Exact Coherent Structures in Turbulent Shear Flows


 Understanding the nature of transition to turbulence has been one of the most important problems in fluid dynamics that has attracted attention since Reynolds' first pipe flow experiments in 1883. It has received increasing attention in recent years with development in nonlinear dynamics and numerical solutions of PDEs. But the progress in understanding the transition threshold and controlling transition to turbulence has been slow due to the highly nonlinear nature and complexity of turbulent shear flows.
 Our approach is through the exact coherent structures (ECS) in plane Couette flow. They come in pairs with upper and lower branches that bifurcate from a neutrally stable streaky flow. The lower branch ECS exhibit an asymptotic structure that consist of O(1) streaks sustained by O(R-1) streamwise rolls and a weak sinusoidal streak instability eigenmode that develops a critical layer structure. These unstable lower branch states have a one-dimensional unstable manifold and may be viewed as the 'backbone' of the phase space boundary separating the basin of attraction of the laminar point from that of the turbulent state. The very low dimensionality of the lower branch unstable manifold suggests new turbulence control strategies.

 The dominant fundamental sinusoidal mode symmetry and the Couette symmetry are imposed to the three-dimensional solutions. Plane Couette flow v=yex is linearly stable at all Reynolds numbers, but turbulence has been observed at R≈350. These 3D traveling wave solutions are typically linearly unstable. Therefore, numerical methods such as DNS are usually not capable of producing them. A nonlinear three-dimensional self-sustaining process (SSP) was introduced to illustrate the underlying physics and also provide a way to compute the solutions. In this process, we select the forcing that is weak yet still sustains the unstable streaks. The SSP theory has led to discovery of linearly unstable traveling wave solutions in plane Couette flow, channel flow and pipe flow. These solutions have the same basic structures that consist of wavy streaks flanked by staggered, counter-rotating, quasi-streamwise vortices. They are called the exact coherent structures (ECS), namely, the steady states or traveling wave solutions of the Navier-Stokes equations.

Top and front views of the no-slip plane Couette flow LOWER branch steady state at (α,γ)=(1.14,2.5) and R=400. Green: isosurface of the streamwise velocity u=min{u(x,y=0,z)}=-0.3332 (low-speed). Top left: isosurfaces of the streamwise vorticity ωx at ±0.8 max{ωx} with red positive and blue negative. Top right: red isosurfaces of Q=∇2p/2 at 0.8 max{Q}=0.0147. Bottom right: yellow isosurface of the streamwise velocity u=min{u(x,y=0,z)}=0.3332 (high-speed).
Top and front views of the no-slip plane Couette flow UPPER branch steady state at (α,γ)=(1.14,2.5) and R=400. Green: isosurface of the streamwise velocity u=min{u(x,y=0,z)}=-0.2622 (low-speed). Top left: isosurfaces of the streamwise vorticity ωx at ±0.6 max{ωx} with red positive and blue negative. Top right: red isosurfaces of Q=∇2p/2 at 0.6 max{Q}=0.0932. Bottom right: yellow isosurface of the streamwise velocity u=min{u(x,y=0,z)}=0.2622 (high-speed).