Finite element approximation of incompressible Navier-Stokes equations with slip boundary condition II

1991 ◽  
Vol 59 (1) ◽  
pp. 615-636 ◽  
Author(s):  
R. Verf�rth
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Stokes Equations ◽  
Finite Element Approximation ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Element Approximation ◽  
Navier Stokes Equations ◽  
Slip Boundary

scholarly journals Involving the Navier–Stokes equations in the derivation of boundary conditions for the lattice Boltzmann method

2011 ◽  
Vol 369 (1944) ◽  
pp. 2184-2192 ◽  
Author(s):  
Joris C. G. Verschaeve
Keyword(s):  
Boundary Condition ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Grid Spacing ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Boltzmann Method

By means of the continuity equation of the incompressible Navier–Stokes equations, additional physical arguments for the derivation of a formulation of the no-slip boundary condition for the lattice Boltzmann method for straight walls at rest are obtained. This leads to a boundary condition that is second-order accurate with respect to the grid spacing and conserves mass. In addition, the boundary condition is stable for relaxation frequencies close to two.

FINITE ELEMENT ERROR ANALYSIS OF SPACE AVERAGED FLOW FIELDS DEFINED BY A DIFFERENTIAL FILTER

2004 ◽  
Vol 14 (04) ◽  
pp. 603-618 ◽  
Author(s):  
ADRIAN DUNCA ◽  
VOLKER JOHN
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Finite Element Approximation ◽  
Flow Fields ◽  
Navier Stokes ◽  
Element Approximation ◽  
Navier Stokes Equations ◽  
Finite Element Approximations ◽  
Three Space ◽  
Element Error

This paper analyzes finite element approximations of space averaged flow fields which are given by filtering, i.e. averaging in space, the solution of the steady state Stokes and Navier–Stokes equations with a differential filter. It is shown that [Formula: see text], the error of the filtered velocity [Formula: see text] and the filtered finite element approximation of the velocity [Formula: see text], converges under certain conditions of higher order than [Formula: see text], the error of the velocity and its finite element approximation. It is also proved that this statement stays true if the L2-error of finite element approximations of [Formula: see text] and [Formula: see text] is considered. Numerical tests in two and three space dimensions support the analytical results.

Variational multi-scale finite element approximation of the compressible Navier-Stokes equations

2016 ◽  
Vol 26 (3/4) ◽  
pp. 1240-1271 ◽  
Author(s):  
Camilo Andrés Bayona Roa ◽  
Joan Baiges ◽  
R Codina
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Finite Element Approximation ◽  
Shock Capturing ◽  
Navier Stokes ◽  
Element Approximation ◽  
Navier Stokes Equations ◽  
Content Type ◽  
Multi Scale ◽  
Non Linear

Purpose – The purpose of this paper is to apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written in conservation form. Even though this formulation is relatively well known, some particular features that have been applied with great success in other flow problems are incorporated. Design/methodology/approach – The orthogonal subgrid scales, the non-linear tracking of these subscales, and their time evolution are applied. Moreover, a systematic way to design the matrix of algorithmic parameters from the perspective of a Fourier analysis is given, and the adjoint of the non-linear operator including the volumetric part of the convective term is defined. Because the subgrid stabilization method works in the streamline direction, an anisotropic shock capturing method that keeps the diffusion unaltered in the direction of the streamlines, but modifies the crosswind diffusion is implemented. The artificial shock capturing diffusivity is calculated by using the orthogonal projection onto the finite element space of the gradient of the solution, instead of the common residual definition. Temporal derivatives are integrated in an explicit fashion. Findings – Subsonic and supersonic numerical experiments show that including the orthogonal, dynamic, and the non-linear subscales improve the accuracy of the compressible formulation. The non-linearity introduced by the anisotropic shock capturing method has less effect in the convergence behavior to the steady state. Originality/value – A complete investigation of the stabilized formulation of the compressible problem is addressed.

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