A path-conservative Osher-type scheme for axially symmetric compressible flows in flexible visco-elastic tubes

Flexible tubes are widely used in modern industrial hydraulic systems as connections between different components like valves, pumps and actuators. For the design and the analysis of the temporal behavior of a hydraulic system, one therefore needs an accurate mathematical model that describes the fluid flow in a compliant duct. Hence, in this paper we want to model the fluid–structure-interaction (FSI) problem given by the axially symmetric flow of a compressible barotropic fluid that flows through flexible tubes made of vulcanized rubber. The material of the tube can be described by using a visco-elastic rheology, which takes into account the strain relaxation of the material. The resulting mathematical model consists in a one-dimensional system of nonlinear hyperbolic partial differential equations (PDE) with non-conservative products and algebraic source terms. To solve this system numerically, we apply the DOT method, which is a generalized path-conservative Osher-type Riemann solver for conservative and non-conservative hyperbolic PDE recently proposed in [23] and [22].We provide numerical evidence that the proposed DOT Riemann solver is well-balanced for the governing PDE system under consideration. The method is compared to available quasi-exact solutions of the Riemann problem in the case of an elastic wall described by the Laplace law. It is also compared to available experimental data and exact solutions obtained in the frequency domain for a linear visco-elastic wall behavior. In all cases under investigation the proposed path-conservative finite volume scheme based on the DOT Riemann solver is able to produce very accurate results.