[rsonmez]

Hello everyone,

I am trying to validate my 1D unsteady heat equation in a commercial CFD program. I have two solid attached to each other.(solid 1 and solid 2) At the interface ( where solids touch each other) material properties are changing.
How I can propose a manufactured solution for that?

So far I tried the followings;
1) Two manufactured solution for solid1 and solid2 so there are two sources for each solid. I made sure that there is continuity of temperature at the interface. I also wrote a code in Python to compare with the software but they only match at very early time.

2) I proposed just one manufactured solution and calculated the source term for both solid. Again, my code and CFD software result matched just very early time step.

Is there any specific approach to manufactured solution when we add it to a commercial CFD program?

Thank you.

[sbaffini]

For steady state conduction in 1D, an analytical solution exists for an arbitrary number of layers, each with its own material and source term. Not sure for unsteady. Is this what you need?

[rsonmez]

Originally Posted by sbaffini

For steady state conduction in 1D, an analytical solution exists for an arbitrary number of layers, each with its own material and source term. Not sure for unsteady. Is this what you need?

Nope, yes there is an exact solution but I want to construct a manufactured solution to verify my code and to verify the case in a CFD software.

[arjun]

Originally Posted by rsonmez

Nope, yes there is an exact solution but I want to construct a manufactured solution to verify my code and to verify the case in a CFD software.

I am confused. For your case, how the manufactured solution be any different than the analytical solution if the PDE has to give unique solution.

[sbaffini]

Originally Posted by rsonmez

Nope, yes there is an exact solution but I want to construct a manufactured solution to verify my code and to verify the case in a CFD software.

I'm not sure if we are on the same page but, just to be clear: a manufactured solution (MS) is an exact solution. It is common to mention MS in verification because exact solutions only exists for a limited number of cases, while MS allows you to to test pretty much anything at the cost of just using a source term.

If for a case there is no known exact solution, you just pick up a a certain function (complex enough to not allow easy cancellations), put it in your equations and the resulting imbalance is your source term. This is MS, roughly speaking. Manufactured indeed comes from the fact that you choose the solution form and the equations give you the source terms needed to make it an actual solution.

But if your case actually has an analytical solution, and this satisfies your testing needs (e.g., if it is steady and you indeed are only interested in a steady approach), that's it, you can use it. The main difference is that you won't have any source terms (unless they were already part of the original problem) and the solution will probably be more relevant and easy to investigate.

So, I guess, you actually meant that you need an unsteady solution, not a steady one?

[sbaffini]

However, a fast search seems to show that analytical unsteady solutions exist for your multi-layer problem, so you are a search away from them

[rsonmez]

Originally Posted by sbaffini

However, a fast search seems to show that analytical unsteady solutions exist for your multi-layer problem, so you are a search away from them

I think, I did not explain myself well enough. Let's think about a PDE that we do not have an exact solution. If I have two different materials in the domain, how I will construct a manufactured solution for that. I have discontinuity in the domain.

Let's say domain length is L = 2.0 cm and at L/2 my conductivity changes. So, region1 from 0 to L/2, conductivity is k1 and region2 from L/2 to L, conductivity is k2. Should I derive two source term for region1 and region2? How I can construct a manufactured solution for this problem so that it will capture the discontinuity at the interface?

[sbaffini]

Ok, the premise was deceiving, as you began with "I am trying to validate", which isn't then your main concern. You, instead, actually want to learn how to obtain a manufactured solution.

STILL, obtaining a manufactured solution is not far from actually looking for a solution, as you need to know certain stuff about your equations (and also weird here, considering there are, literally, books written on the analytical solutions for heat conduction problems).

Long story short: when choosing a manufactured solution, it also embeds the boundary conditions you need to apply. If your code enforces/assumes, as it should, continuity of temperature and fluxes at the interface, so should the manufactured solutions.

As I said, boundary conditions typically come from the chosen solution, and I am not aware of any systematic method to come up with manufactured solutions with certain properties because...

...well, as I previously stated, and it is especially so for this specific problem, that's basically looking for an analytical solution.

Also, let me add, this goes pretty much beyond what the MMS is actually meant to do. Multiple zones are more of a coding aspect than a math one and, again, analytical solutions exist.

EDIT: Let me be more explicit. Yes, you need to autonomously pick up two solutions that will match at the interface together with the fluxes. No, there isn't a systematic method to derive such group of functions because, well, again, that's pretty close to how you obtain the actual analytical solution. Yes, once you have a function for each zone/material, you derive the corresponding source term to use in that zone/material.

[arjun]

Originally Posted by rsonmez

I think, I did not explain myself well enough. Let's think about a PDE that we do not have an exact solution. If I have two different materials in the domain, how I will construct a manufactured solution for that. I have discontinuity in the domain.

Let's say domain length is L = 2.0 cm and at L/2 my conductivity changes. So, region1 from 0 to L/2, conductivity is k1 and region2 from L/2 to L, conductivity is k2. Should I derive two source term for region1 and region2? How I can construct a manufactured solution for this problem so that it will capture the discontinuity at the interface?

Transport phenomenon, Bird Steward, Lightfoot


HEAT CONDUCTION THROUGH COMPOSITE WALLS

This is where description of steady state of your problem. Transient could be derived i believe. At least look at it.

Long time ago, i did write a code to test this and verified that starccm and wildkatze produced the steady state interface temperature given by formula here.