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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 ===== 2008-12-30 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: <latex> e_{\pm} = c_{\pm} u \pm \tau_x u </latex>. * 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 ''e1 P87meantrans''. * 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. {{p87meanlaminar.png}}{{p87mean.png}} {{pode0.png}}{{p87pode0.png}}{{p87meantranse0.png}} {{pode1.png}}{{p87pode1.png}}{{p87meantranse1.png}} {{pode2.png}}{{p87pode2.png}} {{pode3.png}}{{p87pode3.png}}
