This example illustrates the propagation of waves (pressure, velocity etc.) in a flow of incompressible Newtonian fluid through an elastic tube (e.g. blood vessels). The governing equations of this model are based on 1D Navier-Stokes equations. The distensibility of the tube walls is also included. For the details of the model governing equations refer:
1. Hunter,
P.J., M.Eng Thesis, UoA, 1972
2. Smith.
N.P., Ph.D Thesis, UoA 1999
The numerical method used to solve the above mentioned governing equations is two-step Lax-Wendroff finite difference method. Further details of the numerical method can be obtained from reference 2. The geometry of the tube is modeled using 1D cubic Hermite finite elements. Since this finite difference scheme is fully explicit, it is conditionally stable. In other words, once the spatial step is decided there is a maximum time step above which the computational algorithm fails due to amplification of the round-off error.
Here we perturb the system from its initial steady state (constant pressure and zero velocity throughout) by introducing a step increase in the inlet pressure from 12.46 kPa to 14.46 kPa while holding the exit pressure at 12.46 kPa. This perturbation travels at a finite velocity known as 'wave propagation velocity' or 'acoustic velocity'. Since the exit pressure is held at a constant level, the wave moves back and forth and eventually reaches its new steady state.
To create a movie in order to visualise the results of the simulation, use view.com
Simulation of step increase in pressure at the inlet