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STEADY-STATE HYDRODYNAMICS OF HIGH-SPEED VESSELS WITH A TRANSOM STERN 191
Steady-State Hydrodynamics of High-Speed Vessels with a Transom
Stern
Lawrence J.Doctors (The University of New South Wales, Australia)
Alexander H.Day (The University of Glasgow, Scotland)
ABSTRACT
The inviscid linearized near-field solution for the flow past a vessel with a transom stern is developed within the
framework of classical thin-ship theory. However, the innovation in the current approach is that the hollow in the water
behind the stern is represented here by a virtual extension to the usual hull-centerplane source distribution.
In the present work, the near-field solution to the flow using the thin-ship approximation must be computed. This idea
clearly demands a considerable addition to the complexity of the numerical solution which contrasts with the traditional
farfield method employed by Michell (1898).
The computer program functions by iterating the geometry of the hollow until the criterion of atmospheric pressure on
the surface of the hollow is zero. In addition, the iteration procedure includes adjusting the sinkage and trim of the vessel
until it is in equilibrium. The latter component of the computation utilizes an integration of the resulting pressure
distribution over the wetted surface of the vessel in an entirely consistent manner.
Comparison of the theoretical results with a systematic series of twelve models shows excellent correlation with the
towing-tank data. Indeed, the behavior of this approach appears to be much more robust than, for example, that of the
Neumann-Kelvin problem and some fully nonlinear analyses.
INTRODUCTION
Literature Review
Previous work on the subject of prediction of resistance of marine vehicles, such as monohulls and catamarans, has
shown that the trends in the curve of total resistance with respect to speed can be predicted with excellent accuracy, using
the traditional Michell (1898) wave-resistance theory, together with a suitable formulation for the component of frictional
resistance. There have been further enhancements to this wave-resistance theory. These enhancements include the
influences of finite depth and finite width of the canal by Lunde (1951) and Sretensky (1936).
A recent justification for this research, in which linearized free-surface conditions are employed, is the very
encouraging comparisons that were made by Doctors and Renilson (1993) for monohulls and catamarans with closed or
pointed sterns and by Sahoo, Doctors, and Renilson (1999) for monohulls with open or transom sterns.
One difficulty has been that nonlinear and viscous-wave effects are not included and, consequently, the correlation
between theory and experiment has not been sufficiently good for the purpose of practical ship design. For this reason, a
considerable effort has been invested in recent years in the development of fully nonlinear computer codes. The complete
nonlinear kinematic body-boundary condition and the nonlinear free-surface kinematic and dynamic boundary conditions
are satisfied in these programs. There has been excellent progress with such computer programs and they may eventually be
developed to the stage where they can be used for hull-form development.
Currently, the execution time is too long for one to contemplate any realistic optimization of hull forms.
Consequently, any type of optimization, such as the genetic-algorithm method of Doctors and Day (1995) and Day and
Doctors (1997b) is not possible. This is because of the requirement to evaluate the resistance many thousands of times
during the practical design process.
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STEADY-STATE HYDRODYNAMICS OF HIGH-SPEED VESSELS WITH A TRANSOM STERN 192
Figure 1: Definition of the Problem (a) Geometry and Forces
A further point is that more sophisticated computer codes, such as those briefly alluded to above, do not always lead to
more accurate or reliable predictions for sensitive quantities, such as resistance. This is because resistance can be affected
markedly by minor inaccuracies in the computed pressure distribution over the surface of the hull. A revealing study of this
troubling possibility was published by Sahoo, Doctors, and Renilson (1999). It was demonstrated there that more reliable
predictions for the resistance were obtained from the consistent linearized approach, than from a modern nonlinear code,
for a set of fourteen modern high-speed vessels with transom sterns. Indeed, the linearized approach gave predictions which
were within 5% for most of the test cases, while the errors from the competing nonlinear method were typically an order of
magnitude greater.
Further reductions in the errors inherent in the linearized theory can be obtained in a most practical manner by means
of very easily estimated correction factors for the wave resistance and for the frictional resistance. Examples of this
research were published by Doctors (1998a, 1998b, and 1998c).
Current Work
These principles were advanced in the research presented by Doctors and Day (1997). Firstly, transom-stern effects
were included in the theory by accounting for the hollow in the water behind the vessel. This work was essentially a
development of the ingenious and practical approach first presented by Molland, Wellicome, and Couser (1994) and
Couser, Wellicome, and Molland (1998).
Figure 1: Definition of the Problem (b) Centerplane Paneling
There, a simple virtual extension to the hull behind the transom was constructed in the computer program. The wave
resistance for the vessel was deduced on the basis of an application of the Michell integral to the entire model and its
extension, together with an estimate of the hydrostatic resistance due to the existence of the dry transom. The extension was
allowed to grow in length with increasing forward speed of the vessel in a physically plausible manner as detailed by
Doctors and Day (1997).
A more sophisticated enhancement of this work was presented recently by Doctors and Day (2000). The local flow
field was computed on the basis of the linearized theory. This permitted the squat (sinkage and trim) to be determined;
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excellent correlation with experimental data was obtained in this way.
However, the shape of the hollow was still determined in a heuristic manner. It is the purpose of the current research
effort to iterate the shape of this transom-stern hollow in order to improve the accuracy of the approach even further.
THEORY
Definition of the Problem
Figure 1(a) shows a typical arrangement for a vessel traveling at a constant speed U in calm
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STEADY-STATE HYDRODYNAMICS OF HIGH-SPEED VESSELS WITH A TRANSOM STERN 193
water. The x, y, z coordinate system is also depicted. The water is unbounded laterally (in the y direction) as well as having
infinite depth. The components of the forces acting on the vessel are indicated.
The vessel can either be self propelled or be towed. In the former case, the thrust from the propeller or the water jet
acts along a defined line of action relative to the coordinate system attached to the vessel. Thus, the direction and position
of the thrust line vary with the speed of the vessel. In the latter case, the vessel is towed at the specified speed from a
particular point in the hull. Hence, the line of action of the thrust is longitudinal, but the line moves vertically in sympathy
with the sinkage and trim.
Discretization of the Hull
Further details of the problem are provided in Figure 1(b). Here, the centerplane paneling is seen to overlay the hull
and the hollow in the water behind the transom stern. The panels or elements possess a flat facet and a rectangular base.
They are employed, in particular, for the purpose of the numerical calculation of the pressure, or profile, resistance. These
elements are chosen in order to approximate the centerplane area of the hull and the hollow as closely as possible. The
longitudinal and vertical slope of the facets of the elements match the corresponding values of the hull surface in a root-
mean-square sense.
This type of panel is algebraically simpler than the “pyramids” or “tents” which were previously employed by Day and
Doctors (1997b) and Doctors and Day (1997a), for example. The use of flat facets implies a higher level of discontinuity on
the hull surface. On the other hand, numerical convergence tests for wave resistance, based on the two types of panels,
showed that a similar number of panels was required in either case; namely, 40 panels in the longitudinal direction and 8
panels in the vertical direction.
Equations for the Potential
We start in the traditional manner by utilizing the potential whose gradient gives the perturbation velocity. The
potential satisfies the Laplace equation throughout the fluid domain:
(1)
The linearized kinematic free-surface condition, namely
(2)
states that any water particles on the free surface remain there. Here, ζ is the elevation of the free surface measured
upward from its undisturbed position z=0. In addition, the Bernoulli equation provides the linearized dynamic free-surface
condition
(3)
in which g is the acceleration due to gravity. The Rayleigh (1883) artificial viscosity µ, which is assumed to be
vanishingly small and positive, has been introduced in this technique in order to impose the radiation condition, which
states that waves must be propagated downstream. De Prima and Wu (1957) gave a clear description of this concept.
Elimination of ζ from Equations (2) and (3) yields the linearized combined free-surface condition:
(4)
Finally, the “bed” kinematic boundary condition states that
(5)
Potential-Flow Solution
The solution for the flow past the vessel and its transom-stern hollow is obtained by using the equivalent centerplane-
source distribution. This distribution is assembled from panels, as noted earlier, while the panels are constructed from the
elementary Kelvin point source. The potential due to a Kelvin point source of strength Q, obtained by Wehausen and
Laitone (1960, p.484, Equation (13.36)), is
(6)
where we have defined the radial distances
(7)
(8)
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STEADY-STATE HYDRODYNAMICS OF HIGH-SPEED VESSELS WITH A TRANSOM STERN 194
Figure 2: Sample Ship Models (a) Lego Ship Models 7 Figure 2: Sample Ship Models (b) Lego Ship Models 11
and 8 and 12
and the wave term is
(9)
Here, the circular wave number is
the fundamental wave number is
and the complex horizontal wave number is
The first term in Equation (6) can be integrated for a constant-strength-source panel oj and a unit-constant-
longitudinal-slope field panel in the so-called Galerkin manner. The result for the induced longitudinal gradient of the
potential at the field panel is
(10)
in which the weighting factor is given by the formula
and the required arguments in the special function G4 are defined by
The four special functions needed for this analysis are interrelated by spatial integrations as follows:
(11)
(12)
(13)
(14)
The second term in Equation (6) can be integrated in a very similar manner.
The third term can also be integrated with respect to the wavenumber k, as well as with re
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STEADY-STATE HYDRODYNAMICS OF HIGH-SPEED VESSELS WITH A TRANSOM STERN 195
spect to the spatial coordinates, to yield
(15)
in which the argument of the function F4 is given by
The special complex wave function F4 is closely related to the exponential integral and was defined by Doctors and
Beck (1987). The zeroth wave function is
(16)
and the complex exponential integral is
while the other four complex wave functions are
(17)
(18)
(19)
(20)
Modeling of the Hollow
The hollow is considered to be a virtual extension to the hull of the vessel and is modeled by a continuation of the
centerplane source distribution. The determination of the shape of the centerplane of this hollow represents a vital part of
the hydrodynamic problem.
The length of the hollow is initially estimated by the method detailed by Doctors and Day (1997). That is, the
equivalent radial position of a point in the region of the hollow behind the stern is taken to be
(21)
We next state that the effective trajectory of a particle of water on the surface of the hollow is parabolic in nature and
that it has parametric coordinates as follows:
(22)
(23)
in which xtran and rtran are the coordinates of the equivalent springing point of the hollow on the transom girth.
Equations (22) and (23) can be solved by setting r=0 to yield the location of the vertex, or rooster tail, of the hollow
xholl. Hence, the length of the hollow is evaluated as:
(24)
In the current enhanced technique, the initial estimated profile of the hollow is assumed to be defined by the parabola
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that springs longitudinally from the bottom of the transom and meets the vertex on the (undisturbed) free surface.
Forces and Moment on the Vessel
One first assumes that the attitude of the vessel is the same as its static attitude, that is, with zero sinkage and trim. The
required source strength is computed using the standard thin-ship result, namely,
in which b is the local beam. The total gradient of the potential at the field panel i is next computed, by summing
the contributions from the source panels presented in Equations (10) and (15). From this, one can determine the pressure on
the surface of the hull, as follows:
(25)
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STEADY-STATE HYDRODYNAMICS OF HIGH-SPEED VESSELS WITH A TRANSOM STERN 196
where ρ is the density of the water and z is the elevation of the centroid of the field panel.
Next, the three components of the generalized forces on the vessel (pressure resistance, sinkage force, and bow-up
moment) are then found from this pressure distribution:
(26)
in which the generalized surface slope is given by the formula
(27)
Finally, the total resistance can be found by the simple summation
(28)
in which R P is the pressure resistance computed from Equation (26), RF is the frictional resistance estimated from the
1957 International Towing Tank Committee (ITTC) formula, described by Lewis (1988, Section 3.5), and RA is the
correlation resistance.
Equilibrium of the Vessel
At any stage of the iteration for the equilibrium of the vessel, one can estimate the corrections to the sinkage and trim
angle (positive bow down), with respect to the longitudinal center of flotation LCF:
(29)
(30)
The additional symbols introduced here are SP the sinkage pressure force, W the weight of the vessel, Zprop the vertical
component of the propulsion force (equal to zero in the case of the vessel being towed), AW the static waterplane area, zP
the vertical lever arm for the pressure resistance (equal to zero), xP the longitudinal lever arm for the pressure sinkage force
(equal to −LCF), MP the bow-up pressure moment, xprop the longitudinal lever arm for the propulsion force, zprop the
vertical lever arm for the propulsion force, Xprop the longitudinal component of the propulsion force, zF the vertical arm for
the frictional force (measured to the centroid of the wetted surface), zstab the vertical lever arm to the stabilizers, Rstab the
resistance of the stabilizers (zero in the current work), LCB the longitudinal center of buoyancy, and the
longitudinal metacentric height.
The sinkage at the coordinate origin x=0 and the trim are
(31)
(32)
where L is the length of the vessel.
For simplicity, the hydrostatic stiffness coefficients were used for iterating the sinkage and trim of the vessel, as seen in
Equations (29) and (30). The use of the ideally consistent hydrodynamic stiffness coefficients would have posed a
somewhat major computing challenge. Relative convergence of 1×10−4 could be obtained within about eight iterations;
once equilibrium is achieved, there is no error introduced by the simpler approach.
Iteration of the Hollow
The pressure at any point on the hollow is given by a formula of the type of Equation (25). If there are Nholl panels that
lie on the centerplane of the hollow, then there will be the same number of panel collocation points at which we desire the
pressure to be zero (or atmospheric). The Kutta condition is applied by extending this requirement to include the last
column of panels on the surface of the hull next to the transom.
In addition, we must impose the closure conditions. Because the panels are of uniform rectangular shape, we just
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require the source strengths σj along a longitudinal line of panels (z is constant) to sum to zero.
A set of overconstrained linear equations is then set up, in which the pressures are minimized
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STEADY-STATE HYDRODYNAMICS OF HIGH-SPEED VESSELS WITH A TRANSOM STERN 197
in a least-square sense while the closure conditions are satisfied exactly. Next, the secant method is employed in which the
length of the hollow is adjusted until the average pressure on the surface of the hollow is minimized.
The error in the average pressure on the surface of the hollow after the iteration process was typically one percent of
the hydrostatic pressure at the keel of the vessel.
In summary, it can be seen that the methodology used in this work is analogous to that underlying the solution of the
problem of symmetric two-dimensional cavitating flow behind a body possessing a bluff stern, as presented by Newman
(1980, Section 5.13, pp 208–215). However, the following extensions have been introduced:
1. Three-dimensions;
2. Free-surface;
3. Equilibrium of the body;
4. Iteration of cavity shape.
Simplistic Resistance
The simpler and traditional approach to thin-ship resistance has been to utilize the Michell (1898) result for the
resistance applied to the vessel, together with a suitable formulation for the shape of the hollow. Examples are the work of
Day and Doctors (1997a), Day, Doctors, and Armstrong (1997), Doctors and Day (1997), and Doctors (1999). This
approach leads to the following estimate of the total resistance:
(33)
in which R W is the wave resistance, computed as
(34)
where the complex Michell wave function is
(35)
Table 1: Lego Ship Models (Common Data)
Item Symbol Value
Lbow
Length of bow section 0.750 m
B
Waterline beam 0.150 m
T
Draft 0.09375 m
CM
Maximum-section coef. 0.6667
with S0 being the centerplane area. The complex wave function for a panel with center is
(36)
Finally, RH is the so-called hydrostatic resistance, resulting from the imbalance of the hydrostatic pressure owing to
the transom being “dry”:
(37)
in which T tran is the draft at the transom.
This simplistic approach to resistance provides an identical result to that from the near-field approach described earlier
in this paper, in the case of a vessel with no transom.
LEGO SHIP MODEL SERIES
This series of hulls was developed with the intention of studying the hydrodynamics of transom sterns. Doctors
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(1998a) provided the details of the hull segments from which the ship models were assembled. There was a total of seven
segments. The bow and stern segments have parabolic waterplanes. The bow, stern, and parallel-middle-body segments all
possess parabolic cross sections. Figure 2 shows views of four of the test models. Table 1 and Table 2 list the details of all
twelve of these so-called Lego Ship Models.
RESULTS
Numerical Convergence Tests
The wave resistance of a vessel without a transom stern, computed according to Equations (34) to (36), is identical to
that computed
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STEADY-STATE HYDRODYNAMICS OF HIGH-SPEED VESSELS WITH A TRANSOM STERN 198
using the considerably more elaborate near-field solution. This assumes the same discretization of the hull centerplane and
the use of the same mesh in the wave-angle integration. Thus, one has an interesting and powerful check of the computer
coding.
Table 2: Lego Ship Models (Variable Data)
Length of Parallel Middle Body LP (m) Length of Run LR (m) Length L (m) Prismatic Coefficient CP
Ship Model
1 0.000 0.0000 0.7500 0.6666
2 0.000 0.1875 0.9375 0.7290
3 0.000 0.3750 1.1250 0.7499
4 0.000 0.5625 1.3125 0.7290
5 0.750 0.0000 1.5000 0.8332
6 0.750 0.1875 1.6875 0.8494
7 0.750 0.3750 1.8750 0.8499
8 0.750 0.5625 2.0625 0.8275
9 1.500 0.0000 2.2500 0.8888
10 1.500 0.1875 2.4375 0.8957
11 1.500 0.3750 2.6250 0.8928
12 1.500 0.5625 2.8125 0.8735
The four parts of Figure 3 show a test of convergence for Lego Ship Model 7 for four physical parameters of interest.
These parameters are the total resistance RT, the sinkage s, the trim t, and the hollow length Lholl. These parameters have
been rendered dimensionless using the weight of the ship W or its length L, as appropriate. It can be seen that using 40
panels longitudinally and 8 panels vertically is sufficient for the current purpose. In the same vein, one requires 32 points
for half the range of the θ integration in Equation (15) to obtain reasonable convergence of the results.
The annotation Holl=P, K indicates that in this case the pressure condition on the surface of the hollow and the Kutta
condition on the transom were satisfied. Also, Free=Yes states that the model was free to sink and trim. The slenderness
coefficient L/ ∇ 1/3 is also printed on the plots.
Resistance Components
Figure 4 depicts the theoretical computations of the various resistance components referred to above, for four of the
ship models. These show the hydrostatic resistance RH, the pressure resistance RP, the frictional resistance RF, the total
resistance RT, and the total experimental resistance RT,E. In general, the correlation between the theory and the experiments
is good at the higher speeds, which are of practical significance, that is, for a Froude number F of 0.6 or greater.
The disagreement is greatest at the lower speeds, where the transom would in reality be partly wetted, thus reducing
the drag. This phenomenon has been ignored in the current calculations. An indication of the error involved at these low
speeds is just the hydrostatic resistance; it can be observed that subtracting this quantity would bring the theoretical
calculations into line with the experimental data. This process was, in fact, done by Doctors (1998d) in an approximate
manner.
Comparison with Experiments
Figure 5 shows the specific resistance R/W for the four Lego Ship Models. The different methods used are
respectively: the simplified method of Doctors and Day (1997) (Holl=Simp), the simplified method of Doctors and Day
(2000) with the vessel also free to sink and trim (Free=Yes), the current method with the pressure and the Kutta conditions
applied (Holl=P, K), and the current method with the pressure and Kutta conditions applied and also the hollow length Lholl
iterated to minimize this pressure (Holl=P, K, L).
It is noteworthy that the main factor of importance is the need to permit the vessel to sink and trim in order to find its
equilibrium position. The other various enhanced versions of the theory appear to offer little, if any, improved correlation
with the experiments.
We next examine Figure 6, which shows a comparison of the theoretical predictions and experiments for the
dimensionless sinkage s/L for the four Lego Ship Models. The second, third, and fourth methods are considered here. The
first method, of course, does not include sinkage and trim at all. Again, we observe that the simplified method which
permits determining the equilibrium of the vessel, namely that of Doctors and Day (2000), is as effective as the current
more sophisticated approaches.
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STEADY-STATE HYDRODYNAMICS OF HIGH-SPEED VESSELS WITH A TRANSOM STERN 199
Figure 3: Convergence Tests (a) Resistance Figure 3: Convergence Tests (b) Sinkage
Figure 3: Convergence Tests (c) Trim Figure 3: Convergence Tests (d) Hollow Length
Similar sentiments can be expressed with regard to Figure 7, which presents the dimensionless trim t/L for the four
Lego Ship Models. Again, the simplified method which includes the determination of the equibrium of the vessel, is as
accurate as the current methods.
Finally, we present Figure 8, which shows the dimensionless hollow length L holl/L for the four Lego Ship Models. It is
curious to note that there is little significant difference between the length of the hollow computed in the current
hydrodynamic manner and the length of the hollow resulting from the simplified geometric approach of Doctors and Day
(2000), for example. At the very minimum, the general trend of the variation of the length of the hollow with the Froude
number is properly predicted, but would seem to be rather too high.
It is also vital to point out the obvious fact, with regard to Figure 8, that the experimental data is at least as subject to
debate as the theoretical calculations. The length of the hollow was determined by means of a visual observation using a
wire mesh as a reference frame. The accurate location of the rooster tail of the hollow xholl was not possible because of
various factors; it would even seem that the definition of this length itself poses a major difficulty.
CONCLUDING REMARKS
The present research has been instructive in showing that the precise length of the hollow behind the transom stern is
not critical to predicting the resistance, sinkage and trim, of the vessel. In many ways, this is quite remarkable, because the
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STEADY-STATE HYDRODYNAMICS OF HIGH-SPEED VESSELS WITH A TRANSOM STERN 200
flow of the water in the region of the hollow must surely affect the pressure distribution on the hull itself. There should be a
resulting influence on the behavior of the vessel. It now appears that the principal features of the hollow can indeed be
modeled in the simplified manner, allowing one to find (within engineering accuracy) the essential hydrodynamic
performance factors of interest.
Figure 4: Resistance Components (a) Lego Ship Model Figure 4: Resistance Components (b) Lego Ship Model
7 8
Figure 4: Resistance Components (c) Lego Ship Model Figure 4: Resistance Components (d) Lego Ship Model
11 12
Other models for the profile of the hollow have been considered. For example, it is thought that an S-shaped profile
might also be worthy of study. Here, one end of the S corresponds to the springing point from the bottom of the transom
and the other end corresponds to the rooster tail. Both ends of the S would be horizontal in this alternative model.
It is necessary to emphasize that, at least with regard to resistance, the current approach still uses a traditional
estimator for the frictional component. That is, ideally, one might consider utilizing a form factor to correct the frictional
resistance.
Alternatively, and preferably, a more sophisticated approach could be considered. That is, the influence of the actual
geometry of the hull on the frictional resistance would be computed. Such an approach would take into account the non-
zero pressure gradient on the surface of the hull and would be consistent with the current thin-ship theory.
Finally, it would be feasible to incorporate the influence of the deformation of the free-surface on the actual
submerged volume of the vessel. This
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could be effected approximately by means of a local vertical distortion of the sectional geometry. The resulting influence
may well be of similar significance to that of permitting the vessel to find its equilibrium.
Figure 5: Comparison of Resistance (a) Lego Ship Figure 5: Comparison of Resistance (b) Lego Ship
Model 7 Model 8
Figure 5: Comparison of Resistance (c) Lego Ship Figure 5: Comparison of Resistance (d) Lego Ship
Model 11 Model 12
ACKNOWLEDGMENTS
The authors would like to thank the Directorate of Naval Platform Systems Engineering, Department of Defence, for
its support through Contract 9627MZ. They also gratefully acknowledge the assistance of the Australian Research Council
(ARC) Large Grant Scheme (via Grant Number A89917293). The support of this work by The University of New South
Wales is also greatly appreciated.
REFERENCES
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Stern Hulls Using a Slender Body Approach”, International Shipbuilding Progress, Vol. 45, No. 444, pp 331–349 (December 1998)
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Figure 6: Comparison of Sinkage (a) Lego Ship Model 7 Figure 6: Comparison of Sinkage (b) Lego Ship Model 8
Figure 6: Comparison of Sinkage (c) Lego Ship Model Figure 6: Comparison of Sinkage (d) Lego Ship Model
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Figure 7: Comparison of Trim (a) Lego Ship Model 7 Figure 7: Comparison of Trim (b) Lego Ship Model 8
Figure 7: Comparison of Trim (c) Lego Ship Model 11 Figure 7: Comparison of Trim (d) Lego Ship Model 12
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Figure 8: Comparison of Hollow Length (a) Lego Ship Figure 8: Comparison of Hollow Length (b) Lego Ship
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Figure 8: Comparison of Hollow Length (c) Lego Ship Figure 8: Comparison of Hollow Length (d) Lego Ship
Model 11 Model 12
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DISCUSSION
L.Raheja
Indian Institute of Technology, India
It was a nice presentation. It is not a question but more of a comment. It seems that the primary reason or motivation
for using such a simplified model (i.e. centerplane source distribution) is because it is expected or presumed that the wave
drag will be much smaller than the frictional part. I propose that the model could be improved upon by using a
desingularized panel method where the sources could be placed little inside the boundary which would still preserve the
simplified calculation aspect and may work better in the sense of diluting the presumption.
AUTHOR'S REPLY
I appreciate your comment, Since we were getting good results in this case, so we continued with the model.
However, we may improve upon it in subsequent work. Thank you very much for your question.
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