Depth-integrated free-surface flow with a two-layer non-hydrostatic formulation (original) (raw)

Depth-integrated free-surface flow with parameterized non-hydrostatic pressure

International Journal for Numerical Methods in Fluids, 2013

Non-hydrostatic free-surface models can provide better descriptions of dispersive waves by increasing the number of layers at the expense of computational efficiency. This paper proposes a parameterized nonhydrostatic pressure distribution in a depth-integrated two-layer formulation to reduce computational costs and to maintain essential dispersion properties for modeling of coastal processes. The non-hydrostatic pressure at mid flow depth is expressed in terms of the bottom pressure with a free parameter, which is determined to match the exact linear dispersion relation for the water depth parameter up to kd D 3. This reduces the depth-integrated two-layer formulation to a hybrid system with a tridiagonal matrix in the pressure Poisson equation. Linear dispersion relations and shoaling gradients derived from the present model as well as conventional one-layer and two-layer models provide a baseline for performance evaluation. Results from these three models are compared with previous laboratory experiments for wave transformation over a submerged bar, a plane beach, and a fringing reef. The present model provides comparable results as the two-layer model but at the computational requirements of a one-layer model. non-hydrostatic pressure simultaneously . The splitting method consists of a hydrostatic and a non-hydrostatic step with two solution approaches. The fractional step method [5] utilizes the nonhydrostatic pressure only in the non-hydrostatic step, but may introduce splitting errors that affect wave propagation significantly . The projection method also known as the pressure correction technique [7] utilizes the non-hydrostatic pressure in both steps to eliminate the splitting errors in the numerical solution.

Depth‐integrated, non‐hydrostatic model for wave breaking and run‐up

2008

Abstract This paper describes the formulation, verification, and validation of a depth-integrated, non-hydrostatic model with a semi-implicit, finite difference scheme. The formulation builds on the nonlinear shallow-water equations and utilizes a non-hydrostatic pressure term to describe weakly dispersive waves. A momentum-conserved advection scheme enables modeling of breaking waves without the aid of analytical solutions for bore approximation or empirical equations for energy dissipation.

Simulation of nearshore wave processes by a depth-integrated non-hydrostatic finite element model

Coastal Engineering, 2014

This paper presents CCHE2D-NHWAVE, a depth-integrated non-hydrostatic finite element model for simulating nearshore wave processes. The governing equations are a depth-integrated vertical momentum equation and the shallow water equations including extra non-hydrostatic pressure terms, which enable the model to simulate relatively short wave motions, where both frequency dispersion and nonlinear effects play important roles. A special type of finite element method, which was previously developed for a well-validated depth-integrated free surface flow model CCHE2D, is used to solve the governing equations on a partially staggered grid using a pressure projection method. To resolve discontinuous flows, involving breaking waves and hydraulic jumps, a momentum conservation advection scheme is developed based on the partially staggered grid. In addition, a simple and efficient wetting and drying algorithm is implemented to deal with the moving shoreline. The model is first verified by analytical solutions, and then validated by a series of laboratory experiments. The comparison shows that the developed wave model without the use of any empirical parameters is capable of accurately simulating a wide range of nearshore wave processes, including propagation, breaking, and run-up of nonlinear dispersive waves and transformation and inundation of tsunami waves.

Depth-induced wave breaking in a non-hydrostatic, near-shore wave model

Coastal Engineering, 2013

The energy dissipation in the surf-zone due to wave breaking is inherently accounted for in shock-capturing non-hydrostatic wave models, but this requires high vertical resolutions. To allow coarse vertical resolutions a hydrostatic front approximation is suggested. It assumes a hydrostatic pressure distribution at the front of a breaking wave which ensures that the wave front develops a vertical face. Based on the analogy between a hydraulic jump and a turbulent bore, energy dissipation is accounted for by ensuring conservation of mass and momentum. Results are compared with observations of random, uni-directional waves in wave flumes, and to observations of short-crested waves in a wave basin. These demonstrate that the resulting model can resolve the relevant near-shore wave processes in a short-crested wave-field, including wave breaking and wave-driven horizontal circulations.

Further experiences with computing non-hydrostatic free-surface flows involving water waves

International Journal for Numerical Methods in Fluids, 2005

A semi-implicit, staggered ÿnite volume technique for non-hydrostatic, free-surface ow governed by the incompressible Euler equations is presented that has a proper balance between accuracy, robustness and computing time. The procedure is intended to be used for predicting wave propagation in coastal areas. The splitting of the pressure into hydrostatic and non-hydrostatic components is utilized. To ease the task of discretization and to enhance the accuracy of the scheme, a vertical boundary-ÿtted co-ordinate system is employed, permitting more resolution near the bottom as well as near the free surface. The issue of the implementation of boundary conditions is addressed. As recently proposed by the present authors, the Keller-box scheme for accurate approximation of frequency wave dispersion requiring a limited vertical resolution is incorporated. The both locally and globally mass conserved solution is achieved with the aid of a projection method in the discrete sense. An e cient preconditioned Krylov subspace technique to solve the discretized Poisson equation for pressure correction with an unsymmetric matrix is treated. Some numerical experiments to show the accuracy, robustness and e ciency of the proposed method are presented. surface level. We come back to this point later. In the same step, the free-surface condition is reconsidered to assure global mass conservation. From a computational point of view, this improvement implies no requirement of the ÿrst fractional step. Hence, this method has a reduced splitting error. An alternative is proposed by Chen [12], where the pressure Poisson equation is ÿrst solved to obtain the non-hydrostatic pressure and subsequently correct the velocities. Thereafter, the surface elevation followed by the velocity ÿeld is updated. However, the momentum equations must be solved twice per time step, whereas usually the momentum equations are solved once per time step as done in References .

Kinematics and depth-integrated terms in surf zone waves from laboratory measurement

Journal of Fluid Mechanics, 2005

Kinematics of nominally periodic surf zone waves have been measured in the laboratory using LDA (laser Doppler anemometry), above trough level as well as below, for weakly plunging breakers transforming into bores in shallower water. The aim was to determine, through phase-or ensemble-averaging, periodic flow structures in a two-dimensional vertical plane, from large-scale down to small-scale vortical structures. Coherent multiple vortical structures were evident at the initiation of breaking, becoming elongated along the surface during bore propagation. The initial region is likely to become more extensive as waves become more strongly plunging and could explain the difference in turbulence characteristics between plunging and spilling breakers observed elsewhere. Comparison of vorticity magnitudes with hydraulic-jump measurements showed some similarities during the initial stages of breaking, but these quickly grew less as breaking progressed into shallower water. Period-averaged kinematics and vorticity were also obtained showing shoreward mass transport above trough level and undertow below, with a thick layer of vorticity at trough level and a thin layer of vorticity of opposite rotation at the bed. There were also concentrated regions of mean vorticity near the end of the plunging region. Residual turbulence of relatively high frequency was presented as Reynolds stresses, showing marked anisotrophy. Dynamic pressure (pressure minus its hydrostatic component) was determined from the kinematics. The magnitudes of different effects were evaluated through the depth-integrated Reynolds-averaged Navier-Stokes (RANS) equations, which may be reduced to nine terms (the standard inviscid terms of the shallow-water equations conserving mass and momentum with hydrostatic pressure, and six additional terms), assuming that the complex, often aerated, free surface is treated as a simple interface. All terms were evaluated, assuming that a space/time transformation was justified with a slowly varying phase speed, and the net balance was always small in relation to the maxima of the larger terms. Terms due to dynamic pressure and vertical dispersion (due to the vertical variation of velocity) were as significant as the three terms in the inviscid shallowwater equations; terms involving residual turbulence were insignificant. The r.m.s. (root mean square) variation of each along the slope is highly irregular, with the inertia term due to (Eulerian) acceleration always greatest. This is consistent with complex, though repetitive, coherent structures. Modelling the flow with the shallowwater equations, using the surface elevation variation at the break point as input, nevertheless gave a good prediction of the wave height variation up the slope. 280 P. K. Stansby and T. Feng

Modelling of depth-induced wave breaking in a fully nonlinear free-surface potential flow model

Coastal Engineering, 2019

Two methods to treat wave breaking in the framework of the Hamiltonian formulation of free-surface potential flow are presented, tested, and validated. The first is an extension of Kennedy et al. (2000)'s eddyviscosity approach originally developed for Boussinesq-type wave models. In this approach, an extra term, constructed to conserve the horizontal momentum for waves propagating over a flat bottom, is added in the dynamic free-surface condition. In the second method, a pressure distribution is introduced at the free surface that dissipates wave energy by analogy to a hydraulic jump (Guignard and Grilli, 2001). The modified Hamiltonian systems are implemented using the Hamiltonian Coupled-Mode Theory, in which the velocity potential is represented by a rapidly convergent vertical series expansion. Wave energy dissipation and conservation of horizontal momentum are verified numerically. Comparisons with experimental measurements are presented for the propagation of a breaking dispersive shock wave following a dam break, and then incident regular waves breaking on a mildly sloping beach and over a submerged bar.

Numerical Modeling of Wave Propagation, Breaking and Run-Up on a Beach

Lecture notes in computational science and engineering, 2009

A numerical method for free-surface flow is presented at the aim of studying water waves in coastal areas. The method builds on the nonlinear shallow water equations and utilizes a non-hydrostatic pressure term to describe short waves. A vertical boundary-fitted grid is used with the water depth divided into a number of layers. A compact finite difference scheme is employed that takes into account the effect of non-hydrostatic pressure with a few number of vertical layers. As a result, the proposed technique is capable of simulating relatively short wave propagation, where both frequency dispersion and nonlinear shoaling play an important role, in an accurate and efficient manner. Mass and momentum are strictly conserved at discrete level while the method only dissipates energy in the case of wave breaking. A simple wet-dry algorithm is applied for a proper calculation of wave run-up on the beach. The computed results show good agreement with analytical and laboratory data for wave propagation, transformation, breaking and run-up within the surf zone.

Numerical Modeling Of Wave Run-Up In Shallow Water Flows Using Moving Wet/Dry Interfaces

2018

We present a new class of numerical techniques to<br> solve shallow water flows over dry areas including run-up. Many<br> recent investigations on wave run-up in coastal areas are based on<br> the well-known shallow water equations. Numerical simulations have<br> also performed to understand the effects of several factors on tsunami<br> wave impact and run-up in the presence of coastal areas. In all these<br> simulations the shallow water equations are solved in entire domain<br> including dry areas and special treatments are used for numerical<br> solution of singularities at these dry regions. In the present study we<br> propose a new method to deal with these difficulties by reformulating<br> the shallow water equations into a new system to be solved only in the<br> wetted domain. The system is obtained by a change in the coordinates<br> leading to a set of equations in a moving domain for which the<br>...

Two-Layer Shallow Water Equations with Momentum Conservative Scheme for Wave Propagation Simulation

Engineering, Mathematics and Computer Science Journal (EMACS), 2024

In this paper, we discuss the implementation of momentum conservative scheme to shallow water equations (SWE). In shallow water model, the hydrodynamic pressure of the water is neglected. Here, the numerical calculation of mass and momentum conservation was applied on a staggered grid domain. The vertical interval was divided into two parts which made the computation quite efficient and accurate. Our focus is on the performance of the numerical scheme in simulating wave propagation and runup phenomena, where the main challenge is to calculate the wave speed accurately and to count the non-linear term of the model. Here we also considered the wet and dry conditions of the topography. Three benchmark tests were picked out to validate the numerical scheme. A simulation of standing wave was carried out; the results were compared to the linear analytical solution and show a good fit. In addition, a simulation of harmonic wave propagation on a sloping beach was conducted, and the results closely align with the expected values from exact solution. Finally, we carried out a simulation of solitary wave with a sloping topography; and the results were compared to laboratory data. A good agreement was observed between the simulation results and experimental measurements.