The validity of axisymmetric assumptions when investigating pulsatile biological flows

Abstract

Computational fluid simulations of biological flows is increasingly popular due to its inexpense and ability to define the flow throughout the entire domain---both common limiting factors for experimental work. A common assumption has been that both the geometry and the flow field through an aneurysm is axisymmetric; however, investigations into non-biological flows have seen that even with an axisymmetric geometry, non-axisymmetric flow may develop. Idealised geometries are used to investigate these biological flows as it simplifies the model to enable an improved understanding of the effect geometry has on the flow. Additionally this simplification allows the implementation of a computationally cheaper axisymmetric code. We test this axisymmetric assumption by applying Floquet stability analysis to investigate the stability of the flow and thus determine when an axisymmetric aneurysmal flow is unstable to non-axisymmetric instabilities. Dimensions of the model are selected to be consistent with a high risk aneurysm in the human abdominal aorta and Reynolds numbers relevant to aneurysms in large arteries are examined. The presence of three dimensional instabilities has a significant impact on the validity of the assumption of axisymmetry. The maximum streamwise vorticity in the perturbation fields is found to occur at the downstream section of the aneurysm, implying that it is in these areas that the results of axisymmetric simulations differ the most from fully three dimensional flow. References Barkley, D. and Henderson, R. D., Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322 (1996), 215--241. doi:10.1017/S0022112096002777 Brown, P. M., Zelt D. T., Sobolev B., The risk of rupture in untreated aneurysms: The impact of size, gender, and expansion rate. J. Vasc. Surg. 37 (2003), 280--284. doi:10.1067/mva.2003.119 Cowling R., Soria J., Flow Visualisation through Model Abdominal Aortic Aneurysm, Fourth Australian Conference on Laser Diagnostics in Fluid Mechanics and Combustion , The University of Adelaide, South Australia, Australia, 7--9 December 2005, 33--36. Egelhoff C. J., Budwig R. S., Elger D. F., Khraishi T. A., Model studies of the flow in abdominal aortic aneurysms during resting and exercise conditions. J. Biomech , 32 (1999), 1319-1329. doi:10.1016/S0021-9290(99)00134-7 Karniadakis, G. E. and Triantafyllou, G. S., Frequency selection and asymptotic states in laminar wakes. J. Fluid Mech. 199 (1989), 441--469. doi:10.1017/S0022112089000431 Karniadakis, G. E., Israeli, M. and Orszag, S. A., High-order splitting methods for the incompressible Navier-Stokes equations, J. Comp. Phys. 97 (1991), 414--443. doi:10.1016/0021-9991(91)90007-8 Ku, D. N., Blood flow in arteries, Annual review of Fluid Mechanics , 29 (1997), 399--434. doi:10.1146/annurev.fluid.29.1.399 Lasheras J., The Biomechanics of Arterial Aneurysms, Annual Review of Fluid Mechanics , 39 (2007), 293--319 doi:10.1146/annurev.fluid.39.050905.110128 Salsac, A. V., Sparks, S. R., Chomaz, J. M. and Lasheras, J. C., Evolution of the wall shear stresses during the progressive enlargement of symmetric abdominal aortic aneurysms, J. Fluid Mech. , 560 (2006), 19--51. doi:10.1017/S002211200600036X Sheard, G. J., Evans, R. G., Denton, K. M. and Hourigan, K., Undesirable Haemodynamics in Aneurysms, In Proceedings of the IUTAM Symposium on Unsteady Separated Flows and Their Control , Hotel Corfu Chandris, Corfu, Greece, 18--22 June 2007 Sheard, G. J. and Ryan, K., Pressure-driven flow past spheres moving in a circular tube, J. Fluid Mech. 592 (2007), 233--262. doi:10.1017/S0022112007008543 Stedman, 2002, The American HeritageAE Stedmanis Medical Dictionary, Houghton Mifflin Company, Massachusetts. Steinman, D. A., Vorp, D. A. and Ethier, C. R., Computational modelling of arterial biomechanics: Insights into pathogenesis and treatment of vascular disease, J. Vascular Surgery , 37 (2003), 1118--1128. doi:10.1067/mva.2003.122 Waite, L. and Fine, J. (2007). Applied biofluid mechanics . United Stated of America: McGraw-Hill.