Saint-Venant Model Research Articles

Comprehensive mathematical model for efficient and robust control of irrigation canals

Linear control-oriented models are important to represent canal dynamics for designing controllers. This study focuses on hydraulic control structure (gate) modelling to address the complex interdependent behavior inherent in irrigation canals. A comprehensive mathematical model that incorporates the water level with gate-opening to model discharge is introduced for single and multiple canal pool scenarios. The proposed model captures the hydraulic coupling within and among canal pools, a key finding. The model is evaluated extensively under uniform and non-uniform flows across three distinct canals, highlighting the model's applicability to various systems. The uncertainty inherent within the nominal model is also assessed for varying operating conditions and hydraulic parameters. The proposed model is compared with the existing and the Saint-Venant (SV) model, showing improved accuracy in water-level predictions. This advancement in hydraulic modelling contributes to adaptable canal models essential in developing robust controllers to enhance water management in irrigation canals.

Pulsing in the Ahu‘ail‘au pond–spillway system during the 2018 Klauea eruption: a dynamical systems perspective

During the 2018 K $\unicode{x012B}$ lauea lower East Rift Zone eruption, lava from 24 fissures inundated more than 8000 acres of land, destroying more than 700 structures over three months. Eruptive activity eventually focused at a single vent characterized by a continuously fed lava pond that was drained by a narrow spillway into a much wider, slower channelized flow. The spillway exhibited intervals of ‘pulsing’ behaviour in which the lava depth and velocity were observed to oscillate on time scales of several minutes. At the time, this was attributed to variations in vesiculation originating at depth. Here, we construct a toy fluid dynamical model of the pond–spillway system, and present an alternative hypothesis in which pulsing is generated at the surface, within this system. We posit that the appearance of pulsing is due to a supercritical Hopf bifurcation driven by an increase in the Reynolds number. Asymptotics for the limit cycle near the bifurcation point are derived with averaging methods and compare favourably with the cycle periodicity. Because oscillations in the pond were not observable directly due to the elevation of the cone rim and an obscuring volcanic plume, we model the observations using a spatially averaged Saint-Venant model of the spillway forced by the pond oscillator. The predicted spillway cycle periodicity and waveforms compare favourably with observations made during the eruption. The unusually well-documented nature of this eruption enables estimation of the viscosity of the erupting lava.

Tsunami Inundation Modelling in a Built-In Coastal Environment with Adaptive Mesh Refinement: The Onagawa Benchmark Test

In tsunami studies, understanding the intricate dynamics in the swash area, characterised by the shoaling effect, remains a challenge. In this study, we employed the adaptive mesh refinement (AMR) method to model tsunami inundation and propagation in the Onagawa town physical flume experiment. Using the open-source flow solver Basilisk, we implemented the Saint-Venant (SV) equations, Serre–Green–Naghdi (SGN) equations, and a nonhydrostatic multilayer (ML) extension of the SGN equations. A hydraulic bore tsunami-like wave was used as the input boundary condition. The objective was to assess the efficiency of the AMR method with nonhydrostatic tsunami models in overcoming limitations in 2D and quasi-3D models in flume experiments, particularly with respect to improving accuracy in arrival time and run-up detection. The results indicate improved performance of the SGN and SV models in determining tsunami arrival times. The ML model demonstrated enhanced wave run-up simulations on complex built-in terrain. The refined roughness coefficient determined using the ML solver captured the arrival time well in the northern section of the Onagawa model, albeit with a 1 s delay. The AMR method offered a computationally stable solution with an 86.3% reduction in computational time compared to a constant grid. While effective, the nonhydrostatic models entail the use of a great deal of computational resources.

Simulation of 2D Depth Averaged Saint Venant Model of Shatt Al Arab River South of Iraq

Simulation of 2D Depth Averaged Saint Venant Model of Shatt Al Arab River South of Iraq

Characterisation of thermally treated beech and birch by means of quasi-static tests and ultrasonic waves

Wood, being renewable and highly abundant material, with excellent high specific strength and stiffness, has received increasing attention to be used in high performance applications such as the structural element of a battery case in an electric vehicle. For a successful implementation of wood in the automotive sector, it is, therefore, crucial to understand the behaviour of wood during and after temperature exposure and in the event of fire with the presence/absence of oxygen. In this study, the mechanical properties of thermally modified and unmodified European beech and birch in air and nitrogen environments at six different treatment intensities were characterised using compression tests, tensile tests, shear tests and Poisson’s ratio tests. Further, the elastic properties of these wood species were quantified using the ultrasound measurements. The obtained strength and stiffness exhibited mild improvement upon moderate temperature treatment (200 °C), followed by a decrease at elevated temperature levels. This improvement was somewhat more pronounced under nitrogen treatment than under air treatment conditions. Nevertheless, a more noticeable decrease in the material performance was observed in beech compared to birch, occurring at earlier stages of modifications. This study confirms the tension–compression asymmetry of beech and birch where higher Young’s moduli were obtained from tensile than from compression tests for reference and thermally treated beech and birch. The shear moduli obtained from ultrasound for birch were comparable to those obtained from quasi-static tests, whereas there was an overestimation of approximately 11–59% for the shear modulus of beech compared to quasi-static tests. Poisson’s ratios from ultrasound tests corresponded well with those from quasi-static tests for untreated beech and birch, but not for thermally modified samples. The Saint-Venant model can satisfactorily predict the shear moduli of untreated and treated beech wood.

The multi-point hydraulic control method for advanced controller design of the open canal irrigation systems

Abstract Most studies about the automatic control of open canal irrigation systems only focus on the distant downstream water level, which ignores the fact that the offtakes may be located anywhere along the canal. Such a simplified control strategy is likely to result in uncontrollable and inefficient water delivery. Therefore, a multi-point hydraulic control method is proposed, in which a simplified Saint-Venant model is formulated to describe the hydraulic states of multiple controlled points. Then, it is underlined that the controlled points with and without the offtakes may have different control objectives. It is suggested to implement soft constraints to the downstream end when there is no offtake, meaning that moderate water level fluctuations are acceptable. By comparing with the common model predictive control (MPC) controller, where the Integrator Delay model and hard constraint are used for distant downstream water level control, the proposed MPC controller successfully improved the water level control stability before the offtakes and the water supply reliability by 91 and 69.5% under the conventional condition and by 54.9 and 27.1% under the water-deficient condition. Accordingly, the proposed multi-point hydraulic control method shows great potential for the precision irrigation of large irrigation districts.

Numerical Simulation of Hydraulic Jump with the Inclusion of Boussinesq Term

To design a hydraulic structure such as gravity dam, it is crucial to predict the. Location and the length of the hydraulic jump. Numerical model comes in handy to deal with the conditions which are costly to reproduce the laboratories such as the hydraulic jump experiment. In this study, two models (Saint-Venant model and Boussinesq model) are solved numerically to simulate the formation of hydraulic jump in a flat rectangular channel. The performance of the models in reproducing the flow profile of hydraulic jump and its location, under different advection solvers, i.e. the CIP, MUSCL, and upwind scheme are investigated. Numerical results showed that the inclusion of Boussinesq term does not have significant effect on the numerical reproduction of the hydraulic jump in terms of the location and the water surface gradient at the hydraulic jump. Numerical solutions show that the accuracy of the advection term solver and its shock-capturing capability is critical for accurate simulation of hydraulic jump.

Closed-form expressions of shear deformability tensors associated with a new beam model

The application of a recent beam model requires the evaluation of a suitably defined shear deformability tensor in order to ensure both energetic and geometric consistency with the 3D Saint Venant model subjected to non-uniform flexure and torsion. Actually, whatever is the position of the shearing force, the beam model provides the same elastic energy and the same tip displacement of the barycentric axis of the 3D Saint Venant model. The expression of the shear deformability tensor is rather involved since it depends upon three harmonic functions, namely the warping and the warpage functions associated, respectively, with torsion and shear. For this reason, in order to promote its use and shed some light on its properties, we derive explicit expressions of the shear deformability tensor for circular, elliptical and rectangular sections exploiting the relevant analytical formulas of the warping and warpage functions.

Vanishing viscosity limit of the compressible Navier-Stokes equations with finite energy and total mass

Assume the initial data of compressible Euler equations has finite energy and total mass. We can construct a sequence of solutions of one-dimensional compressible Navier-Stokes equations (density-dependent viscosity) with stress-free boundary conditions, so that, up to a subsequence, the sequence of solutions of compressible Navier-Stokes equations converges to a finite-energy weak solution of compressible Euler equations. Hence the inviscid limit of the compressible Navier-Stokes is justified. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., α=1,γ=2).

The exact solution to the Shallow water Equations Riemann problem at width jumps in rectangular channels

Riemann problems at geometric discontinuities are a classic and fascinating topic of hydraulics. In the present paper, the exact solution to the Riemann problem of the one-dimensional (1-d) Shallow water Equations at monotonic width discontinuities is completely determined for any initial condition. This solution is based on the assumption that the relationship between the states immediately to the left and to the right of the discontinuity is a stationary weak solution of the 1-d variable-width Shallow water Equations. Under this hypothesis, it is demonstrated that the solution to the Riemann problem always exists, although there are cases where the solution is triple. This proves that it is possible to define width-jump interior boundary conditions of Saint Venant models that lead to well-posed problems, and that additional physical information is required to pick the relevant Riemann wave configuration among the alternatives when the solution is multiple. The analysis of an existing Finite Volume numerical scheme from the literature, based on flow variables reconstruction with preservation of specific energy and discharge, shows that the algorithm captures the solution with supercritical flow through the width discontinuity when multiple solutions are possible. This suggests that it is possible to change the algorithm accordingly to the structure of the physically relevant solution. Interestingly, the 1-d variable-width Shallow water Equations are formally identical to the 1-d Porous Shallow water Equations, implying that the exact solutions provided in the present paper are relevant for two-dimensional (2-d) Porous Shallow water numerical models aiming at urban flooding simulations.The procedure presented in this paper may be used not only to construct new challenging benchmarks for numerical schemes, but also as a guide for the construction of new algorithms, and for the interpretation of on-field and laboratory data related to transients at rapid channel width variations. The appearance of multiple solutions, which is connected to the hydraulic hysteresis phenomenon observed in the case of supercritical flows impinging a cross-section contraction, requires a criterion for the disambiguation of solutions. This will be the object of future research.

Recovery of a Lamé parameter from displacement fields in nonlinear elasticity models

Abstract We study some inverse problems involving elasticity models by assuming the knowledge of measurements of a function of the displaced field. In the first case, we have a linear model of elasticity with a semi-linear type forcing term in the solution. Under the hypothesis the fluid is incompressible, we recover the displaced field and the second Lamé parameter from power density measurements in two dimensions. A stability estimate is shown to hold for small displacement fields, under some natural hypotheses on the direction of the displacement, with the background pressure fixed. On the other hand, we prove in dimensions two and three a stability result for the second Lamé parameter when the displacement field follows the (nonlinear) Saint-Venant model when we add the knowledge of displaced field solution measurements. The Saint-Venant model is the most basic model of a hyperelastic material. The use of over-determined elliptic systems is new in the analysis of linearization of nonlinear inverse elasticity problems.

Very high order well-balanced schemes for non-prismatic one-dimensional channels with arbitrary shape

Accurate modelling of shallow water flows in canals for realistic scenarios cannot neglect the variations of the size and shape of the cross-section along the canal. In typical situations, especially in view of applications to flows in networks of canals, a 2D model would be too costly, while the standard 1D Saint-Venant model for a rectangular channel with constant breadth would be too coarse. In this paper we derive efficient, high order accurate and robust numerical schemes for a 1.5D model, in which the canal can have an arbitrary cross-section: the shape can vary along the channel and is described by a depth-dependent breadth function σ(x,z), where x is the coordinate along the channel and z represents the vertical direction. Contrary to all previous schemes for this model, we reformulate the equations in a way that avoids the appearance of the moments of σ and of ∂σ/∂x in the source terms. Our numerical schemes are based on the path-conservative approach for dealing with non-conservative products, are well-balanced on the lake at rest solution and can treat wet-try transitions. They can be implemented with any order of accuracy. Schemes up to third order are explicitly constructed and tested, thanks to the CWENO reconstruction technique. Through a large set of numerical tests we show the performance of the new schemes and compare the results obtained with different orders of accuracy.

A Saint-Venant Model for Overland Flows with Precipitation and Recharge

We propose a one-dimensional Saint-Venant (open-channel) model for overland flows, including a water input–output source term modeling recharge via rainfall and infiltration (or exfiltration). We derive the model via asymptotic reduction from the two-dimensional Navier–Stokes equations under the shallow water assumption, with boundary conditions including recharge via ground infiltration and runoff. This new model recovers existing models as special cases, and adds more scope by adding water-mixing friction terms that depend on the rate of water recharge. We propose a novel entropy function and its flux, which are useful in validating the model’s conservation or dissipation properties. Based on this entropy function, we propose a finite volume scheme extending a class of kinetic schemes and provide numerical comparisons with respect to the newly introduced mixing friction coefficient. We also provide a comparison with experimental data.

Estimation of multiple inflows and effective channel by assimilation of multi-satellite hydraulic signatures: The ungauged anabranching Negro river

With the upcoming SWOT satellite mission, which should provide spatially dense river surface elevation, width and slope observations globally, comes the opportunity to assimilate such data into hydrodynamic models, from the reach scale to the hydrographic network scale. Based on the HiVDI (Hierarchical Variational Discharge Inversion) modeling strategy (Larnier et al. (2020)), this study tackles the forward and inverse modeling capabilities of distributed channel parameters and multiple inflows (in the 1D Saint–Venant model) from multisatellite observations of river surface. It is shown on synthetic cases that the estimation of both inflows and distributed channel parameters (bathymetry-friction) is achievable with a minimum spatial observability between inflows as long as their hydraulic signature is sampled. Next, a real case is studied: 871km of the Negro river (Amazon basin) including complex multichannel reaches, 21 tributaries and backwater controls from major confluences. An effective modeling approach is proposed using (i) WS elevations from ENVISAT data and dense in situ GPS flow lines (Moreira (2016)), (ii) average river top widths from optical imagery (Pekel et al. (2016)), (iii) upstream and lateral flows from the MGB large-scale hydrological model (Paiva et al. (2013a)). The calibrated effective hydraulic model closely fits satellite altimetry observations and presents real-like spatial variabilities; flood wave propagation and water surface observation frequential features are analyzed with identifiability maps following (Brisset et al. (2018)). Synthetic SWOT observations are generated from the simulated flowlines and allow to infer model parameters (436 effective bathymetry points, 17 friction patches and 22 upstream and lateral hydrographs) given hydraulically coherent prior parameter values. Inferences of channel parameters carried out on this fine hydraulic model applied at a large scale give satisfying results using noisy SWOT-like data at reach scale. Inferences of spatially distributed temporal parameters (lateral inflows) give satisfying results as well, with even relatively small scale hydrograph variations being inferred accurately on this long reach. This study brings insights in: (i) the hydraulic visibility of the signatures of multiple inflow hydrographs at a large scale with SWOT; (ii) the simultaneous identifiability of spatially distributed channel parameters and inflows by assimilation of satellite altimetry data; (iii) the need for prior information; (iv) the need to further tailor and scale network hydrodynamic models and assimilation methods to improve the fusion of multisource information and potential information feedback to hydrological modules in integrated chains.

Vanishing viscosity limit for a system of balance laws with general type initial data arising from 1D Saint-Venant model

We study the vanishing viscosity limit of a non-strictly hyperbolic system of balance laws known as the 1D Saint-Venant model whose solution admits δ-waves. The balance term depends on both the variables x and t and is unbounded in the space variable x unlike the work of Olenik [Usp Math. Nauk. 12(3), 3–73 (1957) (in Russian)]. We also studied the large time behavior when the viscosity parameter is positive. When the balance term depends only on the time variable, we obtained an explicit formula for the solution.

A conservative saint-venant type model to describe the dynamics of thin partially wetting films with regularized forces at the contact line

This paper deals with the numerical simulation of thin liquid films flowing on partially wetting solid substrates. A 2D Saint-Venant like model is proposed. Its originality lies in the conservative formulation of the capillary forces and in the model used for the disjoining pressure that accounts for the contact line capillary forces. A finite volume scheme is proposed for the resolution of the system and various numerical examples are presented and discussed. In particular, when the mesh resolution is fine enough, the model is proved to be able to predict correctly the spreading of a film with the exact contact angle in the vicinity of the contact line. When the mesh size is larger than the film thickness (which could be the case for many industrial applications), it is of course no longer possible to recover the contact angle. However, the model is proved to correctly predict the spreading of the film. This important feature is related to the thermodynamic consistency of the model in the sense that the latter ensures by construction the decrease of the film total free energy in the absence of external driving forces.

State Estimation in 2D Hydrological Models using Lagrangian Sensors and Low Resolution Elevation Maps

In this research work, the framework to estimate 2D spatio-temporal variation in hydrodynamic variables such as water velocity (m/s) and water level (m) in complex, large scale open channels has been investigated using Lagrangian sensors. The Lagrangian sensors are passive floating platforms, which report its GPS position along with the flow of water. The 2D Saint-Venant model is simulated using HEC-RAS simulation software, the geometrical details for HEC-RAS simulations are obtained using Digital Elevation Map (DEM) of Ravi river, Pakistan. For the system model, the non-linear 2D Saint-Venant model is augmented with a Lagrangian sensor motion model. For state estimation, the GPS position of the Lagrangian sensor along with the upstream water level is assimilated in the augmented model using an Ensemble Kalman Filter (EnKF) with suitable filtering parameters in MATLAB. The hydrodynamic variables and trajectory of the Lagrangian sensor are estimated with low error.

The solution of the Riemann problem in rectangular channels with constrictions and obstructions

Usually, the rapid geometric transitions that are of negligible length with respect to the channel are treated in one-dimensional Saint Venant models as internal boundary conditions, assuming that an instantaneous equilibrium is attained between the flow characteristics through the structure and the flow characteristics in the channel. In the present paper, a different point of view is assumed by considering rapid transients at channel constrictions and obstructions that are caused by the lack of instantaneous equilibrium between the flow conditions immediately upstream and downstream of the structure. These transients are modelled as a Riemann problem, assuming that the flow through the geometric transition is described by a stationary weak solution of the Saint Venant equations without friction. For this case, it is demonstrated that the solution of the Riemann problem exists and it is unique for a wide class of initial flow conditions, including supercritical flows. The solutions of the Riemann problem supplied by the one-dimensional mathematical model compare well with the results of a two-dimensional Shallow Water Equations numerical model when the head loss through the structure is negligible. The inspection of the exact solutions structure shows that the flow conditions immediately to the left and to the right of the geometric discontinuity may be very different from the initial conditions, and this contributes to explain the numerical issues that are reported in the literature for the rapid transients at internal boundary conditions in finite difference models. The solution of the Riemann problem has been coded, and the corresponding exact fluxes have been used as numerical fluxes in a one-dimensional Finite Volume scheme for the solution of the Shallow water Equations. The results demonstrate that spurious oscillations and instability phenomena are completely eliminated, ensuring the robustness of the approach. In the case that the energy loss is not negligible, the exact solutions capture the essential features of the two-dimensional model numerical results, ensuring that the mathematical procedure is generalizable to realistic conditions. This generalization is presented in the final part of the paper.

Numerical characterization of torrential floods in the plain of Saïdia (North-East of Morocco)

The purpose of this study is to characterize torrential floods in the plain of Saidia, caused by water discharges of the rivers (oueds) passing through Ouled Mansor plateau, by determining the following parameters: hydrographs, watershed properties, and hazards maps. Hydrographs of torrential floods are elaborated for different return periods (5,10, 20, 50, and 100 years), based on an hourly disaggregation of rainfall and on the construction of a local Digital Elevation Model (DEM) in the city of Saidia, using the Storm Water Management Model (SWMM). The watersheds are extracted from the SRTM digital elevation model (DEM) (30 m resolution), while Iber code is used to elaborate hazard maps based on the 2D Saint-Venant shallow water model using finite volumes. The hydrographs maxima of the different oueds for different return periods show good correlation with the values recorded by the Agence du Bassin Hydraulique de Moulouya (ABHM). Overall, the obtained hazard maps are consistent with the flooded areas derived by the ABHM. The results show that there are three longitudinally variable runoff areas. For a better illustration of the aforementioned zones, three cross-sectional profiles A-A, B-B, C-C were carried out. The water accumulation is mainly due to the presence of two depressions that have been highlighted by other works. The maximum height of the runoff is in the central area of Saidia, which is a city developed for tourism. Thus, the realization of a hydraulic structure to deviate the water towards the Moulouya River is necessary. The design of the structure is based on the hydrographs obtained in this work. The parameters obtained play an important role in defining flood susceptibility zones in the study region. The results are a key input for the definition of flood mitigation measures in the area and are a useful tool for future urban planning and development of the city of Saidia.

Steady Flow Profile Analysis of Ciliwung River Using Standard Step Method Simultaneous Procedures

The aim of this paper is to examine steady flow profiles over an inclined channel. Different steady flow profiles may appear depending on the type of flow, i.e. subcritical or supercritical. Here, the steady Saint-Venant model is adopted, its solution can be obtained through the application of the numerical integration method (trapezoid). When the water levels upstream and downstream of the channel are known, water level along the channel can be obtained as the solution of a non-linear system of equations, which is then solved by the Newton iteration. This method is known as the standard step method of simultaneous procedure. Furthermore, simulation is done using the measurement data at the downstream part of Ciliwung River. Field measurement data were collected in terms of the water level along the river, from TB Simatupang to PA. Manggarai. The result of numerical simulation is shown a good agreement with field measurement data.