Section: New Results

Modelling of free surface flows

• Participants: Umberto Bosi, Mathieu Colin, Maria Kazolea, Mario Ricchiuto

• Corresponding member: Maria Kazolea

  This year we continued our work on free surface flow modelling. We can dived our work on four main axis. First, we presented a depth-integrated Boussinesq model for the efficient simulation of nonlinear wave-body interaction [4]. The model exploits a €˜unified€™ Boussinesq framework, i.e. the fluid under the body is also treated with the depth-integrated approach. The unified Boussinesq approach was initially proposed by Jiang [60] and recently analysed by Lannes [67]. The choice of Boussinesq-type equations removes the vertical dimension of the problem, resulting in a wave-body model with adequate precision for weakly nonlinear and dispersive waves expressed in horizontal dimensions only. The framework involves the coupling of two different domains with different flow characteristics. Inside each domain, the continuous spectral/hp element method is used to solve the appropriate flow model since it allows to achieve high-order, possibly exponential, convergence for non-breaking waves. Flux-based conditions for the domain coupling are used, following the recipes provided by the discontinuous Galerkin framework. The main contribution of this work is the inclusion of floating surface-piercing bodies in the conventional depth-integrated Boussinesq framework and the use of a spectral/hp element method for high-order accurate numerical discretization in space. The model is verified using manufactured solutions and validated against published results for wave-body interaction. The model is shown to have excellent accuracy and is relevant for applications of waves interacting with wave energy devices. The outcome of this work is the phd thesis of Uberto Bosi.[1].

  Second, a detailed analysis of undular bore dynamics in channels of variable cross-section is performed and presented in [5] . Two undular bore regimes, low Froude number (LFN) and high Froude number (HFN), are simulated with a Serre Green Naghdi model, and the results are compared with the experiments by Treske (1994). We show that contrary to Favre waves and HFN bores, which are controlled by dispersive non-hydrostatic mechanisms, LFN bores correspond to a hydrostatic phenomenon. The dispersive-like properties of the LFN bores is related to wave refraction on the banks in a way similar to that of edge waves in the near shore. A fully hydrostatic asymptotic model for these dispersive-like bores is derived and compared to the observations, confirming our claim.

  An other part of our last year's work was focused on wave breaking for Boussineq-type equations [3]. The aim of this work was to develop a model able to represent the propagation and transformation of waves in nearshore areas. The focus is on the phenomena of wave breaking, shoaling and run-up. These different phenomena are represented through a hybrid approach obtained by the coupling of non-linear Shallow Water equations with the extended Boussinesq equations of Madsen and Sorensen. The novelty is the switch tool between the two modelling equations: a critical free surface Froude criterion. This is based on a physically meaningful new approach to detect wave breaking, which corresponds to the steepening of the wave's crest which turns into a roller. To allow for an appropriate discretization of both types of equations, we consider a finite element Upwind Petrov Galerkin method with a novel limiting strategy, that guarantees the preservation of smooth waves as well as the monotonicity of the results in presence of discontinuities.We provide a detailed discussion of the implementation of the newly proposed detection method, as well as of two other well known criteria which are used for comparison. An extensive benchmarking on several problems involving different wave phenomena and breaking conditions allows to show the robustness of the numerical method proposed, as well as to assess the advantages and limitations of the different detection methods.

  We also continue to work on the modelling of free surface flows by investigating a new family of models, derived from the so-called Isobe-Kakinuma models. The Isobe–Kakinuma model is a system of Euler–Lagrange equations for a Lagrangian approximating Luke's Lagrangian for water waves. In [23], We consider the Isobe–Kakinuma model for two-dimensional water waves in the case of the flat bottom. We show theoretically the existence of a family of small amplitude solitary wave solutions to the Isobe–Kakinuma model in the long wave regime. We have also performed numerical computations for a toy system included large amplitude solitary wave solutions. Our computations suggest the existence of a solitary wave of extreme form with a sharp crest. This models seems very promising for future research.