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Effect of Viscous Damping on the Response of Floating Bodies M. brownie, I. Graham2, X. Zheng~ (~University of Newcastle upon Tyne, United Kingdom) (Imperial College of London, United Kingdom) ABSTRACT. The prediction of hydrodynamic damping flue to vortex shedding on a three-dimensional body in waves is discussed. A method of matching precalculated results for separated oscillatory flow past an isolated edge at each vortex shedding region of the outer potential flow field of the body in waves has been extended to three-dimensional floating bodies by two different methods. In the first method the pressure field due to a local separated flow computed by a discrete vortex method, is applied at all vortex shedding edges of the hull. In the second method the effect of vortex shedding at an edge is represented directly in the frequency domain panel method by special edge panels. A dipole density on each of these edge panels is evaluated by making a comparison between the far field effect of the panel an.` the far field of the appropriate separated vortex flow modelled by the vortex method. NOTATION. A,B,C,M added mass, damping, restoring and mass coefficient matrices vortex force body motions = ~je~it~t Keulegan-Carpenter number = 211U/COLB matching length scale LB, LV body and vortex length scales l barge beam 1V p vortex panel length q velocity amp' itude s surface dist.rlce U velocity scale z complex coordinate in cross-sectional plane = x + iy vortex strength (circulation) internal edge angle complex coordinate in the transformed plane dipole strength density source density velocity potential, sum of component potentials fje~i(0t, j=1...6, for surge, sway, heave, roll pitch, and yaw. frequency of motion Hj Kc TRODUCOl'lON. Prediction of the forces and responses induced by waves on floating bodies is usually based on linear potential flow theory (1), which is quite adequate for the prediction of many types of response . The damping arising from linear potential flow is associated with the waves that radiate out from the body. However certain types of motion for which the first order radiation damping is fairly small are not well predicted. In these cases non-linear effects must be included in the analysis. The most important non-linearities may be accounted for by second order potential flow effects and separation. Both of these are significant for example in low frequency sway motion. In the present paper we are concerned with incorporating separation effects alone into a linear potential method. This is found to greatly improve the prediction of, in particular, toll motions of many types of floating body. The effect of neglecting separation is conspicuously evident in the case of roll, particularly near resonance, when the response can be greatly overpredicted by linear theory (2), see figure 1. In engineering application of linear potential theory it is usual practice to incorpc'~ ate empirical coefficients into the calculation to represent the effects of viscous damping. Good agreement between predicted and experimental results can be obtained for most standard hull sections. The results of the semi-empirical method (3) to predict wave and viscous damping are also shown in figure 1. However prediction of the damping for non-standard hull shapes is more difficult. Other semi-empirical meth ads include experimental determ ination of the individual components of the roll damping (4). use of the concept of crossflow drag on a barge hull and its appendages (5), and work described in references (6 - 10). In the case of a barge or ship hull the dominant non-linear component at toll resonance is due to viscous, i.e. separation, effects. These effects, often described by the general term 'viscous damping' are primarily due to vortex shedding from the hull and its appendages. A two- dimensional model incorporating theoretically the effect of separation has been developed for barge roll (6). This model used a local solution of separated flow past an oscillating edge which was representative of the bilge of the hull and its relative motion. This local flow field was computed 149

l ~ 81 6~ go a , 11a 11 \\ ·t \e ./ ~ W(Hx) Fig. 1. Roll amplitude against frequency for a rectan- gular cylinder in beam waves (Salvesen et al.(2)~. wave and viscous damping; - - -, wave damping only; ·, experiment. as a universal semi-infinite edge flow dependent only on the edge angle (typically 90° ) and was then applied by a matching process to the flow field around the hull. The method used to compute the flow field was the discrete vortex method (11) which gives good flow and force predictions for vortex shedding from sharp edges in oscillatory flow. The flows induced on floating bodies are essentially high Reynolds number and oscillatory. In many cases separation is fixed by the geometry of the body. Flows of this nature have a strong vortex structure that is easily modelled by the discrete vortex method. Numerical solution of the Navier-Stokes equations by finite differences or elements could also be used but is generally m ore expensive because unless a very fine grid is used these methods do not capture the thin shear lay ers in the flow field. The discrete vortex method provides a time domain solution for the separated flow. The Kutta-Joukowski condition is used in the inviscid case to specify the strength of vorticity shed from the edges. Later refinements of the method have shown that rounded edges, bilge keels and viscous effects can also be incorporated successfully,( 12). Since prediction of the response of a floating body requires a large range of input amplitudes, directions and frequencies to be computed, for practical use it is essential to have a method which minimises the computer time for each case. The use of a matched inner flow field as above provides this economy since the separated flow need only be calculated once for all edge sections of the body having the same included angle. This ' inner' flow is then matched to the ' ou er' linearised potential flow at the bit be or salient region of separation so that one time consuming vortex calculation can be used to provide results for a complete range of motions for any given body. The outer flowfield associated with the incident wave field and general motion of the body is solved with a full bound ary integral method. Since this r.`ethod is based on inviscid potential flow theory which cannot model flow separation, the flow field is singular along the lengths of bilges and other shedding 'edges. In the case of a three-dimensional body in a general incident wave field a quasi-two-dimensional assumption is made for the vortex shedding locally. The outer flow field is fully three dimensional but it is assumed that the separated flow is generated within a formation region sufficiently close to the edge on the scale of the body that it may be considered to be locally two- dimensional. Hence the inner separated flow field is matched on a strip theory basis. This approach is justified at long continuous edges but clearly breaks down at those points where sudden changes in geometry occur. The method also assumes that the flows associated with different edges do not interact and also that the wave making effects of the vortices are insignificant. The first assumption is justified for sufficiently small amplitudes of motion, although the results of the method are in reasonable agreement with measured data up to moderately large roll angles. The second assumption is less certain, see for example ( 13). It should be noted here that the entire method is theoretically based and does not require the input of any empirical data. Also the linearised free surface boundary condition is applied so that the method is not limited to the low frequency approximation. The sep arated flow field inducer! by oscillatory flow past an edge commonly consists of pairs of vortices, one pair shed during each flow cycle. These vortices are of alternate sign and often form a closely coupled pair moving away from the edge under their mutually induced velocity fields. Other modes of shedding also occur. A single vortex may convect along a surface under the induced velocity field of its image c Id for rounded edges two vortex pairs may be shed per cycle, ( 12). This method which matches the vc tex calculations to the outer flow past a general body has been described in detail in ( 14). The main feature is that the singularity at each edge in the potential flow past the body cross-section is evaluated by applying a conformal transformation to the section which opens the perimeter out into a straight line, figure 2. The force induced by the ~ ortex shedding on the body is calculated from an integral over this transformed cross-section. This procedure is reasonably straightfc rward to implement for simple hull shapes but becomes complicated and difficult to specify for many other type; of floating structure. For this reason an alternat ive approach which is the subject of th s paper has been developed. In the new method the effect of vortex shedding at an edge of the body is mode:led directly in the potential flow calculation by dipole panels representing the vortices along each edge. The dipole panels satisfy the free surface boundary condition and may also be considered as representing the force induced by vortex 150

Real planeTransformed plane Infinite wedge :_ _,_ z = L-~] L= all ~us matched to US 1 ~-(all)-~' ~ _ _ _ UB Rolling barge Fig. 2. Matching the local and exterior flows. shedding at an edge. Because of the convenience of working in the frequency domain, as with the matching method, the non-linear vortex force is Fourier analysed into a fundamental component at the input frequency and harmonics which are all considerably smaller (10% or less) than the fundamental. It is therefore justifiable to neglect them when primary motions such as roll resonance are being considered. However the same would not necessarily be true where second order potential effects are important such as in low frequency response to waves. The dipole panels are therefore evaluated in the same way as the body surface singularity panels in terms of a complex amplitude at the input frequency. The length of the panels however depends on the amplitude of the motion or waves and is calculated by considering the separation condition at the edge. COMPUTATION OF THE FLOW F1~;LD. Oscillatory flow about a body ma, be characterized by the Keulegan-Carpenter number, Kc. At small Kc the maximum displacement of the fluid particles in the undisturbed flow is small in comparison with the scale of the body. Thus vortices may only move away from its edges under the influence of the velocity field of other vortices shed from those same edges, and hence the shedding from any one edge may become independent of the shedding from other edges. In these circumstances, the local flow becomes analogous to the local flow about an infinite wedge. The discrete vortex analysis of shedding from an isolated edge (11) carried out for a series of infinite wedges of varying internal angle showed that the complex force due to vortex shedding could be related to the vortex strengths and positions by FV = -i P d/dt { ~ JO ( (j ~ ~ J ) } = 1/2 p u2 L KC(26-~)I(3~-23) Aft) (1) where ~ is a dimensionless time-dependent force function which can be Fourier analysed into a fundamental and higher harmonics of the input frequency and U and L are length scales defined in the matching process. Abode Surface Panels. The flow field is described by a velocity potential which satisfies Laplace's equation. For each wave frequency the total potential may be written in terms of the separate potentials for the undisturbed incident wave, the wave scattered by the body considered to be fixed and rigid, and the waves radiated by the body in its six degrees of freedom, where it is understood that the real part only of the potential is considered: 6 do e + ~ (j Hj + ¢7 edict (2) j=1 The potential satisfies the linearised free surface condition `,,2 ¢,j + g 6~/by = 0, j = 0 ... 7, Y=0 (3) where y is the vertical axis. A radiation condition, and a normal velocity condition on the body and sea bed are also satisfied. In the new method the potentials were represented by a mixture of sources and dipoles: tj(q)=l/4nl{oi(q) G(q',q) - 2 fj(q) dG(q~q)/dn}ds B j = 1... 7 (4) The integral is over the points q lying on the wetted surface of the body B with normal n. ~ ( q ) are the source densities and G(q',q) are suitable Green's functions ( 15). The potentials are evaluated by solving the boundary integral equations on the surface of the body numerically ( 16). The equations of motion of a freely floating body can conveniently be expressed as: ~ { - c32 ( Mjk + Ajk ) - i°Bjk + Cjk } Ilk = f; (5) k The force coefficients and the exciting forces fj can be calculated from the linearised form of the Bernoulli's equation using the potentials given I,y the boundary integral method. _ Vortex (Dipole) Panels. We assume that the body has a number of well defined edges from which vortex shedding will take place. Cases where separation is taking place from a moderately curved continuous 151

surface will not be considered in this analysis. Then the solution for the above potential flow will have the local behavi our: With this assumption the complex potential W takes the form, at large distances ~ from the edge: '`:, ~ (2-) (6) W -> - ( i / 2~( ) £ Id ( (j c where ~ is the included angle of the edge and s is the surface distance measured perpendicularly to the edge. For simplicity we will now consider the case of a body for which all the edges are right angles so that ~ = ~/2 Hence the surface speed in the vicinity of the edge is q = c I s I -1/3 + O(1) terms (7) where c can be evaluated from the two adjacent panels (1 and 2 ) as: c = 2/3 ( ~2 - Of ) i ( Is2 12/3 + 1 s1 12/3 ) t;8) The viscous flow response to this singularity in q is for separation to occur resulting in the shedding of vortex pairs from the edge each flow cycle. As in eqn.(1) vortex shedding of this type has been shown ( 17) to exert a force on the body in two-dimensional flow: FV = -iPd/dt { ~rj((j~(Ij) } where Fj is the circulation of each shed vortex and (j is its location in a plane obtained by apply ing the local Schwartz-C:hristoffel transformation, Z = k `2-~/~ (10) in the plane z = x + i y normal lo the edge and local to it, figure 2. (Ij is the image of (j in; the plane surface which is the transformed Goody surface in the ~ plane. The complex potential in the ~ plane in the vicinity of the edge due to this array of shed vortices is W = i/2 £rj { log(-) - log(-) } (11) We now make the assumption, as in the prey ous matching method ( 17), that the amplitude of the body motion or of the incident waves is sufficiently small for the vortex shedding to be considered as a local phenomenon at each edge. That is the length scale Lv associated with the vortex shedding ( for example the typical distance of the centre of a vortex from the edge when it detaches and a new vortex starts to form ) is much smaller than the typical length scale of the body cross-section LB. This is a formal restriction, but having made it, the method will be used practically for amplitudes as large as can be justified by the results. for LV2/3 << ~ << LB2/3 (12) This is the potential of a vortex dipole of strength: pL = _ Ad, Fj ( (j _ (IJ ) (13) located at the edge ~ = 0. The shed vortices are therefore modelled by a dipole panel in the physical plane whose potential at large distances in the transformed ~ plane including the effect of its image is equal to that of eqn.(13). It is convenient in the present panel method to represent the dipole by a piecewise constant distribution of source dipole density aligned in the direction normal to the panel surface, and hence to the vortex dipole direction, with the panel lying along the external bisector of the edge angle. This dipole may also be considered through eqns. (9) and ( 13) as being proportional to the impulse J. FV dt exerted by the vortex shedding at the edge. Because of this the total dipole strength may be obtained directly from the vortex force computations in the form of eqn. ( 1). However the length of the vortex panels remains as an apparently disposable parameter in the numerical method. Since the intention is to derive a method which can be applied in the frequency domain, the dipole density is taken to be proportional to the component of the impulse at the fundamental frequency, which as discussed earlier is the dominant part. The length of the panel is kept constant through the flow cycle. [he length can be assigned in a number of ways and it is not yet clear which is optimal. It could, for example, be made equal to the characteristic distance LVof the vortices from the edge during the formation process . This is not always easy to specify and in the results described here the length is prescribed instead by requiring the Kutta-Joukowski separation condition to be satisfied in the mean sense over the flow cycle at each edge. Hence we obtain for a right angle edge, after some algebra, the source dipole density ,u' length 1Vp given by: ~ = x2 ( b1 cos cot - al sin lot ) / 4Cd 1Vp2/3 (14) and 1Vp = ~ X ( al2 + bl2 )1/2 / 4~,~ ~3/4 (15) where X = 3/2 Icl, c being defined in eqn.(8) in terms of the potential 4? on panels either side of the edge. al and hi are the Fourier coefficients of the fundamental component of the vortex force 152

FV, defined by: 2 a1 = ~/~ ~ FV COS At dt and 14 o 12 2~ b1 = W/~ |FV sin of at (16) 10 o Having specified each vortex panel in this 8 way in terms of the attached flow potential on the . adj acent panels to each segment of an assumed 6 shedding edge, the surface potential calculation is repeated with the vortex effect incorporated. The pressure field can then be computed from 4 . Bernoulli's equation over the panels on the surface of the body only, ie. excluding the vortex panels. 2 RESULTS. O Both methods have been tested for cases of O barges for which the hulls take the simple form of boxes with plane rectangular sides and right angle edges. Experimental data contain ing sufficient information to compute these cases however are scarce. Three cases have been studied, the first and second being cases of barge response in beam seas and the third a forced roll case. The first case was a barge with a beam to draught ratio of 10.0 and a length to beam ratio of approximately 3.3. Experimental data was obtained for a model of the barge, at scale 1:30, with rounded bilge corners with radius of curv ature equal to 18~o of the draught. The results (18) for roll in beam seas ate compared with computed predictions using the first (matching) method in figure 3. In the second case study (8), a barge with a beam to draught ratio of 7.6 and a length to beam ratio of 3.0 was used. Experimental results for both sharp-edged and rounded bilge corners were obtained. The rounded ones had a radius of curvature of 38% of the draught. The damping was significantly reduced when the bilge corners were rounded. These experimental results and those computed by the first method for the sh~.rp edged case are shown in figure 4. In the third case results of a forced roll experiment ( 19) on a barge of beam to draught of 21.0 are compared with the computed results of the second numerical method. The experiment was on a barge spanning the test section of a tank whereas the computation was for a barge of the same beam to draught ratio and a length to beam of 6.14. The results for the damping coefficients for different wave amplitudes are shown in figure 5 and for the added mass in figure 6 DISCUSSION. The vortex force for each degree e of freec~c!m is predicted to have components in phase with the velocity and with the acceleration of the body. Therefore, the method accounts for vise ous contributions to both the damping an ~ added mass terms, including the effects of cross-coupling.In ;1*\~< ~ . . . · ~ 16 20 · , 4 8 12 Fig. 3. Case study 1; Roll RAG against wave period. , wave damping only; viscous damping included, - 0 -, Hw = 3 m,-- -, Hw = 5 m; experiment, - /\ - , Hw = 3 m,-x-, Hw = 5 m. 16 i 1 . so _ 12 · O lo: _ 0 8 4 o Ir y~ , . . O 4 ~At 8 T (B) · · · · _ 12 Fig. 4. Case study 2; Roll RAO against wave period in beam seas. , wave damping only; viscous damp- ing included, - · - , Hw = 0.90 m,- - -, Hw = 1.74 m; experiment, sharp edged bilges,-~-, Hw = 0.90 m,-x-, Hw = 1.74 m. round edged bilges, - 0 - , Hw =0.9Om -+-OH,,, = 1.74 m. 153

3~ 2 1 O . 0 x cob ~ /@ lX: e~\ ~ ~ \ \ 'A \ \ \ a,' . Wy2 ~ 0 0~5 1 0 1 5 Fig. 5. Forced roll damping coefficient against fre- quency of motion. - - - wave damping only; viscous damping included, - · - ,8 = 0.05 red., - + - , ~ = 0.10 red.,-x-, ~ = 0.20 red.; experiment,(~), a = 0.05 rad.,E, ~ = 0.10 rad.,O, o~ = 0.20 red. 6 4 2 o °x~ I Eve \L N ~-~ In- X~X- `e, e ~ ~ . . . . O 0 5 1~0 Fig. 6. Forced roll added mass moment of inertia against frequency of motion. - - - potential flow only; viscous effects included, - · - , ~ = 0.05 red., - + - , = 0.10 red.,-x-, ~ = 0.20 red.; experiment,(D, = 0.05 rad.,ffl, ~ = 0.10 rad.,~, ~ = 0.20 red. the first roll response case the results of the first method show good agreement between the predicted and measured roll responses through the resonance region. In reality the computed results are slightly on the high side because ~ the model barge had rounded edges. This s evident in figure 4, where the model barge for the second roll response case was fitted alternately with rounded and sharp ed<~,,es. However, both figures show a dramatic improvement in the prediction of roll when viscous effects are included in the calculation in comparison with the predictions of potential theory alone. Both sets of results also show that the non-linear dependence of the roll amplitude on the wave height is well predicted. The fact that the vortex forces are critic ally dependent on the relative velocity of the fluid in the immediate vicinity of the shedding edges, being very sensitive to the position of the roll gentle and the barge geometry has been demonstrated before (6). For the first barge, which has a relatively high centre of gravity and less rounded bilge corners, the potential flow calculation greatly overpredicts the roll response. For the second barge, with its centre of gravity approximately at the level of the mean free surface and with well rounded bilges, the potential flow calculation gives results which are in quite good agreement with experiment. In this case the effects of vortex shedding from a rounded edge are small and a relatively large loll amplitude is obtained. However when this barge was fitted with sharp-edged bilges, vortex shedding became important and the response at resonance was considerably reduced to a value in reasonable agreement with the theory including vortex shedding effects and much low' r than that predicted by potential theory alone. The assumptions made abe at the location of the roll centre when using forced roll data are therefore very important. Figure 5 shows the comparison of damping and figure 6. the added mass for forced roll predicted by the second ( vortex panel ) method compared with experimental data ( 19). This data is for a rectangular barge of finite length. 'Jajce potential theory result without vortex damping is shown calculated by the present thriee dimensional potential panel method. the agreement between the predicted and measured damping coefficients is very good when the vortex panels are incorporated in the computation. The added mass is less well predicted. Disagreement here is somewhat unexpected because the vortex force contributes relatively a considerably smaller part to the overall ad led mass than it does to the damping. Any discrepancy / ~would therefore be expected to be more clearly w\/ 29 apparent in the damping rather than added mass coefficients. However Vugts has commented t 19) 15 that in this particular experiment it was difficult to measure the added mass very accurately. It should also be emphasised that the computation was carried out for a three-dimensional barge, but without computing vortex effects from the edges around the two ends, whereas the experiment was for a barge which spanned the tryst tank. She vortex damping is likely to be less effected by this difference than the added mass. 4

RAGES. 6 4 O _ cat to x -1~ Axe WY2 0 05 10 15 - Fig. 7. Vortex/dipole panel length against frequency. -· - , ~ = 0.05 red., - + - , ~ = 0.10 red.,-x-, ~- 0.20 red. Results for three different forced roll amplitudes have been plotted and it is apparent that the vortex panel method does predict the increase in hydrodynamic damping with amplitude quite accurately. Figure 7 shows how the vortex panel k;~gth 1V p v aries with frequency and amplitude. The fact that the length changes very little with frequency is not unexpected since it is only the effect of the free surface condition which will cause such a change. The variation of 1V p with amplitude is approximately proportional to (amplitude)] /4 which would be expected from the variation of the vortex length scale Lv, (17). CONCLUSIONS. It has been shown that the motions of sharp-edged rectangular body floating freely in waves are well predicted by the first ( matching ) method. The results suggest that the n,on- linearities in response are largely due to vortex shedding from the body. The second ( vortex panel ) method has been used to predict damping of a barge due to forced roll and the results are similarly in good agreement with the measured data. The second method is the easier to set up and apply to general shapes of floating body and is therefore being developed into a full motion program. ACKNOWLEDGh/ENTS. The authors gratefully acknowledge the financial support of the SERC through the Marine Technology Directorate Ltd. 1. Newman, J.N. Marine Hydrodynamics. MIT Press. 1977. 2. Salveson, N. Tuck, E.O. and Faltinsen, O.M. 'Ship motions and sea loads.' Trans .C',NAME. !78. 1970, p421. Tanaka, N. 'A study on bilge keels. (Part 4. On the eddy making resistance to the rolling of a ship's hull). J. Soc. Nav. Arch. Japan 109, 1981, p205. Ikeda, Y., Himeno, Y. and Tanaka, N. 'A prediction method for ship roll damping'. Dept. Nav. Arch. Univ. of Osaka Prefecture Rep., 1978, p00405. Kaplan P., Jiang C.-W. and Bentson J. 'Hydrodynamic analysis of barge platform systems in waves'. RINA. Spring Meeting, Paper No. 8, 1982 Bealman P.W., Downie M.J. and Graham J.M.R. 'Calculation method for separated flows with application to oscillatory flow past cylinders and the roll damping of barges'. P i. 14th Symp. Naval Hydrodynamics, Ann Arbor, Michigan, 1982 7. Denise J.P. ' On the roll motion of barges', RINA Paper W10 1982 8. Brown D.T., Eatock-Taylor R. and Patel M.H. 'Barge motions in random seas - a comparison of theory and experiment'. J. Fluid Mech. 12~), 1983, p385. 9. Barge Motion Research Project. N o b 1 e Denton and Associates Ltd. London, 1978 10. DeBord F., Purl J., Mlady J., Wisch D. and Zahn P. 'Measurement of full scale barge motions and comparisons with model test and mathematical model predictions'. Tran s . SNAME 95, 1987, p319. 11. Graham J.M.R. ' The forces on sharp-edged cylinders in oscillatory flow at low Y~eulegarl Carpenter numbers'. J. Fluid Mech97, 1, 19E,0, p331. Braathen A. and Faltinsen O.M. 'Interaction between shed vorticity, free surface waves and forced roll motion of a two-dimensional floating body'. Fluid Dynamics Research 3 1988, pl90. Cozens P.D. 'Numerical modelling of the roll damping of ships due to vortex shedding'. PhD. Thesis, Univ. of London, 1987. Downie M.J., Graham J.M.R. and Zheng X, 'The influence of viscous effects on the motion of a body floating in waves'. Proc. OMAE. Conf., Houston, Texas, 19'v0 John F. 'On the motion of floating bodies, Part II'. Commun. Pure Appl. Maths. 3~ 1980, p45. 16. Zheng X. 'Prediction of motion and wave loads of mono- and twin-hulled ships in waves.' PhD. Thesis. Univ. of Glasgow, 1988. 17. Downie M.J., Bearman P.W. and Graham J.MR. 'Effect of vortex shedding on the coupled rol. response of bodies in waves.' J. Pluid Mech. 189, 1988, p243. 18. Standing R.G. Private communication, 1984. 19. Vugts J.H. 'The hydrodynamic coefficients for swaying, heaving and rolling cylinders in a free surface'. Netherlands Ship Res. Cents TNO. Rept. INS, 1970 12. a 14. 155

DISCUSSION Choung M. Lee Pohang Institute of Science and Technology, Korea It is a very interesting work to simulate the bilge vortices by doublet distribution. It may work very well for the zero-speed case; however, when the body as a forward speed, the bilge keel acting as a low-aspect ratio wing might have a quite different types of vortex sheddings. How would you approach such a problem? AUTHORS' REPLY We agree that the vortex shedding is different in the case of a body undergoing the combined motion of roll and forward speed, since the vorticity is now convected along the hull surface as well as away from it. Work is in progress to extend the model to this case and we hope to be able to report further on it at a future date. DISCUSSION Targut Sarpkaya Naval Postgraduate School, USA It will be appreciated if the authors would comment on the similarities and the fundamental differences in the boundary conditions of the physical experiments and the numerical simulations? What is the actual motion of the barge model in the experiments. Thank you. AUTHORS' REPI,Y The calculations were carried out for two basic flow configurations. The first was a barge freely floating in beam waves, and the second was a barge undergoing forced roll in otherwise quiescent water. In both cases the potential flow part of the calculation was for a three dimensional barge. The matched edge technique has been applied to both cases, the forced roll calculations being reported in a previous paper. The viscous edge panel technique has so far been applied to the case of forced roll only. The results of the calculations were compared with experimental results, in the first case, obtained using a model barge floating in waves, and in the second case, with two dimensional results obtained with a rectangular section spanning a channel and undergoing forced roll. There are of course some important differences between the computed and experimental flows, particularly in the case of forced roll. The computations have assumed perfectly two dimensional flow. In the freely floating case there will be some end effects which are not accounted for by the theory. Similarly, in the case of forced roll, the vortex shedding in the experiment will be affected by the presence of the end walls and will not be entirely two dimensional. Also, in both cases, the vortices approaching the free surface may be subject to strong three dimensional instabilities. Nevertheless we believe that the results are reasonably representative of the major effects of vortex shedding in the flow regimes considered and demonstrate a considerable improvement on those obtained using potential flow theory alone. 156