# Application of a Local Dynamic Model of Large Eddy Simulation to a Marine Propeller Wake
## 4 Results and Discussion
### PSD of Force  
For any time signal \(x(t)\), this signal can be any physical quantity that changes with time.When analyzing the energy of the signal, it is regarded as the unit resistance applied to the resistance, that is, \(R=1\ \Omega\). The current acts on the resistor. Based on this, the energy attribute of this unit resistance is regarded as the energy attribute of this signal. Therefore, the total energy of this signal is  
\[E=\lim_{T\rightarrow\infty}\int_{-T}^{T}I^{2}Rdt=\lim_{T\rightarrow\infty} \int_{-T}^{T}x^{2}(t)dt \tag{43}\]  
At the same time, energy can also be expressed in the frequency domain as  
\[E=\frac{1}{2\pi}\int_{-\infty}^{\infty}|X(\omega)|^{2}d\omega\quad with\quad X (\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt \tag{44}\]  
Figure 8: Comparison of physical (lines) and numerical (dots) experiments: (**a**) Thrust coefficients (hollow dots represent results of RANS-kEpsilon model, solid dots represent results of LES-LDKSGS model). (**b**) Ten times torque coefficients (hollow dots represent results of RANS-kEpsilon model, solid dots represent results of LES-LDKSGS model). (**c**) Thrust, torque and efficiency (for Case7-16, note Case33-42 have similar results with constant \(n\) and same \(J\) value.  
called the Fourier transform or spectral function of \(x(t)\). And the average power of the signal is  
\[P=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}x^{2}(t)dt \tag{45}\]  
Of course, for different signals, the limits in the above two definitions do not necessarily exist, so energy signals and power signals can be distinguished. In other words, if the first limit exists, it is called an energy signal, and if the second limit exists, it is called a power signal. A signal can be neither an energy signal nor a power signal, but it cannot be both an energy signal and a power signal. In many engineering and technical problems, it is necessary to calculate the power spectrum of a time series, which is defined as the square of the absolute value of the spectral function. The calculation of the power spectrum follows the following steps:  
* Step1: cut off a tail or add some 0s to the original data to make the sequence length \(N=2^{m}\), then the time series \(f(t)\) is given as discrete samples \(f_{j}=f(j\Delta t);j=0,1,\cdots,N-1\);
* Step2: use the appropriate \"data window\" \(g(t/T)\) to correct \(f_{j}\), i.e., \(u_{j}=f_{j}g_{j}\);
* Step3: fast algorithm to calculate the discrete Fourier transform \(U_{k}\) of the sequence of real numbers \(u_{j}\), that is \(U_{k}=\sum_{j=0}^{N-1}u_{j}W_{N}^{-jk};k=0,1,\cdots,N-1\);
* Step4: get power spectrum estimate \(P(k\Delta s)\simeq\frac{2\Delta t}{N}|U_{k}|^{2};k=0,1,\cdots,\frac{N}{2}\) where \(\Delta s=1/T\) and \(T=N\Delta t\).  
The power spectral density (PSD) of \(K_{T}\) is shown in Figure 10.  
Figure 9: Time history of force coefficients for LES-LDKSCS (black line) and RANS-kEpsilon (red line, case 3): (**a**) Thrust coefficient at \(J=0.889\). (**b**) Torque coefficient at \(J=0.889\). (**c**) Pressure force coefficient at \(J=0.889\). (**d**) Viscous force coefficient at \(J=0.889\). (**e**) Thrust coefficient at \(J=3\).