Reply to Comment on 'Critical behaviour in the relaminarization of localized turbulence in pipe flow'

This is a Reply to Comment arXiv:0707.2642 by Hof et al. on Letter arXiv:physics/0608292 which was subsequently published in Phys Rev Lett, 98, 014501 (2007). In our letter it was reported that in pipe flow the median time $\tau$ for relaminarisation of localised turbulent disturbances closely follows the scaling $\tau\sim 1/(Re_c-Re)$. This conclusion was based on data from collections of 40 to 60 independent simulations at each of six different Reynolds numbers, Re. In the Comment, Hof et al. estimate $\tau$ differently for the point at lowest Re. Although this point is the most uncertain, it forms the basis for their assertion that the data might then fit an exponential scaling $\tau\sim \exp(A Re)$, for some constant A, supporting Hof et al. (2006) Nature, 443, 59. The most certain point (at largest Re) does not fit their conclusion and is rejected. We clarify why their argument for rejecting this point is flawed. The median $\tau$ is estimated from the distribution of observations, and it is shown that the correct part of the distribution is used. The data is sufficiently well determined to show that the exponential scaling cannot be fit to the data over this range of Re, whereas the $\tau\sim 1/(Re_c-Re)$ fit is excellent, indicating critical behaviour and supporting experiments by Peixinho & Mullin 2006.Comment: 1 page, 1 figur

Hydromagnetic Taylor--Couette flow: wavy modes

We investigate magnetic Taylor--Couette flow in the presence of an imposed axial magnetic field. First we calculate nonlinear steady axisymmetric solutions and determine how their strength depends on the applied magnetic field. Then we perturb these solutions to find the critical Reynolds numbers for the appearance of wavy modes, and the related wavespeeds, at increasing magnetic field strength. We find that values of imposed magnetic field which alter only slightly the transition from circular--Couette flow to Taylor--vortex flow, can shift the transition from Taylor--vortex flow to wavy modes by a substantial amount. The results are compared against onset in the absence of a magnetic field.Comment: 12 pages, 8 figures. To appear in J. Fluid Mech. To appear in J. Fluid Mec

Optimization of the magnetic dynamo

In stars and planets, magnetic fields are believed to originate from the motion of electrically conducting fluids in their interior, through a process known as the dynamo mechanism. In this Letter, an optimization procedure is used to simultaneously address two fundamental questions of dynamo theory: "Which velocity field leads to the most magnetic energy growth?" and "How large does the velocity need to be relative to magnetic diffusion?" In general, this requires optimization over the full space of continuous solenoidal velocity fields possible within the geometry. Here the case of a periodic box is considered. Measuring the strength of the flow with the root-mean-square amplitude, an optimal velocity field is shown to exist, but without limitation on the strain rate, optimization is prone to divergence. Measuring the flow in terms of its associated dissipation leads to the identification of a single optimal at the critical magnetic Reynolds number necessary for a dynamo. This magnetic Reynolds number is found to be only 15% higher than that necessary for transient growth of the magnetic field.Comment: Optimal velocity field given approximate analytic form. 4 pages, 4 figure

Invariant solutions of minimal large-scale structures in turbulent channel flow for Re_τ up to 1000

Understanding the origin of large-scale structures in high Reynolds number wall turbulence has been a central issue over a number of years. Recently, Rawat et al. (J. Fluid Mech., 2015, 782, p515) have computed invariant solutions for the large-scale structures in turbulent Couette flow at Reτ ≃ 128 using an over-damped LES with the Smagorinsky model to account for the effect of the surrounding small-scale motions. Here, we extend this approach to an order of magnitude higher Reynolds numbers in turbulent channel flow, towards the regime where the large-scale structures in the form of very-large-scale motions (long streaky motions) and large-scale motions (short vortical structures) energetically emerge. We demonstrate that a set of invariant solutions can be computed from simulations of the self-sustaining large-scale structures in the minimal unit (domain of size Lx = 3.0h streamwise and Lz = 1.5h spanwise) with midplane reflection symmetry at least up to Reτ ≃ 1000. By approximating the surrounding small scales with an artificially elevated Smagorinsky constant, a set of equilibrium states are found, labelled upper- and lower-branch according to their associated drag. It is shown that the upper-branch equilibrium state is a reasonable proxy for the spatial structure and the turbulent statistics of the self-sustaining large-scale structures

On the transient nature of localized pipe flow turbulence

International audienceThe onset of shear flow turbulence is characterized by turbulent patches bounded by regions of laminar flow. At low Reynolds numbers localized turbulence relaminarizes, raising the question of whether it is transient in nature or becomes sustained at a critical threshold. We present extensive numerical simulations and a detailed statistical analysis of the lifetime data, in order to shed light on the sources of the discrepancies present in the literature. The results are in excellent quantitative agreement with recent experiments and show that turbulent lifetimes increase super-exponentially with Reynolds number. In addition, we provide evidence for a lower bound below which there are no meta-stable characteristics of the transients, i.e. the relaminarization process is no longer memoryless. Copyright © Cambridge University Press 2010

Drag reduction in pipe flow by optimal forcing

In most settings, from international pipelines to home water supplies, the drag caused by turbulence raises pumping costs many times higher than if the flow were laminar. Drag reduction has therefore long been an aim of high priority. In order to achieve this end, any drag reduction method must modify the turbulent mean flow. Motivated by minimization of the input energy this requires, linearly optimal forcing functions are examined. It is shown that the forcing mode leading to the greatest response of the flow is always of m=1 azimuthal symmetry. Little evidence is seen of the second peak at large m (wall modes) found in analogous optimal growth calculations, which may have implications for control strategies. The model's prediction of large response of the large length-scale modes is verified in full direct numerical simulation of turbulence ($Re=5300$, $Re_\tau\approx 180$). Further, drag reduction of over 12% is found for finite amplitude forcing of the largest scale mode, m=1. Significantly, the forcing energy required is very small, being less than 2% of that by the through pressure, resulting in a net energy saving of over 10%.Comment: 6 page

Surfing the edge: Finding nonlinear solutions using feedback control

Many transitional wall-bounded shear flows are characterised by the coexistence in state-space of laminar and turbulent regimes. Probing the edge boundarz between the two attractors has led in the last decade to the numerical discovery of new (unstable) solutions to the incompressible Navier--Stokes equations. However, the iterative bisection method used to achieve this can become prohibitively costly for large systems. Here we suggest a simple feedback control strategy to stabilise edge states, hence accelerating their numerical identification by several orders of magnitude. The method is illustrated for several configurations of cylindrical pipe flow. Traveling waves solutions are identified as edge states, and can be isolated rapidly in only one short numerical run. A new branch of solutions is also identified. When the edge state is a periodic orbit or chaotic state, the feedback control does not converge precisely to solutions of the uncontrolled system, but nevertheless brings the dynamics very close to the original edge manifold in a single run. We discuss the opportunities offered by the speed and simplicity of this new method to probe the structure of both state space and parameter space

Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos

We propose a general strategy for determining the minimal finite amplitude isturbance to trigger transition to turbulence in shear flows. This involves constructing a variational problem that searches over all disturbances of fixed initial amplitude, which respect the boundary conditions, incompressibility and the Navier--Stokes equations, to maximise a chosen functional over an asymptotically long time period. The functional must be selected such that it identifies turbulent velocity fields by taking significantly enhanced values compared to those for laminar fields. We illustrate this approach using the ratio of the final to initial perturbation kinetic energies (energy growth) as the functional and the energy norm to measure amplitudes in the context of pipe flow. Our results indicate that the variational problem yields a smooth converged solution providing the amplitude is below the threshold amplitude for transition. This optimal is the nonlinear analogue of the well-studied (linear) transient growth optimal. At and above this threshold, the optimising search naturally seeks out disturbances that trigger turbulence by the end of the period, and convergence is then practically impossible. The first disturbance found to trigger turbulence as the amplitude is increased identifies the `minimal seed' for the given geometry and forcing (Reynolds number). We conjecture that it may be possible to select a functional such that the converged optimal below threshold smoothly converges to the minimal seed at threshold. This seems at least approximately true for our choice of energy growth functional and the pipe flow geometry chosen here.Comment: 27 pages, 19 figures, submitted to JF