## 1 Introduction
With the implantation angle of the BMHV increased, the degree of asymmetry of the blood flow and the leaflet motion is increased, which can lead to unbalanced forces imposed on BHMV. Coronary flow rate is directly affected by the size, direction and time evolution of the vortex in the sinus, which are sensitive to the orientation of the valve. In term of implantation position, it is found that under similar physiological conditions, the von-mises stresses of the BMHV placed at the root of aorta with axisymmetric sinus are 47% higher than stresses of the BMHV placed at the aorta with axisymmetric bulb. In addition, the leaflet thickness has a significant effect on the flow of the blood [21]. When the leaflet thickness is reduced by 0.1 mm, the equivalent stress is 75% higher than the allowable equivalent stress. Therefore, thinner leaflet cannot be used to replace the diseased aortic valves. Research shows that there are a large number of vortices in the cardiac and large arteries, which play fundamental roles in the normal physiology. Due to vortices, there is a proper balance between blood motion and stresses on the surrounding tissues [22], and the length of the leaflet can vastly affect the transmitral vortex formation, possibly due to altered flow-wall interaction [23]. The detail of the vortex formation including its deformation and breakdown can be described precisely by numerical simulation [24; 25]. In order to fully reveal the fundamental flow characteristics of the BMHV at different working conditions, several researchers have begun to focus on dynamic characteristics of the BMHV [26]. In recent studies, the transient flow details of the blood in the ascending aorta have been presented by a fully coupled simulation method [27]. Results show that the gradient of trans valvular pressure difference produced by On-X valve is relatively small, which can reduce cardiac external work. Trajectories of blood substance are estimated by Lagrangian particle tracking method [28], which can quantify the movement of the leaflet by tracking the position of the leaflet tip during the whole closing process. In addition, the reverse leakage flow passing through the hinge gap is studied, to reduce damage to blood substances [29; 30]. The leakage flow has a maximum speed of 4.7 m/s in the simplified axisymmetric sinus during the early diastolic phase. However, the flow field, with an effective spatial resolution of 167 \(\upmu\)m and temporal resolution of 594 \(\upmu\)s, is measured by two-component PIV. Higher speed leakage is observed t the hinge and the measured average speed can reach as high as 5.7 m/s [31]. To aid clinical decision making and the understanding of pathophysiology, the Lagrangian particle method can be used to improve insight into the transport mechanics of the downstream flow of the valve [32]. Based on this, the pulsating flow in BMHV is simulated by the smoothed particle hydrodynamic method [33]. The advantage of the smoothed particle hydrodynamic method is that it is a Lagrangian method, which can avoid the accuracy loss caused by the grid distortion when the deformation is extremely large. Results show that the non-physiological flow pattern and the main vertical structure have an important influence on the shear stress of blood components. The full three-dimensional geometry of the MHV with moving leaflet in a typical human heart cycle is simulated. The simulation can essentially reproduce the pressure distribution of the left ventricle and the aorta. From the simulation results, wall shear stress and vorticity can be derived to further understand the performance of the valve. Cavitation is the process of the formation, development and collapse of vapor or gas cavities in the liquid or at the liquid-solid interface when the local pressure in the liquid decreases [34; 35]. In early studies, researchers aimed at developing a physical model, supported by experimental observations, described the formation and growth of microbubbles seen in patients with mitral mechanical heart valve [36].