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The derivation of shallow water models from Navier–Stokes equations is revisited yielding a class of two-layer shallow water models. An improved velocity profile is proposed, based on the superposition of an inviscid fluid and a viscous layer inspired by the Interactive Boundary Layer interaction used in aeronautics. This leads to a new friction law which depends not only on velocity and depth but also on the variations of velocity and thickness of the viscous layer. The resulting system is an extended shallow water model consisting of three depth-integrated equations: the first two are mass and momentum conservation in which a slight correction on hydrostatic pressure has been made; the third one, known as von Kármán equation, describes the evolution of the viscous layer. This coupled model is shown to be conditionally hyperbolic, and a Godunov-type finite volume scheme is also proposed. Several numerical examples are provided and compared to the Multi-Layer Saint-Venant model. They emphasize the ability of the model to deal with unsteady viscous effects. They illustrate also the phase-lag between friction and topography, and even recover possible reverse flows.

International audience; h i g h l i g h t s • Highly dispersive potential flow model for water wave propagation. • Linear dispersion relation accurate in very deep water conditions (kh up to 100). • Accurate prediction of wave kinematics (orbital velocities) in deep water. • Validation of linear shoaling properties of the model. • Good prediction of reflected and transmitted waves on a Roseau-type bottom profile. a b s t r a c t The properties and accuracy of the linearized version of the fully dispersive and nonlinear wave model developed in Yates and Benoit (2015) and Raoult et al. (2016) are analyzed for both flat and variable bottom bathymetries. This model considers only a single layer of fluid and uses a basis of orthogonal Chebyshev polynomials to project the vertical structure of the potential. This approach results in an exponential convergence rate with the maximum degree of the Chebyshev polynomial, denoted N T , while only first-and second-order derivatives in space need to be evaluated. For the constant water depth case, the linear dispersion relation of the model is derived analytically, and expressions are established for N T ranging from 2 to 15. The analysis shows a rapid increase in accuracy in the deep water range with increasing N T. For instance, the relative error in the calculated wave celerity (in comparison with Stokes' analytical solution) remains smaller than 2.5% for deep water cases with kh up to 100 using N T ≥ 9 (k and h are the representative wavenumber and water depth, respectively). The wave kinematics, vertical profiles of the horizontal and vertical orbital velocities, converge to the Stokes profiles for kh up to 60 when using a sufficiently high value of N T. The vertically-averaged relative errors of the horizontal and vertical velocities remain below 6% and 3%, respectively, for kh up to 60 when using N T ≥ 11. The presented model shows better dispersive properties in deep water than several high-order Boussinesq-type models. For variable bottom bathymetries, the shoaling properties of the model are studied numerically, exhibiting good agreement with results from Stokes linear theory in the case of mild bottom slopes, using a sufficiently high value of N T with respect to the offshore relative water depth. For an offshore water depth of kh = 10 (i.e. more than 3 times the deep water limit), accurate wave heights in shallow water (kh = 0.25) are obtained with N T = 6 (or higher). Finally, the linear version of the model is validated with comparisons to analytical solutions of the reflection and transmission coefficients of regular waves over Roseau-type bathymetric profiles. Two bottom profiles are considered, including one with a steep slope, whose maximum value reaches about 1:0.7 (i.e. an angle of about 54.9 deg.). Using N T = 7, small differences (

International audience; Abstract With the objective of modeling coastal wave dynamics taking into account nonlinear and dispersive effects, a highly accurate nonlinear potential flow model was developed. The model is based on the time evolution of two surface quantities: the free surface position and the free surface velocity potential. A spectral approach is used to resolve vertically the velocity potential in the domain, by decomposing the potential using an orthogonal basis of Chebyshev polynomials. With this approach, a wide range of relative water depths can be simulated, as demonstrated here with the propagation of nonlinear regular waves over a flat bottom with kh = 2π and 4π (where k is the wave number and h the water depth). The model is then validated by comparing the simulation results to experimental data for four non-breaking wave test cases: (1) nonlinear dynamics of a wave train generated by a piston-type wavemaker in constant water depth, (2) shoaling of a regular wave train on beach with constant slope up to the breaking point, (3) propagation of regular waves over a submerged bar, and (4) propagation of nonlinear irregular waves over a barred beach. The test cases show the ability of the model to reproduce well nonlinear wave interactions and the dynamics of higher-order bound and free harmonics. The simulation results agree well with the experimental data, confirming the model's ability to simulate accurately nonlinear and dispersive effects for non-breaking waves.

This paper presents a parallel algorithm for the finite-volume discretisation of the Poisson equation on three-dimensional arbitrary geometries. The proposed method is formulated by using a 2D horizontal block domain decomposition and interprocessor data communication techniques with message passing interface. The horizontal unstructured-grid cells are reordered according to the neighbouring relations and decomposed into blocks using a load-balanced distribution to give all processors an equal amount of elements. In this algorithm, two parallel successive over-relaxation methods are presented: a multi-colour ordering technique for unstructured grids based on distributed memory and a block method using reordering index following similar ideas of the partitioning for structured grids. In all cases, the parallel algorithms are implemented with a combination of an acceleration iterative solver. This solver is based on a parabolic-diffusion equation introduced to obtain faster solutions of the linear systems arising from the discretisation. Numerical results are given to evaluate the performances of the methods showing speedups better than linear.

International audience; We report on the development and validation of a new Numerical Wave Tank (NWT) solving fully nonlinear potential flow (FNPF) equations, as a more efficient variation of Grilli et al.'s NWT [Grilli et al., A fully nonlinear model for three-dimensional overturning waves over arbitrary bottom, International Journal for Numerical Methods in Fluids 35 (2001) 829-867], which was successful at modeling many wave phenomena, including landslide-generated tsunamis, rogue waves, and the initiation

Dredging operations of navigation channels and harbours are regularly planned in order to maintain the nautical depth and to ensure the navigation safety. Depending on the quality of the dredged material (fixed by the Oslo convention, 1972), dredged sediments can be released into the sea or stored at the Earth’s surface to be cleaned up. The present study deals with release operations. We simulate the complete process (generation and settling of a sediment plume, propagation of a density current on the bottom) using a two-phase flow model. We show that in this case, the sediment (or solid) phase strongly differs from the motion of the water (or fluid) phase. Comparisons between numerical results and experiments are carried out in order to illustrate the effects of sediment diameter, initial concentration, and ambient current on the plume dynamics and on the density current. We obtain a correct agreement with experiments for the specific release case.

International audience; [1] Shoreline location and incident wave energy, observed for almost 5 years at Torrey Pines beach, show seasonal fluctuations characteristic of southern California beaches. The shoreline location, defined as the cross-shore position of the mean sea level contour, retreats by almost 40 m in response to energetic winter waves and gradually recovers during low-energy summer waves. Hourly estimates of incident wave energy and weekly to monthly surveys of the shoreline location are used to develop and calibrate an equilibrium-type shoreline change model. By hypothesis, the shoreline change rate depends on both the wave energy and the wave energy disequilibrium with the shoreline location. Using calibrated values of four model free parameters, observed and modeled shoreline location are well correlated at Torrey Pines and two additional survey sites. Model free parameters can be estimated with as little as 2 years of monthly observations or with about 5 years of ideally timed, biannual observations. Wave energy time series used to calibrate and test the model must resolve individual storms, and model performance is substantially degraded by using weekly to monthly averaged wave energy. Variations of free parameter values between sites may be associated with variations in sand grain size, sediment availability, and other factors. The model successfully reproduces shoreline location for time periods not used in tuning and can be used to predict beach response to past or hypothetical future wave climates. However, the model will fail when neglected geologic factors are important (e.g., underlying bedrock limits erosion or sand availability limits accretion).

A series of experiments were conducted in a wind-wave tank facility in Marseilles (France) to study the effects of preexisting swell conditions (represented by long mechanically-generated waves) on wind-wave growth with fetch. Both monochromatic and irregular (JONSWAP-type) long wave conditions with different values of wave steepness have been generated in the presence of a constant wind forcing, for several wind velocities. A spectral analysis of temporal wave signals combined with airflow measurements allowed to study the evolution of both wave systems with the aim of identifying the interaction mechanisms transportable to prototype scale. In particular, a specific method is used to separate the two wave systems in the measured bimodal spectra. In fetch-limited conditions, pure wind-wave growth is in accordance with anterior experiments, but differs from the prototype scale in terms of energy and frequency variations with fetch. Monochromatic long waves are shown to reduce the energy of the wind-waves significantly, as it was observed in anterior laboratory experiments. The addition of JONSWAP-type long waves instead results in a downshift of the wind-wave peak frequency but no significant energy reduction. Overall, it is observed that the presence of long waves affects the wind-wave energy and frequency variations with fetch. Finally, in the presence of JONSWAP-type long waves, variations of wind-wave energy and peak frequency with fetch appear in close agreement with the wind-wave growth observed at prototype scale both in terms of variations and nondimensional magnitude.

We consider multi-dimensional extensions of Maxwell’s seminal rheological equation for 1D viscoelastic flows. We aim at a causal model for compressible flows, defined by semi-group solutions given initial conditions, and such that perturbations propagate at finite speed. We propose a symmetric hyperbolic system of conservation laws that contains the Upper-Convected Maxwell (UCM) equation as causal model. The system is an extension of polyconvex elastodynamics, with an additional material metric variable that relaxes to model viscous effects. Interestingly, the framework could also cover other rheological equations, depending on the chosen relaxation limit for the material metric variable. We propose to apply the new system to incompressible free-surface gravity flows in the shallow-water regime, when causality is important. The system reduces to a viscoelastic extension of Saint-Venant 2D shallow-water system that is symmetric-hyperbolic and that encompasses our previous viscoelastic extensions of Saint-Venant proposed with F. Bouchut.

We present the system of linear equations to be solved to evaluate the reflection coefficient at a sponge layer boundary with damping forces ρ f S L = − ρ β u , where β is the sponge layer function. The case of 2D waves is discussed in details and different sponge layer functions are proposed. The linear system is solved using the finite element method (FEM) at low computational cost compared to the target simulations. This method should enable the design of efficient sponge layer for any full Navier–Stokes solvers. The linear model results solved with a FEM solver are compared here to SPH simulations with good agreement. The linear model is then used to determine the reflection coefficient for different power sponge functions, length and dissipation coefficient using the FEM solver. The linear model solved by a simple FEM solver can be used to evaluate the suitable dissipation coefficients for waves ranging from shallow to deep water for a given sponge layer length. The coefficient depends on the non-dimensional frequency for Ω = ω d / g 2 . A table of suitable parameters for different sponge layer functions is provided over a large range of non-dimensional frequency and sponge layer length. A 3D application of waves is also presented with 45° incidence angle.

This paper aims at validating pressure and velocity measurements conducted in two physical scale models of an ogee spillway crest operating at heads largely greater than the design head. The design head of the second model is 50% smaller than the one of the first model. No pier effect or air venting is considered in the study. The velocity field is measured by Bubbles Image Velocimetry. The relative pressure along the spillway crest is measured using pressure sensors. Comparison of measured velocities between both spillways indicates low scale effects, the scaled-profiles collapsing in most parts of the flow. By contrast, measurements of relative pressure along the spillway crest differ for large heads. A theoretical velocity profile based on potential flow theory and expressed in a curvilinear reference frame is fitted to the velocity measurements, considered as reference, for extrapolating the velocity at the spillway crest. Comparing the extrapolated velocity at the spillway crest and the velocity calculated from the relative pressure considering a potential flow finally emphasizes that bottom pressure amplitudes seem overestimated for the larger spillway, while an averaging effect might operate for the pressure measurements on the smaller spillway.

The modeling of wave breaking dissipation in coastal areas is investigated with a fully nonlinear and dispersive wave model. The wave propagation model is based on potential flow theory, which initially assumes non-overturning waves. Including the impacts of wave breaking dissipation is however possible by implementing a wave breaking initiation criterion and dissipation mechanism. Three criteria from the literature, including a geometric, kinematic, and dynamic-type criterion, are tested to determine the optimal criterion predicting the onset of wave breaking. Three wave breaking energy dissipation methods are also tested: the first two are based on the analogy of a breaking wave with a hydraulic jump, and the third one applies an eddy viscosity dissipative term. Numerical simulations are performed using combinations of the three breaking onset criteria and three dissipation methods. The simulation results are compared to observations from four laboratory experiments of regular and irregular waves breaking over a submerged bar, irregular waves breaking on a beach, and irregular waves breaking over a submerged slope. The different breaking approaches provide similar results after proper calibration. The wave transformation observed in the experiments is reproduced well, with better results for the case of regular waves than irregular waves. Moreover, the wave statistics and wave spectra are predicted well in general, and in particular for regular waves. Some differences are observed for irregular wave cases, in particular in the low-frequency range. This is attributed to incomplete absorption of the long waves in the numerical model. Otherwise, the wave spectra in the range $[0.5f_p,\: 5f_p]$ are reproduced well, before, inside, and after the breaking zone for the three irregular wave experiments., 25 pages, 14 figures, 5 tables. J. Ocean Eng. Mar. Energy (2019)

For reasons of efficiency and accuracy, water wave propagation is often simulated with potential or inviscid models rather than Navier-Stokes solvers, but for wave-induced flows, such as wave-structure interaction, viscous effects are important under certain conditions. Alternatively, general purpose Navier-Stokes (CFD) models can have limitations when applied to such free-surface problems when dealing with large amplitude waves, run-up, or propagation over long distances. Here we present an Arbitrary Lagrangian-Eulerian (ALE) algorithm with special care to the time-stepping and boundary conditions used for the free-surfaces, integrated into Code_Saturne, and we test its capabilities for modeling a variety of water wave generation and propagation benchmarks, and finally consider interaction with a vertical cylinder. Two variants of the mesh displacement computation are proposed and tested against the discrete Geometric Conservation Law (GCL). The more robust variant, for highly curved or sawtoothed free-surfaces, uses a Compatible Discrete Operator scheme on the dual mesh for solving the mesh displacement, which makes the algorithm valid for any polyhedral mesh. Results for standard wave propagation benchmarks for both variants show that, when care is taken to avoid grids with excessive numerical dissipation, this approach is effective at reproducing wave profiles as well as forces on bodies.

International audience; The numerical modelling of two-phase mixture flows with high density ratios (e.g. water/air) is challenging. Multiphase averaged models with volume fraction representation encompass a simple way of simulating such flows: mixture models with relative velocity between phases. Such approaches were implemented in SPH (Smoothed Particle Hydrodynamics) using a mass-weighted definition of the mixture velocity, but with limited validation. Instead, to handle high density ratios, a mixture model with a volumetric mixture velocity is developed in this work. To avoid conservation issues raised by the discretization of the relative material displacement contribution in the volume fraction equation, a formulation on phase volumes is derived following a finite volume reasoning. Conservativity, realizability, limit behaviour for single-phase flow are the leading principles of this derivation. Volume diffusion is added to prevent development of instabilities due to the colocated nature of SPH. This model is adapted to the semi-analytical SPH wall boundary conditions. Running on GPU, this approach is successfully applied to the separation of phases in a settling tank with low to high density ratios. An analytical solution on a two-phase mixture Poiseuille flow is also used to check the accuracy of the numerical implementation. Then, a Rayleigh-Taylor instability test case is performed to compare with multi-fluid SPH. Finally, a comparison with experimental and numerical data is made on a sand dumping case; this highlights some limits of this mixture model.

International audience; This paper presents the development and validation of a three-dimensional numerical wave tank devoted to studying wave-structure interaction problems. It is based on the fully nonlinear potential flow theory, here solved by a boundary element approach and using unstructured triangular meshes of the domain's boundaries. Time updating is based on a second-order explicit Taylor series expansion. The method is parallelized using the Message Passing Interface (MPI) in order to take advantage of multi-processor systems. For radiation problems, with cylindrical bodies moving in prescribed motion, the free-surface is updated with a fully Lagrangian scheme, and is able to reproduce reference results for nonlinear forces exerted on the moving body. For diffraction problems, semi-Lagrangian time-updating is used, and reproduces nonlinear effects for diffraction on monopiles. Finally, we study the nonlinear wave loads on a fixed semi-submersible structure, thereby illustrating the possibility to apply the proposed numerical model for the design of offshore structures and floaters.

Air entrainment within water is a common feature of flows over hydraulic works – spill over a dam, wave breaking on a dike, etc. – and its accurate modeling is a key to better design such structures. The Smoothed Particle Hydrodynamics (SPH) method appears as a natural way to model such highly distorted flows. To avoid computationally prohibitive costs related to the full discretization of bubbles or drops, a mixture model for high density ratio flows relying on a volume-based formulation with relative velocity between phases has first been developed and validated in Fonty et al. (Proceedings of 13th international SPHERIC workshop. Galway, Ireland, 2018; Int J Multiph Flow 111:158–174, 2019. https://doi.org/10.1016/j.ijmultiphaseflow.2018.11.007 ). Instead of having a once and for all assigned phase as in multifluid SPH, each particle now carries both phases through their respective volume fractions. In the present work, in order to handle practical air entrainment application cases, the open boundary formulation described in Ferrand et al. (Comput Phys Commun 210:29–44, 2017. https://doi.org/10.1016/j.cpc.2016.09.009 ) is adapted to this mixture model. Then, after introducing turbulence through a $$k{-}\epsilon$$ model, a specific closure on the air bubbles relative velocity is proposed including a Stokesian drag term and turbulent diffusion. This model is then applied to two cases of air entrainment: a stepped spillway for interfacial aeration and a plunging jet for local aeration. Finally a 3D industrial test case of discharge-control structure at the La Coche power plant (France) is considered. While valuable insights are obtained for the volume fraction field, further investigations are required to improve the modeling of the flow dynamics.

We present a numerical model for the simulation of 3D poly-dispersed sediment transport in a Newtonian flow with free surfaces. The physical model is based on a mixture model for multiphase flows. The Navier-Stokes equations are coupled with the transport and deposition of the particle concentrations, and a volume-of-fluid approach to track the free surface between water and air. The numerical algorithm relies on operator-splitting to decouple advection and diffusion phenomena. Two grids are used, based on unstructured finite elements for diffusion and an appropriate combination of the characteristics method with Godunov's method for advection on a structured grid. The numerical model is validated through numerical experiments. Simulation results are compared with experimental results in various situations for mono-disperse and bi-disperse sediments, and the calibration of the model is performed using, in particular, erosion experiments. (C) 2018 The Authors. Published by Elsevier Ltd.

Nearshore wave modeling over spatial scales of several kilometers requires balancing the fine-scale modeling of physical processes with the model’s accuracy and efficiency. In this work, a fully nonlinear potential flow model is proposed as a compromise between simplified linear, weakly nonlinear or weakly dispersive models and direct CFD approaches. The core of present approach is the use of a series representation for the velocity potential. This series contains prescribed vertical functions and allows the determination of the velocity potential in terms of unknown horizontal functions. The resulting dimensionally reduced model retains the structure of the Hamiltonian water wave system Zakharov (1968), Craig & Sulem (1993), avoiding the solution of the Laplace problem for the potential. Instead, a numerically convenient linear system of horizontal equations needs to be solved at each step in the temporal evolution. No simplifications concerning the deformation of the physical boundaries are introduced, apart from the typical requirement of a smooth, non-overturning free surface and seabed. The main limitation of this formulation is its inability to account for wave breaking. The treatment of this process is the subject of the present work. Two different techniques are implemented in the present model. Simulation results are compared to laboratory measurements for two test cases: (1) shoaling and breaking of regular waves over a barred bathymetry Beji & Battjes (1993) and (2) shoaling and breaking of regular waves on a plane beach Ting & Kirby (1994).

In order to address the question of the SPH (Smoothed Particle Hydrodynamics) Laplacian conditioning, a spectral analysis of this discrete operator is performed. In the case of periodic Cartesian particle network, the eigenfunctions and eigenvalues of the SPH Laplacian are found on theoretical grounds. The theory agrees well with numerical eigenvalues. The effects of particle disorder and non-periodicity conditions are then investigated from numerical viewpoint. It is found that the matrix condition number is proportional to the square of the particle number per unit length, irrespective of the space dimension and kernel choice.

Local scour has been identified as the main factor that causes failures of structures in offshore engineering. Numerous research efforts have been devoted to local scour around offshore pipelines in the past. In this paper, a finite element numerical model is established for simulating local scour below offshore pipelines in steady currents. The flow is simulated by solving the unsteady Reynolds-averaged Navier–Stokes equations with a standard k − ϵ turbulent model closure. A sand slide scheme is proposed for the scour calculation, and bed load is considered in the proposed scour model. To account for changes in bed level, the moving mesh method is adopted to capture the water–sediment interface bed, and the change of bed level is calculated by solving Exner–Polya equation. All the equations are discretised within the two-step Taylor–Galerkin algorithm in this paper. It is found that the sand slide model works well for the simulation of the scour, and the numerical results are shown to be in good agreement with the available experimental data.

International audience; In this work, a weakly compressible smoothed particle hydrodynamics (WCSPH) multiphase model is developed. The model is able to deal with soil-water interactions coupled in a strong and natural form. A Regularized Bingham Plastic constitutive law including a pressure-dependent Mohr-Coulomb yield criterion (RBPMC-αμ) is proposed to model fluids, soils and their interaction. Since the proposed rheology model is pressure-sensitive, we propose a multiphase diffusive term to reduce the spurious pressure resulting from the weakly compressible flow hypothesis. Several numerical benchmarks are investigated to assess the robustness and accuracy of the proposed multiphase SPH model.

A previously unreported internal tidal bore, which evolves into solitary internal wave packets, was observed in Port Susan, Puget Sound, and the timing, speed, and amplitude of the waves were measured by CTD and visual observation. Acoustic Doppler current profiler (ADCP) measurements were attempted, but unsuccessful. The waves appear to be generated with the ebb flow along the tidal flats of the Stillaguamish River, and the speed and width of the resulting waves can be predicted from second-order KdV theory. Their eventual dissipation may contribute significantly to surface mixing locally, particularly in comparison with the local dissipation due to the tides. Visually the waves appear in fair weather as a strong foam front, which is less visible the farther they propagate.

Saint-Venant equations can be generalized to account for a viscoelastic rheology in shallow flows. A Finite-Volume discretization for the 1D Saint-Venant system generalized to Upper-Convected Maxwell (UCM) fluids was proposed in Bouchut and Boyaval (M3AS 23(08): 1479–1526, 2013, [6]), which preserved a physically-natural stability property (i.e. free-energy dissipation) of the full system. It invoked a relaxation scheme of Suliciu type for the numerical computation of approximate solutions to Riemann problems. Here, the approach is extended to the 1D Saint-Venant system generalized to the finitely-extensible nonlinear elastic fluids of Peterlin (FENE-P). We are currently not able to ensure all stability conditions a priori, but the scheme is fully computable. And, using numerical simulations, it may help understand the famous High-Weissenberg number problem (HWNP) well-known in computational rheology.

In this work we discuss a way to compute the impact of free-surface flow on nonlinear structures. The approach chosen relies on a partitioned strategy that allows us to solve the strongly coupled fluid–structure interaction problem. It is then possible to re-use the existing and validated strategy for each sub-problem. The structure is formulated in a Lagrangian way and solved by the finite element method. The free-surface flow approach considers a Volume-Of-Fluid (VOF) strategy formulated in an Arbitrary Lagrangian–Eulerian (ALE) framework, and the finite volume is used to discrete and solve this problem. The software coupling is ensured in an efficient way using the Communication Template Library (CTL). Numerical examples presented herein concern the 2D validation case but also 3D problems with a large number of equations to be solved.

This paper assesses some recent trends in the novel numerical meshless method smoothed particle hydrodynamics, with particular focus on its potential use in modelling free-surface flows. Due to its Lagrangian nature, smoothed particle hydrodynamics (SPH) appears to be effective in solving diverse fluid-dynamic problems with highly nonlinear deformation such as wave breaking and impact, multi-phase mixing processes, jet impact, sloshing, flooding and tsunami inundation, and fluid–structure interactions. The paper considers the key areas of rapid progress and development, including the numerical formulations, SPH operators, remedies to problems within the classical formulations, novel methodologies to improve the stability and robustness of the method, boundary conditions, multi-fluid approaches, particle adaptivity, and hardware acceleration. The key ongoing challenges in SPH that must be addressed by academic research and industrial users are identified and discussed. Finally, a roadmap is propose...

International audience; We propose a unified framework to derive thin-layer reduced models for some shallow free-surface flows driven by gravity. It applies to incompressible homogeneous fluids whose momentum evolves according to Navier-Stokes equations, with stress satisfying a rheology of viscous type (i.e. the standard Newtonian law with a constant viscosity, but also non-Newtonian laws generalized to purelyviscous fluids and to viscoelastic fluids as well). For a given rheology, we derive various thin-layer reduced models for flows on a rugous topography slowly varying around an inclined plane. This is achieved thanks to a coherent simplification procedure, which is formal but based on a mathematically clear consistency requirement between scaling assumptions and the approximation errors in thedifferential equations. The various thin-layer reduced models are obtained depending on flow regime assumptions (either fast/inertial or slow/viscous). As far as we know, it is the first time that the various thin-layer reduced models investigated here are derived within the same mathematical framework. Furthermore, we obtain new reduced models in the case of viscoelastic non-Newtonian fluids,which extends [Bouchut & Boyaval, M3AS (23) 8, 2013].

International audience; We present a second order residual distribution scheme for scalar transport problems in shallow water flows. The scheme, suitable for the unsteady cases, is obtained adapting to the shallow water context the explicit Runge-Kutta schemes for scalar equations [1]. The resulting scheme is decoupled from the hydrodynamics yet the continuity equation has to be considered in order to respect some important numerical properties at discrete level. Beyond the classical characteristics of the residual formulation presented in [1] and [2], we introduce the possibility to iterate the corrector step in order to improve the accuracy of the scheme. Another novelty is that the scheme is based on a precise monotonicity condition which guarantees the respect of the maximum principle. We thus end up with a scheme which is mass conservative, second order accurate and monotone. These properties are checked in the numerical tests, where the proposed approach is also compared to some finite volume schemes on unstructured grids. The results obtained show the interest in adopting the predictor corrector scheme for pollutant transport applications, where conservation of the mass, monotonicity and accuracy are the most relevant concerns.

A classical incompressible algorithm for Smoothed Particle Hydrodynamics (ISPH) is analyzed in terms of critical time step for numerical stability. For this purpose, a theoretical linear stability analysis is conducted for unbounded homogeneous flows, leading to an analytical formula for the maximum CFL (Courant-Friedrichs-Lewy) number as a function of the Fourier number. This gives the maximum time step as a function of the fluid viscosity, the flow velocity scale and the SPH discretization size (kernel standard deviation). Importantly, the maximum CFL number at large Reynolds number appears twice smaller than with the traditional Weakly Compressible (WCSPH) approach. As a consequence, the optimal time step for ISPH is only five times larger than with WCSPH. The theory agrees very well with numerical data for two usual kernels in a 2-D periodic flow. On the other hand, numerical experiments in a plane Poiseuille flow show that the theory overestimates the maximum allowed time step for small Reynolds numbers.

Smoothed Particle Hydrodynamics (SPH) has been used in the simulation of fluid flows for several years. Recently, SPH saw its range of applications extended to the simulation of real-world engineering applications which often concern the simulation of turbulent three-dimensional flows. However, there is a significant lack in the investigation of such turbulent flows even for simplified applications. Most three-dimensional SPH simulations that analyze properties of a turbulent flow examine the decay of isotropic turbulence, where no solid walls influence the flow. In the present paper several three-dimensional turbulent channel flows are simulated using the SPH method to address this gap in literature. The first set of simulations investigates a quasi direct numerical simulation (DNS) of a channel with reduced size. This enables evaluation of SPH as a pure Navier–Stokes solver by avoiding the introduction of turbulence models. The results show the correct reproduction of the turbulent statistics except for some fluctuations in the near wall region. The second set of simulations attempts a large eddy simulation (LES) of a standard size channel. It is shown that the present SPH formulation fails to reproduce the correct turbulent statistics and the cause for this failure is analyzed with the aide of the Taylor–Green vortex flow and attributed to incorrect reproduction of velocity–pressure interactions by the collocated and large stencil SPH discretisation.

A fast finite volume solver for hydrostatic multi-layered shallow water flows with mass exchange is investigated. In contrast to many models for multi-layered hydrostatic shallow water flows where the immiscible suppression is assumed, the present model allows for mass exchange between the layers. The multi-layered shallow water equations form a system of conservation laws with source terms for which the computation of the eigenvalues is not trivial. For most practical applications, complex eigenvalues may arise in the system and the multi-layered shallow water equations are not hyperbolic anymore. This property makes the application of conventional finite volume methods difficult or even impossible for those methods that require in their formulation the explicit computation of the eigenvalues. In the current study, we propose a finite volume method that avoids the solution of Riemann problems. At each time step, the method consists of two stages to update the new solution. In the first stage, the multi-layered shallow water equations are rewritten in a non-conservative form and the intermediate solutions are calculated using the method of characteristics. In the second stage, the numerical fluxes are reconstructed from the intermediate solutions in the first stage and used in the conservative form of the multi-layered shallow water equations. The proposed method is simple to implement, satisfies the conservation property and is suitable for multi-layered shallow water equations on non-flat topography. The proposed finite volume solver is verified against several benchmark tests and it shows good agreement with analytical solutions of the incompressible hydrostatic Navier–Stokes equations. The method is conservative by construction and preserves the mass to the machine precision. The performance of the method is also demonstrated by comparing the results obtained using the proposed finite volume method to those obtained using the well-established kinetic method.

International audience; We present a comparison between two distinct numerical codes dedicated to the study of wave energy converters. Both are developed by the authors, using a boundary element method with linear triangular elements. One model applies fully nonlin-ear boundary conditions in a numerical wavetank environnment (and thus referred later as NWT), whereas the second relies on a weak-scatterer approach in open-domain and can be considered a weakly nonlinear potential code (referred later as WSC). For the purposes of comparison, we limit our study to the forces on a heaving submerged sphere. Additional results for more realistic problem geometries will be presented at the conference. INTRODUCTION Among the marine renewable energy sources, wave energy is a promising option. Despite the great number of technologies that have been proposed, currently no wave energy converter (WEC) has proven its superiority over others and become a technological solution. Usual numerical tools for modeling and designing WECs are based on boundary elements methods in linear potential theory [1-4]. However WECs efficiency relies on large amplitude motions [5], with a design of their resonance frequencies in the wave excitation. Linear potential theory is thus inadequate to study the behavior of WEC in such configuration.

Wave-induced boundary layer (BL) flows over sandy rippled bottoms are studied using a numerical model that applies a one-way coupling of a "far-field" inviscid flow model to a "near-field" large eddy simulation (LES) Navier–Stokes (NS) model. The incident inviscid velocity and pressure fields force the LES, in which near-field, wave-induced, turbulent bottom BL flows are simulated. A sediment suspension and transport model is embedded within the coupled flow model. The numerical implementation of the various models has been reported elsewhere, where we showed that the LES was able to accurately simulate both mean flow and turbulent statistics for oscillatory BL flows over a flat, rough bed. Here we show that the model accurately predicts the mean velocity fields and suspended sediment concentration for oscillatory flows over full-scale vortex ripples. Tests show that surface roughness has a significant effect on the results. Beyond increasing our insight into wave-induced oscillatory bottom BL physics, sophisticated coupled models of sediment transport such as that presented have the potential to make quantitative predictions of sediment transport and erosion/accretion around partly buried objects in the bottom, which is important for a vast array of bottom deployed instrumentation and other practical ocean engineering problems.

International audience; The numerical simulation of three-dimensional dam break flows is discussed. A non-hydrostatic numerical model for free-surface flows is considered, which is based on the incompressible Navier-Stokes equations coupled with a volume-of-fluid approach. The numerical results obtained for a variety of benchmark problems show the validity of the numerical approach, in comparison with other numerical models, and allow to investigate numerically the non-hydrostatic three-dimensional effects, in particular for the usual test cases where hydrostatic approximations are known analytically. The numerical experiments on actual topographies, in particular the Malpasset dam break and the (hypothetical) break of the Grande-Dixence dam in Switzerland, also illustrate the capabilities of the method for large-scale simulations and real-life visualization.

Le modele original propose par les auteurs (CHAZEL et al., 2009) pour les vagues en zone cotiere est mis en œuvre et valide sur deux cas experimentaux. Nous considerons d’abord les experiences de DINGEMANS (1994), etudiant la propagation de vagues regulieres au-dessus d’une barre trapezoidale immergee. Les resultats numeriques demontrent une excellente capacite du modele a reproduire les effets de levee, les interactions fortement non-lineaires, ainsi que la generation puis la propagation de composantes harmoniques d’ordres eleves apres la barre. Ensuite, nous verifions la capacite du modele a propager des vagues irregulieres non-deferlantes sur une bathymetrie plus complexe (essai 26 de BECQ-GIRARD et al., 1999). L’analyse et la comparaison des spectres de variance montrent que les resultats du modele sont egalement en tres bon accord avec les mesures experimentales. Abstract: The new model recently proposed by the authors (CHAZEL et al., 2009) to simulate waves in the coastal zone is applied and validated on two experimental cases. Firstly the set of experiments by DINGEMANS (1994) is considered, with the propagation of regular waves over a submerged trapezoidal bar. The numerical results show the excellent capabilities of the model to reproduce the shoaling effects, as well as the generation and the further propagation of higher harmonics after the bar. Then we check the ability of the model to propagate irregular non-breaking waves over a more complex bottom profile (experiment 26 by BECQ-GIRARD et al., 1999). The analysis and comparison of variance spectra show that the model results are also in very good agreement with experimental measurements. Keywords: Waves, Nonlinear waves, Dispersive waves, Coastal waves, Wave models, Boussinesq-type model, Two-layer modelling technique.

International audience; The purpose of this work is to develop advanced numerical tools for modeling two-way fully nonlinear interactions of ocean surface waves (irregular waves in the general situation) with submerged structures undergoing large amplitude motion, that could represent Wave Energy Converters (WECs). In our modeling approach, an existing two-dimensional Numerical Wave Tank (NWT), based on potential flow theory, is extended to include a submerged horizontal cylinder of arbitrary cross-section. The mathematical problem and related numerical solution are first introduced. Then, conservation of volume and conservation of energy are checked, respectively, in the case of a circular cylinder in a prescribed large amplitude motion and in the case of a circular cylinder in a free motion. Interactions between waves and a submerged circular cylinder computed by the model are then compared to mathematical solutions for two situations: a cylinder in prescribed motion and a freely moving cylinder.

International audience; The prediction of solid bodies transport (such as algae, debris, sediment grains, or corrosion deposits) is a necessary requirement in many industrial or environmental processes. The physical processes involved cover a wide range of processes, from tidal flow to turbulent eddies and particle drag. A stochastic model was therefore developed to link the different scales of the physical processes where it was assumed that the particles are dilute enough that they do not affect the flow or the motion of other particles while being large enough that each particle does not follow exactly the fluid motions (i.e., macro-particles). The stochastic model is built in such a way that it uses Reynolds-averaged fluid properties to predict trajectories of individual particles. This model was then tested using experimental measurements obtained for isotropic particles released in semi-homogeneous turbulence. The turbulent flow was generated using a pair of oscillating grids and was characterized using particle image velocimetry measurements. The trajectories of the particles were measured using a pair of high resolution cameras. The comparison between the experimental data and different numerical models gives satisfactory results.

A family of explicit algebraic models for Reynolds stresses and passive scalar fluxes is proposed and a few variants are discussed, according to the selected set of model constants. Validation is attempted against direct numerical simulation in a steady closed-channel flow for two values of the Reynolds number, indicating the ability of the model to reasonably predict the anisotropy tensor and the scalar flux components when considering wall functions to account for near-wall effects. The production of turbulent kinetic energy is well predicted, while the neutral turbulent Prandtl number has values in agreement with the literature. The proposed model is then compared with an experiment on a heated cylinder with slight temperature variations, with satisfactory results. The proposed model can be applied to various 2D simply-sheared flows involving a passive scalar, i.e. if stratification effects are negligible.

We propose a new reduced model for gravity-driven free-surface flows of shallow elastic fluids. It is obtained by an asymptotic expansion of the upper-convected Maxwell model for elastic fluids. The viscosity is assumed small (of order epsilon, the aspect ratio of the thin layer of fluid), but the relaxation time is kept finite. Additionally to the classical layer depth and velocity in shallow models, our system describes also the evolution of two scalar stresses. It has an intrinsic energy equation. The mathematical properties of the model are established, an important feature being the non-convexity of the physically relevant energy with respect to conservative variables, but the convexity with respect to the physically relevant pseudo-conservative variables. Numerical illustrations are given, based on a suitable well-balanced finite-volume discretization involving an approximate Riemann solver., Part of this work was completed while Sebastien Boyaval was an academic host at MATHICSE- ASN chair (EPFL). SB would like to thank Prof. Marco Picasso and Prof. Jacques Rappaz for this invitation

International audience; Les travaux présentés concernent le développement d'un code de simulation avancé permettant de décrire le mouvement d'un corps immergé sous l'action des vagues, avec des mouvements de grande amplitude. A terme, cet outil est destiné à modéliser le comportement de certains types de Systèmes Récupérateurs d'Energie des Vagues (SREV) immergés. Dans cette étude on adopte une approche potentielle pour la partie hydrodynamique, dans un cadre 2DV (i.e. dans un plan vertical), correspondant au cas d'un canal à houle numérique. Le modèle utilisé pour la génération et la propagation des vagues est un modèle potentiel complètement non-linéaire, fondé sur une méthode d'éléments de frontières d'ordre élevé, développé par Grilli et ses collaborateurs depuis une vingtaine d'années. Ce modèle, déjà largement validé sur différents types d'applications océaniques et côtières, a été modifié pour prendre en compte la présence d'un corps rigide immergé, fixe ou mobile, avec le calcul des efforts de pression hydrodynamique s'exerçant sur le corps. Une méthodologie spécifique a été développée pour résoudre le problème couplé hydrodynamique/mécanique. Nous présentons deux cas de validation pour lesquels les résultats du modèle numérique sont comparés à des résultats d'autres modèles mathématiques : théorie analytique de WU (1993) sur le cas d'un cylindre en mouvement circulaire imposé et théorie linéaire d'EVANS et al. (1979) sur le cas d'un cylindre soumis à des forces externes (forces de rappel, force représentant l'extraction d'énergie des vagues,...), en plus des efforts hydrodynamiques.

Extensive numerical simulations of fetch-limited growth of wind-driven waves are analysed within two approaches: a ‘traditional’ wind-speed scaling first proposed by Kitaigorodskii (Bull. Acad. Sci. USSR, Geophys. Ser., Engl. Transl., vol. N1, 1962, p. 105) in the early 1960s and an alternative weakly turbulent scaling developed recently by Badulin et al. (J. Fluid Mech.591, 2007, 339–378). The latter one uses spectral fluxes of wave energy, momentum and action as physical scales of the problem and allows for advanced qualitative and quantitative analysis of wind-wave growth and features of air–sea interaction. In contrast, the traditional approach is shown to be descriptive rather than proactive. Numerical simulations are conducted on the basis of the Hasselmann kinetic equation for deep-water waves in a wide range of wind speeds from 5 to 30 m s −1 and for the ideal case of fetch-limited growth: permanent wind blowing perpendicularly to a straight coastline. Two different wave input functions, Sin, and two methods for calculating the nonlinear transfer term Snl (Gaussian quadrature method, or GQM, a quasi-exact method based on the use of Gaussian quadratures, and the discrete interaction approximation, or DIA) are used in the simulations. Comparison of the corresponding results firstly shows the relevance of the analysis of wind-wave growth in terms of the proposed weakly turbulent scaling, and secondly, allows us to highlight some critical points in the modelling of wind-generated waves. Three stages of wind-wave development corresponding to qualitatively different balance of the source terms, Sin, Sdiss and Snl, are identified: initial growth, growing sea and fully developed sea. Validity of the asymptotic weakly turbulent approach for the stage of growing wind sea is determined by the dominance of nonlinear transfers, which results in a rigid link between spectral fluxes and wave energy. This stage of self-similar growth is investigated in detail and presented as a consequence of three sub-stages of qualitatively different coupling of air flow and growing wind waves. The key self-similarity parameter of the asymptotic theory is estimated to be αss = 0.68 ± 0.1.Further prospects of wind-wave modelling in the context of the presented weakly turbulent scaling are discussed.

Laboratory experiments were performed to study the dynamics of three- dimensional mechanically generated waves propagating over an oblique current in partial opposition. The flow velocity varied along the mean wave direction of propagation with an increasing trend between the wave-maker and the centre of the tank. Tests with regular wave packets traversing the area of positive current gradient showed that the concurrent increase of wave steepness triggered modulational instability on otherwise stable wave trains and hence induced the development of very large amplitude waves. In random directional wave fields, the presence of the oblique current resulted in a weak reinforcement of wave instability with a subsequent increase of the probability of occurrence of extreme events. This seems to partially compensate the suppression of strongly non-Gaussian properties due to directional energy distribution.

Hafenanlagen werden oft in Gebieten errichtet,wo die Wassertiefen gering sind, insbesondere inÄstuaren. Es ist dann notwendig, Ausbaggerungsarbeitendurchzuführen, um Schiffen die Zufahrtzu ermöglichen. Das ausgebaggerte Sedimentwird in einer vorher definierten Deponierungszonefreigesetzt. Wegen der Konzentration derSedimentwolke, die sich während der Freisetzungbildet (zu Beginn mehr als 350g/l) entsteht einEinfluss auf die Umwelt. Die chemische Zusammensetzungdes Wassers wird direkt beeinflusst.Nach der Freisetzung schlagen sich die Sedimenteunter einer sehr hoch konzentrierten Wolkenieder. Auf diesen Schritt des Niederschlags folgtein Einfluss auf die Sohle durch Bildung einesSuspensionsstroms.Der Zweck dieses Beitrags ist das Studium diesesPhänomens mittels numerischer Simulationenunter Verwendung eines Zwei-Phasen-Modellsfür den Sedimenttransport (Barbry et al., 2000 ;Chauchat et al., 2008 ; Nguyen et al., 2009). Esbasiert auf der Lösung der Erhaltungs- und Impuls-Gleichungen für jede Phase (Wasser undPartikel) und integriert dann die Interaktion zwischenPartikeln und Wasser. Ein Vergleich mitden experimentellen Konfigurationen von Villaretet al. (1997, 1998), ohne Strömung, zeigt, dassdie verschiedenen Schritte des Phänomens rechtgut ohne Modellierung, The port structures are often established in areaswhere water levels are fairly low, especially in estuaries.It is then necessary to perform dredging toallow ships accessing to the docks. The dredgedsediment is released over a predefined depositzone. Because of the concentration of the sedimentcloud appearing during the release, morethan 350 g/l at the beginning, there is an impacton the environment. The chemical composition ofwater is affected directly.After the release, the sediments settle under acloud of very high concentration. This settling stepis followed after impact on the bed by the formationof turbidity current. The purpose of this work is tostudy the phenomenon via numerical simulationsby using a two-phase model of sediment transport[Barbry et al., 2000 ; Chauchat et al., 2008 ;Nguyen et al., 2009]. It is based on the solving ofconservation and momentum equations for eachphase (water and particles) and then integratesthe interaction between the particles and the water.A comparison with the experimental configurationof Villaret et al. (1997, 1998), without current,shows that the different steps of the phenomenonare quite reproduce without turbulence modelling., La gestion des clapages (ou rejets en mer des produits de dragage) constitue une problématique environnementale et économique importante pour les estionnaires des voies navigables et des infrastructures portuaires. Moins coûteuses qu’un stockage à terre, ces opérations peuvent induire des nuisances sur l’environnement notamment en provoquant un accroissement local de la turbidité et en ensevelissant les habitats de la faune aquatique. Le phénomène de clapage comporte principale ment trois phases [Boutin, 2000]: la phase de chute (soumise aux courants) ; l’impact sur le fond et la génération d’un courant de densité ; la propagation des courants de densité et la décantation des sédiments.

International audience; For geophysical flows, the Saint-Venant system is often a good approximation of the Navier-Stokes equa- tions and so it is widely used for the simulations of shallow water flows and gravity waves. But the Saint- Venant system fails to represent the non-hydrostatic effects such as those associated for example with dispersive waves. To overcome this limitation, we propose an extended version of the Saint-Venant sys- tem and an associated finite volume scheme for the numerical simulations. The proposed model has sev- eral interesting properties e.g. it admits an energy balance and it preserves the dynamics of solitary waves. Several numerical results are given and especially comparisons between simulations and exper- imental measurements.

In this paper, a second order space discontinuous Galerkin (DG) method is presented for the numerical solution of inviscid shallow water flows over varying bottom topography. Novel in the implementation is the use of HLLC and kinetic numerical fluxes in combination with a dissipation operator, applied only locally around discontinuities to limit spurious numerical oscillations. Numerical solutions over (non-)uniform meshes are verified against exact solutions; the numerical error in the $L_2$-norm and the convergence of the solution are computed. Bore-vortex interactions are studied analytically and numerically to validate the model; these include bores as "breaking waves'' in a channel and a bore traveling over a conical and Gaussian hump. In these complex numerical test cases, we correctly predict the generation of potential vorticity by non-uniform bores. Finally, we successfully validate the numerical model against measurements of steady oblique hydraulic jumps in a channel with a contraction. In the latter case, the kinetic flux is shown to be more robust.