## 1 Introduction
[37] examined the contribution of fluid viscous dampers to the fatigue life of offshore wind turbines, showing the importance of damping systems in mitigating vibration-induced fatigue. Moreover, [PERSON] and [PERSON] [38] explored self-heating phenomena in viscous dampers under different loading conditions, using CFD simulations to enhance the design of damping systems under dynamic loads. [PERSON] et al. [39] investigated the coupling effects of temperature and pressure on the mechanical behavior of viscous fluid dampers, highlighting the impact of environmental factors on damping performance. [PERSON] and [PERSON] [40] focused on identifying the parameters of generalized models for fluid viscous dampers, improving the precision of CFD simulations in predicting damping behavior. [PERSON] et al. [41] compared different numerical models for fluid viscous dampers, providing a deeper insight into the optimization of damping systems through CFD-based approaches. Finally, [PERSON] et al. [42] applied CFD modeling and full-scale dynamic testing to better understand the performance of viscous dampers, demonstrating the importance of detailed fluid dynamics simulations in the design of efficient damping systems. Although these studies primarily focus on longitudinal vibrations and the performance of dampers in different mechanical systems, they provide valuable insights into the use of CFD to model and optimize vibration damping, offering a broader perspective on the application of fluid-based damping techniques. The concepts and methods presented in these publications also served as guidance for the authors on how to approach the issue of torsional vibration damping in the context of CFD. The widespread popularity of viscous torsional vibration dampers can be attributed to their simple construction (Figure 1) and considerable operational durability. Another key factor is their ability to maintain efficiency across a wide range of engine rotational speeds, facilitating smoother transitions through critical speed ranges. As noted by [PERSON] et al. [43], viscous dampers filled with silicone oil (silicone TVDs) or those utilizing a silicone-rubber combination (silicone-rubber TVDs) effectively reduce vibration amplitudes across the entire range of tested speeds. Conversely, in the absence of a damper or when a rubber damper (rubber TVD) is used, a significant increase in vibration amplitudes can be observed. In summary, previous studies [43; 44] indicate that thermodynamic and hydrodynamic modeling of viscous torsional vibration dampers plays a key role in determining their efficiency and predicting operational characteristics under steady-state conditions. Emphasis is placed on the critical role of temperature-dependent viscosity, which can influence the pressure distribution and velocity fields within the lubrication zones [45]. Equally important are detailed analyses of the temperature distribution within the damping fluid, as overheating or insufficient heat dissipation from the oil films can lead to viscosity changes and consequently a reduction in damping capacity [46]. In light of these issues, it is essential to account for both spatial and temporal variations in the oil film parameters within the viscous damper. This aspect is analyzed in detail later in the article, along with an evaluation of potential improvements in design solutions. The symbols used in the article are collected and described in Table 1. \begin{table}
\begin{tabular}{l l} \hline \hline  & **Coordinate systems** \\ \hline \(x,y,z\) & Cartesian coordinate system with the \(z\)-axis coinciding with \\  & the axis of rotation of the damper \\ \(r,\varphi\) & Polar coordinate system for the \((x,y)\) plane, where \\  & \(x=-r\sin(\varphi)\) and \(y=r\cos(\varphi)\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Table of symbols used in this study. Figure 1: Design solutions for the crank–piston system with mounted torsional vibration dampers: (**a**) twin-rod crank system, (**b**) single-rod crank system.