Computation of steady flow in a two-dimensional arterial model

The Navier-Stokes equations are solved numerically for steady flow through a double-branched two-dimensional section of a three-dimensional model of a canine aorta for which experimental data is available. The numerical scheme involves transforming the physical coordinates to a curvilinear boundary-fitted coordinate system and performing finite-difference computations in the transformed system. Shear stress at the wall is calculated for a Reynolds number of a 1,000 with branch-to-main aortic flow rate ratios as a parameter. The results are compared with the aforementioned experimental data and show reasonable qualitative agreement.     

Characteristics of secondary flow in steady and pulsatile flows through a symmetrical bifurcation

Steady and pulsatile flow in a glass model simulating an arterial bifurcation was investigated by flow visualization techniques. Secondary flow generated at the bifurcation has a similar pattern to a vortex, called the horseshoe vortex, produced around a wall-based protuberance in a circular tube. The same flow disturbance was clearly observed during the decelerating phase of pulsatile flow. The vortex produces a stagnation point on the top and bottom wall just upstream from the bifurcation apex. When aluminium dust was suspended in the test fluid perfusing the blood vessel model, particles deposited over an area spreading from the stagnation point to the lateral corners of the bifurcation. Comparison between the present results and topographical patterns of atherosclerosis reported in the literature suggests that it is in such low shear regions that lipid deposition tends to occur most.     

Hypoxic pulmonary vasoconstriction during steady and pulsatile flow in ferrets

We examined the effects of hypoxia and pulsatile flow on the pressure-flow relationships in the isolated perfused lungs of Fitch ferrets. When perfused by autologous blood from a pump providing a steady flow of 60 ml/min, the mean pulmonary arterial pressure rose from 14.6 to 31.3 Torr when alveolar PO2 was reduced from 122 to 46 Torr. This hypoxic pressor response was characterized by a 10.1-Torr increase in the pressure-axis intercept of the extrapolated pressure-flow curves and an increase in the slope of these curves from 130 to 240 Torr X l-1 X min. With pulsatile perfusion from a piston-type pump, mean pulmonary arterial pressure increased from 17.5 to 36.3 Torr at the same mean flow. This hypoxic pressor response was also characterized by increases in the intercept pressure and slope of the pressure-flow curves. When airway pressure was raised during hypoxia, the intercept pressure increased further to 25 +/- 1 Torr with a further increase in vascular resistance to 360 Torr X l-1 X min. Thus, in contrast to the dog lung, in the ferret lung pulsatile perfusion does not result in lower perfusion pressures during hypoxia when compared with similar mean levels of steady flow. Since the effects of high airway pressure and hypoxia are additive, they appear to act at or near the same site in elevating perfusion pressure.     

Disorder distal to modeled stenoses in steady and pulsatile flow

Measurements of the velocity and energy spectra were made in the distal region of modeled stenoses in a rigid tube with both steady and pulsatile water flows. Reynolds numbers of 318–2540 and a pulsatile flow frequency parameter of 15 were employed. The effects of the degree of stenosis, the stenosis geometry and the presence or absence of the downstream confining wall on the development of flow disturbances were investigated. Visualization of the distal flow patterns in stenotic and free jets illustrated the existence of complex fields which included vortex shedding, highly turbulent regions, and recirculation zones. Significant flow disorder was created by a mild stenosis in pulsatile, but not in steady, flow. Nondimensionalization employing the stenosis diameter and flow velocity in the throat of the constriction correlates the vortex shedding frequency and energy spectra within a limited postestenotic region.     

Finite element analysis of two dimensional steady flow in model arterial bifurcation

The purpose of the investigation reported in this paper is to determine theoretically the fluid dynamic field in models of common iliac arterial bifurcation and to identify the flow features which might influence the predominant occurrence of atherosclerotic lesions at such sites. This has been accomplished by numerically simulating fluid flow through 90 degrees symmetric bifurcations with branch-to-trunk area ratios of 0.8-1.414 and for Reynolds numbers ranging from 100 to 400. The analysis predicts flow reversal along the outer wall in models with area ratios over unity for high Reynolds number range, while no flow reversal occurred in models with area ratio below unity; a low shear zone along the outer wall and high shear stresses at the divider lip. Adverse pressure gradients are observed along the outer wall downstream of the corner point, the magnitudes increased with Reynolds number for a given branch to area ratio. Biological implication of the results is discussed with specific reference to the sites of atherosclerotic lesions found in man for these geometries.     

Pressure drop and flow rate measurements in a human aortic bifurcation cast for steady and pulsatile flow

Pressure drop and flow rate measurements in a rigid cast of a human aortic bifurcation under both steady and physiological pulsatile flow conditions are reported. Integral momentum and mechanical energy balances are used to calculate impedance, spatially averaged wall shear stress and viscous dissipation rate from the data. In the daughter branches, steady flow impedance is within 30% of the Poiseuille flow prediction, while pulsatile flow impedance is within a factor of 2 of fully developed, oscillatory, straight tube flow theory (Womersley theory). Estimates of wall shear stress are in accord with measurements obtained from velocity profiles. Mean pressure drop and viscous dissipation rate are elevated in pulsatile flow relative to steady flow at the mean flow rate, and the exponents of their Reynolds number dependence are in accord with available theory.     

Comparison of steady and pulsatile flow near the ventral and dorsal walls of casts of human aortic bifurcations

Steady and pulsatile flows were passed through casts of human aortic bifurcations and, by means of a laser Doppler anemometer, fluid velocities were measured at selected sites near the ventral and dorsal walls. At these sites, in the vicinity of the bifurcation, the influence of secondary flow is significant and therefore an appreciation of the phasic variation of secondary flow patterns is important. Results are presented comparing the flow direction in both steady and pulsatile flow at sites in three casts. The common features of the flow at these sites were the persistence of the flow direction during the accelerating and decelerating phases of the pulsatile cycle, and the consistently smaller angle (measured from the inlet centerline) of the pulsatile flow direction as compared to the angle of the flow direction in steady flow.     

Comparison of pressure losses in steady non-Newtonian flow through experimental tapered and cylindrical arterial prostheses

The use of an arterial prosthesis with a tapered lumen has several important advantages; for example, improved stability of flow, increased wall shear and better matching of its size with that of the host vessel. Tapering may, however, lead to increased energy losses, particularly if the angle of taper is large and the flow is high. This study is concerned with the determination of pressure drop for steady and laminar converging flow through rigid wall models of tapered arterial grafts. The angles of taper examined ranged from 0.5° to 1.0°. Aqueous solutions of polyacrylamide, with non-Newtonian viscous properties similar to those of blood, were used. The pressure drops across the tapered tubes were measured and the data were measured and the data were related to the pressure loss in cylindrical tubes of equivalent dimensions. Expressions for the ratio of the pressure drop in a tapered tube to that in a cylindrical tube for steady flow of a power law fluid were derived; there was good agreement between the predicted and the measured pressure drop ratios over a wide range of flows. The results of this study may be applied to the design of tapered arterial grafts. The pressure losses to be expected in tapered bypass grafts having various dimensions can easily be computed.     

Gas transport in serpentine microporous tubes under steady and pulsatile blood flow conditions

A serpentine gas exchange unit was built with cylindrical tubular microporous membranes featuring periodic arcs with a fixed curvature ratio (ratio of tube radius to radius of curvature) of 1/14 and circular angles between 30 and 360 deg. Oxygen transfer was measured under steady and pulsatile blood flow conditions in vitro and ex vivo to assess the design features which most effectively augment gas transfer. Under steady blood flow conditions, oxygen transfer increased with circular angles beyond 70 deg. Under pulsatile conditions, a wide range of geometrical and fluid mechanical parameters could be combined to enhance gas transfer performance, which eventually depended upon the secondary Reynolds number and the Womersley parameter.     

A turbulence model for pulsatile arterial flows

Difficulties in predicting the behavior of some high Reynolds number flows in the circulatory system stem in part from the severe requirements placed on the turbulence model chosen to close the time-averaged equations of fluid motion. In particular, the successful turbulence model is required to (a) correctly capture the "nonequilibrium" effects wrought by the interactions of the organized mean-flow unsteadiness with the random turbulence, (b) correctly reproduce the effects of the laminar-turbulent transitional behavior that occurs at various phases of the cardiac cycle, and (c) yield good predictions of the near-wall flow behavior in conditions where the universal logarithmic law of the wall is known to be not valid. These requirements are not immediately met by standard models of turbulence that have been developed largely with reference to data from steady, fully turbulent flows in approximate local equilibrium. The purpose of this paper is to report on the development of a turbulence model suited for use in arterial flows. The model is of the two-equation eddy-viscosity variety with dependent variables that are zero-valued at a solid wall and vary linearly with distance from it. The effects of transition are introduced by coupling this model to the local value of the intermittency and obtaining the latter from the solution of a modeled transport equation. Comparisons with measurements obtained in oscillatory transitional flows in circular tubes show that the model produces substantial improvements over existing closures. Further pulsatile-flow predictions, driven by a mean-flow wave form obtained in a diseased human carotid artery, indicate that the intermittency-modified model yields much reduced levels of wall shear stress compared to the original, unmodified model. This result, which is attributed to the rapid growth in the thickness of the viscous sublayer arising from the severe acceleration of systole, argues in favor of the use of the model for the prediction of arterial flows.     

In-vitro pulsatile flow visualization studies in a pulmonary artery model

In-vitro pulsatile flow visualization studies were conducted in an adult-sized pulmonary artery model to observe the effects of valvular pulmonic stenosis on the flow fields of the main, left and right pulmonary arteries. The flow patterns revealed that as the degree of stenosis increased, the jet-type flow created by the valve became narrower, and it impinged on the far (distal) wall of the left pulmonary artery further downstream from the junction of the bifurcation. This in turn led to larger regions of disturbed turbulent flow, as well as helical-type secondary flow motions in the left pulmonary artery, compared to the right pulmonary artery. The flow field in the main pulmonary artery also became more disturbed and turbulent, especially during peak systole and the deceleration phase. The flow visualization observations have been valuable in helping to conduct further quantitative studies such as pressure and velocity field mapping. Such studies are important to understanding the fluid mechanics characteristics of the main pulmonary artery and its two major branches.     

A hybrid one-dimensional/Womersley model of pulsatile blood flow in the entire coronary arterial tree

Using a frequency-domain Womersley-type model, we previously simulated pulsatile blood flow throughout the coronary arterial tree. Although this model represents a good approximation for the smaller vessels, it does not take into account the nonlinear convective energy losses in larger vessels. Here, using Womersley's theory, we present a hybrid model that considers the nonlinear effects for the larger epicardial arteries while simulating the distal vessels (down to the 1st capillary segments) with the use of Womersley's Theory. The main trunk and primary branches were discretized and modeled with one-dimensional Navier-Stokes equations, while the smaller-diameter vessels were treated as Womersley-type vessels. Energy losses associated with vessel bifurcations were incorporated in the present analysis. The formulation enables prediction of impedance and pressure and pulsatile flow distribution throughout the entire coronary arterial tree down to the first capillary segments in the arrested, vasodilated state. We found that the nonlinear convective term is negligible and the loss of energy at a bifurcation is small in the larger epicardial vessels of an arrested heart. Furthermore, we found that the flow waves along the trunk or at the primary branches tend to scale (normalized with respect to their mean values) to a single curve, except for a small phase angle difference. Finally, the model predictions for the inlet pressure and flow waves are in excellent agreement with previously published experimental results. This hybrid one-dimensional/Womersley model is an efficient approach that captures the essence of the hemodynamics of a complex large-scale vascular network. The present model has numerous applications to understanding the dynamics of coronary circulation.     

Structure of pulsatile flow in a model of elastic cerebral aneurysm

This study examines the effect of aneurysmal wall elasticity on the structure of flow within an elastic aneurysm during pulsatile flow. We visualized flow structure in a model of an elastic saccular aneurysm located at the bifurcation of the anterior cerebral artery and extending to the anterior communicating artery, and measured changes in the diameter of the aneurysm wall during pulsatile flow using particle imaging velocimetry (PIV). We similarly measured these features during steady flow by PIV and found that dilation of the aneurysmal wall absorbed the dynamic energy within the aneurysm. Accordingly, aneurysm wall elasticity functions as a biocompatible reaction that relieves wall shear stress acting on the vascular wall during pulsatile flow, and should thus inhibit the development and rupture of an aneurysm.     

A model of pulsatile flow in a uniform deformable vessel.

Simulations of blood flow in natural and artificial conduits usually require large computers for numerical solution of the Navier-Stokes equations. Often, physical insight into the fluid dynamics is lost when the solution is purely numerical. An alternative to solving the most general form of the Navier-Stokes equations is described here, wherein a functional form of the solution is assumed in order to simplify the required computations. The assumed forms for the axial pressure gradient and velocity profile are chosen such that conservation of mass is satisfied for fully established pulsatile flow in a straight, deformable vessel. The resulting equations are cast in finite-difference form and solved explicitly. Results for the limiting cases of rigid wall and zero applied pressure are found to be in good agreement with analytical solutions. Comparison with the experimental results of Klanchar et al. [Circ. Res. 66, 1624-1635 (1990]) also shows good agreement. Application of the model to realistic physiological parameter values provides insight as to the influence of the pulsatile nature of the flow field on wall shear development in the presence of a moving wall boundary. Specifically, the model illustrates the dependence of flow rate and shear rate on the amplitude of the vessel wall motion and the phase difference between the applied pressure difference and the oscillations of the vessel radius. The present model can serve as a useful tool for experimentalists interested in quantifying the magnitude and character of velocity profiles and shearing forces in natural and artificial biologic conduits.     

Comparison of LES of steady transitional flow in an idealized stenosed axisymmetric artery model with a RANS transitional model

In this study, two different turbulence methodologies are investigated to predict transitional flow in a 75% stenosed axisymmetric experimental arterial model and in a slightly modified version of the model with an eccentric stenosis. Large eddy simulation (LES) and Reynolds-averaged Navier-Stokes (RANS) methods were applied; in the LES simulations eddy viscosity subgrid-scale models were employed (basic and dynamic Smagorinsky) while the RANS method involved the correlation-based transitional version of the hybrid k-ε/k-ω flow model. The RANS simulations used 410,000 and 820,000 element meshes for the axisymmetric and eccentric stenoses, respectively, with y(+) less than 2 viscous wall units for the boundary elements, while the LES used 1,200,000 elements with y(+) less than 1. Implicit filtering was used for LES, giving an overlap between the resolved and modeled eddies, ensuring accurate treatment of near wall turbulence structures. Flow analysis was carried out in terms of vorticity and eddy viscosity magnitudes, velocity, and turbulence intensity profiles and the results were compared both with established experimental data and with available direct numerical simulations (DNSs) from the literature. The simulation results demonstrated that the dynamic Smagorinsky LES and RANS transitional model predicted fairly comparable velocity and turbulence intensity profiles with the experimental data, although the dynamic Smagorinsky model gave the best overall agreement. The present study demonstrated the power of LES methods, although they were computationally more costly, and added further evidence of the promise of the RANS transition model used here, previously tested in pulsatile flow on a similar model. Both dynamic Smagorinsky LES and the RANS model captured the complex transition phenomena under physiological Reynolds numbers in steady flow, including separation and reattachment. In this respect, LES with dynamic Smagorinsky appeared more successful than DNS in replicating the axisymmetric experimental results, although inflow conditions, which are subject to caveats, may have differed. For the eccentric stenosis, LES with Smagorinsky coefficient of 0.13 gave the closest agreement with DNS despite the known shortcomings of fixed coefficients. The relaminarization as the flow escaped the influence of the stenosis was amply demonstrated in the simulations, graphically so in the case of LES.     

Differential effects of pulsatile versus steady flow on coronary endothelial membrane potential

Steady shear stress stimulates transient hyperpolarization coupled to calcium-sensitive potassium (KCa) channels and sustained depolarization linked to chloride-selective channels. Physiological flow is pulsatile not static, and whereas in vivo data suggest phasic shear stress may preferentially activate KCa channels, its differential effects on both currents remain largely unknown. To determine this interaction, coronary endothelial cells were cultured in glass capillary flow tubes, loaded with the voltage-sensitive dye bis-(1,3-dibutylbarbituric acid)trimethine oxonol, and exposed to constant or pulsatile shear stress. The latter was generated by a custom servoperfusion system employing physiological pressure and flow waveforms. Steady shear induced a sustained depolarization inhibited by the Cl- channel blocker DIDS. Even after exposure to steady flow, subsequent transition to pulsatile shear stress further stimulated DIDS-sensitive depolarization. DIDS pretreatment "unmasked" a pulsatile flow-induced hyperpolarization of which magnitude was further enhanced by nifedipine, which augments epoxygenase synthesis. Pulse-shear hyperpolarization was fully blocked by KCa channel inhibition (charybdotoxin + apamin), although these agents had no influence on membrane potential altered by steady flow. Thus KCa-dependent hyperpolarization is preferentially stimulated by pulsatile over steady flow, whereas both can stimulate Cl--dependent depolarization. This supports studies showing greater potency of pulsatile flow for triggering KCa-dependent vasorelaxation.