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
Islamabad, Pakistan
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
…show more content…
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 [5].

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……..[6]

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

Related

  • Guide to Unix Using Linux Fourth Edition Chapter 1 Review Question Solutions
    2614 words | 11 pages
  • Midterm 2 Solutions
    1890 words | 8 pages
  • Solution of Strategic Marketing Problems Chapter 2
    1445 words | 6 pages
  • Ch. 1 the Equity Method of Accounting for Investments Solutions
    8173 words | 33 pages
  • Mw Petroleum-Case Solution 2
    1588 words | 7 pages
  • | the Occurrence of Osmosis and Diffusion in Artificial and Living Cells
    3653 words | 15 pages
  • Soc 101 Scientific Method Week 2
    1086 words | 5 pages
  • Quantitative Methods for Business Cha 2—Introduction to Probability PROBLEM
    1757 words | 8 pages
  • Lamarsh Solution Chap7
    2306 words | 10 pages
  • Justification of a Scheme of Work
    1597 words | 7 pages