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Numerical Simulation of a Weakly Nonlinear Model for Water Waves with Viscosity

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Abstract

The potential flow equations which govern the free-surface motion of an ideal fluid (the water wave problem) are notoriously difficult to solve for a number of reasons. First, they are a classical free-boundary problem where the domain shape is one of the unknowns to be found. Additionally, they are strongly nonlinear (with derivatives appearing in the nonlinearity) without a natural dissipation mechanism so that spurious high-frequency modes are not damped. In this contribution we address the latter of these difficulties using a surface formulation (which addresses the former complication) supplemented with physically-motivated viscous effects recently derived by Dias et al. (Phys. Lett. A 372:1297–1302, 2008). The novelty of our approach is to derive a weakly nonlinear model from the surface formulation of Zakharov (J. Appl. Mech. Tech. Phys. 9:190–194, 1968) and Craig and Sulem (J. Comput. Phys. 108:73–83, 1993), complemented with the viscous effects mentioned above. Our new model is simple to implement while being both faithful to the physics of the problem and extremely stable numerically.

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Maria Kakleas & David P. Nicholls

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  1. Maria Kakleas
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  2. David P. Nicholls
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Correspondence to David P. Nicholls.

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Kakleas, M., Nicholls, D.P. Numerical Simulation of a Weakly Nonlinear Model for Water Waves with Viscosity. J Sci Comput 42, 274–290 (2010). https://doi.org/10.1007/s10915-009-9324-y

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  • DOI: https://doi.org/10.1007/s10915-009-9324-y

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