Unsteady Effects on the Flow Across Tilting Disk Valves

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Moshe RosenfeldAssociate Professor

e-mail: [email protected]

Idit AvrahamiGraduate Student


Shmuel EinavProfessor


Faculty of Engineering,Tel Aviv University,

Tel Aviv 69978, Israel

Unsteady Effects on the FlowAcross Tilting Disk ValvesThe present study simulates numerically the flow across two-dimensional tilting diskels of mechanical heart valves. The time-dependent Navier-Stokes equations are soassess the importance of unsteady effects in the fully open position of the valve.cases with steady or physiological inflow conditions and with fixed or moving valvesolved. The simulations lead into mixed conclusions. It is obvious that steady inflowthat account for vortex shedding only cannot model realistic physiological cases. In cwith imposed physiological inflow, the details of the flow field for fixed and moving vamight differ in the fully open position as well, although the gross features are qsimilar. The fixed valve case consistently results in safe estimations of several cquantities such as the axial force, the maximal shear stress on the valve, or the tranvular pressure drop. Thus, fixed valve simulations can provide useful information fodesign of prosthetic heart valves, as long as the properties in the fully open positionare sought. @DOI: 10.1115/1.1427696#

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IntroductionSurgical replacement of diseased heart valves by prosthese

routine heart operation. Among the mechanical heart val~MHV !, the most successful are the tilting disk and bileafletsigns. These rigid heart valves produce significant disturbancethe flow that might provoke several clinical problems. The moserious problems are hemolysis, thrombus formation, tissue ogrowth, large transvalvular pressure drop, damage to the endolial tissue and regurgitation due to improper closure of the leafl@1#.

The fluid dynamics associated with MHV play a critical roledetermining the longevity of the replacement. The importancecomprehensive knowledge of the flow field across MHV has pduced a large volume of studies over the last three decades.ultimate aim of these studies is to aid in the design of optimMHV from the hemodynamical point of view. The problem ihowever, very complex. Not only is the flow unsteady at relativhigh Reynolds numbers, but it also occurs within a complexometry of the atrium and the ventricle, not to mention the coplications introduced by the elastic walls and the motion ofvalve. The list of complications can be considerably expandmaking the study of the flow across MHV extremely difficult@2#.

The traditional tools of research were experimental, uspartly in-vivo @see for example@3–5## and mainly in-vitro tech-niques@6–9#. Numerical simulations were added to complemethe experimental studies@see for example:@2,10–16##.

Employing computational fluid dynamics~CFD! methods tosimulate the flow field across MHV might result in improved dsigns. CFD simulations can provide a detailed description ofunsteady flow field~including regions that are difficult to accesexperimentally! and might be extensively used for optimizationfor isolating factors that in many cases cannot be separatedperimentally. Yet, the numerical simulation of the full problembeyond the routine capabilities of existing computationalsources. Therefore, simplified models are needed to allow scomputations to be employed in the design routine of MHV. Comon simplifications include two-dimensional approximatio@10,13–14#, steady flow approximations@15,17–18#, simplifica-tions of the configuration@13,18#, or assuming fixed rather thamoving valves@14,19#.

Two-dimensional approximations considerably reduce the

Contributed by the Bioengineering Division for publication in the JOURNAL OFBIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Divsion December 1, 1998; revised manuscript received August 16, 2001. AssoEditor: S. E. Rittgers.

Copyright © 2Journal of Biomechanical Engineering

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quired computational resources. This approximation excludesability of taking into account 3-D effects, such as cross flows aspiral vortices@15,16#. Yet, it includes, at least to the first approxmation, the main flow features such as the creation of strong slayers near the edges of the valve, the shedding of large-svortices downstream of the valve, or the simultaneous creatiovortices during the deceleration phase of the physiological flSeveral works did consider the three-dimensional case for ste@15# or unsteady@11,19# flows across single tilting disk or bileafleaortic valves. Only qualitative description of global propertiesthe flow could be studied in these works because existing thdimensional solutions are rather limited in the total numbermesh points~less than 50,000 nodes!.

Most studies considered fixed valve conditions because ofdifficulty in simulating moving valve conditions. This simplification was justified because over large parts of the cycle, bothaortic and the mitral valves are open in a fixed position. Anotsimplification was in the modeling of the inflow conditions. Rcent studies agree that steady state simulation cannot approxicorrectly the time-dependent flow over heart valves. However,simplification, in some studies, the time dependent equations wsolved to simulate steady inflow conditions. The studies of Wand Yoganathan@6# and Hanle et al.@7# led to the belief thatconstant inflows can represent realistic pulsating flows. Roshet al. @20# supported this argument by finding a correlation btween the maximal steady and unsteady values for several typvalves. Huang et al.@13# and King et al.@18# justified the use ofsteady inflow with the magnitude of the peak physiological infloas the ‘‘worst case scenario.’’ In their time-dependent solutiothe flow was found unsteady,~although the inflow was steady!,because of vortices that were shed from the valve. However, silations with pulsating inflow are closer to realistic physiologicconditions than those with the steady inflow@11,14,19#. Blacket al. @14# compared steady versus pulsatile flow for a twdimensional model of a bileaflet valve in the aortic position. Thpresented preliminary numerical results that exhibited very diffent velocity field and streamlines pattern at the end systole.

Simulations with moving valves are very rare. The most notaare the fascinating simulations resulting from the group of PrPeskin,@e.g.,@12##. They simulated a complete natural heart eploying a specially developed algorithm that incorporates the eticity of the wall fibers ~but at a low Re number!. These fullythree-dimensional calculations require very large computatioresources that do not allow routine calculations for design p

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poses. Moreover, the Cartesian meshes do not allow the resolof the finer viscous details of the flow. Kiris et al.@11# simulatedthe motion of a three-dimensional aortic tilting disk valve usingchimera schemeand employing a mesh of 47,000 nodes andtime-steps per cardiac cycle. Steady and unsteady flow calctions were carried out to demonstrate the capabilities of theirmerical method to solve problems of biofluid interest.

One of the main issues that apparently had not been answsatisfactorily is the effect of unsteadiness on the global featurethe flow across MHV, as well as on several important desquantities, such as the pressure drop and shear stress in theity of the valve. The duration of the opening or closing of both taortic and mitral valves takes only a small fraction of the cardcycle ~of the order of 10 ms!. This raises the question whether thflow in the fully open position of the valve can be calculatwithout taking into consideration the opening and the closphases, i.e., by assuming that the mechanical valve stays in aposition during the entire cycle. If the answer is positive, the tof numerical simulations can be substantially simplified and thcan be applied to routine design and optimization procedures

In the present work we mainly consider the flow across tiltidisk mitral valves by using a mitral waveform in physiologicalike inflow cases. The structure of the paper is as follows. Tnumerical model that includes the computational domain andboundary conditions; the mesh and other numerical detailswell as the validation of the numerical model, are given in tnext section. Details on the results of different simulations~twosteady inflow cases and two physiological inflow cases, one wa fixed valve and the other one with a moving valve! are given inResults. In the Discussion, general flow features, as well asimportance of vortex shedding and the motion of the valve onflow in the fully open position of the valve, are elaborated.

The Numerical ModelIn the present study we use a two-dimensional approximat

We also assume that the flow field in the vicinity of the valvenot severely affected by the details of the surrounding geomeEven though the anatomical details are complex, we use achannel-like approximation of the enclosure. In fact, in many pvious studies of MHV a constant diameter channel was [email protected].,@6,13,21,22##.

An unsteady, laminar, incompressible and Newtonian flowassumed. The governing Navier-Stokes flow equations are:

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Computational Domain and Boundary Conditions. To testthe various sources of unsteadiness, three different configurawere studied:~i! fixed valve and with steady inflow~two inflowvelocities were simulated!; ~ii ! fixed valve; and~iii ! moving valvecases with pulsatile~physiological-like! inflow.

The two-dimensional computational domain is shown in FigThe valve is placed inside a straight channel of heightH. Thehinge of the valve is located at a distance ofH from the upstream

Fig. 1 The computational domain „not to scale …

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boundary, approximately one-third from the lower edge of tvalve. The upstream portion of the channel represents the atrwhile the region downstream of the valve represents the left vtricle. The downstream boundary is placed at a distance of 3H,larger than used by Huang et al.@13# to eliminate the influence ofthe downstream boundary. Typical maximal opening and closangle of the Bjork-Shiley valve areu50 deg andu557 deg, re-spectively. In the present model, the motion of the valve isstricted to the range of 8 deg,u,57 deg. The closing angle walimited to 8 deg to represent the clearance between the leafletthe housing as well as to reduce unnecessary computational cThe moving valve case was very costly in terms of CPU timethe final closing phase of the cycle, when the gap became vsmall. In the fixed valve cases, the fully open position was simlated (u557 deg).

No-slip and no-penetration boundary conditions were specion the upper and bottom walls of the channel and on the valvall the cases. In the upstream boundary, either a steady axialocity or a physiological mitral waveform as shown in Fig. 2 wimposed. The mitral physiological-like waveform was basedthe experiments of Reul@23# with modifications in the regurgita-tion phase. These modifications were required to be able to cthe valve in a 2-D fluid-structure interaction analysis we pformed in another study. A uniform velocity profile was assumat the upstream boundary@24#. On the downstream boundarystress-free conditions were imposed. The Reynolds numbers bon the peak and mean inflow velocity were Re54200 and 800,respectively. The beat rate was fixed at 75 beats/min.

In the moving valve case, the motion of the valve was prespfied as shown in Fig. 2. The imposed motion was based onstudy of Cheon and Chandran@25#. In real cases the motion of thvalve is determined by the inflow rate. Therefore, the presspecification of the valve motion independent of the flow is yanother approximation that was introduced to avoid the calction of the fluid-structure interaction problem.

In all the cases, the solution was started from a zero veloinitial condition and it was advanced in time until a periodic slution was established. This required a simulation of up tot54 s in the steady inflow cases. In the case of physiologicalflow, it required 5–6 full cycles~4–5 s!.

Mesh and Numerical Details. Typical meshes used in thclosed and open positions of the valve are shown in Fig. 3~a! andFig. 3~b! ~only every second node is plotted in each directio!.Mesh points were clustered around the valve and along the wIn most simulations the mesh consisted of 20,000 nodes, althomeshes with 5,000, 40,000, and 80,000 nodes were employewell in the mesh-independence tests. In the moving valve case

Fig. 2 The physiological inflow waveform and the specifiedmotion of the mitral valve

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moving mesh approach was used during the opening and cloof the valve. Re-meshing was employed whenever necessaprevent the mesh of becoming too distorted.

A commercial finite element package was used to solveunsteady incompressible and laminar Navier-Stokes equat~FIDAP, Fluent Inc, Evanston!. An Euler backward implicit time-advancing algorithm was employed together with a projectmethod for solving the pressure field. A segregated iterative sousing the GMRES method was employed for solving the discequations. Quadrilateral elements were used and streamlinewinding was activated to smooth the solution at the highernumber cases.

Validation. Time-step and mesh independence tests wereformed in order to validate the numerical model. The tests wmade on the fixed valve case with physiological inflow.

To check mesh independence, the same flow case was slated on four meshes with 5000, 20,000, 40,000, and 80,nodes. The number of mesh intervals was systematically chanfrom mesh to mesh, keeping the same clustering of the nodes.axial velocity distribution on a cross section located 2.8H fromthe inlet is plotted in Fig. 4 for these four meshes at the timet50.2 s. The solution is converged for the two finest meshes.coarser mesh of 20,000 nodes yields an accurate solution asexcept in the finest details of the flow as Fig. 5 demonstratesthe latter figures, the streamlines and vorticity field of theseferent mesh cases is shown.

Similarly, time-step independence tests were conducted for1280, 2560, and 5120 steps per cycle for the mesh with 20,nodes. The axial velocity distribution for these four time-stepsshown in Fig. 6 at the same cross section and time. Verytime-steps are necessary to obtain a fully time-step conve

Fig. 3 Typical meshes in the vicinity of the valve in the closed„a… and open „b… positions „only every second mesh node isplotted in each direction …

Fig. 4 Mesh-independence test of the axial velocity distribu-tion at a cross section downstream of the valve

Journal of Biomechanical Engineering

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40,000isnegedsolution. Observation of the vorticity and streamlines for the vaous time-steps, Fig. 7, reveals that the flow field downstreamthe valve is strongly affected by the dependence of the propation velocity of the vortices on the time-step. This affects ttemporal location of the vortices, a fact that has a very promineffect on the velocity profile at any given section. However, tglobal properties of the flow field~such as the number and size othe vortices! is less sensitive to the time-step size. Even the coaest time-step predicts very well the global flow field, in particulin the vicinity of the valve.

Therefore, for practical reasons a mesh with 20,000 nodes1280 time-steps per cycle were used in the simulations, yieldinreasonable compromise between accuracy and computationasources~the error is less than 5% relative to the finest mesh afines time-step used!.

Results

Fixed Valve and Steady Inflow. A steady inflow with a fixedvalve at the fully open position ofu557 deg was first solved. Twoinflow velocity magnitudes were simulated. In one case the inflvelocity was equal to the mean velocity of the mitral wavefor

Fig. 6 Time-step independence test of the axial velocity distri-bution at a cross section downstream of the valve

Fig. 7 The flow field dependence on the time-step „at the timeof tÄ0.2 s…

Fig. 5 The flow field dependence on the spatial resolution „atthe time of tÄ0.2 s…

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given in Fig. 2 and the Reynolds number was Re5800. In theother case, the peak velocity of the mitral waveform was uresulting in an inflow Reynolds number of Re54200. The time-history of the transverse velocity component leeward of the upedge of the valve is plotted in Fig. 8 for the first case (5800). After the transients die out, a periodic variation is foufor the Re5800 case with a dominant frequency off 55.5 Hz or aStrouhal number of St5H f /U51.18, whereU is the inflow ve-locity. For the case of Re54200, a Strouhal number of St51.15was found~the shedding frequency wasf 523 Hz!. Those resultsare in good agreement with the findings of Huang et al.@13#, whoobtained for Re5500 and 1000 Strouhal numbers of 1.14 a1.02, respectively. The small differences might be attributed todifference in the shape of the valve. In the present calculationgeneric planar valve was assumed while in the simulationsHuang et al.@13# the Bjork-Shiley convexo-concave/monostrtilting disk was employed.

The streamlines in the vicinity of the valve are shown in Fig~left column! for four equally spaced time intervals along a singcycle of the Re5800 case. Very similar flow structures were founfor the higher Reynolds case as well. Obviously, the valve, eveits fully open position, poses a significant obstruction to the floThe two jets generated between the valve tips and the wall cra vortical flow consisting of large vortices next to the valve anrow of vortices along each wall. To get a clearer picture ofvortex dynamics, the vorticity field is also given in Fig. 9 for thsame time instants~right column!. A large attached vortex is foundover the entire cycle lee of the lower side of the valve. This voris generated by the roll up of the shear layer emanating fromlower edge of the valve. The vortex has a size comparable toobstruction size and it is partly attached to the upper tip ofvalve. Leeward of the valve, a large region of low velocityfound.

The shear layer emanating from the upper tip of the valve rup into a vortex as well~Fig. 9~a!!. However, this vortex increase

Fig. 8 Time history of the transverse velocity component atthe point „1.7H, 0.6H… downstream of the valve „steady inflowcase …

Fig. 9 Time sequence of the streamlines and the vorticity forone vortex shedding cycle „steady inflow case …


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rapidly in size until it is shed downstream~Fig. 9~b!! by the fastflow that is established between the upper tip of the valve andwall. The lower shear layer drags counter-vorticity from the waThis shear layer feeds the vortex that has been shed fromupper side, making it stronger and larger in size. The shed vortmove toward the lower wall and propagate downstream, creatirow of vortices and a wavy core flow. Each of these vorticoriginates from another shedding cycle. The core flow drags vticity out of the upper wall in the form of shear layers that roll uas well and form the upper row of vortices. However, these vtices are considerably smaller and weaker. The whole systemvortices propagates downstream, creating the unsteady andodic flow that characterizes the flow field downstream of tvalve.

Huang et al.@13# presented the streamlines for four intervaalong one shedding cycle of a 2-D model of the Bjork-Shiltilting disk at Re51000~see their Fig. 5!. The agreement betweetheir results and the present simulations is striking. In additionvalidating the present numerical results, it indicates that the mmechanisms for the generation of the vortical flow pattern areshear layers emanating from the valve tips. The flow charactetics away from the vicinity of the valve are insensitive to thReynolds number as well as to the shape of the valve.

Fixed Valve and Physiological Inflow. In the second case, atime varying physiological inflow is specified at the entrancethe channel. The fundamental time scale is specified by thetime of Tper50.8 s, approximately four times larger than the naral vortex shedding period of the steady case of Re5800. Largevariations in the flow rate are enforced by the specified inflwaveform, see Fig. 2. The peak inflow velocity is 63 cm/s, whthe mean velocity is only 11 cm/s. The Womersley numbera isa5H/2A2p/nTper520.2, wheren is the kinematic viscosity (n50.035 cm2/s). These large time-variations have a dominantfect on the flow field, as shown in Fig. 10, by the time historythe transverse velocity at the same point as in Fig. 8.

The large variations in the flow field can be also observedFig. 11, where a time sequence of the streamlines and the vortis presented for one complete cycle. Although the main interesthe present study is devoted to the fully open position of the vathe entire cycle results are presented to account for differenthat originate from the opening or closing of the moving valcase~see the next section!. The global vortical pattern in the vi-cinity of the valve is fairly similar to the steady inflow caseslong as the flow rate is large enough and positive~the flow is fromthe atrium to the ventricle!. However, the details vary. A largeattached vortex is created by the shear layer emanating fromlower tip of the valve. This vortex is closer to the valve than in tsteady inflow cases and it is strong enough to induce seconvortices on the lower side of the valve. Similar to the stea

Fig. 10 Time history of the transverse velocity component atthe point „1.7H, 0.6H… downstream of the valve „pulsatile inflowcase with a fixed valve …


Journal of Biomec

Fig. 11 Time sequence of the streamlines and the vorticity for one cardiac cycle „pul-satile inflow case with a fixed valve …

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inflow cases, vortex shedding is observed from the upper tip ofvalve only. Two vortices are shed in the acceleration phase ofdiastole. During the reversed flow phase, small vortices are gerated upstream of the valve at both tips. However, due toshort duration of the reversed flow, these vortices have a vshort life span, they are small in size and remain attached.

The most noticeable difference between the present tidependent inflow case and the steady inflow cases is evident ideceleration phase~end of the diastole!, when a number of largevortices are generated in the pulsating flow, Fig. 11(e–g) and(a). Most of the vortices develop downstream of the valve,vortices may be found on the upstream part as well during regitation ~Fig. 11(g)!. In contrast to the vortices generated by votex shedding, these vortices occupy almost the entire width ofchannel and they seem to be created simultaneously ratherbeing generated one in each cycle. In the former case of voshedding, the vortices are generated by the rolling up of shlayers that originate from the tips of the valve. We believe thatlarge vortices of the pulsating inflow are a result ofvorticitywaves. The generation mechanism of vorticity waves can be fouin @26–29#. The group velocity of the vortex wave is large so thseveral vortices develop even in the short time of the deceleraphase, see Figs. 11(d–e).

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Moving Valve and Physiological Inflow. The third case con-siders the model with a moving valve and a physiological infloThe instantaneous streamlines and the vorticity for this casegiven in Fig. 12. During the opening of the valve~Figs. 12(a–d)!,strong shear layers develop from both the upper and lowerbecause the gaps between the valve tips and the walls are smthan in the fully open valve case. Although the opening timeshort, the strong shear layers roll up fast into two large tip voces. The upper vortex is shed and another large vortex is forthere until the valve fully opens. The sequence of the generaof vortices and their propagation is similar to the fixed valve caindicating that the opening of the valve does not have a significinfluence on the global flow properties.

In the fixed valve case~with pulsatile inflow!, the lower shearlayer bursts almost vertically up to the upper region of the valacting as a barrier that does not allow the vortices from the lopart of the valve to increase in size. In the moving valve cahowever, the lower shear layer remains attached to the wallquite a long distance. It detaches gradually from the wall onlyx.1.5H. This allows relatively large motion of the vortices atached to the downstream part of the valve. In this case as wquite large stationary regions are found in the downstream viciof the valve in its fully open position~Figs. 12(c–e)!.

Fig. 12 Time sequence of the streamlines and the vorticity for one cardiac cycle„pulsatile inflow case with a moving valve …


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On the upper tip, a large vortex is developed in the acceleraand part of the deceleration phases of the diastole, but it doesseem to affect properties on the valve itself~Fig. 12(b–d)!. Thisvortex grows until it is shed byt/Tper50.44. In the moving valvecase, the core flow has a larger amplitude, but a smaller wlength and it extends further downstream than in the fixed vacase, indicating that the motion of the valve augments the voity wave.

In the fixed valve case, two vortices have been observedstream of the valve during regurgitation~Fig. 11(f )!. In the mov-ing valve case, no vortices on the upstream face of the valvecreated. Near the lower tip of the valve, the gap is very sm~smaller than on the upper tip!, enforcing most of the mass flowrate to flow through the upper gap. Thus, neither a shear layera vortex can be generated on the lower part of the valve. Althomost of the flow passes through the upper gap, the relative veity between the upper tip of the valve and the flow is small bcause of the closing velocity of the valve. Thus, a strong enoshear layer that can roll up into a vortex does not develop neartip as well.

Discussion

General Flow Features. The main motivation of the presenstudy was to find out the relative importance of various unsteness sources on the flow field during the fully open position otwo-dimensional model of a tilting disk valve. The aim of thstudy is to simplify, as much as possible, the numerical modrequired for simulating flow across MHV, without neglecting mjor flow phenomena occurring in the fully open position of tvalve. Three major sources of unsteadiness were identified inflow across MHV:~i! natural vortex shedding;~ii ! time variationsimposed by the time-dependent inflow and;~iii ! the motion of thevalve. The numerical simulations were chosen to test the cobution of each one of these sources.

The flow field obtained in all these cases had several commproperties. In all the cases a shear layer is created from eachthe valve. The upper shear layer rolls up into a vortex that is shwhile the lower shear layer is relatively stable. It rolls up intovortex but it is not shed~except in the case of the moving valvwhere a very weak vortex is shed!. The flow field in the case ofvortex shedding is inherently unsteady, even in the case oflower steady inflow velocity with a fixed valve position.

Another common feature is the creation of multiple vorticdownstream of the valve. The origin of these vortices, howedepends on the driving flow conditions. In the cases of natuvortex shedding, two moving rows of relatively small vortices agenerated along the walls. Every vortex in a row originates franother shedding cycle. In the pulsating inflow cases, in addito vortex shedding, several vortices of the order of the widththe channel are created simultaneously in the deceleration pof the diastole. The generation mechanism of these latter vortmay be related tovorticity wavesas described by@26–29#. This isessentially an inviscid mechanism that generates a wavy corethat propagates downstream with a large group velocity, creaseveral vortices almost simultaneously. These waves oweexistence to the nonzero vorticity gradient developed by the vathat acts like an obstruction to the flow. These waves are phcally similar to Rossby wavesin the atmosphere, but on a mucsmaller scale. They can also be described asTollmein-Schlichtingwavesof sufficiently large amplitude@29#. Vorticity waves aregenerated also as a result of moving boundaries@29#. It is reason-able, therefore, to assume that vorticity waves are augmentethe motion of the valve. Indeed, in the moving valve case, a mpronounced wavy core flow with a smaller wavelength has bfound in the present simulations, e.g., Figs. 11(d–e) and12(d–e). These two mechanisms, i.e., natural vortex sheddand vorticity waves, coexist in the pulsating inflow cases and tboth mainly affect the flow farther away from the valve.


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Black et al. @14# found in a pulsatile flow multiple vorticesdownstream of a model of an aortic bileaflet valve at end systIn this case three large vortices were generated and their dostream extent and size were limited. Although they did not mtion it, we assume that the vortices in this case were also creby vorticity waves.

The flow in the vicinity of the valve, however, is not significantly affected by these vortex generation mechanisms. In allcases, the flow in the vicinity of the valve is dominated byprimary vortex leeward of the valve and one or more smasecondary vortices. In the steady inflow cases, the vortex sysis weak and quite extensive low velocity regions are found. Inpulsating case with the fixed valve, strong vortices develop duthe barrier generated by the lower shear layer that preventsexpansion of the vortices. In the moving valve case, the vorsystem leeward of the valve exhibits larger regions of low velocduring the fully open position. A similar large, but locally confined, vortex was observed experimentally throughout most ofcardiac cycle by Chandran et al.@30# in the case of an aorticvalve. Kiris et al.@11# reported similar flow characteristics in theanalysis of a moving aortic tilting disk valve. In the latter 3-case two vortices were found as well at the upper and lowerof the valve, possibly indicating that the formation of shear laydominated vortices is independent of the dimensionality ofproblem.

In the process of opening and closing of the valve, the detailthe flow in the vicinity of the valve differ significantly from thefixed valve case. However, we are not interested in these vshort phases of the cardiac cycle. Our interest is focused onfully open position only.

Significance of Vortex Shedding. Huang et al. @13# esti-mated from their steady inflow simulations a vortex shedding fquency of 25 Hz, assuming a peak velocity of 60 cm/s. Tleaves ample time for several (480 ms/40 ms'12) vortex shed-ding cycles that might, according to their claim, dominate the flfield. In the present calculations, only two or three sheddcycles were found during the diastole for the physiologic inflowaveform cases~Figs. 11(a–d) and 12(a–d)!. Therefore, a moresuitable velocity scale would be the mean velocity duringdiastole, for example. In the present case, the mean velo~based on the diastole time! is 20 cm/s, resulting in an estimatioof three vortex shedding cycles, similar to that observed inpresent time-dependent simulations.

The fluctuations in the transverse velocity induced by vorshedding are one order of magnitude lower than the variatifound for the imposed physiological inflow, e.g., Figs. 8 and 1In Figs. 13 and 14 the normal force on the valve and the sh

Fig. 13 The normal force on the valve


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stress on the leading edge of the valve are shown for the csimulated in the present work. In all the cases, the variatidue to the shedding of vortices are at least one order of matude smaller than in the pulsating inflow cases. Lei et al.@17#showed experimentally for a 3-D case that helical vorticare shed from a tilting disk valve. Therefore, they as wellKing et al. @19# concluded that 3-D models should be employto yield physiologically significant results. We assume, howevthat these helical vortices are not a significant physiologcontribution to the flow. The present simulations show thvortex shedding is only a second-order phenomenon and it hsignificantly smaller effect on the global unsteady propertiesthe flow than the variations imposed by the pulsating infloVortex shedding may, however, potentially activate entrapplatelets that were exposed to high flow stresses around thelets @31#. The free emboli that might be generated wouldconvected downstream, increasing the risk of systemic embConsequently, although vortex shedding has a smaller effecthe global flow field, it may result in severe hemodynamdamages.

The mean normal force and the mean shear stress on theing edge of the steady inflow case with Re5800 significantlyunderestimate the physiological waveform cases, indicatingthe steady incoming flow with a flow rate equal to the mean ofphysiological waveform is not representative. This is a resultthe significant steady-streaming nonlinear effects of the vort~exhibited, for example, by the vorticity waves!. If the steady flowsimulation is based on the peak velocity (Re54200), however,the mean values are closer to the peak values obtained inunsteady incoming flow cases. The mean values of the norforce and the shear stress, however, still deviate significantly~thesteady inflow case overestimates the mean magnitude!. In eithercase, the steady flows do not realistically represent the pulsaflow field. If an accurate representation is needed~not just maxi-mal value, for example!, the simulation of physiological inflowcases is required.

Significance of Valve Motion. The next issue is whether thinclusion of the valve motion~its opening and closing! is indeedrequired for design purposes of thefully openposition.

There are certainly similarities in the global flow features of tfixed and moving valve cases with the physiological inflow.both cases, the flow field is dominated by large vortices generby vorticity waves. Vortices are shed from the upper shear layand a system of attached vortices can be found in the leewardof the valve. Yet, there are several differences in the finer deof the flow. The core flow of the moving valve case has a laramplitude and a smaller wavelength. In this case, vortices s

Fig. 14 The shear stress variation at the leading edge of thevalve


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from the lower shear layer as well, although they are very weIn the fixed valve case, on the other hand, the system of attacvortices lee of the valve is stronger. In the closing and especiin the opening phases of the valve, the dissimilarities betweenfixed and the moving valve cases are, naturally, the largest.the present study does not consider these very short phases ocardiac cycle.

In the design of MHV, the main interest is focused on the proerties in the vicinity of the valve. Figure 13 presents the timvariation of the normal force exerted by the flow on the valvethe various cases solved. In the fully open phase, the variationthe force of the fixed and moving valve cases are in phase, bulatter case experiences weaker forces by as much as 50%. Ta result of the stronger vortices exist leeward of the valve infixed valve case. These stronger vortices induce a smaller preson the leeward side of the valve, increasing the total force onThus, it seems safe to use the fixed valve case for structural ling computations, for example.

Several studies indicated that the maximal shear stress isways found at the leading edge of the valve@6,10,13#. Figure 14presents the shear stress on the leading edge of the valve~thelower upstream corner! for the cases solved in the present studThe variations in the shear stress of the two pulsating inflow caare in phase, but again the moving valve case results in usuasmaller shear stress in the fully open position.

Another quantity of interest is the transvalvular pressure drFig. 15. In the steady inflow cases, the mean pressure dropfound to be 580 dyne/cm2 with very small fluctuations~an ampli-tude of less than 70 dyne/cm2! for the case with the mean velocitincoming flow (Re5800) while for the steady inflow case witthe peak velocity (Re54200), the mean and oscillating components are 15,15062000 dyne/cm2. The peak pressure drop of thfixed valve case with physiologic waveform is 26,500 dyne/cm2,significantly larger than the corresponding average6800 dyne/cm2. In the moving mesh case, the peak pressure d~in the fully open phase of the valve! is 21,300 dyne/cm2, whilethe mean is only 4000 dyne/cm2. These results are in good agrement with several experimental measurements. YoganathanCorcoran@21# found for a steady flow with Re51200 a pressuredrop of 670 dyne/cm2, while for a pulsatile flow Burstow et al@5#, Hanle et al.@7#, and Yoganathan and Corcoran@21# found amaximal systolic pressure drop of 15,000, 17,000, a26,000 dyne/cm2, respectively. The corresponding findings for thmean pressure drop of the last two studies were 70008000 dyne/cm2, respectively.

It should be noted that although the steady and the pulsainflow cases have the same mean inflow as the Re5800 steady

Fig. 15 The transvalvular pressure drop variation


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case, the mean pressure drop of the pulsating cases is moreten times larger. This larger pressure drop is a result of nonlinsteady streaming effects@32#. The strong vortices generated in thpulsating inflow cases transfer fluctuating components intomean flow, increasing the mean pressure drop~together with othermean quantities!. The steady streaming effects that significanalter the mean flow, redemonstrate that pulsating inflow caseslarge fluctuating components differ significantly from the equivlent steady inflow cases. Therefore, steady inflow simulations winflow velocity that is equal to the mean of the physiologic wavform cannot represent accurately physiological cases. On the ohand, if the peak incoming flow is used as the steady inflowlocity, a better agreement can be obtained with the peak valuethe pulsating cases. Moreover, there is still a large agreementthe mean values of the pulsating case. Thus, using steady inwith the peak velocity cannot represent the entire cycle. To harealistic description of the entire cardiac cycle, the time-dependpulsating inflow case should be simulated.

In the design of MHV, critical design parameters such aslargest stresses, occur during the closing and opening of the vIn any design of MHV, these stresses should not be ignoMoreover, in these phases, the motion of the valve cannoneglected and the conclusions of the present study do not aYet, the fully open valve phase might be also of importancebecause it experiences the largest stresses, but because of thtively long duration of the loading~about half of the entire cardiacycle!.

Concluding RemarksThe present study clearly demonstrates that the assumption

steady inflow is not a reasonable approximation of physiologflow cases, if not only the peak flow is of interest. Yet, the resuwith the pulsating inflow waveform lead into mixed conclusionThe details of the flow field across fixed and moving valves infully open position might differ, although the gross featuresquite similar. Fortunately, the fixed valve case consistently resin safe estimations of several critical quantities such as the foon the valve, the maximal shear stress on the valve or the trvalvular pressure drop. Therefore, it may indicate that fixed vasimulations can provide useful information for the design of mchanical heart valves and there is no need to simulate the mcomplex problems of moving valves, as long as only the propties in the fully open position are sought.

The major weakness of the present study lies in the approxition of the enclosing geometry as a straight channel and inassumption of a two-dimensional flow. The approximation ofenclosing geometry as a straight channel affects the flow fiespecially in the mitral position and away from the valve. Thefore a more realistic simulation should include realistic shapethe atrium and the ventricle. However, as long as the propertiethe vicinity of the valve only are required, the present geomesimplifications seem to be reasonable.

In realistic cases, the flow is undoubtedly three-dimensioExisting three-dimensional solutions are rather limited in the tonumber of mesh points~20,000-50,000 nodes!. Our mesh refine-ment tests revealed that at least 20,000 mesh points are requirresolve properly thetwo-dimensionalcase. Thus, existing threedimensional simulations may not properly resolve the finer detof the flow. To our estimate, the minimal number of mesh poiin three-dimensional simulations should be of the order o3105 points. Consequently, routine three-dimensional simulatifor design purposes still have to wait for stronger computersmore efficient CFD codes.

In the present study, a laminar flow was assumed, althotransitional or turbulent flows were observed in the deceleraphase in large blood vessels@33#. The numerical simulation ofsuch cases, which also include unsteady, vortical and reveflow regions, is impractical because of lack of reliable turbulemodels. The use of existing models, that were validated for c


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siderably simpler cases, could introduce modeling errors lathan the variations originating from the unsteady effects. Thesumption of laminar flow does introduce inaccuracies, but welieve that the qualitative conclusions regarding the unsteadyfects are valid in realistic cases as well.

AcknowledgmentThis research was partially supported by a grant from G.I.F.,

German-Israeli Foundation for scientific Research and Devement and by Israel Science Foundation.

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Unsteady Effects on the Flow Across Tilting Disk Valves

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