The Reynolds number dependent flow resistance of heterogeneous rough surfaces is largely unknown at present. The present work provides novel reference data for spanwise-alternating sandpaper strips as one idealised case of a heterogeneous rough surface. Experimental data are presented and analysed in direct comparison with drag measurements of homogeneous sandpaper surfaces and numerical simulations. Based on the homogeneous roughness data, the related challenges and sensitivities for the evaluation of roughness functions from experiments and simulations are discussed. A hydraulic channel height is suggested as an alternative measure for the drag impact of rough surfaces in internal flows. For the investigated heterogeneous roughness, it is found that turbulent flow does not exhibit a fully rough flow behaviour, indicating that the assignment of an equivalent sand grain height as commonly applied for homogeneous roughness is not possible. A prediction of the drag behaviour of rough strips based on an average between rough and smooth drag curves appears promising, but requires further refinement to capture the impact of turbulent secondary flows and spatial transients linking smooth and rough surface parts. While turbulent secondary flow induced by the roughness strips yield significant spanwise variation of the mean velocity profile for the investigated rough strips, we show that the spanwise averaged velocity profiles collapse reasonably well with a smooth or homogeneous rough wall flow. This allows to extract a global roughness function from the spanwise averaged flow field in good agreement with the one deduced from global pressure drop measurements.

Despite decades of research, a universal method for prediction of roughness-induced skin friction in a turbulent flow over an arbitrary rough surface is still elusive. The purpose of the present work is to examine two possibilities; first, predicting equivalent sand-grain roughness size $k_s$ based on the roughness height probability density function and power spectrum (PS) leveraging machine learning as a regression tool; and second, extracting information about relevance of different roughness scales to skin-friction drag by interpreting the output of the trained data-driven model. The model is an ensemble neural network (ENN) consisting of 50 deep neural networks. The data for the training of the model are obtained from direct numerical simulations (DNS) of turbulent flow in plane channels over 85 irregular multi-scale roughness samples at friction Reynolds number $Re_\tau =800$. The 85 roughness samples are selected from a repository of 4200 samples, covering a wide parameter space, through an active learning (AL) framework. The selection is made in several iterations, based on the informativeness of samples in the repository, quantified by the variance of ENN predictions. This AL framework aims to maximize the generalizability of the predictions with a certain amount of data. This is examined using three different testing data sets with different types of roughness, including 21 surfaces from the literature. The model yields overall mean error 5 %–10 % on different testing data sets. Subsequently, a data interpretation technique, known as layer-wise relevance propagation, is applied to measure the contributions of different roughness wavelengths to the predicted $k_s$. High-pass filtering is then applied to the roughness PS to exclude the wavenumbers identified as drag-irrelevant. The filtered rough surfaces are investigated using DNS, and it is demonstrated that despite significant impact of filtering on the roughness topographical appearance and statistics, the skin-friction coefficient of the original roughness is preserved successfully.

We adapt the classical definition of locally stationary processes in discrete time (see e.g. Dahlhaus, ‘Locally stationary processes’, in Time Series Analysis: Methods and Applications (2012)) to the continuous-time setting and obtain equivalent representations in the time and frequency domains. From this, a unique time-varying spectral density is derived using the Wigner–Ville spectrum. As an example, we investigate time-varying Lévy-driven state space processes, including the class of time-varying Lévy-driven CARMA processes. First, the connection between these two classes of processes is examined. Considering a sequence of time-varying Lévy-driven state space processes, we then give sufficient conditions on the coefficient functions that ensure local stationarity with respect to the given definition.

Direct numerical simulations (DNS) are used to systematically investigate the applicability of the minimal-channel approach (Chung et al., J. Fluid Mech., vol. 773, 2015, pp. 418–431) for the characterization of roughness-induced drag on irregular rough surfaces. Roughness is generated mathematically using a random algorithm, in which the power spectrum (PS) and probability density function (p.d.f.) of the surface height can be prescribed. Twelve different combinations of PS and p.d.f. are examined, and both transitionally and fully rough regimes are investigated (roughness height varies in the range $k^+ = 25$–100). It is demonstrated that both the roughness function (${\rm \Delta} U^+$) and the zero-plane displacement can be predicted with ${\pm }5\,\%$ accuracy using DNS in properly sized minimal channels. Notably, when reducing the domain size, the predictions remain accurate as long as 90 % of the roughness height variance is retained. Additionally, examining the results obtained from different random realizations of roughness shows that a fixed combination of p.d.f. and PS leads to a nearly unique ${\rm \Delta} U^+$ for deterministically different surface topographies. In addition to the global flow properties, the distribution of time-averaged surface force exerted by the roughness onto the fluid is calculated. It is shown that patterns of surface force distribution over irregular roughness can be well captured when the sheltering effect is taken into account. This is made possible by applying the sheltering model of Yang et al. (J. Fluid Mech., vol. 789, 2016, pp. 127–165) to each specific roughness topography. Furthermore, an analysis of the coherence function between the roughness height and the surface force distributions reveals that the coherence drops at larger streamwise wavelengths, which can be an indication that very large horizontal scales contribute less to the skin-friction drag.

Recent developments in neural networks have shown the potential of estimating drag on irregular rough surfaces. Nevertheless, the difficulty of obtaining a large high-fidelity dataset to train neural networks is deterring their use in practical applications. In this study, we propose a transfer learning framework to model the drag on irregular rough surfaces even with a limited amount of direct numerical simulations. We show that transfer learning of empirical correlations, reported in the literature, can significantly improve the performance of neural networks for drag prediction. This is because empirical correlations include ‘approximate knowledge’ of the drag dependency in high-fidelity physics. The ‘approximate knowledge’ allows neural networks to learn the surface statistics known to affect drag more efficiently. The developed framework can be applied to applications where acquiring a large dataset is difficult but empirical correlations have been reported.

Inhomogeneous rough surfaces in which strips of roughness alternate with smooth-wall strips are known to generate large-scale secondary motions. Those secondary motions are strongest if the strip width is of the order of the half-channel height and they generate a spatial wall shear stress distribution whose mean value can significantly exceed the area-averaged mean value of a homogeneously smooth and rough surface. In the present paper it is shown that a parametric forcing approach (Busse & Sandham, J. Fluid Mech., vol. 712, 2012, pp. 169–202; Forooghi et al., Intl J. Heat Fluid Flow, vol. 71, 2018, pp. 200–209), calibrated with data from turbulent channel flows over homogeneous roughness, can capture the topological features of the secondary motion over protruding and recessed roughness strips (Stroh et al., J. Fluid Mech., vol. 885, 2020, R5). However, the results suggest that the parametric forcing approach roughness model induces a slightly larger wall offset when applied to the present heterogeneous rough-wall conditions. Contrary to roughness-resolving simulations, where a significantly higher resolution is required to capture roughness geometry, the parametric forcing approach can be applied with usual smooth-wall direct numerical simulation resolution resulting in less computationally expensive simulations for the study of localized roughness effects. Such roughness model simulations are employed to systematically investigate the effect of the relative roughness protrusion on the physical mechanism of secondary flow formation and the related drag increase. It is found that strong secondary motions present over spanwise heterogeneous roughness with geometrical height difference generally lead to a drag increase. However, the physical mechanism guiding the secondary flow formation, and the resulting secondary flow topology, is different for protruding roughness strips and recessed roughness strips separated by protruding smooth surface strips.

We present data from direct numerical simulations of flow through channels containing large, longitudinal, surface-mounted, rectangular ribs at various spanwise spacings, which lead to secondary flows. It is shown that appropriate modifications to the classical log-law, predicated on a greater wetted surface area than in a plane channel, lead to a log-law-like region in the spanwise-averaged axial mean velocity profiles, even though local profiles may be very different. The secondary flows resulting from the presence of the ribs are examined and their effects discussed. Comparing our results with the literature we conclude that the sense of the secondary flows is largely independent of the particular rib spacing whether normalised by channel depth or rib width. The strength of the secondary flows, however, is shown to depend on the ratio of rib spacing to rib width and on Reynolds number. Topological features of the secondary flow structure are illustrated via a critical point analysis and shown to be characterised in all cases by a free stagnation point above the centre of the rib. Finally, we show that if the domain size is chosen as a ‘minimal channel’ size, rather than a size which allows adequate development of the usual outer layer flow structures, the secondary flows can be affected and this leads inevitably to differences in the near-rib flows so that for ribbed channels, unlike plain channels, it is unwise to use minimal domains to identify details of the near-wall flow.

The history and etymology of Old Scandinavian hinn is a disputed matter. One question concerns whether hinn as a contrastive demonstrative indicating ‘the other (one)/the former (one)’ and hinn as a pre-adjectival article, both of which to some extent are still found in present-day Icelandic, are related or not. Another issue concerns the fact that hinn has no immediate parallel in Germanic outside Scandinavia, which has led scholars to assume that it is a Proto-Scandinavian innovation. This paper argues that Old Scandinavian possessed two hinn words with separate backgrounds, one stemming directly from an anciently inherited distal demonstrative, and one from an innovated proximal demonstrative. However, the innovation was no more founded on common Germanic material than the former hinn was. Instead, it arose from the reinforcement of an ancient precursor. This precursor is traceable in early Icelandic enn, which was used as a pre-adjectival article and as a primitive post-nominal definiteness marker.

Turbulent flow over a surface with streamwise-elongated rough and smooth stripes is studied by means of direct numerical simulation (DNS) in a periodic plane open channel with fully resolved roughness. The goal is to understand how the mean height of roughness affects the characteristics of the secondary flow formed above a spanwise heterogeneous rough surface. To this end, while the statistical properties of roughness texture as well as the width and spacing of the rough stripes are kept constant, the elevation of the smooth stripes is systematically varied in different simulation cases. Utilizing this variation, three configurations – representing protruding, recessed and an intermediate type of roughness – are analysed. In all cases, secondary flows are present and the skin friction coefficients calculated for all the heterogeneous rough surfaces are meaningfully larger than what would result from the area-weighted average of those of homogeneous smooth and rough surfaces. This drag increase appears to be linked to the strength of the secondary flow. The rotational direction of the secondary motion is shown to depend on the relative surface elevation. The present results suggest that this rearrangement of the secondary flow is linked to the spatial distribution of the spanwise-wall-normal Reynolds stress component, which carries opposing signs for protruding and recessed roughness.

The Bulge Asymmetries and Dynamical Evolution (BAaDE) survey aims to explore the complex structure of the inner Galaxy and Galactic Bulge, by using the 43 GHz receivers at the Karl G. Jansky Very Large Array (VLA) and the 86 GHz receivers at the Atacama Large Millimeter/submillimeter Array (ALMA) to observe SiO maser lines in red giant stars. The goal is to construct a sample of stellar point-mass probes that can be used to test models of the gravitational potential, and the final sample is expected to provide at least 20,000 line-of-sight velocities and positions. A possible bias between the VLA and the ALMA SiO maser lines is explored, and the 86 GHz SiO line-peak velocities agree using either of the four sampled lines. Additionally, the SiO maser velocities agree with the OH maser derived velocities.

The BAaDE (Bulge Asymmetries and Dynamical Evolution) project is an SiO maser survey of the Galactic Plane. About 19,000 sources have been observed at 43 GHz with the VLA, and the production of spectra for each of these sources is well underway. The primary goal of the project is to collect line-of-sight velocities for all the detected masers in the sample to probe Galactic dynamics. With an expected detection rate of over 60% we should collect over 11,000 velocities to probe the Galactic potential. The survey is also a large sample of infrared sources to explore the different evolved stellar populations within the Milky Way. So far we discern three distinct groups in the BAaDE sample: the main group containing oxygen-rich, evolved stars with a high SiO maser detection rate, a much smaller population of carbon-rich evolved stars, and finally a group of likely young stellar objects with no maser emission. These populations are separated using 2MASS and MSX color-color diagrams, and we find a particularly useful cut between the young and evolved objects using the MSX [D] –[E] color. Identification of these populations will isolate BAaDE’s evolved star sample, and will more tightly define the region in IR color-color diagrams where SiO masers occur yielding a better understanding of these kinematical probes. Using our color-divisions we can also study the distribution of each of the populations within the Galactic Plane.

The Bulge Asymmetries and Dynamical Evolution (BAaDE) survey aims to use circumstellar SiO maser line-of-sight velocities as probes for the Galactic gravitational potential and dynamical structure. The SiO masers are detected at a high rate in specific color-selected MSX infrared sources. Furthermore, the SiO maser properties and line ratios, in combination with infrared spectral energy distributions and location in the Galaxy, will statistically yield detailed information on population and evolution of low- to intermediate-mass evolved stars in the Galaxy.

Secondary flows can develop in turbulent boundary layers that grow over surfaces with spanwise inhomogeneities. In this article, we demonstrate the formation of secondary flows in both experimental and numerical tests and dissect the instantaneous structure and topology of these secondary motions. We show that the formation of secondary flows is not very sensitive to the Reynolds number range investigated, and direct numerical simulations and experiments produce similar results in the mean flow as well as the dispersive and turbulent stress distributions. The numerical methods capture time-resolved features of the instantaneous flow and provide insight into the near-wall flow structures, that were previously obscured in the experimental measurements. Proper orthogonal decomposition was shown to capture the essence of the secondary flows in relatively few modes and to be useful as a filter to analyse the instantaneous flow patterns. The secondary flows are found to create extended regions of high Reynolds stress away from the wall that comprise predominantly sweeps similar to what one would expect to see near the wall and which are comparable in magnitude to the near-wall stress. Analysis of the instantaneous flow patterns reveals that the secondary flows are the result of a non-homogeneous distribution of mid-size vortices.

We study several kinds of subschemes of mixed characteristic models of Shimura varieties which admit good (partial) toroidal and minimal compactifications, with familiar boundary stratifications and formal local structures, as if they were Shimura varieties in characteristic zero. We also generalize Koecher’s principle and the relative vanishing of subcanonical extensions for coherent sheaves, and Pink’s and Morel’s formulas for étale sheaves, to the context of such subschemes.

We report on the Bulge Asymmetries and Dynamic Evolution (BAaDE) survey which has observed 19 000 MSX color selected red giant stars for SiO maser emission at 43 GHz with the VLA and is in the process of observing 9 000 of these stars with ALMA at 86 GHz in the Southern sky. Our setup covers the main maser transitions, as well as those of isotopologues and selected lines of carbon-bearing species. Observations of this set of lines allow a far-reaching catalog of line-of-sight velocities in the dust-obscured regions where optical surveys cannot reach. Our preliminary detection rate is close to 70%, predicting a wealth of new information on the distribution of metal rich stars, their kinematics as function of location in the Galaxy, as well as the occurrence of lines and line ratios between the different transitions in combination with the spectral energy distribution from about 1 to 100 μm. Similar to the OH/IR stars, a clear kinematic signature between disk and bulge stars can be seen. Furthermore, the SiO J = →10 (v=3) line plays a prominent role in the derived maser properties.

We show that the automorphic étale cohomology of a (possibly noncompact) PEL-type or Hodge-type Shimura variety in characteristic zero is canonically isomorphic to the cohomology of the associated nearby cycles over most of their mixed characteristics models constructed in the literature.

Using quasi-simultaneous observations of 86 stars with known SiO maser emission, we searched for systematic differences between the strengths of the 43 and 86 GHz v=1 maser lines. Although for individual stars there is wide scatter between the line strengths spanning nearly an order of magnitude, there is no evidence of a systematic difference between these line strengths for the entire sample.