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**Figure 16.** Node distributions for Hybrid-mesh.

#### 5.3.1. Impacts of Near-Field Node Distributions

In this part, impacts of near-field node distributions are studied by changing chordwise spacings and layer-wise growth rates for C-mesh, H-mesh, O-mesh, and Hybrid-mesh. The near-chord refinement is performed by changing the number of nodes in the chordwise direction for structured meshes and chord-wise spacings for Hybrid-mesh with *RG* around <sup>√</sup>2. For structured meshes, a reference setting is coarsened or refined to perform parametric studies. The quantitative relation between a coarsened or refined case and the reference case can be expressed as:

$$\log\_{R\_G} \frac{N\_{c\_i}}{N\_{c\_0}} = \log\_{R\_G} N\_{c\_{i0}} = i, \quad (i = -2, -1, \dots) \tag{27}$$

where *i* denotes the level of refinement with respect to the reference case, and *Nci* is the number of nodes along the chord. Taking C-mesh for example, the rudder is wrapped by denser cells in chord-wise directions, as shown in Figure 13. As for cells in the boundary layer region of the rudder, much attention should be paid to capture violently changing velocity gradients near the wall. The layer-wise growth rate is investigated to provide a detailed view of mesh resolutions in boundary layer regions.

Cells around the rudder with varying chord-wise spacings for C-mesh are shown in Figure 17. As the chord-wise spacing decreases in Table 7, the number of cells with large aspect ratios in the wake increases, which introduces the difficulty of convergence, especially under large angles of attack. The accuracy of *CD* is slightly improved with denser chord-wise node distributions, while variations of *CL* are even smaller. Larger deviations are obtained for *CM* for small *α*, probably due to measurement errors in experiments induced by small absolute values. For *α* = 4.18◦ and *α* = 8.22◦, better mesh resolution along the rudder lead to lower precision, which are related to convergence problems due to the fact that small flow features are being solved. Since calculation results tend to stabilize without diffusing after log*RG Nci*<sup>0</sup> <sup>=</sup> 2 and 5.48 <sup>×</sup> 105 mesh quadrilaterals show relatively high accuracy, the case with log*RG Nci*<sup>0</sup> = 3 is selected for the following simulations on C-mesh, and corresponded uncertainty estimations are shown in Table 8.


**Table 7.**

Comparison

 of results of C-mesh based on various chord-wise

 spacings with that of

experimental

 benchmark

 data adopted from [25] in relative differences

**Figure 17.** Cells around the rudder with varying chord-wise spacings for C-mesh.

**Table 8.** Uncertainty estimations for C-mesh (log*RG Nci*<sup>0</sup> = 2, 3, 4, 5, *α* = 15.2◦) with varying chordwise spacings based on mesh dependency.


For H-mesh, besides chord-wise nodes, nodes distributed along with radial directions in the refinement region also count in near-field node distributions. Therefore, chord-wise nodes and radial nodes near the rudder are both refined for H-mesh. Due to discrepancies in refinement regions in Table 9, the NOC of H has a more rapid growth rate compared with C-mesh. Generally, variations of the accuracy of hydrodynamic coefficients are consistent with the increase in the number of cells, and relatively greater improvements can be observed for *CD* and *CM* taking log*RG Nci*<sup>0</sup> = 2 as the dividing line. For the NOC less than 3.00 × <sup>10</sup>5, C-mesh is a better choice for predictions of *<sup>α</sup>* < 8.22◦ when C-mesh and H-mesh share the same order of magnitude. Considering that near-field node distributions of H-mesh are able to capture more precise flow details around the rudder with denser nearfield node distributions, H-mesh with fine mesh resolutions can be applied with enough computational resources. In this part, to balance the accuracy and efficiency, log*RG Nci*<sup>0</sup> = <sup>2</sup> is selected for following simulations on H-mesh, and corresponded uncertainty estimations are shown in Table 10.


**Table 9.**

Comparison

 of results of H-mesh based on various chord-wise

 spacings with that of

experimental

 benchmark

 data adopted from [25] in relative differences


**Table 10.** Uncertainty estimations for H-mesh (log*RG Nci*<sup>0</sup> = 1, 2, 3, 4, *α* = 15.2◦) with varying chord-wise spacings based on mesh dependency.

For O-mesh in Table 11, the errors of *CL* and *CD* are larger than C and H-mesh because the mesh type fails to capture the wake after the trailing edge. Further, variations of layer-wise growth rates are incapable of improving the situation. Therefore, from the perspective of efficiency, further investigations are not carried out on O-mesh considering its obvious defects.

Various chord-wise spacings are applied for Hybrid-mesh, and the results are in Table 12. Compared with that of structured meshes, the calculation accuracy of *CL* is as good as C and H-mesh, but that of *CD* is worse than C and H-mesh while still better than O-mesh. *CL* tends to be stable when the NOC is more than 2.5 × <sup>10</sup>5, and *CD* results are gradually improved with smaller chord-wise spacings. In the following part, a chord-wise spacing of 2.23 × <sup>10</sup>−<sup>4</sup> *<sup>c</sup>* is selected, and validation for *CL* is achieved as indicated in Table 13.



**Table 12.** Comparison of results of Hybrid-mesh based on various chord-wise spacings with that of experimental benchmark data adopted from [25] in relative differences.



**Table 13.** Uncertainty estimations for Hybrid-mesh (Δ*<sup>c</sup>* <sup>=</sup> 5.88 <sup>×</sup> <sup>10</sup>−<sup>4</sup> *<sup>c</sup>*, 4.46 <sup>×</sup> <sup>10</sup>−<sup>4</sup> *<sup>c</sup>*, 3.15 <sup>×</sup> <sup>10</sup>−<sup>4</sup> *<sup>c</sup>*, 2.23 <sup>×</sup> <sup>10</sup>−<sup>4</sup> *<sup>c</sup>*, *<sup>α</sup>* <sup>=</sup> 15.2◦) with varying chord-wise spacings based on mesh dependency.

No obvious variations can be observed for both *CL*, *CD* and *CM* in Table 14. The insensitivity to layer-wise growth rates for C-mesh is probably caused by good chord-wise mesh resolutions. A rate of 1.15 encounters convergence problems, while results obtained from *r* -1.10 are stable. Therefore, *r* = 1.10 is selected.

For H-mesh in Table 15, the accuracy of *CL* and *CD* has subtle improvements within 1%, while variability of *CM* is slightly larger. The trend shows that a slower growth rate of mesh height leads to better results, but, remarkably, too small rates may reduce calculation efficiencies and contribute to mesh quadrilaterals with too large aspect ratios, which are disadvantageous to the convergence of CFD results. In this case, *r* = 1.05 is well balanced in accuracy and computation time.

For Hybrid-mesh in Table 16, the connection between the structured region and the unstructured region is influenced by layer-wise growth rates. When *r* = 1.03, relatively large aspect ratios in boundary layer regions increases the difficulty of convergence. Therefore, *r* = 1.10 is selected as a value of transition of convergent conditions.

#### 5.3.2. Impacts of Far-Field Node Distributions

With selected near-field node distributions, nodes in far-field regions need to be determined to match variations of cell sizes near wall boundaries to present global meshing strategies. For structured meshes, similar to Section 5.3.1, nodes in different directions are changed according to characters of their topological structures, which can be expressed as:

$$\log\_{\mathbb{R}\_{\mathcal{G}}} \frac{N\_{r\_i}}{N\_{r\_0}} = \log\_{\mathbb{R}\_{\mathcal{G}}} N\_{r\_{i0}} = i, \quad (i = -2, -1, \dots, 4) \tag{28}$$

$$\log\_{R\_G} \frac{N\_{w\_i}}{N\_{w\_0}} = \log\_{R\_G} N\_{w\_{i0}} = i, \quad (i = -2, -1, \dots, 4) \tag{29}$$

where *Nri* and *Nwi* denote the number of nodes in radial and wake directions. For Hybridmesh, cell sizes in far-field regions are varied to change mesh properties.





**Table 16.** Comparison of results of Hybrid-mesh based on various layer-wise growth rates with that of experimental benchmark data adopted from [25] in relative differences.


Number of cells against radial nodes vary over a considerable extent from nearly 3.00 × 105 to more than 1.00 × <sup>10</sup><sup>6</sup> for C-mesh and H-mesh, as shown in Tables <sup>17</sup> and 18. However, such variation has few impacts on magnitudes of any hydrodynamic coefficients, indicating that sufficient mesh resolution along the rudder chord has successfully solved the flow in regions with great gradients. In other words, relatively few nodes in radial directions are capable of ensuring the calculation accuracy. Therefore, the reference case already achieved mesh independence for both C-mesh and H-mesh along with radial directions, and the case is still utilized for benchmark in the following studies.

Similar to impacts of nodes in radial directions, variations of *CL*, *CD* and *CM* against changing cell numbers induced by cell sizes in wake directions are very small, as shown in Tables 19 and 20. Compared with the trend of accuracy variations in Section 5.3.1, impacts of node distributions in the near-field on CFD results are stronger than that in the far-field, which is reasonable considering the decaying disturbance of the object on the flow field with increased distances from the wall boundary. As long as mesh resolutions in the near-field are handled carefully, fewer mesh elements are needed in the far-field. Mesh elements in unstructured regions for Hybrid-mesh are changed as shown in Table 21, and the magnitude of results variations are the same as C-mesh and H-mesh. Further decreasing element sizes of unstructured meshes are memory-consuming, and divergent cases occur when the size is below 0.0625 *c*. Therefore, the body element size of 0.125 *c* is selected.

Tables 22–26 show uncertainty estimations for C-mesh, H-mesh and Hybrid-mesh with varying far-field node distributions. Most uncertainty quantities in Tables 22–25 are relatively small, corresponding to the fact that further increase in radial or wake directions has little impact on the calculation accuracy.



**Table 18.** Comparison of results of H-mesh based on various radial nodes to experimental benchmark data adopted from [25] in relative differences with 150 nodes in radialdirectionsforreferencesetting.

 *E***)**


**Table 19.** Comparison of results of C-mesh based on various wake nodes with that of experimental benchmark data adopted from [25] in relative differences with 300 nodesinwakedirectionsforreferencesetting.




**Table 21.** Comparison of the results of Hybrid-mesh based on various element sizes with that of experimental benchmark data adopted from [25] in relative differences.



**Table 22.** Uncertainty estimations for C-mesh with varying radial node distributions (log*RG Nci*<sup>0</sup> = −2, −1, 0, 1) based on mesh dependency.

**Table 23.** Uncertainty estimations for H-mesh with varying radial node distributions (log*RG Nci*<sup>0</sup> = −2, −1, 0, 1) based on mesh dependency.


**Table 24.** Uncertainty estimations for C-mesh with varying wake node distributions (log*RG Nci*<sup>0</sup> = −2, −1, 0, 1) based on mesh dependency.


**Table 25.** Uncertainty estimations for H-mesh with varying wake node distributions (log*RG Nci*<sup>0</sup> = −2, −1, 0, 1) based on mesh dependency.


**Table 26.** Uncertainty estimations for Hybrid-mesh (Element size = 0.17678 *c*, 0.12500 *c*, 0.08839 *c*, 0.06250 *c*, *α* = 15.2◦) with varying far-field node distributions based on mesh dependency.


#### *5.4. Impacts of Mesh Types*

Simulations of different mesh types with varying mesh properties are conducted in Sections 5.1–5.3. Generally, the accuracy of *CL* is sufficient (within 5% deviations compared with experimental data) for engineering applications except for O-mesh, while discrepancies of *CD* between CFD and EFD results are relatively large under high angles of attack. Physically, the lift and drag coefficient are nondimensional forms of lift and drag force, respectively, while the former is mainly generated by the pressure difference between the windward and leeward surfaces, the inviscid component, and the latter primarily contributed by the frictional shear stress, the viscous component. The precise prediction of viscous stress relies on a more exact forecasting model in regions of separated flow, along with an advanced understanding of rudder hydrodynamics, the development of the discretization, and numerical methods [38]. In other words, it is inevitable to observe excessive error of *CD* with the RANS method and limited computational resources. However, selecting the appropriate mesh type can improve the accuracy of both *CL* and *CD* to a certain extent in the current framework. Therefore, based on the simulation results obtained above, the impacts of different mesh types on rudder hydrodynamics are summarized in this section. To evaluate the efficiency and accuracy of those mesh types, the calculation accuracy of *CD* versus changing numbers of cells in the near-field and far-field is shown in Figures 18 and 19, while blue dotted lines connect minimum errors under each *α* and red lines mark selected mesh settings for each type. Since there is a common trend between impacts of radial and wake distributions on calculation accuracy for C-mesh and H-mesh, only radial node distributions are selected to perform the estimation.

In Figure 18, for the NOC below 3.00 × 105, C-mesh shows higher accuracy than other mesh types. However, H-mesh with better mesh resolution around the rudder achieves better performance under large angles of attack. What's more, it can be foreseen that a further increase in nodes may lead to more accurate results if computational power is not an issue. According to the tendency of Hybrid-mesh, the accuracy of *CD* remained stable, and smaller chord-wise spacing cannot make up for defects of unstructured mesh in solving rudder flows.

In Figure 19, even though variations of *CD* of Hybrid-mesh are slightly larger than Cmesh and H-mesh, results obtained from these mesh types settle in stable levels with fixed chord-wise spacings. Although making further efforts in far-field mesh resolutions can still improve the accuracy, the consequent mesh elements with small size may dramatically increase computation time, as well as bring difficulties with convergence. C-mesh and H-mesh with NOC around 5.00 × <sup>10</sup><sup>5</sup> are selected, which are about 2.00 × <sup>10</sup><sup>5</sup> less than Hybrid-mesh in the number of cells and show higher accuracy especially under high rudder angles, proving that structured meshes have higher efficiency in computations than hybrid meshes.

**Figure 18.** Calculation deviations of *CD* compared with experimental benchmark data adopted from [25] varying with numbers of cells, while changing chord-wise spacings in the near-field (blue dotted lines connect minimum errors under each *α* and red lines mark selected mesh settings for each type).

Based on the principle of balancing efficiency and accuracy, recommended mesh properties for four mesh types investigated in Sections 5.1–5.3 are presented in Table 27, and performance of these settings are compared with that of other codes (Langley Research Center [19]) in Table 28, which coincide with H-mesh best. The results of these CFD codes are achieved using the same block-structured C-mesh in a domain of 500 *c* around the profile and 500 *c* behind it, which is much larger than the domain in this paper. Hence, the appearance of differences in Table 27 is reasonable, and relatively good results are achieved by the current method efficiently. Structured meshes of C-type and H-type can achieve accurate *CL* predictions, while O-mesh fails to capture flow features in wake directions because of the difficulties of actualizing clustering grid points in the wake while maintaining a reasonable spacing at the trailing edge [5]. The concentration degree of mesh elements in regions with a large change of solution gradients for H-mesh is greater than that of C-mesh, contributing to higher accuracy in *CD* predictions when the NOC is above 3.00 × <sup>10</sup>5. Some unexpected divergent cases can be observed for C-mesh and H-mesh, and no general rules can be derived to explain why that happens, though it probably relates to

the processing mechanism of the solver. In systematic parameter studies, it is recommended to fine-tune mesh properties to obtain convergent results. Hybrid-mesh can be generated more simply with less manual intervention, and calculation results are acceptable but still inferior to C-mesh and H-mesh. However, the high adaptability of unstructured meshes to complex geometries extends the scope of application of Hybrid-mesh, which means that such a meshing strategy can be easily transformed into sophisticated configurations like high-lift rudders.

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**Figure 19.** Calculation deviations of *CD* compared with experimental benchmark data adopted from [25] varying with numbers of cells, while changing radial node distributions in near-field (blue dotted lines connect minimum errors under each *α* and red lines mark selected mesh settings for each type).


**Table 27.** Recommended mesh properties for different mesh types.

**Table 28.** Comparison of mesh types with recommended settings to that of other CFD codes adopted from [19] in relative differences.


#### **6. Further Validations**

To further validate the proposed mesh strategy, more 2D rudder profiles are studied in this section, i.e., NACA 0006, NACA 0009, NACA 0015, and NACA 0018, and the calculated force coefficients are compared with corresponding experimental data. The test conditions are shown in Table 29.

**Table 29.** More profiles studied.


By applying H-mesh and Hybrid-mesh with recommended domain size and nodes distributions, obtained *CL* and *CD* are compared with EFD results before stall for tested rudder profiles as shown in Figure 20. For profiles with relatively small thickness ratios, such as NACA 0006 and NACA 0009, Hybrid-mesh failed to obtain credible force coefficients. As for H-mesh, in such cases, the mesh type can achieve acceptable *CL* calculations, although the scope of *α* which leads to convergent solutions is limited. Nevertheless, from the perspective of structural strength, thin profiles are generally not used for rudder designs, so no further efforts are made to modify current mesh strategies to adapt thin profiles in this paper. For NACA 0015 and NACA 0018, when *α* < 5◦, some divergence may occur for H-mesh, while Hybrid-mesh shows good convergence. After that, CFD results for H-mesh and Hybrid-mesh agree with EFD ones well, while H-mesh shows

better accuracy for *α* around 15◦, demonstrating that the proposed meshing strategy can be used for other rudder profiles.

**Figure 20.** Comparison of results of H and Hybrid mesh with experimental benchmark data adopted from [39,40] for NACA 0006, NACA 0009, NACA 0015 and NACA 0018.
