Alessandro Valiani

Finite volume method for simulating extreme flood events in natural channels

The need for mitigating damages produced by extreme hydrologie events has stimulated the European Community to fund several projects. The Concerted Action on Dam-break Modelling workgroup (CADAM) performed a considerable work for the development of new codes and for the adequate verification of their performance. In the context of the CADAM project, a new 2D computer code is developed, tested and applied, as described in the present paper. The algorithm is obtained through the spatial discretisation of the shallow water equations by a finite volume method, based on the Godunov approach. The HLL Riemann solver is used. A second order accuracy in space and time is achieved, respectively by MUSCL and predictor-corrector techniques. The high resolution requirement is ensured by satisfaction of TVD property. Particular attention is posed to the numerical treatment of source terms. Accuracy, stability and the reliability of the code are tested on a selected set of study cases. A grid refinement analysis is performed. Numerical results are compared with experimental data, obtained by the physical modelling of a submersion wave on a portion of the Toce river valley, Italy, performed by ENEL-HYDRO and considered as representative of a real life flood occurrence.

Fourth-order balanced source term treatment in central WENO schemes for shallow water equations

The aim of this work is to develop a well-balanced central weighted essentially non-oscillatory (CWENO) method, fourth-order accurate in space and time, for shallow water system of balance laws with bed slope source term. Time accuracy is obtained applying a Runge-Kutta scheme (RK), coupled with the natural continuous extension (NCE) approach. Space accuracy is obtained using WENO reconstructions of the conservative variables and of the water-surface elevation. Extension of the applicability of the standard CWENO scheme to very irregular bottoms, preserving high-order accuracy, is obtained introducing two original procedures. The former involves the evaluation of the point-values of the flux derivative, coupled with the bed slope source term. The latter involves the spatial integration of the source term, analytically manipulated to take advantage from the regularity of the free-surface elevation, usually smoother than the bottom elevation. Both these procedures satisfy the C-property, the property of exactly preserving the quiescent flow. Several standard one-dimensional test cases are used to verify high-order accuracy, exact C-property, and good resolution properties for smooth and discontinuous solutions.

Hydrodynamic instability of multiple four-wave mixing

In the regime of normal dispersion and low-frequency detunings (or high powers), four-wave mixing is shown to undergo a hydrodynamic type of instability. Such instability involves the formation of shocks (steep fronts) from smooth initial data that are regularized through the appearance of trains of fast oscillations, which exhibit solitonlike behavior, colliding elastically.

A comparison between bottom-discontinuity numerical treatments in the DG framework

Abstract In this work, using an unified framework consisting in third-order accurate discontinuous Galerkin schemes, we perform a comparison between five different numerical approaches to the free-surface shallow flow simulation on bottom steps. Together with the study of the overall impact that such techniques have on the numerical models we highlight the role that the treatment of bottom discontinuities plays in the preservation of specific asymptotic conditions. In particular, we consider three widespread approaches that perform well if the motionless steady state has to be preserved and two approaches (one previously conceived by the first two authors and one original) which are also promising for the preservation of a moving-water steady state. Several one-dimensional test cases are used to verify the third-order accuracy of the models in simulating an unsteady flow, the behavior of the models for a quiescent flow in the cases of both continuous and discontinuous bottom, and the good resolution properties of the schemes. Moreover, specific test cases are introduced to show the behavior of the different approaches when a bottom step interact with both steady and unsteady moving flows.

Thermal Instability of a Power-Law Fluid Flowing in a Horizontal Porous Layer with an Open Boundary: A Two-Dimensional Analysis

A two-dimensional analysis of the onset of thermal convective instability in a horizontal porous layer with open upper boundary is carried out. The saturating fluid is non-Newtonian of power-law behaviour, and its flow is represented through a suitable extension of Darcy’s law. A model of temperature-dependent viscosity is employed where the consistency index is considered as variable, while the power-law index is assumed to be constant. Numerical data for the neutral stability and for the critical values of a modified Darcy–Rayleigh number have been obtained. The feasibility of an experimental validation of the theoretical results predicted by the stability analysis is discussed in detail. An experimental set-up based on a Hele-Shaw cell is described, and preliminary results relative to the onset of convective cells are described. Observed hysteretic effects and deviations from the rheological model are identified as potential sources of uncertainty.

Momentum balance in the shallow water equations on bottom discontinuities

This work investigates the topical problem of balancing the shallow water equations over bottom steps of different heights. The current approaches in the literature are essentially based on mathematical analysis of the hyperbolic system of balance equations and take into account the relevant progresses in treating the non-conservative form of the governing system in the framework of path-conservative schemes. An important problem under debate is the correct position of the momentum balance closure when the bottom elevation is discontinuous. Cases of technical interest are systematically analysed, consisting of backward-facing steps and forward-facing steps, tackled supercritical and subcritical flows; critical (sonic) conditions are also analysed and discussed. The fundamental concept governing the problem and supported by the present computations is that the energy-conserving approach is the only approach that is consistent with the classical shallow water equations formulated with geometrical source terms and that the momentum balance is properly closed if a proper choice of a conventional depth on the bottom step is performed. The depth on the step is shown to be included between the depths just upstream and just downstream of the step. It is also shown that current choices (as given in the literature) of the depth on (or in front of) the step can lead to unphysical configurations, similar to some energy-increasing solutions.

Well balancing of the SWE schemes for moving-water steady flows

In this work, the exact reproduction of a moving-water steady flow via the numerical solution of the one-dimensional shallow water equations is studied.A new scheme based on a modified version of the HLLEM approximate Riemann solver (Dumbser and Balsara (2016) [18]) that exactly preserves the total head and the discharge in the simulation of smooth steady flows and that correctly dissipates mechanical energy in the presence of hydraulic jumps is presented. This model is compared with a selected set of schemes from the literature, including models that exactly preserve quiescent flows and models that exactly preserve moving-water steady flows.The comparison highlights the strengths and weaknesses of the different approaches. In particular, the results show that the increase in accuracy in the steady state reproduction is counterbalanced by a reduced robustness and numerical efficiency of the models. Some solutions to reduce these drawbacks, at the cost of increased algorithm complexity, are presented. The exact numerical simulation of steady flow governed by the SWE is studied.A new energy-balanced formulation of the HLLEM approximate Riemann solver is given.Well-balanced and energy-balanced schemes by literature are compared.The comparison highlights strengths and weaknesses of the different approaches.

Numerical methods for hydraulic transients in visco-elastic pipes

Abstract In technical applications involving transient fluid flows in pipes, the convective terms of the corresponding governing equations are generally negligible. Typically, under this condition, these governing equations are efficiently discretized by the Method of Characteristics. Only in the last years the availability of very efficient and robust numerical schemes for the complete system of equations, such as recent Finite Volume Methods, has encouraged the simulation of transient fluid flows with numerical schemes different from the Method of Characteristics, allowing a better representation of the physics of the phenomena. In this work, a wide and critical comparison of the capability of Method of Characteristics, Explicit Path-Conservative (DOT solver) Finite Volume Method and Semi-Implicit Staggered Finite Volume Method is presented and discussed, in terms of accuracy and efficiency. To perform the analysis in a framework that properly represents real-world engineering applications, the visco-elastic behaviour of the pipe wall, the effects of the unsteadiness of the flow on the friction losses, cavitation and cross-sectional changes are taken into account. Analyses are performed comparing numerical solutions obtained using the three models against experimental data and analytical solutions. In particular, water hammer studies in high density polyethylene pipes, for which laboratory data have been provided, are used as test cases. Considering the visco-elastic mechanical behaviour of plastic materials, 3-parameter and multi-parameter linear visco-elastic rheological models are adopted and implemented in each numerical scheme. Original extensions of existing techniques for the numerical treatment of such visco-elastic models are introduced in this work for the first time. After a focused calibration of the visco-elastic parameters, the different performance of the numerical models is investigated. A comparison of the results is presented taking into account the unsteady wall-shear stress, with a new approach proposed for turbulent flows, or simply considering a quasi-steady friction model. A predominance of the damping effect due to visco-elasticity with respect to the damping effect related to the unsteady friction is confirmed in these contexts. Moreover, all the numerical methods show a good agreement with the experimental data and a high efficiency of the Method of Characteristics in standard configuration is observed. Finally, three Riemann Problems are chosen and run to stress the numerical methods, taking into account cross-sectional changes, more flexible materials and cavitation cases. In these demanding scenarios, the weak spots of the Method of Characteristics are depicted.

Efficient analytical implementation of the DOT Riemann solver for the de Saint Venant–Exner morphodynamic model

Abstract Within the framework of the de Saint Venant equations coupled with the Exner equation for morphodynamic evolution, this work presents a new efficient implementation of the Dumbser-Osher-Toro (DOT) scheme for non-conservative problems. The DOT path-conservative scheme is a robust upwind method based on a complete Riemann solver, but it has the drawback of requiring expensive numerical computations. Indeed, to compute the non-linear time evolution in each time step, the DOT scheme requires numerical computation of the flux matrix eigenstructure (the totality of eigenvalues and eigenvectors) several times at each cell edge. In this work, an analytical and compact formulation of the eigenstructure for the de Saint Venant–Exner (dSVE) model is introduced and tested in terms of numerical efficiency and stability. Using the original DOT and PRICE-C (a very efficient FORCE-type method) as reference methods, we present a convergence analysis (error against CPU time) to study the performance of the DOT method with our new analytical implementation of eigenstructure calculations (A-DOT). In particular, the numerical performance of the three methods is tested in three test cases: a movable bed Riemann problem with analytical solution; a problem with smooth analytical solution; a test in which the water flow is characterised by subcritical and supercritical regions. For a given target error, the A-DOT method is always the most efficient choice. Finally, two experimental data sets and different transport formulae are considered to test the A-DOT model in more practical case studies.

Mathematical study of linear morphodynamic acceleration and derivation of the MASSPEED approach

Abstract Morphological accelerators, such as the MORFAC (MORphological acceleration FACtor) approach ( Roelvink, 2006 ), are widely adopted techniques for the acceleration of the bed evolution, which reduce the computational cost of morphodynamic numerical simulations. In this work we apply an acceleration to the one-dimensional morphodynamic problem described by the de Saint Venant–Exner model by multiplying all the spatial derivatives related to the mass or momentum flux by an acceleration factor  ≥ 1 which may be different for each equation. The goal is to identify the best combination of the accelerating factors for which (i) the bed responds linearly to hydrodynamic changes; (ii) a decrease of the computational cost is obtained. The sought combination is obtained by studying the behavior of an approximate solution of the three eigenvalues associated with the flux matrix of the accelerated system. This approach allows to derive a new linear morphodynamic acceleration technique, the MASSPEED (MASs equations SPEEDup) approach, and the a priori determination of the highest possible acceleration for a given simulation. In this new approach both mass conservation equations (water and sediment) are accelerated by the same factor, differently from the MORFAC approach where only the sediment mass equation is modified. The analysis shows that the MASSPEED gives a larger validity range for linear acceleration and requires smaller computational costs than that of the MORFAC approach. The MASSPEED approach is then implemented using an adaptive approach that applies the maximum linear acceleration similarly to the implementation of the Courant–-Friedrichs–-Lewy stability condition. Finally, numerical simulations have been performed in order to assess accuracy and efficiency of the new approach. Results obtained in the long-term propagation of a sediment hump demonstrate the advantages of the new approach. The validation of the method is performed under steady or quasi-steady flow conditions, whereas further investigation is needed to extend morphological accelerators to fully unsteady flows.