# Dynamic coupling between fluid sloshing and vessel motion

Department of Mathematics
University of Surrey
UK

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 Dynamic coupling between fluid sloshing and vehicle motion The coupled motion between shallow water sloshing in a moving vehicle and the vehicle dynamics is considered. The movement of the vessel is restricted to horizontal motion. Motivated by the theory of Cooker (1994), a new derivation of the coupled problem in the Eulerian fluid representation is given. The aim is to simulate the nonlinear coupled motion numerically, but the nonlinear coupling causes difficulties. These difficulties are resolved by transforming to the Lagrangian represention of the fluid motion. In this representation an explicit, robust, simple numerical algorithm, based on the Stoermer-Verlet method, is proposed. Numerical results of the coupled dynamics are presented. The forced motion (neglecting the coupling) leads to quite complex fluid motion, but the coupling can be a stabilising influence. H. Alemi Ardakani & T.J. Bridges. Dynamic coupling between shallow-water sloshing and horizontal vehicle motion. Euro. J. Appl. Math. 21 479-517 (2010)   EJAM Website M.J. Cooker. Water waves in a suspended container, Wave Motion 20 385-395 (1994).

 Liquid-vessel coupling for a rotating vessel The coupled liquid-vessel motion of a rotating vessel and shallow water sloshing is considered. The equations for the fluid are the rotating shallow water equations derived, and these equations are coupled to an equation for the rotational motion of the vehicle. New equations are derived, starting with a variational formulation and the shallow-water approximation. As a test case the "pendulum slosh" problem is studied. In the pendulum slosh problem the vehicle is a pendulum with the pendulum bob containing fluid. The coupling changes the natural frequencies of the rigid body pendulum and the fluid motion in a fixed vessel. Numerical simulations are reported. M.R. Turner, H. Alemi Ardakani & T.J. Bridges. Instability of sloshing motion in a vessel undergoing pivoted oscillations, J. Fluids Struct. 52 166-180 (2014).   JFS website

 Cooker's sloshing experiment Cooker's sloshing experiment is a prototype for studying the dynamic coupling between fluid sloshing and vessel motion. It involves a container, partially filled with fluid, suspended by two cables and constrained to remain horizontal while undergoing a pendulum like motion. Cooker (1994) reported experimental results and proposed a theoretical model with linear vessel motion coupled to the linear shallow water equations. Cooker's theory identified a curious resonance. Yu (2010) extended the theory by developing a linear theory for the coupled motion with a fully two-dimensional model for the fluid, without any restriction on the depth. This extended theory showed a significant change in the natural frequencies when the depth was no longer shallow. However, resonances do not appear in Yu's theory. In this paper the theory for Cooker's sloshing experiment is extended in two directions. First, a fully nonlinear model is derived for the vessel motion, coupled to the Euler equations for the fluid, showing that the vessel motion is modelled by a nonlinear pendulum forced by the fluid motion. Secondly, we show that there is an important internal 1:1 resonance that explains the resonance noted by Cooker and extends the resonance to the case of fully two-dimensional fluid motion. The implications of the resonance for the fluid dynamics, and for the nonlinear coupled dynamics near the resonance are also discussed. H. Alemi Ardakani & T.J. Bridges. Nonlinear vessel motion and resonance in Cooker's sloshing experiment.   - new website -->

 Compendium of references on dynamic coupling B.H. Adee & I. Caglayan. The effects of free water on deck on the motions and stability of vessels, In Proc. Second Inter. Conf. Stab. Ships and Ocean Vehicles, pp. 413-426. Berlin: Springer (1982). H. Alemi Ardakani. Rigid-body motion with interior shallow-water sloshing, PhD Thesis, University of Surrey (2010). H. Alemi Ardakani & T.J. Bridges. Dynamic coupling between shallow water sloshing and horizontal vehicle motion, Europ. J. Appl. Math. 21 479-517 (2010). H. Alemi Ardakani & T.J. Bridges. The Euler equations in fluid mechanics relative to a rotating-translating reference frame Technical Report, Department of Mathematics, University of Surrey (2010). H. Alemi Ardakani & T.J. Bridges. Symplecticity of the Stormer-Verlet algorithm for coupling between the shallow water equations and horizontal vehicle motion, Technical report, University of Surrey (2010). I. Caglayan & R.L. Storch. Stability of fishing vessels with water on deck: a review, J. Ship Research 26 106-116 (1982). M.J. Cooker. Water waves in a suspended container Wave Motion 20 385-395 (1994). O.M. Faltinsen & A.N. Timokha. Sloshing, Cambridge University Press (2009). J.T. Feddema, C.R. Dohrmann, G.G. Parker, R.D. Robinett, V.J. Romero & D.J. Schmitt. Control for slosh-free motion of an open container, IEEE Control Systems Magazine 17 29-36 (1997). J. Gerrits. Dynamics of liquid-filled spacecraft, PhD thesis, University of Groningen, Netherlands (2001). M. Grundelius & B. Bernhardsson. Control of liquid slosh in an industrial packaging machine, IEEE Int. Conf. Control Appl., Kohala Coast, Hawaii, 6 pages (1999). R.A. Ibrahim. Liquid Sloshing Dynamics Cambridge University Press (2005). T. Ikeda & N. Nakagawa. Non-linear vibrations of a structure caused by water sloshing in a rectangular tank, J. Sound & Vibration 201 23-41 (1997). J. Laranjinha, J.M. Falzarano & C.G. Soares. Analysis of the dynamical behaviour of an offshore supply vessel with water on deck In the Proc. 21st Inter. Conf. Offshore Mechanics and Arctic Eng. (OMAE02), Paper No. OMAE2002-OFT28177, ASME (2002). S.J. Lee, H.H. Kim, D.H. Lee, J.W. Kim & Y.H. Kim. The effects of LNG-tank sloshing on the global motions of LNG carriers, Ocean Engineering 34 10-20 (2007). S.K. Lee, S. Surendran & G. Lee. Roll performance of a small fishing vessel with live fish tank, Ocean Engineering 32 1873-1885 (2005). P.-C. Lui & Y.K. Lou. Dynamic coupling of a liquid-tank system under transient excitation, Ocean Engng. 17 263-277 (1990). C. Prieur & J. de Halleux. Stabilization of a 1-D tank containing a fluid modeled by the shallow water equations, Systems & Control Letters 52 167-178 (2004). S. Rebouillat & D. Liksonov. Fluid-structure interaction in partially filled liquid containders: A comparative review of numerical approaches, Computers & Fluids 39 739-746 (2010). P. Ruponen, J. Matusiak, J. Luukkonen & M. Ilus. Experimental study on the behavior of a swimming pool onboard a large passenger ship Marine Technology 46 27-33 (2009). M.P. Tzamtzi & N.D. Kouvakas. Sloshing control of tilting phases of the pouring process, Inter. J. Math. Phys. Eng. Sciences 1 175-182 (2007). M.P. Tzamtzi & N.D. Kouvakas. Robustness of a robot control scheme for liquid transfer, in Novel Algortithms and Techniques in Telecommunications, Automation and Industrial Electronics, Edited by T. Sobh et al., Springer-Verlag 154-161 (2008). A.E.P. Veldman, J. Gerrits, R. Luppes, J.A. Helder & J.P.B. Vreeburg. The numerical simulation of liquid sloshing on board spacecraft, J. Comp. Physics 224 82-99 (2007). J. Yu. Effects of finite water depth on natural frequencies of suspended water tanks, Stud. Appl. Math. 125 373--391 (2010).