Subcritical Transition to Turbulence in Shear Flows

Abstract

The theme of this thesis is transition to turbulence in linearly stable shear flows. Laminar-turbulent intermittency and hysteresis are typical feature of these flows at transitional Reynolds numbers, characterizing a subcritical scenario. Subcritical transition is a dynamically rich phenomenon in nature and has been scrutinized for over a century. The critical Reynolds number for the transition and its physical mechanisms are fundamental problems in fluid dynamics and are still not fully understood. These questions are addressed in this thesis by performing direct numerical simulations of Taylor-Couette flow in linearly stable regimes: counterrotating narrow-gap regime (in the plane Couette limit) and corotating quasi-Keplerian regime. The essential contributions in this study can be summarized as follows. In the first part of the thesis, a highly efficient parallel DNS code for turbulent Taylor-Couette flow has been developed. The Navier-Stokes equations are discretized in cylindrical coordinates with the spectral Fourier-Galerkin method in the axial and azimuthal directions, and high-order finite differences in the radial direction. Time is advanced by a second-order, semi-implicit projection scheme, which requires the solution of five Helmholtz/Poisson equations, avoids staggered grids and renders very small slip velocities. Nonlinear terms are computed with the pseudospectral method. The code is parallelized using a hybrid MPI-OpenMP strategy, which is simple to implement, reduces inter-node communications and is more efficient compared to a flat MPI parallelization. A strong scaling study shows that the hybrid code maintains very good scalability up to 3X10^4 processor cores and thus allows to perform simulations at higher Re with higher resolutions than previously feasible.In the second part, this code is used to study the subcritical transition to turbulence in plane Couette flow. Here turbulent spots can grow and form continuous stripes, yet in the stripe-normal direction they remain interspersed by laminar fluid. By performing direct numerical simulations in a long narrow domain, it is observed that individual turbulent stripes are transient. In agreement with recent observations in pipe flow it is found that turbulence becomes sustained at a distinct critical point once the spatial proliferation outweighs the inherent decaying process. By resolving the asymptotic size distributions close to criticality, the feature of scale invariance at the onset of turbulence is demonstrated and the critical exponents are obtained. These results shows that the transition is a continuous phase transition and may belong to the directed percolation universality class. Third, subcritical hydrodynamic turbulence is probed in Taylor-Couette flow (TCF) in the quasi-Keplerian regime at Re up to 10^5. Whether hydrodynamic turbulence exists in linearly stable quasi-Keplerian flows is strongly debated and controversial results were reported. To avoid the axial end-wall effects in experiments, DNS simulations of axially periodic TCF have been conducted. By analyzing the temporal evolution of perturbation kinetic energy, secondary instability is identified and it causes the flow break down to turbulence. However, the arising turbulence eventually decays. The effects of Earth rotation may contribute to the observed turbulence in experiments. In the final part of the thesis, linear stability and transient energy growth have been studied in the plane Couette flow with system rotation perpendicular to the wall. It is found that wall-normal external system rotation causes linear instability. At small rotation rates, the onset of linear instability scales inversely with the rotation rate and the optimal transient growth in the linearly stable region is slightly enhanced, ~Re^2. At large rotation rates, the transient growth is significantly inhibited.