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The numerical investigation of Lagrangian and Eulerian coherent structures for the near wake structure of a hovering Drosophila

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Abstract

An integrated simulation of a Drosophila wing–body combination in hovering flight has been carried out in order to analyze the Lagrangian and Eulerian coherent structures. A parallel unstructured finite volume method based on an arbitrary Lagrangian–Eulerian (ALE) formulation has been initially validated for a flapping rectangular plate and then employed to solve the incompressible unsteady Navier–Stokes equations around a Drosophila wing–body combination. A robust mesh deformation algorithm based on indirect radial basis function method is utilized at each time level while avoiding remeshing. The time variation of the three-dimensional Eulerian coherent structures around a flapping Drosophila in hovering flight is analyzed in the near wake with \(\lambda _{2}\)-criterion. Meanwhile, the Lagrangian coherent structures are investigated using finite-time Lyapunov exponent fields. In addition, the instantaneous velocity vectors and particle traces are presented along with the aerodynamic parameters including the force, moment and power for a wing–body combination. Furthermore, a wing-only configuration is also investigated in order to show the body effects on aerodynamic loads. The numerical simulations are used to gain insight into the near wake topology as well as their correlations with the aerodynamic force generation. The present fully coupled ALE algorithm is shown to be sufficiently robust to deal with large mesh deformations seen in flapping wings and reveals highly detailed near wake topology which is very useful to study physics in biological flights and can also provide an effective tool for designing bio-inspired MAVs and MFIs.

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References

  1. Anderson, W.K., Bonhaus, D.L.: An implicit upwind algorithm for computing turbulent flows on unstructured grids. Comput. Fluids 23, 1–21 (1994)

    Article  MATH  Google Scholar 

  2. Aono, H., Liang, F., Liu, H.: Near- and far-field aerodynamics in insect hovering flight: an integrated computational study. J. Exp. Biol. 211, 239–257 (2008)

    Article  Google Scholar 

  3. Bai, P., Cui, E., Li, F., Zhou, W., Chen, B.: A new bionic MAVs flapping motion based on fruit fly hovering at low Reynolds number. Acta Mech. Sin. 23, 485–493 (2007)

    Article  MATH  Google Scholar 

  4. Balay, S., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: Petsc users manual. anl-95/11. Mathematic and computer science division, Argonne National Laboratory (2004)

  5. Batina, J.T.: Unsteady Euler airfoil solutions using unstructured dynamic meshes. AIAA J. 28, 1381–1388 (1990)

    Article  Google Scholar 

  6. Birch, J.M., Dickinson, M.: The influence of wing-wake interactions on the production of aerodynamic forces in flapping flight. J. Exp. Biol. 206, 2257–2272 (2003)

    Article  Google Scholar 

  7. Bomphrey, R.J., Nakata, T., Phillips, N., Walker, S.M.: Smart wing rotation and trailing-edge vortices enable high frequency mosquito flight. Nature 544, 92–95 (2017)

    Article  Google Scholar 

  8. Bos, F.M.: Numerical simulations of flapping foil and wing aerodynamics. Ph.D. Thesis, Delft University of Technology, Delft (2009)

  9. Bos, F.M., van Oudheusden, B.W., Bijl, H.: Wing performance and 3-D vortical structure formation in flapping flight. J. Fluids Struct. 42, 130–151 (2013)

    Article  Google Scholar 

  10. Brunton, S.L., Rowley, C.W.: Fast computation of finite-time Lyapunov exponent fields for unsteady flows. Chaos 20, 017503 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chorin, A.J.: Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  12. Conti, C., Rossinelli, D., Koumoutsakos, P.: GPU and APU computations of finite time Lyapunov exponent fields. J. Comput. Phys. 231, 2229–2244 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dai, H., Luo, H., Doyle, J.F.: Dynamic pitching of an elastic rectangular wing in hovering motion. J. Fluid Mech. 693, 473–499 (2012)

    Article  MATH  Google Scholar 

  14. Dai, M., Schmidt, D.P.: Adaptive tetrahedral meshing in free-surface flow. J. Comput. Phys. 208, 228–252 (2005)

    Article  MATH  Google Scholar 

  15. de Boer, A., van der Schoot, M.S., Bijl, H.: Mesh deformation based on radial basis function interpolation. Comput. Struct. 85, 784–795 (2007)

    Article  Google Scholar 

  16. Dickinson, M.H., Götz, K.G.: Unsteady aerodynamic performance of model wings at low Reynolds numbers. J. Exp. Biol. 174, 45–64 (1993)

    Google Scholar 

  17. Dickinson, M.H., Lehmann, F.-O., Sane, S.P.: Wing rotation and the aerodynamic basis of insect flight. Science 284, 1954–1960 (1999)

    Article  Google Scholar 

  18. Dickson, W.B., Polidoro, P., Tanner, M.M., Dickinson, M.H.: A linear systems analysis of the yaw dynamics of a dynamically scaled insect model. J. Exp. Biol. 213, 3047–3061 (2013)

    Article  Google Scholar 

  19. Dickson, W.B., Straw, A.D., Dickinson, M.H.: Integrative model of Drosophila flight. AIAA J. 46, 2150–2164 (2008)

    Article  Google Scholar 

  20. Dong, H., Liang, Z.: The wing kinematics effects on performance and wake structure produced by finite-span hovering wings. In: AIAA Fluid Dynamics Conference and Exhibition, AIAA paper 2008-3819, Seattle, Washington (2008)

  21. Eken, A., Sahin, M.: A parallel monolithic algorithm for the numerical simulation of large-scale fluid-structure interaction problems. Int. J. Numer. Methods Fluids 80, 687–714 (2016)

    Article  MathSciNet  Google Scholar 

  22. Eken, A., Sahin, M.: A parallel monolithic approach for fluid-structure interaction in a celabral aneurysm. Comput. Fluids 153, 61–75 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Eldredge, J.D., Chong, K.: Fluid transport and coherent structures of translating and flapping wings. Chaos 20, 017509–14 (2010)

    Article  MathSciNet  Google Scholar 

  24. Ellington, C.P.: The aerodynamics of hovering insect flight. III. Kinematics. Philos. Trans. R. Soc. B 305, 41–78 (1984)

    Article  Google Scholar 

  25. Ellington, C.P., Van Den Berg, C., Willmott, A.P., Thomas, A.L.R.: Leading-edge vortices in insect flight. Nature 384, 626–630 (1996)

    Article  Google Scholar 

  26. Engels, T., Kolomenskiy, D., Schneider, K., Sesterhenn, J.: FluSI: a novel parallel simulation tool for flapping insect flight using a fourier method with volume penalization. SIAM J. Sci. Comput. 38, S3–S24 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Epps, B.P.: Review of vortex identification methods. AIAA Paper 2017–0989, 1–22 (2017)

    Google Scholar 

  28. Erzincanli, B., Sahin, M.: An arbitrary Lagrangian–Eulerian formulation for solving moving boundary problems with large displacements and rotations. J. Comput. Phys. 255, 660–679 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Erzincanli, B., Sahin, M.: The numerical simulation of the wing kinematic effects on near wake structure and aerodynamic performance in hovering drosophila flight. Comput. Fluids 122, 90–110 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Falgout, R., Baker, A., Chow, E., Henson, V., Hill, E., Jones, J., Kolev, T., Lee, B., Painter, J., Tong, C., Vassilevski, P., Yang, U.M.: Users manual, HYPRE High Performance Preconditioners, UCRL-MA-137155 DR. Center for Applied Scientific Computing, Lawrence Livermore National Laboratory (2004)

  31. Green, M.A., Rowley, C.W., Haller, G.: Detection of Lagrangian coherent structures in three-dimensional turbulence. J. Fluid Mech. 572, 111–120 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Guventurk, C., Sahin, M.: An arbitrary Lagrangian–Eulerian framework with exact mass conservation for the numerical simulation of 2D rising bubble problem. Int. J. Numer. Methods Eng. 112, 2110–2134 (2017)

    Article  MathSciNet  Google Scholar 

  33. Haller, G.: A variational theory of hyperbolic Lagrangian coherent structures. Physica D Nonlinear Phenomena 240, 137–162 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Haller, G., Yuan, G.: Lagrangian coherent structures and mixing in two-dimensional turbulence. J. Phys. D 147, 352–370 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  35. Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. Johnson, A.A.: Dynamic-mesh CFD and its applications to flapping-wing micro-aerial vehicles. In: The 25th Army Science Conference (ASC), Orlando, Florida (2006)

  37. Johnson, A.A., Tezduyar, T.: Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces. Comput. Methods Appl. Mech. Eng. 119, 73–94 (1994)

    Article  MATH  Google Scholar 

  38. Johnson, A.A., Tezduyar, T.: Advanced mesh generation and update methods for 3D flow simulations. Comput. Mech. 23, 130–143 (1999)

    Article  MATH  Google Scholar 

  39. Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. Comput. Mech. 20, 359–392 (1998)

    MathSciNet  MATH  Google Scholar 

  40. Kim, H.J., Beskok, A.: Quantification of chaotic strength and mixing in a micro fluidic system. J. Micromech. Microeng. 17, 2197–2210 (2007)

    Article  Google Scholar 

  41. Kim, J., Kim, D., Choi, H.: An immersed-boundary finite-volume method for simulations of flow in complex geometries. J. Comput. Phys. 171, 132–150 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  42. Kweon, J., Choi, H.: Sectional lift coefficient of a flapping wing in hovering motion. Phys. Fluids 22, 071703 (2010)

    Article  Google Scholar 

  43. Lehmann, F.O., Dickinson, M.H.: The changes in power requirement and muscle efficiency during elevated force production in the fruit fly Drosophila melanogaster. J. Exp. Biol. 200, 1133–1143 (1997)

    Google Scholar 

  44. Lehmann, F.O., Sane, S.P., Dickinson, M.H.: The aerodynamic effects of wing-wing interaction in flapping insect wings. J. Exp. Biol. 208, 3075–92 (2005)

    Article  Google Scholar 

  45. Lekien, F., Coulliette, C., Mariano, A.J., Ryan, E.H., Shay, L.K., Haller, G., Marsden, J.: Pollution release tied to invariant manifolds: a case study for the coast of Florida. Physica D 210, 1–20 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  46. Liang, Z., Dong, H.: Unsteady aerodynamics and wing kinematics effect in hovering insect flight. In: Proceeding of 47th AIAA Aerospace Sciences Meeting and Exhibit with New Horizons Forum, AIAA paper 2009-1299 (2009)

  47. Lipinski, D., Mohseni, K.: Flow structures and fluid transport for the hydromedusae Sarsia tubulosa and Aequorea victoria. J. Exp. Biol. 212, 2436–2447 (2009)

    Article  Google Scholar 

  48. Lipinski, D., Mohseni, K.: A ridge tracking algorithm and error estimate for efficient computations of Lagrangian coherent structures. Chaos 20, 017504 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Liu, H., Kawachi, K.: A numerical study of insect flight. J. Comput. Phys. 146, 124–156 (1998)

    Article  MATH  Google Scholar 

  50. Mittal, R., Dong, H., Bozkurttas, M., Najjar, F.M., Vargas, A., von Loebbecke, A.: A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries. J. Comput. Phys. 227, 4825–4852 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  51. Oner, E., Sahin, M.: A parallel adaptive viscoelastic flow solver with template based dynamic mesh adaptation. J. Non Newtonian Fluid Mech. 234, 36–50 (2016)

    Article  MathSciNet  Google Scholar 

  52. Peng, J., Dabiri, J.O.: The upstream wake of swimming and flying animals and its correlation with propulsive efficiency. J. Exp. Biol. 211, 2669–2677 (2008)

    Article  Google Scholar 

  53. Ramamurti, R., Sandberg, W.C.: A three-dimensional computational study of the aerodynamic mechanisms of insect flight. J. Exp. Biol. 205, 1507–1518 (2002)

    Google Scholar 

  54. Rendall, T.C.S., Allen, C.B.: Effcient mesh motion using radial basis functions with data reduction algorithms. J. Comput. Phys. 205, 6231–6249 (2009)

    Article  MATH  Google Scholar 

  55. Saad, Y.: A flexible inner-product preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14, 461–469 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  56. Sadlo, F., Peikert, R.: Efficient visualization of Lagrangian coherent structures by filtered AMR ridge extraction. IEE Trans. Vis. Comput. Graph. 13, 1456–1463 (2007)

    Article  Google Scholar 

  57. Sahin, M.: Parallel large-scale numerical simulations of purely-elastic instabilities behind a confined circular cylinder in a rectangular channel. J. Non Newtonian Fluid Mech. 195, 46–56 (2013)

    Article  Google Scholar 

  58. Sane, S.P., Dickinson, M.H.: The control of flight force by a flapping wing: lift and drag production. J. Exp. Biol. 204, 2607–2626 (2001)

    Google Scholar 

  59. Shadden, S.C., Dabiri, J.O., Marsden, J.E.: Lagrangian analysis of fluid transport in empirical vortex ring flows. Phys. Fluids 18, 047105 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  60. Sun, M., Du, G.: Lift and power requirements of hovering insect flight. Acta Mech. Sin. 19, 458–469 (2003)

    Article  Google Scholar 

  61. Sun, M., Tang, J.: Lift and power requirements of hovering flight in Drosophila virilis. J. Exp. Biol. 205, 2413–2427 (2002a)

    Google Scholar 

  62. Sun, M., Tang, J.: Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion. J. Exp. Biol. 205, 55–70 (2002b)

    Google Scholar 

  63. Suzuki, K., Minami, K., Inamuro, T.: Lift and thrust generation by a butterfly-like flapping wing-body model: immersed boundary-lattice Boltzmann simulations. J. Fluid Mech. 767, 659–695 (2015)

    Article  MathSciNet  Google Scholar 

  64. Thomas, P.D., Lombard, C.K.: Geometric conservation law and its application to flow computations on moving grids. AIAA J. 17, 1030–1037 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  65. Weis-Fogh, T.: Quick estimates of flight fitness in hovering animals including novel mechanism for lift production. J. Exp. Biol. 59, 169–230 (1973)

    Google Scholar 

  66. Weldon, M., Peacock, T., Jacobs, G.B., Helu, M., Haller, G.: Experimental and numerical investigation of the kinematic theory of unsteady separation. J. Fluid Mech. 611, 1–11 (2008)

    Article  MATH  Google Scholar 

  67. Wilson, M.M., Peng, J., Dabiri, J.O., Eldredge, J.D.: Lagrangian coherent structures in low Reynolds number swimming. J. Phys. Condens. Matter 21, 204105 (2009)

    Article  Google Scholar 

  68. Wu, D., Yeo, K.S., Lim, T.T.: A numerical study on the free hovering flight of a model insect at low Reynolds number. Comput. Fluids 103, 234–261 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Aeronautics and Astronautics, Istanbul Technical University, 34469, Maslak, Istanbul, Turkey

    Ezgi Dilek & Mehmet Sahin

  2. Faculty of Aeronautics and Astronautics, Kocaeli University, 41285, Kartepe, Kocaeli, Turkey

    Belkis Erzincanli

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  1. Ezgi Dilek
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  2. Belkis Erzincanli
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  3. Mehmet Sahin
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Correspondence to Mehmet Sahin.

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Communicated by Jeff D. Eldredge.

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Dilek, E., Erzincanli, B. & Sahin, M. The numerical investigation of Lagrangian and Eulerian coherent structures for the near wake structure of a hovering Drosophila. Theor. Comput. Fluid Dyn. 33, 255–279 (2019). https://doi.org/10.1007/s00162-019-00492-0

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