Transition of Planar Couette Flow at Infinite Reynolds Numbers

An outline of the state space of planar Couette flow at high Reynolds numbers ($\mathrm{Re}<{10}^5$) is investigated via a variety of efficient numerical techniques. It is verified from nonlinear analysis that the lower branch of the hairpin vortex state (HVS) asymptotically approaches the primary (laminar) state with increasing Re. It is also predicted that the lower branch of the HVS at high Re belongs to the stability boundary that initiates a transition to turbulence, and that one of the unstable manifolds of the lower branch of HVS lies on the boundary. These facts suggest HVS may provide a criterion to estimate a minimum perturbation arising transition to turbulent states at the infinite Re limit.

Figure 1

Amplitudes of representative Fourier modes for the HVS in PCF are plotted against Re for stream- and spanwise extents of the periodic box $(L_x,L_z)=(2\pi ,\pi )$. All the amplitudes of the HVS algebraically decrease with increasing Re. In particular, gradual attenuation in the streak component $|u_0(y,z)-\overline{u}(y)|$ of the HVS (thick solid curve) contrasts well with that of NBW (thin solid curve), which is indicated to be kept at order 1 (as verified in Ref. 11).

Figure 2

$E(t)$ is half the norm of the wall-normal component of the perturbation at time $t$. Trajectories starting around ${\mathit{u}}_H$ separate towards either the laminar (dotted) or the turbulent (solid) states due to a small additive perturbation onto the initial state. Time development is carried out under the periodic boundary conditions with $(L_x,L_z)=(2\pi ,\pi )$ by imposing the reflection symmetry on the perturbation.

Figure 3

Trajectories starting from initial states, ${\mathit{u}}_{(a,b)}$ for a different set of parameters $(a,b)$, separate towards either the laminar ${\mathit{u}}_L$ (grey dashed curves) or the turbulent states (black solid curves), which are obtained at Re$=1000$. In the case of $\eta >0$, the initial state does not satisfy the reflection symmetry. Some of trajectories starting around ${\mathit{u}}_H$ at Re$=1000$ (red upward triangle) with a particular set of $(a,b)$ trace a part of the BB as “watershed,” which divides the whole space into the basins of the turbulent and laminar attractors. This BB is outlined by a heteroclinic orbit [23] connecting ${\mathit{u}}_H$ and ${\mathit{u}}_N$ at Re$=1000$ (red downward triangle). For reference, ${\mathit{u}}_H$ and ${\mathit{u}}_N$ obtained at Re$=300,10000$ are plotted as well (grey triangles), both of which approach towards $\xi =0$ with increasing Re.