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.

Information theory (IT) provides tools to estimate causality between events, in various scientific domains. Here, we explore the potential of IT-based causality estimation in turbulent (i.e. chaotic) dynamical systems and investigate the impact of various hyperparameters on the outcomes. The influence of Markovian orders, i.e. the time lags, on the computation of the transfer entropy (TE) has been mostly overlooked in the literature. We show that the history effect remarkably affects the TE estimation, especially for turbulent signals. In a turbulent channel flow, we compare the TE with standard measures such as auto- and cross-correlation, showing that the TE has a dominant direction, i.e. from the walls towards the core of the flow. In addition, we found that, in generic low-order vector auto-regressive models (VAR), the causality time scale is determined from the order of the VAR, rather than the integral time scale. Eventually, we propose a novel application of TE as a sensitivity measure for controlling computational errors in numerical simulations with adaptive mesh refinement. The introduced indicator is fully data-driven, no solution of adjoint equations is required, with an improved convergence to the accurate function of interest. In summary, we demonstrate the potential of TE for turbulence, where other measures may only provide partial information.

We investigate the discontinuities arising at non-conforming (or non-conformal) interfaces in spectral element method (SEM) simulations. The derivate terms are by definition discontinuous and interface instabilities are usually not visible with a conformal mesh and sufficient resolution. Using the adaptive mesh refinement (AMR) technique the initial coarse mesh is progressively refined according to an error indicator or estimator. In our case, the spectral error indicator (SEI) is adopted. This leads to non-conformal interfaces, where hanging nodes are introduced through h-refinement implemented in the SEM code Nek5000. We consider the flow in a three-dimensional periodic straight pipe and use the turbulent kinetic energy budget as an indicator for assessing discontinuities (wiggles). They involve first and second-order derivatives and represent a fixed point in the statistical analysis of this canonical flow. Looking at the results, we observe that our AMR implementation does not affect the interface discontinuities. The jumps in derivatives are uniquely related to an inadequately resolved mesh. Relying on an error-driven approach, the SEI produces a mesh that allows computing the TKE budgets in excellent agreement with the literature and ensures saving in grid points by a factor of 2.

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.

The turbulent external flow around a three-dimensional stepped cylinder is studied by means of direct numerical simulations with the adaptive mesh refinement technique. We give a broad perspective of the flow regimes from laminar to turbulent wake at, which is the highest ever considered for this flow case. In particular, we focus on the intermediate Reynolds number that reveals a turbulent wake coupled with a stable cylinder shear layer (subcritical regime). This flow shows a junction dynamics similar to the laminar, where no hairpin vortex appears around the edges, and just two horseshoe vortices are visible. A new stable vortex in the form of a ring, which coils around the rear area, is also identified. In the turbulent wake, the presence of three wake cells is pointed out: the large and small cylinder cells together with the modulation region. However, the modulation dynamics varies between the subcritical and turbulent regimes. A time-averaged, three-dimensional set of statistics is computed, and spatially coherent structures are extracted via proper orthogonal decomposition (POD). The POD identifies the (long-debated) connection between the N-cell and the downwash behind the junction. Furthermore, as the Reynolds number increases, the downwash phenomenon becomes less prominent. Eventually, a reduced-order reconstruction with the most energetically relevant modes is defined to explain the wake vortex interactions. This also serves as a valuable starting point for simulating the stepped cylinder wake behaviour within complex frameworks, e.g. fluid-structure interaction.

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 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.