# Application of a Local Dynamic Model of Large Eddy Simulation to a Marine Propeller Wake
## 4 Results and Discussion
### Vortex Structures  
Since the coherent structures (CS) in turbulence has been widely regarded as vortices, the question of what constitutes vortices in turbulence has become meaningful. The objective definition of vortices should allow the use of the concept of vortex dynamics to induce CS, and be able to use CS to explain the formation and evolution of vortices in turbulent phenomena. CS can also be used to develop various turbulence models and control strategies for turbulent phenomena [36]. Intuitively, a vortex is considered a flow tube, with its surface composed of vortex lines [37]. However, the existence of vortex tubes cannot represent the existence of vortices, such as vortex tubes in laminar pipe flow, which are not vortices in any sense. [PERSON] (1979) defined a vortex as multiple particles of matter rotating around a common center, [PERSON] et al. (1990) as a region of complex eigenvalues of \(\
abla u\), [PERSON] et al. (1988) as a region containing both a positive second invariant of \(\
abla u\) and low pressure, and [PERSON] and [PERSON] [36] defined it in terms of the eigenvalues of the symmetric tensor (\(S^{2}+\Omega^{2}\)), in which \(S\) and \(\Omega\) are symmetrical and asymmetrical parts of the velocity gradient tensor \(\
abla u\). [PERSON] (1991) used the DNS database in 1991 to draw a conclusion that the low-pressure criterion can effectively capture the vortex structure in the turbulent boundary layer, but in more general cases, it is not possible to clearly give a certain pressure level to identify all vortex regions in the flow. Therefore, a single pressure threshold cannot reveal all phenomena. The methods of vortex identification available are listed in Table 5.  
The size of vortices in viscous fluids depends on the selected standard threshold. We can identify the vortex core, and the necessary condition for its existence is that there is a net vorticity and the geometry of the vortex core is Galilean invariant. These two necessary conditions cannot lead to a definite identification method, and can only serve as a prerequisite for the possible existence of vortices. [PERSON] and [PERSON] [36] attempted to develop an objective method for identifying vortex core in any flow. Figures 15 and 16 show the CS of vortices in propeller wake under different criterions. The iso-contour of \(\lambda_{2}=0\) is shown in Figure 15. Since it is a 5-blade propeller, Figure 15 only shows the instantaneous planes of 8 successive time units within one phase angle (\(72^{\circ}\)) respectively from top to bottom. The vortex area angle on the blade surface has been marked red on the figure, and the angle increases first and then decreases. Other phase angles have similar results and will not be repeated for simplicity. It can be clearly seen that the interaction between tip vortex helix and trailing edge vortex becomes blurred after about \(1D\) downstream of the propeller. After \(2D\), hub vortex, blade tip vortex and trailing edge vortex meet to further blur the wake. In the near field of wake, some vortex cores with small radius are mainly formed, and the diameter of vortex core increases in the far field. The vortex core radius at the center of the propeller shaft increases more downstream. Figure 16a shows a 3D axis side view of \(\lambda_{2}=0\), and Figure 16b,c are iso-contours of vorticity magnitude and pressure. It is obvious that the pressure criterion is insufficient.  
\begin{table}
\begin{tabular}{c c} \hline
**No.** & **Criterion for Vortex Identification** \\ \hline
1 & Local pressure minimum \\
2 & Pathline and streamline \\
3 & Vorticity magnitude \\
4 & Existence of complex eigenvalues of \(\
abla u\) \\
5 & Second invariant \(Q\) of \(\
abla u\) and kinematic vorticity number \(N_{k}\) \\
6 & Eigenvalues (\(\lambda_{1}>\lambda_{2}>\lambda_{3}\)) of (\(\mathsf{S}^{2}+\Omega^{2}\)) \\ \hline \end{tabular}
\end{table}
Table 5: Vortex identification technology.  
Figure 15: Vortex identifier (case 1): one phase vortex core from top to bottom and from up to down.