500 research outputs found
Reply to comments on “Depth–energy and depth–force relationships in open channel flows II: Analytical findings for power-law cross sections” by A. Valiani, V. Caleffi [Adv. Water Resour. 32 (2009) 213–224]
No full textReply to comments on “Depth–energy and depth–force relationships in open channel flows II: Analytical findings for power-law cross sections” by A. Valiani, V. Caleffi [Adv. Water Resour. 32 (2009) 213–224
Dam break in power-law cross-section channels with different upstream/downstream widths
No full textThe dam break problem is studied in a channel characterised by a power-law cross-section, with different widths upstream and downstream of the dam. This research is the extension of a previous study by Valiani and Caleffi (2019), designed for rectangular cross-sections, to different and more complex geometries. A 1D a-SWE (one dimensional augmented Shallow Water Equation) system is developed, which has been shown to capture the full range of possible solutions.
To address abrupt changes in the section, the classic SWE system is augmented with a third equation consisting of the time-invariance of the width scale. The idea of the augmented system was introduced by LeFloch and Thanh (2011) for the unit-width SWE, using the bed elevation as an additional variable, and by Valiani and Caleffi (2019) for rectangular narrowing/widening cross-sections, using the channel width as an additional variable.
The numerical model is a Finite Volume Method, second-order accurate in space and time, using a path conservative scheme to evaluate the numerical flux at the cell interfaces. A nonlinear path is adopted, which is shown to be optimal for capturing both the sharp contact wave at the dam and the moving shock(s) downstream of the dam.
The comparison of the numerical results with the analytical results for different solution patterns of the solution supports the confidence in the reliability of the model
Crollo diga in alvei di sezione qualunque con brusche variazioni di sezione
No full textIl crollo diga viene trattato nel caso di sbarramento su alveo di sezione qualunque. Il lavoro è concepito per indagare i casi in cui esista una discontinuità geometrica nella sezione dello sbarramento, e rappresenta l'estensione dei risultati di Valiani & Caleffi (2019) per sezione rettangolare. L'applicazione del modello matematico è effettuata su sezioni che seguono una legge di potenza, per una duplice motivazione. In primo
luogo, questa legge è un ottimo compromesso tra una ragionevole generalità ed una notevole semplicità, risultando per le pratiche applicazioni idonea a cogliere gli aspetti essenziali della geometria dell'alveo per corsi d'acqua naturali (Valiani & Caleffi, 2009). In secondo luogo, questa assunzione sulla geometria consente
di confrontare la soluzione numerica con soluzioni di riferimento che mantengono una struttura quasi analitica, permettendo verifiche stringenti degli schemi numerici
Hydraulic jump in diverging channels
No full textThe classical problem of the hydraulic jump in diverging channels is revisited and reformulated, in terms of linear and angular momentum conservation, at the integral scale. The former balance is used in its well known form, while the latter is expressed in the simplest meaningful formulation, taking into account of the moments exerted by momentum flux, by pressure force in each cross section and on lateral walls, and by vertical stresses. The whole flow is supposed to be ideally divided in a mainstream, conveying the total discharge, and in a roller, exerting stresses on the mainstream.
The scheme is the axially-symmetric counterpart of the 2D plane scheme formulated by Valiani [Linear and angular momentum conservation in hydraulic jump, JHR, 35(3), pp.323-352, 1997]. Gravity forces are supposed to be exactly counteracted by bottom reactions. Outside the jump, the inviscid solution is assumed to minimize the degrees of freedom of the problem. The obtained system of two conservation laws, together with the appropriate boundary conditions, is not analytically solvable due to the highly nonlinear relationship between the vertical length scale (depth) and the longitudinal length scale (radius), but a simple numerical solution gives the sequent depths and their positions, as functions of the non-dimensional discharge and of the downstream/upstream energy ratio of the flow. Taking into account the uncertainties in defining the downstream cross section of the jump, the comparison between the numerical solution and a selected set of laboratory data shows the reliability and accuracy of the proposed scheme
Linear and angular momentum conservation for the hydraulic jump in converging channels
No full textThis note is the final completion of a previously published work concerning the integral conservation of linear and angular momentum in the steady hydraulic jump in a linearly diverging channel [Valiani, A., Caleffi, V. (2011). Linear and angular momentum conservation in hydraulic jump in diverging channels. Adv. Water Res. 34(2), 227–242]. The same reasoning is applied to a linearly converging channel, and the theoretical framework, which is almost completely the same, is shown to remain valid. Using a proper mechanical scheme, an analytical solution is obtained for the free surface profile of the flow. This solution allows the determination of the sequent depths and their
positions. Thus, the length of the jump, which is assumed to be equal to the length of the roller, is also found. The mainstream and roller thicknesses can also be derived. This model may be used to derive the average shear stress exerted by the roller on the mainstream and the related exact expression for the total power loss in the jump, allowing to demonstrate the internal consistency of the proposed conceptual scheme
Analysis of a boundary layer of a granular mixture flowing past a plate at zero incidence
No full textThe present paper reports experiments on the flow field of a grain-water mixture around
a flat, thin plate at zero incidence. The velocity measurements are performed using a
Particle Image Velocimetry (PIV) technique. The Proper Orthogonal Decomposition
(POD) analysis reveals that the degree of organisation of the flow field increases with
the Reynolds number. The displacement thickness of the boundary layers generally
increases downstream and increases slightly with the Reynolds number, which is based
on the length of the plate. The vorticity normal to the plane of the flow has a maximum
value at the leading edge and is almost invariant with respect to the Reynolds number;
additionally, the non-dimensional profiles in the direction normal to the plate show selfsimilarity
in the streamwise direction for a single test, and the profiles are almost
coincident for all tests. The flow divergence is assumed to be an indicator of the
variation of the sediment volume concentration; it indicates an increment of the
sediment volume concentration near the walls of the plate and a spatial periodicity
downstream that is triggered for relatively large Reynolds numbers. The spatial correlation analysis allows the evaluation of the integral length scales that are
successively utilised in modelling the non-local rheology of the mixture. The velocity
profiles have been modelled based on Savage’s model and Bagnold’s experiments, with
further modifications from Ertaş and Halsey (2002) that are represented by pseudoturbulence
modelling of the flow field. Vortices have been detected according to the lambda-2-
criterion given by Jeong and Hussain (1995). The statistics of the vortices indicate that
no preferential size is selected and that at a high Reynolds number, the most energetic
vortices develop near the leading edge
A one-dimensional augmented Shallow Water Equations system for channels of arbitrary cross-section
No full textThis work provides a new formulation of the one-dimensional augmented Shallow Water Equations system for open channels and rivers with arbitrarily shaped cross sections, suitable for numerical integration when discontinuous geometry is encountered. The additional variable considered can be the bottom elevation, a reference width, a shape coefficient, or a vector containing these or other geometric parameters. The appropriate numerical method, which is well suited to coupling with the mathematical one, is a path-conservative method, capable of reconstructing the behaviour of physical and geometrical variables at the cell boundaries, where the discrete solution of hyperbolic systems of equations is discontinuous. A nonlinear path suitable for the shallow water context is adopted. The resulting model is shown to be well-balanced and accurate to the second order and is further validated against analytical solutions related to channels with power-law cross sections, specifically for dam break patterns over a variable-width channel and the run-up dynamics of long water waves over sloping bays
A 2D local discontinuous Galerkin method for contaminant transport in channel bends
No full textIn this work, a third-order local discontinuous Galerkin (LDG) method is applied to the numerical integration of a two-dimensional, depth-integrated mathematical model escribing the flow hydrodynamics and the transport of a passive contaminant in open-channel bends.
The mathematical model is described in Begnudelli et al. (2010) [14] and treats the main physical aspects of the flow in curved channels (including bottom shear, momentum dispersion, scalar dispersion and turbulent diffusion) with a homogeneous degree of complexity and exhaustiveness. The numerical integration is performed using the scheme presented in Caleffi and Valiani (2012) [27], which is extended in this work to allow the treatment of diffusion terms. The capability of the model to correctly and efficiently treat computational domains with curved boundaries while preserving the exact well-balancing property is of fundamental importance for the correct simulation of the free-surface flow in laboratory flumes and real channels.
Three test cases are used to validate the model. The first case consists of a problem, with an analytical solution, related to the tracer convection–diffusion in a uniform flow. This test is selected to highlight the excellent resolution obtainable using an LDG method. The other test cases, consisting of comparisons
between numerical results and laboratory data, are selected to verify the capability of the model to reproduce real world phenomena. Both steady and unsteady tracer dispersion are taken into account.
The results show the potentiality of the high-order accuracy models when applied to engineering problems that are characterized by inherently complex physical phenomena. The results also confirm the possibility of using extremely coarse grids, which leads to a high computational efficiency, in these
real-world applications. Moreover, the good agreement between the experiments and simulations is a further confirmation of the suitability of two-dimensional depth-averaged models for the study of the convection–dispersion of momentum and pollutants in curved channels and rivers.
Finally, the reconstruction of the solution by polynomial shape functions, which are typical of the LDG schemes, allows the streamline curvature, which is necessary to evaluate the diffusion coefficients, to be computed in a self-consistent manner, without the use of arbitrary reconstructions
Closure to "Case study: Malpasset dam-break simulation using a two-dimensional finite volume method" by Alessandro Valiani, Valerio Caleffi, and Andrea Zanni
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A comparison between different approaches for the numerical treatment of bottom discontinuities in a DG perspective
No full textThe use of Discontinuous Galerkin (DG) numerical schemes for the Shallow Water Equations (SWE) integration is greatly increased in the last decade. The efforts of many researchers were initially devoted to conceive techniques for the exact preservation of the motionless state over non-flat bottom. Recently, such efforts are mainly oriented to the proper treatment of the bottom discontinuities and to the exact preservation of the moving-water steady flows.
In this work, in the unified context consisting of third-order accurate DG-SWE schemes, a comparison between five numerical treatments of the bottom discontinuities is presented.
We consider three widespread approaches that perform well if the motionless state has to be preserved. First, a simple technique, which consists in a proper initialization of the bed elevation that imposes the continuity of the bottom profile is taken into account [Kesserwani and Liang, INT J NUMER METH ENG, 86, 47-69, 2011]. Then, we consider the hydrostatic reconstruction method [Audusse et al., SIAM J SCI COMPUT, 25, 2050-2065, 2004] and a path-conservative scheme based on a linear integration path [Parés, SIAM J NUMER ANAL, 44, 300-321, 2006].
We also consider two further approaches (both based on mechanical principles) which are promising for the preservation of a moving-water steady state. A model is obtained modifying the hydrostatic reconstruction as suggested in [Caleffi and Valiani, ASCE JEM, 135(7), 684-696, 2009]. This method is characterized by a correction of the numerical flux based on the local conservation of the total energy. The last model is obtained improving the path-conservative scheme using a non-linear path.
Several test cases are used to verify the accuracy, the well-balancing, the behavior in simulating a quiescent flow and the resolution in simulating unsteady flows of the models. A specific test case is also introduced to highlight the difference between the five schemes when a steady moving flow interacts with a bottom step
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