Derivations for LowMachNavierStokes Jacobians

[tradowsk]

Created LaTeX document detailing the derivation of the analytical Jacobians for low_mach_navier_stokes.C, merged into GRINS in PR #183. Makefile included

Set up directory structure as well.

[pbauman]

Instead of a standalone document, could you please add a section to the existing document and use \input{} to add your stuff?

[tradowsk]

Do you want me to create a new part in the document? Because technically, I can split up my doc into multiple parts to fit the chapter structure you have already implemented.

[tradowsk]

Made requested changes.

Formatting seems to have some issues though. For some reason, it looks like everything get progressively shifted rightward. Can be clearly seen in the energy residual section 9.4

[pbauman]

Correct.

Other "parts" I'm envisioning are input options, solvers, etc.

[tradowsk]

Still have that formatting issue...

[pbauman]

We can deal with formatting later. Doing a more fine grained pass now.

[pbauman]

The test function should have an index. I like i.

[pbauman]

In fact, I would do something like F_{p,i} to indicate the ith equation

[pbauman]

OK, all the line notes I made in the pressure section also apply to the momentum and energy sections as well. After addressing those, I think we're good.

[tradowsk]

Can you check this commit to make sure I have the notation correct? I would rather get it down right before I do the rest of the document.

[tradowsk]

For the momentum and energy equations, we have an issue with notation.
If I follow a..h for vectors and i,j,... for dofs, I'm going to overlap due to the number of terms.

F_U = -\rho u_a u_{a,b} \phi^b + P \phi^{c,c} - \mu \left [u_{d,e} \phi_{d,e} + u_{g,f} \phi_{f,g} - \frac{2}{3} u_{h,h} \phi_{i,i} \right ] + \rho g_j \phi_j

I can just use m,n,... for dofs, but then i would no longer be for the test function, which is what you wanted.
I can also use the outer parts of the alphabet for vectors (i.e. a..h , r..z) and the inner for dofs.

Which would you prefer?

[pbauman]

I can also use the outer parts of the alphabet for vectors (i.e. a..h , r..z) and the inner for dofs.

This.

[tradowsk]

So for momentum, I have

F_{U,i} &= -\rho u_a u_{a,b} \phi_i^b + P \phi_i^{c,c} - \mu \left [u_{d,e} \phi_i^{d,e} + u_{g,f} \phi_i^{f,g} - \frac{2}{3} u_{h,h} \phi_i^{r,s} \right ] + \rho g_t \phi_i^t

Is this okay? I am unsure about the repeated i index on phi.

[pbauman]

The way you have it written, r and s are free indices.

[tradowsk]

Whoops, forgot to add the dirac delta into the .tex file

F_{U,i} &= ... - \frac{2}{3} u_{h,h} \delta_{r,s} \phi_i^{r,s} ...

Better?

[pbauman]

Getting there.

The momentum equation is a vector equation, so you should have a free index on the vector component appearing in each term. E.g.

F_{U,i}^a = -rho u_{a,b} u_b \phi_i^a (no sum on a)

So I suggest using a as the free index in all the terms and having a (no sum on a) clause at the end.

[tradowsk]

Closer?

 F_{U,i}^a &= -\rho u \cdot \nabla u \cdot \phi_i^a + P (\nabla \cdot u)(\nabla \cdot \phi_i^a) - \mu \left [\nabla u : \nabla \phi_i^a + (\nabla u)^T : \nabla \phi_i^a - \frac{2}{3} (\nabla \cdot u) I : \nabla \phi_i^a \right ] + \rho \mathbf{g} \cdot \phi_i^a \\
 F_{U,i}^a &= -\rho u_b u_{a,b} \phi_i^a + P u_{c,c} \phi_i^{a,a} - \mu \left [u_{a,d} \phi_i^{a,d} + u_{e,a} \phi_i^{a,e} - \frac{2}{3} u_{f,f} \delta_{a,g} \phi_i^{a,g} \right ] + \rho g_h \phi_i^a \ (no \ sum \ on \ a)

[pbauman]

@tradowsk Have you made all the changes you wanted to make here?

[tradowsk]

Let me do one more pass through tonight before you merge it, but I think it should be all set.

[tradowsk]

Fixed a typo, so I think it is all set now content-wise. The formatting needs some attention though.