**4. CFD analysis of toy submarine USS Dallas**

The mesh of the domain Ω (see **Figure 5**) for toy submarine USS Dallas has 260,100 cells (=*Nx Ny N*<sup>z</sup> = 100 51 51 = 260,100) with *g* = 1, which gives a cell size of about 8.2 mm on the boundaries AB and CD, and a cell size of about 21.7 mm on the boundaries AD and BC. The mesh in the domain Ω<sup>1</sup> has 57,500 cells (=*Nx Nr N<sup>θ</sup>* = 100 115 50 = 575,000) with *g* = 1.1, which gives the distance between SB and SF1 (along any one of the rays) as *l* = 52 mm with the size of the cell adjacent to SF1 as 4.3 mm and in the circumferential direction an angle of 7.2 deg. The length of wall adjacent cell (*l*1) is 0.001 mm. The discretization of the domains Ω and Ω<sup>1</sup> are shown in **Figure 6** for the toy submarine.

The comparison of the drag forces obtained using CFD as reported by Alam *et al*. (2015) with present CFD calculations are recorded in **Table 4** showing very good match (within 0.2% difference). The drag values obtained by Eqns. (2.3), (2.4) and (2.5) were reported by Alam

**Figure 6.** Mesh for toy submarine USS Dallas (No. of nodes 835,100; no. of cells 956,664). (a) Mesh in the full domain for toy submarine USS Dallas. (b) Mesh in the Ω<sup>1</sup> for toy submarine USS Dallas.


**Table 4.** Comparison of drag forces of toy submarine USS Dallas at *<sup>U</sup>* = 0.5 m/s (*ReL* = 1.75 105 ).

*et al*. (2015). The mesh convergence study is not reported here because the present mesh with about 1 million cells is much finer than 0.69 million cells used in Alam *et al*. (2015).

#### **4.1. CFD analysis of AUV Cormoran**

The mesh of the domain Ω (see **Figure 5**) for AUV Cormoran has 26,010 cells (= *Nx Ny N*<sup>z</sup> = 100 51 51 = 260,100) with *g* = 1, which gives a cell size of about 33 mm on the boundaries AB and CD, and a cell size of about 92.3 mm on the boundaries AD and BC. The mesh in the domain Ω<sup>1</sup> has 575,000 cells (= *Nx Nr N<sup>θ</sup>* = 100 115 50 = 575,000) with *g* = 1.1, which gives the distance between SB and SF1 (along any one of the rays) as *l* = 52 mm with the size of the cell adjacent to SF1 as 4.3 mm and in the circumferential direction an angle of 7.2 deg. The length of wall adjacent cell (*l*1) is 0.001 mm. The discretization of the domains Ω and Ω<sup>1</sup> are shown in **Figure 7**.

**Figure 7.** Mesh for AUV Cormoran. (a) Mesh in the full domain for AUV Cormoran(No. of nodes 835,100; no. of cells 956,664). (b) Mesh in the Ω<sup>1</sup> for AUV Cormoran (c) panels on AUV Cormoran. (d) Panels on AUV Cormoran and free surface (one half shown) (No. of panels = 3500 on SB, = 1500 on free surface, = 5000 total). (e) Surface wave pattern generated by AUV Cormoran (*U* = 1 m/s). (f) Surface wave pattern generated by AUV Cormoran in top view (*U* = 1 m/s). (g) Surface wave pattern generated by AUV Cormoran in front view (*U* = 1 m/s).


**(a) Comparison with experiments**

**Table 5.** Comparison of drag coefficients of AUV Cormoran.

The experimental drag forces as reported by Alvarez *et al*. (2008) are for snorkeling depth of submergence (*z* = �0.05 m, see **Figure 5**) which will induce wave making drag in addition to viscous drag. In order to capture the wave-making component of drag, a free surface boundary condition is specified on the boundary AD (see **Figure 5** and **7b**). The dimensions of the free surface are 2 *L* in the *x*-direction and *L* in the *y*-direction. The surface is discretized with 50 equispaced panels in the *x*-direction, which gives a panel size of 28.4 mm and with 30 equispaced panels in the *y*-direction, which gives a panel size of 35.5 mm, making a total of 1500 panels. On the surface SB, in the *x*-direction, for 0 ≤ *x* ≤ *Ln* number of panels are 40 with a g = 1.1, in between *Ln* ≤ *x* ≤ *Ln + Lm*, 10 with g = 1 and between *Ln* + *Lm* ≤ *x* ≤ *L*, 20 with g = 1.1 making a total of 70 panels and in the circumferential direction, 50 equispaced panels, each making an angle of 7.2 deg. The surface wave pattern generated by AUV Cormoran for *U* = 1 m/s is shown in **Figures 7(e), 7(f)** and **7(g)**.

The comparisons of the drag forces are recorded in **Table 5** for four forward speeds. As can be seen, the present calculations are very accurate.
