Approximate Solutions to the 2-D Unsteady Navier-Stokes System with Free Surface

2007 ◽  
pp. 109-119 ◽  
Author(s):  
Marcello Guidorzi ◽  
Mariarosaria Padula
Keyword(s):  
Free Surface ◽  
Stokes System ◽  
Approximate Solutions ◽  
Navier Stokes

INHOMOGENEOUS INCOMPRESSIBLE VISCOUS FLOWS WITH SLOWLY VARYING INITIAL DATA

2016 ◽  
Vol 17 (5) ◽  
pp. 1121-1172
Author(s):  
Jean-Yves Chemin ◽  
Ping Zhang
Keyword(s):  
Initial Data ◽  
Stokes System ◽  
Large Data ◽  
Viscous Flows ◽  
Approximate Solutions ◽  
Navier Stokes ◽  
Decay Properties ◽  
Optimal Decay ◽  
Whole Space ◽  
Well Posed

The purpose of this paper is to provide a large class of initial data which generates global smooth solution of the 3D inhomogeneous incompressible Navier–Stokes system in the whole space $\mathbb{R}^{3}$. This class of data is based on functions which vary slowly in one direction. The idea is that 2D inhomogeneous Navier–Stokes system with large data is globally well-posed and we construct the 3D approximate solutions by the 2D solutions with a parameter. One of the key point of this study is the investigation of the time decay properties of the solutions to the 2D inhomogeneous Navier–Stokes system. We obtained the same optimal decay estimates as the solutions of 2D homogeneous Navier–Stokes system.

scholarly journals Numerical approximation of the 3D hydrostatic Navier–Stokes system with free surface

2019 ◽  
Vol 53 (6) ◽  
pp. 1981-2024
Author(s):  
Sebastien Allgeyer ◽  
Marie-Odile Bristeau ◽  
David Froger ◽  
Raouf Hamouda ◽  
V. Jauzein ◽  
...  
Keyword(s):  
Free Surface ◽  
Shallow Water ◽  
Numerical Approximation ◽  
Stokes System ◽  
Type Systems ◽  
Navier Stokes ◽  
Finite Volume Scheme ◽  
List Type ◽  
Water Type ◽  
Discrete Scheme

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 propagation, tsunami runup and complex 3D hydrodynamics in a raceway.

scholarly journals Some preliminary observations on a defect Navier–Stokes system

2019 ◽  
Vol 347 (10) ◽  
pp. 677-684 ◽  
Author(s):  
Amit Acharya ◽  
Roger Fosdick
Keyword(s):  
Stokes System ◽  
Navier Stokes

scholarly journals Asymptotic analysis of an optimal control problem for a viscous incompressible fluid with Navier slip boundary conditions

2021 ◽  
pp. 1-21
Author(s):  
Claudia Gariboldi ◽  
Takéo Takahashi
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Boundary Conditions ◽  
Control Problem ◽  
Stokes System ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Slip ◽  
Slip Boundary ◽  
Navier Slip Boundary Conditions

We consider an optimal control problem for the Navier–Stokes system with Navier slip boundary conditions. We denote by α the friction coefficient and we analyze the asymptotic behavior of such a problem as α → ∞. More precisely, we prove that if we take an optimal control for each α, then there exists a sequence of optimal controls converging to an optimal control of the same optimal control problem for the Navier–Stokes system with the Dirichlet boundary condition. We also show the convergence of the corresponding direct and adjoint states.

scholarly journals New asymptotic profiles of nonstationary solutions of the Navier–Stokes system

2007 ◽  
Vol 88 (1) ◽  
pp. 64-86 ◽  
Author(s):  
Lorenzo Brandolese ◽  
François Vigneron
Keyword(s):  
Stokes System ◽  
Navier Stokes
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