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The Shallow Water Effect on Linear Derivatives
Copyright © IFAC Manoeuvring and Control of Marine Craft, Brijuni, Croatia, 1997
THE SHALLOW WATER EFFECT ON LINEAR DERIVATIVES
D. Clarke
University of Newcastle upon Tyne, UK
Abstract
Ship manoeuvring studies require that the derivatives in the equations of motion are known for the particular ship. This is accomplished by calculation or experiment in deep water, but there are problems in shallow water, since the derivatives become functions of water depth, as well as ship geometry. A simple semi-empirical correction has been previously published, but has now been found to be in error. Revised equations are given here for the case of hulls with elliptical body sections and new equations are suggested for hulls with rectangular body sections. These equations can then be used to scale the deep water derivatives to values appropriate to a particular shallow water condition. Keywords: Ship Control, Manoeuvrability, Conformal Mapping.
In this paper, the horizontal added mass coefficient is reconsidered by utilising the results of conformal mapping techniques. By drawing on previous work (page, 1913; Richmond, 1924; Lockwood-Taylor, 1930; Kennard, 1967), formal expressions have been derived by Clarke (1995a; 1995b; 1995c) for the case ofbody sections having flat plate, circular, elliptical and elliptical with fiat plate shapes. These results were shown to be self consistent, and also showed that the earlier work of Sheng (1981) was in error.
l. INTRODUCTION
The simulation of any ship manoeuvring problem is critically dependent upon an accurate knowledge of the derivatives required in the equations of motion. The derivatives may be fairly readily calculated for the case of very deep water, but are much more difficult to estimate in shallow water. However, the ratio of the value of the derivative in shallow water to that in deep water, is much easier to establish than either of their absolute values. Empirical formulae have been published previously by Clarke et al. (1983), allowing the shallow to deep water ratio of the four acceleration derivatives and the four velocity derivatives to be found for any water depth to draught ratio. The manner in which these formulae were originally established by Sheng (1981) is now rather obscure, but it is known that they depended upon the experimental determination of the horizontal added mass coefficient of an elliptical body section in shallow water, using the once popular electrolytic analogy method of solving the Laplace equation. It has been found subsequently that this determination of the horizontal added mass coefficient was numerically incorrect.
Nevertheless, the expressions for the shallow water corrections, produced by Sheng (1981) and reproduced by Clarke et al. (1983), may continue to be used, provided that the corrected expression for the added mass, in the form of six coefficients derived in this paper, is used as an alternative. It was also possible to utilise the added mass coefficients calculated by Flagg and Newman (1971), who used the previously published results of Gurevich (1940), which were made more accessible by Sedov (1965). These results can be easily expressed in the same form as those of Sheng (1981), which could give rise to empirical shallow water corrections applicable to rectangular section
117
hulls, as well as those for elliptical hull fonDS. It must be remembered that the added mass of a rectangular section in deep water is not equal to unity, as is the case of an elliptical section, but is a function of the beam to draught ratio.
derivatives being functions of the ratio f/J. Sheng (1981) showed that this ratio of the added mass in deep water to that in shallow water f/J, for elliptic body sections, could be represented by
2. MODWICATION OF LINEAR DERIVATIVES (3)
where
By using the tw
with F
=[ ~ -1] ;
an, bn , cn are given in Table l.
By assuming that the local beam to draught ratio b / T could be described by the relationship,
N/ Yv'
=H
=H
(~)2 ffc H)x'2 dX ,
(4) Sheng (1981) was able to integrate all
G)2 fJ:?:)dX'
I
the I
expressions for the derivatives, except Yr and N v
,
and express them as ratios compared with their deep water values. These were as follows,
Nv' G)2 fJ~c;)X'dX' =H
Yr' = (~)2 fJH +(:?: )X']dX' H
Nr' (~)2 fJHX'+ (:c; )X'2]dX' = H
(1)
It is clear that the derivatives defined in Equation (1) are functions of the added mass coefficient CH ,
which varies with the body section shape along the hull, and in this case with water depth also. If we let the added mass coefficient in deep water be indicated by CH 00 , and that in shallow water as CH , then we can define the ratio between them as f/J, so that
(5)
where Ko , K} and K2 are given in Equation (3). Although the foregoing analYSis is acceptable within the limitations of the slender-body strip method approach, is has been found by Clarke (l995b), that the coefficients an' bn , cn are in error in Equation
(2)
Now since in the case of elliptical body sections, the deep water value is CH 00 =l.0, substitution of Equation (2) into Equation (1) results in the
118
3.1. Circular Body Sections.
(3) for the added mass ratio
However, in shallow water this technique is not appropriate. The boundary conditions imposed by the proximity of the sea bed to the body, must be satisfied by adding an infinite row of similar horizontal strips, above and below the central strip, which is defined by the area between the sea bed and the image sea bed, as shown in Fig. 1.
3. ADDED MASS OF CYLINDERS IN SHALLOW WATER
For instance, a semi-circular body section in shallow water can be represented by an infinite row of circular cylinders, with a spacing equal to twice the water depth. By using a row of dipoles or dipole distributions placed at the cylinder centres, a complex potential can be found which gives the flow around approximately circular cylinders (Clarke, 1995a). In the case of the row of dipoles, the vertical diameter of the body gets smaller than the horizontal diameter as the water depth reduces. This problem is overcome by using dipole distributions instead of the single dipoles. However, although the vertical and horizontal diameters are then the same, the final shape has an area greater than a circle. There is no exact solution to this problem.
The determination of the added mass coefficients of the body sections along the length of the hull of a ship, is a vital step in calculating the forces and moments acting upon that ship. The added masses of the body sections in deep water may be found by confonnal mapping techniques, where the body section is mirrored above the water line. Usually this double body is then mapped onto a unit circle in another complex plane, and the added mass is found simply from the residue of the mapping function (Clarke 1972).
Image Cylinder
3. 2. Elliptical Body Sections. Second Image Strip
The problem of detennining added mass was approached from an alternative direction by Clarke (1995b), using a Schwarz-Christoffel mapping technique. The heavily shaded quarter of the central strip, shown in Fig. I , can be mapped onto the upper half on another complex plane and hence onto a semi-infinite strip in the complex potential plane, see Fig. 2. The relationships between these three complex planes allow the complex potential to be determined, and consequently the added mass of the central cylinder to be found as a function of the beam to draught ratio and the draught to depth ratio.
Image Cy linder
Cylinder
The Schwarz-Christoffel mapping technique is usually applied only to polygonal shapes, but it was pointed out by Page ( 1913) that, under certain circumstances, curved boundaries can also be considered. Using this approach, Richmond (1924) developed the necessary transformations in the solution of the electrostatic field around a row of nearly circular cylinders. One benefit of the method is that allows the treatment of elliptical as well as circular cylinders, but a drawback is that the cylinders are only nearly elliptical or circular for water depth to draught ratios greater than 2.0.
Sea Bed Image Cylinder
Image Cylinder
The potential flow within the shaded and notched semi-infinite strip in the z-plane, shown in Fig. 2, can be determined by finding the mapping function which transforms it onto the semi-infinite strip in the w-plane.
Fig. 1. Infinite Row of Horizontal Strips to Achieve Correct Boundary Conditions.
119
z-
plane
z
= 2Pw +
2Qcosh
_l(COShW) .Ja' (7)
t =-
00
By means of contour integration around the central cylinder, it is possible to determine the added mass coefficient CH in terms of the depth to draught ratio and the parameter A. , which controls the beam to draught ratio, as follows
x t
= 00
2 loge [sec (7r T _1)] 2H 1- A.
t - plane
is
C -00
=----"-----:---=7r T I )2 ( 2H I-A.
o
B
A
1
a
(8)
The eccentricity of the section may be found by solving the following equation
r 00
BI 2 = ~~
w = %i
iv
1t.t't,-= ~O
~
T 7rT
w - plane _ _ _ _ _ _ _ _ _.!...t.:; = r=.-.:::oo::...
10
ge
[sec (;; I~A.) +
2H
1
tan(7rT_l_) . 2H1 - A.
C
(9)
B
A
Ot = 1
t =a
u t
It is interesting to note that when A. = 0, Equation (9) then reduces to BI2T = 0, which is the case of a vertical flat plate. The added mass given by Equation (8) then reduces to the following closed fonn solution,
= 00
Fig. 2. Successive Complex Planes.
(N)] (;;r
210ge [sec
In the z-plane a row of similar oval shapes are equally spaced along the y-axis, with the central oval shape placed at the origin O. The boundary of the shaded strip in the z-plane must be mapped onto the real axis in the t-plane, with the shaded area of the strip being mapped onto the upper half of the tplane.
CH =
(10)
This expression for a vertical flat plate is the same as given by Lockwood-Taylor (1930), Sedov (1965) and Kennard (1967), and is the only exact result known for a shallow water added mass problem ..
TItis is achieved by the modified SchwarzChristoffel transformation used by Richmond (1924), which is dz dt
The ratio of the added mass in deep water to that in shallow water is plotted in Fig. 3, using Equation (3) as derived by Sheng (1981). Using Equations (8,9 and 10) the added mass ratio is shown in Fig. 4, determined by the conformal mapping procedure described in this paper for elliptically shaped bodies, having a range of beam draught ratios, including the exact result for the flat plate (BI2T = 0).
-==P=-= + -===,=,Q=-= -1),Ji a),Ji ,
J(t
J(t -
(6)
where P and Q are constants which control the dimensions in the z-plane. By integration and transformation between the z, t and w planes, Clarke (1995b) showed that the relationship between the zplane and the w-plane, which effectively determines the complex potential of the flow, may be written as,
The discrepancy between the empirical equation of Sheng (1981) and the conformal mapping results described here, is quite apparent when comparing Figs. 3 and 4.
120
1.0 0.9 0.8
1.0
~
~ :-:::::: R ~ ~ " ~ 1\.~ ~
'"
0.7 0.6 0.5
~""" r\." r-..." \ ~ ~ 0 r\.\ \ 1 \ \'\ fs\ ,\1\\'(0'-1.
0.8
In
0.4
" \. \
"~
"
,
~
0.5
\
~
0.2
O. I
0.0
0.0
''\ \.
~
~O
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 l.0
ill
!\~~
~:\ ~ '<2.'?-, ~ ~~ ~~ ~C!i '1'
0.3
0.1
\
\.
0.4
"~
0.2
~
0.6
1\\,\
0.3
0.7
"\
~ ~\
--"
I""r--.. I'-....
0.9
~1
~2
~3
~4
~5
~6
~7
~8
~9
Fig. 3. Added Mass of an Ellipse in Shallow Water (Sheng 1981)
Fig. 5. Added Mass of a Semi-Circle in Shallow Water (Clarke, 1995a&b)
l.0
1.0
~ r--...' ~ r--.....
0.9 0.8 0.7 0.6 0.5 0.4 0.3
0.2 0.1 0.0
0.9
"\ ~ ~ ~ ~ I'-... ~ ~"\ '" r--...." ~~ \\ 1\\, r\.' ~~ ~'o
" ", \ 1\\ \ ro \ ~~1 '\
0.8 0.7 0.6
j\.\ \ 1\
0.5
\\ \
\~ \. r\
1.0
T/H
T/H
0.4
\\ \
0.3
0:' ,\ \ \
,
.... ~ ~
t- r--.....
~ ~ ~ ~ ~ ........
~~
"'-~
"" ~ 0- I\.. ',., ~ '\ ~\ ~ N~ ~ \ ~ ~ ~ 1\\ ~ ~1\.\ ,\
"
0.1
~
\
\
~ ~ ,\i\ ~ ~~
0.2
\\ '" ~~ ~
~
~
0.0
~
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
T/H
T/H Fig. 4. Added Mass of an Ellipse in Shallow Water (Clarke 1995b)
Fig. 6. Added Mass of a Rectangle in Shallow Water (Flagg and Newman 1971)
By plotting the results against draught to depth, then for very shallow water the results tend to zero for T / H = 1.0. This is much more convenient the usual manner of plotting, where the graph goes to infinity in shallow water. A comparison with the circular case using distributed dipoles (Clarke, 1994a) may be seen in Fig. 5, by utilising the previously derived elliptical result for (BI2T = 1.0). For T I H> 0.5, the two results diverge, indicating that the shallow water added masses will be overestimated in the general elliptical case. Also shown is the added mass ratio determined from Equation 3 for BIT = 2.0, the circular case. The error in the added mass data of Sheng (1981) is quite obvious here.
3.3 Rectangular Body Sections.
For the purpose of comparison, ratio of added mass in deep water to that in shallow water for a rectangular section is shown in Fig. 6. This result has been derived from Flagg and Newman (1971). It may be immediately seen, in comparison with Fig. 3, that the ratio of the added masses for the rectangle and the ellipse are remarkably similar. It must be remembered, however, that the added mass of the rectangle in deep water is not unity, and therefore Equations 3 and 4 cannot be used without modification, since the assumption was made there that the added mass in deep water was unity.
121
Clarke, D (1995a). Calculation of the Added Mass of Circular Cylinders in Shallow Water. Technical Report, No. M2 (1), Hydromechanics Research Group, University of Newcastle upon Tyne.
4. NEW COEFFICIENTS A new fit to the data plotted in Fig. 4 was obtained, using Equation 3 and a least squares fitting process. The new coefficients will supersede those given previously by Sheng (1981) and plotted in Fig. 3.
Clarke, D (1995b). Calculation of the Added Mass of Elliptical Cylinders in Shallow Water. Technical Report, No. M2 (2), Hydromechanics Research Group, University of Newcastle upon Tyne.
Table 1.
ao bo al bl Cl
a2
Sheng Fig. 3
New Fit Fig. 4
0.0775 ..().0110 ..().0643 0.0742 -0.0113 0.0342
0.0774 ..().0151 "().0125 0.1674 ..().0199 0.0431
Clarke, D (1995c). Calculation of the Added Mass of Elliptical Cylinders with Vertical Fins in Shallow Water. Technical Report, No. M2 (3), Hydromechanics Research Group, University of Newcastle upon Tyne. Flagg, c., N. and J. N. Newman (1971). Sway Added Mass Coefficients for Rectangular Profiles in Shallow Water. Journal of Ship Research, 15, 257-265. Gurevich, M. I. (1940). Added Mass of a Lattice Consisting of Rectangles (in Russian). Prokl. Mat. i Mech ., 39, 1, 97-115.
In both cases the curve fit is only applicable for TIH values between 0 and 0.8. Even the new fit is not very reliable for small BIT and it would be advantageous in future to derive a different curve fit, which makes use of the fact that all the curves must pass through the corner points (1.0,0) and (0, 1.0). This is not a feature of the fitting which is currently applied with Equation 3.
Kennard, E. H. (1967). Irrotational Flow of Frictionless Fluids, mostly of Invariable Density. David Taylor Model Basin, Report 2299, 108-112. Lockwood- Taylor, 1. (1930). Some Hydrodynamical Inertia Coefficients. Philosophical Magazine, Series 7, 9, 55, 161-183.
5. CONCLUSIONS
By calculating the added mass coefficients of the elliptical body sections in shallow water, it has been possible to amend the previously published empirical equations for the shallow water corrections to the linear derivatives used in ship manoeuvring studies. It is therefore proposed that the formulae derived within this paper are used in future, for the solution of ship manoeuvring problems in shallow water, in place of those published previously.
Page, W. M. (1913). Some Two-Dimensional Problems in Electrostatics and Hydrodynamics.
Proceedings of the London Mathematical SoCiety, Series 2,11,313-328. Richmond, H. W. (1924). On the Electrostatic Field of a Plane or Circular Grating formed of Thick Rounded Bars. Proceedings of the London Mathematical SoCiety, Series 2, 22,389-403 . Sedov, L. I. (1965). Two-Dimensional Problems in Hydrodynamics and Aerodynamics, p.116-118. Interscience Publishers, New York.
REFERENCES Clarke, D (1972). A Two-Dimensional Strip Method for Surface Ships: Comparison of Theory with Experiments on a Segmented Tanker Model.
Sheng, Z. Y. (1981). Shallow Water Effect on Hydrodynamic Derivatives of Ship Hull.
Journal of Mechanical Engineering &ience,
Proceedings of 2nd Symposium on Manoeuvrability. Marine Design and Research
14, 7, (Supplementary Issue), 53-61.
Institute of China, Shanghai, China.
Clarke, D., P. Ged.Iing and G. Hine (1983). The Application of Manoeuvring Criteria in Hull Design using Linear Theory. Transactions of
the Royal institution of Naval Architects, 125, 45-68.
122
THE SHALLOW WATER EFFECT ON LINEAR DERIVATIVES
D. Clarke
University of Newcastle upon Tyne, UK
Abstract
Ship manoeuvring studies require that the derivatives in the equations of motion are known for the particular ship. This is accomplished by calculation or experiment in deep water, but there are problems in shallow water, since the derivatives become functions of water depth, as well as ship geometry. A simple semi-empirical correction has been previously published, but has now been found to be in error. Revised equations are given here for the case of hulls with elliptical body sections and new equations are suggested for hulls with rectangular body sections. These equations can then be used to scale the deep water derivatives to values appropriate to a particular shallow water condition. Keywords: Ship Control, Manoeuvrability, Conformal Mapping.
In this paper, the horizontal added mass coefficient is reconsidered by utilising the results of conformal mapping techniques. By drawing on previous work (page, 1913; Richmond, 1924; Lockwood-Taylor, 1930; Kennard, 1967), formal expressions have been derived by Clarke (1995a; 1995b; 1995c) for the case ofbody sections having flat plate, circular, elliptical and elliptical with fiat plate shapes. These results were shown to be self consistent, and also showed that the earlier work of Sheng (1981) was in error.
l. INTRODUCTION
The simulation of any ship manoeuvring problem is critically dependent upon an accurate knowledge of the derivatives required in the equations of motion. The derivatives may be fairly readily calculated for the case of very deep water, but are much more difficult to estimate in shallow water. However, the ratio of the value of the derivative in shallow water to that in deep water, is much easier to establish than either of their absolute values. Empirical formulae have been published previously by Clarke et al. (1983), allowing the shallow to deep water ratio of the four acceleration derivatives and the four velocity derivatives to be found for any water depth to draught ratio. The manner in which these formulae were originally established by Sheng (1981) is now rather obscure, but it is known that they depended upon the experimental determination of the horizontal added mass coefficient of an elliptical body section in shallow water, using the once popular electrolytic analogy method of solving the Laplace equation. It has been found subsequently that this determination of the horizontal added mass coefficient was numerically incorrect.
Nevertheless, the expressions for the shallow water corrections, produced by Sheng (1981) and reproduced by Clarke et al. (1983), may continue to be used, provided that the corrected expression for the added mass, in the form of six coefficients derived in this paper, is used as an alternative. It was also possible to utilise the added mass coefficients calculated by Flagg and Newman (1971), who used the previously published results of Gurevich (1940), which were made more accessible by Sedov (1965). These results can be easily expressed in the same form as those of Sheng (1981), which could give rise to empirical shallow water corrections applicable to rectangular section
117
hulls, as well as those for elliptical hull fonDS. It must be remembered that the added mass of a rectangular section in deep water is not equal to unity, as is the case of an elliptical section, but is a function of the beam to draught ratio.
derivatives being functions of the ratio f/J. Sheng (1981) showed that this ratio of the added mass in deep water to that in shallow water f/J, for elliptic body sections, could be represented by
2. MODWICATION OF LINEAR DERIVATIVES (3)
where
By using the tw
with F
=[ ~ -1] ;
an, bn , cn are given in Table l.
By assuming that the local beam to draught ratio b / T could be described by the relationship,
N/ Yv'
=H
=H
(~)2 ffc H)x'2 dX ,
(4) Sheng (1981) was able to integrate all
G)2 fJ:?:)dX'
I
the I
expressions for the derivatives, except Yr and N v
,
and express them as ratios compared with their deep water values. These were as follows,
Nv' G)2 fJ~c;)X'dX' =H
Yr' = (~)2 fJH +(:?: )X']dX' H
Nr' (~)2 fJHX'+ (:c; )X'2]dX' = H
(1)
It is clear that the derivatives defined in Equation (1) are functions of the added mass coefficient CH ,
which varies with the body section shape along the hull, and in this case with water depth also. If we let the added mass coefficient in deep water be indicated by CH 00 , and that in shallow water as CH , then we can define the ratio between them as f/J, so that
(5)
where Ko , K} and K2 are given in Equation (3). Although the foregoing analYSis is acceptable within the limitations of the slender-body strip method approach, is has been found by Clarke (l995b), that the coefficients an' bn , cn are in error in Equation
(2)
Now since in the case of elliptical body sections, the deep water value is CH 00 =l.0, substitution of Equation (2) into Equation (1) results in the
118
3.1. Circular Body Sections.
(3) for the added mass ratio
However, in shallow water this technique is not appropriate. The boundary conditions imposed by the proximity of the sea bed to the body, must be satisfied by adding an infinite row of similar horizontal strips, above and below the central strip, which is defined by the area between the sea bed and the image sea bed, as shown in Fig. 1.
3. ADDED MASS OF CYLINDERS IN SHALLOW WATER
For instance, a semi-circular body section in shallow water can be represented by an infinite row of circular cylinders, with a spacing equal to twice the water depth. By using a row of dipoles or dipole distributions placed at the cylinder centres, a complex potential can be found which gives the flow around approximately circular cylinders (Clarke, 1995a). In the case of the row of dipoles, the vertical diameter of the body gets smaller than the horizontal diameter as the water depth reduces. This problem is overcome by using dipole distributions instead of the single dipoles. However, although the vertical and horizontal diameters are then the same, the final shape has an area greater than a circle. There is no exact solution to this problem.
The determination of the added mass coefficients of the body sections along the length of the hull of a ship, is a vital step in calculating the forces and moments acting upon that ship. The added masses of the body sections in deep water may be found by confonnal mapping techniques, where the body section is mirrored above the water line. Usually this double body is then mapped onto a unit circle in another complex plane, and the added mass is found simply from the residue of the mapping function (Clarke 1972).
Image Cylinder
3. 2. Elliptical Body Sections. Second Image Strip
The problem of detennining added mass was approached from an alternative direction by Clarke (1995b), using a Schwarz-Christoffel mapping technique. The heavily shaded quarter of the central strip, shown in Fig. I , can be mapped onto the upper half on another complex plane and hence onto a semi-infinite strip in the complex potential plane, see Fig. 2. The relationships between these three complex planes allow the complex potential to be determined, and consequently the added mass of the central cylinder to be found as a function of the beam to draught ratio and the draught to depth ratio.
Image Cy linder
Cylinder
The Schwarz-Christoffel mapping technique is usually applied only to polygonal shapes, but it was pointed out by Page ( 1913) that, under certain circumstances, curved boundaries can also be considered. Using this approach, Richmond (1924) developed the necessary transformations in the solution of the electrostatic field around a row of nearly circular cylinders. One benefit of the method is that allows the treatment of elliptical as well as circular cylinders, but a drawback is that the cylinders are only nearly elliptical or circular for water depth to draught ratios greater than 2.0.
Sea Bed Image Cylinder
Image Cylinder
The potential flow within the shaded and notched semi-infinite strip in the z-plane, shown in Fig. 2, can be determined by finding the mapping function which transforms it onto the semi-infinite strip in the w-plane.
Fig. 1. Infinite Row of Horizontal Strips to Achieve Correct Boundary Conditions.
119
z-
plane
z
= 2Pw +
2Qcosh
_l(COShW) .Ja' (7)
t =-
00
By means of contour integration around the central cylinder, it is possible to determine the added mass coefficient CH in terms of the depth to draught ratio and the parameter A. , which controls the beam to draught ratio, as follows
x t
= 00
2 loge [sec (7r T _1)] 2H 1- A.
t - plane
is
C -00
=----"-----:---=7r T I )2 ( 2H I-A.
o
B
A
1
a
(8)
The eccentricity of the section may be found by solving the following equation
r 00
BI 2 = ~~
w = %i
iv
1t.t't,-= ~O
~
T 7rT
w - plane _ _ _ _ _ _ _ _ _.!...t.:; = r=.-.:::oo::...
10
ge
[sec (;; I~A.) +
2H
1
tan(7rT_l_) . 2H1 - A.
C
(9)
B
A
Ot = 1
t =a
u t
It is interesting to note that when A. = 0, Equation (9) then reduces to BI2T = 0, which is the case of a vertical flat plate. The added mass given by Equation (8) then reduces to the following closed fonn solution,
= 00
Fig. 2. Successive Complex Planes.
(N)] (;;r
210ge [sec
In the z-plane a row of similar oval shapes are equally spaced along the y-axis, with the central oval shape placed at the origin O. The boundary of the shaded strip in the z-plane must be mapped onto the real axis in the t-plane, with the shaded area of the strip being mapped onto the upper half of the tplane.
CH =
(10)
This expression for a vertical flat plate is the same as given by Lockwood-Taylor (1930), Sedov (1965) and Kennard (1967), and is the only exact result known for a shallow water added mass problem ..
TItis is achieved by the modified SchwarzChristoffel transformation used by Richmond (1924), which is dz dt
The ratio of the added mass in deep water to that in shallow water is plotted in Fig. 3, using Equation (3) as derived by Sheng (1981). Using Equations (8,9 and 10) the added mass ratio is shown in Fig. 4, determined by the conformal mapping procedure described in this paper for elliptically shaped bodies, having a range of beam draught ratios, including the exact result for the flat plate (BI2T = 0).
-==P=-= + -===,=,Q=-= -1),Ji a),Ji ,
J(t
J(t -
(6)
where P and Q are constants which control the dimensions in the z-plane. By integration and transformation between the z, t and w planes, Clarke (1995b) showed that the relationship between the zplane and the w-plane, which effectively determines the complex potential of the flow, may be written as,
The discrepancy between the empirical equation of Sheng (1981) and the conformal mapping results described here, is quite apparent when comparing Figs. 3 and 4.
120
1.0 0.9 0.8
1.0
~
~ :-:::::: R ~ ~ " ~ 1\.~ ~
'"
0.7 0.6 0.5
~""" r\." r-..." \ ~ ~ 0 r\.\ \ 1 \ \'\ fs\ ,\1\\'(0'-1.
0.8
In
0.4
" \. \
"~
"
,
~
0.5
\
~
0.2
O. I
0.0
0.0
''\ \.
~
~O
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 l.0
ill
!\~~
~:\ ~ '<2.'?-, ~ ~~ ~~ ~C!i '1'
0.3
0.1
\
\.
0.4
"~
0.2
~
0.6
1\\,\
0.3
0.7
"\
~ ~\
--"
I""r--.. I'-....
0.9
~1
~2
~3
~4
~5
~6
~7
~8
~9
Fig. 3. Added Mass of an Ellipse in Shallow Water (Sheng 1981)
Fig. 5. Added Mass of a Semi-Circle in Shallow Water (Clarke, 1995a&b)
l.0
1.0
~ r--...' ~ r--.....
0.9 0.8 0.7 0.6 0.5 0.4 0.3
0.2 0.1 0.0
0.9
"\ ~ ~ ~ ~ I'-... ~ ~"\ '" r--...." ~~ \\ 1\\, r\.' ~~ ~'o
" ", \ 1\\ \ ro \ ~~1 '\
0.8 0.7 0.6
j\.\ \ 1\
0.5
\\ \
\~ \. r\
1.0
T/H
T/H
0.4
\\ \
0.3
0:' ,\ \ \
,
.... ~ ~
t- r--.....
~ ~ ~ ~ ~ ........
~~
"'-~
"" ~ 0- I\.. ',., ~ '\ ~\ ~ N~ ~ \ ~ ~ ~ 1\\ ~ ~1\.\ ,\
"
0.1
~
\
\
~ ~ ,\i\ ~ ~~
0.2
\\ '" ~~ ~
~
~
0.0
~
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
T/H
T/H Fig. 4. Added Mass of an Ellipse in Shallow Water (Clarke 1995b)
Fig. 6. Added Mass of a Rectangle in Shallow Water (Flagg and Newman 1971)
By plotting the results against draught to depth, then for very shallow water the results tend to zero for T / H = 1.0. This is much more convenient the usual manner of plotting, where the graph goes to infinity in shallow water. A comparison with the circular case using distributed dipoles (Clarke, 1994a) may be seen in Fig. 5, by utilising the previously derived elliptical result for (BI2T = 1.0). For T I H> 0.5, the two results diverge, indicating that the shallow water added masses will be overestimated in the general elliptical case. Also shown is the added mass ratio determined from Equation 3 for BIT = 2.0, the circular case. The error in the added mass data of Sheng (1981) is quite obvious here.
3.3 Rectangular Body Sections.
For the purpose of comparison, ratio of added mass in deep water to that in shallow water for a rectangular section is shown in Fig. 6. This result has been derived from Flagg and Newman (1971). It may be immediately seen, in comparison with Fig. 3, that the ratio of the added masses for the rectangle and the ellipse are remarkably similar. It must be remembered, however, that the added mass of the rectangle in deep water is not unity, and therefore Equations 3 and 4 cannot be used without modification, since the assumption was made there that the added mass in deep water was unity.
121
Clarke, D (1995a). Calculation of the Added Mass of Circular Cylinders in Shallow Water. Technical Report, No. M2 (1), Hydromechanics Research Group, University of Newcastle upon Tyne.
4. NEW COEFFICIENTS A new fit to the data plotted in Fig. 4 was obtained, using Equation 3 and a least squares fitting process. The new coefficients will supersede those given previously by Sheng (1981) and plotted in Fig. 3.
Clarke, D (1995b). Calculation of the Added Mass of Elliptical Cylinders in Shallow Water. Technical Report, No. M2 (2), Hydromechanics Research Group, University of Newcastle upon Tyne.
Table 1.
ao bo al bl Cl
a2
Sheng Fig. 3
New Fit Fig. 4
0.0775 ..().0110 ..().0643 0.0742 -0.0113 0.0342
0.0774 ..().0151 "().0125 0.1674 ..().0199 0.0431
Clarke, D (1995c). Calculation of the Added Mass of Elliptical Cylinders with Vertical Fins in Shallow Water. Technical Report, No. M2 (3), Hydromechanics Research Group, University of Newcastle upon Tyne. Flagg, c., N. and J. N. Newman (1971). Sway Added Mass Coefficients for Rectangular Profiles in Shallow Water. Journal of Ship Research, 15, 257-265. Gurevich, M. I. (1940). Added Mass of a Lattice Consisting of Rectangles (in Russian). Prokl. Mat. i Mech ., 39, 1, 97-115.
In both cases the curve fit is only applicable for TIH values between 0 and 0.8. Even the new fit is not very reliable for small BIT and it would be advantageous in future to derive a different curve fit, which makes use of the fact that all the curves must pass through the corner points (1.0,0) and (0, 1.0). This is not a feature of the fitting which is currently applied with Equation 3.
Kennard, E. H. (1967). Irrotational Flow of Frictionless Fluids, mostly of Invariable Density. David Taylor Model Basin, Report 2299, 108-112. Lockwood- Taylor, 1. (1930). Some Hydrodynamical Inertia Coefficients. Philosophical Magazine, Series 7, 9, 55, 161-183.
5. CONCLUSIONS
By calculating the added mass coefficients of the elliptical body sections in shallow water, it has been possible to amend the previously published empirical equations for the shallow water corrections to the linear derivatives used in ship manoeuvring studies. It is therefore proposed that the formulae derived within this paper are used in future, for the solution of ship manoeuvring problems in shallow water, in place of those published previously.
Page, W. M. (1913). Some Two-Dimensional Problems in Electrostatics and Hydrodynamics.
Proceedings of the London Mathematical SoCiety, Series 2,11,313-328. Richmond, H. W. (1924). On the Electrostatic Field of a Plane or Circular Grating formed of Thick Rounded Bars. Proceedings of the London Mathematical SoCiety, Series 2, 22,389-403 . Sedov, L. I. (1965). Two-Dimensional Problems in Hydrodynamics and Aerodynamics, p.116-118. Interscience Publishers, New York.
REFERENCES Clarke, D (1972). A Two-Dimensional Strip Method for Surface Ships: Comparison of Theory with Experiments on a Segmented Tanker Model.
Sheng, Z. Y. (1981). Shallow Water Effect on Hydrodynamic Derivatives of Ship Hull.
Journal of Mechanical Engineering &ience,
Proceedings of 2nd Symposium on Manoeuvrability. Marine Design and Research
14, 7, (Supplementary Issue), 53-61.
Institute of China, Shanghai, China.
Clarke, D., P. Ged.Iing and G. Hine (1983). The Application of Manoeuvring Criteria in Hull Design using Linear Theory. Transactions of
the Royal institution of Naval Architects, 125, 45-68.
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