Week 5.3 - Mid term project - Stability analysis in the unsteady problem

Week 5.3 - Mid term project - Stability analysis in the unsteady problem

AIM : To determine the impact of change in CFL number on the unsteady state solution.

The CFL or Courant Number comes into picture where the problem is time marching.

The CFL Number is given as;

`CFL = alpha(Δt)/( Δx^2)`

`alpha` = thermal diffusivity, `m^2/s`

Δt = time interval

Δx = grid size

1. Variation in CFL Number for Explicit Method

Explicit Methods, for stability, is dependent on the CFL Number. In case of the Explicit method, with the value of thermal diffusivity 1e-3, following variation in CFL Number is noted.

Time step (dt)	Grid size	CFL Number
0.1	0.00173	0.0576
0.4	0.00173	0.2304
0.43	0.00173	0.2477
0.44	0.00173	0.2534
0.5	0.00173	0.2880
1	0.00173	0.576

Upto timestep 0.43, the solution is obtained, but as soon as it is increased to 0.44, the CFL Number gets 0.2534, and the solution starts to diverge. With further increase in timestep, above 0.44, the solution is unstable.