Young Researchers Workshop: Kinetic descriptions in theory and applications

Statistical mechanics of the laminar-turbulent phase transition and the connection to directed percolation

Hong-Yan Shih

University of Illinois at Urbana-Champaign

In this talk, I describe the recent progress made in understanding the transition between laminar and turbulent flow in pipes and other shear flows. This transition has remarkable super-exponential behavior in the lifetime of turbulent puffs as a function of Reynolds number. My work uses direct numerical simulation of Navier-Stokes equations for the pipe flow to show that transitional turbulence is dominated by two collective modes: a longitudinal mode for small-scale turbulent fluctuations whose anisotropy induces an emergent large-scale azimuthal mode (so-called zonal flow) that inhibits anisotropic Reynolds stress [1]. This activation-inhibition interaction leads to stochastic predator-prey-like dynamics, from which it follows that the transition to turbulence belongs to the directed percolation universality class [1]. Finally, I show how predator-prey dynamics arises by deriving phenomenologically an effective field theory of the transition from a coarse-graining of the Reynolds equation [2]. Thus I show that the transition between laminar and turbulent flow in a wide class of transitional turbulence systems ranging from pipes, convection and magnetohydrodynamics is actually a non-equilibrium phase transition that is in the universality class of directed percolation and reggeon field theory. I also discuss the relationship between the super-exponential scaling in the transitional pipe flow, the extreme value statistics and the universal critical fluctuation distribution. [1] H.-Y. Shih, T.-L. Hsieh and N. Goldenfeld. Nature Physics 12, 245 (2016) [2] N. Goldenfeld and H.-Y. Shih, Journal of Statistical Physics 167 (3-4), 575 (2017).