# Numerical study and CFD Simulations of Incompressible Newtonian Flow by Solving steady Navier-stokes equations using Newton’s method

K. M.Helal,

#### DOI NO:

https://doi.org/10.26782/jmcms.2015.01.00005

#### Keywords:

Navier-Stokesequations,CFD simulation,finite elementmethod,Newton-Raphsonmethod,

#### Abstract

Inthis paper, incompressible Newtonian flow is numerically studied byapproximatingthesolutionof the steady Navier-Stokes equationsin two dimensional case.Computational Fluid Dynamics (CFD)simulationsare carriedout byusing thefiniteelement method.Newton’smethodis applied to solvetheNavier-Stokes equationswherethe finite element solutions of Stokes equations is considered as the initial guess to obtainthe convergenceof Newton’s sequence.The numericalsimulations are presentedin termsofthe contours ofvelocity, pressure and streamline. All themeshes andsimulations areimplementedonthegeneralfinite element solver FreeFem++.Atwo-dimensionalbenchmark flow was computedwith posteriori estimates.It hasalsobeen established thatthe free accesssolverFreeFem++ can provide a reasonable good numerical simulationsof complicated flow behavior.

#### Refference:

II.Becker, E., Carey, G. and Oden, J.(1981).Finite Elements. AnIntroduction,Vol.1,Prentice Hall, Inc., Englewood Cliffs, New Jersey.

III.Borges, L. (2006). Domain Decomposition Methods for Non-NewtonianFluids, PhD Thesis, IST, Lisbon.

IV.Brezis,H.(2011).Functional Analysis,Sobolev Spaces and Partial Differential Equations,Springer

V.Carey, G. F.and Oden,J.T.(1986).Finite elements. Vol.VI. Fluidmechanics,TheTexasFinite Element Series,VI. Prentice Hall, Inc.,Englewood Cliffs,NewJersey.

VI.Galdi, G. P. (1994). An Introduction to the Mathematical Theory of theNavier-Stokes Equations: Linearized Steady Problems, SpringerTracts inNatural Philosophy, Vol. 38,Springer, New York.

VII.Ghia, U., Ghia, K. N. and Shin, C. T. (1982). High-resolutionsforincom pressible flowsusingthe Navier-Stokes equations and a multigridmethod, J. Comput. Phys. 48, 387-411.

VIII.Hecht,F. (2012). FreeFem++, Version 3.23,http://www.freefem.org/ff++.Kim,S. D., Lee,Y. H. and Shin,B. C.(2006).Newton’s Method for theNavier-StokesEquations with Finite-Element Initial Guess of Stokes Equation,Computers and Mathematics with applications,51, 805-816.

IX.Quarteroni, A. and Valli, A.(1994).Numerical Approximation of Partial Differential Equations, Springer-Verlag.

X.Girault,V. and Raviart, P. A.(1986).Finite Element Approximation of theNavier-stokes Equations,Computational Mathematics,Springer-Verlag,Berlin.

XI.Temam, R.(1984).Navier-Stokes Equations. Theory and Numerical Analysis,3rded.,North-Holland, Amsterdam.

Author(s) : K. M. Helal