Transition to turbulence in a rotating channel

Abstract

Direct numerical simulation is used to determine the flows that occur as the Reynolds number, Re, is increased in a plane channel undergoing system rotation about a spanwise axis. (Plane Poiseuille flow occurs for zero rotation rate and low Re.) A constant system rotation speed of 0.5, non-dimensionalized with respect to the bulk streamwise velocity and channel full width, is used throughout. The spectral numerical method solves the three-dimensional, time-dependent, incompressible Navier-Stokes equations using periodic boundary conditions in the streamwise and spanwise directions. On increasing the Reynolds number above the temporally periodic wavy vortex regime, near Re = 4Rec (Rec = 88.6 is the critical Re for development of vortices), a second temporal frequency, ω2, occurs in the flow that corresponds to slow, constant, spanwise motion of the vortices, superposed on the much faster, constant, streamwise motion of the wavy vortex waves. Curiously, ω2 is always frequency locked with the wavy vortex frequency ω1 for the parameter range explored, although the locking ratio varies. At the slightly higher Re of 4.1 Rec, ω2 is replaced by a new frequency ω′2 that corresponds to a modulation of the wavy vortices like that seen in modulated wavy Taylor vortex flow. However, unlike the Taylor-Couette geometry, the modulation frequency here can become frequency locked with the wavy vortex frequency. Increasing Re further to Re = 4.2 Rec results in the appearance of a second incommensurate modulation frequency ω3, yielding a quasi-periodic three-frequency flow, although there are only two frequencies (ωω′2 and ω3) present in the reference frame moving with the travelling wave associated with ω1. At still higher Re (Re = 4.5 Rec), weak temporal chaos occurs. This flow is not turbulent however. Calculations of the instantaneous largest Lyapunov exponent, λ(t), and the spatial structure of small perturbations to the flow show that the chaos is driven by spanwise shear instability of the streamwise velocity component. At the highest Re of 6.7 Rec considered, quasi-coherent turbulent boundary layer structures occur as transient, secondary streamwise-oriented vortices in the viscous sublayer near the inviscidly unstable (high-pressure) wall. Calculations of λ(t) and the spatial structure of small perturbations to the flow show that the coherent structures are not caused by the local growth of small disturbances to the flow.