Received: 15 September 1998/Accepted: 1 April 1999

E. V. ErmanyukLavrentyev Institute of HydrodynamicsSiberian Division of Russian Academy of ScienceAv. Lavrentyev 15, Novosibirsk, 630090, RussiaE-mail: ermanyuk@hydro.nsc.ru

The present study has been supported by the grant for young scientistsof Siberian Division of Russian Academy of Science. The authorexpress thanks to Prof. A. A. Korobkin and to Prof. I. V. Sturova forvaluable discussions and comments. Thanks are due to Dr. V. A.Kostomakha and Dr. N. V. Gavrilov for their help in experiments anddata processing. Further, the author express thanks to the reviewersfor helpful and contructive comments. The manuscript of this paperwas prepared during the authors stay as a visiting researcher at theResearch Institute for Applied Mechanics, Kyushu University. Thefriendly help and guidance of Prof. M. Ohkusu and Prof. M. Kashiwagiis gratefully acknowledged.

Experiments in Fluids 28 (2000) 152159 ( Springer-Verlag 2000

The use of impulse response functions for evaluation of added mass anddamping coefficient of a circular cylinder oscillating in linearly stratified fluid

E. V. Ermanyuk

Abstract The damped horizontal oscillations of a circularcylinder in linearly stratified fluid are studied experimentally.The cylinder is fixed to the lower end of a physical pendulumwith variable restoring moment. The impulse response func-tion of the pendulum in time domain is recorded andconverted to the frequency response function using Fouriertransform. The density stratification is shown to have astrong effect on frequency-dependent hydrodynamiccoefficients (added mass and damping). The data obtainedare compared with available theoretical predictions. Theapplicability of a simplicistic method implying approximationof impulse response functions by analytical functions isdiscussed.

List of symbolst timeu frequencyo fluid densityg gravity accelerationDo density variation over depthH depth of fluidD diameter of the cylinderN BruntVaisala frequencyr(t) impulse response functionDR(u) D amplitude of frequency response functionh(u) phase of frequency response function

k added massj damping coefficient

1IntroductionInternal waves generated by an oscillating body in a densitystratified fluid have been a topic of much interest during pastthree decades (see, Mowbray and Rarity 1967; Hurley 1969;Appleby and Crighton 1986; Ivanov 1989; Makarov et al. 1990;Hurley 1997; Hurley and Keady 1997). The cases of simplebody geometry (circular and elliptical cylinders) and exponen-tial (at small vertical scales linear) density distribution overdepth are the most commonly studied ones. As a result of thesestudies, the details of the spatial structure of internal wavesemitted by harmonic oscillators are quite well understood.However, the integral quantities (such as added mass anddamping coefficient) characterizing the fluid-body system asa mechanical oscillator have been considered only by Hurley(1997) (inviscid case) and Hurley and Keady (1997) (approx-imate viscous solution). When the fluid is assumed to beinviscid, the energy dissipation is associated solely with theradiation of internal waves by an oscillating body. Hurley(1997) reveals some curious features of fluid-body interactionin this case.

He found that for the frequency range below theBruntVaisala frequency, added mass is zero and damping isnon-zero, and, conversely, for frequencies higher than theBruntVaisala frequency, added mass is non-zero and damp-ing is zero. Moreover, the dependencies of dynamic coefficientson frequency are of universal character for any direction ofbody oscillation. The study presented in Hurley (1997) hasbeen generalized by Hurley and Keady (1997) to take viscouseffects into account. They argue that for sufficiently largeReynolds numbers (which are normally encountered inexperiments), the inviscid solutions do hold approximately forthe viscous case. However, when the frequency of oscillations islower (higher) than the BruntVaisala frequency, the value ofadded mass (damping) in viscous fluid is expected to benon-zero. Note that no experimental confirmation of theabove-mentioned results has been reported in literature so far.

The goal of the present study is to evaluate experimentallyadded mass and damping coefficient for a circular cylinderoscillating horizontally in a linearly stratified fluid. To do this,we use impulse response function. This basic tool is wellknown in different fields of physics. It is known that forany linear system its response to a unit impulse in the timedomain is related by Fourier transform with the response to

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Fig. 1. Scheme of the experimental installation

harmonical forcing in frequency domain. This idea serves asa background of the present study.

The fundamentals concerning this approach as applied tothe theory of transient ship motions in surface waves have beenthoroughly considered by Cummins (1962) (for an overview,see Wehausen 1971). A distinctive physical feature of thisproblem is the memory effect which is evident from thesimple observation that wave disturbance generated by themotion of a body at any time instance persists in future andtherefore acts on the further character of the body motion.Mathematically, the memory effect is taken into account bythe convolution integral introduced into the description of thebody motion in time domain. The importance of the mem-ory effect has been emphasized in early theoretical study bySretenskii (1937). Sretenskii studied damped oscillations ofa body floating on a free surface. The calculated decay of freeoscillations was found to have much more complicatedcharacter than that of a simple damped linear oscillator.

Physically, the wave motion in a stratified fluid has much incommon with surface waves. For a particular type of stratifica-tion (two-fluid system with interface between layers) theapproach used by Sretenskii is further developed by Akulenkoand Nesterov (1987) and Akulenko et al. (1988). The analysis isrestricted to the case of vertical oscillations of thin 2D bodies.The results of these theoretical studies are partially supportedby quantitative measurements of damped oscillations of a bodyat the interface of miscible fluids reported by Pylnev andRazumeenko (1991). The analysis of experimental data in theirpaper was based on the approximation of experimental curvesby simple analytical functions.

Larsen (1969) presents a theoretical and experimental studyof damped oscillations of a sphere initially displaced from theequilibrium position in a linearly stratified fluid. Theory andexperiments are found to be in good agreement. In particular,Larsen argues that the observed damping can be explainedentirely by the radiation of energy by internal waves, while theviscous effects are practically negligible. However, no quantit-ative measurement of frequency dependent dynamic coeffi-cients was presented in his study.

It should be noted that there exists an extensive literatureon the experimental evaluation of inertial and damping forcecoefficients in the case of harmonical oscillations of a circularcylinder in homogeneous fluid. These coefficients areshown to be dependent on two non-dimensional parameters:KeuleganCarpenter number and Stokes number (see, Sar-pkaya 1986). In the present study, we are concerned with theinfluence of stratification which is governed by another basicparameter, namely, the ratio between the frequency ofoscillations of a cylinder and the BruntVaisala frequency.This ratio has the physical sense of the Froude number.

2Experiments

2.1Experimental installationThe experiments were carried out in a test tank [0.14 m wide,0.32 m deep and 1 m long]. A scheme of the experimentalinstallation is shown in Fig. 1. The test tank 1 was filled withlinearly stratified fluid to the depth H\0.28 m. The total

variation of density over depth was Do\0.022 g/cm3. Thecorresponding BruntVaisala frequency was N\J[gdo/dz\0.88 rad/s. The linearity of the density distribution waschecked by the conductivity probe 2.

A weak solution of glycerine in water was used to create thedensity stratification. Let us make a brief note on the physicalproperties of such a solution. Detailed reference tables showingthe dependence of the glycerinewater solutions dynamicviscosity on concentration and temperature may be found inVargaftik (1963). It is known that the dynamic viscosity ofa weak glycerinewater solution varies almost linearly withconcentration. In the present experiments, the volume concen-tration of the glycerine in the solution gradually increasedfrom zero at the free surface up to 8.8% close to the bottom ofthe test tank. Correspondingly, the dynamic viscosity variedfrom 1.14]10~3 kg/(m s) up to 1.45]10~3 kg/(m s). Here, thefirst value corresponds to the dynamic viscosity of pure waterat the temperature T\15C which was fairly constant (towithin ^1C) throughout the experiments. It is interesting tonote that for the weak glycerinewater solution the relativedrop of the dynamic viscosity Dg/g due to a slight increase oftemperature DT in the vicinity of T+15C appears to beindependent from concentration, being equal to 2.4% per 1C.The same value of Dg/g takes place in pure water.

The diameter D of the circular cylinder 3, which was fixed atthe lower streamlined end of the physical pendulum 4, wasequal to 3.7 cm. The centre of the cylinder was submerged tothe depth 0.5H. The volume of the immersed streamlined partof the pendulum was less than 1% of the cylinder volume sothat its influence on the fluid-body interaction may be safelyneglected. The pendulum had two wedge-shaped supports ofsteel, 5. Each wedge contacted a horizontal cylinder made ofsteel and oriented normally to the rib of the wedge. Two pointsof contact defined the axis of pendulum rotation. This designallowed to reduce the friction at the supports to a minimum.

The pendulum had a micrometric screw 6 with a nut 7. Thevariation of vertical distance z0 between the gravity center ofthe nut and the axis of pendulum rotation allowed the restoringmoment of the pendulum to be changed. As result, thefrequency of damped oscillations could be varied. The horizon-tal beam of the pendulum was equipped with a pre-tensionedrubber membrane 8. The impulse of force needed to excite freeoscillations was introduced by dropping a steel ball which was

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initially held by an electric magnet 9. As the characteristicperiod of free oscillations of the pendulum in experiments wasof the order of several seconds, the impact of a ball on themembrane produced practically ideal impulse loading. Toensure a single impulse, the initial position of a ball was chosenso that it hit the membrane slightly off-center and then jumpedaway.

The motion of the pendulum was measured by a displace-ment sensor 10. The sensor was made of a resistive wave gauge.A wave gauge having two vertical wires was placed in a smallvessel containing a conductive liquid. A light shield made ofnon-conductive material and connected to the horizontal beamof the pendulum by a thin needle was free to move between thewires. Static and dynamic performance of the sensor was testedand proved to be both linear and practically instant. The dragat the shield was evaluated from the records of oscillations ofthe pendulum in air and found to be negligibly small. Theadvantage of the described design is the absence of mechanicalCoulombs friction. Additionally, it eliminates the need for anyelectric feeders connected to the moving parts of the installa-tion. These precautions were taken as the present experimentsimply the measurement of motions governed by very smallforces. Let us also note that the diameter of the needleconnecting the shield to the pendulum was minimized in orderto avoid the effects due to surface tension. At this point it ispertinent to add some comments concerning the effects of thelubrication flow in the gaps between the ends of the cylinderand the walls of the test tank. The area of the cylindercross-section was approximately equal to the area of the shieldused in the displacement sensor. The thickness of the oscilla-

tory boundary layer at the shield is of order d\Jl/u (see, forexample, Hurley and Keady (1997)). In the studied frequencyrange, this value is comparable with the width of the gapsbetween the ends of the cylinder and the walls of the test tankwhich were equal to 0.5 mm. Consequently, we can assume thatthe drag due to the lubrication flow in the gaps is comparablewith the drag at the shield. Therefore, it can be safely neglected.

The ends of the test tank were equipped with the wavebreakers 11. Each wave breaker consisted of two perforated flatplates spaced at 2 cm from each other. The plate placed at theweather side was perforated with bigger orifices of about 3 cmin diameter while the second plate had smaller orifices of about1.5 cm in diameter. The performance of the wave breaker wasfound to be very effective.

The geometrical and dynamical parameters of the pendulumwere chosen at the design stage so as to ensure a sufficientlywide range of the oscillation frequencies. The actual valueswere carefully measured. The moment of inertia of thependulum about the axis of rotation is J0\1.12]106 g cm2 withan accuracy of 0.5%. The total moment of inertia (includingnut 7 of mass m\188 g) is J\J0]mz20 . The mass of thependulum (without nut) is M\706 g.

To measure the pendulums restoring moment coefficient,a static calibration in situ was performed. To do this, thependulum was loaded with standard calibrated weights placedin a light bowl 12 suspended at distance 60 cm from the axis ofrotation of the pendulum. The angle of static inclination wasmeasured by the displacement sensor. A typical load used forcalibration varied between 0.5 g for small z0 (about 3 cm)and 0.01 g for large z0 (the pendulum was close to neutral

equilibrium at z0+17 cm). At the second stage of calibration,a known angle deflection was applied to the pendulum bymeans of a micrometric mechanism 13. The mechanismallowed to produce a known vertical displacement with anaccuracy of 0.01 mm. The displacement was applied ata distance of 60 cm from the axis of rotation of the pendulum.The measurement of the restoring moment coefficient wasperformed for all values of z0 used in the experiments. Theaccuracy of this procedure was about 0.5%. The studied rangeof the restoring moment coefficient is between 9]10~3 and0.3 Nm/rad.

It should be noted that the angular inclination of thependulum did not exceed 0.5 in the experiments or 0.7during static calibration. The distance between the axis ofrotation of the pendulum and the centre of the cylinder wasb\60 cm. Thus, the linear horizontal displacement of thecylinder center in experimental runs was less than 0.14 D.

To complete the description of the experimental installation,it should be noted that it was located in a place protected fromair currents and vibrations in order to minimize any distur-bance which could affect such a sensitive system.

2.2Mathematical modelThe mathematical model of the problem can be readilyformulated following Cummins (1962).

For any stable linear system, if r(t), the response to a unitimpulse, is known, the response of the system to an arbitraryforce f (t) is

x(t)\=:0

r(q) f (t[q) dq (1)

Let us assume that the dynamic system is exposed to the actionof a harmonic force

F(iu)\f0e*ut (2)

Substituting Eq. (2) in Eq. (1) one obtains

x(t)\f0e*utR(iu) (3)

where the complex frequency response function R(iu) isdefined as Fourier transform of the impulse response function

R(iu)\=:0

r(q)e~*uq dq

The complex frequency response function can be separatedinto real and imaginary parts as R (iu)\Rc(iu)[iRs(iu),where

Rc\=:0

r(q) cos uq dq

Rs\=:0

r(q) sin uq dq

Furthermore, one can introduce the amplitude DR D\J[Rc(u)]2][R s(u)]2 and the phase h\arctan(Rs/Rc) of thefrequency response function. Let us note that the last repres-entation seems to be more convenient for an experimentalistsince the physical sense of the data obtained can be checked. Inparticular, when the energy is radiated from the system, thephase angle must fall in the range between 0 and n.

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Assuming that the angle u of pendulum inclination is small(u\bx, where x is horizontal displacement of the center of thecylinder), one can write the equation of the horizontal motionof the cylinder center in the frequency domain as follows:

(J/b2]k(u))x(]j(u)xR ]cx\f0e*ut (4)

Here, k(u) is the added mass of the cylinder, j(u) is thedamping coefficient, c is restoring force coefficient, overdotdenote the differentiation with respect to time. Let us note thatc is a linear function of z0 . This property was used to check thestatic calibration data. Combining Eqs. (3) and (4), and usingthe linearity of the system, one can write the expressions foradded mass and damping coefficient

k(u)\cu2 A1[

DR(0) DDR(u) D

cos(h(u))B[J

b2(5)

j(u)\cu

DR(0) DDR(u) D

sin(h(u)) (6)

Here DR(0) D denotes the amplitude of the frequency responsefunction at zero frequency. Once the system is proved to belinear, the normalization of DR(u) D by DR(0) D allows us to useFourier transforms of the experimental impulse responsefunctions at any value of the impulse loading. Thus, there is noneed to perform any direct measurement of the actual value ofthe impulse.

Theoretically, the above-mentioned procedure allows us toevaluate the dependencies k(u) and j(u) for 0\u\R fromone record of damped oscillations. In practice, the reliableestimates may be obtained within a finite frequency range inthe vicinity of the resonant frequency u*, where u* corres-ponds to the maximum of DR(u) D . In the present problem, theevaluation of this frequency range is of special interest as onecan expect a quite different physical behavior of the hydro-dynamic coefficients at the frequencies when damping isconditioned by wave or viscous phenomena.

Along with the method described above, we also applya simple approach to the evaluation of hydrodynamic coeffi-cients using the least squares approximation of the experi-mental impulse response function by an exponentiallydecaying sine function. Mathematically, the method is basedon the minimization of the integral

=:0

(r(t)[ra(t, A, k, X))2 dt, (7)

where ra(t, A, k, X)\Ae~kt sin(Xt); the values A, k, X areunknown variables. Once there variables are determined, then,using a rough assumption that Eq. (4) does hold true fortransient responses, we can determine the added mass anddamping coefficient as follows:

k(X)\c

X2]k2[

Jb2

(8)

j(X)\2kc

X2]k2(9)

The choice of the approximating function in Eq. (7) is,strictly speaking, arbitrary. There is no rigorous theoreticalproof that an exponentially decaying sine function may serveas a meaningful approximation for an actual response function

of a complicated dynamical system with memory. Moreover,it is rather well known that the law of damped oscillationsof a wave-emitting body has different asymptotics at smalland large time-scales (see, Huskind 1947; Wehausen 1971;Akulenko et al. 1988). For this reason, it is not evident that theapproximations of the experimental records of dampedoscillations by simple analytical functions can give reliableestimates of dynamically important frequency-dependentparameters such as added mass and damping. However, withsome variations, this technique has been widely used by manyauthors (see, for example, Pylnev and Razumeenko 1991) as aneffective tool for data analysis. The advantage of this method isthat the comparison of experimental records with theirapproximations gives an explicit illustration of the qualitativedifference between the response of a simple linear dampedoscillator and the response of a complicated system understudy. A quantitative comparison between the results obtainedfollowing the two methods mentioned is one of the goals of thepresent study.

2.3Data aquisition system and data analysisAn IBM computer with sampling software and an 8-bit A/Dconvertor was used to record the time histories of dampedoscillations of the pendulum (impulse response functions). Thesampling frequency was set at 12 Hz. Although the resolutionof the A/D conversion was relatively low, the position of thecylinder centre could be measured with the resolution of0.02 mm what proved to be sufficient for the purposes of thepresent study. The recording system was initiated before theaction of the impulse. For the initial condition of the timehistory (system at rest) the value of the pendulum inclinationwas set zero. Then this part of the time history was cut awayand the standard algorithm of the fast Fourier transform wasapplied to the impulse response function. In order to geta reasonably good frequency resolution, the history of themotion was recorded during a time interval of about 1520periods of damped oscillations. The data analysis was conduc-ted according to expressions given by formulas (5)(8) withthe static calibration data used as input for the value of therestoring force coefficient c. Finally, the plots of k and j versusu (or the value of these coefficients at the frequency X, whenthe method of least square approximations was in use) wereobtained and the experiment was repeated at another value ofthe restoring force coefficient.

Let us make a brief note concerning the experimentalevaluation of DR(0) D introduced in Sect. 2.2. The algorithm ofthe fast Fourier transform applied to the digitally representedcurve r(t) yields DR D and h for a set of discrete values ui. ForiP1 the values DR(ui) D represent a smooth curve havinga resonant peak at a certain frequency uk\u* . The valueDR(0) D corresponds to zero frequency ui/0\0. However, anexperimental record r(t) normally contains a small additiveterm due to zero drift or, simply, due to the discrete nature ofanalog-to-digital conversion. As a result, the value DR(0) D fallsout of a smooth curve DR(ui) D . To evaluate DR(0) D , onecan make an extrapolation. In the present study we takeDR(0) D+DR(u1) D as the first approximation. Practice showsthat this is sufficient. The unaccuracy introduced by thisestimate is noticeable only in the immediate vicinity of u1 . So,

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for the final dependencies k(ui) and j(ui) the points corre-sponding to the lowest frequencies, where i\0. . . 3, were nottaken into account.

2.4Dimensionless parametersThe dynamic interaction of an oscillating circular cylinder witha linearly stratified fluid depends on the physical properties ofthe fluid, the law of the body motion and the geometric setup ofthe problem. For a cylinder oscillating harmonically with theamplitude a, it is convenient to introduce the following set ofthe most important non-dimensional parameters: a/D theKeuleganCarpenter number, DJu/l the Stokes number,u/N the normalized frequency which represents a version ofthe Froude number.

In the present study, we consider only the case of lowKeuleganCarpenter numbers (a/D\0.14). The tests havedemonstrated that the magnitude of the experimental impulseresponse functions is related linearly to the magnitude of theimpulse excitation. Correspondingly, the estimated values ofthe added mass and damping coefficient do not depend on a/Dboth for homogeneous and stratified fluid to within theaccuracy of the experiments.

The fact that the spatial structure of internal waves depends

both on DJu/l and u/N is clear from the theoreticalconsiderations and experimental data discussed in Makarovet al. (1990) and in Hurley and Keady (1997). However, forthe range of parameters studied in the present experiments,we can expect that the effects predicted for the inviscid casein Hurley (1997) are dominant. The laboratory study ofdamped oscillations of a sphere in a linearly stratifiedfluid presented in Larsen (1969) also suggests thedominant role of the inviscid scenario. Therefore, thenormalized frequency u/N is considered in the present studyas the basic dynamically important parameter. The dimension-less added mass and damping coefficients are introduced asfollows

Ck\k/o0W

Cj\j/o0WN (10)

where o0 is the fluid density at the depth corresponding to thecylinder center (in experiments o0\1.011 g/cm3) and W is thevolume of the submerged cylinder.

For homogeneous fluids, the fluid-body interactionis governed by viscous effects; the actual value of theBruntVaisala frequency is zero what makes normalization(10) inappropriate. In that case, we use normalization (10)formally by adopting the value of N which took place in theexperiments with the stratified fluid. Thus, the denominatoro0WN is common both for stratified and homogeneousfluid what is convenient for the explicit quantitative com-parison.

Mention should be also made of the nondimensional ratioD/H characterizing the geometry of the problem. In the presentexperiments D/H\0.132. This value is considered sufficientlysmall to match, in practical terms, the assumption of infinitefluid depth adopted in Hurley (1997) and Hurley and Keady(1997). Some comments concerning the influence of thelimited fluid depth are given in Sect. 2.5.

2.5ResultsAs one of the major objectives of the present study is tocompare the dynamics of body oscillations in stratified andhomogeneous fluid, a series of experiments in homogeneousfluid (water) has been conducted. The experimental responsefunctions for this case are found to have very slight deviationfrom an exponentially decaying sine function. The memoryeffect is weak. The frequency-dependence is detectable only forthe damping coefficient, while the added mass coefficient isconstant being equal to the theoretical value for an infiniteideal fluid (Ck\1) to within the experimental accuracy. Thecomplete quantitative results and discussion for this series arepresented below along with the data obtained for the stratifiedfluid.

Two distinct types of the impulse response functions wereobserved in the experiments with the stratified fluid. Toillustrate the difference between the dynamic system con-sidered and a simple linear damped oscillator, the experi-mental records (solid lines) are shown in Figs. 2 and 3 alongwith the least square approximations of these curves byexponentially decaying sine functions (dashed lines). Asthe experimental system is found to be linear, the responsefunctions r(t) and their approximations ra(t) are normalizedby the maximum magnitude of the experimental responser.!9

for each test and plotted versus nondimensional timetN/2n.

When the frequency u* corresponding to the peak value ofthe Fourier image of the impulse response function is below theBruntVaisala frequency N (i.e. u*/N\1), the damping isessentially conditioned by the radiation of internal waves.A typical experimental record for this case is shown in Fig. 2for z0\15 cm, c\0.154 N/m. As it is easy to see from thisfigure, there is an important difference between the behavior ofthe experimental system and the behavior of a simple lineardamped oscillator. The time interval between the momentscorresponding to ra(t)\0 is constant. In contrast, the experi-mental response r(t) develops over time in such a way that thesucceeding intervals between the moments corresponding tor(t)\0 gradually decrease, what implies a strong dependenceof k and j on frequency. Qualitatively, one can readily say thatk(R)[k(0) as the inertial properties of such a system at t]0and t]R are governed by the added mass coefficientscorresponding to u]R and u]0, respectively (see, Huskind1947). The variation of the characteristic time taken by eachcycle of the damped oscillations implies that the cylindermotion generates the wave field which consists of a widespectrum of elementary wave disturbances. As a result, theimpulse response function recorded at u*/N\1 contains veryrich information on the wave phenomena in a wide range ofnon-dimensional frequencies u/N. In other words, the systemhas long memory in that case.

Let us note that for very small values of the restoring forcecoefficient c\0.022 N/m a critical damping is observed, i.e.the pendulum, being disturbed, reaches a certain maximuminclination and then approaches its equilibrium positionmonotoneously. The logarithmic plot of such a curve showsthat for t]R the disturbance decays as t~1.

For u*/N[1 the damping is mainly conditioned by viscouseffects. However, the presence of stratification affects the

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Fig. 2. Example of experimental impulse response function atu

*/N\1 (z

0\15 cm, c\0.154 N/m, u

*/N\0.74)

Fig. 3. Example of experimental impulse response function atu

*/N[1 (z

0\10 cm, c\0.394 N/m, u

*/N\1.11)

Fig. 4. Non-dimensional added mass coefficient (Ck vs. u/N)

Fig. 5. Non-dimensional damping coefficient (Cj vs. u/N)

initial part of the time history. A typical record of dampedoscillations for this case is shown in Fig. 3, for z0\10 cm,c\0.394 N/m. The difference between the experimental curve(solid line) and its least squares approximation (dashed line) isquite appreciable only for the magnitude of the first oscillation.

It is well known that the harmonic oscillations of a body inthe linearly stratified fluid do not produce propagating internalwaves when u/N[1. However, the process of damped oscilla-tions is both non-stationary and time-dependent. As a result,a certain portion of the energy is radiated by internal waveseven in the case when u*/N[1. The comparison of the curvesshown in Fig. 3 suggests that the effective generation of a wavedisturbance, which serves as an additional damping factor,takes place only during the first period of damped oscillations.The difference between the curves decreases with increasingu*/N; it practically vanishes when u*/NP2. The overallcharacter of the impulse response function shown in Fig. 3implies that it contains mainly the information about theviscous effects while the information on the effects due tostratification is mainly confined within the first cycle ofoscillations. In this case the experimental system has a shortmemory.

The dependencies Ck(u/N) and Cj(u/N) obtained from theexperimental records of impulse response functions according

to the methods presented in Sect. 2.2 are shown in Figs. 4 and 5,respectively, along with theoretical curves by Hurley (1997)

(for u/N\1, Ck\0 and Cj\J1[(u/N)2, for u/N[1, Cj\0and Ck\J1[(N/u)2. The values obtained in the experimentswith homogeneous fluid are marked by crosses. Other symbolscorrespond to the tests conducted at different values of u*/Nas specified in Table 1. The theoretical value of the nondimen-sional added mass coefficient for ideal unbounded homogene-ous fluid (Ck\1) is plotted in Fig. 4 by a dashed line. Theresults obtained in the experiments with homogeneous fluidagree qualitatively with the asymptotical solution for a cylinder

oscillating in viscous fluid at a/D@1 and DJu/lA1 (see,Landau and Lifshitz 1959). The asymptotic theory predictsCk\1 and j\u0.5. A quantitative disagreement betweenasymptotic theory and experiments for j(u) may be attributedto the relatively low value of DJu/l in the experiments(25\DJu/l\40). The effects of the lubrication flow in thegaps between the tank walls and the cylinder ends are believedto be negligibly small as discussed in Sect. 2.1.

In the case of the stratified fluid, the qualitative andquantitative agreement between theoretical predictions by

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Table 1. Legend for Figs. 4 and 5

u*

/N Symbol

Stratified fluid 0.57 s0.74 n0.85 0.93 e1.11 d1.4 m

Method of least square approximations jHomogeneous water ]

Fig. 6. Non-dimensional amplitudes of frequency response functionsfor experimental response (solid line) and its approximation (dashline) ( DR(u/N) D/ DR(u

*/N) D and DR

a(u/N) D/ DR(u

*/N) D). Test

conditions correspond to Fig. 2

Hurley (1997) and the present experiments is found to be verygood. The comparison of the data obtained in the experimentswith stratified and homogeneous fluid shows that the presenceof stratification can drastically increase the damping ofoscillations and reduce the added mass at u/N\1. It isinteresting to note also that for u/N[1 the damping coefficientCj takes somewhat lower values than in the case of homogene-ous water. However, it should be kept in mind that these effects

are dependent both on u/N and DJu/l. A significantvariation of the second parameter can essentially change thecharacter of the fluid-body interaction (see, Hurley and Keady1997).

A careful examination shows that the tests conducted atu*/N+0.8 give more reliable information within a widerfrequency range than the other tests. As one can derive frommultiplying Eq. (4) by xR , integrating and taking an average overone period of oscillation, the mean power radiated withinternal waves is related to the damping coefficient by

P\12 ja2u2 (11)

Substituting Hurleys (1997) result for j, we obtainP\12oWa2u2JN2[u2. For the constant amplitude ofoscillation, the last expression takes a maximum value at

u/N\J2/3+0.8. In the experiments, for the maximumradiated power we obtain a maximum information about thesystem in the frequency domain.

The results of the analysis performed with the use of leastsquare approximations are also shown in Figs. 4, 5 by blacksquares. As one can see, the results provided by two differentmethods are in good agreement. For the experiments conduc-ted at u*/N\1 this fact seems surprising because the impulseresponse function and its approximation in this frequencyrange show quite different behaviors as illustrated in Fig. 2. Itis interesting to compare the Fourier transforms of thesecurves. For the curves shown in Fig. 2 the correspondingnon-dimensional dependencies DR(u/N) D/DR(u*/N) D andDRa(u/N) D/DR(u*/N) D for the experimental response functionand its approximation are shown in Fig. 6 by solid and dashedlines, respectively. The curves are different in details as theadded mass and damping coefficient of the actual physicalsystem are frequency-dependent while the Fourier-image of theexponentially decaying sine function reflects the propertiesof the approximating system having constant dynamic para-meters. In spite of this important difference, the curvesDR(u/N) D/DR(u*/N) D and DRa(u/N) D/ DR(u*/N) D have commonasymptotics for u/N]0 and u/N]R; the location and

magnitude of the resonant peaks also coincide. Analogousbehavior is observed for other samples of the experimentalresponse functions. This fact may be of interest for theoreticalstudies of the systems having memory effects.

An interesting property of Hurleys (1997) result is that thedependencies Ck(u/N) and Cj(u/N) are the same for anydirection of oscillations of a cylinder. The present experimentsare focussed on the study of the horizonal oscillations.However, there exists a non-direct proof that the dependenciesCk(u/N) and Cj(u/N) are the same for any direction ofoscillations. The internal wave pattern of St. Andrew cross forpure horizontal or vertical oscillations of a cylinder in linearlystratified fluid is well known (see Mowbray and Rarity 1967;Hurley 1969; Appleby and Crighton 1986; Ivanov 1989;Makarov et al. 1990). The characteristic feature of this patternis that the waves propagate within four stripes which areinclined at the angle a\arcsin(u/N) to the horizontal.Gavrilov and Ermanyuk (1997) present an experimental studyof internal waves generated by a circular cylinder whose centerdescribes a circular path of small radius. Such a motion may berepresented as a sum of horizontal and vertical oscillations ofequal amplitude shifted in phase by 90. It is shown that in thiscase the waves propagate within only two stripes. For theclockwise direction of the cylinder motion the internal wavesare confined within a band which goes through the first and thethird quadrants of the Cartesian coordinate system; the originof the system is taken in the center of the cylinder trajectory. Itmeans that the wave fields produced by vertical and horizontaloscillations are practically identical so that they can canceleach other or be summed up depending on their relative phaseshifts in the different regions of space. It is known that forpurely horizontal or vertical oscillations exactly the quarter oftotal energy is radiated along each stripe of the wave pattern.This fact is evident from the symmetry of the problem. Thus,the cancellation of waves within two stripes observed inGavrilov and Ermanyuk (1997) implies that the total energyradiated with waves is identical for vertical and horizontaldirections of oscillations. The identity of the damping coeffi-cients follows from Eq. (11). Let us note that the dampingcoefficient j(u) is related with the value k(u)[k(R) by the

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Kramers-Kronig relation (see, Kotik and Mangulis 1962;Wehausen 1971). For u]R the stratified fluid acts asa homogeneous one. Consequently, for an unbounded strat-ified fluid the value k(R) is identical for any direction ofoscillations. Therefore, the dependencies k(u) and j(u) areuniversal for the vertical and horizontal directions of oscilla-tions. Moreover, as far as the linear effects are concerned, theperturbation introduced by the cylinder oscillations of ampli-tude a at the angle to the horizon t can be represented asa sum of perturbations due to horizontal and vertical oscilla-tions with the amplitudes a cos t and a sin t, respectively.Correspondingly, we can conclude that the total energyradiated with waves (which depends on the amplitudesquared) is the same for any direction of oscillations asa2(cos2 t]sin2t)\a2. Finally, the above discussion impliesthat the values of the dynamic coefficients k(u) and j(u) donot depend on the direction of the cylinder oscillations in theunbounded stratified fluid with N\const.

Let us make a brief note concerning the influence of thelimited fluid depth. In the present experiments the ratioD/H\0.132 is sufficiently small so that the effect of flowcontraction on the characteristics measured in the homogene-ous fluid is negligible. However, in the stratified fluid internalwave rays emitted by the oscillating cylinder at u/NO1 can bereflected by free surface and bottom and fall back on thecylinder (reflection at the butt-ends of the test tank wascancelled by the wave breakers). A simple consideration ofinternal wave ray geometry shows that this phenomena takes

place when cos aOD/H, what implies u/NPJ1[(D/H)2. Forthe present experimental conditions it means that the effectof limited depth is very weak as it may manifest itself only inthe immediate vicinity of the BruntVaisala frequency when0.99Ou/NO1.

Experiments are under way to study the influence of thelimited depth on the characteristics of a body oscillating ina stratified fluid.

3SummaryThe damped horizontal oscillations of a circular cylindersubmerged in stratified fluid with a linear density profile arestudied experimentally. The experimental time-histories ofresponses to an impulse excitation (impulse response func-tions) are analysed by Fourier-transforming the problem fromtime- to frequency-domain. It is observed that the impulseresponse functions have different behavior depending on theratio between the frequency of oscillations and BruntVaisalafrequency. When the time-history of damped oscillations isgoverned by the wave phenomena, the memory effects arestrong so that the evaluation of frequency-dependent dynamiccoefficients can, in principle, be performed based on a singleexperimental realization of the impulse response function.

The measurements demonstrate large effects of stratificationon the added mass and damping coefficients. The results ofexperiments confirm theoretical results by Hurley (1997). Forthe frequency of oscillations below the BruntVaisala fre-quency the damping is much greater than in homogeneousfluid, being primarily conditioned by the radiation of internalwaves. The added mass of a circular cylinder reduces to verylow values and amounts to a few percent of its value for

homogeneous fluid. As the frequency of oscillations increasesabove the BruntVaisala frequency, the values of added massand damping coefficient asymptotically approach the valuestypical for homogeneous fluid.

A simple analysis of experimental records implying the useof least square approximations of the impulse responsefunctions by exponentially decaying sine functions is tested.The results are found to be in good agreement with a strict(from the theoretical point of view) analysis using the Fouriertransform. It is shown that the Fourier images of the impulseresponse functions and their approximations have commonpeak values and common asymptotics at u]0 and u]Rwhile being different in details.

ReferencesAkulenko LD; Nesterov SV (1987) Oscillations of a rigid body at

interface of two fluid. Izvestiya Akademii Nauk SSSR, MekhanikaTverdogo Tela, No. 5: 3440 (in Russian)

Akulenko LD; Mikhailov SA; Nesterov SV; Chaikovsky AA (1988)Numerical-analytical investigation of rigid body oscillations oninterface of two fluid. Izvestiya Akademii Nauk SSSR, MekhanikaTverdogo Tela, No. 4: 5966 (in Russian)

Appleby JC; Crighton DG (1986) Non-Boussinesq effects in thediffraction of internal waves from an oscillating cylinder. QuartJ Mech Math 39: 209231

Cummins WE (1962) The impulse response function and shipmotions. Schiffstechnik 9: 101109

Gavrilov NV; Ermanyuk EV (1997) Internal waves generated bycircular translational motion of a cylinder in a linearly stratifiedfluid. J Appl Mech Tech Phys 38: 224227

Hurley DG (1969) Emission of internal waves of vibrating cylinders.J Fluid Mech 36: 657672

Hurley DG (1997) The generation of internal waves by vibrating ellipticcylinders. Part 1. Inviscid solution. J Fluid Mech 351: 105118

Hurley DG; Keady G (1997) The generation of internal waves byvibrating elliptic cylinders. Part 2. Approximate viscous solution.J Fluid Mech 351: 119138

Huskind MD (1947) Methods of hydrodynamics in the problems ofship seakeeping in waves. Trudy TSAGI, No. 603: 174 (in Russian)

Ivanov AV (1989) Internal waves generation by an oscillation source.Izvestiya Akademii Nauk SSSR, Fizika Atmosfery I Okeana, 25:8489 (in Russian)

Kotik J; Mangulis V (1962) On the KramersKronig relations for shipmotions. Int Shipbuilding Prog 9: 351368

Landau LD; Lifshitz EM (1959) Fluid mechanics. Oxford: PergamonLarsen LH (1969) Oscillations of a neutrally buoyant sphere in

a stratified fluid. Deep-Sea Res 16: 587603Makarov SA; Nekhlyudov VI; Chashechkin Yu D (1990) Space

structure of two-dimensional monochromatic internal wave paketsin exponentially-stratified fluid. Izvestiya Akademii Nauk SSSR,Fizika Atmosfery I Okeana, 26: 744754 (in Russian)

Mowbray DE; Rarity BSH (1967) A theoretical and experimentalinvestigation of the phase configuration of internal waves of smallamplitude in a density stratified fluid. J Fluid Mech 28: 116

Pylnev Yu V; Razumeenko Yu V (1991) Investigation of dampedoscillations for a float of a special shape deeply imbedded intohomogeneous and stratified fluid. Izvestiya Akademii Nauk SSSR,Mekhanika Tverdogo Tela, No. 4: 7179 (in Russian)

Sarpkaya T (1986) Force on a circular cylinder in viscous oscillatoryflow at low Keulegan-Carpenter numbers. J Fluid Mech 165: 6171

Sretenskii LN (1937) On the damping of vertical oscillations of thegravity center of floating bodies. Trudy TSAGI, No. 330: 112 (inRussian)

Vargaftik NB (1963) Handbook on the thermophysical properties ofgases and liquids. Moscow: Gos. Izdat. Fiz-mat.lit (in Russian)

Wehausen JV (1971) The motion of floating bodies. Ann Rev FluidMech 3: 237268

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