Abstract
The Kutta Joukowski (KJ) theorem, relating the lift of an airfoil to circulation, was widely accepted for predicting the lift of viscous high Reynolds number flow without separation. However, this theorem was only proved for inviscid flow and it is thus of academic importance to see whether there is a viscous equivalent of this theorem. For lower Reynolds number flow around objects of small size, it is difficult to measure the lift force directly and it is thus convenient to measure the velocity flow field solely and then, if possible, relate the lift to the circulation in a similar way as for the inviscid KJ theorem. The purpose of this paper is to discuss the relevant conditions under which a viscous equivalent of the KJ theorem exists that reduces to the inviscid KJ theorem for high Reynolds number viscous flow and remains correct for low Reynolds number steady flow. It has been shown that if the lift is expressed as a linear function of the circulation as in the classical KJ theorem, then the freestream velocity must be corrected by a component called mean deficit velocity resulting from the wake. This correction is small only when the Reynolds number is relatively large. Moreover, the circulation, defined along a loop containing the boundary layer and a part of the wake, is generally smaller than that based on inviscid flow assumption. For unsteady viscous flow, there is an inevitable additional correction due to unsteadiness.
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References
Batchelor F R S. An Introduction to Fluid Dynamics. Cambridge: Cambridge University Press, 1967
Anderson J. Fundamentals of Aerodynamics, Mcgraw-Hill Series in Aeronautical and Aerospace Engineering. New York: McGraw-Hill Education, 1984
Noca F, Shiels D, Jeon D. A comparison of methods for evaluating time-dependent fluid dynamic forces on bodies, using only velocity fields and their derivatives. J Fluids Struct, 1999, 13: 551–578
Sharma S D, Deshapande P J. Kutta-Joukowsky expression in viscous and unsteady flow. Exp Fluids, 2012, 52: 1581–1591
Wu J C. Theory for aerodynamic force and moment in viscous flows. AIAA J, 1981, 19: 432–441
Saffman P G. Vortex Dynamics. New York: Cambridge University Press, 1992
Howe M S. 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 Math, 1995, 48: 401–425
Bai C Y, Wu Z N. Generalized Kutta-Joukowski theorem for multivortex and multi-airfoil flow (a lumped vortex model). Chin J Aeronaut, 2014, 27: 34–39
Bai C Y, Li J, Wu Z N. Generalized Kutta-Joukowski theorem for multivortex and multi-airfoil flow with vortex production (general model). Chin J Aeronaut, 2014, in press
Wu J C, Lu X Y, Zhuang L X. Integral force acting on a body due to local flow structures. J Fluid Mech, 2007, 576: 265–286
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Li, J., Xu, Y. & Wu, Z. Kutta-Joukowski force expression for viscous flow. Sci. China Phys. Mech. Astron. 58, 1–5 (2015). https://doi.org/10.1007/s11433-014-5574-2
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DOI: https://doi.org/10.1007/s11433-014-5574-2