User talk:Arthurv~enwiki

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Linearized Euler equations

Hello Arthurv. Thank you for your contributions to the Euler equations article. However, I have some problems with the full jacobians you added, since:

the given jacobians are for a special equation of state used, so they are less general than the rest of the article,
the symbol H appears both as enthalpy and as one of the fluxes,
some other variables are also named not in according with the rest of the article (E, V).

Are you willing to change these? If not, I can do it. You can respond either below, or otherwise on my talk page if you wish. Best regards, Crowsnest (talk) 09:00, 26 March 2008 (UTC)[reply]

Variables

Thanks for the comments. I've posted on your talk page, but i'll post here as well. I do see that the variables can duplicate (E,V, H), however enthalpy is usually given as H in many textbooks, as well as E for total energy...I guess there's a global shortage of symbols =/

However you're free to change it if you wish - my opinion is that it would be less confusing to change the symbols for the fluxes (F,G,H) than enthalpy (which is usually given as H) since many textbooks only demonstrate the Euler equations in 1 or 2 dimensions, not 3 dimensions as I've included. Thanks again for your help,

Arthurv (talk) 12:16, 26 March 2008 (UTC)[reply]

Hi Athurv. I agree that the fluxes should be renamed, rather than the enthalpy. I will give it a try.

Further, it is common on Wikipedia to react directly under the question or last reaction to a question (unless the owner of the talk page requests otherwise). Indentation can be used to indicate on which (level of) remarks and questions the new contribution is made. See WP:TALK. Crowsnest (talk) 16:58, 26 March 2008 (UTC)[reply]

Ok, got it. I'm thinking of expanding the section on the solutions of the Euler equations being waves (by showing how the linearized euler equations can be transformed into wave equations) but it's more convenient to only show it in 1 dimension. Do you think it belongs in the article or should it go elsewhere? Arthurv (talk) 18:02, 26 March 2008 (UTC)[reply]

I renamed the fluxes and jacobians, to prevent conflicting variable names. But on inspection of the jacobians, I think they may contain some typographical errors: there must be certain symmetries between x, y and z-directions, and some entries seem inconsistent with others. Can you further add a reference to the article, where you found the jacobians? In my opinion, an extension with waves in 1D-Euler will fit nicely into the article (perhaps another sub-section of "Linearized Euler equations ?). Crowsnest (talk) 21:20, 26 March 2008 (UTC)[reply]

Sorry, the typos were due to me copying, pasting and modifying the first matrix LaTeX code to make the 2nd and 3rd matrices! I've made some corrections and tried to make the notation a bit neater to make it easier to spot errors. Thanks for notifying me. My primary source is E.F Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics 2nd edition (I've added it to the references). That's where I got the x-direction jacobian from, however I derived the y- and z-direction matrices myself. Since the transformation to wave equations involves the jacobians, as you suggested, I'll move the "shock waves" section under the Linearized equations section. Arthurv (talk) 01:23, 27 March 2008 (UTC)[reply]

Great. But I would not move the shock waves under the Linearized Euler equations, since they are essentially non-linear. Crowsnest (talk) 07:58, 27 March 2008 (UTC)[reply]

Ok, I'll move shock waves after Linearized equations? Because the shock waves develop once the characteristic lines intersect. Arthurv (talk) 10:02, 27 March 2008 (UTC)[reply]

I made much bigger errors on Wikipedia than the typographical errors you left in the jacobians.

About the shock waves: if I understand correctly, you intent to apply the method of characteristics to the 1D Euler equations in the form with the jacobians. Is that with linearized or with non-linear jacobian? Because on that it depends very much where to put the shock waves. If you intent to use the non-linear jacobian in your characteristics determination, the shock waves come afterwards (but then my question is: what do you mean with linearized jacobians?). If you use a linearized jacobian, I would put the shock waves first. Crowsnest (talk) 19:11, 27 March 2008 (UTC)[reply]

I did some resectioning, maybe that is of any help to you. Crowsnest (talk) 20:06, 27 March 2008 (UTC)[reply]

Thanks for your assistance - I'm trying to get my notes organized before I put them on. Getting the Jacobians transforms the Euler equations from nonlinear to quasi-linear, since the jacobian is still a function of the state variables. If taken against a reference state, as you said, they become linear and can be used to transform the equations into wave equations. This is because the Jacobian is diagonalizable, with real, linearly independent eigenvectors (as long as sound speed is positive). I'll put it up shortly. Arthurv (talk) 00:35, 28 March 2008 (UTC)[reply]