×

Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models. (English) Zbl 1388.76073

Summary: Optimal control theory is used to minimize the total mean drag for a circular cylinder wake flow in the laminar regime (\(R\)e=200). The control parameters are the amplitude and the frequency of the time-harmonic cylinder rotation. In order to reduce the size of the discretized optimality system, a proper orthogonal decomposition (POD) reduced-order model (ROM) is derived to be used as state equation. We then propose to employ the trust-region proper orthogonal decomposition (TRPOD) approach, originally introduced by M. Fahl [Trust-region methods for flow control based on reduced order modeling, Dissertation, Univ. Trier, Trier (2000; Zbl 1151.93001)], to update the reduced-order models during the optimization process. A lot of computational work is saved because the optimization process is now based only on low-fidelity models. A particular care was taken to derive a POD ROM for the pressure and velocity fields with an appropriate balance between model accuracy and robustness. The key enablers are the extension of the POD basis functions to the pressure data, the use of calibration methods for the POD ROM and the addition in the POD expansion of several non-equilibrium modes to describe various operating conditions. When the TRPOD algorithm is applied to the wake flow configuration, this approach converges to the minimum predicted by an open-loop control approach and leads to a relative mean drag reduction of 30% at reduced cost.

MSC:

76D55 Flow control and optimization for incompressible viscous fluids
76D25 Wakes and jets
76M25 Other numerical methods (fluid mechanics) (MSC2010)

Citations:

Zbl 1151.93001
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Gunzburger, M. D., Flow Control (1995), Springer: Springer New York · Zbl 0875.76490
[2] Gunzburger, M. D., Introduction into mathematical aspects of flow control and optimization, (Lecture Series 1997-05 on Inverse Design and Optimization Methods (1997), Von Kármán Institute for Fluid Dynamics) · Zbl 0274.76024
[3] Gunzburger, M. D., Adjoint equation-based methods for control problems in incompressible, viscous flows, Flow, Turbul. Combust., 65, 249-272 (2000) · Zbl 0996.76024
[4] Booker, A. J.; Dennis, J. E.; Frank, P. D.; Serafini, D. B.; Torczon, V.; Trosset, M. W., A rigorous framework for optimization of expensive functions by surrogates, Struct. Optim., 17, 1, 1-13 (1999)
[5] Antoulas, A. C., Approximation of Large-Scale Dynamical Systems (2005), SIAM · Zbl 1112.93002
[6] Ito, K.; Ravindran, S. S., A reduced-order method for simulation and control of fluid flows, J. Comp. Phys., 143, 403-425 (1998) · Zbl 0936.76031
[7] J.L. Lumley, Atmospheric turbulence and wave propagation, in: A.M. Yaglom, V.I. Tatarski (Eds.), The Structure of Inhomogeneous Turbulence, 1967, pp. 166-178.; J.L. Lumley, Atmospheric turbulence and wave propagation, in: A.M. Yaglom, V.I. Tatarski (Eds.), The Structure of Inhomogeneous Turbulence, 1967, pp. 166-178.
[8] Sirovich, L., Turbulence and the dynamics of coherent structures, Quarter. Appl Math., XLV, 3, 561-590 (1987) · Zbl 0676.76047
[9] K.E. Willcox, Reduced-order aerodynamic models for aeroelastic control of turbomachines, Ph.D. Thesis, Massachusetts Institute of Technology, 2000.; K.E. Willcox, Reduced-order aerodynamic models for aeroelastic control of turbomachines, Ph.D. Thesis, Massachusetts Institute of Technology, 2000.
[10] J. Burkardt, M.D. Gunzburger, H.-C. Lee, Centroidal Voronoi Tessellation-Based Reduced-Order Modeling of Complex Systems, Tech. rep., Florida State University, 2004.; J. Burkardt, M.D. Gunzburger, H.-C. Lee, Centroidal Voronoi Tessellation-Based Reduced-Order Modeling of Complex Systems, Tech. rep., Florida State University, 2004. · Zbl 1111.65084
[11] Rowley, C. W., Model reduction for fluids, using balanced proper orthogonal decomposition, Int. J. Bifurcat. Chaos, 15, 3, 997-1013 (2005) · Zbl 1140.76443
[12] Ma, X.; Karniadakis, G. E., A low-dimensional model for simulating three-dimensional cylinder flow, J. Fluid Mech., 458, 181-190 (2002) · Zbl 1001.76043
[13] Noack, B. R.; Afanasiev, K.; Morzyński, M.; Tadmor, G.; Thiele, F., A hierarchy of low-dimensional models for the transient and post-transient cylinder wake, J. Fluid Mech., 497, 335-363 (2003) · Zbl 1067.76033
[14] B.R. Noack, G. Tadmor, M. Morzyński, Low-dimensional models for feedback flow control. Part I: empirical Galerkin models, in: 2nd AIAA Flow Control Conference, Portland, Oregon, USA, June 28-July 1, 2004, AIAA-Paper 2004-2408 (invited contribution).; B.R. Noack, G. Tadmor, M. Morzyński, Low-dimensional models for feedback flow control. Part I: empirical Galerkin models, in: 2nd AIAA Flow Control Conference, Portland, Oregon, USA, June 28-July 1, 2004, AIAA-Paper 2004-2408 (invited contribution).
[15] M. Bergmann, L. Cordier, J.-P. Brancher, Optimal rotary control of the cylinder wake using POD reduced order model, Phys. Fluids 17 (9) (2005) 097101:1-097101:21.; M. Bergmann, L. Cordier, J.-P. Brancher, Optimal rotary control of the cylinder wake using POD reduced order model, Phys. Fluids 17 (9) (2005) 097101:1-097101:21.
[16] Graham, W. R.; Peraire, J.; Tang, K. T., Optimal control of vortex shedding using low order models. Part 1. Open-loop model development, Int. J. Numer. Meth. Eng., 44, 7, 945-972 (1999) · Zbl 0955.76026
[17] Ravindran, S. S., Reduced-order adaptive controllers for fluid flows using POD, J. Sci. Comput., 15, 4, 457-478 (2000) · Zbl 1048.76016
[18] Ravindran, S. S., Adaptive reduced-order controllers for a thermal flow system, SIAM J. Sci. Comput., 23, 6, 1925-1943 (2002) · Zbl 1026.76015
[19] M. Hinze, S. Volkwein, Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control, Tech. rep., Preprint SFB609, Technische UniversitSt Dresden, 2004.; M. Hinze, S. Volkwein, Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control, Tech. rep., Preprint SFB609, Technische UniversitSt Dresden, 2004. · Zbl 1079.65533
[20] Williamson, C. H.K., Vortex dynamics in the cylinder wake, Ann. Rev. Fluid. Mech., 28, 477-539 (1996) · Zbl 0899.76129
[21] Tokumaru, P. T.; Dimotakis, P. E., Rotary oscillatory control of a cylinder wake, J. Fluid Mech., 224, 77-90 (1991)
[22] Lu, X.-Y.; Sato, J., A numerical study of flow past a rotationally oscillating circular cylinder, J. Fluids Struct., 10, 829-849 (1996)
[23] Chou, M. H., Synchronization of vortex shedding from a cylinder under rotary oscillation, Comput. Fluids, 26, 755-774 (1997) · Zbl 0909.76070
[24] Baek, S. J.; Sung, H. J., Numerical simulation of the flow behind a rotary oscillating circular cylinder, Phys. Fluids, 10, 4, 869-876 (1998)
[25] Mahfouz, F. M.; Badr, H. M., Flow structure in the wake of a rotationally oscillating cylinder, J. Fluids Eng., 122, 2, 290-301 (2000) · Zbl 0979.76079
[26] Baek, S.-J.; Sung, H. J., Quasi-periodicity in the wake of a rotationally oscillating cylinder, J. Fluid Mech., 408, 275-300 (2000) · Zbl 0967.76029
[27] Cheng, M.; Chew, Y. T.; Luo, S. C., Numerical investigation of a rotationally oscillating cylinder in mean flow, J. Fluids Struct., 15, 981-1007 (2001)
[28] Cheng, M.; Liu, G. R.; Lam, K. Y., Numerical simulation of flow past a rotationally oscillating cylinder, Comput. Fluids, 30, 365-392 (2001) · Zbl 1052.76051
[29] Choi, S.; Choi, H.; Kang, S., Characteristics of flow over a rotationally oscillating cylinder at low Reynolds number, Phys. Fluids, 14, 8, 2767-2777 (2002) · Zbl 1185.76086
[30] Protas, B.; Wesfreid, J.-E., Drag force in the open-loop control of the cylinder wake in the laminar regime, Phys. Fluids, 14, 2, 810-826 (2002) · Zbl 1184.76437
[31] Fujisawa, N.; Ikemoto, K.; Nagaja, K., Vortex shedding resonnance from a rotationally oscillating cylinder, J. Fluids Struct., 12, 1041-1053 (1998)
[32] Fujisawa, N.; Kawaji, Y.; Ikemoto, K., Feedback control of vortex shedding from a circular cylinder by rotational oscillations, J. Fluids Struct., 15, 23-37 (2001)
[33] S. Goujon-Durand, J.-E. Wesfreid, P. Jenffer, Contrôle actif du sillage autour d’un cylindre oscillant, in: 15th French Congress of Mechanics, Nancy, September 3-7, 2001.; S. Goujon-Durand, J.-E. Wesfreid, P. Jenffer, Contrôle actif du sillage autour d’un cylindre oscillant, in: 15th French Congress of Mechanics, Nancy, September 3-7, 2001.
[34] Thiria, B.; Goujon-Durand, S.; Wesfreid, J.-E., The wake of a cylinder performing rotary oscillations, J. Fluid Mech., 560, 123-147 (2006) · Zbl 1122.76003
[35] Abergel, F.; Temam, R., On some control problems in fluid mechanics, Theoret. Comput. Fluid Dyn., 1, 303 (1990) · Zbl 0708.76106
[36] Bewley, T. R., Flow control: new challenges for a new Renaissance, Prog. Aerosp. Sci., 37, 21-58 (2001)
[37] T.R. Bewley, The emerging roles of model-based control theory in fluid mechanics, in: Advances in Turbulence IX, 2002, ninth European Turbulence Conference.; T.R. Bewley, The emerging roles of model-based control theory in fluid mechanics, in: Advances in Turbulence IX, 2002, ninth European Turbulence Conference.
[38] He, J.-W.; Glowinski, R.; Metcalfe, R.; Nordlander, A.; PTriaux, J., Active control and drag optimization for flow past a circular cylinder. Part 1. Oscillatory cylinder rotation, J. Comp. Phys., 163, 83-117 (2000) · Zbl 0977.76021
[39] Homescu, C.; Navon, I. M.; Li, Z., Suppression of vortex shedding for flow around a circular cylinder using optimal control, Int. J. Numer. Meth. Fluids, 38, 43-69 (2002) · Zbl 1007.76019
[40] Protas, B.; Styczek, A., Optimal rotary control of the cylinder wake in the laminar regime, Phys. Fluids, 14, 7, 2073-2087 (2002) · Zbl 1185.76304
[41] Bewley, T. R.; Moin, P.; Temam, R., DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms, J. Fluid Mech., 447, 179-225 (2001) · Zbl 1036.76027
[42] Y. Chang, Approximate models for optimal control of turbulent channel flow, Ph.D. thesis, Rice University, 2000.; Y. Chang, Approximate models for optimal control of turbulent channel flow, Ph.D. thesis, Rice University, 2000.
[43] Wei, M.; Freund, J. B., A noise-controlled free shear flow, J. Fluid Mech., 546, 123-152 (2006) · Zbl 1222.76084
[44] N. Alexandrov, J.E. Dennis Jr., R.M. Lewis, V. Torczon, A trust region framework for managing the use of approximation models in optimization, Icase report 97-50.; N. Alexandrov, J.E. Dennis Jr., R.M. Lewis, V. Torczon, A trust region framework for managing the use of approximation models in optimization, Icase report 97-50.
[45] M. Bergmann, L. Cordier, J.-P. Brancher, On the power used to control the circular cylinder drag by rotary oscillations, Phys. Fluids 18 (8) (2006) 088103:1-088103:4.; M. Bergmann, L. Cordier, J.-P. Brancher, On the power used to control the circular cylinder drag by rotary oscillations, Phys. Fluids 18 (8) (2006) 088103:1-088103:4.
[46] Noack, B. R.; Eckelmann, H., A global stability analysis of the steady and periodic cylinder wake, J. Fluid Mech., 270, 297-330 (1994) · Zbl 0813.76025
[47] Barkley, D.; Henderson, R. D., Three-dimensional Floquet stability analysis of the wake of a circular cylinder, J. Fluid Mech., 322, 215-241 (1996) · Zbl 0882.76028
[48] Cordier, L.; Bergmann, M., Proper orthogonal decomposition: an overview, (Lecture series 2002-04 on post-processing of experimental and numerical data (2002), Von Kármán Institute for Fluid Dynamics)
[49] Graham, W. R.; Peraire, J.; Tang, K. T., Optimal control of vortex shedding using low order models. Part 2: model-based control, Int. J. Numer. Meth. Eng., 44, 7, 973-990 (1999) · Zbl 0955.76026
[50] M. Fahl, Trust-region methods for flow control based on reduced order modeling, Ph.D. Thesis, Trier University, 2000.; M. Fahl, Trust-region methods for flow control based on reduced order modeling, Ph.D. Thesis, Trier University, 2000.
[51] J. Nocedal, S.J. Wright, Numerical Optimization, Springer Series in Operations Research, 1999.; J. Nocedal, S.J. Wright, Numerical Optimization, Springer Series in Operations Research, 1999.
[52] M. Bergmann, L. Cordier, Control of the circular cylinder wake by Trust-Region methods and POD Reduced-Order Models, Tech. rep., INRIA.; M. Bergmann, L. Cordier, Control of the circular cylinder wake by Trust-Region methods and POD Reduced-Order Models, Tech. rep., INRIA. · Zbl 1388.76073
[53] E. Arian, M. Fahl, E.W. Sachs, Trust-Region Proper Orthogonal Decomposition for Flow Control, Icase report 2000-25.; E. Arian, M. Fahl, E.W. Sachs, Trust-Region Proper Orthogonal Decomposition for Flow Control, Icase report 2000-25.
[54] Conn, A. R.; Gould, N. I.M.; Toint, P. L., Trust-Region Methods (2000), SIAM: SIAM Philadelphia · Zbl 0958.65071
[55] Toint, P. L., Global convergence of a class of trust-region methods for nonconvex minimization in Hilbert space, IMA J. Numer. Anal., 8, 2, 231-252 (1988) · Zbl 0698.65043
[56] 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.; 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.
[57] Noack, B. R.; Papas, P.; Monkewitz, P. A., The need for a pressure-term representation in empirical galerkin models of incompressible shear-flows, J. Fluid Mech., 523, 339-365 (2005) · Zbl 1065.76102
[58] Galletti, B.; Bruneau, C.-H.; Zannetti, L.; Iollo, A., Low-order modelling of laminar flow regimes past a confined square cylinder, J. Fluid Mech., 503, 161-170 (2004) · Zbl 1116.76344
[59] C. Kasnakoglu, Reduced order modeling, nonlinear analysis and control methods for flow control problems, Ph.D. Thesis, Ohio State University, 2007.; C. Kasnakoglu, Reduced order modeling, nonlinear analysis and control methods for flow control problems, Ph.D. Thesis, Ohio State University, 2007.
[60] Gunes, H.; Sirisup, S.; Karniadakis, G. E., Gappy data: To Krig or not to Krig?, J. Comp. Phys., 212, 358-382 (2006) · Zbl 1216.76062
[61] Willcox, K., Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition, Comput. Fluids, 35, 2, 208-226 (2006) · Zbl 1160.76394
[62] Rempfer, D.; Fasel, H. F., Evolution of three-dimensional coherent structures in a flat-plate boundary layer, J. Fluid Mech., 260, 351-375 (1994)
[63] Couplet, M.; Basdevant, C.; Sagaut, P., Calibrated reduced-order POD-Galerkin system for fluid flow modelling, J. Comp. Phys., 207, 192-220 (2005) · Zbl 1177.76283
[64] Galletti, B.; Bottaro, A.; Bruneau, C.-H.; Iollo, A., Accurate model reduction of transient and forced wakes, Eur. J. Mech. B/Fluids, 26, 3, 354-366 (2007) · Zbl 1150.76019
[65] Favier, J.; Cordier, L.; Kourta, A., Accurate POD reduced-order models of separated flows, Phys. Fluids, 8, 3, 259-265 (2007)
[66] Deane, A. E.; Kevrekidis, I. G.; Karniadakis, G. E.; Orszag, S. A., Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders, Phys. Fluids, 3, 10, 2337-2354 (1991) · Zbl 0746.76021
[67] Afanasiev, K.; Hinze, M., Adaptive control of a wake flow using proper orthogonal decomposition, (Shape Optimization and Optimal Design. Shape Optimization and Optimal Design, Lecture Notes in Pure and Applied Mathematics, vol. 216 (2001), Marcel Dekker) · Zbl 1013.76028
[68] Protas, B.; Wesfreid, J.-E., On the relation between the global modes and the spectra of drag and lift in periodic wake flows, C.R. MTcanique, 331, 49-54 (2003) · Zbl 1178.76128
[69] C. Homescu, L.R. Petzold, R. Serban, Error Estimation for Reduced Order Models of Dynamical Systems, Tech. rep., Lawrence Livermore National Laboratory, 2003.; C. Homescu, L.R. Petzold, R. Serban, Error Estimation for Reduced Order Models of Dynamical Systems, Tech. rep., Lawrence Livermore National Laboratory, 2003. · Zbl 1096.65076
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.