scholarly journals A priori and a posteriori estimates for three-dimensional Stokes equations with nonstandard boundary conditions

2011 ◽  
Vol 28 (4) ◽  
pp. 1178-1193 ◽  
Author(s):  
Hyam Abboud ◽  
Fida El Chami ◽  
Toni Sayah
Keyword(s):  
Boundary Conditions ◽  
Stokes Equations ◽  
A Priori ◽  
Three Dimensional ◽  
A Posteriori ◽  
A Posteriori Estimates

A TRUNCATED FOURIER/FINITE ELEMENT DISCRETIZATION OF THE STOKES EQUATIONS IN AN AXISYMMETRIC DOMAIN

2006 ◽  
Vol 16 (02) ◽  
pp. 233-263 ◽  
Author(s):  
Z. BELHACHMI ◽  
C. BERNARDI ◽  
S. DEPARIS ◽  
F. HECHT
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Stokes Problem ◽  
A Priori ◽  
Three Dimensional ◽  
Angular Variable ◽  
A Posteriori ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Good Agreement

We consider the Stokes problem in a three-dimensional axisymmetric domain and, by writing the Fourier expansion of its solution with respect to the angular variable, we observe that each Fourier coefficient satisfies a system of equations on the meridian domain. We propose a discretization of this problem which combines Fourier truncation and finite element methods applied to each of the two-dimensional systems. We give the detailed a priori and a posteriori analyses of the discretization and present some numerical experiments which are in good agreement with the analysis.

Thermodynamically Admissible Motion Past Rigid Obstacles With Isentropic or Nonisentropic Fluid–Solid Interfaces

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
Gerald G. Kleinstein
Keyword(s):  
Boundary Conditions ◽  
Stokes Equations ◽  
A Priori ◽  
Three Dimensional ◽  
Jump Condition ◽  
Navier Stokes ◽  
Heat Conducting ◽  
Material Interfaces ◽  
Material Interface ◽  
Solid Interface

The motion of a fluid in a defined domain is called thermodynamically admissible if it satisfies the global system of the principles of balance of continuum mechanics and the principle of entropy or its equivalent differential system, consisting of differential equations and jump conditions. In an earlier publication, we have shown that the motion of a three-dimensional rigid body in an irrotational viscous and heat-conducting fluid violates the entropy jump condition, referred to as the Clausius–Duhem jump condition. Such a motion is thermodynamically inadmissible and could not persist. In a more recent publication, we have demonstrated that if the fluid–solid interface is isentropic, boundary conditions at a material interface, such as the no-slip condition and the continuity of the temperature, follow directly from the Clausius–Duhem jump condition. It is the purpose of this analysis to extend this methodology for the derivation of boundary conditions at isentropic material interfaces to nonisentropic material interfaces. We show that if the boundary conditions at the fluid–solid interface are a priori selected to satisfy the Clausius–Duhem jump condition, the resulting motion as described by the solution of the Navier–Stokes equations—whether the interface is isentropic or nonisentropic—is thermodynamically admissible.

Modélisation tridimensionnelle de l'écoulement au voisinage d'un aménagement portuaire

1989 ◽  
Vol 16 (6) ◽  
pp. 829-844
Author(s):  
A. Soulaïmani ◽  
Y. Ouellet ◽  
G. Dhatt ◽  
R. Blanchet
Keyword(s):  
Free Surface ◽  
Computational Analysis ◽  
Stokes Equations ◽  
A Priori ◽  
Three Dimensional ◽  
Free Surface Flows ◽  
Solution Algorithm ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Surface Flows

This paper is devoted to the computational analysis of three-dimensional free surface flows. The model solves the Navier-Stokes equations without any a priori restriction on the pressure distribution. The variational formulation along with the solution algorithm are presented. Finally, the model is used to study the hydrodynamic regime in the vicinity of a projected harbor installation. Key words: free surface flows, three-dimensional flows, finite element method.

Toward Defining Objective Criteria for Assessing the Adequacy of Assumed Axisymmetry and Steadiness of Flows in Rotating Cavities

2005 ◽  
Vol 128 (4) ◽  
pp. 708-716 ◽  
Author(s):  
G. D. Snowsill ◽  
C. Young
Keyword(s):  
Boundary Conditions ◽  
A Priori ◽  
Three Dimensional ◽  
Cyclic Symmetry ◽  
Acceptable Solution ◽  
Inappropriate Use ◽  
Flow Features ◽  
Computational Fluid Dynamics Cfd ◽  
Aero Engines ◽  
Multiple Reference

The need to make a priori decisions about the level of approximation that can be accepted—and subsequently justified—in flows of industrial complexity is a perennial problem for computational fluid dynamics (CFD) analysts. This problem is particularly acute in the simulation of rotating cavity flows, where the stiffness of the equation set results in protracted convergence times, making any simplification extremely attractive. For example, it is common practice, in applications where the geometry and boundary conditions are axisymmetric, to assume that the flow solution will also be axisymmetric. It is known, however, that inappropriate imposition of this assumption can lead to significant errors. Similarly, where the geometry or boundary conditions exhibit cyclic symmetry, it is quite common for analysts to constrain the solutions to satisfy this symmetry through boundary condition definition. Examples of inappropriate use of these approximating assumptions are frequently encountered in rotating machinery applications, such as the ventilation of rotating cavities within aero-engines. Objective criteria are required to provide guidance regarding the level of approximation that is appropriate in such applications. In the present work, a study has been carried out into: (i) The extent to which local three-dimensional features influence solutions in a generally two-dimensional (2D) problem. Criteria are proposed to aid in decisions about when a 2D axisymmetric model is likely to deliver an acceptable solution; (ii) the influence of flow features which may have a cyclic symmetry that differs from the bounding geometry or imposed boundary conditions (or indeed have no cyclic symmetry); and (iii) the influence of unsteady flow features and the extent to which their effects can be represented by mixing plane or multiple reference frame approximations.

scholarly journals RIP CURRENTS ON A BARRED BEACH

2012 ◽  
Vol 1 (33) ◽  
pp. 38
Author(s):  
Andrea Ruju ◽  
Pablo Higuera ◽  
Javier L. Lara ◽  
Inigo J. Losada ◽  
Giovanni Coco
Keyword(s):  
Boundary Conditions ◽  
Stokes Equations ◽  
Numerical Study ◽  
Three Dimensional ◽  
Wave Generation ◽  
Dimensional Structure ◽  
Navier Stokes ◽  
Rip Currents ◽  
Mean Flow ◽  
Forcing Mechanisms

This work presents the numerical study of rip current circulation on a barred beach. The numerical simulations have been carried out with the IH-FOAM model which is based on the three dimensional Reynolds Averaged Navier-Stokes equations. The new boundary conditions implemented in IH-FOAM have been used, including three dimensional wave generation as well as active wave absorption at the boundary. Applying the specific wave generation boundary conditions, the model is validated to simulate rip circulation on a barred beach. Moreover, this study addresses the identification of the forcing mechanisms and the three dimensional structure of the mean flow.

A Priori and A Posteriori Estimates of Conforming and Mixed FEM for a Kirchhoff Equation of Elliptic Type

2017 ◽  
Vol 17 (2) ◽  
pp. 217-236 ◽  
Author(s):  
Asha K. Dond ◽  
Amiya K. Pani
Keyword(s):  
Finite Element ◽  
Convergence Rates ◽  
Mixed Finite Element ◽  
A Priori ◽  
Kirchhoff Equation ◽  
Helmholtz Decomposition ◽  
Elliptic Type ◽  
A Posteriori ◽  
A Posteriori Estimates ◽  
Expanded Mixed Finite Element

AbstractIn this article, a priori and a posteriori estimates of conforming and expanded mixed finite element methods for a Kirchhoff equation of elliptic type are derived. For the expanded mixed finite element method, a variant of Brouwer’s fixed point argument combined with a monotonicity argument yields the well-posedness of the discrete nonlinear system. Further, a use of both Helmholtz decomposition of $L^{2}$-vector valued functions and the discrete Helmholtz decomposition of the Raviart–Thomas finite elements helps in a crucial way to achieve optimal a priori as well as a posteriori error bounds. For both conforming and expanded mixed form, reliable and efficient a posteriori estimators are established. Finally, the numerical experiments are performed to validate the theoretical convergence rates.

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