## 1 Introduction  
Hybrid rocket motor technology has advanced over the past decades, mainly trough ground research [1; 2; 3] but also flight testing [4], and it is now recognized as a potential alternative to the standard liquid and solid ones for multiple aerospace applications, for example sounding rockets [5], small launch vehicles [6; 7] and suborbital space tourism [8]. Safety, reliability, architecture simplicity, throttleability [9; 10; 11] and re-ignition capability are the main advantages compared with the more conventional rocket motor technologies [12; 13]. Moreover, the possibility of using green propellants (e.g., high-test peroxide, HTP) makes hybrid rockets an eco-friendly option compared with solid rocket motors and classical storable toxic liquids, increasing their attractiveness [14; 15]. As a result of these advantages, hybrid rocket engines are expected to be increasingly adopted in the future. Furthermore, recent research addressed the main drawbacks (i.e., low fuel regression rate and low combustion efficiency) typical of the use of hybrid motors, and different effective solutions have been developed to improve performance to acceptable levels [16; 17], at least for some applications. For instance, the use of paraffin-based fuels has been proposed to considerably increase the regression rate values [18]. These types of fuels, called liquefying fuels, melt on the surface, and form a thin liquid layer where droplets are detached and entrained by the oxidizer gas flux, as described by the theory [19; 20; 21] and confirmed by experiments [22]. This added mass transfer mechanism leads to a higher regression rate [23] compared with classical diffusion-limited fuels [24; 25; 26] (e.g., hydroxyl-terminated polybutadiene, HTPB, or high-density polyethylene, HDPE). Additionally, recent studies examined the effects of additives to paraffin-based fuels, and the findings demonstrate the possibility of increasing the fuel thermo-mechanical properties (tensile strength increase over 40%) and tailor the fuel regression rate level (reductions up to over 50%) [27; 28; 29]. Devices such as post chambers (PC) [30], mixers and diaphragms [31; 32; 33] have been studied to enhance the mixing of the reactants by increasing the turbulence level in the combustion chamber, thus improving the combustion efficiency (above 90%). One of the most effective means to improve both the regression rate and combustion efficiency is through vortex injection [34; 35]. This type of injection forces the oxidizer flux to enter the combustion chamber with both tangential and axial velocity components, hence generating unique helical streamlines through the grain port. The vortex oxidizer injection can be placed at the head-end of the motor [36] or at the aft-end [37]. The resulting flow field increases the convective heat flux and the turbulence magnitude inside the combustion chamber. Thus, much higher values of the regression rate (several times' increase) and efficiency (above 90%) can be achieved [38; 39; 40]. The vortex intensity inside the chamber is quantified by the swirl number (\(SN\)). In each cross-section, the \(SN\) is defined as the ratio between the axial flux of the swirl momentum and the axial flux of the axial momentum [41]. \[SN=\frac{\int\rho v_{z}v_{\theta}rdA}{R_{max}\int\rho v_{z}^{2}dA} \tag{1}\]  
Since both the fuel regression rate and the combustion efficiency are related to the vortex intensity inside the combustion chamber, these variables are, likewise, linked to the \(SN\) parameter. Hence, vortex injection provides an additional advantage: the possibility of tailoring the fuel regression rate by adjusting the \(SN\) inside the grain port. Regression rate tailoring is of particular importance as it permits varying the fuel radial ablation velocity, and thus the optimization of the combustion chamber design.