Low Reynolds Number Modeling of Pulsatile Flow in a Moderately Constricted Geometry Jonathan Ryval1∞, Anthony G. Straatman1∗, David A.Steinman2+ 1
Advanced Fluid Mechanics Research Grp,. The University of Western Ontario, London, Ontario, N6A 5B8 2
Imaging Research Labs, Robarts Research Institute, London, Ontario, N6A 5K8
Email:
[email protected],
[email protected],
[email protected]
Low Reynolds number turbulence modeling is of particular relevance for studies of blood flow dynamics in diseased arteries. In this study, a straight, three-dimensional tube with a 50% diameter reduction was analyzed under both steady and pulsatile flow conditions. The unresolved turbulence was modeled using the k-ω model. The results were compared to published results and recirculation lengths are found to be well-reproduced, but discrepancies were found in the core velocity magnitude downstream of the constriction. Pulsatile flow simulations showed good velocity profile comparisons but the turbulence intensity was over predicted.
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ABSTRACT
the carotid bifurcation, e.g. [1]. Understanding the flow characteristics here are paramount to predicting stresses on artery walls and formed elements in the blood. It is also important for the simulated pulsations to follow the actual predicted flow rate in the artery (Figure 1.) so that the numerical results are physiologically relevant.
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1. INTRODUCTION Investigation of the flow in a stenosed (constricted) geometry is of interest because of its significance in relation to human vascular disease. The goal of this project was to use the commercial software FLUENT (version 6.0) to explore laminar and turbulent pulsatile flow in stenosed geometries. The human vasculature that this research is concerned with is the carotid artery. The carotid artery carries blood to the head and brain and is of particular importance because of its high likelihood to constrict due to plaque build up, which can lead to strokes. It has been thoroughly established through experimental work that various degrees of stenosis lead to unusual and potentially turbulent flow through
Figure 1. Pulsating flow rate (in mL/min) found in the normal carotid artery [2] Since no physiologically pulsatile flow experiments currently exist in the literature, FLUENT’s accuracy was verified by comparing simulations to steady [3,4] and sinusoidally pulsatile [5] experimental results obtained from a moderately constricted model.
2. GEOMETRY The three dimensional model under investigation was based on that described by Ahmed and Giddens [3]. The geometry is a constricted, axisymmetric tube (see Figure 2). The diameter (D) of the tube is 5.08 cm and the constriction is a 50% reduction in
∞
M.E.Sc. Candidate, U. Western Ontario
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Associate Professor, Dept. of Mech. & Mat. Eng., U. Western Ontario
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Associate Professor, Dept. of Medical Biophysics, U. Western Ontario
Figure 2. Stenosed Geometry (50% Diameter Reduction) The computational mesh consisted of approximately 143,000 nodes, which was found to provide sufficient spatial resolution. Nodes were concentrated near the walls and in the vicinity of the constriction where flow gradients were expected to be high.
3. NUMERICAL FORMULATION For the present simulations the fluid was considered incompressible and Newtonian. The fluid properties of blood were specified in the simulations; density ρ = 1 g/cm3 and kinematic viscosity ν = 0.035 cm/s2. The walls of the vessel were considered rigid to simplify the problem and to match experimental data. Furthermore, when stenoses develop in human vasculature, the vessel walls in the vicinity of the stenosis are usually relatively solid. The flow was considered to be both laminar and turbulent, i.e. a range of Re was solved using both a laminar and turbulent solver. Under turbulent conditions, the flow was assumed to be mainly lowRe and thus the k-ω model was employed to model the unresolved turbulence. At the inlet a user defined velocity profile was imposed; Poiseuille for steady flow and Womersley flow for pulsatile simulations. When the flow was considered turbulent, the inlet turbulence was based on a low turbulence intensity of 2% and a correspondingly low specific dissipation rate, which was calculated from the intensity and the hydraulic diameter. These conditions were in accordance with the experiments of Ahmed and Giddens [5]. The
pressure was defined at the exit and no-slip/zeropenetration conditions were employed at the walls.
4. RESULTS 4.1 Steady Flow FLUENT has four variations of the k-ω model: a standard version, a shear flow correction version, a transitional flow correction version, and the option of using both the shear flow correction with the transitional corrections. Figure 3 shows results of the predicted recirculation lengths behind the stenosis using the laminar solver and all k-ω model combinations for steady flow. The laminar solver was accurate for low Re (i.e. Re < 250), but for higher Re, the laminar solver yields recirculation lengths that are much longer than those which have been reported experimentally [3]. In fact, figure 3 shows that the recirculation length predicted using the laminar solver increases nearly linearly with Re, in agreement with previous numerical work by Ghalichi et al. [4]. Considering the turbulent simulations, at low Reynolds numbers, eg, 100 through 250, all versions of the turbulence model yield predictions that closely resemble the laminar solution. Worth noting is that FLUENT’s transitional version of the k-ω model appears to be incapable of returning a reasonably correct solution above Re = 250 for this geometry and set of flow conditions. 12.00
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diameter, which corresponds to a 75% reduction in cross-sectional area. For the CFD simulation, the entrance length was set to 4D and the exit length to 16D. Only a short entrance length is required since the flow entering is virtually fully developed based on the imposed inlet boundary condition. The exit length is longer so that the flow can return to a nearly fully developed state such that the outlet boundary does not influence activity occurring upstream.
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Figure 3. Recirculation lengths with variations of FLUENT’s k-ω turbulence model for steady flow simulations The shear flow correction version over predicts the extent of the recirculation region, but not nearly as poorly as the transitional model at Reynolds numbers below 1000. The standard k-ω model
without any correction options can be seen to most closely match the recirculation lengths published in [3] over the full range of Re considered. Although the recirculation length is very similar to that found experimentally, when the numerical and published experimental velocity profiles are compared discrepancies can be seen downstream of the stenosis For example, at a Reynolds number of 500, it can be seen in figures 4 and 5 that while the reattachment points in the flow roughly correspond, the numerical results show the flow is overly damped downstream and returns to a fully developed state too quickly.
4.2 Pulsatile Flow Since Ahmed and Giddens have published data for sinusoidally pulsating flow at low Reynolds number in the same geometry as used for steady flow experiments it seemed prudent to continue by attempting to replicate these experiments. The flow is described in [5,6] as an “approximately” sinusoidal centerline upstream Reynolds number of 600, varying between 200 and 1000. The frequency parameter (Womersley number) is 7.5 and the frequency of the pulse is 0.05Hz. The sinusoidal pulse for the numerical simulations can be seen in figure 6.
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Figure 4. Velocity Profiles Re = 500 (Left to Right locations: Z = 0.0, Z = 1.0, Z = 2.5 where Z is the distance downstream from the throat of the stenosis normalized by the diameter, the experimental results from [3] are the dots and the numerical results are the lines.) 1
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Figure 5. Velocity Profiles Re = 500 (Left to Right locations: Z = 4.0, Z = 5.0, Z = 6.0, see caption of figure 4 for further details) These results correspond to those reported by Ghalichi et al in [4] where FIDAP was used to simulate the flow in a two dimensional, axisymmetric version of the geometry.
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Figure 6. Upstream Reynolds Number for Pulsatile Flow Although the frequency of this pulse is much smaller than that which would be typically found in a physiological flow owing to the use of a scaled-up model in the experiments, the Womersley number is in the physiological range. Three pulse cycles were simulated to damp initial transients and ensure acceptable cycle-to-cycle convergence. Twenty, forty, and eighty time steps per cycle were used in preliminary simulations to determine the influence of time step size. It was found that the results were well-converged using forty steps per cycle. In order to make comparisons between the experimental results published and those simulated herein, the transient results were averaged over certain time intervals (ie. the acceleration phase of the pulse, and the peak phase) as in [6]. Comparing velocity profiles found in [5], once again as in the steady flow results, it can be seen that while the measured and computed near-stenosis velocity profiles agree well, the core velocities are overdamped downstream of the stenosis (figures 7, 8, 9).
intervals that the numerical simulations greatly over predict the intensity compared to that reported. A comparable trend can still be observed between the experimental results and the numerical ones; turbulence develops at the throat (Z = 0) and then a larger peak occurs downstream. The trend of the numerical simulation seems to be an exaggeration of that which is published.
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Figure 7. Centre Line Velocity at Sine Wave Peak (The dots are from the experimental results published by Ahmed [6] and the solid lines are those generated numerically)
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Figure 8. Velocity Profiles at Sine Wave Peak (Left to Right locations: Z = 0.0, Z = 1.0, Z = 2.5, the dots are the experimental results from [5])
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5. SUMMARY
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Figure 9. Velocity Profiles at Sine Wave Peak (Left to Right locations: Z = 4.0, Z = 5.0, Z = 6.0, see caption of figure 8 for further details) Of particular importance in these simulations are turbulent quantities. Ahmed reports turbulence intensities in [6] over two time intervals: the acceleration phase of the pulse and the peak phase. Comparisons of these are shown in figures 10 and 11, respectively. It can be clearly seen during both time
It has been demonstrated for the presented geometry that FLUENT’s transitional k-ω turbulence model as well as the standard laminar solver are unable to return an experimentally produced solution for steady, fully developed low Reynolds number flow. Using the standard k-ω turbulence model, experimental recirculation lengths results have been shown to be reproducible. However, the downstream velocity profiles do not correspond well The pulsatile results show better correlation for the velocity profiles, but the standard k-ω model over predicts the turbulence generated in these simulations.
6. REFERENCES [1] Steinman DA, Poepping TL, Tambasco M, Rankin RN, Holdsworth DW (2000), Flow Patterns at the Stenosed Carotid Bifurcation: Effect of Concentric versus Eccentric Stenosis. Ann Biomed Eng. Apr; 28(4):415-423. [2] Smith RF, Rutt BK, Fox AJ, Rankin RN, Holdsworth DW (1996), Geometry Characterization of Stenosed Human Carotid Arteries. Acad Radiol 3:898-911 [3] Ahmed S, Giddens D (1983), Velocity Measurements in Steady Flow Through Axisymmetric Stenoses at Moderate Reynolds Numbers. J. Biomech 16:505-516 [4] Ghalichi F, Deng X, De Champlain A, Douville Y, King M, Guidoin R (1998), Low Reynolds Number Turbulence Modeling of Blood Flow in Arterial Stenoses. Biorheology 35:281-294 [5] Ahmed S, Giddens D (1984), Pulsatile Poststenotic Flow Studies with Laser Doppler Anemometry. J. Biomech 17: 695-705 [6] Ahmed S (1998), An Experimental Investigation of Pulsatile Flow Through a Smooth Constriction. Exper Therm Fluid Sci 17: 309-318 [7] FLUENT 6.0 User Guide, 2001, Chapter 6
