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OCR for page 780
Radiation Loads on a Cylinder Oscillating in Pycnocline
E Ermanyuk, N Gavrilov, I Sturova
(Lavrent,vev Institute of Hydrodynamics, Russia)
ABSTRACT
The results of experimental and theoretical studies
of the mdintion loads (added mass and damping) on
n horizontal circular cylinder oscillating in n trot
Ned fluid are presented Two cases of the density
distribution over depth are considered: a) linearly
stratified fluid of limited depth, b) two layers of mis
cible fluids with the region of high density gradient
(pJcnocline) in between
The experimental technique for evaluation of the
frequency dependent added mass and damping is
based on the Fourier analysis of damped horizontal
oscillations of the cylinder in strntifled and homage
neons fluids
The linear 2 D wave radiation probl
ered theoretically both for horizontal
of three Infers (the upper and lower Infers are
homogeneous and the middle layer is linearly
strntifled), what provides an approximate model
of the experimental conditions The Boussinesq
npprcximation is used By introducing n Green
function, the boundary integml equation for the
disturbance pressure is formulated Comparison
of the theoretical and experimental data is presented
INTRODUCTION
The internal waves generated by n body oscillat
ing in n stmtifled fluid h we been studied in some
details both theoretically and experimentally A
comprehensive bibliography on this problem is given
-- T rner (1973), StepanJnnts et cl (1987), Voisin
(1991) Recent progress in the technique of inter
nal wave vi ualizati:n is described by Sutherland et
cl (1999) How ver, it should be emphasized that
only few authors have been concerned with evaluate
tion of hydrodynamic loads acting on n body in n
strntifled fluid despite the fn t that this problem has
an engineering counterpart relevant to the predic
tion of the I w speed and I w frequency motions of
marine structures and deep submersibles in real sea
environment which is characterized by the presence
of vertical density gradient A bibliographic survey
on this problem has been presented -- E rmanyuk &
Sturova (1996)
Let us note that continuous density distribu
tion over depth which is normally observed in
nature causes some specific properties of internal
w wes The fundamental characteristic of n strntifled
fluid with density distribution p(y) is the buoyancy
(Brunt Vnisaln) frequency defined as
N(y) = ~
where p is the gravity acceleration and y axis is di
rected vertically upwards The maximum Brunt
Vnisaln frequency for n given density distribution
over depth defines the upper cut fl frequency for in
ternal w e effects Thus, in contrast with infinite
frequency spectrum of surface waves, the frequency
spectrum of internal w wes is finite Moreover, when
the oscillation frequency of n body is I wer, equal or
higher than the buoyancy frequency, the equations of
fluid motion are of hyperbolic, parabolic or elliptic
type, respectively, what, once gain, is in contrast
with free surface problems which are described by
the Lnplace equation The major part of the stud
ies on the body oscillations in continuously strut
idled fluid is pe formed within the model of ideal,
uniform :, strntifled (the Brunt Vnisaln frequency is
assumed to be constant), Boussinesq fluid of ink
nite extent in particular, the tim~domnin analy
sis of damped oscillations of n sphere and circular
cylinder is given in Larsen (1969b) and the solo
tions in frequency domain Ewe been given -- Lay
& Lee (1981) for vertical oscillations of n spheroid
and by HurieJ (1997) for nrbitrarJ dire ted oscillate
tions of an elliptic cylinder Mention should be made
of the study performed -- Gorodt ov & Teodorovich
(1986) To evaluate the p wer radiated with inter
nal w wes by an oscillating body, they h we used the
Green fun tion approach while the body geometry
has been m: dsllsd -- the distribution of singularities
borrowed from the solution of the pe tinent problem
in homogeneous fluid Their results are found to be
in dis greement with the results by HurieJ (1997)
and Lay & Lee (1981) H wever, the disagreement
is entirely caused -- the form of the surface source
distribution For a correct form of the source, the for
mulas presented in Gorodtsov & Tsodorovich (1986)
will give a correct a timats The result presented in
Hurled (1997) h we been sxpsrimentally confirmed
OCR for page 781
-- ErmanJuk (2000) for the horizontal oscillations
of n circular cylinder
Let us note that, strictiJ speaking, oscillations of
n 5: d:, in n stratified fluid can excite su face w wes
H wever, the maximum buoyancy frequency is nor
mnllJ much lower than the typical frequency of sig
nificnnt surface wave effect For this reason, once
we are concerned with the don mic e' ts due to in
ternal w wes, the effects at free surface can be safely
neglected so that the free surface itself can be con
sidered as n rigid lid
The present paper deals with experimental and
theoretical investigation of the force coefi cients
(added mass and damping) for n circular cylinder
oscillating in n linearly stratified fluid of finite depth
and in n pycnocline of finite thickness
IMPULSE RESPONSE TECHNIQUE
The technique used in the present study is simi
Ian to the one described in Ermanyuk (2000) (see,
also Cummins (1962) for theoretical background and
Kerwin & Nnritn (1965) for pioneering experiment )
Foil wing this approach, we make use of experi
mental records of damped oscillations of n cylinder
(impulse response functions) in fluid and Fourier
transform the problem from time to frequency
domain in order to evaluate the frequency dependent
added mass and damping coefi cient
Let us assume that n 5: d:, pe forming small oscil
lotions in n continuously stratified fluid can be ide
Sized as n linear sy tem it is well kn wn that, once
the response of any stable linear system to n unit
impulse r(t) is known, the response of the system to
an nrbitrarJ force f (t) may be written as the convo
lotion integral
ad)= I
r(r)f(t r)dr (1)
For the particular case of harmonic excitation, the
equation of body motion in
reduces to the second order ii
tion with frequency depends
~ ?(~)]~ + N(~)~ + cow =
mins 1962)
Here, M is inertia of n body, q(h~) is
one degree of freedom
near difierentinl equal
at coefi cients (Cum
fo exp(f t) (a)
the added mass,
N(h~) is damping coefi cient, c is restoring force coef
ficient, an overdot indicates difierentintion with re
spect to time t Combining (2) and (3) and using
linearity of the system one can write the formulas
for frequency dependent coefi cients
/(hl)=Cl1 ~iCOS[9(hl)]}/hl2 M (4)
N(h~)=cR~sin[9(h~)]/h3, He= R(0) / R(h3) (5)
Here 11(0) denotes the amplitude of the frequency
response function at zero frequency As the ah ye
considerations are applicable for the unit impulse em
citation, the use of experimental records obtained for
an nrbitrarJ value of impulse necessitates the nor
mnlisation of JI(h~) by 11(0)
The above described approach can be applied to
prove the identity of Lnrsen's (1969b) and Hurled s
(1997) solutions to the problem on the vertical oscil
lotions of n circular cylinder in unbounded stratified
fluid with N = coast Consider n cylinder of dinm
eter D and the mass per unit length m floating at
the horizon of -e :-:.1 equilibrium so that m = p,S,
where pa, is the fluid density at the depth correspond
ing to the cylinder center, S = xrD2/4 is the cams
sectional area Assume that the cylinder undergoes
small harmonical vertical oscillations The restoring
force coefi cient can be evaluated from hJdrostntics
as c = gSdp/dz Then, for non dimensional added
mass and damping coefi cients, defined as
air) = / v~r)J:t rear I)
Jo Con = q/p,S, Cz = N/p,SN (6)
In the particular case of harmonic force F(fh3) = the equations (4), (5) yield
To exp(fhtt) one obtains
x(t) = foexp(fhtt)Tt(fhl) (2) ~ Q2 [ ~2] ' ~ ~2' N ( )
where the complex frequency response function
Tt(f(AJ) is defined as Fourier transform of the impulse
response fun tion
[~(f(A3)=T(.((A3) fTt6((A3)=; r(r)exp( f~r)dr
F rthermore, we can introduce the amplitude
To = [ .] + . .] ) and the phase 9 =
nrctan(1l6/1l,) of the frequency response function
As it is found -- Larsen (1969b), the time hi tory
:(t) of damped oscillations of n cylinder, which was
initially held at the vertical distance Jo from the
horizon of -e :-:.1 buoyancy and then released with
zero initial velocity, is described -- the function
h(t) = :(t)/:o = Jo(Nt) e: Jo is the Bssssl func
tion As follow from (1), the unit impulse response
function r(t) is related to the unit top response
function h (a) by time difisrsntiation r(t) = h(t)
Thus, the impulse response function of a cylinder
OCR for page 782
in uniform :, stratified fluid is r(t) = NJi(Nt) The
Fourier tm form of the function NJi(Nt) (see, e g
Ditkin & Prudnikov 1961) is
[~6 = ~/~ ~ 1
516=0, 51,= (172 1) I/2(17+V~)
(~>1)
By substituting these expressions in (7) one can
obt in
CHICO, C>=~/~ (~<1)
(8)
C=, C>=0 (~>1) (9)
what exactly coincides with Hurled s (1997) results
for z circular cylinder in similar manner, it can be
sh wn that Lzrsen's (1969b) solution for z sphere
coincidm with the one given in Lay & Lee (1981)
It is important to note that zdded mass q(h~)
and damping coed cient N(h~) are interrelated by the
Kramers Kronig relations The derivation of these
relations is presented in detail in Landau & Lit hits
(1958) The relevance of the Kramers Kronig relay
tions to the theory of ship motions was first recog
nized -- Kotik & Mangulis (1962) Foll wing Lain
dau & Lit hits (1958) we can write, in our notations,
?(~)
?(CO) = 7r`.l; N(~) 2 2 (10)
i(h)= 2h32`,/; 1~(o) ( )1 do i(0)
(11)
Here Woo) is the limit of zdded mass at hi ~ co
The above expressions are different from those ones
given in WeEzusen (1971) -- the presence of the last
term in (11) As discussed in Landau & Lit hits
(1958) this term is to be zdded when the function
q(h~) + fi(h~)/h~ has z simple pole at hi ~ O For
many problems of 5: d:, oscillations in surface waves
there is no singularity at zero frequency (see, Kotik &
Mzngulis 1962, WeEzusen 1971) However, the pole
at zero frequency does occur for Hurled s (1997) so
lotion There are reasons to believe that the similar
situation may tzLe place in other 2 D problems of
body oscillations in continuously stratified fluid
The validity of formulas (10), (11) for Hurled s
(1997) solution can be verified by the residue cal
culus TzLing into account the finite spectrum of
internal w es (12 < 1), eq (10) can be rewritten in
non dimensional form as toll ws
Cr(l:) Cr(ca = rr J o Of ,,2
Keeping in mind that ~ (hi) and ~ (hi) /h~ are even
~ uL cv, rmpectivelJ, the integml
in (12) can be represented as the integml along the
unit circle C by substituting a= cm c and
the trigonometric functions in complex for
~l; ~82 C22=
1 / (zag 2z2 + 1) do
f JO 'l + 2 (1 2172) Z2 + 1] Z
It can be easily verified that for O
is one simple pole at z = 0 inside L
When 12 = 0, two dditional poles of
der appear at the contour C in the
The residue at z = 0 is I while t
z = if ore zero Correspondingly, f:r
the zdded mass coed cient is Cal = 0 (for circular
cylinder Cr(ca) = 1) When 12 > 1, in addition
to the pole at z = 0, there ore four poles corre
sponging to the roots of the biquadratic equation
29 + 2 (I 2122) 22 + I = 0 The calculation of the
residues at these poles gives Hurled s (1997) expres
sion for the zdded mass coed cient (9) The validity
of the eq (11) which in the present problem tzLes the
form
expressing
Cz (17) = 2172 ~l; [C
r(r) 1] 82 C22 + I
can be proven in z similar manner it should be also
noted that the presence of the singularity at zero
frequency in the function q(h~) + f i(h3)/h) necessi
tatm the correction of .- ~ popular formula for
the zdded mass coed cient which is of use in the the
ore of su face wa:vm According to WeEzusen (1971),
this formula looks as
rod
/ [? (hi) ~ (CO)] do, = 0
Jo
(13)
Using the prope to that ~ (hi) is even function, the
integral in (13) can be evaluated viz residue calculus
as the integral for the function ~ (hi) + f i(h3)/h) in
the upper half plane along the closed contour which
consists of the real axis (the path of integmtion goes
around the pole at hi = 0 -- z semicircle of infiniteiJ
small mdius) and the infiniteiJ la go semicircle in
non dimensional form, for Hurled s (1997) solution
the integration Jioids
`,/; [Cal (hi) Cal (co)] dO = x/2
Similar result (i g the nonhero value of the integml
(12) (13)) may 59 expected for other 2 D problem in
OCR for page 783
continuously stratified fluid. Let us note that in 3-D
problem the Eqs. (10), (11) and (13) can be used
in their standard form (Wehausen 1971) since the
singularity at zero frequency does not occur.
EXPERIMENTAL ARRANGEMENT
The experiments were carried out in a test tank
(0.15m wide, 0.32m deep and 2m long). The scheme
of the experimental installation is shown in Fig. 1.
The damped oscillation tests were performed with
the help of a cross-shaped pendulum. A circular
cylinder of diameter D = 3.7cm was attached to the
lower end of the pendulum. The gaps between the
ends of the cylinder and the side walls of the tank
were equal to 0.5mm. The volume of the immersed
streamlined part of the pendulum was less than 1/
of the cylinder volume.
nut n screw
ball o sensors 1
at, ,, 11 1 ·-
l Yn
=
~ I I I I I I L
~ L test tack
H
1 1
11
pendulums |
cylinder
v _
L
b
P(Y)
Figure 1.
The upper part of the pendulum had a micrometric
screw with a nut of mass me = 188g. The variation
of the vertical coordinate of the nut In allowed to
change the restoring moment of the pendulum. The
distance between the point of rotation of the pendu-
lum and the center of the cylinder was b = 60cm.
As the angular deflection of the pendulum in exper-
iments did not exceed 0.5°, the horizontal oscilla-
tions of the cylinder center could be with high accu-
racy considered as rectilinear. The maximum mag-
nitude of the horizontal displacement of the cylinder
in experimental runs did not exceed 0.14D. The mo-
ment of inertia of the pendulum Io (without nut) was
measured with the accuracy of 0.5/, the measured
value being I = 1.12 x 106g cm2. The total moment
of inertia is I = Io + mrly2. Correspondingly, the
value of inertial term in the equation of the rectilin-
ear motion (3) and formulas (4), (5) is M = I/b2.
The oscillations of the pendulum were induced by
dropping a steel ball on a pre-tensioned rubber mem-
brane attached to the end of the horizontal bar of the
pendulum. The history of damped oscillations was
measured by an electrolytic sensor whose output was
sampled at 20Hz with a 12-bit analog-to-digital con-
vertor. Let us note that the sensor produced very low
liquid friction proportional to the first power of ve-
locity. The final results were corrected for this value.
The tests of damped oscillations in air demonstrated
that all other kinds of frictional losses in the sys-
tem including the friction at supports were negligi-
bly small. To evaluate the restoring force coefficient
c, a static calibration ire site was performed by load-
ing a light bowl at the end of the horizontal bar of
the pendulum with standard calibrated weights. De-
pending on the vertical coordinate of the nut, typical
full load used for calibration varied from 0.5g down
to O.Olg. The accuracy of this procedure was about
0.57. Because of the high sensitivity of the experi-
mental system, special care was taken to protect it
from mechanical vibrations and air currents. To pre-
vent the reflection of waves at the ends of the test
tank, we used two types of wave-absorbing devices.
In the case of linear stratification the wave energy of
incident internal waves was effectively dissipated by
perforated flat plates installed parallel to the end of
the test tank. In experiments with a pycnocline the
wave-energy absorber represented a 'sandwich' set of
two perforated flat plates combined with an opaque
plate inclined at small angle to horizon. The perfor-
mance of the wave absorbers proved to be sufficiently
effective.
A weak solution of glycerine (linear stratification)
or sugar (pycnocline) in water was used to produce a
prescribed density distribution in the present exper-
iments. Linear stratification was created by slowly
filling the test tank with several layers of fluid having
a prescribed density difference between the layers.
For different values of fluid depth H. the thickness
of one layer was about 1.5 . 2cm. The layered struc-
ture eventually disappeared due to diffusion so that
two days after filling the tank the density distribu-
tion was perfectly linear. The linearity of the den-
sity distribution was checked by the measurements of
conductivity by a probe calibrated over the samples
of solutions of known density. These data were used
to evaluate the Brunt-Vaisala frequency.
A smooth density profile with a pycnocline was
created by filling the test tank with two layers of mis-
cible fluids. Owing to diffusion, initial sharp interface
between the layers evolved into a smooth density pro-
file. In the coordinate system with the origin taken
at free surface and y-axis directed vertically upwards
the measured density distribution over depth fitted
the following approximation
p~y)=pO~1 - -2tanh:2˘~/~> i')~\
OCR for page 784
OCR for page 786
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OCR for page 789
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OCR for page 791
Representative terms from entire chapter:
added mass
Pa = 2 ~ e = P
where h is the depth of the upper Inter, f is the char
act ritic thickness of the pJcnocline, PI and /2 are
the fluid densities in the upper and lower Inters, re
affectively Let us note that according to theoretical
solution for the problem of did ion of n weak admix
tore (see, for example, Landau & Lit hits 1987) the
density profile is described by error function H w
ever, in Taylor serim for error function and for ho
perbolic tangent the first two terms coincide while
the third terms differ by only 25~o As result, both
functions provide good approximations to the exper
imental data H wever, from practical point of vi w,
the use of n simple anal tical function, such as ho
perbolic tangent, is more convenient in accordance
with theoretical predictions the characteristic thick
nerd of the PJ cnocline increases with time as f ~ tt/2
Owing m - -- low rat of glycerine and sugar diffusion
in water, the characteristic time scale of this gr wth
is measured -- days There was no detectable in
crease of f within few hours needed to perform n
series of experiments The experiments were per
formed at e = 0 009
For each set of experimental conditions we
recorded about n dozen of impulse response func
tions at different valum of the re toring force
coed cient c Theoretically, n single realization of
impulse response function allow to evaluate q(h~)
and N(h~) for any frequency O < hi < co in practice,
n reliable estimate of the frequency dependent
coed cients may be obtained in n certain frequency
range in the vicinity of the frequency his correspond
ing to the resonant peak of P(h~) To study the
whole frequency range of interest, it is necessary to
perform n series of experiments for n set of his and
match the results at n common plot so that the data
obt .=- d at different his overlap The variation of
his can be easy attained -- variation of the restoring
force coed cient c
THEORETICAL ANALYSIS
It is assumed that the :=-_ :d incompressible fluid
occupies the region x < co, No < y = y+EI~ < Elf
and there are three Inters: homogeneous upper and
low r ones and n linearly stratified middle one Thus,
density stratification in an undisturbed tat p(y)
takes the form
P(Y) = 1 ~ [1 sb/H2]
/2
where 11~, ET2, ET3 = ET ET2 are the depths of the
upper, middle and lower Inters, :-_ tiveiJ
The circular cy linger is situated entirely within the
middle layer and undergoes mall oscillations in the
two possible degrees of freedom (surge and he we)
with n frequency Lo The radiation and difiraction
problems for n circular cylinder located beneath n
pJcnociine in n constant density layer h we been con
ridered -- Sturova (1999) The reciprocity relations
are derived for the solutions of these problems
Assuming the perturbed oscillatory motion of fluid
P(')(x bit) = plRe[exp(ih3t)~ fpf (A A)]
)=t
where the superscript s = 1, 2, 3 denotes the values
in the upper, middle and I wer Inters, respectiveiJ
In the homogeneous fluid layers the functions pa t)
p(3) satisfy the Lnplace equation
Z\pft)=o (o
the distance between the center of the cylinder and
the upper boundary of the middle layer
Omitting the hydrostatic term, the hydrodynamic
forces F = (Fl, Fz) acting on the oscillating cylinder
can be determined by integrating the pressure over
the contour S
2
Fu = I, f rut (k = 1, 2)
)=t
rug = h~2qu: forum = Pl, J P>2)~Uds (21)
YE Nu: are the added mass and damping coed
crents, respectively
To solve the formulated problem, let us use the
method of singularities in terms of unkn wn source
distribution o>(x) over contour S. the fluid pressure
in the middle layer can be represented in the form
2)(x) = J o,(x~)G(2)(x, x~)ds
x= (x,b), X, = ( I,YI)
Here G(2)(x,x~) is the Green function in the mid
die layer for the problem under consideration This
function can be determined from the solution of the
foil wing system of equations
AG(~) = 0 (0 < y < Al) (22)
52G(2) 625 G() =47r62f(x Xl) ( 11z
obt fn the infinite svstem of linear enuation for un
.fi cients am(tm)
.o
2 ~ amAm~ =
2 m=t
27r
h,2
27r
2~ 2~
Am~= / sinmr / sin~Px
Jo Jo
/ SG(2) cosr5G(2)\
sinr— 2 — IdPdr
:~ 6 Yt J
r2~ r2~
Bm~= / cosmr / cos~Px
Jo Jo
( i SG(2) C°S75G(2))d~d
S~ 6 yt
r = zrctan[x~/(y~ + h)], f~ is the Kroneker deltz
(29)
Since the exprmsions for Am~, Bm~ zre rather
cumbersome, let us describe the basic principles of
their evaluation The integrals in the exprmsions for
the imaginarJ pz ts of Am~, Bm~ can be evaluated
znalJticallJ bJ using the foll wing tzbulated formu
las (Abram witz & Stegun 1964)
r2~
/ cos mz cos(X6 cos z) cos(xsin z)dz =
JO
~rlm cos mf Jm(q)
r2~
/ cos mzsin(X 6 cos z) cos(xsin z)dz =
JO
~rlm sin mf Jm (q)
J sin mz cos(x 6 cos z) sin(X sin z)dz =
o
~rlm cos mf Jm(q)
.,/; sin mz sin(X 6 cos z) sin(X sin z)dz =
~rlm sin mf Jm(q)
where f = zrctan 6, q = X/12, X = ak, Im = I
( I)m, Jm zre the Bersel fun tions of the first I
(31)
(32)
(33)
(34)
cind
evaluating the
Am~, Bm~ For evaluation we use well kn wn ex
pansions (Abmm witz & Stegun 1964)
cos[qsin(r + f)] = Jo(q) + 2 ~ Jzv(q) cos[2~(r+ f)]
(35)
.o
sin[qsin(r + f)] = 2 ~ Jzv+r(q) sin[(2~ + I)(r + f)]
v=o
The summation i
(36)
.... , , ;o the
number of term which zll wed to ensure the rela~
tive inaccumcJ of order 10 Z in the calculations of
integrals for each wwe mode in Am~ Bm~
The solution of the sJstems (29), (30) mzJ be trun
eating the infinite series zt z finite number of term,
which depends on the desired zccuracJ
Once the coefi cient am tm zre obtained, we can
determine zll the cEzmcteristic of the fiuid motion
In the far field, the fiuid motion represents z super
position of infinite number of w e modes Thus, for
ex mple, the pressure in the middle IzJer zt x ~ co
has the form
n (351. (36) was executed UD ~
co
i2) = ~ ~ Er (k,)K(k,, y) exp(+f k,x)
.=t
~o
P i2) = ~ Ez (k,)K(k,, y) exp(+f k,x)
Et(k) = 4~r2~6~(k) ~ amJm(q)(lmcm + I+Sm)
m=t
co
E2(k) = 4f7r2~6~(k) ~ tmJm(q)(l+Cm + ImCm)
m=t
Cm = (h, 6t~h6) cos mf, Sm
= (h6+ 6t~h,)sinmf
h, = cos k 6h, h6 = sin k 6h
The pressure in the upper znd I wer IzJers can be
determined using the form of the eigenfunctions of
the given problem (see, for example, Sturova 1999)
According to (21), the hJdrodJnamic load is
~2~ ;~,:~ =
m=t
2~ 2~
tm / stn mr / sm PG dPdr (37)
JO JO
h~2~22 fh3~22 =
.o :2~ 2~
p~a ~tmJ cosmr cosPG(2)dPdr (38)
of the hJdrodJnamic
zero
The kn wn zpproach to zpprcximate solution of
the present problem uses the expressions for am, tm
which zre borr wed from the solution for zn infinite
homogeneous fiuid: ar = tr = h32/(27r), am =
bm = 0 (m > 2) Under this assumption, the damp
ing coed cient can be determined from the imag
inarJ part of the Green fun tion (26) in nddi
tion, taking into account (31) (34), the integration
in (37), (38) can be performed anal tically For n
layer of n linearly stratified fluid bounded -- rigid
horizontal lids the npprcximate solution is given by
Gorodt ov & Teodorovich, 1986 How ver success
fund the above mentioned npprcximation has been
used in the study of the surface w wes, it cannot pro (2) X~ smam(D + ;? )
vide n consistent result for the hydrodynamic lo ding P' _, m m
on n body oscillating in n linearly stratified fluid
This fn t can be convenientiJ illustrated for uni
form y stratified fluid of infinite e tent The total
hydrodynamic lo d acting on n circular cylinder in
this case has been determined by Hurled, (1997) (see
(8), (9))
?~t = ?22 = 0, Ntt = N22 = 7rp~s
?~t = Y22
(hi < N)
= 7rpl~ ~/h3, Nt
(hi > N)
An interesting property of this solution is the fact
that the horizontal and vertical loads coincide The
npprcximate solution for diagonal damping coed
cients takes the form (Gorodt ov & Teodorovich
1986)
Ntt = 41rpt~2hd2~iN2
N22 = 47rp~2(N2 :~,2~372yN2 (IS < N
(39)
=22=0
(40)
(41)
tween (39), (40) and (41), (42)
When Lo > N. one can use the integral equation
(25) with the Green function
e cylinder
We seek the solution of the problem (14) (20) in
the form
In this case, the fluid
tion of x, while for vertical
.o
Pit) = ~ smamFm (0 < y < B ) (43)
m=t
( Bz < y < 0) (44)
Pi3) = ~ mTm ( B < y < B2)
m=t
1 rho
Fm(~ A)= ~ / km Icoshk(y B )
(45)
sin kxA(k)dk
Dm(~>y) = ( I)'x
JO km le ~u(f +~) sin kid k (y + h > 0)
l)m+l ~ km t3~u(f +~) sin kid k (y + h < 0)
;7m(~>Y) = ( 1)'`,/; km t[B(k)3]U(f+~)+
C(k)3 ~U(f+~)] sin kink
Tm(~ A)= ( I)! J km tcosEk(y+B)x
sin kxW(k)dk
The fun tions A(k), B(k), C(k), W(k) determined
from the matching conditions at the boundaries of
fluid layers (18), (19) are equal to
G(2) = 2~[1nP `.l; T(k) t)~(k)dk] ( ) cost B Z2(k)' B(k)= 2Z (k) Zi(k)e
here C(k)= ( i3)Z3(k)3~U(6 82)
~2 = (A ~1)2 + :2(y yl)2 :2 = 6Z 2Z (k)
(ok) = cosh[k:(y + Or + Bz)](l ~ tots)+
~sinh[k:(y + Or + H2)](t3 to)+
+exp( k H2)cosh[k (y Vl)](l 7tr)(l to)
T(k) = (1+~2t~t3) sinh(k H2)+~(tr+t3) cosh(k:Bz)
H wever, for circular cylinder the method of mul
tipole expansions is more effective The standard
application of this method is ferment d, for exam
pie, in Sturova, 1999 Since the solution is rather
cumbersome, we shall consider bel w the horizontal
W (k) = cosh kB Zz (k)
Zi(k) = (1+ t3)e~(82 6) ( I)m(l tt3)3~u(~ 82)
Zz (k) = ~r(tr+t3 ) cosh (~kH2 )+ (1+~2tlt3 ) sinh(:kB z)
Z3(k) = (1 :~)3 ~u~ ( 1)m(l +.~t ) ~u~
The unkn wn con tant Sm are to be determined
from the boundarJ condition (20) nt the contour S
This condition cnn be written as n sJstem of linenr
equntions bJ consecutive multiplication bJ sin ~P nnd
integrntion over P from O to 2~r The integrnls which
mericallJ
The constants Sm be
culate the pressure at
Ration (21) When hi
has real value and is c
mass
~Pt
?~t :~2
where
sing detern
contour S
> N. the hydrodynamic load
letermined only -- the added
' (m 1)l (Xm + Ym)~ ?2t = 0
Xm = 2r(m)1m `.l; (72 Coz2 P + sin2 p)m/2
sin (m arctan ) do
~ [ (I 7) (I + 7) ]
,.o
/ x ~7m(q)[B(x) + C(x)]dx
Jo
P(m) is gamma~function
Since the Boussinmq approximation is used, in
the limit case of high frequency oscillations of a
cylinder the problem reduces to the description of
the cylinder motion in the layer of constant density
and the full depth B ~ + B At G~ ~ co () ~ 1), the
used approach reduces to the standard method of
multipole expansions
NUMERICAL RESULTS
When LO < N. the investigation of the accuracy of
numerical results depending on the number of terms
Nl in expansions (27), (28) and the number of in
ternal wave modes N2 in the Green function (26) is
of considerable inter t The non dimensional added
mass MI I and damping Let coed cient for horizontal
oscillations of the cylinder at BI = B3 = 0, Bz/a =
5, F/~ = 2 are presented in Table for three values
of the non dimensional frequency 12 = 0 3; 0 6; 0 9
The toll wing notations are introduced
M)) = PZZ L>> = N. (j = 1,2)
Let us note that the non dimensional coed cients
M)) and L>> are identical with the coed cients
Cal and Cv introduced in (6), being different only
-- the definition of the reference density For a
weakly stratified fluid, this difference can be safely
neglected Also presented are the complex values
of the first three coed cients am = 27ram/hd2 it
of ~~ show that they are essentially different from
those used in the approximate solution Both real
and imaginary parts of ~~ depend on frequency
At certain frequencies, the absolute values of real
and imaginary part are close to each other The
modules of the two successive coed cients cry and
~3 h we the same order of magnitude as the
modulus of ~~
The necessity to take into account a large num
her of internal w e modes in investigation of their
propagation in the layer of linearly stratified fluid has
been emphasized -- Larsen (1969a) This e' t can
be explained -- the absence of any viscous effects in
the model considered The inclusion of viscous ef
fects into analysis of the wave motion should lead to
better convergence of the solution with the increase
of N2
The hydrodynamic loads for horizontal and verti
cal oscillations of the cylinder are sh wn in Fig 2 in
numerical calculations at LO < N we use Nl = 20
and N2 = 200 At LO > N. 15 term in the expan
sion (43) (45) are taken into account The solution
(39), (40) is sh wn in Fig 2 -- solid line it is appar
ent that the hydrodynamic loads essentially depend
on the type of oscillations what is especially notable
at I w frequencies The increase of the added mass
at I w frequency horizontal oscillations can be ex
plained by blocking With the increase of the strut
Ned layer thickness, the numerical solutions gradu
ally approach to the dependencies (39), (40) what
is most pronounced for the case of vertical oscillate
tions The hydrodynamic lo ds are also fected by
the presence of sup cientiJ deep homogeneous layers
Let us note that in the presence of homogeneous
layers the added massm for horizontal and vertical
oscillations of the cylinder practically coincide at
hi > N in all the cases presented The limit values
M)) at G~ ~ co are essentially different from I only
for B = B3 = 0, Bz/a = 5, h/n = 2 These values
MII = 1 3326 and M22 = 1 165 are sh wn by the
dash lines in Fig 2
(: ompari.on with experimental rezultz
The results of the theoretical and experimental
evaluation of the added mass and damping coed
cients of the cylinder in the linearly stratified fluid
of limited depth are shown in Figs 2, 4 for a layer
of linearly stratified fluid and in Figs 5, 6 for pJcno
cline in numerical calculations the thickness of the
middle layer H2 was taken equal to 5
N1
N2
Q
0.3
0.6
0.9
Q m
0.3 2
3
1
0.6 2
3
0.9 2
3
10
100 1 100
Ml1
0.626
-0.036
-0.079
Realm
0.981
0.005
0.037
0.581
0.004
0.065
0.547
0.014
0.015
Ll1
0.465
0.452
0.141
Imorm
-0.929
-0.002
-0.118
-0.456
0.002
0.018
-0.042
0.075
-0.156
Ml1
0.630
-0.034
-0.079
Realm
0.983
0.005
0.041
0.585
0.004
0.068
0.551
0.017
0.015
Ll1
0.463
0.453
0.142
Imorm
-0.920
-0.002
-0.106
-0.455
0.002
-0.014
-0.043
0.077
-0.154
Ml1
0.626
-0.034
-0.079
Realm
0.978
0.005
0.031
0.581
0.004
0.063
0.550
0.014
0.014
Table
20
200
Ll1
0.467
0.453
0.143
Imorm
-0.927
-0.001
-0.122
-0.453
0.002
-0.021
-0.040
0.079
-0.157
Ml1
0.624
-0.034
-0.079
Realm
0.977
0.005
0.028
0.580
0.004
0.061
0.549
0.013
0.014
1.
1
to
~r 1
=~
0.5
The solid lines in Figs.3-6 correspond to analyt-
ical solution (8), (9) for the unbounded exponen-
tially stratified fluid. In non-viscous stratified fluid
the generation of internal waves is the sole physical
mechanism of the energy dissipation. The numeri-
cal simulation gives zero damping coefficient when
Q > 1. As the depth of stratified layer decreases,
the calculated values of the damping coefficient at
Q < 1 also decrease. The physical reason for this be-
havior is the effect of the wave-guide. In particular,
when Q ~ 1, the internal waves with nearly vertical
vector of the group velocity undergo multiple reflec-
tions when travelling between the cylinder and the
bottom. As result, a certain portion of wave energy
is 'trapped' instead of being effectively radiated.
Ll1
0.468
0.453
0.143
Imorm
0.928
-0.001
-0.126
-0.452
0.003
-0.023
-0.039
0.079
-0.158
j=1 j=2 hick H1/a Hz/a H3/a |
· 2 0 5 0 1
V ~ 2 100 5 100 1
O · 25 0 50 0
Hurley (1997)
Figure 2.
Ml1
0.624
-0.034
-0.079
Realm
0.976
0.005
0.027
0.581
0.004
0.062
0.549
0.013
0.013
.5 Q 2
Ll1
0,467
0.454
0.143
Imorm
0.926
-0.001
-0.122
-0.452
0.003
-0.023
-0.039
0.079
-0.158
Ejj~
0.51
n :
0.5 Q
In experiments, dissipation of energy is due to
combination of wave and viscous effects. The con-
tribution of the wave damping to the total value of
Cx can be roughly represented as the difference be-
tween the values of Cx measured in the stratified and
homogeneous fluids at the same H/D. The exper-
iments conducted in homogeneous fluid show that
Cx increases when H/D decreases. Having this in
mind, one can note reasonably good agreement be-
tween the numerical and experimental data for suf-
ficiently large Q. Numerical results for the added
mass coefficient Cll shown in Figs.3, 5 seem to cap-
ture well the main experimentally observed effects at
Q > 1. However, at low Q the behavior of numerical
and experimental data is quite different. It seems
cat '' 1/65 CO !"- ::
1 ++-+- I've ~
00 O,0 O,5 1,0 1 5 Q 2,O
O 1 0 O 1 5 1 1 0 Q 1 1 5 Figure 6.
Figure 3.
2-
Cp
1~
o
HID
+ 1.65
~ · x 2.19
+\° ~ 3.24
\\
~ a~ AA a A
A ~ - ~~X AX _ ~ _
a'
O1 0 O1 5
115 l
Ci,
1 1°
0,0 -
Exp.(^ · o)
Num.(+ x a)
l
1 1°
Figure 4.
E~p.~.o. ~ a)
Num.(+ x i)
. . .
me.
O1 5 ~:
l l l
1 15 Q TO
dID
0~43
0 0~78
+ · 1~24
x ~ 1~62
3' ' 2~62
0 8 god ~N<~.~,~9~4 .
.
O. 0 O1 5 1 1 0 Q 1 1 5
Figure 5.
likely, that some additional theoretical analysis is to
be done in future to establish the exact asymptotic
at Q ~ 0.
In experiments, for low values of the pycnocline
thickness (5 < D), the dissipation of energy due
to internal wave radiation is low. Correspondingly,
the damping coefficient is small and Cat is close to
1 (i.e. the value it takes in homogeneous fluid). As
the thickness of the pycnocline increases, the depen-
dencies of the added mass and damping approach
asymptotically those ones corresponding to the case
of infinitely deep exponentially stratified fluid.
CONCLUSION
To our knowledge, this report presents the first
experimental and numerical investigation of the hy-
drodynamic loads acting on a 2-D body oscillating in
a continuously stratified fluid of limited depth. The
density stratification is shown to have a strong ef-
fect on the frequency-dependent hydrodynamic coef-
ficients (added mass and damping). As the thickness
of linearly stratified layer decreases, the portion of
energy radiated with internal waves also decreases.
The results obtained for stratification with a
smooth pycnocline allow to estimate the ranges of
applicability of the most popular theoretical ideal-
izations, namely, the model of two-layer fluid with
an interface and the model of exponentially strat-
ified unbounded fluid, for the description of a real
stratification observed in natural conditions.
The future research in the field will be aimed to
the investigation of a 3-D problem.
Acknowledgments:
This research has been supported by Council
"Leading Scientific Schools", grant N 00-15-96162,
Pussian Foundation of Basic P search, grant N
0001 00812, SD PAS integrate Project N 1 2000
and grant N 6 for young scientists
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