High-order incompressible Navier-Stokes equations solver for blood flow

Abstract

A high-order finite difference solver was written to solve the incompressible Navier-Stokes (NS) equations and was applied to analyze the blood flow. First, a computer code was written to solve incompressible Navier- Stokes equations using the exact projection method/fractional step scheme. A fifth-order weighted essentially nonoscillatory (WENO) spatial operator was applied to the convective terms of Navier-Stokes equations. The diffusion term was solved by using a sixth-order compact central difference scheme. A fractional step scheme in conjunction with the third-order Runge-Kutta total variation diminishing (RK TVD) scheme was used for the time discretization. At this stage, non-Newtonian effects and the pulsatile nature of the flow were not included. The developed Newtonian flow code was tested using benchmark problems for incompressible flow, namely, the driven cavity flow, Couette flow, Taylor-Green vortex problem, double shear layer problem, and skewed cavity flow. The results were compared with existing published experimental data in order to build confidence that the computer code was working properly in the simple blood flow conditions, i.e., as a Newtonian fluid. In the second stage, the backward-facing step was analyzed for Newtonian steady and pulsatile flow, and for non-Newtonian steady and pulsatile flow. The results were compared with experimental data and found to be in agreement. In the third stage, the computer program was extended to three dimensions. Flow through an infinite long pipe and through a 90-degree bend was carried out. The velocity profile in the pipe and at different locations of the bend was obtained, and the numerical values indicate good agreement with analytical and experimental values.