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Using adjoint-based optimization to study kinematics and deformation of flapping wings

Published online by Cambridge University Press:  21 June 2016

Min Xu
Affiliation:
Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA
Mingjun Wei*
Affiliation:
Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA
*
Email address for correspondence: mjwei@nmsu.edu

Abstract

The study of flapping-wing aerodynamics faces a large control space with different wing kinematics and deformation. The adjoint-based approach, by solving an inverse problem to obtain simultaneously the sensitivity with respect to all control parameters, has a computational cost independent of the number of control parameters and becomes an efficient tool for the study of problems with a large control space. However, the adjoint equation is typically formulated in a fixed fluid domain. In a continuous formulation, a moving boundary or morphing domain results in inconsistency in the definition of an arbitrary perturbation at the boundary, which leads to ambiguousness and difficulty in the adjoint formulation if control parameters are related to boundary changes (e.g. the control of wing kinematics and dynamic deformation). The unsteady mapping function, as a traditional way to deal with moving boundaries, can in principle be a remedy for this situation. However, the derivation is often too complex to be feasible, even for simple problems. Part of the complexity comes from the unnecessary mapping of the interior mesh, while only mapping of the boundary is needed here. Non-cylindrical calculus, on the other hand, provides a boundary mapping and considers the rest of domain as an arbitrary extension from the boundary. Using non-cylindrical calculus to handle moving boundaries makes the derivation of the adjoint formulation much easier and also provides a simpler final formulation. The new adjoint-based optimization approach is validated for accuracy and efficiency by a well-defined case where a rigid plate plunges normally to an incoming flow. Then, the approach is applied for the optimization of drag reduction and propulsive efficiency of first a rigid plate and then a flexible plate which both flap with plunging and pitching motions against an incoming flow. For the rigid plate, the phase delay between pitching and plunging is the control and considered as both a constant (i.e. a single parameter) and a time-varying function (i.e. multiple parameters). The comparison between its arbitrary initial status and the two optimal solutions (with a single parameter or multiple parameters) reveals the mechanism and control strategy to reach the maximum thrust performance or propulsive efficiency. Essentially, the control is trying to benefit from both lift-induced thrust and viscous drag (by reducing it), and the viscous drag plays a dominant role in the optimization of efficiency. For the flexible plate, the control includes the amplitude and phase delay of the pitching motion and the leading eigenmodes to characterize the deformation. It is clear that flexibility brings about substantial improvement in both thrust performance and propulsive efficiency. Finally, the adjoint-based approach is extended to a three-dimensional study of a rectangular plate in hovering motion for lift performance. Both rigid and flexible cases are considered. The adjoint-based algorithm finds an optimal hovering motion with advanced rotation which has a large leading-edge vortex and strong downwash for lift benefit, and the introduction of flexibility enhances the wake capturing mechanism and generates a stronger downwash to push the lift coefficient higher.

Type
Papers
Copyright
© 2016 Cambridge University Press 

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References

Andersen, A., Pesavento, U. & Wang, Z. 2005 Unsteady aerodynamics of fluttering and tumbling plates. J. Fluid Mech. 541, 6590.Google Scholar
Anderson, J. M., Streitlien, K., Barrett, D. S. & Triantafyllou, M. S. 1998 Oscillating foils of high propulsive efficiency. J. Fluid Mech. 360, 4172.Google Scholar
Berman, G. J. & Wang, Z. J. 2007 Energy-minimizing kinematics in hovering insect flight. J. Fluid Mech. 582, 153168.Google Scholar
Bewley, T. R., Moin, P. & Temam, R. 2001 DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms. J. Fluid Mech. 447, 179225.Google Scholar
Collis, S. S., Ghayour, K., Heinkenschloss, M., Ulbrich, M. & Ulbrich, S.2001 Towards adjoint-based methods for aeroacoustic control. AIAA Paper 2001-0821.Google Scholar
Culbreth, M., Allaneau, Y. & Jameson, A.2011 High-fidelity optimization of flapping airfoils and wings. AIAA Paper 2011-3521.Google Scholar
Dong, H. & Liang, Z.2010 Effects of ipsilateral wing-wing interactions on aerodynamic performance of flapping wings. AIAA Paper 2010-71.Google Scholar
Dong, H., Mittal, R. & Najjar, F. 2006 Wake topology and hydrodynamic performance of low aspect-ratio flapping foils. J. Fluid Mech. 566, 309343.Google Scholar
Eldredge, J. D., Toomey, J. & Medina, A. 2010 On the roles of chord-wise flexibility in a flapping wing with hovering kinematics. J. Fluid Mech. 659, 94115.Google Scholar
Ghommem, M., Hajj, M. R., Mook, D. T., Stanford, B. K., Beran, P. S., Snyder, R. D. & Watson, L. T. 2012 Global optimization of actively morphing flapping wings. J. Fluids Struct. 33, 210228.Google Scholar
Guilmineau, E. & Queutey, P. 2002 A numerical simulation of vortex shedding from an oscillating circular cylinder. J. Fluids Struct. 16 (6), 773794.Google Scholar
Hirt, C., Amsden, A. A. & Cook, J. 1974 An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 14 (3), 227253.Google Scholar
Hunt, J. C. R., Way, A. & Moin, P.1988 Eddies, stream, and convergence zones in turbulent flows. Tech. Rep. CTR-S88. Center for Turbulence Research.Google Scholar
Jameson, A. 2003 Aerodynamic shape optimization using the adjoint method. In VKI Lecture Series on Aerodynamic Drag Prediction and Reduction, von Karman Institute of Fluid Dynamics, Rhode St Genese, pp. 37.Google Scholar
Jones, K. & Platzer, M.1997 Numerical computation of flapping-wing propulsion and power extraction. AIAA Paper 97–0826.CrossRefGoogle Scholar
Jones, M. & Yamaleev, N. K.2013 Adjoint-based shape and kinematics optimization of flapping wing propulsive efficiency. AIAA Paper 2013-2472.Google Scholar
Lee, B. J., Padulo, M. & Liou, M.-S.2011 Non-sinusoidal trajectory optimization of flapping airfoil using unsteady adjoint approach. AIAA Paper 2011-1312.Google Scholar
Li, L., Sherwin, S. & Bearman, P. W. 2002 A moving frame of reference algorithm for fluid/structure interaction of rotating and translating bodies. Intl J. Numer. Meth. Fluids 38 (2), 187206.Google Scholar
Liang, Z., Dong, H. & Wei, M.2010 Computational analysis of hovering hummingbird flight. AIAA Paper 2010-555.Google Scholar
Milano, M. & Gharib, M. 2005 Uncovering the physics of flapping flat plates with artificial evolution. J. Fluid Mech. 534, 403409.Google Scholar
Mittal, R., Dong, H., Bozkurttas, M., Najjar, F., Vargas, A. & Von Loebbecke, A. 2008 A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries. J. Comput. Phys. 227 (10), 48254852.Google Scholar
Moubachir, M. & Zolesio, J.-P. 2006 Moving Shape Analysis and Control: Applications to Fluid Structure Interactions, Pure and Applied Mathematics, vol. 277. Chapman & Hall/CRC.Google Scholar
Nadarajah, S. & Jameson, A.2000 A comparison of the continuous and discrete adjoint approach to automatic aerodynamic optimization. AIAA Paper 2000-667.Google Scholar
Nadarajah, S. K. & Jameson, A. 2007 Optimum shape design for unsteady flows with time-accurate continuous and discrete adjoint method. AIAA J. 45 (7), 14781491.Google Scholar
Pesavento, U. & Wang, Z. J. 2004 Falling paper: Navier–Stokes solutions, model of fluid forces, and center of mass elevation. Phys. Rev. Lett. 93 (14), 144501.Google Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. 1996 Numerical Recipes in Fortran 90, 2nd edn. Cambridge University Press.Google Scholar
Protas, B. & Liao, W. 2008 Adjoint-based optimization of pdes in moving domains. J. Comput. Phys. 227 (4), 27072723.Google Scholar
Trizila, P., Kang, C.-K., Aono, H., Shyy, W. & Visbal, M. 2011 Low-Reynolds-number aerodynamics of a flapping rigid flat plate. AIAA J. 49 (4), 806823.Google Scholar
Tuncer, I. H. & Kaya, M. 2005 Optimization of flapping airfoils for maximum thrust and propulsive efficiency. AIAA J. 43 (11), 23292336.Google Scholar
Vanella, M., Fitzgerald, T., Preidikman, S., Balaras, E. & Balachandran, B. 2009 Influence of flexibility on the aerodynamic performance of a hovering wing. J. Expl Biol. 212 (1), 95105.Google Scholar
Wang, Q. & Gao, J. 2013 The drag-adjoint field of a circular cylinder wake at Reynolds numbers 20, 100 and 500. J. Fluid Mech. 730, 145161.Google Scholar
Wei, M. & Freund, J. B. 2006 A noise-controlled free shear flow. J. Fluid Mech. 546, 123152.Google Scholar
Williamson, C. H. K. & Roshko, A. 1988 Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2, 355381.Google Scholar
Xu, M. & Wei, M.2013 Using adjoint-based approach to study flapping wings. AIAA Paper 2013-839.Google Scholar
Xu, M. & Wei, M.2014 A continuous adjoint-based approach for the optimization of wing flapping. AIAA Paper 2014-2048.Google Scholar
Xu, M., Wei, M., Li, C. & Dong, H. 2015 Adjoint-based optimization of flapping plates hinged with a trailing-edge flap. Theor. Appl. Mech. Lett. 5, 14.Google Scholar
Xu, M., Wei, M., Yang, T. & Lee, Y. S. 2016 An embedded boundary approach for the simulation of a flexible flapping wing at different density ratio. Eur. J. Mech. (B/Fluids) 55, 146156.CrossRefGoogle Scholar
Yang, T., Wei, M. & Zhao, H. 2010 Numerical study of flexible flapping wing propulsion. AIAA J. 48 (12), 29092915.Google Scholar
Yin, B. & Luo, H. 2010 Effect of wing inertia on hovering performance of flexible flapping wings. Phys. Fluids 22, 111902.CrossRefGoogle Scholar