Genesis of Streamwise-Localized Solutions from Globally Periodic Traveling Waves in Pipe Flow

The aim in the dynamical systems approach to transitional turbulence is to construct a scaffold in phase space for the dynamics using simple invariant sets (exact solutions) and their stable and unstable manifolds. In large (realistic) domains where turbulence can coexist with laminar flow, this requires identifying exact localized solutions. In wall-bounded shear flows, the first of these has recently been found in pipe flow, but questions remain as to how they are connected to the many known streamwise-periodic solutions. Here we demonstrate that the origin of the first localized solution is in a modulational symmetry-breaking Hopf bifurcation from a known global traveling wave that has twofold rotational symmetry about the pipe axis. Similar behavior is found for a global wave of threefold rotational symmetry, this time leading to two localized relative periodic orbits. The clear implication is that many global solutions should be expected to lead to more realistic localized counterparts through such bifurcations, which provides a constructive route for their generation.