mathematical analysis of wave making resistance of ship

International Mathematical Forum, Vol. 15, 2020, no. 7, 343 - 367

HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/imf.2020.91264

Mathematical Analysis of Wave Making Resistance

of Ship Using the Potential Based - Panel Method

Mohammed Nizam Uddin1, A.N.M. Rezaul Karim2,*,

Farzana Sultana Rafi3, and Salma Afroz4

1,3,4 Department of Applied Mathematics, Noakhali Science and Technology

University, Bangladesh * Corresponding author

2 Department of Computer Science and Engineering

International Islamic University

Chittagong, Bangladesh

This article is distributed under the Creative Commons by-nc-nd Attribution License.

Copyright © 2020 Hikari Ltd.

Abstract

Background and Objective: The effectiveness of a ship highly depends on the trim

change. In very shallow waters, sinkage and trim put a higher limit on the speed in which

ships operate. It is, therefore, important to consider the effects of sinkage and trim when

calculating steady ship waves. A ship model that can improve wave resistance from

sinking and trimming. The aim of the analysis is to provide an overview of the wave

resistance of ships using the source panel method. Material and Methods: The potential

theory-based program is a modified panel method used in the calculation of the resistance

to wave-making. Results: A numerical program is designed to optimize the ship hull

based on the resistance to wave formation. Attempts also have been made to determine

the speed of a ship at water level in the series 60 Hull. Conclusion: It is concluded that

the higher-order panel method can produce large gains in computation speed and

accuracy compared to the base method, and its further development is recommended.

Keywords: Resistance, Ship Hull, Hess and Smith Panel Method, Dawson’s Method,

Finite Difference Method, Sinkage and Trim

__________________________________________

Received: May 1, 2020; Accepted: September 15, 2020; Published: October 16, 2020

344 M.N. Uddin, A.N.M. Rezaul Karim, F.S.Rafi and S. Afroz

1. Introduction

Ships typically travel at a mean onward velocity, and their vibration movements in

waves are superimposed on a stable flow sector. The solution to the steady-state

problem is a matter of interest in itself, especially in the measurement of the

resistance of waves in calm water. The problem of the movement of the ship in the

waves can be considered as a superposition in these two particular cases, but

correlations between the stable and oscillatory flow fields make a difficult extra

common problem.

The ship's hull is a streamlined structure designed to generate desirable pressure

gradients to meet the lowest resistance to frontward move. Ship hull shapes are not

easy to symbolize by mathematical expressions. As a result of the difficulty of the

ship hull, no mathematical logic has been developed to reflect the commercial ship

hull accurately. Therefore, it is demanding to improve an existing hull shape as per

necessity. In the past, Ship design scholar basically depicted the hull type using the

ship’s offset table. The advantage of using the offset table is that it is comparatively

easy to deal with and realize. To overcome these issues, the scientist began using the

Bezier curve to illustrate the hull. The Bezier curve is straightforward to use and

reflects geometry well. Yet there are some significant flaws in this curve. The degree

of bezier curve adds precision to the system, while raising the computing energy and

the likelihood of creating computational noise in the measurement simultaneously. In

addition to these, the Bezier curve needs extra control segmentation with vertices to

describe the curve appropriately and, therefore, combining two Bezier curves is

reasonably complicated. The spline curve was implemented to solve these issues.

There is a wide variety of splines available. The most famous of them is the cubic

spline. The Spline is a mathematical statement that is continuous piecewise, and a

polynomial equation determines its form. The B-spline has recently been very well

known for its surface demonstration. This may provide a simple mathematical

meaning to the hull of the ship. In the process, the surface points are very close to the

core points of the B-spline. As a consequence, it provides significantly detailed

knowledge regarding the ship's hull. Rogers [1,2] explored a comprehensive

mechanism about how to create a hull using a B-spline board.B-spline related surface

modelling approach was used by Park and Choi [3] throughout the optimization

process. Since the B spline represents geometry very well, It has a few disadvantages.

B-spline curves are a transformational type. In addition, such curves, the conic parts

can not be described. An updated B-spline variant known as the Non-Uniform

Logical B-spline (NURBS) is implemented to address perceived errors.

Under NURBS, the consumer has greater jurisdiction over the control scores

because they are controlled by weights. One researcher often uses hull representations

focused on NURBS. Kim et al [4,5] developed a CFD-based optimization technique

in which the geometry input is NURBS surface. Ping et al.[6] developed a new

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Mathematical analysis of wave making resistance 345

design of hulls focused on the NURBS sheet. Parametric simulation is implemented

with the aid of B-spline and NURBS. A design-oriented parametric concept

vocabulary is adopted such that ship hulls differ easily and smoothly.

Harries [7] also came up with a technique based on both regional and global

parameters like different coefficients, principal dimensions, et cetera, which used

parametric modelling. He utilized an array of B-spline curves in the surface

generation to denote sectional curves of the ship hull. Later, Abt et al. [8] modified

the above method by using a parametric modelling method that is efficient and quick

in the production of multiple hulls during the period of change in the hull. Ping et al .

[6] implemented a more rapid method to creating and variating hulls focused on

parametric hull production. Han et al. [9] used parametric hull modeling to

characterize the fairness-optimized B-spline shape function.

Various panel methods that are used in the calculation of flow and resistance to

flow, which is important in the determination of wave resistance. Hess and Smith

developed a panel method focused on the Rankine formula of 1964. [10]. The

original panel by Hess and Smith consisted of a number of quadrilateral panels for the

discretization of the body surface. Each of the panels consisted of a constant source

of density was believed. There are efforts that have been put over the past decades to

develop the exactness and competence of the method. The resistance to wave-

making expected by the second-order solution demonstrates more intimate

cooperation with the test than that of the first-order solution [11].

A mathematical model is defined as "a collection of equations based on

quantitative description of a real-world phenomenon, it is created in the hope that the

predicted behavior will resemble the actual one" [12]. There are various scientists or

mathematicians who have achieved a higher configuration of wave resistance. But, to

understand the solution, there is some question. The approach that has been addressed

has been interpreted in an exact way of computation or comprehension.

This article used a de-singularized higher-order panel technique (focused on NURBS)

to compare the numerical results against other numerical results or analytical

solutions. The approach is considered appropriate for 3D potential flow problems as

well as accurate when giving rapid numerical solutions.

2. Materials and Methods

Mathematical modeling is a method of expression by means of equations and

geometry that is used to explain events that take place on Earth. The purpose of this

paper is to examine what can be done to calculate the shapes of ship hulls, which will

have less wave-making resistance than those developed by ordinary drafting

techniques. The work done in this case has been reviewed, and there is evidence that

it has enormous potential to improve mathematical methods.

346 M.N. Uddin, A.N.M. Rezaul Karim, F.S. Rafi and S.Afroz

First, the mathematical techniques used so far are not flexible enough to be

applied to many practical cases. Second, the wave-making resistance formula on

which these techniques are based has not produced precise predictions of

experimental results, and this has cast doubt on the validity of computations derived

from it. A component of the total resistance working on a ship, which may be reduced

by an efficient design, is the resistance due to the wave-making.

a. Hess and Smith Panel Method

The system used by Hess and Smith [13] was the first genuinely functional (in

practice) panel system, which combined with a formulation of the boundary layer.

Panel methods represent numerical models that simplify the properties and physics

behind the flow of air over an aero-plane or aircraft. One thing neglected in the

calculation is the viscosity of air. Viscosity has the net effect on the wing,

summarized when the flow leaves the edge of the wing smoothly.

The problem is for considering a steady flow of a perfect fluid in a 3-D body. If the

surface of the body is represented by S, the equation form of S would be F(x,y,z) =0

(1)

The start of flow is shown as a uniform stream with a unit magnitude and is denoted

by the constant V, while Vx, Vy, and Vz are the components of three axes

1VVVV 21

2

z

2

y

2

x

(2)

The velocity of the fluid at any point is shown as a negative gradient with a potential

function,, that must satisfy the set condition. That is, it must be according to the

Laplace equation in the region exterior to S, that is, Rand must have 0 normal

derivatives on S, approaching the right uniform stream potential at infinity.

Symbolically

0 (3)

0gradnn 0F

S

(4)

0FgradF

gradFn

(5)

Where identifies the Laplacian operator and n denotes the perpendicular vector to

the tangent vector at any point of the surface S.

zyx zVyVxV (6)


It is suitable to mark mentioned below

(7)

Where

)zVyVxV( zyx (8)

this is the standard flow potential and is a potential for disruption by the body.

, then, satisfy

0 (9)

0F0FS

Vngradnn

(10)

222 zyxfor0 (11)

The potential function ( ) will be represented as the potential of the density

distribution of a source over the exterior surface S. given the three-dimensional co-

ordinates, the potential at a point p and a unit point source q on the exterior surface S

will be given by (p) = 1/{r(p, q)}.

r(p, q) represents the distance between p and q

on the exterior surface S. extrapolating from the formula, then, the potential at p due

to a source density allotment (q) on the exterior surface S is

S

dS)q,p(r

)q()z,y,x(

(12)

The possibility of velocity derivatives can be searched in point P on S

VndS

)q,p(r

)q(

n)p(2

n SS

(13)

Dividing the surface (S) into several N panels and the equation (13) on i-th panel can

be disclosed by the equation (10) as

V)i(n)i(dS

)q,p(r

1

n)i(2

N

ij1j

j (14)

348 M.N. Uddin, A.N.M. Rezaul Karim, F.S. Rafi and S.Afroz

The above equation seems a 2-D Fredholm integral equation over the surface S. The

solution of the equation for velocity component at any flow point is got by

differentiating the equation in all coordinate directions and adding the onset flow

component.

zijziyijyixijxi

jzijyijxij

VnVnVn

dS)j,i(r

1

zndS

)j,i(r

1

yndS

)j,i(r

1

xndS

)q,p(r

1

n

b. Dawson’s Method

The wave resistance problem lends itself quite well to an entirely theoretical

treatment, an appropriate mathematical model for the wave generation and

propagation being known. Since this model in its general form is too complicated to

permit a direct solution by analytical means, the work of many hydro dynamic and

mathematicians during the last decades was focused on devising simplifications that

would lead to a tractable mathematical problem and on the other hand would retain

enough realism to be useful in practice. The main simplification required, and often

motivated by certain assumptions on the hull shape or dimensions, was a linearization

of the nonlinear boundary conditions to be imposed at the water surface. A whole

series of linearized formulations has been proposed, applicable to thin ships, slender

ships, flat ships, fast ships, etc.

Almost none of these has been found to be sufficiently accurate for normal ships.

Motivated by increasing the speed of computers and the advancement of numerical

methods, around 1976/1977, some researchers independently proposed closely related

linearised theories claimed to be valid asymptotically for slow ships. One of these,

the method proposed in 1977 by Charles Dawson [14], has been found, In general, to

offer relatively practical results and to be very effective and versatile.

Since 1980 several authors have proposed further improvements. Dawson's method

nowadays can be considered mature; it has been implemented in several different

forms and is available at many institutes all over the world. One member of this

family is the code DAWSON developed by the author in 1986-1988, which has been

used in the plan of practical ship at the Maritime Research Institute Netherlands

(MARIN) since 1986 and has meanwhile been applied to several hundreds of

practical cases. This approach has resulted in significant improvements to the design

process of a ship's hull form at MARIN, which now is characterized by detailed pre-

optimization using the flow predictions, prior to any model test. Recent advanced

techniques for visualizing the computed flow permit to obtain detailed insight in the

behavior of the flow and its relation with the hull form, in a way not achievable in

experiments. Thus the many wave pattern calculations have taught us a lot about

wave-making and have permitted a substantial further improvement of ship hull


forms. Even so, several shortcomings of the predictions by linearised methods have

come up. Quantitative wave resistance predictions have often been impossible. In

general, mainly the wave pattern around the forebody was useful, stern flows being

poorly modeled. The predicted wave pattern displays systematic deviations, and

certain important effects of the hull form are absent. All this makes the Dawson type

of methods a tool requiring substantial experience in order to judge the quality of the

results and to deduce recommendations for hull form modifications - there still is an

appreciable amount of intuition and art involved.

c. Finite Difference Method

Laplace Equation appears as the second-order partial differential equation (PDE) in

different areas of engineering like electricity, fluid flow, and the condition of steady

heat. To get the solution to the equation, the satisfaction of the boundary of the

domain is a requirement. If a function on the part of the boundary is specified, the

specified part is called Dirichlet boundary. However, when the ordinary derivative of

the function is the one that is specified on a part of the boundary, the part is known as

Neumann boundary.

For the case of an ordinary differential equation, the Finite-Difference Method

(FDM) [15-20] discretises the partial deferential equation through the replacement of

partial derivatives with approximates or finite-difference. The scheme can be

illustrated using the Laplace equation below.

The shown figures show division of a two-dimensional region into small regions

where points x and y directions are increased, given as yandx in fig.1.

Fig. 1 Finite gap between x and y

The Nodal point is identified by the numbering scheme i and j, when i mean x

increment and j means y increment, as in fig. 2

350 M.N. Uddin, A.N.M. Rezaul Karim, F.S. Rafi and S. Afroz

Fig. 2 The 5-point stencil of the Laplace equation

A finite-difference equation seems to be appropriate for the internal nodes of a static

two-dimensional system, which can be achieved by Considering the equation of

Laplace to the point i, j as

0y

T

x

T2

2

2

2

(15)

The 2nd derivatives can be estimated at the nodal point (i, j) (resulting from the

Taylor series) as

2

j1,iji,j,1i

2

2

Δx

T2TT

x

T

and

21JI,JI,1ji,

2

2

Δy

T2TT

y

T

(16)

Equation (14) then gives

2

j1,iji,j,1i

Δx

T2TT +

21JI,JI,1ji,

Δy

T2TT =0 (17)

If yx , the rough computation of Laplace’s equation may be expressed as

04TTTTT ji,1ji,1ji,j1,ij1,i

3. Implementation

c. Modeling of the problem

The speed of a ship is controlled by a motion equation which balances the external

force and the moment of inertia on the ship. Modeling the suggested problem is as

follows. At first, a ship was thought to be moving at a constant speed, U, where

water’s depth is infinite, and in a direction against the x-axis (negative) – look at

figure 1below. In such a case, There is aextension of the z-axis and y-axis upwards

and to the starboard in that order. .Further, there is an onsideration of the beginning of

the co-ordinate system in an unbroken incident flow. The origin is at an uninterrupted


free surface at amidships, that allows the continuous flow of events; it takes a form of

a fluxing towards the x-axis (+ve) are considered for the model.

The potential speed, , is calculated by adding the double model velocity potential,

0, and velocity after the effect of free surface wave, which is the perturbed velocity

potential 1.

Fig. 3 Description of a coordinate system outline

The summation of the velocity potential, is given as the 0 + 1, which is the double

model velocity plus the perturbed potential velocity. 1 represents the impact of the

free surface wave

10 (18)

Now a ship's issue can be generated by defining the equation Laplace

0)( 10

2 (19)

The conditions below are boundary

Hull boundary: The component of normal speed must be zero.

0n)( 10 (20)

Where, n is normal to the surface of the hull in the outer direction

(a) Open surface: The potential on speed must meet the dynamic features and the

kinematic properties on the surface

2U2

1g (21)

on z =

x

z y

o U


0zyyxx (22)

on z =

Removing from equations (21) and (22)

0g2

1

2

1zyyxx on z = (23)

A radiation condition will be imposed so that the waves at the free surface end

upstream after disturbances so that the model will be actualized with known

conditions. Following linearization, The condition of the surface boundary (23) may

lastly be exposed as

112

1z1111112

1 g2 on z = 0. (24)

Laplace equation solution in relation to the limit conditions (20) & (24) and Radiation

state for movement surrounding the stern of the cruiser [21].

d. Free surface condition linearization

Equation 23, representing free surface, is non-linear and should be fulfilled on an

actual surface. It can be linearized by neglecting the non-linear term 1 and using 0

about the double model. The 1, that is, perturbation potential is assumable to be

smaller than 0, the potential of the double body.

The 0 is equivalent to the solution of limitation since the Froude value leaning

towards zero, Whereby the free surface functions as a plane of reflection. In this case,

a boundary state of reflection with uninterrupted free surface should be implemented

at this stage.

0z0 on z = 0 (25)

From the equation no 23

0g2

1

2

1zy

2

z

2

y

2

xyx

2

z

2

y

2

xx (26)

For equation (26), Substituting equation (18),


0g2

1

2

1

z10y

2

z10

2

y10

2

x10y10

x

2

z10

2

y10

2

x10x10

(27)

Expanding equation (27),

0gg

2222

1

2222

1

2222

1

2222

1

z1z0

y

2

z1z1z0

2

z0

2

y1y1y0

2

y0

2

x1x1x0

2

x0y1

y

2

z1z1z0

2

z0

2

y1y1y0

2

y0

2

x1x1x0

2

x0y0

x

2

z1z1z0

2

z0

2

y1y1y0

2

y0

2

x1x1x0

2

x0x1

x

2

z1z1z0

2

z0

2

y1y1y0

2

y0

2

x1x1x0

2

x0x0

(28)

Using equation (25) and ignoring the nonlinear terms of 1 The state of free-surface

equation (28) to be linear regarding the double model solution 0.

0g2

1

2

1

2

1

2

1

z1y

2

y0

2

x0y1x

2

y0

2

x0x1

yy1y0x1x0y0xy1y0x1x0x0

y

2

y0

2

x0y0x

2

y0

2

x0x0

on z = 0 (29)

If F is any kind of function then,

llyyxx FFF

lF means the first-order differentiation in the symmetry panel z = 0 alongside a

double-model platform potential streamline 0 . The equation (29) can therefore be

drawn up as

0g2

1

2

1z1l

2

l0l1ll1l0l0l

2

l0l0 on z = 0 (30)

Simplifying the equation above,


0g z1ll0l0l1ll1l0l1ll0l0ll0l0l0 on z = 0

0g z1ll0l0l1ll1

2

l0l1ll0l0ll0

2

l0 on z = 0

ll0

2

l0z1l1ll0l0ll1

2

l0 g2 on z = 0 (31)

Again

ll0l1l0ll1

2

l0ll1l0l0

yy1

2

y0yy0y1x0x1y0

xx1

2

x0xy0y1x0x1x0

yy1y0xy1x0y0yy0y1x0x1y0

y0xy1x0xx1x0xy0y1x0x1x0

yy0y1yx0x1y0xy0y1xx0x1x0

yy0y0xy0x0y1yx0y0xx0x0x1

y

2

y0

2

x0y1x

2

y0

2

x0x12

1

2

1

(32)

c. Potential Theory

The term denotes an old, well-developed, and elegant mathematical theory that is

used in finding a solution of 0φ2 (33)

The solution to the equation can be viewed in different ways. One of the ways that

are most acquainted with aerodynamicists is the view of singularities. The view of

singularities is the algebraic functions that tend to satisfy the equation and when

combined help to develop flow fields of the fluid. The superposition of the solutions

can be used since the equation is linear. The aerodynamicists are familiar with three

singularities, namely double, vortex, and the point source. Many classic examples of

having singularities on the inside of a body, but arbitrary body shapes do not have the

property. For such arbitrary body shapes, a more sophisticated approach is used to

determine the potential flow. From the development of the theory by mathematicians,

Any one of the following (34) or (35) patterns is adopted

φ on Σ +κ {Dirichlet Problem} (34)


nφ

on Σ+κ {Neuman Problem} (35)

According to the potential flow theory, it is not right to specify both subjectively.

Others might have a mixed boundary situation,

aφ +bn

φ

in Σ+κ. Since Neumann Problem matches to the problem where there is

a specification of the flow via the surface, which is commonly zero, it can be

identified using the analysis provided above. On the other hand, the Dirichlet

Problem can only be identified as design as it matches the case of the aerodynamic. In

the case, there is a specification of the surface pressure distribution, Although here

the distribution of pressure has been tried to match the size of the body. There is a

variety of problem formulation in the linear theory, which makes it necessary to

undertake analysis procedures that are like Dirichlet problems. However, equation

(35) should also be used.

d. Second derivative along the streamline direction

An operation known as the upstream or backward finite gap is applied to fulfil the

state of radiation using the 2nd derivative in the case of dual body rationale. At the

front and rearmost points of control at the free surface, a two-point is utilized. Later

on, 3 or 4 point operations are utilized in the rest of the control points.

Representation the derivative of f(x, y) with the streamline direction l and the

considerations of (x, y, and z) global co-ordinates is mentioned below:

dy

d

d

)j,i(df

dy

d

d

)j,i(df)j,i(f

dy

)j,i(df

dx

d

d

)j,i(df

dx

d

d

)j,i(df)j,i(f

dx

)j,i(df

)j,i(f

vu

v)j,i(f

vu

u)j,i(f

dl

)j,i(df

y

x

y2ij

2ij

ij

x2ij

2ij

ij

l

The differentiations of x, y, f are as follows

)4i(6

))3i(x2)2i(x9)1i(x18)i(x11(

)3i(2

)2i(x)1i(x4)i(x3(

)2i()1i(x)i(x

)1i()i(x)1i(x

d

dx

1111

111

11

11


)4i(6

))3i(y2)2i(y9)1i(y18)i(y11(

)3i(2

)2i(y)1i(y4)i(y3(

)2i()1i(y)i(y

)1i()i(y)1i(y

d

dy

1111

111

11

11

)4i(6

))f2f9f18f11(

)3i(2

ff4f3(

)2i(ff

)1i(ff

d

df

j,3ij,2ij,1ij,i

j,2ij,1ij,i

j,1ij,i

ijj,1i

)4i(6

))3j(x2)2j(x9)1j(x18)j(x11(

)3i(2

)2j(x)1j(x4)j(x3(

)2i()1j(x)j(x

)1i()j(x)1j(x

d

dx

1111

111

11

11

)4j(6

))3j(y2)2j(y9)1j(y18)j(y11(

)3j(2

)2j(y)1j(y4)j(y3(

)2j()1j(y)j(y

)1j()j(y)1j(y

d

dy

1111

111

11

11

)4i(6

))f2f9f18f11(

)3i(2

ff4f3(

)2i(ff

)1i(ff

d

df

3j,i2j,i1j,ij,i

2j,i1j,ij,i

1j,ij,i

j,i1j,i


dy

dy

dy

dx

dx

dx

dyy

dxx

d

dyy

dxx

d

dy

dx

d

d

yx

yx

dy

dx

yy

yx

dy

dx

dy

dx

yy

xx

dy

dx

d

d1

yx

yx

e. Hull Coefficients of Form

The hull module determines geometric hull coefficients of form as well as the meshed

hull surface area. The total resistance is calculated when the hull coefficients and the

S.A are passed towards the Resistance Module. The coefficients calculated using the

Hull Module – the Max Section, B.T., Prismatic, and Volumetric.

The hull geometric coefficients and Corresponding Equation:

LAC

xp

; pC Prismatic Coefficient

3vL

C

; vC Volumetric Coefficient

BTCx

; xC Max Section

f. Solving the Free Surface Problem

Rankine sources represent 0 and 1 as the potentials of velocity; which are spread

over the surface of double model S0 and the free surface S1 without interruptions

correspondingly.


dSr

1Ux)z,y,x(

0S

00

0

(36)

dSr

1dS

r

1)z,y,x(

0S

0

1S

11

01

(37)

222

1

222

0

z)yy()xx(r

)zz()yy()xx(r

To acquire the double model’s past flow, a numerical solution is required. The

solution of the double model is the closest answer to the question of the free surface.

A free surface problem is roughly resolved by the rigid wall when the number of

Froude is limited to zero.

Equation (31), which is the free surface boundary condition, entails having the slope

of the velocity potential alongside the stream-wise direction, 1, and differentiating the

along double model corresponding streamlines. The velocity is determined by

y12

y0

2

x0

y0

x12

y0

2

x0

x0l1

It should be noted that this format of differentiation estimates the direction of flow on

a free surface by the double model.

The double model surface and the free surface are split into M0 and M1 panels,

respectively. Their strengths’ sources are then assumed to be constant at the control

points of the panels. Using equation (31), the derivatives 1l and 1ll can be expressed

at the free surface’s ith panel.

01 M

1j

00

M

1j

11l1 )ij(L)j()ij(L)j()i( (38)

01 M

1j

00

M

1j

11ll1 )ij(CL)j()ij(CL)j()i( (39)

1 1

0 0

S S 12y0

2x0

y0

12y0

2x0

x01

S S 02y0

2x0

y0

02y0

2x0

x00

dSr

1

ydS

r

1

x)ij(L

dSr

1

ydS

r

1

x)ij(L

(40)

1N

1N

knk )j,ni(Le)ij(CL (k = 0, 1) (41)

Where, en indicates an operator at N-point upstream.


ji0

ji)i(2 1

z1 (42)

Using equations (38), (39) and (42) in equation (31), the equations for 1 and 0 can

be written as

Mathematical Analysis of Wave Making Resistance 357

)i()i()i(g2

)ij(L)j()ij(L)j()i()i(2)ij(CL)j()ij(CL)j()i(

ll0

2

l01

M

1j

M

1j

0011ll0l0

M

1j

M

1j

0011

2

l0

1 01 0

Rearranging above equation,

11

M

1j

00

M

1j

11 M~1i)i(B)i(g2)ij(A)j()ij(A)j(01

(43)

Where

)ij(L)i()i(2)ij(CL)i()ij(A

)ij(L)i()i(2)ij(CL)i()ij(A

1ll0l01

2

l01

0ll0l00

2

l00

(44)

)i()i()i(B ll0

2

l0 (45)

Replacing equation (37) with equation (20)

101

M

1j

00

M

1j

11 MM~1Mi0)ij(N)j()ij(N)j(01

(46)

where

11

00

S 1z

1y

1x

S 11

S 0z

0y

0x

S 00

dS)r

1(

zn)

r

1(

yn)

r

1(

xndS)

r

1(

n)ij(N

dS)r

1(

zn)

r

1(

yn)

r

1(

xndS)

r

1(

n)ij(N

(47)

To obtain the answer of equations (43) and (46), the iterative method is applied. The

double model solution 0 can be used to represent the source distribution over the hull

surface at the first approximation.


00 (48)

Thus, the simultaneous equation for the first-order approximation is derived from the

equation (43)

)i(B)i(g2)ij(A)j( )1(

1

M

1j

1

)1(

1

1

(49)

The solution of the first hypothesis enters the downstream flow through the surface of

the hull.

1M

1j

1

)1(

1n )ij(N)j()i(v (50)

)i(v4

1)i( n0

(51)

It nearly meets the boundary condition represented by equation (46). Taking into

consideration, the potential of the velocity 0 produced by 0, the 2nd order

approximation of the value 1 is resultant from the simultaneous equation.

11 M

1j

00

)2(

1

M

1j

1

)2(

1 )ij(A)j()i(B)i(g2)ij(A)j( (52)

4. Experimental result and discussion

a. Series of 60 Hull

Ship hull is designed based on of its resistance to waves by developing a numerical

programme. Series 60 is one of the most commonly used types of hull for researchers

to study ship hull optimization. The optimization of the 60 hull sequence is done at its

design level ( Fn = 0.316).

Fig. 4 Series 60 of a hull (7090) Fig. 5 Free surface of the 60 hull series (70 90)


Fig. 6 Series 60 of a hull (7016) Fig. 7 Free surface of the 60 hull series

(7016)

Fig. 8 Wave shape on 60 hull series Fig. 9 Wave shape on 60 hull series

at nF = 0.22 at

nF = 0.25

Fig. 10 Wave outline of 60 hull series Fig. 11 Wave outline of 60 hull series

at nF = 0.25 at

nF = 0.26

Fig. 12 Resistance of 60 (16) hull series Fig. 13 Resistance of 60 (19) hull series


Fig. 14 60 CB

= 0.6 meshed hull series Fig. 15 60 C B = 0.65 meshed hull series

Fig. 16 Comparison of C

W of experimental, calculated and optimized Series 60

Hull

b. Resistance

There are three components of the resistance of a ship, namely, its form, friction on

the water, and the wave drag. The assumption in model testing is that the measured

wave drag or residual is the same as the whole ship because of the model at Froude

similitude. The resistance caused by friction is calculated from the Cf correlation line

of the entire ship with the result of the model test form factor. The process is used

because there is a mismatch between the Reynolds numbers of the model as models

are run at Froude simulation results. A ship's total resistance is shown below.

R=0.5C T ρ SU2

(53)

CT

indicates the total coefficient of resistance, while S means the surface’s area and

U means the speed of a ship. The overall coefficient of resistance is determined as

C T = (1+k) wf CC (54)


Cfindicates the friction and C

w wave resistance coefficients known as,

2

ff

0.5ρ.5

RC (55)

Cw

=2

w

0.5ρ.5

R (56)

Cf or the resistance caused by friction of the ship is a dependent factor on the

Reynolds number. The number is non-dimensional related to the viscous forces and

inertia forces.

Rv

ULe (57)

Both model test correlation lines and a boundary layer code can be used to obtain the

frictional resistance. A popular model test correlation line:

2e

f2Rlog

0.075C

(58)

The above popular ITTC 1957 line is usually modified to cater to the viscous

resistance form drag using a modified form factor, I + k. The use of the factor of form

drag is common for fuller ships because their residual resistance at low Froude

numbers is more than zero. There are various ways to determine the form factor.

Watanabe gives a formula:

K = - 0.095 + 25.6

T

B

B

L

C2

B

(59)

Since the wave drag becomes negligible at the low speed, a low-speed model text

(F.R. <0.15) may be used to determine the form factor. Using the correlation line, the

type factor (1+k) is calculated, which is a ratio of the overall flow to frictional

resistance.

f

T

C

C(1+k) (60) (60)

There is a third method for the determination of the form factor known as Prohaska’s

Method, which relies on the model test. The method does not rely on running the test

at a low Froude number for the drag to be at zero. When using the Prohaska Method,

Here it is assumed that the wave resistance coefficient remains proportional to Fr4,

f

4r

1f

T

C

Fkk1

C

C (61)

364 M.N.Uddin, A.N.M. Rezaul Karim, F.S.Rafi and S.Afroz

When the Prohaska Method is plotted as f

4

r

C

F vs.

f

T

C

C, the results of the data being

measured is a strain line with slope ki, and the point of intersection of the line and x-

axis being 1+k

Fig. 17 Plotted by Prohaska method

Viscous drag refers to the combination of form drag and friction. From Prohaska

Method, the friction resistance may be got from the 2D integral boundary. The

pressure drag that comes from the flow is separated from exterior helps in the

calculation of the form factor while the non-dimensional resistance (C) of waves is

influenced by the Froude number related to gravity.

gL

UFr (62)

There is a need to estimate the wave drag as correctly as possible because it is the

main resistance when the Froude numbers are higher and the viscous drag exists at

low Froude numbers.

c. The wave-making resistance calculation:

The Bernoulli equation will be used in its linearized version, which should be

consistent with the condition of a linearized Dawson free surface boundary. The

equation will help calculate the hull surface pressure from the perturbation potential.

z1z0y1y0x1x0

2

z0

2

y0

2

x0

2

0

1000

2

0

1000

2

0

1010

2

0

2

0

222gz2U2

1pp

.2gz2U2

1pp

.2

1gzU

2

1pp

)()(2

1gzU

2

1pp

U2

1p

2

1gzp


Now the pressure co-efficient

z1z0y1y0x1x0

2

z0

2

y0

2

x0

2

22

0

p 222gz2UU

1

U2

1

ppC

Similarly, the wave profile from equation (21)

)22U(g2

1)y,x( y1y0x1x0

2

y0

2

x0

2

Fig. 18 Resistance Vs Speed

If the pressure remains constant inside the surface of the hull, the resistance of the

wave can be measured.

ixip

2/M

1i222

w

w Sn)i(CL

1

LU2/1

RC

0

Where Si is the panel surface area and nxi denotes the x-component.

5. Conclusion

Desingularized higher-order panel technique is helpful in comparing numerical

results against analytical solutions. The approach is significant and regarded suitable

in the determination of potential flow problems in three-dimension, giving rapid

numerical solutions. In the paper, a method for series 60 hull based on wave-making

resistance has been implemented, and waves profiles and patterns are analyzed.

A brief overview of the variables of hull design, speed regime and the different

shapes of the planing hull is intended to be given to the builder. To obtain profound

realization of the nature of the boat at hand and the task at hand and the Method of

366 M.N.Uddin, A.N.M. Rezaul Karim, F.S.Rafi and S.Afroz

prediction to be observed. It is necessary to search for a long-term solution that is

based on physical phenomena, instead of mathematical, numerical or empirical

approximations. It is anticipated that a successful analytical approach based on the

physics following the planing phenomenon of three-dimensional hull-shaped would

give complete results.

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Received: May 1, 2020; Accepted: September 15, 2020; Published: October 16, 2020

mathematical analysis of wave making resistance of ship

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