## 4 Numerical Model  
In this study, Converge CFD commercial software was employed with the adaptive mesh refinement (AMR) algorithm to accurately capture velocity and temperature gradients within the density-based transient CFD solver. Base mesh sizes of 3 mm, 4 mm, and 5 mm were utilized to investigate mesh independence for Sandia's PIV validation case, thereby ensuring the accuracy of the fuel mixture prediction. For each base mesh size, local embedding and AMR techniques were applied with corresponding minimum cell sizes of 0.375 mm, 0.5 mm, and 0.625 mm, and maximum cell sizes of 0.75 mm, 1 mm, and 1.25 mm, respectively. An illustration of the refined mesh views for different base sizes is provided in Figure 4, where a three-level refinement was applied in the injection region. Figure 3: Sandia Single Cylinder Research Engine [6] Injector Valve Lift Profile. Figure 2: Sandia Single Cylinder Research Engine [6] CFD Geometric Model (**Left**) Tumble Adaptor Illustration (**Right**). A three-level automatic refinement strategy was implemented based on velocity and temperature gradients in both the PIV validation and combustion CFD models. A semi-implicit numerical solver was selected to solve the transport equations, ensuring proper convergence and stability of the solutions. Hydrogen combustion was modeled using the POLIMI_H2 CO_1412 reaction mechanism developed by Chemical Reaction Engineering and Chemical Kinetics (CRECK) Group at Politecnico di Milano [23; 24], specifically tailored for detailed hydrogen oxidation and consisting of 32 species and 173 elementary reactions. This mechanism, denoted as POLIMI_H2 CO_1412, has been validated against experimental data for ignition delay times and laminar flame speeds in hydrogen and syngas mixtures under a wide range of pressures and equivalence ratios. Hydrogen injection was modeled to occur between \(-140\)\({}^{\circ}\)CA and \(-70\)\({}^{\circ}\)CA. A total of 70 mg of hydrogen per cycle was injected at a fuel temperature of 303.15 K. The intake port pressure was set to 2.65 bar, and the intake air temperature at the boundary was defined as 320 K. To establish lean combustion conditions, the global equivalence ratio was adjusted to achieve a lambda (\(\lambda\)) value of 2.1 across all simulations. Wall boundary conditions were defined using adiabatic assumptions, with fixed wall temperatures of 510 K for the piston and cylinder head, and 460 K for the liner. The intake port wall temperature was set to 340 K. These consistent boundary and initial conditions were used throughout the optimization process to ensure a controlled and comparable evaluation of injector design effects. The Stiff Arrhenius Gas-phase Equilibrium (SAGE) detailed chemical kinetics solver [25] was utilized to resolve the full set of chemical reaction equations during the combustion event within Converge CFD. To ensure numerical stability and accuracy, a density-based compressible solver was employed, with the time step automatically adjusted based on a CFL number maintained between 1 and 2, thereby enabling convergence through a variable time-step approach. The reliability of the SAGE solver coupled with Converge CFD for hydrogen combustion modeling has also been demonstrated in previous research conducted by [PERSON] et al. [14], where similar operating conditions were used to investigate knock formation and combustion characteristics in a hydrogen-fueled direct injection engine. Moreover, the extended Zeldovich mechanism [26] was employed to model thermal NO\({}_{\kappa}\) formation, capturing high-temperature nitrogen oxidation pathways accurately, which also showed validated performance in the same study [14]. In this study, the RNG k-\(\epsilon\) turbulence model was employed to simulate the hydrogen-air mixture behavior due to its proven accuracy in capturing turbulent jet evolution in hydrogen direct-injection engines. Notably, Argonne National Laboratory has  
Figure 4: Mesh Independence Study of PIV Validation Cases Mesh Illustration.