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\c{\bf   SPECTRAL SIMULATIONS OF VOLCANIC ERUPTIONS AND TSUNAMI WAVES }

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\c{        First A. Uthor$^*$ and Second A. U. Thor $^{\dag}$        }

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\c{\vbox{\halign{ \qquad\lft{#}\qquad    &\qquad    \lft{#} \cr
 \llap{$^*$} Center for Lava Mechanics   &  \llap{$^{\dag}$} Center for Abrupt Wave Mechanics \cr
 \ University of St Helens               &  \ Atlantis University                 \cr
 \ Portland, Oregon 96454, USA           &  \ Heraclitos, 12345 Crete, Greece     \cr
 \ john.doe@crater.edu                   \cr
}} }  
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\noindent 
Vortex induced vibrations (VIV) of marine cables, usually termed strumming, 
increases fatigue due to the large hydrodynamic forces. VIV is a critical factor in 
the design of underwater cable systems, drilling risers, and offshore platforms. 
It is therefore important to understand and be able to predict from first principles 
the hydrodynamic forces and motion of cables and beams caused by flow induced 
vibrations. While there is a substantial data base for cable dynamics derived 
from field experiments (see [1] and references therein) there are very few 
laboratory experiments to study cable flows under controlled conditions. 
Simulations from first principles have only recently become possible [2]. 

In this paper we present a numerical study of flows past flexible cables and 
beams at lock-in and non-lock-in states. In particular, we are concentrating on 
the aspects of transition to turbulence in the wake and its effects on the motion of 
the structure. Transition to turbulence in the wake of a fixed cylinder occurs at 
Reynolds number between 250 and 400, as has been established by experimental results 
and our previous simulation studies. However, the transition process changes 
fundamentally if the cylinder is flexible and vibrates freely. In this numerical 
study, we consider flow past flexible beams and cables undergoing free 
oscillations subject to near lockin excitation. We investigate different 
vibrating conditions, corresponding to varying the bending stiffness (beams), 
tension (cables) and the mass ratio parameter, in order to determine the 
new transition mechanisms. In general, cables tend to promote wake transition 
whereas beams tend to delay transition compared to the fixed cylinder behavior.

The simulations are based on a spectral element method reformulated in 
body-fitted coordinates for this problem [3]. The flow equations are the incompressible 
Navier-Stokes equations while the equation that describes the motion of the cable 
for free vibrations is the wave equation with a forcing term due to the fluid 
forces on the cable. The three-dimensional Navier-Stokes equations are solved 
using a parallel spectral element/Fourier method. Spectral elements are used 
to discretize the $x$-$y$ planes, while a Fourier expansion is used in the 
$z$-direction, i.e. along the cable. Each Fourier mode is assigned to a separate 
processor allowing efficient parallel computation. Typically, more than one 
million degrees of freedom are employed with 128 nodes along the cable axis.

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{\bf References}
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[1] H. Lamb, {\it Hydrodynamics}, Dover, New York, 1945.

[2] D.J. Newman and G.E. Karniadakis, ``Simulations and models of flow over a 
flexible cable: Standing wave patterns," {\it Proc. ASME/JSME Fluids Eng. Conf.}, 
August 1995, Hilton Head, SC.

[3]  J.L. Author, ``Cable vibrations from shed vortices,'' 
{\it J. Cable Mech.} {\bf 25}, 111--222, 1965.

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{\bf Keywords}: dynamics, vibrations, waves

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