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NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 553
Numerical Prediction of Scale Effects in Ship Stern Flows with
Eddy-Viscosity Turbulence Models
L.Eça (Instituto Superior Técnico, Portugal)
M.Hoekstra (Maritime Research Institute, Netherlands)
ABSTRACT
This paper presents a numerical investigation of scale effects on ship stern flows. The most popular algebraic, one-
equation and two-equation eddy-viscosity turbulence models are successfully applied to the calculation of the flow around
the Mystery tanker from model up to full scale Reynolds number. It is shown that the choice of the turbulence model does
have an influence on the results (notably the near-wake field), but the differences caused by changing the turbulence
model tend to diminish with the increase of the Reynolds number.
1 INTRODUCTION
One of the main advantages of Computational Fluid Dynamics (CFD) over traditional model testing is the (potential)
capability to predict Reynolds number effects on the flow field around a ship. But before this advantage can be exploited,
two major tasks arise. The first is to change the potential capability to a true capability by making sure that the numerical
method can cope with the extreme requirements posed by flow simulations at full scale Reynolds number. The second
task is to go through the verification and validation processes.
Only a few attempts to predict scale effects with CFD have been reported. One such attempt is [1], presenting
results, however, which are not numerically convincing. The present authors have shown to be more successful in
computing ship stern flows from model up to full scale Reynolds numbers, [2], with verification of numerical errors, [3]
and [4]. Unfortunately, there are virtually no reliable experimental data available for full scale ship stern flows, so that the
validation process is obstructed. The best thing to do is then to increase the level of confidence by validating at model-
scale Reynolds number and showing that well-known trends for Rn increasing are systematically reproduced.
In the last two decades, a huge effort has been made to validate CFD predictions at model scale Reynolds number.
However, the present status is far from being completely satisfactory. Notably the accurate prediction of the axial velocity
field in regions of high streamwise vorticity has proved to be difficult. Some success has been claimed for second
moment closures, e.g. [5]. But if one aims at selecting a turbulence closure that is numerically robust at model and full
scale Reynolds numbers, eddy-viscosity models are still the only reasonable choice.
In this paper we present a numerical investigation into the prediction of scale effects with eddy-viscosity turbulence
models, including algebraic, one-equation and two-equation models. Two main goals are considered:
• Investigate which turbulence models are numerically robust from model up to full scale Reynolds numbers,
without the need of further tuning.
• Evaluate the influence of the Reynolds number on the differences between solutions obtained with different eddy-
viscosity turbulence models.
With achieving these goals, we expect to increase the confidence in the use of CFD at full scale Reynolds numbers,
using turbulence models that have been originally developed for thin shear layers at model scale Reynolds numbers.
The paper is organised in the following way: section 2 gives a brief description of the turbulence models and their
numerical implementation. The results of application to the flow around the Mystery tanker at Reynolds numbers from
model scale up to full scale Rn are presented and discussed in section 3. Section 4 summarizes the conclusions of the paper.
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NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 554
2 TURBULENCE MODELS
In order to cover a broad spectrum of eddy-viscosity turbulence models, we have considered the following models:
• Algebraic models
– Cebeci & Smith, [6], (CS)
– Baldwin & Lomax [7], (BL)
• One-equation models
– Spalart & Allmaras, [8], (SA)
– Menter, [9], (MT)
• Two-equation models
– k-ε models
Two-layer model, [10], (KE-TL)
Chien's low-Rn model, [11], (KE)
– k-ω models
Standard, [12], (KW)
Menter, [13], (KWM)
– q-ζ model, [14], (QZ)
The one-equation model of Baldwin & Barth, [15], and the SST version of Menter's k-ω model, [13], were also
tested. However, results of these models will not be included in the present paper. The Baldwin & Barth model showed a
very poor behaviour in the initial test runs, while the SST version of Menter's k-ω does not perform better than the other k-
ω models tested.
It is possible to improve the quality of the predictions of ship stern flows with the one-equation and two-equation
turbulence models using a simple correction to the production term of the transport equations, [16]. However, in this
paper we will adopt the standard versions of the models.
2.1 Algebraic Models
The two algebraic models are well-known and based on a two-layer definition of the eddy-viscosity, vt, where the
eddy-viscosity is obtained from the minimum of its values in the two layers. In the inner-layer, both models use the
mixing-length approach with the Van Driest damping function in the near-wall region.
In the Cebeci & Smith model, the eddy-viscosity in the outer region is obtained from
(1)
where qe stands for the velocity at the edge of the viscous region and δ* is the displacement thickness, which is an
integral parameter defined for 2-D boundary-layer flows, and γk is the intermittency factor, which is given by
(2)
where δ is the thickness of the viscous region.
In a ship stern flow calculation, performed in a curvilinear coordinate system, (ξ, η, ζ)1 some assumptions have to be
made to compute qe, δ* and γk. We determine these quantities from information along each grid line normal to the wall
and the viscous layer thickness is calculated from the total head. Details on the calculation of qe, δ* and γk are given in
[16].
The main advantage of the Baldwin & Lomax model over the Cebeci & Smith model is the absence of δ and δ* from
the definition of the length scale in the outer region. In the Baldwin & Lomax model, (vt)0 is given by
(3)
where
(4)
Fmax is the maximum of the function
(5)
and ymax is the value of yn where Fmax occurs. Udif is the difference between the maximum and minimum values of q
along an η grid line, |W| is the magnitude of the vorticity vector, A−=26 and is the nondimensional distance to the wall
in wall coordinates. Fkleb is the equivalent of γk, and is given by
the authoritative version for attribution.
(6)
Ccp=1.6, C2=0.25 and Ckleb=0.3.
Although the calculation of Fwake and ymax seems to be straightforward, it is not. The values of these two parameters
are directly related to the vorticity, which is inversely proportional to the grid line distance in the physical space. This
dependency of F on the vorticity makes its numerical calculation extremely sensitive to
1ξ is a stream wise coordinate, η is a coordinate normal to the ship surface and ζ is a girthwise coordinate.

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NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 555
the mean flow field, which depends on vt and is determined iteratively. To avoid undershoots and overshoots of the eddy-
viscosity, the spatial variation of ymax must be limited. In the present implementation, the following limiters are adopted
is the ymax value of a given ξ line at stream wise stationi and
where stands for the ymax value on the
previous upstream station for the same ξ line.
It is our experience that it is very hard to converge the eddy-viscosity field when the limiters of ymax are turned off.
On the other hand, if the limiters are always turned on there is a risk of ymax being determined by the upstream flow
instead of the local flow quantities. Therefore, the apparent advantages of the Baldwin & Lomax model in the
determination of (vt)0 are severely compromised by the numerical difficulties found in the determination of ymax.
Details of the adjustment of the two models to make them suited for application to wakes are given in [16].
2.2 One-Equation models
2.2.1 Spalart & Allmaras
The new generation of one-equation turbulence models is based on a transport equation for the eddy viscosity rather
than for the turbulence kinetic energy. The Spalart & Allmaras model proposed in [8] solves the following transport
equation:
(7)
where
(9)
(8)
Here and in the remainder of this paper S represents the rate of strain squared.
The eddy-viscosity is obtained from
The model constants are:
The transport equation of (7), includes the term which does not contribute to the stability of its numerical
solution. Therefore, equation (7) has been re-written as
(10)
In the wake, the distance to the wall is computed from
(11)
where yn is the distance measured along the normal grid lines and x−xte is the distance to the ‘trailing edge' of the
ship measured along the symmetry plane.
2.2.2 Menter
The one equation model proposed by Menter in [9] derives the following transport equation from the k-ε model:
(12)
The eddy-viscosity is given by
(13)
and
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(14)
The model constants are:

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NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 556
2.2.3 Boundary Conditions
In a ship stern flow calculation we need to specify boundary conditions at the six boundaries of the domain: inlet,
outlet, ship surface, ship symmetry plane, free surface2 and external boundary. The boundary conditions are specified in
the same way for both one-equation turbulence models. At the ship surface the turbulent quantities are zero and symmetry
conditions are applied on the free surface and at the ship's symmetry plane. At the outlet, the streamwise derivative of the
turbulent quantities is assumed to be zero. Dirichlet boundary conditions are imposed for the turbulent quantities on the
external boundary and are fixed by the value on the border of the inlet plane. The present calculations were performed
only for the aft part of the ship; the inlet plane is located at mid-ship, and so there is already a viscous region in the inlet
plane. A standard straightforward procedure was adopted to define the turbulent quantities at the inlet plane: vt is
calculated with the Cebeci & Smith algebraic model and the turbulent quantities are then derived from the known eddy-
viscosity.
2.3 Two-equation Models
2.3.1 Two-layer k- ε
The two-layer k-ε model presented by Chen and Patel in [10] solves two transport equations in the outer flow region.
In the near-wall region only the equation for the turbulence kinetic energy, k, is solved. The value of ε is derived from an
algebraic length scale. The near-wall model is equivalent to the one-equation model of Wolfshtein, [17].
The eddy-viscosity is obtained from
(15)
and k and ε for a steady flow are obtained from the solution of the equations:
(16)
and
(17)
In the near-wall region, ε is determined from
(18)
with
(19)
and
(20)
The fµ function defined by the Wolfshtein one-equation model is
(21)
The standard k-ε constants are cµ=0.09, C1=1.44, C2=1.92, σk=1 and σε=1.3.
The key feature of this two-layer model is the determination of the boundary between the inner and outer layers
which is often defined by a criterion based on y−. However, with y− it is difficult to establish a criterion which is
insensitive to the Reynolds number. In our approach, the inner-layer region is defined by the following criteria:
The first criterion would be the natural choice to border the inner-layer region. However, in the iterative
determination of the eddy-viscosity field it may lead to excessively large regions, which provoke numerical convergence
problems. Therefore, we have added the second criterion which originates from the knowledge on flat plate boundary
layers, where the ‘fully-turbulent' region starts at This approach does not guarantee that the fµ is close to 1
at the edge of inner-layer. Therefore, in the outer-layer the ε transport equation is solved but fµ is still obtained from (21).
As in the one-equation models, (yn)*, defined by equation (11), is used in the wake to represent the distance to the
wall.
2.3.2 Chien's k- ε model
The low Reynolds k-ε model proposed by Chien, [11], does not distinguish between inner and outer layers and is
directly applicable in the near-wall region. The eddy-viscosity is obtained from equation (15). The k and ε transport
the authoritative version for attribution.
equations of this model are:
(22)
2In the present calculation we have used the double-model approximations and so the free surface is asymmetry plane.

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NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 557
and
(24)
(23)
The near-wall damping functions are given by:
(25)
The model constants are
2.3.3 Standard k- ω model
The k-ω model has been proposed by Wilcox, [12]. It obtains the eddy-viscosity from
(26)
and k and ω are obtained from the solution of the equations:
(27)
and
(28)
The ω equation can be integrated down to the wall and the model constants are:
2.3.4 Menter's k- ω model
Menter's version of the k-ω model, [13], is a blending between the k-ε and the k-ω models. The objective of the
model is to solve the ω equation in the near-wall region, which does not require extra damping functions, whereas in the
outer region of the flow the ε transport equation is solved.
As in the standard k-ω model, the eddy-viscosity is computed from (26) and the k transport equation is given by (27).
The ω transport equation is re-written as:
(29)
where the constants of the model, for convenience symbolically denoted by are obtained from
The set of constants is the one of the standard k-ω model and the set of constants has been derived from the k-ε
model and is given by:
The blending function, F1 is given by
(30)
with
(32)
(31)
and
2.3.5 q- ζ model
The q-ζ model is proposed in [14]. It is a two-equation model derived from the k-ε model with the objective of
having turbulent quantities which go to zero at the wall. The two turbulent quantities of the model, q and ζ are related to k
the authoritative version for attribution.
and ε by
(33)
The transport equations of q and ζ are derived from the transport equations of k and ε with the relations:
(35)
(34)
In the present implementation of the method, Chien's low Reynolds version of the k-ε model was adopted to obtain
the transport equations of q and ζ. The eddy-viscosity is given by

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NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 558
and
(36)
(37)
The near-wall damping functions are given by:
(38)
(39)
(40)
The model constants are
2.3.6 Boundary Conditions
As in the one-equation models, boundary conditions are required for the transport equation of the turbulent quantities
at the six boundaries of the computational domain of a ship stern flow calculation. At the symmetry plane of the ship and
at the free surface, symmetry boundary conditions are applied to all turbulent quantities. Zero streamwise derivatives are
applied at the outlet boundary. At the external boundary, we have applied Dirichlet boundary conditions. The values of
the different turbulent quantities were set to obtain vt=0.01v and assuming that ω∞=U∞/L3. At the inlet plane, the turbulent
quantities are fixed with a similar procedure as adopted for the one-equation models. However, in the two-equation
models we need extra information because we have two turbulent quantities to define.
We determine the k (or q) profiles from
(41)
with
(42)
where uτ is the friction velocity. The second turbulent quantity is obtained from k (or q) and the values of vt obtained
from the Cebeci & Smith algebraic model.
In the outer region, yn≥0.15δ, an Hermite cubic interpolation is used to obtain the turbulent quantities, assuming zero
derivatives at the external boundary.
At the ship surface, k, q and ζ are zero. In Chien's formulation of the k-ε model, ε is also zero at the wall. However,
the wall boundary condition for ω asks for a few more words. According to [12], ω behaves in the vicinity of the wall as:
(43)
where
Although almost everyone reports that the near-wall behaviour is a strong point of the model, the value of ω at the
wall actually tends to infinity! Only in [12] it is recognised that the ω boundary condition at the wall may lead to
numerical difficulties when fine grids are used in the near-wall region. The most popular implementation of the ω
boundary condition at the wall is suggested by Menter in [13], which just fixes the wall value by
(44)
where ∆y1 is the distance of the first grid node to the wall. Obviously, this condition is just equation (43) multiplied
by 10, and can be criticized for the following reasons:
the authoritative version for attribution.
1. The factor 10 is completely arbitrary4.
2. It is based on the ω solution without viscous corrections in a region where viscous effects are dominant.
3. It is clearly grid dependent, because the distance of the first grid node to the wall appears explicitly in the
definition of ωw.
Two alternative wall boundary conditions are proposed by Wilcox in [12]. The first one is to calculate ω directly
from equation (43) for yn<2.5 instead of solving the ω transport equation in the near-wall region. The second one is to
derive ωw from the skin friction velocity using a ‘slightly rough wall' boundary condition. It has already been shown in
[18] that the latter approach leads to unsatisfactory results for
3U is the undisturbed flow velocity and L is the ship length.
∞
4Theoretically ω should be infinite.

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NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 559
the skin friction coefficient for the flow on a flat plate. Therefore, only the first option is left to avoid the use of a grid-
dependent boundary condition. In this paper we shall compare two alternatives: i) adopt equation (43) with Nω including
viscous corrections to define the near-wall values of ω, BC1; and ii) obtain ω at the wall from equation (44), BC2.
3 RESULTS AND DISCUSSION
3.1 General
All calculations were carried out with the computer code PARNASSOS, [19], which solves the Reynolds Averaged
Navier-Stokes equations in their complete form [20]. The test case in this paper is the flow around the Dyne (Mystery)
tanker which has earlier been subject of comparative computations in the Gothenburg and Tokyo Workshops, [21] and
[22]. Two main reasons justify this choice: the flow is sufficiently complex to test the accuracy of the turbulence models,
and, in particular, it exhibits, at least at model scale Reynolds number, the so-called ‘hook shape' of the isolines of the
axial velocity at the end of the stern, as a result of the existence of strong bilge vortices. A detailed numerical verification
study has been performed with PARNASSOS for this test case, both at model scale Reynolds number, [3], and full scale
Reynolds number, [4], which permits the selection of a grid with sufficient resolution.
In the present study, five different Reynolds numbers have been considered: 5×106, 2×107, 108, 5×108 and 2×109,
with the Reynolds number defined by
A Cartesian coordinate system is introduced with the x axis along the undisturbed stream, the z axis vertical positive
pointing upwards and y completing a right-hand system. The origin of the coordinate system is located on the forward
perpendicular at the ship symmetry plane on the keel line. All the variables presented are made non-dimensional using U∞
and L as the velocity and length reference scales.
The computational domain covers only the flow field near the stern. The inlet and outlet plane are x constant planes.
The inlet plane is located at x=0.5L and the outlet plane at x=1.25L. The external boundary is an elliptical cylinder, given
by:
The remaining boundaries are the free surface, plane z=0.056L, the symmetry plane of the ship, y=0, and the hull
surface.
The volume grids were created with a proprietary elliptic PDE grid generator, based on the GRAPE approach [23].
The number of grid nodes in the streamwise and girthwise direction is the same for the five Reynolds numbers: Nξ=161
and Nζ=41. The number of grid nodes in the normal direction, Nη, increases with Rn. Nη=81 for the lowest Rn and 10 grid
lines are added each time Rn is increased, which leads to Nη=121 for Rn=2×109. The grid line spacing in the normal
direction is defined by one-dimensional stretching functions, which are tuned to obtain a maximum value of y− at the first
layer of grid nodes away from the ship surface of approximately 0.5.
Five significant flow parameters were selected to compare the different numerical solutions:
• Friction resistance coefficient,
• Pressure resistance coefficient,
• Wake fraction, Wf:
• Maximum cross-stream velocity at x=0.989L, (Vw)max, with
• Minimum axial velocity component in the flow field,
The maximum cross-stream velocity at x=0.989L is related to the bilge vortex intensity and identifies the
existence of streamwise flow separation. The integrals included in the definitions of and Wf are evaluated with
Gaussian quadrature rules assuming a bi-linear variation of the unknowns between the grid nodes. The area Ω for the
calculation of the wake fraction is the propeller disc, which has been located at x=0.989L with the axis of the propeller at
z=0.0166L; the disc radius is R=0.015L, while a zero hub radius has been assumed.
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NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 560
3.2 Wall Boundary Condition for
To investigate the influence of the numerical implementation of the wall boundary condition of ω, we have
calculated the flow at Rn=5×106 with Menter's version of the k-ω model with the two options considered: BC1, which
obtains ω from the theoretical value for y−<2.5 and BC2, which is based on an ad hoc definition of a finite value at the
wall. Table 1 presents the five selected flow quantities and the mean and maximum values of vt obtained with BC1 and
BC2.
Table 1: Comparison of solutions obtained with Menter's k-ω model using different implementations of the ω wall boundary
condition.
Variable BC1 BC2
1.944 1.612
0.911 0.878
0.609 0.565
Wf
0.343 0.365
(Vw)max
−0.075 −0.076
(vt)med×104 0.265 0.257
4
(vt)max×10 1.582 1.504
The differences obtained between the two solutions are certainly not negligible. As one might expect, the friction
resistance coefficient, exhibits the largest difference. However, the limiting streamlines of both calculations, which
are depicted in figure 1, are similar.
Figure 1: Limiting streamlines for Menter's k-ω model with different wall boundary conditions.
At the propeller plane, x=0.989L, there are significant differences between the isolines of the axial velocity, as shown
in figure 2. With BC1 the speed is definitely lower in the inner wake than with BC2.
Figure 2: Axial velocity isolines at x=0.989L for Menter's k-ω model with different wall boundary conditions.
These results show that the flow prediction is clearly dependent on the numerical implementation of the ω wall
boundary condition. They suggest that the ω behaviour at the wall can hardly be seen as a strong point of the k-ω models.
From the results it is not clear which is the best choice, BC1 or BC2, however, as discussed above, the results of BC2 are
inherently grid-dependent. Therefore, we will adopt BC1 for the remaining calculations with the k-ω models.
3.3 Scaling Effects
the authoritative version for attribution.
The results of the five selected flow quantities and the maximum value of vt are given in table 2 for the five Reynolds
numbers and for the various turbulence models tested. ∆ stands for the maximum difference between the predictions of
the different turbulence models at a given Reynolds number. The differences between predictions with different
turbulence models come out as appreciable at model scale Reynolds number but tend to diminish with the increase of the
Reynolds number. An exception is found in the maximum cross-stream velocity at the propeller plane; it is the only one
of the selected flow variables which does not change monotonically with the Reynolds number.
Figure 3 presents the friction resistance coefficients given in table 2. We have tried to compare the results with two
friction lines, the ITTC line,

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NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 561
Table 2: Comparison of solutions obtained with the several turbulence models at different Reynolds numbers.
Variable Rn CS SA MT KE-TL KE QZ KW KWM ∆
5×106 1.573 1.816 1.606 1.561 1.598 1.445 1.939 1.944 0.499
2×107 1.269 1.469 1.335 1.280 1.315 1.197 1.546 1.547 0.350
108 1.022 1.177 1.098 1.059 1.082 0.988 1.213 1.221 0.233
5×108 0.846 0.965 0.916 0.885 0.900 0.824 0.981 0.983 0.141
2×109 0.732 0.825 0.791 0.767 0.776 0.713 0.841 0.843 0.130
5×106 0.598 0.788 0.741 0.713 0.687 0.655 0.923 0.911 0.325
2×107 0.527 0.720 0.648 0.638 0.645 0.614 0.829 0.817 0.320
108 0.466 0.649 0.583 0.580 0.604 0.589 0.744 0.732 0.278
5×108 0.423 0.605 0.547 0.543 0.570 0.575 0.690 0.675 0.267
2×109 0.402 0.578 0.535 0.542 0.565 0.570 0.648 0.638 0.236
5×106 0.528 0.632 0.619 0.563 0.576 0.532 0.616 0.609 0.104
Wf
2×107 0.462 0.547 0.535 0.493 0.512 0.479 0.532 0.530 0.085
108 0.397 0.452 0.454 0.421 0.437 0.415 0.451 0.449 0.057
5×108 0.347 0.393 0.393 0.369 0.380 0.365 0.384 0.386 0.046
2×109 0.315 0.352 0.353 0.334 0.341 0.330 0.345 0.345 0.038
5×106 0.209 0.280 0.290 0.287 0.266 0.263 0.347 0.343 0.138
(Vw)max
2×107 0.199 0.301 0.289 0.286 0.284 0.271 0.378 0.367 0.179
108 0.203 0.302 0.276 0.262 0.284 0.268 0.396 0.377 0.193
5×108 0.206 0.294 0.252 0.229 0.256 0.244 0.395 0.369 0.189
2×109 0.208 0.263 0.228 0.227 0.229 0.228 0.336 0.331 0.128
5×106 −0.012 −0.059 −0.068 −0.028 −0.015 −0.007 −0.075 −0.075 0.063
2×107 −0.004 −0.055 −0.040 −0.007 −0.016 −0.011 −0.065 −0.064 0.061
108 0.000 −0.035 −0.001 0.000 −0.001 0.000 −0.041 −0.041 0.041
5×108 0.000 −0.005 0.000 0.000 0.000 0.000 −0.027 −0.034 0.034
2×109 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
(vt)max ×104 5×106 2.327 1.627 1.259 1.113 1.095 1.046 1.492 1.582 1.281
2×107 1.979 1.338 1.026 0.927 0.965 0.889 1.318 1.361 1.090
108 1.647 1.093 0.845 0.798 0.823 0.742 0.931 1.141 0.905
5×108 1.374 0.917 0.711 0.660 0.708 0.603 0.763 0.862 0.800
2×109 1.173 0.780 0.626 0.556 0.586 0.505 0.700 0.756 0.668
and the Schoenherr line,
Since our computation domain covers the aft half of the hull only, we have estimated the equivalent plate friction of
the aftbody as
where Sw is the wetted surface of the ship included in the computational domain, which is assumed to be half of the
total wetted surface. The predictions of all the turbulence models exhibit the correct trend with the increase of the
Reynolds number, but there is a clear difference in slope.
As a further relevant result, we have plotted the mean wake fraction, Wf, as a function of the Reynolds number in
figure 4. It is interesting to note that in both figures 3 and 4 there is good agreement between the SA model and the two k-
ω models, KW and KWM.
The calculated limiting streamlines at Rn=5×106, Rn=108 and Rn=2×109 are illustrated in figures 5 to 7 for the
models CS, SA, MT, KW, KWM and KE.
As in the previous results, the differences between the predictions of the various turbulence models tend to diminish
with the increase of Rn. Once more, the
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NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 562
results of the SA, KW and KWM models are very similar. At full scale, Rn=2×109, the CS, MT and KE also show good
agreement.
Figure 3: Friction resistance coefficient of the aft-body,
as a function of the Reynolds number for the
various turbulence models tested.
Figure 5: Limiting streamlines at Rn=5×106.
Figure 4: Wake fraction, Wf, as a function of the
Reynolds number for the several turbulence models tested.
The axial velocity isolines at the propeller plane, x=0.989L, at the same three Reynolds numbers, 5×106, 108 and
2×109, are presented in figures 8 to 10. The turbulence models included are again the CS, SA, MT, KW, KWM and KE.
The U1 isolines exhibit a drastic influence of Rn. At model scale, the typical ‘hook shape' does appear for the k-ω models,
and to some extent, for the one-equation models5 SA and MT. However, at x=0.989L, the ‘hook shape' tends to disappear
with the increase of the Reynolds number. At Rn=2×109, none of the predictions exhibits a ‘hook shape' and differences
between the results of the various models, including the algebraic CS model, are rather small.
This effect of Rn is related to the stretching of the bilge vortex, generated within the ship boundary layer which
reduces its thickness with the increase of Rn. Figures 11 to 13 illustrate the cross-stream velocity field at x=0.989L for the
same three Reynolds numbers. The plots include the CS, MT and KWM models. Although there are some differences
between the predictions of the three models even at Rn=2×109,
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5As mentioned before, these prediction can be easily improved with a simple correction to the production term of the transport
equation of the turbulent quantity.

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NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 563
the stretching of the bilge vortex with the increase of Rn is clear for the three models.
Figure 7: Limiting streamlines at Rn=2×109.
Figure 6: Limiting streamlines at Rn=108.
We should note that the present results do not imply that the typical ‘hook shape' of the axial velocity isolines
disappears with the increase of Rn, it just appears further downstream. Figure 14 presents the velocity field at x=1.1L
obtained with the KWM model for Rn=2×109. At this location, the bilge vortex is almost axisymmetric and the U1 plot
shows the typical ‘hook shape', which is understandbly weaker than at the propeller plane at model scale, because the
bilge vortex has not only rolled up in the near wake but it has also diffused.
4 CONCLUSIONS
We have presented results of a numerical investigation of scaling effects in ship stern flows using algebraic, one-
equation and two-equation turbulence models. The turbulence models were all implemented without any special tuning
dependent on the Reynolds number.
For the two-equation k-ω models, we have pointed out the deficiencies of a widely accepted numerical
implementation of the wall boundary condition of ω.
The results of the calculation of the flow around the Dyne (Mystery) tanker at five Reynolds numbers, 5×106, 2×107,
8, 5×108 and 2×109, suggest the following conclusions:
10
• It is possible to simulate numerically ship stern flows from model up to full scale Reynolds numbers with the
most popular eddy-viscosity turbulence models, including algebraic, one-equation and two-equation models.
• In global terms, the predictions exhibit the same trend in the flow field with the increase of the Reynolds number
for all the turbulence models.
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at
Rn=5×106.
x=0.989L obtained
Figure 6: Axial velocity isolines
at
at x=0.989L obtained at Rn=108.
Figure 7: Axial velocity isolines
NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS
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at
Rn=2×109.
x=0.989L obtained
Figure 8: Axial velocity isolines
at
Rn=5×106.
Figure 9: Transverse velocity
field at x=0.989L obtained at
NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS
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Rn=108.
Figure 10: Transverse velocity
field at x=0.989L obtained at
Rn=2×109.
Figure 11: Transverse velocity
field at x=0.989L obtained at
NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS
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NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 567
Figure 12: Velocity field at x=1.1L obtained with Menter's k-ω model at Rn=2×109. Above: axial velocity, U1, isolines.
Below: cross-stream velocity field.
• The discrepancies between flow fields obtained with different turbulence models at a given Reynolds number
tend to decrease with the increase of the Reynolds number.
• Although the performance of the k-ω models seems to be very encouraging, the predictions depend on the
numerical implementation of the ω boundary condition at a solid surface. Some implementations advised in the
open literature are unacceptable.
The present results reinforce the need for reliable experimental data at full scale Reynolds number for validation
purposes. All eddy-viscosity turbulence models, used here, were essentially developed for boundary-layers at moderate
Reynolds numbers.
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NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 568
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