Twenty-Fourth Symposium on Naval Hydrodynamics (2003)

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.

Twenty-Fourth Symposium on Naval Hydrodynamics Fukuoka, Japan, July 8-13, 2002 Nonlinear Free-Surface Effects on the Resistance and Squat of High-SpeecI Vessels with a Hansom Stern Lawrence ]. Doctors (The University of New South Wales, Australia) Alexander H. Day (The Universities of Glasgow and Strat;hclyde, Scotland) Abstract The inviscid linearized near-field solution for the flow past a vessel with a transom stern is de- veloped within the framework of classical thin-ship theory. The hollow in the water behind the stern is represented here by a virtual extension to the usual hull-centerplane source distribution. The shape and length of this hollow are permitted to change in a realistic manner with increasing forward speed of the vessel, as well as with any consequent sinkage and trim that the vessel might suffer. To this end, the near-field solution to the flow using the thin-ship approximation must be computed, in contrast with the traditional far-field method developed by Michell (1898~. The computer program includes the facility to adjust the sinkage and trim of the vessel un- til it is in equilibrium. The latter feature of the computation utilizes an integration of the resulting pressure distribution over the wetted surface of the vessel in an entirely consistent manner. Developments reported in this paper are the inclusion of nonlinear free-surface effects, by intro- ducing a vertical straining or distortion of the hull, in order to account for the changing submerged wetted volume, resulting from the profile of the dis- turbed free surface. In addition, enhancements to the analysis, due to the influence of viscosity, by in- corporating the displacement thickness on the hull surface, are considered here. Finally, a semi-empirical theory for the water flow within the transom-stern hollow at low speeds is introduced here. This theory can approximately predict the nature of the partially ventilated flow and its resultant "back-pressure" on the transom. Introduction Previous Work Previous work on the subject of prediction of resistance of marine vehicles, such as monohulls and catamarans, has shown that the trends in the curve of total resistance with respect to speed can be predicted with excellent accuracy, using the tra- ditional linearized Michell (1898) wave-resistance theory, together with a suitable formulation for the component of frictional resistance. A recent justification for this research, in which linearized free-surface conditions are em- ployed, is the very encouraging comparisons that were made by Doctors and Renilson (1993) for monohulls and catamarans with closed or poirated sterns and by Sahoo, Doctors, and Renilson (1999) for monohulls with opera or transom sterns. Of course, there has also been much devel- opment of extensive computer codes that attempt to model, in an approximate manner, the nonlinear and viscous-wave effects. There has been excellent progress with such computer programs and they may eventually be developed to the stage where they can be used for hull-form development. Unfortunately, the execution time is too long for one to contemplate any realistic optimization of hull forms using such complex computer programs. Consequently, any realistic type of optimization is not feasible. This is because of the necessity to evaluate the object function of the vessel resistance many times during the practical design process. A further point is that more sophisticated computer codes, such as those briefly alluded to

z :` ~ l An S'> I ZTP G_ RP+~d ~ 1 _ . Hollow y ~ anon z . BY A r it, . Figure 1: Definition of the Problem (a) Geometry and Forces above, do not always lead to more accurate or re- liable predictions for numerically sensitive quanti- ties, such as resistance. This is because resistance can be affected markedly by minor inaccuracies in the computed pressure distribution over the sur- face of the hull. A revealing study of this troubling possibility was published by Sahoo, Doctors, and Renilson (1999~. It was demonstrated there that more reliable predictions for the resistance were obtained from the consistent linearized approach, than from a modern nonlinear code, for a set of fourteen modern high-speed vessels with transom sterns. Indeed, the linearized approach gave pre- dictions which were within 5% for most of the test cases, while the errors from the competing nonlin- ear method were typically an order of magnitude greater. In recent years, further improvements to this work, which dates back to the research published by Doctors and Day (1997), have been considered. That initial paper showed that a simple, but ef- fective, heuristic model of the shape of the hollow behind the transom stern would suffice to represent the flow. This model properly estimated the form of the hollow as a function of such parameters as the geometry of the transom stern and the forward speed of the vessel. This approach was a devel- opment of the original concept for the transom- stern hollow introduced by Molland, Wellicome, and Couser (1994~. Doctors and Day (2000a) included near-field effects, so that the sinkage and trim (squat) of the vessel could also be estimated, demonstrating very encouraging agreement with experiments. Later, z Inside hollower Transom I ' / == ———~] .. I.\ ..... _ Paneling Y _ ~- ~ _ Hull surface Profile ~~ t ~ I Boundary layer Figure 1: Definition of the Problem (b) Centerplane Paneling Doctors and Day (2000b) added a considerable el- ement of sophistication, by including an algorithm to iterate the shape of the transom-stern hollow so that the pressure on the surface of the hollow would be atmospheric. This approach involves ap- olYin~ the Kutta condition to the edge of the tran- som stern and also adjusting the length of the hull to minimize the deviations of the pressure from at- mospheric. Current Work The primary aim of the current work is to study the possibly of incorporating nonlinear wave effects in the analysis of the vessel. The logic is that for most practical cases of interest for a high- speed vessel, the elevation of the generated waves is small compared to their length; this should permit us to continue to employ the linearized free-surface boundary conditions, thus representing a consider- able advantage. On the other hand, the wave elevation (to- gether with the local sinkage of the vessel) can be relatively large in terms of the local draft of the vessel. Thus, it is proposed to include corrections, or distortions, to the hull geometry, so that the lon- gitudinal distribution of the immersed volume will be correct. Some initial results from this part of the work have already been presented by Doctors and Day (2002~. A secondary aim here is to implement an im- provement to the viscous model of the flow past the hull. That is, the influence of the boundary-layer 2

will be included by means of adding the displace- ment thickness to the hull. It would be anticipated that the outcome of this change is to increase the pressure drag on the vessel, because of the greater effective local beam. Finally, a tertiary aspect of the study is to develop a semi-empirical model for the water flow withers the transom-sterrz hollow at low speeds. The need for this improvement was made apparent from previous work on this subject, where it was recog- nized that assuming that the water was fully sep- arated, even at low speeds, resulted in predictions for the resistance which were impractically high. Theory Definition of the Problem Figure lta) shows a typical arrangement for a vessel traveling at a constant speed U in calm water. The x,y,z coordinate system is also de- picted. The water is unbounded laterally (in the y direction) as well as having infinite depth. The components of the forces acting on the vessel are indicated. The vessel is defined by means of the local beam bin, z). The vessel can either be self propelled or be towed. In the former case, the thrust from the pro- peller or the water jet acts along a defined line of action relative to the coordinate system attached to the vessel. Thus, the direction and position of the thrust line vary with the speed of the vessel. In the latter case, the vessel is towed at the specified speed from a particular point in the hull. Hence, the line of action of the thrust is longitudinal, but the line moves vertically in sympathy with the sinkage and trim. Discretization of the Hull Figure lobe shows how the centerolane pan- eling is used to represent the hull and the hollow in the water behind the transom stern. The panels or elements possess a flat facet and a rectangular base. They are employed, in particular, for the purpose of the numerical calculation of the pressure, or pro- file, resistance. These elements are chosen in order to approximate the centerplane area of both the hull and the hollow as closely as possible. This type of panel is algebraically simpler than the "pyramids" or "tents" which were pre- viously employed by Day and Doctors (1997) and Doctors and Day (1997), for example. The use of flat facets implies a higher level of discontinuity on the hull surface. However, numerical tests indicate that the required number of panels is not at all excessive. Potential-Flow Solution We start in the classical manner by utiliz- ing the potential function whose gradient gives the perturbation velocity. The potential satisfies the Laplace equation throughout the fluid domain. In addition, the standard linearized free-surface kine- matic and free-surface dynamic conditions are to be satisfied. The solution for the flow past the vessel and its transom-stern hollow is obtained by using the equivalent centerplane-source distribution. This distribution is assembled from panels, as noted ear- lier, while the panels themselves are constructed from the elementary Kelvin point source. The potential due to a Kelvin point source, obtained by Wehausen and Laitone (1960, p. 484, Equa- tion (13.36~), is the starting point of the analysis. Since the hull panel is assumed to be flat, it is equivalent to a constant-source distribution. Hence it is a straightforward matter to analytically integrate this influence over the area of the source panel. Similarly, this influence is also integrated analytically over the unit-constant-longitudinal- slope field panel in the so-called Galerkin manner. The result for the induced longitudinal gra- dient of the potential at the field panel was pub- lished in detail by Doctors and Day (2000a and 2000b) and will not be repeated here. It was nec- essary for this purpose to define special wave func- tions, which are closely related to the exponential integral of a complex argument. In this way, it is a possible to express the final result, in which one is only required to numerically integrate with respect to the wave angle 8. Furthermore, because of the abovemen- tioned Galerkin approach, the integrand is ex- tremely well behaved. Hence, an appropriate num- ber of points in the wave-angle integration, over the range—7r/2 < ~ < 7r/2, is just 64. 3

Symbol Theory Field Michell field integral Linear Near-field with no squat NL-1 Near-field with squat NL-2 Near-field with squat and hull distortion NL-3 Near-field with squat, hull dis- tortion and hull-pressure correc- t~on Table 1: Five Wave Theories Nonlinear Wave Effects With respect to the importance or otherwise of nonlinear wave effects, the five different theories employed for this current work are: The "Field" approach, which is based on the Michell integral together with a transom-stern hydrostatic drag correction. This method, of course, is too elementary to predict sinkage and trim. It was referred to as the simplistic approach by Doctors and Day (2000b). 2. The "Linear" near-field approach in which the actual pressure is computed, with the vessel fixed. Nevertheless, the sinkage and trim can still be computed, using the forces together with the hydrostatic stiffnesses of the vessel in sinkage and trim. 3. A partly nonlinear approach, denoted by "NL-1", in which the vessel attitude is prop- erly iterated. 4. A more nonlinear approach, introduced here and denoted by "NL-2", in which the hull is also strained, or distorted, according to the formulas: x' — x _ , A' = Y. z' = z—(. (1) (2) (3) The primed coordinates indicate their new val- ues and ~ is the local elevation of the free sur- face. 5. A further modification, denoted by "NL-3", in which the hull-surface pressure p is cor- rected so that the new pressure p' is zero on Symbol Theory Simple Simple friction line using vessel speed V Friction line using RMS velocity on hull surface V & 5* Friction line using RMS velocity on hull surface, which has been thickened by the boundary-layer displacement thickness Table 2: Three Viscous Theories Local Power Coeff- Reynolds Law accent Number n C2 Rod < 3 x 106 7 0.3709 > 3 x 106 9 0.2716 Table 3: Displacement Thickness the (strained) free surface, as required by the physics of the problem: I'd, y, z) = pox, y, z) - pox, y, 0) . (4) These different approaches are summarized Enhanced Viscous Effects In previous work by the authors, the fric- tional resistance RF was estimated using only a suitable friction line based on a semi-empirical method, such as the 1957 International Towing Tank Committee (ITTC) formula, described by Lewis (1988, Section 3.5~. In the present effort, we consider the three approaches for incorporating viscous effects which are summarized in Table 2. These approaches are: 1. The abovementioned simple friction line. 2. A modified approach, in which the root-mean- square velocity on the surface of the hull is uti- lized in place of the speed of the vessel in order to compute both the effective Reynolds num- ber and the resulting frictional force. This ap- proach can only be implemented in a method that involves the near-field computation. 4

- - Item Symbol Value Length of bow section Lbow 0.750 m Waterline beam B 0.150 m Draft T 0.09375 m Maximum-section coef. CM 0.6667 Table 4: Lego Ship Models (Common Data) 3. A further modified approach, in which the esti- mated boundary-layer displacement thickness [* is added to the local hull beam b, in order to generate a hull, with an effective local beam b", for the purpose of computing the potential- flow solution. This idea has been used in the past in order to include some viscous ejects in an otherwise potential-flow analysis. The concept is shown in Figure lobe. Thus, the effective local beam is given by the formula: b" = b + 25* . (5) To this end, the approach outlined by Dun- can, Thom, and Young (1970, pp 311 to 319), was implemented. This method provides the following equations: u/uOO = 516 = 5*15 = Chin ~ cIR-2/(n+3) 1/(n + 1), (6) (7) (8) where u is the velocity within the boundary layer, uOO is the velocity at the edge of the boundary layer, ~ is the coordinate normal to the hull surface, ~ is the boundary-layer thick- ness, ~ is the longitudinal coordinate measured back from the stem at the relevant waterline, and Ret is the corresponding local Reynolds number. The two constants, C2 and n in Equations (6) to (8), were obtained from boundary-laver measurements made by Smith and Walker (1958). Appropriate values depend on the Reynolds number and are listed in Table 3. Low-E'roude-Number Regime The assumption that the water separates cleanly at the sharp edge of the transom stern has Length ~ ~ = rat 1 0.000 0.0000 0.7500 0.6666 2 0.000 0.1875 0.9375 0.7290 3 0.000 0.3750 1.1250 0.7499 4 0.000 0.5625 1.3125 0.7290 5 0.750 0.0000 1.5000 0.8332 6 0.750 0.1875 1.6875 0.8494 7 0.750 0.3750 1.8750 0.8499 8 0.750 0.5625 2.0625 0.8275 9 1.500 0.0000 2.2500 0.8888 10 1.500 0.1875 2.4375 0.8957 11 1.500 0.3750 2.6250 0.8928 12 1.500 0.5625 2.8125 0.8735 Table 5: Lego Ship Models (Variable Data) been shown to be a reasonable approach for model- ing the physical situation at most Froude numbers of practical interest. Indeed, if one were interested only in the power requirements of such vessels in their principal operational mode, it would proba- bly be unnecessary to be concerned about the fact that this idealized model predicts a resistance equal to the hydrostatic drag at vanishingly low speeds. From a practical point of view, of course, it is often the case that a high-speed vessel must also be capable of operating at low speeds for extended periods of times, in order to conserve fuel. With this in mind, we shall now build upon the basic ideas introduced by Doctors (1998c). Figure lobe illustrates the possibility of the transom hollow being partly filled with water in the low-Froude-number regime. We shall assume that the water is essentially stagnant and that the level of the water is below that of the surrounding sea, as a consequence of the hydrodynamic suction created by the flow behind the transom. The elevation of the stagnant water in the hollow is derived from the Bernoulli equation as (troll = CPU /29, (9) where Cp is the (negative) pressure coefficient that would be generated behind the transom stern in a Is

Figure 2: Towing-Tank Models (a) Lego Ship Models 6 and 8 double-body version of the hull and g is the accel- eration due to gravity. The hydrostatic pressure force acting on the surface of the transom stern is estimated on the basis of the relative elevation of the calculation point. This contribution to the re- sistance (positive aft) is: o RH,1 = Pg Jo b(xtran' Z)(Z—(hO11) dz, (10) —1 tray in which p is the water density and Than is the draft at the transom; this is located at the longitudinal coordinate Fran Forces and Moment on the Vessel One first assumes that the attitude of the vessel is the same as its static attitude, that is, with zero sinkage and trim. The source strength is computed using the standard thin-ship approach. The total gradient of the potential at the field panel is next computed, by summing the con- tributions from the individual source panels. From this, one can determine the pressure on the surface of the hull, using the linearized Bernoulli equation. Next, the three components of the general- ized forces on the vessel (pressure resistance, sink- age force, and bow-up moment) are found from this pressure distribution, in the normal manner. All these calculations were implemented using the standard consistent linearized approximations for areas and normal vectors. Finally, the total resistance can be found by 6 ~ ~ ~ Model 12 Model 10 Figure 2: Towing-Tank Models (b) Lego Ship Models 10 and 12 the simple summation of the pressure resistance RP, the frictional resistance RF discussed earlier in this paper, and the correlation resistance RA. For experiments performed in the towing tank, we can assume that RA is zero, since the model hull is hydraulically smooth. Equilibrium of the Vessel At any stage of the iteration for the equilib- rium of the vessel, one can estimate the corrections to the sinkage and trim angle (positive bow down), with respect to the longitudinal center of flotation LCF: [sLCF = (Sp + W—Zprop)/(p9Aw) ~ (11) 5p = ~—zpRp + (up—LCF)Sp—Mp— —(xprop—LCF)Zprop + + ZpropXprop— —ZF(RF + RA)—ZstabRstab + + (LCB—LCF)W]/(WGML) ~ (12) The symbols introduced here are SO the sinkage pressure force, W the weight of the vessel, Zprop the vertical component of the propulsion force (equal to zero in the case of the vessel being towed), Aw the static waterplane area, zp the vertical lever arm for the pressure resistance (equal to zero), up the longitudinal lever arm for the pressure sinkage force (equal to—LCF), Mp the bow-up pressure moment, xprop the longitudinal lever arm for the propulsion force, Zprop the vertical lever arm for the propulsion force, Xprop the longitudinal com- ponent of the propulsion force, OF the vertical arm

14 - xl~2 12 - 10 - 1 2l ~ Curve . . 0 0 0 8- . ~ _ ~ 6- O- 2- O- _ Data Exp NL—1 NL—1 NL—1 NL—1 : _ 32 32 16 _ 32 . _ Grid 20X10 dox05 40X10 ~ (OX10 ~ I-' ox / Series = Lego Mao Model = 8 L = 2~069 m B/L = 0.07Z73 Visc = Simule _ 1 1 1 1 1 - 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 F Figure 3: Convergence Tests (a) Resistance Data Exp NL—1 NL—1 NL—1 NL—1 . 32 32 16 _ 32 . Grid 20xlO dOx~ (OX1O dOxlO _ ~ '~~Oc /o° Model = 8 ~ ° L. = 2.063 m /° B/L = 0.072713 ~hc¢~° Visc —Simple 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 Figure 3: Convergence Tests (c) Trim for the frictional force (measured to the centroid of the wetted surface), Cab the vertical lever arm to the stabilizers, Rs~ab the resistance of the stabiliz- ers (zero in the current work), LCB the longitudi- nal center of buoyancy, and GM the longitudinal metacentric height. The sinkage at the coordinate origin x = 0 and the trim are simply s = sLCF—3 ·LCF, (13) t = —L,B, (14) where L is the length of the vessel. For simplicity, the hydrostatic stiffness coef- ficients were used for iterating the sinkage and trim of the vessel, as seen in Equations (11) and (12~. The use of the ideally consistent hydrodynamic 8— xlo-s 6— - 2— O— Serisa = Logo Model = 8 L = 2063 m B/L = 0.0727~3 Visc = Simple 1 1 1 0 0.1 02 0.3 0.4 F oooo° Curve 0 0 0 0 _ Ne ~2 32 16 ~2 Grid 20X10 40x05 {OX10 {OX10 0.5 0.6 0.7 0.8 0.9 Figure 3: Convergence Tests (b) Sinkage stiffness coefficients would have posed a somewhat major computing challenge. Relative convergence of 1 x 1O-4 could be obtained within about eight iterations; once equilibrium is achieved, there is no error introduced by the simpler approach. Lego Ship Mode! Series This series of hulls was developed with the intention of studying the hydrodynamics of tran- som sterns. Doctors (1998a, 1998b, and 1998c) pro- vided the details of the hull segments from which the ship models were assembled. There was a total of seven segments. The bow and stern segments have parabolic waterplanes. The bow, stern, and parallel-middle-body segments all possess parabolic cross sections. Figure 2 shows views of four of the test models. Table 4 and Table 5 list the details of all twelve of these so-called Lego Ship Models. Results Numerical Convergence Tests The three parts of Figure 3 show a test of convergence for Lego Ship Model 8, for three phys- ical parameters of interest. These parameters are the total resistance RT, the sinkage s, and the trim t. These parameters have been rendered dimen- sionless using the weight of the ship W or its length L, as appropriate. The abscissa in the plots is the Froude number F. It can be seen that using 40 7

o.ll 0.05 G 0.2 - 0.15 0.o5l , Rr/W RH/W RP/W RF/W Rr/W Data Exp NL-1 NL-1 NL-1 NL-1 Serisa = Lego Model = 6 L = 1.688 m B/L = 0.08889 0 ° Visc = Simple on cow W --~ in: - - - - - - - - - - - - 1 1 ~, 1 1 0 0~2 0.4 0.6 0.8 1 F Figure 4: Resistance Components (a) Lego Ship Model 6 o 1 . t Rr/W RH/W RP/W RF/W Rr/W, . Data Exp NL-1 NL-1 NL—1 . NL-1 Series = Lego Model = 10 L = Z4~8 m B/L = 0.06154 Visc = Simple o o __~> ~ ~~O—o _ _ ~ ~~ _ _ _ r . _~— 1 1 1 1 1 1 1 ~ 0 0.1 0~ 0.3 Ql 0.5 0.6 0.7 0.8 0 0.1 0~2 0.~3 F Figure 4: Resistance Components (c) Lego Ship Model 10 panels longitudinally and 10 panels vertically is suf- ficient for the current purpose. Also, one requires only 32 points for each quadrant of the wave-angle ~ integration. The model length L and the beam- to-length ratio B/L are also printed on the plots. Resistance Components Figure 4 depicts the theoretical computa- tions of the various resistance components referred to earlier, for four of the ship models. These show the hydrostatic resistance RH, the pressure resis- tance RP, the frictional resistance RF, the total re- sistance RT, and the total experimental resistance. In general, the correlation between the theory and the experiments is good at the higher speeds, which are of practical significance, that is, for a Froude number F of 0.5 or greater. These results all per- 02 - Curve 0 0 0 ~ 0.15— 0.1 - 0.05 - O- _ o Data , RT/W E=p RH/W NL—1 RP/W NL—1 RF/W NL—1 RT/W NL—1 Series = Lego Model = 8 L = 2~063 m B/L = 0.072713 Visc = Simple =:~ I' - --__ ~ 0.1 02 0.13 0.4 0.5 0.6 0.7 0.8 0.9 F Figure 4: Resistance Components (b) Lego Ship Model 8 0~2- Curve ~ ~ ~ Data ~ Series = L`ego O O O O RT/W RAP Model = 12 RH/W NL—1 L = 2~813 m 0.15 - ~ RP/W NL—1 B/L = 0.05~ ~~__~ RF/W NL—1 Visc = Simple _ RT/W NL—1 0.1— Q05— O- - 9~ 1 1 1 0.4 0.5 F of 0.6 0~7 Figure 4: Resistance Components (d) Lego Ship Model 12 fain to the theory NL-1, corresponding to the sim- plest near-field theory incorporating squat. The disagreement between theory and exper- iment is greatest at the lower speeds, where the transom would in reality be partly wetted, thus re- ducing the drag. This phenomenon has been ig- nored in the current calculations. An indication of the error involved at these low speeds is just the hydrostatic resistance; it can be observed that sub- tracting this quantity would bring the theoretical calculations into line with the experimental data. This process was done in an approximate manner by Doctors (1998c). Nonlinear Wave Effects Figure 5 shows the specific resistance R/W 8

0.16 - 0.12— 0.04 Data Exp Field Linear NL—1 NL—2 NL—~ Series = Lego Model= 6 1.688 no B/L = 0.08889 Visc = Simple=,°' in',,'' 54~,,' ~ — ~ 0.16 0.12 0.08 0.04 - O— ,_ 7~ _ ~ 1 1 1 1 0 0~2 0.` 0.6 0.8 1 0 Figure 5: Comparison of Resistance (a) Lego Ship Model 6 oL.- Data Exp Field Linear NL—1 NL—2 . NL—3 Series = Lego Model = 10 L = Z4~8 m B/L = 0.06154 Visc = Simple 0.16 - 0.12 - 0.08 - QOl O— 0.6 0.7 0.8 0 - ~TT 0 0.1 02 0.S 0.4 0.5 F 1 Figure 5: Comparison of Resistance (c) Lego Ship Model 10 for the four Lego Ship Models. The different meth- ods used are listed in Table 1. It can be noted that either the Field method or the simplest nonlinear method NL-1 is consistently the best. However, it is noteworthy that nonlinear method NL-2, which incorporated hull distortion, is superior for the slen- derest models. These are Model 10 and Model 12. The dimensionless sinkage s/L is plotted in Figure 6. It is quite clear that the two superior methods are the Field method and the simplest nonlinear method NL-1. Indeed, it is disappointing to observe that the more sophisticated nonlinear methods, namely NL-2 and NL-3, are quite poor in terms of predicting linkage. Similar comments can be made about the nensionless trim t/L in Figure 7. As for the case Data Exp Field Linear NL—1 NL—2 NL—3 Series = Logo Model= 8 L = 206~3 m B/L = 0.0727~3 Visc = Simple I' K~i o O.~ 1 1 1 1 1 1 02 0.23 0.4 Q5 0.6 0.7 0.8 0.9 F Figure 5: Comparison of Resistance (b) Lego Ship Model 8 Data Exp Field Linear NL—1 NL—2 _ NL—3 Series = LeB° Model = 12 L = 2813 m B/L = 0.053 Visc = Simple ~ ," 0.1 0~2 0.13 0.4 0.5 0.6 0 F Figure 5: Comparison of Resistance (d) Lego Ship Model 12 of linkage, the two more sophisticated nonlinear theories predict excessively high values of the trim. It is gratifying, however, to see that quite reliable results can again be obtained from either of the linear or the simpler nonlinear theory NL-1. Enhanced Viscous Effects We now turn to the question of improving the correlation between the numerical predictions and the experiments, by adding some degree of so- phistication to the modeling of the viscous effects. In order to carefully study these effects, which are evidently quite small, we will first ex- amine in Figure 8 the impact on the pressure resis- tance Rp. To further clarify the phenomena, the vessel is not free to sink and trim in Figure 8. For 9

28 - X10 a 24 - 20 - 16 - 12] 8- o 20 - xlo-3 1° 16 - _ 12 - 8— I_ O- _ Curve | Data . o o o o Exp Linear NL—1 NL—2 NL—3 Series = Lego frv~~ ~Model = 6 J r L = 1.688 m 7/ ~ B/L = 0.08889 f | Visc = Simple ,,. 1,-1 ~ ~ o O O _ . 0.4 Figure 6: Comparison of Sinlcage (a) Lego Ship Model 6 Curve Data o o o o Exp _ Linear NL—1 NL—2 NL—3 roll 6/ Series = Lego j~ ~0~o° ~0_-- / v, - , ~ _~r~ ~ ~~ L = 2.4~8 m B/L = 0.06154 Visc = Simple , ^~ it, f ~ O OOZE 0 0.1 02 0.S 0.4 F l 0.5 0.6 0.7 Figure 6: Comparison of Sinkage (c) Lego Ship Model 10 the sake of brevity, the results for just two models, Lego Ship Model 6 and Model 8, are presented. The influence of adding the displacement thickness to the ship hull in order to generate the ef- fective hull, as detailed in Equation (5), does indeed increase the pressure (or potential-flow) resistance of the vessel. Furthermore, this increase operates in the hoped-for direction, in that the final result for the predicted total resistance will increase. Thus, these predictions will be brought more into line with the experimental data plotted in Figure 5. Nevertheless, the effects are indeed small and only amount to around one percent of the pressure resistance. Hence, the favorable impact of the dis- placement thickness on the total resistance will be even less on a relative basis. Data Exp Linear NL—1 NL—2 NL—3 _- 1 14 - X10 ~ 12— 10 - 8— 6— I_ 2— O— 0.8 0 Series = Lego Model = 8 ~ _ '' I! ~ ~ ~ f / f ~—~2—=~ of -cam ~ ~ L = 2063 m ,,—~ ~Q~,/ B/L = 0.07273 ~ Visc = Simple 1 1 1 1 1 1 1 1 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 F Figure 6: Comparison of Sinkage (b) Lego Ship Model 8 Data Exp Linear NL—1 NL—2 NL—3 Series = Lego ~ Jo\/ Model = 12 vA> 0OOO g ,~1 At, . f '_,_ ~—/~4' out B/L —0.05333 O Jo - °~ Visc = Simple ~ 1 1 1 1 1 0.1 0.2 0.3 0.4 0.5 0.6 0 F Figure 6: Comparison of Sinkage (d) Lego Ship Model 12 We now fix our attention onto the more real- istic case of a vessel which is permitted to sink and trim. These results are plotted in Figure 9. As in Figure 8, we present the results for only two Lego Ship Models. It can now be seen that any minor ef- fects of the displacement thickness on the pressure resistance are even less for this case. The reason for this is presumably that any slight increase in the general level of the pressure (as noted in Figure 8), will (in the real case) lift the vessel slightly out of the water. That is, the sinkage is marginally less. As a consequence, a smaller part of the hull is immersed and the final increase in the pressure re- sistance, due to the displacement thickness, is now essentially negligible. Figure 9 also differs from Figure 8 in that 10

60 - X10 50 - 40 - 30 - 20: 10 - 0] Data Exp Linear NL—1 NL—2 _ NL—:3 ~—_/ rJ I ;,~{~-o =-o-=oo-~ f ~ ~ Leo Series = Lego ~ Model = 6 0 0 0 oo~o L = 1.688 m B/L = 0.08889 vim = Timely 0 0.2 0.4 0.6 0.8 1 F Figure 7: Comparison of Trim (a) Lego Ship Model 6 40 xlO~3 Curve 0 0 0 0 30 20 - ~ _ 10— O— —10 - Data Exp Linear NL—1 NL—2 NL—:3 / ~ ,/ in_,/ ~ ~ /x /~ -I 0 ,'~°Series= Lego ,c~ Model— 10 B/L = 0.06154 Visc = Simple 1 1 1 1 1 1 0 0.1 02 0.13 0.4 0.5 0.6 0.7 0 F Figure 7: Comparison of Trim (c) Lego Ship Model 1O three curves are plotted rather than just two. The additional curve represents the intermediate case of allowing only for the variation in the velocity of the water past the hull, without the influence of the displacement thickness. Ideally, the result- ing slight difference in frictional drag will alter the equilibrium condition for the vessel, and hence the computed pressure resistance. The data shows that this effect is truly quite negligible. Low-Ffoude-Number Regime Finally, we consider the matter of the low- Froude-number theory for the assumed stagnant water in the transom hollow, as discussed earlier. The four parts of Figure 10 show results of these computations for the four Lego Ship Mod- 60 - X10 ~ 50 - 40 - 30 - 20 - 10 - O— —10 - 0 0.1 02 0.3 0.4 Q5 0.6 0.7 0.8 0.9 F Data LinEeXaPr /r ~ W/' NL—1 ~ NL—2 / Series = Lego _ NL—~ ~/ Model = 8 __ -I L = 2063 m W B/L = 0.07273 Visc = Simple 1 1 1 1 1 1 1 Figure 7: Comparison of Trim (b) Lego Ship Model 8 30— X10-3 25— 20— 15— 10— 5— O— —5— Data Exp Linear NL—1 NL—2 NL—:3 of 0 0.1 02 0.S 0.4 F Figure 7: Comparison of Trim (d) Lego Ship Model 12 f ~ f \. Series = LeB° Model = 12 L = 2.813 m B/L = 0.05~ ^~ ,~J Visc = Simple ~/ ~ , /,~ _~ 1 0.5 0.6 0.7 els under study here. It was recognized that these predictions for the drop in the water level, given by Equation (9), depend on the value of Cp, and hence the form of the hull. To this end, one can consult Hoerner (1965, Figure 21, p. 3-12~. This figure shows the base-pressure coefficient for bullet- like bodies in an unbounded flow. Typical values of Cp lie between—0.2 and—0.1, depending on the length-to-diameter ratio for this body. There are indications that the pressure coefficient can be somewhat more negative for other bluff bodies. The principal observation is that the theoret- ical model does predict the correct tendencies very well. It can be seen that the use of a pressure coeffi- cient of—0.3 works well for Lego Ship Model 6 and Model 10, for which the vessel is almost parallel- sided near the stern. On the other hand, the use of 11

40 X 10-3 Series = Lego Model= 6 L = 1.688 m B/L = 0.08889 Data = Linear /' Free = No /' / Curve | Visa / Simple Van* - l l l 1 0 0 ~ 0.4 0.6 0.8 F 36 - _ E 32 - A: 28 - 24 20 - Figure 8: Viscous Ejects without Squat (a) Lego Ship Model 6 52 xio-3 48 - 44 - 40- 36- :32] 28- 24— 20 - _ _ Series = Lego Model = 6 L = 1.688 m B/L = 0.08889 ~/ Data = NL—1 Free = Yes l 0 02 0.4 F 0.6 0.8 1 Figure 9: Viscous Effects with Squat (a) Lego Ship Model 6 a pressure coefficient of—0.2 works well for Lego Ship Model 8 and Model 12, where there is a strong closing in of the hull ahead of the transom. Concluding Remarks The work presented in this paper has shown: 1. The nonlinear theories failed on the most part to improve the correlation between the predic- tions and the experimental data. Generally speaking, either the field approach or the sim- plest nonlinear theory provided very workable correlation. That is, the idea of applying dis- tortion to the hull body in order to generate a more correct longitudinal distribution of the immersed hull volume usually gave worse pre- 1 12 32 - xlo-s 28 - ~ 24- R 20- 16 - 12 - 8- Series = IRgo Model= 8 L = 2063 m B/L = 0.07273 Data = Linear Free = No | Curve | Visa | — 1~ 1 0 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 F Figure 8: Viscous Effects without Squat (b) Lego Ship Model 8 4.5— X10 ~ 40— 35— ~ 30- 20— 15 - 10 - 5— Series = Len Model = 8 L = 2~063 m B/L = 0.07273 of Data = NL—1 Free = Yes i/ Curve Vise Simple ~ ..~ ~. Y Van* l l l l l l l 0 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 F Figure 9: Viscous Effects with Squat (b) Lego Ship Model 8 dictions for resistance, linkage, and trim. I. The viscous-correction model marginally im- proved the accuracy for the prediction of resis- tance, by increasing the values of these predic- tions. Hence, the correlation with model tests could be made better. Because the boundary layer is relatively thinner for a prototype ves- sel, this effect can be ignored at full scale. It should be emphasized here that the cor- rection given by Equation (5) could be made normal to the hull surface, rather than in the transverse direction. Similarly, one could de- fine ~ in Equation (7) as the distance around the waterline from the stem at the relevant draft, rather than as the longitudinal distance. Both these changes would slightly increase the impact of the displacement thickness on the pressure resistance.

0.1 0.08 - 0.04 - /~-' it' 0 0~ 0.4 0.6 Lea = Lego Model = 6 L = 1.688 m B/L = 0.08889 vise = v & 6* 1 0.8 1 Figure 10: Model for lYansom-Stern Hollow (a) Lego Ship Model 6 0.16- o _ Curare Data C Series = Lego LD Exp Model = 10 NL—1 —0.2 L = 2~4~8 m B/L = 0.06154, Visa = Vim* 0.04 - O- _ 0 0.1 o 1 1 02 0.3 0.4 0.5 0.6 0.7 0.8 F Figure 10: Model for liansom-Stern Hollow (c) Lego Ship Model 10 3. A frictional-resistance form factor of unity has been used throughout. It is clear from a study of Figure 4 that a factor slightly greater than unity, which increases with the beam-to-length ratio, would greatly improve the numerical predictions for the total resistance. 4. The low-Froude-number theory for the tran- som flow was able to predict the correct trends in the resistance curve at low speeds. This the- ory provides predictions for the total resistance that are vastly superior to those resulting from the simpler assumption of the fully-ventilated transom stern. Regarding the low-Froude-number theory, it should be worthwhile to investigate this matter fur- ther. In particular, it would appear that near- perfect correlation with towing-tank experiments 0.16 - Curve Data _ o o o o Exp NL—~ 02 0.12- ~ NL-1 —.S 0 / NL—1 Full / 0.08- ~ o.o4_ ~ , _ o 0 0.1 02 0.S 0.4 0.5 F Series = Lego Model = 8 L = Z0613 m B/L = 0.07273 Visa = V & 6* 1 0.6 0.7 0.8 0.9 Figure 10: Model for Tr~nsom-Stern Hollow (b) Lego Ship Model 8 Ql6- Curve Data — Series = Lego o o o o Exp Model = 12 _ NL-1 - ~ L = 2~813 m 0.12 - ~ NL—1 —.S B/L = 0.051333 NL—1 Full Visa = V & 6* 0.08 - 0.04- ~ o- 1~1 0 0.1 0.2 0.S 0.4 F i< - 0~ I_ 1 0.5 Q6 0.7 Figure 10: Model for Trar~som-Stern Hollow (d) Lego Ship Model 12 will be obtained, if one could just develop a sim- ple approximate dependence of the transom-stern pressure coefficient on the form of the ship hull (the longitudinal rate of change of the sectional area) just ahead of the transom stern. Further insight into this matter was pro- vided by Oving (1985), who claimed that the beam-to-draft ratio of the transom plays an impor- tant role in answering this question. An alterna- tive approach, using computational-fluid mechan- ics (CFD), may also provide understanding of the transom-stern-hollow flow. Finally, the matter of the towing-tank exper- iments should be raised. To this end, additional tests are planned in order to gauge the accuracy of some of the experimental data presented here.

Acknowledgments The authors would like to acknowledge the assistance of the Australian Research Council (ARC) Discover-Project Grant Scheme (via Grant Number DP0209656~. The in-kind support of this work by The Uni- versity of New South Wales and The Universities of Glasgow and Strathclyde is also greatly appreci- ated. References DAY, A.H. AND DOCTORS, L.J.: "Resistance OR timization of Displacement Vessels on the Basis of Principal Parameters", J. Ship Research, Vol. 41, No. 4, pp 249-259 (December 1997) DOCTORS, L.J.: "Modifications to the Michell Integral for Improved Prediction of Ship Resis- tance", Proc. Twenty-Seventh Israel Conference on Mechanical Engineering, Technion, Haifa, Is- rael, pp 502-506 (May 1998) DOCTORS, L.J.: "Intelligent Regression of Resis- tance Data for Hydrodynamics in Ship Design", Proc. T~uenty-Second Symposium on Naval Hydro- dynamics, Washington, DC, pp 33-48, Discussion: 49 (August 1998) DOCTORS, L.J.: "An Improved Theoretical Model for the Resistance of a Vessel with a Transom Stern", Proc. Thirteenth Australasian Fluid Me- chanics Conference (13 AFMCJ, Monash Univer- sity, Melbourne, Victoria, Vol. 1, pp 271-274 (De- cember 1998) DOCTORS, L.J. AND DAY, A.H.: "Resistance Pre- diction for Transom-Stern Vessels", Proc. Fourth International Conference on Fast Sea Transporta- tion (FAST '97J, Sydney, Australia, Vol. 2, pp 743- 750 (July 1997) DOCTORS, L.J. AND DAY, A.H.: "The Squat of a Vessel with a Transom Stern", Proc. Fif- teenth International Workshop on Water Waves and Floating Bodies (15 IWWWFB), Caesarea, Is- rael, pp 40-43 (February-March 2000) DOCTORS, L.J. AND DAY, A.H.: "Steady-State Hydrodynamics of High-Speed Vessels with a Tran- som Stern", Proc. Twenty-Third Symposium or Naval Hydrodynamics, Val de Reuil, France, pp 12- 1-12-14, Discussion: p 12-15 (September 2000) DOCTORS, L.J. AND DAY, A.H.: "Nonlinear Effects on the Squat of a Vessel with a Tran- som Stern", Proc. Seventeenth International Workshop on Water Waves and Floating Bodies (17 IWWWFBJ, Cambridge, England, 4 pp (April 2002) DOCTORS, L.J. AND RENILSON, M.R.: "The In- fluence of Demihull Separation and River Banks on the Resistance of a Catamaran", Proc. Second In- ternational Conference on Fast Sea Transportation (FAST '93J, Yokohama, Japan, Vol. 2, pp 1231- 1244 (December 1993) DUNCAN, W.J., THOM, A.S., AND YOUNG, A.D.: Mechanics of Fluids, Edward Arnold (Publishers) Ltd. London, 725+xiv pp (1970) HOERNER, S.F.: Fluid-Dynamic Drag, Ho- erner Fluid Dynamics, Brick Town, New Jersey, 438+xiv pp (1965) Lewis, E.V. (ED. ) Principles of Naval Architec- ture: Volume II. Resistance, Propulsion and Vibra- tion, Society of Naval Architects and Marine Engi- neers, Jersey City, New Jersey, 327+vi pp (1988) MICHELL, J.H.: "The Wave Resistance of a Ship", Philosophical Magazine, London, Series 5, Vol. 45, pp 106-123 (1898) MOLLAND, A.F., WELLICOME, J.F., AND COUSER, P.R.: "Theoretical Prediction of the Wave Resistance of Slender Hull Forms in Catama- ran Configurations", University of Southampton, Department of Ship Science, Report 72, 24+i pp (March 1994) OVING, A.J.: "Resistance Prediction Method for Semi-Planing Catamarans with Symmetrical Demi- hulls", Maritime Research Institute Netherlands (MARIN), Wageningen, 79+i pp (September 1985) SAHOO, P.K., DOCTORS, L.J., AND RENILSON, M.R.: "Theoretical and Experimental Investi- gation of Resistance of High-Speed Round-Bilge Hull Forms", Proc. Fifth International Conference on Fast Sea Transportation (FAST '99J, Seattle, Washington, pp 803-814 (August-September 1999) SMITH, D.W. AND WALKER, J.H.: "Skin-Friction Measurements in Incompressible Flow", National Advisory Committee for Aeronautics, Tech. Note 4231, 67+i pp (March 1958) WEHAUSEN, J.V. AND LAITONE, E.V.: "Surface Waves", Er7cyclopedia of Physics: Fluid Dynam- ics III, Ed. by S. Flugge, Springer-Verlag, Berlin, Vol. 9, pp 445-814 (1960) 4

DISCUSSION A.F. Molland University of Southampton, United Kingdom I agree with authors that are difficult to make significant changes to basic results of linearised thin ship (Michell) theory. Transom stem correction is important as it influences trim. Current work at Southampton is using a panel method t1] to find sinkage and trim based on a hull waterline elevation from thin ship theory. Results to date are encouraging with the panel code giving the full surface pressure and velocity distribution. Have the authors made wave cuts for these models to make sure that they are getting the correct resistance for the correct reason? If wave form is not correct then this will cause problems for investigations of wash effects. Would the authors concur that a viscous form factor approach for resistance estimate is more appropriate for their method given implicit limitations of thin ship approach? What choices would the authors make for a prior prediction of the resistance and squat of a transom stern vessel and what confidence levels would they ascribe to that prediction? L1] Turnock, S.R., "Palisupan: user guide and technical manual," Ship Science Report No. 100, University of Southampton, 1997. AUTHORS' REPLY I would like to thank Dr Molland for his interesting questions. Dr Molland has done much original research into the hydrodynamics of high-speed vessels, in particular catamarans, so it is a pleasure to receive a considered question from him. I understand that much of our work parallels that of Molland and his colleagues; in particular, the hydrodynamic model for the transom-stern hollow that we currently use was inspired by his early contributions to this field. At the time of the Fukuoka conference, wave cuts had not been made for our models, although these were at the planning stage. Since then, I have tested a series of catamaran models in the Ocean Basin at the Australian Maritime College. In these tests, various demi-hull spacings, water depths, and model speeds were tested. In each case, the wave elevation along a set of eight longitudinal wave cuts was recorded. While the analysis of these results is still at an early stage, we have demonstrated that the associated wave-elevation-prediction computer program provides wave profiles which are verified by the experiments within a few percent. Differences are mainly in the phasing of the waves rather than in their elevation and, hence, the energy density in the wave system and the resulting wave resistance is expected to be predicted well - as demonstrated by the total resistance predictions presented in our paper. We agree that a viscous form factor, applied to a traditional friction line, can be used to improve the correlation between theory and experiment. Generally speaking, we have found that a viscous form factor between 1.05 and 1.10 and a wave-resistance form factor between 0.90 and 0.95 provide extremely close correlation, which can be within 5% for practical hull forms. In conclusion, experience over a five-year period indicates that this correlation can be achieved with the straightforward application of the Michell (1898) theory, provided the transom hollow is included in the formulation. The far more complicated near-field calculation provides predictions of sinkage and trim, which are probably sufficient for practical purposes, but the accuracy in percentage terms is not as high as that for the Michell far-field resistance predictions.