TV-HLL Solver for One-Dimensional Fluid Flow Inside Elastic Vessels

Abstract

The TV-HLL solver for the numerical solution of fluid flow inside elastic vessels is constructed in this paper. The system of partial differential equations describing the fluid flow inside collapsible tubes is presented and solved numerically by using the finite volume method. The evaluation of the numerical flux at the interfaces is performed by using the TV-HLL scheme which is derived from the previous studies presented for solving the Euler equations Tchuen, 2014, 2015). Indeed, the present scheme is obtained by following the Toro-Vázquez splitting, and using the HLL algorithm with modified wave speeds for the pressure system. An essential feature of the TV-HLL scheme is to associate two systems of differential equations, called the advection system and the pressure system. The proposed scheme preserves the non-negativity of essentially nonnegative quantities (cross section). It has been applied to some one-dimensional numerical tests dealing with the fluid flow through the male urethra and the blood flow inside cerebral vessels.

A new approach for numerical-diffusion control of flux-vector-splitting schemes for viscous-compressible flows

“…However, they also had limitations like low-frequency post-shock fluctuations in the cases of slowly moving shocks, carbuncle phenomenon, odd–even decoupling and kinked Mach stem in certain cases (Xie et al , 2015). In addition, several numerical flux schemes were also developed by combining the merits of various FVS and FDS schemes (Sun and Takayama, 2003; Kapen and Ghislain, 2014; Kapen and Tchuen, 2015, 2016; Tchuen et al , 2016; Kapen and Tchuen, 2017; Kapen and Ghislain, 2019).…”

“…The flux-splitting scheme of Toro and Vázquez is referred to as the TV-AWS scheme. It may be noted that the TV-AWS scheme was designed for accurate capturing of isolated stationary contact discontinuities and it inspired the development of several new schemes (Kapen and Ghislain, 2014; Kapen and Tchuen, 2015; Xie et al , 2015; Kapen and Tchuen, 2016, 2017; Kalita and Sarmah, 2019; Kapen and Ghislain, 2019). The tests reveal that the new methodology enhances the accuracy of the parent schemes despite the modifications being very simple.…”