Twenty-Third Symposium on Naval Hydrodynamics (2001) / Chapter Skim
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Radiation Loads on a Cylinder Oscillating in Pycnocline
Radiation Loads on a Cylinder Oscillating in Pycnocline
Pages 780-791
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From page 780... ...
= ~ where p is the gravity acceleration and y axis is di rected vertically upwards The maximum Brunt Vnisaln frequency for n given density distribution over depth defines the upper cut fl frequency for in ternal w e effects Thus, in contrast with infinite frequency spectrum of surface waves, the frequency spectrum of internal w wes is finite Moreover, when the oscillation frequency of n body is I wer, equal or higher than the buoyancy frequency, the equations of fluid motion are of hyperbolic, parabolic or elliptic type, respectively, what, once gain, is in contrast with free surface problems which are described by the Lnplace equation The major part of the stud ies on the body oscillations in continuously strut idled fluid is pe formed within the model of ideal, uniform :, strntifled (the Brunt Vnisaln frequency is assumed to be constant) , Boussinesq fluid of ink nite extent in particular, the tim~domnin analy sis of damped oscillations of n sphere and circular cylinder is given in Larsen (1969b)
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From page 781... ...
Foil wing this approach, we make use of experi mental records of damped oscillations of n cylinder (impulse response functions) in fluid and Fourier transform the problem from time to frequency domain in order to evaluate the frequency dependent added mass and damping coefi cient Let us assume that n 5: d:, pe forming small oscil lotions in n continuously stratified fluid can be ide Sized as n linear sy tem it is well kn wn that, once the response of any stable linear system to n unit impulse r(t)
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From page 782... ...
Z2 + 1] Z It can be easily verified that for O is one simple pole at z = 0 inside L When 12 = 0, two dditional poles of der appear at the contour C in the The residue at z = 0 is I while t z = if ore zero Correspondingly, f:r the zdded mass coed cient is Cal = 0 (for circular cylinder Cr(ca)
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From page 783... ...
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.
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From page 784... ...
To study the whole frequency range of interest, it is necessary to perform n series of experiments for n set of his and match the results at n common plot so that the data obt .=- d at different his overlap The variation of his can be easy attained -- variation of the restoring force coed cient c THEORETICAL ANALYSIS It is assumed that the :=-_ :d incompressible fluid occupies the region x < co, No < y = y+EI~ < Elf and there are three Inters: homogeneous upper and low r ones and n linearly stratified middle one Thus, density stratification in an undisturbed tat p(y) takes the form P(Y)
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From page 785... ...
~Uds (21) YE Nu: are the added mass and damping coed crents, respectively To solve the formulated problem, let us use the method of singularities in terms of unkn wn source distribution o>(x)
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From page 786... ...
, (30) mzJ be trun eating the infinite series zt z finite number of term, which depends on the desired zccuracJ Once the coefi cient am tm zre obtained, we can determine zll the cEzmcteristic of the fiuid motion In the far field, the fiuid motion represents z super position of infinite number of w e modes Thus, for ex mple, the pressure in the middle IzJer zt x ~ co has the form n (351.
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From page 787... ...
= 7rpl~ ~/h3, Nt (hi > N) An interesting property of this solution is the fact that the horizontal and vertical loads coincide The npprcximate solution for diagonal damping coed cients takes the form (Gorodt ov & Teodorovich 1986)
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From page 788... ...
) at G~ ~ co are essentially different from I only for B = B3 = 0, Bz/a = 5, h/n = 2 These values MII = 1 3326 and M22 = 1 165 are sh wn by the dash lines in Fig 2 (: ompari.on with experimental rezultz The results of the theoretical and experimental evaluation of the added mass and damping coed cients of the cylinder in the linearly stratified fluid of limited depth are shown in Figs 2, 4 for a layer of linearly stratified fluid and in Figs 5, 6 for pJcno cline in numerical calculations the thickness of the middle layer H2 was taken equal to 5
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From page 789... ...
Numerical results for the added mass coefficient Cll shown in Figs.3, 5 seem to capture well the main experimentally observed effects at Q > 1. However, at low Q the behavior of numerical and experimental data is quite different.
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From page 790... ...
CONCLUSION 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)
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From page 791... ...
I.V.Sturova, 'Diffraction and radiation problems for the circular cyiiDder in stratified fluid', Fluid DJD, Vol 34, Up 81 94, 1999 B.R.Sutherland, S.B.Dalziel, G.O.Hughez, P.F.Linden, 'Visualization and measurement of iDterDai w wes -- synthetic schtieren Part I Vertically oscillating cylinder' J Fluid Mech, Vol 390, Up 93 126, 1999 J.S.Turner, 'BUOJaDCJ Eflects in Fluids' Cam bridge; Cambridge UDiv Press, 1973 B.Voizin, Internal w ve generation in uniformly stratified fluids Part I Green s function and point sources', J Fluid Mech, Vol 231, Up 439 480, 1981 J.V.Wehauzen, 'The motion of floating bodies', ADD P v Fluid Mech, Vol 3, Up 237 268, 1971 J.-H.Wu, X.-H.Wu, S.-M Li, 'A theory of wwe diffraction and radiation by a large body in stratified ocean (111) Boundary element method', J HydrodyD, -e- A, Vol 5, Up 74 80, 1990
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