Rill flow is a major driver of soil erosion and a threat to watershed stability. As concentrated runoff evolves into gullies and steep step-pool channels, understanding its hydrodynamics becomes essential, especially once erosion rates diminish. In these conditions, strong vertical velocity gradients invalidate shallow-water assumptions. This work develops a section-averaged, non-hydrostatic rill-flow model that incorporates bed-form-related flow resistance while excluding sediment transport. A second-order finite-volume scheme with semi-implicit time stepping, solved through a Newton-Raphson method, is implemented. The model is evaluated using analytical solutions and laboratory data, showing satisfactory performance. Sensitivity analyses highlight the importance of accurate rill-width characterization. Experimental observations reveal rill widening at pools and the presence of bank cavities linked to secondary recirculating cells; incorporating these features significantly improves model accuracy. Studying steady flows in step-pool systems represents an initial step toward a comprehensive non-hydrostatic erosive flow model.

This data descriptor presents a detailed dataset of open-channel flow interacting with arrays of obstacles, originating from a laboratory experimental campaign involving Gaussian- and log-normal-shaped obstacles arranged sequentially in a flume, as well as a solitary dune profile. Six different array configurations were tested. Both steady and unsteady flow conditions were implemented resulting in a wide range of experiments. Data on free surface flow level, flume bed level, and head pressure data were systematically extracted using a methodology specifically designed for this purpose. The dataset is supplemented with schematic representations of the data and panoramas images of the experiments, including characteristics of turbulence and wave transformation phenomena. To the best of our knowledge, this is the first database in the literature to comprehensively examine the interaction of flow with various obstacle arrays under both steady and unsteady flow conditions. Combined with the detailed description of the employed methodology, this data descriptor is expected to promote significant usage of the dataset, benefiting the open-channel flow modeling community.

Simulating unsteady free surface flows with full three-dimensional (3D) Navier-Stokes (N-S) equations is computationally expensive, so simplified models beyond shallow flow approximations are needed for large-scale river and ocean dynamics. In this study, we focus on a Vertically Averaged and Moment (VAM) equations approach for modeling unsteady free surface flows. The VAM equations are a quasi-3D modeling method derived from a variational approximation of the N-S equations, and therefore are not limited by the shallow flow approximation. The VAM models can approximate the N-S equations in a variety of free surface flow problems with good accuracy, but there is a lack of systematic consideration of these methods in the literature, which has slowed their adoption and development. Here, the VAM equations are presented, and several advances are introduced. For instance, a novel and efficient numerical method that easily enables the transformation of a Saint Venant solver into a VAM solver is presented. The nonhydrostatic pressure field is determined by a projection method inspired by Chorin's technique, notably without requiring iterative processes. Key physical aspects of the VAM model are then systematically highlighted, namely form drag over bedforms, frequency dispersion of water waves under nonlinear conditions as in Favre waves and collision of surges, turbulence modeling in the canonical case of uniform flow, breaking waves including dam-break flows and hydraulic jumps, and morphodynamic modeling, which encompasses geomorphic dam-break waves, erosion of dikes due to overtopping, and the generation and migration of antidunes in fluvial streams.

Overland flow resulting from the rainfall-runoff transformation is an important hydrological process in agricultural and urban watersheds, occurring in the form of a thin fluid sheet moving on rough and relatively steep terrain. Mixed flows involving moving critical points are frequent, especially in urban drainage, but a method to deal with these flows is so far not available. In this work a new and robust solution method for the computation of equilibrium mixed flows is presented, resulting in solutions not available so far. Based on these results, the convergence features to mixed flows of a dynamic wave model developed are investigated choosing suitable tailwater boundary conditions. The new solution method developed for mixed flows reveals that pseudo-uniform flow profiles are closer to dynamic wave profiles than kinematic profiles. It allowed the definition of a new kinematic-wave truncation of the momentum equation, the pseudo-kinematic wave, suitable for unsteady rainfall-runoff modeling. The new shallow water wave approach proposed is closer to dynamic waves than the standard kinematic approach and permits to simulate unsteady overland flows more accurately over a wider range of conditions, e.g. for kF02 > 5 and F0 < 2. The dynamic wave model presented is compared with experiments, other computational solutions, and two new analytical solutions developed, one for steady flows and another for unsteady flows. The steady mixed flow profiles are compared with experiments and results of the dynamic wave model, detailing the formation of critical points. Dynamic, kinematic and pseudo-kinematic waves are extensively compared for flow conditions where the kinematic wave is invalid. Finally, the roll wave development, which is not detailed so far in overland flow under rainfall, is considered to settle an upper validity limit of the new pseudo-kinematic wave approach.

In this paper, we explore the formation of alternate river bars in the presence of submerged vegetation, modelled as uniformly spaced rigid cylinders. We analyse the stability of the erodible bed by coupling the Exner equation for bed evolution with the continuity and momentum equations for the fluid phase. Through linear and weakly nonlinear stability analyses, we predict the equilibrium wavelength and amplitude of vegetated alternate bars. The computational results hinge on two key parameters: the vegetation aspect ratio (vegetation height to diameter ratio) and the vegetation packing density (dimensionless frontal area per unit volume). We present streamwise flow velocity profiles for different vegetation aspect ratios and vegetation packing densities. We find that the equilibrium wavelength decreases with higher vegetation aspect ratio and vegetation packing density. Vegetation reduces the minimum channel aspect ratio required for the braided channel formation and the threshold channel aspect ratio for the alternate bar formation. The equilibrium amplitude increases with vegetation aspect ratio but reduces with vegetation packing density, eventually reaching a constant value. The predicted alternate bar wavelength and amplitude align with field observations in the Arc River, Hooge Raam River, Alpine Rhine River and Isère River.

We consider a steady water flow in a channel where a vertical grid is clogged by a rectangular patch of aquatic vegetation. Laboratory experiments are conducted to observe the decrease in the water table within the patch, as well as the subsequent head loss, as a function of the patch length, the flowrate and the upstream water height. Various models of porous media are used to produce theoretical formulae describing the water surface profile within the patch, among which the Barree-Conway model proves to perform reasonably well. However, for large enough Froude numbers a seepage face or water chute appears past the vegetation patch while non-hydrostatic effects become important. Under the latter condition, the accuracy of our analytic solution is less satisfactory, while remaining accurate enough for practical purposes. As confirmed by numerical simulations, with the specific aquatic plants used in our experiments the porous flow is not Darcian, so that the seepage and head loss could not be explained by the exact Polubarinova-Kochina theory.

Accurate flow models are crucial for simulating shallow water hydrodynamics, particularly in predicting and mitigating the impacts of extreme events involving free-surface flows. Many of these extreme scenarios in river environments involve fluid dynamics with significant dynamic pressures, invalidating the use of standard Saint-Venant-type models. This study presents a robust and accurate novel alternative based on the Reynolds-averaged Navier-Stokes (RANS) equations solved through variational methods. Despite their potential, variational methods have been underutilized in the literature, and their application has been limited to low-level expansions. Moreover, they are rarely validated against experiments that simulate complex flows. This study addresses both challenges. First, a general mathematical framework is developed for the variational RANS (VR) model of arbitrary high-level. The VR level III model is presented and is solved numerically using a robust finite volume-finite difference solver for turbulence flow modeling. Second, an extensive experimental program was conducted to validate this new flow modeling tool, focusing on two challenging flow scenarios. The first scenario involves the propagation of turbulent breaking waves over an irregular, uneven bathymetry-conditions similar to those observed during bedform development in riverine environments. This scenario involved the experimental characterization of unsteady surges over an array of obstacles in series. The second scenario investigated sill-controlled released discharges, similar to those occurring in estuary inlets with sediment bars. Comparisons between the new experimental data and the predictions from the VR level III model reveal the model's accuracy and robustness, making it a highly suitable tool for simulating free-surface flows.

Sill-controlled flows involving the acceleration across a crest or control section with significant flow curvature are characterized by the Euler equations of an inviscid fluid. The problem is of both theoretical and practical interest in fluid mechanics research, given the role of such flows in water discharge measurements structures, underwater oceanic currents, river flows and shallow bar-built estuary inlets, among others. While three-dimensional computations are feasible, vertically averaged solutions are sought to gain efficiency. The vertically averaged representation of sill overflows based on the Serre–Green–Naghdi (SGN) theory is limited to very shallow flows, and a good averaged approach for large overflows is currently not available. High-level Green–Naghdi (GN) theory forms a hierarchy of theories of increasing accuracy based on expanding the kinematic field and vertically-averaging in a weighted-residual sense. These theories have been successfully applied to ocean research but so far, they have not been applied to flow in open channels and structures. In this work, the high-level Green–Naghdi theories, of which SGN equations are the lowest level possible (GN level I theory), are formulated and newly applied to the sill-controlled flow problem. High-level GN theory is compared with detailed experiments from the literature and new ones conducted, and with a new fully non-linear vertically resolved potential flow solver which uses a x – ψ mapping. It was found that the GN expansions are convergent in the sill problem, in contrast to former perturbation solutions, which are asymptotic. The GN level V theory was found to be in good agreement with experiments for the sill-controlled flow problem, which excellently reproduced the free surface, bottom pressure, and vertical distributions of velocity and pressure. The GN level V theory was found to be applicable for relatively wide range of overflows up to E / R = 3, where E is the minimum specific energy and R the sill crest radius of curvature. From a practical viewpoint, large improvements were observed when using GN level II theory, instead of GN level I (e.g. the SGN theory), producing good results up to E / R = 1.75, which is the minimum level thus recommended in practice.

This study presents a comprehensive dataset comprising multiple data packages derived from laboratory experiments on steady and unsteady hydraulic jumps interacting with a large-scale Gaussian-shaped bed obstacle in an open-channel flume. The primary objective was to accurately measure the impact of hydraulic jump on the free surface and the bed pressure along the obstacle, ensuring the transferability of the results. A multi-process method was followed: designed experiments were recorded, images were postprocessed, and water level data were digitalized. For steady conditions, the bed pressure along the obstacle were measured by piezometers. The repository data are organized and provided in a single package, supplemented by a second package containing panoramas for each experimental time instant and graphical representations of the data, facilitating rapid evaluation of the outcomes. This study provides versatile data that can be utilized in various ways, particularly for fluvial model validation and studying turbulence-driven phenomena in open-channel flows. The detailed methodology presented herein can contribute to the advancement of enhanced laboratory techniques to study similar flow problems.

A spillway is a hydraulic structure of major importance in dam safety, and its current analysis usually involves a hybrid approach combining CFD modeling with experimental research, either using well-known WES design charts or conducting new model experiments in the laboratory. Flow over spillway crests involves fluid accelerations, making irrotationality an adequate simplification of the Navier–Stokes (NS) equations. However, an efficient tool using this method is currently lacking for spillway flow, particularly for ogee spillway flow. This work focuses on this aspect of the problem, and a new method for computing irrotational flow solutions over ogee spillways is proposed by developing flow net computational solutions. The proposed method entails a new iterative procedure in the complex potential plane where free surface pressures are exactly set to zero, contrary to other methods, and an automatic determination of the critical point, the unknown energy head, and the free surface profile. The model generates solutions efficiently in only a few seconds on a personal workstation, permitting a fast estimate of spillway flow operation, and is thus an effective complement to experimental and NS-CFD modeling. The solutions produced are compared with observations of a high operational head equal to five times the design head of the ogee crest, resulting in reasonable agreement. The application of the new model to investigate the limitations of analytical equations used in spillway flow, like Jaeger’s theory, establishes limits for its use by relating its curvature parameter to the spillway chute slope.

Critical flow in irrotational motion is important in theoretical hydrodynamics and dam hydraulics. Therefore, the commented paper is relevant in theory and practice. It deals with an approximation for critical flow based on a set of simplified irrotational flow equations in the gravity field. The underlaying model equations were found by the discussers to strongly rely on Jaeger’s work. Therefore, some important aspects need a detailed clarification. Jaeger’s velocity profile was determined here by a possibly novel procedure starting from the irrotational flow relations in the complex potential plane. It was shown that, though not perfect, a theory assuming critical crest conditions gives consistent estimates of the discharge coefficient, crest flow depth, bottom pressure head, and velocity profile. A new method for computing the flow profile over an ogee crest is presented by simultaneous determination of the discharge coefficient and the real critical point position using the Bélanger–Böss theorem, resulting a physically based determination of the critical point in spillway flow. It is demonstrated that Jaeger’s curvature parameter K is not a universal value, such that neither the current comment nor the discussed paper are therefore “free” from empirical parameters.

The hydraulic characteristics (such as velocity profiles, near-bed velocity profile, bed shear stress, and resistance coefficients) of shallow flows over rough surfaces were investigated using numerical simulations. A novel method is presented to simulate shallow flows over rough surfaces in a two-dimensional (2D) numerical domain, where the physical numerical domain represents bed topography. Results reveal that the model can accurately predict spatially averaged velocity profiles, turbulence characteristics, shear stresses, and uniform flow depths. The analysis identified two distinct flow regions based on mean and turbulent flow profiles. Results show that the turbulent shear stress profiles provide a more accurate estimation of the bed shear stresses. Resistance coefficients (friction factor or Manning’s roughness coefficient) vary with Froude number and submergence ratio (depth divided by roughness height).

Transitional free surface flow profiles with a critical point occur with weak vorticity and viscosity effects and thus can be modeled with an irrotational flow approach. Important examples in hydraulic engineering include flow over low obstacles, transition structures in canals, and flow over high spillways. While solving Laplace’s equation is relatively simple, the determination of the unknown free surface and energy head of the flow is challenging. Both hydraulic quantities need to be iterated before solving the Laplace equation. Former models iterated the energy head on a trial-and-error basis, assuming that the linked free surface profile is smooth, i.e., free of waves. The iteration of the free surface for a given head is frequently accomplished using the Newton–Rapshon method, which is difficult for the challenging case of spillway flow, giving no solution in some cases. An alternative method of computing irrotational flow profiles in transitional flows involving a critical point is proposed in this work. The model contains three elements: mapping of Laplace’s equation to directly track the streamlines, determination of the critical point and unknown energy head using a critical flow condition for irrotational flows, and determination of the water surface position using an exact analytical solution. The proposed model is favorably compared with experimental data from different sources and CFD results, indicating a reasonable agreement.

Culverts are ubiquitous hydraulic structures used in cross-drainage and agricultural water systems, especially at farm outlets in irrigation systems. Gate control adds hydraulic complexity to culvert water flow, yet it has hardly been studied. This study has two objectives: (1) to classify the operational conditions of gated culverts, and (2) to establish an experimental data set for analyzing discharge relationships based on that classification. A theoretical inlet-outlet control-based classification resulted in 22 inlet control and 19 outlet control cases, organized in 4 inlet and 5 outlet control groups. A total of 230 laboratory tests were conducted. The laboratory setup reproduced the conditions of inlet submerged with outlet in free surface (one of four inlet control groups) and all conditions except inlet in free surface with outlet submerged (four of five outlet control groups) of the inlet and outlet control groups, respectively. Comparison with existing classification schemes for gated culverts confirmed the suitability of the proposed scheme and provided further insight into flow classification. The discharge grouped by narrow ranges of gate openings was found to follow a power-law relationship of head with an exponent between 0 and 1. Flow transition regions between groups of conditions were examined, suggesting parabolic transition curves. The gate opening was found to be the dominant variable, resulting in a direct proportional effect on discharge, i.e., data corresponding to greater gate openings enveloped those with smaller openings. The slope's influence could not be discerned within the slope range tested. Discharge estimation based on a dimensionless, empirical model produced reasonable results, although its case-dependent nature requires further research to develop operational hydraulic models for determining discharge through gated culverts. Application of the proposed classification scheme could help develop these hydraulic tools with sufficient accuracy.

The computation of three-dimensional unsteady non-hydrostatic flows over large domains and/or for long simulation times is frequently conducted in research and practice using approximate methods to avoid the cost of a fully 3D solution. Among these methods are the shallow-water perturbation theories of St. Venant and Boussinesq, and the theory of directed fluid sheets by Green and Naghdi. The latter theory is not limited to shallow-water flows, and it is essentially a weighted-average residual method of Galerkin type, further developed by Shields and Webster for maritime hydraulics. The method is in essence equivalent to the vertically-averaged and moment (VAM) equations developed by Steffler and Jin for open channel flows, although this has not been recognized in the literature. A general framework for constructing VAM models of high-order, by linking maritime and open channel flow developments, is not available in the literature. In this work, a generalized framework for designing weighted-average residual equations for free-surface flow is presented based on the Kantorovich and Krylov method. The development of physically sound expansions for the hydrodynamic variables, and the construction of general systems of VAM equations to determine the unknowns in the expansions by selecting suitable weighting functions, is discussed in detail. The approach produces high-order models, thereby generalizing Steffler and Jin's development. A hierarchy of high-order VAM models is demonstrated to progressively converge to the exact dispersion relation of periodic waves by increasing the vertical resolution. These models are not limited by any shallowness assumption and exhibit more accurate wave dispersion properties compared to the Serre-Green-Naghdi equations. Computational results show that the VAM equations produce an accurate prediction of dam-break waves and dispersive wave effects over submerged bars.

The hydraulic jump is a phenomenon which frequently appears in nearshore, estuarine and riverine regions; it plays a major role for example in the development and migration of sand waves in mobile beds, and in controlling the propagation of infra-gravity water waves from the nearshore into the river through bar-built, shallow estuaries. Although the prediction of these natural flows becomes crucial in many circumstances, it is difficult because hydraulic jumps are complex flows, and their modeling is far from simple. The steady-state hydraulic jump may be undular or broken, and their physical properties differ significantly. Undular jump profiles are dominated by the deviation of the fluid pressure from hydrostatic conditions, with turbulence playing a secondary role. In contrast, broken hydraulic jumps are determined by the differential advection of momentum due to breaking, and turbulence stresses. Between both types, there is a complex transition involving the interaction of non-hydrostaticity and turbulence, which is difficult to mimic with depth-averaged models. Simulations of these flows are typically conducted by resorting to vertically-averaged models either using the Saint-Venant or the Serre-Green-Naghdi equations, but none of them are fully satisfactory. In this work, an alternative approach for hydraulic jumps is proposed by developing a variational Reynolds-Averaged Navier-Stokes (RANS) model which uses Kantorovich-Krylov expansions of the flow variables, and a novel depth-averaged, eddy viscosity approach suitable for non-hydrostatic flows. The model is validated through results of simulations of both undular and broken jumps compared against detailed experiments. New experiments were conducted in a simplified bar-built shallow river inlet; the variational RANS model was found to reproduce well the flow profile over the bar, the hydraulic jump position and dynamic pressures over the bar surface.