## 2 Method and Validation
### Numerical Model  
As illustrated in Figure 1, the flow channel is divided into three parts to generate mesh. Relatively coarse meshes are selected at the inlet and outlet parts, and relatively dense meshes are used at the vicinity of valve leaflet. The mesh re-generation method when the leaflets are moved is shown below.  
When the difference between the latest calculation result and the last one is less than the set convergence criterion, it is considered that the current time step calculation converges and enters the next time step calculation. The detailed calculation process during the mesh updating process at each time step is shown in Figure 2. In order to reveal fully the flow characteristics of the blood in the MHV, the software FLUENT is used in this work. The inlet boundary condition is pressure inlet and the outlet boundary condition is pressure outlet. A no-slip boundary condition was imposed on the walls of the leaflets. The blood is considered as an incompressible Newtonian fluid with the density of \(1.06\times 10^{3}\) kg/m\({}^{3}\) and the dynamic viscosity of \(3.5\times 10^{-3}\) Pa-s. The pressure of saturated steam is \(6.343\times 10^{3}\) Pa, which is consistent with water at 37.5 \({}^{\circ}\)C. Flow is assumed to be laminar and the boundary conditions at heart rate of 70 bpm are loaded by means of the segmental application method.  
Figure 1: Structure of mechanical heart valve and grid partition.  
Figure 2: Calculation process of fluid flowing in mechanical heart valve.  
When the blood flows through the MHV, it follows the laws of mass conservation, momentum conservation and energy conservation. Equations are as follows:  
The equation of continuity is expressed as:  
\[\frac{\partial p}{\partial t}+\rho\frac{\partial u_{i}}{\partial x_{i}}=0 \tag{1}\]  
\(p\) is the pressure of the blood; \(t\) is the time; \(\rho\) is the density of the blood; \(i\) is the direction of the blood flow and \(i=1,2,3\).  
The moment equation is expressed as:  
\[\rho\frac{\partial u_{j}}{\partial t}+\rho u_{i}\frac{\partial u_{j}}{ \partial x_{i}}=-\frac{\partial p}{\partial x_{j}}-\frac{\partial\tau_{ij}}{ \partial x_{i}}+\rho g_{j} \tag{2}\]  
\(p\) is the pressure of the blood; \(g\) is the acceleration of gravity; \(\tau_{ij}\) is viscous stress tensor and can be expressed as follows:  
\[\tau_{ij}=-\mu\bigg{(}\frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_ {i}}{\partial x_{j}}\bigg{)}+\frac{2}{3}\delta_{ij}\mu\frac{\partial u_{k}}{ \partial x_{k}} \tag{3}\]  
\(\mu\) is dynamic viscosity.  
The energy equation is expressed as:  
\[\rho C_{v}\frac{\partial T}{\partial t}+\rho C_{v}u_{i}\frac{\partial T}{ \partial x_{i}}=-p\frac{\partial u_{i}}{\partial x_{j}}+C_{p}\frac{\mu}{ \Pr}\frac{\partial^{2}T}{\partial x_{i}^{2}}-\tau_{ij}\frac{\partial u_{j}} {\partial x_{i}} \tag{4}\]  
\(T\) is the temperature of the blood; \(P_{r}\) is Prandtl number; \(C_{v}\) is the specific heat at constant volume; \(C_{p}\) is the specific heat at constant pressure.  
The \(k-\varepsilon\) equation of the turbulence model in the turbulent viscosity coefficient method is selected to solve the problem of the turbulent flow in the valve. The transport equations of turbulent kinetic energy and turbulent dissipation rate in the standard \(k-\varepsilon\) turbulence model are as follows:  
\[\frac{\partial}{\partial x_{i}}(\rho ku_{i})=\frac{\partial}{\partial x_{j}} \bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{k}}\bigg{)}\frac{\partial k}{ \partial x_{j}}\bigg{]}+G_{k}+G_{b}-\rho\varepsilon \tag{5}\]  
\[\frac{\partial}{\partial x_{i}}(\rho\varepsilon u_{i})=\frac{\partial}{ \partial x_{j}}\bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{\varepsilon}} \bigg{)}\frac{\partial\varepsilon}{\partial x_{j}}\bigg{]}+C_{1\varepsilon} \frac{\varepsilon}{k}(G_{k}+C_{3\varepsilon}G_{b})-C_{2\varepsilon}\rho\frac {\varepsilon^{2}}{k} \tag{6}\]  
\(k\) is the turbulent kinetic energy; \(\varepsilon\) is the turbulence dissipation; \(G_{k}\) and \(