Studying edge geometry in transiently turbulent shear flows

AbstractIn linearly stable shear flows at moderate Reynolds number, turbulence spontaneously decays despite the existence of a codimension-one manifold, termed the edge, which separates decaying perturbations from those triggering turbulence. We statistically analyse the decay in plane Couette flow, quantify the breaking of self-sustaining feedback loops and demonstrate the existence of a whole continuum of possible decay paths. Drawing parallels with low-dimensional models and monitoring the location of the edge relative to decaying trajectories, we provide evidence that the edge of chaos does not separate state space globally. It is instead wrapped around the turbulence generating structures and not an independent dynamical structure but part of the chaotic saddle. Thereby, decaying trajectories need not cross the edge, but circumnavigate it while unwrapping from the turbulent saddle.

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Model-based Coherent-structure Control of Turbulent Shear Flows Using Low-dimensional Vortex Models

33rd AIAA Fluid Dynamics Conference and Exhibit ◽

10.2514/6.2003-4261 ◽

2003 ◽

Cited By ~ 13

Author(s):

Mark Pastoor ◽

Rudibert King ◽

Bernd Noack ◽

Andreas Dillmann ◽

Gilead Tadmor

Keyword(s):

Coherent Structure ◽

Shear Flows ◽

Turbulent Shear ◽

Structure Control ◽

Model Based ◽

Turbulent Shear Flows ◽

Low Dimensional

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Direct numerical simulation of stratified homogeneous turbulent shear flows

Journal of Fluid Mechanics ◽

10.1017/s0022112089000765 ◽

1989 ◽

Vol 200 ◽

pp. 563-594 ◽

Cited By ~ 195

Author(s):

T. Gerz ◽

U. Schumann ◽

S. E. Elghobashi

Keyword(s):

Heat Flux ◽

Prandtl Number ◽

Richardson Number ◽

Shear Flows ◽

Water Channel ◽

Vertical Direction ◽

Turbulent Shear ◽

Turbulent Heat ◽

Moderate Reynolds Number ◽

Turbulent Shear Flows

The exact time-dependent three-dimensional Navier-Stokes and temperature equations are integrated numerically to simulate stably stratified homogeneous turbulent shear flows at moderate Reynolds numbers whose horizontal mean velocity and mean temperature have uniform vertical gradients. The method uses shear-periodic boundary conditions and a combination of finite-difference and pseudospectral approximations. The gradient Richardson number Ri is varied between 0 and 1. The simulations start from isotropic Gaussian fields for velocity and temperature both having the same variances.The simulations represent approximately the conditions of the experiment by Komori et al. (1983) who studied stably stratified flows in a water channel (molecular Prandtl number Pr = 5). In these flows internal gravity waves build up, superposed by hot cells leading to a persistent counter-gradient heat-flux (CGHF) in the vertical direction, i.e. heat is transported from lower-temperature to higher-temperature regions. Further, simulations with Pr = 0.7 for air have been carried out in order to investigate the influence of the molecular Prandtl number. In these cases, no persistent CGHF occurred. This confirms our general conclusion that the counter-gradient heat flux develops for strongly stable flows (Ri ≈ 0.5–1.0) at sufficiently large Prandtl numbers (Pr = 5). The flux is carried by hot ascending, as well as cold descending turbulent cells which form at places where the highest positive and negative temperature fluctuations initially existed. Buoyancy forces suppress vertical motions so that the cells degenerate to two-dimensional fossil turbulence. The counter-gradient heat flux acts to enforce a quasi-static equilibrium between potential and kinetic energy.Previously derived turbulence closure models for the pressure-strain and pressure-temperature gradients in the equations for the Reynolds stress and turbulent heat flux are tested for moderate-Reynolds-number flows with strongly stable stratification (Ri = 1). These models overestimate the turbulent interactions and underestimate the buoyancy contributions. The dissipative timescale ratio for stably stratified turbulence is a strong function of the Richardson number and is inversely proportional to the molecular Prandtl number of the fluid.

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