# Solution of 2-D Incompressible Navier Stokes Equations with Artificial Compressibility Method Using Ftcs Scheme

4433 words 18 pages
SOLUTION OF 2-D INCOMPRESSIBLE NAVIER STOKES EQUATIONS WITH ARTIFICIAL COMRESSIBILITY METHOD USING FTCS SCHEME

IMRAN AZIZ
Department of Mechanical Engineering College of EME
National University of Science and Technology
Imran_9697@hotmail.com

Abstract— The paper deals with the 2-D lid-driven cavity flow governed by the non dimensional incompressible Navier-Stokes theorem in the rectangular domain. Specific boundary conditions for this case study have been defined and the flow characteristics pertaining to the scenario have been coded in MATLAB using artificial compressibility method and FTCS scheme. The results are compared successfully with an authentic research paper by Ghia, Ghia & Shin.

Keywords: Navier

Discretized P equation :

pi,jn+1-pi,jn∆t+1/β2[ ui+1,jn-ui-1,jn2∆x+vi,j+1n-vi,j-1n2∆y ]= 0

Once the pressure component is computed for the time level n+1, the velocity equations can be solved by using the pressure value calculated above as

Discretized U equation:

ui,jn+1-ui,jn∆t+ui+1,j2n-(ui-1,j2)n2∆x+pi+1,jn-pi-1,jn2∆x
+ ui,j+1n*vi,j+1n-ui,j-1n*vi,j-1n2∆y

=(1/Re)[ ui+1,jn-2ui,j+ui-1,jn∆x2+ui+1,jn-2ui,j+ui-1,jn∆y2 ]

Discretized V equation:

vi,jn+1-vi,jn∆t+ vi,j+12n-vi,j-12n2∆y+pi,j+1n-pi,j-1n2∆y

+ui+1,jn*vi+1,jn-ui-1,jn*vi-1,jn2∆x

=(1/Re)[ vi+1,jn-2vi,j+vi-1,jn∆x2+vi+1,jn-2vi,j+vi-1,jn∆y2 ]

Boundary conditions for Pressure:

The pressure at the boundary conditions will come from the pressure at the internal points by using the equation approximated as Neuman boundary condition on all four walls. For pressure, Neuman boundary conditions are applied at top, bottom and left, right walls. The normal derivative ∂p∂n is given by two points having boundary in the middle .

Pi,j+1-Pi,jdy=0 => Pi,j+1=Pi,j
As the point Pi,j+1 is out of domain so it is approximated as Pi,j-1.

Vorticity:

The vorticity at a fluid point is defined as twice the angular velocity and is mathematically expressed as

Vorticity(Ω)=2ω= ∇×V

For two-dimensional flow, it reduced to
Ωz=∂u∂y-∂v∂x
In descritized form

Ωz=Ui,j+1n-Ui,j-1 n2∆y-Vi+1,jn-Vi-1,jn2∆x……..

The vorticity equation is solved in Matlab in order to see the rotational effect of flow particles.