An efficient and accurate non‐hydrostatic model with embedded Boussinesq‐type like equations for surface wave modeling

A novel approach that embeds the Boussinesq‐type like equations into an implicit non‐hydrostatic model (NHM) is developed. Instead of using an integration approach, Boussinesq‐type like equations with a reference velocity under a virtual grid system are introduced to analytically obtain an analytical form of pressure distribution at the top layer. To determine the size of vertical layers in the model, a top‐layer control technique is proposed and the reference location is employed to optimize linear wave dispersion property. The efficiency and accuracy of this NHM with Boussinesq‐type like equations (NHM‐BTE) are critically examined through four free‐surface wave examples. Overall model results show that NHM‐BTE using only two vertical layers is capable of accurately simulating highly dispersive wave motion and wave transformation over irregular bathymetry. Copyright © 2008 John Wiley & Sons, Ltd.     

Depth‐integrated free‐surface flow with a two‐layer non‐hydrostatic formulation

This paper presents the derivation of a depth‐integrated wave propagation and runup model from a system of governing equations for two‐layer non‐hydrostatic flows. The governing equations are transformed into an equivalent, depth‐integrated system, which separately describes the flux‐dominated and dispersion‐dominated processes. The depth‐integrated system reproduces the linear dispersion relation within a 5 error for water depth parameter up to kd = 11, while allowing direct implementation of a momentum conservation scheme to model wave breaking and a moving‐waterline technique for runup calculation. A staggered finite‐difference scheme discretizes the governing equations in the horizontal dimension and the Keller box scheme reconstructs the non‐hydrostatic terms in the vertical direction. An semi‐implicit scheme integrates the depth‐integrated flow in time with the non‐hydrostatic pressure determined from a Poisson‐type equation. The model is verified with solitary wave propagation in a channel of uniform depth and validated with previous laboratory experiments for wave transformation over a submerged bar, a plane beach, and fringing reefs. Copyright © 2011 John Wiley & Sons, Ltd.     

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

An implicit finite difference model in the σ co‐ordinate system is developed for non‐hydrostatic, two‐dimensional vertical plane free‐surface flows. To accurately simulate interaction of free‐surface flows with uneven bottoms, the unsteady Navier–Stokes equations and the free‐surface boundary condition are solved simultaneously in a regular transformed σ 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 σ 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 flows 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 flow problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Boussinesq cut‐cell model for non‐linear wave interaction with coastal structures

Boussinesq models describe the phase‐resolved hydrodynamics of unbroken waves and wave‐induced currents in shallow coastal waters. Many enhanced versions of the Boussinesq equations are available in the literature, aiming to improve the representation of linear dispersion and non‐linearity. This paper describes the numerical solution of the extended Boussinesq equations derived by Madsen and Sørensen (Coastal Eng. 1992; 15 :371–388) on Cartesian cut‐cell grids, the aim being to model non‐linear wave interaction with coastal structures. An explicit second‐order MUSCL‐Hancock Godunov‐type finite volume scheme is used to solve the non‐linear and weakly dispersive Boussinesq‐type equations. Interface fluxes are evaluated using an HLLC approximate Riemann solver. A ghost‐cell immersed boundary method is used to update flow information in the smallest cut cells and overcome the time step restriction that would otherwise apply. The model is validated for solitary wave reflection from a vertical wall, diffraction of a solitary wave by a truncated barrier, and solitary wave scattering and diffraction from a vertical circular cylinder. In all cases, the model gives satisfactory predictions in comparison with the published analytical solutions and experimental measurements. Copyright © 2007 John Wiley & Sons, Ltd.     

An efficient curvilinear non‐hydrostatic model for simulating surface water waves

An efficient curvilinear non‐hydrostatic free surface model is developed to simulate surface water waves in horizontally curved boundaries. The generalized curvilinear governing equations are solved by a fractional step method on a rectangular transformed domain. Of importance is to employ a higher order (either quadratic or cubic spline function) integral method for the top‐layer non‐hydrostatic pressure under a staggered grid framework. Model accuracy and efficiency, in terms of required vertical layers, are critically examined on a linear progressive wave case. The model is then applied to simulate waves propagating in a canal with variable widths, cnoidal wave runup around a circular cylinder, and three‐dimensional wave transformation in a circular channel. Overall the results show that the curvilinear non‐hydrostatic model using a few, e.g. 2–4, vertical layers is capable of simulating wave dispersion, diffraction, and reflection due to curved sidewalls. Copyright © 2010 John Wiley & Sons, Ltd.     

A depth‐integrated non‐hydrostatic finite element model for wave propagation

This paper presents a two‐dimensional finite element model for simulating dynamic propagation of weakly dispersive waves. Shallow water equations including extra non‐hydrostatic pressure terms and a depth‐integrated vertical momentum equation are solved with linear distributions assumed in the vertical direction for the non‐hydrostatic pressure and the vertical velocity. The model is developed based on the platform of a finite element model, CCHE2D. A physically bounded upwind scheme for the advection term discretization is developed, and the quasi second‐order differential operators of this scheme result in no oscillation and little numerical diffusion. The depth‐integrated non‐hydrostatic wave model is solved semi‐implicitly: the provisional flow velocity is first implicitly solved using the shallow water equations; the non‐hydrostatic pressure, which is implicitly obtained by ensuring a divergence‐free velocity field, is used to correct the provisional velocity, and finally the depth‐integrated continuity equation is explicitly solved to satisfy global mass conservation. The developed wave model is verified by an analytical solution and validated by laboratory experiments, and the computed results show that the wave model can properly handle linear and nonlinear dispersive waves, wave shoaling, diffraction, refraction and focusing. Copyright © 2013 John Wiley & Sons, Ltd.     

Three‐dimensional modeling of non‐hydrostatic free‐surface flows on unstructured grids

A three‐dimensional numerical model is presented for the simulation of unsteady non‐hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics‐based scheme, which simulates sub‐critical and super‐critical flows. Three‐dimensional velocity components are considered in a collocated arrangement with a σ‐coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term to ensure exact mass conservation. The unstructured grid in the horizontal direction and the σ coordinate in the vertical direction facilitate the use of the model in complicated geometries. Solution of the non‐hydrostatic equations enables the model to simulate short‐period waves and vertically circulating flows. Copyright © 2015 John Wiley & Sons, Ltd.     

A new σ‐transform based Fourier‐Legendre‐Galerkin model for nonlinear water waves

This paper presents a new spectral model for solving the fully nonlinear potential flow problem for water waves in a single horizontal dimension. At the heart of the numerical method is the solution to the Laplace equation which is solved using a variant of the $\sigma$‐transform. The method discretizes the spatial part of the governing equations using the Galerkin method and the temporal part using the classical fourth‐order Runge‐Kutta method. A careful investigation of the numerical method's stability properties is carried out, and it is shown that the method is stable up to a certain threshold steepness when applied to nonlinear monochromatic waves in deep water. Above this threshold artificial damping may be employed to obtain stable solutions. The accuracy of the model is tested for: (i) highly nonlinear progressive wave trains, (ii) solitary wave reflection, and (iii) deep water wave focusing events. In all cases it is demonstrated that the model is capable of obtaining excellent results, essentially up to very near breaking.     

An implicit three‐dimensional fully non‐hydrostatic model for free‐surface flows

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 accuracy in time and space and is unconditionally stable; and it can be effectively used to model 3D surface wave motions. Copyright © 2004 John Wiley & Sons, Ltd.     

A high‐order Petrov–Galerkin finite element method for the classical Boussinesq wave model

A high‐order Petrov–Galerkin finite element scheme is presented to solve the one‐dimensional depth‐integrated classical Boussinesq equations for weakly non‐linear and weakly dispersive waves. Finite elements are used both in the space and the time domains. The shape functions are bilinear in space–time, whereas the weighting functions are linear in space and quadratic in time, with C⁰‐continuity. Dispersion correction and a highly selective dissipation mechanism are introduced through additional streamline upwind terms in the weighting functions. An implicit, conditionally stable, one‐step predictor–corrector time integration scheme results. The accuracy and stability of the non‐linear discrete equations are investigated by means of a local Taylor series expansion. A linear spectral analysis is used for the full characterization of the predictor–corrector inner iterations. Based on the order of the analytical terms of the Boussinesq model and on the order of the numerical discretization, it is concluded that the scheme is fourth‐order accurate in terms of phase velocity. The dissipation term is third order only affecting the shortest wavelengths. A numerical convergence analysis showed a second‐order convergence rate in terms of both element size and time step. Four numerical experiments are addressed and their results are compared with analytical solutions or experimental data available in the literature: the propagation of a solitary wave, the oscillation of a flat bottom closed basin, the oscillation of a non‐flat bottom closed basin, and the propagation of a periodic wave over a submerged bar. Copyright © 2008 John Wiley & Sons, Ltd.     

A σ‐coordinate three‐dimensional numerical model for surface wave propagation

A three‐dimensional numerical model based on the full Navier–Stokes equations (NSE) in σ‐coordinate is developed in this study. The σ‐coordinate transformation is first introduced to map the irregular physical domain with the wavy free surface and uneven bottom to the regular computational domain with the shape of a rectangular prism. Using the chain rule of partial differentiation, a new set of governing equations is derived in the σ‐coordinate from the original NSE defined in the Cartesian coordinate. The operator splitting method (Li and Yu, Int. J. Num. Meth. Fluids 1996; 23 : 485–501), which splits the solution procedure into the advection, diffusion, and propagation steps, is used to solve the modified NSE. The model is first tested for mass and energy conservation as well as mesh convergence by using an example of water sloshing in a confined tank. Excellent agreements between numerical results and analytical solutions are obtained. The model is then used to simulate two‐ and three‐dimensional solitary waves propagating in constant depth. Very good agreements between numerical results and analytical solutions are obtained for both free surface displacements and velocities. Finally, a more realistic case of periodic wave train passing through a submerged breakwater is simulated. Comparisons between numerical results and experimental data are promising. The model is proven to be an accurate tool for consequent studies of wave‐structure interaction. Copyright © 2002 John Wiley & Sons, Ltd.     

A numerical method for the determination of weakly non‐hydrostatic non‐linear free surface wave propagation

A numerical method is described that may be used to determine the propagation characteristics of weakly non‐hydrostatic non‐linear free surface waves over a general, bottom topography. In shallow water of constant undisturbed depth, such waves are equivalent to the familiar cnoidal waves characterized by sharp crests and relatively flat troughs. For a certain range of parameters, these propagate without change of form by virtue of the weakly non‐hydrostatic balance in the vertical momentum equation. Effectively, this counters the tendency for the non‐linearity in a purely hydrostatic theory to lead to a continuously deforming surface wave profile. The realistic representation furnished by cnoidal wave theory of free surface waves in the shallow near‐shore zone has led to its utilization in evaluating their propagation characteristics. Nonetheless, the classic analytical theory is inapplicable to the case of wave propagation over a sloping beach or off‐shore sand bar topography. Under these conditions, a local change in form of the surface wave profile is anticipated before the waves break and knowing this is required in order to evaluate fully the propagation process. The efficacy of the numerical method is first demonstrated by comparing the solution for water of constant depth with the evaluation of the analytical solution expressed in terms of the Jacobian elliptic function cn. The general method described in the paper is then illustrated by experiments to determine the change in profile of weakly non‐hydrostatic non‐linear surface waves propagating over bed forms representative of those found in shallow coastal seas. Copyright © 2001 John Wiley & Sons, Ltd.     

Internal waves in a two‐layer system using fully nonlinear internal‐wave equations

In order to understand the nonlinear effect in a two‐layer system, fully nonlinear strongly dispersive internal‐wave equations, based on a variational principle, were proposed in this study. A simple iteration method was used to solve the internal‐wave equations in order to solve the equations stably. The applicability of the proposed numerical computation scheme was confirmed to agree with linear dispersion relation theoretically obtained from variational principle. The proposed computational scheme was also shown to reproduce internal waves including higher‐order nonlinear effect from the analysis of internal solitary waves in a two‐layer system. Furthermore, for the second‐order numerical analysis, the balance of nonlinearity and dispersion was found to be similar to the balance assumed in the KdV theory and the Boussinesq‐type equations. Copyright © 2009 John Wiley & Sons, Ltd.     

Simulation of non‐hydrostatic,density‐stratified flow in irregular domains

A numerical model has been developed for simulating density‐stratified flow in domains with irregular but simple topography. The model was designed for simulating strong interactions between internal gravity waves and topography, e.g. exchange flows in contracting channels, tidally or convectively driven flow over two‐dimensional sills or waves propagating onto a shoaling bed. The model is based on the non‐hydrostatic, Boussinesq equations of motion for a continuously stratified fluid in a rotating frame, subject to user‐configurable boundary conditions. An orthogonal boundary fitting co‐ordinate system is used for the numerical computations, which rely on a fourth‐order compact differentiation scheme, a third‐order explicit time stepping and a multi‐grid based pressure projection algorithm. The numerical techniques are described and a suite of validation studies are presented. The validation studies include a pointwise comparison of numerical simulations with both analytical solutions and laboratory measurements of non‐linear solitary wave propagation. Simulation results for flows lacking analytical or laboratory data are analysed a posteriori to demonstrate satisfaction of the potential energy balance. Computational results are compared with two‐layer hydraulic predictions in the case of exchange flow through a contracting channel. Finally, a simulation of circulation driven by spatially non‐uniform surface buoyancy flux in an irregular basin is discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

A horizontally curvilinear non‐hydrostatic model for simulating nonlinear wave motion in curved boundaries

A horizontally curvilinear non‐hydrostatic free surface model that embeds the second‐order projection method, the so‐called θ scheme, in fractional time stepping is developed to simulate nonlinear wave motion in curved boundaries. The model solves the unsteady, Navier–Stokes equations in a three‐dimensional curvilinear domain by incorporating the kinematic free surface boundary condition with a top‐layer boundary condition, which has been developed to improve the numerical accuracy and efficiency of the non‐hydrostatic model in the standard staggered grid layout. The second‐order Adams–Bashforth scheme with the third‐order spatial upwind method is implemented in discretizing advection terms. Numerical accuracy in terms of nonlinear phase speed and amplitude is verified against the nonlinear Stokes wave theory over varying wave steepness in a two‐dimensional numerical wave tank. The model is then applied to investigate the nonlinear wave characteristics in the presence of dispersion caused by reflection and diffraction in a semicircular channel. The model results agree quantitatively with superimposed analytical solutions. Finally, the model is applied to simulate nonlinear wave run‐ups caused by wave‐body interaction around a bottom‐mounted cylinder. The numerical results exhibit good agreement with experimental data and the second‐order diffraction theory. Overall, it is shown that the developed model, with only three vertical layers, is capable of accurately simulating nonlinear waves interacting within curved boundaries. Copyright © 2011 John Wiley & Sons, Ltd.     

A semi‐implicit finite element model for non‐hydrostatic (dispersive) surface waves

The objective of this research is to develop a model that will adequately simulate the dynamics of tsunami propagating across the continental shelf. In practical terms, a large spatial domain with high resolution is required so that source areas and runup areas are adequately resolved. Hence efficiency of the model is a major issue. The three‐dimensional Reynolds averaged Navier–Stokes equations are depth‐averaged to yield a set of equations that are similar to the shallow water equations but retain the non‐hydrostatic pressure terms. This approach differs from the development of the Boussinesq equations where pressure is eliminated in favour of high‐order velocity and geometry terms. The model gives good results for several test problems including an oscillating basin, propagation of a solitary wave, and a wave transformation over a bar. The hydrostatic and non‐hydrostatic versions of the model are compared for a large‐scale problem where a fault rupture generates a tsunami on the New Zealand continental shelf. The model efficiency is also very good and execution times are about a factor of 1.8 to 5 slower than the standard shallow water model, depending on problem size. Moreover, there are at least two methods to increase model accuracy when warranted: choosing a more optimal vertical interpolation function, and dividing the problem into layers. Copyright © 2005 John Wiley & Sons, Ltd.     

Depth‐integrated,non‐hydrostatic model with grid nesting for tsunami generation,propagation, and run‐up

Tsunamis generated by earthquakes involve physical processes of different temporal and spatial scales that extend across the ocean to the shore. This paper presents a shock‐capturing dispersive wave model in the spherical coordinate system for basin‐wide evolution and coastal run‐up of tsunamis and discusses the implementation of a two‐way grid‐nesting scheme to describe the wave dynamics at resolution compatible to the physical processes. The depth‐integrated model describes dispersive waves through the non‐hydrostatic pressure and vertical velocity, which also account for tsunami generation from dynamic seafloor deformation. The semi‐implicit, finite difference model captures flow discontinuities associated with bores or hydraulic jumps through the momentum‐conserved advection scheme with an upwind flux approximation. The two‐way grid‐nesting scheme utilizes the Dirichlet condition of the non‐hydrostatic pressure and both the horizontal velocity and surface elevation at the inter‐grid boundary to ensure propagation of dispersive waves and discontinuities across computational grids of different resolution. The inter‐grid boundary can adapt to bathymetric features to model nearshore wave transformation processes at optimal resolution and computational efficiency. A coordinate transformation enables application of the model to small geographic regions or laboratory experiments with a Cartesian grid. A depth‐dependent Gaussian function smoothes localized bottom features in relation to the water depth while retaining the bathymetry important for modeling of tsunami transformation and run‐up. Numerical experiments of solitary wave propagation and N‐wave run‐up verify the implementation of the grid‐nesting scheme. The 2009 Samoa Tsunami provides a case study to confirm the validity and effectiveness of the modeling approach for tsunami research and impact assessment. Copyright © 2010 John Wiley & Sons, Ltd.     

Depth‐averaged non‐hydrostatic extension for shallow water equations with quadratic vertical pressure profile: equivalence to Boussinesq‐type equations

We reformulate the depth‐averaged non‐hydrostatic extension for shallow water equations to show equivalence with well‐known Boussinesq‐type equations. For this purpose, we introduce two scalars representing the vertical profile of the non‐hydrostatic pressure. A specific quadratic vertical profile yields equivalence to the Serre equations, for which only one scalar in the traditional equation system needs to be modified. Equivalence can also be demonstrated with other Boussinesq‐type equations from the literature when considering variable depth, but then the non‐hydrostatic extension involves mixed space–time derivatives. In case of constant bathymetries, the non‐hydrostatic extension is another way to circumvent mixed space–time derivatives arising in Boussinesq‐type equations. On the other hand, we show that there is no equivalence when using the traditionally assumed linear vertical pressure profile. Linear dispersion and asymptotic analysis as well as numerical test cases show the advantages of the quadratic compared with the linear vertical non‐hydrostatic pressure profile in the depth‐averaged non‐hydrostatic extension for shallow water equations. Copyright © 2017 John Wiley & Sons, Ltd.     

Three‐dimensional numerical modelling of free surface flows with non‐hydrostatic pressure

A three‐dimensional numerical model is developed for incompressible free surface flows. The model is based on the unsteady Reynolds‐averaged Navier–Stokes equations with a non‐hydrostatic pressure distribution being incorporated in the model. The governing equations are solved in the conventional sigma co‐ordinate system, with a semi‐implicit time discretization. A fractional step method is used to enable the pressure to be decomposed into its hydrostatic and hydrodynamic components. At every time step one five‐diagonal system of equations is solved to compute the water elevations and then the hydrodynamic pressure is determined from a pressure Poisson equation. The model is applied to three examples to simulate unsteady free surface flows where non‐hydrostatic pressures have a considerable effect on the velocity field. Emphasis is focused on applying the model to wave problems. Two of the examples are about modelling small amplitude waves where the hydrostatic approximation and long wave theory are not valid. The other example is the wind‐induced circulation in a closed basin. The numerical solutions are compared with the available analytical solutions for small amplitude wave theory and very good agreement is obtained. Copyright © 2002 John Wiley & Sons, Ltd.