A mathematical model for blood flow through an arterial bifurcation

By introducing the finite element technique, a study of blood flow through an arterial bifurcation is presented in this paper. The blood is represented by a modified model of thixotropic power-law fluids, for which the parametric values for blood, both in normal and pathological states, have already been established. The results for the velocity profiles, pressure and wall shear stress distributions are elucidated and discussed for normal old and diseased states. The separation and reattachment points are also located for different values of the Reynolds number and the flow behaviour index (n) of the model representing the blood. The analysis identifies low shear stress zones behind the stenosis along the outer wall and high shear stresses downstream of the apex. The increasing percentage of the stenosis and the increasing values of the Reynolds number facilitate the high shear stress zones, whereas the thixotropy of the blood depicts an inbuilt mechanism of reducing high shear stresses as well as flow reversal regions.     

Characterization of a sudden expansion flow chamber to study the response of endothelium to flow recirculation

In order to simulate regions of flow separation observed in vivo, a conventional parallel plate flow chamber was modified to produce an asymmetric sudden expansion. The flow field was visualized using light reflecting particles and the size of the recirculation zone was measured by image analysis of the particles. Finite element numerical solutions of the two and three-dimensional forms of the Navier-Stokes equation were used to determine the wall shear stress distribution and predict the location of reattachment. For two different size expansions, numerical estimates of the reattachment point along the centerline of the flow chamber agreed well with experimental values for Reynolds numbers below 473. Even at a Reynolds number of 473, the flow could be approximated as two-dimensional for 80 percent of the chamber width. Peak shear stresses in the recirculation zone as high as 80 dyne/cm2 and shear stress gradients of 2500 (dyne/cm2)/cm were produced. As an application of this flow chamber, subconfluent bovine aortic endothelial cell shape and orientation were examined in the zone of recirculation during a 24 h exposure to flow at a Reynolds number of 267. After 24 h, gradients in cell orientation and shape were observed within the recirculation zone. At the location of reattachment, where the wall shear stress was zero but the shear stress gradients were large, cells plated at low density were still aligned with the direction of flow. No preferred orientation was observed at the gasket edge where the wall shear stress and shear stress gradients were zero. At higher cell densities, no alignment was observed at the separation point.(ABSTRACT TRUNCATED AT 250 WORDS)     

Coherence analysis of the electrical activity of the rabbit brain in the presence of a hunger dominant

Abnormal uptake of atherogenic substances and lipid infiltration have been believed to contribute to the localized genesis and development of atherosclerosis, as well as to late failures of synthetic arterial prostheses. To verify the theoretical prediction that accumulation of lipoproteins on the luminal surface of arterial walls occurs in the regions of disturbed flow, we have carried out an in vitro mass transfer experiment to test the effect of a pseudo steady recirculation flow on the uptake of 3H-7-cholesterol by the arterial wall at a surgically created stenosis. It was found that, as predicted by the theory, in the flow field of the stenosis the uptake of labeled cholesterol reached a maximum around the reattachment point of the vortex distal to the stenosis, where the wall shear stress was lowest (zero). This value of the highest uptake rate was almost twice the average, whereas the uptake level was at a minimum at the stenosis itself where the wall shear stress was highest. The lowest uptake was only 60% of the average. These results provide strong support to our hypothesis, based upon the theory that, in addition to the flow induced changes to the biologic function of endothelial cells, the disturbed flow with slow recirculation itself favors the accumulation of atherogenic lipoproteins at the blood-endothelium boundary, therefore playing an important role in the localized pathogenesis and development of atherosclerosis.     

Flow patterns in dog aortic arch under a steady flow condition simulating mid-systole

To elucidate the possible connection between blood flow and localized pathogenesis and the development of atherosclerosis in humans, we studied the flow patterns and the distribution of fluid axial velocity and wall shear stress in the aortic arch in detail. This was done by means of flow visualization and high-speed cinemicrographic techniques, using transparent aortic trees prepared from the dog. Under a steady flow condition at inflow Reynolds numbers of 700-1600, which simulated physiologic conditions at early- to mid-systole, slow, spiral secondary, and recirculation flows formed along the left anterior wall of the aortic arch and at the entrance of each side branch adjacent to the vessel wall opposite the flow divider, respectively. The flow in the aortic arch consisted of three major components, namely, an undisturbed parallel flow located close to the common median plane of the arched aorta and its side branches, a clockwise rotational flow formed along the left ventral wall, and the main flow to the side branches, located along the right dorsal wall of the ascending aorta. Thus, looking down the aorta from its origin, the flow in the aortic arch appeared as a single helical flow revolving in a clockwise direction. Regions of low wall shear stress were located along the leading edge of each side branch opposite the flow divider where slow recirculation flows formed, and along the left ventral wall where slow spiral secondary flows formed. If we assume that the flow patterns in the human aortic arch well resemble those observed in the dog, then it is likely that atherosclerotic lesions develop preferentially at these sites of low wall shear stress in the same manner as in human coronary and cerebral arteries.     

Prediction of hemolysis in turbulent shear orifice flow

This study proposes a method of predicting hemolysis induced by turbulent shear stress (Reynolds stress) in a simplified orifice pipe flow. In developing centrifugal blood pumps, there has been a serious problem with hemolysis at the impeller or casing edge; because of flow separation and turbulence in these regions. In the present study, hemolysis caused by turbulent shear stress must occur at high shear stress levels in regions near the edge of an orifice pipe flow. We have computed turbulent shear flow using the low-Reynolds number k-epsilon model. We found that the computed turbulent shear stress near the edge was several hundreds times that of the laminar shear stress (molecular shear stress). The peak turbulent shear stress is much greater than that obtained in conventional hemolysis testing using a viscometer apparatus. Thus, these high turbulent shear stresses should not be ignored in estimating hemolysis in this blood flow. Using an integrated power by shear force, it is optimal to determine the threshold of the turbulent shear stress by comparing computed stress levels with those of hemolysis experiments or pipe orifice blood flow.     

Stability Analysis of Velocity Profiles in Water-Hammer Flows

This paper performs linear stability analysis of base flow velocity profiles for laminar and turbulent water-hammer flows. These base flow velocity profiles are determined analytically, where the transient is generated by an instantaneous reduction in flow rate at the downstream end of a simple pipe system. The presence of inflection points in the base flow velocity profile and the large velocity gradient near the pipe wall are the sources of flow instability. The main parameters that govern the stability behavior of transient flows are the Reynolds number and dimensionless timescale. The stability of the base flow velocity profiles with respect to axisymmetric and asymmetric modes is studied and its results are plotted in the Reynolds number∕timescale parameter space. It is found that the asymmetric mode with azimuthal wave number 1 is the least stable. In addition, the results indicate that the decrease of the velocity gradient at the inflection point with time is a stabilizing mechanism whereas the migration of the inflection point from the pipe wall with time is a destabilizing mechanism. Moreover, it is shown that a higher reduction in flow rate, which results in a larger velocity gradient at the inflection point, promotes flow instability. Furthermore, it is found that the stability results of the laminar and the turbulent velocity profiles are consistent with published experimental data and successfully explain controversial conclusions in the literature. The consistency between stability analysis and experiments provide further confirmation that (1) water-hammer flows can become unstable; (2) the instability is asymmetric; (3) instabilities develop in a short (water-hammer) timescale; and (4) the Reynolds number and the wave timescale are important in the characterization of the stability of water-hammer flows. Physically, flow instabilities change the structure and strength of the turbulence in a pipe, result in strong flow asymmetry, and induce significant fluctuations in wall shear stress. These effects of flow instability are not represented in existing water-hammer models.     

LES and RANS Studies of Oscillating Flows over Flat Plate

Oscillatory flows over a flat plate are studied by using Large Eddy Simulation (LES) and Reynolds-Average Navier-Stokes (RANS) methods. A dynamic subgrid scale (SGS) model is employed in LES, while the Saffman's turbulence model in RANS. The mean velocity profile, the turbulence intensity, and the wall shear stress are computed and compared with earlier experimental and numerical works. The results indicate that the flow behaviors are quite different during the accelerating and decelerating phases of the oscillating cycle. The transition from laminar to turbulent is also investigated as a function of the Reynolds number, R, which represents the square of the ratio of the oscillation amplitude at free stream to the thickness of the Stokes layer at the plate. The present results both from LES and RANS show that the transition occurs in the range of 5 × 104 < R < 5 × 105. The evolution of the flow structure in the Stokes layer during the transition from laminar to turbulent is clearly demonstrated from the numerical results. The friction coefficient of the amplitude of oscillatory surface shear stress varies as R?0.5 with a phase angle of 45° in laminar regime and transition to R?0.23 with a phase angle of about 10° in turbulence regime. These variations in the surface shear stress with the Reynolds number are in excellent agreement with the earlier experimental results of Kamphuis and the numerical results of Blondeaux. The excellent agreement between the LES and RANS demonstrated that Saffman's turbulence model, as originally intended by Saffman, is applicable for unsteady flows.     

Flow visualization and 1- and 3-D laser-Doppler-anemometer measurements in models of human carotid arteries

Pulsatile flow, wall distensibility, non-Newtonian flow characteristics of blood in flow separation regions, and high/low blood pressure were studied in elastic silicon rubber models having a compliance similar to human vessels and the same surface structure as the biological intima models of (1) a healthy carotid artery model, (2) a 90% stenosis in the ICA, and (3) 80% stenosis in both the internal and external carotid arteries. Flow was visualized for steady flow and pulsatile studies to localize flow separation regions and reattachment points. Local velocity was measured with a 1-, 2-, or 3-D laser-Doppler-anemometer (LDA). Flow in the unstenosed model was Re = 250. In the stenosed models, the Re number decreased to Re = 180 and 213 under the same experimental conditions. High velocity fluctuations with vortices were found in the stenosed models. The jet flow in the stenosis increased up to 4 m/s. With an increasing bifurcation angle, the separation regions in the ECA and ICA increased. Increased flow (Re = 350) led to an increase in flow separation and high velocity shear gradients. The highest shear stresses were nearly 20 times higher than normal. The 90% stenosis created high velocity shear gradients and velocity fluctuations. Downstream of the stenoses, eddies were found over the whole cross-section. In the healthy model a slight flow separation region was observed in the ICA at the branching cross-section whereas in the stenosed models, the flow separation regions extended far into the ICA. We conclude that a detailed understanding of flow is necessary before vascular surgery is performed especially before artificial grafts or patches are implanted.     

Testing Block Probes for Wall Shear Stress Measurement in Water Flows

The authors developed and tested two types of sublayer pressure probe or blocks, to indirectly measure the average modulus and direction of the wall parallel component of the shear stress tensor, usually referred to as wall shear stress, or boundary shear, in water flows. The local shear stress has been investigated by means of a Preston tube for different Reynolds numbers, in a smooth rectangular section duct. The observed data have been used to calibrate the blocks. The blocks’ design is a direct outcome of the Dexter yaw meter, developed to function in an air flow. The observed pressure field associated with a given shear stress is compared to its theoretical counterpart, based on the assumption that the velocity profile satisfies a two-dimensional law of the wall, for smooth boundary.     

Analysis of the axial flow field in stenosed carotid artery bifurcation models--LDA experiments

Laser Doppler anemometer (LDA) experiments were performed to gain quantitative information on the differences between the large-scale flow phenomena in a non-stenosed and a stenosed model of the carotid artery bifurcation. The influence of the presence of the stenosis was compared to the effect of flow pulse variation to evaluate the feasibility of early detection of stenosis in clinical practice. Three-dimensional Plexiglass models of a non-stenosed and a 25% stenosed carotid artery bifurcation were perfused with a Newtonian fluid. The flow conditions approximated physiological flow. The results of the velocity measurements in the non-stenosed model agreed with the results from previous hydrogen-bubble visualization. A shear layer separated the low-velocity area near the non-divider wall from the high-velocity area near the divider wall. In this shear layer, vortex formation occurred during the deceleration phase of the flow pulse. The instability of this shear layer dictated the flow disturbances. The influences of the mild stenosis, located at the non-divider wall, was mainly limited to the stability of the shear layer. No disturbances were found downstream of the stenosis near the non-divider wall. Using a pulse wave with an increased systolic deceleration time, the velocity distribution showed an extended region with reversed flow, a more pronounced shear layer and increased vortex strength. From these measurements it is obvious that the influence of the presence of a mild stenosis, mainly limited to the stability of the shear layer, can hardly be distinguished from the effects of a variation of the flow pulse. From this it can be concluded that methods for detection of mild stenosis, using solely the large-scale flow phenomena, as can be measured by ultrasound or MRI techniques, will hardly have any clinical relevance.     

Initiation of Motion of Calcareous Sand

This paper examines the initiation of motion of four natural and five sieved calcareous sand samples in unidirectional flow. Flume experiments yield the sediment transport rate as a function of bed shear stress up to bed-form development. Reference-based criteria are supplemented by visual observations to determine the critical shear stress. The results are compared with published data for rounded and irregular particles in terms of the median sieve size and the corresponding nominal and equivalent diameters as functions of particle Reynolds number. The comparison shows that the critical shear stresses of the irregular particles are higher than the Shields curve in the hydraulically smooth flow regime and lower in the rough turbulent flow regime.     

Direct Numerical Simulation of Turbulent Flow in a Square Duct: Analysis of Secondary Flows

A direct numerical simulation of turbulent flow in a square duct was performed for a Reynolds number based on bulk streamwise velocity and duct height equal to 4,440. The mechanism by which secondary flows are generated in a square duct was investigated. Two counterrotating secondary flows occur around the duct corner. These secondary flows were found to play a key role in momentum transfer between the corner and center of the duct. A conditional quadrant analysis was performed in the local maximum and minimum regions of the wall shear stress in order to characterize the pattern of the mean secondary flows.     

Investigation of Twin Vortices near the Interface in Turbulent Compound Open-Channel Flows Using DNS Data

The direct numerical simulation of turbulent flows in a compound open channel is described. Mean flows and turbulence structures are provided, and are compared with numerical and measured data available in the literature. The simulated results show that twin vortices are generated near the interface of the main channel and the floodplain and that their maximum magnitude is about 5% of the bulk streamwise velocity. Near the interface, the simulated wall shear stress reaches a maximum, contrary to experimental data. A quadrant analysis shows that both sweeps and ejections become the main contributor to the production of Reynolds shear stresses near the interface. Through the conditional quadrant analysis, it is demonstrated how the directional tendency of dominant coherent structures determines the production of Reynolds shear stress and the pattern of twin vortices near the interface. In addition, the time-dependent characteristics of three-dimensional vortical structures in a compound open-channel flow were investigated using direct numerical simulation (DNS) data.     

Radiofluorinated L-m-tyrosines: new in-vivo probes for central dopamine biochemistry

PURPOSE: Recent information indicates that large, sustained wall shear stress gradients are a dominant hemodynamic parameter associated with the location and severity of atherosclerosis and myointimal hyperplasia. This study computes the spatial values of wall shear stresses and their gradients for three carotid artery bifurcation geometries. METHODS: A computational fluid dynamics program was used to solve the transient two-dimensional partial differential equations that describe fluid flow. Blood was treated as both a Newtonian and a non-Newtonian incompressible fluid. Solutions for the velocities, wall shear stresses, and wall shear-stress gradients were obtained for three carotid bifurcation geometries: a normal carotid bifurcation (similar to a primarily reconstructed carotid endarterectomy), a patch-reconstructed carotid endarterectomy, and a gradually tapered, low-angle carotid bifurcation (no carotid bulb). RESULTS: Computed velocity profiles closely match published experimental ones. Disturbed flow velocities are largest in the bulb segment of the normal carotid bifurcation. Peak and minimum wall shear stresses and peak shear stress gradients occurred in the lateral internal carotid artery wall. These were binodal in the normal or primarily reconstructed carotid artery, localized at the distal end of the patch-reconstructed carotid bifurcation, and minimal in the smooth, tapered carotid bifurcation. Wall shear stresses and their gradients were slightly higher for non-Newtonian than Newtonian fluids in the normal carotid artery but were similar in the other two geometric configurations. CONCLUSION: These results indicate that flow disturbances in general and wall shear stress gradients in particular are markedly reduced in carotid artery bifurcations that are smooth and gradually tapered and do not have a bulb. Abrupt geometric wall changes such as those occurring in the normal carotid bulb and at the distal end of a patch-reconstruction after carotid endarterectomy are harbingers of disturbed flow and high wall shear stress gradients. These results suggest that carotid endarterectomy reconstruction geometry characterized by a gradually tapered internal carotid artery may minimize the hemodynamically induced component of early myointimal hyperplasia and thrombosis and late atherosclerotic restenosis.     

Coupling of U-Tube and Shear Layer Oscillations in an Interconnected Flow Compartment

Results of experiments conducted in a 2?m high flume at large Reynolds numbers are reported in this paper. The flume was partitioned into two compartments. Flow entered the bottom of the upstream test compartment as a wall jet, at jet Reynolds number ranging from 11,000 to 170,000. Periodic oscillations of the free surface in the two compartments resembling the oscillatory flow in a liquid-filled U-tube, and large coherent structures formed above the potential core of the wall jet were observed. Coupling of the U-tube oscillations and vortex shedding is attributed to fluid-dynamic and fluid-resonant feedback processes. For test compartment length, Lc = 0.8?m, fluid-resonant feedback was found to be dominant, and the shear layer was observed to oscillate at the natural frequency of the two-compartment, U-tube system. The observed U-tube oscillations are initiated by the oscillations of the shear layer at a frequency equal to the subharmonic component for the U-tube. The flow oscillations were generally weaker for Lc = 1.2 and 2.0?m with oscillation frequencies governed by fluid-dynamic feedback, verified from a comparison with the results from a previously reported study.     

Computational analysis of flow in a curved tube model of the coronary arteries: effects of time-varying curvature

The flow through a curved tube whose radius of curvature varies with time was studied in order to better understand flow patterns in coronary arteries. A computational flow model was constructed using commercially available software. The artery model featured a uniform circular cross section, and the curvature was assumed to be constant along the tube, and in one plane. The computational model was verified with the use of a dynamically similar in vitro apparatus. A steady uniform velocity was prescribed at the entrance at a Reynolds number of 300. Two sets of results were obtained: one in which the curvature was held constant at the mean, maximum and minimum radii of curvature (quasistatic), and another in which the curvature was varied sinusoidally in time at a frequency of I Hz (dynamic). The results of the dynamic analysis showed that the wall shear rates varied as much as 52% of the static mean wall shear rate within a region of 10 tube diameters from the inlet. The results of the dynamic analysis were within 6% of the quasistatic predictions. Realistic modeling of the deforming geometry is important in determining which locations in the coronary arteries are subjected to low and oscillating wall shear stresses, flow patterns that have been associated with atherogenesis.     

Influence of Inlet Shear on Structure of Wake behind a Square Cylinder

The force coefficients and the frequency of vortex shedding in the wake of a square cylinder exposed to a uniform shear flow and the flow structure around it were numerically investigated. The Reynolds number defined on the basis of cylinder width was in the range of 250–1,500. The shear parameter, namely the transverse velocity gradient, which is nondimensionalized using the obstacle width and the average incoming velocity, was varied between 0 and 0.2. Analyses were performed for a number of flow parameters using various combinations of Reynolds number and shear parameters. Results show that mean and root-mean-square values of drag coefficient initially decrease up to certain values of the shear rate and then increase with increase in shear parameter. The root-mean-square values of lift coefficient show a similar behavior. The Strouhal number decreases uniformly with increase in shear parameter. At higher shear rates, the von Kármán vortex street comprising alternating vortices breaks, and the far wake shows mainly clockwise vortices.     

Combined MRI and CFD analysis of fully developed steady and pulsatile laminar flow through a bend

A combined MR and computational fluid dynamics (CFD) study is made of flow in a simple phantom laboratory flow rig consisting of a 180 degree bend with straight entry and exit sections. The aim was to investigate the potential of the use of MRI-linked CFD simulations for in vivo use. To this end, the experiment was set up for both steady and pulsatile laminar flow conditions, with Reynolds and Dean numbers and Womersley pulsatility parameter representative of resting flow in the human aorta. The geometrical images of the pipe and the velocity images at entry to the bend were used as boundary conditions for CFD simulations of the flow. The CFD results for both steady and pulsatile cases compared favorably with velocity images obtained at exit from the bend. Additional information such as pressure and wall shear stress, which either could not be measured adequately via MRI, or could not be measured at all, was also extracted from the simulation. Overall, the results were sufficiently promising to justify pursuing subsequent in vivo studies.     

Effect of Shape on Incipient Motion of Large Solitary Particles

The aim of this study is to investigate the effect of shape and size of solid particles on their initiation of motion in open channel flows. Initial motions of 22 solitary particles having different shapes and sizes were observed in a tilting flume of rectangular cross section. A smooth fixed bed and an obstructing element of smaller height with respect to the particle size was used throughout the experiments. The ratio of the height of the obstructing element to the height of the particle was kept constant at 1/5. By either changing the slope of the tilting flume or the discharge, or both, a range of shear stress values was obtained. Various equations and graphical representations in terms of dimensionless bed shear stress, grain Reynolds number, and the ratio of flow depth to grain diameter were presented to determine the flow conditions corresponding to the initiation of motion of solitary particles of given shapes. The experiments have revealed that critical flow conditions are dependent not only on the particle size and shape but also on the ratio of flow depth to grain diameter.