# Application of a Local Dynamic Model of Large Eddy Simulation to a Marine Propeller Wake
## 5 Summary and Conclusions  
In the framework of continuum mechanics, assuming that the fluid is single-phase incompressible Newtonian fluid with uniform constant density water, the flow through a marine propeller is studied in physical space by solving general N-S equations and using local dynamic \(k\)-equation subgrid-scale model LES-LDKSGS according to different space-time resolution and dynamic description levels. If the commutation error generated by the differentiation of the filter applied in space can be ignored, then the filtered momentum equation introduces a nonlinear term that can be represented by a subgrid tensor. This nonlinear term can be decomposed, and based on the separability assumption of turbulence at large and small scales, direct simulation calculations can be carried out for anisotropic large scales with high energy in turbulence, while turbulence at small scales is modeled  
Figure 16: Vortex identifier (case 1): (**a**) 3D contour of \(\lambda_{2}=0\). (**b**) Contours of magnitude of vorticity (**c**) Contours of pressure. using the SGS model. The small-scale turbulence perturbations are modeled through eddy viscosity and diffusivity assumptions, which are themselves smoothed out in the flow. Within the applicable scope of the self-similarity hypothesis, when a dynamic \(k\)-equation SGS model is used to close the general N-S equations for flow, the LES-DKSGS model is obtained, and this model does not assume the local balance of SGS energy generation and dissipation, but directly simulates and calculates SGS to account for the nonlocal and historical effects completely ignored by the LES-DASGS model. Therefore, it can be optimistically expected that the LES-DKSGS model will provide better vortex viscosity predictions than the LES-DASGS model. However, in order to avoid numerical instability, the LES-DKSGS model adopts a local averaging technique for model coefficient calculation, which also has certain drawbacks. Using a local dynamic k-equation model (LES-LDKSGS), as no averaging technique is introduced in the calculation of model coefficients, vortex viscosity predictions can be directly calculated locally and does not cause numerical instability. The wake of a five-bladed, right-handed marine propeller with variable pitch, no skew and no rake was studied. There are all 42 simulations included in this paper. From these numerical experiments, we mainly draw the following conclusions:  
1. For mean thrust and torque coefficients LES-LDKSGS (used in our article) results show good agreements with the WT ([PERSON] et al. 2006) results, but less than RANS-kEpsilon results (using the same mesh as in LES-LDKSGS simulation, note there is actually no need of too refinement grid for RANS-kEpsilon simulation) under the same computing cost, which implies that if your concern is only for mean force of the propeller the LES method may be not your best choice considering cost;
2. The results of LES-LDKSGS (used in our article) with a lower computational cost is comparable to the results of LES-DASGS [3], which shows the superiority of the LES-LDKSGS model than the LES-DASGS model to some extent;
3. The evolution mechanism of propeller wake is well captured by the LES-LDKSGS model, including the interaction mechanism between the tip vortex and the trailing edge vortex that dominate the evolution of propeller wake from near to far fields, although this mechanism actually depends on the geometry and operating conditions of the propeller;
4. The vortex CSs of the propeller wake, including the size of the vortex core, are visualized using different criterions to give more insight. All authors contributed to the work. [PERSON] performed the numerical computations and data analysis; [PERSON] guided the whole research project and edited/proof-read the paper. All authors have read and agreed to the published version of the manuscript.