# Application of a Local Dynamic Model of Large Eddy Simulation to a Marine Propeller Wake
## 1 Introduction  
With today's largest computers, it is not sustainable to apply direct numerical simulation (DNS) to numerically solve the complex flow such as flows around a propeller or a wind turbine. Realizing the limitations of classical statistical methods based on the Reynolds averaged Navier-Stokes (RANS) equations, a better solution may be large eddy simulation (LES). LES has lower computational cost than DNS and is more accurate than RANS simulation. Application of LES of rotor-like devices (such as propeller, wind turbine and helicopter) is challenging because of the complex geometry involved and the meshing strategy balancing accuracy and cost accepted. In this paper a local dynamic \(k\)-equation subgrid-scale (SGS) model is applied to study the flow over a David Taylor Model Basin marine propeller 4381 (DTMB p4381) from a theoretical and applied point of view. In reference [1], a high fidelity large eddy simulation of a seven blade submarine propeller was carried out on a computational grid consisting of 840 million points, which combined with the acoustic model. The large eddy model uses the wall adaptive local eddy viscosity (WALE) model [2]. This kind of calculation cost is too high, and if there is no in-depth and rigorous discussion of the concrete LES model details, the results obtained are to some extent meaningless, or we can adopt a lower calculation cost model or strategy to achieve similar results. The literature [3] used 181 million unstructured grids composed of hexahedron elements only to conduct large eddy simulation for marine five blade propeller, and the use of dynamic Smagorinsky model was not described in detail. The billions of computing grids still hinder most researchers from practicing. We may obtain comparable conclusions through lower computational costs. Literature [4] used large eddy simulation to study the flow around a counter rotating propeller in uniform free streaming. The large eddy model used the dynamic Smagorinsky model proposed by [PERSON] et al. [5] and modified by [PERSON] [6]. Using a computational grid with two levels of refinement, the finegrid has approximately 19.3 million grid units and the coarse grid has about 7.7 million. The results indicate that the consistency of the load and velocity fields around the propeller between the two grids and the experimental results are encouraging. Reference [7] uses large eddy simulation method to predict the flow around ship propeller in forward and backward operation modes, and the large eddy model also uses the dynamic Smagorinsky model. The minimum grid size is \(1.7\times 10^{-3}\) times the diameter of the propeller, with a total grid count of approximately 13 million. The simulated thrust and torque coefficients, circumferential average speed, and root mean square fluctuation of speed were compared with experimental data, and good consistency was observed. Reference [8] used large eddy simulation to simulate the flow around a marine propeller and verified it with particle image velocimetry (PIV). The large eddy model uses dynamic large eddy simulation, delayed discrete eddy simulation (DDES) and stress mixed eddy simulation. The numerical results are compared with the experimental data to verify the effectiveness of the methods used. In recent years, more and more researchers have used PIV testing to measure and analyze the velocity information of propeller wake, as it can measure a large range of areas in a relatively short time. Reference [9] conducted experimental research on a small unmanned aerial vehicle propeller with serrated trailing edges, using a high-resolution digital PIV system to measure the flow field and quantify the detailed flow structure around the propeller. Reference [10] used PIV measurements of the wake flow field of a large oil tanker to validate two-dimensional PIV measurements and reconstruct three-dimensional flow fields by overlaying two-dimensional measurements, and studied the influence of rotating propellers on average velocity and turbulence characteristics.