## 3 Results and Discussion
### Extraction and Analysis of Streak Structures  
In turbulent boundary layers, streak structures are generally clearly present in the region where \(y^{+}<30\), and vortex structures mainly exist in the form of quasi-streamwise vortices in the near-wall region where \(y^{+}<60\). In the region where \(40<y^{+}<100\), they will elevate to become hairpin vortices [8]. Therefore, self-sustaining mechanisms typically occur in the region where \(y^{+}<60\). In early visualizations of wall-parallel flow, the contribution of streamwise vortices and streak structures to the spanwise spectrum of the streamwise fluctuating velocity is equal [48]. Therefore, the study of streak structures is also one of the key points in exploring turbulent structures within the boundary layer. In order to observe the difference in streak structures in different near-wall areas, a plane with clear streak structures (\(y^{+}=8.8\)) was selected for shooting. To better observe the streak structures in the turbulent boundary layers of different wall surfaces, the Proper Orthogonal Decomposition (POD) method was utilized to extract the highest energy-containing low-order modes for each condition and compare them. The Proper Orthogonal Decomposition, introduced by [PERSON] [49], is a mathematical technique that can extract coherent structures from flow fields. The principle involves decomposing the original data into multiple spatial modes (called eigenmodes) and the corresponding time evolution coefficients (sequences) for each mode, which are mutually orthogonal. The order of the modes is sorted according to the energy they capture, from high to low, with the energy being given by the eigenvalues corresponding to the eigenmodes. The expansion in these eigenmodes optimally captures the energy of the data. Modal decomposition techniques provide a powerful means of identifying an effective low-dimensional coordinate system for capturing the dominant flow mechanisms, with the reduction in system order corresponding to the selection of an appropriate (reduced basis) coordinate system to represent fluid flow [50]. Consider the velocity field as a superposition of POD modes:  
\[u(t,x)=\sum\limits_{j=1}^{N}a_{j}(t)\varphi_{j}(x) \tag{4}\]  
Here, \(N\), \(a_{j}(t)\) and \(\varphi_{j}(x)\) represent the total number of snapshot modes, the time coefficient of the \(j\)-th mode, and the modal characteristic value, respectively. This section uses  
Figure 5: (a) Distribution of the integrated turbulence intensity; (b) distribution of the Reynolds shear stress. the POD method to perform modal decomposition on 4000 instantaneous flow fields for each wall, and analyzes the first five modes under two flow fields for the four walls. Figure 6 shows the energy distribution of POD modes under two flow fields. The lower the order, the higher the energy content ratio. The first-order mode with the highest energy content ratio is the main flow structure. In the clean water flow field, there is a significant difference in the first-order modes between different walls. Starting from the second order, the energy content ratio gradually approaches each other, and by the fifth order, the energy content ratios of the four walls are very close. Therefore, the riblet walls mainly affect the first-order mode with the highest energy content ratio. Among them, the 30\({}^{\circ}\) convergent riblet wall increases the energy content ratio of the first-order mode, while the streamwise riblet and the 30\({}^{\circ}\) convergent riblet walls successively decrease it. This reflects that the drag-reduction riblet walls can effectively suppress the development of large-scale structures, and the decrease in the energy content ratio means that its influence on the flow field is weakened. In the two-phase flow field, the energy content ratios of the first three modes for the four walls are all significantly reduced, and the differences between them become smaller.