Unified length scale of spray structure by unlike impinging jets

In bi-propellant thrusters, impinging type injectors are widely used to deliver propellants to a combustion chamber. By impinging the jet streams of fuel and oxidizer, the spray spreads while the two liquids mix. To design the injectors, several correlations related to injection conditions have been proposed (e.g., Rupe factor), and practically utilized over the last half-century. However, the physical meanings of the past correlations are not well understood, because the essential scale of the spray structure is elusive. In this paper, we derive the global length scale of the spray produced by impinging injectors of unlike doublet, fuel-oxidizer-fuel triplet, and oxidizer-fuel-oxidizer triplet in a consistent manner. The unified length scale is found representing the spray width ratio of oxidizer to fuel evidenced by comprehensive cold-flow tests including several past studies, covering various parameters such as injector types, nozzle diameters, physical properties of working liquids, and injection velocities. Finally, we clearly provide the physical meaning based on practical correlations in a phenomenological sense.