Spectral-finite element method for solving three-dimensional unsteady Navier-Stokes equations

1993 ◽  
Vol 9 (1) ◽  
pp. 27-38 ◽  
Author(s):  
Cao Weiming ◽  
Guo Benyu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Spectral Finite Element Method ◽  
Spectral Finite Element ◽  
Element Method

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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