## 1 Introduction
Two-dimensional simulations have been mostly focused on the early stages of the breakup process, without investigating the droplet fragmentation and the subsequent mist development [13; 14]. On the other hand, to make them affordable in the context of preliminary analyses, 3D computations have been most often conducted by solving the Euler equations, while neglecting the viscous effects [15; 16]. Typically, compressible two-phase solvers based on a five-equation model [17] are used to computationally study shock-droplet interactions, where the gas-liquid interface is modeled using the volume-of-fluid (VOF) approach [18]. The governing equations consist of two continuity equations for the two different phases, the volume fraction advection equation, and the mixture momentum and energy equations. As far as turbulence modeling is concerned, either the under-resolved direct numerical simulation (DNS) or the large eddy simulation (LES) approach is usually followed [6; 16]. However, these expensive CFD models are generally limited to the early stages of the aerobreakup process, investigating the droplet surface instabilities, without examining the later-stages fragmentation and the liquid mist evolution. The main goal of the present work is the computational evaluation of the droplet aerobreakup under the SIE regime following a relatively lighter approach, where the viscous effects are not neglected. To investigate the multiscale features of the phenomenon with a viable computational cost, the mean turbulent flow field is simulated by solving the compressible unsteady Reynolds-averaged Navier-Stokes (URANS) equations endowed with a two-equation turbulence model. In contrast to our previous work [19], the newly proposed URANS model is supplied with the scale-adaptive simulation (SAS) method for unsteady turbulent flows [20]. Being able to dynamically adjust to the resolved unsteady turbulent structures, the SAS-URANS model represents an attractive alternative to the existing expensive numerical methods for transient two-phase flows. Moreover, the CFD modeling approach is enriched by introducing a Lagrangian method for tracking the sub-particles that originate from the breaking droplet. The simulations are conducted with the commercial solver Ansys Fluent, which is widely used for industrial CFD applications [1; 21]. The mean-flow-governing equations are solved employing a finite volume (FV) numerical technique [13; 22], where the transient tracking of the air-water interface is approximated through the VOF methodology. Basically, a hybrid two-phase model combining the VOF approach and the discrete phase model (DPM) is implemented in the CFD solver, with the former model trackingthe gas-liquid interface to describe the deformation and fragmentation of the parent body, and the latter model allowing for tracking the child particles. The combination of CFD with DPM represents a powerful tool to simulate two-phase flows of engineering interest, which has recently received considerable attention. For instance, the analysis of a combined CFD-DPM method for predicting the performance of finely atomized sprays for dust suppression has been performed in [23], while a two-stage multi-phase model for close-coupled gas atomization has been developed in [24], by combining an Eulerian model for primary atomization and a Lagrangian particle tracking approach to predict the particle size distribution resulting from secondary atomization. Following a similar but different approach, a VOF-to-DPM model has been recently employed for the under-resolved DNS of the aerodynamic droplet breakup in [11]. In the present work, instead, the combined VOF-Lagrangian model is developed and tested in the framework of the SAS-URANS approach. The overall computational modeling procedure for simulating the droplet aerobreakup is validated against reference data that are provided by both experimental studies [8] and high-fidelity numerical simulations [15]. The rest of this manuscript is organized as follows.