# Application of a Local Dynamic Model of Large Eddy Simulation to a Marine Propeller Wake
## 2 Numerical Experiments
### LES Models
The establishment of this model does not assume a local balance between SGS energy generation and dissipation, but instead directly calculates SGS. Therefore, the model can consider non-local and temporal effects,which are ignored in LES-DASGS model. It is optimistic to expect LES-DKSGS model to provide better calculation results than LES-DASGS model. Using the same procedure, the equation for \(c_{v}\) can be derived as  
\[\frac{\partial\bar{u}_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}(\overline {u_{i}u_{j}})=-\frac{\partial\bar{p}}{\partial x_{i}}+v\frac{\partial}{ \partial x_{j}}(\frac{\partial\bar{u}_{i}}{\partial x_{j}}+\frac{\partial\bar {u}_{j}}{\partial x_{i}}) \tag{15}\]3  
with \(M_{ij}=\bar{\Delta}(\widetilde{k_{SGS}^{0.5}\bar{S}_{ij}})-\bar{\Delta}K^{0. 5}\bar{S}_{ij}\) and \(K=\frac{1}{2}L_{ij}+\bar{k}_{SGS}\). The TKE \(K\) at the test filter level is obtained from trace of Equation (21). Equation (27) has the same form as the Equation (23), so \(c_{v}\) is also determined using the least-square method, that is  
\[\frac{\partial\bar{u}_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}(\overline {u_{i}u_{j}})=-\frac{\partial\bar{p}}{\partial x_{i}}+v\frac{\partial}{ \partial x_{j}}(\frac{\partial\bar{u}_{i}}{\partial x_{j}}+\frac{\partial\bar {u}_{j}}{\partial x_{i}}) \tag{15}\]4  
The following equation can be obtained based on the relationship between the dissipation rate between the grid scale filter (\(\varepsilon\)) and the test filter (\(\mathcal{E}\))  
\[\frac{\partial\bar{u}_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}(\overline {u_{i}u_{j}})=-\frac{\partial\bar{p}}{\partial x_{i}}+v\frac{\partial}{ \partial x_{j}}(\frac{\partial\bar{u}_{i}}{\partial x_{j}}+\frac{\partial\bar {u}_{j}}{\partial x_{i}}) \tag{15}\]5  
with \(\varepsilon=v(\frac{\partial\bar{u}_{i}}{\partial x_{j}}\frac{\partial\bar{ u}_{i}}{\partial x_{j}}-\frac{\partial\bar{u}_{i}}{\partial x_{j}}\frac{ \partial\bar{u}_{j}}{\partial x_{j}})\) and \(\mathcal{E}=v(\frac{\partial\bar{u}_{i}}{\partial x_{j}}\frac{\partial\bar{u}_ {i}}{\partial x_{j}}-\frac{\partial\bar{u}_{i}}{\partial x_{j}}\frac{\partial \bar{u}_{j}}{\partial x_{j}})\). Then, we can further obtain  
\[\frac{\partial\bar{u}_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}(\overline {u_{i}u_{j}})=-\frac{\partial\bar{p}}{\partial x_{i}}+v\frac{\partial}{ \partial x_{j}}(\frac{\partial\bar{u}_{i}}{\partial x_{j}}+\frac{\partial\bar {u}_{j}}{\partial x_{i}}) \tag{15}\]6  
with \(G=K^{1.5}/\bar{\Delta}-(\widetilde{k_{SGS}^{1.5}/\bar{\Delta}})\). The Equation (30) is a scalar equation for one unknown \(c_{\varepsilon}\) and can be solved directly as \(c_{\varepsilon}=\mathcal{F}/G\). The model coefficients in this LES-DKSGSmodel is also evaluated by using a local averaging technique to avoid numerical instability and share the same defects as LES-DASGS. To address this issue, [PERSON] and [PERSON] [23] proposed a local dynamic \(k\)-equation subgrid-scale (LES-LDKSGS) model where the model coefficients are calculated locally without using any averaging techniques or causing any numerical instability. As shown in Table 1, we consider scales with different energy levels. The filtered kinetic energy characteristic length scale \(\lambda\) is unknown and is related to the dissipation rate, and there is a relationship \(\lambda<\bar{\Lambda}\) due to relationship \(\overline{u_{i}u_{i}}>\bar{u_{i}}\bar{u_{i}}\). Term \(\overline{u_{i}u_{i}}\) contains SGS kinetic energy, while \(\bar{\Lambda}\) is related to the rate of SGS kinetic energy generation, and there is a clear separation between the scales of SGS kinetic energy generation and dissipation. To accurately model the generation and dissipation of kinetic energy in SGS, additional information on energy transfer at these two length scales must be provided, but this information is unknown. Therefore, we further assume that the energy transfer that occurs on a small scale is determined by the strain rate on a large scale and requires bidirectional energy transfer.