In this Letter we show that a bifurcation cascade and fully sustained turbulence can share the phase space of a fluid flow system, resulting in the presence of competing stable attractors. We analyze the toroidal pipe flow, which undergoes subcritical transition to turbulence at low pipe curvatures (pipe-to-torus diameter ratio) and supercritical transition at high curvatures, as was previously documented. We unveil an additional step in the bifurcation cascade and provide evidence that, in a narrow range of intermediate curvatures, its dynamics competes with that of sustained turbulence emerging through subcritical transition mechanisms.

A new experimental database of adverse-pressure-gradient (APG) turbulent boundary layers (TBLs) obtained through hot-wire anemometry and oil-film interferometry covering a momentum–loss Reynolds number 450<Reθ<23450 and Clauser pressure-gradient-parameter range up to β≈2.4 is presented. Both increasing and approximately constant β distributions with the same upstream history are characterised. Turbulence statistics are compared among the different pressure-gradient distributions with additional numerical and experimental zero-pressure-gradient (ZPG) TBL data. Cases at approximately constant β, which can be considered as canonical representations of the boundary layer under a certain pressure-gradient magnitude, exhibit skin-friction and shape-factor curves consistent with the ones proposed by Vinuesa et al. (2017). These curves show a similar scaling behaviour as those proposed by Nagib et al. (2007) for ZPG TBLs. The pre-multiplied power-spectral density is employed to study the differences in the large-scale energy content throughout the boundary layer. Two different large-scale phenomena are identified, the first one related to the pressure gradient and the second one (also present in high-Re ZPG TBLs) due to the Reynolds number. Recently proposed scaling laws by Kitsios et al. (2016) and Maciel et al. (2018) are tested over a wider Reynolds-number range and for different β cases. The mean velocity and streamwise velocity fluctuation profiles are found to be dependent on the upstream development. The mean velocity profile is found to be self-similar only in the outer region, in agreement with classical theory. The mean and higher-order statistics of the new APG TBL database are made available under www.flow.kth.se.

The present investigation aims at finding out whether there are pipe vibrations in the higher Reynolds number range at the Long Pipe Facility at the CICLoPE facility and to quantify their amplitude and frequency. Since vibrations are natural to any wind-tunnel facility, similar vibration measurements have also been performed in an established high-quality wind-tunnel facility, viz. the Minimum Turbulence Level (MTL) wind tunnel at KTH Royal Institute of Technology, in order to provide reference data. Results affirm that the amplitudes observed in Willert et al. (J Fluid Mech 826, 2017) are most likely the result of an amplification due to the optical set-up that is attached to the window plug, rather than vibrations of the pipe structure.

Very-large-scale structures in pipe flows are characterized using an extended Proper Orthogonal Decomposition (POD)-based estimation. Synchronized non-time-resolved Particle Image Velocimetry (PIV) and time-resolved, multi-point hot-wire measurements are integrated for the estimation of turbulent structures in a pipe flow at friction Reynolds numbers of 9500 and 20000. This technique enhances the temporal resolution of PIV, thus providing a time-resolved description of the dynamics of the large-scale motions. The experiments are carried out in the CICLoPE facility. A novel criterion for the statistical characterization of the large-scale motions is introduced, based on the time-resolved dynamically-estimated POD time coefficients. It is shown that high-momentum events are less persistent than low-momentum events, and tend to occur closer to the wall. These differences are further enhanced with increasing Reynolds number.

Extensive combined experimental and theoretical investigations of the linear evolution of three-dimensional (3D) Tollmien-Schlichting (TS) instability modes of 3D boundary layers developing on a swept airfoil section have been carried out. The flow under consideration is the boundary layer over an airfoil at 350 sweep and an angle of attack of +1.5 degrees. At these conditions, TS instability is found to be the predominant one. Perturbations with different frequencies and spanwise wavenumbers are generated in a controlled way using a row of elastic membranes. All experimental results are deeply processed and compared with results of calculations based on theoretical approaches. Very good quantitative agreement of all measured and calculated stability characteristics of swept-wing boundary layers is achieved.

Extensive combined experimental and theoretical investigations of the linear evolution of unsteady (in general) Cross-Flow (CF) and three-dimensional (3D) Tollmien-Schlichting (TS) instability modes of 3D boundary layers developing on a swept airfoil section have been carried out. CF-instability characteristics are investigated in detail at an angle of attack of -5 degrees when this kind of instability dominates in the laminar-turbulent transition process, while the 3D TS-instability characteristics are studied at an angle of attack of +1.5 degrees when this kind of instability is predominant in the transition process. All experimental results are deeply processed and compared with results of calculations based on several theoretical approaches. For the first time, very good quantitative agreement of all measured and calculated stability characteristics of swept-wing boundary layers is achieved both for unsteady CF- and 3D TS-instability modes for the case of a boundary layer developing on a real swept airfoil. The first part of the present study (this paper) is devoted to the description of the case of CF-dominated transition, while the TS-dominated case will be described in detail in a subsequent second part of this investigation.

The effect of a streamwise pressure gradient on the wake developed by wall-attached square ribs in a turbulent boundary layer is investigated experimentally. Favourable-, adverse- and zero-pressure-gradient conditions (FPG, APG and ZPG, respectively) are reproduced at matched friction Reynolds number and non-dimensional rib height. Flow-field measurements are carried out by means of Particle Image Velocimetry (PIV). Turbulence statistics are extracted at high resolution using an Ensemble Particle Tracking Velocimetry approach. Modal analysis is performed with Proper Orthogonal Decomposition (POD). We demonstrate that a non-dimensional expression of the pressure gradient and shear stress is needed to quantify the pressure-gradient effects in the wake developing past wall-attached ribs. We suggest the Clauser pressure-gradient parameter beta, commonly used in the literature for the characterization of turbulent boundary layers under the effect of a pressure gradient, as a suitable parameter. The results show that, in presence of an adverse pressure gradient, the recirculation region downstream of the rib is increased in size, thus delaying the reattachment, and that the peak of turbulence intensity and the shed eddies are shifted towards larger wall-normal distances than in the ZPG case. The observed changes with respect to the ZPG configuration appear more intense for larger magnitude of beta, which are more likely to be obtained in APG than in FPG due to the reduced skin friction and increased displacement thickness.

In this work the modal decomposition of convective heat transfer distributions in turbulent flows is explored. The organization and thermal footprint of the turbulent flow features generated downstream of wall-proximity two-dimensional square ribs immersed in a turbulent boundary layer are investigated experimentally. This study employs modal decomposition to investigate whether this analysis can allow identifying which characteristics of the flow topology are responsible for the Nusselt-number augmentation, aiming to uncover the underlying physics of heat-transfer enhancement. Heat transfer and flow velocity measurements are performed at a Reynolds number (based on the free-stream velocity and rib side-length) equal to 4600. Square ribs are tested for two different gap spacings from the wall (0.25 and 0.5 ribs side-length) and in wall-attached configuration. A low-thermal-inertia heat transfer sensor coupled with high-repetition-rate Infrared (IR) thermography is designed to study the unsteady variation of the convective heat-transfer coefficient downstream of the obstacles. Flow-field measurements are performed with non-time-resolved Particle Image Velocimetry (PIV). A modal analysis with Proper Orthogonal Decomposition (POD) is applied to both convective heat-transfer maps and velocity-fields. The comparison of the Nusselt-number spatial modes of the clean turbulent boundary layer configuration and of the configurations with the ribs shows a variation of the spatial pattern associated with oscillations with strong spanwise coherence, opposed to the thin elongated streaks which dominate the convective heat transfer in the clean turbulent boundary layer. In configurations where the convective heat transfer is enhanced by coherent structures located close to the wall, similar eigenspectra are observed for both flow field and convective heat transfer modes. The results of the modal analysis support a picture of a direct relation between the coherence of near-wall flow features and heat-transfer augmentation, providing a statistical evidence for the fact that near-wall coherent eddies are extremely efficient in enhancing heat transfer.

When the Organizing Commitee of CADE began to choose the program of CADE-92, it was decided that D-modules would be a central topic at this conference.

The theory of D-modules is quite recent. It began in the late sixties and at first was considered to be quite abstract and difficult. Over the years the situation improved with the development of the theory and its applications. The organizers felt that it was time to try to introduce it to a larger audience interested in differential equations and computer algebra, since the theory of D-modules offers an excellent way to effectively handle linear systems of analytic PDEs.

Once this decision was made it was natural to ask Bernard Malgrange to be the “invité d'honneur” at CADE-92, with the task of lecturing about D-modules in a way adapted to an audience interested in effectivity. This was natural because Bernard Malgrange is not only one of the most famous mathematicians in this field, but also because he is perhaps the true originator of this direction. It is generally admitted that D-module theory began in the early seventies with the fundamental work of I. N. Berstein and of the Japanese school around M. Sato, but in fact Bernard Malgrange introduced the basic concepts ten years ago for the constant coefficients case (see his 1962 Bourbaki report “systèmes différentiels à coefficients constants”), and later for the general case (see his lectures at Orsay Cohomologie de Spencer (d'après Quillen)).

Many recent investigations on the scale interactions in wall-bounded turbulent flows focus on describing so-called amplitude modulation, the phenomenon that deals with the influence of large scales in the outer region on the amplitude of the small-scale fluctuations in the near-wall region. The present study revisits this phenomenon regarding two aspects, namely the method for decomposing the scales and the quantification of the modulation. First, the paper presents a summary of the literature that has dealt with either or both aspects. Second, for decomposing the scales, different spectral filters (temporal, spatial or both) and empirical mode decomposition (EMD) are evaluated and compared. The common data set is a well-resolved large-eddy simulation that offers a wide range of Reynolds numbers spanning Re-theta = 880-8200. The quantification of the amplitude modulation is discussed for the resulting scale components. Particular focus is given to evaluate the efficacy of the various filters to separate scales for the range of Reynolds numbers of interest. Different to previous studies, the different methods have been evaluated using the same data set, thereby allowing a fair comparison between the various approaches. It is observed that using a spectral filter in the spanwise direction is an effective approach to separate the small and large scales in the flow, even at comparably low Reynolds numbers, whereas filtering in time should be approached with caution in the low-to-moderate Re range. Additionally, using filters in both spanwise and time directions, which would separate both wide and long-living structures from the small and fast scales, gives a cleaner image for the small-scales although the contribution to the scales interaction from that filter implementation has been found negligible. Applying EMD to decompose the scales gives similar results to Fourier filters for the energy content of the scales and thereby for the quantification of the amplitude modulation using the decomposed scales. No direct advantage of EMD over classical Fourier filters could be seen. Potential issues regarding different decomposition methods and different definitions of the amplitude modulation are also discussed.