Fig. 3 shows a complex configuration known as a rotor-stator system, used to study aerodynamic flows and heat transfer. The air is injected into the space using two cylindrical inlet tubes. The spacing between the two discs, also known as the rotor-stator spacing, is an essential parameter that is precisely adjusted within a range of 0.02 to 0.1. The air passes through two tubes before reaching the turning disc. These tubes perform a vital role in regulating the air flow, and preparing the flow before it reaches the rotating disc. In this context, the Reynolds numbers associated with the two air jets and the rotation are fundamental parameters for characterising the stream flow. The Reynolds number of the jet is determined by taking into account parameters such as the injected air flow, the air jet diameter and the kinematic viscosity of the air. The following factors can have a considerable impact on the dynamics of fluid flow in this particular system:

Spacing between rotor and stator G=HRd

Discrete transport equations refer to the mathematical equations that describe the behaviour of fluids in a system. In this context, these equations are solved on a computational domain, which is subdivided into numerous tetrahedral elements. The choice of a tetrahedral mesh (Fig. 4) has several advantages. Tetrahedral elements provide a better representation of the complex geometry of disc surfaces and offer greater flexibility for adjusting the mesh size. In the specific case of the spacing between two discs, it is crucial that the mesh is very fine. This means that the mesh size needs to be reduced around the walls in our tests in most cases in order to detect details and variations in the flow near the disc surfaces. In particular, it is important to have enough points in the boundary layer, which is the area where viscosity effects are dominant. To ensure an accurate representation of this limit layer, it is essential to ensure that the first layer of wall cells in the vertical orientation has a value of less than 5. Y+ is a dimensionless parameter used to characterise the resolution of the boundary layer, and a value of less than 5 guarantees an extremely precise reflection of the free flow regime in the vicinity of the walls. To achieve these objectives, it is necessary to adjust the mesh size according to the parameters that influence the flow, such that the air flow introduced, the spacing between the discs (G) and the rotor rotation speed. By modifying these parameters, it is important to adapt the mesh size accordingly to ensure adequate resolution of the boundary layer.

Problem Boundary Specification defined as follows:

The flow characteristics are influenced by the velocities of the two jets, noted U1j and U2j.

The Reynolds approach takes a statistical approach to exploring the characteristics of physical quantities. It subdivides these quantities into two distinct components: a mean value and a fluctuation. The Reynolds decomposition procedure is then used to integrate these two aspects into the Navier-Stokes flow equations. As far as the mean quantities are concerned, here are the equations derived from them: (1)u¯j∂u¯i∂xj=−1ρ∂p¯∂xi+ν∂2u¯i∂xj∂xj−∂(ui′uj′¯)∂xj

(2)uj¯∂T¯∂xj=−∂∂xj(λρcp∂T¯∂xj−T′uj′¯)where Rij=ui′uj′¯ and T′uj′¯ are called the terms relating to Reynolds constraints and turbulent heat transport. The number of unknowns in the problem then exceeds the number of equations available, and it becomes imperative to modelling turbulent tensors using turbulence models, in order to solve the problem consistently and establish a link between the fluctuating field and the mean field. For more details on turbulent flows and turbulence models [17], we encourage the reader to refer to Schiestel et al.’s book [18].

Our approach is centred on a model using a 2nd order closure with a low Reynolds number derived from the model described by [19] and which is sensitive to the rotational effects described by [20]. The RSM model presented here offers an in-depth explanation of the flow field. Milan Đorđević et al. [21] has stated that RSM predictions are generally more accurate in the turbulent region. The general Reynolds tensor R_ij is written as:

(3)uj¯∂ui′uj′¯∂xj=Pij+Dij−εij+Oij+Ωij+Φijwhere Pij, Ωij, ϕij, εij, Oij simultaneously represent the following terms: The diffusion term, the pressure-deformation correction term and the dispersion term.

Fig. 5 shows a detailed graphical representation of the lines for different values of the rotation parameter. The area of the rotor and stator, delineated by solid lines, is clearly defined. At the core of this intricate configuration, two air jets converge toward the rotor, initiating the formation of a flow that radially propagates along the inner wall of the gap. When these two air jets come into contact, they generate a radial flow that is subsequently captured within a well-defined recirculation region. An interesting observation is that the size of this recirculation zone remains constant, irrespective of variations in the rotational Reynolds number. These findings align with the conclusions drawn from references [20,22], suggesting that the size of this recirculation zone primarily varies with the spacing parameter G. This stability in the dimension of the recirculation zone has significant implications for our overall understanding of the complex flows characterizing this specific configuration. These results underscore the importance of spacing parameter G as a key factor in the dynamics of flows and the convective transfer in the area under study.

For Reω=2.35×105 and beyond and r/Rd ratio of 0.62, the flow becomes centrifugal with streamlines parallel to the rotor. In cases of high rotation rates (Reω=4.02×105 and Reω=5.04×105) and at r/R_d values of 0.84 and 0.67, a large recirculation zone is observed near the stator, corresponding to an inflow of fluid from the outer region. Reference [22] noted the flow takes on a tangential orientation starting at r=0.186 and r=0.124 for a rotation rate of N=1.237 and N=2.47. For a Reynolds number and beyond an r/R_d ratio of 0.62, the flow takes on a centrifugal configuration with streamlines moving parallel to the rotor. In situations of high rotation rates (Reω=4.02×105 and Reω=5.04×105), at r/Rd points of 0.84 and 0.67, a large recirculation zone can be observed near the stator, corresponding to an intake of fluid from the outer region.

Fig. 6 provides a detailed analysis of the radial mean velocity fields for different rotation values within our system. These data reveal significant variations in velocity near the impact points of the air jets, indicating a change in the direction of airflow at these specific locations. These variations are not random but result from the dynamic interaction between the air jets, the rotor, and the stator. When these jets interact with the rotor and stator, their trajectories undergo noticeable adjustments, resulting in velocity fluctuations. The negative velocities observed in the contours suggest regions where the airflow is moving in the opposite direction to the main flow direction. These regions may be associated with changes of direction or downshifts in the airflow, particularly near the edges of the rotor or stator. These negative velocities are often the result of phenomena such as air recirculation or vortex formation, which can significantly affect the overall flow dynamics and influence heat transfer in the system under study.

Fig. 7 of our study highlights the axis to axis distribution of radial and tangential velocities, considering non-zero rotation parameters, at three distinct positions for r/D (4.8, 5, 6.73, 6.92, 7.11). Examination of the characteristics of the radial and tangential velocity profiles enables us to identify different outflow structures. A stream with two clearly defined limiting levels, divided by a vigorously rotating, inviscid liquid, is categorized as a Batchelor [23] stream. Conversely, if the stream has only one boundary layer, with virtually zero tangential velocity at the distance of this boundary layer, it is referred to as a Stewartson [24] stream. These distinctions underline the variability of flow phenomena as a function of rotation parameters and positions in the configuration under consideration.

For Reω=2.35×105 and G=0.04 (see Fig. 7a1), the radial elements of the velocity, noted at various radial locations (r/D = 4.8, 5, 6.73, 6.92 and 7.11), show a parabolic profile similar to that obtained in the case of two parallel plates. The maximum peak of this magnitude is reached at the midpoint of the two disks for Z/H = 0.5. However, a negative radial velocity is observed for r/D = 6.92 and 7.11, corresponding to fluid re-entry. At r/D = 6.73, in the middle of the rotor between the two jets, a symmetrical radial velocity is observed.

Tangential velocity values (Fig. 7a2), which describe the flow in the vicinity of the rotor, are notably high. However, they gradually decrease with distance from the rotating disk in the direction normal to the rotor.

At the stator surface, characterized by Z/H = 1, these values ultimately reach zero. This development highlights a significant variation in flux within the system, with initially high tangential values near the rotor gradually declining to negligible values at the stator surface. At the stator surface, characterized by Z/H = 1, these values ultimately reach zero. This development highlights a significant variation in flux within the system, with initially high tangential values near the rotor gradually declining to negligible values at the stator surface.

Velocities in the circumferential direction are characterized by a single frontier zone close to the rotor. As one moves increasingly away from this boundary zone, the circumferential velocity decreases, finally reaching zero when Z/H = 1. The flow phenomena identified in this configuration are therefore similar to Stewartson-type [24].

The analysis of heat exchange is based on a study of the local convection coefficient. Fig. 8 presents the variation in Nur for different values of the rotation parameter Reω=2.35×105, Reω=4.02×105 and Reω=5.04×105. For a constant jet Reynolds number of Rej=16×103 the evolution of the Nusselt number shows a significant variation along the disc. However, this variation shows a relatively low dependence on the number (Re_ω), particularly for small values (when r/D≤6.4). The local thermal convection coefficient distribution exhibits peaks at positions corresponding to r/D = 1.6, r/D = 5 and r/D = 6.6, suggesting high rates of heat transfer. This increase in heat transfer is that it results from increased Radial displacement within the boundary layer, particularly near the rotor, creating higher levels of shear stress. This increase in shear stress promotes more efficient heat transfer. When moving further from the point of impact and exceeding values of radius r/D ≥ 7.6, an increase in thermal convection coefficients is observed in relation to the parameters r and Re_ω in response to the increased influence of rotation. This observation can be attributed to a decline in radial in the vicinity of the stator and the expansion of the distance in the radical direction speed variation in the vicinity of the rotor. This leads to greater levels of the tangential deformation rate proximate to the rotor, which in turn promotes improved heat exchange in this region. In our study, an in-depth comparison of our results with those presented in reference works [6,25] reveals a significant observation related to the Nusselt number. Our meticulous analysis shows a remarkable constancy of the Nusselt number over the entire rotor surface. This observation conclusively suggests that our experimental configuration achieves full rotor cooling. The work of [6,25] played a crucial role in establishing a baseline for our study. The concordance of the results obtained reinforces the validity of our numerical approach. The observation of a constant Nusselt number underlines a homogeneous heat transfer over the rotor surface, highlighting the effectiveness of our experimental methodology. This stability of the Nusselt number can be attributed to a multitude of factors, including the specific design of the experimental set-up, the rigorously controlled operating conditions, as well as the specific thermal properties of the materials employed. In sum, our study makes a substantial contribution to the understanding of heat transfer in this particular configuration, confirming and extending the findings of previous work. These results provide a solid foundation for practical applications requiring optimal thermal management.

In Fig. 9, keeping the rotational Reynolds number constant at Reω=5.04×105, we proceeded to vary the jet Reynolds number values, varying them from Rej=16×103 to 55.46×103. In the range of values where (r/D ≤ 4.4) is relatively low, it is essential to note that the introduction of air into the process has a significant impact on improving local heat transfer, this area is sensitive to variations in air flow, which is of crucial importance in optimising heat transfer. In the immediate vicinity of the interaction region, precisely when the value of r/D is in the range 4.4 to 6.8, the influence of the airflow becomes significant, creating a zone characterised by intensive heat transfer. This zone is characterised by fluctuations in the amplitude of heat transfer peaks, closely correlated with the jet’s relative Reynolds number, Re_j. The study identified three well-defined zones, separated by characteristic radii. The first of these zones is a low-radius region (r/D ≤ 4.4) where heat transfer is relatively unaffected by rotational effects and airflow. The second zone, located near the point of impact, is mainly governed by the influence of the injected airflow, making it the main factor controlling heat transfers. Finally, the third zone, at the periphery, is characterised by the predominance of rotational effects in heat exchange.

The study, as shown in Fig. 10, focuses on the interaction between (Re_ω) and (Re_j), and their impact on the average Nusselt number, a crucial indicator of heat transfer. The principal focus of this research is to increase our comprehension of the cooling mechanisms of the rotating disc and their repercussions on a global scale. The results reveal a clear trend: the mean Nusselt number increases proportionally with both Reω and Rej, a crucial relationship for assessing cooling efficiency. These findings at the global level confirm previous local observations, reinforcing our understanding of heat exchange. It is essential to note that increasing the flow of fresh air injected by the two jets has a direct impact on the overall cooling of the rotating disc, suggesting that airflow management can be an effective means of regulating the cooling process, with important practical implications for rotating systems.