The Propagation of Shock Waves in Incompressible Fluids: The Case of Freshwater
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The Propagation of Shock Waves in Incompressible Fluids: The Case of Freshwater Andrea Mentrelli · Tommaso Ruggeri
Received: 19 December 2013 / Accepted: 17 January 2014 © Springer Science+Business Media Dordrecht 2014
Abstract In this paper we investigate the basic features of shock waves propagation in freshwater in the framework of a hyperbolic model consisting of the one-dimensional Euler equations closed by means of polynomial equations of state extracted from experimental tabulated data available in the literature (Sun et al. in Deep-Sea Res. I 55:1304–1310, 2008). The Rankine–Hugoniot equations are numerically solved in order to determine the Hugoniot locus representing the set of perturbed states that can be connected through a k-shock to an unperturbed state. The results are found to be consistent with those previously obtained in the framework of the EQTI model by means of a modified Boussinesq equation of state. Keywords Incompressible fluids · Shock waves in water · Rankine–Hugoniot conditions
1 Introduction Incompressibility is a useful idealization for those materials which exhibit very high resistance to volume change. For this reason, even though fully incompressible materials do not exist in nature, mathematical modelling of incompressible fluids has been given considerable attention during the past decades. In the case of a purely mechanical framework, namely when there is no change in temperature, a material is said to be incompressible if the changes of its specific volume or, equivalently, its density, are negligible. In this case, a broad literature is available concerning qualitative analysis as well as numerical methods for building the solutions of incompressible model equations as limit of the solutions of compressible ones, as the sound velocity goes to infinity [1–4].
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A. Mentrelli ( ) · T. Ruggeri Department of Mathematics & Research Center of Applied Mathematics (CIRAM), University of Bologna, via Saragozza 8, 40123 Bologna, Italy e-mail: [email protected] T. Ruggeri e-mail: [email protected]
A. Mentrelli, T. Ruggeri
In contrast to the purely mechanical case, the non isothermal case is not even well defined and several definitions of incompressibility, leading to different models, have been proposed over the years [5–9]. For a discussion of the limitations of these models, see [10–15]. In order to unify the treatment of compressible and incompressible fluids, the first step to be taken is to choose the pressure, instead of the density, as unknown field variable. As the aim of this paper is to study the propagation of shock waves, we consider the Euler system of equations in which viscosity and heat conductivity are neglected. The resulting system of conservation laws of mass, momentum and energy is the following: ∂ρ ∂ρvi + = 0, ∂t ∂xi ∂ ∂ρvj + (ρvi vj + pδij ) = 0 (j = 1, 2, 3), ∂t ∂xi ∂(ρε + 12 ρv 2 ) ∂ 1 2 + ρε + ρv + p vi = 0, ∂t ∂xi 2
(1)
where ρ, v ≡ (vi ), p and ε are, respectively, the density, the velocity, the pressure and the internal energy d
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