1,983 research outputs found
An energy-consistent depth-averaged Euler system: derivation and properties
In this paper, we present an original derivation process of a non-hydrostatic
shallow water-type model which aims at approximating the incompressible Euler
and Navier-Stokes systems with free surface. The closure relations are obtained
by aminimal energy constraint instead of an asymptotic expansion. The model
slightly differs from thewell-known Green-Naghdi model and is confronted with
stationary andanalytical solutions of the Euler system corresponding to
rotationalflows. At the end of the paper, we givetime-dependent analytical
solutions for the Euler system that are alsoanalytical solutions for the
proposed model but that are not solutionsof the Green-Naghdi model. We also
give and compare analytical solutions of thetwo non-hydrostatic shallow water
models
A multilayer Saint-Venant system with mass exchanges for Shallow Water flows. Derivation and numerical validation
The standard multilayer Saint-Venant system consists in introducing fluid
layers that are advected by the interfacial velocities. As a consequence there
is no mass exchanges between these layers and each layer is described by its
height and its average velocity. Here we introduce another multilayer system
with mass exchanges between the neighborhing layers where the unknowns are a
total height of water and an average velocity per layer. We derive it from
Navier-Stokes system with an hydrostatic pressure and prove energy and
hyperbolicity properties of the model. We also give a kinetic interpretation
leading to effective numerical schemes with positivity and energy properties.
Numerical tests show the versatility of the approach and its ability to compute
recirculation cases with wind forcing.Comment: Submitted to M2A
A 2D model for hydrodynamics and biology coupling applied to algae growth simulations
Cultivating oleaginous microalgae in specific culturing devices such as
raceways is seen as a future way to produce biofuel. The complexity of this
process coupling non linear biological activity to hydrodynamics makes the
optimization problem very delicate. The large amount of parameters to be taken
into account paves the way for a useful mathematical modeling. Due to the
heterogeneity of raceways along the depth dimension regarding temperature,
light intensity or nutrients availability, we adopt a multilayer approach for
hydrodynamics and biology. For free surface hydrodynamics, we use a multilayer
Saint-Venant model that allows mass exchanges, forced by a simplified
representation of the paddlewheel. Then, starting from an improved Droop model
that includes light effect on algae growth, we derive a similar multilayer
system for the biological part. A kinetic interpretation of the whole system
results in an efficient numerical scheme. We show through numerical simulations
in two dimensions that our approach is capable of discriminating between
situations of mixed water or calm and heterogeneous pond. Moreover, we exhibit
that a posteriori treatment of our velocity fields can provide lagrangian
trajectories which are of great interest to assess the actual light pattern
perceived by the algal cells and therefore understand its impact on the
photosynthesis process.Comment: 27 pages, 11 figure
Layer-averaged Euler and Navier-Stokes equations
In this paper we propose a strategy to approximate incompressible hydrostatic
free surface Euler and Navier-Stokes models. The main advantage of the proposed
models is that the water depth is a dynamical variable of the system and hence
the model is formulated over a fixed domain.The proposed strategy extends
previous works approximating the Euler and Navier-Stokes systems using a
multilayer description. Here, the needed closure relations are obtained using
an energy-based optimality criterion instead of an asymptotic expansion.
Moreover, the layer-averaged description is successfully applied to the
Navier-Stokes system with a general form of the Cauchy stress tensor
Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems
From the free surface Navier-Stokes system, we derive the non-hydrostatic
Saint-Venant system for the shallow waters including friction and viscosity.
The derivation leads to two formulations of growing complexity depending on the
level of approximation chosen for the fluid pressure. The obtained models are
compared with the Boussinesq models
Data Assimilation for hyperbolic conservation laws. A Luenberger observer approach based on a kinetic description
Developing robust data assimilation methods for hyperbolic conservation laws
is a challenging subject. Those PDEs indeed show no dissipation effects and the
input of additional information in the model equations may introduce errors
that propagate and create shocks. We propose a new approach based on the
kinetic description of the conservation law. A kinetic equation is a first
order partial differential equation in which the advection velocity is a free
variable. In certain cases, it is possible to prove that the nonlinear
conservation law is equivalent to a linear kinetic equation. Hence, data
assimilation is carried out at the kinetic level, using a Luenberger observer
also known as the nudging strategy in data assimilation. Assimilation then
resumes to the handling of a BGK type equation. The advantage of this framework
is that we deal with a single "linear" equation instead of a nonlinear system
and it is easy to recover the macroscopic variables. The study is divided into
several steps and essentially based on functional analysis techniques. First we
prove the convergence of the model towards the data in case of complete
observations in space and time. Second, we analyze the case of partial and
noisy observations. To conclude, we validate our method with numerical results
on Burgers equation and emphasize the advantages of this method with the more
complex Saint-Venant system
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