The Linearised Dam-Break Problem

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• Contents

A dam is a barrier that separates two horizontal layers of fluid from each other. Generally, a dam is a man made structure that is used to hold a body of water in place, forming a reservoir or lake, which can then be used to support local communities. Dams can be used in order to supply water to local communities, or act as a flood defence. They can also be used to supply water for farmland and to allow hydro-power electricity, for further examples, see [via DIALOG. (2015)]. The failure of a dam will generally have a catastrophic impact on the surrounding environment. As a consequence the study of the hydrodynamical effects of dam failure is of principal significance. The problem of a dam failure is modelled mathematically as the classical dam-break problem. It is developed to model the situation where a mass of fluid, lying on an impermeable bed, is held in place by a solid boundary (the dam) so that there is a change in the fluid surface elevation across the dam site. The dam-breaks at an initial time and the aim is to determine the behaviour of the subsequent hydrodynamic flow; that is, to determine the fluid velocity field and, in particular, the free surface behaviour. The dam-break flow is formulated mathematically by the conservation of mass and momentum equations for the fluid flow, together with appropriate boundary and initial conditions. This leads to a free boundary nonlinear evolution problem; it is therefore convenient to make some reasonable approximations to reduce the difficulty in analysing the dam-break problem. As this is a water wave problem, viscous effects are ignored and the fluid is taken to be incompressible. It is also assumed that the dam disappears instantly at the initial time. These assumptions are standard in dam-break problems, see for example [Ungarish et al. (2014)], [Yilmaz et al. (2013)], [Korobkin and Yilmaz (2009)], [Ancey et al. (2008)], [Fernandez-Feria (2006)], [Hunt (1983)], [Ostapenko (2003)]…

Throughout this monograph we consider a fluid which is incompressible and inviscid. Further, the only external force acting on the fluid is gravity. Specifically, we consider the situation when a body of fluid is initially at rest above an impermeable rigid and horizontal boundary, and is bounded above by a free surface, which initially is stationary, and represents a transition from one uniform depth to another uniform depth. The initial displacement of the free surface varies only in one horizontal space dimension. This problem is generally referred to as the two-dimensional dam-break problem. The spatial domain is specified by the Cartesian coordinate system (x, z), with z meaning distance vertically upwards and x meaning distance horizontally, whilst t ≥ 0 represents time. A fixed position in the spatial domain will be denoted as $\underset{\_}{r}=\left(x,z\right)=x\underset{\_}{i}+z\underset{\_}{k}$, where $\underset{\_}{i}=\left(1,0\right)$ and $\underset{\_}{k}=\left(0,1\right)$ represent unit vectors in the x and z directions respectively. The velocity field of the fluid is denoted as $\underset{\_}{u}=\left(u\left(x,z,t\right),w\left(x,z,t\right)\right)=u\left(x,z,t\right)\underset{\_}{i}+w\left(x,z,t\right)\underset{\_}{k}$, the free surface of the fluid is located at z = η(x, t) and the impermeable base is set at z = −h₀. The region occupied by the fluid at time t ≥ 0 will be denoted as D(t), where

In this chapter we consider a linearised form of [IBVP], which corresponds to a dam with a small step height and slope. To this end, we introduce appropriate scalings for ϕ(x, z, t) and η(x, t) from which we formulate the linearised dam-break problem. Applying the Fourier transform to our linearised problem, we obtain exact solutions for the scaled fluid velocity potential $\overline{\varphi }\left(x,z,t\right)$ and scaled free surface displacement $\overline{\eta }\left(x,t\right)$.

In this chapter we consider the free surface displacement $\overline{\eta }\left(x,t\right)$, as given by (3.30), for x ∈ ℝ as t → 0. We begin by approximating I(x, t) in (3.33) as t → 0 when x ≥ 0, and then construct the approximation to $\overline{\eta }\left(x,t\right)$, for x ∈ ℝ as t → 0, via (3.32) and (3.34). We obtain an approximation to $\overline{\eta }\left(x,t\right)$ for x ∈ ℝ as t → 0 which consists of an outer region and two inner regions, which are O(t²) neighbourhoods of the corners in the initial data, at x = 0 and x = β. The inner regions show incipient jet formation near x = β and incipient collapse near x = 0.

In this chapter we consider $\overline{\eta }\left(x,t\right)$, as given by (3.30), in the far fields as ∣x∣→∞ for t = O(1). We begin by approximating I(x, t) in (3.33) as x→∞ with t = O(1), and we find the approximation has three distinct asymptotic regions as x→∞. We then construct the approximation to $\overline{\eta }\left(x,t\right)$, for t = O(1) as ∣x∣ →∞, via (3.32) and (3.34).

In this chapter we consider $\overline{\eta }\left(x,t\right)$, as given in (3.30), as t → ∞ with x ∈ ℝ. The natural spatial coordinate as t → ∞ is $y=\left(\frac{x}{t}\right)$. We now construct an approximation to $\overline{\eta }\left(y,t\right)$ as t → ∞, uniformly for y ∈ ℝ. We find that the approximation consists of four outer regions, two of which exhibit oscillatory behaviour and two that show an exponentially small disturbance in the far field conditions. We then find inner region approximations for $\overline{\eta }\left(y,t\right)$, which connect the oscillatory regions to the exponentially decaying regions, which are in terms of Airy functions and their integrals and connect the oscillatory regions to the exponentially decaying regions.

In this chapter we give a summary of the exact solution for the free surface displacement $\overline{\eta }\left(x,t\right)$ for (x, t) ∈ ℝ×ℝ⁺ and the asymptotic structure of $\overline{\eta }\left(x,t\right)$ as t → 0 and t → ∞, as determined in Chap. 3, Chap. 4 and Chap. 6 respectively. Illustrations of the asymptotic structure are given in each case and graphs of the approximations in each case are also shown.

In this chapter we give a numerical evaluation of the exact free surface solution. We apply Simpson’s rule to $\overline{\eta }\left(x,t\right)$ and graph the numerical evaluations, which show excellent agreement with the asymptotic structure summarised in Chap. 7. In order to efficiently numerically evaluate $\overline{\eta }\left(x,t\right)$, we recall that

In this chapter we consider the situation when the free surface displacement is modelled by the linearised shallow water theory. The free surface is governed by the one-dimensional wave equation, subject to initial conditions. We find the free surface solution in terms of D’Alemberts general solution and we graph the solutions for various times. Comparisons with results in Chaps. 7 and 8 show that the shallow water theory gives the general shape of the free surface, whilst remaining monotone and oscillation free.

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• Bibliography
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