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OCR for page 687
Ship Wake Detectability
in the Ocean Turbulent Environment
A. Benilov, G. Bang (Stevens Institute of Technology, USA)
A. Safray, I. Tkachenko ( stitute of Oceanology, Russian Academy of Sciences, Russia)
AB STRACT
The tabule t shuctxe of ship wrke md ocem
upper icy r is presented m this tudy We discuss
results of 1) th ory on fhe txbulent ship-wrke md
th ocem upper Icyer t xbulerw4 mder s xface wa~ss
effects includmg wa~s brerkmg -- m crurlyticcl
st dy; 2) based on k~ t xbule t clo xc c couple 3-D
non-sterdy mmerical model (wrke + ppx kyer)
which mcludes the wa~s b~v44king; 3) expximental
ir~sstigation p~eliminary remits) in the Davidson
Lctorctory towing tmk on ship-wrke detection by
mecs xmg t xbulence m-situ
The fheoretical crurly is of fhe ship-wrke
txbulence uses th shear-fiee model, seff-modeling
md Kokmogorov's hypodhesis for th pmpose of
clos xc The envi ommentcl t xbulence in the oce m
upper kyer hcs beff~ formokted by the css mption of
hori ontally miform hyd ody rmic fleld md k ~
g o p model mder fhe :xistence of smface we ss
md its breking BcY4d on k 8 tmbulent closmv4 c
couple 3-D non-stecdy m mericcl model (wrke +
upper kyx) which mcludes fhe waws brerking h~s
been sed, in fhe shear-flv4e cpprocch, to cxry o t
m mericclly fhe ship-wrke detectability m the ocem
txbule t envi omment Bodh of the crurlyticcl md
m mericcl remits show fhe 3-D truct xc of fhe ship
wrke for dffferent wind conditions cod ship peeds,
md the detection ~a ge on the oce m s xface md fhe
detectability in depfh of fhe ship wak in terms of
ship parrmetxs mddhe wmd speed
The experimentcl study is destir~sd to srify fhe
th oreticcl md m mericcl prediction on detectability
of tmbulent ship wrke mder dffferent experimentcl
conditiom The wrke t xbulence sigmfficmtly
exceds fhe m~tmal lew41 of fluctuatiom in th t mk
md fhe v~brction noise produced by the towing
sy tem The wrke t xbulent spech ms he s w 11
exp~essed Kolomogov's r mg The ship-wrke
txbulence is w 11 detectable md Kolmogorov's
rmge cm be identified e sn for fhe mo t ~emote
location of the probe (~ I O L~, L~ is fhe ship ienf h)
es w 11 mder fhe r mdom s xface wa~s conditioa
1. INTRODUCTION
1.1. Stdp Wake. Th physiccl mechmism of ship
wrke in the ocem is c remit of fhe tmbulence
ger~xated by movmg ship The tmbulent diffusion
deflr~4s th region of the t xbulent wske th~t is
sp~ecding in time Nmdascher (1965) tudied 6he
wrke of x41f-propelled bodies He fomd 6~t the
wrke width hcs c powsr hw behavior Field
mecs xements for ship wrkes he s been mcde by
Milg cm et cl (1993) md they fo md the wrke widdh
hcs c powsr hw of x behavior where x is c
di tcrw4 from 6he ship Hock t~a & Ligtelijn (1991)
mecs xed 6he maximum value of t xbulence intensity
in eah cross-section of 6he wrke of c Sm lo g ship
model The remit shows 6~t 6he tabulent kmetic
er~srgy hcs c x csymptotic behavior The result
of mecsmv4ments dor~s by Milg rm et c1(1993),
Hoeksh c & Ligtelijn (1991) cg ee wi6h the text books
w itten by Birkhoff & Zx mtor~sllo (1957, chapter
14), md Temrskes & L mley (1990, chaptx 4)
Dommermubh et cl (1996) performed mmerical
Ixge sddy simoktions on tmbulent fiee-suface
flows They obtaim4d probability distributions of
slocity field m the wrke md compxed the ~esults
wi6h experimental mecs xements
A other cspect of ship wrke detection is 6he
s mfa ce nom miform ity of s xfa ct mts The phy iccl
mechmism cllowing the detection of ship-wrke on
th o s m s xface is c ~esult of the diff sion of 6he
smface~cti s subst mce in the t xb 4ent region of 6he
wrke Peltn4r et cl 1991, Benilov 1994, 1997,
Zikmm md Miloh 1996) The tabulent diffusion
forms the s xface nommfformities of this substance es
wsll es the cssocicted nommiformities of the Yxface
tff~sion, which effecti sly mppress 6he centimeter
b md of s xface wa~ss th~t is respons~ble for the ~adar
l
OCR for page 688
imcge of c wake Poulter et cl, 1994) The
di tribution fetu es of fhe varimce md mem
g cdient of th su face-ative substance are the cause
of c specffic imcg of the far wake that looks like c
"rcikoad h cck" (Kdilg cm et cl ,1993)
When fhe see su face is moderctely wmd
roug)~ened, typicclly by winds of 2 5 to 7 5 m/s (5 to
15 k ots), sy thetic cpertu e ~adar (SAR) imcges of
ship wakes, obtamed fiom ci craft or spacecraft,
of en cppear es c long, narrow, dark sheck cgainst c
brighter backg oumd The dark sheck imcges c m
have ienf hs of tens of kilometers et the low r end of
th wind rmge The spacecmft imcge of th oce m
suface with the ship wake imcge shows fnat fhe
visible iength of the ship-wake imcge recches ctou
I 00 km (Naval Ai W uf~e C nter Ai cmft Division,
Warmmster, PA, 1992/ m Bemlov (1994, 1997c))
1.2. The Upper Layer Turbulence. The tubule t
upper hy r of th oce m hcs c complex verticcl
shu tme defned by i flu nces of dffferent phy iccl
mech misms such es energy cod momentum h msfer,
th presence of su face waves md their breckmg, fhe
tu bule t energy produ tion by me m she flow, fhe
wave motions of th fluid, md fhe effect of fhe
Coriolis force
In conbast to ctmo pheric boumdary hyers over
I md, whe~e me m velocity she is th mcin sou e of
tubulence energy, th tubulence in the upper hy r
of th ocem is gowxned not only by me m velocity
she, but clso by su face waves Th t mspo t of
momentum, hect, moistme md sclt occu s across fhe
ci -see i terface cod is effected by fhe oce m surface
waves The~efore suface waves phy m importmt
role in fhe ci -see interaction ystem
The tmbulent motion m the upper oce m is c highly
pecffm example of tmbulence in c liquid whose fie
su face is su ject to wmd frictioa Th result of fhis
cction is fhe fcrmation of wa~s, pme d if cunents
mdtubulence, whichlecdto sho g ve ticclmi mg of
fhe surface kyer ~ conbc t to boumdary hy rs et c
solid wall whe~e me m velocity sh ar is fhe mcm
sou e of fhe tubue t energy, th tmbulence in th
upper ccc m is govemed m mmy ~e pects by th
rutu e of waves The totcl me m ctmospheric tress
not only indu es oce m cu rents th ough th action of
th shear t~ess clone but also supplies momentum to
g owing su face waves A part of fhe momentum md
energy is tr m fened di~ectly fiom the wind to d if
cunents, while moth r put goes i to surface waves
Wmd waves contain c considemble cmou t of
momentum md energy md fhey redi tobute fhe
momentum md energy over g ect distances md
supply energy to d fft cu rents md tmbulence by
th ir brecking Th wave breaking mectes c highly
tmbulent envi omment withm fhe top few meters of th
ocecn, md th wave dissipation by the breaking
i tensffies tubulence in fhe oce m mi cd hy r
~renarm et cl, 1992) Wave b~eaking pro id s c
mechmism for injection of both moment m md
tmbulent kinetic energy fi om th mface winds to th
water E perimentcl ~esults mdicate th~t fhe rektive
energy th~t is lo t from th wave motion du to c
smgle breaking lies between 102 md 10 ~ ( M Iville
md Rcpp, 1985) The energy t m fened p r u it time
fi om th wind to th water su face is m ord r of pme
d if cune ts Kitaygorodskiy, Mi opolskiy, 1968;
Kitcygorodskiy, 1970) Therefcre, the tmbulence of
6he upper ccc m is nou ished by the energy supplied
fiom th wa~s Consequ tly, th tmbulence
characteri tics sh mid dep nd on 6he tate of 6he oce m
s lrface
A moving ship leaves c long wake hail behind
md mckes it poss~ble to monitor the t~avelmg ships
long rmge by mems of radar systems Tubulent
sensors me clso ~ole to detect "m situ" the wake
tubulence m the case of complex enviromme t
situ~tion
Smce 6he properties of c ship wake me expected to
depend on th peed md size of 6he ship, 6hecreticcl
tudy should be provided to crurlye 6he wake properly
~ 6he oc m, m my natmcl ctmo pheric md ccc mic
fectmes intemct with 6he wake Th ~efore, th rutu cl
oce m tmbu ence should be studied to identffy 6he ship
2. TBEORETICAL MODEL
2.1. Wake Turbulence. A ship t~aveling with c
con t mt speed is considered es m active sou cc of
tubulence, md the tmbulence is developed within
th boumdary that is g owing in time md
characteri:D:s 6he sccle of the tmbulence To descobe
th dynamic behavior of the tmbulence m the ship
wake, 6he following assumptions are mcde Benilov,
1994, 1997~):
I. The wake tu bulent kinetic energy
sig if c mtly exceeds 6he upper hy r
tubulence 6~t redu es the tubulent wake
problem to th tubulent region development
inanon-tubule tliquid
2. The mcm souce of tubulence is c movmg
ship that me ms that cll i te~actiom betw en
th wake tu bu ence md enviromment do not
2
OCR for page 689
conhibue in the wak dynamics md allows
us to red ~ e She problem to She shear-fiee
tubule t model
3. The characteristic scales of ch mge along
moving di ection L, kinetic energy kw.
tubule t mi ing length t md the peed of
ship Us are mch that L >> e J: W / UD
These as mmptiom red ~ e She wake problem to m
axisymmetriccl md non- tationary model of
developme t of c ce tam cylind i al turbulent region
with the axis of ymmetry located on the oce m
su face et vertical coordinate z = 0 The axis of
symmeby in She phne (z, y) hr. She coordinate
(0,0), where y is She hori onbl t msverse
coordinate For She wake problem, the time t c m be
convened to She longitudinal coordinate x, She
di lance fiom She ship in th di ection along She
wake, by t msformi g x = Ust
The change of kinetic energy, kw, in time md
space w may describe withm She hamessork of She
lliE equ tion in c she free approximation, where
se f-modeling md Kolmogorov's hypotheses are used
for the pu pose of closme The Meg ction gives She
solution in th following form
r (t) = awl +—)= ~
j t) 2 a (I t ) +2
t.= 72217'
where o is the ship beam, She corbt mt C i
invari mt of the problem equaled
(1)
s m
C = reify: = constant, O < t < co, (2)
~ is She eiger~lu of She boundary valu problem
which is en algebraic fu tion of clo He parameters
This solution gives the sin of She turbulent wake m
time, md lliE of She wake, i,,, is m proportion with
roe Becmse She closure parameter of She turbulent
scale is urJmow, the eigenr tlu of She boundary
valu problem, /, c m be obtained by mecsmements
The mecsoxement done by Milgam et cl (1993)
gives ~ = 8 The parameter C depends on th ship
turbulence coefhcient St that characterizes She
efficiency of the ship propeller to gerexate She wake
turbulence The lintl solution of the problem gives
expressions for IKE md dissipation rate in the wake
area, md the wake sin Benilov md B mg, l 999)
2.2. Upper Layer Turbulence. Th atmospheric
action on She oce m suxfae results m the energy md
momentum fluxes These fluxes gerexate the oce m
me m flow, surface waves md smcll-sccle turbulence,
which pltv m impo tan roll m the upper oce m
dynamics Me m shear flow is the one of She main
son es of the smell scale turbulence, in 6 is respect,
th turbulence of the oce m upper It.sx should be
similar to the classical sh ar tmbuence when She
me m shear energy produ tion dominates The effect
of She souffle wave on ant olence appears two ways
The first one is wave t ret mg which prod s es
sig iflc mt energy flu to the smell-scale turbulence
md She momentum flu to the me m flow on She
su face The second one is the local vorticity
produ tion c used by She in lability of the suxfae
waves Bemlov et al 1993) T is effect should also
be taken i to account in the balance of momentum
md turbulent kinetic energy of th oce m upper By r
The turbulence itself effected by She me m shear flow,
wave motion md wave breckmg may reveal
conhibuions of these energy t toes though She
tubule t kinetic energy md dissipation rate Melville
1985, 1994, 1996; Benilov 1997) it will create She
sub-hyer in She upper oce m where th energy
balance takes different forms
Thus the theoretical model of She dynamics of
oce m upper layer ht. to include equations which
describe the me m flow, turbulent kinetic energy,
i ~ on betw en turbulence md surface waves,
wave breaking, cod turbulent mi ing le gth The
boundary conditions have to describe th flu es of
momentum md energy produ ed by the wave
breaking es well es Demo phxic action on th oce m
su face
To describe dynamic behavior of She turbulence in
th upper layer, She Whom hyd odynamic field is
as umed to be u if mm in horizontal planes, which
me ms all statistical characteristics of the turbulence
md efface waves are fu non of vertical coordinate
md time The momentum flu from th atmospheric
boundary layer to She oce m, Ha, hits two components
which are the direct moment m flu to She wind
cunent, [c . md the momentum flu to the surface
3
OCR for page 690
Wb s, Tw The moment m fl~ Sw cbm be expressed
bS b combim~tion of sww 6he moment m flux for 6he
Wb g owth, bmd Twb . 6he moment m fl~ produced
by Wb breaking Smce Twb finully goes to the wind
curtent, 6he tott I moment m fl~ to the wind c trent
inthe tpper Ot bm, [c. is [c = [cd + [wb
In g r~4rbl, they tbati fy the ir~4quality
|~Cd | S |~C | s |~ |
(3)
The eru4rgy fl~ to the meam flow Cb n be expressed b S
qC C D ( od + TWb )i D . (4)
where t s = t (O) is the s tib e d fft olocity This
er~4rgy fl~ blSO bPPebtS to be b remit of bodh 6he
di~ect b tion of the wmd on the oceam s rLb e bmd 6he
Wb breaking
The p~esence of 6he fiee s rLb e mder pow rf 4
comporu4nt of water motion, s tib e Wb s bmd Wb
breaking, makes b distmction betw en pper Iby r
t tb tlence bmd wall t tb 4ence in the constamt
fiiction s tb-ibyer This meamr that 6he remit of Wbli
t tb tlence Cb mot be bpplied to th oct m tpper
Iby r
The meam moment m per mit breb of 6he s rLb e
Wb bmd the eru4rgy density of 6he Wb motion cbm
be obtamed bS
MW = 3 Ct'
EW = P4 P C0 = 4 CoMw'
(5)
(6)
where CO=g/t O(t) thephase olocityofthes tib e
Wb bt 6he spectrbl peak fiequ~4ncy t (t), d is 6he
Phillips' con tamt A simple physical situation is to be
fo tud when 6he wind blows stt udily o 4r b lb tge breb
for b long eno ~gh period Under 6his condition, 6he
wind shesshas magmt tde p~u3 bmd it is t msmitted
to 6he mderlying water m b sttti tiCblly
homog ru40 ts Wb feld Following Long et-
Higgim (1969) w ertimate moment m bmd em4rgy
fl~os ~ b bmd q b produced by the Wb breaking bS
some fibction of 6he tOtal moment m MW bmd E,, To
fud o tt 6hese fl~os, let Twb bmd qwb be
propo tional to 6he Wb moment m bmd energy
decay per chatacteristic Wb fiequ~oncy tDo bS
following
3 b=Y1t oMw. {b =Y?t oEw. C)
where y~ bmd yt bte mmttical constamts
rep~esenting 6he proportiom~lity of th Wb
moment m bmd eru4rgy spent m 6he Wb breaking,
bmd
Y3 = (4 / 3)Y,
(8)
Using the ertimate for f 41y de oloped Wb s (Co =
COb CO~ /U¢~ = 31 ) bmd th c t tomaty vulu~4s d =
O 01 bmd (P~ / PW) =103, th m merit~l vulue of y
cbm be obtt iru4d;
Y~ = C = 3 x I 0 4
(9)
This is th highest vulue thtt yl cbm hb o bet~ se
th moment m fl~ fiom bir to Wb °.Tw. hbr been
taken bS the tOtal fl~ fiom bir to water, p~u?
Therefore,mmericalestimatesof y, bmd y, cbmbe
written bW
r, S3X10 4, y~S4xlO 4 (10)
The mmmericai vahr4s fo md he~e bre m good
bgeement with the Long et-Higgins' b taged
ertimate (1969) They cab3lated 6he lelati o erxirgy
lo t from the Wb o motion d~ro to th breaking bS
10 The ewperimenttl ertimate of the single
breaking e o t Melville bmd Rupp, 1985) shows this
q mtity bS 10 ~ 10 This dist tepb y mby be
explairu4d by the mtermittt e of the Wb o breakmg
As b mebr te of the Wb o breakmg intermittence, 6he
re!xti o bteb occipied by 6he Wb o bre~mg e onts
mby be b cepted Th n, 6he e timate of Melville bmd
Rupp mfltiplied by th intermittence valmo
10 ~10 will take rame order of mag it3de
wi6h the Long et-Higgins' ertimate bmd 6he ertimate
in (4 44) The fl~os induced by Wb o b~eaking cbm
bho be written m the mequality forms bS
~ b <3xlO t oMw=10 PwP
4
(I 1)
OCR for page 691
q b 4 x 10 alO Ew = 10 pwpc3 (12)
Typical ocem condition cbm be bpplied to fud bm
e dmple of m merical values Twb bmd qWb Td g
b phase v locity of th Wb b3 CO ~ IOt s which
conesponds to the moderate wind condition,
~wb S O. IN/m ~ qwb < IW/m . (l3)
This estimate shows th moment m flux mduced by
Wb b~eaki g in b fully dev loped situation can be
compabble with th tOtal value of the moment m
flux from the btmo phere, which is
Z~ = p~u3 = p~Cu~ = 0 IN/m3, (14)
where C~ ~ 10 3 is 6he d bg coefhcient of 6he oceam
smfb e The hig)~est estimate of 6he energy flux due
to 6he Wb b~eaking cbm be compb ed with equation
(4) 6~t shows 6he energy flux goes to the meam flow
The hig)~est e timate Of qc hbS 6he common form
?~=PaU*ai =pau-aiak~=k~cupai ~ (15)
where kd is 6he coefhcient of the wmd s xib e d if
bmd its customb y estimate is kd ~ 1/30 In the f lly
dev loped situation, the ratio betw en qc bmd qw
cbm be fo md
{' ~104 3 ~k~C P~ ~ u~ ~ = 3xlO 3 (16)
Pw(Cd J
This estimate shows th~t the energy flux produced by
th Wb breakmg sigwifcbmtly ew eds the erxxgy
flux to the meam flow In quasi-steady mproach of
th upper oceam tmbulence, 6he energy flux qc
rep~esents the energy i flux from the xfb WbV s
to 6he t xtulence Th refore, the estimate (16) shows
th~t th Wb breking play bm importamt role m the
formmg 6he upper kyer t xbulence ~egime It is
b3 mmed that th s xib e Wb s bre fully or blmo t
dev loped, bmd 6he wind condition chmges slow
enough to adapt 6he steady bpproacb This
bwomption reduces the n mb x of urdmown s aLb e
ww qw O it giv s Iw = IWb bmd
qw =qWb ~ 6he cbse of dev loped Wb s, 6he
moment m flux produced by Wb breakmg mby
hav same order of magmitude wi6h the moment m
flux fiom the btmo ph re Therefore, the simple t
hypodhesiS m be made b3 p~u~ = Tw = ~Wb The4
th moment m flux to the wind cmrent [c becomes
~c = ~wb = pau~a = PwU~w (17)
The tmbulent kinetic energy, k, in the oceam upper
Iby r mby hbV dismepb ies with regmbr t xbule t
models becbmse the potentibl motion due to 6he
3mfb e Wb s has vxy shong impact on dy dmic
behavim Vb io d bttempts to deriv the equation of
t xbule t kinetic energy with presence of 6he
pote tibl Wb component m the mdom v locity
feld of the ppx oceam hbV been undertaken by
Benilov (1973, 1997b), Benilov bmd Lozo tski
(1976) bmd Kitbigmodski bmd L mley (1983) in 6he
sub-layer of comtamt fiiction, the tabJent kmetic
energy budget wi6h the p~esence of s aLb e Wb is
]~k = P~kl]~3Vz]~3 (k + Okv) (18)
vr(a~3i ~ 3
where k is 6he Wb kinetic energy, a is the ratio of
th t xbulent pramdd n mbers, a = Pr Pr~ The
t msport equation for 6he dissipation cbm be tdken m
th form of k 3 tmbulence theory wi6h bdditiorurl
txm nV that rep~esents 6he Wb soxce of
dissipation increcse:
~ 3v~] 3b+C~ kv~(] 3i +
where Pr~ is 6he t xb dent pramdtl m mber, Cr bmd Cz
b e constmts Thei typiCbl values b e Pr~=1 3,
Cr=1 44 bmd C =1 92 ~offmb m, 1989) Smce 6he
Wb motion becomes 3mbller in dep6h,11v tO bt 6he
location fb enough fiom the oceam smfb e From
th se remits 6he t xtulent diff sion of the t xbule t
kmetic energy bmd 6he Wb kmetic energy is
domi mt in 6he ramge of depth where 6he Wb
motion is vigorous, or O < X3 < LW v where LW V
is 6he thickmess of the Wb -t xtulent sub-kyer ~
this kyer, energy produced by 6he meam beb flow
s
OCR for page 692
may be neglected since turbulent energy produced by
the wave breaking exceeds the mean shear effect
significantly in the vicinity of the ocean surface. And
the c; can be obtained as an eigenvalue of the
problem,
~=_(3CV ) 712/3-
Then £~0) can be found as
£(0) = CVk / (0) = CVkV/ (0) =
1(0) 2xLa3/2
49/i,BgCo ~ 1.2xlO-4Co.
(20)
Below the wave-turbulent sub-layer there is a
transitional diffusive sub-layer in the range
];w-v < X3 < ];w where ];w is the lower boundary
of the transitional diffusive sub-layer. In this region,
the turbulent diffusion still exceeds the mean shear
contribution in the turbulent kinetic energy budget
but the effect of wave motion becomes insignificant.
The diffusive turbulent sub-layer is a transition zone
between the wave-turbulent surface layer and the
layer where mean shear flow controls turbulent
regime. By changing the variable from X3 to
Z = X3 - LW-V, where z = X3 - 1;w-v we have the
expressions for the TKE and the dissipation rate as
k(Z) = kit + ~ J
where
(22)
£(Z) = £W VY = £~0~(1+ ~ ) ~
(23)
Lit = 6RQW-v£(o) ~ (24)
k(0) = (:C Qw v ) (25)
Qw-v is the energy flux at z=O, and R. vat, v2 are
algebraic functions of the closure constants. The
solutions show that the turbulent diffusion
mechanism gives the power laws for k and £ . And
the turbulent mixing length becomes a linear function
of the vertical coordinate. The exponents V1, V2
can be found as vat = 8.462, v2 = 4.974 from the
standard values of the closure constants.
Following after the transitional diffusive sub-
layer, there is an another sub-layer located in the
,n3
v.
ad
Ls
lot ~
10
_
—~ fVoirth
/forlarge\;
with heav
tall w
lobre
Ives
breal
lo 0 5 Lo 1.5
Figure 1. Ship Wake Detection Range
under the Various Conditions
allele ;s
akmg
.
_
2.0
Us /Ua
range 1;w < x3 < 1; where the mean shear
production of turbulent energy is dominant. In terms
of classical steady turbulent boundary layer problem,
this sub-layer corresponds to the logarithmic
turbulent sub-layer(Monin and Yaglom, 19874.
2.3. Ship Wake Identification. The combination of
the ship-wake and the upper layer turbulence gives
the detection range on the ocean surface and the
detectability in depth of the ship wake in terms of
ship parameters and the wind speed.
2.3a. Surface Detection Range. The detection range
xd can be obtained from the turbulent kinetic energy
of the ship wake along the x axis and the surface
turbulent kinetic energy of the ocean upper layer.
Introducing the notation that kw is the kinetic energy
of the ship wake along the x axis, which gives the
maximum value in the cross-section of the wake, and
ke is the surface kinetic energy of the environmental
turbulence, the detection range is a solution of the
equation
k = k . <264
After substituting ST = 14.25, ~ = 8, PK7 =
Cv =0~09, ,0=10 2 and Co=Ua
' K
, the
detection range xd can be found from the expression
, -1.25
—= 1 2.3 6 —0.07 . (274
6
OCR for page 693
This is the detection range based on fully developed
waves. It is shown in Figure 1 with the case of no
wave breaking. Since the turbulent kinetic energy of
the natural ocean surface ke (O) has been formulated
from the hypothesis that all the energy transferred
from atmosphere transports to the wave breaking,
ke (O) can be expected to be the highest value that
the ocean can have.
In the case of weak wind, we may apply the
condition of momentum flux continuity on the air-
water interface to find the environmental turbulence.
It can be seen that, for the weak wind, the two upper
sub-layers can be neglected because there is no wave
breaking. Therefore, the wall turbulence is developed
practically from the surface in the case of weak wind
with no wave breaking. The friction velocity u*w in
the water is
u = j Pa ~ u (28)
Pw
According to the theory of wall turbulence, the
turbulent kinetic energy does not depend on the depth
and is proportional to u2w . As an estimate for our
application, the kinetic energy can be assumed to
have the same order of magnitude of u2w. It gives
the detection range of the ship wake in the case of
weak wind with no breaking as
~1.25
Xd=394.62 U.. ~ _0.
Ls Ua J
.07. (29)
According to Beaufort wind scale and specifications,
(29) corresponds to light breeze wind with speed of
2.4~4.4m/s (4~6knots). Under this situation, small
waveless are generated and there is no wave breaking.
Whereas (27) corresponds to a strong wind with
speed of 11.4~13.8m/s (22~27knots). This condition
generates large waves and white foam crests. The
ship wake detection range for those two conditions
are shown in Figure 1.
2.3b. Ship Wake Identification in Depth. The ship
wake identification below the ocean surface can be
made by comparing the turbulent kinetic energy of
the wake and environmental ocean turbulence. If we
let
ED (m2/s2)
x3(m)
7
-0.02
O _
-
10
1D
20
25
30 _
Figure 2. Detectability of Ship Wake
by Kinetic Energy in Depth (with
Wave Breaking)
kD = kw—ke (30)
where kD is the detectable kinetic energy. Then, if
kD is positive, the wake can be detected. The
detectable kinetic energy profile at the specific
distance x along the wake axis can be obtained as
ED = Us2 ( 1 + 14.25—~
—2.55 x 10 Ua t1-
1- —
x3
at + 14.25 ~
~ Ls )
~ .
-4.974
x3
0.062U 2
a
(31)
Equation (31 ) shows the detectable kinetic energy
distribution everywhere inside the wake. This is valid
only for fully developed waves, which corresponds to
a strong breeze wind. A simple estimate can be made
by assuming the ship speed Us = lOm/s, the ship
beam a = 1 Om, ship length Ls = 1 OOm and wind
OCR for page 694
constant friction layer. The results are shown in the
Figure 3. The ship parameters are the same as fully
developed case, but a wind speed of 4m/s is applied.
o
10
x3 (m
20
30
40 -
k (m2iS2'
0.00 0.01
0.02 0.03
Figure 3. Detectability of Ship Wake
by Kinetic Energy in Depth (without
Breaking)
velocity Ua = lOm/s . The results are shown in
Figure 2. It shows the detectable kinetic energy
distribution at various distances from the ship.
For the case of weak wind with no wave breaking,
kD can be written as
kD=U2(1+14.25—) x
..~1-
x
N2.91 l2 (5. 12)
X3
a(1 + 14.25—)
—10 6Ua .
The turbulent kinetic energy of environmental
turbulence for the weak wind case is constant through
the depth of a statistically steady layer where the
Coriolis force is not effective, which is called the
3. NUMERICAL MODEL
Based on k-£ turbulent closure (Mohammadi
and Pironneau, 1994) a couple 3-D non-steady
numerical model (wake + upper layer) which
includes the wave breaking, has been used to carry
out numerically the ship wake detectability in the
ocean turbulent environment. The previous
theoretical analysis shows that the shear-free
turbulence can be applied to the couple model. Using
this result as an assumption, we significantly simplify
the governing equations of the model. Galerkin's
finite elements method was applied to solution of the
turbulence transport equations. We present results of
numerical experiments, which have been carried out
using the software created for above discussed the
numerical k-£ - turbulent model of the ship wake in
the turbulent ocean environment. Table 1 presents the
conditions of the experiments: the ship speed Us and
the wind speed Ua . The results show the 3-D
structure of the ship wake for different wind
conditions and ship speeds (Figures 4, 54. Figure 4
shows the body of the wake for calm situation (a -
theoretical result) and within environmental
turbulence (b- numerical experiment). Here, Ro is the
wake radius; Ls is the ship length; a is the ship beam.
At the left of Figure 5, different colors show the
surface manifestation of the wake for different winds,
the condition that the wake TKE meets the surface
environmental TKE associated with the wind speed
defines the wake boundary. The side view at the right
of Figure 5 shows the wake cross-section along the
wake axis by the equi-kinetic energy isolines. In the
ranges of the wind speed Ua =3 - 10m/s and ship
speed Us = 3 - 10m/s the wake has the surface range
of detectability up to 4 km in x-direction (Ls = 100
m, a = 10 m), and the submerged range also up to
over 4 km. The surface range decreases with the
growth of wind speed. The maximum length of the
submerged range is 3-4 km and descends deeper with
the growth of wind speed reaching about one ship
beam of depth below the free surface for Us = Ua =10
m/s. The maximum of the wake width is about 70 m.
The depth of wake penetration in the ocean exceeds
30 m for the condition of the numerical model. Thus,
the total scales of the detectable body of the wake
may reach 3-4 km length, 70 m width, and 30 m
depth. The turbulent kinetic energy k(x, 0, 0) in
8
OCR for page 695
normalized form conforms the theoretical and a)
Milgrem's power law that well approximates
experimental data.
Table 1. Conditions of the numerical
experiments
use 3 7
m/s
Ua, 3, 5, 3, 5,
m/s 7,10 7,10
a)
As
~:~ ~
.
.......
~~...~ 9....
· ~ : . .
b)
N
~3
3, 5,
_ 7, 10
k(d, ,t)
k (O,O)
Figure 4. The Body of the Wake within
Environmental Turbulence (Ua=5m/s). Equi-Kinetic
Energy Isolines. 3D Side View.
9
60
55 _
45 _
4 _
no
b)
l~qr.rA9~t
-1
3
4 _
10 20
I ~ ~ ~ ~ I
Ua~Om,&
Unarm,&
: U - m,&
UEF7m,&
UEFOm,6
33 4C
I ~ ~ ~ ~ ~
1
O6
55
45
4
35
3
25
2
15
05
n
~ lo:
33 40
him In 300 ~X) ~
1wm ~
ha
300 To
Figure 5. Ship-Wake Equi-Kinetic Energy Isolines
(a - Top View, b - Side View )
1
OCR for page 696
4. EXPERIMENTAL STUDY
The experimental study is destined to verify the
theory of turbulent ship-wake parameters in different
kind of experimental conditions. The diagram (Figure
6) shows experimental setup. The ship-model and the
hot film anemometer (HFA) probe are mounted on
the carriage at the fixed distance L between them.
The carriage moves with steady speed U (towing
speed). The model with propeller generates a
turbulent ship-wake. The hot film probe moving with
the same speed measures the turbulence in ship-wake
cross-section at the distance L from the ship-model.
This distance varies from test to test. There are three
basic measurement systems: Propeller Rotation
Controller (PRC), HE Anemometer (A), and Data
Acquisition System (DAS). The PRC manages the
propeller rotation to produce a given value of the
driving force, F. during the test. The Anemometer
measures turbulent velocity fluctuations. The DAS
collects data from the Anemometer and PRS. The
model scales are 30cm x 160 cm. The signal
produced by the driving force, F. controls the motor
(propeller rotation) keeping the driving force, F. to be
equal zero during the test. The towing speed has a
constant value during the test. The probe support
construction gives opportunity to put the HE sensor at
an arbitrary location in the wake area and turbulent
upper layer. The ship-wake detection by this
experimental setup can be done for different wave
conditions including wave breaking (Figures 7 - 134.
The signal of the wake is detected at the distances of
the ship up to approximately L ~10 Ls (Ls is the ship
length) with correspondences to the actual ship speed
up to 20 knots. Figure 7 presents an example of the
ship wake turbulence in this experiment in terms of
u'-spectrums, Suu, versus frequency, f, for two the
ship speeds Us = 0.63 m/s (the actual ship speed 10
knots) and Us = 1.26 m/s (the actual ship speed 20
knots) at the distance from the ship - model 1.63 m
that corresponds one ship length L. The wake
turbulence significantly exceeds the natural level of
fluctuations in the tank and the vibration noise
produced by the towing system. The turbulent
spectrums have well expressed Kolomogov's range
that allows estimating the dissipation rate. Figure 3c
is an example of the wake turbulent spectrums at
different distances x from the ship model (x from 1
ship length L to 10 L). One can see that even for the
most remote location of the probe (~ 10 L) the ship-
wake turbulence is well detectable and Kolmogorov's
range can be identified. Figure 13 shows the
spectrum of wake turbulence at the distance 5.4 L
from the ship model and the spectrum environmental
turbulence in the case of running spectral waves. The
model speed corresponds to the actual ship speed 20
knots, and the surface waves correspond to the actual
developed wind waves for the wind speed about 12
m/s. It can be seen that the wake turbulence for that
location still significantly exceed the level of the
environmental noise and has the well identified
Kolmogorov's range. Thus the turbulent sensors can
also detect the wake turbulence in the case of
complex environment situation.
. . . . . . . . ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ , 5 >
..... .... ~5 ~~]
Figure 6. Experimental setup.
A5594. 101
0.001 ~
~ -5/3
0.0001 ~ -
0.00001
).000001
0000001
0000001
0.01 0.1
1
f, Hz
10 100
Figure 7. Examples of ship-wake
turbulence for two values of actual ship
speed.
10
OCR for page 697
Wake Turbulence
o.ooo
0.0000
0.00000
0.000000
0.0000000
0.00000000
0.0000
0.00000
_ ~
- 0
_ ~
- Q
.=
= ~ __
- I=
_ ~
- ~4
= ~
_ ~
- VO
_ .~
Rlln
166"
108
315"
315"
43'4"
43'4"
Us=20 knots
model speed 4.2 ft/s
1 1 1 1 1 1 1 11 1
1 1 1 1 1 1 11
0.01 0.1 1
o
.~
vc
o
Run
265
268
274
----- 280
f, Hz
Figure 8. Turbulent spectrums along the wake
axis at different distances from the shin model.
0.0001 ~
— ~~.:Y,,.~,~,,
wake axis
~3 t 4~:
/
Noise
1 1 1 1 1 1 111 1 1 1
10 100
0.0004*(xA(-5/3))
L-4.2 m, v-68
-~ (at ~~
/,,~>,~-,,,,,6 ·-
~ ~~ r
0.000 1 *(xA(-5/3))
environmental noise
f, Hz
Figure 9. Turbulent spectrums in the wake cross section at the
distance L = 4.2 m from the model. model speed Us =68 calls.
11
OCR for page 698
0.0001
Su
L - 8 m, v-202 cm/s
0 ~ 0023 ok (xAt -5/3 ))
_ _ is tam
environmental noise ~,.-
0.000001
1
10
Su
0.00001
0.000001
f, Hz
FigurelO. Turbulent spectrums in the wake cross section at the
distance L = 8 m from the model, model speed U s =202 cm/s .
~2~ i\ L-13 m, v- 202
I..,
or
....
0.0006 * (xA(-5/3 ))
- .~
~~,
009 A
/ 7 /
/Environmental noise
10
~ ~~...~. `_ _ ~
;ssss~,,,.,.,.~ ~ 4... N. . ~
~~ ·~N
Wake axis
20
f, Hz
30
Figurell. Turbulent spectrums in the wake cross section at the
distance L = 13 m from the model. model speed Us =202 calls.
12
OCR for page 699
su
0.00001
0.000001
0.0001
0.00001
0.000001
L- 13m,v68cm/s
.~ Wake boundary
_ ........ - ' ,
_~ I. .s
,,,~ ~ ~ ~~ - - 1 .: _
W Lke axis nvlronmenta
I noise
- I'd ~0.000065*(XA(-5/3)) / 1
~~ Am, ,, AS
-A (}~2 / A" =~
_ ~ ~ . = ~
0.0000 15 ~ (xA(-5/3)) 0
. 1 .,
f, Hz
Figurel2. Turbulent spectrums in the wake cross section at the
distance L = 13 m from the model. model speed Us =6X cm/s.
Run
background 312
wake 288
=---------------------------------------~
Spectral waves 1", 1 Hz
L =13 m, Us = 126 cm/s
f -5/3
0.01 0.1 1 10
f, Hz
Figure 13. Wake turbulence under spectral waves conditions at
distance L=13 m from the model, model speed 126 cm/s, wave
elevation variance 2.54 cm, spectral peak frequency 1 Hz.
13
(f)
u'u
S
OCR for page 700
5. CONCLUSION
Theoreticcl base, m mericcl modelmg md
experimentcl st dy on ship wake detection in 6he
oce m tmbulent envi omment hcs been developed
A11 6hese studies demonshate 6hat 6he ship-wake
tubulence is w 11 detectable md Kolmogorov's
r mge c m be ide tifled
Th tmbulent semors cm detect 6he wake
tubulence m the case of complex enviromme t
situ~tions cod et sig iflc mt dist mces fi om c ship
In case of shong wmd conditions th re is
sig iflc mt submerged wake body that c m be
detecte d by 6he tu bulent sens ors
We exp~ess ou deep g ctitude to the ADD of
R public of Korec for support of 6his work
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Bemlov, A Yo, 1973, "The Tubulence Generction
in the Ou m by Su face Waves", Izv Ac Sci USSR
Atm Ocem Phys, V 9, No3, ppl60-164
(Tr mskted m Engdish)
Benilov, AY, Lozovatsky, ID, 1975 Spectral
Models of the O e mic Tubulence, ~ Monog cph:
R search of the Tubulent Structme of the O e m,
The See Hyd ophysiccl in titute cd AN USSR
Sevastop o I, pp 102 - 112
Benilov, AY, md Lozovatsky, ID, 1977 Semi
empiriccl Medhods of 6he Tubulence Desuiption in
th Ocecn, in Monog cph: The Tubulence md
Diff sion of the Ingredie ts in 6he Sea, The Co-
O dirution Center of the COMECON (SEV,
I formation Bulletin, Vol. 5, Moscow, pp 89-97
Bemlov, A Yo, T. G M Kee md A S Saficy, 1993,
"On 6he Vort :x instability of Linear Su face Wave",
In: N mericcl Methods in Lcminar & Tubulent
Flow, Vol. V111, part2, Pmeridge P'ess, U. K, pp
1323-1334
Benilov, AY, 1994, A Discussion of th Tubule t
Nctme md Poss~ble Ccuses of the Ship Wake Rcdar
Imcge TR-SIT-DL-9409-20704, Stevens Imtitute of
Techmology, Hoboken, NJ, 49 pp
Benilov, AY, 1997(c), Ship - Wcke Tubulence, in:
NUMER CAL METHODS in LAMNAR &
TURBULENT FLOW, vol. 10, Edited by: C Tcylor,
University of Sw msec, U. K, Pineridge Press
Benilov, A Y. 1997(b), The Ou m Upper Lcy r
Wi6h the Presence of 6he Suface Waves md Thei
Breckmg, TR-SIT-DL-9707, Stevens Instit te of
Techmology, Hoboken, NJ, 54 pp
Bemlov, AY md GC Bmg, 1999, Ship Wake
D tection in the O m Upper Lcyer, TR-SIT-DL-
9904, Stevens in tit te of Techmology, Hobok n, NJ,
111 PP
Birkhoff G md E H. Zarmtonello, 1957, "Jets,
Wckes md Cc ities", Academic, S m Diego, Cclff
D enann, W. M, K K Kahmc, E A Termy, M A
Donelm, md S A Kitcygorodskiy, 1992,
"Observations of 6he erJurxeme t of kinetic energy
dissipation benec6h brekmg wmd wa~s", Edited by
M L Banner md R H. G imshow, Spri ger Verkg
NewYork,pp 95-101
Hoekshc M, J. Th Ligtelijn, 1991, "Macro wake
fectu es of r mge of ships", Techmiccl R port 410461 -
I-PV, Mar it ime ~ e search Instit te Nether kmds ,
Wcgemogen, The Netherl mds
Hoffmm Klms A, 1989, "Computatiom~l Fluid
Dynamics For E gineers", A Publication of
E gi ering Edu ction System, Austm, Texas
Kitcygorodskiy, S A md Yu Z. Miropolskiy, 1968,
"Dissipation of Tubule t Energy m th Sufae L~yer
offheOu m",lzv Acad Sci USSRAtmospheric md
O emic Phy ics, Vol. 4, No 6, Tmnskted m E gdish)
Kitcygorodskiy, S A, J. A Lumley, 1983, "Wave-
Tubulence I teractiom in fhe Upper Ouzu Part 1:
The Eurgy Bckmu of th ~te~acti g Fields of
Suface Wind Waves md Wmd Indued The
Dimemiom~l Tubuence", J. of Physiccl
O e mog cphy, Vol. l 3, No 11, pp 1977-1987
Longmet-Higgms, M S ,1969, "O fhe Wave B'eaki g
md the Equibbrium Spectrum of wind-Genemted
Waves", Proc Roy Soc, London, Vol. A310, No
1501, pp 151-159
Melville, W. K, mdR. J. Rcpp, 1985, Momentum
flu inb~ecking waves, Nctue, 317, 514-516,1985
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Melv~lle, W. K, 1994, "E ergy dissipation by
breakmg waves", J. Phys O emog, 24, 2041 -
Melv~lle, W. K, 1996, "The role of surface-wave
Fimd Mechamcs Vol. 28 279°n32Am~l R view of
Milg cm J. H. R D Pelt:D:r, md O. M G ffhn
1993, "S pp~ession of Short See Waves in Ship
Wckes: M csurements md Observations", J.
Geophysics Rs Vol. 98,NoC4,pp7103-7114
Mohammcdi B. md O. Pirom m, l994, "Analysis of
K Eps ion Turbulent Model", Johm Wiley & Sons,
Monm A S. A M Y~dom, 1965, "Statistical
Hyd omechmics: Turbulence M chmics Part 1",
(1967, Part 2) Moscow, Fi matgi, pp 639 Fff6h
prmtmg, 1987, Stati ticcl Fluid Mech~mcs:
Mechmms of Turbulence Vol. 1 -2 Th M T Press)
Nmdascher, E, 1965, "Flow in the Wcke of Self-
Propelled Bodies md R Icted Sources of
Turbulence", J. Fl id Mech, V 22, No 1, pp 625-
Phillips O. M, 1966, "The Dynamics of th Upper
O e m", Ccmbridge Umversity P'ess
Poulter E M, M J. Smith md J. A M G'egor,
1994, Microwave backscatter from the see surface
Bragg scattering by short g avity waves" Jourm~l of
Geophysiccl Research, vol. 99, No Ci, pp 7929-
7943
Stewart, R 1985, "Method of Sctellite
O mog cphy, Univ of Cclifomic Press, Berkley,
Temmekes, H. md J. L L mley, 1990 "A Fi t
Course in Turbulence", 13~ Printing, MT Press,
Ccmbridge, MA
15
OCR for page 702
DISCUSSION
D Se itsky
Davidson Labomtory
Stevens Petit te of Tech olo :, USA
The mthors present m excellent overview
of the t rbulent properties of ship wakes; oce m
upper layer turbulence; Ed their interaction
Further, She results of their recent model tests of
ship wake turbulence Ed wave t rbulence
measured separately Ed in comhinsrion (ship
model rummmg in heed sees) are also presented
These experimental results particularly the
combination of ship wake Ed wave turbulence
fields) appear to be unique Ed constit te c
valuable contribution to She literature
I have two questions for She mthors
Concerning the turbulent wake of the ship clone,
it would be useful ff the mfhors would id tify
th rektive conh~butions of ship hull form;
propulsor; Ed ship generated waves (which may
break) to the total turbulent wake field Visual
examination of the surface wake aft of c ship
gives the impression Nat the propulsor wake
may dominate so Nat the particulars of She ship
geometry may only be import mt m defming the
propulsor thrust Ed the concentrated kinetic
energy it imparts into c diameter of fluid which
is subst mticlly smeller th m She beam of the ship
In my event it would be used I to under t Ed She
mcke-up of the term kw in their Equation (26)
Ed how it relates to ship form, propeller
performance (especially cavitation effects),
breaking of ship generated waves, au
entrainment Ed speed Also, since that equation
implies that the turbulent fields due to ship wake
Ed waves are independent prope ties that are
directly additive, does the model date indeed
support this assumption?
The second que non concerns the turbulent
properties of waves used in the mfhors' model
te ts These waves are generated mechanically;
have no atmospheric wind on then surface; are
probably devoid of au entry ment, Ed likely
are not contimmously breckmg What is the
mthors' opinion on the characteristics of model
wave turbulence fields vs those m f 11-sccle
wind generate d breaking wa ve .?
At THOR S REPLY
None received
OCR for page 703
DISCUSSION
K Volick
Russi m Academy of Sciences, Russia
he reviewed pap r consists of th ee
sections: crurlyticcl, mmmerical, Ed exp rimentcl,
united by c common idea to show the ship wake
turbulence against c background t rbulence of
the upper oce m
he theoretical section is based on the
assumption that the shear-free axisymmetriccl
unsteady turbulent motion with some scaling
parameters spreads into c nonturbulent fluid
hen the Kolmogorov's self-similarity solutions
are used to describe the cylind iccl turbulent
wake controlled by the ship beam, turbulence
coefficient, Ed velocity, es w 11 es by m
experimentally determined eigenvalue of the
con esponding boundary problem
he model of turbulence in the upper oce m
is chosen es c q csistecdy horizontally uniform
r mdom hyd odynamic field with account of
su face waves (breaking, in general) Ed me m
shear flow he letter factors allow mfhors to
separate three t rbulent Icyers in the oce m
medium se tical profile: wave-pert rbed
subsu face region, tr msitiom3l diffusive
interkyer, Ed underlying classical logarithmic
me m-shear half-space When using quite general
estimates for the uppermost layer, he clove
vertical truct re is presented in m explicit form
he result of performed crurlyticcl
estimates, based on comparison of He wake Ed
the medium turbulence, is presented in the form
of technical formulas Ed diagrams containing
ship Ed envu onrnenr characteristics
Further in the section devoted to mmmericcl
simulation, She mthors directly model She wake
upper oce m interact ion by t rbu lent h msp ort
equations Ed present graphs all. c 3D shucture
of She ship wake depending on impo t mt
controlling parameters of the source Ed
en- nomnent
he experimental study with c propelled
ship model m the Ictomtory tank, closing the
paper, should, on mthors' opinion, verify the
developed theory Ed mmmericcl cclcubtions in
fact She wake turbulence spectra are rather
thoroughly measured by c hot fit anemometer
across Ed along She t mk behind the model he
anemometer sensitivity mm out ~~1lic~ ml.
high to measure very week turbulence si-rLtls
even against She background of specially
em mated mu facewaves
All the th ee approaches usedby mthors of
the reviewed paper do not overlap some compact
concept of the ship turbulent wake in She retl
oce m en- nomnent to m equal extend but rasher
complement each other For example, no layered
struct re of turbulence, developed analytically, is
show evidently in Humeri 31 Ed laoorcto y
experiments However, the method of
presentation used in She paper has ow
cdv mesa es to make perhaps the pattern of
studied phenomenon m ore voluminous
Some questions c m also arise, when
recdmg the article First, it ..-m~ld be desuable to
take the see wave directionality into account m
the model of upper turbulence, in parallel to the
wake axial scale, Though this task is not too
simple Also She technical formulas derived c m
be readily simplified, for example, by lowering
the accuracy in mmmericcl factors Ed exponents,
es well es by exp mdmg or approximating some
terms, et
Second, on my opinion, it is import mt to
widen the description of mmmericcl model Ed
the presentation of calculated data, m particular,
to show both the simplifications based on the
developed theory Ed c possible restructuring of
the wake es it penetrates Through She wave-
perturbed zone mto deeper Icyers
let the experimental section, it is not clear
how w re He su face waves excited in He tank,
Ed in general She extensive wave modeling
would be of crucial importance in view of the
main problem posed m She work An exphrmtion
of He experiment scaling seems also to be not
superfluous Besides, c general accent on
mmmerics Ed experiments (maybe m oral
presentation) cm redishibute She review d
matericlmme mmehicclly
As c whole, She discussed study is w 11-
grounded Ed comprehensive, its results are
reliable Ed intere ting to researchers, e pecially
such es data on She mmmericclly modeled
deepened ship wake
AUTHOR'S REPLY
None received
Representative terms from entire chapter:
kinetic energy
