BIGLOBAL STABILITY ANALYSIS OF STENOTIC FLOW Robin Pitt, Spencer Sherwin Department of Aeronautics, Imperial College London, SW7 2AZ, United Kingdom robin.pitt@imperial.ac.uk, s.sherwin@imperial.ac.uk Vassilis Theofilis Universidad Polit´ ecnica de Madrid, Escuela Tecnica Superior Ingenieros Aeron´ auticos, Departamento de Motopropulsi´on y Termofluidodin´ amica Pza. Cardenal Cisneros, 3 E-28040 Madrid, Spain vassilis@torroja.dmt.upm.es ABSTRACT A BiGlobal stability analysis technique based on spec- tral/hp element technology is discussed, and applied to the biomedically important problem of the flow through a con- stricted channel. Results for both steady flows and the Floquet analysis of time periodic flow are presented. In the steady case the onset of three-dimensionality is investigated, and the effect of stenosis extent is examined. In the periodic case preliminary results concerning the initial bifurcation to asymmetry are presented. INTRODUCTION Atherosclerosis, the formation of plaques within the ar- terial wall, continues to be a major cause of death in the developed world. The associated narrowing, or stenosis, of the artery causes a significant reduction in the blood flow supplied to downstream vessels. A further life threaten- ing condition may occur if the plaque ruptures. This can cause thrombosis of the affected vessel or particles to be- come lodged in smaller vessels, possibly inducing myocardial infarction or stroke. While the fluid mechanical factors contributing to the initiation of sclerotic regions (the process of atherogenesis) are fairly well understood (Caro et al., 1971, Nerem and Cornhill, 1980), those playing a part in plaque rupture are less so. Study of the flow within relatively highly occluded artery models is therefore motivated, in order to characterise the flow conditions that arise in the immediate neighbour- hood of the plaque. Physiological flow conditions within most large healthy arteries are normally pulsatile in nature but typically oper- ate in a laminar unsteady flow regime. However, the solution of flow in stenotic geometries, such as that shown in Figure 1, provides an interesting additional fluid mechanical chal- lenge since, for a given flow rate, the local Reynolds number of the flow increases as the inverse of the vessel diameter reduction. Therefore even at moderate levels of occlusion transitional Reynolds numbers are possible. Numerically, the high local velocities at the stenosis and the need for a fine discretisation results in a greatly reduced time-step when considering the CFL stability restriction associated with an explicit treatment of the advection op- erator. The resulting high computational time can make a thorough investigation of the many parameters involved such as geometry, inflow waveform and Reynolds number, prohibitively expensive. A BiGlobal stability analysis (Theofilis, 2003) can alter- natively be employed to study the laminar instabilities and transitions occurring within the stenosis. Unlike classical stability analysis where a one-dimensional base flow is con- sidered and the other two spatial directions are harmonically expanded, in the BiGlobal stability analysis both the basic state and the amplitude functions of small-amplitude distur- bances superimposed upon the basic state are non-periodic two-dimensional functions; the third spatial direction is con- sidered homogeneous and expanded harmonically in Fourier wavenumbers β. The method is thus suited to investigation of the stability of flows with homogeneity of geometry in one dimension, for example channel, cylinder or axisymmet- ric geometries. In this paper, the geometry we consider is a plane channel geometry, infinite in the z -direction, with a prescribed contraction and subsequent re-expansion in the y -direction. In this paper we first present a brief introduction to the numerical method involved in this BiGlobal stability analy- sis, before demonstrating how the stability of stenotic flows is affected by the Reynolds number, by the contraction ra- tio of the stenosis and by the addition of pulsatility to the inflow. These results are given in terms of the value of the dominant eigenvalue, the shape of the dominant eigenmode, and the spanwise wavenumber β. NUMERICAL METHODOLOGY We take as the governing equations for arterial flow the incompressible Newtonian Navier-Stokes equations ∂u ∂t = −N(u) − 1 ρ ∇p + 1 Re ∇ 2 u in Ω (1) together with the continuity requirement ∇· u =0 in Ω (2) where u is the three dimensional velocity field, ρ and p are the fluid density and pressure respectively, and Re is the Reynolds number Re = UD/ν. For our purposes the length