CFD analysis of the hull form of a manned submersible for minimizing resistance

Abstract

A manned submersible is an underwater vehicle that is capable of operating independently. During the design process, estimating resistance and power is key to choosing the right propeller. The shape improvement of the manned submersible has a significant impact on the submersible's energy utilization and endurance. Using computational fluid dynamics (CFD), it is possible to accurately calculate the forces acting on manned submersibles. A CFD flow analysis framework has been formulated and the process applied to circular bodies of different lengths. To validate the results of the numerical analysis, the results are compared with experimental results available in the literature. The validated CFD framework was used to conduct the flow analysis on manned submersibles of different sizes and shapes to minimize the resistance. The optimized shape resistance is further utilized for thruster selection and to estimate the descent speed, ascent speed, descent time and ascent time for a 6000 m rated vehicle.

Appendices

Appendix 1: AUV hull shape—Myring equation

A Myring-type body is used to model the AUV hull shapes. The front side, nose section, back side, and tail section are connected by a middle cylindrical section. The parameters used to generate the AUV hull shape are a, b, c, d, n, and θ. The typical shape of the AUV hull is shown below (Fig. 15).

Fig. 15

AUV hull shape based on Myring equation

The nose section is described by a modified semi-elliptical radius equation:

$$r\left(x\right)= \frac{1}{2} d{\left[1-{\left(\frac{x-a}{a}\right)}^{2}\right]}{^{\frac{1}{n}}}.$$

The tail section is described by a cubic equation:

$$r\left(x\right)= \frac{1}{2} d-\left(\frac{3d}{2{c}^{2}}-\frac{\mathrm{tan}\theta }{c}\right){\left(x-a-b\right)}^{2}+\left(\frac{d}{{c}^{3}}-\frac{\mathrm{tan}\theta }{{c}^{2}}\right){\left(x-a-b\right)}^{3},$$

where x is the axial distance to the nose tip, a, b and c are the length of the nose, middle and tail sections, respectively, d is the middle hull diameter, n is the index of the nose shape, θ is the tail semi-angle.

Appendix 2: Descent and ascent scheme of the manned submersible

See Fig. 16.

Fig. 16

Descent and ascent scheme of the manned submersible

Appendix 3: Compressibility of spherical pressure hulls and syntactic foam

• 1.
  $${\mathrm{Seawater \; compressibility }C}_{\mathrm{sw}}=\frac{\left(\frac{4}{3}\right) \uppi {R}_{0}^{3} (1-\frac{{\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{D}}})}{\mathrm{h}}.$$
• 2.
  $${\mathrm{Pressure \; hull \; compressibility }C}_{\mathrm{hull}} =\frac{4 \uppi }{3h}\left\{{\left({R}_{\mathrm{m}}+t/2\right)}^{3}- {\left[{R}_{\mathrm{m}}+\frac{t}{2}- (1-\nu )\frac{{\mathrm{PR}}_{\mathrm{m}}^{2}}{2\mathrm{Et}}\right]}^{3}\right\}.$$
• 3.
  $$\begin{aligned} & \mathrm{The \; amount \; of \; sea}-\mathrm{water \; that \; must \; be \; discharged}\\ & \;\mathrm{ from \; VBT \; or \; submerged \; weight \; of \; drop \; weights \; that}\\ & \;\mathrm{must \; be \; released \; to \; attain \; neutral \; buoyancy }= {C}_{\mathrm{sw}}- {C}_{\mathrm{hull}},\end{aligned}$$

  where ${\rho }_{\mathrm{s}}$ is the density of seawater at the surface (1025 kg/m³), ${\rho }_{\mathrm{D}}$ is the density of seawater at the operating depth (1054 kg/m³ at 6000 m), R_o is the outer radius of the sphere, R_m is the mean radius of the sphere, t is the thickness of the pressure hull, h is the submerged depth, P is the hydrostatic pressure at the submerged depth “h”, E is the elastic modulus of pressure hull material, and ν is the Poisson’s ratio.

Parameters	Unit	SPH	VBT	PPT
Mean radius of the sphere	mm	1090	360	336
Thickness of the sphere	mm	80	30	26
Elastic modulus	GPa	115	115	115
Poisson's ratio	–	0.3	0.3	0.3
Operating depth	m	6000	6000	6000
Operating pressure	MPa	60.6	60.6	60.6
Density at the surface	kg/m3	1025	1025	1025
Density at the operating depth	kg/m3	1054	1054	1054
Change in volume per meter depth of sea-water	m3/m	2.4876E−05	8.9619E−07	7.2864E−07
Change in volume per meter depth of unstiffened sphere	m3/m	7.3075E−06	2.3417E−07	2.0383E−07
Ballast change per unit depth	m3/m	1.7568E−05	6.6203E−07	5.2481E−07
Ballast change at operating depth	m3	0.10541	0.00397	0.00315
Number of pressure hull, VBT, PPT	–	1	2	3
Ballast change at operating depth	m3	0.1054	0.0079	0.0094
Ballast to be adjusted to attain neutral buoyancy		0.1228 m3 or 129.43 kg
Syntactic foam
Volume of syntactic foam in air/at surface of water	m3	15.25
Reduced volume of syntactic foam at 6000 MSW	m3	14.96
Change in volume	m3	0.290
Reduction in buoyancy	kg	305.66
Buoyancy due to increase in density at 6000 m	kg	136.59
Net buoyancy reduced	kg	169.07
Total ballast to be adjusted to attain neutral buoyancy		298.50 kg

Appendix 4: Initial mesh height and boundary layer thickness calculation

• 1.
  $${\mathrm{Re}}=\frac{\rho \mathrm{VL}}{ \upmu }$$
• 2.
  $$\nu =\frac{\mu }{\rho },$$
• 3.
  $${y}^{+}=\frac{\rho \cdot {U}_{\tau } \cdot \Delta {y}_{1}}{\mu },$$
• 4.
  $${U}_{\tau }= \sqrt{\frac{{\tau }_{\mathrm{w}}}{\rho }},$$
• 5.
  $${\tau }_{\mathrm{w}}= \frac{1 }{2}{C}_{\mathrm{f}}\rho {V}^{2},$$
• 6.
  $${C}_{\mathrm{f}}=0.0079 {\mathrm{Re}}^{-0.25}(\mathrm{for \; internal \; flows}),$$
• 7.
  $${C}_{\mathrm{f}}=0.058 {\mathrm{Re}}^{-0.2}(\mathrm{for \; external \; flows}),$$
• 8.
  $$\delta =0.37 {x}_{\mathrm{cr}}{\mathrm{Re}}^{-0.2},$$

  where Re is the Reynolds number, $\nu$ is the kinematic viscosity = 1.003 × ${10}^{-6}$ m²/s, μ is the dynamic viscosity in N-s/m², ρ is fluid density = 1025 kg/m³, V is the velocity of vehicle/fluid in m/s, L is the characteristic length in m, ${y}^{+}$ is the dimensionless number of boundary layer thickness, ${U}_{\tau }$ is the frictional velocity in m/s, ${\tau }_{\mathrm{w}}$ is wall shear stress in N/m², $\Delta {y}_{1}$ is the estimated first cell height in m, ${C}_{\mathrm{f}}$ is the skin friction coefficient, δ is the boundary layer thickness in mm, ${x}_{\mathrm{cr}}$ is the critical length of the boundary layer in mm.

Appendix 5: Estimation of viscous drag based on ITTC-1957 and power

• 1.
  $${R}_{\mathrm{f}}=\frac{1}{2}{C}_{\mathrm{f}}\rho {A}_{\mathrm{w}}{V}^{2},$$
• 2.
  $${C}_{\mathrm{f}}=\frac{0.075}{{({\mathrm{log}}_{10}\mathrm{Re}- 2)}^{2}},$$

  where ${R}_{\mathrm{f}}$ is the viscous drag in N, ${C}_{\mathrm{f}}$ is the skin friction coefficient, ρ is the density of fluid = 1025 kg/m³, V is the velocity of vehicle/fluid in m/s, ${A}_{\mathrm{w}}$ is the wetted surface area in m².

• 3.
  $$\mathrm{Quasi \; propulsive \; coefficient }\left(\mathrm{QPC}\right)=\frac{\mathrm{useful \; power}}{\mathrm{power \; delivered \; to \; the \; propeller} \; \mathrm{thruster}},$$
• 4.
  $$\mathrm{QPC}= {\eta }_{\mathrm{H}}{\eta }_{\mathrm{P}},$$
• 5.
  $$\mathrm{Propulsive} \; \mathrm{Coefficent} \left(\mathrm{PC}\right)= {\eta }_{\mathrm{H}}{\eta }_{\mathrm{P}}{\eta }_{\mathrm{M}},$$
• 6.
  $${\eta }_{\mathrm{P}}= {\eta }_{\mathrm{O}}{\eta }_{\mathrm{R}},$$
• 7.
  $${\eta }_{\mathrm{O}}={(0.9)}^{n},$$
• 8.
  $${\eta }_{\mathrm{M}}=\frac{\mathrm{Propulsive} \; \mathrm{power}}{\mathrm{Shaft} \; \mathrm{power}},$$

  where ${\eta }_{\mathrm{H}}$ is the hull efficiency = 1.00 for a well-designed stern and propeller (Allmendinger 1990), ${\eta }_{\mathrm{P}}$ is the propeller efficiency, ${\eta }_{\mathrm{O}}$ is the open water propeller efficiency $=0.73 @3 \mathrm{knots}$, ${\eta }_{\mathrm{R}}$ is the relative rotational efficiency = 0.95 to 1.00 for submersibles (Allmendinger 1990), ${\eta }_{\mathrm{M}}$ is the machiner efficiency = 0.95 to 0.99 (it depends on the shaft length and number of bearings) (Allmendinger 1990), n is the speed of the submersible in knots.

Appendix 6: Different thruster models and their details

Tecnadyne
S. No	Models	Input power	Bollard thrust generated	Thrust generated at 3 knots speed	Effective or useful power	Weight in air	Diameter of propeller	Ratio between effective power/Input power
		kW	kgf	kgf	kW	kg	mm	---
1	Model 260	0.350	5.40	3.81	0.06	0.9	76	0.16
2	Model 280	0.350	6.10	4.72	0.07	1.0	115	0.20
3	Model 300	0.500	8.20	5.44	0.08	1.0	88	0.16
4	Model 521	0.500	10.40	7.14	0.11	1.8	121	0.21
5	Model 540	0.500	10.00	6.67	0.10	2.3	150	0.20
6	Model 560	0.975	17.30	11.79	0.17	2.3	121	0.18
7	Model 580 - DD	0.975	17.30	12.61	0.19	1.9	150	0.19
8	Model 1020	1.10	25.00	17.24	0.25	4.8	153	0.23
9	Model 1040	1.25	25.00	17.01	0.25	4.3	203	0.20
10	Model 1060 - DD	2.20	48.00	38.10	0.56	6.8	181	0.25
11	Model 1080 - DD	2.20	48.00	34.99	0.51	*	203	0.23
12	Model 2020	6.20	118.00	81.65	1.20	12.2	246	0.19
13	Model 2020-DD	6.00	118.00	86.02	1.27	19.0	246	0.21
14	Model 2040	5.20	85.00	60.00	0.88	10.0	254	0.17
15	Model 8020	12.60	230.00	163.29	2.40	26.3	305	0.19
16	Model 8040	12.10	172.00	121.11	1.78	23.5	339	0.15
*Data is not available in the public domain							Average	0.20

Innerspace Corporation
S. No	Models	Input power	Bollard thrust generated	Thrust generated at 3 knots speed	Effective or useful power	Weight in air	Diameter of propeller	Ratio between effective power/Input power
		kW	kgf	kgf	kW	kg	mm	---
1	1002H - 14150	6.00	139.00	121.11	1.78	27.7	236	0.30
2	1002H - 14300	9.50	190.00	165.11	2.43	27.7	236	0.26
3	1002H - 14550	12.50	228.00	197.31	2.90	27.7	236	0.23
4	1004B - 3150	1.50	24.00	20.64	0.30	6.4	109	0.20
5	1004B - 3300	2.70	36.00	30.96	0.46	6.4	109	0.17
6	H106 - 9150	4.30	64.00	57.61	0.85	11.3	236	0.20
7	H106 - 9300	7.48	93.00	82.55	1.21	11.3	236	0.16
							Average	0.22

Forum Energy Technologies
S. No	Models	Input power	Bollard thrust generated	Thrust generated at 3 knots speed	Effective or useful power	Weight in air	Diameter of propeller	Ratio between effective power/Input power
		kW	kgf	kgf	kW	kg	mm	---
1	SPE - 75	1.62	26.00	18.96**	0.28	3.3	144	0.17
2	SPE - 180	2.50	45.00	32.81**	0.48	5.9	178	0.19
3	SPE - 250	6.20	100.00	72.90**	1.07	13.0	246	0.17
**Calculated based on an open water efficiency of 0.73							Average	0.18

Appendix 7: Flow analysis plots in sway and heave directions

See Figs. 17, 18 and 19.

Fig. 17

Flow trajectories during sway motion (at 1 knot) for models 1, 2, 3 and 4

Fig. 18

Flow trajectories during descent motion (at 1 knot) for models 1, 2, 3 and 4

Fig. 19

Flow trajectories during ascent motion (at 1 knot) for models 1, 2, 3 and 4