Postgraduate Research Conference 2009
The Prize Winners
PRESENTATION: Numerical Simulation of Physiological Flows
student: Matthew Yates
supervisor: Dr Tim Craft and Prof Hector Iacovides
Abstract
The numerical simulation of physiological flows is necessary to aid the development of future medical treatments. Whilst these flows are geometrically simple, they exhibit a number of complex phenomena such as flow separation, re-circulation and transition to turbulence. In addition, the forces imparted by the fluid can cause the elastic walls of the artery to deform. An accurate simulation of physiological flow must take these factors into account. This paper describes the numerical simulation of steady flow through constricted (stenosed) and dilated (aneurismal) arteries. The effects of wall compliance have been accounted for with the development of a finite-volume based solid-body stress analysis code.
Simulation of Flow through a Rigid Walled Stenosis
Before fully coupled fluid-structure interaction (FSI) simulations could be performed it was
necessary to validate the flow solver for the rigid walled case. Simulations were performed with
the in-house CFD code STREAM. This is a finite-volume based code which solves the steadystate,
two-dimensional Navier-Stokes equations. The physical geometry matched Model M2
used in the experiments of Young and Tsai [1]. Both laminar and turbulent simulations were
performed; the latter used the low-Reynolds-number form of the Launder-Sharma turbulence
model.
At low Reynolds numbers the laminar solver was able to accurately predict the pressure drop and the separation and re-attachment points. Converged laminar solutions were only achievable for Reynolds numbers below 700. However, it can be seen from Figure 2 that the predicted pressure drop begins to deviate from the experimental data at Reynolds numbers greater than 300. This coincides with the experimentally observed point of transition to turbulence. Turbulent calculations were in good agreement with the experimental data at Reynolds numbers greater than 1500. The results indicate that the flow solver is capable of accurately predicting the flow field in a rigid walled tube with a severe axisymmetric stenosis for all but a small range of transitional Reynolds numbers.
 Figure 1. Laminar separation and re-attachment |
 Figure 2. Normalized pressure drop
|
Simulation of Flow through an Elastic Walled Stenosis
To account for the effects of wall compliance the flow solver was coupled to a newly developed
finite-volume based stress analysis code. This code solves the two-dimensional elasticity
equations with thermo-elastic constitutive relations. This allows for an accurate representation
of the entire coupled system; many methods use simplifications to either fluid or solid model.
The method is implemented in a manner entirely consistent with the CFD code to allow simple
and efficient coupling. The use of a single numerical method and a single grid over both fluid
and solid domains removes the need for inefficient and inaccurate software interfaces to interpolate and transfer data between the two domains. This is in contrast to many existing FSI
solvers which use finite-elements and finite-volumes on the solid and fluid domain respectively.
To validate the newly developed FSI solver the case of flow through an elastic walled tube was simulated and the results were compared with the experimental data of Stergiopulos et al [2]. Due to a lack of information regarding the material properties of the elastic tube, a number of parametric tests were conducted by varying the elastic modulus and wall thickness. The most accurate prediction of the deformation at the throat of the stenosis was obtained by setting the elastic modulus to 1.3 MPa and the wall thickness to 0.2 mm. It can be seen from Figure 3 that the throat diameter reduces almost linearly, until a critical point at which the tube undergoes collapse. Numerical simulations were not performed beyond the point of collapse as the axisymmetric model would not be capable of predicting such an asymmetric process. As with the rigid walled case, converged solutions were not achievable for a range of intermediate Reynolds numbers due to the difficulties associated with capturing transition to turbulence.
 Figure 3. Normalized throat diameter |
 Figure 4. Volumetric flow rate
|
Simulation of Flow through a Rigid Walled Aneurysm
Another important category of physiological flow is the dilated or aneurismal artery. Before
FSI simulations could be performed it was again necessary to validate the flow solver for the
rigid walled case. Simulations have been performed and the results have shown good
agreement with the experimental data of Asbury et al [3]. Future attention will be focused
towards fully coupled FSI simulations of flow through elastic walled aneurysms.
Figure 5. Flow streamlines and velocity vectors through a rigid walled aneurysm.
References
[1] Young, D.F. and F.Y. Tsai, Flow characteristics in models of arterial stenoses - 1 Steady flow. Journal of Biomechanics, 1973. 6: 395 - 410.[2] Stergiopulos, N., et al., Steady flow tests and demonstration of collapse on models of compliant axisymmetric stenoses. Advances in Bioengineering, 1993. BED-26: 455 - 458.
[3] Asbury, C.L., et al., Experimental investigation of steady flow in rigid models of abdominal aortic aneurysms. Annals of Biomedical Engineering, 1995: 29 - 39.