Coastal Engineering 69 (2012) 42–66
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Coastal Engineering journal homepage: www.elsevier.com/locate/coastaleng
An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations M. Kazolea a, A.I. Delis b,⁎, I.K. Nikolos c, C.E. Synolakis a a b c
Environmental Engineering Department, Technical University of Crete, University Campus, Chania, Crete 73100, Greece Department of Sciences, Division of Mathematics, Technical University of Crete, University Campus, Chania, Crete, Greece Department of Production Engineering & Management, Technical University of Crete, University Campus, Chania, Crete 73100, Greece
a r t i c l e
i n f o
Article history: Received 1 November 2011 Received in revised form 16 May 2012 Accepted 21 May 2012 Available online 23 June 2012 Keywords: Boussinesq-type equations Finite volumes Unstructured meshes Well-balancing Solitary waves Regular waves Runup
a b s t r a c t We present a high-order well-balanced unstructured finite volume (FV) scheme on triangular meshes for modeling weakly nonlinear and weakly dispersive water waves over slowly varying bathymetries, as described by the 2D depth-integrated extended Boussinesq equations of Nwogu, rewritten here in conservation law form. The FV scheme numerically solves the conservative form of the equations following the median dual node-centered approach, for both the advective and dispersive part of the equations. For the advective fluxes, the scheme utilizes an approximate Riemann solver along with a well-balanced topography source term upwinding. Higher order accuracy in space and time is achieved through a MUSCL-type reconstruction technique and through a strong stability preserving explicit Runge–Kutta time stepping. Special attention is given to the accurate numerical treatment of moving wet/dry fronts and boundary conditions. The model is applied to several examples of non-breaking wave propagation over variable topographies and the computed solutions are compared to experimental data. The presented results indicate that the presented FV model is robust and capable of simulating wave transformations from relatively deep to shallow water, providing accurate predictions of the wave's propagation, shoaling and runup. © 2012 Elsevier B.V. All rights reserved.
1. Introduction In coastal and ocean engineering, accurate simulations of nonlinear and dispersive water waves in realistic environments are important and have largely replaced laboratory experiments for the design of coastal structures. In return, mathematical and numerical modeling has advanced in the past decade as to provide the means of accurately predicting near-shore wave processes such as shoaling and runup, diffraction, refraction and harmonic interaction. Significant research effort has been expanded into advancing important simulation issues which include, the validity of specific mathematical models in nearshore zones as well as in deeper waters, wave breaking, transitions between sub and supercritical flows, frequency dispersion and accurate numerical treatment of natural topographies and wetting/drying processes. To this end and to avoid the complexity and computational resources needed to solve the full Navier–Stokes equations, the depthaveraging assumption has been used to simplify the equations so that numerical models can be of practical use in design. Using mass and momentum conservation equations where the velocity in the vertical direction is assumed negligible, depth-averaged models (also called to as depth-integrated) have gained a lot of popularity in terms of applicability ⁎ Corresponding author. Tel.: + 30 2821037751. E-mail addresses:
[email protected] (M. Kazolea),
[email protected] (AI. Delis). 0378-3839/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2012.05.008
and development. The depth-averaged nonlinear shallow water equations (NSWE) is one of the most applied models in this category. Although the application of the NSWE appears to be able to model important aspects of wave propagation phenomena and the general characteristics of the runup process, they are not appropriate for deeper waters where frequency dispersion effects may become more important than nonlinearity. On the other hand, Boussinesq-type equations introduce dispersion terms and are more suitable in waters where dispersion begins to have an effect on the free surface, and thus have become an increasingly important predictive tool in coastal engineering. Boussinesq-type equations are essentially two-dimensional models that avoid free surface boundary issues, by explicitly eliminating the vertical coordinate, yet retaining some vertical flow structure. Under the assumption that nonlinearity and frequency dispersion are weak, and in the same order of approximation, Peregrine (Peregrine, 1967) derived the so called standard Boussinesq equations for variable depth using the free surface displacement and the depth averaged velocity as dependent variables. The standard Boussinesq equations, written in terms of the depth averaged velocity, break down when the depth is greater than one fifth of the equivalent deep water wavelength, and, as such, they are limited to relatively shallow water. In addition, the weakly nonlinear assumption limits the largest possible wave height that can be accurately modeled. Considerable effort has been made in the past two decades to extend the validity and applicability of the standard Boussinesq equations to
M. Kazolea et al. / Coastal Engineering 69 (2012) 42–66
deeper water depths (or to shorter waves). As such, several alternative formulations with improved dispersive properties have been presented. The extended Boussinesq formulation of Nwogu (Nwogu, 1994) has received much attention in recent years for modeling wave propagation from deep to shallow water. Nwogu's equations are formulated in terms of the surface elevation and horizontal velocity at a specific depth chosen to minimize wave propagation errors from the linear theory. Other widely used equation sets have been derived under the weak nonlinearity and dispersion assumption by Madsen and Sørensen (1992) and also by Beji and Nadaoka (1996) and possess equivalent dispersion properties. These extended models give a more accurate representation of the phase and group velocities in intermediate water with water depth to wavelength ratio up to 1/2, and sometimes are referred to as low-order enhanced Boussinesq-type equations. Moreover, significant effort has been made in recent years into advancing the nonlinear and dispersive properties of Boussinesq-type models by including highorder nonlinear and dispersion terms, we refer for example to Bingham et al. (2009), Gobbi et al. (2000), Lynett and Liu (2004) and Madsen et al. (2002, 2003, 2006) among others, which in turn are more difficult to integrate and thus require substantially more computational effort in their numerical integration. From the numerical point of view, until recently the finite difference (FD) method was the predominant method for solving Boussinesq-type equations, see, for example Beji and Nadaoka (1996), Fuhrman and Bingham (2004), Fuhrman and Madsen (2008), Madsen and Sørensen (1992), Madsen et al. (2002), Nwogu (1994), and Wei and Kirby, (1995). The popularity of the FD method can be attributed to the ease in which higher-order derivatives can be approximated and to the well structured resulted linear systems, which can be efficiently solved (e.g. tri-diagonal ones). However, one major drawback of the FD approach is that, for 2D simulations, one has to use structured spatial meshes, even for irregular domains, which can lead to loss of accuracy. Some negative aspects of this shortcoming can be avoided using curvilinear coordinates (Shi et al., 2001), but considerable work may be needed to generate such grids for arbitrary geometries, and accuracy may be limited due to mapping problems. Another problem with the FD method is the correct treatment of the boundary conditions. The use of unstructured spatial meshes for 2D complex geometries, where the mesh size is adapted to local features such as, depth profile and complex boundaries, has been put forward as a strategy to obtain more cost-effective models. In Sørensen et al. (2004) it was estimated that the potential reduction factor, compared to structured meshes, is of the order of 10–20. The most natural candidates for unstructured methods are finite element (FE) methods and finite volume (FV) methods. The use of FE methods in the solution of extended Boussinesq-type models has increased in the last ten years with promising results in terms of accuracy and efficiency, see for example Antunes do Carmo et al. (1993), Engsig-Karup et al. (2008), Eskilsson and Sherwin (2006), Eskilsson et al. (2006), Li et al. (1999), Lopes et al. (in press), Sørensen et al. (2004), Walkey and Berzins (2002), Woo and Liu (2004) Zhong and Wang (2008). The main disadvantage of the FE methodology is that usually higher-order spatial derivatives present even in the low-order extended Bousssinesq equations must be reduced. For example, third-order derivatives in low-order extended models are usually reduced by introducing auxiliary variables to the system of equations, resulting in additional equations to be solved. Although relatively efficient schemes with higher-order accuracy can be derived within the FE framework, significant complexities and stability issues may also arise, especially on arbitrary grids. FV methods usually require significantly less computational effort than FE ones, while non-linear advection terms and topography source terms can be more easily treated when compared to FD methods. For example, the advantages of the FV method for numerically approximating the NSWE are well known also in terms of the topography and wet/ dry front treatment, we refer to Delis et al. (2011), Toro and GarciaNavarro (2007) and references therein for comprehensive reviews. FV
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schemes have been applied to solve the NSWE for a wide range of applications, like flood propagation, dam-break flows, bore propagation as well as to long wave propagation and runup, see for example Brocchini and Dodd (2008), Brocchini et al. (2001), Brufau et al. (2002), Delis and Kazolea (2011), Delis et al. (2008), Guinot (2003), Hubbard and Dodd (2002), Li and Raichlen (2002), Marche et al. (2007), Nikolos and Delis (2009), Titov and Synolakis (1995, 1998), and Toro (2001) among many others. Specifically, Godunov-type FV schemes based on Riemann solvers have the advantage of solving the integral form of the nonlinear equations as fully conservative schemes with intrinsic shock capturing properties as well as with correctly incorporating the bed topography and treating accurately advancing wet/dry fronts (Brocchini et al., 2001; Delis and Kazolea, 2011; Delis et al., 2008, 2011; Hubbard and Dodd, 2002; LeVeque, 2002; Marche et al., 2007; Toro, 2001; Toro and Garcia-Navarro, 2007). For computing the topography source term within the FV framework, considerable progress has been made and as such several numerical and mathematical treatments have been proposed for balancing the flux gradient and the source term, in order to properly compute stationary or almost stationary solutions. This property is known as wellbalancing and is currently a very active subject of research, we refer to Delis et al. (2011) and references therein. The appearance of dry areas, due to initial conditions or as a result of the water motion is also an important problem. As such, the necessity to handle wetting and drying moving boundaries (e.g. shoreline motion Briganti and Dodd, 2009) is a challenge that has attracted much attention. Several approaches have been proposed in different models and numerical schemes, using the NSWE equations (Titov and Synolakis, 1995) and, for Boussinesqtype equations, (Fuhrman and Madsen, 2008; Lynett et al., 2002) have used an extrapolation technique to allow the models to handle moving boundaries. Other techniques have been developed as to avoid dealing with wet/dry interfaces by excluding the dry cells from the computational domain (Brocchini et al., 2001, 2002), by artificially wetting dry cells (Hubbard and Dodd, 2002) and by modeling the shore as porous or slotted (Kennedy et al., 2000). Very recently, and in one spatial dimension, classical FV schemes (of the Godunov type) have been modified to solve enhanced Boussinesqtype equations. In these modifications the FV method is used to solve the nonlinear shallow water part of the equations, while the dispersive terms are discretized by FD schemes resulting in hybrid FV/FD schemes (we refer to Borthwick et al. (2006), Cienfuegos et al. (2006, 2007), Dutykh et al. (2011), Erduran (2007), Erduran et al. (2005), Kazolea and Delis (in press), Lynett et al. (2002), Ma et al. (2012), Roeber and Cheung (submitted for publication), Roeber et al. (2010), Shi et al. (2012), Shiach and Mingham (2009) Soares-Frazão and Guinot (2008)). These hybrid schemes, which combine the FV and FD methodology have been introduced for Boussinesq-type equations as to incorporate the flexibility and shock-capturing capabilities of the FV approach into dispersive wave models. This approach is particularly useful for short and long wave interactions as the solution can be easily turned into entirely FV solution of the NSWE by removing the higherorder Boussinesq terms, if needed. Further, this hybrid approach has been extended to two space dimensions but for uniform structured grids (Kim et al., 2009; Tonelli and Petti, 2009, 2010). Although for structured grids hybrid FV/FD schemes are relatively simple to implement, they can severely restrict the modeling when dealing with 2D irregular geometries, similar to the FD method. For coastal flooding over complex topography and wave interaction with coastal structures, this leads to a loss of accuracy or to the use of excessively refined grids. One approach to reduce the effect of a structured grid was presented in Ning et al. (2008), where the cut-cell approach was used to solve the 2D Madsen and Sørensen (1992) equations by a Godunov-type second order FV scheme. The use of unstructured meshes provides geometrical flexibility and, in addition, mesh resolution can be more easily refined where needed, for example in shallow regions or near structures. A first attempt to apply the unstructured FV methodology
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M. Kazolea et al. / Coastal Engineering 69 (2012) 42–66
on Nwogu's extended equations was presented in El Asmar and Nwogu (2006), where a low order staggered FV scheme was presented. The aim of the current work is to present the development, application and potentials of a FV scheme for the numerical integration of extended Boussinesq-type equations on unstructured triangular meshes. As such, Nwogu's extended equations are chosen here but formulated in a conservation law form, incorporating the time-derivative component of the dispersion terms into the vector of conserved variables with a flux term that is identical to that of the NSWE. The resulting equations are then numerically solved employing a higher-order, node-centered, Godunov-type FV technique that utilizes the approximate Riemann solver of Roe with the topography source term discretized in an upwind manner as to provide a well-balanced scheme. Higher order spatial accuracy in the nonlinear part of the equations is achieved through a MUSCLtype reconstruction technique applied on the physical variables and the dispersive terms are discretized by consistent FV approximations. Temporal accuracy is achieved through a strong stability preserving explicit Runge–Kutta time stepping. An accurate conservative treatment of moving wet/dry fronts and shoreline motion is integrated in the proposed scheme. As such, no special algorithms are needed to be implemented, e.g. extrapolation or exclusion of dry cells, to accurately compute shoreline movements. With the exemption of El Asmar and Nwogu (2006) and to the best of our knowledge, this is the first attempt to numerically solve enhanced Boussinesq-type equations on unstructured meshes by a higher-order FV scheme which exploits the advantages of the FV approach and incorporates state of the art discretizations for the topography and for the treatment of wet/dry fronts.
Eqs. (1)–(3) can model weakly non-linear and weakly dispersive water waves in variable water depth and were derived under the assumption that 2
:¼ A=h
