FTCS Tutorial -1

Computational Fluid Dynamics

Problem Statement : 

 Given a rod of Length  L ,with boundary conditions,boundary conditions as follows :  

   $T(0,t) = 0^{\circ}C $

   $T(1,t) = 0^{\circ}C $

Also the initial conditions are as follows:

  $T(x,0) = 0^{\circ}C $

Find the Temperature distribution for the complete length of the rod for a time period of 300s.

Governing Equations:

The physical equation is as follows from the below heat conduction equation.

$ \frac{\delta T}{\delta t} = \alpha \frac{\delta ^2 T}{\delta x^2}$ 

Now we are applying FTCS scheme to the above equation we get: 

$T_{i}^{n+1} = T_{i}^{n} + \alpha \Delta t \frac{(T_{i+1}^n - 2T_{i}^n + T_{i-1}^n)}{(\Delta x)^2}$

Pseudo code :

1.  Initialize the variables $\alpha ,\Delta t,T,\Delta x,N_x,L$.
   (Note here T is a matrix with $N_x$ columns ,and $20/(\Delta t) +1= N_y$ rows)
2.  For n = 2 to $N_y$ execute the statements 3 and 4.
3.  For i = 2 to $N_x-1$ execute statement 4.
4.  $T[i][n+1] = T[i][n] + \frac{\alpha \Delta t}{(\Delta x)^2} (T[i+1][n] - 2T[i][n] + T[i-1][n] )$
5.  end

Stability of scheme : 

Let $\gamma = \alpha \frac{\Delta t}{ \Delta x^2}$.

Scheme is stable if $\gamma$ < $\frac{1}{2} $
Derivation : 
$$e(n,x) = A_n  e^{ikx}$$
Here $k$ is the wave number , $i = (-1)^{0.5}$
Therefore substituting this in the finite difference scheme  we get
$$ \frac{A_{n+1}}{A_{n}} = 1 + \frac{\alpha \Delta t}{\Delta x^2}\left ( e^{i k \Delta x} -2 + e^{-i k \Delta x}   \right )$$

$$ \frac{A_{n+1}}{A_{n}} = 1 + \frac{\alpha \Delta t}{\Delta x^2}\left (-4 sin^2 (k \frac{\Delta x}{2} )\right )$$


$$ \left \vert \frac{A_{n+1}}{A_{n}} \right \vert   \leq 1$$
$$  -1 \leq 1 + \frac{\alpha \Delta t}{\Delta x^2} \left (-4 sin ^2  (k \frac{\Delta x}{2} ) \right ) $$
Finally we get the required criterion. 

MATLAB Run : 

Web App Run:

MATLAB /Web App:

Code Link : https://github.com/RAAKASH/Intro-to-CFD-

Web App : https://aakashr.shinyapps.io/CFD-Assignments/