Experimental and model study of swirling fluid flow in a converging channel as a simulation of blood flow in the heart and aorta

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Abstract

The study of swirling flows in channels corresponding to the static approximation of flow channels of the heart and major vessels with a longitudinal-radial profile zR2 = const and a concave streamlined surface at the beginning of the longitudinal coordinate has been carried out. A comparative analysis of the flow structure in channel configurations zRN = const, where N = –1; 1; 2; 3, in the absence and presence of a concave surface was carried out. The numerical modelling was compared with the results of hydrodynamic experiments on the flow characteristics and the shape of the flow lines. The numerical model was used to determine the velocity structure, viscous friction losses and shear stresses. Numerical modelling of steady-state flows for channels without a concave surface showed that in the channel zR2 = const there is a stable vortex flow structure with the lowest viscous friction losses. The presence of a concave surface of sufficient size significantly reduces viscous friction losses and shear stresses in both steady state and pulsed modes.

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About the authors

Ya. Е. Zharkov

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

Email: agorodkov@bk.ru
Russian Federation, Moscow

Sh. T. Zhorzholiani

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

Email: agorodkov@bk.ru
Russian Federation, Moscow

А. А. Sergeev

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

A. 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

Russian Federation, Moscow

References

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Supplementary files

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2. Fig. 1. Geometric approximation of the shape of the left atrium (a) and left ventricle (b) using a statically shaped channel with a longitudinal-radial profile zR2 = const with a concave surface.

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3. Fig. 2. Scheme of the experimental setup (a) and geometric configuration of the numerical model (b, c): a) Ht – height of the water column in a water tank, He – channel height, D – cut-off diameter, d – channel outlet diameter; b) Pin – specified static pressure at the inlet to the channel; (c) α is the angle of fluid velocity relative to the normal of the source surface. Physical modeling for various geometric configurations of the channel was carried out in water, the pressure drop in the channel was created using a column of water in the reservoir, and the swirl of the liquid was organized using a rotating blade. The flow rate was recorded at the output of the experimental channel; the flow lines were visualized by introducing dye into the channel using two needles: measuring and control. In Fig. 2 b, c shows that the numerical simulation was carried out under conditions identical to the experimental ones. The static pressure boundary condition at the channel inlet was varied depending on the modeling requirements. The creation of the azimuthal component of the fluid velocity was carried out by parameterizing the angle α between the velocity vector at the channel entrance to the normal of the input surface.

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4. Fig. 3. Dependence of flow characteristics on the height of the liquid column in the tank for channels of various shapes: Cone – dependence of flow rate for a conical-shaped channel. N = 1, 2, 3 – flow rate dependence for channels zRN = const with the specified exponent. It can be seen that the flow characteristics for channels zRN = const have little difference. The flow characteristics of a conical channel are 4% lower.

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5. Fig. 4. Photographs of streamlines in the absence (a) and in the presence (b) of swirl for the channel zR2 = const. It can be seen that in the absence of the azimuthal component of the flow velocity, the shape of the fluid flow line follows the shape of the channel, while in its presence the shape has the form of a converging spiral.

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6. Fig. 5. Experimental dependence of the number of turns of the line on the height of the measuring needle in the channel slit (a) and photographs of streamlines with the camera positioned above the channel (b-d): The control needle is located on the right, the measuring needle is on the left; a jet of dye from above visualizes the channel boundary. From Fig. 5 it can be seen that the number of turns of the spiral current line increases with increasing height of the measuring needle. According to Fig. 5d shows that visual tracking of the number of turns in the extreme position of the measuring needle is difficult.

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7. Fig. 6. Flow characteristics during numerical and physical modeling for different channel shapes (a), flow velocity structure for conical channels (b) and zR2 = const shape (c): Cone – flow dependence for a conical channel. N = 1, 2, 3 – flow rate dependence for channels zRN = const with the specified exponent. It can be seen that the consumption characteristics are in good agreement. According to Fig. 6b, c it is clear that the decrease in flow rate for channels zRN = const is associated with high hydrodynamic resistance in the zone of maximum narrowing of the channel.

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8. Fig. 7. Dependence of the number of turns of the streamline on the height of the needle during physical and numerical experiments (a), three-dimensional visualization of numerically determined streamlines for heights of 3, 5 and 7 mm (b). It can be seen that the number of turns of spiral current lines in numerical and physical modeling has a satisfactory agreement.

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9. Fig. 8. Cut lines in the axial projection of the numerical model (a), distribution of azimuthal velocity in the radial projection for channels zRN = const at N = 1; 2 and 3 (b, c, d): Blue line – cut at the zero height of the channel, green curve – cut at a height of –3 cm, green curve – cut at a height of –6 cm. It can be seen that a stable vortex structure is formed only in zR2 = const.

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10. Fig. 9. Integral viscous friction losses for channels zRN = const of different orders of degree (a), distribution of viscous friction losses in channels at N = 1; 2 and 3 (b, c, d) It can be seen that in a channel with exponent N = 1, a vortex structure is not formed. In channels with an exponent N = 2, 3, the flow is of a vortex nature, while in the channel N = 2 the loss values in the region of vortex formation are lower than in the case of N = 3.

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11. Fig. 10. Integral of viscous forces along the streamlined surface of the channels zRN = const with an exponent from 1 to 3 (a), distribution of viscous forces along the streamlined surfaces of the channels (b–d). It can be seen that in a channel with an exponent N = 2 there is no clearly defined region of a local maximum of viscous forces along the channel surface, and the integral of viscous forces is minimal among all the cases considered.

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12. Fig. 11. An example of a geometric configuration of a channel with an exponent N = 2 with a concave surface with parameters of the overall surface height h = 5 mm and the ratio of the overall radial size of the concave surface to the entrance radius of the channel k = 0.5.

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13. Fig. 12. Distribution of losses due to viscous friction (a) and structure of azimuthal velocities (b) for different sizes of the concave surface. It can be seen that with an increase in the additional volume created by the concave surface, the flow becomes vortex with the expansion of the vortex formation region, which helps reduce losses in this zone.

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14. Fig. 13. Specific losses due to viscous friction for various overall dimensions of the concave surface: k is the ratio of the overall radius of the concave surface to the maximum radius of the channel, h is the overall height of the concave surface. It can be seen that with an increase in the additional volume created by the concave surface, the integral losses due to viscous friction decrease. At maximum volume, viscous friction losses become lower than in the absence of a concave surface.

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15. Fig. 14. Calculated oscillogram of the static pressure pulse at the entrance to the channel during dynamic modeling, simulating the systole of the cardiac cycle.

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16. Fig. 15. Values of maximum shear stresses arising in a second-order channel depending on time in the presence and absence of a concave surface of maximum volume: solid line - maximum shear stresses in a channel without a concave surface, broken line - maximum shear stresses in a channel in its presence. Red lines mark the time points to represent the shear stress structure. It can be seen that the presence of a concave surface of maximum volume contributes to a twofold reduction in the maximum shear stresses in the channel.

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17. Fig. 16. Values of maximum shear stresses arising in a second-order channel depending on time in the presence and absence of a concave surface of maximum volume: solid line - maximum shear stresses in a channel without a concave surface, broken line - maximum shear stresses in a channel in its presence. Red lines mark the time points to represent the shear stress structure. It can be seen that the presence of a concave surface of maximum volume contributes to a twofold reduction in the maximum shear stresses in the channel.

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18. Fig. 17. Maximum values of time integrals of shear stresses in channels in the presence and absence of a concave surface (a), structure of the integral of the action of shear stresses in these channels (b, c). It can be seen that the presence of a concave surface makes it possible to reduce the effect of shear stresses by half in the near-wall zone, and also significantly reduces their effect in the area of formation of vortex motion.

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