Sources and sinks, Vortex, Doublet, Rankine half body,
Rankine’s oval,
Flow past load waterline of a 174m –Barge.
by
Sai Madhav Tikkani
Phani Raja Sekhar Reddy Padala
3rd year, B.Tech
Ocean Engineering and Naval Architecture
Under the guidance of
Dr. N.Datta
Department of Ocean Engineering and Naval Architecture
Indian Institute of Technology, Kharagpur
Summer 2013
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CERTIFICATE
This is to certify that the project entitled “Flow past load waterline of a Barge” is a
bonafide record of the work carried out by PHANI RAJA SEKHAR REDDY PADALA
(Roll no. 11NA30012), under my supervision and guidance for fulfillment of the
requirements for the 2nd year summer project in Ocean Engineering & Naval
Architecture during the Academic session 2012-2013 in the Department of Ocean
Engineering & Naval Architecture, Indian Institute of Technology, Kharagpur.
Asst.Prof. Nabanita Datta
Assistant Professor
Dept. of OE & NA
IIT KHARAGPUR
July 2013
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CERTIFICATE
This is to certify that the project entitled “Flow past load waterline of a Barge” is a
bonafide record of the work carried out by SAI MADHAV TIKKANI (Roll no.
11NA30018), under my supervision and guidance for fulfillment of the requirements for
the 2nd year summer project in Ocean Engineering & Naval Architecture during the
Academic session 2012-2013 in the Department of Ocean Engineering & Naval
Architecture, Indian Institute of Technology, Kharagpur.
Asst.Prof. Nabanita Datta
Assistant Professor
Dept. of OE & NA
IIT KHARAGPUR
July 2013
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ACKNOWLEDGEMENTS
We take this opportunity to express my gratitude to my project guide, Asst.Prof. Nabanita
Datta, Assistant Professor in the department of Ocean Engineering and Naval
Architecture, Indian Institute of Technology, Kharagpur, for her esteemed guidance,
persistent motivation, constant encouragement and helpful support for the completion of
my project, and for helping support for the completion of project, and for helping me to
look further deep into my area of work and develop an interest in it.
I thank my colleagues who made me strive for the betterment in my work. I also thank
my parents who gave me emotional support throughout the year of work.
(PHANI RAJA SEKHAR REDDY PADALA)
(SAI MADHAV TIKKANI)
Date :
Place :
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Nomenclature
m – Source Strength
v - Velocity of the fluid
u - Velocity of the Uniform Stream
ѳ - Angle of inclination
ω – Strength of Vorticity
Ʌ - Source/Sink Strength
Ѱ – Stream Function
Ф – Scalar Potential
Figures
Fig: 1.1 Slope of a Streamline
Fig: 2.1 Source and Sink Flow
Fig: 2.2 Vortex Flow Depiction
Fig: 2.3 Rankine Half Body Flow Depiction
Fig: 2.4 Rankine Oval Depiction
Fig: 3.1 Sources and Sink Flow Streamline Flow
Fig: 3.2 Vortex Streamline Flow
Fig: 3.3 Rankine Half Body Streamlines
Fig:3.4 Rankine Oval Streamlines
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Matlab Figures
Fig: 4.1 Streamline Flow of a source
Fig: 4.2 Streamline Flow of a sink
Fig: 4.3 Potential Flow of a Source
Fig: 4.4 Streamline Flow of 3D Source
Fig: 4.5 Potential Flow of 3D Source
Fig: 4.6 Streamline Flow of 2D Vortex
Fig: 4.7 Streamline Flow of 2D Rankine Half Body
Fig: 4.8 3D Rankine Half Body
Fig: 4.9 Rankine Oval
Fig: 4.10 174 m Barge
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Contents
1. Introduction
1.1 Velocity Potential
1.2 The Stream Function
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2. Problem Formulation
2.1 Source and Sink Flows
2.2 Vortex Flow
2.3 Rankine Half Body
2.4 Rankine oval
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3. Analysis Methodology of Various Flows
3.1 Source and Sink Flows
3.2 Vortex Flow
3.3 Rankine Half Body
3.4 Rankine oval
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4. Results
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5. Discussion
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6. Reference
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Abstract
The primary goal of our current study is to find potential flow past a barge of given length
and breadth i.e., the VELOCITY DISTRIBUION around the barge given.It is essential to find
flow velocity distribution around the barge which translates to pressure distribution after
using Be noulli’s e uation which inte n
is used fo calculating the FRICTIONAL
RESISTANCE (using the boundary layer theory) on the barge by fluid flow around it.Here in
problem solution barge is modelled equivalently as a rankine oval which is stationary in a
fluid moving in the opposite direction with the velocity of equal and opposite to that of
barge in problem, which is a valid modelling procedure used in standard CFD models for
analyzing fluid flow past a given body.The modeling is performed in matlab.We have used
one source sink pair to model barge so, our model well suits for any barge given.
1. Introduction
1.1 The Velocity Potential
In potential flow the velocity field v is irrotational. This means that
Vorticity = = v = 0 (1)
When v = 0 the rate of rotation of an infinitesimal element of fluid is zero. From vector
calculus we know that if a velocity field is irrotational then it can be expressed as the gradient
of a "scalar potential" :
v = -(irrotational flow)
Where is the "velocity potential." Using the definition of the operator
In Cartesian coordinates
V1 = -
�
�

V2 = 
In cylindrical coordinates:
�
vr = -
�
v= -
�
�
V3 = -
�
�
�
��
vz = -

�
�
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1.2 The Stream Function
The stream function can be defined for any two-dimensional flow, whether the flow is
irrotational or not, compressible or incompressible. Two-dimensional means that at least one
of the velocity components is zero (in other words, at most two of the velocity components
are nonzero).
Some flow types for which the stream function is useful, and the accompanying definitions
of the stream function, are:
Flow in Cartesian coordinates, with v3 = 0:
V1 = -

�
�
V2 = 

�
�
Flow in cylindrical coordinates with VZ = 0:
v=
��
�
vr = -
�
�
Flow in cylindrical coordinates with v= 0:
v r= 
�
�

vz = -
�
�
Lines of constant (streamlines) are perpendicular to lines of constant  (velocity
potential lines).
On a streamline, is constant, so that
d= • ds =
�
�
dx1 +
�
�
dx2
-(1)
The stream function does not change
(i.e. d= 0) because the displacement
ds = 1dx1 + 2dx2
is taken along a streamline. Making use of equations
V1 = -
�
�
; V2 = 
�
�

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Equation (1) becomes
V2 dx1 -
v1 dx2 = 0
Or, after rearrangement,
-(2)
Equation (2) gives the slope of the streamline, as depicted in Figure,
Fig: 1.1 Slope of a Streamline
On a line of constant velocity potential, is constant. Therefore,
dФ= Ф• ds =
�
�
dx1 +
�
�
dx2
….. (3)
In equation (3), the displacement ds occurs along a line of constant velocity potential. Making
use of equations
V1 = -
�
�
; V2 = - 
�
�
,,
Equation (1) becomes v1dx1 – v2dx2 = 0
or, after rearrangement,
--- (4)
Equation (4) gives the slope of a line of constant velocity potential. Since the slopes of a
streamline
(Equation (2)) and of a line of constant velocity potential (equation (4)) are negative
reciprocals of one another, streamlines and velocity potential lines must be mutually
perpendicular.
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2. Problem Formulation
Our objectives are to evaluate the expressions for Velocity Potential and Stream Function and
Location of Stagnation points and Body shape of the Various Flows. To arrive at the load water
line of given barge ship, represent flow past that load water line
2.1 Source and Sink Flows.
In two dimensions, a source is a line (into the page) from which fluid flows
outward, and a sink is a line at which fluid flows inward and is removed. For these flows,
vθ= 0 and Vr = Q'/(2r)
Q' is the total volumetric flow rate outward from the source, per unit depth into the page. Q'
> 0 for a source, Q' < 0 for a sink. The flow pattern in a sink and source is shown below in the
figure
Fig: 2.1 Source and Sink Flow
2.2 Potential Vortex Flow.
If the velocity potential and stream function for the source/sink flow are interchanged, we get
a new flow which is called Potential Vortex Flow.
The velocities for this new flow are Vr = 0 and V= Г/(2r)
Fig: 2.2 Vortex Flow
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2.3 Rankine half-body
It is the superposition of a uniform stream of constant speed U and a source of strength m.
The Streamline flow of the Rankine half body is shown in the figure below
Fig: 2.3 Rankine Half Body
2.4 Rankine oval
It is the superposition of a uniform stream of constant speed U and a source and sink of
strength Ʌ.The Streamline Flow of the Rankine Oval is shown in the figure below.
Fig: 2.4 Rankine Oval
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3. Analysis Methodology
3.1 Source and Sink Flows.
For these flows the tangential and radial velocities are Vѳ= 0 and Vr = Q'/(2πr) respectively.
�
We know that, vr = -
,we get
�
vr
= -
�
Q'/(2r)




= - Q'/(2) lnr + f()

�
= - Q'/(2) lnr
The function of integration f () was set to zero since
v= -
�
= 0 implies that at most
��
f can be an arbitrary constant. Similarly, the stream function can be found using equations







vr = -
�
��
Q'/(2r)
Ѱ= - Q'/(2)  + f(r)


Ѱ= - Q'/(2) 
Where similar arguments can be made to set f(r) = 0.
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The velocity potential lines and streamlines are illustrated in the Figure below. Again, we see
that the curves of and are orthogonal, as required.
Fig: 3.1 Sources and Sink Flow
3.2 Vortex Flow
The velocities for vortex flow are Vr = 0 and V= Г/(2r). By comparison with a source flow, we
realize that Ѱ and Ф lines for a source shall be interchanges for a vortex. So we get,

 = - Г/(2) Ѱ= - Г/(2) lnr
Fig: 3.2 Vortex Flow
Superposition
The expressions for Ф and Ѱ derived for the basic flows such as the uniform flow, the source
and the sink and the free vortex, can be put to practical use by superposition. By superposition
we mean the algebraic combination of the Ф or Ѱ functions of two or more basic flows. The
combined expressions then can represent potential flow over complex geometries.
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3.3 Rankine half-body
Fig: 3.3 Rankine Half Body
Consider a source located in a uniform flow in the negative x-direction.
The stream function for a source is
Ѱ=

And for a uniform flow is Ѱ= -Ur sin
The stream function in the flow field due to the combination of the basic flow is
Ѱ= -Ur sin


We note that the Ѱ=0 streamline corresponds to the curve
The typical values of the radial distance from the source are
for ѳ = 0
for ѳ = ±π/2
∞
for ѳ = π
The thickness of the half body is
And its maximum value is
y = r sinѳ =
ymax =
at ѳ= π
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Stagnation point
The point on the body where both the velocity components are zero is called the stagnation
points. Physically it means that the flow issuing out of the source exactly counter-balances the
uniform flow at S. The point S is located at a distance
from the source .At higher angles,
the source flow diverts the uniform flow resulting in the bending of the streamlines. Very far
away from the body, where the influence of the source is minimal, the streamline tends to
remain parallel with x axis
3.4 Rankine oval
The source and sink are each of strength Ʌ placed on the x axis on either side of the origin at a
distance 2m apart.
The velocity of the uniform flow is U. The angles subtended by a point P(x, y) respectively
from the source and the sink are ѳ1 and ѳ2
The location of Stagnation points are noted by setting u=0 on Ѱ=0 line. The result is
Fig:3.4 Rankine Oval
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θ1 = tan-1(
√
) ; θ2 = tan-1(
)
The stream function for the source, sink and uniform flow at P(x, y) is
Ѱ=
2–
1 -
U0 y
or
Ѱ=
[tan-1(
) - tan-1(
)] - U0 y
The velocity component in the x-direction is
u=
�
�
[
=
]-U0
-
And in y-direction
v=
�
�
=
[
-
]
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4. Results
The shapes of different flow were plot using MATLAB for a given source strength of
5 m2 /sec, using the velocity potential and stream functions of respective flows.
4.1 Source and sink
Depiction of Streamline flow of a source and a sink is shown in the figure below
.
Fig: 4.1 Streamline Flow of a source
Fig 4.2 Streamline Flow of a sink
The shape of Potential flow for a source is concentric circle as shown below
Fig 4.3 Potential Flow of a Source
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The Streamline flow and Potential flow for a 3D source is shown below
Fig: 4.4 Streamline Flow of 3D Source
Fig: 4.5 Potential Flow of 3D Source
4.2 2-Dimensional Vortex
Fig 4.6 Streamline Flow of 2D Vortex
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2D Rankine Half Body
Fig: 4.7 Streamline Flow of 2D Rankine Half Body
3D Rankine Half Body
Fig: 4.8 3D Rankine Half Body
Rankine oval
Streams horizontal velocity
Streams vertical velocity
Source/Sink strength
Source/Sink x distance
Source/Sink y distance
50
0
50
0.4m
0
Fig: 4.9 Rankine Oval
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Finally, we made a barge of length 174 m with a source and sink pair and a uniform
stream of given strength.
The Streams horizontal velocity 5
The Streams vertical velocity 0
Distance between stagnation points 174
Breadth of bulk carrier 23
Length of doublet required =166.8403
Source strength_ = 114.8768
Fig: 4.10 174 m Barge
Discussion
We initially analyzed the Velocity potential and Stream functions for various flows such
as source, sink, vortex flow, rankine half body and rankine oval and extended the same to
3D. Using the equations we plotted the shapes of the above flows. We extended this
theory to produce a 174m barge, with a source sink pair and a uniform flow. Then we
started programming for ships with fine form which will be our future topic of project.
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References
1. Fluid Mechanics, A.K.Mohanty
2. Fluid Mechanics, Fox and McDonald
3. Marine Hydrodynamics, MIT Lecture Notes
4. Marine Hydrodynamics, John Nicholas Newman
5. NPTEL Cources,Fluid Mechanics,
Prof.Gautam Biswas and Prof.S.K.Som
3. http://web.mit.edu/fluids- modules/www/potential_flows/LecturesHTML/
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