# Application of a Local Dynamic Model of Large Eddy Simulation to a Marine Propeller Wake
## 2 Numerical Experiments
### Governing Equations of Flow
The most accepted theory about turbulence is based on the energy cascade concept developed by [PERSON] [17; 18], according to which turbulence is composed of eddies of different scales. Large eddies break up and transfer their energy to small eddies in a chain process. At the smallest eddy scale, molecular viscosity can effectively dissipate turbulent kinetic energy and convert it into heat. The smallest eddy is characterized by the Kolmogorov micro length scale \(\eta=(\
u^{3}/\varepsilon)^{0.25}\) and time scale \(t_{\eta}=(\
u/\varepsilon)^{0.5}\) with \(\
u\) the molecular kinematic viscosity and \(\varepsilon\) the average rate of dissipation of turbulent kinetic energy (TKE). The largest eddy is proportional to the size of the geometry involved, also known as the integral length scale. Turbulence DNS simulation based on the energy cascade concept must use a very small time step to solve the whole spectrum of turbulence scales contained in time and space. This time step is determined by the Courant number \(Co\), to ensure that \(Co<1\), resulting in a very large number of grid points (proportional to Reynolds number, \(\propto\)\(Re^{3}\), \(dx_{i}<\eta\)). Due to the extremely high and unacceptable computational cost of DNS simulation, it is currently not widely used to solve industrial problems, and this situation may also change in the future with the development of computer technology. To reduce significant computational costs, statistical analysis is used to simplify the resolution of turbulence. The time-dependent nature of turbulence and its wide range of time scales indicate that statistical averaging techniques can be used to approximate random perturbations. At present, the most popular method for solving turbulence in industry is RANS simulation [19], where statistical averaging is based on an appropriate time, and the requirement for the number of grid points is not as high as DNS and LES simulation. With the development of computer technology in recent years, the use of LES to solve industrial problems is becoming increasingly acceptance. The statistical average of LES simulation is different from that of RANS simulation based on spatial average [20]. There are three Reynolds averaging techniques [19], i.e., time averaging (TA), spatial averaging (SA) and ensemble averaging (EA).They are shown as followings respectively  
\[\tilde{\phi}(\mathbf{x})=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{t}^{t+T}\phi( \mathbf{x},t)dt \tag{8}\]  
\[\tilde{\phi}(t)=\lim_{V\rightarrow\infty}\frac{1}{V}\int_{V}\phi(\mathbf{x},t)dV \tag{9}\]  
\[\tilde{\phi}(\mathbf{x},t)=\lim_{N\rightarrow\infty}\frac{1}{N}{\sum_{i=1}^{N}} \phi(\mathbf{x},t) \tag{10}\]  
where, \(N\) is the number of individual experiments. TA is the average of physical quantities over a time period and is suitable for steady-state turbulence. The average value of steady-state turbulence over time period \(T\) does not change over time. SA represents the average of physical quantities on a spatial volume \(V\), suitable for homogeneous turbulence. EA is the overall average over a certain time, applicable to any type of turbulence. EA is the average of many identical quantities at a certain time and is suitable for any type of turbulent flows. If \(T\) is sufficiently large compared to the turbulent perturbation time scale, the Equation (8) can be replaced by  
\[\frac{\partial(\rho u_{i})}{\partial t}+(\rho u_{i}u_{j})_{,j}=\tau_{ij,j}+ \rho f_{i}^{b} \tag{2}\]0  
LES simulation began with the famous [PERSON] model model in the 1960s. Meteorologist [PERSON] wanted to express the influence of three-dimensional subgrid turbulence cascading to small scales mechanism on large-scale synoptic quasi two-dimensionalatmospheric or ocean movement. This mechanism was described by [PERSON] in 1926 and formally proposed by the famous mathematician [PERSON] in 1941. In the 1970s, theoretical physicist [PERSON] developed an important concept of spectral eddy viscosity.