Momentum balance in the shallow water equations on bottom discontinuities
Introduction
The numerical integration of shallow water equations (SWE) with source terms has been intensively investigated in recent years due to the significant advances in computational fluid dynamics in finite difference methods, finite volume methods, discontinuous Galerkin methods, and so forth (Caleffi, Valiani, 2009, Caleffi, Valiani, 2012, Caleffi, Valiani, Bernini, 2006, Caleffi, Valiani, Li, 2016). Particular interest has been devoted to discontinuous solutions, typically due to the physics of stationary jumps and moving bores generated by the hyperbolic nature of the homogeneous problem, in light of the more general Rankine–Hugoniot theory of inviscid shock propagation. A further source of discontinuity is due to the possible discontinuous profile of the bottom elevation, which introduces singularities in the term related to the topography.
From recent works on proper numerical methods for SWE on discontinuous bed elevations, the first contribution may be considered (Alcrudo and Benkhaldoun, 2001). The authors analysed the similarity solutions of the Riemann problem over a step, evidencing the existence of a standing discontinuity over the step and studying a relevant number of possible solutions. They imposed total head conservation across the step and highlighted the need for an additional kinetic energy term to properly take the energy dissipation into account, when necessary. This work has the merit of highlighting the wide variety of solutions that can be found, which are considerably more numerous than the classical case of the Riemann problem over a flat horizontal bottom. Moreover, this work demonstrates the potential complexity of any numerical method that has the ambition to incorporate the variability of such a number of situations.
The study LeFloch and Thanh (2011), which is a deepening of the previous study LeFloch and Thanh (2007), investigated the Riemann problem over a discontinuous bed elevation and provided a detailed analysis of the possible solutions. This work is likely the most complete study on this topic. More specifically, the authors investigated the existence and uniqueness conditions for such solutions, showing the uniqueness in the non-resonant regime and the existence of multiple solutions in the resonant regime. The eigensystem analysis provides two genuinely hyperbolic characteristic fields and one linearly degenerate field. Regarding the linearly degenerate field, it is shown that across a discontinuity, the bottom elevation must remain constant or the discontinuity must be stationary. This result means that over a discontinuous bed, the propagation celerity must be zero. The jump relations on the discontinuity provide the constancy of the unit discharge and the conservation of the total head of the flow, which are exactly the assumptions adopted herein.
The investigation is extended in Han and Warnecke (2014), where the exact Riemann solutions for the shallow water equations with a bottom step is completed, including the dry bed problem, which can occur in specific circumstances.
The conservation of the discharge and of the total head are successfully applied in 1D modelling of open-channel hydraulics, both for the unit width channel and in a channel of arbitrary shape (Murillo, Garcia-Navarro, 2013, Murillo, Garcia-Navarro, 2014, Navas-Montilla, Murillo, 2015).
A different approach is presented in Bernetti et al. (2008), which presents the exact solution of the Riemann problem for shallow water equations with a step-like bottom. The solution was obtained by solving the system that included an additional equation for the bottom geometry and then using the principles of conservation of mass and momentum across the step. By writing the Rankine–Hugoniot condition for the standing wave across the step, it is shown that the resulting solution is unique and satisfies the principle of the dissipation of energy across the shock wave. Because all the methods establish the momentum integral principle across the step, the key step is estimating the force exerted by the step on the fluid. In the framework of SWE and therefore assuming a hydrostatic distribution of pressure, it is possible to express the step force as a function of an equivalent, conventional depth on the step. In the case of a forward-facing step, such a depth is evaluated, as in all the other similar methods, using the depth just in front of the step measured from the centroid of the step. In the case of a backward-facing step, which is not explicitly described, it can be argued that the depth must be evaluated again as the depth just in front of the step measured from the centroid of the step. More specifically, the phrase “in front of” means the upstream flow depth minus half the step height in the forward-facing step case, and it means the downstream flow depth minus half the step height depth in the backward-facing step case. This choice is similar to the other methods described in the following (Cozzolino, Morte, Covelli, Giudice, Pianese, 2011, Rosatti, Begnudelli, 2010, Rosatti, Fraccarollo, 2006, Rosatti, Murillo, Fraccarollo, 2008), even if the formal position of the momentum balance appears slightly different.
A strictly closed approach to the last one is that from Rosatti and Fraccarollo (2006), which investigated the numerical computation of one-dimensional, unsteady, free surface flows over a mobile bed. They considered a strong interaction between the flow and the erodible bottom, taking non-conservative terms in the momentum equation into account and solving the related Riemann problem. The technique was named AWB (approximately well balanced), and it was applied to schematic and experimental test cases. At the simplest order of approximation, they considered a backward-facing step as the typical scheme concerning the flow across two consecutive computational cells. The force exerted by the step on the fluid is proportional to the height of the step and to a “proper” water depth “in front of” the step.
The word “proper” means that the depth must be measured from the centroid of the step. The term “in front of” means that the downstream (or right) depth must be considered if the step is facing backward and that the upstream (or left) depth must be considered if the step is facing forward. The force is clearly exerted on the fluid in the flow direction in the former case, whereas it is opposite to the flow direction in the latter case. The context of this work is broader than that of the present work because (as in Rosatti et al. (2008)) a mobile bed is considered, but the same procedure was also used for a fixed bed elevation (Rosatti and Begnudelli, 2010).
In particular, Rosatti and Begnudelli (2010) thoroughly analysed the Riemann problem for the one-dimensional shallow water equations from theoretical and numerical perspectives. The analysis of the wave at the step leads to a non-conservative crossing of the step in terms of total head. The momentum balance on the step is closed using an integral momentum balance that is very similar to that proposed herein. The only difference is the estimate of the depth on the step, which is performed in Rosatti and Begnudelli (2010) using the same technique proposed by Rosatti and Fraccarollo (2006) and Bernetti et al. (2008). However, the simple reasoning proposed here does not require a new formulation of the Riemann invariants, and a proper choice of the special depth satisfies both momentum and energy conservation requirements.
A recent contribution to the problem which given by Cozzolino et al. (2011), which analysed the shallow water equations on bottom discontinuities. In this work, the hydrostatic-like pressure distribution at the step, according the concepts of Bernetti et al. (2008) and Rosatti and Fraccarollo (2006), is also analysed and discussed in the framework of the path-conservative theory of Maso et al. (1995). They rebut the preservation of the total head through the bed step (Alcrudo, Benkhaldoun, 2001, Caleffi, Valiani, 2009, Caleffi, Valiani, Li, 2016, LeFloch, Thanh, 2011), claiming that experimental evidence shows that the dissipation on the step is a well-known phenomenon. They consider that the integral momentum balance, taking into account the force that the bed exerts onto the flow, conflicts with the total head conservation hypothesis. In Cozzolino et al. (2011), they explicitly make reference to the flow detachment and reattachment and eddy recirculation cells, which subtract mechanical energy from the mean flow.
By contrast, the hypothesis supported here is that the dissipation is evident, but it is not automatically incorporated in the shallow water scheme. If a proper source term is not added, then the total head is conserved, and integral momentum conservation can also be preserved by simply selecting a proper depth on the step.
Caleffi and Valiani (2009) proposed a finite volume WENO scheme, which is fourth-order accurate in space and time, for the numerical integration of shallow water equations with the bottom slope source term. The method for managing bed discontinuities is based on a suitable reconstruction of the conservative variables at the cell interfaces, coupled with a correction of the numerical flux based on the local conservation of total energy. Properly selected test cases show the efficiency of the method in treating discontinuities. The method finds a flux correction on bed steps that takes into account the force exchanged between the step itself and the fluid, and the flux balancing is achieved not only in still water conditions but, more generally, also for each steady-state dynamic condition. The analytical treatment of the step that is proposed here is conceived as a deepening of the force estimate on the step itself.
In Caleffi et al. (2016), using an unified framework consisting of a third-order accurate discontinuous Galerkin scheme, five different numerical methods applied to the free-surface shallow flow simulation on bottom steps are compared. The role that the treatment of bottom discontinuities plays in the preservation of specific asymptotic conditions is examined. In particular, three widespread approaches based on the motionless steady state balancing (Audusse, Bouchut, Bristeau, Klein, Perthame, 2004, Kesserwani, Liang, 2011, Parés, 2006) are compared with two approaches (Caleffi, Valiani, 2009, Caleffi, Valiani, Li, 2016) that are based on the preservation of a moving-water steady state. The fundamental findings support the concept that the well balancing of a moving steady state (rather than a well-balanced model in the case of still water) significantly improves the overall behaviour of the schemes. This perspective is successful both in the framework of the classical finite volume approach and in the framework of the path-conservative schemes. Finally, in the context of such path-conservative schemes, the fundamental concept of total head conservation on the contact discontinuity over the step provides optimal results in reconstructing well-balanced solutions.
Section snippets
Shallow water equations with source terms: discrepancies between the physics and the scheme
The present approach is completely based on the necessity of simultaneously satisfying the SWE as balance laws with source terms on bed discontinuities and the constraints implicated in such balance laws, in other words, to respect the schematized physics that we are assuming.
Referring to the unit-width channel, the standard form of SWE without friction is considered: where: Y(x, t), q(x, t) and z(x, t) are the depth, the specific discharge and the
Some analytical tools
Some definitions and some analytical results are presented here, which are taken from classical hydraulics and from Valiani and Caleffi (2008).
Let Y, Yc be the depth and critical depth, E, Ec be the specific energy and the critical specific energy, and F, Fc be the total force and the critical total force per unit width. The main definitions and their non-dimensional counterparts (η, Γ, Φ as the non-dimensional depth, energy and total force, respectively) are summarized here (ρ is fluid
Subcritical and supercritical flows over bottom discontinuities
The computations are organized as follows. A precise value of depth is supposed to exist as the only one that allows for momentum conservation. It can be considered as being measured from a bed elevation that is an intermediate value between zL and zR; the centroid of the step can be the natural reference point, but this is not a necessary assumption in the following because in the total head conservation, only zL and zR appear directly, while in the integral momentum balance, only the step
Conclusions
This work consist of a review of the SWE behaviour on a discontinuity in bottom elevation. This review is focused on the differences between the SWE scheme and real-world hydraulics on backward- and forward-facing steps. In the classical limits of the SWE scheme, the pressure distribution is hydrostatic on each vertical, and the flow velocity is uniform on each vertical (by definition of SWE). Using the classical results of LeFloch and Thanh (2011), the conservation of generalized Riemann
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