Lower Branch Unstable Manifold


 The steady states and traveling waves capture the key statistics of coherent structures and play an important role in turbulence transition. However, the motions are best understood by thinking about a phase space. The system evolves in time by moving around in the phase space. The laminar state has a constant velocity field that is smooth and steady in time. Hence, it is represented by a single point in phase space. We use the Channelflow code to look at the time evolutions of turbulent shear flows in phase space and study their stability, eigenspectra, eignevalues and other properties on the exact coherent structures.

 In plane Couette flow, the energy is injected through the drag at the moving walls and consumed at small scales by the viscous dissipation. For steady states or traveling waves, the energy input rate balances the energy dissipation rate, while in the turbulent solutions they vary chaotically in time. The exact coherent structures not only capture some key characteristics observed in the near-wall region of turbulent shear flows, but also are closely connected to turbulence.

 The figure on the left shows a projection of the multi-dimensional phase space onto the two-dimensional energy input-dissipation (I-D) plane for no-slip plane Couette flow at (α,γ,R)=(1.14,2.5,400). The blue orbit is a turbulent trajectory that starts near the unstable upper branch state indicated by the red marker on the plot with I=D=3.04. The corresponding lower branch state is indicated by the green marker with I=D=1.43, and the laminar flow is at I=D=1. The solution was computed for 2000 time units.

 In phase space the turbulent solution wanders around the corresponding upper branch fixed point for most of the time, and the mean and RMS velocity profiles of the turbulent solution are remarkably well described by the upper branch fixed point.

 The right figure shows the mean velocity and RMS velocity fluctuation profiles at (α,γ,R)=(1.14,2.5,400) in no-slip plane Couette flow. Blue: u rms, red: v rms, green: w rms. The upper branch fixed point is solid, and the turbulent averages over 2000 time units is dashed. This suggests that the upper branch solution forms the "backbone" for the turbulent attractor in phase space.

(α,γ,R)=(1,2,1000). Starting from near the linear unstable lower branch fixed point, in one direction, the flow decays back to the linearly stable but nonlinearly unstable laminar flow while it shoots to a turbulent state in the other direction. The decay of the perturbed lower branch state back to the laminar flow. It follows a standard two-step evolution. First, the fundamental mode v1, with its critical layer structure, disappears and the flow relaxes to an x-independent state that consists of streamwise rolls and streaks, and then it slowly decays back to the laminar flow on a long viscous time scale.

 The stability analysis of the lower branch exact coherent states indicates that these states are distinguished not only by their asymptotic structure but also by their stability characteristics. The eigenmode analysis of the three-dimensional lower branch steady state in plane Couette flow, up to R=12,000 shows that there is a single, real unstable eigenvalue for (α,γ)=(1.14,2.5) illustrated in the right figure.
 There is an extra complex conjugate pair near the onset Reynolds number Rsn≈218. This state is most unstable at R≈342, and then the unstable eigenvalue steadily decreases with increasing R at an asymptotic rate of R-0.48. Thus the lower branch states become less unstable with increasing R. This is not true for the upper branch states which develop new bifurcations and unstable modes as R increases.
 The stability analysis confirmed that in the one-period domain with fundamental wavenumbers, the lower branch state is an unstable equilibrium with a one-dimensional unstable manifold. It represents a saddle in phase space and a turbulent state could spend a substantial amount of time in its neighborhood. Its stable manifold locally splits the phase space into two parts, the laminar side and the turbulent side. Transition to turbulence only requires a perturbation on the turbulent side of the stable manifold of the lower branch states.
 Therefore, the lower branch coherent states may be viewed as the "backbone" of the separatrix - the phase space boundary separating the basin of attraction of the laminar point from that of the turbulent attractor, and may be the key states controlling transition to turbulence.

The lin-lin and log-log plots of the single unstable eigenvalue of the lower branch state. The asymptotic scaling of the unstable eigenvalue is approximately R-0.48. The upper branch is indicated in red dashed.