Modeling rough stenoses by an immersed-boundary method

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Abstract

A pulsatile laminar flow of a viscous, incompressible fluid through a stenosed artery was simulated by an immersed-boundary method. The method allows the use of a simple (rectangular) computational domain in order to simulate a flow around a complex geometry obstacle with surface irregularities (roughness). The influence of the shape and the surface roughness on the flow resistance was explored. The obtained numerical results were validated by comparison with published experimental and numerical results. We show that the surface irregularities have no significant influence on the flow resistance across an obstacle for a physiological range of Reynolds numbers. Notwithstanding, an accurate representation of irregularities allows investigation of the near-wall effects of a realistic flow such as fluid recirculation. We show that a detailed study of flow patterns in the immediate vicinity of the irregular surface can be performed using the immersed boundary method.

Introduction

Cardiovascular flows in a complex geometry, such as a stenosis in blood vessels or artificial valves and stents (a tiny metal tube used to prop open arteries after angioplasty), are commonly accompanied by separation, stagnation, recirculation and secondary vortex motion. In many cases, a blood is considered as a Newtonian fluid and computerized analysis of such flows is based on solving the Navier–Stokes equations. A natural flow through stenosed blood vessel is three-dimensional, time dependent and occurs in a very complex geometry whose shape is asymmetric and whose surface may contain rough irregularities which resemble small valleys. These irregularities make it very difficult to perform experimental and numerical simulations of such flows, although some attempts to study flows through occluded vessels have been made (Andersson et al., 2000; Back et al., 1984; Johnston and Kilpatrick, 1991). In most sophisticated investigations, a blood vessel with a three-dimensional geometry was considered, while an irregular stenosis shape was approximated by a mathematically smooth surface. Flow recirculation and stagnation regions occur in the vicinity of vessel's surface irregularities. A flow in these regions are characterized by the so-called, long “residence times”. Analysis of recirculation regions near irregular walls is of primary biomedical interest because it is commonly accepted that near-wall residence time of blood cells is an important predictor of various pathologies.

A numerical study of the fundamental hydrodynamic effects in complex geometries is a challenging task when discretization of the Navier–Stokes equations in the vicinity of complex geometry boundaries being the most complex problem. The use of boundary-fitted, structured or unstructured grids in finite-element methods help to deal with this problem; however, the numerical algorithms implementing such grids are usually inefficient in comparison to those using simple rectangular meshes. This disadvantage is particularly pronounced for simulating non-steady incompressible flows when the Poisson equation for the pressure has to be solved at each time step. Iterative methods used for complex meshes have low convergence rates, especially for fine grids. On the other hand, very efficient and stable algorithms for solving Navier–Stokes equations in rectangular domains have been developed. These algorithms use fast direct methods for solving the Poisson's equation for the pressure (Swarztrauber, 1974). These difficulties led to the development of approaches which allow formulate complex geometry flows on simple rectangular domains. These algorithms allow extend the problems that can be tackled numerically.

One approach is based on the immersed boundary (IB) method as introduced by Peskin (1971) during the early seventies. At present IB-based methods are considered to be a powerful tool for simulating complex flows. References of different immersed-boundary methods can be found in Balaras (2004), Fadlun et al. (2000), Kim et al. (2001), Moin (2002), Tseng and Ferziger (2003). In the present study, we applied a direct forcing immersed-boundary method developed in Kim et al. (2001) for simulating time-dependent flows through a stenosis with severe surface irregularities.

Section snippets

Numerical method

IB methods were originally developed to reduce simulating complex geometry flows to those defined on simple (rectangular) domains. To understand the method, consider a flow of an incompressible fluid around an obstacle Ω (S is its boundary) placed into a rectangular domain (Π). The flow is governed by the Navier–Stokes and incompressibility equations with the no-slip boundary condition on S.1

Flow through stenoses

Numerical investigation of blood flows in occluded vessels is an excellent tool for physicians in diagnosing blood vessel lesions. In early works on flow through stenoses, the stenosis shape was represented by a smooth mathematical function, such as a cosine curve. In reality, the stenosis geometry is much more complex. Usually its shape is asymmetric and the surface contains rough irregularities resembling small valleys and ridges. Back et al. (1984) published data on the geometrical contour

Smooth vs. irregular stenosis

This section presents the numerical results of simulations of flows for three geometrically different stenoses (Fig. 4). We can see from the figure that stenosis surface irregularity leads to complex flow patterns with recirculation/stagnation regions. Correct resolving of flows inside these regions is important for predicting the near-wall locations with long fluid residence times.

Pulsatile flows through rough stenoses

A realistic blood flow is pulsatile in nature. The waveform of the average axial velocity is mainly relevant for simulations, while different forms of spatially averaged inlet flow are applied to model the systole and diastole blood circulation cycles (Berger and Jou, 2000; Park, 1989). In this section, we simulate a pulsatile flow driven through a stenosis by sinusoidally time-varying pressure dropΔp(t)=Δp0[1+γpsin(ωt)],where Δp0 is a prescribed pressure difference across a computational

Conclusions

We employed a direct forcing immersed-boundary method to simulate flows through complex geometry arterial stenoses. The method allows the use of a simple (rectangular) computational domain in order to simulate a flow around complex geometry obstacles. Different geometrical models of stenoses, including the case of a very irregular (rough) surface, have been considered. Our observations showed that modeling a realistic geometry is important because surface irregularities may affect the dynamics

Acknowledgements

The authors wish to thank Prof. Helge Andersson for turning their attention on the irregular stenosis case and for useful comments. This research was supported by Binational United States Israel Science Foundation Grant 2001-150 and in part by CEAR of The Hebrew University of Jerusalem. The work of NN was also supported in part by Russian Foundation for Basic Research under the Grant 02-01-00492.

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