## Exact coherent structures in channel flow.(English)Zbl 0987.76034

Summary: Exact coherent states in no-slip plane Poiseuille flow are calculated by homotopy from free-slip to no-slip boundary conditions. These coherent states are unstable traveling waves. They consist of wavy low-speed streaks flanked by staggered streamwise vortices closely resembling the asymmetric coherent structures observed in the near-wall region of turbulent flows. We show that the traveling waves arise from a saddle-node bifurcation at a sub-turbulent Reynolds number with wall-normal, spanwise and streamwise dimensions smaller than but comparable to $$50^+$$, $$100^+$$ and $$250^+$$, respectively. These coherent solutions come in pairs with distinct structure and instabilities, and there is a three-dimensional continuum of such exact coherent states.

### MSC:

 76F06 Transition to turbulence 76D05 Navier-Stokes equations for incompressible viscous fluids 76M22 Spectral methods applied to problems in fluid mechanics
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