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A nonlinear dynamic model for unsteady separated flow control and its mechanism analysis

  • Guoping Huang (a1), Weiyu Lu (a1), Jianfeng Zhu (a1), Xin Fu (a1) and Jinchun Wang (a1)...


In the analysis of the interaction between external periodic excitation and unsteady separated flow in controlling the flow separation, a new nonlinear approximate model has been established. This model is used to describe the typical chaotic and coherent characteristics of a separated flow such as small- or large-scale vortices, the injection, and the dissipation of the kinetic energy based on a simulation of a simplified cross-direction motion of free shear flows. This study presents an appropriate treatment to simulate the external periodic excitation and uses the maximum Lyapunov exponent to evaluate the degree of flow ordering in the different control states. The results of the nonlinear model are compared with experimental and numerical results, showing that the nonlinear model could be used to effectively explain the behaviours of chaotic flows and investigate the rules for controlling separated flows. In addition, as shown in the nonlinear approximate model, the self-synchronization of unsteady flow separation and periodic excitation has been analysed. Initially, the research provided an explanation of the self-synchronization mechanism, which cites that the effects of the separated flow control are independent of the phase difference between the periodic excitation and the unsteady flow. The characteristics of unsteady separated flow control have also been presented in this model, where the corresponding large eddy simulation (LES) was used for separated flows in a curved diffuser. The proper orthogonal decomposition (POD) method was used to analyse the difference between separated vortical structures with or without periodic excitation. The results showed that the model and the simulation had the same mechanism of flow control as for the separated flows. The periodic excitation transforms the original chaotic flow into a relatively ordered flow and decreases the magnitude of the chaotic unstable vortices, rather than completely eliminating the vortices, while flow mixing is reduced, inducing less energy loss.


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Akhtar, I., Marzouk, O. A. & Nayfeh, A. H. 2009a A van der Pol–Duffing oscillator model of hydrodynamic forces on canonical structures. J. Comput. Nonlinear Dynam. 4, 17241732.
Akhtar, I., Nayfeh, A. H. & Ribbens, C. J. 2009b On the stability and extension of reduced-order Galerkin models in incompressible flows. A numerical study of vortex shedding. Theor. Comput. Fluid Dyn. 23, 213237.
Albarè, D. P. & Provansal, M. 1995 Quasi-periodic cylinder wakes and the Ginzburg–Landau model. J. Fluid Mech. 291, 191222.
Alobaidi, G., Smith, C. J. & Mallier, R. 2014 Waves on a Stuart vortex. Appl. Maths Comput. 227, 370383.
Amitay, M. & Cannelle, F. 2006 Evolution of finite span synthetic jets. Phys. Fluids 18, 645666.
Aref, H. 1987 Stirring by chaotic advection. J. Fluid Mech. 143, 121.
Aubry, N., Holmes, P. & Lumley, J. L. 1988 The dynamic of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115173.
Benettin, G., Galgani, L., Giorgilli, A. & Strelcyn, J. M. 1980 Lyapunov characteristic exponents for smooth dynamical systems; a method for computing all of them: Part I: theory, Part II. Numer. Applications Meccanica 15, 930.
Berkooz, G., Holmes, P. & Lumley, J. L. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 53, 321575.
Chatterjee, A. 2000 An introduction to the proper orthogonal decomposition. Curr. Sci. 78, 809817.
Chen, W., Liu, Y. & Hu, H.2014 Suppression of vortex shedding from a circular cylinder by using a traveling wave wall. AIAA Paper 2014-0399.
Collis, S. S., Joslin, R. D., Seifert, A. & Theofilis, V. 2004 Issues in active flow control: theory, control, simulation, and experiment. Prog. Aerosp. Sci. 40, 237289.
Dano, B. & Liburdy, J.2006 Vortical structure of a 45 degree inclined pulsed jet in crossflow. AIAA Paper 2006-3543.
Dimotakis, P.1989 Turbulent free shear layer mixing. AIAA Report 1989-0262.
Dušek, J., Fraunié, P. & Gal, P. L. 1994 Local analysis of the onset of instability in shear flows. Phys. Fluids 6, 172186.
Dyke, M. V. 1982 An Album of Fluid Motion. The Parabolic Press.
Feng, L. H., Wang, J. J. & Pan, C. 2011 Proper orthogonal decomposition analysis of vortex dynamics of a circular cylinder under synthetic jet control. Phys. Fluids 23, 014106.
Gal, P. L., Nadim, A. & Thompson, M. 2001 Hysteresis in the forced Stuart–Landau equation: application to vortex shedding from an oscillating cylinder. J. Fluids Struct. 15, 445457.
Gaster, M. & Jordinson, R. 1975 On the eigenvalues of the Orr–Sommerfeld equation. J. Fluid Mech. 72, 121133.
Glezer, A. & Amitay, M. 2002 Synthetic jets. Annu. Rev. Fluid Mech. 34, 503529.
Gmelin, C., Steger, M., Wassen, E., Thiele, F., Huppertz, A. & Swoboda, M.2010 Unsteady RANS simulations on flow control in a compressor cascade using pulsed jets at the blade. AIAA Paper 2010-4588.
Greenblatt, D. & Wygnanski, I. J. 2000 The control of flow separation by periodic excitation. Prog. Aerosp. Sci. 36, 487545.
Grosch, C. E. & Salwen, H. 1978 The continuous spectrum of the Orr–Sommerfeld equation. Part 1. The spectrum and the eigenfunctions. J. Fluid Mech. 87, 3354.
Gross, A.2005 Simulation of active flow control for a low pressure turbine blade cascade. AIAA Paper 2005-869.
Gursul, I., Cleaver, D. J. & Wang, Z. 2014 Control of low Reynolds number flows by means of fluid–structure interactions. Prog. Aerosp. Sci. 64, 1755.
Hassan, A.2004 Oscillatory and Pulsed Jets for Improved Airfoil Aerodynamics – A Numerical Simulation. AIAA Paper 2004-0227.
Hecklau, M., Wiederhold, O., Zander, V., King, R., Nitsche, W. & Swoboda, M. 2011 Active separation control with pulsed jets in a critically loaded compressor cascade. AIAA J. 49, 17291739.
Ho, C. M. & Huerre, P. 2003 Perturbed free shear layers. Annu. Rev. Fluid Mech. 16, 365422.
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.
Kim, J. 2009 Control of turbulent boundary layers. Phys. Fluids 15, 10931105.
Kim, J. & Bewley, T. R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383417.
Kovacic, I. & Brennan, M. J. 2011 The Duffing Equation: Nonlinear Oscillators and their Behaviour. Wiley.
Ku, W. L., Girvan, M. & Ott, E. 2014 Dynamical transitions in large systems of mean field-coupled Landau–Stuart oscillators: extensive chaos and clumped states. Physics 25, 614617.
Lepicovsky, J., Ahuja, K., Brown, W. H. & Morris, P. J. 1986 Acoustic control of free jet mixing. J. Propul. Power 2, 323330.
Lima, R. & Pettini, M. 1990 Suppression of chaos by resonant parametric perturbations. Phys. Rev. A 41, 726733.
Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130141.
Lumley, J. L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Wave Propagation, pp. 166176. Nauka.
Miles, J. W. 1962 A note on the inviscid Orr–Sommerfeld equation. J. Fluid Mech. 13, 427432.
Monir, H. E., Tadjfar, M. & Bakhtian, A. 2013 Tangential synthetic jets for separation control. J. Fluids Struct. 45, 5065.
Nayfeh, A. H., Marzouk, O. A., Arafat, H. N. & Akhtar, I. 2005 Modeling the Transient and Steady-State Flow Over a Stationary Cylinder. (2005 ASME Design Engineering Technical Conferences) , pp. 15131523. ASME.
Nishioka, M., Asai, M. & Yoshida, S. 1987 Control of flow separation by acoustic excitation. AIAA J. 28, 19091915.
Noack, B. R., Afanasiev, K., Morzynski, 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.
Olinger, D. J. 1993 A low-dimensional model for chaos in open fluid flows. Phys. Fluids A 5, 19471951.
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689703.
Roberts, F. A.1985 Effects of Periodic Forcing on Mixing in Turbulent Shear Layers and Wakes. AIAA Report 85-0570.
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.
Saric, W. S., Reed, H. L. & Kerschen, E. J. 2002 Boundary-layer receptivity to freestream disturbances. Annu. Rev. Fluid Mech. 34, 291319.
Schmid, P. J. 2008 Dynamic Mode Decomposition of numerical and experimental data. J. Fluid Mech. 656, 528.
Schmid, P. J., Li, L., Juniper, M. P. & Pust, O. 2010 Applications of the dynamic mode decomposition. Theor. Comput. Fluid Dyn. 25, 249259.
Seifert, A., Greenblatt, D. & Wygnanski, I. J. 2004 Active separation control: an overview of Reynolds and Mach numbers effects. Aerosp. Sci. Technol. 8, 569582.
Sinha, S.1999 Active Flexible Walls for Efficient Aerodynamic Flow Separation Control. AIAA Paper 99-3123.
Sirovich, L. & Kirby, M. 1987 Low-dimensional procedure for the characterization of human faces. J. Opt. Soc. Am. A 4, 519524.
Skamnakis, D. & Papailiou, K. 2005 Flow stability analysis and excitation using pulsating jets. Comptes Rendus Mecanique 333, 628635.
Skop, R. A. & Balasubramanian, S. A. 1995 A nonlinear oscillator model for vortex shedding from a forced cylinder part 1: uniform flow and model parameters. Intl J. Offshore Polar Engng 5, 4.
Smith, B. L. & Glezer, A. 1998 The formation and evolution of synthetic jets. Phys. Fluids 10, 22812297.
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech. 4, 121.
Stuart, J. T. 1967 On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29, 417440.
Sun, L. 2002 Numerical studies on excited Stuart–Landau model. Chinese J. Appl. Mech. 19, 6164.
Tan, B. T., Damodaran, M. & Willcox, K. E. 2004 Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition. AIAA J. 42, 15051516.
Theofilis, V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 26, 249315.
Thompson, M. C. & Gal, P. L. 2004 The Stuart–Landau model applied to wake transition revisited. Eur. J. Mech. (B/Fluids) 23, 219228.
Wang, L., Li, L. & Fu, S. 2014 Numerical investigation of active flow control on a pitching NACA 0015 airfoil using detached-eddy simulation. Procedia Engng 79, 4954.
Wu, C., Xie, Y. & Wu, J. 2003 ‘Fluid roller bearing’ effect and flow control. Acta Mechanica Sin. 19, 476484.
Wu, C. J., Wang, L. & Wu, J. Z. 2007 Suppression of the von Kármán vortex street behind a circular cylinder by a travelling wave generated by a flexible surface. J. Fluid Mech. 574, 365391.
Wu, J. Z., Ma, H. Y. & Zhou, M. D. 2006 Vorticity and Vortex Dynamics, vol. 11, pp. 2135. Springer.
Wu, X. H., Wu, J. Z. & Wu, J. M.1991 Streaming effect of wall oscillation to boundary layer separation. AIAA Paper 91-0541.
Wygnanski, I. J. & Petersen, R. A. 1987 Coherent motion in excited free shear flows. AIAA J. 25, 201213.
Wygnanski, I.2004 The variables affecting the control of separation by periodic excitation. AIAA Paper 2004-2505.
Ye, T., Cattafesta, L. & Mittal, R.2006 Adaptive control of separated flow. AIAA Paper 2006-1401.
You, D. & Moin, P. 2006 Large-eddy simulation of flow separation over an airfoil with synthetic jet control. 59th Annual Meeting of the APS Division of Fluid Dynamics. American Physical Society.
Zheng, X. Q., Zhou, X. B. & Zhou, S. 2005 Investigation on a type of flow control to weaken unsteady separated flows by unsteady excitation in axial flow compressors. J. Turbomach. 127, 489496.
Zhu, J., Huang, G., Fu, X., Fu, Y. & Yu, H. 2013 Use of POD method to elucidate the physics of unsteady micro-pulsed-jet flow for boundary layer flow separation control. In ASME Turbo Expo 2013: Turbine Technical Conference and Exposition, GT2013-95266.
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