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Gibson research blog, spring 2009
Let us see how this format works. For now I will just start writing. I hope that greater flexibility in the structure (multiple pages and links) will make it easy to reorganize as the material gathers.
State space of HKW cell
Basis sets for HKW cell
I have been dissatisfied for quite a while with the projection basis sets I've used for plotting periodic orbits in the HKW cell. We don't have good equilibria there, so we can't easily do the translational-basis set we used for the narrow cell in the JFM08 paper. I have previously used a POD basis from the data of a single periodic orbit (P87p89). The plots were good, in that they showed turbulent trajectories lying in the same regions of state space as the periodic orbits, and forming similar pretty-well-defined shapes in whatever projection we looked at. But Predrag never liked this since it involved POD and averages, and to me, it was rhetorically weak to use a basis based on periodic orbits to argue that the periodic orbits live where turbulence is.
So, I decided to compare the orbit-POD basis to a translational basis based on a single velocity field (for Predrag) and a turbulent-data POD basis (for me). It turns out they're all remarkably similar.
Some key points:
All results here are confined to the S-invariant subspace as defined in our JFM08 paper.
Unlike the narrower γ=1.5 cell, in the HKW γ=1.67 cell, the streamwise phase of the roll/streak system never changes, at least in my observation. If the high-speed (red) streak starts in the middle of the box, it stays in the middle of the box, for thousands of time units.
Thus, unlike the narrow cell, a translational basis need not include spanwise (z) half-cell shifts in order to (roughly) span the region of state space that a turbulent trajectory explores. That gives just a two-d basis:
Since we don't have a good equilibrium to form a translational basis, I used the mean of the P87p89 orbit (pictured below). In the figures these basis elements are labeled
e0 P87meantrans and
The turbulent POD basis was constructed from about two hundred samples over a trajectory of several thousand time units. The basis clearly reflects the fixed z-phase of the roll-streak structures.
The POD basis sets were calculated without removing the mean first, and the mean is far from the origin, so the first (zeroth) POD mode corresponds to the mean of the data. This facilitated comparison with the translational basis.
The POD calculations also explicitly enforced symmetry under half-cell x translations. E.g. the averages were computed over both P87p9 and
P87p89, and similarly for the samples of the the turbulent trajectory.