A second-order semi-implicit hybrid scheme for one-dimensional Boussinesq-type waves in rectangular channels (original) (raw)
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Physics of Fluids, 2021
Undular bores, also termed dispersive shock waves, generated by an initial discontinuity in height as governed by two forms of the Boussinesq system of weakly nonlinear shallow water wave theory, the standard formulation and a Hamiltonian formulation, two related Whitham–Boussinesq equations, and the full water wave equations for gravity surface waves are studied and compared. It is found that the Whitham–Boussinesq systems give solutions in excellent agreement with numerical solutions of the full water wave equations for the positions of the leading and trailing edges of the bore up until the onset on modulational instability. The Whitham–Boussinesq systems, which are far simpler than the full water wave equations, can then be used to accurately model surface water wave undular bores. Finally, comparisons with numerical solutions of the full water wave equations show that the Whitham–Boussinesq systems give a slightly lower threshold for the onset of modulational instability in ter...
Undular bores and secondary waves - Experiments and hybrid finite volume modelling
Journal of Hydraulic Research, 2003
Secondary free-surface undulations (Favre waves), appearing for example after the opening of a sluice gate or at the head of a bore, cannot be reproduced by numerical models based on the hydrostatic pressure assumption. The Boussinesq equations take into account the extra pressure gradients but are difficult to integrate due to the high-order derivative terms. The paper describes the physics of wave initiation and proposes a demonstration of the Boussinesq equation based on relatively wider assumptions than usually adopted. A linear stability analysis is developed in finite-difference frame to highlight some potential source of numerical instabilities. These conclusions are transposed in a new hybrid finite-volume / finite-difference scheme, which reveals a better accuracy in period and amplitude when evaluated against experiments.
Irregular wave propagation with a 2DH Boussinesq-type model and an unstructured finite volume scheme
European Journal of Mechanics - B/Fluids, 2018
The application and validation, with respect to the transformation, breaking and run-up of irregular waves, of an unstructured high-resolution finite volume (FV) numerical solver for the 2D extended Boussinesq-type (BT) equations of Nwogu (1993) is presented. The numerical model is based on the combined FV approximate solution of the BT model and that of the nonlinear shallow water equations (NSWE) when wave breaking emerges. The FV numerical scheme satisfies the desired properties of wellbalancing, for flows over complex bathymetries and in presence of wet/dry fronts, and shock-capturing for an intrinsic representation of wave breaking, that is handled as a shock by the NSWE. Several simulations and comparisons with experimental data show that the model is able to simulate wave height variations, mean water level setup, wave run-up, swash zone oscillations and the generation of near-shore currents with satisfactory accuracy.
Numerical modeling of solitary waves by 1-D Madsen and Sorensen extended Boussinesq equations
ISH Journal of Hydraulic Engineering, 2015
In this paper, finite element modeling of one-dimensional extended Boussinesq equations derived by Madsen and Sorensen is presented for simulation of propagating regular waves. In order to spatially discretize the finite element equations, method of weighted residual Galerkin approach is used. Discretization of third-order derivative in momentum equation is performed by introducing of an auxiliary equation, which makes it possible to use linear finite element method. Adams-Bashforth-Moulton predictor-corrector method is used for time integration. Regular wave trains are simulated using the proposed numerical scheme. For validation of the developed code, the model is applied to several examples of wave propagation over the computational domain and the obtained results of the current computations are compared against the experimental measurements. In all cases, the proposed model has proved very suitable for simulating the propagation of wave indicating favorable agreements with experimental data.
Hybrid finite-volume finite-difference scheme for the solution of Boussinesq equations
International Journal for Numerical Methods in Fluids, 2005
A hybrid scheme composed of ÿnite-volume and ÿnite-di erence methods is introduced for the solution of the Boussinesq equations. While the ÿnite-volume method with a Riemann solver is applied to the conservative part of the equations, the higher-order Boussinesq terms are discretized using the ÿnitedi erence scheme. Fourth-order accuracy in space for the ÿnite-volume solution is achieved using the MUSCL-TVD scheme. Within this, four limiters have been tested, of which van-Leer limiter is found to be the most suitable. The Adams-Basforth third-order predictor and Adams-Moulton fourth-order corrector methods are used to obtain fourth-order accuracy in time. A recently introduced surface gradient technique is employed for the treatment of the bottom slope. A new model 'HYWAVE', based on this hybrid solution, has been applied to a number of wave propagation examples, most of which are taken from previous studies. Examples include sinusoidal waves and bi-chromatic wave propagation in deep water, sinusoidal wave propagation in shallow water and sinusoidal wave propagation from deep to shallow water demonstrating the linear shoaling properties of the model. Finally, sinusoidal wave propagation over a bar is simulated. The results are in good agreement with the theoretical expectations and published experimental results. Copyright ? 2005 John Wiley & Sons, Ltd.
Journal of Computational Physics, 2014
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.
Boussinesq Modelling of Solitary Wave Propagation, Breaking, Runup and Overtopping
Coastal Engineering Proceedings, 2011
A one-dimensional hybrid numerical model is presented of a shallow-water flume with an incorporated piston paddle. The hybrid model is based on the improved Boussinesq equations by Madsen and Sorensen (1992) and the nonlinear shallow water equations. It is suitable for breaking and non-breaking waves and requires only two adjustable parameters: a friction coefficient and a wave breaking parameter. The applicability of the model is demonstrated by simulating laboratory experiments of solitary waves involving runup at a plane beach and overtopping of a laboratory seawall. The predicted free surface profiles, maximum runup and overtopping volumes agree very well with the measured values.
Journal of Fluid Mechanics, 2003
The objective of this paper is to discuss and analyse the accuracy of various velocity formulations for water waves in the framework of Boussinesq theory. To simplify the discussion, we consider the linearized wave problem confined between the stillwater datum and a horizontal sea bottom. First, the problem is further simplified by ignoring boundary conditions at the surface. This reduces the problem to finding truncated series solutions to the Laplace equation with a kinematic condition at the sea bed. The convergence and accuracy of the resulting expressions is analysed in comparison with the target cosh-and sinh-functions from linear wave theory. First, we consider series expansions in terms of the horizontal velocity variable at an arbitrary z-level, which can be varied from the sea bottom to the still-water datum. Second, we consider the classical possibility of expanding in terms of the depth-averaged velocity. Third, we analyse the use of a horizontal pseudo-velocity determined by interpolation between velocities at two arbitrary z-levels. Fourth, we investigate three different formulations based on two expansion variables, being the horizontal and vertical velocity variables at an arbitrary z-level. This is shown to have a remarkable influence on the convergence and to improve accuracy considerably. Fifth, we derive and analyse a new formulation which doubles the power of the vertical coordinate without increasing the order of the horizontal derivatives. Finally, we involve the kinematic and dynamic boundary conditions at the free surface and discuss the linear dispersion relation and a spectral solution for steady nonlinear waves.
On uniformly accurate high-order Boussinesq difference equations for water waves
International Journal for Numerical Methods in Fluids, 2006
A new accurate ÿnite-di erence (AFD) numerical method is developed speciÿcally for solving highorder Boussinesq (HOB) equations. The method solves the water-wave ow with much higher accuracy compared to the standard ÿnite-di erence (SFD) method for the same computer resources. It is ÿrst developed for linear water waves and then for the nonlinear problem. It is presented for a horizontal bottom, but can be used for variable depth as well. The method can be developed for other equations as long as they use Padà e approximation, for example extensions of the parabolic equation for acoustic wave problems. Finally, the results of the new method and the SFD method are compared with the accurate solution for nonlinear progressive waves over a horizontal bottom that is found using the stream function theory. The agreement of the AFD to the accurate solution is found to be excellent compared to the SFD solution. 926 Y. TOLEDO AND Y. AGNON makes them very attractive. In general, HOB models have an algebraic dispersion relation, which approximates the exact (transcendental) dispersion relation of the fully dispersive waterwaves theory.
Two-dimensional Boussinesq equations applied to channel flows: deducing and applying the equations
RBRH
A basic hypothesis adopted for theoretical formulation of fluid flows is the hydrostatic pressure distribution. However, many researchers have pointed out that this simplification can lead to errors, in cases such as dam break flow. Discrepancy between computational solution and the experiment is attributed to the pressure distribution. These findings are not new, but it is not presented any formulation in the literature that considers the non-hydrostatic pressure distribution in 2D flow. This article deduces the Boussinesq Equations as an evolution of the Shallow Water Equations with the hypothesis of non-hydrostatic pressure distribution in the vertical direction. XYZ Orthogonal Cartesian System is used, considering the influence of channel bed slope and head losses of flow. It is presented the non-hydrostatical correction in the Boussinesq equation in two dimension using Fourier series. The solution uses Runge-Kutta Discontinuous Galerkin Method and the formulation is applied to ...