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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 the authoritative version for attribution. 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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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 the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line 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 the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line RADIATION LOADS ON A CYLINDER OSCILLATING IN PYCNOCLINE 791 Russian Foundation of Basic Research, grant N 0001–00812, SD RAS Integrate Project N 1–2000 and grant N 6 for young scientists. REFERENCES M.Abramowitz & I.A.Stegun, ‘Handbook of Mathematical Functions', NBS, 1964. W.E.Cummins, ‘The impulse response function and ship motions'. Schiffstechnik, Vol. 9, pp 101–109, 1962. V.A.Ditkin & A.P.Prudnikov, ‘Integral Transforms and Operational Calculus'. Gos. Izd. Fiz. -Mat. Lit. Moscow, 1961, (in Russian). E.V.Ermanyuk, ‘The use of impulse response functions for evaluation of added mass and damping coefficient of a circular cylinder oscillating in linearly stratified fluid', Exp. Fluids, Vol. 28, pp 152–159, 2000. E.V.Ermanyuk & I.V.Sturova, ‘Hydrodynamic loads on a body in a stratified fluid', Proc. 1st Int. Conf. Marine Industry, Varna, 1996. V.A.Gorodtsov & E.V.Teodorovich, ‘Study of internal waves in the case of rapid horizontal motion of cylinders and spheres', Fluid Dyn., Vol. 17, pp 893–898, 1982. V.A.Gorodtsov & E.V.Teodorovich, ‘Energy characteristics of harmonic internal wave generators', J. Appl. Mech. Tech. Phys., Vol. 27, pp 523–529, 1986. D.G.Hurley, ‘The generation of internal waves by vibrating elliptic cylinders. Part 1. Inviscid solution', J. Fluid Mech., Vol. 351, pp 105–118, 1997. J.E.Kerwin & H.Narita, ‘Determination of ship motion parameters by a step response technique', J. Ship Research, Vol. 9, pp 183–189, 1965. J.Kotik & V.Mangulis, ‘On the Kramers-Kronig relations for ship motions', Int. Shipbuilding Prog., Vol. 9, pp 351–368, 1962. R.Y.S.Lai & C.-M.Lee, ‘Added mass of a spheroid oscillating in a linearly stratified fluid', Int. J. Engng. Sci. Vol. 19, pp 1411–1420, 1981. L.D.Landau & E.M.Lifshitz, ‘Statistical Physics'. Addison Wesley Pub., 1958. L.D.Landau & E.M.Lifshitz, ‘Fluid Mechanics'. Butterworth-Heineman, 1987. Landau, L.D. & Lifshitz, E.M. 1987 L.H.Larsen, ‘Internal waves incident upon a knife edge barrier', Deep—Sea Res., Vol. 16, pp 411–419, 1969a. L.H.Larsen, ‘Oscillations of a neutrally buoyant sphere in a stratified fluid', Deep-Sea Research, Vol. 16, pp 587–603, 1969b. R.M.Robinson, ‘The effect of a vertical barrier on internal waves', Deep-Sea Res., Vol. 16, pp 421–429, 1969. Yu.A.Stepanyants, I.V.Sturova, E.V.Teodorovich, ‘The linear theory of the generation of the surface and internal waves', Itogi Nauki i Tekhniki. Mekhanika Zhidkosti i Gaza, Vol. 21, pp 92–179, 1987, (in Russian). I.V.Sturova, ‘Diffraction and radiation problems for the circular cylinder in stratified fluid', Fluid Dyn., Vol. 34, pp 81–94, 1999. B.R.Sutherland, S.B.Dalziel, G.O.Hughes, P.F.Linden, ‘Visualization and measurement of internal waves by ‘synthetic schlieren'. Part 1. Vertically oscillating cylinder', J. Fluid Mech., Vol. 390, pp 93–126, 1999. J.S.Turner, ‘Buoyancy Effects in Fluids'. Cambridge; Cambridge Univ. Press, 1973. B.Voisin, ‘Internal wave generation in uniformly stratified fluids. Part 1. Green's function and point sources', J. Fluid Mech., Vol. 231, pp 439–480, 1981. J.V.Wehausen, ‘The motion of floating bodies', Ann. Rev. Fluid Mech., Vol. 3, pp 237–268, 1971. J.-H.Wu, X.-H.Wu, S.-M.Li, ‘A theory of wave diffraction and radiation by a large body in stratified ocean (III) Boundary element method', J. Hydrodyn., Ser. A, Vol. 5, pp 74–80, 1990. the authoritative version for attribution.