Dynamics of a Body with a Sharp Edge in a Viscous Fluid

    2018, Vol. 14, no. 4, pp.  473-494

    Author(s): Mamaev I. S., Tenenev V. A., Vetchanin E. V.

    This paper addresses the problem of plane-parallel motion of the Zhukovskii foil in a viscous fluid. Various motion regimes of the foil are simulated on the basis of a joint numerical solution of the equations of body motion and the Navier – Stokes equations. According to the results of simulation of longitudinal, transverse and rotational motions, the average drag coefficients and added masses are calculated. The values of added masses agree with the results published previously and obtained within the framework of the model of an ideal fluid. It is shown that between the value of circulation determined from numerical experiments, and that determined according to the model of and ideal fluid, there is a correlation with the coefficient $\mathcal{R} = 0.722$. Approximations for the lift force and the moment of the lift force are constructed depending on the translational and angular velocity of motion of the foil. The equations of motion of the Zhukovskii foil in a viscous fluid are written taking into account the found approximations and the drag coefficients. The calculation results based on the proposed mathematical model are in qualitative agreement with the results of joint numerical solution of the equations of body motion and the Navier – Stokes equations.
    Keywords: Zhukovskii foil, Navier – Stokes equations, joint solution of equations, finitedimensional model, viscous fluid, circulation, sharp edge
    Citation: Mamaev I. S., Tenenev V. A., Vetchanin E. V., Dynamics of a Body with a Sharp Edge in a Viscous Fluid, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp.  473-494

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    [1] Borisov, A. V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, R&C Dynamics, Institute of Computer Science, Izhevsk, 2005, 576 pp. (Russian)  mathscinet
    [2] Vetchanin, E. V. and Kilin, A. A., “Control of the Motion of an Unbalanced Heavy Ellipsoid in an Ideal Fluid Using Rotors”, Nelin. Dinam., 12:4 (2016), 663–674 (Russian)  mathnet  crossref  mathscinet  zmath
    [3] Zhukovsky, N. E., Collected Works, v. 1, Gostekhizdat, Moscow, 1937, 490 pp. (Russian)
    [4] Kozlov, V. V., “On the Problem of Fall of a Rigid Body in a Resisting Medium”, Mosc. Univ. Mech. Bull., 45:1 (1990), 30–36  mathscinet  zmath; Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1990, no. 1, 79–86 (Russian)  zmath
    [5] Kochin, N. E., Kibel, I. A., and Roze, N. V., Theoretical Hydrodynamics, Wiley, New York, 1964, 583 pp.  mathscinet
    [6] Lamb, H., Hydrodynamics, 6th ed., Dover, New York, 1945, 768 pp.  adsnasa
    [7] Sedov, L. I., A Course in Continuum Mechanics, v. 2, Physical Functions and Formulations of Problems, Wolters-Noordhoff, Groningen, 1972, 330 pp.
    [8] Tenenev, V. A., Vetchanin, E. V., Ilaletdinov, L. F., “Chaotic Dynamics in the Problem of the Fall of a Screw-Shaped Body in a Fluid”, Nelin. Dinam., 12:1 (2016), 99–120 (Russian)  mathnet  crossref  mathscinet  zmath
    [9] Chaplygin, S. A., “On the Pressure Exerted by a Plane-Parallel Flow upon an Obstructing Body (Contribution to the Theory of Aircraft”, The Selected Works on Wing Theory of Sergei A. Chaplygin, Garbell Research Foundation, San Francisco, 1956, 1–16  mathscinet; Mat. Sb., 28:1 (1910), 120–166 (Russian)  mathnet
    [10] Chaplygin, S. A., “On the Action of a Plane-Parallel Air Flow upon a Cylindrical Wing Moving within It”, The Selected Works on Wing Theory of Sergei A. Chaplygin, Garbell Research Foundation, San Francisco, 1956, 42–72  mathscinet
    [11] Andersen, A., Pesavento, U., and Wang, Z. J., “Analysis of Transitions between Fluttering, Tumbling and Steady Descent of Falling Cards”, J. Fluid Mech., 541 (2005), 91–104  crossref  mathscinet  zmath  adsnasa
    [12] Belmonte, A., Eisenberg, H., and Moses, E., “From Flutter to Tumble: Inertial Drag and Froude Similarity in Falling Paper”, Phys. Rev. Lett., 81:2 (1998), 345–348  crossref  adsnasa
    [13] Birkhoff, G., Hydrodynamics: A Study in Logic, Fact, and Similitude, Princeton Univ. Press, Princeton, 2015, 202 pp.  mathscinet
    [14] Bizyaev, I. A., Borisov, A. V., Kozlov, V. V., and Mamaev, I. S., Fermi-Like Acceleration and Power Law Energy Growth in Nonholonomic Systems, 2018, 33 pp., arXiv: 1807.06262 [nlin.CD]  mathscinet  zmath
    [15] Bizyaev, I. A., Borisov, A. V., and Kuznetsov, S. P., The Chaplygin Sleigh with Friction Moving due to Periodic Oscillations of an Internal Mass, 2018, 14 pp., arXiv: arXiv:1807.06340 [nlin.CD]  zmath
    [16] Bizyaev, I. A., Borisov, A. V., and Mamaev, I. S., “The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration”, Regul. Chaotic Dyn., 22:8 (2017), 955–975  mathnet  crossref  mathscinet  zmath  adsnasa
    [17] Blasius, H., “Funktionentheoretische Methoden in der Hydrodynamik”, Z. Math. Phys., 58 (1910), 90–110
    [18] Borisov, A. V., Kozlov, V. V., and Mamaev, I. S., “Asymptotic Stability and Associated Problems of Dynamics of Falling Rigid Body”, Regul. Chaotic Dyn., 12:5 (2007), 531–565  crossref  mathscinet  zmath  adsnasa
    [19] Borisov, A. V., Kuznetsov, S. P., Mamaev, I. S., and Tenenev, V. A., “Describing the Motion of a Body with an Elliptical Cross Section in a Viscous Uncompressible Fluid by Model Equations Reconstructed from Data Processing”, Tech. Phys. Lett., 42:9 (2016), 886–890  crossref  adsnasa; Pis'ma Zh. Tekh. Fiz., 42:17 (2016), 9–19 (Russian)
    [20] Borisov, A. V. and Mamaev, I. S., “On the Motion of a Heavy Rigid Body in an Ideal Fluid with Circulation”, Chaos, 16:1 (2006), 013118, 7 pp.  crossref  mathscinet  zmath  adsnasa
    [21] Borisov, A. V., Mamaev, I. S., and Vetchanin, E. V., “Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation”, Regul. Chaotic Dyn., 23:4 (2018), 480–502  mathnet  crossref  mathscinet  adsnasa
    [22] Borisov, A. V., Vetchanin, E. V., and Kilin, A. A., “Control Using Rotors of the Motion of a Triaxial Ellipsoid in a Fluid”, Math. Notes, 102:3–4 (2017), 455–464  mathnet  crossref  mathscinet  zmath; Mat. Zametki, 102:4 (2017), 503–513 (Russian)  crossref  zmath
    [23] Brendelev, V. N., “On the Realization of Constraints in Nonholonomic Mechanics”, J. Appl. Math. Mech., 45:3 (1981), 351–355  crossref  mathscinet  zmath; Prikl. Mat. Mekh., 45:3 (1981), 481–487 (Russian)  mathscinet  zmath
    [24] Brown, C. E. and Michael, W. H., “Effect of Leading-Edge Separation on the Lift of a Delta Wing”, J. Aeronaut. Sci., 21:10 (1954), 690–694  crossref  zmath
    [25] Childress, S., Spagnolie, S. E., and Tokieda, T., “A bug on a Raft: Recoil Locomotion in a Viscous Fluid”, J. Fluid Mech., 669 (2011), 527–556  crossref  mathscinet  zmath  adsnasa
    [26] Eldredge, J. D., “Numerical Simulation of the Fluid Dynamics of $2D$ Rigid Body Motion with the Vortex Particle Method”, J. Comput. Phys., 221:2 (2007), 626–648  crossref  mathscinet  zmath  adsnasa
    [27] Mason, R. J., Fluid Locomotion and Trajectory Planning for Shape-Changing Robots, PhD Dissertation, California Institute of Technology, Pasadena, Calif., 2003, 264 pp.
    [28] Kasper, W., Aircraft Wing with Vortex Generation, U.S. Patent No. 3 831 885 (27 Aug 1974)
    [29] Karapetyan, A. V., “On Realizing Nonholonomic Constraints by Viscous Friction Forces and Celtic Stones Stability”, J. Appl. Math. Mech., 45:1 (1981), 30–36  crossref  mathscinet  zmath; Prikl. Mat. Mekh., 45:1 (1981), 42–51 (Russian)  mathscinet  zmath
    [30] Kirchhoff, G., Vorlesungen über mathematische Physik, v. 1, Mechanik, Teubner, Leipzig, 1876, 466 pp.
    [31] Klenov, A. I. and Kilin, A. A., “Influence of Vortex Structures on the Controlled Motion of an Above-Water Screwless Robot”, Regul. Chaotic Dyn., 21:7–8 (2016), 927–938  mathnet  crossref  mathscinet  zmath  adsnasa
    [32] Kozlov, V. V., “Realization of Nonintegrable Constraints in Classical Mechanics”, Sov. Phys. Dokl., 28 (1983), 735–737  mathscinet  zmath  adsnasa; Dokl. Akad. Nauk SSSR, 272:3 (1983), 550–554 (Russian)  mathscinet  zmath
    [33] Kozlov, V. V. and Ramodanov, S. M., “The Motion of a Variable Body in an Ideal Fluid”, J. Appl. Math. Mech., 65:4 (2001), 579–587  crossref  mathscinet  zmath; Prikl. Mat. Mekh., 65:4 (2001), 592–601 (Russian)  mathscinet  zmath
    [34] Kozlov, V. V. and Ramodanov, S. M., “On the Motion of a Body with a Rigid Hull and Changing Geometry of Masses in an Ideal Fluid”, Dokl. Phys., 47:2 (2002), 132–135  mathnet  crossref  mathscinet  adsnasa; Dokl. Akad. Nauk, 382:4 (2002), 478–481 (Russian)
    [35] Kozlov, V. V. and Onishchenko, D. A., “Motion of a Body with Undeformable Shell and Variable Mass Geometry in an Unbounded Perfect Fluid: Special Issue Dedicated to Victor A. Pliss on the Occasion of His 70th Birthday”, J. Dynam. Differential Equations, 15:2–3 (2003), 553–570  crossref  mathscinet  zmath  adsnasa
    [36] Kutta, W. M., “Auftriebskräfte in strömenden Flüssigkeiten”, Illustr. aeronaut. Mitteilungen, 6 (1902), 133–135
    [37] Kuznetsov, S. P., “Plate Falling in a Fluid: Regular and Chaotic Dynamics of Finite-Dimensional Models”, Regul. Chaotic Dyn., 20:3 (2015), 345–382  mathnet  crossref  mathscinet  zmath  adsnasa
    [38] Michelin, S. and Llewellyn Smith, S. G., “An Unsteady Point Vortex Method for Coupled Fluid-Solid Problems”, Theor. Comput. Fluid Dyn., 23:2 (2009), 127–153  crossref  zmath
    [39] Mittal, R., Dong, H., Bozkurttas, M., Najjar, F. M., Vargas, A., and von Loebbecke, A., “A Versatile Sharp Interface Immersed Boundary Method for Incompressible Flows with Complex Boundaries”, J. Comput. Phys., 227:10 (2008), 4825–4852  crossref  mathscinet  zmath  adsnasa
    [40] Mougin, G. and Magnaudet, J., “The Generalized Kirchhoff Equations and Their Application to the Interaction between a Rigid Body and an Arbitrary Time-Dependent Viscous Flow”, Int. J. Multiph. Flow, 28:11 (2002), 1837–1851  crossref  zmath
    [41] Nelson, R., Protas, B., and Sakajo, T., “Linear Feedback Stabilization of Point-Vortex Equilibria near a Kasper Wing”, J. Fluid Mech., 827 (2017), 121–154  crossref  mathscinet  adsnasa
    [42] Pollard, B., Berkey, T., and Tallapragada, Ph., “Direct Tail Actuation vs Internal Rotor Propulsion in Aquatic Robots”, ASME 2017 Dynamic Systems and Control Conference (Tysons, Va., USA, Oct 11–13, 2017), v. 1, DSCC2017-5208, V001T30A008, 6 pp.
    [43] Ramodanov, S. M. and Tenenev, V. A., “Motion of a Body with Variable Distribution of Mass in a Boundless Viscous Liquid”, Nelin. Dinam., 7:3 (2011), 635–647 (Russian)  mathnet  crossref  mathscinet
    [44] Ramodanov, S. M., Tenenev, V. A., and Treschev, D. V., “Self-Propulsion of a Body with Rigid Surface and Variable Coefficient of Lift in a Perfect Fluid”, Regul. Chaotic Dyn., 17:6 (2012), 547–558  mathnet  crossref  mathscinet  zmath  adsnasa
    [45] Tallapragada, Ph., “A Swimming Robot with an Internal Rotor As a Nonholonomic System”, Proc. of the American Control Conference (Chicago, Ill., USA, July 1–3, 2015), 657–662
    [46] Tanabe, Y. and Kaneko, K., “Behavior of a Falling Paper”, Phys. Rev. Lett., 73:10 (1994), 1372–1375  crossref  adsnasa
    [47] van der Waerden, B. L., Mathematische Statistik, Grundlehren Math. Wiss., 87, 3rd ed., Springer, Berlin, 2012, 360 pp.  mathscinet
    [48] Vetchanin, E. V. and Gladkov, E. S., “Identification of Parameters of the Model of Toroidal Body Motion Using Experimental Data”, Nelin. Dinam., 14:1 (2018), 99–121 (Russian)  mathnet  crossref  mathscinet
    [49] Vetchanin, E. V. and Kilin, A. A., “Controlled Motion of a Rigid Body with Internal Mechanisms in an Ideal Incompressible Fluid”, Proc. Steklov Inst. Math., 295 (2016), 302–332  mathnet  crossref  mathscinet  zmath; Tr. Mat. Inst. Steklova, 295 (2016), 321–351 (Russian)  mathscinet  zmath
    [50] Vetchanin, E. V. and Kilin, A. A., “Control of Body Motion in an Ideal Fluid Using the Internal Mass and the Rotor in the Presence of Circulation around the Body”, J. Dyn. Control Syst., 23:2 (2017), 435–458  crossref  mathscinet  zmath
    [51] Vetchanin, E. V., Mamaev, I. S., and Tenenev, V. A., “The Self-Propulsion of a Body with Moving Internal Masses in a Viscous Fluid”, Regul. Chaotic Dyn., 18:1–2 (2013), 100–117  mathnet  crossref  mathscinet  zmath  adsnasa
    [52] Woolsey, C. A. and Leonard, N. E., “Underwater Vehicle Stabilization by Internal Rotors”, Proc. of the American Control Conference (San Diego, Calif., 1999), 3417–3421
    [53] Fedorov, Yu. N. and García-Naranjo, L. C., “The Hydrodynamic Chaplygin Sleigh”, J. Phys. A, 43:43 (2010), 434013, 18 pp.  crossref  mathscinet  zmath
    [54] Karavaev, Yu. L., Kilin, A. A., and Klekovkin, A. V., “Experimental Investigations of the Controlled Motion of a Screwless Underwater Robot”, Regul. Chaotic Dyn., 21:7–8 (2016), 918–926  mathnet  crossref  mathscinet  zmath  adsnasa

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