Fabian Waleffe

Homotopy of exact coherent structures in plane shear flows

Three-dimensional steady states and traveling wave solutions of the Navier–Stokes equations are computed in plane Couette and Poiseuille flows with both free-slip and no-slip boundary conditions. They are calculated using Newton’s method by continuation of solutions that bifurcate from a two-dimensional streaky flow then by smooth transformation (homotopy) from Couette to Poiseuille flow and from free-slip to no-slip boundary conditions. The structural and statistical connections between these solutions and turbulent flows are illustrated. Parametric studies are performed and the parameters leading to the lowest onset Reynolds numbers are determined. In all cases, the lowest onset Reynolds number corresponds to spanwise periods of about 100 wall units. In particular, the rigid-free plane Poiseuille flow traveling wave arises at Reτ=44.2 for Lx+=273.7 and Lz+=105.5, in excellent agreement with observations of the streak spacing. A simple one-dimensional map is proposed to illustrate the possible nature of ...

Exact coherent structures in channel flow

Exact coherent states in no-slip plane Poiseuille flow are calculated by homotopy from free-slip to no-slip boundary conditions. These coherent states are unstable travelling waves. They consist of wavy low-speed streaks flanked by staggered streamwise vortices closely resembling the asymmetric coherent structures observed in the near-wall region of turbulent flows. The travelling waves arise from a saddle-node bifurcation at a sub-turbulent Reynolds number with wall-normal, spanwise and streamwise dimensions smaller than but comparable to 50 + , 100 + and 250 + , respectively. These coherent solutions come in pairs with distinct structure and instabilities. There is a three-dimensional continuum of such exact coherent states.

Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence

Forced turbulence in a rotating frame is studied using numerical simulations in a triply periodic box. The random forcing is three dimensional and localized about an intermediate wavenumber kf. The results show that energy is transferred to scales larger than the forcing scale when the rotation rate is large enough. The scaling of the energy spectrum approaches E(k)∝k−3 for k<kf. Almost all of the energy for k<kf lies in the two-dimensional (2D) plane perpendicular to the rotation z-axis, and thus the large-scale motions are quasi-2D with E(k)≈E(kh,kz=0), where kh and kz are, respectively, the horizontal and vertical components of the wavevector. The large scales consist of cyclonic vortices. Possible mechanisms responsible for the two-dimensionalization are discussed. The development of the 2D spectrum E(kh,kz=0)∝kh−3 is analogous to the dynamics of β-plane turbulence leading to the Rhines spectrum E(ky,kx=0)∝ky−5.

Generation of slow large scales in forced rotating stratified turbulence

Numerical simulations are used to study homogeneous, forced turbulence in three-dimensional rotating, stably stratified flow in the Boussinesq approximation, where the rotation axis and gravity are both in the z ˆ-direction. Energy is injected through a three-dimensional isotropic white-noise forcing localized at small scales. The parameter range studied corresponds to Froude numbers smaller than an O (1) critical value, below which energy is transferred to scales larger than the forcing scales. The values of the ratio N / f range from ≈1/2 to ∞, where N is the Brunt–Vaisala frequency and f is twice the rotation rate. For strongly stratified flows ( N / f [Gt ]1), the slow large scales generated by the fast small-scale forcing consist of vertically sheared horizontal flow. Quasi-geostrophic dynamics dominate, at large scales, only when 1/2 [les ] N / f [les ] 2, which is the range where resonant triad interactions cannot occur.

Lower Branch Coherent States in Shear Flows: Transition and Control

Lower branch coherent states in plane Couette flow have an asymptotic structure that consists of O(1) streaks, O(R(-1)) streamwise rolls and a weak sinusoidal wave that develops a critical layer, for large Reynolds number R. Higher harmonics become negligible. These unstable lower branch states appear to have a single unstable eigenvalue at all Reynolds numbers. These results suggest that lower branch coherent states control transition to turbulence and that they may be promising targets for new turbulence prevention strategies.

Polymer drag reduction in exact coherent structures of plane shear flow

Recently discovered traveling-wave solutions to the Navier–Stokes equations in plane shear geometries provide model flows for the study of turbulent drag reduction by polymer additives. These solutions, or “exact coherent states” (ECS), qualitatively capture the dominant structure of the near-wall buffer region of shear turbulence, i.e., counter-rotating pairs of streamwise-aligned vortices flanking a low-speed streak in the streamwise velocity. The optimum length scales for the ECS match well the length scales of the turbulent coherent structures and evidence suggests that the ECS underlie the dynamics of these structures. We study here the effect of viscoelasticity on these states. The changes to the velocity field for the viscoelastic ECS, where the FENE-P model calculates the polymer stress, mirror the modifications seen in experiments of fully turbulent flows of polymer solutions at low to moderate levels of drag reduction: drag is reduced, streamwise velocity fluctuations increase while wall-normal ...

On some dyadic models of the Euler equations

Katz and Pavlovic recently proposed a dyadic model of the Euler equations for which they proved finite time blow-up in the H 3/2+∈ Sobolev norm. It is shown that their model can be reduced to a dyadic model of the inviscid Burgers equation. The inviscid Burgers equation exhibits finite time blow-up in H α , for a > 1/2, but its dyadic restriction is even more singular, exhibiting blow-up for any α > 0. Friedlander and Pavlovic developed a closely related model for which they also prove finite time blow-up in H 3/2+∈ . Some inconsistent assumptions in the construction of their model are outlined. Finite time blow-up in the H α norm, for any a > 0, is proven for a class of models that includes all those models. An alternative shell model of the Navier-Stokes equations is discussed.

Spectrum of the Jacobi Tau Approximation for the Second Derivative Operator

It is proved that the eigenvalues of the Jacobi tau method for the second derivative operator with Dirichlet boundary conditions are real, negative, and distinct for a range of the Jacobi parameters. Special emphasis is placed on the symmetric case of the Gegenbauer tau method where the range of parameters included in the theorems can be merged and characteristic polynomials given by successive order approximations interlace. This includes the common Chebyshev and Legendre tau and Galerkin methods.

Transition Threshold and the Self-Sustaining Process

The self-sustaining process is a fundamental and generic three-dimensional nonlinear process in shear flows. It is responsible for the existence of non-trivial traveling wave and time-periodic states. These states come in pairs, an upper branch and a lower branch. The limited data available to date suggest that the upper branch states provide a good first approximation to the statistics of turbulent flows. The upper branches may thus be understood as the “backbone” of the turbulent attractor while the lower branches might form the backbone of the boundary separating the basin of attraction of the laminar state from that of the turbulent state. Evidence is presented that the lower branch states tend to purely streaky flows, in which the streamwise velocity has an essential spanwise modulation, as the Reynolds number R tends to infinity. The streamwise rolls sustaining the streaks and the streamwise undulation sustaining the rolls, both scale like R −1 in amplitude, just enough to overcome viscous dissipation. It is argued that this scaling is directly related to the observed R −1 transition threshold. These results also indicate that the exact coherent structures never bifurcate from the laminar flow, not even at infinity. The scale of the key elements, streaks, rolls and streamwise undulation, remain of the order of the channel size. However, the higher x-harmonics show a slower decay with R than naively expected. The results indicate the presence of a warped critical layer.

A new property of a class of Jacobi polynomials

Polynomials whose coefficients are successive derivatives of a class of Jacobi polynomials evaluated at x = 1 are stable. This yields a novel and short proof of the known result that the Bessel polynomials are stable polynomials. Stability-preserving linear operators are discussed. The paper concludes with three open problems involving the distribution of zeros of polynomials.