Fluid flows in channels of variable cross-section are widely used in various technical applications, namely, in heat exchangers and cooling systems, in elements of chemical and drilling equipment, nuclear reactors, etc. Very often, local constrictions in channels are used as passive heat transfer intensifiers: their location with a certain step on the channel walls leads to destruction of the boundary layer and provides intensive mass and heat transfer between the main flow and the low-velocity flow regions near the wall.
Studies of viscous fluid flow in the area of canal narrowing are also important for in-depth understanding of hemodynamics of the human cardiovascular system [1‐4]. Pronounced local narrowing (stenosis) of the artery is one of the most common diseases that can lead to ischemic stroke, long-term disability or even death. In addition, stroke survivors remain at a high risk of recurrent ischemic episodes. This risk is associated both with the possible growth of the tissues of the inner layer of the vessels (intima) downstream of the constriction, and, accordingly, the expansion of the affected area of the artery, and with the entrainment of tissue particles directly from the constriction area, followed by blockage of the vessels.
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Most often, the low parietal shear stress is called among the main reasons for progression of atherosclerosis and growth of the intima thickness [1‐8]. It was also established that, in addition to the magnitude of shear stresses, the growth of the inner layer of affected vessel wall is under significant action of the dynamics of this parameter, namely, the high gradients of the wall friction stress correlate in time and space with the accelerated increase in the intima thickness. In addition, at high values of surface friction, as well as its gradients, characteristic of stenotic vessels, there is a high probability of detachment of parts of atherosclerotic plaques with possible subsequent blockage of the smaller vessels downstream.
The occurrence of abnormal (high or low) values of surface friction is associated with hemodynamics in the area of narrowing of the artery caused by atherosclerotic changes. The pulsating blood flow in large vessels not affected by atherosclerosis is most often laminar. The flow in the lumen of a stenotic vessel is characterized by an increase in the flow velocity and in the friction stress on the wall. The flow separation region is formed downstream of the local constriction. Within this region, the skin friction is significantly lower than outside it. In addition, the results of clinical studies of the flow structure in arteries with various blood flow disorders have shown that flow around an obstacle in form of a pronounced local narrowing of the vessel may be accompanied by the loss of stability of the shear layers and local turbulence of flow even at very moderate Reynolds numbers of the flow ahead of the narrowing [9‐11].
Studies of hydrodynamic instabilities and turbulence transition based on the rigid models of large blood vessels with stenosis were carried out both for stationary (with respect to the flow rate) streams and for pulsating flows with various variants of the form of stenosis and its degree - the fraction of the vessel cross-section blocked by stenosis. As a rule, the blood-simulant fluid was assumed to be Newtonian; the justification of acceptability of this approximation is given, for example, in [12, 13]. The first experimental [14‐16] and computational [17, 18] studies in this direction were restricted to the case of axisymmetric shape of narrowing, with the greatest interest in the flow through a clinically significant stenosis (75% degree or so). The results presented in these papers for a stationary-in-average flow through a hemodynamically significant stenosis of a cosine profile generally showed that the critical Reynolds number of the loss of stability of laminar flow lies above Re = 500 (calculated from the flow-average velocity and the channel diameter D upstream of the constriction section), and the conditions for transition to local turbulence downstream of the stenosis arise at values of Re close to 1000. In particular, in [18], using the direct numerical simulation method and the computational three-dimensional analysis based on the methods of linear and nonlinear theory of hydrodynamic stability, it was found that the turbulence transition downstream of 75% stenosis (with the total length of 2D) is of the rigid nature with a fairly pronounced hysteresis interval in the Reynolds number 690 < Re < 722, where the upper value, calculated according to the linear theory of stability, determines the conditions for the absolute instability of stationary laminar flow.
The strong influence of even a small deviation from the axisymmetric geometry on the conditions of transition to local turbulence downstream of the stenosis was demonstrated in computational study [19]. The authors of this study, when performing direct numerical simulation of a steady-state flow in the channel with an axisymmetric 75% stenosis (also of length 2D), obtained the solutions with a stable laminar flow regime, both at Re = 500 and at Re = 1000. However, with the introduction of an eccentricity of 5% of the channel diameter in the solution at Re = 1000, local transition to turbulence downstream of the stenosis was observed at a distance of approximately five channel diameters. This was confirmed by the appearance of an inertial interval in the velocity fluctuation spectra. Further downstream, the velocity fluctuations rapidly decayed and the flow became laminar again. It was also shown that the laminar-turbulent transition downstream of the stenosis was accompanied by significant temporal and spatial friction stress gradients. In [20], the same authors presented the results of calculations performed for the 75% stenosis with a 5% eccentricity in the case of superposition of harmonic flow-rate pulsations with the amplitude of the 67% cycle-average flow rate; in this case, the cycle-average Reynolds number was equal to approximately 600. The calculations showed that in the flow acceleration phase an accelerating vortex is formed in the region downstream of the stenosis. This vortex is destroyed in the maximum velocity phase with the formation of a turbulent spot downstream. The highest level of turbulent fluctuations was observed in the beginning of the flow deceleration phase.
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In [21] harmonically pulsating flow in a nominally square (H × H) channel with a one-sided 50% cosine-shaped stenosis of length 2H was studied numerically at the oscillation-period-average Reynolds number varied on the interval from 1000 to 2000. The calculations were carried out using the large eddy simulation method (Large Eddy Simulation, LES) with subgrid viscosity estimation using the Smagorinsky model; the grids covered the computational domain with the total length of 20 channel diameters and contained about 0.5 million elements. It was found that the maximum turbulent velocity fluctuations are observed near the channel wall on which stenosis is formed at a distance of up to six channel diameters. The presence of a pronounced inertial interval in the velocity fluctuation spectra in this region was demonstrated.
In [22] flow downstream of an asymmetric 70% stenosis with the highest eccentricity of the round throat was experimentally studied using the PIV (Particle Image Velocimetry) method in the case of pulsating (non-harmonic) flow at the peak Reynolds number equal to 1803; the cycle-average value of this parameter was equal to 650. In [22] measurements on a facility that ensures the full development of laminar flow in the supply round tube were performed for three variants of stenosis of the similar geometry: two rigid and one deformable under the action of the flow; in the first (basic) variant, non-deformable one-sided stenosis was characterized by the total length of 2D and a cosine profile in the plane of symmetry. The results of measurements, given in [22], performed mainly in the plane of symmetry of the test section, quantify the effect of variations in the stenosis patterns on the velocity profiles averaged over many cycles for different phases of the cycle (briefly, “phase averaging”), as well as on the phase-averaged field of the production term in the balance equation for the turbulence kinetic energy and the time dependences of the individual components of this equation in the region of the largest values of the production term.
In [23], flow in a model of blood vessel with one-sided 70% stenosis was numerically studied at the Reynolds number Re = 1803 that corresponds to the peak value of this parameter in the experimental work [22]. The solution was obtained by the LES method using the Germano–Lilly dynamic subgrid viscosity model on a grid containing about 4.5 million cells. In three-dimensional flow, the following zones downstream of the channel narrowing were distinguished: a jet flow zone, an extended recirculation region and secondary flows which, in the neighborhood of the stenosis throat, have the form of a pair of Dean vortices characteristic of the flow in a curvilinear channel, and a four-vortex structure in the recirculation region. An analysis of the field of the kinetic energy of turbulence developed as a result of instability of flow in the mixing layer at the boundary of the recirculation region, as well as of the field of turbulent shear stresses, was carried out. It was shown that the high level of turbulent stresses is observed in a section with the length of about four gauges, approximately in the same place, where the high values of the production term were observed in experiments [22] in the phase of the maximum flow rate. Downstream, the flow becomes laminar again.
The results of numerical study of the effect of variations in the stenosis shape (10 variants) on laminar-turbulent transition in a coronary artery model are presented in recent paper [24]. The three-dimensional time-dependent calculations were carried out on grids containing about 5 million elements with the use of the ANSYS Fluent package based on the Navier–Stokes equations with the approximation of convective terms by the second-order upwind scheme, in fact, by means of the ILES method (Implicit Large Eddy Simulation). The degree of stenosis for all its forms remained constant and was equal to 73%; the time dependence of the flow rate was set according to physiological data, with the peak Reynolds number of about 850. It was shown that the length of the jet flow which is formed during the passage of the stenosis throat significantly depends on the shape of stenosis. The indicators of turbulence transition, observed in almost all cases, except for axisymmetric stenosis, and identified by an increase in the velocity fluctuation amplitude, begin to manifest themselves in the jet flow dispersion region in which the maximum instantaneous value of the longitudinal velocity component is half the maximum instantaneous velocity of the jet in the constriction region. Further downstream, flow relaminarization is observed: the velocity profiles acquire the characteristic parabolic shape. The location of the flow turbulence region was determined depending on the shape of stenosis and the spectral characteristics of flow velocity pulsations in this region were analyzed.
The foregoing makes it possible to draw the following conclusions. In healthy vessels, the laminar flow regime is mainly preserved. This regime can be violated due to various vascular diseases leading to vascular curvature or local narrowing (stenosis). The deposition of sclerotic plaques on the vessel walls causes the appearance of flow separation areas which contribute to the further aggravation of atherosclerosis. Most researchers associate these processes with low frictional stress on the wall and its large spatial or temporal gradients in the flow separation region downstream of the constriction (stenosis). Flow separation can initiate the development of local flow turbulence and, as a result, an increase in the intensity of fluctuations of the dynamic flow parameters. These phenomena have been actively studied in recent decades using various experimental and computational methods. However, in general, the problems of the development of laminar-turbulent transition in the areas of arterial stenosis, that are very diverse in the geometry and its hemodynamic consequences, have not been studied in detail. The questions of the influence of irregular input perturbations which can develop, for example, during the passage of blood flow through the area of bifurcation of the vascular bed located upstream of the stenosis, also remain open.
The present study covers the results of the first stage of investigation which is generally aimed at improving understanding of the structure of the separated flow and the transition to turbulence in post-stenotic blood flows and involves the use of a combined approach: combination of experimental studies and numerical simulation. The presented results were obtained by studying an average stationary flow in a canal with an asymmetric narrowing of the cosine profile at the degree of narrowing of 70% and the Reynolds number Re = 1800, which is in the range of physiological values being assessed from the maximum blood flow rate in the human femoral artery during the period of heart contractions. In this stage of study, the authors used the currently traditional approach to simulation of post-stenotic blood flow, according to which the elasticity of vessel walls is not taken into account.
1 EXPERIMENTAL EQUIPMENT AND MEASUREMENT TECHNIQUE
The experimental studies were carried out using the facility whose scheme is shown in Fig. 1. The motion of fluid in the test section 1 of the facility is provided by the static pressure created by the pressure tank 4 mounted on a rigid frame. Liquid from the pressure tank enters the storage tank 6 through the test section a, from storage tank 6 it returns to the pressure tank with the help of submersible pump 7. The required fluid flow in the test section is set using orifice 10 located in the drain pipeline and is controlled by direct measurements of the weight of liquid passing through the test section during a given period of time. A constant liquid level in the pressure tank is maintained by means of overflow device 5.
Fig. 1.
Facility for investigation of steady-state flow through the area of narrowing (stenosis) of the artery: 1—the area of the studied flow; 2—transparent box; 3—upstream section of a smooth straight pipe; 4—pressure tank; 5—overflow system; 6—vessel; 7—submersible pump; 8 and 9—shut-off valves; 10—drain pipeline with a consumable washer; and 11—coordinate device.
Test section 1 is the pipe section with an asymmetric narrowing (asymmetric stenosis, Fig. 2), whose geometry is given by the expression [22]
$$\begin{gathered} \frac{{d{\text{(}}x{\text{)}}}}{D} = \left( {1 - \frac{S}{{200}}} \right) - \frac{S}{{200}}\cos \left( {\frac{{2x\pi }}{L}} \right),\quad - \frac{L}{2} \leqslant x \leqslant \frac{L}{2}, \\ \frac{{c{\text{(}}x{\text{)}}}}{D} = \frac{1}{2} - \frac{{d{\text{(}}x{\text{)}}}}{{2D}},\quad - \frac{L}{2} \leqslant x \leqslant \frac{L}{2}. \\ \end{gathered} $$
Fig. 2.
Geometry of the test section simulating asymmetric stenosis of the artery and the view of the computational grid in the longitudinal median section.
Here, D is the pipe diameter ahead of and behind the narrowing section, d is the local diameter of the flow section at the stenosis site, S = (1 – dmin/D) × 100%, L is the length of the narrowing area, S = 45%, and L = 2D (in the presented experiments, the inner diameter of the pipe D = 17 mm). The area of the minimum flow cross-section (x = 0) in the stenotic area is equal to 30.25% of the cross-sectional area of the pipe outside the stenosis, i.e., the case of 69.75% stenosis (rounded, 70%) is considered.
Distilled water was used as the working fluid. The experiments were carried out under conditions of stationary isothermal flow at the Reynolds number Re = 1800 ± 25 calculated from the flow-average velocity 〈U〉 and the diameter D. In the experiments, the indicated scattering of the Reynolds numbers is mainly due to variation in the temperature of the working fluid maintained at the level of 20 ± 0.5°С. The flow rate before the experiments was measured by the weight method with an uncertainty of ±1%. The length of the upstream section of straight smooth pipe, which was equal to 120D, was chosen from the condition of its sufficiency to achieve (at given Reynolds number) the state of developed laminar flow ahead of entering the test section with the velocity profile corresponding to the Poiseuille solution.
The instantaneous velocity vector fields downstream of the constriction area were measured using the SIV (Smoke Image Velocimetry) technique [25] based on capturing the flow pattern with an Evercam 2000-4M high-speed camera in a light sheet created by a SSP-ST-532-NB-5-5-LED-VAC continuous-wave laser. Polyamide particles 5 μm in diameter were used as tracers. The camera and laser were mounted on a coordinate device 11, which ensured their movement relative to the test section. The photography was carried out in the xy plane which coincides with the median longitudinal section of the test section (the plane of geometric symmetry). To reduce the error in the measurement results caused by the difference in the refractive indices of the working fluid and air, the test section was placed in a box filled with glycerol and made in the form of a rectangular parallelepiped.
The results of preliminary measurements of the flow velocity profiles ahead of the narrowing area (x/D = –10) in two mutually perpendicular cross-sections showed their satisfactory agreement with the Poiseuille profile. The level of root-mean-square fluctuations of the flow velocity in this cross-section of the upstream pipe was about 0.5% of the flow-average velocity. The measured spectrum of pulsations testified that the amplitude distribution over frequencies was approximately uniform. Presumably, the source of these pulsations is the irremovable low-amplitude vibrations of the platform (floor) on which the facility frame is fixed and oscillations of the free surface of liquid in pressure tank 4 (see Fig. 1).
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2 NUMERICAL SIMULATION TECHNIQUE
The flow in the model of a blood vessel with stenosis of the same geometry was calculated by the large eddy simulation method with the artificial introduction of input velocity pulsations (of “turbulent content”). The need to specify the inlet turbulent content in the calculations that involves a comparison with the measurement data on the facility described above stemmed from the results of the initial comparisons of the measured velocity profiles behind the stenosis with the data of numerical simulation performed earlier [23] under the assumption that there were no input disturbances.
Similarly to [23], the calculations were carried out using the hydrodynamic “finite-volume” general-purpose code ANSYS CFX (version 18.2) of the second-order spatial and temporal discretization. The Reynolds number Re was assumed to be equal to 1803. The possibility of specifying the input turbulent content was provided by using the hybrid zonal RANS/LES approach, which provides for the activation of a synthetic turbulence generator at the boundary between the RANS zone calculated on the basis of the Reynolds-averaged Navier–Stokes equations and the LES-zone, calculations in which are carried out according to the large eddy simulation method. In the present calculations, the RANS zone located in front of the entrance to the narrowing section was only one pipe diameter and within its limits the velocity distribution differed only slightly from that specified at the entrance to the computational region of the Poiseuille profile. The k–ω turbulence SST model was used as the closing RANS model. The calculation of flow in the LES zone was carried out using the Smagorinsky model for estimating the subgrid viscosity, with the Smagorinsky coefficient reduced by several times relative to the standard value.
In addition to the parabolic velocity distribution, at the entrance to the computational domain (the entrance to the RANS zone), the local turbulence intensity Tu = 1% was set. This value was chosen based on the results of preliminary parametric calculations from the condition of acceptable agreement between the experimental and calculated data on the length of the separation zone behind the stenosis (discussion and illustrations will be given below).
At the outlet boundary of the computational domain located at a distance of 20D from the narrowing point, the constant pressure condition was set; the specified length of the computational domain is sufficient for the output boundary condition to have no significant effect on the flow near the stenosis. The no-slip condition was implied on the pipe walls.
The calculations were carried out on a grid consisting of about 4.5 million hexahedral sells. In Fig. 2 we have reproduced a fragment of the computational grid in the longitudinal cross-section of the model. In the narrowing section and in the entire exit section, the longitudinal grid step was uniform and amounted to 0.04D, while in the transverse directions the maximum step was equal to 0.02D. At the inlet section, the longitudinal step of the grid gradually decreased to 0.04D when approaching the beginning of narrowing. The normalized time step was 0.006ts, where ts = D/〈U〉 is the characteristic time scale of the problem; the taken time step provided local values of the Courant number less than unity over the entire LES zone (the necessary condition of stability of the explicit numerical scheme). The second-order central scheme was used to approximate the convective terms of the equations of motion. The time sample used to obtain the average flow characteristics was accumulated over 1050ts; the previous time interval, covering about 600ts, was sufficient to reach the statistically steady-state flow regime. The calculations were carried out on the cluster “Polytechnic–RSC Tornado” of the supercomputer center “Polytechnic” (http://www.scc.spbstu.ru).
3 RESULTS AND DISCUSSION
The actual velocity was measured in the plane of symmetry for the cross-sections of test section with the following coordinates: x/D = 2.0, 2.57, 3.13, 3.7, 4.27, 4.84, 5.41, 6.0, 6.55, 7.12, 7.69, and 8.27 (for the cross-sections x/D < 2 the measurements were not carried out due to opacity of the material from which the insert imitating arterial stenosis was made). Based on the set of measured profiles of the time-average longitudinal velocity, the two-dimensional field is constructed by applying the interpolation procedure. This field is shown in Fig. 3 in comparison of the calculated data obtained in the present study with the artificial introduction of turbulent content at the entrance to the narrowing section and those obtained in [23] for zero input disturbances. We can clearly see the effect of taking into account input disturbances. This effect leads to significant reduction in the length of the separation (recirculation) zone behind the stenosis as compared to the case of zero disturbances and, as a result, to the much better agreement with the measurement results, including the length of the high-speed jet formed when flow crosses the throat of stenosis.
Fig. 3.
Field of the averaged longitudinal flow velocity U/〈U〉 in the plane of symmetry of the model of blood vessel with stenosis: (a) experiment; (b) calculation with the introduction of input disturbances; and (c) calculation without input disturbances [20].
The maximum flow velocity in the high-speed jet is higher than the flow-rate-average velocity ahead of the constriction by more than four times. Further downstream, the jet loses its intensity, is smeared, and at x/D ≈ 5 the flow velocity in the jet becomes comparable with the flow-average velocity 〈U〉. According to the results of measurements and calculations with the introduced input disturbances, the longitudinal dimension of the recirculation region normalized to the base diameter D is equal to 4.1 ± 0.1; being normalized to the stenosis height, this corresponds to 7.4. It should be noted that the longitudinal dimension of the recirculation region was determined as the difference between the coordinates of the points of flow reattachment and separation, taking into account the fact that separation occurs immediately downstream of the throat of stenosis. We also point out that the solution obtained with zero input disturbances gives the position of the flow reattachment point x/D = 5.3 [23].
In Fig. 4 we have reproduced the fields of the time-average velocity components in three cross-sections of the vessel model obtained in calculations with the introduced input disturbances. It can be seen that the jet with relatively high local velocities formed in the area of stenosis is also characterized by the presence of intense transverse (secondary) flow in the form of a paired vortex (Fig. 4a). This pair of vortices, similar to the Dean vortices in curvilinear tubes, is formed in the upstream part of the stenosis, where the flow occurs along curvilinear streamlines, corresponding to the geometry of the stenosis. In turn, the paired vortex that has arisen in the stenosis induces a secondary flow (of opposite circulation) in the reverse flow zone behind the stenosis.
Fig. 4.
Calculated longitudinal velocity field with superimposed transverse velocity vectors in three channel cross-sections downstream of the stenosis: (a–c) correspond to x/D = 2, 4, and 10, respectively.
The transverse flow abruptly transforms in the vicinity of the end of the reverse flow zone (Fig. 4b) and then almost completely degenerates. At a distance of about 10D from the center of the stenosis, the average longitudinal velocity distribution again acquires the axisymmetric form with the maximum velocity in the center of the vessel (Fig. 4c).
As an example, in Fig. 5 we have given a comparison of the calculated and experimental velocity profiles at two distances from the stenosis (x/D = 2 and 6.55). We can see that, in general, the results are in adequate agreement with each other.
Fig. 5.
Comparison of calculated (curves ) and experimental (dots) flow velocity profiles in the symmetry plane downstream of the constriction area: (a) and (b) correspond to x/D = 2 and 6.55, respectively.
The most interesting feature of the flow under consideration consists in the formation of a local turbulence region behind the stenosis. To illustrate this feature, in Fig. 6 we have reproduced the Q-criterion isosurfaces constructed from the numerical simulation data for two calculation variants: with zero input disturbances [23] and with introduction of disturbances. The constructed isosurfaces are colored according to the absolute values of the local velocity and clearly visualize the domain of existence of different-scale turbulent vortex structures that arise as a result of the manifestation of hydrodynamic instabilities inherent in the jet shear flow formed downstream of the stenosis. In the variant of calculations with superimposed input disturbances, transition to the flow state with three-dimensional turbulent formations occurs almost immediately downstream of the stenosis. In the second variant, the formation of quasi-two-dimensional vortices resulting from the Kelvin–Helmholtz instability can be clearly traced and the region of turbulence transition is displaced downstream.
Fig. 6.
Visualization of vortex structures in the channel with a local asymmetric constriction: Q-criterion isosurfaces colored by the velocity modulus based on the results of calculations without imposition (a) and with imposition (b) of input disturbances.
In Fig. 7 we have plotted the experimental profiles of the root-mean-square pulsations of the longitudinal component of the flow velocity urms measured in the plane of symmetry of the test area at different distances from the stenosis throat. It can be seen that already in the first of the examined cross-sections (x/D = 2) the velocity fluctuations are very large with the peak value of urms being about 90% of the flow-average velocity 〈U〉, or about 27% of the flow-average velocity in the stenosis throat, 〈US〉 = 3.3〈U〉. The same level of the maximum values of urms is also observed in two subsequent cross-sections (x/D = 2.55 and 3.13). At the same time, in all three cross-sections, the peak values of urms are located at y/D ≈ 0.6, where the maximum gradients of the average velocity are also observed (see Fig. 5). Starting from the cross-section x/D = 3.5, the level of velocity fluctuations sharply decreases and at x/D > 6 the fluctuation distribution becomes almost uniform over the flow core, being characterized by the value urms/〈U〉 ≈ 0.2.
Fig. 7.
Experimental profiles of root-mean-square values of fluctuations of the longitudinal component of the flow velocity.
The examples of comparison of the calculated and experimental profiles of root-mean-square pulsations of the longitudinal velocity presented in Fig. 8 testify to the general consistency of the nature of compared distributions and intensity levels of pulsations measured and calculated at different distances from the stenosis. The observed differences in the pulsation intensity are apparently due to some shift in the experiments of the plane of light sheet from the plane of symmetry of the channel, which has a greater effect on the magnitude of the measured values of urms.
Fig. 8.
Comparison of calculated (curves) and experimental (dots) profiles of root-mean-square velocity fluctuations: (a) and (b) correspond to x/D = 2 and 6.55, respectively.
For the general estimate of the degree of flow turbulence at various distances from the throat of stenosis, we introduce the characteristic turbulence intensity as the ratio of two values, namely, the maximum mean square pulsation of the longitudinal velocity over a cross-section and the maximum averaged flow velocity over the cross-section Umax. The use of the maximum velocity in the cross-section for normalization of the velocity pulsations is due to the fact that it is just Umax that determines the characteristic value of the transverse velocity gradient in the mixing layer downstream of the stenosis, which, in turn, determines the development of hydrodynamic instabilities and turbulence transition. In Fig. 9 we have illustrated the change in the introduced turbulence intensity downstream of the stenosis; the graph is plotted using the experimental data. From this figure we can see that the velocity pulsation intensity reaches a maximum at x/D ≈ 3.2, then the pulsation intensity reduces and practically does not change at x/D ≥ 8.
Fig. 9.
Variation in the characteristic intensity of velocity fluctuations downstream of the constriction.
In Fig. 10 we have compared the field of the tangential Reynolds stresses u′v′/〈U〉2 calculated from the numerically resolved components of the fluctuating motion with the results of measurements obtained by the SIV method. It can be seen that both in the calculation and in the experiment, the turbulent shear stress is significant in magnitude only in the region x/D < 5. This is an additional evidence of turbulence localization in the near wake behind the stenosis. The area with the maximum Reynolds stresses is located in the calculated and experimental field almost at the same place. At the same time, the maximum stresses predicted in the calculations are by approximately 25% higher than those measured.
Fig. 10.
Reynolds stress fields u′v′/〈U〉2 in the plane of symmetry: (a) numerical simulation with the introduction of input disturbances and (b) experiment.
In Fig. 11 we have reproduced the experimental oscillograms of the longitudinal component of the flow velocity at the points corresponding to the coordinates of the maxima of the root-mean-square velocity fluctuations in the channel cross-sections (see Fig. 7). As can be seen from the figure, flow downstream of the stenosis is accompanied by relatively low-frequency flow velocity pulsations caused by the formation and motion of vortex structures of different scales in the mixing layer behind the constriction. The highest pulsation amplitude is observed at x/D = 3.13. This complies with the data in Fig. 9. Further downstream, the pulsation amplitude noticeably decreases (Fig. 11).
Fig. 11.
Oscillograms of the longitudinal component of the flow velocity at different distances from the constriction.
In Fig. 12 we have reproduced the experimental spectra of flow velocity fluctuations at points corresponding to the maximum value of urms in the channel cross-section (see Fig. 7) for two cross-sections behind the stenosis (x/D = 3.13 and 7.69). In the given spectra we can distinguish an inertial interval whose length decreases with increase in the distance from the stenosis. However, the difference in the pulsation frequency on the boundaries of this interval by less than an order of magnitude does not make it possible to draw a reliable conclusion about the law of variation of the pulsation amplitude in this interval. Nevertheless, the nature of the presented spectra indicates local flow turbulence with signs of subsequent laminarization as the distance from the stenosis increases.
Fig. 12.
Spectrum of pulsations of the longitudinal component of the flow velocity at points located at different distances from the stenosis: curves 1 and 2 correspond to x/D = 3.13 and 7.69, respectively.
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SUMMARY
The computational and experimental study of the flow in a model of blood vessel with an asymmetric narrowing of the cosinusoidal profile with the narrowing degree of 70% and the Reynolds number Re = 1800 was carried out. This Reynolds number is in the range of physiological values estimated from the maximum blood flow rate in the human femoral artery during the period of heart contractions. The experiment was carried out using the optical SIV method, which makes it possible to measure the instantaneous flow velocity vector fields with high spatial and temporal resolution. The three-dimensional numerical solution was obtained by the large eddy simulation method.
The experimental data obtained from measurements in the longitudinal central channel section (plane of symmetry) definitely indicate the existence of a region downstream of the stenosis with sufficiently developed local turbulence. This region is by 30–40% longer than the separation (recirculation) zone that forms behind the throat of stenosis. Already in the first of the examined cross-sections located at a distance of two calibers (channel diameters) from the throat of stenosis, the longitudinal velocity pulsations are very large, with a peak value of the root mean square velocity pulsation urms that amounts to about 90% of the flow-rate-average velocity in the areas ahead of and downstream of the narrowing. The same level of the maximum urms values is also observed on the subsequent one and a half calibers. Further downstream, the level of the velocity pulsations sharply decreases, and in the last examined cross-sections (6–8 gauges from the stenosis throat), the pulsation distribution becomes almost uniform along the flow core, being characterized by the value of urms of about 20% of the flow-rate-average velocity. The time oscillograms and spectra of velocity fluctuations in the last cross-sections clearly indicate degeneration of the high-frequency components. This is a sign of the subsequent flow relaminarization.
Numerical eddy-resolving simulation, which has both comparative and supporting value in relation to the experiment, was carried out with taking into account the irregular input disturbances recorded in flow upstream of the test section, that simulates the stenosis in the experimental setup, and presumably caused by vibrations of the floor and structural elements of the facility. The input disturbances are taken into account by artificially introducing an input synthetic turbulence into the calculation model. The turbulence was added to the input velocity distribution in form of the Poiseuille profile. The calculated data obtained for the time-average longitudinal velocity field and the most interesting components of the Reynolds stress tensor are in the satisfactory agreement with the results of measurements by the SIV method in the channel symmetry plane. As a result of taking into account irregular perturbations at the channel inlet, the calculated length of the separation zone reduced by approximately 25%, as compared with the value predicted earlier in calculations [23] with no input perturbations, and, in agreement with the experiment, amounted to a value of about 4 channel diameters. The general structure of the mean flow downstream of the stenosis is similar to that predicted in [23] and is characterized by the presence of a high-speed jet flow zone, a relatively extended recirculation region, and significant secondary flows in the form of a pair of eddies near the throat of stenosis and two pairs of eddies in the recirculation region.
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Translated by E.A. Pushkar