**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30172

##### A Nonlinear ODE System for the Unsteady Hydrodynamic Force – A New Approach

**Authors:**
Osama A. Marzouk

**Abstract:**

**Keywords:**
reduced-order model,
wake oscillator,
nonlinear,
ODEsystem

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1331863

**References:**

[1] J. H. Lienhard, "Synopsis of lift, drag and vortex frequency for rigid circular cylinders," College of Engineering Research Division bulletin 300, Washington State University, 1966.

[2] C. H. K. Williamson, "Vortex dynamics in the cylinder wake," Annual Reviews of Fluid Mechanics, vol. 28, pp. 477-539, 1996.

[3] B. M. Sumer and J. Fredse, Hydrodynamics around Cylindrical Structures, World Scientific, 2006.

[4] C. A. J. Fletcher, Computational Techniques for Fluid Dynamics, Volume II, Second Edition, Germany: Springer-Verlag, 1991.

[5] K. A. Hoffmann and S. T. Chiang, Computational Fluid Dynamics for Engineers, Volume II, Kansas: Engineering Education System, 1993.

[6] T. Cebeci, J. P. Shao, F. Kafyeke, and E. Laurendeau, Computational Fluid Dynamics for Engineers: From Panel to Navier-Stokes Methods with Computer Programs, Germany: Springer, 2005.

[7] R. D. Henderson, "Details of the drag curve near the onset of vortex shedding," Physics of Fluids, vol. 7(9), pp. 2102-2104, 1995.

[8] G. Birkhoff and E. H. Zarantonello, Jets, Wakes, and Cavities, New York: Academic Press, 1957.

[9] R. E. D. Bishop and A. Y. Hassan, "The lift and drag forces on a circular cylinder oscillating in a flowing fluid," Proceedings of the Royal Society Series A, vol. 277, pp. 51-75, 1964.

[10] R. T. Hartlen and I. G. Currie, "Lift-oscillator model of vortex vibration," Journal of Engineering Mechanics, vol. 96, pp. 577-591, 1970.

[11] I. G. Currie, R. T. Hartlen, and W. W. Martin, "The response of circular cylinders to vortex shedding," IUTAM/IAHK Symposium on Flow- Induced Structural Vibrations, Karlsruhe, Germany, August 14-16, pp. 128-142, 1972.

[12] R. A. Skop and O. M. Griffin, "A model for the vortex-excited resonant response of bluff cylinders," Journal of Sound and Vibration, vol. 27(2), pp. 225-233, 1973.

[13] W. D. Iwan and R. D. Blevins, "A model for vortex induced oscillation of structures," Journal of Applied Mechanics, vol. 41(3), pp. 581-586, 1974.

[14] R. D. Blevins, "Flow Induced Vibration of Bluff Structures," Ph.D. Dissertation, Dynamics Laboratory, California Institute of Technology, Pasadena, California, 1974.

[15] R. Landl, "A mathematical model for vortex-excited vibrations of bluff bodies," Journal of Sound and Vibration, vol. 42(2), pp. 219-234, 1975.

[16] I. G. Currie and D. H. Turnbull, "Streamwise oscillations of cylinders near the critical Reynolds number," Journal of Fluids and Structures, vol. 1, pp. 185-196, 1987.

[17] S. Balasubramanian and R. A. Skop, "A nonlinear oscillator model for vortex shedding from cylinders and cones in uniform and shear flows," Journal of Fluids and Structures, vol. 10(3), pp. 197-214, 1996.

[18] S. Krenk and S. R. K. Nielsen, "Energy balanced double oscillator model for vortex-induced vibrations," Journal of Engineering Mechanics, vol. 125(3), pp. 263-271, 1999.

[19] N. W. Mureithi, S. Goda, and H. Kanki, "A nonlinear dynamics analysis of vortex-structure interaction models," Journal of Pressure Vessel Technology, vol. 123(4), pp. 475-479, 2001.

[20] A. H. Nayfeh, F. Owis, and M. R. Hajj, "A model for the coupled lift and drag on a circular cylinder," ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Chicago, IL, September 2-6, 2003.

[21] M. Facchinetti, E. De Langre, and F. Biolley, "Coupling of structure and wake oscillators in vortex-induced vibrations," Journal of Fluids and Structures, vol. 19, pp. 123-140, 2004.

[22] L. Qin, "Development of Reduced-Order Models for Lift and Drag on Oscillating Cylinders with Higher-Order Spectral Moments," Ph.D. Dissertation, Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, Virginia, 2004.

[23] O. A. Marzouk, A. H. Nayfeh, H. Arafat, and I. Akhtar, "Modeling steady-state and transient forces on a cylinder," Journal of Vibration and Control, vol. 13(7), pp. 1065-1091, 2007.

[24] O. M. Griffin, R. A. Skop, and G. H. Koopmann, "The vortex-excited resonant vibrations of circular cylinders," Journal of Sound and Vibration, vol. 31(2), pp. 235-249, 1973.

[25] R. A. Skop and O. M. Griffin, "On a theory for the vortex-excited oscillations of flexible cylindrical structures," Journal of Sound and Vibration, vol. 41(3), pp. 263-274, 1975.

[26] R. H. M. Ogink and A. V. Metrikine, "A wake oscillator with frequency dependent tuning coefficients for the modeling of VIV," 27th International Conference on Offshore Mechanics and Arctic Engineering, Estoril, Portugal, June 15-19, 2008.

[27] W.J. Kim and N. C. Perkins, "Two-dimensional vortex-induced vibration of cable suspensions," Journal of Fluids and Structures, vol. 16(2), pp. 229-245, 2002.

[28] O. A. Marzouk and A. H. Nayfeh, "Differential/Algebraic Wake Model Based on the Total Fluid Force and Its Direction, and the Effect of Oblique Immersed-Body Motion on ÔÇÿType-1- and ÔÇÿType-2- Lock-in," 47th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, Florida, January 5-8, 2009.

[29] R. D. Blevins, Flow-Induced Vibration, First Edition, New York: Van Nostrand Reinhold, 1977.

[30] R. Gopalkrishnan, "Vortex-Induced Forces on Oscillating Bluff Cylinders," Ph.D. Dissertation, Department of Ocean Engineering, Massachusetts Institute of Technology, Massachusetts, 1993.

[31] J. Carberry, J. Sheridan, and D. Rockwell, "Controlled oscillations of a cylinder: forces and wake modes," Journal of Fluid Mechanics, vol. 538, pp. 31-69, 2005.

[32] X. Lu and C. Dalton, "Calculation of the timing of vortex formation from an oscillating circular cylinder," Journal of Fluids and Structures, vol. 10(5), pp. 527-541, 1996.

[33] S. Dong and G. E. Karniadakis, "DNS of flow past a stationary and oscillating cylinder at Re=10000," Journal of Fluids and Structures, vol. 20(4), pp. 519-531, 2005.

[34] O. A. Marzouk and A. H. Nayfeh, "Physical interpretation of the nonlinear phenomena in excited wakes," 27th ASME Wind Energy Symposium, Reno, Nevada, January 7-10, 2008.

[35] A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, New York:Wiley, 1979.

[36] O. A. Marzouk and A. H. Nayfeh, "New wake models with capability of capturing nonlinear physics," 27th International Conference on Offshore Mechanics and Arctic Engineering, Estoril, Portugal, June 15-19, 2008.

[37] R. E. D. Bishop and A. Y. Hassan, "The lift and drag forces on a circular cylinder in a flowing fluid," Proceedings of the Royal Society Series A, vol. 277, pp. 32-50, 1964.

[38] C. Farell, "Flow around circular cylinders: fluctuating loads," Journal of the Engineering Mechanic, vol. 107, pp. 565-587, 1981.

[39] P. R. Spalart and S. R. Allmaras, "A one-equation turbulence model for aerodynamic flows," 30th AIAA Aerospace Sciences Meeting, Reno, Nevada, January 6-9, 1992.

[40] S. E. Rogers, D. Kwak, and C. Kiris, "Steady and unsteady solutions of the incompressible Navier-Stokes equations," AIAA Journal, vol. 29(4), pp. 603-610, 1991.

[41] F. M. Owis and A. H. Nayfeh, "Numerical simulation of 3-D incompressible, multi-phase flows over cavitating projectiles," European Journal of Mechanics - B/Fluids, vol. 23(2), pp. 339-351, 2004.

[42] A. Roshko, "On the development of turbulent wakes," NACA Technical Note 2913, 1953.

[43] S. K. Jordan and J. E. Fromm, "Oscillatory drag, lift, and torque on a cylinder in a uniform flow," Physics of Fluids, vol. 15, pp. 371-376, 1972.

[44] M. Braza, P. Chassaing, and H. Ha Minh, "Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder," Journal of Fluid Mechanics, vol. 165(2), pp. 79-130, 1986.

[45] C. H. K. Williamson, "Oblique and parallel models of vortex shedding in the wake of a circular cylinder at low Reynolds numbers," Journal of Fluid Mechanics, vol. 206, pp. 579-627, 1989.

[46] P. K. Stansby and A. Slaouti, "Simulation of vortex shedding including blockage by the random-vortex and other methods," International Journal for Numerical Methods in Fluids, vol 17(11), pp. 1003-1013, 1993.

[47] D. Shiels, A. Leonard, and A. Roshko, "Flow-induced vibration of a circular cylinder at limiting structural parameters," Journal of Fluids and Structures, vol. 15, pp. 3-21, 2001.

[48] H. Q. Zhang, U. Fey, B. R. Noack, M. K┬¿onig, and H. Eckelmann, "On the transition of the cylinder wake," Physics of Fluids, vol. 7(4), pp. 779-794, 1995.

[49] C. Y. Zhou, R. M. C. So, and K. Lam, "Vortex-induced vibrations of an elastic circular cylinder," Journal of Fluids and Structures, vol. 13, pp. 165-189, 1999.

[50] Y. Wang, J. Yang, and X. Li, "CFD Analysis of unsteady flows around a new cell-truss spar and the corresponding vortex-induced motions," 27th International Conference on Offshore Mechanics and Arctic Engineering, Estoril, Portugal, June 15-19, 2008.

[51] L. Kaiktsis, G. S. Triantafyllou, and M. ┬¿O zbas, "Excitation, inertia, and drag forces on a cylinder vibrating transversely to a steady flow," Journal of Fluids and Structures, vol. 23(1), pp. 1-21, 2007.

[52] H. M. Blackburn and R. D. Henderson, "A study of two-dimensional flow past an oscillating cylinder," Journal of Fluid Mechanics, vol. 385, pp. 255-286, 1999.

[53] J. F. Ravoux, A. Nadim, and H. Haj-Hariri, "An embedding method for bluff body flows: interactions of two side-by-side cylinder wakes," Theoretical and Computational Fluid Dynamics, vol. 16, pp. 433-466, 2003.

[54] C. Norberg, "Fluctuating lift on a circular cylinder: review and new measurements," Journal of Fluids and Structures, vol. 17, pp. 57-96, 2003.

[55] Z. C. Zheng and N. Zhang, "Frequency effects on lift and drag for flow past an oscillating cylinder," Journal of Fluids and Structures, vol. 24(3), pp. 382-399, 2008.

[56] R. Clift, J. R. Grace, and M. E. Weber, Bubbles, Drops and Particles, New York: Academic Press, 1978.

[57] C. Wieselsberger, "Neuere Feststellungen ber die Gesetze des Flssigkeits- und Luftwider-stands," Physikalische Zeitschrift, Vol. 22, pp. 321-328, 1921.