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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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References

Kelly, S. D. , and Xiong, H. , 2010, “ Self-Propulsion of a Free Hydrofoil With Localized Discrete Vortex Shedding: Analytical Modeling and Simulation,” Theor. Comput. Fluid Dyn., 24(1), pp. 45–50. [CrossRef]
Kanso, E. , 2010, “ Swimming in an Inviscid Fluid,” Theor. Comput. Fluid Dyn., 24, pp. 201–207. [CrossRef]
Tallapragada, P. , and Kelly, S. D. , 2013, “ Dynamics and Self-Propulsion of a Spherical Body Shedding Coaxial Vortex Rings in an Ideal Fluid,” Regular Chaotic Dyn., 18, pp. 21–32. [CrossRef]
Tallapragada, P. , and Kelly, S. D. , 2013, “ Reduced-Order Modeling of Propulsive Vortex Shedding From a Free Pitching Hydrofoil With an Internal Rotor,” American Control Conference, June 17–19, pp. 615–620.
Tallapragada, P. , and Kelly, S. D. , 2012, “ Up a Creek Without a Paddle: Idealized Aquatic Locomotion Via Forward Vortex Shedding,” ASME Paper No. DSCC2012-MOVIC2012-8863.
Tallapragada, P. , 2015, “ A Swimming Robot With an Internal Rotor as a Nonholonomic System,” American Control Conference, July 1–3, pp. 657–662.
Tallapragada, P. , and Kelly, S. D. , 2015, “ Self-Propulsion of Free Solid Bodies With Internal Rotors Via Localized Singular Vortex Shedding in Planar Ideal Fluids,” Eur. Phys. J.: Spec. Top., 224(17), pp. 3185–3197. [CrossRef]
Kutta, W. M. , 1902, “ Auftriebskrafte in stromenden Flussigkeiten,” Ill. Aeronaut. Mitt., 6, pp. 133–135.
Milne-Thomson, L. M. , 1966, Theoretical Aerodynamics, Dover, New York.
Kelly, S. D. , and Tallapragada, P. , 2012, “ Symmetries and Constraints in Aquatic Propulsion Via Vortex Shedding,” Ninth International Conference on Flow Dynamics.
Kelly, S. D. , Fairchild, M. J. , Hassing, P. M. , and Tallapragada, P. , 2012, “ Proportional Heading Control for Planar Navigation: The Chaplygin Beanie and Fishlike Robotic Swimming,” American Control Conference, pp. 4885–4890.
Kai, T. , and Kimura, H. , 2006, “ Theoretical Analysis of Affine Constraints on a Configuration Manifold—Part I: Integrability and Nonintegrability Conditions for Affine Constraints and Foliation Structures of a Configuration Manifold,” Trans. Soc. Instrum. Control Eng., 42(3), pp. 212–221. [CrossRef]
Chaplygin, S. A. , 2008, “ On the Theory of the Motion of Nonholonomic Systems: The Reducing-Multiplier Theorem,” Regular Chaotic Dyn., 13(4), pp. 369–376 [Mat. Sb., 28(1), pp. 303–314 (1911)]. [CrossRef]
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Fedonyuk, V. , and Tallapragada, P. , 2015, “ The Stick–Slip Motion of a Chaplygin Sleigh With a Piecewise Smooth Nonholonomic Constraint,” ASME Paper No. DSCC2015-9820.
Fedonyuk, V. , and Tallapragada, P. , “ Stick-Slip Motion of the Chaplygin Sleigh With a Piecewise Smooth Nonholonomic Constraint,” ASME J. Comput. Nonlinear Dyn. (submitted).
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Osborne, J. M. , and Zenkov, D. V. , 2005, “ Steering the Chaplygin Sleigh by a Moving Mass,” American Control Conference.
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Shashikanth, B. N. , 2005, “ Poisson Brackets for the Dynamically Interacting System of a 2D Rigid Cylinder and N Point Vortices: The Case of Arbitrary Smooth Cylinder Shapes,” Regular Chaotic Dyn., 10(1), pp. 1–14. [CrossRef]
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Pollard, B. , and Tallapragada, P. , “ An Aquatic Robot Propelled by an Internal Rotor,” IEEE/ASME Trans. Mechatronics (submitted).

Figures

Grahic Jump Location
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

Grahic Jump Location
Fig. 3

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

Grahic Jump Location
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

Grahic Jump Location
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.

Grahic Jump Location
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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