1 Introduction

A major aspect in the design of an underwater vehicle, is the need to achieve stipulated motion control and manoeuvring capability. This in turn requires adequate sizing and location of the vehicle’s control surfaces. Therefore, during the early stages of its design, it is necessary to have a good estimate of the vehicle’s linear manoeuvring coefficients, or derivatives. This makes it possible to determine the required size and location of the appendages to achieve the desired level of motion stability and controllability.

Various empirical methods exist to estimate these coefficients, including the influence of appendages, which are presented in [1]. The reference also provides an introduction to the equations of motions required for the prediction of the trajectory of the vehicles and the relevant coefficients within the equations. The equations are developed about the coordinate system shown in Fig. 1 (see [1] for definitions).

Fig. 1
figure 1

Coordinate system for the submarine

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The required linear coefficients are [1]:

Horizontal Plane: \({Y}_{v}, {N}_{v}, {Y}_{r}, {N}_{r}\)

Vertical Plane: \({Z}_{w}, {M}_{w}, {Z}_{q}, {M}_{q}\)

For an axisymmetric body, the relationships between the coefficients in the horizontal plane and the vertical plane are given by Eqs. 1–4 [1].

$${Y}_{v}={Z}_{w}$$
(1)
$${N}_{v}={-M}_{w}$$
(2)
$${Y}_{r}={Z}_{q}$$
(3)
$${N}_{r}={M}_{q}$$
(4)

The coefficients \({Y}_{v}\), \({N}_{v}\), \({Z}_{w}\) and \({M}_{w}\) for the bare axisymmetric body including the influence of the Length (L) to Diameter (D) ratio (i.e. L/D ratio) are presented in [2] and [3]. They conducted numerical and experimental studies in which the results show a good agreement. The expressions for \({Y}_{v}\) and \({N}_{v}\) for the bare axisymmetric body are given as a function of L/D ratio in [1].

Empirical methods to take into account the influences of appendages and the propulsor on all the linear coefficients are also given in [1]. Unfortunately, however, empirical methods do not exist for the linear rotary coefficients for a bare hull, and hence it was recommended to set both \({Y}_{r}\) and \({N}_{r}\) to zero, [1], in the absence of any other information.

Thus, the aim of the current work is to determine the values of \({Y}_{r}\) and \({N}_{r}\) for an axisymmetric body as functions of L/D to improve the empirical estimate of these values for a fully appended submarine. Note that if \({Y}_{r}\) and \({N}_{r}\) can be estimated for an axisymmetric body, then \({Z}_{q}\) and \({M}_{q}\) can also be obtained using Eqs. 3 and 4 respectively.

2 Methodology

2.1 Geometry

This study used a full scale Suboff bare hull geometry [4] having a length, L, of 60 m with an original L/D ratio of 8.57 (see Fig. 2). The original L/D was varied from 6 to 14 which covers the practical range of submarine hulls.

Fig. 2
figure 2

Full scale Suboff bare hull geometry [4]

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In order to keep the Reynolds number (Re) constant, the variation in L/D was obtained by changing the diameter, D. Two different methods were used to change the diameter, as shown in Fig. 3.

Fig. 3
figure 3

Suboff bare hulls having various values of Length to Diameter (L/D) ratios: a Different Parallel Middle Body (DPMB) hull; and b Constant Parallel Middle Body (CPMB) hull

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For the first method, referred to as the Different Parallel Middle Body (DPMB), the size of the parent hull was changed whilst keeping geometric similarity (GEOSIM) to achieve the desired value of the diameter, D. Then, the length of the Parallel Middle Body was altered such that the overall length of the submarine was returned to the original value of 60 m, as shown in Fig. 3a. Thus, for larger values of L/D a GEOSIM was developed with reduced length and diameter compared to the parent. The length was then increased to the desired value of 60 m by increasing the length of the Parallel Middle Body. For smaller values of L/D a larger GEOSIM was developed with the desired diameter, and the length of the Parallel Middle Body reduced to give the desired overall length of 60 m.

In the second method, referred to as the Constant Parallel Middle Body (CPMB), the diameter was varied by simply multiplying all the ordinates along the length of the submarine by the value required to achieve the desired overall diameter, as shown in Fig. 3b.

The geometric characteristics of the DPMB and CPMB hulls are outlined in Table 1. The mass was calculated based on the volume with density,ρ, of 999 kg/m3. The mass was non-dimensionalised (m') using Eq. 5.

Table 1 Geometric characteristics of the Different Parallel Middle Body (DPMB) and Constant Parallel Middle Body (CPMB) hulls. (LCG: Longitudinal centre of Gravity)
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$${m}^{^{\prime}}=\frac{mass}{0.5\rho {L}^{3}}$$
(5)

2.2 Coordinate system, rotational motion and test cases

The body fixed axis system on the Suboff geometry is shown in Fig. 4. The origin is at midships on the plane of symmetry. Symbols used are as defined in [1].

Fig. 4
figure 4

Body fixed horizontal coordinate frame with its origin located at midships

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The hydrodynamic forces (X and Y) and moment (N) acting on the body in the horizontal plane were computed with respect to the body-fixed coordinate system with the origin at midships (see Fig. 4). The positive directions along the x and y axes were defined as being forward and to starboard respectively. The forces and moment were non-dimensionalised using Eq. 6. The density and kinematic viscosity adopted in the computations were 999 kg/m3 and 8.91 × 10−7m2s respectively.

$${X}^{^{\prime}} \mathrm{\,and\, }{Y}^{^{\prime}}=\frac{X \mathrm{and }Y}{0.5\rho {L}^{2}{{V}_{\infty }}^{2}}\, {N}^{^{\prime}}=\frac{N}{0.5\rho {L}^{3}{{V}_{\infty }}^{2}}$$
(6)

The rotational motion was conducted at zero drift angle (β), and fixed radius (R) in the horizontal plane, as shown in Fig. 5. The radius is defined as the length between the rotation origin (O) and the body fixed origin (midships). The vehicles velocity, V∞, of 7.42 m/s was constant for all simulations, giving a Reynolds number (Re) of 5.0 × 108. The rotation rate was obtained using Eq. 7.

Fig. 5
figure 5

Schematic of rotational motion at a fixed radius (R) in the horizontal plane

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$$r=\frac{{V}_{\infty }}{R}$$
(7)

The non-dimensional rotational rate (r′) was obtained using Eq. 8.

$${r}^{^{\prime}}=\frac{r.L}{{V}_{\infty }}=\frac{L}{R}$$
(8)

Table 2 Test cases which cover a range of non-dimensional rotation rates (r′) from 0.1 to 0.4. The Computational Fluid Dynamic (CFD) rotational motion simulations were run for each of the geometry configurations listed in Table 1 and for each of the test cases given in Table 2.

Table 2 Test cases involving a range of rotation rates (r′)
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2.3 Numerical setup

The present study used the commercial viscous flow solver ANSYS Fluent v18.1 to solve the RANS equations during computations. These adopted the SST k–ω turbulence model with curvature correction [5], as [6] and [7] reported that the results obtained with the SST k–ω model are inferior to those obtained using curvature correction. The simulations were performed in steady state using the Moving Reference Frame (MRF) method, which enabled rotational flow around a stationary vehicle. Figure 6 shows the boundary conditions applied to the computational domain. The domain was meshed using an unstructured polyhedral grid with inflation layers consisting of polyhedral prisms. The thickness of the inflation layers on the vehicle surface were calculated based on the estimations presented in [8]. A detailed discussion of the numerical technique used is given in [9].

Fig. 6
figure 6

Boundary conditions of the computational domain for rotational motion

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A grid dependence study was carried out to ensure minimising the error due to insufficient grid resolution. The grid sizes on the surface of the DPMB Suboff hull were reduced by \(\sqrt{2}\), giving a number of grid densities defined as coarse, medium, medium fine, fine and very fine grids (see Table 3). The hydrodynamic loads (Y′ and N′) on the hull were predicted for r′ = 0.4 and L/D = 6. The differences in the loads at each grid density compared to those obtained with the very fine grid are given in Table 3. It was deemed that the fine grid is capable of providing a grid-dependent prediction. Thus, the fine grid was used for the rest of the study.

Table 3 Grid dependence study showing the percentage difference of the hydrodynamic loads
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The CFD rotational motion simulations were validated against the experimental data [10] using a model scale Suboff bare hull having a length of 4.356 m. The rotating arm experiment was conducted in the Naval Surface Warfare Center Carderock Division (NSWC-CD) Rotating Arm Basin having a diameter of 79.2 m and a depth of 6.1 m. The experimental uncertainty of Y’ and N’ measurements were 4.9% and 4.1%, respectively [6]. The influence of the sting on the measured force and moment in the experiment was reported to be 4.8% and 5.0%, respectively [11]. The grid discretisation uncertainty for Y′ and N′ was 1.9% and 0.3% with the safety factor of 1.25 [12]. Thus, the overall validation uncertainty [6] for Y′ and N′ was 7.04% and 6.47%, respectively. Table 4 shows the CFD prediction and experimental measurements for the hydrodynamic loads (Y′ and N′) on the Suboff at β = 10.2 and 16.5 degrees and Re = 6.5 × 106. Overall, the hydrodynamic load predictions show a good agreement with the experimental data within the validation uncertainty.

Table 4 Hydrodynamic force and moment coefficients for the Suboff in steady turn at β = 10.2 and 16.5 degrees and Re = 6.5 × 106
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3 Results

3.1 Introduction

The validated simulation model outlined in Sect. 2.3 was used to obtain the values of the force and moment on the body as functions of yaw velocity for the range of values of L/D. It is worth noting that it was not possible to achieve successful solutions for the DPMB geometries for values of L/D of 6 and 7. The reason for this is not clear. However, as the results for the two geometries for the other values of L/D were similar, and as the main range of interest was for values of L/D greater than 7, this was not pursued further.

3.2 Force and moment as functions of yaw velocity

The value of the non-dimensional force, Y′, and non-dimensional yaw moment, N’, as functions of non-dimensional yaw rate, r′, are given in Figs. 7 and 8 respectively for the parent hull for which L/D = 8.57. Also shown on these figures are the best fit equations, and the estimate of the slope at r′ = 0, which are the ‘derivatives’, \({Y}_{r}^{^{\prime}}\) and \({N}_{r}^{^{\prime}}\) respectively. These are referred to as the ‘linear coefficients’.

Fig. 7
figure 7

Y′ as a function of r′ for the parent hull (L/D = 8.57)

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Fig. 8
figure 8

N′ as a function of r′ for the parent hull (L/D = 8.57)

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Note that although values were obtained at r′ = 0.4, these were not used to obtain the sway force derivative, \({Y}_{r}^{^{\prime}}\), as the relationship between Y′ and r′ is highly non-linear. By definition, the linear coefficient is the slope at r′ = 0. On the other hand, the relationship between N′ and r′ is much more linear. In fact, the magnitude of \({Y}_{r}^{^{\prime}}\) for the axisymmetric body is actually quite small. For a fully appended submarine the value is dominated by the appendages [1].

The corresponding plots of force and moment as functions for other values of L/D are given in [9].

3.3 Linear coefficients as functions of L/D ratio

The linear coefficients obtained from the results of force and moment as functions of yaw velocity for each value of L/D are given in Table 5 for the DPMB and CPMB geometries.

Table 5 Linear coefficients as functions of L/D
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The linear coefficients for sway force and yaw moment are plotted as functions of the L/D ratio in Figs. 9 and 10 respectively.

Fig. 9
figure 9

\({Y}_{r}^{^{\prime}}\) as a function of L/D

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Fig. 10
figure 10

\({N}_{r}^{^{\prime}}\) as a function of L/D

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Also plotted in Figs. 9 and 10 are the best fits to the data, given by Eqs. 9 and 10.

$${Y}_{r}^{^{\prime}}=0.01{\left(\frac{L}{D}\right)}^{-2}+0.00018$$
(9)
$${N}_{r}^{^{\prime}}=-5{\left(\frac{L}{D}\right)}^{-4}-0.0003$$
(10)

Note that in determining the best fit for both the coefficients, more emphasis was placed on the results for L/D ratios greater than 7, as they are more representative of modern submarines. In particular, the result for \({Y}_{r}^{^{\prime}}\) at L/D = 6 was not included when developing the best fit equation, as it would distort the estimate for the more important range of L/D ratios.

3.4 Mass as functions of L/D ratio

The mass for each of the configurations are given in Table 1. The non-dimensional mass, \({m}^{^{\prime}}\), is plotted as a function of L/D in Fig. 11 for the two shapes.

Fig. 11
figure 11

\({m}^{^{\prime}}\) as a function of L/D

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The best fit to this data is given by Eq. 11.

$${m}^{^{\prime}}=0.8{\left(\frac{L}{D}\right)}^{-1.8}$$
(11)

4 Discussion

Equations 9 and 10 were developed to estimate the values of the linear rotary coefficients, \({Y}_{r}^{^{\prime}}\) and \({N}_{r}^{^{\prime}}\), for an unappended axisymmetric body as functions of the L/D ratio. Methods to take into account the influence of appendages on these coefficients in order to predict the values for a realistic underwater body are discussed in [1].

The generation of these coefficients used force and moment values as functions of yaw velocity obtained numerically, as discussed in Sect. 2.3.

The basis hull form used was the Suboff geometry, details of which are given in [1] and [4]. Two different approaches were used to generate the hull forms for the different L/D ratios, as discussed in Sect. 2. The differences in the values of the linear coefficients for these two different shapes were minimal, meaning that it is likely that Eqs. 9 and 10 can be applied to a range of typical modern submarine shapes.

When developing the expression for \({Y}_{r}^{^{\prime}}\) considerable weighting was placed on the results for L/D ≥ 7. It was noted that the resulting equation did not fit the single result obtained for L/D = 6, which was considerably different to the trend of the results for higher values of L/D. The reason for this is not clear. However, as very few modern submarines have values of L/D < 7 this limitation was not considered to be important. An enlargement of Fig. 9 with the scale altered to exclude the value of \({Y}_{r}^{^{\prime}}\) at L/D = 6 is given as Fig. 12. In this figure the small differences in the values of \({Y}_{r}^{^{\prime}}\) for the two different shapes are clearly seen.

Fig. 12
figure 12

\({Y}_{r}^{^{\prime}}\) as a function of L/D

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However, it is important to note that the coefficient \({Y}_{r}^{^{\prime}}\) is small, and does not have a major influence on the manoeuvring of the submarine. In addition, it will be modified considerably by the presence of appendages, which will have a major effect on its value for the actual vehicle shape.

Finally, in the equations of motion, \({Y}_{r}^{^{\prime}}\) is always combined with the mass [1], resulting in the term \(\left({Y}_{r}^{^{\prime}}-{m}^{^{\prime}}\right)\), further reducing the importance of an accurate prediction of the value of \({Y}_{r}^{^{\prime}}\). The relative size of \({Y}_{r}^{^{\prime}}\) and \({m}^{^{\prime}}\) for the range of L/D ratios investigated is given in Table 6. As seen from this table, the value of \({m}^{^{\prime}}\) is always substantially greater than the value of \({Y}_{r}^{^{\prime}}\) for the unappended body. Thus, it is considered that the estimate of \({Y}_{r}^{^{\prime}}\) using Eq. 9 is adequate.

Table 6 Relative magnitude of \({Y}_{r}^{^{\prime}}\) and \({m}^{^{\prime}}\) as functions of L/D
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On the other hand, the linear coefficient: \({N}_{r}^{^{\prime}}\) (and \({M}_{q}^{^{\prime}}\) in the vertical plane, which can be obtained from \({N}_{r}^{^{\prime}}\) using Eq. 4) has a major influence on the manoeuvring of a submarine as demonstrated by [13] and reported in [1]. As seen from Fig. 10 the best fit (Eq. 10) provides a very good estimate of the values of \({N}_{r}^{^{\prime}}\) for the axisymmetric body as a function of L/D for both shapes considered. This can be modified to take into account the effect of the appendages, as discussed in [1].

5 Concluding remarks

A validated numerical technique has been used to determine the values of the linear rotary coefficients, \({Y}_{r}^{^{\prime}}\) and \({N}_{r}^{^{\prime}}\), for an unappended axisymmetric underwater body for a range of L/D ratios. The parent hull was the well-known Suboff geometry, and two different methods were used to obtain the geometries for the different values of L/D. The results from the two different geometries were relatively similar.

Expressions were developed to estimate the values of the linear rotary coefficients as functions of the L/D ratio. These are given in Eqs. 12 and 13.

$${Y}_{r}^{^{\prime}}=0.01{\left(\frac{L}{D}\right)}^{-2}+0.00018$$
(12)
$${N}_{r}^{^{\prime}}=-5{\left(\frac{L}{D}\right)}^{-4}-0.0003$$
(13)

In addition, a method to estimate the non-dimensional mass, \({m}^{^{\prime}}\), of an axisymmetric underwater body was developed, as given in Eq. 14.

$${m}^{^{\prime}}=0.8{\left(\frac{L}{D}\right)}^{-1.8}$$
(14)

Equations 12, 13 and 14 are valid for the range: 6 ≤ L/D ≤ 14. For values of L/D < 7 Eq. 12 may not be particularly accurate. However, \({Y}_{r}^{^{\prime}}\) has a small value, and does not have a major contribution to the manoeuvring of a submarine. In addition, it is represented in the equations in conjunction with the mass as \(\left({Y}_{r}^{^{\prime}}-{m}^{^{\prime}}\right)\). The value of \({m}^{^{\prime}}\) is much greater than the value of \({Y}_{r}^{^{\prime}}\), reducing the importance of an accurate prediction of the value of \({Y}_{r}^{^{\prime}}\).

Thus, it is considered that Eqs. 12, 13 and 14 are adequate for the prediction of these coefficients for an unappended axisymmetric underwater body, especially at the early design stage.