**4.1 Rudder parameters**

The rudders parameters include root chord, half span, aerofoil and backswept, which are shown in Figure 13. As defined in the [23], the chord is denoted by*C* , the distance from the leading edge to trailing edge in a given two-dimensional section. The chord is measured in parallel with the section at the root of the rudder. In general, the chord can vary along the span, in which case the geometric mean chord,*C* , is used in computations unless noted[21]. The *C* is defined based on Figure 14 as

$$
\overline{C} = \frac{\mathbf{C}\_{\iota} + \mathbf{C}\_{r}}{2} \tag{10}
$$

Fig. 13. Rudders parameters

Fig. 14. Foil section and hydrodynamic force

Hydrodynamic Characteristics of the Main Parts of a Hybrid-Driven Underwater Glider PETREL 51

than other sections which is shown in the Figure 18. The NACA0012 section with angle of


> NACA0008 NACA0012 NACA0016 NACA0020 NACA0025


NACA0008 NACA0012 NACA0016 NACA0020 NACA0025

stall about 20° and a higher L/D was adopted by the Hybrid glider PETREL.

Fig. 15. CFD meshing results


> 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Drag Coefficient

 *C* D

Fig. 16. The relationship of profile lift coefficient and angle of attack

Fig. 17. The relationship of profile drag coefficient and angle of attack

Lift Coefficient

*C*L

The semi-span, denoted by / 2 *b* , measures the distance from the rudder root to tip along the line perpendicular to the root section. The span, in this work, is twice as long as the root-totip distance for an isolated plan. The hydrodynamic forces including lift and drag acted on the aerofoil is shown in Figure 14 and can be expressed as

$$\mathcal{L} = \mathbf{1} \{ \mathcal{D}\rho \mathcal{C}\_{\perp} A V^{\perp} \tag{11}$$

$$D = \mathbf{1} \{ \mathbf{2} \, \rho \mathbf{C}\_{\mathrm{o}} A V^{z} \tag{12}$$

Here, ρ is the density of the water; *CL* is the lift coefficient; *CD* is the drag coefficient; *A* is the area of rudder; *V* is the velocity of water; α is the angle of attack. The rudderpost location is expressed by *P* , which is shown in Figure 14.

#### **4.2 Foil section**

The geometry of a rudder is mainly defined by the two-dimensional foil section. The symmetrical foil sections are generally used by the underwater vehicles. Many types of the foil sections are proposed by many countries to improve the hydrodynamic performance. The famous foil sections series include NACA series, HEЖ series, ЦАГИ series, and JFS series [21], among which the four-digit NACA sections are most widely used for underwater vehicle rudders in that it provides the higher lift and the lower drag. The four-digit NACA section series is a low velocity foil sections series, and have a bigger radius of leading edge and a plumpy head section, which is suitable for the rudder of underwater vehicles at low velocity. In this work, the four digit NACA00×× section was used, where the ×× denote the thickness-to-chord ratio. The lift coefficient and drag coefficient of the foil sections can be calculated as

$$C\_{\perp} = \frac{L}{1/2 \,\rho V^2 \mathcal{C}}$$

$$C\_{\rm D} = \frac{D}{1/2 \,\rho V^2 \mathcal{C}}$$

Here, *L* is the profile lift, *D* is the profile drag, *C* is the chord. The NACA0008, NACA0012, NACA0016, NACA0020 and NACA0025 are usually used for the rudders of miniature underwater vehicles, their hydrodynamic characteristics were calculated by using computational fluid dynamics. According to the most often adopted velocity of the autonomous underwater vehicles and the velocity of PETREL in AUV mode, the calculation velocity was determined as 2m/s. An example of CFD meshing result is shown in figure 15, where the unstructured mesh was adopted and the wall of section was made dense. The calculating results were shown in the Figure 16~ Figure 18

The relationship of lift coefficient and angle of attack is illustrated in Figure 16, where we can see that there was a bigger angle of stalling and bigger maximal lift coefficient when the section becomes much thicker. From the figure 17 we can see that the thinner wing section has a lower drag cofficient when the angle of attack is small, but the thicker wing section has a lower drag cofficient when the angle of attack is bigger than a certain critical angle of attack. The NACA0008 section has the maximal L/D and NACA0025 has the minimal L/D than other sections which is shown in the Figure 18. The NACA0012 section with angle of stall about 20° and a higher L/D was adopted by the Hybrid glider PETREL.

Fig. 15. CFD meshing results

50 Autonomous Underwater Vehicles

The semi-span, denoted by / 2 *b* , measures the distance from the rudder root to tip along the line perpendicular to the root section. The span, in this work, is twice as long as the root-totip distance for an isolated plan. The hydrodynamic forces including lift and drag acted on

> <sup>2</sup> 1 2 *L C AV* = ρ

<sup>2</sup> 1 2 *D C AV* = ρ

The geometry of a rudder is mainly defined by the two-dimensional foil section. The symmetrical foil sections are generally used by the underwater vehicles. Many types of the foil sections are proposed by many countries to improve the hydrodynamic performance. The famous foil sections series include NACA series, HEЖ series, ЦАГИ series, and JFS series [21], among which the four-digit NACA sections are most widely used for underwater vehicle rudders in that it provides the higher lift and the lower drag. The four-digit NACA section series is a low velocity foil sections series, and have a bigger radius of leading edge and a plumpy head section, which is suitable for the rudder of underwater vehicles at low velocity. In this work, the four digit NACA00×× section was used, where the ×× denote the thickness-to-chord ratio. The lift coefficient and drag coefficient of the foil sections can be

> <sup>L</sup> <sup>2</sup> 1 2 *<sup>L</sup> <sup>C</sup>*

<sup>D</sup> <sup>2</sup> 1 2 *<sup>D</sup> <sup>C</sup>*

Here, *L* is the profile lift, *D* is the profile drag, *C* is the chord. The NACA0008, NACA0012, NACA0016, NACA0020 and NACA0025 are usually used for the rudders of miniature underwater vehicles, their hydrodynamic characteristics were calculated by using computational fluid dynamics. According to the most often adopted velocity of the autonomous underwater vehicles and the velocity of PETREL in AUV mode, the calculation velocity was determined as 2m/s. An example of CFD meshing result is shown in figure 15, where the unstructured mesh was adopted and the wall of section was made dense. The

The relationship of lift coefficient and angle of attack is illustrated in Figure 16, where we can see that there was a bigger angle of stalling and bigger maximal lift coefficient when the section becomes much thicker. From the figure 17 we can see that the thinner wing section has a lower drag cofficient when the angle of attack is small, but the thicker wing section has a lower drag cofficient when the angle of attack is bigger than a certain critical angle of attack. The NACA0008 section has the maximal L/D and NACA0025 has the minimal L/D

ρ*V C* <sup>=</sup>

ρ*V C* <sup>=</sup>

is the density of the water; *CL* is the lift coefficient; *CD* is the drag coefficient; *A* is

α

*<sup>L</sup>* (11)

*<sup>D</sup>* (12)

is the angle of attack. The rudderpost

(13)

(14)

the aerofoil is shown in Figure 14 and can be expressed as

the area of rudder; *V* is the velocity of water;

location is expressed by *P* , which is shown in Figure 14.

calculating results were shown in the Figure 16~ Figure 18

Here, ρ

**4.2 Foil section** 

calculated as

Fig. 16. The relationship of profile lift coefficient and angle of attack

Fig. 17. The relationship of profile drag coefficient and angle of attack

Hydrodynamic Characteristics of the Main Parts of a Hybrid-Driven Underwater Glider PETREL 53

parameters Tip chord *Ct* Root chord*Cr* Semi-span / 2 *b* section Value 125mm 200mm 120mm NACA0012

The hinge moment is produced by a hydrodynamic force about the hinge line of a control surface. It makes an impact on maneuverability of the underwater vehicle in that the hinge moment must be overcome during steering. The bigger hinge moment will make the turning

The hydrodynamic performance of three dimension rudders at different angles of attack

velocity of rudders become slowly and make the control action slow-witted.

was simulated by using CFD methods. The inlet velocity was set to be 2m/s.

Fig. 20. Pressure distribution chart when angle of attack is 20°

0 0.2 0.4

0.6

*C*L,*C*D,*L*/10

*D*

0.8 1

1.2

presssure *P/Pa*

2.6×103 1.4×103 2.6×102 -5.2×10<sup>2</sup> -1.7×10<sup>3</sup> -2.5×10<sup>3</sup> -4.1×10<sup>3</sup> -5.2×10<sup>3</sup>

Fig. 21. *C*<sup>L</sup> ,*C*<sup>D</sup> and /10 *L D* variation curve with different angles of attack

L/10D

*C*<sup>L</sup> *C*<sup>D</sup>

0 4 8 12 16 20 24 28 32 36 40 angle of attack (°)

α

Table 7. The parameters of the rudder

**4.4 Hinge moment analysis** 

Fig. 18. The relationship of L/D and angle of attack

#### **4.3 Area of rudder calculation**

The area of rudder as an important parameter for maneuverability of the underwater vehicle is related to the size and shape of the body. The area of rudder can be design by cut and try method, master model method and empirical formula design method. For the high maneuverable ship, the control surfaces can be designed according to Det Norske Veritas, (DNV) rudder sizing rules [24].

$$Area = \frac{DL}{100} [1 + 25(\frac{B}{L})^2] \tag{15}$$

Here, *D* is the diameter of the vehicle, *L* is the length of the vehicle, *B* is the width of the vehicle, and *B D*= for revolution body. It suggested 30% increase in area if rudders in front of the propeller, and then increased by an additional 50% to match empirical data from other underwater vehicles by the DNV rules. The turn diameter induced by single rudder is about triple-length of the vehicle in terms of the design by DNV rules. The rudder design for the hybrid glider PETREL is shown in Figure 19 and the parameters of the rudder shown in table 7.

Fig. 19. The photo of the rudder of PETREL


Table 7. The parameters of the rudder

#### **4.4 Hinge moment analysis**

52 Autonomous Underwater Vehicles


The area of rudder as an important parameter for maneuverability of the underwater vehicle is related to the size and shape of the body. The area of rudder can be design by cut and try method, master model method and empirical formula design method. For the high maneuverable ship, the control surfaces can be designed according to Det Norske Veritas,

<sup>2</sup> [1 25( ) ] <sup>100</sup>

*L*

*DL B Area*

= +

Here, *D* is the diameter of the vehicle, *L* is the length of the vehicle, *B* is the width of the vehicle, and *B D*= for revolution body. It suggested 30% increase in area if rudders in front of the propeller, and then increased by an additional 50% to match empirical data from other underwater vehicles by the DNV rules. The turn diameter induced by single rudder is about triple-length of the vehicle in terms of the design by DNV rules. The rudder design for the hybrid glider PETREL is shown in Figure 19 and the parameters of the rudder shown in

NACA0008 NACA0012 NACA0016 NACA0020 NACA0025

(15)


**4.3 Area of rudder calculation** 

(DNV) rudder sizing rules [24].

Fig. 19. The photo of the rudder of PETREL

table 7.

Fig. 18. The relationship of L/D and angle of attack

*L/D*

The hinge moment is produced by a hydrodynamic force about the hinge line of a control surface. It makes an impact on maneuverability of the underwater vehicle in that the hinge moment must be overcome during steering. The bigger hinge moment will make the turning velocity of rudders become slowly and make the control action slow-witted.

The hydrodynamic performance of three dimension rudders at different angles of attack was simulated by using CFD methods. The inlet velocity was set to be 2m/s.

Fig. 20. Pressure distribution chart when angle of attack is 20°

Fig. 21. *C*<sup>L</sup> ,*C*<sup>D</sup> and /10 *L D* variation curve with different angles of attack

Hydrodynamic Characteristics of the Main Parts of a Hybrid-Driven Underwater Glider PETREL 55

1 <sup>2</sup> 2 *D VC A* = ρ

ρ

velocity of the vehicle in m/s, *A* is the reference area in m², *CD* is the drag coefficient

Figure 24 shows the overall drag of the two models in the glide mode. The propeller in this mode doesn't rotate. The overall drags of two models are calculated by CFD firstly and then are fitted by the semi-empirical formulae (16). The drag coefficients of two models are respectively 0.32 and 0.26. The average relative error of overall drag between CFD and semiempirical formulae is 4.7%. The overall drag increase 21%-26% with the propeller shroud compared with the model two according to the CFD computation results, so the shroud greatly increased the drag of the hybrid in glide mode. The drag components of the mode1 at the speed of 0.5m/s without angle of attack are shown in Fig. 25. The drag on the body, rudders and wings is mainly viscous forces, while the drags on the propeller, shroud and GPS antenna pole are primarily the pressure forces,. As shown in Figure 26, the propeller and its shroud make up over 30% of total resistance and the percentage will increase with the increment of the velocity. The reason for the high percentage is because of the great pressure drags on the shroud in the glide mode. The local velocity streamline diagram near the shroud of model one shown in the Figure 27. In the Figure, we can see that in *v* and *P*in are the velocity and pressure inside the shroud of water, out *v* and *P*out are the velocity and pressure outside the shroud of water. Because the propeller doesn't rotate in the glide mode, the velocity of water inside the shroud is slower than that outside the shroud, so there exits out in *v v* > . According to the Bernoulli equation there was *P P* in out > , so a pressure force *f* is produced by the pressure difference. The percentage of the shroud drag to total resistance is 26%-35% at the different speed due to the pressure force in the

*<sup>D</sup>* (16)

is the density of water in kg/m³,*V* is the

(a) model 1 (b) model 2

The drag on the vehicle can be expresses as equation (16).

(dimensionless). The reference area *A* of the PETREL is 0.096m2.

Fig. 23. The models studied in the paper

**5.2 Effect of shroud on the glide drag** 

Where, *D* is the force of drag in Newton,

glide mode.

Fig. 22. Hinge moment with different angles of attack

The pressure distribution of the rudder is illustrated in Figure 20, where we can find that there is higher pressure on the front flow face and was local higher pressure area on the back flow face of the tail, that means there exist roundabout flow at the tail of the rudder. Figure 21 shows the relationship between lift, drag and angle of attack. The relationship between *L/D* and angle of attack is also illustrated in the figure 21, the *L/D* value reduces ten times for the same scale with other two curves. It can be known that the maximal lift to drag ratio was about 8° and the angle of stall about 34°, so the angles of stall of three dimensional rudders are greater than two-dimension section. The hinge moment of rudders with different axis of rudder position is shown in Figure 22, where we can seen that the hinge moment varied with the angle of attack. The hinge moments are little while 0.4 *P c* = for the rudder we design no matter how the angle of attack changed.

#### **4.5 Results and discussion**

Aiming at the key problems of the rudder design for autonomous underwater vehicle,the hydrodynamic characteristic of the NACA00xx series section at different angles of attack were simulated when velocity was 2m/s by using the two-dimensional computational fluid dynamics (CFD). For the rudder we design, the stall angle is about 34° for the three dimensional rudders and about 20° for the two-dimensional foil section, so the angle of stall of three dimensional rudders are greater than two-dimension foil section. The area of the rudder of PETREL was calculated using the DNV rules;The hinge moments are little when 0.4 *P c* = for the rudder we design no matter how the angle of attack changed.
