Generation of Hairpin Vortices and Computation of Associated Invariant Solutions in Plane Poiseuille Flow*

Hairpin vortices and their role in the dynamics of wall-bounded flows have

been a subject of study for decades, yet there is a general lack of

systematic methods for generating hairpins under controlled conditions.

We present a novel method for constructing ``hairpin seeds,'' flow fields

that generate hairpin vortices under time integration in plane Poiseuille

flow conditions.

By fixing the size of the computational domain and the

magnitude of the hairpin seeds in wall units, we define a family of hairpin seeds

parametrized by Reynolds number, and summarize the associated simulation data for

a range of Reynolds numbers.

We then present an associated traveling wave solution, computed from an initial

guess taken from the hairpin seed simulation data.

Numerical continuation reveals that related solutions exist for a wide

range of Reynolds numbers, with rich bifurcation structure across both

upper and lower branches, including the emergence of periodic orbits and

symmetry-breaking bifurcations.

The relatively simple spatial structure of these solutions combined with

the low dimensionality of their unstable manifolds suggests that they play

an important role in the dynamics of plane Poiseuille flow