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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
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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.
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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)
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Mathematical analysis of wave making resistance 347
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)
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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
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Mathematical analysis of wave making resistance 349
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
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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
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Mathematical analysis of wave making resistance 351
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
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352 M.N. Uddin, A.N.M. Rezaul Karim, F.S. Rafi and S. Afroz
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),
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Mathematical analysis of wave making resistance 353
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,
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354 M.N. Uddin, A.N.M. Rezaul Karim, F.S. Rafi and S. Afroz
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)
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Mathematical analysis of wave making resistance 355
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
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356 M.N. Uddin, A.N.M. Rezaul Karim, F.S. Rafi and S. Afroz
)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
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Mathematical analysis of wave making resistance 357
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.
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358 M.N. Uddin, A.N.M. Rezaul Karim, F.S. Rafi and S. Afroz
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.
Page 17
Mathematical analysis of wave making resistance 359
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.
Page 18
360 M.N. Uddin, A.N.M. Rezaul Karim, F.S. Rafi and S. Afroz
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)
Page 19
Mathematical analysis of wave making resistance 361
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
Page 20
362 M.N. Uddin, A.N.M. Rezaul Karim, F.S. Rafi and S. Afroz
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)
Page 21
Mathematical analysis of wave making resistance 363
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)
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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
Page 23
Mathematical analysis of wave making resistance 365
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
Page 24
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