**3. Results and Discussion**

## *3.1. E*ff*ect of Di*ff*erent Flow Rate Conditions*

Figure 4a shows the blood flow patterns by velocity vector distribution of a BMHV at different flow rate conditions (5 L/min, 10 L/min, 15 L/min, 20 L/min, and 25 L/min), where the color represents the velocity magnitude [23,24]. The flow patterns developed in all cases were similar to each other because of the same opening degree of 80◦. Because of the nonphysiologic geometries of mechanical valves, a three-jet configuration through the central and lateral orifices can show a relatively higher velocity and a low-velocity region located at the whole surfaces and trailing edges of both leaflets. The spread region of the trailing edges of the leaflets increases with the flow rate. As the flow rate changed from 5 L/min to 25 L/min, the maximum velocity increased from 0.25 m/s to 1.21 m/s in the central orifices and increased from 0.22 m/s to 0.95 m/s in the lateral orifices, respectively. Higher velocities at the central and lateral orifices were accompanied by higher level of velocity gradient in the vicinity of leaflet surfaces [25,26].

Figure 4b shows the vortex structure of the flow field at different flow rate conditions by means of the vortex core region, where the color represents the value of the Q criterion. In all cases, the vortex core region was mainly formed in the low-velocity region near both leaflet edges. As the flow rate changed from 5 L/min to 25 L/min, the value of the Q criterion in the vortex core region increased rapidly, and the maximum Q value increased from 1.65 <sup>×</sup> 10<sup>4</sup> s−<sup>2</sup> to 3.24 <sup>×</sup> 105 s−2. As the flow rate increased to more than 15 L/min, the velocity of the mainstream region increased and the vortices shed downstream from the trailing edges of the leaflets.

**Figure 4.** *Cont*.

**Figure 4.** Velocity distribution and Q criterion of vortex core region of the 80◦ opening degree bileaflet mechanical heart valve (BMHV). (**a**) Velocity vector distribution and (**b**) Q criterion of vortex core region.

In order to study the pressure pulsation characteristics in the vicinity of the leaflets under different flow conditions, two monitoring lines perpendicular to the flow direction (Y direction) were set at the leading edge and trailing edge of the leaflets of the BMHV, respectively, as shown in Figure 4b. One hundred monitoring points were set on every monitoring line, which were evenly distributed in the central and lateral orifices. The transient flow calculation results showed that the pressure value became stable after the first calculation cycle. The pressure value data in the second calculation cycle at each monitoring point were recorded, and the time-average pressure pulsation at each point were calculated by Equation (5).

$$p' = \left(\int\_{T}^{2T} |p - p\_m| dt\right) / \Delta T \tag{6}$$

where *T* is the time of one calculation cycle of 300 ms, from the 200th ms to the 500th ms in one cardiac cycle. *p*' is the time-average pulsating pressure, which represents the average pressure pulsation range at a certain point.

The average pressure pulsation at all monitoring points under five different flow conditions were measured and calculated, and the coefficients of pressure pulsation were calculated by equation (4). The coefficient distribution of pressure pulsation at the leading edge and trailing edge monitoring lines under different flow conditions are shown in Figures 5 and 6, respectively.

**Figure 5.** Coefficient distribution of pressure pulsation at the leading edge monitoring lines under different flow conditions at 80◦ opening degree. (**a**) The complete view. (**b**) The local enlarged view.

**Figure 6.** Coefficient distribution of pressure pulsation at the trailing edge monitoring lines under different flow conditions at 80◦ opening degree. (**a**) The complete view. (**b**) The local enlarged view.

Figure 5a is the complete view of the coefficient distribution of pressure pulsation at the leading edge monitoring lines at flow rates from 5 L/min to 25 L/min. Figure 5b is the local enlarged view. Under different flow conditions, the level of pressure pulsation coefficient near the wall and the leading edge of the leaflets was higher. The level of the pressure pulsation coefficient in central and lateral orifices was relatively lower. As the flow rate increased, the distribution of the pressure pulsation coefficient on the leading edge monitoring line showed a trend of gradual decline, and the maximum value of pressure pulsation coefficient near the leading edge decreased from 0.014 to 0.003. The results show that the lower the flow velocity on the leading edges of the leaflets, the larger the coefficient of pressure pulsation. The flow velocity near the leading edges of the leaflets increased with the increased flow rate. The vortices fell off downstream, and the influence of the pressure pulsation decreased.

Figure 6a is the complete view of the coefficient distribution of pressure pulsation at the trailing edge monitoring lines at flow rates from 5 L/min to 25 L/min. Figure 6b is the local enlarged view. Under different flow conditions, the coefficient of pressure pulsation near the trailing edge of the leaflets was relatively higher (the maximum value, equal to 8.4), which was reached at the trailing edges of the leaflets. At the main flow region, the coefficients of pressure pulsation in central and lateral orifices were close to zero. As the flow rate increased from 5 L/min to 15 L/min, the distribution of the pressure pulsation coefficient near the trailing edge decreased rapidly, with the maximum value decreasing from 8.4 to 2.1. As the flow rate increased from 20 L/min to 25 L/min, the maximum value of pressure pulsation coefficient gradually decreased from 1.4 to 1.2. The results show that the high-pressure pulsation region was mainly formed in the vicinity of the trailing edges of leaflets. As the flow rate increased, the coefficient of pressure pulsation showed a decreasing trend. Additionally, the level of the pressure pulsation coefficient near the trailing edges was much larger than the data obtained near the leading edges of leaflets. It could be considered that one of the main reasons for the result in leaflet vibration was the pressure pulsation located at the trailing edges of leaflets under low-velocity conditions.

#### *3.2. E*ff*ect of Di*ff*erent Fully Opening Angles of Leaflets*

In the same way, the pressure pulsation characteristics in the vicinity of the leaflets under different fully opening angle conditions, from 75◦ to 85◦, were studied. Figure 7a shows the blood flow patterns by the velocity vector distribution of a BMHV at different opening degree conditions, where the color represents the velocity magnitude. All cases were simulated under the same flow rate condition of 5 L/min. The leaflets act as an obstruction to the blood flow through the BMHV, and coupled with the relatively high-velocity three-jet through the central and lateral orifices. As the leaflet opening degree increased from 75◦ to 85◦, the maximum velocity decreased from 0.28 m/s to 0.22 m/s and from 0.24 m/s to 0.19 m/s in the central and lateral orifices, respectively. The results showed that, with the increase of the opening angle, the velocity distribution was more uniform and the average level of flow velocity decreased. Figure 7b shows the vortex structure of the flow field under different leaflet opening angle conditions by means of vortex core region, where the color represents the value of the Q criterion. In all cases, the vortex core region mainly formed in the vicinity of the surfaces of both leaflets. As the leaflet opening degree increased from 75◦ to 85◦, the range of the vortex core region near the leaflets was gradually enlarged, and the maximum Q value decreased from 2.02 <sup>×</sup> 104 s−<sup>2</sup> to 1.13 <sup>×</sup> 104 s−2. The results showed that, because of the larger opening angle and the lower average velocity, the vortices were more likely to stay at the surfaces of the leaflets than to shed downstream from the trailing edges.

**Figure 7.** Velocity vector distribution and Q criterion of the vortex core region of the different opening angle bileaflet mechanical heart valves (BMHVs) at a flow rate of 5 L/min. (**a**) Velocity vector distribution. (**b**) Q criterion of vortex core region.

Figure 8 shows the coefficient distribution of pressure pulsation at the leading edge monitoring lines at different fully opening leaflet angles from 75◦ to 85◦. Under different fully opening angle conditions, the coefficient distribution of pressure pulsation at the leading edge monitoring lines were similar. The level of the pressure pulsation coefficient near the wall and the leading edge was relatively higher, and relatively lower at the central and lateral orifices. As the fully opening angle increased from 75◦ to 85◦, the level of the pressure pulsation coefficient in the region of the leading edges and the lateral orifices slightly decreased, and the maximum value of the pressure pulsation coefficient near the leading edge decreased from 0.02 to 0.017. Under the same conditions, the pressure pulsation coefficient in the central orifice gradually increased and the minimum value of the pressure pulsation coefficient at the central orifice increased from 0.007 to 0.011. The results showed that, at the same flow rate condition, the level of the pressure pulsation coefficient near the leading edges slightly decreased with the increased leaflet fully opening angle. It could be inferred that the main influencing factor of

the pressure pulsation coefficient level near the leading edges was the different velocity distribution. The lower the flow velocity, the larger the coefficient of pressure pulsation. Since the level of the pressure pulsation coefficient near the leading edges was very low in all cases, it could be considered that the change of the leaflet fully opening angle had little effect on pressure pulsation and the potential vibration of leaflets leading edges.

**Figure 8.** Coefficient distribution of pressure pulsation at the leading edge monitoring lines under different fully opening angle conditions at a flow rate of 5 L/min.

Figure 9 is the coefficient distribution of pressure pulsation at the trailing edge monitoring lines at different fully opening leaflet angles from 75◦ to 85◦. In all three cases, the distribution of the pressure pulsation coefficient at the trailing edge monitoring lines were similar. The pressure pulsation coefficient near the trailing edge was relatively higher. At the main flow region, the pressure pulsation coefficients in the central and lateral orifices were close to zero. As the fully opening angle increased from 75◦ to 85◦, the level of the pressure pulsation coefficient near the trailing edges gradually increased, and the maximum value of the pressure pulsation coefficient increased from 6.6 to 9.5. Additionally, in comparison to another two cases, the pressure pulsation coefficient in the central orifice showed a nonuniform distribution with the under 75◦ opening angle condition. The results showed that at the same flow rate condition, the level of the pressure pulsation coefficient near the trailing edges gradually increased as the leaflet fully opening angle increased. It could be inferred that the main influencing factor on the pressure pulsation coefficient level near the trailing edges was the different vortex structure distribution. The larger leaflet fully opening angle led to the larger vortex core region and a higher level of pressure pulsation coefficient. The nonuniform distribution of the pressure pulsation coefficient indicated that the nonuniform velocity distribution might be induced by the narrow central orifice under a relatively lower leaflet opening angle condition. Thus, considering the pressure pulsation and the flow uniformity, the recommended setting of leaflet fully opening angle was about 80◦.

**Figure 9.** Coefficient distribution of pressure pulsation at the trailing edge monitoring lines under different fully opening angle conditions at a flow rate of 5 L/min.

#### **4. Conclusions**

In this study, the pressure pulsation characteristics induced by unsteady blood flow in the BMHV under different flow rates and leaflet fully opening angle conditions were analyzed. The conclusions can be listed as follows.

In regard to blood flow through a BMHV, the main influencing factor of the pressure pulsation coefficient level was the different velocity distribution. In all cases, the vortex core region and low-velocity region were mainly formed in the leading and trailing edges of leaflets. The lower the flow velocity on the edges of leaflets, the larger the coefficient of pressure pulsation. As the flow rate increased, the flow velocity near the edges of the leaflets increased and the coefficient of pressure pulsation decreased; the vortices then fell off downstream, and the influence of pressure pulsation decreased.

The level of the pressure pulsation coefficient near the trailing edges was much larger than the data obtained near the leading edges of leaflets, which indicated that one of the main reasons for leaflet vibration was the pressure pulsation located at the trailing edges of leaflets under low-velocity conditions. At the same flow rate and different leaflet fully opening angle conditions, the level of the pressure pulsation coefficient near the leading edges was close to 0, which indicated that the change in leaflet fully opening angle had little effect on the pressure pulsation and the potential vibration of leaflet leading edges.

The range of the vortex core region increased with the increased leaflet fully opening angle. The relatively lower the velocity in the vortex core region, the larger the level of the pressure pulsation coefficient. Meanwhile, the nonuniform velocity distribution could have been induced by the narrow central orifice under the relatively lower leaflet opening angle condition. Thus, considering the pressure pulsation and the flow uniformity, the recommended setting of leaflet fully opening angle was about 80◦.

In this study, because the influence of the change of pressure and velocity boundary conditions to the phenomenon of leaflet fluttering are not well understood, a model of leaflets in fixed positions was adopted to simplify the calculation, which may have led to deviation between the simulation results and the actual situation. In addition, only one of the velocity profile monitoring lines was compared and validated by the experimental results. Considering the periodic boundary conditions, the simulation of the whole cardiac cycle and more detailed data are required for full validation in future work.

**Author Contributions:** Conceptualization, X.-g.-X. and S.-x.-L.; methodology, X.-g.-X.; software, X.-g.-X. and L.Z.; validation, T.-y.-L. and X.-g.-X.; formal analysis, T.-y.-L.; investigation, T.-y.-L. and X.-g.-X.; resources, S.-x.-L.; data curation, T.-y.-L. and C.L.; writing—original draft preparation, X.-g.-X. and C.L.; writing—review and editing, X.-g.-X. and C.L.; visualization, X.-g.-X.; supervision, S.-x.-L.; project administration, X.-g.-X.; funding acquisition, S.-x.-L.

**Funding:** This research was funded by National Natural Science Foundation of China, grant number 51569012, and the Fundamental Research Funds for Colleges and Universities in Gansu province, grant number 20146302.

**Conflicts of Interest:** The authors declare no conflict of interest.
