The present work is targeted at performing a strong scaling study of the high-order spectral element fluid dynamics solver Nek5000. Prior studies such as [5] indicated a recommendable metric for strong scalability from a theoretical viewpoint, which we test here extensively on three parallel machines with different performance characteristics and interconnect networks, namely Mira (IBM Blue Gene/Q), Beskow (Cray XC40) and Titan (Cray XK7). The test cases considered for the simulations correspond to a turbulent flow in a straight pipe at four different friction Reynolds numbers Reτ = 180, 360, 550 and 1000. Considering the linear model for parallel communication we quantify the machine characteristics in order to better assess the scaling behaviors of the code. Subsequently sampling and profiling tools are used to measure the computation and communication times over a large range of compute cores. We also study the effect of the two coarse grid solvers XXT and AMG on the computational time. Super-linear scaling due to a reduction in cache misses is observed on each computer. The strong scaling limit is attained for roughly 5000 − 10, 000 degrees of freedom per core on Mira, 30, 000 − 50, 0000 on Beskow, with only a small impact of the problem size for both machines, and ranges between 10, 000 and 220, 000 depending on the problem size on Titan. This work aims at being a reference for Nek5000 users and also serves as a basis for potential issues to address as the community heads towards exascale supercomputers.

In this work we report the results of DNSs and LESs of the turbulent flow through hexagonal ducts at friction Reynolds numbers based on centerplane wall shear and duct half-height Reτ,c ≃ 180, 360, and 550. The evolution of the Fanning friction factor f with Re is in very good agreement with experimental measurements. A significant disagreement between the DNS and previous RANS simulations was found in the prediction of the in-plane velocity, and is explained through the inability of the RANS model to properly reproduce the secondary flow present in the hexagon. The kinetic energy of the secondary flow integrated over the cross-sectional area 〈K〉yz decreases with Re in the hexagon, whereas it remains constant with Re in square ducts at comparable Reynolds numbers. Close connection between the values of Reynolds stress uw¯ on the horizontal wall close to the corner and the interaction of bursting events between the horizontal and inclined walls is found. This interaction leads to the formation of the secondary flow, and is less frequent in the hexagon as Re increases due to the 120∘ aperture of its vertex, whereas in the square duct the 90∘ corner leads to the same level of interaction with increasing Re. Analysis of turbulence statistics at the centerplane and the azimuthal variance of the mean flow and the fluctuations shows a close connection between hexagonal ducts and pipe flows, since the hexagon exhibits near-axisymmetric conditions up to a distance of around 0.15DH measured from its center. Spanwise distributions of wall-shear stress show that in square ducts the 90∘ corner sets the location of a high-speed streak at a distance zv+≃50 from it, whereas in hexagons the 120∘ aperture leads to a shorter distance of zv+≃38. At these locations the root mean square of the wall-shear stresses exhibits an inflection point, which further shows the connections between the near-wall structures and the large-scale motions in the outer flow.

A set of accurate quadrature rules applicable to a class of integrable functions with isolated singularities is designed and analysed theoretically in one and two dimensions. These quadrature rules are based on the trapezoidal rule with corrected quadrature weights for points in the vicinity of the singularity. To compute the correction weights, small-size ill-conditioned systems have to be solved. The convergence of the correction weights is accelerated by the use of compactly supported functions that annihilate boundary errors. Convergence proofs with error estimates for the resulting quadrature rules are given in both one and two dimensions. The tabulated weights are specific for the singularities under consideration, but the methodology extends to a large class of functions with integrable isolated singularities. Furthermore, in one dimension we have obtained a closed form expression based on which the modified weights can be computed directly.

Swirl-switching is a low-frequency oscillatory phenomenon which affects the Dean vortices in bent pipes and may cause fatigue in piping systems. Despite thirty years worth of research, the mechanism that causes these oscillations and the frequencies that characterise them remain unclear. Here we show that a three-dimensional wave-like structure is responsible for the low-frequency switching of the dominant Dean vortex. The present study, performed via direct numerical simulation, focuses on the turbulent flow through a 90 • pipe bend preceded and followed by straight pipe segments. A pipe with curvature 0.3 (defined as ratio between pipe radius and bend radius) is studied for a bulk Reynolds number Re = 11 700, corresponding to a friction Reynolds number Re τ ≈ 360. Synthetic turbulence is generated at the inflow section and used instead of the classical recycling method in order to avoid the interference between recycling and swirl-switching frequencies. The flow field is analysed by three-dimensional proper orthogonal decomposition (POD) which for the first time allows the identification of the source of swirl-switching: a wave-like structure that originates in the pipe bend. Contrary to some previous studies, the flow in the upstream pipe does not show any direct influence on the swirl-switching modes. Our analysis further shows that a threedimensional characterisation of the modes is crucial to understand the mechanism, and that reconstructions based on 2D POD modes are incomplete.

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