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**An implicit three-dimensional fully non-hydrostatic model for free-surface flows.**
*(English)*
Zbl 1060.76621

Summary: An implicit method is developed for solving the complete three-dimensional (3D) Navier-Stokes equations. The algorithm is based upon a staggered finite difference Crank-Nicholson scheme on a Cartesian grid. A new top-layer pressure treatment and a partial cell bottom treatment are introduced so that the 3D model is fully non-hydrostatic and is free of any hydrostatic assumption. A domain decomposition method is used to segregate the resulting 3D matrix system into a series of two-dimensional vertical plane problems, for each of which a block tri-diagonal system can be directly solved for the unknown horizontal velocity. Numerical tests including linear standing waves, nonlinear sloshing motions, and progressive wave interactions with uneven bottoms are performed. It is found that the model is capable to simulate accurately a range of free-surface flow problems using a very small number of vertical layers (e.g. two-four layers). The developed model is second-order accurate in time and space and is unconditionally stable; and it can be effectively used to model 3D surface wave motions.

### MSC:

76M20 | Finite difference methods applied to problems in fluid mechanics |

76D27 | Other free boundary flows; Hele-Shaw flows |

76D05 | Navier-Stokes equations for incompressible viscous fluids |