Lower Branch Coherent States in Shear Flows: Transition and Control

Lower branch coherent states in plane Couette flow have an asymptotic structure that consists of $O(1)$ streaks, $O(R^{-1})$ streamwise rolls and a weak sinusoidal wave that develops a critical layer, for large Reynolds number $R$. Higher harmonics become negligible. These unstable lower branch states appear to have a single unstable eigenvalue at all Reynolds numbers. These results suggest that lower branch coherent states control transition to turbulence and that they may be promising targets for new turbulence prevention strategies.

Figure 1

Amplitude of $x$-Fourier modes for a 3D steady state in plane Couette flow vs $R$ for $(\alpha ,\gamma )=(1,2)$. Top to bottom: $O(1)$ streak $u_0(y,z)-\overline{u}(y)$, $O(R^{-0.9})$ fundamental mode $|w_1|$, $O(R^{-1})$ streamwise rolls ($v_0$, $w_0$) and $o(R^{-1})$ $|{\mathit{v}}_2|$ and $|{\mathit{v}}_3|$. Continued beyond $R=6168$ by dropping all harmonics. $R_{\mathrm{sn}}\approx 164$ is the turning point where lower and upper branch solutions coalesce.

Figure 2

Contours of $v_0(y,z)$ (solid, top) and $|v_1(y,z)|$ (solid, bottom) both with contours of $u_0(y,z)=[-2:2]/3$ (dashed) for $(\alpha ,\gamma ,R)=(1,2,50171)$. The critical layer $u_0(y,z)=0$ is shown as a bold solid curve in both plots. Top (bottom) wall at $y=1$ ($-1$) moves into (out of) the page so no-slip boundary conditions impose $u_0(\pm 1,z)=\pm 1$ and $v_0(\pm 1,z)=0$.

Figure 3

Contours of $|v_1|$ for $(\alpha ,\gamma )=(1,2)$ at $R=50171$ (solid) and 12 637 (dashed) stretched by $R^{1/3}$ factors along curves normal to $u_0$ contours to match $|v_1|$ contours at $R=3079$ (dash-dot line). The (almost overlapping) black and yellow solid curves show $u_0(y,z)=0$ at the 3 $R$’s.

Figure 4

The single unstable eigenvalue of the lower branch state $(\alpha ,\gamma )=(1.14,2.5)$ as function of $R$. Asympotic scaling is $\approx O(R^{-0.48})$. There is an extra complex conjugate pair near the onset $R_{\mathrm{sn}}\approx 218$.

Figure 5

Energy input/dissipation rate starting near the lower branch fixed point $(\alpha ,\gamma ,R)=(1,2,1000)$ on its unstable manifold. In one direction, the flow goes to turbulence while in the other direction it relaminarizes. The dot spacing is $△t=5$. The blue marker at (1,1) is the laminar flow, green marker at (1.35,1.35) is the lower branch state and the red marker at (3.89,3.89) is its upper branch brother.