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FREE SURFACE VISCOUS FLOW COMPUTATION AROUND A TRANSOM STERN SHIP BY CHIMERA OVERLAPPING SCHEME 441
Free Surface Viscous Flow Computation Around A Transom Stern
Ship By Chimera Overlapping Scheme
C.W.Lin, S.Percival
(Naval Surface Warfare Center, Carderock Division, USA)
ABSTRACT
The focus of this paper is to investigate the capability of a numerical scheme that incorporates a free surface
boundary treatment and a sinkage/trim calculation into a viscous flow computation around a surface ship. Utilizing this
scheme computations are performed on two selected transom stern ships. The ability to compute the wet, partially wet, or
dry transom is developed, including a new grid topology for transom sterns. Computational results on two transom-stern
ships are consistent with available model experimental data. The shape of the wave profile along the hull is generally
predicted well, while the magnitude of wave elevation still needs to be improved. The wave pattern is also well
correlated, except that wave crests and troughs are not as sharp as the model measurements. The computed total resistance
is in good correlation with model-measured values. Although two different types of grid spacing and distribution
techniques are investigated, a further study is needed to determine a more effective grid system for free surface viscous
flow computations.
INTRODUCTION
Computing the flow around a surface ship moving steadily in calm water has been a challenging research task for
years. The complexity of the flow physics around a surface ship has generally required two separate numerical
approaches, one to compute the wave elevation on the free surface and a second to compute the viscous boundary layer
around ship hull. The baseline equation to compute the free surface elevation is usually based on potential flow theory.
The numerical scheme can use either a complicated Green function or use the panel method with simple Rankine sources.
The governing equations to compute the viscous boundary layer on a ship hull surface are based on the Navier-Stokes
equations. The numerical methods to compute viscous flow have been developed from a simplified boundary layer
approximation to a variety of different formulations of Navier-Stokes equations. In this paper, a combined numerical
technique based on computing the free surface viscous flow around a realistic ship hull is developed. It is based on
solving the Reynolds Averaged Navier-Stokes (RANS) equations using a computational scheme that also calculates the
water surface elevation generated by a moving ship hull on the free surface boundary.
Another challenging task is to be able to compute flow around a surface ship with various types of ship geometry. In
addition to the necessary appendages required to operate a ship, designers have developed a variety of hull shapes to meet
the functional requirements of a ship, either for commercial benefits or military advantages. Transom stern ships are the
focus of this paper. Transom stern ships have been used for years, especially for high-speed ship designs. The
hydrodynamic performance of this type of ship has been investigated in various model test facilities. However, the flow
around the transom stern is very complicated and difficult to compute due the discontinuity of the hull geometry.
It is especially challenging to compute the free surface elevation around the transom stern. The issue of wet/dry
transom condition is a major concern for a transom-stern ship design. In this paper, a numerical technique is developed to
handle this complicated flow phenomena. The numerical technique includes a new grid topology for transom stern flow
calculations. In this way the wet/dry transom condition can be computed.
A special topic of free surface flow computation is to calculate the sinkage and trim of the vessel. The quantities of
sinkage and trim depend on the shape of underwater geometry and the ship speed. Model test results have shown a
difference in measured resistance between fixed models and free to sink/trim models. Since the sinkage and trim data are
not known in advance, the free surface flow
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FREE SURFACE VISCOUS FLOW COMPUTATION AROUND A TRANSOM STERN SHIP BY CHIMERA OVERLAPPING SCHEME 442
computation has to calculate them as part of the flow solutions. The numerical scheme in this paper includes this
capability.
The validation discussed in this paper is the comparison of computational results with a set of experimental model
data. For free surface viscous flow calculations, two series of experimental measurements are chosen, DTMB Model 5365
(R/V Athena) and 5415 (a surface combatant hull selected by ONR for CFD validation). The comparisons include total
resistance coefficients, sinkage/trim, wave profiles, and wave pattern in the free surface. The numerical treatments for
computing free surface elevation and sinkage/trim are discussed in detail. The computational grid topology used to handle
transom stern geometry is described, as well as grid spacing, grid distribution, and computational grid domain. A limited
grid study on flow computations is briefly discussed.
NUMERICAL SCHEME
Numerical calculation of viscous flow around a surface ship must include two major physical mechanisms:
computation of the water elevation at the free surface and computation of the viscous boundary layer around the ship hull.
Until recently, the free surface wave elevation generated by a moving ship has been handled by the potential flow
approximation, which assumes no viscosity. The calculation of the viscous boundary layer is performed by the double-
body approximation, which ignores the effect of the free surface. Based on this kind of decoupled numerical treatment,
the free surface viscous flow around a moving ship can be affordably solved. The interaction between viscous boundary
layer and free surface wavemaking is assumed to be minimal. As for practical engineering calculations, this assumption
remains valid to support ship design efforts. However, due to recent advances in CFD numerical schemes and computer
CPU power, this decoupling of the numerical methodology may not be necessary. One can compute the viscous boundary
layer and the free surface wave elevation together in an affordable manner. This is achieved by solving the RANS
equations with careful treatment of free surface boundary.
The fundamental RANS equations in the free surface flow computations remain the same as those used for
calculations of double-body flow, with the exception of the pressure term which has to include static head from free
surface boundary. The numerical treatment of double-body RANS computations can be generally adopted in the free
surface RANS calculations without any modifications. The numerical scheme for this double-body RANS computation in
FREE98 has been in development for years and will not be discussed in this paper (Lin et al 1995, Lin et al 2000). The
main numerical effort here is to handle the movement of the free surface boundary. Several numerical methods have been
developed to compute this free surface boundary, and they can generally be classified in three ways: surface fitting,
surface tracking, and surface capturing. The efforts of all these methods are to compute free surface elevation by
satisfying both kinematic and dynamic boundary conditions required in the free surface. The kinematic boundary
condition forces water particles on the free surface to remain in the boundary surface all the time. The dynamic condition
satisfies constant atmospheric pressure on the free surface boundary. FREE98 has adopted the surface fitting method
developed by Farmer (Farmer et al 1993). A volume grid comprising the computational domain is developed similar to
that for a double body calculation, with the difference that it extends above the still waterline. The lines of this initial grid
that are normal to the free surface are fitted with B-spline curves, which will be used to track the free surface elevation
during computations. At the beginning of the computation, the grid is moved to the still water line by redistributing the
grid points along the B-spline curves. The RANS equations are then solved for this current grid. Based on the new
computed flow variables, the water surface elevations in the free surface boundary are updated to satisfy both the
kinematic and the dynamic boundary conditions. For the fully nonlinear condition, the free surface must move with the
flow boundary and the boundary conditions must be applied on this distorted free surface. The water elevation on the free
surface is computed by integrating the free surface kinematic equation, which is derived by treating the free surface as a
material surface required by free surface kinematic boundary condition. Through the interior of the free surface
computational domain, all derivatives are computed using the second order central difference. On the boundary a second
order central stencil is used along the boundary tangent and a first order one-sided difference stencil is used in the
boundary normal direction. Background dissipation must be added to prevent decoupling oscillation of the solution. This
numerical dissipation should be scaled down to the minimum value within the viscous layer, which is similar to the
double-body RANS flow computation. Through a series of numerical experiments, the numerical dissipation is controlled
to keep it as small
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FREE SURFACE VISCOUS FLOW COMPUTATION AROUND A TRANSOM STERN SHIP BY CHIMERA OVERLAPPING SCHEME 443
as possible. It has been found that numerical dissipation in the free surface computation has significant effect on the
dispersion of the wave system. Excessive dissipation may have better numerical convergence, but gives a wrong phase
angle of the wave system. Once the free surface update is completed, the pressure is adjusted on the free surface. The
updated free surface serves as new boundary values for the bulk RANS flow computations. The coupling is established by
computing a bulk flow solution and then using the bulk flow as a boundary condition for the free surface computation.
The free surface elevation is updated and its present values are used as a boundary condition for the pressure on the bulk
flow. This therefore completes a numerical iteration of free surface RANS flow calculation.
SINKAGE AND TRIM COMPUTATION
A surface ship moving at a constant speed will find equilibrium in sinkage and trim due to the balance of dynamic
forces generated by the motion. Model tests are performed in the same manner; the model is free to sink and trim, and the
values of sinkage and trim are measured at different speeds. The sinkage and trim changes the underwater shape of the
hull and the resulting hydrodynamic performance. It is therefore important to include the sinkage and trim in the
computations.
The sinkage and trim can be computed simultaneously with the free surface RANS flow computations. At each
iteration, the forces and moments acting on the underwater geometry are used to compute the change in sinkage and trim
through a simple hydrostatic calculation based on the ship's waterplane geometry. Using the initial waterplane instead of
calculating the current waterplane saves a substantial amount of computations. The assumption is the waterplane area will
not change substantially due to sinkage and trim and therefore this imposes a limitation.
The sinkage and trim is implemented by moving the hull (and the local grid) relative to the global grid, rather than
trying to move the mean free surface reference. All the free surface elevations then need to be recalculated to fulfill both
kinematic and dynamic boundary conditions in the new free surface locations. The current version of FREE98 can
activate the calculation of sinkage and trim at the beginning or after a certain number of free surface RANS computations.
The results shown in this paper are obtained by activating the sinkage and trim from the beginning of the computation and
performing one sinkage and trim calculation for every 10 iterations of free surface RANS computations.
HULLFORM GEOMETRY
The two hullforms in this paper were chosen for their transom stern shape and the availability of model test data.
Model 5365 (R/V Athena) is a high-speed, transom stern displacement ship. Model 5415 is a surface combatant hull
selected by ONR for CFD validation. Dimensions for the hullforms are shown in table 1 and bodyplan plots of stations 0
to 20 are shown in figures 1 and 2. The only appendages included in either the computations or the model tests are shown
in figures 1 and 2. Model 5365 has a centerline skeg and model 5415 has an integrated skeg and sonar dome.
Figure 1. DTMB Model 5365
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Figure 2. DTMB Model 5415

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FREE SURFACE VISCOUS FLOW COMPUTATION AROUND A TRANSOM STERN SHIP BY CHIMERA OVERLAPPING SCHEME 444
Table 1. Hullform dimensions
5365 5415
Length (m) 5.960 5.724
Beam (m) 0.836 0.764
Draft (m) 0.183 0.248
CHIMERA OVERLAPPING GRID
Grid Topology
Free surface viscous flow computations involve two major physical parameters, Reynolds number and Froude
number. To accurately compute viscous flow phenomena due to the effect of Reynolds number, fine grid resolution is
needed in the boundary layer. Conversely, to resolve the free surface wave systems due to Froude number effect requires
relatively fine grid resolution outside the boundary layer. The chimera approach has the capability to handle the necessary
grid refinement to meet both requirements without using an impractically large number of points. The computational
domain is divided into free surface and double body grid components, which are connected to each other with an
interpolation interface. By decoupling the topology of the two grids, enough flexibility is gained to generate a suitable
grid for a viscous free surface calculation.
In addition, free surface flow computation around a transom stern is a challenge. Very large pressure and velocity
gradients exist at the transom, requiring fine grid resolution and high grid quality. Furthermore, the free surface elevation
moves up-and-down with the transition from wet to dry transom. Getting the computational grid to move with the free
surface around the discontinuous edge of the transom is a topological challenge. The decoupling of the two grid
components enables a suitable grid block topology to be developed.
Figure 3. Perspective view of overall grid topology
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FREE SURFACE VISCOUS FLOW COMPUTATION AROUND A TRANSOM STERN SHIP BY CHIMERA OVERLAPPING SCHEME 445
A reverse “C” topology, a combination of an “H” topology at the bow and an “O” topology at the stern, was used for
the grid at the free surface boundary. The “H” topology simplifies setting the upstream boundary condition and the “O”
topology is efficient and provides for a relatively uniform, rectangular grid around the stern where grid quality is most
critical. In the transverse plane the grid sweeps down from the free surface around to the plane of symmetry in “O”
fashion. Forward of the stern the computational domain is split equally between the free surface and double body grid
components along a 45-degree line. In the aft region this dividing 45-degree line is rotated about a vertical axis at the
stern in order to follow the “O” topology. A perspective view of the overall topology is shown in figure 3. In this and
succeeding grid figures only every 2nd point is shown for clarity. Although this figure depicts the grid for model 5415 at
20 knots, the topology for all the grids in this paper is identical.
Figure 5. Topology of the free surface grid at transom
Figure 4. A transverse cut of the grids at Station 1
The grid on the hull surface is split between the two grid systems at approximately half the wetted girth length.
Ideally the double body grid would encompass most of the hull surface. However, some allowance must be made to
provide room for the free surface grid to redistribute itself as it moves up and down the hull. In order to simplify the
double body grid, the sonar dome on model 5415 is modeled with a third grid system. A transverse cut of all three grids at
station 1 is shown in figure 4.
The most common approach to gridding a transom stern ship has been to extrude a block from the transom face aft to
the outer bound. Amongst the many trade-offs of this approach, there are two major disadvantages for free surface
computations. Continuing the boundary layer spacing aft off the transom results in extreme variations in cell size and
distribution. The second major disadvantage is that the transom block has nowhere to move in the case of a dry transom.
The “O” topology in conjunction with chimera grids solves both of these problems. The cell sizes are fairly uniform and
the distributions are smooth. Figure 5 is a blow-up of the stern area in Figure 3., showing just the free surface block. The
grid lines that move normal to the free surface travel down the transom and forward under the hull, thus allowing the free
surface to do the same. An added benefit of this topology is that, because of its relative simplicity, the grid is easily
generated.
Extent of Domain and Grid Distribution, Size
To compute accurately the wave system generated around a moving ship, especially the diverging wave system,
requires a suitable domain and grid refinement in the free surface. Normally, in double body RANS, the distance to the
outer bound of the computational domain is based on the hull
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FREE SURFACE VISCOUS FLOW COMPUTATION AROUND A TRANSOM STERN SHIP BY CHIMERA OVERLAPPING SCHEME 446
length. For free surface computations, larger Froude numbers mean longer wavelengths in the wave system, which require
larger computational domains to cover several wave components of the generated wave system. It was decided to set the
distance to the outer bound at 2 characteristic wavelengths (Lw=2πFr2/L). In an attempt to define the grid resolution it was
decided to set the maximum dimension of any cell within one wavelength of the hull to be no larger than Lw/20 and any
cell anywhere to be no larger than Lw/10. 20 points should be enough to satisfactorily define a wave and 10 points should
give a rough definition at the outer bound.
The most difficult dimension to achieve this criterion is normal to the hull, where the boundary layer requires
extremely small spacing. Two approaches were tried using different distribution functions for the free surface block. The
first grid for Model 5365, hereafter referred to as Grid 1, used a single two-sided hyperbolic tangent distribution (TANH)
from the hull to the outer bound. The initial spacing normal to the wall was sized to y+ value of 10. The growth rate of the
grid spacing was a constant 10% for the first wavelength away from the hull and then asymptotically approaches 0% at
the outer bound, where the grid spacing is Lw/10. The effect of this distribution is to concentrate a large fraction of the
total the points inside the boundary layer.
A single two-sided Monotonic Rational Quadratic Spline distribution from the hull to the outer bound was used for
subsequent grids of Model 5365, referred to as Grid 2, and for both 5415 grids. The initial spacing normal to the wall was
sized to y+ value of 1. The initial growth rate of the grid off the wall is extremely large and this results in a more
uniformly spaced grid, with far fewer points inside the boundary layer.
For the double body blocks in all grids a TANH distribution was used normal to the hull. Grid 1 was intended to be
relatively coarse and the dimension on the free surface normal to the hull was chosen to be 61. The approximate total
number of grid points was 500,000. Grid 2 and both 5415 grids were sized to 111 points and refined in the other
dimensions as well. The approximate total number of grid points was 1.3 million for each grid.
EXPERIMENTAL DATA
Model 5365 was tested in the bare hull condition with the centerline skeg in place. Model tests were conducted at
two different facilities, one at David Taylor Model Basin (Jenkins 1984), and the other at the National Maritime Institute
(Gadd and Russell 1981). The model experiments were conducted in two modes: with the model free to sink and trim, and
with the model rigidly restrained at the static draft. Total resistance of the model was measured with the floating girder
that is attached to the carriage. The bow and stern sinkage of the model were measured with two displacement transducers
that were mounted at the bow and stern of the model. The wave elevation along the side of the model was recorded
manually, using a grease pencil to mark the water height at 21 locations. Subsequently, the distance between the marked
heights and the calm water surface was measured, taking into account the sinkage and trim of the model. In addition, a set
of measurements of the stern wave elevations was made on Model 5365 in the fixed condition at a Froude number of
0.48. The stern wave heights were obtained using thin rods that were manually adjusted until their tips just touched the
water surfaces, once the model reached a steady state condition. The accuracy of the measured total resistance was
reported to be ±1.5%. No systematic uncertainty analysis was done at the time of this experiment.
The experimental data for 5415 was collected during several tests spanning many years (Ratcliffe 1999). The
original bare hull resistance tests were done in 1982. Around 1990 measurements of the free surface wave heights around
the model were obtained using stereophotogrammetric techniques. Two Hasselblad metric still cameras were mounted on
a ceiling-mounted fixture above the Carriage I basin. As the model was towed down the basin, sequences of photographs
were taken, synchronized by strobe lights. The water surface was “seeded” using computer punch card chips that were
distributed on the surface before each run. An array of calibrated target grid points penetrated the free surface in the far
field. These grid points were also imaged in the stereo pairs and served as control points during the analysis of the
photographs. The resultant stereo photographs were subsequently “measured” by the National Ocean Survey over a 76.2
mm X 76.2 mm grid. The data was obtained at model speeds of 4.01 and 6.02 knots, corresponding to Froude numbers of
0.28 and 0.41.
Surface wave height measurements were made on the bow and stern waves in 1996–7. The measurements were
obtained on the bare hull model as it was being towed in the Carriage 3 Towing Basin. For these measurements, a
dynamic wave height probe called a “whisker probe” was used. At each data collection point, x, y, and z values were
recorded
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FREE SURFACE VISCOUS FLOW COMPUTATION AROUND A TRANSOM STERN SHIP BY CHIMERA OVERLAPPING SCHEME 447
via a voltage reading from the probe and traverse encoders. The trim conditions for the two speeds were reportedly the
same as the 1982 test.
Wave profile measurements along the length of the hull were obtained on the bare hull model during experiments
run in 1997. The model was free to sink and trim at each speed. The wave profile traces were drawn on the model and
recorded with reference to the drawn waterline. The trim conditions for the two speeds were reportedly the same as the
1982 test.
The whisker probe wave data was combined with the stereophotographic data by the authors to obtain a more
complete picture of the wave pattern. Due to many unknowns the accuracy of this data cannot be quantitatively established.
RESULTS AND ANALYSIS
Model 5365
Flow computations using Grid 1 are performed for Fn=0.35, 0.48, and 0.65 in fixed condition. It is obvious this Grid
1 is too small for Fn =0.65 and marginal for Fn=0.48. Figures 6–8 show the wave patterns using Grid 1 for these three
Froude numbers in a fixed condition. These wave patterns also show the far field boundary condition is well implemented
at the outer boundary of the computational domain. Computed wave profiles along the hull for these three Froude
numbers are compared with experimental measured data, shown in Figures 9–11.
Flow computations are performed on Grid 2 for Fn=0.28, 0.35, 0.48, and 0.65 in both fixed and sink/trim modes.
Wave patterns on these three grids for four Froude numbers are shown in Figures 12–15 for fixed condition and Figures 16–
19 for sink/trim condition. Since no measurement is available, these wave patterns are shown to examine their
relationship with a wide range of different Froude numbers qualitatively. Comparing with wave patterns from Grid 1
calculations, only minor differences are found except that the wave pattern in Grid 1 is truncated by the outer boundary.
Only minor differences are found on the wave patterns between fixed and sink/trim conditions. The computed wave
profiles are compared with the experimental measurements in both fixed and sink/trim modes, which are shown
respectively in Figures 20–27. Generally, the wave profiles near bow area are well predicted comparing with both sets of
experimental data. The trough of wave profile at the mid/aft portion of the hull is predicted consistent with Gadd's
experimental data and under predicted comparing with Jenkins's data.
A comparison of the measured and computed stern wave elevations in the fixed condition at Fn=0.48 is shown in
Figures 28–29. The computed elevations from Grid 2 show better correlation with experimental data than Grid 1. Since
the measured data is rather limited, the comparison here is only quantitatively validated.
Forces and moments are calculated along with computations of flow variables in the process of flow computations.
They are obtained by integrating the shear stress and pressure on the wall surface. The total resistance of a surface ship is
thus computed as the x-component of total integrated force acting on the ship wetted hull surface. Table 2 shows the
computed total resistance compared with available measured data for Model 5365 computation. For the fixed condition,
both calculations on Grid 1 and Grid 2 are mostly over-predicted compared with Jenkins's measurements, while the result
from Grid 2 computation is closer to the model measured data. However, the prediction trend is consistent with
experimental trend. In Table 3 the computed results for sink/trim condition correlate well with the model test data with
the exception of Fn=0.65. In Tables 4 and 5, the computed sinkage and trim for Model 5365 computation using Grid 2 are
listed along with the model-measured data. It is found that both sinkage and trim computations have consistent trends
with model experimental data. An increased positive trim (bow up) is obtained from Fn=0.28 to 0.65, but the magnitude is
over-predicted. The sinkage calculations are well correlated in magnitude with experimental data at low speed, but over-
predict at both higher Froude numbers.
Model 5415
For model 5415 calculations, two computational grids were generated for two different ship speeds, 20 and 30 knots.
Only the calculation on the fixed-at-the-measured-sink/trim condition is performed. The computed wave patterns for both
ship speeds are shown in Figures 30–31 along with the measured wave contours. In general, computed wave patterns are
consistent with the measured patterns for both bow and stern wave systems. The phase angle of both wave systems is well
correlated. The magnitude of wave elevation is however under-predicted and the wave crests and troughs are not as sharp
as measured data. This may be due to the problem of having not enough grid resolution in those areas of crest/trough
formations. The wave profiles along the hull are
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FREE SURFACE VISCOUS FLOW COMPUTATION AROUND A TRANSOM STERN SHIP BY CHIMERA OVERLAPPING SCHEME 448
shown in Figures 32–33 for both speeds respectively. They are well correlated with experimental data, except being under-
predicted in the bow wave elevation. The trend of wave elevation profile is well predicted. The total resistance for both
ship speeds are shown in Table 6 and correlate well with the model measurements.
CONCLUSION
Computations of free surface viscous flow around two transom stern ships are performed to investigate the capability
of a numeric scheme developed for surface ship hydrodynamic performance prediction. The inclusion of free surface
boundary treatment into the viscous flow computation is first described. The technique to handle wet/dry transom
computation is developed, which includes a new grid topology for transom stern flow calculations. The capability to
compute sinkage and trim for a moving surface ship is important, which is one of the focuses in the paper. Computational
results on two transom-stern ships are consistent with available model experimental data. The shape of the wave profile
along the hull is generally predicted well, while the magnitude of wave elevation still needs to be improved. The wave
pattern is also well correlated, except that wave crests and troughs are not as sharp as the model measurements. The
computed total resistance is in good correlation with model-measured values. Although two different types of grid
spacing and distribution techniques are investigated, a further study is needed to determine a more effective grid system
for free surface viscous flow computations. In addition, more systematic comparisons between numerical flow
computations and model experimental measurements are needed so that the applications of numerical flow tools into ship
design effort can finally be accomplished.
REFERENCES
Lin, C.W., Percival, S., Gotimer, E.H., “Viscous Drag Calculations for Ship Hull Geometry”, Ninth International Conference on Numerical Methods
in Laminar and Turbulent Flow, Atlanta, 1995.
Lin, C.W., Percival, S., Fisher, L., “Validation of Computational Forces and Moments on an Appended Body”, International Maritime Association of
Mediterranean IX Congress, Italy, 2000.
Farmer, J., Martinelli, L., Jameson, A., “A Fast Multigrid Method for solving the Nonlinear Ship Wave Problem with a Free Surface,” 6th
International Conference on Numerical Ship Hydrodynamics, Iowa, 1993.
Jenkins, D.S., “Resistance Characteristics of the High Speed Transom Stern Ship R/V Athena in the Bare Hull Condition, Represented by DTNSRDC
Model 5365,” DTNSRDC-84/024, June 1984, David W. Taylor Naval Ship Research and Development Center, Bethesda, MD.
Gadd, G.E., & Russell, M.J., “Measurements of the Components of Resistance of a Model of R.V. ‘Athena',” NMI R119, October 1981, National
Maritime Institute.
Ratcliffe, T.J., (1999) “Model 5415,” http://www50.dt.navy.mil/5415/, (1 May 2000).
Table 2. Comparison of Resistance Coefficient for Model 5365 at Fixed Condition
Froude # CT x 1000
Jenkins Grid 1 Grid 2
0.28 4.774 n/a 5.102
0.35 4.239 5.428 4.795
0.48 4.437 5.987 5.085
0.65 4.219 5.727 3.784
Table 3. Comparison of Resistance Coefficient for Model 5365 at Sink/Trim Condition
Froude # CT x 1000
Jenkins Grid 2
0.28 5.531 5.432
0.35 5.030 5.020
0.48 5.774 5.516
0.65 4.924 3.962
Table 4. Comparison of Sinkage for Model 5365
Froude # ∆Tm/L x 100
Jenkins Grid 2
0.28 0.105 0.135
0.35 0.200 0.209
0.48 0.245 0.389
0.65 0.095 0.251
Table 5. Comparison of Trim for Model 5365
Froude # (∆Tf−∆Ta)/L x 100
Jenkins Grid 2
0.28 0.060 0.224
0.35 0.095 0.392
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0.48 1.240 1.249
0.65 1.760 2.250
Table 6. Comparison of Resistance for Model 5415
Froude # CT x 1000
Ratcliffe Computed
0.28 4.14 4.541
0.41 7.01 7.218

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Fig 8. Wave Pattern for Fr #=0.65
Fig 6. Wave Pattern for Fr #=0.35
Fig 10. Comparison of Wave Profile for Fr #=0.48
Fig 7. Wave Pattern for Fr #=0.48
Fig 9. Comparison of Wave Profile for Fr #=0.35
Fig 11. Comparison of Wave Profile for Fr #=0.65
FREE SURFACE VISCOUS FLOW COMPUTATION AROUND A TRANSOM STERN SHIP BY CHIMERA OVERLAPPING SCHEME
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Fig 14. Wave Pattern for Fr #=0.48
Fig 12. Wave Pattern for Fr #=0.28
Fig 21. Comparison of Wave Profile for Fr #=0.35
Fig 13. Wave Pattern for Fr #=0.35
Fig 22. Comparison of Wave Profile for Fr #=0.48
Fig 20. Comparison of Wave Profile for Fr #=0.28
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Fig 17. Wave Pattern for Fr #=0.35
Fig 15. Wave Pattern for Fr #=0.65
Fig 24. Comparison of Wave Profile for Fr #=0.28
Fig 16. Wave Pattern for Fr #=0.28
Fig 25. Comparison of Wave Profile for Fr #=0.35
Fig 23. Comparison of Wave Profile for Fr #=0.65
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Fig 18. Wave Pattern for Fr #=0.48
Fig 27. Comparison of Wave Profile for Fr #=0.65
Fig 28. Comparison of Wave Pattern for Fr #=0.48
Fig 19. Wave Pattern for Fr #=0.65
Fig 26. Comparison of Wave Profile for Fr #=0.48
Fig 29. Comparison of Wave Pattern for Fr #=0.48
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Fig 31. Comparison of Wave Pattern for Fr #=0.41
Fig 30. Comparison of Wave Pattern for Fr #=0.28
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Fig 33. Comp of Wave Profile for Fr #=0.41
Fig 32. Comp of Wave Profile for Fr #=0.28
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DISCUSSION
Gabor Karafiath
Naval Surface Warfare Center, Carderock Div., USA
1. The authors must be congratulated for devising the ingenious gridding method that enables the combined free
surface viscous flow computation. This methodology was also used by the authors to perform free surface viscous flow
calculations to study the flow behind a stern flap and help confirm the scale effect that occurs between model size and
ship size. (Karafiath et al 1999) In that particular case, the use of the Chimera gridding was a great asset with regard to the
efficient numerical geometry definition of the stern flap.
2. In general, the current commercial practice associated with a ship hull form development is to use either the free
surface potential flow computation or the double body RANS calculation for viscous flow. The combined free surface
viscous flow calculation, either the one presented by the authors or one of the few other emerging free surface viscous
flow codes, is generally not used because of the newness of the codes and because of the associated extra cost of the
calculation. Could the author's comment on the turn around time, grid preparation and calculation effort that is required
for the combined free surface viscous flow calculation?
3. Given that there is a somewhat greater cost for this new calculation method, what do we gain? Could we see a
comparison of the wave height prediction relative to the equivalent free surface potential flow code prediction and also a
comparison of the boundary layer thickness relative to the prediction from the double body RANS code?
4. In Figure 30 and 31, we see a comparison of the predicted free surface viscous wave field to the model
measurements using the whisker probes. As characteristic of many similar comparisons, the predictions are very smooth
in nature whereas the model measurements have a great deal of high frequency content that is not captured in the
prediction. My observation of many model tests conducted in very still calm water is that the flow field around the model
is in reality an unsteady flow with a significant temporal flow variation tendency near the transom and that the high
frequency content in the wave field is real. Could the author's comment on the resolution of this problem of a steady state
prediction for phenomena that has some temporal variation?
Karafiath, G., Cusanelli, D.S. and Lin, C, W. “Stern Wedges and Stern Flaps for Improving Powering—U.S. Navy
Experience” Transactions of the Society of Naval Architects and Marine Engineers, 1999, Baltimore, Maryland.
AUTHOR'S REPLY
None received.
DISCUSSION:
Y.Tahara
Oskaka Prefecture University, Japan
(1) As the authors mentioned in the paper, Most recent high-speed fine ships as well as Model 5415 have transom
stern in order to obtain wide waterplane area to secure sufficient stability. However, the wide transoms tend to increase
disturbance on transom wave fields, and that results in increase of hull resistance. The present authors and others (Iwasaki
et al., 1996; Tahara et al., 1997) had carried out investigation on transom flow and wave fields using computational and
experimental models. In the work, it appeared that transom wave field can be classified as the following 3 types: (A) with
dead water zone right after stern end; (B) with no dead water zone, but wave breaking in near wake region; and (C) with
neither dead water zone nor wave breaking in near wake region, i.e., free surface is smoothly continuous from the stern
end. In your paper, the above are simply referred to dry/wet conditions.
My question is how accurately your numerical method can predict the above (A) through (C) for ship models
considered in your work.
(2) For the above-mentioned type (A) transom wave condition, the significant bubble entrainment is usually
observed in the measurements. The effects must be included in free-surface boundary conditions in order to accurately
predict the wave field. Currently, inclusion of the effects may not be focused in your work; however, I would like to know
if you have prospect or suggestion for feasible numerical treatment to include the effects.
REFERENCES:
Iwasaki, Y., Tahara, Y., Okuno, T., Himeno, Y. and Yamano, T., “Studies on Relationship between Water Surface behind Stern and Stern End Form of
Fine Ships,” J. of Society of Naval Architects of Japan, Vol. 180, 1996, pp. 13–20 [Japanese].
the authoritative version for attribution.

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None received.
Century, Hong Kong, June 1998, pp. 83–92 [English].
AUTHOR'S REPLY
FREE SURFACE VISCOUS FLOW COMPUTATION AROUND A TRANSOM STERN SHIP BY CHIMERA OVERLAPPING SCHEME
456
Naval Architects, No. 227, 1997, pp. 7–19 [Japanese]; also, Proceedings of the 2nd Conference for New Ship & Marine Technology into 21st
Tahara, Y. and Iwasaki, Y., “A Study of Transom-Stern Free-Surface Flows by 2-D Computational and Experimental Models,” J. Kansai Society of