Modelling blood flow in large arteries using the finite volume method
De Villiers, Anna Magdalena.
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The purpose of this research is the development of a one–dimensional (1D) computer code that models blood flow through large arteries. There are many of these models in literature, the majority is solved with the finite element method. The problem is analogous to a compressible liquid in a pipe network. Methods to solve the pipe network flow problem have evolved over the years. One of these methods, which can handle discontinuities and branching naturally to solve the blood flow problem, was used in this research. The blood flow problem can be modelled by solving mass flow, momentum conservation and the interaction between the blood flow and the arterial wall. In essence we are looking at two problems in two time scales, namely mass flow and the propagation of the pressure pulse. The mass flow rate of the blood is not very fast – it takes a blood particle approximately one minute to travel to the organs and back. Everytime the heart beats, it sends a 'shockwave' through the system. These waves or pulses propagate at speeds at least three orders higher than the blood flow. When these pressure waves reach a discontinuity or branch in the arterial network, part of the wave is reflected. The method used for this study discretises the partial differential equations by using a staggered grid and the finite volume method. An iterative method similar to the Semi Implicit Method for Pressure Linked Equations (SIMPLE) was used to solve the discretised equations. By using the characteristic system, characteristic variables that are constant along characteristic lines can be derived. These variables represent forward and backward travelling wave fronts. By expressing the boundary conditions in terms of these variables, rather than in terms of flow, area and pressure, we can prescribe non–reflecting boundary conditions. This way pressure waves can travel out of the computational domain unhindered. Discontinuities and branching are handled naturally because of the staggered grid discretisation. A computer code was written in Octave to solve the discretised equations for a number of test cases. The results show that when a small input pressure wave is prescribed, the solution behaves linearly. When a large input pressure wave is prescribed the solution behaves nonlinearly. The non–reflecting boundary conditions work perfectly for the linear test case, but a small portion of the outgoing wave is reflected for the non–linear test case. Discontinuities and branching were handled satisfactorily with the code for a number of test cases.
- Engineering