Optimizing the shape and speed performance of the boats

Optimizing the shape and speed performance of the boats A report submitted to Modelling Week and Study Group Meeting on Industrial Problems-2013 by

AWL Pubudu Thilan1, Baikuntha Nath Sahu2, Darshan Lal3, E. Suresh Reddy3, K.M Abdushukoor3, Mary Reena KE3 (1. Lecturer in Mathematics, University of Ruhuna 2. I. MSc. in Physics, NIT Rourkela, 3. Research Scholar, NIT Calicut)

Under the supervision of NGA Karunathilake

University of Kelaniya, Sri Lanka

Organized by

and Department of Mathematics NIT Calicut

Industrial Mathematics Group IIT Mumbai

Modelling Week and Study Group Meeting on Industrial Problems December, 2013

Acknowledgement First of all we would like to give our sincere thanks to Mr. N.G.A. Karunathilaka, Senior Lecturer of the Department of Mathematics, University of Kelaniya, Sri Lanka, for guiding and encouraging us in numerous ways to achieve a successful outcome. Also thank to Dr. Aleksandra GRM, University Ljubljana for his time and effort devoted to help us in overcoming the difficulties during our work. We gratefully acknowledge Dr. M.K. Aberathne, Senior Lecturer of Department of Mathematics, University of Ruhuna, Sri Lanka and Mr. Jagath Adipola, Manager, Barramundi Boatyard (PVT) Ltd., Sri Lanka for their contributions. We would like to thank to the organizing committee of the Modelling Week and Study Group Meeting on Industrial Problems 2013, for giving us the opportunity to carry out our research work and allowing us to use resources of the Department of Mathematics, National Institute of Technology Calicut. Finally we would like to thank all those who helped us in many ways to complete this work.

Abstract Most of design of boats have V shape of the main hull. When the boat is having a V hull it is good in stability but not good in maximum velocity. Problem with this type is, when it is loaded the hull sink more into the water due to shape of the hull which resists the boat to go faster. However, boats with flat bottom goes faster than with same engines and loads of V shape boats. In this study we carried out a qualitative study to determine optimum depth of the boat to have more stability and high maximum velocity. According to our numerical simulation results, we can qualitatively conclude that the maximum velocity of the boat can be increased by changing the depth of the boat.

Contents Content Details

Page No

Acknowledgement .............................................................................................................. ii Abstract .............................................................................................................................. iii List of Figures ..................................................................................................................... v 1. Introduction ................................................................................................................... 1 2. Design of the boat and mathematical model .............................................................. 5 2.1 Stability of the boat ........................................................................................................... 3 2.2 Restoring Momentum ........................................................................................................ 6 2.3 Point of center of gravity ................................................................................................... 6 2.4 Point of center of buoyancy ............................................................................................... 8 2.5 Stability function ............................................................................................................... 8 2.6 Boat motion ....................................................................................................................... 9

3. Numerical Simulation ................................................................................................. 11 4. Concluding Remarks .................................................................................................. 13 A. Maxima Code ............................................................................................................. 15 Bibliography

List of Figures 1.1

One day sailing boats . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2.1

A cross section of the boat . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.2

Force of buoyancy equal force of gravity . . . . . . . . . . . . . . . . .

5

2.3

Force of buoyancy equal force of gravity . . . . . . . . . . . . . . . . .

5

2.4

Center of gravity for mechanical equilibrium . . . . . . . . . . . . . .

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2.5

Center of gravity for rotated state . . . . . . . . . . . . . . . . . . . . .

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2.6

Point of center of buoyancy . . . . . . . . . . . . . . . . . . . . . . . .

9

2.7

Force on the boat at maximum speed . . . . . . . . . . . . . . . . . . . 10

3.1

Stability function and velocity function . . . . . . . . . . . . . . . . . 12

v

Chapter 1 Introduction The analysis of the dynamics of a boat advancing into the sea and the corresponding numerical simulations is an interesting and challenging problem. From an engineering point of view it is useful to evaluate the dynamic behavior of a hull in as many configurations as possible. This can be a helpful tool for boat design, where it could be used to improve the hull performance. This task can be accomplished in two (complementary) ways: experimentally in a towing tank or numerically by Computational Fluid Dynamics (CFD) simulations [3]. Not surprisingly, experimental measurements are usually very expensive since they require the construction of as many scale-models as the number of configurations to be analysed and also technical equipments and facilities are particularly onerous. Furthermore it is possible to reproduce only a limited range of configurations due to practical limitations with the experimental set-up. For these reasons, numerical computations are becoming more and more relevant in this field and much effort is being devoted in the industrial and academic research to make them as accurate as possible. When a numerical simulation approach is used, it is easy to switch from a model to another one and the spectrum of configurations that it is possible 1

2 to analyze is a priori unlimited. On the other hand, the physical problem is very complex and the most advanced mathematical models have to be used to simulate the problem correctly [1]. In this work we consider a ribs (one day sailing boat) produced by Barramundi Boatyard (PVT ) Ltd. Sri Lanka (see Figure 1.1). The V-shape bottom of the hull shape of the boat is designed in order to achieve the maximum stability of the boat in the sea. However, empirical results prove that the V-shape hull bottom drastically reduces the maximum speed of the boat compare to the plat bottom hull. The company wants to optimize the shape of the bottom hull in order to achieve the maximum speed keeping the maximum stability of the boat in the sea.

Figure 1.1 One day sailing boats

Throughout this work we construct a mathematical model for the description of the stability of the boat and reshape the bottom hull cutting the edge of the V-shape bottom in order to achieve maximum speed with keeping the stability. In what follows, in the Chapter 2 we introduce our design of the boat and the mathematical model based on the stability of the boat and the dynamic of the boat. In the Chapter 3 we present the numerical simulation. The limitations of the model and rooms for the improvements are also mentioned in the Chapter 4.

Chapter 2 Design of the Boat and the Mathematical Model 2.1 Stability of the boat As the first approach we approximate the topography of the hull through a two dimensional geometry, taking a cross section of the boat. Also we assume that the width of the top of the boat (2b) and width of the bottom of the boat (2a) are fixed. We want to determine the optimal depth of the boat (h) that provides the stability and the maximum velocity to the boat (see Figure 2.1).

Figure 2.1 A cross section of the boat

3

2.1 Stability of the boat

4

When a boat is floating at rest in calm water, it is acted upon by two sets of forces: 1. the downward force of gravity, and 2. the upward force of buoyancy. The force of gravity is the result of a combination of all downward forces including the weights of all parts of the boat’s structure, all the equipment, cargo, fuel and personnel. This combined force is the weight of the boat, and may be considered as a single force which acts downward through a single point called the center of gravity (G). This force is equal to the boat’s displacement and usually lies on the boat’s centerline near the boat’s middle section under normal trim conditions. The force of buoyancy is also a combined force which results from the pressure of seawater on all parts off the boat’s hull below the waterline. At almost every point on the submersed hull, the pressure of the seawater acting on the hull can be broken into two perpendicular components, one in a horizontal direction and another in a vertical direction. Horizontal pressures on the hull of a boat cancel each other under normal conditions, as there are equal forces acting in opposite directions. However, the vertical pressures add up to a single force of buoyancy equal to the boat’s displacement in tons that may be regarded as acting vertically upward through a single point called the center of buoyancy (B). The center of buoyancy is the geometric center of the boat’s underwater body and lies on the centerline and usually near the middle section when the boat is on an even keel. Its vertical height above the keel is usually a little more than half the draft of the vessel [2]. When a boat is at rest on an even keel in calm water, the forces of buoyancy (B) and gravity (G) are equal and opposite, and lie in the same vertical line (see Figure 2.2).

2.1 Stability of the boat

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Figure 2.2 Force of buoyancy equal force of gravity A boat may be disturbed from rest by conditions which tend to make it heel over to an angle. These include such things as wave and wind action, forces during a turn, shifting of weights or location of weights off-center. When a disturbing force exerts an inclining moment to the vessel, the boat’s underwater body changes shape. The center of the underwater volume is shifted in the direction of the heel which causes the center of buoyancy to relocate off of the vessel’s centerline (originally at B) and move to the geometric center of the new underwater body (at B’). As a result, the lines of action of the forces of buoyancy and gravity are no longer acting in the same vertical line, but are separated thereby creating what is called a moment that wants to restore the boat back onto an even keel as shown below.

Figure 2.3 Force of buoyancy equal force of gravity When boat is rotated away from such an equilibrium orientation, then volume

2.2 Restoring momentum

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of displaced water will change then shape in such away on to shift then the center of buoyancy moves to the other side of the center of gravity, reversing thereby then direction of the moment of force to restore the equilibrium.

2.2 Restoring momentum If the load is fixed, then the buoyancy force B is a constant. Then, the restoring momentum is given by M = (GB sin α)B.

(2.1)

In here, sin α ≪ 1 and then sin α ≈ 1. It indicates that M ∝ GB. As a consequence of that we get maximum resorting momentum for the largest GB. Hence, the stability is directly related to the length of GB and it can mathematically be denoted as fStability ∝ GB.

(2.2)

Since h is included in GB, we can qualitatively argue that fStability ∝ fST(h) .

(2.3)

Finally our objective is reduced to get the hoptimum = Maximize{ fST(h) }. h∈R

(2.4)

2.3 Point of center of gravity In x′ y′ coordinate system (see Figure 2.4), the coordinates of the center of gravity ′ ). By using the definition of center of the boat at mechanical equillibrium is G(0, YG

2.3 Point of center of gravity

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′ of gravity, its y coordinate YG can be written down as: ∫ ∫ y dm dx dy ′ YG = ∫ ∫ dm dx dy ∫ h ∫ h ρ ρ y y dy + h(2b)ρ + dy 0·ρ+ cos ϕ cos ϕ 0 0 = ∫ h ∫ h ρ ρ 2aρ + dy + 2aρ + dy 0 cos ϕ 0 cos ϕ 2 cos ϕbh + h2 = . 2(a + b) cos ϕ + 2h

Figure 2.4 Center of gravity for mechanical equilibrium

For a rotated state (see Figure 2.5), new coordinates of the center of gravity can easily be obtained in the use of the transformation,       cos α sin α   0  XG           .   =       − sin α cos α YG′  YG

2.4 Point of center of buoyancy

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Figure 2.5 Center of gravity for rotated state

2.4 Point of center of buoyancy The coordinates of the point of center of buoyancy (XB , YB ) are defined as: ∫ ∫ xρF dx dy XB = ∫ ∫ , ρF dx dy ∫ ∫ yρF dx dy ∫ ∫ . YB = ρF dx dy

(2.5)

(2.6)

Two integral are difficult to evaluate for this geometry by hand. Therefore, the evaluation was carried out using symbolic manipulation software known as Maxima.

2.5 Stability function Since we know (XB , YB ) and (XG , YG ), with the use of Pythagorean theorem we can calculate the distance GB as GB =

√ (XB − XG )2 + (YB − YG )2 .

(2.7)

2.6 Boat motion

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Figure 2.6 Point of center of buoyancy We have already shown that stability is directly proportional to the GB distance. Therefore, for the purpose of getting different stabilities, one can change GB distance. At the same time, GB depends on the height of the boat (h) and hence it can be used to get different stabilities. For instance, by increasing (h), the distance between two points G and B can be increasedease and it will increase the stability as well. Therefore, we can qualitatively argue that the stability can be improved by changing the depth of the boat. However, while increasing h for more stability it will automatically increase the surface area of the boat under water or immersed area (A). It will destroy the simplicity of the approach for more stable boat formation. The reason is that large immersed area will increase the drag force or the force against the motion and it will reduce the speed of the boat.

2.6 Boat motion In the absent of wind forces, only two forces are acting on the boat at the maximum speed (See Figure 2.7). One is the force generated by the engine (H) and the other

2.6 Boat motion

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one is the drag force (FD ). Applying Newton’s second law, we have F = ma H − FD = m(0) FD = H 1 CD AV 2 = H 2 2H 1 V2 = CD A √ 2H 1 V = . CD A1/2

(2.8)

According to the equation (2.8), the maximum velocity of the boat (V) is a function of immersed area (A). The type of relationship indicates that having large immersed area will reduce the maximum speed of the boat as we expected. The other important point is that h is included in (A) and we can qualitatively argue that the increment of h will increase A and it will automatically reduce the speed of the boat. Therefore, optimization problem becomes as hoptimum = Maximize{ fST , VMax }. h∈R

Figure 2.7 Force on the boat at maximum speed

(2.9)

Chapter 3 Numerical Simulation We have qualitatively argued that the maximum velocity and the stability of the boat are both functions (h). To confirm our argument, we did a numerical simulation using a program written using Maxima software (see Appendix). In the simulation, test data were taken for following: • engine force (H) • top width of the boat (2b) • bottom width of the boat (2a) • drag coefficient (CD ) • inclined angle of the boat (ϕ) • rotation angle of the boat (α) • draft values d1 and d2

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12 The stability function and the maximum velocity function were obtained on the same plot for 0.2 ≤ h ≤ 1.5.

Figure 3.1 Stability function and velocity function

Chapter 4 Concluding Remarks As far as V shape boats are concerned they have more stability but less maximum velocity. The reason behind this story is the increment of drag force in parallel with immersed area. Our intention here is to get an optimum value for h to have more stability and high maximum velocity. Numerical simulation (see Figure 3.1) shows us that while increasing the value of h, the stability of the boat becomes increasing and maximum velocity becomes decreasing. The intersection point of stability curve and the maximum velocity curve, suggest us an optimum value for h to have more stability and the high maximum velocity. Hence, we can qualitatively proposed an optimum value for the depth of the boat (h) based on this kind of numerical simulation. In similar capacity, other dimensions of the boat can also be considered to propose an optimal geometry for a boat. In this study, we considered only a cross sectional area of the boat and it reduced our problem into two dimension. However, in actual sense it is not realistic. As a future work we are going to propose a more realistic model taking into consideration the real geometry of a boat. In addition to that some factors like wind velocity, velocity of the fluid will also be taken into consideration in the 13

14 proposed model. The other problem we faced during this study was the lack of exact data. If exact data of factors like engine capacity, inclined angle of the boat, top width of the boat and bottom width of the boat are avialble, we would propose a more realistic model.

Bibliography 1. Lombardi, M., Parolini, N., and Rozza, G., (2010). Numerical simulation of sailing boats: dynamics, FSI, and shape optimization. 2. Megel, J and Kaliva, J. (2008). On the buoyancy force and the metacentre,UFR de Physique, Universit de Bordeaux 1, 351 cours de la Libration, 33405 Talence cedex, France. 3. Percival, S., Hendrix, D., and Noblesse, F., (2001). Hydrodynamic optimization of ship hull forms, David Taylor Model Basin, MD 20817-5700, USA.