). PAESD6 0376-0421
56. B. Protas, T. R. Bewley, and G. Hagen, "A comprehensive framework for the regularization of adjoint analysis in multiscale PDE systems," J. Comput. Phys. 195, 49 (2004). JCTPAH 0021-9991
57. M. Braza, P. Chassaing, and H. Ha Minh, "Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder," J. Fluid Mech. 165, 79 (1986). JFLSA7 0022-1120
58. R. D. Henderson, "Nonlinear dynamics and pattern formation in turbulent wake transition," J. Fluid Mech. 352, 65 (1997). JFLSA7 0022-1120 10.1017/S0022112097007465
59. M. D. Gunzburger, "Reduced-order modeling, data compression and the design of experiments," Second DOE Workshop on Multiscale Mathematics, 20-22 July 2004, Broomfield, CO.