A framework is presented for the spectral-element code Nek5000, which has been, and still is, widely used in the computational fluid dynamics (CFD) community to perform high-fidelity numerical simulations of transitional and high Reynolds number flows. Despite the widespread usage, there is a deficiency in having a comprehensive set of tools specifically designed for conducting simulations using Nek5000. To address this issue, we have created a unique framework that allows, inter alia, to perform stability analysis and compute statistics of a turbulent flow. The framework encapsulates modules that provide tools, run-time parameters and memory structures, defining interfaces and performing different tasks. First, the framework architecture is described, showing its non-intrusive approach. Then, the modules are presented, explaining the main tools that have been implemented and describing some of the test cases. The code is open-source and available online, with proper documentation, to-run instructions and related examples.

The three-dimensional turbulent flow around a Flettner rotor, i.e. an engine-driven rotating cylinder in an atmospheric boundary layer, is studied via direct numerical simulations (DNS) for three different rotation speeds (α). This technology offers a sustainable alternative mainly for marine propulsion, underscoring the critical importance of comprehending the characteristics of such flow. In this study, we evaluate the aerodynamic loads produced by the rotor of height h, with a specific focus on the changes in lift and drag force along the vertical axis of the cylinder. Correspondingly, we observe that vortex shedding is inhibited at the highest α values investigated. However, in the case of intermediate α, vortices continue to be shed in the upper section of the cylinder (y/h>0.3). As the cylinder begins to rotate, a large-scale motion becomes apparent on the high-pressure side, close to the bottom wall. We offer both a qualitative and quantitative description of this motion, outlining its impact on the wake deflection. This finding is significant as it influences the rotor wake to an extent of approximately one hundred diameters downstream. In practical applications, this phenomenon could influence the performance of subsequent boats and have an impact on the cylinder drag, affecting its fuel consumption. This fundamental study, which investigates a limited yet significant (for DNS) Reynolds number and explores various spinning ratios, provides valuable insights into the complex flow around a Flettner rotor. The simulations were performed using a modern GPU-based spectral element method, leveraging the power of modern supercomputers towards fundamental engineering problems.

Large-scale coherent structures in incompressible turbulent pipe flow are studied for a wide range of Reynolds numbers (Re-tau = 180, 550, 1000, 2000 and 5200). Employing the Karhunen-Loeve decomposition and a novel approach based on the Voronoi diagram, we identify and classify statistically coherent structures based on their location, dimensions and Re-tau. With increasing Re-tau, two distinct classes of structures become more energetic, namely wall-attached and detached eddies. The Voronoi methodology is shown to delineate these two classes without the need for specific criteria or thresholds. At the highest Re-tau, the attached eddies scale linearly with the wall-normal distance with a slope of approximately l(y) similar to 1.2y/R, while the detached eddies remain constant at the size of l(y) approximate to 0.26R, with a progressive shift towards the pipe centre. We extract these two classes of structures and describe their spatial characteristics, including radial size, helix angle and azimuthal self-similarity. The spatial distribution could help explain the differences in mean velocity between pipe and channel flows, as well as in modelling large and very-large-scale motions (LSM and VLSM). In addition, a comprehensive description is provided for both wall-attached and detached structures in terms of LSM and VLSM.

We investigate the global instability mechanism of the flow past a three-dimensional stepped cylinder. A comprehensive study is performed for different diameter ratios of the two joined cylinders ranging from to. Independently of, the spectrum of the linearised Navier-Stokes operator reveals a pair of complex conjugate eigenvalues, with Strouhal number. The initial transition is triggered by a two-dimensional mechanism of the larger cylinder only, not affected by the presence of the junction and the smaller cylinder . The structural sensitivity analysis is used to identify where the instability mechanism acts. The onset of transition is solely localised in the large cylinder wake (L cell), where the wavemaker has two symmetric lobes across the separation bubble. When the Reynolds number increases, a second and a third unstable pair of complex conjugate eigenvalues appears. They are localised in the small cylinder (S) wake and modulation (N) region. For any, the appearance of unstable eigenmodes resembling the three cells S-N-L in the wake is observed. The nonlinear simulation results support this finding, in contrast with the previous classification of the laminar vortex shedding in direct (L-S) and indirect (L-N-S) modes interaction Lewis & Gharib (Phys. Fluids, vol. 4, 1992, pp. 104-117). This result indicates that each cell undergoes a supercritical Hopf bifurcation for any. As approaches, the modal linear stability results also show an unstable eigenmode in the wake of the small cylinder resembling a new modulation cell, named N2, similar to the N cell but mirrored with respect to the junction plane.

We perform the Karhunen-Loéve decomposition of the data from direct numerical simulations pertaining to incompressible turbulent pipe flow at various Reynolds numbers, in order to identify large-scale coherent structures. A novel approach based on the Voronoi diagram is introduced to estimate the energy distribution along the radial direction as a function of the geometrical properties of the modes. In contrast to previous classifications, no user-defined criterion, threshold or ad-hoc separation of the eddies is required since the two most energetic branches are inherently present as the Reynolds number increases. Details about the Voronoi analysis are provided, together with a comprehensive validation and comparison with previous classifications at low Reynolds numbers. We discuss the results and potential of the presented approach at Reτ = 180 and 5200, commenting on the two classes of coherent structures with a varying and constant size in the radial direction appearing at Reτ = 5200.

The boundary-layer stability on a section of a rotating wind turbine blade with an FFA-W3 series aerofoil at a chord Reynolds number of 3 × 105, with varying rotation and radii, is studied with direct numerical simulations and linear stability analyses. Low rotation does not significantly affect transition in the outboard blade region. The relative insensitivity to rotation is due to a laminar separation bubble near the leading edge, spanwise-deformed by a primary self-excited instability, promoting the secondary absolute instability of the Kelvin–Helmholtz (KH) vortices and rapid transition. Moderate increases in rotation, or moving inboard, stabilise the flow by accelerating the attached boundary layer and possibly inducing competition between cross-flow and KH modes. This delays separation and transition. Initially, for high rotation rates or radial locations close to the hub, transition is delayed. Nevertheless, strong stationary and travelling cross-flow modes are eventually triggered, spanwise modulating the KH rolls and shifting the transition line close to the leading edge. Cross-flow velocities as high as 56 % of the free stream velocity directed towards the blade tip are reached at the transition location. For radial locations farther from the hub, the effective angle of attack is decreased, and cross-flow transition occurs at lower rotation rates. The advance or delay of the transition line compared with a non-rotating configuration depends on the competing rotation effects of stabilising the attached boundary layer and triggering cross-flow modes in the separation flow region.

The coherent structures arising in the turbulent flow around a three-dimensional stepped (or step) cylinder are studied through direct numerical simulation. This geometry is widespread in many applications and the junction substantially modifies the wake behaviour, generating three main cells. The mechanisms of vortex connections on the junction are difficult to be captured and interpreted. We thus use a high-order spectral -element methodology (SEM), and the adaptive mesh refinement technique (AMR) to adequately resolve each region of the domain, capturing the smallest turbulent scales. In this way, we can analyse the vortical interactions on the junction via the A2-criterion and understand the evolution of the train of hairpins, which appears only when the cylinder shear layer gets unstable. Together with the hairpins, four horseshoe and edge vortices coexist on the flat junction surface. A complete picture of the vortices' evolution in time is provided. To extract the large-scale, and most energetic, structures in the wake we perform a three-dimensional proper orthogonal decomposition (POD) of the flow. The first six POD modes correspond to three travelling modes which identify the large (L), the small (S) and the modulation (N) cells. The ReD trend shows that these cells persist at higher Reynolds numbers with a larger separation between the vortex shedding frequencies fN and fL. At the same time, the downwash POD mode gets less strong with a more intense and localised modulation region which affects a more extended portion of the large cylinder wake.

The simulation of turbulent flows requires high spatial resolution in potentially a priori unknown, solution -dependent locations. To achieve adaptive refinement of the mesh, we rely on error indicators. We assess the differences between an error measure relying on the local convergence properties of the numerical solution and a goal-oriented error measure based on the computation of an adjoint problem. The latter method aims at optimizing the mesh for the calculation of a predefined integral quantity, or functional of interest. This work follows on from a previous study conducted on steady flows in Offermans et al. (2020) and we extend the use of the so-called adjoint error estimator to three-dimensional, turbulent flows. They both represent a way to achieve error control and automatic mesh refinement (AMR) for the numerical approximation of the Navier-Stokes equations, with a spectral element method discretization and non-conforming h-refinement.The current study consists of running the same physical flow case on gradually finer meshes, starting from a coarse initial grid, and to compare the results and mesh refinement patterns when using both error measures. As a flow case, we consider the turbulent flow in a constricted, periodic channel, also known as the periodic hill flow, at four different Reynolds numbers: Re = 700, Re = 1400, Re = 2800 and Re = 5600. Our results show that both error measures allow for effective control of the error, but they adjust the mesh differently. Well-resolved simulations are achieved by automatically focusing refinement on the most critical regions of the domain, while significant saving in the overall number of elements is attained, compared to statically generated meshes. At all Reynolds numbers, we show that relevant physical quantities, such as mean velocity profiles and reattachment/separation points, converge well to reference literature data. At the highest Reynolds number achieved (Re = 5600), relevant quantities, i.e. reattachment and separation locations, are estimated with the same level of accuracy as the reference data while only using one-third of the degrees of freedom of the reference. Moreover, we observe distinct mesh refinement patterns for both error measures. With the spectral error indicator, the mesh resolution is more uniform and turbulent structures are more resolved within the whole domain. On the other hand, the adjoint error estimator tends to focus the refinement within a localized zone in the domain, dependent on the functional of interest, leaving large parts of the domain marginally resolved.

The global stability of the flow in a spatially developing 180∘-bend pipe with curvature δ=R/Rc=1/3 is investigated by performing direct numerical simulations to understand the underlying transitional mechanism. A unique application of the adaptive mesh refinement technique is used during the stability analysis for minimizing the interpolation and quadrature errors. Independent meshes are created for the direct and adjoint solutions, as well as for the base flow extracted via selective frequency damping. The spectrum of the linearized Navier-Stokes operator reveals a pair of complex conjugate eigenvalues, with frequency f≈0.233. Therefore, the transition is attributed to a Hopf bifurcation that takes place at Reb,cr=2528. A structural sensitivity analysis is performed by extracting the wavemaker. We identify the primary source of instability located on the outer wall, θ≈15 downstream of the bend inlet. This region corresponds to the separation bubble on the outer wall. We thus conclude that the instability is caused by the strong shear resulting from the backflow, similar to the 90-bend pipe flow. We believe that understanding the stability mechanism and characterizing the base flow in bent pipes is crucial for studying various biological flows, like blood vessels. Hence, this paper aims to close the knowledge gap between a 90-bend and toroidal pipes by investigating the transition nature in a 18-bend pipe flow.