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\"Exploring the concept of homotopy from topology, different kinds of homotopy-based methods have been proposed for analytically solving nonlinear differential equations, given by approximate series solutions. Homotopy-Based Methods in Water Engineering attempts to present the wide applicability of these methods to water engineering problems. It solves all kinds of nonlinear equations, namely algebraic/transcendental equations, ordinary differential equations (ODEs), systems of ODEs, partial differential equations (PDEs), system of PDEs, and integro-differential equations using the homotopy-based methods\"-- Provided by publisher.

Image velocimetry has proven to be a promising technique for monitoring river flows using remotely operated platforms such as Unmanned Aerial Systems (UAS). However, the application of various image velocimetry algorithms has not been extensively assessed. Therefore, a sensitivity analysis has been conducted on five different image velocimetry algorithms including Large Scale Particle Image Velocimetry (LSPIV), Large-Scale Particle Tracking Velocimetry (LSPTV), Kanade–Lucas Tomasi Image Velocimetry (KLT-IV or KLT), Optical Tracking Velocimetry (OTV) and Surface Structure Image Velocimetry (SSIV), during low river flow conditions (average surface velocities of 0.12–0.14 m s − 1 , Q60) on the River Kolubara, Central Serbia. A DJI Phantom 4 Pro UAS was used to collect two 30-second videos of the surface flow. Artificial seeding material was distributed homogeneously across the rivers surface, to enhance the conditions for image velocimetry techniques. The sensitivity analysis was performed on comparable parameters between the different algorithms, including the particle identification area parameters (such as Interrogation Area (LSPIV, LSPTV and SSIV), Block Size (KLT-IV) and Trajectory Length (OTV)) and the feature extraction rate. Results highlighted that KLT and SSIV were sensitive to changing the feature extraction rate; however, changing the particle identification area did not affect the surface velocity results significantly. OTV and LSPTV, on the other hand, highlighted that changing the particle identification area presented higher variability in the results, while changing the feature extraction rate did not affect the surface velocity outputs. LSPIV proved to be sensitive to changing both the feature extraction rate and the particle identification area. This analysis has led to the conclusions that for surface velocities of approximately 0.12 m s − 1 image velocimetry techniques can provide results comparable to traditional techniques such as ADCPs. However, LSPIV, LSPTV and OTV require additional effort for calibration and selecting the appropriate parameters when compared to KLT-IV and SSIV. Despite the varying levels of sensitivity of each algorithm to changing parameters, all configuration image velocimetry algorithms provided results that were within 0.05 m s − 1 of the ADCP measurements, on average.

Understanding the Earth's magnetic field evolution requires examining the fluid flow at the core‐mantle boundary that drives the changes over different timescales. The inversion process to derive core‐surface flow velocities from secular variation data encounters non‐uniqueness issues, necessitating a priori assumptions that yield different flow solutions. In this work, we investigate the Earth's core‐surface flows over the last 3,300 years using the SHAWQ‐family archeomagnetic model (Campuzano et al., 2019, ; Osete et al., 2020, ) to invert for time‐dependent purely toroidal and tangentially geostrophic solutions. We apply the constraints as regularization terms that let us reproduce different large‐scale flows at the core surface. We evaluate the frozen‐flux hypothesis and we show that the temporal averaging range in archeomagnetic models reliably captures the long‐term behavior of core‐surface flows over time. Then, we use these core flow models to analyze different global phenomena, such as the episodes of large‐scale eastward and westward flow and the exchange of angular momentum between the fluid core and the mantle that contributes to the Length of the Day variations at millennial timescales.

Two-dimensional free-surface flow past a submerged rectangular disturbance in an open channel is considered. The forced Korteweg–de Vries model of Binder et al. (Theor Comput Fluid Dyn 20:125–144, 2006) is modified to examine the effect of varying obstacle length and height on the response of the free-surface. For a given obstacle height and flow rate in the subcritical flow regime an analysis of the steady solutions in the phase plane of the problem determines a countably infinite set of discrete obstacle lengths for which there are no waves downstream of the obstacle. A rich structure of nonlinear behaviour is also found as the height of the obstacle approaches critical values in the steady problem. The stability of the steady solutions is investigated numerically in the time-dependent problem with a pseudospectral method.

The International Geomagnetic Reference Field (IGRF) has been regularly updated since its inception in 1965. Every recent iteration contains an estimate of the geomagnetic secular variation (SV), for the intermediate years between iterations. We submit a candidate model for the geomagnetic secular variation (SV) for the period 2025–2030 for the 14 th generation of the IGRF. Given the recent evidence in the geomagnetic SV record for core surface waves, we forecast SV based on the periodic behaviour of core surface flow acceleration. We obtain an advective core surface flow model, in terms of poloidal and toroidal flow coefficients, from spatial gradients of SV geomagnetic virtual observatory data from the low-Earth orbiting CHAMP and Swarm missions from January 2001 to January 2010, and April 2014 to January 2024, respectively. From these, we calculate the flow acceleration coefficients from the first time-derivative. This assumes the flow is spatio-temporally simple, without imposing any physical constraints on its geometry. We fit each acceleration coefficient with a sinusoidal function, which is used to extrapolate 6 years into the future. These sinusoidally varying acceleration time series are integrated over time to obtain the core flow coefficients, which are then used to predict the average advected SV over the 5-year IGRF period. We recreate previous IGRF predictions using our CHAMP-based flows to validate our methodology, which we find to outperform previous IGRF iterations, and use the Swarm-based flows to forecast the SV for IGRF-14. Our Swarm-based model predicts sudden changes in SV—also known as geomagnetic jerks—in 2024 in the Equatorial Pacific, and in 2028 in the region around central Africa. Although the IGRF SV is a snapshot over a 5-year period, allowing for periodic behaviour offers potential improvements over other methods of prediction. Graphical Abstract

Well‐developed karst aquifers consist of highly conductive conduits and a relatively low permeability fractured and/or porous rock matrix and therefore behave as a dual‐hydraulic system. Groundwater flow within highly permeable strata is rapid and transient and depends on local flow conditions, i.e., pressurized or nonpressurized flow. The characterization of karst aquifers is a necessary and challenging task because information about hydraulic and spatial conduit properties is poorly defined or unknown. To investigate karst aquifers, hydraulic stresses such as large recharge events can be simulated with hybrid (coupled discrete continuum) models. Since existing hybrid models are simplifications of the system dynamics, a new karst model (ModBraC) is presented that accounts for unsteady and nonuniform discrete flow in variably saturated conduits employing the Saint‐Venant equations. Model performance tests indicate that ModBraC is able to simulate (1) unsteady and nonuniform flow in variably filled conduits, (2) draining and refilling of conduits with stable transition between free‐surface and pressurized flow and correct storage representation, (3) water exchange between matrix and variably filled conduits, and (4) discharge routing through branched and intermeshed conduit networks. Subsequently, ModBraC is applied to an idealized catchment to investigate the significance of free‐surface flow representation. A parameter study is conducted with two different initial conditions: (1) pressurized flow and (2) free‐surface flow. If free‐surface flow prevails, the systems is characterized by (1) a time lag for signal transmission, (2) a typical spring discharge pattern representing the transition from pressurized to free‐surface flow, and (3) a reduced conduit‐matrix interaction during free‐surface flow. Key Points Design of a hybrid model that considers unsteady and nonuniform flow The model considers the transition between free‐surface and pressurized flow Considering free‐surface conduit flow is significant for karst characterization

Two-dimensional open channel flow past a rectangular disturbance in the channel bottom is considered in the case of supercritical flow, where the dimensionless flow rate is greater than unity. The response of the free surface to the height and length of a rectangular disturbance is investigated using the forced Korteweg–de Vries model of Michalski et al. (Theor Comput Fluid Dyn 38:511–530, 2024). A rich and complex structure of solutions is found as the length of the disturbance increases, especially in the case of a negative disturbance. As the length of the disturbance is decreased, some solutions approach those of the well-studied point forcing approximation, but there are other solutions, for a negative disturbance, that are not predicted by the point forcing model. The stability of steady solutions is then considered numerically with established pseudospectral methods.

Recently, the authors considered a thin steady developed viscous free-surface flow passing the sharp trailing edge of a horizontally aligned flat plate under surface tension and the weak action of gravity, acting vertically, in the asymptotic slender-layer limit (J. Fluid Mech., vol. 850, 2018, pp. 924–953). We revisit the capillarity-driven short-scale viscous–inviscid interaction, on account of the inherent upstream influence, immediately downstream of the edge and scrutinise flow detachment on all smaller scales. We adhere to the assumption of a Froude number so large that choking at the plate edge is insignificant but envisage the variation of the relevant Weber number of $O(1)$. The main focus, tackled essentially analytically, is the continuation of the structure of the flow towards scales much smaller than the interactive ones and where it no longer can be treated as slender. As a remarkable phenomenon, this analysis predicts harmonic capillary ripples of Rayleigh type, prevalent on the free surface upstream of the trailing edge. They exhibit an increase of both the wavelength and amplitude as the characteristic Weber number decreases. Finally, the theory clarifies the actual detachment process, within a rational description of flow separation. At this stage, the wetting properties of the fluid and the microscopically wedge-shaped edge, viewed as infinitely thin on the larger scales, come into play. As this geometry typically models the exit of a spout, the predicted wetting of the wedge is related to what in the literature is referred to as the teapot effect.

The small-scale velocity gradient is connected to fundamental properties of turbulence at the large scales. By neglecting the viscous and non-local pressure Hessian terms, we derive a restricted Euler model for the turbulent flow along an undeformed free surface and discuss the associated stable/unstable manifolds. The model is compared with the data collected by high-resolution imaging on the free surface of a turbulent water tank with negligible surface waves. The joint probability density function (p.d.f.) of the velocity gradient invariants exhibits a distinct pattern from the one in the bulk. The restricted Euler model captures the enhanced probability along the unstable branch of the manifold and the asymmetry of the joint p.d.f. Significant deviations between the experiments and the prediction are evident, however, in particular concerning the compressibility of the surface flow. These results highlight the enhanced intermittency of the velocity gradient and the influence of the free surface on the energy cascade.

The sedimentary structure of tailings is of high significance to the engineering design and safety management of tailings dams. However, due to a lack of accurate measurement techniques for the flow field and hydrodynamic conditions of tailings reservoirs, it is challenging to study the complicated sedimentary structure of tailings dams from the perspective of fluid mechanics. This study focuses on developing a large-scale particle image velocimetry (LSPIV) system in a 20 m long and 2 m wide deposition model flume to measure the flow field characteristics during the ore-drawing process accurately. According to the surface flow field characteristics measured by LSPIV, the tailings in the flume can be divided into three zones, namely the fan-shaped zone, channel zone, and laminar flow zone. Then, a simple method for estimating the flow rate of the slurry was proposed using the surface velocities measured by LSPIV. The flow rate of iron tailings slurry in the flume displays a decreasing trend along the flow direction. The variation of the flow rate of tailings slurry can be described by an exponential function. After the deposition of tailings slurry, the sedimentary characteristics of tailings are investigated, and the distribution of iron tailings particles is discussed in combination with the flow field of the tailings slurry. The LSPIV system can be applied to further deposition model tests of different slurry concentrations, discharge flow rates, and tailings compositions to investigate the effects of these factors on the tailings flow and deposition.