## 2 Materials and Methods
### Numerical Schemes and Boundary Conditions
#### 2.4.3 Numerical Implementation  
Since the blood flow in the aortic region is momentum-dominated, we used the Newtonian fluid model for blood, with dynamic viscosity and density values of 0.0035 Pa - s and 1056 kg/m\({}^{3}\), respectively [39]. The simulations accounted for transient flow and initially covered two cardiac cycles, equivalent to 1.6 s. We employed an explicit version of the Crank-Nicolson Adams-Bashforth scheme for time-stepping, with a time-step size of \(\Delta t=1\times 10^{-5}\,\mathrm{s}\), resulting in a maximum CFL number below 0.6 for all simulations. The linear system of equations given in Equations (1)-(4) for the flow velocity, pressure, and the location of the structure was solved via the flexible generalized minimal residual method (FGMRES) algorithm [40] with a projection method-based pre-conditioner [35]. The convergence of the solver was controlled by the relative residual tolerance, which was taken as \(10^{-12}\) for all simulations, and this led to approximately 10 iterations for each time step and a maximum system residual below \(10^{-6}\) for the continuity, velocity, and pressure of the system. We conducted grid convergence studies to determine the consistency and accuracy of our numerical results. By generating four distinct mesh resolutions corresponding to an \(N\times N\times N\) grid, with a one-level local refinement at a ratio of four, we compared the results for \(N\) values of 58, 68, 78, and 88. Our refinement approach ensured that the background Cartesian grid contained approximately one structural mesh node for each grid cell, leading to maximum element edge lengths for the finest grid at 0.5, 0.43, 0.37, and 0.33 mm for each respective \(N\) value. Considering the sensitivity of the IB method due to the coupling of Eulerian and Lagrangian variables, defining convergence for the numerical method is challenging [41]. Even minor alterations in the simulations, such as in grid spacing or time-step size, can significantly impact flow velocities and subsequent displacements. Our accuracy assessment involved comparing the average fluid velocity near peak systole across the aforementioned grid sizes. Figure 5 represents this average fluid velocity, ranging from a time of \(t=0.05\) s, to a final time of \(t=0.2\) s. Across all tested grid resolutions, we observed consistent values. Though convergence is not explicit, and since the flow is transient, we expected that the average flow velocity would show convergence as \(t\) approached infinity. While our results are fairly well resolved for large-scale flow measurements, peak flow velocities in coarser cases exceed those in finer ones. These discrepancies in peak flow velocities stem from the numerical scheme employed to couple the Lagrangian and Eulerian variables in our IB method [42]. Such differences tend to diminish with grid refinement, as corroborated by our grid convergence observations. For greater accuracy, the mesh factor (\(M_{\text{FAC}}\)) should be below one, meaning that the structural mesh must be refined correspondingly when the fluid grid is refined. However, due to the substantial computational costs of the \(N=88\) case--which comprises roughly 6 million fluid mesh elements and 4.25 million structure mesh elements--we chose \(N=78\) as the basis for our model. We carried out our simulations at the National Center of High-Performance Computing of Turkey (UHeM) using the IBAMR 0.9.0 software framework [43]. The computing cluster was equipped with an InfiniBand(r)-based network, AMD EPYC 7742 2.24 GHz CPUs, and 256 GB of RAM. Each typical simulation took approximately 240 h of wall clock time on one node with 128 cores, equivalent to roughly 30,000 CPU hours. Comprehensive Quantification of Transvalvular Hemodynamic, Wall Shear Stress-Based Indices, and Helicity Descriptors  
In this section, a comprehensive analysis is conducted to quantify various hemodynamic indices [11], WSS-based indices, and helicity descriptors pertinent to transvalvular flow.