Research on Numerical Algorithms for the Three-Dimensional Navier-Stokes Equations. II. Dissipative Finite Element.

1981 ◽  
Author(s):  
A. J. Baker
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Numerical Algorithms ◽  
Navier Stokes ◽  
Navier Stokes Equations

scholarly journals A convergent FV–FE scheme for the stationary compressible Navier–Stokes equations

2020 ◽  
Author(s):  
Charlotte Perrin ◽  
Khaled Saleh
Keyword(s):  
Finite Element ◽  
Numerical Scheme ◽  
Space Dimension ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Convergence Result ◽  
Ideal Gas ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Industrial Software

Abstract In this paper we prove a convergence result for a discretization of the three-dimensional stationary compressible Navier–Stokes equations assuming an ideal gas pressure law $p(\rho )=a \rho ^{\gamma }$ with $\gamma> \frac{3}{2}$. It is the first convergence result for a numerical method with adiabatic exponents $\gamma $ less than $3$ when the space dimension is 3. The considered numerical scheme combines finite volume techniques for the convection with the Crouzeix–Raviart finite element for the diffusion. A linearized version of the scheme is implemented in the industrial software CALIF3S developed by the French Institut de Radioprotection et de Sûreté Nucléaire.

A Finite Element Analysis of the Navier-Stokes Equations Applied to High Speed Thin Film Lubrication

1987 ◽  
Vol 109 (1) ◽  
pp. 71-76 ◽  
Author(s):  
J. O. Medwell ◽  
D. T. Gethin ◽  
C. Taylor
Keyword(s):  
Finite Element ◽  
High Speed ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
The Finite Element Method ◽  
Thin Film Lubrication ◽  
Element Analysis ◽  
Navier Stokes Equations ◽  
Film Lubrication

The performance of a cylindrical bore bearing fed by two axial grooves orthogonal to the load line is analyzed by solving the Navier-Stokes equations using the finite element method. This produces detailed information about the three-dimensional velocity and pressure field within the hydrodynamic film. It is also shown that the method may be applied to long bearing geometries where recirculatory flows occur and in which the governing equations are elliptic. As expected the analysis confirms that lubricant inertia does not affect bearing performance significantly.

Pseudo Compressible Mixed Interpolation Finite Element Method for Solving Three Dimensional Navier-Stokes Equations

2013 ◽  
Author(s):  
Shrinivas G. Apte ◽  
Brian H. Dennis
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Compressibility Factor ◽  
Navier Stokes ◽  
Implicit Time ◽  
Navier Stokes Equations ◽  
Linear Solver ◽  
Element Method

A pseudo compressible finite element method for solving three dimensional incompressible Navier-Stokes equations is presented. A physical problem discretized using tetrahedral elements with linear and quadratic interpolation functions for pressure and velocity variables respectively is then marched in time by using implicit time marching scheme based on finite differencing. The possible formation of indefinite matrix due to incompressibility constraint is avoided by inserting an artificial/pseudo time dependent term (Chorin, 1974) into the continuity equation that is eliminated when steady state is reached. This definite matrix system can then be solved using standard pre-conditioners and iterative solvers. Solutions for pressure driven flows obtained using this method are validated with the ones obtained from a standard problem of flow over a cylinder and also with numerical benchmark case of a 3-D laminar flow around an obstacle. An object oriented C++ program was developed which uses exact integrals of shape functions in its calculations rather than numerical integrations. This program was tested with different values of artificial compressibility factor (β), Reynolds numbers (Re) and grid sizes (number of Elements) and time steps (dt). The effect of these parameters on the the number of linear solver iterations required for convergence is studied efficiently using the non-dimensional numbers Pseudo Compressibility Number (PCN) and Elemental Reynolds Number (ERe). Although the relationship between the linear solver performance and these two non dimensional numbers remain complicated, it is found that there exists an optimum range of PCN as a function of ERe for which the solution convergence can be obtained with the minimum number of iterations.

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