# Application of a Local Dynamic Model of Large Eddy Simulation to a Marine Propeller Wake
## 2 Numerical Experiments  
### Governing Equations of Flow  
Fluid dynamics is controlled by three conservation laws of classical mechanics, namely, the conservation of mass, momentum and energy. Partial differential equations (PDEs) can be derived from these conservation laws and simplified in appropriate cases. These conservation laws are derived under the assumption of continuum mechanics, and we can then express the physical characteristics of flow (including velocity, pressure, density, etc.) as time-dependent scalars or vectors in three-dimensional space \(\mathcal{R}^{3}\), such as pressure \(p(t,\mathbf{x})\) and velocity \(\mathbf{u}(t,\mathbf{x})\). For liquids, such as the water used in this article, the continuity assumption always holds in practice.Flow can be represented in the Eulerian formulation as a time history of flow properties at each fixed point in a space domain. According to three laws of conservation, the continuity, momentum and energy equations in an arbitrary material volume \(\mathbf{V}(t)\) with surface \(\mathbf{S}(t)\) are derived [13], they (totally called general N-S equations) are as follows, herein [PERSON] summation convention (1916) is used. \[\frac{\partial\rho}{dt}+(\rho u_{i})_{,i}=0 \tag{1}\]  
\[\frac{\partial(\rho u_{i})}{\partial t}+(\rho u_{i}u_{j})_{,j}=\tau_{ij,j}+ \rho f_{i}^{b} \tag{2}\]  
\[\frac{\partial(\rho E)}{\partial t}+(\rho u_{i}E)_{,i}=(u_{i}\tau_{ij})_{,j}+ \rho u_{i}f_{i}^{b}+\rho q+(kT_{,i})_{,i} \tag{3}\]  
where, \(\mathbf{\tau}\) is the stress tensor, \(f^{b}\) is the body force per mass acting on a material particle, \(E=e+0.5u_{i}u_{i}\) is the total energy per unit mass, \(e\) is the internal energy per mass, \(0.5u_{i}u_{i}\) is the kinetic energy per mass of a material particle, \(q\) is the rate of heat added to each material particle per mass. If \(s\) is the heat flux per area through the surface \(\mathbf{S}(t)\), heat diffusion is governed by [PERSON]'s law, i.e., \(s=k\mathbf{n}\cdot\text{grad}\,T\), wherein \(k\) is the thermal conductivity and \(T\) is the temperature. Assuming Newtonian fluid, we can get the simplest constitutive relation, that is, the relationship between stress tensor and fluid motion [14]  
\[\tau_{ij}=-p\delta_{ij}+2\mu(e_{ij}-\frac{1}{3}\Delta\delta_{ij}) \tag{4}\]  
where, \(e_{ij}=0.5(u_{i,j}+u_{j,i})\), \(\Delta=e_{ii}=div\mathbf{u}\), \(p\) is the pressure, \(\delta_{ij}\) is the Kronecker delta, \(\mu\) is the dynamic molecular-viscosity coefficient, \(e_{ij}\) is the rate of strain tensor. When dealing with continuous medium Newtonian fluid, the influence of molecular fluctuation can be smoothed out through viscosity and represented by \(\mu\). To take assumptions that flow is incompressible (Mach number \(Ma=|\mathbf{u}|/U_{sound}\ll 1\), \(U_{sound}\sim\)1.4 km/s in water at 15 \({}^{\circ}\)C, depending on the amount of dissolved air), and the density of the water is constant, i.e., \(\rho_{w}=const. (\rho_{w}=10^{3}\) kg/m\({}^{3}\), \(t=4\) \({}^{\circ}\)C). Equations (1)-(3) can be simplified as  
\[u_{i,i}=0 \tag{5}\]  
\[\rho\frac{\partial u_{i}}{\partial t}=-p_{,i}+\mu u_{i,ij}+\rho f_{i}^{b} \tag{6}\]  
\[\rho\frac{\partial E}{\partial t}=(u_{i}\tau_{ij})_{,j}+\rho u_{i}f_{i}^{b}+ \rho q+(kT_{,i})_{,i} \tag{7}\]  
Equations (6) was first derived by [PERSON] (1823, Frenchman), [PERSON] (1831), [PERSON] (1843) and [PERSON] (1845, Englishman). Since meteorologist [PERSON], numerical schemes have allowed us to solve motion equations in a deterministic manner after providing initial conditions and specifying boundary conditions [15]. When deriving the general N-S equations, it was not assumed that the flow was laminar or turbulent, but laminar flow was stable, and turbulence was chaotic and diffusive, leading to rapid mixing, time-dependent, and three-dimensional (3D) vorticity perturbations with a broad scale range of time and length [16].