Computational analysis of confined jet flow and mass transport in a blind tube.

A computational analysis of confined nonimpinging jet flow in a blind tube is performed as an initial investigation of the underlying fluid and mass transport mechanics of tracheal gas insufflation. A two-dimensional axisymmetric model of a laminar steady jet flow into a concentric blind-end tube is put forth and the governing continuity, momentum, and convection-diffusion equations are solved with a finite element code. The effects of the jet diameter based Reynolds number (Re(j)), the ratio of the jet-to-outer tube diameters (epsilon), and the Schmidt number (Sc) are evaluated with the determined velocity and contaminant concentration fields. The normalized penetration depth of the jet is found to increase linearly with increasing Re(j) for epsilon = O(0.1). For a given epsilon, a ring vortex that develops is observed to be displaced downstream and radially outward from the jet tip for increasing Re(j). The axial shear stress profile along the inside wall of the outer tube possesses regions of fixed shear stress in addition to a local minimum and maximum in the vicinity of the jet tip. Corresponding regions of axial shear stress gradients exist between the fixed shear stress regions and the local extrema. Contaminant concentration gradients develop across the ring vortex indicating the inward diffusion of contaminant into the jet flow. For fixed epsilon and Sc and Re(j) approximately 900, normalized contaminant flow rate is observed to be approximately twice that of simple diffusion. This model predicts modest net axial contaminant transport enhancement due to convection-diffusion interaction in the region of the ring vortex.     

Relationship between mass transfer coefficient and liquid flow velocity in heterogenous biofilms using microelectrodes and confocal microscopy

The relationship between local mass transfer coefficient and fluid velocity in heterogenous biofilms was investigated by combining microelectrodes and confocal scanning laser microscopy (CSLM). The biofilms were grown for up to 7 days and consisted of cell clusters separated by interstitial channels. Mass transfer coefficient depth profiles were measured at specific locations in the cell clusters and channels at average flow velocities of 2.3 and 4.0 cm/s. Liquid flow velocity profiles were measured in the same locations using a particle tracking technique. The velocity profiles showed that flow in the open channel was laminar. There was no flow at the top surface of the biofilm cell clusters but the mass transfer coefficient was 0.01 cm/s. At the same depth in a biofilm channel, the flow velocity was 0.3 cm/s and the mass transfer coefficient was 0.017 cm/s. The mass transfer coefficient profiles in the channels were not influenced by the surrounding cell clusters. Local flow velocities were correlated with local mass transfer coefficients using a semi-theoretical mass transfer equation. The relationship between the Sherwood number (Sh,) the Reynolds number (Re,) and the Schmidt number (Sc) was found using the experimental data to find the dimensionless empirical constants (n1, n2, and m) in the equation Sh = n(1) + n(2)Re(m) Sc(1/3). The values of the constants ranged from 1.45 to 2.0 for n(1), 0.22 to 0.28 for n(2), and 0.21 to 0.60 for m. These values were similar to literature values for mass transfer in porous media. The Sherwood number for the entire flow cell was 10 when the bulk flow velocity was 2.3 cm/s and 11 when the bulk flow velocity was 4.0 cm/s. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 681-688, 1997.     

Pulsatile blood flow and gas exchange across a cylindrical fiber array

The pulsatile blood flow and gas transport of oxygen and carbon dioxide through a cylindrical array of microfibers are numerically simulated. Blood is modeled as a homogeneous Casson fluid, and hemoglobin molecules in blood are assumed to be in local equilibrium with oxygen and carbon dioxide. It is shown that flow pulsatility enhances gas transport and the amount of gas exchange is sensitive to the blood flow field across the fibers. The steady Sherwood number dependence on Reynolds number was shown to have a linear relation consistent with experimental findings. For most cases, an enhancement in gas transport is accompanied with an increase in flow resistance. Maximum local shear stress is provided as a possible indicator of thrombosis, and the computed shear stress is shown to be below the threshold value for thrombosis formation for all cases evaluated.     

The role of the Womersley number in pulsatile blood flow a theoretical study of the Casson model.

The purpose of this Note is to clarify the meaning of the Womersley number alpha in pulsatile blood flow in small vessels. In particular. we explain why the use of alpha as aperturbation parameter to obtain approximate solutions of the Casson model (frequently used in the literature) is not appropriate. Using the techniques of dimensional analysis and scaling, we show that alpha is the product of the Reynolds and Strouhal numbers. Since the latter is very small for physiological flows, the result is that alpha < 1 even at relatively high values of the Reynolds number (i.e., for non-negligible inertia) and we validate our perturbation theory results by comparison with a numerical integration of the full model. Although this analysis is based on the Casson model, our method has general validity and may be applied to other models which describe more accurately the rheological properties of blood.     

Blood flow in abdominal aortic aneurysms: pulsatile flow hemodynamics

Numerical predictions of blood flow patterns and hemodynamic stresses in Abdominal Aortic Aneurysms (AAAs) are performed in a two-aneurysm, axisymmetric, rigid wall model using the spectral element method. Physiologically realistic aortic blood flow is simulated under pulsatile conditions for the range of time-averaged Reynolds numbers 50< or =Re(m)< or =300, corresponding to a range of peak Reynolds numbers 262.5< or =Re(peak) < or = 1575. The vortex dynamics induced by pulsatile flow in AAAs is characterized by a sequence of five different flow phases in one period of the flow cycle. Hemodynamic disturbance is evaluated for a modified set of indicator functions, which include wall pressure (p(w)), wall shear stress (tau(w)), and Wall Shear Stress Gradient (WSSG). At peak flow, the highest shear stress and WSSG levels are obtained downstream of both aneurysms, in a pattern similar to that of steady flow. Maximum values of wall shear stresses and wall shear stress gradients obtained at peak flow are evaluated as a function of the time-average Reynolds number resulting in a fourth order polynomial correlation. A comparison between predictions for steady and pulsatile flow is presented, illustrating the importance of considering time-dependent flow for the evaluation of hemodynamic indicators.     

To what extent does a minimal atherosclerotic plaque alter the arterial wall shear stress distribution? A model study by an electrochemical method

An electrochemical surface shear stress measurement was applied to a model of very thin unilateral arterial stenosis (height of 1/8 of the model pipe diameter with very smooth surface). Three dimensional wall shear stress distribution was measured under steady flow field from a relatively low Reynolds number, Re = 270, to a high Reynolds number, Re = 1200. There was a characteristic high and low wall shear distribution pattern around the stenosis. There were also remarkable high shear stress areas on the opposite wall and both side walls of the stenosis. It was clearly shown that three dimensional structure of the flow field, hence, the wall shear stress distribution, is affected by a minimal change on the arterial wall.     

A numerical simulation of steady flow fields in a bypass tube.

Steady flow in a complete by-pass tube was simulated numerically. The study was to consider a complete flow field, which included both the by-pass and the host tubes. The changes of the hemodynamics were investigated with three parameters: the inlet flow Reynolds number (Re), anastomotic angle (alpha) and the position of the occlusion in the host tube. The baseline flow field was set up with Re=200, alpha=45 degrees and the centered position of occlusion. The parametric study was then conducted on combination of Re=100, 200, 400, alpha=35 degrees, 45 degrees, 60 degrees, 75 degrees, 90 degrees and three occlusion positions: left, center and right. It was found that in the baseline case, large slow/recirculation flows could be seen in the host tube both upstream and downstream of the occlusion. The separation points were on the opposite walls to the junctions. Recirculation zones were also found near the toe and in the proximal outer wall of the by-pass tube. Their sizes were about one diameter of the tube or smaller. In some cases, pairing vortices could be seen in the host tube upstream of the occlusion. The shear rate distribution associated with the flow fields was presented. The flow pattern obtained was agreeable to those observed experimentally by other investigators. The difference of the flow fields between a complete bypass and simple anastomosis was discussed. The present numerical code provides a preliminary simulation/design tool for bypass graft flows.     

Combined MR imaging and numerical simulation of flow in realistic arterial bypass graft models

We report methods for (a) transforming a three-dimensional geometry acquired by magnetic resonance angiography (MRA) in vivo, or by imaging a model cast, into a computational surface representation, (b) use of this to construct a three dimensional numerical grid for computational fluid dynamic (CFD) studies, and (c) use of the surface representation to produce a stereo-lithographic replica of the real detailed geometry, at a scale convenient for detailed magnetic resonance imaging (MRI) flow studies. This is applied to assess the local flow field in realistic geometry arterial bypass grafts. Results from a parallel numerical simulation and MRI measurement of flow in an aorto-coronary bypass graft with various inlet flow conditions demonstrate the strong influence of the graft inlet waveform on the perianastomotic flow field. A sinusoidal and a multi harmonic coronary flow waveform both with a mean Reynolds number (Re) of 100 and a Womersley parameter of 2.7 were applied at the graft inlet. A weak axial flow separation region just distal to the toe was found in sinusoidal flow near end deceleration (Re = 25). At the same location and approximately the same point in the cycle (Re = 30) but in coronary flow, the axial flow separation was stronger and more spatially pronounced. No axial flow separation occurred in steady flow for Re = 100. Numerical predictions indicate a region in the vicinity of the suture line (where there is a local narrowing of the graft) with a wall shear magnitude in excess of five times that associated with fully developed flow at the graft inlet.     

Numerical simulation of steady flow in a model of the aortic bifurcation.

The governing equations of steady flow of an incompressible viscous fluid through a 3-D model of the aortic bifurcation are solved with the finite element method. The effect of Reynolds number on the flow was studied for a range including the physiological values (200 < or = Re < or = 1600). The symmetrical bifurcation, with a branch angle of 70 degrees and an area ratio of 0.8, includes a tapered transition zone. Secondary flows induced by the tube curvature are observed in the daughter tubes. Transverse currents in the transition zone are generated by the combined effect of diverging and converging walls. Flow separation depends on both the Reynolds number and the inlet wall shear.     

Flow-induced wall shear stress in abdominal aortic aneurysms: Part II--pulsatile flow hemodynamics

In continuing the investigation of AAA hemodynamics, unsteady flow-induced stresses are presented for pulsatile blood flow through the double-aneurysm model described in Part I. Physiologically realistic aortic blood flow is simulated under pulsatile conditions for the range of time-average Reynolds numbers 50< or =Re(m) < or =300. Hemodynamic disturbance is evaluated for a modified set of indicator functions which include wall pressure (p(w)), wall shear stress (tau(w)), Wall Shear Stress Gradient (WSSG), time-average wall shear stress (tau(w)*), and time-average Wall Shear Stress Gradient WSSG*. At peak flow, the highest shear stress and WSSG levels are obtained at the distal end of both aneurysms, in a pattern similar to that of steady flow. The maximum values of wall shear stresses and wall shear stress gradients are evaluated as a function of the time-average Reynolds number resulting in a fourth order polynomial correlation. A comparison between numerical predictions for steady and pulsatile flow is presented, illustrating the importance of considering time-dependent flow for the evaluation of hemodynamic indicators.     

Flow-induced wall shear stress in abdominal aortic aneurysms: Part I--steady flow hemodynamics

Numerical predictions of blood flow patterns and hemodynamic stresses in Abdominal Aortic Aneurysms (AAAs) are performed in a two-aneurysm, axisymmetric, rigid wall model using the spectral element method. Homogeneous, Newtonian blood flow is simulated under steady conditions for the range of Reynolds numbers 10 < or =Re < or =2265. Flow hemodynamics are quantified by calculating the distributions of wall pressure (p(w)), wall shear stress (tau(w)), Wall Shear Stress Gradient (WSSG). A correlation between maximum values of hemodynamic stresses and Reynolds number is established, and the spatial distribution of WSSG is considered as a hemodynamic force that may cause damage to the arterial wall at an intermediate stage of AAA growth. The temporal distribution of hemodynamic stresses in pulsatile flow and their physical implications in AAA rupture are discussed in Part II of this paper.     

Pulsatile flow in a constricted channel.

A nonuniform channel is used as a simple model of a constricted arterial vessel. Flow patterns have been calculated for pulsatile flow with both sinusoidal and nonsinusoidal flow rates for a range of Reynolds number, Re, and Strouhal number, St. The results show that even for relatively low frequency flows a strong vortex wave will be generated with a complex wall shear stress distribution and peak values much greater than those found in steady or unsteady parallel flow. The vortex wave increases in strength with increasing Re and St, with its total length and wavelength independent of Re but inversely proportional to St. The form of the imposed flow rate is found to have an important effect on the flow and the shear stress distribution.     

Application of the synthetic jet concept to low Reynolds number biosensor microfluidic flows for enhanced mixing: a numerical study using the lattice Boltzmann method

The concept of macro scale synthetic jets has been applied to the low Reynolds number (Re=10), two-dimensional channel flows which may be found in biosensor microfluidic systems. The current numerical investigation utilizes a hybrid approach of the lattice Boltzmann (LB) method for flow field computations and a finite-difference, convection-diffusion equation for passive scalar transport. The study presents the modified main channel flow results for various wall jet geometries (derived from synthetic jets), jet inlet conditions, scaling issues and Reynolds numbers. The results indicate limited effects due to jet cavity-slot geometry, and that the forced jet imparts momentum to the channel flow thus enhancing fluid mixing.     

Computational fluid dynamics modeling of steady-state momentum and mass transport in a bioreactor for cartilage tissue engineering

Computational fluid dynamics (CFD) models to quantify momentum and mass transport under conditions of tissue growth will aid bioreactor design for development of tissue-engineered cartilage constructs. Fluent CFD models are used to calculate flow fields, shear stresses, and oxygen profiles around nonporous constructs simulating cartilage development in our concentric cylinder bioreactor. The shear stress distribution ranges from 1.5 to 12 dyn/cm(2) across the construct surfaces exposed to fluid flow and varies little with the relative number or placement of constructs in the bioreactor. Approximately 80% of the construct surface exposed to flow experiences shear stresses between 1.5 and 4 dyn/cm(2), validating the assumption that the concentric cylinder bioreactor provides a relatively homogeneous hydrodynamic environment for construct growth. Species mass transport modeling for oxygen demonstrates that fluid-phase oxygen transport to constructs is uniform. Some O(2) depletion near the down stream edge of constructs is noted with minimum pO(2) values near the constructs of 35 mmHg (23% O(2) saturation). These values are above oxygen concentrations in cartilage in vivo, suggesting that bioreactor oxygen concentrations likely do not affect chondrocyte growth. Scale-up studies demonstrate the utility and flexibility of CFD models to design and characterize bioreactors for growth of tissue-engineered cartilage.     

The effect of Reynolds number on the propulsive efficiency of a biomorphic pulsed-jet underwater vehicle

The effect of Reynolds number on the propulsive efficiency of pulsed-jet propulsion was studied experimentally on a self-propelled, pulsed-jet underwater vehicle, dubbed Robosquid due to the similarity of its propulsion system with squid. Robosquid was tested for jet slug length-to-diameter ratios (L/D) in the range 2-6 and dimensionless frequency (St(L)) in the range 0.2-0.6 in a glycerin-water mixture. Digital particle image velocimetry was used for measuring the impulse and energy of jet pulses from the velocity and vorticity fields of the jet flow to calculate the pulsed-jet propulsive efficiency, and compare it with an equivalent steady jet system. Robosquid's Reynolds number (Re) based on average vehicle velocity and vehicle diameter ranged between 37 and 60. The current results for propulsive efficiency were compared to the previously published results in water where Re ranged between 1300 and 2700. The results showed that the average propulsive efficiency decreased by 26% as the average Re decreased from 2000 to 50 while the ratio of pulsed-jet to steady jet efficiency (η(P)/η(P, ss)) increased up to 0.15 (26%) as the Re decreased over the same range and for similar pulsing conditions. The improved η(P)/η(P, ss) at lower Re suggests that pulsed-jet propulsion can be used as an efficient propulsion system for millimeter-scale propulsion applications. The Re = 37-60 conditions in the present investigation, showed a reduced dependence of η(P) and η(P)/η(P, ss)on L/D compared to higher Re results. This may be due to the lack of clearly observed vortex ring pinch-off as L/D increased for this Re regime.     

Pulsatile flow and oxygen transport past cylindrical fiber arrays for an artificial lung: computational and experimental studies

The influence of time-dependent flows on oxygen transport from hollow fibers was computationally and experimentally investigated. The fluid average pressure drop, a measure of resistance, and the work required by the heart to drive fluid past the hollow fibers were also computationally explored. This study has particular relevance to the development of an artificial lung, which is perfused by blood leaving the right ventricle and in some cases passing through a compliance chamber before entering the device. Computational studies modeled the fiber bundle using cylindrical fiber arrays arranged in in-line and staggered rectangular configurations. The flow leaving the compliance chamber was modeled as dampened pulsatile and consisted of a sinusoidal perturbation superimposed on a steady flow. The right ventricular flow was modeled to depict the period of rapid flow acceleration and then deceleration during systole followed by zero flow during diastole. Experimental studies examined oxygen transfer across a fiber bundle with either steady, dampened pulsatile, or right ventricular flow. It was observed that the dampened pulsatile flow yielded similar oxygen transport efficiency to the steady flow, while the right ventricular flow resulted in smaller oxygen transport efficiency, with the decrease increasing with Re. Both computations and experiments yielded qualitatively similar results. In the computational modeling, the average pressure drop was similar for steady and dampened pulsatile flows and larger for right ventricular flow while the pump work required of the heart was greatest for right ventricular flow followed by dampened pulsatile flow and then steady flow. In conclusion, dampening the artificial lung inlet flow would be expected to maximize oxygen transport, minimize work, and thus improve performance.     

Experimental analysis of the influence of stenotic geometry on steady flow.

We studied the flow behavior under steady flow conditions in four models of cylindrical stenoses at Reynolds numbers from 150 to 920. The flow upstream of the constrictions was always fully developed. The constriction ratios of the rigid tubes (D) to the stenoses (d) were d/D = 0.273; 0.505; 0.548; 0.786. The pressure drop at various locations in the stenotic models was measured with water manometers. The flow was visualized with a photoelasticity apparatus using an aqueous birefringent solution. We also studied the flow behavior at pulsatile flow in a dog aorta with a constriction of 71%. The flow through stenotic geometries depends on the Reynolds number of the flow generated in the tube and the constriction ratio d/D. At low d/D ratios, (with the increased constriction), the flow separation zones (recirculation zones, so-called reattachment length) and flow disturbances increased with larger Reynolds numbers. At lower values, eddies were generated. At high Re, eddies were observed in the pre-stenotic regions. The pressure drop is a function of the length and internal diameter of the stenosis, respective ratio of stenosis to the main vessel and the Reynolds numbers. At low Re-numbers and low d/D, distinct recirculation zones were found close to the stenosis. The flow is laminar in the distal areas. Further experiments under steady and unsteady flow conditions in a dog aorta model with a constriction of 71% showed similar effects. High velocity fluctuations downstream of the stenosis were found in the dog aorta. A videotape demonstrates these results.     

Local hemodynamics affect monocytic cell adhesion to a three-dimensional flow model coated with E-selectin.

Monocyte adhesion to the endothelium depends on concentrations of receptors/ligands, local concentrations of chemoattractants, monocyte transport to the endothelial surface and hemodynamic forces. Monocyte adhesion to the inert surface of a three-dimensional perfusion model was shown to correlate inversely with wall shear stress, but was also affected by flow patterns which influenced the near-wall cell availability. We hypothesized that (a) under the same flow conditions, insolubilized E-selectin on the model's surface may mediate adhesive interactions at higher wall shear stresses, compared to an uncoated model, and (b) pulsatile flow may modify the adhesion profile obtained under steady flow. An axisymmetric flow model with a stenosis and a sudden expansion produced a range of wall shear stresses and a separated flow with recirculation and reattachment. Pre-activated U937 cells were perfused through the model under either steady (Re = 100, 140) or pulsatile (Remean = 107) flow. The velocity field was characterized through computational fluid dynamics and validated by inert particle tracking. Surface E-selectin greatly increased cell adhesion in all regions at Re = 100 and 140, compared to an uncoated model under the same flow conditions. In regions where the cells near the wall were abundant (taper and stenosis), adhesion to E-selectin correlated with the reciprocal of local wall shear stress when flow was steady. Pulsatile flow distributed the adherent cells more evenly throughout the coated model. Hence, characterizing both the local hemodynamics and the biological activity on the vessel wall is important in leukocyte adhesion.     

Effect of geometric variations on pressure loss for a model bifurcation of the human lung airway

Characteristics of pressure loss (ΔP) in human lung airways were numerically investigated using a realistic model bifurcation. Flow equations were numerically solved for the steady inspiratory condition with the tube length, the branching angle and flow velocity being varied over a wide range. In general, the ΔP coefficient K showed a power-law dependence on Reynolds number (Re) and length-to-diameter ratio with a different exponent for Re≥100 than for Re<100. The effect of different branching angles on pressure loss was very weak in the smooth-branching airways.