City University of London EX2010_Computing_2021-22

The temperature distribution in a control volume of a rectangular shape can be modelled by the following PDE

???? ???? ??-?? ??-??

???? + ?? ???? = ??! ????- + ??- ???? – + ??

where ??, ??, ??!,-, and ?? are temperature, convective velocity, thermal diffusivities and heat source term, respectively. We want to solve for the temperature distribution in a scenario where the length of the plate is considerably larger than its width (i.e., ?? » ??) and the temperature gradient in ??-direction can be neglected. Therefore, the above equation reduces to

???? ???? ??-??

???? + ?? ???? = ??! ????- + ??.

Consider the convective velocity and the initial condition to be approximated by

?? = ?? + ?? ??????(0.5 ????) + ?? cos(4????),

??$ = ??.

The heat source is located at the middle of the domain, which is modelled with the following equation

)%*/- !

?? = ?? ??%$.’( $.$’ , .

Here you are asked to write a code to solve this problem with implicit Euler scheme in time and central difference scheme in space. You may use built in solver of MATLAB ( backslash). Assume the boundary conditions to be periodic in streamwise direction (??. ??. , ??) and take the base parameters as ??! = 0.02 ??-/??, ?? = 0.6 , ?? = 0.05, ?? = 0.01, ?? = 0.05, and ?? = 20-??. The default values of grid space and time steps are ??? = 0.01 ?? and ??? = 0.005 ??, where ?? = 4 ?? and ??./0 = 100 ??.

At the end of your work, you need to provide a zipped folder that contains all the functions that you have implemented in addition to a main script which is the interface that runs the whole simulation (you must load the variables directly in the main script, do not allow the user to fix them on the fly). All coding needs to be explained by appropriate comments. Also, you must provide a PDF file, which contains the following results and discuss them.

1. Draw the distribution temperature along the domain at following time instants: ?? = 0, 1, 5, 10, 20, 50, 100 ??. Discuss the results and the main character of the observed physical behaviour.

2. Investigate the impact of advective velocity distribution by turning off the coefficients ?? and ??, i.e., ?? = 0 and ?? = 0.

3. Investigate the impact of diffusion transport mechanism by changing the value of ??!.

The range of values is arbitrary.

4. Investigate the impact of the source term by changing the value of ??. The range of values is arbitrary.

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