Develop a MATLAB code for solving the 2D, incompressible Navier-Stokes equations

1) Develop a code for solving the 2D, incompressible Navier-Stokes equations in 

generalized curvilinear coordinates using the artificial compressibility method. Use a three-point, second order accurate finite difference to discretize the convective and viscous fluxes (as you have done in your homework) in conjunction with scalar, fourth- difference, third-order accurate artificial dissipation for stability. Advance the governing equations in time using an explicit, four stage Runge-Kutta time stepping. 

2) Apply the code to simulate steady, laminar flow in a square (L×L) driven cavity show 

upper wall is moving at constant speed U. The Reynolds number for this flow is defined as Re=UL/υ. Carry out simulations for Re=100, 200, and 400 and compare your computed results with the benchmark solutions of Ghia et al. (J. comp physics, 48, 387- 411, 1982). 

3) Using successively finer grids show that your method converges to a grid independent solution. Use both uniform and stretched grids. For the latter case, stretch the grid near the four walls of the cavity using a geometric series stretching. 

4) Plot the streamlines and contours of vorticity for each Re. Also plot the velocity at the 

centerline and compare it to the one reported by Ghia et al. 

5) Explore the effect of artificial dissipation on the accuracy of the solution. 

6) Write a detailed report presenting your work. Your write up should follow the format of a standard journal paper with an abstract, introduction (with review of previous work on the cavity flow), presentation of the numerical method, description of the flow case and grid generation, presentation and discussion of your results, summary and conclusions, and references. Make figures in a professional manner and use them in your report. 

Hint: Try to set the pressure of one point (im,jm) to zero, e.g., for all (i,j) P(i,j)=P(i,j)-P(im,jm), to improve convergence. 

Bonus points: Investigate the effect of local time stepping and CFL number (10 points) and implicit residual smoothing (20 points) on the rate of convergence of the time-marching algorithm. 

Report Format 

Please write-up findings of your study in a report using the following guidelines. The report will be graded out of 20 points in total. Five of the 20 points is on the overall format and presentation of the report. Any report that does not follow these guidelines will therefore be graded for less than full credit. 

Title Page – A page with the title of your study, your name, course info. and date. 

Introduction – Background information and a short summary of the problem you studied (~1-2 pages). Include the cases you conducted in a well-organized table or set of tables. Make sure to include a caption(s) with your table(s). 

Method of Solution – A summary of the difference equations used, sketch of stencils, implementation of boundary conditions. Please DO NOT list your code in this section. Include a listing of the computer program as an appendix. (~ 2-4 pages) 

Discussion of Results – Include all plots and discussion in this section. When writing your discussion, please use the following rules of thumb. (~5-10 pages) 

• If you include a figure, make sure to refer to it in the discussion, e.g., “Figure 1 shows the overall heat release as a function of time for cases 1 and 2.” There should be no figures in your report that are not discussed. If the word “figure” is the first word of a sentence, such as the previous example, then spell it out. If the word “figure” is used in the middle of a sentence then use the abbreviation “Fig. “. The same rules apply for the word “equation”, e.g., “Substituting Eq. (1) into Eq. (4) results in the following.” vs. “Equation (5) summarizes the governing equations.” The word “Table” always is spelled out. 
• Be quantitative with your discussion, e.g., “Comparing Figs. (3) and (4), it is clear that the error is reduced by 35% using the Crank-Nicholson method.” DO NOT use statements such as: “Using the higher order centered difference stencil reduced the error a lot.” or “Much more time is required using an implicit method than the explicit method.”. These statements are too qualitative. Be quantitative by specifying how much something has changed. 
• Please keep your figures to a minimum by plotting several results on one plot. For instance, if you are comparing different numerical methods then group your results by plotting results at the time, spatial location or both. Credit will be deducted for reports that have excessive figures. 
• Please make sure that all plots and tables have numbered captions. The caption should contain short distinct descriptions and should be complete sentences. Please do not place discussion in the captions. 
• On plotting figures ..... 

o All figures should have axis labels (with units), numbered major tick marks, minor tick marks and 

a legend. Each line should have a distinct line type. Color is optional. o Please make sure minor tick marks are chosen for easy reading of the plots. Choose minor tick 

marks as 0.25, 0.2, 1, etc. DO NOT choose values such as 0.23, 0.56, etc. o A major tick mark and a label should end of each axis. o Make sure all labels on the axes and the legend are clear and large enough to be easily read. 

Summary and Conclusions - Briefly summarize your findings and any conclusions. Please do not simply cut-n- paste from the discussion. Try to put your finding into some larger relevance with regard to the theory. Did the results agree well with the theory always? What appears to be the largest limitation? (< 1 page). 

Appendices – Include any lengthy hand calculations and/or code listing as an appendix.