4c. Exact solutions: Periodic orbits
Equilibria are steady states of the Navier-Stokes flow. As -by definition- they do not move, they cannot be “turbulent.” “Turbulent” dynamics is captured by the (infinity of) unstable periodic solutions, 3D movies that repeat exactly after some finite time T.
The next three movies show three new periodic orbits computed by our group (viewing any one will suffice to get the idea):
| [MOVIE] T=75.348 periodic orbit (1 MB) | Computations were performed with Viswanath's
Newton - hookstep - GMRES search algorithm from close recurrences in a single
u(x,t) trajectory. Each orbit lies within the S-invariant subspace of HKW cell and is a true periodic orbit (no x or z translation after a completed period). |
| [MOVIE] T=88.905 periodic orbit (1.5 MB) | |
| [MOVIE] T=102.285 periodic orbit (1.3 MB) |
The first orbits of this type within the S-invariant subspace were computed by Kawahara & Kida, while the first “relative periodic” orbits in the full state-space of plane Couette were computed by Viswanath.
What is striking about these periodic orbit simulations is how visually similar they are to the turbulent dynamics --indeed, given only a segment of periodic orbit movie we would have no way of guessing that the dynamics is exactly periodic in time.
Now that we have many exact solutions of Navier-Stokes equations: how do we visualize them, fit them together?
