337
1 INTRODUCTION
The interest in hydrodynamics of floating, fully
submerged bodies is increasing due to the
development of floating facilities for research and
extraction of marine resources, maintenance of
underwater pipelines, production platforms and so
on. That revives interest of researchers on one
fundamentalproblemoffloatingsubmergedbodies‐
the
effect of free surface depending on body
immersion.Thisproblemandactualityarisefromthe
fact that usually underwater bodies and appliances
aretestedmainlyfordeepwater,wheretheresistance
and all other forces and moments have a viscous
nature,i.e.wave‐makingandinducedwaveforcesare
ignored.
Such bodies, however the shallow water
inevitably has to navigate in close proximity of free
surface that cannot be considered asʺdeep
immersionʺ. In these regimes the free surface
substantially alters the pressure distribution in the
hull,thereforeallhydrodynamicforcesandmoments
on the body. The pressure field around
the body
affects the nearby free surface and generates waves
which, in consequence change the pressure
distributiononthebody.Withthe floating bodythe
wave system is moving that must beʺfedʺ with
energyfromtheforcesworkonthebody.Thisaddsto
the wave resistance component and
all the other
forcesandmomentsonthebody.
The motion close the free surface, except
occasionallyduringthesomeoperations,takes place
inparticularwhensailinginthefairwaywithlimited
depth. This further complicates the situation. Upon
movementofshallowwateralsoresultinextraforces
and moments due
to the influence of the bottom,
causingaccelerationoftheflowaroundthehulland
the spread of the wave system, and accordingly the
Hydrodynamics of DARPA SUBOFF Submarine at
Shallowly Immersion Conditions
D.V.Efremov&E.M.Milanov
BulgarianShipHydrodynamicsCentre,Varna,Bulgaria
ABSTRACT:Recently,thesubmarinemissionsareoftenevolvingintooperatingtolittoralareas,whichrequire
operatinginshallowwater.Suchshallowwateroperationsstronglycontrastwiththetraditionalonesdueto
the effect of a close to free water surface expressed mainly by surface suction
force. This influence is
particularlyimportantforsubmarinemaneuverabilityaccountingforrestrictedareaavailable.Thepredictionof
submarine behavior in similar conditions requires adequate mathematical model and understanding of the
additionalhydrodynamicloadgeneratednearthesurfaceregion.
The paper is aimed for better understanding of these issues and relating
to development of a submarine
simulationmodel,theexperimentalprogramoftowingandPMMcaptivetestsofDARPASuboffsubmarine
modelwereconductedatatowingtank.Theinfluenceofphenomenonsuchaseffectofaclosetofreesurface
and Froude number at hydrodynamic forces and moments including control surfaces
effectiveness were
investigatedandalsowasestimateddirectionalstabilityofmotioninhorizontalplane.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 13
Number 2
June 2019
DOI:10.12716/1001.13.02.09
338
pressure redistribution. The significance of the
problem is determined by the specifics of the
underwaterunitcontrol(inhabitedanduninhabited)‐
namely,thereductioninsteeringperformanceorloss
ofcontrolcanleadtomaterialorhumanlosses.
Theabovefeaturesarenotonlyimportantforthe
water resistance during body
motion, hence for the
propulsive qualities, but at least for its
maneuverability.Ifinthestudyoftheresistanceitis
possibletoassumethatbodymotionisstationaryona
straight trajectory, in the general case of a real
maneuvering control we should consider the
curvilinearmotionwithan
angleofdriftandangular
velocity. In this hypothesis of stationarity is now
untenableduetothepresenceofaccelerations.Course
stabilitybeneaththefreesurfaceissubjecttoresearch
in[6],[10],[11].Inthiscasethemaneuveringqualities
oftheobjectchangesasaresultoftheactionofthree
main
factors:
dueconsidered wave makingimpact not only on
thelongitudinalforceofresistance,butalsoonthe
lateral force and moment, caused bydrift and
angularvelocityofthebody;
asa result of the influence of the free surface on
the control means characteristics, which are
generally symmetrical lifting surfaces with small
aspectratio;
frequencyofoscillationaffectshydrodynamicload
onthebodyandinparticulargeneratedbycontrol
surfaces forces. In conjunction with the wave
effects, this may lead to oscillator modes of
movement, and in marginal cases‐to the
bifurcationnatureofthemotion
controlprocess.
Figure1. Lift coefficient on the spheroid close to the free
surface[12]
Figure2.SurfacesuctionforceofSuboffgeometrywithsail
only,H/D=1,5[13]
Regardingtheconsideredfactorsinfluenceonthe
hydrodynamics of underwater body close the free
surface and the bottom in the literature is available
researchonsingleparametersofthephenomenonata
fixedgeometryofthebody.Thisnecessitatespossible
globalapproachtotheproblem, possibly accounting
for the interference effects.
As a rule, further
underwaterbodieshavebeenstudiedmainlyfordeep
immersion in the basic operating mode, where the
resistance and all other forces and moments have a
viscous nature, i.e. wave making and induced him
wave forces are ignored or regarded as small
[1],[3],[4]. In recent years, there
are studies on the
influenceofthefreesurface,experimental,butmainly
withnumericalsimulations[2],[5].However,theyare
limitedonly to some of the effects‐the free surface
[2],[5]oronthebottomoftheflowvolume‐[7],[8].
2 OBJECTOFRESEARCH
Object of present investigations is axisymmetric
streamlinedelongated
referencebody,developed for
the Defense Advanced Research Projects Agency
(DARPA) Suboff Project. Based on the submarine
geometry(model5470),describedbyGrovesetal.[9],
the 3D vehicle model was generated and physical
reinforced plastic model have been manufactured –
Figure3.
Figure3.BSHCDARPASuboffmodel–fullyappended
Principal hull and control surfaces data of the
underwatervehiclearegiveninTable1.
Table1.Mainparticularofthehullandappendages
_______________________________________________
HulldataSymbolValues
_______________________________________________
Lengthoverall,mLOA4,356
Diametermoulded,mD0,508
Rudderlateralarea,m2AR0,0814
Volumeofdisplacement,m3
0,718
Wettedsurface,m2SWA6,338
Numberpropellers[–]1
Numberrudders[–]4
_______________________________________________
ControlsurfacesdataSymbolValues
_______________________________________________
Rudderarea,m2AR0,0814
Rudderheight,mhR0,17
Ruddermeanchord,mbR0,184
AspectratioλR0,72
RudderprofileNACA0020
_______________________________________________
3 EXPERIMENTALSET‐UPANDTESTPROGRAM
Thetestswerecarried out in the deepwatertowing
tank of BSHC which main dimensions are: 200 m
length x 16 m breadth x 6.5 m depth. To meet
requirements of submerged body tank testing a
project for the modernization of the
existing Planar
Motion Mechanism was implemented. This creates
possibility to carry out captive maneuvering tests to
339
determine the course stability derivatives of
submerged objects. Specially developed telescopic
struts are rigidly connected to the two pair of two‐
componentloadcellswithstraingagesforvaryingthe
depthofimmersionoftheunderwaterbody–Figure
4.
Figure4.ModelinstalledonthePMMviastruts&loadcells
Theset‐upofDARPASuboffmodelexperimental
maneuveringinvestigationsallowstovarythedepth
of body immersion and by towing carriage – the
linearspeedofmotion.Dueconsiderationonlytothe
idealized case – course stability in the horizontal
plane,assumingthedominantroleofthehullforces,
without
considering the wave resistance and wave
pattern,thesuboffmodelisinverted.Itisavoidsthe
influenceofthestruts–theset‐upofsailbetweentwo
struts.
Inviewofpreviousstudiesontheinfluenceofthe
free surface on the submarine behavior, are selected
depthsrangingfrom
deepimmersion,wherethereis
no such effect, to the critical relative submersion
depth, corresponding to the critical Froude number
(
h
Fn ≈1)relatedtotheinfluenceofshallowwater.
/
h
F
nvgH
where V – carriage speed [m/s];g‐gravitational
acceleration [m/s
2
]; H‐submergence depth of
centrelineaxistothefree‐surface[m];
Thus BSHC experimental investigations were
performed at three depth immersion of the vehicle
model,describedbydataandpictureincorporatedin
the Table 2. The model vertical distance from water
freesurfacewasmesuredfromthemodelcenterline.
The
scope of work consists of captive model
experimentsusingscalegenericsubmergedmodelto
quantifytheaxialresistance(dragforce),lateralforce
(lift) for a submarine operating close to the free
surface. Measurements are conducted at a constant
forwardspeed,rangesubmergencedepths,andfully
appended hull configurations. The tank
test matrix
hasbeenimplementedintwobasic modes ofmodel
towing:staticanddynamic–Table3.
Table2Testdepthsrelatedtothemodelcenterline
_______________________________________________
H3H2H1
_______________________________________________
Hm0,508 1,016 1,27
H/D [‐]1 2 2,5
Rn10
6
8,95 8,98 9,03
Fn
H [‐]1,035 0,732 0,655
_______________________________________________
Table3.Coursestabilitymodeltestsprogram
_______________________________________________
STATICTESTS
_______________________________________________
TestMode Speed Immersion Driftangle
U[m/s]
[deg]
Staticdrift 2,31 H1,H2,H3 ‐ 4,‐2,0;2;4;
_______________________________________________
DYNAMICTESTS
_______________________________________________
TestMode Speed Immersion Yawrate
U[m/s]r’[‐]
Pureyaw 2,31 H1,H2,H3 0,04; 0,08;
0,12; 0,16;
_______________________________________________
4 COURSESTABILITYANALYSIS
An analysis of the stability characteristics of the
submarine in horizontal plane was realized making
use of dynamic stability indices based on the
linearizedequationsofmotionandonhydrodynamic
derivatives,obtainedbyaforementionedPMMtests.
Accordingto[12],[13]theunderwatervehiclecan
be
consideredinherentlydirectionallystableifaftera
disturbance from steady‐state straight motion it
resumes its steady motion on another straight
direction with controls fixed in zero or neutral
position.BelowananalysisofDARPASuboffcourse
stability as a function of body immersion is
performed.Thisistakeninto
accountfortwoaspects
ofvehiclecoursestability,namely:
inherenttobodygeometrycoursestability;
vehicleresponsetoimpulseexternalactions.
Linear“sway‐yaw”modelofmaneuveringmotion
inhorizontalplanehasbeenconsidered.Therequired
linear hydrodynamic derivatives were obtained by
processingthecaptivePMMtanktestsdata.
4.1 Coursestabilityinhorizontalplane
Obtained by model PMM tests DARPA Suboff at
different submergence ratio H/D hydrodynamic
derivativesoflinearizedswayandyawequationsof
motion are given in Table 4. Data from other three
sourcesareincluded,also.
340
Table4.Non‐dimensionalstabilityderivativesinhorizontal
plane*10
3
_______________________________________________
H/D Nvʹ Yrʹ Yvʹ Nrʹ Gh
[‐] [‐] [‐] [‐] [‐] [‐]
_______________________________________________
DTMB[13] 6 ‐13.65‐12.85‐27.83‐4.44 ‐0.419
Autosub[17] 3 ‐0.45 12.64 ‐28.45‐5.35 1.037
NPS[16] n.a. ‐7.4 ‐43.2 ‐100 ‐160.800
_______________________________________________
2.5 1.5 10.8 ‐44.2 4.00.845
BSHC 2 1.6 7.2‐45.7 4.20.774
1.52.5 5.7‐59.5 4.20.631
_______________________________________________
DARPASuboffinherent course stability has been
estimated by means of well‐known criterion G
h
[12],[13]:
)](/[)(1 mxNYmYNG
grvrvh
(1)
where:
rvrv
YYNN ,,, ‐firstorderderivativesofside
forceandyawmomentinrespecttoswayvelocity
v
andyawrate
r
.
Calculatedvaluesofcriterion
h
G fortherealized
testcases,includingdeeplysubmerged[13]aregiven
byFigure5.
Figure5.Effectofnon‐dimensionalimmersionH/DonGh
We can conclude that this underwater vehicle is
directionallystableclosetothefreewatersurfaceand
unstable in deep water conditions. It should be
pointedthatthissubmarinefeatureisinherenttothe
given body geometry: hull & sail & stern controls.
Obviously, the operational course stability in both
planes
will depend on the effectiveness of control
surfaces. By the way, to obtain quicker submarine
response to control actions in submarine design is
observedatendencytokeep small course instability
indeepwaterconditions.
4.2 Impulseresponseinhorizontalplane
To investigate the behavior of underwater vehicle
when impulse
change in motion parameters takes
placethelinearanalysisapproach[14],[15]toanopen
loopcontrolsystemdescribedbythesetoftwolinear
differential equations [13] is applied. Let’s suppose
that at the moment the ship was subject to instant
externalswayandyawdisturbance:
00
)( vv
;
00
)( rr
where:
0
v and
0
r ‐values of the initial impulse
actions.
The Laplace transformation of the linearized
equationsofmotionandsolutionofthesystemwith
respecttothesideandyawvelocitiesyields:
)()()(
144113310
babababapvpV
)(/
34430
pQbabar
(2)
)()()(
233213310
babababaprpR
)(/
21120
pQbabav
(3)
Theexpressions (2) & (3) have a formof rational
functions:
)(
)(
)(
pN
pM
pF
In above expression the numerator and
denominator are polynomials of the parameter p in
whichthedenominatorisofpowern.
Original of similar function can be found using
Heavisideexpansiontheorem:
i
i
p
pp
n
i
i
e
pN
pppM
f
1
)(
))((
)(
Than by inverse transform we can obtain
expressions for the catamaran response by drift
velocity and yaw rate changes when instantaneous
disturbance has been applied. Separating above
characteristics as four combination of response to
particulardisturbancesinswayandyaw,namely:
0
)(
v
v
;
0
)(
r
v
;
0
)(
v
r
;
0
)(
r
r
Thenwehave:
)).([(
1)(
21
211331
210
p
p
epepbaba
ppv
v
)])((
21
1441
pp
eebaba
(4)
)).([(
1)(
21
3443
210
p
p
eebaba
ppr
v
(5)
)).([(
1)(
21
2112
210
p
p
eebaba
ppv
r
(6)
341
)).([(
1)(
21
211331
210
p
p
epepbaba
ppr
r
)])((
21
2332
pp
eebaba
(7)
TimehistoriesofBSHCDARPASuboffresponseto
small disturbances in sway and yaw velocities,
describedbysolutionintimeofequations(4)to(7)for
3differentdistancesfromfreewaterareillustratedby
Figure6andFigure7.
Figure6. DARPA Suboff response to sway‐to‐sway and
yaw‐to‐yaw instantaneous disturbance close to the free
surface
Figure7. DARPA Suboff response to yaw‐to‐sway and
sway‐to‐yaw instantaneous disturbance close to the free
surface
Basedonthisitcanbeconcludedthat:
Inthecaseof“yaw‐to‐yaw”(r0/r)disturbancethe
motion is critically damped, i.e. the yaw rate
decreasesuniquelytozero;
In other cases after impulse disturbances the
motion is underdamped, i.e. responses
increase/decrease (with sign change) and then
go
tozero.
5 CLOSESURFACEEFFECTINFLUENCEONTHE
SUBOFFRUDDEREFFICIENCY
When considering the vehicle controllability and
operational course stability it is of interest to have
information about steering forces, generated as a
result of control surfaces deflection. In our case we
had study the influence of submergence
ratio on
verticalruddersforces.Thus,hydrodynamicloadson
Suboffverticalruddersclosetothefreesurfacehave
been investigated. Calculations for three non‐
dimensional submergence depths at one flow rate
were performed, and the data for the two
hydrodynamics characteristic parameters of rudder
efficiencywerederived‐dragandliftingforce.
Ratio
ofimmersionisdefinedas:
H
*
=HR/hR
where: H
R‐ submergence depth of centreline axis to
thefree‐surface,[m];
R
h
‐rudderheight,[m]
The results of performed numerical study by
meansofsoftwareʺANSYS“offreesurfaceeffecton
ruddercharacteristicsindicatethatwithreductionof
the depth of immersion the changes of the drag are
minimal. This beneficial phenomenon, however, is
compensatedbythenoisyreductioninliftingforce‐
the most important factor determining the rudderʹs
effectiveness.Inthiscase,itcanbeconcludedthatthe
performance of the rudder is getting worse as it
approaches the free surface. The factor that may be
decisivefortheseresultswouldbetheʺsuctionʺeffect.
Itwouldbegivenriseto
byincreasingthevelocityin
the area between the rudder and the free surface,
followed by reduced dynamic pressure. Rudder
hydrodynamicload(dragandlift)dependencyonits
immersionfortherangedeflectionanglesisshownon
Figure8and9.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.6 0.8 1.0 1.2
CD
Fnh
0[deg]
5[deg]
15[deg]
35[deg]
Figure8.Depthinfluenceontherudderdrag
342
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.6 0.8 1.0 1.2
CL
Fnh
0[deg]
5[deg]
15[deg]
35[deg]
Figure9.Depthinfluenceontherudderlift
6 CONCLUSIONS
In this study, a DARPA Suboff submarine
hydrodynamicsforthecaseofclosefreewatersurface
operation was investigated. The hydrodynamic
coefficients of the maneuvering model have been
determined for the three immersions of vehicle by
means of specially adopted PMM for underwater
bodies tank testing. In such conditions,
the course
stability in horizontal plane has been estimated and
compared with other research data. Similarly, the
submarineresponsetoimpulsedisturbancescloseto
thefreesurfacewas calculated.Specialattention has
been paid to the effectiveness of submarine control
surfacesinshallowlyimmersedconditions.
Intheframeofthe
recentprojecttheresearchwill
continue with development of 6DOF maneuvers
simulation model for above discussed conditions,
whichprocessshallbe supported byadditionaltank
testsandnumericalstudies.
ACKNOWLEDGEMENTS
This work is performed under Project DM07/6 “
Hydrodynamics of Underwater Body Close to the Free
Surface”, funded by Bulgarian National Science Fund at
“Competition for financial support for projects of junior
researchers–2016”.
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