On the Calculation of a Flapping Flexible Airfoil in the Flow of Viscous Incompressible Fluid

The study of motion of deformable bodies in a liquid flow is a topical task in engineering and biomechanics. Modelling of these problems as a two-way coupled interaction is problematic. In this case, at each time step, the hydrodynamic equations must be solved taking into account the motion of the body, which deforms and moves under the action of the hydrodynamic forces that depend on the motion of the body. One way to solve these problems is to use the splitting method when the equations of hydrodynamics and dynamics are solved separately by successive iterations. However, this approach does not always lead to a solution. In particular, when there is a significant difference in timescales of fluid hydrodynamic and body dynamics, the splitting method requires an unattainably small time step. In this paper, the meshless numerical method eliminating these problems is applied. The method allows combining the equations of flow vorticity from the surface of the body in the viscous fluid, and the equations of solid body dynamics into an integrated system of linear equations. This way all the unknowns including the body motion parameters are calculated over one time step, without splitting them into the hydrodynamic and the dynamic components. The examples of numerical solutions to the problems of rigid and flexible airfoils flapping in a viscous fluid flow are presented. The airfoil flexibility is modelled by flexible coupling of several non-deformable sections. The numerical solutions are compared with the experimental results. It is shown that the flexibility of the airfoil can have a significant influence on the flow and the resulting propulsive force of the flapping airfoil.

References

[1] Godoy-Diana R., Aider J.L., Wesfreid J.E. Transitions in the wake of a flapping foil. Physical Review E – Statistical, Nonlinear, and Soft Matter Physics, 2008, vol. 77, iss. 1, no. 016308.

[2] Marais C., Thiria B., Wesfreid J.E., Godoy-Diana R. Stabilizing effect of flexibility in the wake of a flapping foil. Journal of Fluid Mechanics, 2012, vol. 710, pp. 659–669.

[3] Richter T., Wick T. Finite elements for fluid–structure interaction in ALE and fully Eulerian coordinates. Computer Methods in Applied Mechanics and Engineering, 2010, 199(41–44), pp. 2633–2642.

[4] Fuchiwaki M., Nagata T., Tanaka K. Dynamic forces acting on elastic heaving airfoils based on the bending stiffness considerations. Proceedings of the ASME 2014 4th Joint USEuropean Fluids Engineering Division Summer Meeting FEDSM2014, August 3–7, 2014, Chicago, Illinois, USA, code 109724.

[5] Quinn D. B., Lauder G. V., Smits A.J. Maximizing the efficiency of a flexible propulsor using experimental optimization. Journal of Fluid Mechanics, 2015, vol. 767, pp. 430–448.

[6] Mysa R.C., Venkatraman K. Intertwined vorticity and elastodynamics in flapping wing propulsion. Journal of Fluid Mechanics, 2015, vol. 787, pp. 175–223.

[7] Michelin S., Llewellyn Smith S.G. Resonance and propulsion performance of a heaving flexible wing. Physics of Fluids, 2009, vol. 21, iss. 7, no. 071902, doi:10.1063/1.3177356.

[8] Andronov P.R., Guverniuk S.V, Dynnikova G.Ia. Vikhrevye metody rascheta nestatsionarnykh gidrodinamicheskikh nagruzok [Vortex methods for unsteady hydrodynamic loads]. Moscow, MSU publ., 2006. 184 p.

[9] Andronov P.R., Grigorenko D.A., Guverniuk S.V., Dynnikova G.Ia. Chislennoe modelirovanie samovrashcheniia plastin v potoke viazkoi zhidkosti [Numerical simulation of autorotation plates in a viscous flow]. Izvestiia Rossiiskoi akademii nauk. Mekhanika zhidkosti i gaza [A Journal of Russian Academy of Sciences. Fluid Dynamics]. 2007, no. 5, pp. 47–60.

[10] Dynnikov Y.A., Dynnikova G.Y. Application of Viscous Vortex Domains Method for Solving Flow-Structure Problems. Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics, 2013, pp. 877–882.

[11] Dynnikov Ia.A. Energoeffektivnost’ samodvizheniia deformiruiushchikhsia podvodnykh ob"ektov [Energy efficiency of self-propulsion submarines deformable objects]. V sbornike Trudy konferentsii-konkursa molodykh uchenykh, 13–15 oktiabria 2010. [In Proceedings of the conference-contest of young scientists, 13–15 October 2010]. Moscow, MSU publ., 2011, pp. 124–127.

[12] Dynnikova G.Ia. Lagranzhev podkhod k resheniiu nestatsionarnykh uravnenii Nav’e–Stoksa [Lagrangian approach to the non-stationary Navier-Stokes equations]. Doklady Akademii nauk [Reports of the Academy of Sciences]. 2004, vol. 399, no. 1, pp. 42–46.

[13] Ogami Y., Akamatsu T. Viscous flow simulation using the discrete vortex model – the Diffusion Velocity Method. Computers and Fluids, 1991, vol. 19 (3/4), pp. 433–441.

[14] Moreva V.S., Marchevsky I.K. Vortex element method for 2D flow simulation with tangent velocity components on airfoil surface. ECCOMAS 2012 – 6th European Congress on Computational Methods in Applied Sciences and Engineering: Book of proceedings, Vienna, 2012. 14 p.