Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-26T12:21:45.460Z Has data issue: false hasContentIssue false
  • English
  • Français

Vortex force maps for three-dimensional unsteady flows with application to a delta wing

Published online by Cambridge University Press:  13 August 2020

Juan Li Open the ORCID record for Juan Li [Opens in a new window] ,
Xiaowei Zhao Open the ORCID record for Xiaowei Zhao [Opens in a new window]  and
Michael Graham Open the ORCID record for Michael Graham [Opens in a new window]
Juan Li
Affiliation:
School of Engineering, The University of Warwick, CoventryCV4 7AL, UK
Xiaowei Zhao*
Affiliation:
School of Engineering, The University of Warwick, CoventryCV4 7AL, UK
Michael Graham
Affiliation:
Department of Aeronautics, Imperial College, LondonSW7 2BY, UK
*
†Email address for correspondence: xiaowei.zhao@warwick.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The unsteady forces acting on a body depend strongly on the local flow structures such as vortices. A quantitative understanding of the contribution of these structures to the instantaneous overall force is of fundamental significance. In the present study, a three-dimensional (3-D) vortex force map (VFM) method, extended from a two-dimensional (2-D) one, is used to provide better insight into the complex 3-D flow dynamics. The VFM vectors are obtained from solutions of potential equations and used to build the 3-D VFMs where the critical regions and directions associated with significant positive or negative contributions to the forces are identified. Using the existing velocity/vorticity field near the body, these VFMs can be used to obtain the body forces. A decomposed form of the force formula is also derived to separate the correction term contributed from the uncaptured vortices (close to or far away from the body). The present method is applied to the starting flow of a delta wing at high angle of attack, where LEVs are enhanced and stabilized by an axial flow effect. The analogy between the normal force of a slender delta wing and that of a 2-D flat plate with a steadily growing span is demonstrated via the VFM analysis. We find, for this application, that the force evolution exhibits some similar behaviour to a 2-D airfoil starting flow and, surprisingly, the force contribution mainly comes from the conical vortex sheet rather than the central core. Moreover, a quantitative understanding of the influence of LEVs in different evolution regimes on the body force is demonstrated.

JFM classification

Vortex Flows: Vortex shedding Wakes/Jets: Separated flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A36
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Akbari, M. H. & Price, S. J. 2003 Simulation of dynamic stall for a NACA 0012 airfoil using a vortex method. J. Fluids Struct. 17 (6), 855–874.CrossRefGoogle Scholar
Ayoub, A. & McLachlan, B. G. 1987 Slender delta wing at high angles of attack – a flow visualization study. AIAA Paper, 87-1230.Google Scholar
Birch, J. M. & Dickinson, M. H. 2001 Spanwise flow and the attachment of the leading-edge vortex on insect wings. Nature 412, 729–733.CrossRefGoogle Scholar
Dickinson, M. H. & Gotz, K. G. 1993 Unsteady aerodynamic performance of model wings at low Reynolds numbers. J. Expl Biol. 174, 45–64.Google Scholar
Earnshaw, P. B. & Lawford, J. A. 1966 Low-Speed Wind-Tunnel Experiments on a Series of Sharp-Edged Delta Wings, Aeronautical Research Council reports and memoranda, vol. 3424. H.M. Stationery Office.Google Scholar
Ellington, C. P., van Den Berg, C., Willmott, A. P. & Thomas, A. L. R. 1996 Leading-edge vortices in insect flight. Nature 384, 626–630.CrossRefGoogle Scholar
Graham, J. M. R. 1983 The lift on an aerofoil in starting flow. J. Fluid Mech. 133, 413–425.CrossRefGoogle Scholar
Graham, W. R., Pitt Ford, C. W. & Babinsky, H. 2017 An impulse-based approach to estimating forces in unsteady flow. J. Fluid Mech. 815, 60–76.CrossRefGoogle Scholar
Howe, M. S. 1995 On the force and moment on a body in an incompressible fluid, with application to rigid bodies and bubbles at high Reynolds numbers. Q. J. Mech. Appl. Maths 48, 401–425.CrossRefGoogle Scholar
Hsieh, C. T., Kung, C. F., Chang, C. C. & Chu, C. C. 2010 Unsteady aerodynamics of dragonfly using a simple wing–wing model from the perspective of a force decomposition. J. Fluid Mech. 663, 233–252.CrossRefGoogle Scholar
Kang, L. L., Liu, L. Q., Su, W. D. & Wu, J. Z. 2018 Minimum-domain impulse theory for unsteady aerodynamic force. Phys. Fluids 30 (1), 016107.CrossRefGoogle Scholar
Lee, J. J., Hsieh, C. T., Chang, C. C. & Chu, C. C. 2012 Vorticity forces on an impulsively started finite plate. J. Fluid Mech. 694, 464–492.CrossRefGoogle Scholar
Li, C., Dong, H. & Zhao, K. 2018 A balance between aerodynamic and olfactory performance during flight in Drosophila. Nat. Commun. 9 (1), 3215.CrossRefGoogle ScholarPubMed
Li, G. J. & Lu, X. Y. 2012 Force and power of flapping plates in a fluid. J. Fluid Mech. 712, 598–613.CrossRefGoogle Scholar
Li, J. & Wu, Z. N. 2015 Unsteady lift for the Wagner problem in the presence of additional leading/trailing edge vortices. J. Fluid Mech. 769, 182–217.CrossRefGoogle Scholar
Li, J. & Wu, Z. N. 2016 A vortex force study for a flat plate at high angle of attack. J. Fluid Mech. 801, 222–249.CrossRefGoogle Scholar
Li, J. & Wu, Z. N. 2018 Vortex force map method for viscous flows of general airfoils. J. Fluid Mech. 836, 145–166.CrossRefGoogle Scholar
Luckring, J. M. 2019 The discovery and prediction of vortex flow aerodynamics. Aeronaut. J. 123 (1264), 729–803.CrossRefGoogle Scholar
Ma, B. F., Wang, Z. J. & Gursul, G. 2017 Symmetry breaking and instabilities of conical vortex pairs over slender delta wings. J. Fluid Mech. 832, 41–72.CrossRefGoogle Scholar
Mangler, K. W. & Smith, J. H. B. 1959 A theory on the flow past a slender delta wing with leading edge separation. Proc. R. Soc. 251 (1265), 200–217.Google Scholar
Mao, X. & Sorense, J. N. 2018 Far-wake meandering induced by atmospheric eddies in flow past a wind turbine. J. Fluid Mech. 846, 190–209.CrossRefGoogle Scholar
Muijres, F. T., Johansson, L. C., Barfield, R., Wolf, M., Spedding, G. R. & Hedenstrom, A. 2008 Leading-edge vortex improves lift in slow-flying bats. Science 319, 1250–1253.CrossRefGoogle ScholarPubMed
Noca, F., Shiels, D. & Jeon, D. 1997 Measuring instantaneous fluid dynamic forces on bodies, using only velocity fields and their derivatives. J. Fluids Struct. 11, 345–350.CrossRefGoogle Scholar
Pitt Ford, C. W. & Babinsky, H. 2013 Lift and the leading-edge vortex. J. Fluid Mech. 720, 280–313.CrossRefGoogle Scholar
Polhamus, E. C. 1966 A concept of the vortex lift of sharp-edge delta wings based on a leading-edge-suction analogy. NACA Tech. Note, D-3767.Google Scholar
Polhamus, E. C. 1971 Predictions of vortex lift characteristics by a leading-edge suction analogy. J. Aircr. 8, 193–198.CrossRefGoogle Scholar
Rao, A., Minelli, G., Basara, B. & Krajnovic, S. 2018 On the two flow states in the wake of a hatchback Ahmed body. J. Wind Engng Ind. Aerodyn. 173, 262–278.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Wu, J. C. 1981 Theory for aerodynamic force and moment in viscous flows. AIAA J. 19, 432–441.CrossRefGoogle Scholar
Wu, J. Z., Lu, X. Y. & Zhuang, L. X. 2007 Integral force acting on a body due to local flow structures. J. Fluid Mech. 576, 265–286.CrossRefGoogle Scholar
Xia, X. & Mohseni, K. 2013 Lift evaluation of a two-dimensional pitching flat plate. Phys. Fluids 25, 091901.CrossRefGoogle Scholar
Zhang, J. 2017 Footprints of a flapping wing. J. Fluid Mech. 818, 1–4.CrossRefGoogle Scholar
Zhang, M., Wang, Y. K. & Fu, S. 2017 Generation mechanism and reduction method of induced drag produced by interacting wingtip vortex system. J. Mech. 34 (02), 231–241.CrossRefGoogle Scholar