**5. Results and Discussions**

With the strategies and numerical methods illustrated in Section 4, systematic simulations are conducted to study the impacts of mesh properties on hydrodynamic performance of the NACA 0012 rudder profile in this section. For each case with different mesh settings, errors with respect to benchmark data [19,25] are displayed in tabular form. Besides, uncertainty estimations are conducted for cases with varying cell sizes. For colored error tables in this section, the color legends are marked based on the values of relative differences. The

smallest difference is the darkest green and the largest one is the darkest red for each mesh group, while other colors between the darkest green to red show the transition. The impacts of Reynolds numbers are presented in Section 5.1, and Section 5.2 discusses the selection of domain sizes taking the C-mesh as an example, and impacts of node distributions and mesh types are discussed in Sections 5.3 and 5.4.

#### *5.1. Impacts of Reynolds Numbers*

Before conducting parametric investigations, the Reynolds number should be checked first. The Reynolds number (*Re*) is defined as the ratio of inertial forces to viscous forces, which represents the viscous similarity among flow patterns and can be expressed as:

$$\text{Re} = \frac{\rho l L\_{\infty} L}{\mu} \tag{26}$$

where *U*∞ is the inflow velocity, *L* is the characteristic length, and *μ* is the viscosity coefficient. For rudder hydrodynamics investigations, due to the limited model size and capacity of the test facility, *Re* of tests in wind tunnels and towing tanks has to be scaled. Analogously, model-scale rudders are preferred by CFD methods since a fine mesh and a large domain are needed for full-scale simulations. In most cases, the similarity law of viscosity and gravity can not be satisfied simultaneously. Generally, experiments or numerical simulations are conducted under a *Re* which is large enough to guarantee that the turbulence develops sufficiently. A high *Re* ensures a fully turbulent flow and reduces the effect of the laminar-turbulent transition, achieving an increase of prediction accuracy compared to that of an analysis at low *Re* with the same mesh.

Cases with larger *Re* show less calculation efficiency seeing that more cells around boundaries are required. The height of the first row of cells Δ*s* varies with *Re*, as shown in Figure 8, while increasing *Re* leads to decreasing heights. When discretizing domains into multiples mesh elements, the aspect ratio for a single element should be around 1 to ensure mesh quality. When it comes to cells near walls, aspect ratios are connected with Δ*s* and chord-wise spacings Δ*c*. As shown in Figure 9, more nodes along the wall are distributed with smaller Δ*s* to keep aspect ratios compliant with the requirement of solvers. Thus, a decrease in *Re* leads to a decrease in the number of cells. To sum up, we suggest that using a minimum *Re* which is achievable in experiments, ensures full-turbulent conditions, and can be solved with fewer mesh elements.

**Figure 8.** Different meshes' height of first layer with varying *Re* and *y*<sup>+</sup> = 1.

**Figure 9.** Impacts of height of first mesh Δ*s* on aspects ratios.

To determine the threshold of the Reynolds number, above which the turbulence develops completely and force coefficients do not change significantly, cases with varying *Re* are simulated in this section. Based on the flat-plate boundary layer theory and the calculation method in [33], the height of meshes' first layer Δ*s* above the rudder surface depends on *y*<sup>+</sup> and *Re*. According to the application scope of the *k*-*ω* SST model, *y*<sup>+</sup> is set as 1 for all cases, and corresponding heights are shown in Figure 8.

Common benchmark wind tunnel tests are carried out at *Re* in the range of 1.00 × <sup>10</sup><sup>5</sup> [34] to 1.00 × <sup>10</sup><sup>7</sup> [25]. As low-Reynolds-number RANS analysis is still challenging [35,36] and high-Reynolds-number simulations may be expensive in computation time, the present work tests the NACA 0012 profile at *Re* in a range of 2.00 × 105∼1.89 × 107. Among these aerodynamic experimental results, limited cases with *Re* = 3.94 × <sup>10</sup>6∼1.89 × 107 and a small Mach number can be taken as validation results of incompressible water simulations, as the compressibility effects of a fluid with a Mach number smaller than 0.20 is small. Table 4 presents the case settings and results of various *Re* under distinct *α* series, as specific values of *α* are listed in [25].

Figure 10 presents calculation results of hydrodynamic coefficients versus the same rudder angles for different *Re*, which indicates that the lift curve rises with an increase of the *Re* while the drag curve decreases. Compared to that of the lift coefficients, the drag coefficients are more sensitive to changes in *Re*. The drag coefficient under 15◦ at *Re* = 1.89 × <sup>10</sup><sup>7</sup> is about half of the value at *Re* = 4.00 × <sup>10</sup>5, whereas the lift coefficient is 1.20 times larger. The differences in lift and drag forces between low and high *Re* increase with an increasing angle of attack. Consistent with findings by Ladson [25] and Molland and Turnock [37], *Re* = 6.00 × <sup>10</sup><sup>6</sup> can be considered as a threshold value for the mesh generation, above which little variation may be found. Moreover, the *y*<sup>+</sup> along the chord for four cases under small and large rudder angles with varying *Re* are shown in Figure 11. *<sup>y</sup>*<sup>+</sup> values for the case with *Re* <sup>&</sup>gt; 5.97 <sup>×</sup> 106 are roughly between 0 and 1, indicating that the *k*-*ω* SST turbulence model is applicable for the case. Since ship rudders in propeller slipstream usually experience full turbulence with large *Re*, low *Re* conditions are not of interest in current studies. In this paper, we set *Re* of the following simulations as 6.00 × <sup>10</sup>6.


**Figure 10.** Impacts of *Re* on lift and drag coefficients.

**Figure 11.** *y*<sup>+</sup> along rudder chord with varying *Re*.

#### *5.2. Impacts of Domain Sizes*

A C-mesh is applied to investigate the effects of the domain size on RANS solutions by conducting simulations with various sizes against varying cell numbers in this section. Test parameters and related relative differences are given in Table 5. *Cin* and *Cout* are doubled for each case compared with that of the previous one, while the narrowest domain extends 7.5 *c* upstream and 15 *c* downstream, which is 16 times smaller than the largest ones, i.e., 120 *c* and 240 *c*. Since equivalent mesh nodes correspond to different spacing variations in a variety of domains, different nodes distributions are applied for each domain to determine reasonable meshing strategies.

Pressure distributions in the whole domain and around the rudder are presented in Figure 12. From Figure 12a–c, regions with great gradient changes near the rudder can be observed even for the smallest domain. Comparing Figure 12d–f, neither high-pressure regions near the stagnation points nor regions near the trailing edges show significant distinctions. The impact of domain size on pressure distributions around the rudder is relatively small, which corresponds to minor differences of *CL* mainly derived from pressure integration.

Compared with that of *CL*, accuracy of *CD* is more sensitive to domain sizes in Table 5. With an increase of the domain size, the accuracy of the prediction of *CD* is more significantly improved than *CL*. Small domain sizes show poor accuracy in *CD* but obtained *CL* is acceptable for engineering applications. The relative differences of *CD* for *α* = 15.20◦ drop from 55.19% to 34.79%, indicating that boundary effects can evidently change viscosity components around the rudder, which mainly contribute to drag forces. On the other hand, larger domains, like 60 *c*/120 *c* and 120 *c*/240 *c*, tend to show high and stable precision with more cells. However, further expansion of the domain size may dramatically increase the number of cells and computation time.


**Table 5.** Comparison of results of C-mesh based on various domain sizes with that of experimental benchmark data adopted from [25] in relative differences.

From the perspective of efficiency, a domain size that eliminates boundary effects and meanwhile requires fewer mesh nodes is desired for following investigations. Table 5 shows that a domain size of *Cin* = <sup>30</sup> *<sup>c</sup>* and *Cout* = <sup>60</sup> *<sup>c</sup>* with about 2.50 × 105 cells achieves a good balance in both accuracy and efficiency, and Table 6 shows that the validation process is achieved for *CL* obtained from the case (*φ*2). This domain size is large enough to obtain sufficiently accurate estimates of lift and drag, and the number of cells is still manageable by common desktop computers. Thus, it is applied for the remainder of the simulations in this paper. Table 5 indicates that even more accurate predictions can be obtained with larger domains. Thus, we suggest applying a large domain when computation power and time are available. If the lift is the only purpose, a small domain size of 15 *c*/30 *c* may be more favorable as it requires less computation power and time than a large domain.

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**Figure 12.** Pressure contours of different domain sizes with *α* = 15.20◦.

**Table 6.** Uncertainty estimations for 4 cases (30 *c*/60 *c*, NOC = 203,144, 251,065, 423,225, 524,605, *α* = 15.2◦) based on mesh dependency.


#### *5.3. Impacts of Node Distributions*

After determining the test *Re* and the domain size, raw meshes are ready for the mesh independence study for different mesh types. The common procedure is to carry out simulations on an initial mesh with a residual error in a range of 1.00 × <sup>10</sup>−4∼1.00 × <sup>10</sup><sup>−</sup>5. After that, refine the mesh globally to around 2 or <sup>√</sup><sup>2</sup> times the initial mesh. Next, run simulations with the refined mesh and compare solutions obtained from the coarse mesh and the fine mesh. Repeat the refinement until the results do not significantly change with a finer mesh. The mesh independence study requires at least 3 solutions to evaluate the convergence of certain inputs [26]. Considering the computation time, it is always better to use the smallest number of cells. In this paper, mesh resolutions are changed by conducting local rather than global refinements, while the grid refinement factor *RG* is still around <sup>√</sup><sup>2</sup> for target parameters.

Since the flow near boundaries of the rudder profile changes more violently than that in the far-field and has larger effects on rudder hydrodynamics, near-field node distributions are studied by fixing far-field node distributions for four mesh types. After that, the impacts of far-field distributions are studied with proper near-field distributions.

In this section, with a domain size of 30 *c*/60 *c*, impacts of near-field node distributions are investigated for four mesh types, as shown in Figures 13–16. Meshes in different regions are categorized into two parts, which are located in the far-field and near the rudder surface. Near-field meshes are defined as those located along the rudder chord and within the boundary layers regions. The domain for C-mesh in Figure 13 consists of a semicircle and a rectangle, and nodes distributed parallel to the radius are defined as in radial directions, and the rudder in O-mesh is surrounded by radial nodes in Figure 15. The nodes extending from the trailing edge to the outlet of the boundary are defined as in wake directions. Patterned after the definitions above, node distributions for H-mesh are shown in Figure 14, and nodes from the leading edge to the inlet are in inlet directions. For Hybrid-mesh in Figure 16, considering quite different mesh generation methods compared with that of structured meshes, meshes in the refinement area near the rudder are treated as near-field meshes while others are in far-field.

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**Figure 13.** Node distributions for C-mesh.

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**Figure 15.** Node distributions for O-mesh.
