Lower Branch Asymptotic Theory


 The transition of laminar flow, with its clean layers of flow tubes, to strongly mixed, irregular turbulent flow is one of the principal problems of modern hydrodynamics. It is certain that this fundamental change in type of motion of the fluid is traceable to an instability in the laminar flow, for laminar flows of themselves would always be possible solutions of the hydrodynamic equations. -- W. Tollmien (1935)

 The transition threshold is the smallest perturbation amplitude that triggers transition from laminar to turbulence. Because of the non-orthogonality of the eigenfunctions of the linearized Navier-Stokes equations, the perturbations can be amplified by a significant factor via transient growth. In shear flows, the linear transient growth primarily results from the redistribution of the streamwise velocity by the streamwise rolls.
 The pipe flow experiments (2003) provide clear evidence that the transition threshold scales like R-1 as R→∞. We provide numerical evidence that such scalings are asymptotically exact as R→∞.
 The root-mean-square (RMS) velocity fluctuation scalings provide a global visual confirmation that streaks approach constant, the streamwise rolls and the fundamental mode scale like R-1. The 2nd harmonic scales like R-3/2 and the 3rd harmonic scales like R-2, in contrast to the R-2 and R-3 scalings respectively (see the figure below). This indicates that a critical layer arises from the singularity u0(y,z)-c=0. It reduces the decay rate of the harmonics.

Mean velocity profiles for lower branch steady states in no-slip plane Couette flow. Left: (α,γ)=(1.14,2.5) for R=400,816,6220 (higher R closer but not converging to laminar), right: (α,γ)=(1,2) for R=400,816,6168. The laminar profile is indicated by the dashed line in both plots.


Velocity fluctuation profiles from top to bottom: streak, 1st harmonic, streamwise rolls, 2nd and 3rd harmonics at (α,γ)=(1,2).
A 3D critical layer of thickness of R-1/3. Contours of |u1|, |v1|, |w1| scaled by R-1/3 in the u0-normal directions at R=3079 (blue), 12637 (red) and 50171 (green) for (α,γ)=(1,2). The thick solid curves show the critical layer u0=0 at the three R's.


 The nonlinear coupling between the fundamental v1, with its critical layer structure, and the streamwise rolls provides a challenge for the development of a full asymptotic theory of the lower branch states that would be able to predict the amplitude scaling of the fundamental mode.

 The figure on the left provides a 3D view of the lower branch steady state at at (α,γ,R)=(1.14,2.5,1000) with green isosurface of the streamwise velocity u=0, and red isosurfaces of the 2nd invariant Q=∇2p/2 at 0.6 max{Q}=0.0033. The streamwise velocity is very weakly dependant on x and the streak is flanked by the staggered, quasi-streamwise rolls. The iso-level of Q decreases as R increases like R-1.

 The lower branch exact coherent structures tend to a relatively simple but nontrivial quasi-2D singular asymptotic state as R→∞, that is not a solution of the Euler equation, and that is not the laminar flow v=yex. So the lower branch states do not bifurcate from the laminar flow, not even at R=∞.