On turbulent spots in plane Poiseuille flow

Turbulence characteristics inside a turbulent spot in plume Poiseuille flow are investigated by analyzing a data base obtained from a direct numerical simulation. The spot is found to consist of two distinct regions - a turbulent area and a wave area. The flow inside the turbulent area has a strong resemblance to that found in the fully developed turbulent channel. Suitably defined mean and rms fluctuations as well as the internal shear layer structures are found to be similar to the turbulent counterpart. In the wave area, the inflexional mean spanwise profiles cause a rapid growth of oblique waves, which break down to turbulence. The breakdown process of the oblique waves is reminiscent of the secondary instability observed during transition to turbulence in channel and boundary layer flows. Other detailed characteristics associated with the Poiseuille spot are presented and are compared with experimental results

A model for the collapse of the edge when two transitions routes compete

The transition to turbulence in many shear flows proceeds along two competing routes, one linked with finite-amplitude disturbances and the other one originating from a linear instability, as in e.g. boundary layer flows. The dynamical systems concept of edge manifold has been suggested in the subcritical case to explain the partition of the state space of the system. This investigation is devoted to the evolution of the edge manifold when a linear stability is added in such subcritical systems, a situation poorly studied despite its prevalence in realistic fluid flows. In particular the fate of the edge state as a mediator of transition is unclear. A deterministic three-dimensional model is suggested, parametrised by the linear instability growth rate. The edge manifold evolves topologically, via a global saddle-loop bifurcation, from the separatrix between two attraction basins to the mediator between two transition routes. For larger instability rates, the stable manifold of the saddle point increases in codimension from 1 to 2 after an additional local saddle node bifurcation, causing the collapse of the edge manifold. As the growth rate is increased, three different regimes of this model are identified, each one associated with a flow case from the recent hydrodynamic literature. A simple nonautonomous generalisation of the model is also suggested in order to capture the complexity of spatially developing flows.Comment: 12 pages, 10 figures, under review in Phys. Rev.

Secondary threshold amplitudes for sinuous streak breakdown

The nonlinear stability of laminar sinuously bent streaks is studied for the plane Couette flow at Re=500 in a nearly minimal box and for the Blasius boundary layer at Re_d*= 700. The initial perturbations are nonlinearly saturated streamwise streaks of amplitude AU perturbed with sinuous perturbations of amplitude AW. The local boundary of the basin of attraction of the linearly stable laminar flow is computed by bisection and projected in the AU – AW plane providing a well defined critical curve. Different streak transition scenarios are seen to correspond to different regions of the critical curve. The modal instability of the streaks is responsible for transition for AU ~ 25%–27% for the considered flows, where sinuous perturbations of amplitude below AW ~ 1%–2% are sufficient to counteract the streak viscous dissipation and induce breakdown. The critical amplitude of the sinuous perturbations increases when the streamwise streak amplitude is decreased. With secondary perturbations amplitude AW ~ 4%, breakdown is induced on stable streamwise streaks with AU ~ 13%, following the secondary transient growth scenario first examined by Schoppa and Hussain [J. Fluid Mech. 453, 57 (2002)]. A cross-over, where the critical amplitude of the sinuous perturbation becomes larger than the amplitude of streamwise streaks, is observed for streaks of small amplitude AU < 5%–6%. In this case, the transition is induced by an initial transient amplification of streamwise vortices, forced by the decaying sinuous mode. This is followed by the growth of the streaks and final breakdown. The shape of the critical AU – AW curve is very similar for Couette and boundary layer flows and seems to be relatively insensitive to the nature of the edge states on the basin boundary. The shape of this critical curve indicates that the stability of streamwise streaks should always be assessed in terms of both the streak amplitude and the amplitude of spanwise velocity perturbations

On the relevance of Reynolds stresses in resolvent analyses of turbulent wall-bounded flows

The ability of linear stochastic response analysis to estimate coherent motions is investigated in turbulent channel flow at friction Reynolds number Re$_\tau$ = 1007. The analysis is performed for spatial scales characteristic of buffer-layer and large-scale motions by separating the contributions of different temporal frequencies. Good agreement between the measured spatio-temporal power spectral densities and those estimated by means of the resolvent is found when the effect of turbulent Reynolds stresses, modelled with an eddy-viscosity associated to the turbulent mean flow, is included in the resolvent operator. The agreement is further improved when the flat forcing power spectrum (white noise) is replaced with a power spectrum matching the measures. Such a good agreement is not observed when the eddy-viscosity terms are not included in the resolvent operator. In this case, the estimation based on the resolvent is unable to select the right peak frequency and wall-normal location of buffer-layer motions. Similar results are found when comparing truncated expansions of measured streamwise velocity power spectral densities based on a spectral proper orthogonal decomposition to those obtained with optimal resolvent modes

Stability of a jet in crossflow

We have produced a fluid dynamics video with data from Direct Numerical Simulation (DNS) of a jet in crossflow at several low values of the velocity inflow ratio R. We show that, as the velocity ratio R increases, the flow evolves from simple periodic vortex shedding (a limit cycle) to more complicated quasi-periodic behavior, before finally exhibiting asymmetric chaotic motion. We also perform a stability analysis just above the first bifurcation, where R is the bifurcation parameter. Using the overlap of the direct and the adjoint eigenmodes, we confirm that the first instability arises in the shear layer downstream of the jet orifice on the boundary of the backflow region just behind the jet.Comment: Two fluid dynamics videos, high-resolution 1024x768 (~80MB), and low resolution 320x240 (~10MB), included in the ancillary file

Recurrent bursts via linear processes in turbulent environments

Large-scale instabilities occurring in the presence of small-scale turbulent fluctuations are frequently observed in geophysical or astrophysical contexts but are difficult to reproduce in the laboratory. Using extensive numerical simulations, we report here on intense recurrent bursts of turbulence in plane Poiseuille flow rotating about a spanwise axis. A simple model based on the linear instability of the mean flow can predict the structure and time scale of the nearly-periodic and self-sustained burst cycles. Rotating Poiseuille flow is suggested as a prototype for future studies of low-dimensional dynamics embedded in strongly turbulent environments

Bypass transition and spot nucleation in boundary layers

The spatio-temporal aspects of the transition to turbulence are considered in the case of a boundary layer flow developing above a flat plate exposed to free-stream turbulence. Combining results on the receptivity to free-stream turbulence with the nonlinear concept of a transition threshold, a physically motivated model suggests a spatial distribution of spot nucleation events. To describe the evolution of turbulent spots a probabilistic cellular automaton is introduced, with all parameters directly fitted from numerical simulations of the boundary layer. The nucleation rates are then combined with the cellular automaton model, yielding excellent quantitative agreement with the statistical characteristics for different free-stream turbulence levels. We thus show how the recent theoretical progress on transitional wall-bounded flows can be extended to the much wider class of spatially developing boundary-layer flows