Research Papers

Integrability of Velocity Constraints Modeling Vortex Shedding in Ideal Fluids

[+] Author and Article Information
Phanindra Tallapragada

Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29634
e-mail: ptallap@clemson.edu

Scott David Kelly

Department of Mechanical Engineering
and Engineering Science,
University of North Carolina at Charlotte,
Charlotte, NC 28233
e-mail: scott@kellyfish.net

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 1, 2016; final manuscript received September 16, 2016; published online December 2, 2016. Assoc. Editor: Zdravko Terze.

J. Comput. Nonlinear Dynam 12(2), 021008 (Dec 02, 2016) (7 pages) Paper No: CND-16-1048; doi: 10.1115/1.4034862 History: Received February 01, 2016; Revised September 16, 2016

A mathematical model that invokes the Kutta condition to account for vortex shedding from the trailing edge of a free hydrofoil in a planar ideal fluid is compared with a canonical model for the dynamics of a terrestrial vehicle subject to a nonintegrable velocity constraint. The Kutta condition is shown to be nonintegrable in a sense that parallels that in which the constraint on the terrestrial vehicle is nonintegrable. Simulations of the two systems' dynamics reinforce the analogy between the two.

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Fig. 1

The Chaplygin sleigh

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Fig. 2

(a) Heading angle θ and (b) path of the center of the Chaplygin sleigh with initial conditions ux = 5 and ω = 5

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Fig. 3

The Joukowski transformation maps a circle of radius rc in the complex ζ plane to the foil in the z plane

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Fig. 4

The body axes are denoted by Xb and Yb, and the velocities of the foil in the body frame of reference are V1 and V2

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Fig. 5

Snapshots of the simulated motion of the foil with initial conditions A = 50 and L = 0. The red dots denote vorticity with a clockwise circulation while blue dots denote vorticity with a counterclockwise circulation.

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Fig. 6

(a) Heading angle θ, (b) path of the center of the foil for various initial conditions, and (c) total circulation due to the point vortices in the fluid. The three graphs shown in the figures correspond to the following cases: (A0=50, L=0) (lowermost graph); (A0=50,Lx=10, Ly=0) (red); and (A0=50,Lx=10, Ly=10) (topmost graph).




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