## 1 Introduction  
Pressure-swirl atomizers (PS) play a unique role in many industrial applications including combustion, spray cooling, spray drying, etc. A relatively simple geometrical design, resistance to clogging, wide spray cone angle and high atomization efficiency are among the favourable parameters. A typical PS atomizer contains tangential entry ports and a swirl chamber with an exit orifice. The liquid is fed via tangential ports into the swirl chamber where it gains high swirl velocity. The swirling liquid is discharged from the exit orifice in the form of a hollow conical liquid sheet which consequently disintegrates into filaments and ligaments. A ratio of the swirl to axial momentum determines the spray cone angle (\(SCA\)). Despite the simple geometry, the internal flow behaviour is complex. The swirling liquid creates a low-pressure zone along a centre line of the swirl chamber where the static pressure usually drops below the ambient pressure, and the air from the surrounding atmosphere is pulled inside the low-pressure zone, so an air-core is formed. The internal vortex behaves as a Rankine vortex since the swirl velocity has its maximum located at the air-core surface which behaves like a virtual solid cylinder [1]. Secondary flow effects, as Gortler vortices, could be also presented in a boundary layer inside the swirl chamber [2; 3]. It is well known that the internal flow directly affects the parameters of the discharged liquid sheet, such as its thickness, stability, velocity, and _SCA_. These parameters consequently determine the sizes of ligaments and droplets. To understand the link between the atomizer's performance and its design, the internal flow must be examined. Some authors used the exact analytical solution to predict the discharge parameters. Simple non-viscous treatment, reviewed in [4; 5], proved to be a useful tool for a basic insight into the flow behaviour, but it lacks accuracy for some atomizer geometries. A better agreement can be achieved when the viscous flow is assumed [6; 7], but still some aspects such as the liquid sheet temporal stability or secondary flow effects, are not resolved and their mathematical description is very extensive. On the other hand, numerical simulation has arisen in recent years due to an increase in computational performance, and many commercial software programs are available in the market. These software tools for numerical simulations of the flow dynamics could be simply applied to predicting the internal flow of the PS atomizer. However, many different geometrical, numerical and physical setup combinations can be used for the same atomizer and operating conditions. The CFD simulation of the internal flow ideally suits for final adjustments of the atomizer geometry, since it allows for fine-tuning of the atomizer's individual parts and captures the internal flow instabilities. Nevertheless, the numerical results should be still validated by an experiment. In the past, many authors performed a CFD simulation of the internal flow of PS atomizers. An overview of some published papers is presented in Table 1. One of the first numerical studies of the PS atomizer was conducted in 1997 by [PERSON] and [PERSON] [8]. They used a 2D axisymmetric geometry with a laminar solver and reported a deviation of discharge coefficient, \(C_{D}\), from experimental data to be less than 3%. A similar numerical setup was later used by [PERSON] [6]. Even this simple 2D model yield better agreement with the experimental data than the analytical viscous solution. Note here that the author used the laminar solver even for values of Reynolds number within the inlet port, \(Re=122\),000. The port-based \(Re\) is defined as \(Re=v_{p}d_{p}\rho_{l}/\mu_{l}\), where \(v_{p}\) is velocity within the inlet port, \(d_{p}\) is the inlet port hydraulic diameter, \(\mu_{l}\) is liquid dynamic viscosity and \(\rho_{l}\) is liquid density.