A comparison of explicit and semi‐implicit finite volume schemes for viscous compressible flows in elastic pipes in fast transient regime

Abstract

AbstractThis paper compares the accuracy and computational efficiency of fully explicit and semi‐implicit 1D and 2D finite volume schemes for the simulation of highly unsteady viscous compressible flows in laminar regime in axially symmetric compliant tubes. There are essentially two main classes of mathematical models that can be used to predict the pressure and velocity distribution along the tube: one class is based on the full compressible Navier‐Stokes equations in an axially symmetric geometry, leading to a two‐dimensional governing PDE system with moving boundaries, and the other class uses a simpler, cross‐sectionally averaged version of the Navier‐Stokes equations, which leads to a non‐conservative PDE system in only one space dimension along the axial direction of the tube. Within the first class of models, the influence of the wall friction on the flow field is directly obtained from first principles, without any further modelling assumptions and is thus valid even for highly unsteady flows. In the second case, only averaged flow quantities are available, and it is well known from previous studies published in the literature that the correct representation of the wall friction needs to be frequency dependent, since the use of a simple steady friction model, like the classical Darcy‐Weisbach law, is not sufficient to reproduce the wall friction effects in highly transient flows. For the cross‐sectionally averaged Navier‐Stokes equations, there are again two main classes of frequency‐dependent wall friction models: convolution integral (CI) models and instantaneous acceleration (IA) models.In this paper we provide a very thorough and critical comparison of all the above‐mentioned methods for the simulation of highly oscillatory flows in rigid and compliant tubes concerning accuracy and computational efficiency. From our numerical results we can conclude that the convolution integral models are significantly superior to instantaneous acceleration models concerning accuracy. Furthermore, the CI models require only a slight computational overhead if they are properly implemented by solving a set of additional ODEs for appropriate auxiliary variables, instead of directly computing the convolution integrals. We also find that semi‐implicit finite volume methods are clearly superior to conventional explicit finite volume schemes concerning computational efficiency, however, providing the same level of accuracy.