Computational Fluid Dynamics Analysis into the Comparison of Resistance Characteristics Between DARPA Suboff and Modified U209 Types of Submarines

Abstract

Submarines are required to have good performance, which is influenced by their type of hull, hull conditions, and operational conditions. This study compares the resistance between a Modified-U209 (U209) submarine and the DARPA Suboff. The former is an older hull geometry with both surface and submerged operation considered, whereas the latter represents a modern nuclear-powered submarine designed for submerged operations only. The two geometries were scaled to give the same usable volume, and all results were non-dimensionalized using this to ensure consistency. A Computational Fluid Dynamics (CFD) method was utilized to predict resistance by employing the Reynolds-averaged Navier–Stokes (RANS) equations. The results show that the total resistance coefficient for the U209 bare hull is approximately 6% higher than the Suboff bare hull. When a casing was added to the U209 geometry the increase in total resistance coefficient was approximately 8%. The addition of the sail resulted in an increase in total resistance coefficient ranging from approximately 4% (Suboff sail added to U209) to approximately 14% (U209 sail added to U209). An existing empirical prediction technique was used to predict the resistance, with the total resistance coefficient predicted being consistently about 5% lower than the values obtained using CFD.

1. Introduction

1.1. Background

Submarines have played a crucial role in naval warfare and strategic defense for decades. Their unique ability to operate underwater, hidden from adversaries, gives them a distinct advantage in reconnaissance, surveillance and covert operations [1]. Their military importance around the world is continuing to grow, with many navies modernizing and increasing the size of their submarine fleets [2].

Submarines are an effective tool in both anti-ship and anti-submarine warfare due to their stealth and surprise tactics [3,4]. Their ability to remain undetected underwater allows them to gather intelligence, track enemy movements and launch attacks with precision. Furthermore, submarines are highly effective in hit-and-run attacks, attrition warfare and single salvo strikes ashore. They excel in missions that require quick strikes and minimal exposure, making them invaluable assets for disrupting enemy supply lines and executing targeted campaigns [5].

Submarines are also crucial for strategic nuclear deterrence, providing a second-strike capability in the event of a nuclear exchange. Their ability to carry and launch nuclear armed ballistic missiles from hidden locations beneath the ocean’s surface makes them a key component of national security strategies for many countries. Because of this, attack submarines are also used extensively to track enemy ballistic missile capable submarines, such that they can be neutralized prior to launching their ballistic missiles in the event of the outbreak of hostilities.

In addition to their military applications, submarines also play a vital role in scientific research and exploration of the ocean depths. Their manned and unmanned missions have led to groundbreaking discoveries and enhanced the understanding of the mysteries hidden beneath the sea. They are vital for studying and documenting marine life, geological formations and hydrothermal vents in the deep ocean [6].

Submarines have evolved from surface ships that are able to submerge from time to time, for example to evade detection or to attack a surface ship (better described as “submersibles”), to those designed to remain submerged throughout their whole mission: the true submarine. As the latter are not required to operate on the surface of the ocean, their hull shapes are quite different from those of the former.

Research into submarine hydrodynamic performance is generally very classified, and it is not possible to publish results without giving away details of a nation’s submarine designs. To assist in conducting submarine hydrodynamics research that could be shared in the public domain, the submarine technology office of the Defense Advanced Research Projects Agency (DARPA) in the US, developed an unclassified submarine shape, referred to as Suboff [7,8]. It was intended that this represents a typical modern (at the time) large nuclear-powered attack submarine—a true submarine. This has been used by many researchers in the field.

Renilson [9] summarized much of the work in this field, and presented detailed information about the geometry of the Suboff. Zheku et al. [10] used the Suboff geometry to assess numerical approaches to captive model tests. Uzen et al. [11] and Utama et al. [12] used the Suboff geometry to study the influence of biofouling on the drag of a submarine. Lyu et al. [13] used the Suboff geometry to investigate microbubble resistance reduction. Kinaci et al. [14] used the Suboff geometry to investigate self-propulsion using CFD and made comparisons with empirical methods. Chase and Carrica [15] used the DARPA Suboff and its propeller E1619 to investigate different turbulence modeling approaches. Byeon et al. [16] also used the DARPA Suboff geometry and propeller E1619 to simulate self-propulsion and investigate the wake velocities behind the submarine. They reported good agreement with experimental results. Delen and Kinaci [17] used the DARPA Suboff model to conduct direct CFD simulations of maneuvering tests.

The small diesel–electric 209 type of submarine (U209) was developed by Howaldtswerke-Deutsche Werft (HDW) in the late 1960s. This was based on type 206 and is generally considered a derivative of wartime designs, whose hull forms are compromises between surface and sub-surface performance. There are many varieties of type 209, with some still being used by a number of nations, including Indonesia. The geometry of the U209 is given in [18].

The design philosophies for these two submarines are quite different. Type 209 is a lot older, and designed with operations on the surface, as well as submerged, in mind. On the other hand, the Suboff is a more modern form with an axisymmetric hull, meant to represent a nuclear-powered submarine, designed purely for operations submerged. Nuclear submarines only spend very limited time on the surface.

1.2. Aim of Present Work

The main aim of the present work is to investigate the influence of geometry, including casing and sail, on the resistance of a submarine. Two basis submarines are used: the DARPA Suboff geometry [7,8] and the U209 geometry [18]. Sufficient details are available in the public domain for both geometries to enable such calculations to be carried out.

In order to make a meaningful comparison between these two geometries their dimensions were scaled to give them both the same usable volume, defined as the same volume in the hull, excluding the casing and the sail. The principal dimensions of the resulting submarines are given in

Table 1. Principal particulars of all cases.

Case	Description	L (m)	D (m)	L/D	LPMB (m)	S (m2)	∇ (m3)
1	Suboff bare hull only	4.356	0.530	8.219	2.250	6.232	0.755
2	U209 bare hull only (no casing/free flood space)	4.484	0.495	9.058	2.793	6.424	0.755
3	Suboff with Suboff sail	4.356	0.530	8.219	2.250	6.379	0.757
4	U209 with casing (no sail)	4.484	0.495	9.058	2.793	6.931	0.822
5	U209 with casing and Suboff sail	4.484	0.495	9.058	2.793	7.073	0.824
6	U209 with casing and U209 sail	4.484	0.495	9.058	2.793	7.453	0.841

. Note that these are all at model scale to make comparison with experimental results easier. However, the trends in the results will apply to a full-scale submarine, regardless of the size.

L is the overall length of the submarine; D is the diameter of the pressure hull; L_pmb is the length of the parallel middle body; S is the total wetted surface area (including casing and sail, where relevant); and ∇ is the total volume (including casing and sail, where relevant). The greater values of S and ∇ for Cases 3–6 compared to Cases 1 and 2 reflects the increase in wetted surface area and volume due to the casing and sail. As these will be free flood spaces, they are not considered to increase the useable volume of the submarine. In each case, the usable volume was kept constant at 0.755m³.

Schematics of the geometries of the six cases are given in Figure 1.

Note that as the Suboff was developed to represent a large nuclear-powered submarine it was designed to operate below the surface only. However, U209 was designed with surface, as well as submerged, operations in mind. Hence, U209 has a different bow shape, and a greater value of L/D than the Suboff. Both these characteristics would suggest that U209 will have a greater resistance when submerged than the Suboff. In addition, the Suboff has no casing, and only a small sail (relative to the size of the hull) whereas U209 has both a casing and a large sail. Again, both these characteristics would suggest that U209 will have a greater resistance.

It is interesting to note that despite the differences in the shapes of the two bare hulls (Case 1 and Case 2), as shown in the schematic diagrams in Figure 2, they have very similar wetted surface areas. An empirical estimation of the wetted surface of a submarine hull, knowing only its diameter, length, and length of parallel middle body was developed by one of the authors [9] and is given in Equation (1).

$$S=2.25D\left(L-L_{PMB}\right)+\pi D{L}_{PMB}$$

(1)

Using Equation (1) the wetted surface of the hull alone can be estimated for both cases. These are compared with the measured values in

Table 2. Wetted surface area of bare hull cases.

Case	Estimated (m2)	Actual (m2)	Difference
1	6.255	6.232	−0.4%
2	6.224	6.424	−3.2%

. The estimate of the wetted surface using Equation (1) for the Suboff is very good, being only 0.41% higher than the actual value. The estimate of the wetted surface using Equation (1) for U209 is not so good, being about 3% too low. However, considering that the hull shape of U209 is quite different to the data used to generate Equation (1), this is still remarkably close. Thus, it is possible to conclude that Equation (1) gives a very good estimate of the wetted surface of a submarine hull (without casing), suitable for the early design stage.

As can be seen from Table 1, in order to achieve the same usable volume, the two geometries had different overall lengths, and wetted surfaces, so to make the results consistent the drag was non-dimensionalized using the usable volume, C_∇, as given in Equation (2).

$$C_{\mathsf{\nabla }}={\displaystyle \frac{R}{0.5\rho V^2{{\mathsf{\nabla }}_u}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}$$

(2)

where R is the drag of the submarine; ρ is the water density; V is the speed of the submarine, and ∇_u is the usable volume.

In addition, as the two geometries have different lengths for the same usable volume, to compare the results using the Reynolds number based on length would be inconsistent. Hence, for this study, the Reynolds number was based on volume, Re_∇, as given in Equation (3).

$${Re}_{\mathsf{\nabla }}={\displaystyle \frac{V{\mathsf{\nabla }}^{1/3}}{\nu }}$$

(3)

where ν is the kinematic viscosity of the water.

2. Method

The investigation was carried out using Computational fluid dynamics (CFD) as a tool for resistance estimation as discussed below. This has been used before by the authors for this purpose [12].

2.1. Background

CFD using the Reynolds-averaged Navier–Stokes (RANS) approach has been used successfully to make predictions of the drag of a ship for many years [19,20,21,22,23]. For this work the implementation of RANS within ANSYS-CFX was utilized. ANSYS 2023 R2 version, developed by ANSYS, Inc., based in Canonsburg, Pennsylvania, USA, is used in this case.

It has been shown that the selection of turbulence models is of most importance as applied to the modeling of wake fields. The k-e turbulence model was used for this work. The k-e turbulence model has been widely used in industrial applications and research because it is relatively economical in terms of CPU time, especially when compared to models like the SST turbulence model [24,25,26,27,28].

It is noted, however, that Byeon et al. [16] compared the predicted results using CFD obtained with various turbulence models to experimental results for a fully appended DARPA Suboff model. They found that for this case, the Reynolds stress turbulence model (RSM) showed the best agreement with the experimental results, although it is worth noting that they appeared to use smaller grid sizes than used in the current study, and that their work was on a fully appended hull form, with aft control surfaces, whereas the aft control surfaces were omitted in the current work to better demonstrate the difference due to hull form and sail configuration.

Park and Seok [29] used the k-ω SST to predict the drag on a fully appended model of the BB2 submarine, with and without mounting struts using a grid size of approximately 2 M. They also reported the total resistance coefficients predicted by six other national research organisations.

Mai et al. [30] used the k-ω SST to predict the hydrodynamic derivatives of a fully appended BB2 submarine. They conducted a grid convergence study for the static drift case with y⁺ = 1. They determined that a grid size of about 3M is appropriate for oblique motion, using the grid convergence index (CGI) method of Roache [31].

2.2. Modeling

The fluid flow field is solved with the help of the RANS solver, which is a component of ANSYS-CFX; Equation (4) and Equation (5), respectively.

$${\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho U_j\right)=0$$

(4)

In Equation (4), ρ is fluid density, t is time, and U_j is the flow velocity vector field.

$${\rho \overline{f}}_i+{\displaystyle \frac{\partial }{\partial x_j}}\left[-\overline{p}{\delta }_{ij}+\mu \left({\displaystyle \frac{{\partial \overline{u}}_i}{\partial x_j}}+{\displaystyle \frac{{\partial \overline{u}}_j}{\partial x_i}}\right)-\overline{\rho u_i^{\prime }{u}_j^{\prime }}\right]-{\rho \overline{u}}_j{\displaystyle \frac{{\partial \overline{u}}_i}{\partial x_j}}=0$$

(5)

The left side of the RANS equation (Equation (5)) represents the change in the mean momentum of the fluid element to the unsteadiness in the mean flow. This change is balanced by the mean body force ($\overline{f}$), the mean pressure field ($\overline{p}$), the viscous stress,$\mu \left({\displaystyle \frac{{\partial \overline{u}}_i}{\partial x_j}}+{\displaystyle \frac{{\partial \overline{u}}_j}{\partial x_i}}\right)$, and apparent stress ($\rho u_i^{\prime }{u}_j^{\prime }$) to the fluctuating velocity field.

The ANSYS-CFX program utilizes a spatial discretization technique known as element-based finite volume, which is complemented by the implementation of a high-resolution strategy to stabilize the convective term. The process of time discretization is accomplished by the utilization of the Second Order Backward Euler scheme. The ANSYS-CFX software, 2023 R2 version, employs a linked solver that effectively solves the hydrodynamic equations, encompassing the variables of velocity components (u, v, w) and pressure (p) as a unified system. Initially, the non-linear equations undergo a process known as linearization using coefficients of iteration. Subsequently, the resulting linear equations are resolved using an Algebraic Multigrid (AMG) solver [32]. The convergence criterion utilized in both codes was established by assessing the residual error in the mass and momentum equations with a predetermined threshold of 10⁻⁴ [33,34] as discussed by Roache [31].

The submarine hull geometries are illustrated in Figure 1. The main particulars of the models are listed in Table 1.

2.3. Domain

The computational domain used is a cylinder shown in Figure 2. It comprises a velocity inlet 2L forward of the submarine and an outlet 5L aft of the submarine. The radius of the domain is L.

The boundary of the domain is defined as a free-slip condition, and the surface of the hull is defined as a fixed boundary with a no-slip condition. The domain size and the boundary conditions are shown in Figure 2.

The inlet flow velocity is prescribed, and the outlet hydrostatic pressure is defined as a function of water level.

2.4. Grid Convergence Independence

In this study, the DesignModeller in ANSYS-CFX was utilized to generate the mesh. Both structured and unstructured meshes were employed to decompose the computational domain. Due to the complex geometry of the hull, a mesh with triangular elements was constructed on its surface. Subsequently, the boundary layer was refined with prism elements by extending the surface mesh nodes. By strategically incorporating inflated tetrahedral elements proximate to the hull and employing an unstructured mesh in the distal region, we have markedly optimized the design. This approach not only enhances structural integrity but also significantly reduces the number of elements required, thereby increasing efficiency and performance. Such innovative techniques are essential for advancing engineering solutions and achieving favorable outcomes (as illustrated in Figure 3).

A fine mesh may always deliver reliable results in CFD, but at the same time, a finer mesh increases the computational cost and time consumption owing to the huge numbers of elements. The mesh size plays a significant part in the calculation operation.

Three mesh sizes were used for each case, for all four speeds. These ranged from 6,362,594 to 42,722,071 depending on the complexity of the geometry. Other than for the three lower speeds for Case 1, these all showed asymptotic convergence with Ri < 1.0 and GCI > 1.0. Hence, the finest mesh was used for the results given in this paper.

2.5. Turbulence Model Study

The selection of an appropriate turbulence model is crucial in accurately determining ship resistance, particularly for vessels where frictional resistance is the dominant component, as in this case, where the ship operates at a low speed (low Reynolds number). To identify the most suitable turbulence model, this study conducted a turbulence model analysis, incorporating the turbulence model of k-ε with y⁺ = 150, k-ω with y⁺ = 0.5, and k-ω SST with y⁺ = 0.5.

To achieve the targeted y⁺ values, adjustments were made to the number of element layers near the submarine surface and the height of the first-layer element. For y⁺ = 150, at a speed of 13.92 knots, 22 layers were used, with a first-layer height of 5.9 × 10⁻⁴. In contrast, for y⁺ = 0.5, 54 layers were used, with a first-layer height of 1.69 × 10⁻⁶. The element shape and the y⁺ distribution can be seen in Figure 4 and Figure 5.

2.6. Validation Against Experimental Results

The method developed above was then applied to the Suboff geometry without a sail and the predictions compared with the experimentally measured results presented in [8]. The comparison is given in Figure 5.

As can be seen from Figure 6, there is generally very good agreement between the CFD predictions (for all turbulence models) and the measured results, which gave confidence in applying this approach to investigate the influence of geometry on resistance.

The percentage difference between each turbulence model and the experimental results (Figure 7) shows that the k-ω model yields the best results, with discrepancies ranging from 3.2% to 9.6% for speeds between 5.92 knots and 13.92 knots. In contrast, the k-ε turbulence model provides moderate results, with differences ranging from 5.6% to 11.3%. The k-ω SST turbulence model, however, produces the largest discrepancies among the three turbulence models, ranging from 6% to 12.6%.

Regarding the running time (Figure 8), the k-ε turbulence model yields the best performance, with a lower running time ranging from 5 to 11 h. In contrast, the other turbulence models require running times between 11 and 25 h.

The k-ε model should be more suitable for high Reynolds numbers, while the k-ω model is more appropriate for low Reynolds numbers, and the k-ω SST model is suitable for both. However, in this case, considering that the percentage differences from the experimental results are not significantly large between the k-ε, k-ω, and k-ω SST models, while the running time differences are quite significant, the k-ε turbulence model was selected for this study.

3. Results and Discussion

3.1. Influence of Hull Shape

Figure 9 is a plot of the resistance coefficient, C_∇, as a function of the Reynolds number, Re_∇, for the two different bare hull geometries. Note that both have been non-dimensionalized using usable volume, as explained in Section 1.2. This presentation is consistent, and hence the results can be directly compared.

As can be seen from Figure 9, the frictional resistance coefficient is about 5.7% greater for the U209 bare hull (Case 2) compared to the Suboff bare hull (Case 1). The U209 has an approximately 3% greater wetted surface area.

The pressure resistance coefficient for the U209 bare hull (Case 2) is approximately 14% higher than that for the Suboff bare hull (Case 1). Again, this is expected as the bow of the U209 hull is much bluffer, and the shape in general is less streamlined with a longer parallel middle body length and hence shorter forward and aft lengths. This is possibly because of the compromise in the design for operations on the surface as well as submerged.

This results in the total resistance coefficient for the U209 bare hull (Case 2) being approximately 6% higher than that for the Suboff bare hull (Case 1).

As expected, the resistance coefficient is lower for higher Reynolds number values. This is consistent across all the results.

3.2. Influence of Casing

Figure 10 is a plot of the resistance coefficient, C_∇, as a function of the Reynolds number, Re_∇, for the U209 without a casing (Case 2) and with a casing (Case 4).

The increase in wetted surface area as a result of the casing is about 8%, which results in an increase in friction resistance of approximately 7%. The increase in pressure resistance coefficient caused by the casing is about 18%, which compares with the value of 15% derived empirically and given in reference [9] for a simple casing.

The overall increase in resistance coefficient as a result of the casing is approximately 8%.

3.3. Influence of Sail

Figure 11 is a plot of resistance coefficient, C_∇, as a function of the Reynolds number, Re_∇, for the Suboff without sail (Case 1) and with a sail (Case 3). Note that, as can be seen from Figure 1, the Suboff sail is small compared to the size of the hull, which corresponds to that of a large nuclear-powered submarine.

The increase in wetted surface area as a result of the addition of the small sail to the Suboff is about 2%, which results in an increase in friction resistance coefficient of approximately 4%. The increase in pressure resistance coefficient due to the sail is approximately 11% of the pressure resistance. Although this percentage increase seems large, it has to be remembered that the pressure resistance is only a very small component of the total resistance, and the pressure resistance of the Suboff is quite small.

The overall increase in resistance coefficient due to the sail on the Suboff being approximately 5%.

Figure 12 is a plot of resistance coefficient, C_∇, as a function of the Reynolds number, Re_∇, for the U209 without a sail (Case 4), with a small “Suboff” sail (Case 5), and with the larger U209 sail (Case 6). In each case, there is the same casing.

The increase in wetted surface area by adding the Suboff sail to U209 with the casing is approximately 2%, and the increase in friction resistance coefficient is approximately 3%. The increase in pressure resistance coefficient when adding the Suboff sail to the U209 with casing is approximately 5%. The increase in total resistance coefficient when adding the Suboff sail to the U209 with casing is approximately 4%.

The increase in wetted surface area by adding the U209 sail to the U209 with only the casing is approximately 7.5%, and the corresponding increase in friction resistance coefficient is approximately 13%. The increase in pressure resistance coefficient when adding the U209 sail to the U209 hull with casing is approximately 23%. The resulting total increase in resistance coefficient for the U209 with the U209 sail, compared to the U209 with the casing alone, is approximately 14%.

3.4. Comparison with Empirical Prediction

The results for all six cases are compared with empirical predictions made using the method described in [9] in

Table 3. Resistance coefficient components (×10⁻³).

	Case 1			Case 2			Case 3			Case 4			Case 5			Case 6
	Emp	CFD	%	Emp	CFD	%	Emp	CFD	%	Emp	CFD	%	Emp	CFD	%	Emp	CFD	%
F	19.5	21.2	−8	19.9	22.4	−11	20.3	22.1	−8	21.5	24.0	−10	22.2	24.6	−10	23.8	27.1	−12
P	2.7	2.1	25	3.3	2.4	36	3.0	2.4	26	4.0	2.9	40	4.3	3.2	37	4.9	3.6	38
Total	22.2	23.3	−5	23.3	24.9	−6	23.3	24.5	−5	25.5	26.9	−5	26.6	27.8	−4	28.7	30.7	−6

and Figure 13 for a volumetric Reynolds number of 6.21 × 10⁶.

As can be seen from Table 3 and Figure 9, the empirical resistance method described in [9] consistently underpredicts the total frictional resistance coefficient by about 10%. The equation for friction resistance used in [9] is given as Equation (6).

$$C_{F_{form}}={\displaystyle \frac{0.075}{{{(\mathrm{l}\mathrm{o}\mathrm{g}}_{10}{R}_e-2)}^2}}\left[1+K_F\right]$$

(6)

where:

$$K_F=C_1{\displaystyle \frac{D}{L}}$$

(7)

The factor $K_F$ accounts for the friction form resistance of the axisymmetric hull form, and was added to account for the effect of L/D, based on CFD data for three axisymmetric hull forms obtained by Leong [35]. The value of $C_1$ used in [9] was 0.3 based on the results from Leong [35]. Further work is required to investigate whether modifications to Equations (6) and (7) are required to improve the empirical estimation of frictional resistance coefficient given in [9].

Note that Equation (6) gives the frictional resistance coefficient, $C_{F_{form}}$, non-dimensionalized using the wetted surface, rather than the usable volume. Equation (8) is used to convert this to C_∇, which is non-dimensionalized using the volume.

$$C_{\mathsf{\nabla }}=C_{F_{form}}\left({\displaystyle \frac{S}{{\mathsf{\nabla }}^{2/3}}}\right)$$

(8)

As noted above, the volume used in Equation (8) is the usable volume, which is the same for each of the submarine hull forms, and hence the results given in the paper are internally consistent and can be compared meaningfully.

Also, as can be seen from Table 3 and Figure 13, the empirical resistance method described in [9] significantly overpredicts the total pressure resistance coefficient by approximately 25–40%.

In [9], the pressure resistance coefficient is obtained from Equation (9).

$$C_P=K_P{C}_{F_{form}}$$

(9)

$K_P$ is given by Equation (10).

$$K_p=\left[{\xi }_{hull}+{\xi }_{PMB}{\left({\displaystyle \frac{L_{PMB}}{L}}\right)}^{n_{PMB}}\right]{\left({\displaystyle \frac{L}{D}}\right)}^n$$

(10)

where $L_{PMB}$ is the length of the parallel middle body, and the values of the constants can be obtained from

Table 4. Values of constants for Equation (10).

Constant	Value
ξhull	4
ξPMB	15
nPMB	3
n	−1.8

.

In Equation (10), the first term in the square brackets represents the pressure resistance on a hull with no parallel middle body and the second term accounts for parallel middle body. The values of the constants given in Table 4 have been obtained from data provided by [35].

Further work is required to investigate whether modifications are required to Equation (10) and/or the constants in Table 4 to improve the empirical estimate of pressure resistance coefficient presented in [9].

However, it is worth noting that the estimate of the total resistance coefficient obtained using the empirical method is consistently only about 5% lower than the values obtained from the present CFD method.

4. Concluding Remarks

The resistance coefficients for two different submarine geometries, scaled to have the same useable volume, were obtained using CFD. One of these is the well-known Suboff geometry, an axisymmetric hull with a parallel middle body, and the other the U209 hull form, designed with surface, as well as submerged, operations in mind.

The total resistance coefficient for the U209 bare hull is approximately 6% higher than that for the Suboff bare hull. This quantifies the advantage, in terms of submerged resistance, of a hull form specifically designed for underwater performance, compared to an older design where operation on the surface was also considered.

When a casing was added to the U209 geometry the increase in total resistance coefficient was approximately 8%. This quantifies clearly the disadvantage, in terms of submerged resistance, of fitting a casing to a submarine. Part of this increase is due to an increase in wetted surface area, and that can be obtained simply by allowing for the increased area. The increase due to pressure resistance is about 18%, which compares with the value of 15% derived empirically and given in [9]. Whilst there may be practical reasons for a casing on a submarine, the designer needs to be aware of the hydrodynamic disadvantage to doing this.

When the Suboff sail was added to the Suboff geometry, the overall increase in total resistance coefficient was approximately 5%.

When the Suboff sail was added to the U209 geometry, with a casing, the overall increase in resistance coefficient was approximately 4%, and when the U209 sail was added instead, the increase in resistance coefficient was approximately 14%. This is because the U209 sail is much larger than the Suboff sail.

Thus, the disadvantage, from a resistance point of view, of fitting a small or a large sail to the submarine hull can be quantified.

The empirical resistance prediction method given in [9] consistently underpredicts the friction resistance coefficient by approximately 10% and overpredicts the pressure resistance coefficient by approximately 25–40%. However, the empirical prediction of the total resistance coefficient is only about 5% too low. More work is required to improve the empirical resistance prediction method.