# Application of a Local Dynamic Model of Large Eddy Simulation to a Marine Propeller Wake
## 2 Numerical Experiments
### LES Models  
Based on different levels of spatial-temporal resolution and dynamic description, LES simulation is directly applied to solve low-frequency spatial modes. The constitutive equations of LES simulation, i.e., the filtered N-S equations, are as follows  
\[\frac{\partial\bar{u}_{i}}{\partial x_{i}}=0;i=1,2,3 \tag{14}\]  
\[\frac{\partial\bar{u}_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}(\overline {u_{i}u_{j}})=-\frac{\partial\bar{p}}{\partial x_{i}}+v\frac{\partial}{ \partial x_{j}}(\frac{\partial\bar{u}_{i}}{\partial x_{j}}+\frac{\partial\bar {u}_{j}}{\partial x_{i}}) \tag{15}\]  
where, \(\bar{p}\) is the filtered pressure. The filtered momentum equation introduces a nonlinear term \(\overline{u_{i}u_{j}}\) which can be represented by a function of \(\overline{u}\) and \(\mathbf{u}^{\prime}\), and then we obtain the nonlinear term:  
\[\overline{u_{i}u_{j}}=\overline{(\bar{u}_{i}+u_{i}^{\prime})(\bar{u}_{j}+u_{ j}^{\prime})}=\overline{\bar{u}_{i}\bar{u}_{j}}+\overline{\bar{u}_{i}u_{j}^{ \prime}}+\overline{\bar{u}_{j}u_{i}^{\prime}}+\overline{\bar{u}_{i}^{\prime} u_{j}^{\prime}} \tag{16}\]  
where, \(\mathcal{C}_{ij}=\overline{\bar{u}_{i}u_{j}^{\prime}}+\overline{\bar{u}_{i}u_ {j}^{\prime}}\) is the cross-stress tensor representing the interactions between large and small scales, and \(\mathcal{R}_{ij}=\overline{u_{i}^{\prime}u_{j}^{\prime}}\) is the Reynolds subgrid tensor reflecting the interactions between SGS. The \(\overline{\bar{u}_{i}\bar{u}_{j}}\) term can further be decomposed as \(\overline{\bar{u}_{i}\bar{u}_{j}}=(\overline{\bar{u}_{i}\bar{u}_{j}}-\bar{u}_ {i}\bar{u}_{j})+\bar{u}_{i}\bar{u}_{j}=\mathcal{L}_{ij}+\bar{u}_{i}\bar{u}_{j}\). The newly introduced Leonard tensor \(\mathcal{L}\) represents large-scale relationships. If the filter commutation error under spatial differentiation can be ignored, the filtered momentum equation can be expressed as:  
\[\frac{\partial\bar{u}_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}(\bar{u}_{ i}\bar{u}_{j})=-\frac{\partial\bar{p}}{\partial x_{i}}+\
u\frac{\partial}{ \partial x_{j}}(\frac{\partial\bar{u}_{i}}{\partial x_{j}}+\frac{\partial\bar{ u}_{j}}{\partial x_{i}})-\frac{\partial\tau_{ij}}{\partial x_{j}};i=1,2,3 \tag{17}\]  
where, \(\mathbf{\tau}\) is the subgrid tensor defined as  
\[\tau_{ij}=\mathcal{L}_{ij}+\mathcal{C}_{ij}+\mathcal{R}_{ij}=\overline{u_{i}u _{j}}-\bar{u}_{i}\bar{u}_{j} \tag{18}\]  
which is the famous Leonard or triple decomposition. LES simulation is based on the separation of large and small turbulent scales, and anisotropic large scales containing higher energy can be directly simulated and calculated, while small scales must be modeled using subgrid scale models. Turbulent small-scale fluctuations are modeled through eddy viscosity and diffusivity assumption. The scale greater than the cut-off length of the filter is called large or resolved scale, while others are called small or subgrid scale (SGS). The SGS model is a statistical subgrid model, and precise modeling of SGS is possible because small scales smaller than the filter cutoff length can be considered isotropic and do not depend on the type of flow and boundary conditions. Large eddy simulation is the decomposition of the nonlinear term in the above equation, and then modeling the unknown term. Various modeling strategies and models already exist, and the most popular SGS model is the algebraic eddy viscosity model, proposed by [PERSON] [21], i.e.,  
\[\tau_{ij}=-2\
u_{SGS}\bar{S}_{ij}+\frac{1}{3}\delta_{ij}\tau_{kk} \tag{19}\]  
where, \(\
u_{SGS}=c_{S}^{2}\bar{\Lambda}^{2}|\bar{S}|\) is the SGS eddy viscosity, \(c_{S}\) is the Smagorinsky constant, \(\bar{S}_{ij}=\frac{1}{2}(\partial\bar{u}_{i}/\partial x_{j}+\partial\bar{u}_{ j}/\partial x_{i})\) is the resolved strain rate tensor with \(|\bar{S}|=(2\bar{S}_{ij}\bar{S}_{ij})^{0.5}\). The constant \(c_{S}\) must be adjusted for different flows. This inconvenience can be skipped by using a dynamic procedure proposed by [PERSON] and et al.