Dwight Barkley

A turbulent–laminar banded pattern in plane Couette flow is studied numerically. This pattern is statistically steady, is oriented obliquely to the streamwise direction, and has a very large wavelength relative to the gap. The mean flow, averaged in time and in the homogeneous direction, is analysed. The flow in the quasi-laminar region is not the linear Couette profile, but results from a non-trivial balance between advection and diffusion. This force balance yields a first approximation to the relationship between the Reynolds number, angle, and wavelength of the pattern. Remarkably, the variation of the mean flow along the pattern wavevector is found to be almost exactly harmonic: the flow can be represented via only three cross-channel profiles as U(x, y, z) ≈ U0(y) + Uc(y) cos(kz) + Us(y) sin(kz). A model is formulated which relates the cross-channel profiles of the mean flow and of the Reynolds stress. Regimes computed for a full range of angle and Reynolds number in a tilted rectangular periodic computational domain are presented. Observations of regular turbulent–laminar patterns in other shear flows – Taylor–Couette, rotor–stator, and plane Poiseuille – are compared.

A turbulent-laminar banded pattern in plane Couette flow is studied numerically. This pattern is statistically steady, is oriented obliquely to the streamwise direction, and has a very large wavelength relative to the gap. The mean flow, averaged in time and in the homogeneous direction, is analysed. The flow in the quasi-laminar region is not the linear Couette profile, but results from a non-trivial balance between advection and diffusion. This force balance yields a first approximation to the relationship between the Reynolds number, angle, and wavelength of the pattern. Remarkably, the variation of the mean flow along the pattern wavevector is found to be almost exactly harmonic: the flow can be represented via only three cross-channel profiles as U(x,y,z) = U_0(y) + U_c(y) cos(kz) + U_s(y) sin(kz). A model is formulated which relates the cross-channel profiles of the mean flow and of the Reynolds stress. Regimes computed for a full range of angle and Reynolds number in a tilted rectangular periodic computational domain are presented. Observations of regular turbulent-laminar patterns in other shear flows -- Taylor-Couette, rotor-stator, and plane Poiseuille -- are compared.

Regular patterns of turbulent and laminar fluid motion arise in plane Couette flow near the lowest Reynolds number for which turbulence can be sustained. We study these patterns using an extension of the minimal flow unit approach to simulations of channel flows pioneered by Jimenez and Moin. In our case, computational domains are of minimal size in only two directions. The third direction is taken to be large. Furthermore, the long direction can be tilted at any prescribed angle to the streamwise direction. We report on different patterned states observed as a function of Reynolds number, imposed tilt, and length of the long direction. We compare our findings to observations in large aspect-ratio experiments.

We explore situations in which certain stochastic and high-dimensional deterministic systems behave effectively as low-dimensional dynamical systems. We define and study moment maps, maps on spaces of low-order moments of evolving distributions, as a means of understanding equations-free multiscale algorithms for these systems. We demonstrate how nonlinearity arises in these maps and how this results in the stabilization of metastable states. Examples are shown for a hierarchy of models, ranging from simple stochastic differential equations to molecular dynamics simulations of a particle in contact with a heat bath.

A computational linear stability analysis of spiral waves in a reaction-diffusion equation is performed on large disks. As the disk radius R increases, eigenvalue spectra converge to the absolute spectrum predicted by Sandstede and Scheel. The convergence rate is consistent with 1/R, except possibly near the edge of the spectrum. Eigenfunctions computed on large disks are compared with predicted exponential forms. Away from the edge of the absolute spectrum the agreement is excellent, while near the edge computed eigenfunctions deviate from predictions, probably due to finite-size effects. In addition to eigenvalues associated with the absolute spectrum, computations reveal point eigenvalues. The point eigenvalues and associated eigenfunctions responsible for both core and far-field breakup of spiral waves are shown.

Turbulent-laminar patterns near transition are simulated in plane Couette flow using an extension of the minimal-flow-unit methodology. Computational domains are of minimal size in two directions but large in the third. The long direction can be tilted at any prescribed angle to the streamwise direction. Three types of patterned states are found and studied: periodic, localized, and intermittent. These correspond closely to observations in large-aspect-ratio experiments.

A spatially confined stability analysis is reported for the cylinder wake at Reynolds numbers 190 and 260. The two three-dimensional instabilities at these Reynolds numbers are shown to be driven by the flow just behind the cylinder.

Turbulent-laminar patterns near transition are simulated in plane Couette flow using an extension of the minimal flow unit methodology. Computational domains are of minimal size in two directions but large in the third. The long direction can be tilted at any prescribed angle to the streamwise direction. Three types of patterned states are found and studied: periodic, localized, and intermittent. These correspond closely to observations in large aspect ratio experiments.