Research Papers: Fundamental Issues and Canonical Flows

A Two-Dimensional Multibody Integral Approach for Forces in Inviscid Flow With Free Vortices and Vortex Production

[+] Author and Article Information
Juan Li, Chen-Yuan Bai

Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China

Zi-Niu Wu

Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China
e-mail: ziniuwu@tsinghua.edu.cn.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received November 16, 2013; final manuscript received September 12, 2014; published online October 8, 2014. Assoc. Editor: Feng Liu.

J. Fluids Eng 137(2), 021205 (Oct 08, 2014) (10 pages) Paper No: FE-13-1671; doi: 10.1115/1.4028595 History: Received November 16, 2013; Revised September 12, 2014

In this paper, we propose an integral force approach for potential flow around two-dimensional bodies with external free vortices and with vortex production. The method can be considered as an extension of the generalized Lagally theorem to the case with continuous distributed vortices inside and outside of the body and is capable of giving the individual force of each body in the case of multiple bodies. The lift force formulas are validated against two examples. One is the Wagner problem with vortex production and with moving vortices in the form of a vortex sheet. The other is the lift of a flat plate when there is a standing vortex over its middle point. As a first application, the integral approach is applied to study the lift force of a flat plate induced by a bounded vortex above the plate. This bounded vortex may represent a second small airfoil at incidence. For this illustrative example, the lift force is found to display an interesting distance-dependent behavior: for a clockwise circulation, the lift force acting on the main airfoil is attractive for small distance and repulsive for large distance.

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Pitt Ford, C. W., and Babinsky, H., 2013, “Lift and the Leading-Edge Vortex,” J. Fluid Mech., 720, pp. 280–313. [CrossRef]
Xia, X., and Mohseni, K., 2013, “Lift Evaluation of a Two-Dimensional Pitching Flat Plate,” Phys. Fluids, 25(9), p. 091901. [CrossRef]
Oterberg, D., 2010, “Multi-Body Unsteady Aerodynamics in 2D Applied to a Vertical-Axis Wind Turbine Using a Vortex Method,” Master thesis, Uppsala University, Uppsala, Sweden.
Smith, F. T., and Timoshin, S. N., 1996, “Planar flows Past Thin Multi-Blade Configurations,” J. Fluid Mech., 324, pp. 355–377. [CrossRef]
Hsieh, C. T., Kung, C. F., and Chang, C. C., 2010, “Unsteady Aerodynamics of Dragonfly Using a Simple Wing-Wing Model From the Perspective of a Force Decomposition,” J. Fluid Mech., 663, pp. 233–252. [CrossRef]
Katz, J., and Plotkin, A., 2001, Low Speed Aerodynamics, Cambridge University, Cambridge, UK, Chap. 6.9.
Crowdy, D., 2006, “Calculating the Lift on a Finite Stack of Cylindrical Aerofoils,” Proc. R. Soc. A., 462(2069), pp. 1387–1407. [CrossRef]
Batchelor, F. R. S., 1967, An Introduction to Fluid Dynamics, Cambridge University, Cambridge, UK.
Crighton, D. G., 1985, “The Kutta Condition in Unsteady Flow,” Annu. Rev. Fluid Mech., 17(1), pp. 411–445. [CrossRef]
Anderson, J., 2010, Fundamentals of Aerodynamics (McGraw-Hill Series in Aeronautical and Aerospace Engineering), McGraw-Hill, NY.
Sharma, S. D., and Deshapande, P. J., 2012, “Kutta–Joukowski Expression in Viscous and Unsteady Flow,” Exp. Fluids, 52(6), pp. 1581–1591. [CrossRef]
Li, J., Xu, Y. Z., and Wu, Z. N., 2014, “Kutta-Joukowski Force Expression for Viscous Flow, Science in China,” Phys., Mech. Astron. (accepted). [CrossRef]
Saffman, P. G., 1992, Vortex Dynamics, Cambridge University, NY.
Milne-Thomson, L. M., 1968, Theoretical Hydrodynamics, Macmillan Education Ltd., Hong Kong.
Eames, I., Landeryou, M., and Lore, J. B., 2008, “Inviscid Coupling Between Point Symmetric Bodies and Singular Distributions of Vorticity,” J. Fluid Mech., 589, pp. 33–56. [CrossRef]
Kanso, E., and Oskouei, B. G., 2008, “Stability of a Coupled Body–Vortex System,” J. Fluid Mech., 600, pp. 77–94. [CrossRef]
Ramodanov, S. M., 2002, “Motion of a Circular Cylinder and N Point Vortices in a Perfect Fluid,” Regular Chaotic Dyn., 7(3), pp. 291–298. [CrossRef]
Shashikanth, B. N., Marsden, J. E., Burdick, J. W., and Kelly, S. D., 2002, “The Hamiltonian Structure of a Two-Dimensional Rigid Circular Cylinder Interacting Dynamically With N Point Vortices,” Phys. Fluids, 14(3), pp. 1214–1227. [CrossRef]
Streitlien, K., and Triantafyllou, M. S., 1977, “Force and Moment on a Joukowski Profile in the Presence of Point Vortices,” AIAA J., 33(4), pp. 603–611. [CrossRef]
Lagally, M., 1922, “Rerechnung der Kriifte und Momente die Str6mende Fliissig-keiten auf ihre Begrenzung Ausiiben,” Z. Angew. Math. Mech., 2(6), pp. 409–422. [CrossRef]
Landweber, L., and Yih, C.-S., 1956, “Forces, Moments, and Added Masses for Rankine Bodies,” J. Fluid Mech., 1(3), pp. 319–336. [CrossRef]
Landweber, L., and Miloh, T., 1980, “Unsteady Lagally Theorem for Multipoles and Deformable Bodies,” J. Fluid Mech., 96(1), pp. 33–46. [CrossRef]
Landweber, L., 1967, “Lagally's Theorem for Multipoles,” Schiflstechnik, 14, pp. 19–21.
Wu, C. T., Yang, F. L., and Young, D. L., 2012, “Generalized Two-Dimensional Lagally Theorem With Free Vortices and Its Application to Fluid-Body Interaction Problems,” J. Fluid Mech., 698, pp. 73–92. [CrossRef]
Bai, C. Y., and Wu, Z. N., 2014, “Generalized Kutta-Joukowski Theorem for Multi-Vortices and Multi-Airfoil Flow (Lumped Vortex Model),” Chin. J. Aeronaut., 27(1), pp. 34–39. [CrossRef]
Bai, C. Y., Li, J., and Wu, Z. N., 2014, “Generalized Kutta-Joukowski Theorem for Multi-Vortices and Multi-Airfoil Flow (General Model),” Chin. J. Aeronaut. (in press) [CrossRef].
Wu, J. C., 1981, “Theory for Aerodynamic Force and Moment in Viscous Flows,” AIAA J., 19(4), pp. 432–441. [CrossRef]
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. Math., 48(3), pp. 401–426. [CrossRef]
Noca, F., Shiels, D., and Jeon, D., 1999, “A Comparison of Methods for Evaluating Time-Dependent Fluid Dynamic Forces on Bodies, Using Only Velocity Fields and Their Derivatives,” J. Fluids Struct., 13(5), pp. 551–578. [CrossRef]
Wu, J. C., Lu, X. Y., and Zhuang, L. X., 2007, “Integral Force Acting on a Body due to Local Flow Structures,” J. Fluid Mech., 576, pp. 265–286. [CrossRef]
Ragazzo, C. G., and Tabak, E. G., 2007, “On the Force and Torque on Systems of Rigid Bodies: A Remark on an Integral Formula due to Howe,” Phys. Fluids, 19(5), p. 057108. [CrossRef]
Chang, C. C., Yang, S. H., and Chu, C. C., 2008, “A Many-Body Force Decomposition With Applications to Flow About Bluff Bodies,” J. Fluid Mech., 600, pp. 95–104. [CrossRef]
Wagner, H., 1925, “Uber die Enstehung des Dynamischen Auftreibes von Tragflugeln,” Z. Angew. Math. Mech., 5(1), pp. 17–35. [CrossRef]
Saffman, P. G., and Sheffield, J. S.,1977, “Flow Over a Wing With an Attached Free Vortex,” Stud. Appl. Math., 57(2), pp. 107–117.
Chow, C. Y., and Huang, M. K., 1982, “The Initial Lift and Drag of an Impulsively Started Aerofoil of Finite Thickness,” J. Fluid Mech., 118(1), pp. 393–409. [CrossRef]
Graham, J. M. R., 1983, “The Initial Lift on an Aerofoil in Starting Flow,” J. Fluid Mech., 133(1), pp. 413–425. [CrossRef]
Sakajo, T., 2012, “Force-Enhancing Vortex Equilibria for Two Parallel Plates in Uniform Flow,” Proc. R. Soc. A, 468(2140), pp. 1175–1195. [CrossRef]


Grahic Jump Location
Fig. 1

Thin airfoil model

Grahic Jump Location
Fig. 2

Impulsively started plate with shedding of a vortex sheet

Grahic Jump Location
Fig. 3

Flat plate of chord length cA, above which there is a bounded vortex (lumped vortex representation of a small airfoil with a chord ca≪cA). (a) The small airfoil has a positive angle of attack so that Γv < 0 and (b) the small airfoil has a negative angle of attack so that Γv > 0.

Grahic Jump Location
Fig. 4

Lift coefficient cl versus the nondimensional distance h¯ for a flat plate subjected to the influence of a bounded vortex above the middle of the plate




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