# Application of a Local Dynamic Model of Large Eddy Simulation to a Marine Propeller Wake
## 2 Numerical Experiments
### LES Models
[5], in which two filters are used to directly solve the model coefficient \(c_{S}^{2}\), namely a grid-scale filter \(\bar{\Lambda}\) and a test filter \(\bar{\Lambda}\) (typically, \(\bar{\Lambda}=2\bar{\Lambda}\)). By applying the test filter, we can obtain  
\[T_{ij}=\overline{\overline{u_{i}u_{j}}}-\bar{u}_{i}\bar{u}_{j} \tag{20}\]  
\[L_{ij}=\overline{\bar{u_{i}u_{j}}}-\bar{u}_{i}\bar{u}_{j} \tag{21}\]  
Assuming that the subgrid stress has self-similarity, we can model \(T_{ij}\) in the same way as modeling \(\tau_{ij}\), that is  
\[T_{ij}=-2\
u_{SGS}\bar{S}_{ij}+\frac{1}{3}\delta_{ij}T_{kk} \tag{22}\]  
with \(\
u_{SGS}=c_{S}^{2}\bar{\Lambda}^{2}|\bar{S}|\), and to combine Equations (19)-(22), we can get an equation for \(c_{S}^{2}\) as  
\[L_{ij}-\frac{1}{3}\delta_{ij}L_{kk}=2c_{S}^{2}M_{ij} \tag{23}\]  
where, \(M_{ij}=\bar{\Lambda}^{2}|\overline{|\bar{S}|\bar{S}_{ij}}-\bar{\Lambda}^{2}| \bar{S}|\bar{S}_{ij}\). Equation (23) for solving one unknown \(c_{S}^{2}\) is an overdetermined equation system. To minimize the error, [PERSON] [6] proposed a least squares method to solve it, which can obtain  
\[\frac{\partial\bar{u}_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}(\overline {u_{i}u_{j}})=-\frac{\partial\bar{p}}{\partial x_{i}}+v\frac{\partial}{ \partial x_{j}}(\frac{\partial\bar{u}_{i}}{\partial x_{j}}+\frac{\partial\bar {u}_{j}}{\partial x_{i}}) \tag{15}\]0  
The dynamic modelling combined with Smagorinsky model is named dynamic algebraic SGS model (LES-DASGS), which can also be called dynamic Smagorinsky model (LES-DSM). The calculation of coefficient \(c_{S}^{2}\) can lead to numerical instability, as the value of the denominator in Equation (24) may become very small at certain points in the flow, and there may be persistent negative values of \(c_{\xi}^{2}\) at some locations. This problem can be solved by taking the local mean of the model coefficients along the direction of homogeneous flow [5]. [PERSON] and et al. [22] proposed an average technique along particle trajectories. [PERSON] and [PERSON] [23] developed another local averaging technique based on the local structure of vorticity. Whether based on direction of homogeneity or local structures, the local averaging technique is an artifact (without any physical justification) to avoid numerical instability and isn't consistent with the dynamic procedure. A true dynamic model should solve the model coefficients locally without using any averaging techniques. However, research by [PERSON] and et al. [24] has shown that when solving flows with homogeneous directions, the results obtained using averaging techniques are the same as those obtained using more rigorous dynamic models, but the computational cost is lower. [PERSON] and et al. [24] also proposed another local dynamic model for inhomogeneous flows. Applying a dynamic \(k\)-equation subgrid-scale model belonging to subgrid viscosity models, we can obtain the LES-DKSGS model, that is [25; 26; 27]  
\[\frac{\partial\bar{u}_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}(\overline {u_{i}u_{j}})=-\frac{\partial\bar{p}}{\partial x_{i}}+v\frac{\partial}{ \partial x_{j}}(\frac{\partial\bar{u}_{i}}{\partial x_{j}}+\frac{\partial\bar {u}_{j}}{\partial x_{i}}) \tag{15}\]1  
where, \(k_{SGS}=\frac{1}{2}(\overline{u_{i}^{2}}-\bar{u}_{i}^{2})\) is the SGS kinetic energy. The three terms to the right of Equation (25) are the production rate, the dissipation rate and the transport rate of \(k_{SGS}\) respectively. SGS tensor \(\tau_{ij}\) can be modeled as  
\[\frac{\partial\bar{u}_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}(\overline {u_{i}u_{j}})=-\frac{\partial\bar{p}}{\partial x_{i}}+v\frac{\partial}{ \partial x_{j}}(\frac{\partial\bar{u}_{i}}{\partial x_{j}}+\frac{\partial\bar {u}_{j}}{\partial x_{i}}) \tag{15}\]2  
where, if the cut-off length of the filter is within the applicable range of the scale similarity assumption, there are \(v_{SGS}=c_{v}k_{SGS}^{0.5}\bar{\Delta}\) and \(\varepsilon=c_{\xi}k_{SGS}^{1.5}/\bar{\Delta}\).