Spontaneous discrete precipitation pattern formation directed by the coupling of solution permeation and solvent evaporation.

Functionally Graded Materials (FGMs) represent a highly advanced solution for the repair of cracked aircraft structures, offering superior performance compared to conventional homogeneous materials. FGMs contribute to more durable, lightweight, and structurally effective repairs, extending service life while maintaining safety and performance standards. This study investigates the effectiveness of functionally graded material (FGM) patches for reinforcing cracked aluminum alloy plates. Using three-dimensional finite element analysis implemented in ABAQUS with a USDFLD subroutine, spatial variations of the Young's modulus were applied according to linear, logarithmic, and exponential gradation laws. The fracture behavior was quantified through J-integral calculations under plane strain conditions. Results show that FGM patches significantly reduce the J-integral at the crack tip compared to unrepaired plates: up to 56% reduction for linear gradation, 52% for logarithmic, and 44% for exponential gradation. Linear gradation is most effective in mitigating longitudinal crack propagation, while logarithmic gradation efficiently reduces local principal stresses. Contour convergence and comparison with homogeneous patch and unrepaired plate benchmarks confirm the reliability of the numerical results. These findings provide mechanistic insight into stiffness-gradient-driven stress redistribution, guiding the optimal design of FGM repairs for enhanced durability.

Bed roughness plays an important role in controlling flow resistance, turbulence and energy dissipation in open-channel flows, particularly in high-velocity hydraulic structures such as spillways. The influence of wart-type roughness elements on supercritical open-channel flow dynamics was investigated through controlled flume experiments, comparing smooth and rough ( S = 20 cm spacing) bed conditions. Velocity profiles, shear stress, and specific energy distributions were measured along the channel using point velocity measurements in a 7 m long, 0.5 m wide Plexiglas flume with a 2% slope at the Ujigawa Hydraulic Laboratory, Kyoto University. Results show that the rough configuration significantly reduces mean velocities compared to the smooth bed, with pronounced velocity deficits near the bed. Wall shear stress ( τ ) in the smooth case ranged from 2.92 to 3.04 Pa, showing a slight Reynolds number (Re) dependence, while the rough case exhibited lower τ (1.88–2.28 Pa) with greater variability, suggesting reduced drag due to flow separation. Non-dimensional shear stress ( τ/(ρv 2 )) was nearly constant for the smooth bed and fully constant for the rough bed, indicating a transition to a fully rough regime. The logarithmic law of the wall was validated, with a consistent von Kármán constant ( κ = 0.41) and reduced intercept ( A = 5.09 vs. 5.5) for the rough bed. Specific energy distributions revealed enhanced dissipation near the rough bed, impacting hydraulic efficiency. These findings are limited to S/K = 5.71 and Fr > 1.8, but highlight the potential of this specific wart-type roughness for energy dissipation and erosion protection in high-velocity structures.

Estimating in a stream’s cross-section the depth-averaged velocity, V, from the free-surface velocity, vsurf, is an efficient, non-invasive hydrometric method. The ratio fv = V/vsurf is typically assumed constant at fv = 0.86 in field applications, despite observations to the contrary. Guidance is, therefore, needed in estimating actual fv-ratios when velocity profile data are absent. This work provides field-verified guidance based on the hydromechanics of the logarithmic velocity law, which shows that fv depends on the scaled resistance measure ‘friction length/depth’, yo/h, with the yo(k) function of the equivalent sand grain roughness, k. The mean-to-surface-velocity ratio in rough-bed streams is estimated from the bed roughness and stream morphology by modifying Nikuradze’s equation, yo = k/30, to yo = ck, with c(h/k) ≥ 1/30, and k ≈ D84—data fit: c ≈ 8.61(h/k)−1.821, ~5 ≤ h/k < ~30. Field-verification of the ratio’s modified hydromechanics, fv = fh/yo, with yo(h/k) evaluated from bed roughness estimated by inspection or sieve analysis shows this ratio holding within ~|10|% error for shallow streamflow over a coarse bed of gravels and rocks, giving submergences of ~5 ≤ h/D84 ≤ ~30; yo = k/30 suits large streams with smooth beds (h/k ≥ ~30, fv ≥ ~0.86). Variable roughness-estimated fv-ratios appear to be more reliable than the fixed default, fv(h/yo ≈ 1000) = 0.86. This flow-gauging concept is based on observable physical characteristics of a monitoring cross-section and facilitates the rating of hard-to-access streams draining small basins in ragged upland terrain.

Evaluation of SoDAR technology and wind profile extrapolation techniques for modern wind energy applications

Direct numerical simulations at a low Mach number (M=0.085) are conducted to investigate the complex interplay between near-wall turbulence and wall pressure fluctuations in curved turbulent boundary layers. The study focuses on two forward-facing chamfered steps with distinct curvatures, each featuring an upstream concave wall blending into a downstream convex wall. Curvature-induced pressure gradients influence Reynolds shear stress distributions and cause the mean velocity profile to deviate from the classical logarithmic law. Building on these fundamental observations of mean flow and turbulence statistics, the curvature effects on wall pressure fluctuations are further examined. Space–time correlations show that the convection velocity of wall pressure fluctuations first increases and then decreases, correlating with pressure gradient variations. A quantitative evaluation of seven recently developed semi-empirical models of wall pressure spectra reveals varying predictive performance across frequency bands, with most exhibiting larger prediction errors in the high-frequency region. Results suggest that accounting only for local pressure gradients is insufficient to accurately model the wall pressure spectra in curved turbulent boundary layers. Instead, boundary-layer parameters capable of more precisely capturing upstream pressure gradient history effects over curved walls with rapid transitions of pressure gradients should be incorporated into the curved wall pressure modeling.

Abstract The mean velocity profile is essential for characterizing near-wall dynamics in turbulent pipe flow and is widely used in engineering applications involving momentum and heat transfer. This study investigates the effects of surface roughness and temperature on the mean velocity profile. Extensive experiments were conducted at the Izmir Katip Celebi University Civil Engineering Flow Loop using water over a range of flow rates, relative roughness values, and temperatures. A numerical model was developed in OpenFOAM using the PIMPLE algorithm, which combines the Pressure Implicit with Splitting of Operators (PISO) and the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE). The simulations were performed using the RNG κ-ɛ turbulence model. The numerical predictions showed good agreement with the experimental results, with an absolute average percentage error (AAPE) of 10.38% for the tested conditions. For smooth pipes, the velocity profile closely matched the classical u + = y + relation in the viscous sublayer and the logarithmic law in the inertial sublayer. For y + > 30, the power-law velocity distribution also fits well, supporting its suitability for modeling fully turbulent regimes. Increased roughness height significantly disrupted the logarithmic region, leading to pronounced deviations in the mean velocity profile. Although both smooth and rough pipes demonstrated similar trends with temperature variation, the roughness Reynolds number had a stronger influence on these deviations. Elevated temperatures caused a noticeable downward shift in the mean velocity profile, particularly in the near-wall region, which was consistent with viscosity-driven changes in wall scaling. Overall, the validated numerical model provides a useful tool for predicting turbulent flow behavior under the investigated conditions.

In this study, we propose a delayed sums method to investigate the convergence rates of partial sums. This approach enables general and systematic treatment of the convergence behavior of partial sums, encompassing and extending classical results such as the law of large numbers, the law of logarithm, and the law of the iterated logarithm, as well as convergence with respect to the general norming factors. By establishing almost sure convergence of appropriately defined delayed sums, the proposed method yields explicit convergence rates across a wide range of probabilistic settings. As a result, many convergence problems that were previously treated in isolation can be analyzed within a single coherent theoretical structure.

This paper revisits the classical logarithmic velocity profile of wall-bounded turbulent shear flows, with particular emphasis on the persistent scatter in its defining parameters—the von Kármán constant (κ) and the additive constant (B). A combined experimental–theoretical framework is developed to extract these parameters across the three canonical flows: plane channel, circular pipe, and zero-pressure-gradient (ZPG) boundary layer. The authors' derivations show that three different partial differential equations governing these flows in the fully developed state behave similarly as Reτ→∞, particularly within the inertial sublayer, thereby supporting the universality of the log-law, where Reτ denotes the friction Reynolds number. The theoretical approach, based on the slope of the mean velocity profile in logarithmic coordinates, is validated using new hot-wire measurements together with literature data, accounting for wall-curvature effects in pipe flow and flow development in ZPG boundary layers. The analysis indicates that, at sufficiently high Reynolds numbers, the inertial sublayer is governed by the log-law, with an estimated von Kármán constant of κ≈1/e≈0.367879⋯. This value is supported by renormalization group theory and exponential self-similarity arguments, reinforcing that κ is not merely a fitting parameter but a consequence of turbulence dynamics. Furthermore, the present study demonstrates that κ and B cannot be specified independently, as multiple κ–B pairs yield equally valid log-law predictions. A linear κB–B relation, κB = 0.4582 B − 0.3134, is established, reducing data scatter and clarifying the cross-dependence of the two parameters. The present findings highlight the robustness of the logarithmic law and the ongoing challenge of defining its universal constants in wall-bounded turbulent flows, while providing a physical framework for developing and calibrating turbulence models.

The hydraulic transportation technology of piped vehicles is a new type of pipeline transportation mode. A concentric annular turbulent flow with different boundaries is formed between the barrel of the piped vehicle and the pipe wall. The study on the annular turbulent flow can provide basic support for the application and promotion of this technology. Therefore, in this paper, the PIV technique was utilized to experimentally investigate the statistical characteristics of the annular turbulent flow in a fully developed smooth concentric annular pipe. The results showed that the position of the maximum velocity in the annular turbulent flow was not at the center but biased towards the barrel wall. Moreover, the smaller the radius ratio, the more it shifted towards the barrel wall. The position of the maximum velocity was independent of the Reynolds number and was a univariate function of the radius ratio; it was obtained by fitting experimental data that rmt*=k0.3491+k0.349. The resistance coefficient of annular turbulence was independent of the radius ratio and was a univariate function of the Reynolds number; it was obtained by fitting experimental data that λ=0.3183Rea0.2487. The shear stress on the barrel wall was greater than that on the pipe wall in annular turbulent flow. Moreover, as the radius ratio increased, the shear stress on the barrel wall decreased, while that on the pipe wall increased. The velocity distribution in annular turbulent flow was divided into an inner region and an outer region. In the inner region, the uc+−yc+ curves were greatly affected by the Reynolds number, and the average gradient increased with the increase in the Reynolds number, while in the outer region, the average gradient of the up+−yp+ curves decreased with the increase in the Reynolds number. The velocity distribution in annular turbulent flow cannot be expressed by a unified relationship. However, at high Reynolds numbers, there existed a region where the velocity distribution satisfied the logarithmic law in the outer region, and the slope of the logarithmic region was greater than that in circular pipe flow and parallel-plate flow.

The riblet surface is a passive turbulence drag reduction technology with promising aerospace application prospects. To investigate the drag reduction effects of riblets under flow conditions more representative of actual aircraft surfaces, this study establishes an adverse pressure gradient environment at moderate-to-high Reynolds numbers. Symmetrically arranged two testing plates with riblets’ surface and smooth surface, hot-wire anemometry is employed to measure the skin friction drag of both plates to get a direct measurement of the drag reduction rate. And the drag reduction mechanism is analyzed through burst events detection and coherent structure’s inclination angle. The measurement results indicate that the adverse pressure gradient itself leads to a reduction in wall friction, and the turbulent boundary layer velocity profile deviates from the standard logarithmic law, rendering the Clauser chart method unsuitable for estimating the friction velocity. The adverse pressure gradient contributes positively to the drag reduction rate of riblets, while the increase in Reynolds number in this experiment has no substantial effect. For the near wall structures, their asymmetrical movement of ejection and sweep and investigated by VITA. The significant decrease in burst frequency and increase in coherent structure inclination angle in the turbulent boundary layer over the riblet surface are identified as the primary reasons for reduced wall friction, with these changes being particularly pronounced under adverse pressure gradient conditions.

This study presents a comprehensive techno-economic assessment of offshore wind projects in the Colombian Caribbean, emphasizing the impact of site-specific parameters on development costs and performance. Wind resource conditions were evaluated in four coastal regions (La Guajira, Magdalena, Atlántico, and Bolívar) using hourly meteorological data from 2015 to 2024, adjusted to 100 m above ground level through logarithmic and power law wind profile models. The analysis included wind speed, bathymetry, distance to shore, distance to substation, foundation type, wind power density (WPD), and capacity factor (Cf). Based on these parameters, annual energy generation was estimated, and both capital expenditures (CAPEX) and operational expenditures (OPEX) were calculated, considering the technical and cost differences between fixed and floating foundations. Results show that La Guajira combines excellent wind conditions (WPD of 796 W/m2 and Cf of 61.5%) with favorable construction feasibility (bathymetry of −32 m), resulting in the lowest CAPEX among the studied regions. In contrast, Magdalena and Atlántico, with bathymetries exceeding 200 m, require floating foundations that more than double the investment costs. Bolívar presents an intermediate profile, offering solid wind potential and fixed foundation feasibility at a moderate cost. The findings confirm that offshore wind project viability depends not only on wind resource quality but also on physical site constraints, which directly influence the cost structure and energy yield. This integrated approach supports more accurate project prioritization and contributes to strategic planning for the sustainable deployment of offshore wind energy in Colombia.

Wall-bounded turbulence modeling is fundamentally limited by the representation of the logarithmic law, yet its parameters such as the von Kármán constant, κ, and intercept, B, and its spatial bounds (ylow+, yhigh+) are often assumed a priori. Here we present a hybrid optimization – physics-informed neural network (PINN) framework that directly addresses this point. The method's novelty is a two-stage process: first, an optimization pre-processing step uses direct numerical dimulation data to calibrate the log-law constants (κ,B) and (ylow+, yhigh+). Next, these values are hard-coded into a PINN, whose loss function constrains the solution to obey both the governing Reynolds-averaged Navier–Stokes equations and the log-law within its identified region. By accurately delineating the log-layer we prevent the PINN from enforcing incorrect physics in the viscous sublayer, which is essential for obtaining physically consistent velocity profiles and their derivatives. This method suggests a more universal and accurate model across various Reynolds numbers, thereby enabling the development of data-informed, physics-grounded turbulence closures.

This study is concerned with the near-wall flow structure over a NACA 0025 aerofoil at a constant chord-based Reynolds number of 100 000 across various angles of attack, where an array of 12 circular-orifice synthetic jet actuators (SJAs) was used to reattach the flow under conditions of flow separation. The SJAs were operated in burst-mode at two distinct momentum coefficients, a 50 % duty cycle and a modulation frequency of 200 Hz, targeting the separated shear layer frequency. Particle image velocimetry was conducted using three side-by-side cameras to capture the velocity fields along the aerofoil surface at the centreline. At zero angle of attack, the velocity profiles exhibited characteristics of a turbulent boundary layer, following the law of the wall in the inner layer while deviating from the logarithmic law in the outer layer. At higher angles of attack, while some logarithmic behaviour could still be detected close to the wall, a wide region of the velocity profiles became predominantly linear, exhibiting a behaviour differing from both a canonical turbulent boundary layer and a turbulent wall jet. The entire shear flow was decomposed into three regions: the boundary layer, the jet layer and the mixing layer that extended between the two. The mixing layer was analysed by applying several scaling laws to the time-averaged velocity components, where it was revealed that the characteristic velocity of the two velocity components is different. An asymptotic solution was obtained under a low spreading rate at infinite Reynolds number, providing a theoretical basis for the experimental observations.

This paper investigates the asymptotic behavior of kernel-based estimators for the error distribution in a first-order autoregressive model with dependent errors. The model assumes that the error terms form an α-mixing sequence with an unknown cumulative distribution function (CDF) and finite second moment. Due to the unobservability of true errors, we construct kernel-smoothed estimators based on residuals obtained via least squares. Under mild assumptions on the kernel function, bandwidth selection, and mixing coefficients, we establish a logarithmic law of the iterated logarithm (LIL) for the supremum norm difference between the residual-based kernel estimator and the true distribution function. The limiting bound is shown to be 1/2, matching the classical LIL for independent samples. To support the theoretical results, simulation studies are conducted to compare the empirical and kernel distribution estimators under various sample sizes and error term distributions. The kernel estimators demonstrate smoother convergence behavior and improved finite-sample performance. These results contribute to the theoretical foundation for nonparametric inference in autoregressive models with dependent errors and highlight the advantages of kernel smoothing in distribution function estimation under dependence.

Abstract Conventional surface layer wind profile models, such as the logarithmic and power laws, tend to provide inaccurate estimates of mean wind speeds at the altitudes relevant to modern wind energy systems. Additionally, these models do not account for wind veering, low-level jets, and thermal stratification of the geostrophic region, which are particularly relevant under stable atmospheric conditions. To enhance the reliability of wind turbine response analyses, more advanced wind profile modelling approaches are needed. In this study, we investigate three analytical wind profile models by comparing the predicted wind speeds to lidar observations recorded under stable conditions at two research platforms in the North Sea. Atmospheric stability is assessed using the ERA5 reanalysis database and validated against sonic anemometer data. While the models perform similarly well at low heights and near-neutral conditions, the logarithmic profile overestimates wind speeds at increased altitudes and stability. The Gryning model, an extended formulation of the logarithmic profile, provides more accurate predictions at large heights and stabilities. A third model, that accounts for the presence of a low-level jet, yields the best results in stable atmospheres and additionally captures the observed veer profiles effectively.

We develop an asymptotic theory of a compressible turbulent boundary layer on a flat plate, in which the mean velocity and temperature profiles can be obtained as exact asymptotic solutions of the boundary-layer equations, which are closed using functional relations of a general form connecting the turbulent shear stress and turbulent enthalpy flux to the mean velocity and enthalpy gradients. The outer region of the boundary layer is considered at moderate supersonic free-stream Mach numbers, when the relative temperature difference across the layer is of order one. A special change of variables allows us to construct the solution in the outer region in the form of asymptotic expansions at large values of the logarithm of the Reynolds number based on the boundary-layer thickness. As a result of asymptotic matching of the solutions for the outer region and logarithmic sublayer, the velocity and temperature defect laws are obtained, which allow us to describe the profiles of these quantities in the outer and logarithmic regions by universal curves known for the boundary layer of an incompressible fluid. Similarity rules for the Reynolds-tensor components and root-mean-square enthalpy fluctuation are given. The recovery and Reynolds-analogy factors are calculated. A friction law is established that is valid under arbitrary wall-heat-transfer conditions.

The law of the iterated logarithm precisely refines the law of large numbers and plays a fundamental role in probability limit theory. The framework of sub-linear expectation spaces substantially extends the classical concept of probability spaces. In this study, we employ a methodology that differs from the traditional probabilistic approach to study the k-iterated logarithm law for weighted sums of stable random variables with the exponent α∈(0,2) within sub-linear expectation space, establishing a highly general form of the k-iterated logarithm law in this context. The obtained results include Chover’s law of the iterated logarithm, as well as the laws for partial sums and moving average processes, thereby extending many corresponding results obtained in classical probability spaces.

Correlations between dislocations in crystals reduce the elastic energy via screening of the strain by the surrounding dislocations. We study the correlations of threading dislocations in GaN epitaxial films with dislocation densities of 5 × 1 0 8 cm −2 and 1 . 8 × 1 0 10 cm −2 by X-ray diffraction (XRD) in reciprocal space and by high-resolution electron backscatter diffraction (HR-EBSD) in real space, where the strain is derived from a cross-correlation analysis of the Kikuchi patterns. The measured XRD curves and HR-EBSD strain and rotation maps are compared with Monte Carlo simulations within one and the same model for the dislocation distributions. The screening of the dislocation strains is modeled by creating pairs of dislocations with opposite Burgers vectors, with the mean distance between dislocations in a pair equal to the screening distance. The pairs overlap and cannot be distinguished as separate dipoles. The HR-EBSD-measured autocorrelation functions of the strain and rotation components follow the expected logarithmic law for distances smaller than the screening distances and become zero for larger distances, which is confirmed by the Monte Carlo simulations. The kink in the plot of the autocorrelation function allows a robust and accurate determination of the screening distance without making any simulation or fit. Screening distances of 2 μ m and 0.3 μ m are obtained for the samples with low and high dislocation densities, respectively. The dislocation strain is thus screened by only 4 neighboring dislocations. In addition, an anisotropic resolution of the HR-EBSD measurements is observed and quantified.

Atopic dermatitis (AD) is a prevalent skin disorder affecting individuals globally, with many patients experiencing a range of symptoms. A pronounced clinical phenomenon associated with AD is the cyclic alternation of two distinct phases in time: inflammation and remission, depending on patients' immune response and skin permeability. Frequent and relatively long inflammatory times lead to symptoms that can severely deteriorate the quality of life for the patient. Through mathematical modeling, we find that patients with similar AD symptoms can be categorized into two phases depending on the skin permeability and immune response that constitute the most clinically relevant parameter plane: the inflammatory time is shorter or longer than the remission time, respectively and the transition between the two phases is of the second-order type. In the parameter plane, a critical threshold curve emerges, which separates the two phases. Computing the frequency and duration of the inflammatory response, we uncover a logarithmic scaling law governing the inflammatory and remission times and discuss its clinical implications. In particular, when the skin condition is managed to be near the phase transition point, the benefits of treatment are more pronounced. However, at this stage, the effectiveness of skincare in reducing flare-ups tends to be less noticeable, making it difficult to evaluate the success of the treatment, largely due to the nature of logarithmic decay in the remission time. Our study provides insights into the mechanisms of AD that can enhance diagnostic accuracy and treatment by understanding the alternation between inflammation and remission periods.