scholarly journals Error analysis of non inf-sup stable discretizations of the time-dependent Navier–Stokes equations with local projection stabilization

2018 ◽  
Vol 39 (4) ◽  
pp. 1747-1786 ◽  
Author(s):  
Javier de Frutos ◽  
Bosco García-Archilla ◽  
Volker John ◽  
Julia Novo
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Convective Term ◽  
Navier Stokes Equations ◽  
Local Projection ◽  
Finite Element Approximations ◽  
Velocity Error ◽  
Mesh Width ◽  
Numerical Studies ◽  
Local Projection Stabilization

Abstract This paper studies non inf-sup stable finite element approximations to the evolutionary Navier–Stokes equations. Several local projection stabilization (LPS) methods corresponding to different stabilization terms are analyzed, thereby separately studying the effects of the different stabilization terms. Error estimates are derived in which the constants are independent of inverse powers of the viscosity. For one of the methods, using velocity and pressure finite elements of degree $l$, it will be proved that the velocity error in $L^\infty (0,T;L^2(\varOmega ))$ decays with rate $l+1/2$ in the case that $\nu \le h$, with $\nu$ being the dimensionless viscosity and $h$ being the mesh width. In the analysis of another method it was observed that the convective term can be bounded in an optimal way with the LPS stabilization of the pressure gradient. Numerical studies confirm the analytical results.

Transport-stabilized Semidiscretizations of the Incompressible Navier—Stokes Equations

2006 ◽  
Vol 6 (3) ◽  
pp. 239-263 ◽  
Author(s):  
L. Angermann
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Sufficient Conditions ◽  
Navier Stokes ◽  
Approximation Technique ◽  
Convective Term ◽  
Navier Stokes Equations ◽  
Velocity Error ◽  
Sufficient Conditions For Convergence ◽  
Upwind Method

AbstractWithin the framework of finite element methods, the paper investigates a general approximation technique for the nonlinear convective term of Navier — Stokes equations. The approach is based on an upwind method of the finite volume type. It has been proved that the discrete convective term satisfies the well-known collection of sufficient conditions for convergence of the finite element solution. For a particular nonconforming scheme, the assumptions have been verified in detail and the estimate of the semidiscrete velocity error has been proved.

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.

Numerical Study of Wave Slamming on an Oscillating Flap

2014 ◽  
Author(s):  
Yanji Wei ◽  
Alan Henry ◽  
Olivier Kimmoun ◽  
Frederic Dias
Keyword(s):  
Stokes Equations ◽  
Numerical Study ◽  
Flow Characteristics ◽  
Ocean Waves ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Numerical Studies ◽  
Wave Slamming ◽  
Volume Method

Bottom hinged Oscillating Wave Surge Converters (OWSCs) are efficient devices for extracting power from ocean waves. There is limited knowledge about wave slamming on such devices. This paper deals with numerical studies of wave slamming on an oscillating flap to investigate the mechanism of slamming events. In our model, the Navier–Stokes equations are discretized using the Finite Volume method with the Volume of Fluid (VOF) approach for interface capturing. Waves are generated by a flap-type wave maker in the numerical wave tank, and the dynamic mesh method is applied to model the motion of the oscillating flap. Basic mesh and time step refinement studies are performed. The flow characteristics in a slamming event are analysed based on numerical results. Various simulations with different flap densities, water depths and wave amplitudes are performed for a better understanding of the slamming.

scholarly journals On mathematical modelling of aeroelastic problems with finite element method

2018 ◽  
Vol 180 ◽  
pp. 02104
Author(s):  
Petr Sváček
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Small Displacement ◽  
Structure Model ◽  
Navier Stokes Equations ◽  
Aeroelastic Response ◽  
Finite Element Approximations ◽  
Element Method

This paper is interested in solution of two-dimensional aeroelastic problems. Two mathematical models are compared for a benchmark problem. First, the classical approach of linearized aerodynamical forces is described to determine the aeroelastic instability and the aeroelastic response in terms of frequency and damping coefficient. This approach is compared to the coupled fluid-structure model solved with the aid of finite element method used for approximation of the incompressible Navier-Stokes equations. The finite element approximations are coupled to the non-linear motion equations of a flexibly supported airfoil. Both methods are first compared for the case of small displacement, where the linearized approach can be well adopted. The influence of nonlinearities for the case of post-critical regime is discussed.

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