5.1.1. Variation of the First Cell Size Normal to the Wall, i.e., y<sup>+</sup> Variation

It is essential to calculate the wall shear stresses accurately as the resistance of a ship at model scale is often dominated by the frictional resistance component. Previous studies performed with SHIPFLOW and other codes [28,39,40] showed that computation of the frictional resistance component is rather sensitive to the turbulence model, and the first cell size normal to the wall. In order to investigate both of them, the second finest grid (g2) is selected as a reference point since the differences between the g1 (finest grid) and g2 in *C*<sup>V</sup> were less than 0.2% (except for one hull). Keeping the same number of cells as g2 in all directions and the position of no-slip grid points identical in longitudinal and circumferential directions, the height of the first cell in the normal direction to the wall was varied. These variations were performed for all test cases using EASM and *k* − *ω* SST turbulence models.

The height of the first cell normal to the wall is non-dimensionalized as <sup>y</sup><sup>+</sup> = (*u*∗*y*)/*ν*, where *y* is the height of the first cell and *ν* is the kinematic viscosity. In SHIPFLOW, the average y<sup>+</sup> is calculated by arithmetic mean

$$(\mathbf{y}^+)\_{d\text{v\,y}} = \frac{1}{N\_{\text{grid}}} \sum\_{i=1}^{N\_{\text{grid}}} \mathbf{y}\_i^+ \,. \tag{8}$$

The height of the first cells for all test cases are adjusted to cover a wide range of (y+)*avg* including the values exceeding the recommended y<sup>+</sup> < 1 for the wall resolved approach. The results of the y<sup>+</sup> variations are presented in Figure 3 as the comparison error of form factors

$$E\%D = (D - S) / (D + 1) \times 100\tag{9}$$

where *D* is the experimentally determined form factor (using the Prohaska method), and *S* denotes the CFD based form factor based on the ITTC-57 line. The comparison error

shows a consistent <sup>±</sup>2.5% of spread throughout the (y+)*avg* range of 0.1 to 1.5. When the fitted curve to all the computed results (dashed black line) is considered, the average *E*%*D* converges to zero as the (y+)*avg* gets smaller, and it is nearly constant for the simulations performed where (y+)*avg* < 0.5. The mean comparison error for the computations where (y+)*avg* <sup>≈</sup> 1.5 is the largest, as expected, since the non-dimensional height y<sup>+</sup> is required to be lower than 1 for the wall resolved approach. However, the *E*%*D* of the simulations with (y+)*avg* <sup>≈</sup> 0.75 and (y+)*avg* <sup>≈</sup> 1 are also in an increasing trend. As presented in Figure 4, the histogram of the acquired y<sup>+</sup> values for the six different first cell heights for the H3 hull reveals that achieving (y+)*avg* < 1 does not guarantee that all (or most) of the y<sup>+</sup> values will be also below 1. As seen in Figure 4, 3%, 35% and 60% of the no-slip cells have y<sup>+</sup> values are higher than 1 for the simulations that resulted in (y+)*avg* of 0.5, 0.73, and 0.95, respectively. Considering the histogram plots of y<sup>+</sup> values of the other test cases as well, it is recommended that the target (y+)*avg* should be smaller than 0.5 in order make sure that nearly all the y<sup>+</sup> values will be smaller than 1.

As observed in Figure 3, CFD based form factors are heavily dependent on the choice of the turbulence model. The form factors obtained by the *k* − *ω* SST model are consistently 1.5 to 2.5 percent higher than what is achieved with EASM, mainly as a result of the calculated *C*<sup>F</sup> being approximately 3% higher with the *k* − *ω* SST model than the EASM for the same grid. These trends are noticeably consistent within the range of 0.1 < (y+)*avg* < 1. Therefore, if the (y+)*avg* is smaller than 0.5, the modeling error is dominated by the choice of turbulence model.

**Figure 3.** Comparison error of all test cases for average *y*<sup>+</sup> variation.
