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Exact coherent states with hairpin-like vortex structure in channel flow. (English) Zbl 1415.76137
Summary: Hairpin vortices are widely studied as an important structural aspect of wall turbulence. The present work describes, for the first time, nonlinear travelling wave solutions to the Navier-Stokes equations in the channel flow geometry – exact coherent states (ECS) – that display hairpin-like vortex structure. This solution family comes into existence at a saddle-node bifurcation at Reynolds number $$\text{Re}=666$$. At the bifurcation, the solution has a highly symmetric quasi-streamwise vortex structure similar to that reported for previously studied ECS. With increasing distance from the bifurcation, however, both the upper and lower branch solutions develop a vortical structure characteristic of hairpins: a spanwise-oriented ‘head’ near the channel centreplane where the mean shear vanishes connected to counter-rotating quasi-streamwise ‘legs’ that extend toward the channel wall. At $$\text{Re}=1800$$, the upper branch solution has mean and Reynolds shear-stress profiles that closely resemble those of turbulent mean profiles in the same domain.

##### MSC:
 76D05 Navier-Stokes equations for incompressible viscous fluids 37N10 Dynamical systems in fluid mechanics, oceanography and meteorology 76D17 Viscous vortex flows 76F06 Transition to turbulence
channelflow
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