Time-averaged shallow water model: asymptotic derivation and numerical validation

The objective of this paper is to derive, from the Navier-Stokes equations in a shallow domain,a new bidimensional shallow water model able to filter the high frequency oscillations that are produced, when the Reynolds number is increased, in turbulent ows. With this aim, the non- dimensional Navier-Stokes equations are time-averaged, and then asymptotic analysis techniques have been used as in our previous works. The small non-dimensional parameter considered, "e", is the quotient between the typical depth of the basin and the typical horizontal length of the domain; and it is studied what happens when "e" becomes small. Once the new model has been justified, by the method of asymptotic expansions, we perform some numerical experiments. The results of these experiments con rm that this new model is able to approximate analytical solutions of Navier-Stokes equations with more accuracy than classical shallow water models, when high frequency oscillations appear. To reach a given accuracy, the time step for the new model can be much larger (even four hundred times larger) than the time step required for the classical models

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

Implementation of the LANS-alpha turbulence model in a primitive equation ocean model

This paper presents the first numerical implementation and tests of the Lagrangian-averaged Navier-Stokes-alpha (LANS-alpha) turbulence model in a primitive equation ocean model. The ocean model in which we work is the Los Alamos Parallel Ocean Program (POP); we refer to POP and our implementation of LANS-alpha as POP-alpha. Two versions of POP-alpha are presented: the full POP-alpha algorithm is derived from the LANS-alpha primitive equations, but requires a nested iteration that makes it too slow for practical simulations; a reduced POP-alpha algorithm is proposed, which lacks the nested iteration and is two to three times faster than the full algorithm. The reduced algorithm does not follow from a formal derivation of the LANS-alpha model equations. Despite this, simulations of the reduced algorithm are nearly identical to the full algorithm, as judged by globally averaged temperature and kinetic energy, and snapshots of temperature and velocity fields. Both POP-alpha algorithms can run stably with longer timesteps than standard POP. Comparison of implementations of full and reduced POP-alpha algorithms are made within an idealized test problem that captures some aspects of the Antarctic Circumpolar Current, a problem in which baroclinic instability is prominent. Both POP-alpha algorithms produce statistics that resemble higher-resolution simulations of standard POP. A linear stability analysis shows that both the full and reduced POP-alpha algorithms benefit from the way the LANS-alpha equations take into account the effects of the small scales on the large. Both algorithms (1) are stable; (2) make the Rossby Radius effectively larger; and (3) slow down Rossby and gravity waves.Comment: Submitted to J. Computational Physics March 21, 200

A weakly non-hydrostatic shallow model for dry granular flows

A non-hydrostatic depth-averaged model for dry granular flows is proposed, taking into account vertical acceleration. A variable friction coefficient based on the $\mu(I)$ rheology is considered. The model is obtained from an asymptotic analysis in a local reference system, where the non-hydrostatic contribution is supposed to be small compared to the hydrostatic one. The non-hydrostatic counterpart of the pressure may be written as the sum of two terms: one corresponding to the stress tensor and the other to the vertical acceleration. The model introduced here is weakly non-hydrostatic, in the sense that the non-hydrostatic contribution related to the stress tensor is not taken into account due to its complex implementation. A simple and efficient numerical scheme is proposed. It consists of a three-step splitting procedure, and it is based on a hydrostatic reconstruction. Two key points are: (i) the friction force has to be taken into account before solving the non-hydrostatic pressure. Otherwise, the incompressibility condition is not ensured; (ii) both the hydrostatic and the non-hydrostatic pressure are taken into account when dealing with the friction force. The model and numerical scheme are then validated based on several numerical tests, including laboratory experiments of granular collapse. The influence of non-hydrostatic terms and of the choice of the coordinate system (Cartesian or local) is analyzed. We show that non-hydrostatic models are less sensitive to the choice of the coordinate system. In general, the non-hydrostatic model introduced here much better reproduces granular collapse experiments compared to hydrostatic models. An important result is that the simulated mass profiles up to the deposit and the front velocity are greatly improved. As expected, the influence of the non-hydrostatic pressure is shown to be larger for small values of the slope

On the relevance of the dam break problem in the context of nonlinear shallow water equations

The classical dam break problem has become the de facto standard in validating the Nonlinear Shallow Water Equations (NSWE) solvers. Moreover, the NSWE are widely used for flooding simulations. While applied mathematics community is essentially focused on developing new numerical schemes, we tried to examine the validity of the mathematical model under consideration. The main purpose of this study is to check the pertinence of the NSWE for flooding processes. From the mathematical point of view, the answer is not obvious since all derivation procedures assumes the total water depth positivity. We performed a comparison between the two-fluid Navier-Stokes simulations and the NSWE solved analytically and numerically. Several conclusions are drawn out and perspectives for future research are outlined.Comment: 20 pages, 15 figures. Accepted to Discrete and Continuous Dynamical Systems. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutyk

Asymptotic Analysis of a Thin Fluid Layer Flow Between Two Moving Surfaces

Financiado para publicación en acceso aberto: Universidade da Coruña/CISUG[Abstract] In this paper we study the behavior of an incompressible viscous fluid moving between two very close surfaces also in motion. Using the asymptotic expansion method we formally justify two models, a lubrication model and a shallow water model, depending on the boundary conditions imposed. Finally, we discuss under what conditions each of the models would be applicable.This work has been partially supported by Ministerio de Economía y Competitividad of Spain, under grant MTM2016-78718-P with the participation of FEDER, and the European Union's Horizon 2020 Research and Innovation Programme, under the Marie Sklodowska-Curie Grant Agreement No 823731 CONMECH. Funding for open access charge: Universidade da Coruña/CISU

Un modelo unidimensional de flujo sanguíneo obtenido mediante el método de desarrollos asintóticos

En este trabajo presentamos un modelo unidimensional del movimiento de un fluido newtoniano a través de un tubo elástico no necesariamente rectilíneo. En particular, el modelo obtenido es aplicable al estudio del flujo sanguíneo. Para obtener el modelo propuesto explotamos el hecho de que el área de la sección transversal del tubo es mucho menor que su longitud, lo que nos permitir´a introducir un pequeño parámetro adimensional y utilizar el método de desarrollos asintóticos. El modelo así obtenido incorpora un nuevo término dependiente de la curvatura de la línea media del tubo, que no hemos encontrado en la literatura, y que se opone al avance del fluido.Ministerio de Educación y Cienci

A new thin layer model for viscous flow between two nearby non-static surfaces

We propose a two-dimensional flow model of a viscous fluid between two close moving surfaces. We show, using a formal asymptotic expansion of the solution, that its asymptotic behavior, when the distance between the two surfaces tends to zero, is the same as that of the the Navier-Stokes equations. The leading term of the formal asymptotic expansions of the solutions to the new model and Navier-Stokes equations are solution of the same limit problem, and the type of the limit problem depends on the boundary conditions. If slip velocity boundary conditions are imposed on the upper and lower bound surfaces, the limit is a solution of a lubrication model, but if the tractions and friction forces are known on both bound surfaces, the limit is a solution of a thin fluid layer model. The model proposed has been obtained to be a valuable tool for computing viscous fluid flow between two nearby moving surfaces, without the need to decide a priori whether the flow is typical of a lubrication or a thin fluid layer problem, and without the enormous computational effort that would be required to solve the Navier-Stokes equations in such a thin domain.Comment: arXiv admin note: substantial text overlap with arXiv:2203.0869