## 7 Non-Premixed Hydrogen Diffusion Combustion
### Gas Jet Model  
Since hydrogen jet combustion in non-premixed diffusion combustion mode relies significantly on high pressure jet characteristics, the characteristics of the gas jet need to be understood. The jet features are equally important, if hydrogen is burned in partially premixed combustion mode, as the hydrogen jet features affect fuel and air mixing as well as fuel stratification. Figure 6 shows a widely accepted turbulent transient gaseous jet model, known as the vortex ball model, proposed by [PERSON] in 1962 [99; 100]. If a round nozzle is used, the gaseous jet in an unconfined environment is assumed to have a conical shape with axisymmetric head vortex and some minor irregularities. Most of the air entrainment is expected to occur in the steady-state region, with Gaussian velocity profiles in the fully developed region upstream of the head vortex [100]. The jet penetration, defined as the axial distance between the jet tip and injector nozzle, alongside with the jet cone angle, were shown to have a strong influence on ambient air entrainment.  
Figure 6: The vortex ball model of turbulent transient gaseous jet. Reproduced from [PERSON] [100].  
As discussed earlier, increasing the injection pressure increases fuel density in reservoir and fuel mass flow rate during the injection, which implies that HPDI of hydrogen has potential to achieve a high engine load. Usually, hydrogen in the jet can be described as an ideal gas and the flow is considered as an isentropic process, indicating no heat transfer and frictionless system [98]. The flow pattern in the core region depends on the pressure ratio across the nozzle--the jet can be considered as underexpanded when sonic condition is reached and flow is choked at nozzle exit. Literature distinguishes the moderately underexpanded and highly underexpanded jets, which occurs when the pressure ratio between the reservoir and ambient exceeds 2 and 3.85, respectively [101]. For a moderately underexpanded jet, a shock normal to the flow direction is formed with repeated diamond-shaped oblique shocks established in the core region [101]. On the other hand, for a highly underexpanded jet, the hydrogen gas will rapidly accelerate upon leaving the nozzle. The large difference in gas pressure after exiting the nozzle results in the formation of expansion waves in a process described by the Prandtl-Meyer expansion fan [102]. These waves grow until the boundary is generated to resist the jet expansion and to reflect the expansion waves as compression shock waves. A barrel-shaped shock with a Mach disc is formed downstream the injector nozzle due to the coalescence of compression shock waves, as shown in Figure 7[100]. The diameter and distance of the Mach disc from the nozzle exit increase with the pressure ratio across the nozzle as well as nozzle diameter [98]. It is noted that the highly supersonic flow and shock boundary prevent any gas exchange within the barrel-shaped shock [102]. Therefore, the formation of a barrel shock can considerably affect the overall fuel and air mixing.  
However, the influence of combustion on the gas jet model under reactive engine conditions still has to be explored. It is also noted that in realistic applications, the hydrogen jet will inevitably interact with the engine chamber wall. Therefore, flame-wall interaction effect also needs to be considered, constituting a more complex scenario than free jet. Literature investigating hydrogen jet interaction with walls is scarce. Examples from liquid fuel jet research include optical investigations studying jet impingement on different wall geometries at a defined distance from the nozzle, for instance a plane flat wall or a confined wall [103; 104].