All of the fields with and without the laminar field subtracted.
Laminar subtracted on the left, laminar omitted on the right:
I could go either way on whether or not to include the laminar solution on top of the eigenvalue.
Dustin or JohnG, can you explain why one would subtract laminar from eigenvectors? Adding a constant to u(x,t) does not change the matrix of state-space velocity gradients A, or its eigenvalues, eigenvectors, or am I missing something? — Predrag Cvitanovic 2009-03-31 07:10
Dustin, in your first try I went through the exercise of guessing which of your eigenvectors correspond to which eigenvalues. Did you recheck my labeling? It's important one gets this correctly to get the right behavior and right symmetries in close passages to equilibria, and in particular for their heteroclinic connections. Also, I do need the least contracting stable eigevalue (pair), JohnG and I believe that it offers a natural axis for local 3D plots. — Predrag Cvitanovic 2009-03-31 07:10
I do believe that your labelling is correct because it corresponds to John's description of the full-space eigenvectors that he listed in his Poincare Section write-up. I have not performed the calculation restricted to the SSS subspace as John has and found the first 30 eigenvectors in that subspace. I am also unsure of the meaning of “2 marginal eigenvalues - do they have nontrivial eigenvectors?” I also am 90% certain that the pair of Re e_11, Ime_11 does form the least contracting eigenvalue pair. — Dustin Spieker 2009-03-31 09:11
Should you scale colors to bring some structure in the leading eigenvector e_1? In principle the most important one in the full state space? Also, Im e_2 is pale compared to Re e_2, but they should be roughly the same intensity, as they span a spiral-out plane. Seems like manually adjusting these colors is a pain; can you scan the picture for redest/bluest, and then use these for a full rescale up to max blue (or red, whichever is larger in the orignal version). — Predrag Cvitanovic 2009-03-31 07:40
I'm curious: what do the the eigenvectors of the two marginal eigenvalues look like? They should point along the two continuous translations, streamwise and spanwise. Perhaps obvious… — Predrag Cvitanovic 2009-03-31 07:10
responding to Dustin 2009-03-31 above: Thanks for confirming that list agrees with JohnG's. For each continuous symmetry there is one marginal eigenvalue whose value is exactly 0; I assume that the code generates two of those to at list 6 significant digits precision. If code alos generates two associated eigenvectors, I am curious what they look like. — Predrag Cvitanovic 2009-04-01 02:54
The reason to remove the laminar flow from the eigenfunction is that if you add two fields that each include the laminar flow, the sum no longer meets the boundary conditions. Instead of having walls moving at +/-1, you'd have walls moving at +/-2. So if you plot both the equilibrium and the eigenfunction with the laminar flow included, you have to mentally subtract the laminar flow from one of them when you think about adding picture A to picture B.
The total velocity field of a small perturbation along an eigenfunction v of equilibrium uEQ is
The previous paragraph says, in effect, that it is misleading to plot both ulam + uEQ and ulam + ε v , because these fields don't sum.
As I see it, there are a few different combinations of the three terms on the right-hand-side of the above equation that do make sense for plotting.
1. I often plot the total velocity fields in movies of flows and of equilibia, e.g. ulam + uEQ because it's helpful for people to see how the rolls drag the high/low speed fluid from the walls into the center, and that high/low near the wall business is contained in the non-zero boundary conditions of the laminar flow.
If I want to illustrate with two plots an equilibrium field and an eigenfunction perturbation, I show either
2. a plot of ulam + uEQ and a plot of ε v, or
3. a plot of uEQ and a plot of ε v
I don't think it ever makes sense to plot ulam + ε v , unless the eigenvector is an eigenvector of the laminar state itself, in which case you would be plotting a total velocity field that meets the boundary conditions.
Everything is much easier if you incorporate the laminar flow into the equations of motion and let the independent variable be the deviation u with Dirichlet boundary conditions. Then the space of allowed fields u is a vector space and you can do sums over fields and remain in the same space. Occasionally, for pedagogical purposes, you might want to add the laminar field back in to make plots and movies so that fields look familiar to people and you have less explaining to do during talks. But otherwise I find it much easier to get rid of laminar once and for all at the beginning.
John Gibson 2009-04-01 14:43 EST