Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

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 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 Ï(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

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 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(iÏ)=f0 exp(iÏt) one obtains (2) where the complex frequency response function R(iÏ) 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 Î¸=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, Âµ (Ï) is the added mass, Î»(Ï) 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(Ï)| 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=ÏcS, where Ïc is the fluid density at the depth corresponding to the cylinder center, S=ÏD2/4 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=gSdÏ/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 Î¶(t) of damped oscillations of a cylinder, which was initially held at the vertical distance Î¶0 from the horizon of neutral buoyancy and then released with zero initial velocity, is described by the function h(t)=Î¶(t)/Î¶0=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 the authoritative version for attribution.

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 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 Âµ (Ï) and damping coefficient Î»(Ï) 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 Âµ (â) is the limit of added mass at Ï â â. 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 Âµ (Ï)+iÎ»(Ï)/Ï has a simple pole at Ï â 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 (â¦â¤1), eq. (10) can be rewritten in non-dimensional form as follows (12) Keeping in mind that Âµ (Ï) and Î»(Ï)/Ï are even and uneven functions of Ï, respectively, the integral in (12) can be represented as the integral along the unit circle C by substituting Î¾=cos Î± and expressing the trigonometric functions in complex form It can be easily verified that for 0<â¦â¤1 there is one simple pole at z=0 inside the unit circle. When â¦=0, two additional poles of the second order appear at the contour C in the points z=Â±i. The residue at z=0 is 1 while the residues at z=Â±i are zero. Correspondingly, for 0â¤â¦â¤1 the added mass coefficient is CÂµâ¡0 (for circular cylinder CÂµ(â)=1). When â¦>1, in addition to the pole at z=0, there are four poles corresponding to the roots of the biquadratic equation z4+2 (1â 2â¦2) 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 Âµ (Ï)+iÎ»(Ï)/Ï 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) the authoritative version for attribution. Using the property that Âµ (Ï) is even function, the integral in (13) can be evaluated via residue calculus as the integral for the function Âµ (Ï)+iÎ»(Ï)/Ï in the upper half-plane along the closed contour which consists of the real axis (the path of integration goes around the pole at Ï=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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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 0.5Â°, 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=1.12Ã106g 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 âsandwich' 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 1.5Ã·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 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) Âµ kj Î»kj 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 Ïj (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)â(19) and the radiation condition in the far field, Î´ 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 Ï<N (Î²2>0), equations (16), (23) are of hyperbolic type. Accordingly, the oscillations of the cylinder generate internal waves. When Ï>N (Î²2<0), for all real values of Ï the mentioned equations are elliptic; the generation of internal waves does not occur. In what follows we consider these cases separately. At Ï<N the Green function in the middle layer can be written in the form where (26) the prime denotes the differentiation with respect to k. The values kl (k1<k2<â¦) given by real-valued positive roots of the equation D(k)=0 represent the wave numbers of internal waves. In the general case at l â â kl â lÏ/(Î²H2). In the form analogous to (26) the Green function has been obtained by Robinson (1969) in the study on the scattering of internal waves at vertical barrier in a channel filled with linearly stratified fluid. The source distribution Ïj(x) should be formulated in terms of continuous functions since the use of point the authoritative version for attribution. singularities leads to the paradox of infinite losses of energy (Gorodtsov & Teodorovich 1986). Introducing the polar coordinate system r, Î¸ with the origin taken in the center of the contour S, let us formulate the following representation for the functions Ïj: (27) (28) for horizontal oscillations and for vertical oscillations. Substituting these relations in Eq. (25), sequentially multiplying it by sin nÎ¸(cos nÎ¸) and integrating over S (note that for a circular cylinder n1=âsin Î¸, n2=âcos Î¸), we

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 786 obtain the infinite system of linear equation for unknown coefficients am(bm) where (29) (30) Ï=arctan[x1/(y1+h)], Î´1n 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= /â¦, =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 10â5 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 â â 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). the authoritative version for attribution. 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=âÏ2/(2Ï), am=

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 787 bm=0 (mâ¥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)â(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 Ï>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)â(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 the authoritative version for attribution. 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 nÎ¸ and integration over Î¸ from 0 to 2Ï. The integrals which

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 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 Ï>N, the hydrodynamic load has real value and is determined only by the added mass where Î(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 Ï â â (Î³ â 1), the used approach reduces to the standard method of multipole expansions. NUMERICAL RESULTS When Ï<N, the investigation of the accuracy of numerical results depending on the number of terms N1 in expansions (27), (28) and the number of internal wave modes N2 in the Green function (26) is of considerable interest. The non-dimensional added mass M11 and damping L11 coefficients for horizontal oscillations of the cylinder at H1=H3=0, H2/ a= 5, h/a=2 are presented in Table for three values of the non-dimensional frequency â¦=0.3; 0.6; 0.9. The following notations are introduced Let us note that the non-dimensional coefficients Mjj and Ljj are identical with the coefficients CÂµ and Cv introduced in (6), being different only by 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 coefficients Î±m=â2Ïam/Ï2. It can be seen that the convergence of the results with the increase of N1 and N2 is good. The analysis of Î±1 shows that they are essentially different from those used in the approximate solution. Both real and imaginary parts of Î±1 depend on frequency. At certain frequencies, the absolute values of real and imaginary parts are close to each other. The modules of the two successive coefficients |Î±2| and |Î±3| have the same order of magnitude as the modulus of |Î±1|. The necessity to take into account a large number of internal wave modes in investigation of their propagation in the layer of linearly stratified fluid has been emphasised by Larsen (1969a). This effect can be explained by the absence of any viscous effects in the model considered. The inclusion of viscous effects 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 vertical oscillations of the cylinder are shown in Fig. 2. In numerical calculations at Ï<N we use N1=20 and N2=200. At Ï>N, 15 terms in the expansion (43)â(45) are taken into account. The solution (39), (40) is shown in Fig. 2 by solid line. It is apparent that the hydrodynamic loads essentially depend on the type of oscillations what is especially notable at low frequencies. The increase of the added mass at low-frequency horizontal oscillations can be explained by blocking. With the increase of the stratified layer thickness, the numerical solutions gradually approach to the dependencies (39), (40) what is most pronounced for the case of vertical oscillations. The hydrodynamic loads are also affected by the presence of sufficiently deep homogeneous layers. Let us note that in the presence of homogeneous layers the added masses for horizontal and vertical oscillations of the cylinder practically coincide at Ï>N in all the cases presented. The limit values Mjj at Ï â â are essentially different from 1 only for H1=H3=0, H2/a=5, h/a=2. These values M11=1.3326 and M22=1.165 are shown by the dash lines in Fig. 2. Comparison with experimental results The results of the theoretical and experimental evaluation of the added mass and damping coefficients 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 pycnocline. In numerical calculations the thickness of the middle layer H2 was taken equal to Î´. the authoritative version for attribution.

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 â¦ 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 â0.036 0.452 â0.034 0.453 â0.034 0.453 â0.034 0.453 â0.034 0.454 0.9 â0.079 0.141 â0.079 0.142 â0.079 0.143 â0.079 0.143 â0.079 0.143 â¦ ReÎ±m ImÎ±m ReÎ±m ImÎ±m ReÎ±m ImÎ±m ReÎ±m ImÎ±m ReÎ±m ImÎ±m m 0.3 1 0.981 â0.929 0.983 â0.920 0.978 â0.927 0.977 â0.928 0.976 â0.926 2 0.005 â0.002 0.005 â0.002 0.005 â0.001 0.005 â0.001 0.005 â0.001 3 0.037 â0.118 0.041 â0.106 0.031 â0.122 0.028 â0.126 0.027 â0.122 0.6 1 0.581 â0.456 0.585 â0.455 0.581 â0.453 0.580 â0.452 0.581 â0.452 2 0.004 0.002 0.004 0.002 0.004 0.002 0.004 0.003 0.004 0.003 3 0.065 â0.018 0.068 â0.014 0.063 â0.021 0.061 â0.023 0.062 â0.023 0.9 1 0.547 â0.042 0.551 â0.043 0.550 â0.040 0.549 â0.039 0.549 â0.039 2 0.014 0.075 0.017 0.077 0.014 0.079 0.013 0.079 0.013 0.079 3 0.015 â0.156 0.015 â0.154 0.014 â0.157 0.014 â0.158 0.013 â0.158 Figure 2. The solid lines in Figs. 3â6 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 â¦>1. As the depth of stratified layer decreases, the calculated values of the damping coefficient at â¦<1 also decrease. The physical reason for this behavior is the effect of the wave-guide. In particular, when â¦ â 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 âtrapped' 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 CÎ» can be roughly represented as the difference between the values of CÎ» measured in the stratified and homogeneous fluids at the same H/D. The experiments conducted in homogeneous fluid show that CÎ» increases when H/D decreases. Having this in mind, one can note reasonably good agreement between the numerical and experimental data for sufficiently large â¦. Numerical results for the added mass coefficient CÂµ shown in Figs. 3, 5 seem to capture well the main experimentally observed effects at â¦>1. However, at low â¦ the behavior of numerical and experimental data is quite different. It seems the authoritative version for attribution.

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 790 likely, that some additional theoretical analysis is to be done in future to establish the exact asymptotic at â¦ â 0. Figure 3. Figure 4. Figure 5. Figure 6. In experiments, for low values of the pycnocline thickness (Î´<D), the dissipation of energy due to internal wave radiation is low. Correspondingly, the damping coefficient is small and CÂµ is close to 1 (i.e. the value it takes in homogeneous fluid). As the thickness of the pycnocline increases, the dependencies of the added mass and damping approach asymptotically those ones corresponding to the case of infinitely deep exponentially stratified fluid. CONCLUSION the authoritative version for attribution. To our knowledge, this report presents the first experimental and numerical investigation of the hydrodynamic 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 effect on the frequency-dependent hydrodynamic coefficients (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 idealizations, namely, the model of two-layer fluid with an interface and the model of exponentially stratified 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 âLeading Scientific Schoolsâ, grant N 00â15â96162,

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 0001â00812, SD RAS Integrate Project N 1â2000 and grant N 6 for young scientists. REFERENCES M.Abramowitz & I.A.Stegun, âHandbook of Mathematical Functions', NBS, 1964. W.E.Cummins, âThe impulse response function and ship motions'. Schiffstechnik, Vol. 9, pp 101â109, 1962. V.A.Ditkin & A.P.Prudnikov, âIntegral Transforms and Operational Calculus'. Gos. Izd. Fiz. -Mat. Lit. Moscow, 1961, (in Russian). E.V.Ermanyuk, âThe 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 152â159, 2000. E.V.Ermanyuk & I.V.Sturova, âHydrodynamic loads on a body in a stratified fluid', Proc. 1st Int. Conf. Marine Industry, Varna, 1996. V.A.Gorodtsov & E.V.Teodorovich, âStudy of internal waves in the case of rapid horizontal motion of cylinders and spheres', Fluid Dyn., Vol. 17, pp 893â898, 1982. V.A.Gorodtsov & E.V.Teodorovich, âEnergy characteristics of harmonic internal wave generators', J. Appl. Mech. Tech. Phys., Vol. 27, pp 523â529, 1986. D.G.Hurley, âThe generation of internal waves by vibrating elliptic cylinders. Part 1. Inviscid solution', J. Fluid Mech., Vol. 351, pp 105â118, 1997. J.E.Kerwin & H.Narita, âDetermination of ship motion parameters by a step response technique', J. Ship Research, Vol. 9, pp 183â189, 1965. J.Kotik & V.Mangulis, âOn the Kramers-Kronig relations for ship motions', Int. Shipbuilding Prog., Vol. 9, pp 351â368, 1962. R.Y.S.Lai & C.-M.Lee, âAdded mass of a spheroid oscillating in a linearly stratified fluid', Int. J. Engng. Sci. Vol. 19, pp 1411â1420, 1981. L.D.Landau & E.M.Lifshitz, âStatistical Physics'. Addison Wesley Pub., 1958. L.D.Landau & E.M.Lifshitz, âFluid Mechanics'. Butterworth-Heineman, 1987. Landau, L.D. & Lifshitz, E.M. 1987 L.H.Larsen, âInternal waves incident upon a knife edge barrier', DeepâSea Res., Vol. 16, pp 411â419, 1969a. L.H.Larsen, âOscillations of a neutrally buoyant sphere in a stratified fluid', Deep-Sea Research, Vol. 16, pp 587â603, 1969b. R.M.Robinson, âThe effect of a vertical barrier on internal waves', Deep-Sea Res., Vol. 16, pp 421â429, 1969. Yu.A.Stepanyants, I.V.Sturova, E.V.Teodorovich, âThe linear theory of the generation of the surface and internal waves', Itogi Nauki i Tekhniki. Mekhanika Zhidkosti i Gaza, Vol. 21, pp 92â179, 1987, (in Russian). I.V.Sturova, âDiffraction and radiation problems for the circular cylinder in stratified fluid', Fluid Dyn., Vol. 34, pp 81â94, 1999. B.R.Sutherland, S.B.Dalziel, G.O.Hughes, P.F.Linden, âVisualization and measurement of internal waves by âsynthetic schlieren'. Part 1. Vertically oscillating cylinder', J. Fluid Mech., Vol. 390, pp 93â126, 1999. J.S.Turner, âBuoyancy Effects in Fluids'. Cambridge; Cambridge Univ. Press, 1973. B.Voisin, âInternal wave generation in uniformly stratified fluids. Part 1. Green's function and point sources', J. Fluid Mech., Vol. 231, pp 439â480, 1981. J.V.Wehausen, âThe motion of floating bodies', Ann. Rev. Fluid Mech., Vol. 3, pp 237â268, 1971. J.-H.Wu, X.-H.Wu, S.-M.Li, âA theory of wave diffraction and radiation by a large body in stratified ocean (III) Boundary element method', J. Hydrodyn., Ser. A, Vol. 5, pp 74â80, 1990. the authoritative version for attribution.