We investigate spanwise-coherent structures in the turbulent flow around airfoils, motivated by their connection with trailing-edge noise. We analyse well-resolved large-eddy simulations (LES) of the flow around NACA 0012 and NACA 4412 airfoils, both at a Reynolds number of 400 000 based on the chord length. Spectral proper orthogonal decomposition performed on the data reveals that the most energetic coherent structures are hydrodynamic waves, extending over the turbulent boundary layers around the airfoils with significant amplitudes near the trailing edge. Resolvent analysis was used to model such structures, using the mean field as a base flow. We then focus on evaluating the dependence of such structures on the domain size, to ensure that they are not an artefact of periodic boundary conditions in small computational boxes. To this end, we performed incompressible LES of a zero-pressure-gradient turbulent boundary layer, for three different spanwise sizes, with the momentum-thickness Reynolds number matching those near the airfoils trailing edge. The same coherent hydrodynamic waves were observed for the three domains. Such waves are accurately modelled as the most amplified flow response from resolvent analysis. The signature of such wide structures is seen in non-premultiplied spanwise wavenumber spectra, which collapse for the three computational domains. These results suggest that the spanwise-elongated structures are not domain-size dependent for the studied simulations, indicating thus the presence of very wide structures in wall-bounded turbulent flows.

    The characteristics of turbulent boundary layers (TBLs) subjected to adverse pressure gradients are analysed through well-resolved large-eddy simulations. The geometries under study are the NACA0012 and NACA4412 wing sections, at 0 and 5 degrees angle of attack, respectively, both of them at a Reynolds number based on inflow velocity and chord length Re_c = 400,000. The turbulence statistics show that adverse pressure gradients (APGs) have a significant effect on the mean velocity, velocity fluctuations and turbulent kinetic energy budget, and this effect is more prominent on the outer region of the boundary layer. Furthermore, the effect of flow history is assessed by means of an integrated Clauser pressure-gradient parameter, β, through the study of cases with matching local values of β and the friction Reynolds number, Re_τ, to isolate this effect. Our results show a noticeable effect of the flow history on the outer region, however the differences in the near-wall peak of the tangential velocity fluctuations appear to be mostly produced by the local APG magnitude. The one-dimensional power-spectral density shows energetic small scales in the outer region of APG TBLs, whereas these energetic scales do not appear in zero-pressure-gradient (ZPG) TBLs, suggesting that small scales near the wall are advected towards the outer layer by the APG. Moreover, the linear coherence spectra show that the spectral outer peak of high-Reynolds-number ZPG TBLs is highly correlated with the near-wall region , unlike APG TBLs which do not show such a correlation. This result, together with the different two-dimensional spectra of APG and high-Reynolds-number ZPG TBLs, suggests different energisation mechanisms due to APG and increase in Reynolds number. To the authors' knowledge, this is the first in-depth analysis of the TBL characteristics over wings, including detailed single-point statistics, spectra and coherence.

The implementation of adaptive mesh refinement (AMR) in the spectral-element method code Nek5000 is used for the first time on the well-resolved large-eddy  simulation (LES) of the turbulent flow over wings. In particular, the flow over a NACA4412 profile with a 5° angle of attack at chord-based Reynolds number Re_c=200,000 is analysed in the present work. The mesh, starting from a coarse resolution, is progressively refined by means of AMR, which allows for high resolution near the wall and wake whereas significantly larger elements are used in the far-field. The resulting mesh is of higher resolution than those in previous conformal cases, and it allows for the use of larger computational domains, avoiding the use of precursor RANS simulations to determine the boundary conditions. All of this with, approximately, 3 times lower total number of grid points if the same spanwise length is used. Turbulence statistics obtained in the AMR simulation show good agreement with the ones obtained with the conformal mesh. Finally, using AMR on wings will enable simulations at Re_c beyond 1 million, thus allowing the study of pressure-gradient effects at high Reynolds numbers relevant for practical applications.

Turbulent boundary layers (TBLs) subjected to various amplitudes of adverse pressure gradients (APGs) are of paramount importance in the field of fluid dynamics due to their implications in countless industrial applications in which the flow develops on a curved surface such as wings or nozzles, among others.

The implementation of adaptive mesh refinement (AMR) in Nek5000 is used for the first time on the simulation of the flow over wings. This is done by simulating the flow over a NACA4412 profile with 5 degrees angle of attack at chord-based Reynolds number 200,000. The mesh is progressively refined by means of AMR which allows for high resolution near the wall whereas significantly larger elements are used in the far-field. The resultant mesh shows higher resolution than previous conformal meshes, and it allows for larger computational domains,which avoid the use of RANS to determine the boundary condition, all of this with, approximately, 3 times lower total number of grid points. The results ofthe turbulence statistics show a good agreement with the ones obtained with the conformal mesh. Finally, using AMR on wings leads to simulations at higher Reynolds numbers (i.e. Rec = 850, 000) in order to analyse the effect of adverse pressure gradients at high Reynolds numbers.

Motivated by the recent analysis by Sano et al. 2019, Phys. Rev. Fluids, vol. 4, p. 094602, of spanwise-coherent structures in the turbulent flow around airfoils and their connection to trailing-edge noise, we carry out a thorough characterisation of such structures in three simulation databases. We analyse two different numerical simulations of incompressible flow in turbulent regime, both at chord Reynolds number of 400,000: a large-eddy simulation for a NACA 0012 profile at zero angle of attack, and a direct numerical simulation for a NACA 4412 airfoil with an angle of attack of 5 degrees. Snapshots of the flow field were analysed using Spectral Proper Orthogonal Decomposition (SPOD), in order to extract the dominant coherent structures of the flow. Focus is given to  the aforementioned spanwise-coherent fluctuations, which two-dimensional disturbances in the computational domain due to the use of periodic boundary conditions. The leading SPOD modes show that the most energetic coherent structures are wavepackets, extending over the whole turbulent boundary layers around the airfoils with significant amplitudes near the trailing-edge. Higher amplitudes are observed in the region of  stronger adverse pressure gradient at the suction side of the NACA 4412 airfoil. To understand how such structures in the turbulent field can be modelled, the linear response of the flow using the singular value decomposition of the linearised resolvent operator was performed, using the mean field as a base flow and considering a locally parallel approximation. Such analysis shows that the leading SPOD modes can be associated to optimal, linearised flow responses, particularly for stations far from the trailing edge; the latter introduces a discontinuity in boundary conditions, and the locally parallel approximation becomes questionable. We then focus on evaluating the dependence of such wavepackets on the domain size, to ensure that these structures are not an artifact of the use of periodic boundary conditions in small computational boxes. To do so, we performed an incompressible LES of a zero-pressure gradient turbulent boundary layer (ZPGTBL), for three different spanwise sizes: L_z=32 δ^*, L_z=64 δ^* and L_z=128 δ^*, where δ^* is a reference displacement thickness in a region of developed turbulent flow, with Reynolds number matching the values in the airfoil simulations. The signature of such wavepackets is seen in non-premultiplied spanwise wavenumber spectra, which reaches, for the three domain sizes, a plateau for spanwise wavelengths going to infinity (or wavenumbers going to zero); this plateau is representative of the spanwise-coherent structures seen in the airfoil simulations. Similar SPOD and resolvent analyses were carried out for the zero spanwise wavenumber of the ZPGTBL, and the same coherent wavepackets were observed for the three domains, with very similar amplitudes. Such wavepackets were also accurately modelled using the optimal resolvent response. These results confirm that the spanwise-elongated structures are not domain-size dependent for the studied simulations, and are thus a feature of turbulent boundary layers.