Help me with an aerodynamics paper

Space shuttle on re-entry from [login to view URL] (accessed 2007) The tiles on a space shuttle formed a crucial heat shield to protect the structure from extreme temperatures during re-entry into the atmosphere. Damage to the tiles in a critical area led to the Columbia space shuttle disaster on February 1 2003. A next-generation reusable spracecraft will use a similar layer of thermal protection tiles. Your tasks are to predict the temperature through the thickness of a tile during the period of re-entry, and to choose a suitable tile thickness based on the temperature at the inner surface. The relevant properties of the tile are: Thermal conductivity 0.0577 W/(m K) Density 144 kg/m3 Specific heat 1259 J/(kg K) For simplicity a 1-D model of the tile will be used, considering variations through the thickness of the tile. The program models the temperature in one dimension, with one end (i.e. tile outer surface) at a prescribed temperature and the other end with a Neumann insulated boundary. As measured data for this new spacecraft are not available yet, we will use measurements taken by NASA on the Space Shuttle, Measured outer surface temperature data are shown below. This will be used as the outer surface boundary condition. Surface temperature Measured temperature at tile #597 (Blanchard et al., Infrared Sensing Aeroheating Flight Experiment: STS-96 Flight Results, AIAA 2001-0352, 39th AIAA Aerospace Sciences Meeting & Exhibit, Jan 2011, Reno, available here] A crude approximation to the measured surface temperature is provided, but this should be improved using the plottemp.m program or otherwise. An incomplete program is provided. You need to add code for the numerical methods, and set the appropriate boundary conditions. Tasks using this program: Download the program file shuttle.m. Examine the structure of the program. Add code in the appropriate place to implement the forward differencing method. Run it using suitable values. For example: [x, t, u] = shuttle(4000, 501, 0.05, 21, 'forward', true); Type 'help shuttle' for instructions on how to run it, or look at the comments at the start of the program. Download the program plottemp.m and the image file temp597.jpg. Edit the code and use it to create the data file [login to view URL], which contains the surface temperature boundary condition data. Edit shuttle.m to read this data file. Add the DuFort-Frankel method. This is implemented in a very similar way to Forward Differencing. Add the Backward Differencing method. See the notes or Matlab example for this. Add the Crank-Nicolson method. This is implemented in a very similar way to Backward Differencing. Change the inner surface to a Neumann boundary to represent zero heat flow, for all methods. Using nx = 21, and tmax = 4000 sec, investigate the stability and accuracy of the Forward Differencing, Backward Differencing, Crank-Nicolson and DuFort-Frankel methods for a suitable range of values of timestep. Create another m-file to do this; you can use the commands listed below as a starting point. For each method, this plots the inner surface temperature at time = 4000 sec against the timestep. i=0; nx = 21; thick = 0.05; tmax = 4000; for nt = 41:20:1001 i=i+1; dt(i) = tmax/(nt-1); disp (['nt = ' num2str(nt) ', dt = ' num2str(dt(i)) ' s']) [~, ~, u] = shuttle(tmax, nt, thick, nx, 'forward', false); uf(i) = u(end, 1); [~, ~, u] = shuttle(tmax, nt, thick, nx, 'backward', false); ub(i) = u(end, 1); end plot(dt, [uf; ub]) ylim([0 200]) legend ('Forward', 'Backward') Select the most appropriate method and suitable timestep to give an efficient solution with good overall accuracy and clarity. You may also wish to implement a similar script to that of step 7 to choose the number of spatial steps nx. Plot the inner surface temperature against time, for 0 to 4000 sec. Do this for a practical range of tile thicknesse.