# Application of a Local Dynamic Model of Large Eddy Simulation to a Marine Propeller Wake
## 2 Numerical Experiments
### Geometry and Performance Characteristics of Propeller  
This study used a five bladder, right-handed, variable pitch, non skew, and no rake marine propeller DTMB 4381, as used in Bridges [29]. Cylindrical cross-sections of the blade at \(r/R=0.2\sim 1\) are airfoils as shown in Figure 1, in which \(R\) is the radius of the propeller rotor disc. The distributions of relative chord length (normalized with propeller diameter \(D\)) and twist angle are shown in Figure 2. The hub was tapered. The forces and moments of a specific geometric propeller can be represented by a series of dimensionless quantities [30], including thrust coefficient \(K_{T}\), torque coefficient \(K_{Q}\), advance coefficient (also called advance ratio) \(J\) and cavitation number \(\sigma\), they are  
\[K_{T}=\frac{T}{\rho n^{2}D^{4}},K_{Q}=\frac{Q}{\rho n^{2}D^{5}},J=\frac{V_{a}} {nD},\sigma=\frac{p_{0}-e}{\rho V^{2}/2} \tag{39}\]  
where, \(e\) is water vapor saturation pressure. The dimensional analysis method can be used to analyze geometrically similar propellers. When a marine propeller operates far from the free water surface and does not cause surface waves, the forces acting on it are easily understood to be related to the diameter \(D\), the advance speed \(V_{a}\), the rotational speed \(n\), the density of the fluid \(\rho\), the viscosity of the fluid \(\mu\) and the static pressure of the fluid at the propeller \((p_{0}-e)\). Therefore, the thrust \(T\) on the propeller is proportional to them, i.e., \(T\propto\rho^{k_{1}}D^{k_{2}}V_{a}^{k_{3}}n^{k_{4}}\mu^{k_{5}}(p_{0}-e)^{k_{6}}\). To ensure that the dimensions on both sides of the equation are the same, we can obtain \((ML)/T^{2}=(M/L^{3})^{k_{1}}L^{k_{2}}(L/T)^{k_{3}}(1/T)^{k_{4}}(M/(LT))^{k_{5} }(M/(LT^{2}))^{k_{6}}\), with the mass  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & **Grid Filter Level** & **Test Filter Level** & \\ \hline \multirow{3}{*}{kinetic energy dissipation rate} & filtered & resolved & filtered & resolved \\  & \(k_{1}=\overline{u_{i}u_{i}}\) & \(k_{2}=\overline{u_{i}}u_{i}\) & \(k_{3}=\overline{u_{i}u_{i}}\) & \(k_{4}=\overline{u_{i}u_{i}}\) & \(k_{5}=\overline{u_{i}}\overline{u_{i}}  
\(M\), the length \(L\) and the time \(T\). Compare the left and right ends of this equation to get \(1=k_{1}+k_{5}+k_{6}\) for \(M\), \(1=-3k_{1}+k_{2}+k_{3}-k_{5}-k_{6}\) for \(L\) and \(-2=-k_{3}-k_{4}-k_{5}-2k_{6}\) for \(T\), then we get \(k_{1}=1-k_{5}-k_{6},k_{2}=4-k_{3}-2k_{5}-2k_{6}\) and \(k_{4}=2-k_{3}-k_{5}-2k_{6}\), which used to get \(T\varpi\rho^{(1-k_{5}-k_{6})}D^{(4-k_{3}-2k_{5}-2k_{6})}V_{a}^{k_{3}}n^{(2-k_{3 }-k_{5}-2k_{6})}\mu^{k_{5}}(p_{0}-e)^{k_{6}}\), then \(T\varpi\rho n^{2}D^{4}(V_{a}/(nD))^{k_{3}}\cdot(\mu/(\rho nD^{2}))^{k_{5}}\cdot( (p_{0}-e)/(\rho n^{2}D^{2}))^{k_{6}}\), so  
\[K_{T}=f^{k_{3}}(\frac{1}{Rn})^{k_{5}}\sigma_{0}^{k_{6}}=f(J,Rn,\sigma_{0}),R_ {n}=\frac{\rho nD^{2}}{\mu},\sigma_{0}=\frac{p_{0}-e}{\rho n^{2}D^{4}} \tag{40}\]  
In the same way we also have \(K_{Q}=g(J,Rn,\sigma_{0})\). The open water efficiency of a propeller is defined as the ratio of thrust horsepower to delivered horse power as  
\[\eta_{0}=\frac{TV_{a}}{2\pi nQ}=\frac{K_{T}}{K_{Q}}\frac{J}{2\pi} \tag{41}\]  
Let \(P_{B}\) the brake power or the maximum continuous rating power which is delivered at the engine coupling flywheel, then the shaft power \(P_{S}\) at the output coupling of the gearbox is \(P_{S}=P_{B}\eta_{1}\) with \(96\%\leq\eta_{1}\leq 98\%\). The power transmitted to the propeller and absorbed by the propeller is defined as the increase in kinetic energy per unit time of the slipstream or the work done by the propeller thrust after deducting the loss of bearings, i.e., \(P_{D}=P_{S}\eta_{S}=2\pi nQ\) with \(98\%\leq\eta_{S}\leq 99\%\), in which the shafting mechanical efficiency \(\eta_{S}\) is usually between \(98\%\) and \(99\%\) depending on the length of the shaft and the number of bearings.