A two‐dimensional vertical non‐hydrostatic σ model with an implicit method for free‐surface flows

Abstract

SUMMARYAn implicit ÿnite di erence model in the co-ordinate system is developed for non-hydrostatic, twodimensional vertical plane free-surface ows. To accurately simulate interaction of free-surface ows with uneven bottoms, the unsteady Navier-Stokes equations and the free-surface boundary condition are solved simultaneously in a regular transformed domain using a fully implicit method in two steps. First, the vertical velocity and pressure are expressed as functions of horizontal velocity. Second, substituting these relationship into the horizontal momentum equation provides a block tri-diagonal matrix system with the unknown of horizontal velocity, which can be solved by a direct matrix solver without iteration. A new treatment of non-hydrostatic pressure condition at the top-layer cell is developed and found to be important for resolving the phase of wave propagation. Additional terms introduced by the co-ordinate transformation are discretized appropriately in order to obtain accurate and stable numerical results. The developed model has been validated by several tests involving free-surface ows with strong vertical accelerations and non-linear waves interacting with uneven bottoms. Comparisons among numerical results, analytical solutions and experimental data show the capability of the model to simulate free-surface ow problems.

A nonhydrostatic version of FVCOM: 1. Validation experiments

“…Setting q′ = 0 within the top layer cell was proposed to avoid such a difficulty in some nonhydrostatic models. This approach, however, can cause a significant phase error for surface wave problems [Yuan and Wu, 2004;Zijlema and Stelling, 2005]. Instead of directly specifying w* at the surface, we reconstruct the gradient of w* in the top layer from the interior values using a fourth-order Lagrangian extrapolation method.…”

“…Since these two methods represent a fairly straightforward augmentation of existing hydrostatic ocean models and are relatively inexpensive in terms of computational costs, they are widely used in the development of nonhydrostatic ocean models. Methods that include the spontaneous momentum and pressure update have also been employed [Stelling and Zijlema, 2003;Yuan and Wu, 2004;Namin et al, 2001]; however, they are primarily used for inviscid surface wave problems.…”

A non-hydrostatic pressure distribution solver for the nonlinear shallow water equations over irregular topography

“…Following ( Yuan and Wu, 2004 ), in this test, we address a solitary wave that propagates in a lab flume, consisting of a horizontal stretch (with initial water depth h 0 = 0.0762 m), a sloping beach (with slope 1:20) and a second horizontal stretch (with initial water depth h 1 = h 0 /2) (see Fig. 4 ).…”

“…The numerical flume is 6 m long and 0.1 m wide ( Yuan and Wu, 2004 ), both upstream and downstream channel ends are open boundaries, while impervious wall conditions are assigned at the lateral sides. According to Yuan and Wu (2004 ), we initially place the peak of the solitary wave at x = x 0 = −0.8 m so that we move the upstream (offshore) channel side from x = 0 to x = −2 m (see Fig.…”

“…According to Yuan and Wu (2004 ), we initially place the peak of the solitary wave at x = x 0 = −0.8 m so that we move the upstream (offshore) channel side from x = 0 to x = −2 m (see Fig. 4 ).…”