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.

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PRACTICAL CFD APPLICATIONS TO DESIGN OF A WAVE CANCELLATION MULTIHULL SHIP 206 Practical CFD Applications to Design of a Wave Cancellation Multihull Ship Chi Yang1, Francis Noblesse2, Rainald LÃ¶hner1, Dane Hendrix2 1 Institute for Computational Sciences and Informatics George Mason University, Fairfax VA 22030â4444, USA 2 David Taylor Model Basin, CD-NSWC 9500 MacArthur Blvd, West Bethesda MD 20817â5700, USA ABSTRACT Four methods of analysisâa nonlinear method based on Euler's equations and three linear potential flow methodsâ are used to determine the optimal location of the outer hulls for a wave cancellation multihull ship that consists of a main center hull and two outer hulls. The three potential flow methods correspond to a hierarchy of simple approximations based on the Fourier-Kochin representation of ship waves and the slender-ship approximation. INTRODUCTION This study considers an illustrative practical application of CFD tools to a simple ship design problem. This simple design case is the wave cancellation multihull ship concept examined in [1], where experimental measurements and theoretical calculations based on Michell's thin-ship approximation are given. The wave cancellation multihull ship that is considered consists of a main center hull and two identical outer hulls centered at with respect to the center of the waterplane of the main hull. The main center hull of the multihull ship considered in [1] and the present study has a length The main hull and the outer hulls are defined in the Appendix. The study considers the elementary design problem of determining the optimal location of the outer hulls with respect to the main center hull, i.e. the optimal values of the two parameters LX and LY, for the purpose of minimizing the wave drag of the ship. Four methods of analysis are used and compared to one another and to experimental data. One of the methods is the nearfield flow calculation method presented in [2] and [3]. This method is based on the Euler equations and the nonlinear free-surface boundary condition. The other three methods are linear potential flow methods that correspond to a hierarchy of simple approximations based on the Fourier-Kochin representation of ship waves [4] and the slender-ship approximation [5,6]. HAVELOCK AND FOURIER-KOCHIN REPRESENTATION OF WAVE DRAG Consider a ship advancing along a straight path, with constant speed U, in calm water of effectively infinite depth and lateral extent. The flow is observed from a Cartesian system of coordinates moving with the ship. The X axis is taken along the path of the ship and points toward the ship bow; i.e., the ship advances in the direction of the positive X axis. The Z axis is vertical and points upward, and the mean free surface is the plane Z=0. The flow appears steady in the translating system of coordinates, and consists of the disturbance flow due to the ship superimposed on a uniform stream opposing the ship's forward speed. The components of the disturbance velocity along the (X, Y, Z) axes are (U, V, W). Thus, the total velocity is given by (UâU, V, W). Nondimensional coordinates and velocities are defined in terms of a characteristic length L, taken and the ship speed U as here as Define the Froude number F and v as the authoritative version for attribution. Here, g is the acceleration of gravity. The drag DW=ÏU2L2CW associated with the wave energy transported by the waves trailing the ship can be determined from the Havelock formula (1) Here, the wavenumber k: is defined in terms of the Fourier variable Î² by (2a)

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PRACTICAL CFD APPLICATIONS TO DESIGN OF A WAVE CANCELLATION MULTIHULL SHIP 207 Furthermore, Sr and Si are the real and imaginary parts of the spectrum function S=S( Î±, Î²). Here, Î± is defined in terms of the Fourier variable Î² by (2b) F2Î±2=k: The relations (2) follow from the dispersion relation with The velocity representation of free-surface flows and the related Fourier-Kochin representation of waves given in [4] define the spectrum function S in terms of the velocity distribution at the ship hull surface (or more generally at a boundary surface that surrounds the ship). Specifically, [4] and [7] define the spectrum function S in terms of a distribution of elementary waves exp[kzâi(Î±xâÎ²y)]: (3a) with (3b) (3c) Here, Î£ stands for the mean wetted ship hull surface (or a boundary surface that surrounds the ship) and is the intersection curve between the surface Î and the mean free-surface plane z=0. Furthermore, and respectively stand for the differential elements of area and arc length of Î£ and Î at the integration point Finally, the amplitude functions A and A Î are defined in terms of the boundary velocity distribution by (3d) (3e) is the unit vector normal to the ship hull surface Î£ ; points inside the mean flow domain Here, (i.e. outside the ship). The unit vectors and are tangent and normal, respectively, to the ship waterline Î; is oriented clockwise (looking down) and points outside the ship, like Finally, are the x and y components of the vector Expressions (3) define the spectrum function S in the Havelock integral (1) in terms of the normal components of the velocity at the ship hull surface Î£ âª Î . and and the tangential components and APPLICATION TO MULTIHULL SHIPS âª Î consists of N component surfaces, i.e If the ship hull surface Î£ (4a) the spectrum function S can be expressed as centered at (4b) with (4c) In the particular case of a ship consisting of three hulls, a center hull centered at (0, 0, 0) and two identical outer hulls centered at (4b) becomes If the outer-hull spectrum functions are assumed to be identical, we have Here, and stand for the spectrum functions associated with the center hull and an outer hull, respectively. the authoritative version for attribution. Define (5a) are defined by (3b) and (3c) in which Î£ , Î are taken as where the spectrum functions and or Î£o, Îo. The real and imaginary parts of the spectrum function S are then given by These expressions yield

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PRACTICAL CFD APPLICATIONS TO DESIGN OF A WAVE CANCELLATION MULTIHULL SHIP 208 We thus have with Here, A, AR and Al are defined as (5b) (6a) The wave drag CW can then be expressed as (6b) Here, and are given by (6c) and represent the wave drags of the center hull and of an outer hull, respectively. The component accounts for interference effects and is defined as (6d) with A, AR and Al given by (5b). The wave-interference coefficient Ci given by (6e) can be defined in terms of the wave drag coefficients and FOURIER-KOCHIN REPRESENTATION OF NEAR-FIELD STEADY SHIP WAVES The velocity representation of free-surface flows and the related Fourier-Kochin representation of waves expounded in [4] and [8] show that, within the framework of potential-flow theory, the velocity field generated by a ship can be (7a) decomposed as Here, and respectively represent a wave component and a local component that are associated with the (7b) decomposition of the Green function G associated with the free-surface boundary condition wâF x 2u =0. The wave component is given by a single Fourier integral. Specifically, the ship hull Î£ (which here stands for the center hull Î£c or the outer hull Î£ o) is divided into a set of patches Î£ p associated with reference points (xp, yp, zpâ¤0) located in the vicinity of Î£ p. The patch reference points (xp, yp, zpâ¤0) attached to the patches Î£p need not lie on p. The size of the patches in the x direction is O(ÏF2) as required by the function Î defined by (9). The wave component at a field point is defined in [7] as (8) Here, Î²c is a large positive real constant, and the Fourier variable Î± and the wavenumber are the functions of the Fourier variable Î² given by (2). Furthermore, the function Îp is defined as (9) with Ï a positive real constant. Summation in (8) is performed over all the N patches Î£ p âª Î p that represent the surface Î£ âª Î , i.e. (9) shows that Îp â 0 as (xp Î¾)/( ÏF 2) â â . Thus, the contribution of a patch Î£p to the wave component is located at a distance O(ÏF2) ahead of the reference point attached to Î£ p. negligible at a point The functions in (8) are defined as (10a) the authoritative version for attribution. (10b) (10c) (10c) with Furthermore, and in (10) are defined as (11a) associated with the patch Î£ p. where and are the real and imaginary parts of the spectrum function

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PRACTICAL CFD APPLICATIONS TO DESIGN OF A WAVE CANCELLATION MULTIHULL SHIP 209 The spectrum function Sp is given by distributions of elementary waves over Î£p âª Îp . Specifically, (3a)â(3c) yield (11b) (11c) (11d) The amplitude functions A Î£ and A Î are given by (3d) and (3e). At a distance O(ÏF2) behind the ship, (9) yields p â 2 and the foregoing representation of the wave component can be simplified as in [7]. This simplified representation, which does not require subdivision of the ship hull (or more generally boundary surface) Î£ âª Î into patches Î£p âª Î p, may be used to extend a nearfield flow past a ship into a farfield region entirely located behind the ship [7]. The more general representation of given here, on the other hand, is valid in the entire flow domain, and thus can be used to compute nearfield waves. SLENDER-SHIP APPROXIMATION The slender-ship approximation expounded in [5] and [6] defines the velocity field generated by a ship explicitly in terms of the speed and shape of the ship: (12) The slender-ship approximation (12) may be regarded as a generalization of the Michell thin-ship approximation. Specifically, the Michell approximation differs from (12) in that it defines in terms of a distribution of sources, with strength 2nx, over the ship centerplane y=0 instead of a distribution of sources of strength nx over the port and starboard sides of the actual ship surface Î£ . In addition, there is no distribution of sources around the waterline Î in Michell's approximation. The slender-ship approximation to the wave component in the decomposition (7a) is given by the Fourier-Kochin representation defined by (8)â(11) with the amplitude functions AÎ£ and A Î in (11c) and (11d) taken as (13) can be effectively evaluated from (12) with the Green function G taken as the simple The local component analytical approximation to the local component GL given in [9] or the simpler approximation (14) Here, r, râ² , râ³ are defined as The approximation (14) yields as dictated by the free-surface boundary condition wâ F2ux=0. At the ship hull surface Î£ , the velocity field given by the slender-ship approximation (12) can be modified using the transformation (15) This transformation yields and thus ensures that the hull boundary condition is satisfied exactly by the velocity distribution at the hull surface. The hull-condition transformation (15) can be applied to any velocity distribution computed at a ship hull surface. In particular, (15) shows that the velocity distribution associated with the trivial velocity distribution is given by This velocity distribution is normal to the ship surface Î£ (the tangential velocity component is null), and evidently satisfies the hull boundary condition The velocity distribution corresponds to the slender-ship approximation (12) and (13). the authoritative version for attribution. FOUR METHODS OF ANALYSIS CW The wave drag coefficient defined by (6a)â(6d) and (5) account for interference effects of the farfield waves generated by the center hull and the two outer hulls. The spectrum functions Sc and So in (5) are defined by (3) in terms of the normal and tangential components of the velocity distributions at the center hull Î£ c and the outer hulls Î£o. The velocity distributions at Î£ c and Î£o are affected by nearfield flow interactions between the center hull and the outer hulls. Thus, the wave drag C W defined by (6a)â(6d), (5), and (3) account for both farfield wave-interference effects and nearfield flow interactions. Evaluation of the wave drag CW requires evaluation of the nearfield velocity distribution at Î£c and Î£o for a range of values of the Froude number F and of the parameters a and b

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PRACTICAL CFD APPLICATIONS TO DESIGN OF A WAVE CANCELLATION MULTIHULL SHIP 210 that define the location of the outer-hull. This computational task is daunting using typical calculation methods but can actually be easily performed using the slendership approximation. Specifically, a single-loop set of computations of the wave drag C W associated with the slender-ship approximation to the nearfield velocity distribution at Î£ c and Î£o for 4 values of the Froude number, 61 values of a and 26 values of b, corresponding to 4Ã61Ã26=6,344 nearfield flow calculations (using 11,525 panels to approximate the three hulls) requires 18 hours of CPU time using an SGI origin 2000 with 4 processors. This approach is identified as method 3 hereafter. Considerable simplifications are obtained if nearfield flow interactions are ignored, i.e. if the velocity distribution at the center hull Î£c is evaluated for the center hull alone (i.e. without the two outer hulls) and the velocity distribution at an outer hull is similarly evaluated for the outer hull alone (i.e. without the center hull and the other outer hull). This approximation, which accounts for farfield wave interference effects but neglects nearfield flow interactions, only requires two nearfield flow evaluations (one for the center hull, and one for an outer hull) per Froude number, a task that can easily be performed using existing calculation methods. The spectrum functions Sc and So defined by (3) and the related functions A, AR and A l given by (5) similarly need be evaluated only once per Froude number. These spectrum functions, like the velocity distributions at the three hulls, are independent of the parameters a and b, which only appear in (6d), within the ânegligible nearfield interactionâ approximation. Thus, this approximation effectively uncouples the outer-hull location parameters (a, b) and nearfield flow calculations. For the purpose of estimating the importance of nearfield flow interactions upon the wave drag CW, the nearfield velocity distribution is evaluated here using the slender-ship approximation already used in method 3. This approach is identified as method 2 hereafter. Thus, comparison of methods 2 and 3 provides insight into the importance of nearfield flow interaction effects. Method 2 corresponds to the first-order slender-ship approximation defined in [5]. Insight into the importance of using a sophisticated nearfield flow calculation method can be gained by comparing method 2 and method 1, which corresponds to the zeroth-order slender-ship approximation in [5] and to the trivial approximation in (3d) and (3e). Thus, no nearfield flow calculation is required in this simplest approximation. Indeed, the zeroth- order slender-ship approximation is associated with the trivial approximations and for the tangential components of the velocity at the ship hull Î£ and water line Î , as previously explained. Methods 1, 2, and 3 are based on the Fourier-Kochin representation of ship waves, i.e. on linear potential flow, and the further simplification associated with the slender-ship approximation. Thus, even method 3 only accounts for effects of nearfield flow interactions in an approximate manner. However, nearfield flow interactions are fully taken into account in the nearfield flow calculation method presented in [2] and [3], which is based on the Euler equations and the nonlinear free-surface boundary condition. This method is identified as method 4. RESULTS OF ANALYSIS Figure 1 depicts the experimental values of the residuary drag coefficient CR given in [1] and the corresponding predictions of the wave drag coefficient C W given by methods 1, 2, 3, and 4 for four arrangements of the outer hulls. These four arrangements of the outer hulls correspond to a= â 0.128, â0.205, â0.256, â 0.385 and b=0.136 (the same value for all four cases). The left column in Fig. 1 compares CR and CW predicted by methods 1 and 2. The right column shows CR and C W given by methods 2 and 3, and by method 4 at F=0.25, 0.3, 0.4, 0.5. Differences between the values of CW predicted by methods 1 and 2 (left column) and between methods 2 and 3 (right column) are fairly small, and these 3 methods yield values of CW that are in fair agreement with the experimental values of CR. In particular, the variation of C R with respect to the Froude number F is well captured by the theory. The values of C W given by method 4 at F=0.25, 0.3, 0.4, 0.5 are in even better agreement with CR on the whole, and are in fairly good agreement with CW predicted by methods 2 and 3. Figs 2a,b,c,d compare the values of C W given by methods 1, 2, 3 for F=0.5, 0.4, 0.3, 0.25 and outer-hull arrangements within the region (16) For a=0.75, the sterns of the outer hulls are aligned with the bow of the main center hull; similarly, the bows of the outer hulls are aligned with the stern of the center hull if a= â0.75. The contour plots of CW depicted in Figs 2aâd are based on calculations for increments in the values of the outer-hull location parameters a and b equal to âa=0.025 and âb=0.01. Thus, the contour plots of CW in Figs 2 correspond to evaluations of CW using methods 1, 2, 3 for 4 values of the Froude number F, 61 values of a and 26 values of b, i.e. 3Ã4Ã61Ã26=19,032 evaluations. Lowest values of C W, corresponding to best outer-hull arrangements, are indicated in blue, and highest values of CW, i.e. worst the authoritative version for attribution.

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PRACTICAL CFD APPLICATIONS TO DESIGN OF A WAVE CANCELLATION MULTIHULL SHIP 211 outer-hull arrangements, are indicated in red. Figs 2 indicate that methods 1, 2, and 3 predict best (blue regions) and worst (red regions) outer-hull arrangements that are in fairly good agreement. Fig. 3 shows the values of the wave drag coefficient CW given by methods 1, 2, 3, 4 at the three best distinct outer-hull arrangements, i.e. for (ak, bk) with k=1, 2, 3, predicted by method 2. This figure shows that, although the values of C W predicted by methods 1, 2, 3, 4 are not identical, these four methods yield The results depicted in Figs 1, 2, and 3 indicate that methods 1 and 2, which are computationally more efficient than methods 3 and 4, may be used for the purpose of determining optimal arrangements of the outer hulls. Method 2 is then used to further study the best and worst outer-hull arrangements. Fig. 4 depicts the variation of the wave drag coefficient CW , for F=0.3 and F=0.5, within the region Consideration of this large region, which greatly extends the region (16) of practical interest for the design of a wave cancellation multihull ship, provides a broader view of the variation of the wave drag coefficient CW within the smaller region (16) examined in Figs 5a,b. Figs 5a,b show the variation of the wave drag coefficient CW predicted by method 2 within the region (16) with âa=0.025 and âb=0.01, as in Figs 2aâd, for 20 values of the Froude number F in the range 0.2147â¤Fâ¤0.5426. Thus, Figs 5a,b present the result of 61Ã26Ã20=31,720 evaluations of CW . As in Figs 2, lowest and highest values of CW , i.e. best and worst outer-hull arrangements, correspond to blue and red regions in Figs 5. The left side of Fig. 6 depicts the variation, with respect to the Froude number F, of the wave drag coefficient and of the wave drag coefficients and corresponding to the best and worst outer-hull arrangements found (using method 2) within the region The right side of Fig. 6 depicts the interference coefficients and defined by (6e) corresponding to the best and worst outer-hull arrangements considered in the left side of Fig. 6. The large differences between and and between and shown in Fig. 6 demonstrate the importance of selecting a favorable outer-hull arrangement and the benefits that can derived from an optimal arrangement. Indeed, the left side of Fig. 6 shows that the ratio approximately varies between 2 and 6. The lower part of Fig. 7 depicts the experimental values of CR given in [1] for four arrangements of the outer hulls and the wave drag coefficient already shown in the left side of Fig. 6, corresponding to the best outer-hull arrangement (which varies with respect to the Froude number) given by method 2. As one expects, the wave drag coefficient is smaller than the experimental values of CR (which do not correspond to optimal arrangements of the outer hulls over the entire range of Froude numbers). The lower part of Fig. 7 also shows the values of the wave drag coefficient C W that correspond to near-best arrangements of the outer hulls. These near-best arrangements are defined here as those outer-hull arrangements for which CW does not exceed by more than 10%. The values of the outer-hull location parameters a and b corresponding to the best and near-best arrangements considered in the lower part of Fig. 7 are depicted in the upper part of Fig. 7. The upper part of Fig. 7 also shows the values of a and b for which experiments are reported in [1] and in the lower part of Fig. 7. CONCLUSION Illustrative practical applications of CFD tools to the wave cancellation multihull ship concept examined in [1] have been summarized. The wave cancellation multihull ship considered in [1] and here consists of a main center hull and two identical outer hulls centered at with respect to the center of the waterplane of the center hull. The elementary design problem of determining the optimal arrangement of the outer hulls with respect to the main center hull, i.e. the optimal values of the two parameters a and b, for the purpose of minimizing the wave drag of the ship has been considered using four methods of analysis. One of the four methods is the nearfield flow calculation method presented in [2] and [3]. This method is based on the Euler equations and the nonlinear free-surface boundary condition. The other three methods are linear potential flow methods that correspond to a hierarchy of simple approximations based on the Fourier-Kochin representation of ship waves [4] and the slendership approximation [5,6]. These four methods of analysis have been compared to one another and to the experimental data given in [1] for the purpose of establishing and validating a practical methodology that can be used for more complex hull-form design problems involving minimization of wave drag. In addition to the problem of selecting optimal outer-hull arrangements considered here, a realistic multihull ship design problem involves optimal selections of the lengths, beams, drafts, and shapes of the main center hull and the outer hulls, which have been taken as in [1] for the purpose (of main interest for the authoritative version for attribution.

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as APPENDIX 212 the present study) of comparing and validating alternative methods of analysis of main interest in the present study. Furthermore, constraints associated with mission requirements, structural considerations, seakeeping, and course keeping must evidently be considered. The three methods based on the Fourier-Kochin representation of ship waves and the slender-ship approximation, especially the methods (called methods 1 and 2) corresponding to the zeroth-order and first-order slender-ship approximations given in [5], provide simple and highly efficient tools. These practical tools have been shown to be adequate for the purpose of determining optimal locations of the outer hulls. Method 4, based on a more refined flow analysis, can then be used effectively to further evaluate the flow at the optimal outer-hull arrangement. The use of a pragmatic approach that relies on a combination of simple and more refined tools evidently is a well- established practice. In particular, the zeroth-order slender-ship approximation (i.e. method 1) had previously been used with success for hull-form optimization in [10] and [11]. Thus, the practical usefulness of this remarkably simple approximation is confirmed in the present study. ACKNOWLEDGEMENTS The work of Yang and LÃ¶hner was partially funded by AFOSR (Dr. Leonidas Sakell technical monitor) and by NRL LCP&FD (Dr. William Sandberg technical monitor). The work of Noblesse and Hendrix was supported by the ILIR program at NSWC-CD. All computer runs were performed on a 128-Processor R10000 SGI Origin 2000 at the Naval Research Laboratory. REFERENCES [1] M.B.Wilson, C.C.Hsu & D.S.Jenkins (1993) Experiments and predictions of the resistance characteristics of a wave cancellation multihull ship concept, 23rd American Towing Tank Conf., 103â112 [2] R.LÃ¶hner, C.Yang, E.OÃ±ate & S.Idelssohn (1999) An unstructured grid-based, parallel free-surface solver, Applied Numerical Mathematics 31: 271â 293 [3] C.Yang & R.LÃ¶hner (1998) Fully nonlinear ship wave calculation using unstructured grid and parallel computing, 3rd Osaka Coll. Advanced CFD Applications to Ship Flow and Hull Form Design, Osaka, 125â150 [4] F.Noblesse (2000) Velocity representation of free-surface flows and Fourier-Kochin representation of waves, s ubmitted [5] F.Noblesse (1983) A slender-ship theory of wave resistance, Jl Ship Research 27:13â33 [6] F.Noblesse & G.Triantafyllou (1983) Explicit approximations for calculating potential flow about a body, Jl Ship Research 27:1â12 [7] C.Yang, R.LÃ¶hner & F.Noblesse (2000) Farfield extension of nearfield steady ship waves, Ship Technology Research 47:22â34 [8] F.Noblesse, X.B.Chen & C.Yang (1999) Generic super Green functions, Ship Technology Research 46:81â92 [9] F.Noblesse, C.Yang & D.Hendrix (2000) Steady free-surface potential flow due to a point source, 15th Intl Workshop on Water Waves & Floating Bodies, Israel [10] J.S.Letcher Jr., J.K.Marshall, J.C.Oliver III & Nils Salvesen (1987) Story & Stripes, Scientific American, 257:34â40 [11] D.C.Wyatt & P.A.Chang (1994) Development and assessment of a total resistance optimized bow for the AE 36, Marine Technology 31:149â160 APPENDIX The outer hulls are considered first. Let 2Lo, 2Bo, and Do stand for the length, beam, and draft of an outer strut, which consists of a parallel midbody and parabolic sharp-ended nose and tail regions. Define a system of coordinates (Xo, Y o, Zo) with origin at the center of the outer strut. The upper part of the strut is given by with defined as These equations hold for The lower part of the strut is given by Thus, the outer hulls are defined by the five parameters Lo, Bo, Do, with and The centers of the water planes of the two outer struts are located at X=âLX and with respect to the center of the water plane of the main center hull. Define the system of coordinates (X, Y, Z) with origin at the the authoritative version for attribution. center of the water plane of the center hull. Thus, the coordinates (X, Y, Z) corresponding to the outer-strut coordinates (Xo, Yo, Zo) are given by for the outer struts centered at . The outer-strut surfaces can be defined in terms of the

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as APPENDIX 213 center-hull coordinates via the foregoing coordinate transformations. The main center hull is now considered. Let and stand for the length, beam, and draft of the center hull. The local beam (widest waterline) of the center hull is given by with defined as with B0 defined as The top waterline of the center hull is given by Thus, the local beam and the top waterline of the main center hull are defined by the six parameters and Every frameline consists of a straight horizontal bottom and a straight (but not vertical) side connected by a portion of circle of radius equal to Ï is the local beam and Ï=1/2. Thus, the hull bottom is defined by , where The hull side is given by where Câ² and Dg are the local slope and depth of the hull side. The circular bilge connecting the hull side and bottom is defined by Here, Ï=tanÎ¸* is defined by the condition of smooth contact between the hull side and bilge for circle, as shown below. Consider the portion of a frameline located in the half plane Yâ¥0. The vectors are normal to the side frameline and tangent to the bilge, respectively. Thus, we have at the contact point Î¸=Î¸* between the side and the bilge. This condition defines C â² as It follows that the local depth Dg of the hull side is given by Thus, the hull side is defined by The foregoing equations fully define the main center hull except for the contact parameter Ï =tanÎ¸*. This parameter is defined below. At the contact point Î¸=Î¸*, the additional condition holds. This condition yields i.e. where Ï=tanÎ¸*. This equation yields We then have with the authoritative version for attribution. Thus, the contact parameter Ï is determined by the 3 parameters and Ï. If we have A0=0 and Ï=0 as expected.

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as APPENDIX 214 Framelines are vertical at some distance above the waterplane Z=0. The straight hull side below the waterplane, inclined at an angle Î¸*=tanâ1 Ï with respect to the vertical, and the vertical upper hull are connected by a circular arc of radius R. Consider the portion of a frameline located in the half plane Yâ¥0. The frameline intersects the waterplane at the point (Y=B0, Z=0). The center of the circular arc that joins the lower and upper framelines is located at the point The contact point between the upper vertical frameline and the circular transitional portion of the frameline is Thus, the center hull above the waterplane Z=0 is defined by The circular transition region between the upper vertical frameline and the hull side below the waterplane is defined by were used above. The radius R may be taken The relations proportional to the beam of the top waterline, i.e. with Î»=1. The six parameters Lc, Dc, B c, that define the main center hull and the five parameters Lo, Bo, Do, and and that define the outer hulls are given by for the wave cancellation multihull ship considered in [1] and the present study. The experimental results given in [1] and reported in Fig. 1 of this study correspond to four locations of the outer hulls given by (âLX,Â±Ly) with Wave cancellation multihull ship model the authoritative version for attribution.

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. APPENDIX Fig. 1. Calculated wave drag and experimental residuary drag 215

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. APPENDIX Fig. 2a. Wave drag coefficient predicted by methods 1,2,3 for F=0.5 Fig. 2b. Wave drag coefficient predicted by methods 1,2,3 for F=0.4 216

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. APPENDIX Fig. 2c. Wave drag coefficient predicted by methods 1,2,3 for F=0.3 Fig. 2d. Wave drag coefficient predicted by methods 1,2,3 for F=0.25 217

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. APPENDIX Fig. 4. Wave drag coefficient for â3â¤aâ¤3 and 0â¤b â¤1 Fig. 3. Wave drag coefficient at three best arrangements of outer hulls 218

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. APPENDIX Fig. 5a. Wave drag coefficient for 10 Froude numbers 219

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. APPENDIX Fig. 5b. Wave drag coefficient for 10 Froude numbers 220

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. APPENDIX Fig. 6. Wave drag for best and worst arrangements of outer hulls Fig. 7. Wave drag and parameters a and b for best and near-best arrangements of outer hulls 221

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as APPENDIX 222 DISCUSSION L.J.Doctors University of New South Whales Australia I would like to express my appreciation to the four authors for a most interesting paper on the subject of multihulls, a matter of interest to many researchers who are aiming to reduce either the overall resistance or the wave resistance (as in the present work). The contour plots of Figure 2, Figure 4, and Figure 5 are an excellent way of presenting the considerable quantity of data showing the influence of longitudinal stagger and lateral offset on the wave resistance. Could the authors kindly explain the background behind the breakdown of the Green function into the three terms in Equation (14)? The first two terms are well known and the (new) third term provides the correct limiting behavior for low and high Froude numbers. However, there seems to be no evidence of wave-like behavior in the third term. Indeed, there would be a family of similar versions of the third term that would exhibit the appropriate limiting behavior. Finally, can the authors indicate whether the effects of sinkage and trim are included in their work and whether they feel these would affect the final predictions for the wave resistance? AUTHOR'S REPLY Thank you for your interest in our work and your questions. The flow field and the Green function are expressed as sum of a wave component and a local component, as is indicated in equations (7a) and (7b). Equation (14) provides an extremely simple approximation to the local component alone, not the wave component. This component is defined by equations (8) through (11aâd) with expressions (3aâe), or expression (13) in the slender-ship approximation. Effects of sinkage and trim have not been included in our calculations, although this could be done without any great difficulty. Our guess is that sinkage and trim would not significantly modify the optimal hull arrangement, but it would be interesting to verify that this is indeed the case. As noted in our reply to the discussion by Profs. Nakatake and Ando, lifting effects are not included in methods 1â3 (thus, only sources are used in these 3 potential-flow methods) but are taken into account in method 4. DISCUSSION K.Nakatake and J.Ando Kyushu University, Japan We congratulate you on your paper to predict the optimal location of the outer hulls for the main hull by applying the rather simple wave drag formulas. We also applied our Rankine source method to a trimaran and confirmed by some experiments that the total wave drag fairly changes according to the location of the outer hulls for the main hull. From Fig. 4, it is interesting to note that the wave drag coefficient changes like a wave contour. Among your methods, method 4 seems to be most exact. In the strict sense, the main hull can be treated as a nonlifting body by method 4, but the outer hull should be treated as a lifting body because the flow around it is not symmetrical with respect to the center plane. Therefore the additional vortex (or doublet) distribution is needed to satisfy the Kutta's condition at the trailing edge of the outer hull. This effect may become larger when the transverse distance between the main hull and the outer hull is small. What do you think of this point? AUTHOR'S REPLY Thank you for your interest in our paper and for providing information about your own work on the effect of hull arrangement upon wave drag. We agree that the outer hulls should in principle be treated as lifting surfaces, and that lifting effects can be expected to be larger if the outer hulls are closer to the center hull. The 3 simple potential-flow methods we have used (methods 1, 2, 3) do not include lifting effects. Method 4, based on the Euler equations, accounts for lifting effects. The relatively good agreement between experimental residuary drag and wave drag predicted by all 4 methods shown in Fig. 1 suggests that lifting effects may not be very important in the present case. the authoritative version for attribution.