Numerical treatment of wave breaking on unstructured finite volume approximations for extended Boussinesq-type equations

A new methodology is presented to handle wave breaking over complex bathymetries in extended two-dimensional Boussinesq-type (BT) models which are solved by an unstructured well-balanced finite volume (FV) scheme. The numerical model solves the 2D extended BT equations proposed by Nwogu (1993), recast in conservation law form with a hyperbolic flux identical to that of the Non-linear Shallow Water (NSW) equations. Certain criteria, along with their proper implementation, are established to characterize breaking waves. Once breaking waves are recognized, we switch locally in the computational domain from the BT to NSW equations by suppressing the dispersive terms in the vicinity of the wave fronts. Thus, the shock-capturing features of the FV scheme enable an intrinsic representation of the breaking waves, which are handled as shocks by the NSW equations. An additional methodology is presented on how to perform a stable switching between the BT and NSW equations within the unstructured FV framework. Extensive validations are presented, demonstrating the performance of the proposed wave breaking treatment, along with some comparisons with other well-established wave breaking mechanisms that have been proposed for BT models. A novel approach is developed to handle wave breaking for the 2D Boussinesq-type equations of Nwogu.It is the first time such a development is considered in the unstructured Finite Volume framework.The method combines numerical approximations of the BT and the non-linear shallow water model.The shock-capturing properties of the scheme correctly simulate the energy dissipation during wave breaking.

[1]  Marco Petti,et al.  Hybrid finite volume – finite difference scheme for 2DH improved Boussinesq equations , 2009 .

[2]  Erika Kristina Lindstrøm.  Run-up of solitary waves , 2011 .

[3]  Philippe Bonneton,et al.  Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes , 2011 .

[4]  A. Bressan.  Hyperbolic Systems of Conservation Laws , 1999 .

[5]  Marco Petti,et al.  Shock-capturing Boussinesq model for irregular wave propagation , 2012 .

[6]  Harry B. Bingham,et al.  Velocity potential formulations of highly accurate Boussinesq-type models , 2009 .

[7]  J. Kirby,et al.  BOUSSINESQ MODELING OF WAVE TRANSFORMATION , BREAKING , AND RUNUP . II : 2 D By , 1999 .

[8]  Pilar García-Navarro,et al.  A numerical model for the flooding and drying of irregular domains , 2002 .

[9]  D. Peregrine.  Long waves on a beach , 1967, Journal of Fluid Mechanics.

[10]  Marcelo H. Kobayashi,et al.  Shock-capturing Boussinesq-type model for nearshore wave processes , 2010 .

[11]  G. D. van Albada,et al.  A comparative study of computational methods in cosmic gas dynamics , 1982 .

[12]  I. A. Svendsen,et al.  The flow in surf-zone waves , 2000 .

[13]  Costas E. Synolakis,et al.  The runup of solitary waves , 1987, Journal of Fluid Mechanics.

[14]  Kwok Fai Cheung,et al.  Depth‐integrated, non‐hydrostatic model for wave breaking and run‐up , 2009 .

[15]  G. Wei,et al.  A Coastal Processes Model Based on Time-Domain Boussinesq Equations. , 1996 .

[16]  Pilar García-Navarro,et al.  Flux difference splitting and the balancing of source terms and flux gradients , 2000 .

[17]  Mario Ricchiuto,et al.  Stabilized residual distribution for shallow water simulations , 2009, J. Comput. Phys..

[18]  O. Nwogu.  Alternative form of Boussinesq equations for nearshore wave propagation , 1993 .

[19]  Philippe Bonneton,et al.  A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part II : boundary conditions and validation , 2007 .

[20]  H. Schäffer,et al.  Higher–order Boussinesq–type equations for surface gravity waves: derivation and analysis , 1998, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[21]  Patrick J. Lynett,et al.  Nearshore Wave Modeling with High-Order Boussinesq-Type Equations , 2006 .

[22]  Okey Nwogu,et al.  Finite volume solution of boussinesq-type equations on an unstructured grid , 2007 .

[23]  Finite volume simulation of waves formed by sliding masses , 2011 .

[24]  I. Nikolos,et al.  An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations , 2012 .

[25]  M. Petti,et al.  Simulation of wave breaking over complex bathymetries by a Boussinesq model , 2011 .

[26]  P. Bonneton,et al.  A new approach to handle wave breaking in fully non-linear Boussinesq models , 2012 .

[27]  M A U R ´ I C I,et al.  A Fully Nonlinear Boussinesq Model for Surface Waves. Part 2. Extension to O(kh) 4 , 2000 .

[28]  Vasily Titov,et al.  Modeling of Breaking and Nonbreaking Long-Wave Evolution and Runup Using VTCS-2 , 1995 .

[29]  Maurizio Brocchini,et al.  A reasoned overview on Boussinesq-type models: the interplay between physics, mathematics and numerics , 2013, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[30]  T. Barth,et al.  Finite Volume Methods: Foundation and Analysis , 2004 .

[31]  Per A. Madsen,et al.  Surf zone dynamics simulated by a Boussinesq type model. Part II: surf beat and swash oscillations for wave groups and irregular waves , 1997 .

[32]  P. A. Madsen,et al.  A new form of the Boussinesq equations with improved linear dispersion characteristics , 1991 .

[33]  Jurjen A. Battjes,et al.  Experimental investigation of wave propagation over a bar , 1993 .

[34]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[35]  Franklin Liu,et al.  Modeling wave runup with depth-integrated equations , 2002 .

[36]  Manuel Jesús Castro Díaz,et al.  Available Online at Www.sciencedirect.com Mathematical and So,snos ~__d,~ot" Computer Modelling the Numerical Treatment of Wet/dry Fronts in Shallow Flows: Application to One-layer and Two-layer Systems , 2022 .

[37]  Nikolaos A. Kampanis,et al.  A robust high‐resolution finite volume scheme for the simulation of long waves over complex domains , 2008 .

[38]  Per A. Madsen,et al.  Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves , 1997 .

[39]  Philippe Bonneton,et al.  A fourth‐order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq‐type equations. Part I: model development and analysis , 2006 .

[40]  Pilar García-Navarro,et al.  Zero mass error using unsteady wetting–drying conditions in shallow flows over dry irregular topography , 2004 .

[41]  Clive G. Mingham,et al.  A temporally second-order accurate Godunov-type scheme for solving the extended Boussinesq equations , 2009 .

[42]  Prabhu Ramachandran,et al.  Approximate Riemann solvers for the Godunov SPH (GSPH) , 2014, J. Comput. Phys..

[43]  I. A. Svendsen.  Mass flux and undertow in a surf zone , 1984 .

[44]  D. Lannes,et al.  A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model , 2010, J. Comput. Phys..

[45]  Ge Wei,et al.  Generation of waves in Boussinesq models using a source function method , 1999 .

[46]  Rolf Deigaard,et al.  A Boussinesq model for waves breaking in shallow water , 1993 .

[47]  J. Kirby,et al.  BOUSSINESQ MODELING OF WAVE TRANSFORMATION, BREAKING, AND RUNUP. II: 2D , 2000 .

[48]  K. Cheung,et al.  Boussinesq-type model for energetic breaking waves in fringing reef environments , 2012 .

[49]  Marco Petti,et al.  Finite volume scheme for the solution of 2D extended Boussinesq equations in the surf zone , 2010 .

[50]  Ge Wei,et al.  A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O(kh)4 , 2000, Journal of Fluid Mechanics.

[51]  B. V. Leer,et al.  Towards the Ultimate Conservative Difference Scheme , 1997 .

[52]  Philippe Bonneton,et al.  Wave-Breaking Model for Boussinesq-Type Equations Including Roller Effects in the Mass Conservation Equation , 2010 .

[53]  Felice D'Alessandro,et al.  The BCI criterion for the initiation of breaking process in Boussinesq-type equations wave models , 2008 .

[54]  T. Barth.  A 3-D upwind Euler solver for unstructured meshes , 1991 .

[55]  Takashi Okamoto,et al.  The Relative Trough Froude Number for initiation of wave breaking: Theory, experiments and numerical model confirmation , 2006 .

[56]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[57]  I. A. Svendsen,et al.  MODELING TURBULENT BORE PROPAGATION IN THE SURF ZONE , 1984 .

[58]  Jean-Antoine Désidéri,et al.  Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes , 1998 .

[59]  Vincent Guinot,et al.  A second-order semi-implicit hybrid scheme for one-dimensional Boussinesq-type waves in rectangular channels , 2008 .

[60]  P. Lax.  Hyperbolic systems of conservation laws , 2006 .

[61]  Thomas M. Smith,et al.  Comparison of Reconstruction Techniques for Unstructured Mesh Vertex Centered Finite Volume Schemes , 2007 .

[62]  I. Nikolos,et al.  An unstructured node-centered finite volume scheme for shallow water flows with wet/dry fronts over complex topography , 2009 .

[63]  Ioannis K. Nikolos,et al.  Performance and Comparison of Cell-Centered and Node-Centered Unstructured Finite Volume Discretizations for Shallow Water Free Surface Flows , 2011 .

[64]  G. Wei,et al.  Time-Dependent Numerical Code for Extended Boussinesq Equations , 1995 .

[65]  A well-balanced shock-capturing hybrid finite volume-finite difference numerical scheme for extended 1D Boussinesq models , 2013 .

[66]  Qiuhua Liang,et al.  Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography , 2009 .

[67]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[68]  K. S. Erduran,et al.  Performance of finite volume solutions to the shallow water equations with shock‐capturing schemes , 2002 .

[69]  J. Boussinesq,et al.  Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. , 1872 .

[70]  Alan George,et al.  Computer Solution of Large Sparse Positive Definite , 1981 .

[71]  T. Barth.  Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations , 1994 .

[72]  R. LeVeque.  Approximate Riemann Solvers , 1992 .

[73]  S. Socolofsky,et al.  A depth-integrated model for weakly dispersive, turbulent, and rotational fluid flows , 2009 .

[74]  P. A. Madsen,et al.  A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry , 1992 .

[75]  Alf Tørum,et al.  Wave‐Induced Forces on Armor Unit on Berm Breakwaters , 1994 .

[76]  Per A. Madsen,et al.  Simulation of nonlinear wave run-up with a high-order Boussinesq model , 2008 .

[77]  George W. Housner,et al.  Numerical Model for Tsunami Run-Up , 1970 .

[78]  Peter Stansby,et al.  Solitary wave transformation, breaking and run-up at a beach , 2006 .

[79]  D. H. Peregrine,et al.  SPILLING BREAKERS, BORES, AND HYDRAULIC JUMPS , 1978 .

[80]  Steven J. Ruuth,et al.  A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods , 2002, SIAM J. Numer. Anal..

[81]  Marcel Zijlema,et al.  Depth-induced wave breaking in a non-hydrostatic, near-shore wave model , 2013 .

[82]  Paul H. Taylor,et al.  From the paddle to the beach - A Boussinesq shallow water numerical wave tank based on Madsen and Sørensen's equations , 2012, J. Comput. Phys..

[83]  Th. V. Karambas,et al.  A breaking wave propagation model based on the Boussinesq equations , 1992 .

[84]  J. A. Zelt.  The run-up of nonbreaking and breaking solitary waves , 1991 .

[85]  P. Liu,et al.  A two-layer approach to wave modelling , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[86]  Per A. Madsen,et al.  Surf zone dynamics simulated by a Boussinesq type model. III. Wave-induced horizontal nearshore circulations , 1998 .

[87]  Stephan T. Grilli,et al.  A high-order adaptive time-stepping TVD solver for Boussinesq modeling of breaking waves and coastal inundation , 2012 .

[88]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .