Entropy consistent and hyperbolic formulations for compressible single- and two-phase flows modeling in both rigid and elastically deformable pipes: Application to Euler, Kapila and Baer-Nunziato equations

Abstract

The mathematical modeling of compressible flows in both rigid and elastic pipes is discussed here. Both single- and two-phase flow modeling are considered in the present paper. First, the derivation of the models through the integration of the 3-D equations over the radially deformable inner pipe cross-section is described. Then, the Coleman-Noll procedure is used in order to formulate constitutive/closure laws consistent with the second law of thermodynamics leading to entropy consistent models. For elastically deformable pipes, an additional closure law relating the elastic variation of the cross-section area with the internal fluid pressure is also considered. The derivation of this closure law, classically referred to as the tube law, is also discussed here. This effective averaging procedure leads to non-conservative terms in the one-dimensional models linked to the pressure effects due to the spatial and temporal cross-section variations. The mathematical properties of the obtained system of partial differential equations, i.e. hyperbolicity, the structure of the waves, the expression of the Riemann invariants and the existence of a mathematical entropy, are then examined. In addition, the consequences of the definition of shock-waves for these models are also discussed. The Euler equations are first considered and analyzed in this context. Then, the approach is extended to more advanced compressible two-phase flow modeling. The two examples considered in the present paper are the Kapila five-equation and the Baer-Nunziato seven-equation models involving non-conservative terms due to the change of volume fraction in the balance equations which are for the first time investigated in this context. The corresponding models are shown to be hyperbolic and the characteristic fields are analyzed. Finally, some shock-tube problems in both rigid or elastic pipes are examined showing agreement with the present mathematical analysis.