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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…*

Primary vortices are the major vortices and occur along the streamlines of fluid flow. Secondary vortices are generated at a place from which the fluid departs already and vacuum is developed, that vacuum causes the flow to circulate back and hence generate secondary vortices. RESULTS 1. Validation of Results for 81×81 grid: Results for 81×81grid with Re=1000 are shown in Figure 1.It contains contours and graphs at Re =1000, with convergence criteria =1e-6, No of iterations = 15000, and No of time steps =50000. Numerical values of velocities for 81x81 grid weren’t found so we made the tables using 129x129 grid at Re =1000. Fig 1a: U velocity distribution along Vertical Center line. Fig 1b: Ghia, Ghia &Shin’s U velocity plots for comparison [7] Fig 1c: V velocity distribution along Horizontal Center line Fig 1d: Results of V velocities from Ghia & Ghia[8] Table-1a: RESULTS OF U VELOCITY ALONG VERTICAL LINE | Grid point number | Corodinate number (y) | U velocity | Ghia, Ghia n Shin[9] | 1 | 0 | 0 | 0 | 8 | .0547 | -0.023 | -0.42735 | 9 | 0.0625 | -0.0243 | -0.42537 | 10 | 0.0703 | -0.0256 | -.41657 | 14 | 0.1016 | -0.032 | -0.38000 | 23 | 0.1719 | -0.0377 | -0.32709 | 37 | 0.2813 | -0.0492 | -.23186 | 59 | 0.4531 | -0.0752 | -0.07540 | 65 | 0.5000 | -0.0815 | 0.03111 | 80 | 0.6172 | -0.0735 | 0.08344 | 95 | 0.7344 | -0.0176 | 0.20673 | 110 | 0.8516 | 0.0279 |

Primary vortices are the major vortices and occur along the streamlines of fluid flow. Secondary vortices are generated at a place from which the fluid departs already and vacuum is developed, that vacuum causes the flow to circulate back and hence generate secondary vortices. RESULTS 1. Validation of Results for 81×81 grid: Results for 81×81grid with Re=1000 are shown in Figure 1.It contains contours and graphs at Re =1000, with convergence criteria =1e-6, No of iterations = 15000, and No of time steps =50000. Numerical values of velocities for 81x81 grid weren’t found so we made the tables using 129x129 grid at Re =1000. Fig 1a: U velocity distribution along Vertical Center line. Fig 1b: Ghia, Ghia &Shin’s U velocity plots for comparison [7] Fig 1c: V velocity distribution along Horizontal Center line Fig 1d: Results of V velocities from Ghia & Ghia[8] Table-1a: RESULTS OF U VELOCITY ALONG VERTICAL LINE | Grid point number | Corodinate number (y) | U velocity | Ghia, Ghia n Shin[9] | 1 | 0 | 0 | 0 | 8 | .0547 | -0.023 | -0.42735 | 9 | 0.0625 | -0.0243 | -0.42537 | 10 | 0.0703 | -0.0256 | -.41657 | 14 | 0.1016 | -0.032 | -0.38000 | 23 | 0.1719 | -0.0377 | -0.32709 | 37 | 0.2813 | -0.0492 | -.23186 | 59 | 0.4531 | -0.0752 | -0.07540 | 65 | 0.5000 | -0.0815 | 0.03111 | 80 | 0.6172 | -0.0735 | 0.08344 | 95 | 0.7344 | -0.0176 | 0.20673 | 110 | 0.8516 | 0.0279 |

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