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RADIATION LOADS ON A CYLINDER OSCILLATING IN PYCNOCLINE 780
Radiation Loads on a Cylinder Oscillating in Pycnocline
E.Ermanyuk, N.Gavrilov, I.Sturova
(Lavrentyev Institute of Hydrodynamics, Russia)
ABSTRACT
The results of experimental and theoretical studies of the radiation loads (added mass and damping) on a horizontal
circular cylinder oscillating in a stratified fluid are presented. Two cases of the density distribution over depth are
considered: a) linearly stratified fluid of limited depth, b) two layers of miscible fluids with the region of high density
gradient (pycnocline) 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 stratified and homogeneous fluids.
The linear 2-D wave radiation problem is considered theoretically both for horizontal and vertical oscillations of a
cylinder. The fluid is assumed to be non-viscous incompressible and composed of three layers (the upper and lower layers
are homogeneous and the middle layer is linearly stratified), what provides an approximate model of the experimental
conditions. The Boussinesq approximation is used. By introducing a Green function, the boundary integral equation for
the disturbance pressure is formulated. Comparison of the theoretical and experimental data is presented.
INTRODUCTION
The internal waves generated by a body oscillating in a stratified fluid have been studied in some details both
theoretically and experimentally. A comprehensive bibliography on this problem is given by Turner (1973), Stepanyants
et al (1987), Voisin (1991). Recent progress in the technique of internal wave visualization is described by Sutherland et
al (1999). However, it should be emphasized that only few authors have been concerned with evaluation of hydrodynamic
loads acting on a body in a stratified fluid despite the fact that this problem has an engineering counterpart relevant to the
prediction of the low-speed and low-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 by Ermanyuk & Sturova (1996).
Let us note that continuous density distribution over depth which is normally observed in nature causes some
specific properties of internal waves. The fundamental characteristic of a stratified fluid with density distribution ρ(y) is
the buoyancy (Brunt-Vaisala) frequency defined as
where g is the gravity acceleration and y-axis is directed vertically upwards. The maximum Brunt-Vaisala frequency
for a given density distribution over depth defines the upper cutoff frequency for internal wave effects. Thus, in contrast
with infinite frequency spectrum of surface waves, the frequency spectrum of internal waves is finite. Moreover, when the
oscillation frequency of a body is lower, equal or higher than the buoyancy frequency, the equations of fluid motion are of
hyperbolic, parabolic or elliptic type, respectively, what, once again, is in contrast with free-surface problems which are
described by the Laplace equation. The major part of the studies on the body oscillations in continuously stratified fluid is
performed within the model of ideal, uniformly stratified (the Brunt-Vaisala frequency is assumed to be constant),
Boussinesq fluid of infinite extent. In particular, the time-domain analysis of damped oscillations of a sphere and circular
cylinder is given in Larsen (1969b) and the solutions in frequency-domain have been given by Lay & Lee (1981) for
vertical oscillations of a spheroid and by Hurley (1997) for arbitrary directed oscillations of an elliptic cylinder. Mention
should be made of the study performed by Gorodtsov & Teodorovich (1986). To evaluate the power radiated with internal
waves by an oscillating body, they have used the Green function approach while the body geometry has been modelled by
the distribution of singularities borrowed from the solution of the pertinent problem in homogeneous fluid. Their results
are found to be in disagreement with the results by Hurley (1997) and Lay & Lee (1981). However, the disagreement is
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entirely caused by the form of the surface source distribution. For a correct form of the source, the formulas presented in
Gorodtsov & Teodorovich (1986) will give a correct estimate. The results presented in Hurley (1997) have been
experimentally confirmed

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RADIATION LOADS ON A CYLINDER OSCILLATING IN PYCNOCLINE 781
by Ermanyuk (2000) for the horizontal oscillations of a circular cylinder.
Let us note that, strictly speaking, oscillations of a body in a stratified fluid can excite surface waves. However, the
maximum buoyancy frequency is normally much lower than the typical frequency of significant surface wave effects. For
this reason, once we are concerned with the dynamic effects due to internal waves, the effects at free surface can be safely
neglected so that the free surface itself can be considered as a rigid lid.
The present paper deals with experimental and theoretical investigation of the force coefficients (added mass and
damping) for a circular cylinder oscillating in a linearly stratified fluid of finite depth and in a pycnocline of finite
thickness.
IMPULSE RESPONSE TECHNIQUE
The technique used in the present study is similar to the one described in Ermanyuk (2000) (see, also Cummins
(1962) for theoretical background and Kerwin & Narita (1965) for pioneering experiments). Following this approach, we
make use of experimental records of damped oscillations of a 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 coefficient.
Let us assume that a body performing small oscillations in a continuously stratified fluid can be idealized as a linear
system. It is well known that, once the response of any stable linear system to a unit impulse r(t) is known, the response
of the system to an arbitrary force f(t) may be written as the convolution integral
(1)
In the particular case of harmonic force F(iω)=f0 exp(iωt) one obtains
(2)
where the complex frequency response function R(iω) is defined as Fourier transform of the impulse response function
Furthermore, we can introduce the amplitude |R|=([Rc]2+[Rs]2)1/2 and the phase θ=arctan (Rs/Rc) of the frequency
response function. For the particular case of harmonic excitation, the equation of body motion in one degree of freedom
reduces to the second-order linear differential equation with frequency-dependent coefficients (Cummins 1962)
(3)
Here, M is inertia of a body, µ (ω) is the added mass, λ(ω) is damping coefficient, c is restoring force coefficient, an
overdot indicates differentiation with respect to time t. Combining (2) and (3) and using linearity of the system one can
write the formulas for frequency-dependent coefficients
(4)
(5)
Here |R(0)| denotes the amplitude of the frequency response function at zero frequency. As the above considerations
are applicable for the unit impulse excitation, the use of experimental records obtained for an arbitrary value of impulse
necessitates the normalization of |R(ω)| by |R(0)|.
The above-described approach can be applied to prove the identity of Larsen's (1969b) and Hurley's (1997) solutions
to the problem on the vertical oscillations of a circular cylinder in unbounded stratified fluid with N=const. Consider a
cylinder of diameter D and the mass per unit length m floating at the horizon of neutral equilibrium so that m=ρcS, where
ρc is the fluid density at the depth corresponding to the cylinder center, S=πD2/4 is the cross-sectional area. Assume that
the cylinder undergoes small harmonical vertical oscillations. The restoring force coefficient can be evaluated from
hydrostatics as c=gSdρ/dz. Then, for non-dimensional added mass and damping coefficients, defined as
(6)
the equations (4), (5) yield
(7)
As it is found by Larsen (1969b), the time-history ζ(t) of damped oscillations of a cylinder, which was initially held
at the vertical distance ζ0 from the horizon of neutral buoyancy and then released with zero initial velocity, is described by
the function h(t)=ζ(t)/ζ0=J0(Nt) where J0 is the Bessel function. As follows from (1), the unit impulse response function r
(t) is related to the unit step response function h(t) by time-differentiation r(t)=h(t). Thus, the impulse response function
of a cylinder
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RADIATION LOADS ON A CYLINDER OSCILLATING IN PYCNOCLINE 782
in uniformly stratified fluid is r(t)=NJ1(Nt). The Fourier-transform of the function NJ1(Nt) (see, e.g. Ditkin & Prudnikov
1961) is
By substituting these expressions in (7) one can obtain
(8)
(9)
what exactly coincides with Hurley's (1997) results for a circular cylinder. In similar manner, it can be shown that
Larsen's (1969b) solution for a sphere coincides with the one given in Lay & Lee (1981).
It is important to note that added mass µ (ω) and damping coefficient λ(ω) are interrelated by the Kramers-Kronig
relations. The derivation of these relations is presented in detail in Landau & Lifshitz (1958). The relevance of the
Kramers-Kronig relations to the theory of ship motions was first recognized by Kotik & Mangulis (1962). Following
Landau & Lifshitz (1958) we can write, in our notations,
(10)
(11)
Here µ (∞) is the limit of added mass at ω → ∞. The above expressions are different from those ones given in
Wehausen (1971) by the presence of the last term in (11). As discussed in Landau & Lifshitz (1958) this term is to be
added when the function µ (ω)+iλ(ω)/ω has a simple pole at ω → 0. For many problems of body oscillations in surface
waves there is no singularity at zero frequency (see, Kotik & Mangulis 1962, Wehausen 1971). However, the pole at zero
frequency does occur for Hurley's (1997) solution. There are reasons to believe that the similar situation may take place in
other 2-D problems of body oscillations in continuously stratified fluid.
The validity of formulas (10), (11) for Hurley's (1997) solution can be verified by the residue calculus. Taking into
account the finite spectrum of internal waves (Ω≤1), eq. (10) can be rewritten in non-dimensional form as follows
(12)
Keeping in mind that µ (ω) and λ(ω)/ω are even and uneven functions of ω, respectively, the integral in (12) can be
represented as the integral along the unit circle C by substituting ξ=cos α and expressing the trigonometric functions in
complex form
It can be easily verified that for 0<Ω≤1 there is one simple pole at z=0 inside the unit circle. When Ω=0, two
additional poles of the second order appear at the contour C in the points z=±i. The residue at z=0 is 1 while the residues
at z=±i are zero. Correspondingly, for 0≤Ω≤1 the added mass coefficient is Cµ≡0 (for circular cylinder Cµ(∞)=1). When
Ω>1, in addition to the pole at z=0, there are four poles corresponding to the roots of the biquadratic equation z4+2 (1–
2Ω2) z2+1=0. The calculation of the residues at these poles gives Hurley's (1997) expression for the added mass
coefficient (9). The validity of the eq. (11) which in the present problem takes the form
can be proven in a similar manner. It should be also noted that the presence of the singularity at zero frequency in the
function µ (ω)+iλ(ω)/ω necessitates the correction of another popular formula for the added mass coefficient which is of
use in the theory of surface waves. According to Wehausen (1971), this formula looks as
(13)
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Using the property that µ (ω) is even function, the integral in (13) can be evaluated via residue calculus as the integral
for the function µ (ω)+iλ(ω)/ω in the upper half-plane along the closed contour which consists of the real axis (the path of
integration goes around the pole at ω=0 by a semicircle of infinitely small radius) and the infinitely large semicircle. In
non-dimensional form, for Hurley's (1997) solution the integration yields
Similar result (i.e. the nonzero value of the integral (13)) may be expected for other 2-D problems in

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RADIATION LOADS ON A CYLINDER OSCILLATING IN PYCNOCLINE 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.
Figure 1.
The upper part of the pendulum had a micrometric screw with a nut of mass mn=188g. The variation of the vertical
coordinate of the nut yn allowed to change the restoring moment of the pendulum. The distance between the point of
rotation of the pendulum and the center of the cylinder was b=60cm. As the angular deflection of the pendulum in
experiments did not exceed 0.5°, the horizontal oscillations of the cylinder center could be with high accuracy considered
as rectilinear. The maximum magnitude of the horizontal displacement of the cylinder in experimental runs did not
exceed 0.14D. The moment of inertia of the pendulum I0 (without nut) was measured with the accuracy of 0.5%, the
measured value being I=1.12×106g cm2. The total moment of inertia is Correspondingly, the value of
inertial term in the equation of the rectilinear 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 membrane 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 convertor. Let us note that the sensor produced very low liquid friction proportional to the
first power of velocity. 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 system including the friction at supports were negligibly small. To evaluate
the restoring force coefficient c, a static calibration in situ was performed by loading a light bowl at the end of the
horizontal bar of the pendulum with standard calibrated weights. Depending on the vertical coordinate of the nut, typical
full load used for calibration varied from 0.5g down to 0.01g. The accuracy of this procedure was about 0.5%. Because of
the high sensitivity of the experimental system, special care was taken to protect it from mechanical vibrations and air
currents. To prevent 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
performance 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 experiments. 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 structure eventually disappeared due to diffusion so that two days
after filling the tank the density distribution was perfectly linear. The linearity of the density 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 miscible fluids.
Owing to diffusion, initial sharp interface between the layers evolved into a smooth density profile. 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
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RADIATION LOADS ON A CYLINDER OSCILLATING IN PYCNOCLINE 784
where h is the depth of the upper layer, δ is the characteristic thickness of the pycnocline, ρ1 and ρ2 are the fluid
densities in the upper and lower layers, respectively. Let us note that according to theoretical solution for the problem of
diffusion of a weak admixture (see, for example, Landau & Lifshitz 1987) the density profile is described by error
function. However, in Taylor series for error function and for hyperbolic tangent the first two terms coincide while the
third terms differ by only 25%. As result, both functions provide good approximations to the experimental data. However,
from practical point of view, the use of a simple analytical function, such as hyperbolic tangent, is more convenient. In
accordance with theoretical predictions the characteristic thickness of the pycnocline increases with time as δ~t1/2. Owing
to the low rate of glycerine and sugar diffusion in water, the characteristic time-scale of this growth is measured by days.
There was no detectable increase of δ within few hours needed to perform a series of experiments. The experiments were
performed at ε=0.009.
For each set of experimental conditions we recorded about a dozen of impulse response functions at different values
of the restoring force coefficient c. Theoretically, a single realization of impulse response function allows to evaluate µ
(ω) and λ(ω) for any frequency 0≤ω≤∞. In practice, a reliable estimate of the frequency-dependent coefficients may be
obtained in a certain frequency range in the vicinity of the frequency corresponding to the resonant peak of |R(ω)|. To
study the whole frequency range of interest, it is necessary to perform a series of experiments for a set of and match
the results at a common plot so that the data obtained at different overlap. The variation of can be easy attained by
variation of the restoring force coefficient c.
THEORETICAL ANALYSIS
It is assumed that the inviscid incompressible fluid occupies the region and there
are three layers: homogeneous upper and lower ones and a linearly stratified middle one. Thus, density stratification in an
undisturbed state takes the form
where H1, H2, H3=H−H2 are the depths of the upper, middle and lower layers, respectively.
The circular cylinder is situated entirely within the middle layer and undergoes small oscillations in the two possible
degrees of freedom (surge and heave) with a frequency ω. The radiation and diffraction problems for a circular cylinder
located beneath a pycnocline in a constant-density layer have been considered by Sturova (1999). The reciprocity
relations are derived for the solutions of these problems.
Assuming the perturbed oscillatory motion of fluid to be steady, the pressure can be written as follows
where the superscript s=1, 2, 3 denotes the values in the upper, middle and lower layers, respectively. In the
homogeneous fluid layers the functions satisfy the Laplace equation
(14)
(15)
Within the Boussinesq approximation the pressure in the middle layer satisfies the equation
(16)
where The impermeability conditions are imposed on the upper and lower boundaries of fluid
(17)
while the fluid motions at the interfaces between the layers should meet the matching conditions
(18)
(19)
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where β2=N2/ω2−1. In the far field the radiation condition is to be satisfied.
At the contour located in the middle layer we should pose the impermeability condition
(20)
where n=(n1, n2) is the inner normal vector at the contour S, a=D/2 is the radius of the cylinder, h is

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RADIATION LOADS ON A CYLINDER OSCILLATING IN PYCNOCLINE 785
the distance between the center of the cylinder and the upper boundary of the middle layer.
Omitting the hydrostatic term, the hydrodynamic forces F=(F1, F2) acting on the oscillating cylinder can be
determined by integrating the pressure over the contour S
where
(21)
µ kj λkj are the added mass and damping coefficients, respectively.
To solve the formulated problem, let us use the method of singularities. In terms of unknown source distribution σj
(x) over contour S, the fluid pressure in the middle layer can be represented in the form
Here G(2) (x, x1) is the Green function in the middle layer for the problem under consideration. This function can be
determined from the solution of the following system of equations
(22)
(23)
(24)
with the boundary conditions identical to (17)–(19) and the radiation condition in the far field, δ is Dirac's delta.
Using Green's theorem, the impermeability condition at the cylinder surface (20) yields the integral equation
(25)
where the symbols pv denote the principal value of the integral. Previously, the boundary integral equation has been
obtained by Gorodtsov & Teodorovich (1982) for uniformly stratified fluid and by Wu et al (1990) for an arbitrary stable
stratification without the Boussinesq approximation.
In the present problem, the principal properties of the solution essentially depend on the frequency of the cylinder
oscillations.
When ω0), equations (16), (23) are of hyperbolic type. Accordingly, the oscillations of the cylinder generate
internal waves. When ω>N (β2<0), for all real values of ω the mentioned equations are elliptic; the generation of internal
waves does not occur. In what follows we consider these cases separately.
At ω

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RADIATION LOADS ON A CYLINDER OSCILLATING IN PYCNOCLINE 786
obtain the infinite system of linear equation for unknown coefficients am(bm)
where
(29)
(30)
τ=arctan[x1/(y1+h)], δ1n is the Kroneker delta.
Since the expressions for Amn, Bmn are rather cumbersome, let us describe the basic principles of their evaluation.
The integrals in the expressions for the imaginary parts of Amn, Bmn can be evaluated analytically by using the following
tabulated formulas (Abramowitz & Stegun 1964)
(33)
(31)
(34)
(32)
where q= /Ω, =ak, Jm are the Bessel functions of the first kind of order m.
The main difficulties arise when evaluating the integrals in the expressions for the real parts of Amn, Bmn. For
evaluation we use well-known expansions (Abramowitz & Stegun 1964)
(36)
(35)
The summation in (35), (36) was executed up to the number of terms which allowed to ensure the relative inaccuracy
of order 10−5 in the calculations of integrals for each wave mode in Amn Bmn.
The solution of the systems (29), (30) may be truncating the infinite series at a finite number of terms, which
depends on the desired accuracy.
Once the coefficients am bm are obtained, we can determine all the characteristic of the fluid motion. In the far field,
the fluid motion represents a superposition of infinite number of wave modes. Thus, for example, the pressure in the
middle layer at x → ∞ has the form
where
The pressure in the upper and lower layers can be determined using the form of the eigenfunctions of the given
problem (see, for example, Sturova 1999).
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According to (21), the hydrodynamic load is
(38)
(37)
The non-diagonal coefficients of the hydrodynamic load on a circular cylinder are zero.
The known approach to approximate solution of the present problem uses the expressions for am, bm which are
borrowed from the solution for an infinite homogeneous fluid: a1=b1=−ω2/(2π), am=

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RADIATION LOADS ON A CYLINDER OSCILLATING IN PYCNOCLINE 787
bm=0 (m≥2). Under this assumption, the damping coefficients can be determined from the imaginary part of the Green
function (26). In addition, taking into account (31)–(34), the integration in (37), (38) can be performed analytically. For a
layer of a linearly stratified fluid bounded by rigid horizontal lids the approximate solution is given by Gorodtsov &
Teodorovich, 1986. However successfully the above-mentioned approximation has been used in the study of the surface
waves, it cannot provide a consistent result for the hydrodynamic loading on a body oscillating in a linearly stratified
fluid. This fact can be conveniently illustrated for uniformly stratified fluid of infinite extent. The total hydrodynamic
load acting on a circular cylinder in this case has been determined by Hurley, (1997) (see (8), (9))
(39)
(40)
An interesting property of this solution is the fact that the horizontal and vertical loads coincide. The approximate
solution for diagonal damping coefficients takes the form (Gorodtsov & Teodorovich 1986)
(41)
(42)
Apparently, there is an essential disagreement between (39), (40) and (41), (42).
When ω>N, one can use the integral equation (25) with the Green function
where
However, for circular cylinder the method of multipole expansions is more effective. The standard application of this
method is presented, for example, in Sturova, 1999. Since the solution is rather cumbersome, we shall consider below the
horizontal oscillations of the cylinder. In this case, the fluid pressure is uneven function of x, while for vertical
oscillations it is even function.
We seek the solution of the problem (14)–(20) in the form
(43)
(44)
(45)
where
The functions A(k), B(k), C(k), W(k) determined from the matching conditions at the boundaries of fluid layers (18),
(19) are equal to
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The unknown constants sm are to be determined from the boundary condition (20) at the contour S. This condition
can be written as a system of linear equations by consecutive multiplication by sin nθ and integration over θ from 0 to 2π.
The integrals which

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RADIATION LOADS ON A CYLINDER OSCILLATING IN PYCNOCLINE 788
cannot be integrated analytically are evaluated numerically.
The constants sm being determined, one can calculate the pressure at contour S and execute integration (21). When
ω>N, the hydrodynamic load has real value and is determined only by the added mass
where
Γ(m) is gamma-function.
Since the Boussinesq 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 H1+H. At ω
→ ∞ (γ → 1), the used approach reduces to the standard method of multipole expansions.
NUMERICAL RESULTS
When ω

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RADIATION LOADS ON A CYLINDER OSCILLATING IN PYCNOCLINE 789
Table
N1 10 20 30
N2 100 100 200 300 300
Ω M11 L11 M11 L11 M11 L11 M11 L11 M11 L11
0.3 0.626 0.465 0.630 0.463 0.626 0.467 0.624 0.468 0.624 0,467
0.6 −0.036 0.452 −0.034 0.453 −0.034 0.453 −0.034 0.453 −0.034 0.454
0.9 −0.079 0.141 −0.079 0.142 −0.079 0.143 −0.079 0.143 −0.079 0.143
Ω Reαm Imαm Reαm Imαm Reαm Imαm Reαm Imαm Reαm Imαm
m
0.3 1 0.981 −0.929 0.983 −0.920 0.978 −0.927 0.977 −0.928 0.976 −0.926
2 0.005 −0.002 0.005 −0.002 0.005 −0.001 0.005 −0.001 0.005 −0.001
3 0.037 −0.118 0.041 −0.106 0.031 −0.122 0.028 −0.126 0.027 −0.122
0.6 1 0.581 −0.456 0.585 −0.455 0.581 −0.453 0.580 −0.452 0.581 −0.452
2 0.004 0.002 0.004 0.002 0.004 0.002 0.004 0.003 0.004 0.003
3 0.065 −0.018 0.068 −0.014 0.063 −0.021 0.061 −0.023 0.062 −0.023
0.9 1 0.547 −0.042 0.551 −0.043 0.550 −0.040 0.549 −0.039 0.549 −0.039
2 0.014 0.075 0.017 0.077 0.014 0.079 0.013 0.079 0.013 0.079
3 0.015 −0.156 0.015 −0.154 0.014 −0.157 0.014 −0.158 0.013 −0.158
Figure 2.
The solid lines in Figs. 3–6 correspond to analytical solution (8), (9) for the unbounded exponentially stratified fluid.
In non-viscous stratified fluid the generation of internal waves is the sole physical mechanism of the energy dissipation.
The numerical simulation gives zero damping coefficient when Ω>1. As the depth of stratified layer decreases, the
calculated values of the damping coefficient at Ω<1 also decrease. The physical reason for this behavior is the effect of
the wave-guide. In particular, when Ω → 1, the internal waves with nearly vertical vector of the group velocity undergo
multiple reflections when travelling between the cylinder and the bottom. As result, a certain portion of wave energy is
‘trapped' instead of being effectively radiated.
In experiments, dissipation of energy is due to combination of wave and viscous effects. The contribution of the
wave damping to the total value of Cλ can be roughly represented as the difference between the values of Cλ measured in
the stratified and homogeneous fluids at the same H/D. The experiments conducted in homogeneous fluid show that Cλ
increases when H/D decreases. Having this in mind, one can note reasonably good agreement between the numerical and
experimental data for sufficiently large Ω. Numerical results for the added mass coefficient Cµ shown in Figs. 3, 5 seem
to capture well the main experimentally observed effects at Ω>1. However, at low Ω the behavior of numerical and
experimental data is quite different. It seems
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RADIATION LOADS ON A CYLINDER OSCILLATING IN PYCNOCLINE 790
likely, that some additional theoretical analysis is to be done in future to establish the exact asymptotic at Ω → 0.
Figure 3.
Figure 4.
Figure 5.
Figure 6.
In experiments, for low values of the pycnocline thickness (δ

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Russian Foundation of Basic Research, grant N 0001–00812, SD RAS Integrate Project N 1–2000 and grant N 6 for
young scientists.
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