Self-similarity in the equation of motion of a ship

International Journal of Naval Architecture and Ocean Engineering.
2014.
Jun,
6(2):
333-346

This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

- Published : June 30, 2014

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INTRODUCTION

If we want to analyze the motion of a body in a fluid, we should analyze the rigid-body dynamics of the body and the fluid dynamics of the surrounding fluid. Traditionally, the potential theory has been used for the analysis of the flow of fluid in studying the motion of ships. Added mass, which is the mass of surrounding fluid moving together with a ship, was treated as part of the virtual mass of that ship, and the wave damping forces were treated as the damping of the body. The main external force comes from incident waves, so gravity wave dynamics had to be solved, and the resultant pressure was integrated over the body surface to form the wave external force.
Currently, in analyzing ship motion, besides the potential theory, Computational Fluid Dynamics (CFD) based on the Navier-Stokes equation is being used; owing to the development of computers. One of the main differences between the potential theory and CFD is the treatment of added mass. In the potential theory, the added mass can be expressed explicitly, so one obtains the added mass, adds this to the mass of the ship, and then analyzes the ship motion. However using CFD, the ship motion analysis is done by first solving the flow of surrounding fluid for a given ship motion; then adding the pressure forces acting on the ship surface into the external forces of the equation of motion. This is because the inertia forces due to added mass cannot be expressed explicitly, so one cannot help but treat it as external force. Thus the difference is whether the added mass can be treated as ship-mass or be included in external forces. Using CFD, an assumption of convergence is made; that is, there is no problem if the time interval for time integration is reduced sufficiently, even if the inertia force due to added mass is treated as external force.
In this study, we revisited the property of added mass, investigated its property in the equation of motion, and the problems that can take place. Our work revealed that the equation of motion has a self-similar structure if the forces due to added mass are treated as external forces; then the solution can diverge. In differential equation, the highest order differential term has the most important role in the equation. We say that the equation has a self-similarity, if the variable to be sought(the highest order differential term) is expressed by forcing terms including that variable itself. In a numerical example, it was shown that the solution can be divergent even if the time interval is reduced sufficiently. This divergent solution may be a false solution due to the self-similarity of the equation, not due to the dynamic property. When using the non-linear potential theory or CFD based on the Navier-Stokes equation for analysis of the surrounding fluid, the reconfiguration technique using ‘pseudo-added-mass’ to resolve the problem of self-similar structure was proposed. The pseudo-added-mass is indeed mass that is pseudo-added in order to ensure a convergence of the solution.
ADDED MASS REVISITED

The concept of added mass or virtual mass has appeared in the literature since 1828, when Friedrich Bessel described the motion of a pendulum in fluid domain. The period was longer than that in the air, and this phenomenon was explained as being due to an increase in the effective mass by the surrounding fluid. Later, the concept of added mass played an important role in the analysis of the motion of a body, and the added mass was obtained analytically for simple shapes. Surely, the concept of added mass has been used in ship motion analysis, but the added mass could only be obtained after the shape of the ship was modified to a less accurate, simpler shape.
The added mass of more ship-like shapes was obtained by
Lewis (1929)
. He used the conformal mapping technique to mathematically represent the shape of the ship cross section; a result that is now called the Lewis form. He used added mass in his analysis of the vibration of a ship.
Hess and Smith (1962)
proposed the numerical method to calculate the potential flow around a 3-dimensional body of arbitrary shape. The added mass of a body floating on a free surface has different characteristics from those of a body in unbounded fluid. The formulations on this problem were made by
Ursell (1949)
, and
John (1949
;
1950)
.
Ursell (1949)
analyzed the motion of a circular cylinder on a free surface by obtaining the added mass using the multi-pole expansion method. Later,
Tasai (1959)
developed the multi-pole expansion method further to obtain the added mass of a Lewis form on a free surface. Now, the added mass of ship-like section could be calculated.
Frank (1967)
proposed a calculation method for added mass of an arbitrary shape, and more developments on this method made it applicable to 3-dimensional shapes.
Newman (1978)
gave a good account of the history and efforts toward representing the motion of a ship.
In order to describe the motion of a body in the fluid domain, the two systems of dynamics are needed: rigid-body and fluid dynamics. For convenience, the explanation will be given for a 1-dimensional problem, but this logic can be applied to 2 and 3-dimensional problems as well. According to rigid-body dynamics, Newton’s law appears in the form:
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FLUID FORCE AND EQUATION OF MOTION

Let us consider the equation of motion relating to the added mass. The focus is on how the fluid force is represented, and on the concept of the treatment of added mass; not on the details of the equation of motion.
- Linear potential

The linear theory for analyzing the motion of a body was developed mainly with the potential theory. There are two main streams of analysis: one for the frequency-domain and the other for the time-domain. The two results are related to each other through a Fourier transform because of their linearity. Let us see a method for expressing the fluid force.
- Frequency domain

In frequency-domain analysis, only the time-harmonic terms are considered; not the transition terms. The hydrodynamic force can be expressed as:
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- Time domain

In time-domain analysis, the impulse-response function is used. It is defined as the time history of the fluid force when the velocity of the body is given as impulse.
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- Non-linear potential

If there is no free surface, the fluid flow has linear properties, while the pressure has non-linear terms. However, the flow of fluid has non-linearity if a free surface exists. Using potential theory, the pressure force can be expressed as:
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- Navier-stokes equation

Let us consider the equation of motion, in which the external force is represented by using the flow from the Navier-Stokes equation. First let us check whether the added mass is included in the pressure force, or not. Although we are hear considering one dimensional flow, the result can be expanded to 2 and 3-dimensional flow without loss of consistency. The Navier-Stokes equation and continuity equation are:
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EXAMPLE OF NUMERICAL CACULATION

In order to investigate conceptually the problem of the added mass, let us simplify the problem to include the important terms only. Dividing Eq. (5) by the virtual mass (sum of the mass and added mass), and simplifying the system forces to include only damping force leads to:
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- RECONFIGURATION OF EQUATION OF MOTION

As seen in the previous sections, we know that the equation of motion should be solved after modifying the equation via the renormalization concept(i.e., so that all the inertia forces appear in the left hand side of equation of motion), when the added mass which is included on the right hand side of the equation is greater than the mass of a body. When the added mass is less than the mass of a body, for example the motion of a deep buoy, there is no big difficulty in calculation of motion except for the bouncing transition solution of the initial stage. However, in the case of the heave and pitch of a shallow draft ship, an instability problem may arise because of a large amount of added mass and moment of inertia. This conclusion can be applied to rotational motion. For the roll of a ship, the added moment of inertia of fluid is usually less than that of the ship, so the solution may be obtained without great difficulty.
In cases where all the pressure forces are treated as external forces like Eq. (3) (i.e. using non-linear potential theory or the Navier-Stokes equation), the stability problem may arise. This is particularly true when the added mass is greater than, or equal to, the mass of a body. This is because the equation of motion has a self-similar structure. To resolve this stability problem, all one has to do is move the added mass term to the left hand side of the equation when the added mass term is expressed explicitly. However, when the added mass cannot be expressed explicitly, one may add the same quantity to both sides to resolve the problem. Here we introduce a method for adding ‘pseudo-added-mass’.
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CONCLUSIONS

The rigid-body dynamics and fluid dynamics of a fluid surrounding a body are self-consistent and seem to be exact. However, when the two dynamics are coupled with each other in order to produce the equation of motion, the resulting equation of motion may have a self-similar structure. Furthermore, when the added mass is greater than the mass of a body, the solution of that equation will diverge instantaneously, and this problem is shown to be inherent.
This study reviewed the structure of various types of equation of motion. The equations using the traditional potential theory resolve this problem of self-similarity by introducing the concept of the added mass, but equations using a non-linear potential or the Navier-Stokes equation have a self-similar structure.
In order to see whether the inherent problem of self-similarity happens only in theory, a numerical study was done. The results showed that the numerical solutions also have the same property as theory, except for the effect of time step size.
With the results obtained in this study, a reconfiguration method using ‘pseudo-added-mass’ was proposed. It may be useful for resolving the divergent phenomenon caused by self-similarity when the implicit added mass is greater than the mass of a body.
In this study, the analysis involved a 1-dimensional case, but its logic can easily be applied to 2 and 3-dimensinal cases. The author hopes that provisions for 2 and 3-dimensional cases will be made in the future.
Acknowledgements

This research has begun with the investigation of the numerical stability of the motion of a ship. The author hopes that this article will give a little clue to refine the numerical schemes for the field of a numerical analysis (CFD and non-linear potential theory) on the ship motion in a fluid. This study is partially supported by Principal Project of KRISO (PES172B).

Frank W.
1967
Oscillation of cylinders in or below the free surface of deep fluids, DTNSRDC
Naval Ship Research and Development Center
Washington D.C.

Hess J.L.
,
Smith A.M.O.
1962
Calculation of non-lifting potential flow about arbitrary three-dimensional bodies
Douglas Aircraft Co. Inc.
Long Beach, California

John F.
1949
On the motion of floating bodies I.
Communications on Pure and Applied Mathematics
2
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13 -
52

John F.
1950
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Simple Harmonic Motions, Communications on Pure and Applied Mathematics
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45 -
101

Landau L.D.
,
Lifshitz E.M.
1959
Fluid mechanics, (English edition) course of theoretical physics, chapter I ideal fluids
Pergamon Press
Oxford

Lewis F.M.
1929
The Inertia of the water surrounding a vibrating ship
37th general meeting of The Society of Naval Architects and Marine Engineers
1 -
20

Longuet-Higgins M.S.
,
Cokelet E.D.
1976
The deformation of steep surface wave on water; I. A numerical method of computation
Proceedings of the Royal Society of London A
350
1 -
26

Newman J.N.
1977
Marine hydrodynamics
MIT Press
Cambridge, Massachusetts

Newman J.N.
1978
The theory of ship motions
Advances in Applied Mechanics
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221 -
283

Parker S.P.
1983
Physics(McGraw-Hill Encyclopedia of)
McGraw-Hill Inc.
New York

Patankar S.V.
1980
Numerical heat transfer and fluid flow
Hemisphere Publishing Corporation
Washington

Tasai F.
1959
On the damping force and added mass of ships heaving and pitching
Journal of Zosen Kiokai
105
47 -
56

Ursell F.
1949
On the heaving motion of a circular cylinder on the surface of a fluid
Quarterly Journal of Mechanics and Applied Mathematics
2
(2)
218 -
231

Citing 'Self-similarity in the equation of motion of a ship
'

@article{ E1JSE6_2014_v6n2_333}
,title={Self-similarity in the equation of motion of a ship}
,volume={2}
, url={http://dx.doi.org/10.2478/IJNAOE-2013-0183}, DOI={10.2478/IJNAOE-2013-0183}
, number= {2}
, journal={International Journal of Naval Architecture and Ocean Engineering}
, publisher={The Society of Naval Architects of Korea}
, author={Lee, Gyeong Joong}
, year={2014}
, month={Jun}