The hydrodymanic coefficients for swaying, heaving and rolling cylinders in a free surface

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  I Introduction --A solution for the ship motion problem at seis requiresthe determination ofthe dynamic equilibrium of forces and moments. It is- generally accepted that for the fluid forces the influence of viscosity and surface tension isof minor importance -compared to pressure and wave effects. At the present state of knowledge this proposi- tion has not been disproved by model or full scale experiments, at least as far as ship motions are con- cerned. Possibly the manoeuvring problem is suffering from viscous effects. 1-t has further been supposed that the whole motion problem can be regarded as linear. Again up to the present state of development this -has -been ¿onflrmed surprisingly, in any-case for engineering purposes and apart from very special objects. - By these circumstances the determination of the hydrodynamic forces acting on the ship's hull forms a linear boundary value problem in potential theory The superposition principle holds and the actual phenom- enon can be split up into the sum olharmonic oscilla- tions of the ship in still water and waves coming in on the restrained ship. The two fields can be investigated entirely separately. Considering only the first field the problem can be stated as the oscillation ola rigid body, moving with a certain specd in the surface of a heavy, ideal fluid. The solution supplies the six transfer func- tions of the ship, which are composed of both rigid body characteristics und hydrodynamic quantities. Un- fortunately the solution of this general three-dimen-sional problem, including fòrward -spccd, ¡s not yet possible. - Two theories have been developed to find an ap- proximate solution: the slender body theory and the strip theory. Both- have- limits as to their validity, theformer giving better results at low frequencies of mo- )Rcport 112 S Nèthorlands Ship ilcsoarchContrc fl40. i Shipbuilding L,boratory Deift Techni,Iogicii Univcrsity. THE HYDRODYNAMIC COEFFICIENTS FOR SWAYING, -HEAVING AND ROLLING- CYLINDERS IN A FREE SURFACE*) - by Ir. J. H. VUGTS **) Summary For various cross-sections the hydrodynamic coefficients of two-dimensional cylinders are determined by forced oscillation tests and by theoretical computations. The.purposeof thisstudy is to check the theoretical basis of the computations for all three possible modes of motion and to establish the influence of section shape in this respect. Theresults show good agreement for heaving and forswaying, while there is-a fair correspondence for rolling. Apart from deviatitins due to experimental inaccuracies appreciable differences between theory and experiment only exist for the coefficients- of those terms hich dissipate energy in sway and roll There viscous effects are distinctly present especially for sharply edged sections in roll The wave exciting terms in the two-dimensional case are measured and compared to calculations as well. So a complete-set of hydro- dynamic quantities for the coupled motions of cylinders in beam waves is presented. tian, the latter at relatively high frequencies of motion. Accepting the basic assumptions the development of the slender body theory -is mathematically much more rigid. It has confirmed-several characteristics- andshown- new lines for the investigations. However, it is ques- tionable whether the results are more correct and itdoes not look very promising for practical purposes either. The strip theory has been developed by physical and intuitive reasoning and is much simpler in use. Moreover, it gives better results in the frequency range, which is important for the longitudinal mot-ions.In the past 15 years most attention has been con centrated upon pitching and heaving. M-uch progress has been made by means of combining elementary two-dimensional solutions for the hydrodynamic forces in a modified strip theory. For many practical applica- tions the-matter may be regarded as solved in this way.Evidence that the hydrodynamic coefficients calculated by two-dimensional potential theory are correct is- scarce, however, although the ultimate resùlts for the ship and for parts of the ship are in good agreement with experiments [1, 2]. It is acceptable that the theo- retical solution will be confirmed in a wide range by suitable experiments. As a matter of fact some experi- mental evidence is available. But the performance atrelatively low and high frequencies of motion and the influence of an accurate representation of section shape are unsettled details. As a contribution to this matter a series of experiments was carried out in heaving with cylinders of 7 different cross-sections; see figure 1. It has to be considered now whether a solution in the- field of the- lateral motions and rolling, which cover a frequency range from very low to rather high, depending on the size and type of ship, can be obtained in a way similar to that in heave and pitch. An analytic three-dimensional solution of the hydrodynamic prob- lem for an arbitrary case does not appear to be possible in the near future. Moreover, it was not established 25J  .252 a priori that for these motions the neglect of viscous and non-linear effects is just as permissible as for the symmetric heaving. Possibly eddy formation plays a more important role. For both reasons a theoretical and experimental investigation of the basic infinitely long cylinder is ofgreat valuc.Cylinders with 5 different crosssections have been oscillated in sway and roll and the measured hydrodynamic coefficients are compared with those, computed by potential theory. Coupling terms of sway into roll and vice versa are included. To complete the picture also the wave exciting forces and moment on the restrained cylinders have been obtained by measurement. They are compared with theoretical results as well. So a complete set of hydro- dynamic quantities is presented, by which the two- dimensional case, that is the motion pattern of infinitely long cylinders in beam waves, can be analysed. 2 Historical development The subject rolling has an important place in the literature since 1860. Attention concentrated especially upon roll damping by determining extinction curves for free floating models and even for actual ships. The value of severaFof these tests may be questioned when they were performed in small basins or at full scale in docks or harbours. Besides the influence of the induced swaying and of the position of the centre of gravity i a certain condition of loading upon the results of the tests does not seem to have been recognized fully. In 1933 Serat [3] used small cylindrical models for extinc- tion experiments to study the effect of different forms and of the position of the centre of gravity on roll damping more fundamentally. Although he used cylin- ders in principle he did not imitate two-dimensional conditions. But in 1937 Baumann [4] did for a large. circular cylinder. He recognized that added mass, moment of inertia and damping in roll were zero for this section, a fact which allowed him to determine the added mass and damping in sway for his freely floating model. He only investigated one frequency of motion, but his results are of a remarkably correct order. Ursell (1949) [5] calculáted the outgoing waves for a forced rolling motion at very low frequencies by potential theory. He found that for a well-rounded rectangle of BIT 2.52 roll damping would vanish. This theoret- ical result vas experimentally verified by McLeod and 1-Isiek [6]. For the first time now the roll axis was fixed in space and situated in the water surfice. Their experi- ments were not fully convincing, but they agreed fairlywell with the predicted results, despite the fact that the tests were carried out at the natural frequency of the cylinders, which was not so low that the condition w -e O of Ursell could simply be considered satisfied. MeLeod iitl Ilsich on the other lind found tlit the wave damping only accounted for 20 to 50 per cent. of the total damping. In heaving some experiments vcre performed in the thirties by Dirnpker [7] and Holstein [8]. They imi- tated two-dimensional conditions for a circle, a wedge a nd a recta agIe i n a sinn Il ta n k . The fornìer author only investigated free oscillations. He determined the damping decrement, while the increase in natural period with respect to the period calculated allowed him to give an indication of the added mass Holstein. on the other hand, also perforiiied forced oscillations and measured the progressing waves. His results may still be useful, although they arc not very accurate andare possibly influenced by wave reflection. Further it is interesting to note that Dimpker founda departure from pure two-dimensional conditions in forced heaving. For a combination of heave amplitude and frequency, exceeding a certain limit, a standing wave system along the length of the cylinder developed. According to his investigations this is a pure hydro- dynamic phenomenon, not depending upon sectionshape, surface tension or accidental circumstances during the test.In 1949 Urscll [9] had also indicated a general way to come to a theoretical solution of (lie boundary value problem in two dimensions. Grim [IO, Il] and Tasai[12, 13] extended this principle from the circular to elliptic cylinders and Lewis-forms, while Porter [14] ultimately formulated the solution for heaving of an arbitrarily shaped cylinder. Now in principle the way was free to investigate the influence of form, of fre- quency of motion, and of the coupling effects between sway and roll in detail. But first the validity of the theoretical approach had to be established by experi-ment. Naturally this was first tried for the most simple case of heaving. The experimental difficulties are verygreat, however, and it is not surprising that only a few experiments of the actual two-dimensional case are known. Tasai [15] measured the wave heights produced by forced heaving cylinders. Porter [14] measured the total vertical force on a heaving circular cylinder and the pressure in a number of points along the contour. Paulling and Richardson [16] carried out the most extensive experiments so far. For fourdifferent sections the vertical force and the pressure in 4 to 6 locations was recorded, both its magnitude and phase. Wave heights were measured as well. The results of these experiments vere such that the theoretical prediction was contirmedsubstantially. Thus for heaving only some details remain to be investigated, as discussed in the introduction. Now time has certainly come to direct the attention again to rolling and.swaying. The situation is quite dif-  fcrcnt with respect to these motions. Theoretical predic- tions for some Lewis-fòrms have been presented by Tasai [:13], but as far as the author knows not a single experimental check in these fields is available. And it has already been stated in the introduction that the validity of the theory cannot automatically be extendedto these cases. lt is especially questioned whether the flow condition when rolling can be described ade- quately by potential theory. As experimental procedure two methods are possible.When the cylinder has aforced oscillation in one mode of motion, either the pressure along the cylinder con- tour or the force required to sustain the motion can be measured. Pressure measurement has the advantage that it allows the most direct and most detailed com- parison with theory. On the other hand itrequires high. accuracy, complicated equipment and extensive anal- ysis The force measurement involves the prcssûre integra- tion over the body surface, so only the overall result can be compared with theoretical predictions. In DeIft experience has been gained with the latter way of testing and an advanced measuring technique has been developed [li]. Therefore the force measure-ment as to amplitude and phase has been accepted to obtain the experimental results. 3 The mathematical model for motions in two dimensions Let Or: be a coordinate system whichisfixedinspace. The r-axis is in the water surface, directed to the starboard side of the section. The z-axis is vertical,positive downwards The origin O is the intersection of the centreline of the sectionand the Waterline. Suppose now that the centre of gravity G of thecylinder is situated in O. The most general way of describing the motion of a linear system is a set of three coupled. equations of motion. In. a formal nota-tion this set can be put down as (ni + a1)i +by;.j (J _f. + 'sz + Cy:Z + + a) + bq+ cçb = Y. (ni + a.)z + b.:± + C: + Ci; + b2q+ C:q,(l + = z. (i + a) + b4/ + c/ + + + c,.,y + + (1.:'+ 1)4 += K. Where in = mass of the cylinder sectioti, ¡ = mass moment of inertia about G, (3.1) 253 a51 = hydrodynamic mass or mass moment of inertia in the. i-mode of motion, a,j = mass coupling coefficient in the i-equation by motion in thej-mode, b5 = damping coefficient against motion in the i-mode, b = damping coupling coefficient in the i-equation by motion in the j-mode, Cil hydrostatic restoring coefficient against a dis- placement in the i-direction, Cli = hydrostatic coupling coefficient in the i-equation by a displacement in the j-direction, Y = horizontal wave force (Y) when freely floating in waves or external force (Y0,0) when forcedly oscillated in still water, z = ditto in the vertical direction, K = ditto, moment about O. The coefficients c1, and c53 can by definition be deter- mined by pure hydrostatics. By simple reasoning the equations (3.1) can be simplified greatly. The horizontal displacement is not opposed by any restoring force, so e» = C:i = C#, = O. The vertical motion is symmetric with respect to the z-axis and. canñot produce any lateral forces or mo- ments; therefore = b, = c, = a,s = b#: C = O. A static heel j does not generate a horizontal force, or c, = O. But due tO differences in the immersed and emerged wedge when heeling about a fixed axis in space, in general C:# O. Then the mathematical model is reduced to (m + a,)j+ b»j' + ay +b,,4' = Y (I ± + bçb + c+ aj + b,j' = K (in + a2)ï +b2 +cz+ a24,i/. + + c,çb + = Z Heave does not influence the coupled. sway-roll motion, as is seen by (3.2), but the reverse need not be true: see (3.3). It will be clear, however, that the 4' and y-components 'in the z-equation may be expected to be extremely small with respect to the z-components. The experimental results will also show this clearly. In a potential flow the sway and roll problem is asym- metric with respect to the z-axis.and these contributions theoretically even vanish. Thus heaving becomes an uncoupled, one degree of freedom motion. For the time being the equation (3.3) is retained, however. The cylinders are harmonically oscillated in one of the three modes of motion y = Y0$)t, z = 4' = O z = z0sinwl; çA = y = O 4' = 4'05mn0)1 z = y = O,  while Y, Z and K are measu red simultaneously i ft amplitude and phase.In principle the cylinders were oscillated in roll about thc point G, coinciding, with 0. But for the rectangle it was very impractical to change the mechanical set-upfor the three draughts investigated. Thereforethe centre of gravity and the point of rotation remained at the highest position and G was no longer situated in O for BIT = 4 and 8. It, is preferable to define the hydro- dynamic quantities a, and C.j always with respectto the Ovz-axes, because they have nothing to do with rigid body characteristics. On the other 'hand body motions are most logically introduced as translations of and rotations about the centre of gravity. Therefore the equations (3.2) and (3.3) are rewritten for aGyz- system with 0G O, but with a1» and, Cj defined for motion of the point 0. This is obtained by trans-. forming. YG = Yo ö-. (J)o G 0 ZG = Z0 KG = K0+0G (hor. forces.). Working this out and dropping the index G for motions.añd exciting forces results in. (m + a»)i +.b)j' + {.a + j. a} + = Yw {I +a+0Ga,.+0G2 a,,+0Ga#}+ +.{b+ 0G b.,,+OG2 + {c + 0G g)4 + {a#+OG a}$ + = .0 +a:)±+ b::+c:zz+(az,+0Gaz,}+ + { b +ö. b.,}. + c4,4 + aj + (3.6) +b:yi = Zw . J Regarding these equations the following remarks are made: - 0G is positive when G is below 0, negative when G is above the water surface. - The right hand sides are expressed as wave forces, which (being hydrodynamic quantities) arc also determined with respect to the Oyz-system; so Kis the wave moment about 0.- When oscillating the moment to sustain the motion is directly measured about G, so the right hand side of (3.5) is to be replaced by K0, only. - The restoring coefficient in (3.5) is equal tomg . GM = ,,zg(OG - 0M) = .i;ig .OG - nig OAt; the latter term, (,,,q.OM) is nothing else than c, so this con- tribution is negative when M is below the watet surface and positive when M is above O. 4 .A theoretical solution of the hydrodynamic problem The hydrodynamic problem arising from the motion of an infinitely long cylinder in the free surface of'an ideal fluid is uniquely deteriruned and solvable. In an ideal fluid, being initially at rest, a velocity potential must exist satisfying Laplace's equation in two dimensions and the respective initial and boundary, conditions. When linearization is considered permissible the har- monic motion of the cylindér becomes of primary im- portance. In that case only the boundary conditions for the velocity potential remain when the transient phenomena have died out. Summarizing, the velocity .potential must satisfy the following requirements in two dimensions Laplace's equation;the linearized free. surface condition; the radiation condition, which states that a wavetrain of constant amplitude progresses from. the cylinder to infinity; every disturbance in the fluid must vanish at in- finite depth; the normal component of the fluid velocity at thecylinder' is equal to the same component of the cylinder velocity, both taken in the nican positionof the cylinder. The underlying assumptions are the fluid is inviscid, incompressible and irrotational; the surface tension may be neglected; the fluid domain is infinitely large; the motion amplitude is small with respect to the. dimensions of the section and. the generated waves have amplitudes which are small with respect to their length. It is well known that these conditions are approximat- ely fulfilled in many problems associated with ship motions, so that (lie theoretical solution is of significant value both qualitatively and quantitatively.. Perhaps some reserve is justified regarding the rolling motion as far as viscous efThcts and small motion amplitudes are concerned. The results of the experiments will have to show to what extent this invalidates the theoretical corn putat ions. The solution has been obtained according to Ursell's } (3.4) (3.5)
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