Chapter 4 X.Pdf

CHAPTER 4 ANALYSIS OF THE GOVERNING EQUATIONS 4.1 INTRODUCTION The mathematical nature of the systems of governing equations deduced in Chapters 2 and 3 is investigated in this chapter. The systems of PDEs that express the quasi-equilibrium approximation are studied in greater depth. The analysis is especially important for the systems whose solutions are likely to feature discontinuities, as a result of strong gradients growing steeper, or because the initial data is already discontinuous. The geomorphic shallow-water flows, such as the dam-break flow considered in Chapter 3, are the paradigmatic example of flows for which discontinuities arise fundamentally because of the initial conditions. Laboratory experiments show that, in the first instants of a sudden collapse of a dam, vertical accelerations are strong and a bore is formed, either through the breaking of a wave (Stansby et al. 1998) or due to the incorporation of bed material (Capart 2000, Leal et al. 2002). Intense erosion occurs in the vicinity of the dam and a highly saturated wave front is likely to form at ttt≡≈0 4 , thg00= , where h0 is the initial water depth in the reservoir. The saturated wave front can be seen forming in figure 3.1(a). Unlike the debris flow resulting from avalanches or lahars, the saturated front is followed by a sheet-flow similar to that 303 encountered in surf or swash zones (Asano 1995), as seen in figures 3.2(b) and (c). The intensity of the sediment transport decreases in the upstream direction as the flow velocities approach fluvial values. While the flow is highly erosive in the wave front region, sediment debulking may result into generalised deposition as the flow velocity decreases. Thus, the solution of the competent system of equations comprises continuous reaches eventually separated by discontinuities. If the collapse of a dam should be idealised as an instantaneous removal of a vertical barrier, initially separating two constant states that extend indefinitely on both up- and downstream directions, as seen in figure 4.1, the dam-break problem is a Riemann problem. Riemann problems admit self-similar solutions relatively to the variable x tgh0 if the hyperbolic equations are homogeneous, i.e., if G = 0. For special cases of the flux vector, F, explicit expressions for the dependent variables, functions of time and spatial co-ordinates, are attainable, as it is the case of the flat- fixed-bed solution for the shallow water equations presented by Ritter in 1887. The latter is generalized and thoroughly described by Stoker (1958), pp. 311-326, 333-341 and 513-522. The importance of explicit theoretical solutions is threefold: i) they are computationally simpler than numerical solutions, ii) they provide an order of magnitude and important phenomenological insights on the behaviour of the system under more general conditions and iii) they provide a way to access the quality of numerical discretization techniques. Stoker’s or Ritter’s solutions have been used to verify the quality of, virtually, all numerical models build ever since Stoker’s reference book was published. They also provided an elemental proof of existence of a weak solution for the shallow water equations and sparkled important theoretical advances on the hydrodynamics of unsteady open-channel flow (e.g. Su & Barnes 1970, Hunt 1983, 1984). The practical use of Stoker’s solution was extended to play the role of a reference situation for the interpretation of experimental results on dam-break flows. As an example, Ritter’s value for the velocity of the dam-break wave front, 2 gh0 , is the reference order of magnitude of the dam-break flood wave propagation, to which every experimental result is compared and at whose light is discussed. Pure hydrodynamic models, Stoker’s solution included, fail to reproduce the characteristic time and length scales of the dam-break flow when morphological impacts are important. Because of this fundamental inadequacy, research in geomorphic dam-break flows has been conducted through a combination of fieldwork, laboratory physical modelling, theoretical analysis and numerical simulation. Research projects like CADAM and IMPACT provided the framework for a number of studies, notably Capart & Young (1998), Fraccarollo & Capart (2002) or Leal et al. (2005), that resulted in major advances, comparable to those proportioned by the landmark works of Dressler (1952, 1954), Whitham (1955) and Stoker, op. cit., in the conceptualisation of the phenomena involved and in the development of simulation capabilities. Especially relevant is the study of Fraccarollo & Capart (2002) whom, in the wake of works by Capart & Young (1998) and Fraccarollo & Armanini (1999), have built a solution for the Riemann problem posed by the homogeneous geomorphic shallow-water equations subjected to initial conditions comprising a jump in the water level. The solution is not explicit because of the strong non-linearity of the closure equations. Numerical computations are unavoidable because the invariants of the simple centred waves can not be explicitly found. 304 y/h0 RL YL RR YR YbL YbR 0 x0/h0 x/h0 FIGURE 4.1. Graphic depiction of the initial conditions for the Riemann problem posed to the geomorphic shallow water equations. The variables, Y, R, and Yb, are, respectively, the water level, the unit mass discharge and the bed elevation. The subscripts L and R stand, respectively, for the left and right states, respectively. The main objectives of this chapter are, in the wake of Fraccarollo & Capart (2002), the development of a weak solution of the Riemann problem for the geomorphic shallow water equations and the description of the main features of the wave structure. Special attention will be devoted to the condition of existence of alternative wave structures, depending on the initial data. The initial values for the Cauchy problem are the left and right states, characterised by the water elevation, Y, bed elevation, Yb, and total mass discharge per unit width, R. Adding to the discontinuity in the water level, the initial discontinuity in the bed elevation is, thus, explicitly addressed. Most of the chapter is dedicated to the study of the characteristic fields for which discontinuities are likely to develop. It is necessary to investigate the fundamental properties of the characteristic fields, notably signal, monotonicity and non-linearity. In addition, because the debate on the role of non-linear algebraic equations, describing important physical phenomena such as sediment transport capacity or bulk flow resistance, is yet to be closed, questions concerning existence and uniqueness of the solution must, thus, be posed. The existence of the solutions for the Riemann problem was proved by Lax (1957) for a finite set of strictly hyperbolic, genuinely non-linear, conservation equations. However, the proof is valid for the case of small discontinuities in the initial data, which is generally not the case in the dam-break problem. Further works by Glimm (1965), the first proof for arbitrary initial values, Smoller (1969), di Perna (1973), Dafermos (1973) or Liu (1974) helped building a library of theoretical results that may be used as a guide to establish the conditions of existence and uniqueness of the Riemann solution of the geomorphic shallow water equations. Considerations on the existence and unicity of the solution of the Riemann problem for the geomorphic shallow water equations are, thus, legitimate and will be addressed. The text is structured so as to highlight the main objectives mentioned above. Wave-like description of wave forms and hyperbolicity are discussed in §4.2.1. The governing equations 305 subjected to analysis are described in §4.2.2, namely in what concerns the embedded hypotheses. The following sections, §4.2.3 to §4.2.5 are dedicated to the mathematical analysis of the system of equations. Special attention is given to the type of hyperbolicity and non-linearity of the system of equations. The properties of each of the characteristic fields are investigated, notably signal, monotonicity and non-linearity. Two possible Riemann wave structures are identified in §4.3. The conditions for the existence of each of the types of solution is discussed in §4.3.2. Entropy-compatible solutions are calculated in §4.4, with shocks determined by the Rankine-Hugoniot jump equations. The existence of Riemann invariants for the simple waves encountered is also discussed. 4.2 MATHEMATICAL ANALYSIS OF THE CHARACTERISTIC FIELDS 4.2.1 Notes on hyperbolicity and non-linear propagation of non-linear hyperbolic waves 4.2.1.1 Source terms and hyperbolicity The generic quasi-linear, autonomous, non-conservative form of the governing equations is AB∂ VVG+∂ = (4.1) tix( ) i ( ) where A e Bi, i = 1…m, are real bounded matrix-valued functions of the dependent variables, m is the dimension of the number of space-like variables, V is the n-dimensional vector of dependent variables and G, the source term, is a n-dimensional vector valued bounded function of the dependent variables. The source terms are of paramount importance in what concerns the quality of the solutions, understood as its physical plausibility and agreement with observations. They are less important for the study of the mathematical properties of the system, a claim that is substantiated next. It should be made clear that the study of the mathematical properties are, in this chapter, restricted to the qualitative discussion of the solutions, namely existence and unicity, and to the study of the nature of the propagation of information, in particular type and number of conditions at the contour of the solution domain. Regarding the existence and unicity of the solutions, if the initial conditions are smooth bounded functions and the components of the matrixes A to Bm are smooth functions of the dependent variables, there is a region around the initial conditions in which the solution exists and is unique provided that the source term, G, is integrable.

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