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Pipelines are used in a huge range of industrial processes involving fluids, and the ability to accurately predict properties of the flow through a pipe is of fundamental engineering importance. Armed with parallel MPI, Arnoldi and Newton--Krylov solvers, the Openpipeflow code can be used in a range of settings, from large-scale simulation of highly turbulent flow, to the detailed analysis of nonlinear invariant solutions (equilibria and periodic orbits) and their influence on the dynamics of the flow., Comment: 5 pages, 4 figures

International audience; Recently, a set of stationary invariant solutions for the large-scale structures in turbulent Couette flow was computed at Reτ ≃ 128 using an over-damped LES with the Smagorinsky model which accounts the effect of the surrounding small-scale motions (Rawat et al. J. Fluid Mech., 2015, 782:515). In this talk, we show that this approach can be extended to Reτ ≃ 1000 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 in the form of a traveling wave can be computed from simulations of the self-sustaining large-scale structures in the minimal unit with midplane reflection symmetry. 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 related wall shear stress. In particular, we will show 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.

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 pages

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 figure

Fully 3-dimensional computations of flow through a long pipe demand a huge number of degrees of freedom, making it very expensive to explore parameter space and difficult to isolate the structure of the underlying dynamics. We therefore introduce a `2+epsilon' dimensional model of pipe flow which is a minimal 3-dimensionalisation of the axisymmetric case: only sinusoidal variation in azimuth plus azimuthal shifts are retained, yet the same dynamics familiar from experiments are found. In particular the model retains the subcritical dynamics of fully resolved pipe flow, capturing realistic localised `puff'-like structures which can decay abruptly after long times, as well as global `slug' turbulence. Relaminarisation statistics of puffs reproduce the memoryless feature of pipe flow and indicate the existence of a Reynolds number about which lifetimes diverge rapidly, provided that the pipe is sufficiently long. Exponential divergence of the lifetime is prevalent in shorter periodic domains. In a short pipe, exact travelling-wave solutions are found nearby to flow trajectories on the boundary between laminar and turbulent flow. In a long pipe, the attracting state on the laminar-turbulent boundary is a localised structure which resembles a smoothened puff. This `edge' state remains localised even for Reynolds numbers where the turbulent state is global., Comment: 21 pages, 9 figures; as accepted, J. Fluid Mech