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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

## Abstract

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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## Figures

Fig. 1

Thin airfoil model

Fig. 2

Impulsively started plate with shedding of a vortex sheet

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.

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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