Experimental Thermal and Fluid Science

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1 Experimental Thermal and Fluid Science 34 (2010) Contents lists available at ScienceDirect Experimental Thermal and Fluid Science journal homepage: Steady and unsteady flow within an axisymmetric tube dilatation Ch. Stamatopoulos a, Y. Papaharilaou b, D.S. Mathioulakis a, *, A. Katsamouris c a Department of Mechanical Engineering, Fluids Section, National Technical University of Athens, Greece b Institute of Applied and Computational Mathematics, FORTH, Crete, Greece c Division of Vascular Surgery, Medical School, University of Crete, Greece article info abstract Article history: Received 23 November 2009 Received in revised form 12 February 2010 Accepted 20 February 2010 Keywords: Tube dilatation Wall shear Vortex Flow separation reattachment The flow field in an axisymmetric tube dilatation is studied employing a 2D PIV system and the commercial numerical code FLUENT. Experiment and numerical predictions are in good agreement providing similar trends and the same flow topology. For the steady case and for Re varying in the range , the recirculation zone length increases with Re, the flow reattachment line being displaced towards the exit of the model. Upstream of this line and a small distance from it, negative velocity maximizes close to the wall as well as the wall shear stress (in absolute value). Downstream of this region, the wall pressure peaks and wall shear takes a local maximum at the model exit. In the rest part of the cavity both wall shear and pressure do not practically vary due to separated flow. The axial velocity on the longitudinal axis of the model does not change streamwise for higher Re (Re = 690), resembling the near field of a jet, entraining fluid from the cavity region. In the unsteady case the flow rate is sinusoidal, the Womersley number is 3.3 and peak Re = 272. During early acceleration, a vortex ring is formed at the proximal part of the cavity and two stagnation points appear on the longitudinal axis of the model approaching each other as time progresses, eventually disappearing when the majority of the fluid particles changes direction. The velocity profile at the exit is most of the cycle blunt compared to the parabolic type profile at the model entrance. In contrast to the steady case, the pressure variation does not exhibit a local peak within the cavity rather varying in a smooth way. Conversely, wall shear stress shows high peaks at the distal end of the dilatation being proportional to the time dependent flow rate. The reattachment line travels along the wall, as well as the local pressure peak which is always located downstream of it. Massless particles released at various locations and time instants within a cycle are not trapped in the recirculation zone, being exposed to varying shear stress values. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction The objective of this work was to examine the flow in a tube dilatation both experimentally and numerically aiming at understanding the fundamental fluid mechanics phenomena which occur in arterial aneurysms. Due to severe implications on human health following an aneurysm rupture, there has been a lot of effort by fluid mechanics researchers to reveal the etiology of the aneurysm formation and eventually its rupture, mainly from the haemodynamic and in a lesser degree from a fluid structure interaction point of view. The time dependent blood flow rate and flow separation which takes place as a result of the sudden expansion of the aneurysm make the flow field being quite complex. The problem becomes even more involved, if transition to * Corresponding author. Address: National Technical University of Athens, School of Mechanical Engineering, Fluids Section, 9 Heroon Polytechniou Ave., Zografos 15710, Athens, Greece. Tel.: ; fax: address: mathew@fluid.mech.ntua.gr (D.S. Mathioulakis). turbulence and relaminarization occurs within each cycle of the periodic flow [1]. Rupture diagnosis of an arterial aneurysm and subsequently the need for surgical treatment are of paramount importance for the clinicians. More particularly, abdominal aneurysms are considered of high rupture risk when their maximum diameter exceeds 5 cm or their expansion rate is greater than 0.5 cm per year [2]. However, according to clinical practice there have been cases that aneurysms have ruptured without following the above criteria [3 5]. There is strong evidence that growth and rupture of arterial aneurysms are related to haemodynamics since they appear at specific locations of the vascular system. Wall shear stresses within the aneurysmal bulge are considered as the predominant factor for arterial wall remodeling whereas static pressures are believed to be responsible for wall weakening, leading eventually to rupture [6]. Due to the increase of the arterial diameter, the flow separates at the proximal edge, resulting in low negative wall shear stresses. In physiological flow waveforms these low stresses are additionally time dependent, triggering a mechanism of blood particle adhesion to the arterial wall, like monocytes and platelets, causing thrombus /$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi: /j.expthermflusci

2 916 Ch. Stamatopoulos et al. / Experimental Thermal and Fluid Science 34 (2010) Nomenclature d D L p d p out p in p w p * p un r r * Re parental tube diameter maximum model diameter length of the model downstream wall pressure wall pressure at the exit of the computational domain wall pressure at the inlet of the computational domain wall pressure nondimensional wall pressure, steady case nondimensional wall pressure, unsteady case radial coordinate normalized radial coordinate Reynolds number t T u x a Dp m s rx s st s w s w x time instant period of a cycle axial velocity longitudinal axis Womersley number maximum pressure difference kinematic viscosity fluid shear stress wall shear stress in straight tube wall shear stress nondimensional wall shear stress cyclic frequency formation [7]. In a recent study [8] on patients with intracranial aneurysms it was demonstrated that low wall shear stress regions are associated with aneurysm growth. However, a comprehensive review study by Lasheras [9] raises among other the intraluminal thrombus formation issue which cancels out the role of flow shear in the expansion of the aneurysm. Furthermore, this study provides a good explanation of the small progress that has been achieved so far in answering questions with regard to aneurysm enlargement and rupture; namely, (a) the exact mechanism of flow shear on endothelium cells function is still unknown, (b) the mechanical properties of the arterial wall being nonlinear and anisotropic are difficult to be determined especially due to remodeling, and (c) the present medical imaging techniques are not able to provide information about the wall thickness and its geometry with adequate accuracy. During the last decade, a number of computational and experimental in vitro studies have been published aiming at providing information on the flow characteristics in either idealized or patient-specific aneurysm shapes. The computational works treat normally blood as a Newtonian fluid, while the flow rate is considered either constant (steady flow) or time dependent. The employed experimental techniques for velocity measurements are mainly 2D Particle Image Velocimetry [10 13] which allows for the detection of coherent vortices translating on a lighted plane as well as Laser Doppler Velocimetry (LDV) for point measurements [14,15]. Normally, the aneurysm model walls are treated as rigid and in few cases these are assumed to be compliant, studying the corresponding fluid structure interaction problem [16 19]. Also, flow visualization is used for qualitative study of the flow in aneurysms [20 23]. In the context of the present work a simple tube dilatation shape was chosen as a first step towards understanding the fundamental mechanisms of blood motion in an aneurysm. Employing a 2D PIV system the velocity field is simultaneously measured at a good number of points on a symmetry plane whereas ensemble averaging is employed in the unsteady case. Numerical analysis is also used revealing valuable information of the flow close to the wall where PIV fails to provide accurate results due to light scattering. The majority of the relevant works in the existing literature are either numerical or experimental while very few of them use both experiment and numerical analysis. Especially, for time dependent flows in aneurysms, no such works have been published before, to the best of our knowledge. Therefore, the present work aims to feel this gap. Moreover, the objective of the present effort is the exploration of the basic flow characteristics which are more or less common in aneurysms, like flow separation and reattachment, vortex formation, shear stress and pressure distributions. Knowing the basic flow trends in a simple geometry it may help our understanding of the flow behavior in nonsymmetric patient-specific models. In order to be able to draw solid conclusions, the examined Re and Womersley numbers were chosen to be relatively low to avoid the appearance of flow instabilities. Wall shear stress and wall static pressure variations along the wall are presented and the differences between steady and unsteady flow fields are revealed. Details of the flow reattachment region are shown as a function of time as well as the moving wall pressure and shear stress peaks. Although the used conditions in this work, namely rigid tube walls, idealized tube dilatation shape, Re and Womersley numbers are not consistent with physiological flows in real aneurysms, the basic flow field characteristics described here are similar with the results of in vitro experiments in realistic aneurysm shapes and their numerical predictions. 2. Experimental method and model The tube dilatation model is transparent, axisymmetric, of elliptical shape in the streamwise direction, made from an elastomer material (Sylgard-184) with a dilatation ratio D/d = 2.46 and elongation ratio L/d = 3.46, where D = 32 mm is the maximum diameter, d = 13 mm is the parental tube diameter and L = 45 mm is the length of the model (Fig. 1). The above geometrical ratios are typical for fusiform type aneurysms [15]. The model was manufactured of two identical half parts which were joined by pressing one against the other. The minimum thickness of the model walls was 4 mm so that it was not deformed when time dependent pressures were applied. The model is connected, upstream and downstream of it, to 600 mm long straight tubes of internal diameter d = 13 mm. This tube length is adequate in order for the flow to be fully developed when entering the model for all the examined cases. More particularly, according to [24] the entrance length for a straight circular tube, assuming laminar steady flow, is given (in tube diameters) by the formula [ (0.0567Re) 1.6 ] 1/1.6. Therefore, for Re = 690, which was the maximum tested Re in this work, the entrance length is 39d = 507 mm. In the unsteady case, this length is time dependent and smaller than the entrance length Fig. 1. Tube dilatation model. Dimensions in mm.

3 Ch. Stamatopoulos et al. / Experimental Thermal and Fluid Science 34 (2010) Fig. 2. Experimental set-up for the unsteady case. that corresponds to peak Re (see [25,26]). Therefore, for the examined peak Re = 272 this length is 234 mm. The flow is established either by gravity when it is steady or sinusoidal for which a piston pump is used (Fig. 2). In the latter case, in order to obtain a smooth flow rate waveform, an air water chamber is installed. The velocity field is measured on a symmetry plane of the model, employing a 2D PIV system (Oxford Lasers). A water glycerine solution (40/60 volume ratio) is used as the working fluid in order to reduce image distortions due to the difference of refraction indices at the fluid solid interface. In order to reduce light scattering, the model is immersed in a bath of the same fluid contained in a plexiglas tank. The viscosity of the solution was measured with a viscometer (Cannon-Ubbelohde) and its density by a hydrometer. The temperature of the fluid was kept constant within ±1 C, so that its kinematic viscosity (8.11 cst) did not vary more than 3.4%. The flow was seeded with 10 lm hollow glass spheres of neutral buoyancy (Potters Industries Inc.) and the flow rate was measured by an electromagnetic flow meter (Carolina Medical, model FM501). In the unsteady flow case, ensemble averaging is used, starting taking images (t = 0) at a prescribed location of the reciprocating pump piston through a noncontact sensor, based on eddy currents principle. Fifty images are averaged for each instant to obtain a good statistical convergence. Cross correlation is performed for the calculation of the particle displacements and 50% overlapping is employed along both directions. The interrogation windows are in size corresponding to an area of mm 2. The time interval between two consecutive images ranges between 0.3 ms and 2.2 ms dependent on the flow rate. respect to the boundary conditions, a uniform velocity profile is considered at the entrance (left side of Fig. 3), fully developed conditions at the exit of the computational domain, no slip conditions at the solid surfaces and symmetry conditions on the longitudinal Fig. 3. Computational domain and a portion of the numerical grid. 3. Numerical procedure The flow field, assumed axisymmetric, was predicted using the commercial code FLUENT, version (Ansys Inc.). Half of the geometric model (based on the experimental one), the used coordinate system (x, r) and a portion of the numerical grid are shown in Fig. 3. The lengths of the straight tubes connected to the aneurysm model are mm, or 52 tube diameters each, being adequate for the flow to be fully developed (see previous paragraph). The computational domain is discretised with 295,972 quadrilateral elements, the numerical grid is unstructured in the bulge area and structured in the straight tubes (Fig. 3). The closest node to the wall is 1% of the tube diameter far from it so that the wall velocity gradients are better calculated. This grid was selected following a grid independence test performing simulations with three different meshes with elements whose numbers varied between 10 5 and Comparing the wall shear stress in the aneurysm bulge and axial velocity distributions, the thinner mesh predictions differed from the coarser one (185,969 elements) by 1% at most, thus assuming that a converged solution was obtained. With Fig. 4. Bulk velocity versus time.

4 918 Ch. Stamatopoulos et al. / Experimental Thermal and Fluid Science 34 (2010) model axis. In the unsteady case, the velocity at the entrance of the computational domain (x = 700 mm) is uniform and time dependent, based on the experimental flow rate (see Fig. 4) given by the formula u = sin(xt ) (m/s), where x is the cyclic frequency. An implicit first order scheme is implemented for the time dependent terms with a time step of T/150, where T = 3 s is the time period. Wall shear stress predictions based on a time step of T/100 deviated from those of T/150 with less than 1% except for two stations, namely at the sharp edges that the bulge is connected to the straight tubes where the deviations did not exceed 6% within the cycle. Since these deviations were only local, it was assumed that the selected time step of T/150 does not affect the predictions of the whole flow field. The numerical solution s convergence for each time step was assumed when the normalized residuals for continuity and momentum equations was less than A time periodic solution was obtained after two cycles. 4. Results and discussion 4.1. Steady case The flow field within the model, under steady inlet flow conditions, is examined for five Reynolds numbers, namely Re = 105, 168, 206, 584 and 690. The common features of the above cases are flow detachment at the entrance of the model and reattachment at its distal end, so that retrograde flow covers significant part of the bulge volume. The magnitude of the negative axial velocity increases with Re, being at most 5% of the inlet bulk velocity for Re = 105, while for Re = 690 this is about three times greater (15%). Normalized (by the bulk velocity) axial velocity contours are depicted in Fig. 5, for various Re, presenting both computational predictions (upper part of each figure) and experimental measurements (lower part). Differences between predictions and measurements are on average lower than 10% of the inlet bulk velocity, being smaller for the higher Reynolds numbers. Increasing Re, the location of the reverse flow maximum is displaced downstream along with the reattachment line. The fluid particles close to the wall, moving upstream from the reattachment region accelerate for a distance of roughly L/4 and then they decelerate. The fluid decelerates around the model longitudinal axis with a rate which is reduced with increasing Re. This is clearly shown in Fig. 6 where axial velocity profiles are presented. Namely, for Re = 206 the velocity along the symmetry axis drops about 10% from the entrance to the exit while for Reynolds 584 and 690 this is essentially constant. The latter two cases resemble the flow field of a free jet being characterized by an inviscid core which for a certain length remains unchanged, whereas far from it the fluid is entrained by the action of viscosity. The influence of Re is depicted in the entrance velocity profiles which tend to become blunt at higher Re and the ratio of the peak to bulk velocity drops from 2 (for Re = 105) to 1.63 (for Re = 690). At the model exit the velocity profile exhibits a discontinuity a small distance from the wall Fig. 5. Axial velocity contours. Steady case. Flow direction from left to right. (a) Re = 105, (b) Re = 168, (c) Re = 206, (d) Re = 584, and (e) Re = 690. Upper part: numerical, lower part: experimental. Fig. 6. Axial velocity profiles. Steady case. (a) Re = 206, (b) Re = 584, and (c) Re = 690.

5 Ch. Stamatopoulos et al. / Experimental Thermal and Fluid Science 34 (2010) Fig. 7. Velocity vectors. Steady case. (a) Re = 206 and (b) Re = 690. Upper part: numerical, lower part: experimental. Fig. 8. Flow reattachment region. Velocity vectors. Steady case. (a) Re = 206 and (b) Re = 690. attributed to local fluid acceleration at the sharp junction of the model to the exit straight tube. The flow entrainment is depicted in Fig. 7, where both velocity components are presented, namely the radial component takes negative values in the area of the formed free shear layer. Approaching the exit of the model this component becomes positive since the flow is curved towards the reattachment region. This region is believed to be a probable site of aneurysm rupture due to the high spatial velocity gradients which affect both the wall shear stress and wall pressure values in a way that the arterial wall strength is reduced [11]. Based on the computational results, the velocity vectors close to the flow reattachment region are shown in Fig. 8 for Re = 206 and 690 as well as the locations of maximum wall pressure and wall shear. It is noteworthy that the pressure peak appears a little downstream of the stagnation point and the wall shear peaks further upstream from these points. In other words, the maximum normal and tangential forces are applied to the wall a certain distance apart which becomes shorter with increasing Re. With regard to the boundary layer thickness, this is reduced with Re, thus increasing the wall shear stress values due to higher velocity gradients. Since the flow impinges on the wall obliquely, the velocity close to the wall takes higher values downstream of the reattachment point compared to upstream symmetric locations, affecting the variation of the wall pressure and shear stress distributions, accordingly (see Figs. 9 and 10). Fig. 9. Wall pressure distribution. Steady case. Re = 105, 168, 206, 584, 690. Fig. 10. Wall shear stress distribution. Steady case. Re = 105, 168, 206, 584, 690.

6 920 Ch. Stamatopoulos et al. / Experimental Thermal and Fluid Science 34 (2010) Nhe normalized wall pressure distribution p based on CFD is shown in Fig. 9, where p ¼ðp w p e Þ=ðp in p e Þ, p w is the wall pressure, p e and p in are the wall pressures at the exit and the inlet of the computational domain, respectively. Although this is almost constant for the major part of the dilatation (in fact there is a small pressure recovery), its spatial gradient is high in the vicinity of the reattachment region where it takes its maximum value. Downstream of this, the pressure drops abruptly and then it is reduced linearly in the exit straight tube, as expected. Increasing Re, the pressure peak increases as well while its location is practically invariant of Re. The normalized wall shear stress s w based on CFD results is shown in Fig. 10. The normalization is based on the wall shear stress (s st ) of the straight tubes connected to the aneurysm model, namely, s w ¼ s w=s st, where s w is the wall shear stress. Due to light scattering at the liquid solid interface, there was some ambiguity with regard to the measured velocities close to the wall, not allowing the correct calculation of the wall shear stresses which were in general underestimated. According to Fig. 10, the wall shear stress takes small and almost constant negative values, due to flow separation, for almost 75% of the model length. However, close to the reattachment region this varies significantly, taking a peak nondimensional positive value of the order of 6 7 (depending on Re) at the junction of the model with the exit straight tube which was sharp (zero radius of curvature). A small distance upstream of the reattachment region, where this stress is negative, its absolute value maximizes. The location of this local maximum for two Re is shown in Fig. 8 and its nondimensional value was at most For Re = 105 and 168 the wall shear stress does not exhibit a local extremum upstream of the reattachment region, increasing monotonically to its peak value at the model exit. Again, as with wall pressure, wall shear stress variations are greater with increasing Re. Similar observations about the wall pressure and wall shear stress variations for much higher Re have been presented in various works like Budwig et al. [27] for a range of Re between 500 and 2000, Ekaterinaris et al. [28] for Re = presenting laminar and turbulent flow simulations and Finol and Amon [29] for the case of two aneurysms in a series. According to Ekaterinaris et al. [28] the basic difference between laminar and turbulent flow is the much higher wall shear stresses at the exit of the dilatation when the flow is treated as turbulent while their longitudinal variation trends are similar to the present results. According to many studies, the small negative wall shear stresses increase the probability of thrombosis or clotting of blood [30] while high wall shear stresses activate platelets which deposit at areas of low wall shear stress [31]. The motion of the fluid particles can be visualized through the streamlines since these coincide with the particle paths for the steady flow case. In Fig. 11 the particles are shown to follow elliptical type paths in the cavity of the aneurysm, the centers of which are displaced downstream with increasing Re. Due to the higher negative fluid velocities at the distal end of the cavity, the paths there approach each other. Another parameter which may cause physiological changes is the fluid stresses far from the wall and their spatial variation. The contours of shear stress s rx normalized with s st are shown in Fig. 12. It is noticeable that these stresses increase close to the reattachment region with increasing Re as well as their radial gradients, verified by the convergence of the contour lines Unsteady case Employing a reciprocating piston, an oscillatory flow of period T = 3 s was established with a practically zero mean. This period was selected, instead of the physiologic correct T = 1 s, in order to be able to analyze the flow in greater detail, based on the limitation of the available PIV system whose maximum rate was five image pairs per second. Therefore, 15 velocity fields were obtained for qffiffiffiffi each cycle for a Womersley number a ¼ 0:5d 2p Tm = 3.3. The peak Reynolds number was 272, based on the maximum velocity at the model entrance, being relatively small, in order to avoid flow instabilities which appear at high Re and especially during the deceleration phase of the cycle. However, these relatively small values of a and Re are consistent with physiological flows like those in brain aneurysms due to their small diameter, compared to the aorta [32]. Moreover, since the aim of this work was to explore the basic flow characteristics in an aneurysm, these low numbers facilitate the extraction of solid conclusions. Fig. 11. Streamlines and axial velocity contours. Steady case. (a) Re = 105, (b) Re = 168, (c) Re = 206, (d) Re = 584, and (e) Re = 690. Upper part: numerical, lower part: experimental.

7 Ch. Stamatopoulos et al. / Experimental Thermal and Fluid Science 34 (2010) The bulk velocity in the straight tube versus time is shown in Fig. 4 indicating the time instants for which the flow field was measured. Due to the sinusoidal flow rate variation, eight time instants are chosen to analyze the flow field characteristics. Six of these instants refer to the backwards motion of the fluid, namely the main flow is directed towards the negative x-axis, during both Fig. 12. Shear stress distribution. (a) Re = 105, (b) Re = 168, (c) Re = 206, (d) Re = 584, and (e) Re = 690. Upper part: numerical, lower part: experimental. Fig. 13. Axial velocity contours. Unsteady case. (a) t = 1 s, (b) t = 1.2 s, (c) t = 1.4 s, (d) t = 1.6 s, (e) t = 1.8 s, (f) t = 2.2 s, (g) t = 2.4 s, and (h) t = 2.6 s. Upper part: numerical, lower part: experimental.

8 922 Ch. Stamatopoulos et al. / Experimental Thermal and Fluid Science 34 (2010) flow acceleration (t = 1.4 s, 1.6 s, 1.8 s) and deceleration (t = 2.2 s, 2.4 s, 2.6 s). Time instants t = 1.0 s and t = 1.2 s refer to the forward motion of the flow during its deceleration phase. During early acceleration (t = 1.4 s), the flow is attached to the model wall, while the fluid around the model central axis moves in the opposite direction, as a continuation of the previous cycle. Consequently, two stagnation points appear on the central axis, the distance of which progressively is reduced, disappearing when the majority of the fluid particles changes direction. The contours of the axial velocity component both measured and predicted are shown in Fig. 13. It has been nondimensionalized by the bulk fluid velocity of the straight tube at the peak of the flow rate. Before flow peak Fig. 14. Axial velocity profiles. Unsteady case. (a) t = 1 s, (b) t = 1.2 s, (c) t = 1.4 s, (d) t = 1.6 s, (e) t = 1.8 s, (f) t = 2.2 s, (g) t = 2.4 s, and (h) t = 2.6 s. Fig. 15. Measured velocity field. Unsteady case. (a) t = 1 s, (b) t = 1.2 s, (c) t = 1.4 s, (d) t = 1.6 s, (e) t = 1.8 s, (f) t = 2.2 s, (g) t = 2.4 s, and (h) t = 2.6 s.

9 Ch. Stamatopoulos et al. / Experimental Thermal and Fluid Science 34 (2010) Fig. 16. Wall pressure distribution. Unsteady case. the flow detaches at the proximal part of the dilatation (t = 1.6 s), forming a vortex, the extent of which increases progressively, covering the major volume of the cavity at the end of the deceleration phase. As a result, the reattachment point moves along the wall, affecting both the wall shear stress and pressure distributions. Comparing the steady with the unsteady velocity fields, they exhibit some similarities only during the deceleration phase in a sense that the tube dilatation is covered by a recirculation zone. However, even in this phase of the cycle, in contrast to steady case, the reattachment line moves downstream instead of upstream, when the flow rate drops, namely when the Reynolds number decreases. Furthermore, the axial velocity variation close to the model axis is stronger, compared to the steady case, especially at the distal end, as the contours of Fig. 13 clearly show. Representative axial velocity profiles are shown in Fig. 14, providing a good description of the flow field time and space variation. A basic difference between the flow at the entrance and the exit of the model is that for the major part of the cycle the velocity profile at the exit is flat (like the case of a convergent nozzle), in contrast to the parabolic type profiles at the inlet. During early acceleration (t = 1.4 s), it is characteristic that the velocity exhibits two maxima at both the inlet and the exit. This is due to the fact that the flow around the central axis does not change immediately direction when the bulk flow reverses. Namely, it takes some time before the flow around the central axis changes sign along the whole length of the model. Therefore, during early acceleration, the major portion of the flow is directed towards the cavity, in contrast to the deceleration phase, where this follows the central axis. The formation of a vortex during acceleration was also detected by Yu et al. [11] for a sinusoidal flow rate waveform, employing a 2D PIV system. However, in cases where physiological waveforms are used, characterized by a faster acceleration, the vortex appears after flow peak [12,33,34], during flow deceleration. It is well documented that impulsively started flows behave as nonviscous for the first time steps since it needs some finite time before the action of viscosity becomes evident [12,35 37]. The measured velocity vectors are shown in Fig. 15 for various time instants, displaying the formation of a vortex at the proximal part of the model, its development and propagation. The flow field Fig. 17. Wall shear stress distribution. Unsteady case.

10 924 Ch. Stamatopoulos et al. / Experimental Thermal and Fluid Science 34 (2010) is shown to be axisymmetric. According to the numerical work of Jamison et al. [38] on fusiform aneurysms, the critical Re for transition to three dimensionality of the flow is a function of the aneurysm geometry and the Womersley number. More particularly, lowering the Womersley number the critical Re is reduced as well. Apparently, in our case this transition did not occur. Fig. 18. Normalized shear stress s rx at (a) model entrance and (b) model exit. Fig. 19. Reattachment region. Unsteady case. (a) t = 1 s, (b) t = 1.2 s, (c) t = 1.6 s, (d) t = 1.8 s, (e) t = 2.2 s, (f) t = 2.4 s, and (g) t = 2.6 s.

11 Ch. Stamatopoulos et al. / Experimental Thermal and Fluid Science 34 (2010) The wall pressure distribution along the wall of the model as well as along the straight tubes connected to it is presented in Fig. 16 in the nondimensional form p un =(p w p d )/Dp, where p d is the time dependent wall static pressure 2.5 dilatation lengths downstream of the model and Dp is the maximum pressure difference within a cycle between this station and a station 2.5 dilatation lengths upstream of the model. In the acceleration phase of the cycle (t = 1.4 s, 1.6 s, 1.8 s) the pressure drops linearly in the straight tubes with a rate which is proportional to flow acceleration. On the other hand, the pressure along the model is almost constant, whereas approaching its distal end it drops. It is reminded that in the steady case quite the opposite occurs in this area with the pressure being increased due to flow reattachment. During deceleration, there is a mild pressure recovery in the dilatation resembling the low Re steady cases. Practically, the pressure in this unsteady case does not seem to vary practically within the model in contrast to wall shear which exhibits local peaks. The same conclusion is drawn in the numerical work of Khanafer et al. [39] where the flow of a non-newtonian fluid in an abdominal aortic aneurysm is predicted under laminar and turbulent flow conditions for maximum Re of the order of Namely, they found that the pressure did not practically vary along the aneurysm wall within each cycle. The wall shear stress, nondimensionalized by the wall shear of the straight tubes at the instant of the maximum flow rate, is presented in Fig. 17. It exhibits extreme values at the entrance and the exit of the model, taking higher absolute values at its exit as well as during the acceleration phase. There is also a shear peak at the moving flow reattachment line (t = 1.4 s, 1.6 s, 1.8 s) of smaller however value compared to the previous ones. The distributions of shear s rx at both edges of the model are shown in Fig. 18 as a function of the normalized radial distance r (by the local bulge radius) from the model longitudinal axis. Fig. 18a corresponds to station x = +20 mm which is 2.5 mm upstream of the right edge of the dilatation and Fig. 18b its symmetric counterpart (x = 22 mm). When t > 1.2 s, bulk flow is directed towards negative x-axis so that Fig. 18a corresponds to the inlet of the model and Fig. 18b to its outlet. It is noticeable that the shear stress amplitude increases in the radial direction, being smaller at the exit due to the blunt velocity shape at this location for the major part of the cycle. The drop of the shear close to the wall is due to the fact that this is located inside the bulge where the flow most of the time is almost stagnant. This radial stress distribution is in contrast with the remark of Yip and Yu [40] that the magnitudes of the shear stresses are uniform in the radial direction for a symmetric aneurysm, unless their flow was a plug type due to the high Womersley numbers that they examined. The central part of the model is characterized by low and oscillating about zero wall shear stresses which according to medical evidence may energize biological mechanisms of thrombus formation [9]. In the numerical work of Khanafer et al., [39] wall shear maximum was predicted to appear at the exit of an aneurysm during systole. The motion of the formed vortex is associated with the propagation of the reattachment line and consequently of the pressure and shear stress maxima. The velocity field around the reattachment region is presented in Fig. 19 along with the locations of pressure and wall shear maximum. It should be noted that the pressure maximum systematically appears downstream of the reattachment point, like in the case of a backward facing step [41] and the shear maximum upstream of it. During each cycle both points travel along the wall which is thus exposed to a cyclic loading. Using the predicted velocity field of 30 time instants in a cycle, the particle paths of 10 massless fluid particles were computed on a Lagrangian basis, integrating the velocity field via a second-order Runge Kutta. The integration was performed by the commercial code TECPLOT 360 which performs a linear interpolation between Fig. 20. Particle paths. Unsteady case. (a) Half cycle. (b) Complete cycle.

12 926 Ch. Stamatopoulos et al. / Experimental Thermal and Fluid Science 34 (2010) (b) The reattachment region is characterized by high gradients of both wall pressure and wall shear. In the rest part of the cavity these quantities are practically unchanged. The fluid moving upstream of the reattachment line close to the wall accelerates for a distance and then it decelerates taking values up to 15% of the inlet bulk velocity. (c) Most of the model wall is exposed to low negative velocities while the axial velocity along the longitudinal model axis of symmetry is almost constant, especially for the higher Re, resembling the flow field of a free jet. (d) The shear stress s rx radial gradients take high values at the exit of the model. (e) The particle paths in the model cavity are of elliptical type shape, the centers of which are displaced downstream with increasing Re. The flow field for the unsteady case (sinusoidal) with a peak Re = 272 and Womersley number 3.3 has the folowing characteristics: the solution time levels. Nine of these particles were released at the entrance of the model along a radius and the tenth at a point in the middle of its length. Starting the integration at the instant that the flow changes sign, the particles are released with velocities which are equal to the local fluid velocities. Their paths are shown in Fig. 20 for one half of the period, namely during acceleration and deceleration, which correspond to 16 time instants. During early acceleration, the particles released at the entrance of the dilatation, move towards the cavity and then they move slightly backwards (Fig. 20a). In the next half of the period, for which the direction of the bulk flow has changed, they exit the model following paths which are at close proximity one with the other (Fig. 20b). It is obvious that none of the released particles approached the cavity walls, or it was trapped in a region close to them. The same conclusion was drawn running several numerical tests by releasing particles at various locations. Therefore, for this particular flow waveform, there is no apparent evidence about the residence time of the fluid particles in the cavity region which could be of some value from the bioengineering point of view. However, each of these particles is exposed to varying shear stresses during each cycle due to spatial and temporal flow variations as depicted clearly in Fig. 21 which might be of clinical importance. Namely, it has been documented that the activation rate of platelets (which are responsible for thrombus formation) depends not only on the level of shear stresses applied on them but on their dynamic loading as well [42]. 5. Conclusions Fig. 21. Normalized shear stress s rx each particle is exposed to. The flow field in an axisymmeric tube dilatation is examined under steady and time varying flow conditions, experimentally using 2D PIV and numerically through the commercial code FLU- ENT. The experimental data compared well with the numerical predictions, show the same flow topology. For the steady case, where five Reynolds numbers are examined, from 105 to 690 the major flow features are: (a) The flow separates at the entrance of the model and it reattaches a small distance upstream of its exit. This distance becomes progressively smaller with increasing Re. (a) During early acceleration the major volume of the fluid is diverted towards the walls of the model, the flow being attached, whereas at its center it moves in the opposite direction in continuation of the previous cycle. As a result the velocity at both the entrance and the exit of the aneurysm peaks off center and two stagnation points appear on the central model axis which approach each other at time progresses, eventually disappearing. (b) Before flow peak a vortex ring is formed at the proximal part of the model which, during deceleration, propagates downstream, covering the whole cavity of the cavity at the end of this phase. (c) The wall pressure varies weakly along the model and it recovers slightly during the deceleration phase. In contrast to the steady case, it does not exhibit any extreme values. (d) Conversely, wall shear stress takes high values close to the model exit, being a function of the flow rate. In the major length of the model, it is almost constant, oscillating with small amplitude about a zero mean. (e) The reattachment line moves downstream, after the formation of the vortex ring, as well as the point of pressure peak which is always located downstream of it. (f) The amplitude of the shear stress at the model entrance and exit increases radially. (g) Massless particles released at various locations and time instants were not found to be trapped in the cavity, being exposed to varying shear stresses which might be of clinical importance, like the platelet activation rate issue. Acknowledgments This paper is part of the 03ED244 research project, implemented within the framework of the Reinforcement Programme of Human Research Manpower (PENED) and co-financed by National and Community Funds (20% from the Greek Ministry of Development-General Secretariat of Research and Technology and 80% from EU European Social Fund). References [1] A. Salsac, S. Sparks, J. Lasheras, Hemodynamic changes occurring during the progressive enlargement of abdominal aortic aneurysms, Ann. Vasc. Surg. 18 (1) (2004) [2] P.M. Brown, D.T. Zelt, B. Sobolev, The risk of rupture in untreated aneurysms: the impact of size, gender, and expansion rate, J. Vasc. 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