## 2 Materials and Methods
### Numerical Schemes and Boundary Conditions
#### 2.4.3 Numerical Implementation
Traditional parameters such as the geometric orifice area, maximum jet velocity, kinetic energy, energy dissipation, and the magnitude of vorticity are studied. WSS-based indices such as time-averaged wall shear stress (TAWSS), oscillatory shear index (OSI), relative residence time (RRT), and transverse wall shear stress (transWSS) are also examined. In order to quantify the helical flow in the whole domain, helicity descriptors (\(h_{1}\), \(h_{2}\),..., \(h_{6}\)) are employed. The geometric orifice area (GOA) at a time instant \(t\) is defined as the region on the aortic cross-sectional plane that is enclosed by the projected curves of the leaflet commissures, representing the opening through which blood flows on the plane of leaflet attachment. Figure 5: Mesh independence analysis; horizontal axis is non-dimensionalized with the maximum radial length of the Valsalva sinus: (**a**) average velocity measurements at arbitrary selected locations (\(\text{I}=0.07\) m and \(\text{II}=0.08\) m); (**b**) for average velocity magnitude profiles over a line at \(y=0.07\) m; (**c**) for average velocity magnitude profiles over the line at \(y=0.08\) m.  
Another key indicator of transvalvular hemodynamics is the peak jet velocity \((V_{\rm max})\) passing the orifice area of the aorta. This metric serves as a basis for estimating the pressure differential across the valve, typically calculated via a simplified form of the Bernoulli equation. The \(V_{\rm max}\) not only provides insights into flow dynamics but also has clinical implications for evaluating valve function and potential stenosis. The next index for assessing transvalvular hemodynamics is the energy dissipation rate, which quantifies the real-time power depletion in the flow of blood. Expressed in Watts, and computationally derived as:  
\[\dot{E}_{\rm diss}(t)=\frac{dE_{\rm diss}}{dt}(t)=\int_{V}\Phi\,dV \tag{14}\]  
In this formula, \(\Phi=\tau\otimes\
abla\mathbf{v}\) symbolizes a dissipation function, where \(\tau\) denotes the viscous shear stress tensor. \(\dot{E}_{\rm diss}(t)\) accounts for the power loss within the Valsalva sinus, extending into the ascending aorta, thereby providing insights into the energy loss associated with blood viscosity and the presence of aortic stenosis. Another crucial hemodynamic variable is the kinetic energy of the blood flow, denoted by \(E_{\rm kin}(t)\), which can be calculated using the formula:  
\[E_{\rm kin}(t)=\frac{1}{2}\rho_{f}\int_{V}\mathbf{v}^{2}\,dV \tag{15}\]  
where the integration is carried out over the volume of the aorta. \(E_{\rm kin}(t)\) serves as an indicator of the blood flow's mechanical energy, which interchanges with both the pressure and viscous dissipation components. The final index for evaluating transvalvular hemodynamics is the mean magnitude of vorticity, assessed within the aortic volume. The vorticity magnitude can be calculated using the expression:  
\[\Omega_{a}(t)=\frac{1}{V}\int_{V}|\mathbf{\Omega}|\,dV \tag{16}\]  
which averages the vorticity over the aortic volume. This quantity is derived from \(|\mathbf{\Omega}|=\sqrt{w_{x}^{2}+w_{y}^{2}+w_{z}^{2}}\), which is the square root of the sum of the squares of the individual components of the vorticity vector \(\mathbf{\Omega}\), defined as the curl of the velocity vector \(\mathbf{v}\). The next critical category of hemodynamic metrics encompasses WSS-based indices. These parameters characterize the viscous forces acting upon the vascular wall by the surrounding fluid, described by the WSS. Mathematically, the magnitude of the WSS is represented as \(\boldsymbol{\tau}_{w}=\text{WSS}=n_{i}\cdot\tau_{ij}\), where \(n_{i}\) is the surface normal vector and \(\tau_{ij}\) denotes the viscous stress tensor. The metrics of the TAWSS, OSI [44], RRT [45], and transWSS [46] were computed based on the subsequent formulas:  
\[