4. State-space visualization
The following movies show a plane Couette velocity field evolving under Navier-Stokes on the right and the corresponding state-space trajectory on the left. The evolution of the field is computed with a CFD algorithm. The cell size, Reynolds number: [Lx, Ly, Lz] = [5.51, 2, 2.51] , Re=400. The fields in this section are within the S-invariant subspace.![[state space / velocity field movie of plane couette flow]](../nb18frame0small.png)
The state-space projection is from the 105 dimensional space of free variables in the CFD algorithm onto a 2d plane (e1,e2) formed from linear combinations of the upper-branch equilibrium and its half-cell translations, with the laminar equilibrium as the origin. For example, e1 = (1 + τx + τz + τxz) uUB, normalized to ||e1|| = 1, where τx is a translation operator that shifts a velocity field by half the cell length in x. The norm and projection operator are defined by the L2 inner product. For example a1(t) = 1/||V|| ∫V u(t) ∙ e1 dV, where V is the cell volume.
Why does this make sense? See the Gibson, Halcrow and Cvitanović paper.
The labeled points are
LM |
Laminar equilibrium |
