Non-linear unsteady aerodynamic model for insect-like flapping wings in the hover. Part 2: Implementation and validation

Abstract The essence of this two-part paper is the analytical, aerodynamic modelling of insect-like flapping wings in the hover for micro-air-vehicle applications. A key feature of such flapping-wing flows is their unsteadiness and the formation of a leading-edge vortex in addition to the conventional wake shed from the trailing edge. What ensues is a complex interaction between the shed wakes, which, in part, determines the forces and moments on the wing. In an attempt to describe such a flow, two novel coupled, non-linear, wake integral equations were developed in the first part of the paper. The governing equations derived were exact, but did not have a closed analytical form. Solutions were, therefore, to be found by numerical methods and implemented in Fortran. This is the theme of the second part of the paper. The problem is implemented by means of vortex methods, whereby discrete point vortices are used to represent the wing and its wake. A number of numerical experiments are run to determine the best values for numerical parameters. The calculation is performed using a time-marching algorithm and the evolution of the wakes is tracked. In this way, both flow field and force data are generated. The model is then validated against existing experimental data and very good agreement is found both in terms of flow field representation and force prediction. The temporal accuracy of the simulations is also noteworthy, implying that the underlying flow features are well captured, especially the unsteadiness. The model also shows the similarity between two-dimensional and three-dimensional flows for insect-like flapping wings at low Reynolds numbers of the order of Re ε 200.

[1]  D. Poling,et al.  The Trailing Edge of a Pitching Airfoil at High Reduced Frequencies , 1987 .

[2]  N. Rott.  Diffraction of a weak shock with vortex generation , 1956, Journal of Fluid Mechanics.

[3]  Adrian L. R. Thomas,et al.  FLOW VISUALIZATION AND UNSTEADY AERODYNAMICS IN THE FLIGHT OF THE HAWKMOTH, MANDUCA SEXTA , 1997 .

[4]  D. Crighton.  The Kutta Condition in Unsteady Flow , 1985 .

[5]  Abe Silverstein,et al.  Experimental Verification of the Theory of Oscillating Airfoils , 1939 .

[6]  M. Dickinson,et al.  Spanwise flow and the attachment of the leading-edge vortex on insect wings , 2001, Nature.

[7]  G. Vatistas,et al.  A simpler model for concentrated vortices , 1991 .

[8]  Quanhua Sun,et al.  Flat-plate aerodynamics at very low Reynolds number , 2004, Journal of Fluid Mechanics.

[9]  L. Rosenhead.  The Formation of Vortices from a Surface of Discontinuity , 1931 .

[10]  J. Gordon Leishman,et al.  Wake Structure Diagnostics of a Flapping Wing MAV , 2005 .

[11]  Mao Sun,et al.  Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion. , 2002, The Journal of experimental biology.

[12]  P. Spalart.  Vortex methods for separated flows , 1988 .

[13]  G.A.N. Ising.  Några luftelektriska iakttagelser i Lappland. , 2022 .

[14]  J. Gordon Leishman,et al.  Principles of Helicopter Aerodynamics , 2000 .

[15]  A. Betz.  Verhalten von Wirbelsystemen , 1932 .

[16]  D. Poling,et al.  The Response of Airfoils to Periodic Disturbances —The Unsteady Kutta Condition , 1984 .

[17]  G. Birkhoff,et al.  Helmholtz and taylor instability , 1962 .

[18]  M. Dickinson,et al.  The aerodynamic effects of wing rotation and a revised quasi-steady model of flapping flight. , 2002, The Journal of experimental biology.

[19]  Kevin Knowles,et al.  Non-linear unsteady aerodynamic model for insect-like flapping wings in the hover. Part 1: Methodology and analysis , 2006 .

[20]  Herbert Wagner.  Über die Entstehung des dynamischen Auftriebes von Tragflügeln , 1925 .

[21]  M. Dickinson,et al.  The control of flight force by a flapping wing: lift and drag production. , 2001, The Journal of experimental biology.

[22]  Adrian L. R. Thomas,et al.  Leading-edge vortices in insect flight , 1996, Nature.

[23]  C. Ellington,et al.  The three–dimensional leading–edge vortex of a ‘hovering’ model hawkmoth , 1997 .

[24]  L. Ting,et al.  Viscous Vortical Flows , 1991 .

[25]  J. Giesing.  Vorticity and Kutta Condition for Unsteady Multienergy Flows , 1969 .

[26]  Dickinson,et al.  THE EFFECTS OF WING ROTATION ON UNSTEADY AERODYNAMIC PERFORMANCE AT LOW REYNOLDS NUMBERS , 1994, The Journal of experimental biology.

[27]  M. Dickinson,et al.  Wing rotation and the aerodynamic basis of insect flight. , 1999, Science.

[28]  J. Gordon Leishman,et al.  Flow Visualization of Micro Air Vehicle Scaled Insect-Based Flapping Wings. , 2005 .

[29]  M. Dickinson,et al.  UNSTEADY AERODYNAMIC PERFORMANCE OF MODEL WINGS AT LOW REYNOLDS NUMBERS , 1993 .

[30]  T. Kármán,et al.  Airfoil Theory for Non-Uniform Motion , 1938 .

[31]  Dean T. Mook,et al.  Perspective: Numerical Simulations of Wakes and Blade-Vortex Interaction , 1994 .

[32]  T. Sarpkaya.  Computational Methods With Vortices—The 1988 Freeman Scholar Lecture , 1989 .