**2. Geometry definition of the hull forms**

The geometry of the UV is axisymmetric and is shown in **Figure 2**. The parameterized shape of the body is given by [16]:

$$\begin{aligned} r(\mathbf{x}) &= r\_{\text{max}} \left( 1 - \left( \frac{L\_n - \mathbf{x}}{L\_n} \right)^{n\_n} \right)^{1/n\_n} \quad \text{for } 0 \le \mathbf{x} \le L\_n\\ r(\mathbf{x}) &= r\_{\text{max}} \quad \text{for } L\_n \le \mathbf{x} \le L\_n + L\_m\\ r(\mathbf{x}) &= r\_{\text{max}} \left( 1 - \left( \frac{\mathbf{x} - L\_n - L\_m}{L\_t} \right)^{n\_t} \right) \text{OR } r\_{\text{max}} + \frac{(\mathbf{x} - L\_n - L\_m)(r\_t - r\_{\text{max}})}{L\_t} \text{ for } L\_n + L\_m \le \mathbf{x} \le L\\ L\_n + L\_m + L\_t &= L \end{aligned} \tag{1}$$

where *r*max is the maximum radius of the body so that the maximum diameter is *dmax* (=2*rmax*) and a middle body length *Lm* has this radius, *r* (*x*) is the variation of radius over the total length

**Figure 2.** Parameterization of the hull geometry. (a) Axisymmetric body without blunt tail. (b) Axisymmetric body with blunt tail.

*L*, *Ln* and *Lt* are the lengths of the nose and the tail respectively and the exponents *nn* and *nt* are associated with the nose and tail shapes respectively.

The shape given by Eq. (1) leads to a conical nose shape (i.e. linear *r*(*x*)) for *nn* = 1 and a conical tail shape for *nt* = 1. For large values of *nn*, the nose shape profile approaches a rectangle (i.e. *r*(*x*) approaches *rmax*) and for large values of *nt*, the tail shape profile also approaches a rectangle. For *nn* < 1 and *nt* < 1, the nose and tail shapes reverse the sign of their curvatures. The volume (∇), a design variable, is

$$\nabla = \pi \int\_0^L r^2(\mathbf{x}) d\mathbf{x} \tag{2}$$

In this work, we have chosen two axisymmetric vehicles, namely, toy submarine USS Dallas (**Figure 3**) and AUV Cormoran (**Figure 4**). The parameters of these five vehicles are given in **Table 2**. It is to be noted that there is no parent hull form for the submarine. Its parameters, given in **Table 2**.

**Figure 3.** Toy submarine USS Dallas.

**Figure 4.** AUV Cormoran.



**Table 2.** The hull parameters.

Toy submarine USS Dallas was studied by [2] who reported drag force obtained from CFD simulation for a speed of 0.5 m/s which translates to *ReL* = 1.75 105 . AUV Cormoran was studied by [16] who reported drag coefficients for a few speeds measured in towing tank.
