How Can Numerical Stability Be Achieved in Unsteady Laminar Flow Equations?

[Aero51]

I took a CFD class last semester (had to leave school though due to personal garbage). I am making a come back this fall and as some extra credit I am trying to numerically solve the unsteady laminar flow equation in a pipe. The equation is

$\dot{U} + U'' + K = 0$
where dots denote the time derivative and primes denote spatial derivatives (in this case the radius, r)

The discretization of the equation is:
$(U^{n+1}_{i}-U^{n}_{i})/\Delta T + (U^{n}_{i+1} - 2U^{n}_i+U^n_{i-1})/(\Delta R)^2 + K$

However, when I try to do the stability analysis I get this really ugly problem:
$CFL = 1-2{\Delta T}/(\Delta R)^2 * (cosh(\Delta R)-1)-K\Delta T e^{i K_m R}$

Any ideas how to eliminate that last euler number to make the stability analysis more feasible?

 

[Aero51]

Update: I am going to try to take the natural log and multiply by its conjugate base and see what happens

 

[verty]

Since you are already retaking this class, I would just look up the answer. You have tried your hardest to solve it, now find the answer. It just isn't worth the energy to get stuck on something like this.

PS. And of course I mean, find the answer and try to understand it, read more online, etc.

 

[Aero51]

I talked it over with my professor, he actually wasn't sure how to deal with the constant term either. However, he did mention that he was certain the stability analysis was only dependent on the unsteady and viscus terms. Also, I had a coefficient negative when it should have been positive...

 

[blue_raver22]

I am glad to hear that you are making a comeback and taking on the challenge of numerically solving the unsteady laminar flow equation in a pipe. The equation you have provided is a common form of the Navier-Stokes equation, which is a fundamental equation in fluid dynamics.

The numerical stability of partial differential equations (PDEs) is an important aspect to consider when solving them numerically. It ensures that the numerical solution accurately represents the behavior of the physical system being modeled. In the case of unsteady PDEs, stability refers to the ability of the numerical method to maintain accuracy over a long time period.

The discretization of the equation you have provided is known as the Forward Time Centered Space (FTCS) method. It is a commonly used method for solving unsteady PDEs, but it does have some limitations when it comes to stability. The CFL (Courant-Friedrichs-Lewy) condition you have mentioned is a way to determine the time step size that will ensure numerical stability. It is based on the maximum signal speed in the system, which in this case is the wave speed.

The term involving the Euler number (e) in your stability analysis is due to the use of the FTCS method. This term can cause instability in the numerical solution, as it introduces a growth factor that can amplify numerical errors. There are various ways to eliminate this term, such as using a different numerical method or modifying the discretization scheme.

One approach you could try is using a different numerical method, such as the Crank-Nicolson method, which is known to be unconditionally stable for linear PDEs. Another option is to modify the FTCS method by introducing a damping term, which can help reduce the growth factor and improve stability.

In conclusion, the numerical stability of PDEs is a crucial aspect to consider when solving them numerically. It is important to carefully select the numerical method and discretization scheme to ensure accurate and stable solutions. I encourage you to continue exploring different methods and approaches to improve the stability of your numerical solution. Good luck with your project!