Numerical study of swirling flows in converging channels with a concave base as an analogy to blood flow in the heart and aorta

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Abstract

The study presents a numerical parametric investigation of flow structures in channels with a longitudinal-radial profile zRN = Const and a spherical dome at the base. The goal of the study was to examine the flow structures in these channels depending on the exponent N of the profile and the height of the dome, to determine the conditions that provide optimal centripetal swirling flow, analogous to blood flow in the heart chambers and major vessels. The investigation was conducted using a comparative analysis of flow structures in channel configurations zRN = Const, carried out in two stages. In the first stage, the convergence parameter N was varied from 1.25 to 2.75 to identify the value that ensures optimal flow conditions. In the second stage, for the established value of N, the dome height was varied from 2.5 mm to 15 mm to identify the beneficial effects associated with its presence. The method of investigation involved numerical modeling in a steady-state regime. The results of the study on the influence of the convergence parameter revealed that the profile zR2 = Const provides optimal conditions for the formation of swirling flow with minimal specific losses and a uniform distribution of velocity gradients. This channel configuration also showed the best agreement with the analytical solutions for Burgers’ vortex, confirming its effectiveness in the static approximation of flows. The parametric investigation of dome height indicated that an optimal dome height of 7 mm contributes to the smoothing of velocity gradients and the reduction of viscous losses due to the optimal enhancement of the centripetal swirling flow scale.

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About the authors

Ya. E. Zharkov

A.N. Bakulev National Medical Research Center for Cardiovascular Surgery

Email: agorodkov@bk.ru
Russian Federation, Moscow

A. V. Agafonov

A.N. Bakulev National Medical Research Center for Cardiovascular Surgery

Email: agorodkov@bk.ru
Russian Federation, Moscow

Alex. Y. Gorodkov

A.N. Bakulev National Medical Research Center for Cardiovascular Surgery

Author for correspondence.
Email: agorodkov@bk.ru
Russian Federation, Moscow

L. A. Bockeria

A.N. Bakulev National Medical Research Center for Cardiovascular Surgery

Email: agorodkov@bk.ru

Academician of the RAS

Russian Federation, Moscow

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Supplementary files

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2. Fig. 1. Geometrical configuration of channels and boundary conditions used for numerical modeling. The geometrical configuration of channels includes three regions with two types of boundary conditions. Region (1) is a part of the channel with a converging bed, the boundary of which is determined by the expression zRN = K + z0. The channel convergence index N was varied from 1.25 to 2.75, the variation region is shown in green. Region (2) corresponded to the entrance slit to the channel and remained unchanged during the numerical modeling. Region (3), shown in purple, is formed by a spherical dome with a height from 0 (no dome) to 1.25 cm. The boundary conditions included static pressures at the entrance and exit of the channel, and the Wall Function condition without taking into account adhesion was used to describe the walls.

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3. Fig. 2. Dependence of integral losses due to viscous friction in a steady state on the channel convergence index zRN = Const. The red round marker indicates the calculated values ​​obtained as a result of numerical modeling of a steady flow. The blue line is the interpolation curve Mod. Akima, approximating the shape of the dependence of integral viscous losses on the channel convergence index N. The black marker reflects the minimum point of the interpolation curve. It is evident from Fig. 2 that the dependence of integral losses due to viscous friction has a minimum point at the convergence index value N = 2.32, which differs from the expected minimum at N = 2.

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4. Fig. 3. Distribution of total viscous losses on the example of channel N = 2 and the main areas of their localization. The figure shows that there are three main zones of loss localization: I) the wall zone of intense constriction; II) the axial zone at the constriction; III) the area of ​​maximum constriction of the channel. The analysis of the causes of the occurrence of loss localization areas was carried out using the assessment of the contribution of the components of the viscous loss tensor to each area.

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5. Fig. 4. Distribution of azimuthal velocity (a), contribution of longitudinal-radial component of viscous losses (b), components of the main diagonal of the tensor (c – d), longitudinal-radial component (e) and distribution of viscous losses (e) in the channel N = 2. Spatial distributions of the parameters illustrate the general nature of the flow using the example of a channel with the convergence index N = 2. The development of the flow after the entrance slit is accompanied by an increase in radial, azimuthal and azimuthal-radial losses, which is a consequence of the increase in the azimuthal velocity of the liquid. In the region of intense narrowing near the channel axis (region II), there is a decrease in the swirling of the liquid with the transformation of the azimuthal velocity into longitudinal. Longitudinal mass transfer of liquid in the channel in the axial zone leads to the formation of a wall flow (region I), which merges with the main flow when approaching region III, where the longitudinal velocity of the liquid flow predominates.

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6. Fig. 5. Dependences of integrals of viscous loss tensor components for channels with convergence indices N = 1.25 – 2.75. This image is the dependence in Fig. 2, decomposed into components of the loss tensor. The integral characteristics of losses show that an increase in the convergence index of 1.25 to 2.75 leads to an increase in azimuth-radial losses, reflecting friction during jet rotation, while longitudinal losses decrease nonlinearly. Analysis of the distributions in Fig. 6 showed that the nature of the change in this type of loss is determined by the region of their localization, depending on the convergence index. Dependences of the integral values ​​of the longitudinal-radial component show the existence of a local maximum in the region of convergence indices N = 1.25 – 1.75. The reasons for its occurrence are shown in Fig. 7.

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7. Fig. 6. Distribution of total viscous losses (a-c) and azimuthal velocities for channels (d-e) for the convergence index N = 1.5, 2.25, 2.75 for analyzing the differences in flow structures in channels with different convergence. Spatial distributions of viscous loss parameters and azimuthal velocity demonstrate a correlation between the vortex intensity and the loss localization regions. In the case of N = 1.5, low azimuthal velocity values ​​lead to high losses in the channel outlet region. High vortex intensity in the channel N = 2.75 leads to the occurrence of high velocity gradients in the channel constriction region. The flow in the channel N = 2.25 demonstrates an intermediate distribution between the two extreme cases.

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8. Fig. 7. Distribution of total viscous losses (a-c) and azimuthal velocities (g-e) in channels N = 1.25, 1.5, 1.75 for analysis of the causes of the local maximum of longitudinal-radial losses in the range of N = 1.25 – 1.75. Spatial distributions of the parameters show the cause of the local maximum of longitudinal-radial losses in the convergence range of N = 1.25 – 1.75. An increase in the length of the channel wall leads to an increase in longitudinal-radial losses in region I. The Coanda effect, which occurs due to the existence of a vortex, is a counteracting factor to this.

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9. Fig. 8. Dependence of viscous losses on the convergence index N, reduced to the channel volume. The dependence of specific viscous losses showed that the most energy-efficient channel is one with a convergence index N = 2.05. The increase in losses with an increase in the convergence index above 2.05 occurs due to an increase in the azimuthal velocity in the vortex. The increase in losses with a decrease in the index below 2.05 is explained by an increase in losses in the areas near the channel walls.

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10. Fig. 9. The root-mean-square deviation of the longitudinal velocity RMS(uz) for channels of different shapes (a) with an analytical solution and with modeling in Comsol Multiphysics. From Fig. 9, a it follows that the longitudinal velocity distribution formed in the channel with a convergence index N = 2 has the best match with the analytical expression according to the root-mean-square deviation criterion, but the value with a convergence index N = 1.75 is only 0.3% higher. In this regard, the comparison included an analysis of the average longitudinal velocity at the outlet for channels N = 1.75 and 2. A comparison of the velocities in the outlet section of the channels showed that the difference in the average velocity calculated by the two methods is 0.5 m/s with a convergence index of N = 1.75 and 0.02 m/s in the case of N = 2, which indicates the identity of the flows obtained by the two methods for a channel with a convergence of N = 2 in the longitudinal velocity parameter.

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11. Fig. 10. Interpolation dependence of total viscous losses (black curve), contributions of the components of the viscous loss tensor (histogram) and the location of the minimum of the interpolation dependence (red marker). It follows from the given image that the dependence of viscous losses on the dome height is not monotonic and has a minimum at a height of 0.7 cm. An analysis of the contributions of radial-azimuthal gradients shows a discrepancy with the results obtained by varying the convergence index: in the case of channels with a height from 0.25 to 0.75, the losses during fluid rotation change insignificantly, but at the same time there is a decrease in the losses of the components of the main diagonal. The reasons for this effect are reflected in the spatial distributions of viscous losses and azimuthal velocities in Fig. 11.

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12. Fig. 11. Distribution of total viscous losses (a, c, d) and azimuthal velocities (b, d, e) for channels with a dome height of H = 0.25, 0.75 and 1.5. Spatial distributions of the parameters of viscous losses and azimuthal velocity show a change in the flow with an increase in the dome height. An increase in the dome height leads to an increase in the scale of the vortex, which reduces the longitudinal gradients in region II. With a greater dome depth, an increase in the scale of the vortex leads to a decrease in the azimuthal velocity, which increases the longitudinal losses in region III. The optimal dome size leads to a uniform distribution of losses throughout the channel.

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