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SUBMARINE MANOEUVRABILITY ASSESSMENT USING COMPUTATIONAL FLUID DYNAMIC TOOLS 820
Submarine manoeuvrability assessment using Computational Fluid
Dynamic tools
D.Bellevre, A.Diaz de Tuesta, P.Perdon
(Bassin d'Essais des Carènes, France)
ABSTRACT
Thanks to the constant increase in computing power, it is now becoming possible to aim at more and more ambitious
results in using Computational Fluid Dynamic. The object of this paper is the description of the implementation of a
calculation tool, which should eventually contribute to the setting up of a quasi-exhaustive data bank of hydrodynamic
coefficients of any submarine, for any maneuvers likely to be studied.
A mesh generation tool developed in order to facilitate the pre-processing stage of CFD calculation is presented.
Then different cases of calculation performed are described, the results are compared to those obtained with towing tank
model tests. The validity of each type of calculation is discussed, with an overview on the actual progress. A
mathematical maneuverability model has been identified from the results obtained through calculation. Simulations
performed with this model are compared to results obtained at sea.
INTRODUCTION
It is now possible to expect rapid results for a wide range of calculations. CFD (Computational Fluid Mechanic) is
considered here as a “numerical towing tank”, which allows to compare the results with model test data.
Although the direct simulations of maneuvers using an unsteady RANSE code is possible, it is very time consuming
and the use of a mathematical model based on coefficients in a quasi steady approach is practically instantaneous and
allows a very wide range of simulations in a short time.
In order to facilitate the pre-processing stage of CFD calculation, a mesh generation tool has been developed at
Bassin d'essais des carènes. This tool automatically provides a 3D mesh when cinematic parameters (drift angle, angle of
attack, rate of turn in horizontal/vertical plane) are given or when changes in the geometry of the submarine (L/D ratio,
number, size or location of the appendages, shape of deck,…) are proposed. This avoids the long and laborious task which
consists in re-mesh the submersible for any minor change in its geometry. Furthermore, using this tool the grid topology
is identical from one case to another so that the numerical results are more reliable, at least for comparison purposes.
This study has been performed on an existing submarine for which model test results were available, and
calculations have been done on the basis of usual captive model test: rudder effectiveness tests, oblique towing tests, and
rotating arm test in both vertical and horizontal planes. The solver used was a commercially available Reynolds Average
Navier-Stokes code (Newtonian homogeneous and incompressible fluid).
MESH GENERATION
The philosophy of the numeric tool used to conduct this study is based on a “modular” conception of the submarine,
so that the shape, position, or even a detail in the mesh of each part of the ship can easily be changed, and re-incorporated
into the mesh of the whole ship. The following paragraphs describe briefly the chronology of manipulations carried out to
obtain this adaptable mesh.
Body and deck
The shapes of the deck and of the body of the submarine are obtained from a CAD file. At this stage, the number,
size and repartition of the cells corresponding to those two parts of the submarine are set up.
Appendages
We mean by “appendages” the direction and dive rudders, and the sail. The geometric data necessary are, on the one
hand, the general characteristics of the appendage (wingspread, chord, relative thickness), and on the other hand the files
of Bezier poles defining the thickness laws at the base and in head of the appendage.
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The number of cells in thickness and in chord is now fixed. The number of cells in wingspread will adapt at best the
revolution mesh of the mother hull.
Combination
The principle for the mesh of the submarine for the purpose of maneuverability calculation is the following: the 2D
mesh of the body is reproduced by symmetry of revolution around the axis of the ship. The different appendages are
afterwards incorporated in the mesh, by locally distorting it. The deck is finally taken into account, by a deformation of
the mesh in the concerned areas.
Maneuvering
To satisfy to the limit conditions of calculation, which are of “Neumann” type, it is necessary to maintain the flow
perpendicular to the outlet of the mesh. So the mesh used for calculation in maneuvering (rotation, incidence) is deformed
to adapt to this necessity. This operation is fairly quick (one or two minutes).
Fig 1: deformation of the mesh for a rotation calculation
NUMERICAL TESTS
Different mesh
The size of the mesh is a preponderant factor in the time required and in the quality of calculations. In order to
determine a satisfying compromise between those two necessities, tests have been performed with three different mesh
sizes.
The smallest one, quite coarse, comprises 400.000 cells for the whole submarine. The intermediate one, 800.000
cells, and the finest one, 1.400.000 cells. These numbers correspond to the case where the symmetry related to the vertical
plane cannot be used (typically the cases of maneuvering in the horizontal plane). For the other cases, the number of cells
is of course reduced by one half.
It is obvious that the more the number of cells is important, the longer the calculations are. First, because each
iteration takes more time, and then because the calculation needs more iterations to converge. That is why the size of the
mesh is a parameter that has to be carefully chosen, before starting of a wide range of calculations.
A mesh that includes the forward dive planes requires the use of the finest mesh resulting in a large number of cells.
Therefore the impact of fins on the quality of the results has to be evaluated.
Fig 2: coarsest mesh
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Fig 3: intermediate mesh
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Fig 4: finest mesh
Fig 5: “forward dive planes” mesh
Pure incidence/pure drift
Calculation cases have been chosen to coincide with the results of captive model tests available.
The following table enumerates the different cases of incidence (vertical plane: ε) and drift (horizontal plane: δ)
performed during this study.
ε δ
values in degrees(°)
−8 0
−6 +/−0.5
−4 +/−1
−3 +/−1.5
−2 +/−2
−1 +/−2.5
0 +/−3
1 +/−4
2 +/−5
3 +/−6
4 +/−8
6
8
The two cases ε=0 and δ=0 correspond to the same calculation. In the same way, the negative values of δ correspond
to the same cases as the positive ones, because of the symmetry of the submarine.
Rotation (vertical and horizontal planes)
As in previous section, the calculations with yaw and pitch rate have been modeled based on rotating arm test which
results were available. During these tests, the radii of rotation were chosen in a geometrical progression between the two
extreme positions of the rotating arm carriage.
Were retained:
R=±11m; ±13.83m; ±17.39m; ±21.97m; ±27.50
Since the model length was 4.33m, we obtained the following non dimensional rates of turn:
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q(or r)=L/R=±0.39; ±0.31; ±0.25; ±0.16
By convention, we use q to design the rate of turn in the vertical plane, and r for the horizontal one. The following
table, exposing the calculation performed, was obtained by combining those values with different incidence (or drift) angle.
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Some of these calculations were performed with different mesh, in order to determine the best compromise in term
of mesh size between precision and rapidity.
Rudder effectiveness
The rudder effectiveness is estimated by studying the hydrodynamic coefficients in a straight line, with no incidence,
for different plane deflection (or flap angles, if the rudder doesn't move entirely around its axes, as in this case). In the
following table, β1 represents the angle of flap of the stern dive planes, and α the rudder angle.
β1 or α (degrees)
0 ± + ± ± ± ±
5 10 15 20 25 30
CONVERGENCE
The following graphs show the convergence of a calculation: the curve representing the evolution of the computed
values of physical data, like the forces along the x-axis must present a zero gradient at the end of the calculations. The
curve “residual” represents the evolution of the difference between the solution computed at stage n, and the one at stage n
+1. Depending on the quality of convergence expected, a maximum value is fixed for those “residual”, above which the
calculation is not considered as being converged.
Some calculations required more attention than others before they converge in a satisfactory way.
Among the main parameters we modified are the values of the relaxation factor used for the calculations. This
relaxation factor represents the proportion of the result of the iteration “n” re-used as a base to compute the iteration “n
+1”. The larger the factor is, the faster the solution converges, if everything is going all right, but also the higher is the
risk of divergence. Indeed, if the first solution computed is radically false, and if the second one is based in a large extent
on the precedent, it is very likely to obtain a result even worse.
The majority of the problems of this type were met for rotation/incidence cases. They were countered by reducing
the relaxation for the first 100 iterations, before setting them back to their usual (default) values.
The missing points on the following graphs, showing the calculations results, correspond to the cases where
calculation did not converge, despite changes in relaxation factors. Most of them have been encountered for rotations in
horizontal plane.
EFFECT OF MESH SIZE
Horizontal plane
For manoeuvring in horizontal plane, the forward dive planes theoretically do not play a leading role. On the other
hand, a very clear difference was observed between the coarsest mesh and the middle one.
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The comparison with model test results showed that the middle mesh gives results closer to the reality. Tests with the
finest mesh still have to be performed. Since it is not possible to use any symmetry in the horizontal plane (which was not
the case for vertical plane motions), the size of the
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SUBMARINE MANOEUVRABILITY ASSESSMENT USING COMPUTATIONAL FLUID DYNAMIC TOOLS 824
required mesh is at least doubled and therefore, still poses memory capacity problems.
Fig 6: Pure drift: comparison between two meshes
The accuracy of the results seems therefore to be directly related to the size of mesh, and in all likelihood, it will not
be judicious to merely compute using the coarsest mesh, even if this would have represented a considerable economy in
calculation time.
Vertical plane
Pure incidence:
It appears that the mesh density has a lesser effect on the result accuracy than in horizontal plane.
The presence or the absence of the front rudders leads to a difference of 18% on the gradient of the curve of heave
force versus incidence, for the pure incidence maneuvering. This results are presented further, in the paragraph
“comparison with model test results” (fig 10).
The following graph shows the curves related the drag coefficient as a function of the incidence angle for different
meshes.
Fig 7: drag coefficient in pure incidence manoeuvring
The results of the coarsest mesh are quite different from those obtained with the three finer meshes. It is more
difficult to obtain precise results for the drag coefficient calculations, because of its nature: the whole drag is the algebraic
sum of different drag forces of great absolute value, and of comparable magnitude, but of different signs. A low relative
variation of these great values has a very sensible effect on the whole sum, which absolute value is very low.
The drag induced by incidence which is more of concern for manoeuvrability purposes does not seem to be much
dependant on the refinement of the mesh.
Low rotation rates:
The difference between the heave force coefficient calculated with the coarse mesh and the finer one is less
important when a rotation is introduced in the manoeuvre.
The following graph shows that even for a low rotation rate, the difference between the two curves related
respectively with the coarse and the “forward plane” mesh is only 10%, against the 18% observed in pure incidence. We
do not have yet any satisfying explanation for this difference, but this will allow us to consider that the coarse mesh is
sufficient as long as the rotation rate remains inferior or equal to 0, 2. It must be noted that during usual situation the non-
dimensional rate of turn of a submarine in the vertical plane does not exceed 0, 2.
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Fig 8: Heave force for rotation in vertical plane (low rotation rates)
Higher rotation rates:
This good agreement between results corresponding to the two mesh remains valid as long as the rotation rates are
not too high. Actually,
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SUBMARINE MANOEUVRABILITY ASSESSMENT USING COMPUTATIONAL FLUID DYNAMIC TOOLS 825
as soon as q>0.25, the curve related to the coarse mesh collapses for the highest incidence angles.
Fig 9: Pitching moment for rotation in vertical plane (higher rotation rates)
Fig 10: pitching moment for rotation in vertical plane (two different mesh)
On the other hand, a very clear improvement is obtained by refining the mesh if we compare the curves related to the
pitching moment, even for low rotation rates. For largest incidences, we observe a notable reduction of the values
obtained with the small mesh which does not exist for model tests, and which reveals a bad repartition of forces due to the
insufficient precision of the cells.
COMPARISON WITH MODEL TESTS DATA
Pure incidence (without rotation)
The slope of the curves resulting of calculation, representing the lift coefficient present a satisfying concordance with
those of model tests in vertical plane.
It is clear that due to the presence of the sail and the deck a submarine is not symmetrical in the vertical plane and
therefore a non zero value is expected for heave force and pitching moment at zero incidence angle. This offset is much
bigger when for model tests results but is to a great extend related to the experimental set-up (presence of struts) and to
the precision limit of strain gauges calibrated for much bigger forces.
Fig 11: Heave force coefficient in incidence
manoeuvring (vertical plane)
Pure drift (without rotation)
As seen in the section “comparison between meshes”, the concordance between model tests and calculation results is
very good, as soon as a sufficient refinement of the mesh is adopted.
The following graphs show the evolution of the yaw moment as a function of drift angle, for the two meshes already
performed, and for the model tests. We see that the slope of the curves is very similar. The shift of values observed is
probably due to errors related to test operating. Actually, since the submarine (without propeller) is symmetrical to the
horizontal plane, it is logical to obtain a zero value of the yaw coefficient for a zero drift angle, as observed for
calculation results, unlike for model test results.
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Fig 12: yaw moment versus drift angle
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Rotation/incidence (vertical plane)
We present here the comparison between the results of model tests and calculations with the finest mesh (“forward
planes” mesh), since we saw before that they are sensibly better than the ones with a coarser mesh.
Fig 13: Heave force coefficient for combination of pitching rate and incidence
Despite the presence of an offset already addressed, it can be seen that the overall prediction of heave force is quite
correct for all combination of rate of turn and incidence. For largest incidence angles (where forces are the highest), we
can notice a slight under estimation of heave forces by calculation.
Fig 14: Pitch moment coefficient for combination of pitching rate and incidence
The same comments can apply to the pitching moment coefficient (figure 14). The higher discrepancy for points at
large turning rates can be explained by the fact that the influence of incidence on pitching moment becomes lower as
turning rate increases (at least for combination of incidence and pitch rate close to the natural situation of a free running
submarine).
Rotation/drift (horizontal plane)
The rotation/drift cases posed most of the convergence problems (that is why some of calculation points are missing
on the graph above). The calculations are more time consuming in the horizontal plane, because of the lack of symmetry,
but this cannot be the only reason for those difficulties. We can suppose that the rotation of the sail in horizontal plane
disrupts strongly the flow, and therefore the calculations. More precisely, the chord of the sail can not be considered as
small in relation to the radius of rotation and therefore the sail acts as a lifting surface in a curved flow.
Fig 15: Sway force coefficient (combination of drift and yaw rate)
The results obtained in the horizontal plane are not as good as they were in the vertical plane. Independently from the
offset, it can be observed that the points do not follow the ideal line. Forces are under estimated for large drift angle.
Looking at the yaw moment coefficient we can observe that the overall tendency is well respected. However, a small
variation of the gradient can be detected between the different rates of turn.
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Fig 16: Pitching moment coefficient (combination of drift and yaw rate)
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Some options have been thought of to improve these results, such as a local refinement of the mesh in “strategic
areas” (connection between rudders and body of the submarine, for example), but they have not been performed yet.
Rudder effectiveness
The gap between test and calculation results lead us to try to have the mesh to coincide better with the real geometry
of the submarine.
Indeed, we observe an overestimation of almost 50% of the lift coefficient related to the rudder orientation.
The pressure distribution on both sides of the starboard stern plane for the original mesh is presented on the
following figure corresponding to 25 degrees deflection of the plane (β1).
On this picture, it can be seen that the pressure distribution over the flap (responsible for the lifting effect) is
regularly spread over the span and is also present in the vicinity of the root of the plane.
Fig 17: calculated pressure on suction side (high) and on pressure side (low) of the right dive rudder, with the finest
mesh.
But the real geometry of those rudders is so that when the rudder's flap is inclined, a gap appears between it and the
submarine's body, letting the fluid balance in a certain measure this pressure difference observed in calculation results.
The easiest way of remedying this problem was to modify a small part of the mesh and transforming the cells
corresponding to the gap into “live” cells, which means that they let the fluid go through. The mesh with live cells
corresponds to the mesh with forward planes with this slight modification.
The following figure shows the results obtained in this way: the depression surface on the suction side is less wide
than the one calculated with the former mesh, like the overpressure surface on the pressure side. The heave coefficient
resulting in the difference of pressure repartition between the two faces will therefore be less important. This effect can be
assimilated to a reduction of the effective span.
Fig 18: calculated pressure on suction side (high) and on pressure side (low) of the right dive rudder, with the “alive
cells” mesh
The results obtained with the new mesh (“live cells” mesh) show a remarkable improvement of the concordance
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between calculation and test (approximately 30%), even if the results are not perfect yet, as you can see on the following
graph. Those types of calculations have not been performed yet for rudders (because of the mesh size problem for
horizontal plane manoeuvring we have already mentioned).
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Fig 19: Rudders effectiveness
PREDICTION OF SUBMARINE BEHAVIOUR
In order to evaluate the benefit of numerical calculation in the prediction of submarine manoeuvrability, a
mathematical model has been derived from the results of the calculation.
All the derivatives usually obtained through captive model tests have been identified from the results of calculations,
except for rudder and stern plane effectiveness derivatives, for which prediction appeared to be poor. Fins effectiveness
derivatives used in the following simulations have been derived from captive model tests.
In the horizontal plane, the lack of accuracy of yaw rate influence on the sway forces lead to a bad estimation of the
stability indices. In this particular case, the submarine is predicted as being course stable though sea trials (and model
tests) indicated slight course instability.
Fig 20: Non dimensional yaw rate versus rudder deflection
On the same figure it can be seen that the estimation of non-dimensional rate of turn for intermediate rudder
deflection is quite correct while greater discrepancy arise for larger deflection angles. In addition to the under estimation
of sway forces for large drift angles, the relatively simple mathematical model used in this case can also explain those
latter discrepancies. More precisely non-linear derivatives used to describe the coupling effect between planes deviation
and local incidence were voluntary omitted.
In the vertical plane, the prediction of the submarine behaviour is globally better. The figure 21 display the
maximum non-dimensional pitch rate obtained during stern plane deflection trials (stern plane angle being average
between deflection to dive and deflection to rise). It can be observed that the behaviour of the submarine is well
represented. The non-symmetry of the behaviour of the submarine in the vertical plane is more important for the
simulation, but sea trials results show some discrepancy
Fig 21: Non dimensional maximum pitch rate vs. mean stern plane angle deflection
A step further was the simulation of six degrees of freedom manoeuvres. For this purpose, a reduced set of out of
plane derivatives has been identified from NS calculation.
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Fig 22: Pitching moment induced by yaw
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The simulation is performed using time histories of actuator positions measured at sea as inputs. The maneuver
considered consists in a stern plane deviation, followed 30 seconds later by a rudder deviation in conjunction with
forward plane deviation.
In addition to rudder and stern plane motion, the propeller is stopped. Such a maneuver, which can be considered as
a recovery maneuver for dive plane jam, emphasizes the well known coupling effects between yaw and pitch.
Fig 23: Simulation inputs (planes deviation)
Motions predicted by the simulation are compared in the time domain to full scale tests. For this complex maneuver,
the dynamic of the submarine is qualitatively well predicted as shown on figure 24 where the rates of turn simulated are
compared to measurements.
Fig 24: Rates of turn
It is clear that for time domain simulation, the small errors encountered during the prediction of forces and
consequently on components of acceleration is magnified by the integration. Therefore, motions calculated during
simulation differ significantly from those measured at sea. On figure 25, a factor two is observed on the pitch angle and
also on the depth changing between simulation and sea trials.
Fig 25: Motions
CONCLUSIONS
NS calculations have shown their ability to provide relevant results in terms of efforts acting on a maneuvering
submarine.
Pure drift and incidence forces are very well predicted by NS calculations even with a relatively simple mesh.
As soon as a rotation is introduced, the quality of the results becomes more dependent on the parameters of the
calculation. This study showed that the use of a finer mesh was absolutely necessary. Vertical plane rotation prediction is
quite satisfying on the basis of the calculations already performed. Horizontal plane rotation would require an even finer
mesh than the vertical plane in order to simulate more accurately the details of the appendages. Indeed, an oversimplified
geometrical description of the planes led to bad results in terms of rudder efficacy prediction. Results have already been
improved thanks to an easy manipulation of the existing mesh. Some more efforts are to be made in this direction.
Relevant comparisons (on a relative basis) between different alternative shapes are already being made and very
interesting results have been obtained which enable the evaluation of novel appendage configurations without having to
engage in costly systematic experimental tests.
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In the end, this study should allow us to obtain accurate absolute values of the hydrodynamic coefficients of the
submarines as well as a description of the mechanism of the generation of these forces on the different elements of the
submarine. Indeed, one of the great advantages of using computational fluid dynamic tools is that in addition to an
estimation of global forces, it provides a precise evaluation of local consequences of some geometrical modifications,
which would
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bring to light phenomena which are so far poorly understood.
To gain confidence in this new tool, calculations are now systematically performed in parallel with captive model
tests. At this stage, the mathematical models used for the purpose of maneuvering simulations are the same as those
developed for captive model tests analysis.
We do think that a careful analysis of the huge quantity of data provided by each NS calculation will be of help to
improve existing empirical prediction methods, and from a more general point of view to improve the understanding of
physics involved in submarine maneuverability.
FUTURE DEVELOPMENT
The present study showed some difficulties for prediction of rudder efficacy derivatives and for yaw induced forces.
To overcome those problems, we plan to perform calculations of a single flapped rudder to determine the necessary
amount of refinement in the mesh to allow for an acceptable prediction. This calculation will be based on an existing set
of test data for which geometrical details such as root gap were explored.
We are confident that horizontal plane forces associated with yaw motions will be correctly calculated once
sufficient mesh sizes will be manageable.
Once the overall prediction of hydrodynamic forces will be considered as being correct, specific aspects are to be
studied. Among those aspects, the influence of test instrumentation (stings, supporting struts, etc..) on towing tank and
rotating arm tests will be considered. Calculations will be performed with and without strut, and for different strut
arrangement.
Also, specific efforts will be devoted to emphasize the influence of propulsion on maneuverability.
One additional perspective for further studies in maneuverability is the direct simulations of maneuvers using the
unsteady flow capabilities of the RANSE codes.
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DISCUSSION
U.Bulgarelli
Instituto Nazionale per Studi ed Esperienze di
Architettura Navale, Italy
To apply CFD to study of maneuverability you should already developed unsteady Navier states code, because the
phenomenon under investigation is completely unsteady. Is this true?
AUTHOR'S REPLY
The quasi steady approach we used to investigate the capacity of RANS code for submarine manoeuvrability studies
has two advantages:
Using CFD calculations as a numerical towing tank gave the opportunity to use an existing set of tools to quickly
derive time domain simulations of manoeuvres from calculation of forces.
Results of calculation were directly comparable to existing model tests data and could therefore lead, to a certain
extend, to validation.
Although unsteady phenomena arise during manoeuvres, the quasi steady approach has demonstrated for a long time
its capacity to provide pertinent simulations of the behaviour of submarines ion conventional manoeuvres. For most cases,
the dynamic of the submarine is relatively slow compared to the dynamic of unsteady phenomena concerned and the sea
trials don't really show some major influence of unsteadiness.
An other problem of unsteadiness, which is not solved, is that the solution of steady flow calculation can not
theoretically be considered as the mean force acting on the body on which separation causes unsteadiness in the flow (as
it would be measured in a towing tank or in a rotating arm facility). The calculations performed here showed however
some concordance with measurements.
DISCUSSION
K-H.Kim
Naval Surface Warfare Center, Carderock
Division, USA
When simulating the deflection of fin/control surface, how do you handle the gap between the body and fins?
AUTHOR'S REPLY
During simulations of rudder effectiveness for a flapped rudder, we discovered large discrepancy between
calculation and model tests. Previous calculations for all movable surfaces didn't show such problems and we went to the
conclusion that the mesh of the geometry of the surface and especially the root (including gaps) should be improved. To
solve this problems, we considered that the cells located at the root are active. The difference between the two meshes is
shown on figure 17 and 18 of our paper. Although encouraging, the results obtained are not completely satisfactory and
more work has to be done in this field. We planned to do more extensive calculation of isolated fins in order to compare
to cavitation tunnel tests. The objective is to define what is the “minimum acceptable” degree of refinement of the mesh
for rudder effectiveness prediction bearing in mind that in our meshing method, the refinement of the mesh in the vicinity
of planes has a great impact on the overall size of the mesh for the whole submarine.
DISCUSSION
I-Y.Koh
Naval Surface Warfare Center, Carderock
Division, USA
How do you plan to do unsteady maneuvering prediction using CFD?
AUTHOR'S REPLY
It is already feasible to perform unsteady manoeuvring prediction using CFD. For this purpose, it is possible to use
the “deforming mesh” option of unsteady RANS codes for which unsteady calculation are performed at each time step, a
new mesh being automatically generated taking into account the dynamic of the submarine as well as the changes in
control surface deflection. Bassin has already used this kind of approach in a simplified way (without
the authoritative version for attribution.
OCR for page 832
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approach and confidence has still
SUBMARINE MANOEUVRABILITY ASSESSMENT USING COMPUTATIONAL FLUID DYNAMIC TOOLS
832
shaft (3D). However, it is clear that, at the moment, the 3D case of a manoeuvring submarine would require too much
solving the equation of the dynamic of the body) for a cycloidal thruster (2D) or for a conventional propeller on a inclined
computation time to be of a practical use. On the other, hand the possibility for validation are reduced for such an
Representative terms from entire chapter:
vertical plane
