An efficient semi-implicit method for three-dimensional non-hydrostatic flows in compliant arterial vessels

TL;DR

This paper introduces a semi-implicit finite difference and finite volume method for simulating three-dimensional non-hydrostatic blood flows in compliant arteries, achieving stability and efficiency for complex geometries.

Contribution

A novel splitting approach for pressure into hydrostatic and non-hydrostatic parts enables robust, efficient, and mass-conservative simulations of arterial blood flow in 3D with complex geometries.

Findings

Method accurately models steady and pulsatile flows.

Algorithm is stable and computationally efficient.

Results agree with analytical solutions and previous studies.

Abstract

Blood flow in arterial systems can be described by the three-dimensional Navier-Stokes equations within a time-dependent spatial domain that accounts for the elasticity of the arterial walls. In this article blood is treated as an incompressible Newtonian fluid that flows through compliant vessels of general cross section. A three-dimensional semi-implicit finite difference and finite volume model is derived so that numerical stability is obtained at a low computational cost on a staggered grid. The key idea of the method consists in a splitting of the pressure into a hydrostatic and a non-hydrostatic part, where first a small quasi-one-dimensional nonlinear system is solved for the hydrostatic pressure and only in a second step the fully three-dimensional non-hydrostatic pressure is computed from a three-dimensional nonlinear system as a correction to the hydrostatic one. The resulting algorithm is robust, efficient, locally and globally mass conservative and applies to hydrostatic and non hydrostatic flows in one, two and three space dimensions. These features are illustrated on nontrivial test cases for flows in tubes with circular or elliptical cross section where the exact analytical solution is known. Test cases of steady and pulsatile flows in uniformly curved rigid and elastic tubes are presented. Wherever possible, axial velocity development and secondary flows are shown and compared with previously published results.