**3. Computational domain and discretization**

The computational domain is shown in **Figure 5** where SB is the body boundary, SF1 is a surface enclosing the subdomain Ω<sup>1</sup> as well as SB and SF is a surface enclosing the entire domain Ω, which encloses Ω1. The domain Ω, bounded by SF and marked ABCD in the *x-z* or the vertical plane (*x-y* or the horizontal plane being similar since the body is axisymmetric), has a length 6.2 *L* (*L* being the length of the body) and a height (i.e. breadth in the *z* direction) of 1.2 *L*. The domain length upstream is 0.7 *L* and the domain length downstream is 4.5 *L*. The domain is large enough to capture the entire viscous-inviscid interaction and the wake development. It should be noted that the domain Ω embeds in it the domain Ω<sup>1</sup> as well as the body surface SB. The mesh of the domain Ω is modeled by a uniform grid such that on the boundaries AB and CD, the number of cells is *Nz* (=*Ny*) and on the boundaries AD and BC, the number of cells is *Nx*. Thus, the domain Ω has a grid of *Nx Ny N*<sup>z</sup> cells in *x* (length), *y* (breadth) and *z* (height) directions of the cuboid Ω.

**Figure 5.** Fluid domain and boundaries for CFD calculations of AUV Cormoran, Afterbody 1 and AUV Autosub.

The subdomain Ω<sup>1</sup> is defined such that a point on SF1 is at a (perpendicular) distance of *l* from SB and is meshed with *Nr* cells over this distance *l* with a graded mesh. The grid point nearest to SB (wall adjacent cell size) is located at a (perpendicular) distance of *l*<sup>1</sup> from it and a successive ratio (*g*), defined as the ratio of successive distances between grid points normal to the body surface, is prescribed. This ensures that

$$l = \sum\_{i=1}^{N\_r} l\_1 g^{i-1} \tag{3}$$

and the mesh over Ω<sup>1</sup> is an H-type structured mesh.

The successive or growth ratio (*g*) should be so chosen that it prevents the wall adjacent cells from being placed in the buffer layer of *y*<sup>þ</sup> ¼ 1. The acceptable distance between the cell centroid and the wall adjacent cells is usually measured in the wall unit *y*<sup>+</sup> . However, it requires some trial and error to determine a suitable value of *l*1. In all calculations, a value of *l*<sup>1</sup> = 0.001 mm was adopted which is found to satisfy the *y*<sup>þ</sup> *<* 1 requirement. The mesh of the domain Ω1, which embeds the body, was modeled by adopting *g* = 1.1 and a grid of *Nx* � *Nr* � *N<sup>θ</sup>* cells in longitudinal, radial and circumferential direction respectively. The interface between Ω<sup>1</sup> and Ω is handled by the CFD solver by constructing interfacial cells.

#### **3.1. Boundary conditions**

The conditions imposed on boundaries of the fluid domain (see **Figure 5**) are:


#### **3.2. Solver parameters**

The commercial CFD solver has been used and the solver parameters are presented in **Table 3**. The velocity components are governed by the momentum equations. The Roe flux algorithm is used for coupling the pressure and velocity terms. Second order upwind scheme is adopted for the discretization of pressure, momentum, turbulent kinetic energy and turbulence dissipation rate. The convergence criterion of 10�<sup>4</sup> is set for velocity components and 10�<sup>6</sup> for continuity, *k* and *ω*. The termination of the program is based on the final steady value of drag. All simulations were run using 3D steady segregated RANS solver. In all the optimization problems treated in this thesis, the number of iterations was fixed at 2000, and the drag force has been obtained by its average over one cycle (between a peak and a trough) preceding the last iteration number.


**Table 3.** Solver parameters and constants used in the study.

This reduces the computational effort significantly because instead of about 4000–6000 iterations for an accurately converged value of drag, one can use only 2000 iterations for all intermediate configurations during the optimization process. The drag of the final optimized configuration, however, is obtained by approximately 4000–6000 iterations for high accuracy.
