Zhiyao Song

A high-resolution method for the depth-integrated solute transport equation based on an unstructured mesh

This paper presents a high-resolution numerical method for solving mass transport problems involving advection and anisotropic diffusion in shallow water based on unstructured mesh. An alternating operator-splitting technique is adopted to advance the numerical solution with advection and diffusion terms solved separately in two steps. By introducing a new r-factor into the Total Variation Diminishing (TVD) limiter, an improved finite-volume method is developed to solve the advection term with significant reduction of numerical diffusion and oscillation errors. In addition, a coordinate transformation is introduced to simplify the diffusion term with the Green-Gauss theorem used to deal with the anisotropic effect based on unstructured mesh. The new scheme is validated against three benchmark cases with separated and combined advection and diffusion transport processes involved. Results show that the scheme performs better than existing methods in predicting the advective transport, particularly when a sharp concentration front is in presence. The model also provides a sound solution for the anisotropic diffusion phenomenon. Anisotropic diffusion has been largely neglected by existing flow models based on unstructured mesh, which usually treat the diffusion process as being isotropic for simplicity. Based on the flow field provided by the ELCIRC model, the developed transport model was successfully applied to simulate the transport of a hypothetical conservative tracer in a bay under the influence of tides. Highlights? New r-factor for TVD limiter is presented to reduce artificial diffusion. ? Green-Gauss theorem is used to approximate the anisotropic diffusion term. ? The method can be incorporated into the ocean models based on unstructured mesh.

On the Fourier approximation method for steady water waves

A computational method for steady water waves is presented on the basis of potential theory in the physical plane with spatial variables as independent quantities. The finite Fourier series are applied to approximating the free surface and potential function. A set of nonlinear algebraic equations for the Fourier coefficients are derived from the free surface kinetic and dynamic boundary conditions. These algebraic equations are numerically solved through Newton’s iterative method, and the iterative stability is further improved by a relaxation technology. The integral properties of steady water waves are numerically analyzed, showing that (1) the set-up and the set-down are both non-monotonic quantities with the wave steepness, and (2) the Fourier spectrum of the free surface is broader than that of the potential function. The latter further leads us to explore a modification for the present method by approximating the free surface and potential function through different Fourier series, with the truncation of the former higher than that of the latter. Numerical tests show that this modification is effective, and can notably reduce the errors of the free surface boundary conditions.

On the universal third order Stokes wave solution

This paper presents a universal third-order Stokes solution with uniform current. This solution is derived on the basis of potential theory by expanding the free surface and potential function in Fourier series and determining the Fourier coefficients by solving a set of nonlinear algebraic equations through the Taylor expansion and perturbation method. The universal solution is expressed upon the still water depth with the still water level as datum and retains a global perturbation parameter. The wave set-up term generated by the self-interaction of oscillatory waves is explicitly included in the free surface function. With the use of different definitions for the wave celerity, different water levels as the datum, different non-dimensional variables as the perturbation parameter, and different treatments for the total head, the universal solution can be reduced to the existing various Stokes solutions, thus explaining the reasons and the physical significance of different non-periodic terms in them, such as the positive or negative constant term in the free surface expression and the time- or space-proportional term in the potential function.

An Efficient Numerical Model of Hyperbolic Mild-Slope Equation

Introduction of an effective wave elevation function, the simplest time-dependent hyperbolic mild-slope equation has been presented and an effective numerical model for the water wave propagation has been established combined with different boundary conditions in this paper. Through computing the effective wave elevation and transforming into the real transient wave motion, then related wave heights are computed. Because the truncation errors of the presented model only induced by the dissipation terms, but those of Lin’s model (2004) contributed by the convection terms, dissipation terms and source terms, the error analysis shows that calculation stability of this model is enhanced obviously compared with Lin’s one. The tests show that this model succeeds to the merit in Lin’s one and the computer program simpler, computational time shorter because of calculation stability enhanced efficiently and computer memory decreased obviously. The presented model has the capability of simulating exactly the location of transient wave front by the speed of wave propagation in the first test, which is important for the real-time prediction of the arrival time of water waves generated in the deep sea. The model is validated against experimental data for combined wave refraction and diffraction over submerged circular shoal on a flat bottom in the second test. Good agreements are gained. The model can be applied to the theory research and engineering applications about the wave propagation in the coastal waters.Copyright

Analytical Model for Surface Saltwater Intrusion in Estuaries

ABSTRACT Ye, R.; Song, Z.; Zhang, C.; He, Y.; Yu, S.; Kong, J., and Li, L., 2017. Analytical model for surface saltwater intrusion in estuaries. Based on the laterally averaged salt transport equation, a physically based surface saltwater intrusion model is presented for estimating the spatial distribution of salinity near the surface of the river along the estuary over high-water slack, low-water slack, and tidal-average conditions. The model is applied to simulate steady-state salinity profiles on the water surface of the Modaomen Estuary of the Pearl River delta in China. The results indicate that this model describes the surface salt profiles reasonably well and is capable of forecasting surface saltwater intrusion.

A non-negative and high-resolution finite volume method for the depth-integrated solute transport equation using an unstructured triangular mesh

This paper proposes a new high-resolution finite volume method for solving the two-dimensional (2D) solute transport equation using an unstructured mesh. A new simple r-factor algorithm is introduced into the Total Variation Diminishing flux limiter to achieve a more efficient yet accurate high-resolution scheme for solving the advection term. To avoid the physically-meaningless negative solutions resulted from using the Green–Gauss theorem, a nonlinear two-point flux approximation scheme is adopted to deal with the anisotropic diffusion term. The developed method can be readily coupled with a two-dimensional finite-volume-based flow models under unstructured triangular mesh. By integrating with the ELCIRC flow model, the proposed method was verified using three idealized benchmark cases (i.e., advection of a circle-shaped solute field, advection in a cyclogenesis flow field and transport of a initially square-shaped solute plume), and further applied to simulate the non-reactive solute transport process driven by irregular tides in the Deep Bay, eastern Pearl River Estuary of China. These cases are also simulated by models using other existing methods, including different r-factor for advection term and the Green–Gauss theorem for diffusion term. The comparison between the results from the new method and those from other existing methods demonstrated the new method could describe advection induced concentration shock and discontinuities, and anisotropic diffusion at high resolution without providing spurious oscillations and negative values.