# A Deep Learning-Based Mapping Model for Three-Dimensional Propeller RANS and LES Flow Fields
## 1 Introduction  
Turbulence, also known as turbulent flow, is a common phenomenon in nature. For example, it exists in smoke in chimneys, the wake behind propellers, waterfalls in rivers, and so on. Turbulence phenomena are also prevalent in various fields of engineering. Despite over a century of study and exploration into turbulence, a comprehensive understanding of its underlying mechanisms remains elusive, and the nature of turbulence continues to be an unsolved challenge in classical physics [1]. Although, in principle, Direct Numerical Simulation (DNS) can accurately simulate turbulence, this approach is often economically challenging due to the high demand for computational resources. Therefore, how to describe the characteristics of turbulence economically has become a research trend [2], with numerical simulation methods having become an essential tool for revealing the essence of turbulent processes. Numerical methods for turbulent flow analysis are typically categorized as Direct Numerical Simulation (DNS) [3], Large Eddy Simulation (LES) [4], and Reynolds-averaged Navier-Stokes (RANS) [5], according to grid resolutions. DNS and LES have gained popularity in both research and engineering thanks to the constantly improving computer capabilities and parallel computing. However, their utility is limited in scenarios involving complex geometries and high Reynolds numbers since they need larger grid sizes. In contrast, despite their lower accuracy compared to DNS and LES, RANS models have been widely employed in engineering due to their user-friendly nature and computational efficiency [6]. Specifically, DNS computes the transient Navier-Stokes equations directly for turbulent flow fields. Although DNS can yield results of high precision, it requires solving different temporal and spatial scales of turbulence throughout the entire flow field, necessitating extremely high grid resolutions in both time and space. Due to limitations in computational resources and costs, this approach is challenging to be applied to real-world engineering problems. LES can perform unsteady-state calculations like DNS, but its computational cost is less than that of DNS because small-scale eddies are not resolved during the computation process. Although this method does not solve all scales of turbulence in the flow field, its relaxation of computational scale can improve the efficiency of turbulence simulation to some extent. RANS is currently the most widely used turbulence numerical simulation method in engineering, owing to its ease of use and efficiency in engineering practice. RANS models, mostly based on the eddy viscosity assumption, tend to yield good results in attached flows [7]. Moreover, empirical parameters in the models are often determined based on specific flow conditions, which increases the uncertainty of the model and affects its applicability. Nowadays, researchers mainly make efforts to explore complex system models combining RANS and LES through the following two approaches. The first approach is based on traditional theoretical model architectures, where an ideal system description is established according to the control equations of physical problems. This type of model usually requires researchers to have a deep understanding of physical processes and to translate them into mathematical models. Most existing turbulence models, such as the k-\(\omega\) model [8] and the Spalart-Allmaras (SA) model [9], are constructed based on this approach. The second approach is the data-driven method, where black-box or gray-box models are directly constructed based on sample data from system simulations or experiments. How to efficiently utilize these large datasets to extract key information from them and to guide the development of fluid mechanics has become the focus of attention for researchers. In recent years, machine learning has been widely applied in various fields and has made significant strides in traditional scientific and engineering domains.