## 1 Introduction  
For over a century, the automotive industry has been firmly entrenched in using carbon-based fuels, such as gasoline and diesel. However, carbon neutrality has been an urgent issue that requires effective reductions in carbon dioxide emissions from the automotive sector worldwide. In the meantime, as vehicle emission regulations become increasingly stringent, the quest for cleaner alternative fuels has become more imperative. In this context, hydrogen has emerged as a promising fuel with the advantages of it having a nitrogen- and carbon-free nature, and the combustion produces only water. These inherent characteristics provide significant advantages over other alternative fuels, particularly in achieving zero carbon emissions and near-zero nitrogen oxide (NO\({}_{\text{x}}\)) production [1; 2; 3; 4]. In addition, the extensive flammability range (4-75% volume fraction in air) of hydrogen allows it to have a lower combustion temperature when operating with lean mixtures. This not only inhibits the formation of NO\({}_{\text{x}}\), but also enhances engine efficiency. Numerousstudies have confirmed that hydrogen internal combustion engines (H\({}_{2}\)ICEs) can provide similar performance and comfort to conventional gasoline engines [5; 6; 7]. In the early phases of research on H\({}_{2}\)ICEs, it was generally common to use intake port injection to form a homogeneous mixture. This method is simple and easy to implement, with low research and development costs. However, because hydrogen has a very low density, using intake port injection causes a decrease in the volumetric efficiency of the engine, resulting in a lower power output [8]. As a relatively new solution, the direct-injection strategy of hydrogen can circumvent the volumetric efficiency loss induced by hydrogen displacement with air. Additionally, compared to port injection, the direct-injection strategy has the potential to help reduce the possibilities of abnormal combustion phenomena, such as backfire, pre-ignition, and knock. Overall, the direct-injection strategy could offer superior efficiency, power density, and safety advantages for H\({}_{2}\)ICEs [9; 10; 11; 12]. However, for direct-injection H\({}_{2}\)ICEs, the quality of the hydrogen-air mixture in the cylinder directly affects the combustion quality, subsequently affecting the power output, fuel economy, and emissions of the engine [13; 14; 15; 16]. Previous studies have indicated that injection timing has a minimal impact on the mixture during intake port injection in gas engines [17]. But direct injection into the cylinder provides more flexibility regarding injection strategies than intake port injection. The geometry and velocity of the jet are determined by the injection pressure and nozzle structure. The formation and concentration distribution of the mixture are primarily influenced by hydrogen injection timing and in-cylinder flow. Improper injection timing will reduce the combustion efficiency of hydrogen, resulting in a large amount of unburned hydrogen in the exhaust [18; 19]. Therefore, optimizing the injection strategy and coupling with the in-cylinder flow to form a suitable mixture is key to developing H\({}_{2}\)ICEs. Currently, methods such as particle image velocimetry (PIV), planar laser-induced fluorescence (PLIF), constant-volume bomb, Schlieren techniques, and numerical simulations can be used to study the mixing characteristics and concentration distribution of hydrogen-air mixtures [20; 21; 22; 23; 24; 25]. [PERSON] et al. [23; 24] studied the mixture distribution under different hole numbers, hole diameters, and nozzle positions using the PLIF method. The results indicated that when hydrogen was injected during the early compression stroke, a relatively uniform mixture was formed. However, when hydrogen was injected during the mid-compression stroke, the mixture formed under different nozzle structures was non-uniform near the cylinder wall, resulting in unfavorable ignition conditions. [PERSON] et al.