In this paper we propose a stable and robust strategy to approximate the 3d incompressible hydrostatic Euler and Navier-Stokes systems with free surface.Compared to shallow water approximation of the Navier-Stokes system, the idea is to use a Galerkin type approximation of the velocity field with piecewise constant basis functions in order to obtain an accurate description of the vertical profile of the horizontal velocity. Such a strategy has several advantages. It allows • to rewrite the Navier-Stokes equations under the form of a system of conservation laws with source terms,• the easy handling of the free surface, which does not require moving meshes,• the possibility to take advantage of robust and accurate numerical techniques developed in extensive amount for Shallow Water type systems.Compared to previous works of some of the authors, the three dimensional case is studied in this paper. We show that the model admits a kinetic interpretation including the vertical exchanges terms, and we use this result to formulate a robust finite volume scheme for its numerical approximation. All the aspects of the discrete scheme (fluxes, boundary conditions,. . . ) are completely described and the stability properties of the proposed numerical scheme (well-balancing, positivity of the water depth,. . . ) are discussed. We validate the model and the discrete scheme with some numerical academic examples (3d non stationary analytical solutions) and illustrate the capability of the discrete model to reproduce realistic tsunami waves.

Hydrodynamics in a high-rate production reactor for microalgae cultivation affects the light history perceived by cells. The interplay between cell movement and medium turbidity leads to a complex light pattern, whose forcing effects on photosynthesis and photoacclimation dynamics are non-trivial. Hydrodynamics of high density algal ponds mixed by a paddle wheel has been studied recently, although the focus has never been on describing its impact on photosynthetic growth efficiency. In this multidisciplinary downscaling study, we first reconstructed single cell trajectories in an open raceway using an original hydrodynamical model offering a powerful discretization of the Navier–Stokes equations tailored to systems with free surfaces. The trajectory of a particular cell was selected and the associated high-frequency light pattern was computed. This light pattern was then experimentally reproduced in an Arduino-driven computer controlled cultivation system with a low density Dunaliella salina culture. The effect on growth and pigment content was recorded for various frequencies of the light pattern, by setting different paddle wheel velocities. Results show that the frequency of this realistic signal plays a decisive role in the dynamics of photosynthesis, thus revealing an unexpected photosynthetic response compared to that recorded under the on/off signals usually used in the literature. Indeed, the light received by a single cell contains signals from low to high frequencies that nonlinearly interact with the photosynthesis process and differentially stimulate the various time scales associated with photoacclimation and energy dissipation. This study highlights the need for experiments with more realistic light stimuli to better understand microalgal growth at high cell densities. An experimental protocol is also proposed, with simple, yet more realistic, step functions for light fluctuations.

We are interested in free surface flows where density variations coming, for example, from temperature or salinity differences, play a significant role in the hydrodynamic regime. In water, acoustic waves travel much faster than gravity and internal waves, hence the study of models arising from compressible fluid mechanics often requires a decoupling between these waves. Starting from the compressible Navier-Stokes system, we derive the so-called Navier-Stokes-Fourier system in an "incompressible" regime using the low-Mach scaling, hence filtering the acoustic waves, neglecting the density dependency on the fluid pressure but keeping its variations in terms of temperature and salinity. A slightly modified low-Mach asymptotics is proposed to obtain a model with thermo-mechanical compatibility. The case when the density depends only on the temperature is studied first. Then the variations of the fluid density with respect to temperature and salinity are considered, and it seems to be the first time that salinity dependency is considered in this low Mach limit. The obtained models conserve the mass of the fluid but not the volume and satisfy the second principle of thermodynamics.

Abstract