# An Analysis of the Factors Influencing Dual Separation Zones on a Plate
## 1 Introduction  
Shock waves constitute a prevalent aerodynamic configuration existing in hypersonic inlets, performing an essential role in decelerating and pressurizing the incoming flow to meet the requirements of the combustion chamber [1]. However, the shock wave itself and the flow separation induced by the shock wave/boundary layer interaction (SWBLI) can lead to issues such as flow distortion, energy loss, and excessive local aerodynamic loads and heat loads [2, 3], which all should be avoided during flight for safety and efficiency.  
Regarding the shock wave/boundary layer interaction phenomenon itself, a rich collection of theoretical models and empirical formulas has been established. [PERSON] pointed out that the separated flow induced by shock wave impingement on the boundary layer belongs to free interaction, where the initial separation pressure rise is solely influenced by incoming flow conditions and remains independent of downstream factors [4]. [PERSON] et al. applied the minimum entropy production principle to counter-symmetric separation zones. Considering both the upstream and downstream conditions of the separation zone, they calculated the separation shock angle and successfully established prediction criteria for the transition of the main flow shock waves from regular reflection to Mach reflection [5]. Beside flow field structures, the scale of separation zones also requires attention. Current mainstream approaches primarily rely on semi-empirical formulas forprediction [6, 7, 8], with a general consensus that separation zone dimensions are related to factors including the incoming flow Mach number, incident shock intensity, and boundary layer thickness.  
In practical flow fields, the SWBLI is not found to be independent but coupled with flow field structures such as background wave systems [9, 10]. Under complex background influences, the separation zone exhibits peculiar phenomena that cannot be fully explained by existing theories of single shock wave/boundary layer interactions (S-SWBLIs). These include sudden increases in the size of merged separation zones [11] and hysteresis during the mergence and re-separation of dual separation zones [12, 13, 14]. The large-scale merged separation zone induces a substantial degradation in both the total pressure recovery coefficient of the inlet and the flow field uniformity, and even leads to an unstart in severe circumstances. Therefore, it is necessary to reveal the underlying mechanisms governing the interaction between dual separation zones to prevent unexpected phenomena. In response, scholars have conducted research on shock wave/boundary layer interaction flow fields containing dual separation zones (DSZs), establishing predictive formulas for dual-separation-zone scales [15, 16] and critical separation criteria for dual separation zones [17]. However, current research on DSZs still fails to completely explain the special phenomena mentioned above, and lacks detailed descriptions of the whole interaction. Also, influencing factors in the mergence and re-separation of DSZs require further analysis.