Conceptual Design and Hydrodynamic Analysis of aHigh-Speed Sealift Adjustable-Length Trimaran

K. J. Maki (AM) , L. J. Doctors (M), R. M. Scher (M), W. M. Wilson (V), S. H. Rhee (AM) ,A. W. Troesch (FL), R. F. Beck (FL)

This paper describes a novel adjustable-length trimaran design for service as a high-speed sealift vessel. Differentnumerical hydrodynamic tools were exercised to assess the performance of the vessel. An extensive model test programthat included powering and seakeeping experiments was conducted to elucidate utility of the numerical predictions.

KEYWORDS: calmwater resistance; seakeeping; multihull;model testing; thin-ship theory; computational fluid dynamics;strip theory

INTRODUCTIONFor ship designers the increasing emphasis on rapid and flex-

ible support of logistic missions represents several distinct tech-nical challenges. Among these, one of the most difficult is thedesire for significantly higher speeds than previous sealift shipclasses, capable of carrying substantial military payloads overtrans-oceanic stage lengths, butwithout prohibitive increases inship size, power, and fuel consumption (especially in view of re-cent trends in oil price). The need for operational flexibility inlogistics in unconventional warfare places a premium on theabil-ity to use less developed, “austere,” ports, with inherent limita-tions on ship size, including stringent constraints on length anddraft. Finally, the developing doctrine of Sea Basing demands theability to transfer a variety of cargoes, especially rolling stock,between ships in open water. If these challenges are to be suc-cessfully addressed, innovative ship configurations are clearly re-quired: accordingly, certain fundamental changes in design meth-ods and philosophy may have to occur. What these changes willbe, and specifically what impacts on design criteria and shipop-erating methods will result, will undoubtedly be a field of contro-versy for some time.

Evolving mission needs for logistics in expeditionary warfare

have not yet converged to a single view. However, several inde-pendent visions of the future, paralleled by recent conceptstud-ies and designs, have included a doubling of transit speed, fromthe conventional 20 to 24 knot regime of present sealift ships, tospeeds exceeding 40 knots; cargo payloads from several hundredto several thousand tons, and stage lengths varying from about2500 to about 6000 nautical miles.

Ship length constraints for austere port operations (quiteapart from limited acquisition budgets and operating costs) de-mand solutions that differ markedly from the conventional,large,sealift ship (with whatever increased power, weight saving, andvariation of form coefficients go with it). While designers strive togenerate concepts that will fulfill future logistic needs, includingsome hull-form types that are quite unprecedented, the state-of-the-art hydrodynamic analysis tools that they rely on are under-developed and unvalidated in these new operating regimes. Thedevelopment of new ship concepts, and new design tools, mustaccordingly advance together.

Recently, US Office of Naval Research (ONR) supportedconcept studies and hydrodynamic tool development for High-Speed Sealift (HSSL) to Austere Ports. The design ground rulescalled for the following major capabilities:

• Unrefueled range: 5000 nautical miles at average speed of43 knots

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• Full mobility, including 43 knot speed, through Sea State 4

• Military payload of 4000 short tons (representing approxi-mately the weight and stowage area for vehicles of a Strykerbattalion task force)

• Austere port access: maximum over-all length 170 m; max-imum draft 6.5 m

• At-sea transfer of vehicle cargo (nominally through SeaState 4)

This paper summarizes the design and hydrodynamic charac-terization of a vessel to meet these mission requirements, and isorganized as follows. First, our most desirable design candidateis described with details about the unique challenges that accom-pany its unconventional form. Next, the experimental measure-ments and numerical predictions used during the design processare introduced and discussed. Then, the calm-water resistanceand seakeeping motion results are presented. Finally, the paperconcludes with a summary of the merit of the different numericaltools in predicting vessel performance.

DESIGN CONCEPTThe Adjustable-Length Trimaran

Clearly, the speed, range, and payload requirements of high-speed sealift present a naval architectural challenge of a highorder, evenwithout the addition of severely constrained lengthand draft for austere port access. With these dimensional con-straints, especially the length constraint, the problem has re-peatedly shown itself to be beyond the reach of conventionalship designs with presently available material, machinery, andfuel technologies. Credible concept-level designs, whether theyhave been of “conventional” (long centerhull) trimaran, catama-ran, or monohull configuration, displacement, heavy-planing, orcushion-borne, donotconverge (carrying payload and fuel for thedesired range) except by exceeding the 170 m austere port lengthconstraint by a considerable margin, and in some cases breakingthe draft constraint as well.

However, suppose we envision a trimaran in which thethree hulls are of approximately equal length (the austere-port-constrained length) and of nearly equal displacements. As theresults presented subsequently will demonstrate, a trimaran ofthis type, with the hulls staggered (instead of directly alongsideone another), can achieve substantial wave resistance advantagescompared with other configurations. With an appropriate staggerfor a given transverse separation of the hulls, wave cancellationcan be achieved and the wave resistance of the trimaran can beless than that of a slender monohull of equivalent displacementand over-all length.

At this juncture, we must acknowledge the paper by Day,Clelland, and Nixon (2003). This paper covered an extensivediscussion and investigation into eight different configurations ofmonohulls and multihulls. In particular, the three authorsstudieda so-called Arrow configuration for a trimaran, which is similarto the subject of the present work, in that the central subhull was

(a) Sidehull (b) Centerhull

Figure 1: Vessel Geometry

located forward of the two sidehulls. They demonstrated both ex-perimentally and theoretically (as we shall here) that the (total)resistance of a trimaran is lower over most of the speed rangeinthis configuration.

Their work was both theoretical and experimental in nature;it was based on earlier work by Tuck and Lazauskas (1998),whose theoretical and numerical investigations were very thor-ough. We must also emphasize that trimarans are, by no means,a new concept. For example, at one of the historic meetingson wave resistance, Narita (1976) considered trimaran layoutsin which the two sidehulls were staggered both forward and aft.They demonstrated that both positive and negative staggersleadto similar reductions in wave resistance.

Returning now to our mission requirements, the over-alllength of the trimaran with sidehull stagger unfortunatelyexceedsthe austere-port-length constraint. That is the dilemma.

An unconventional solution can be proposed, however. Sup-pose that the hulls are structurally connected and locked ina stag-gered configuration during transit, even though the over-all lengthexceeds the austere port constraint. On arrival then, the vesselmechanically alters its configuration for port access by retractingto a more compact length. On departure, the vessel extends andlocks into its transit configuration again. This is not a trivial de-sign problem, but on the other hand it does not appear to be outof reach with present technologies.

Body plans of sidehull and centerhull for our notionaladjustable-length trimaran are shown in Figure 1; principal char-acteristics of the hulls are listed in Table 1. Transom configura-tions permit the installation of a single waterjet propulsor in eachhull, an arrangement considered most suitable for model testing.However, propulsion-machinery selections (for an actual vessel)might ultimately favor single or twin waterjets in one, two,or allthree subhulls.

Design Challenges of the Adjustable-LengthTrimaranLocking, Retraction and Extension The principal design chal-lenge is apparent: mechanical systems required for structural con-nection of the retractable hull must be capable of supporting the

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Table 1: Nominal Particulars of HSSL ALT Subhulls

Item Symbol UnitsCenter-

hullSide-hull

Displacement mass ∆ t 5827 5964Waterline length L m 168.1 168.6Waterline beam B m 10.03 10.04Draft T m 6.510 6.510Waterplane-area coefficient CWP 0.7770 0.8091Maximum section coefficient CM 0.7860 0.7863Block coefficient CB 0.5174 0.5279Prismatic coefficient CP 0.6583 0.6714Slenderness coefficient L/∇1/3 9.424 9.375

design sea loads, and still be able to retract and extend under someother (presumably less severe) set of loads. This is not completelywithout precedent. Locking and bearing mechanisms sized tocarry loads (of magnitudes appropriate for primary ship struc-tures) have been used in commercial marine applications suchas integrated and articulated tug-barge (ATB) systems. Simi-larly, mechanical systems for retraction and extension of large ob-jects, under substantial forces, have been developed for other ma-rine and offshore applications, such as jack-up drilling rigs, off-shore structure launching appliances, and heavy-lift derricks andcranes. However, the additional challengefor a high-speed shipapplication is to provide the required system capabilities (lock-ing, retraction, and extension) at an acceptable weight penalty.

Figure 2 shows a concept for an adjustable-length trimaranwith a retractable centerhull section. The centerhull is lockedin the extended position for transit, with locking mechanisms,broadly similar to the devices that are used on integrated tugbarges: that is, sliding wedges or rams, sized to take the structuralloads required for open ocean operation at high speed. For accessto an austere port, the locks would be disengaged, and the center-hull would be retracted along a system of guides which need onlytake the loads required in port, sheltered waters at relatively lowspeeds. Based on current practice and commercial systems usedin integrated tug barges, a structural locking system can bede-veloped at a reasonable weight, provided that appropriate designloads can be established.

In seakeeping model tests that will be described below, forcesand moments were measured at the connections between the portand starboard cross structures and the sides of the centerhull. Thehighest irregular wave case tested was a representative SeaState 7spectrum (Bretschneider, 7.5 m significant wave height, 15 sec-ond modal period), in head seas, at 25 knots. Based on a pre-liminary analysis of those results it is considered likely that ver-tical bending moments and shear forces will govern the designof the locking system, at least in a configuration where the cen-terhull which retracts under the cross structure between the twoside hulls. (Conceptually, there are other ways of performing thisretraction. For example, the port and starboard cross-structuresand hulls could slide fore and aft along the sides of the centerhull.

Figure 2: Adjustable-Length Trimaran Concept with RetractableCenterhull Section Extended and Retracted

In fact, such an arrangement would permit at least one geomet-ric advantage in terms of freeboard constraint on the centerhull.However, in other important respects, especially structural conti-nuity and arrangements for cargo stowage and access, a conceptwith the centerhull connected underneath the cross structure of a“catamaran section” seems more reasonable.)

A concept for locking the centerhull is shown schematicallyin Figure 3. Notionally, the locking system consists of opposedpairs of horizontal sliding wedges driven (for example) by ajackscrew. To lock the centerhull, the wedges are driven outboard andengage corresponding “buckets” fixed in the cross structure. Twoassemblies, separated longitudinally, transmit the vertical bend-ing moment and shear, and also resist longitudinal forces andtorsion of the centerhull about the longitudinal axis. One pairof locking wedges is located near the aft end of the centerhull,

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Figure 3: Adjustable-Length Trimaran Concept; Section Showinga Possible Location of Locking and Retraction Systems

and engages with buckets located near the mid-length of the crossstructure. The other pair is located near the mid-length of thecenterhull, and engages the cross-structure as far forwardas prac-ticable. Thus, the lever arm between the two sets of locks is ap-proximately half the length of the centerhull.

In oblique seas and during maneuvers, of course, there arealso significant lateral bending moments and side forces actingon the centerhull. These forces also have to be transmitted by thelocking wedges, although primarily as loads along the axis of theactuating screw, rather than as shear in the wedges. Model test re-sults in oblique seas have only been obtained in irregular waves upto 4 m significant height. Consequently, firm conclusions on theappropriate design loads cannot be drawn yet. However, if ver-tical bending and shear loads do indeed govern the design of thelocking members, as is expected, it appears that an opposed pairof locking wedges of the type shown in Figure 3 would have tobe capable of transmitting an extreme load of about 8300 tonnes.This is hardly a small load, and it reflects to a great extent theinfluence of high speed. However, it is within the shear capacityof practical forgings and weldments, not much different in sizefrom those currently used on large articulated tug-barges.In ad-dition, while ATB coupling systems must incorporate bearings topermit rotation under load, the locking wedges of the trimarando not, and this simplifies the design of the system. The dimen-sions of the wedge (diameter or largest dimension of the cross-section) would be on the order of 1500 mm, and the materialthickness around the perimeter of the section about 120 mm. In-ternal diaphragms could also be required inside the wedge tohelpmaintain the desired distribution of contact forces, and totransferforces to the jack screw.

Apart from locking the centerhull in its extended position atsea, the system must also provide a means of retracting and ex-tending the centerhull. Guides for this motion may take the formof rails with wheeled carriages or slides. The loads on the guidesare expected to be modest by comparison with the loads on thelocking system, but certainly not negligible. The effects of weightdistribution and trim, which can vary widely depending on fueland cargo loads, will be at least one possible driver of load re-quirements on the guides, even in calm water. Types of actuators

to move the centerhull could include rack and pinion (similar tosystems used for jack-up drill rigs), traction wheels, winches andcables, ball screws, or other linear actuators. Some care isobvi-ously needed with regard to buoyancy and trim in an adjustable-length vessel. Just because the entire trimaran is in staticequi-librium with the sections locked in a certain position, it does notfollow that when the locks are released each section will still be inequilibrium individually. Thus, the guide system may be carryingboth a force and a moment when the locks are released, and therewould be situations when the centerhull would tend to move alongthe guide simply because of an excess of buoyancy or a trimmingmoment in one section or the other.

Weights for the locking and retraction systems were includedin estimates for the adjustable-length trimaran, and amount tosomewhat over 250 tonnes, including additional foundations.This is not an insignificant weight. However, the penalty hastobe compared with the added propulsion machinery and fuel thatwould be associated attempting to meet speed and range require-ments on an over-all length limited to 170 m, instead of 50 percentlonger.

Arrangement In the arrangement implied by Figures 2 and 3,the centerhull contains fuel tankage and propulsion machineryspaces, but for reasons that should be apparent no cargo or habit-ability spaces. The cross-structure of the catamaran section con-tains all the cargo stowage areas, cargo access arrangements, ac-commodations, and ship control spaces, in addition to propulsionmachinery rooms and most of the auxiliary machinery. There isno superstructure on the centerhull section, of course, becauseof the need to retract it under the cross-structure. This limita-tion, in effect, restricts the freeboard of the centerhull to the wetdeck height of the catamaran section. This in turn can have aconsiderable influence on performance and seakeeping, as dis-cussed in subsequent sections. The solution to excessive wetnessof the centerhull forward, without excessive height of the catama-ran section, remains a practical challenge for design trade-offs.[In this connection it is noted that according to Michell’s theory,staggering the centerhull aft offers the same magnitude of wave-resistance advantage as does forward stagger. Aft stagger resultsin a rather strange looking planform (Figure 4) and possiblycer-tain additional machinery-arrangement and maneuvering issuesas well, because of the unusual longitudinal location of theout-board propulsors. From the purely seakeeping perspective,how-ever, it does allow the higher-freeboard catamaran sectionto beforward, even if this may only mean that a bow-wetness concernhas been exchanged for wet-deck slamming.]

Other Design Issues In principle, the adjustable-length tri-maran may (and probably will) have propulsion machinery lo-cated in all three hulls. Intakes and exhausts from centerhullpropulsion engine(s) present an unusual arrangement and gas-plume problem. It can be assumed that retraction and extensionoperations will be performed with the center engines secured, asthis would only be done at low speeds in sheltered water. Evenso,on arrival the center engines and machinery spaces would still berather hot, and the machinery and enclosures would still be radiat-

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Figure 4: Adjustable-Length Trimaran with Aft Stagger of theCenterhull

ing. Several alternative approaches for air management have beenconsidered, of which the most promising is to route intake and ex-haust for the centerhull engines through vertical trunks incorpo-rated in the catamaran section. This allows a conventional exhaustlocation with respect to the catamaran superstructure. However itdemands the design of a watertight seal detail (at or near thelevelof the wet deck of the catamaran section). These seals must bemade effective when the centerhull section is extended, andthencan be “broken” before disengaging the locks and retractingthecenterhull section. The development of seals for these arrange-ments is not a trivial design problem, but it is considered feasible.

Distributive systems and personnel access to the centerhullpresent another peculiar challenge to the designer. Important sys-tems that have to be able to connect the centerhull and catamaransections are electric power, fuel fill and transfer, and internal com-munications, control and monitoring. The centerhull and catama-ran sections can (and probably will) have independent ballast andbilge systems, firemain, seawater cooling, and other auxiliary sys-tems such as lubrication oil, fire extinguishing, compressed air,and ventilation. However, there are sound reasons to keep theelectric generating plant entirely in the catamaran section, and inthis case electric power to the centerhull will have to be trans-ferred by internal “festoon” or cable and reel systems. Signalconnections that must be hard-wired may also use a similar fes-toon or cable and reel, or possibly some wireless connections canbe incorporated. Fuel fill and transfer systems, due to largehosesizes, may be somewhat more difficult, however, systems can beconnected after the hull is extended, and disconnected prior to re-tracting. Personnel and material access arrangements presumablywill require a system of double watertight hatches to trunks, withlocations provided for use in the extended and retracted hull posi-tions. Details of these systems have not been developed, buttheyare considered feasible.

The adjustable-length trimaran is an interesting vehicle forthe extremely challenging HSSL speed, range, payload, and aus-tere port access constraints. At least to the level of a conceptdesign weight estimate, it meets the basic HSSL mission require-ments. In exchange for this, however, the concept presents sev-eral unusual design challenges. Apart from primary structuralloads and locking arrangements, which are obviously key areasfor further analysis, the low freeboard of the centerhull (assuming

stagger with the centerhull forward) is a leading concern. Whilewetness of the centerhull is not necessarily an intractabledesignissue, it will require careful attention. The relatively small reservebuoyancy and breadth forward has been observed (during modeltests) to permit wave impacts to occur on the forward surfacesof the cross structures. Also, asymmetric pitch responses (bowdown peaks substantially greater than bow up) of the kind notedfor tumblehome wave-piercing forms in general, may become alimitation in severe seas states. Further hydrodynamic researchand the development of appropriate analytic tools, as discussedbelow, will be extremely helpful if this concept is to be furtherdeveloped.

DESIGN ANALYSISIn this work, several different numerical methods were

used to predict resistance and seakeeping performance of theadjustable-length trimaran. Resistance was predicted with aprogram that uses thin-ship theory, and a code that solvesthe unsteady Reynolds-Averaged Navier-Stokes (RANS) equa-tions. The seakeeping performance is simulated using a velocity-potential formulation and strip theory. The adjustable-length tri-maran represents a unique case in terms of the importance of hullinteractions, forces and moments, and the geometry and propor-tions of both the below water and above water portions of thehull and cross structure. In order to make a quantitative assess-ment of the accuracy of the performance predictions, a suitablebenchmark was needed. Accordingly, a large model test programwas conducted at the Maritime Research Institute Netherlands(MARIN).

ExperimentsA 1-to-34 scale model was used for experiments in the pow-

ering and seakeeping basins. Calm-water resistance was mea-sured with the center and sidehulls arranged in different con-figurations, and the motion and inter-subhull force componentswere measured with the model self-propelled in a range of seastates. The resulting dataset comprehensively describes the majorelements of hydrodynamic performance of an adjustable-lengthtrimaran. It also provides unique guidance for the design ofunique structural and mechanical systems required to make anadjustable-length trimaran feasible.

Resistance Test Program The three individual hulls were con-structed out of wood and connected together using aluminumbeams which allowed for easy reconfiguration. The model wasoperated in six different loading configurations of varyingbeam,length, and displacement, as listed in Table 2 and viewed in Fig-ure 5. Essentially, this consisted of three different staggers ofthe sidehulls and two different offsetss2/2 of the centerplane ofthe sidehulls from the centerplane of the centerhull. During theplanning phase of the experiments, we had intended that the draftshould be identical for all test configurations. Unfortunately, thefreeboard proved to be too small for the reduced dimensions ofConfiguration 4 and Configuration 5; as a result, an excessiveamount of spray impacted on the bridging cross structure con-necting the three subhulls. This deficiency in the design of thetrimaran could easily be corrected in any planned extensionto the

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(a) Configuration 1

(b) Configuration 2

(c) Configuration 3

(d) Configuration 4

(e) Configuration 5

(f) Configuration 6

Figure 5: Bottom View of the 6 Hull Configurations

Figure 6: Image of Model in Configuration 4

project. The final test matrix operated Configurations 1 to 3 and 6in the heavier departure-displacement, and Configurations4 and 5in the lighter arrival-displacement.

The model was attached to the carriage and towed with avertical heave staff and gimbal so that the model was free toheave and pitch. The longitudinal location of the tow point cor-responded to the longitudinal center of buoyancy of the trimaran(which varied depending on the stagger of the sidehulls). Sandstrips were applied to the bow of each hull to stimulate turbulence.An image of the model, arranged in Configuration 4, is shown inFigure 6 where details of the bow geometry and the location ofthe sand strips can be seen.

The same models were used for the seakeeping tests whichrequired the installation of a water jet in each hull. The water jetopenings were covered for the resistance tests so that the hullswere effectively bare.

Seakeeping Test Program The seakeeping test program wasconducted in MARIN’s Seakeeping and Manoeuvring Basin. Weconcentrated our interests regarding motions on Configuration 6,because this is the heavy-load long-range fuel-efficient mode forwhich the adjustable-length trimaran was specifically designed. Itis this configuration that would pertain to the vast majorityof theoperational time and that would present the principal concern forthe operators of the vessel. The three hulls were connected us-ing force frames so that the six degree-of-freedom inter-subhullforces could be measured, and a portion of the hull structurewasattached in the bow region to facilitate and measure under-deckslamming. The inertial properties of the seakeeping model aresummarized in Table 3.

A water jet was installed in each hull so that the model couldbe tested in a self-propelled condition. The nozzles on the outerhull jets were steerable and controlled by an autopilot to keep themodel on course.

The wave test conditions for the model tests with respect tomotions fell into two broad categories: regular waves and irreg-ular waves. For each category, the model was operated in dif-ferent combinations of two forward speeds, 25 and 43 kn, wave-heading angles between 0 and 180 degrees, and wave steepnessor sea state. The wave conditions are noted in Table 4. Becauseof the vast quantity of data that was collected, we shall constrainour discussion in this paper to just the two cases of a 90-degreeheading and a 150-degree heading. The reason for this choiceisthat we firstly desired to compare the regular-wave analysiswiththe irregular-wave analysis. This eliminates the cases of the 0,120, 180, and 285-degree headings. Secondly, the cases of 0 and180-degree heading are less interesting, because some modes ofmotion are (ideally) zero.

The regular seas tests at the 150 degree heading used wave-lengths spanning one-half to six times the overall ship length. Thebeam seas tests were conducted at wavelengths between one toseven times the ship beam. All regular seas tests were done witha wave amplitude of 1 m, though repeat tests were done near fre-quencies of large response. For the 150 degree heading, addi-

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Table 2: Details of Model Experiments for Resistance

Config-uration

OverallLengthL (m)

OverallBeamB (m)

SidehullStaggerr2 (m)

SidehullOffset

s2/2 (m)

DraftT (m)

Displace-ment∆ (t)

SpeedRangeU (kn)

1 168.6 56.0 0.0 23.0 6.51 17,760 20–562 210.6 56.0 −42.5 23.0 6.51 17,760 20–603 253.1 56.0 −85.0 23.0 6.51 17,760 20–544 168.2 40.0 0.0 15.0 5.19 12,340 20–545 210.2 40.0 −42.5 15.0 5.19 12,340 20–486 253.1 40.0 −85.0 15.0 6.51 17,760 20–58

Table 3: Details of Seakeeping Model

Item Symbol Units ValueConfiguration 6Longitudinal center-of-gravity from transom LCG m 109.68Vertical center-of-gravity above keel KG m 10.00Mass radius of gyration about longitudinal axis kxx m 12.92Mass radius of gyration about lateral axis kyy m 51.00Mass radius of gyration about vertical axis kzz m 54.90

Table 4: Details of Model Experiments for Seakeeping

Config-uration

Displace-ment∆ (t)

SpeedU (kn)

Headingβ (degrees)

WaveTypes∗

6 17,760 43 0 I6 17,760 43 90 R & I6 17,760 43 120 I6 17,760 43 150 R & I6 17,760 43 180 R & I6 17,760 25 180 I6 17,760 43 285 R6 17,760 43 300 R & I

∗ R (Regular), I (Irregular)

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tional waves of varying amplitude, namely 0.5 and 0.25 m, weretested at the most interesting ship-to-wavelength ratio ofunity.Additional beam seas tests were conducted at the large-responsefrequency corresponding to the wavelength-to-beam ratio of 1.6.For this repeat frequency, an amplitude of 0.5 m was used.

The irregular seas results that are analyzed in this paperwere generated using Pierson-Moskowitz-type spectra. Thewavepower spectrum of a given test is defined by the significant waveheight, and the peak period. We thoroughly tested the conditionthat simulates Sea State 4, because the mission requirements man-date that the vessel operate fully in this environment. Accord-ingly, the largest portion of the test matrix was attributedto a sig-nificant wave height and peak-wave period corresponding of 2mand 8.8 s, values that emulate Sea State 4. Additional irregularseas conditions were tested using peak periods that ranged from12 to 31 s, and significant wave heights up to 7.5 m.

Numerical ToolsResistance Using Thin-Ship Theory In the current work, wedescribe an exhaustive investigation into the resistance of a can-didate for a high-speed sealift trimaran. The purpose of theeffortwas twofold: to consider a suitable trimaran design and to test ouravailable computational tools. Computations were performed in a“blind fashion”, in that the experimental results for the resistancewere not initially made available to those persons performing thecalculations. Of course, the opportunity was later provided tolearn from the comparisons of theory and experiment and, conse-quently, to improve the software.

The software developed by Doctors and Day (1997), Doctors(1999), and Doctors and Scrace (2003), based on extensions tothe Michell (1898) theory, represented one focus of our computa-tional work. The other principal computational focus was basedon computational-fluid dynamics (CFD) as described later inthispaper.

The current version of our thin-ship software contains a num-ber of enhancements, as follows:

• The program can handle any number of subhulls up to six, byemploying additional centerplane source distributions. Thetransverse velocities induced by one subhull should be cor-rected for by the use of laterally-directed centerplane dipoledistributions; however, this is ignored for the sake of con-siderable simplification to the coding of the software. It canbe shown that our simplification is reasonable, provided therelative lateral spacing between the subhulls of the multihullis sufficiently large. We admit that in the current project,we have been burdened with a stringent design requirementon the overall beam of the vessel. This implies that we aretesting this simplifying assumption to the limit.

• An enhanced model for the transom-stern flow has been un-der development for 11 years now. The current model allowsfor a realistic extension to the vessel by means of a virtual ex-tension to the vessel in order to represent the presence of thetransom-stern hollow. This has been described by Doctorsand Beck (2005), Maki, Doctors, Beck, and Troesch (2006),

and Doctors (2007). In this way, the effective hydrodynamiclength of the subhulls grows with increasing vessel speed ina physically realistic manner that has been substantiated bynumerous towing-tank experiments.

• The transom-stern model includes a procedure for estimat-ing the progressive unwetting or ventilation of the transom.This feature permits a practical estimate of the transom-sternor hydrostatic drag suffered by such vessels. This proce-dure has been detailed by Doctors, Macfarlane, and Young(2007). In that later publication, the previous very exten-sive experiments on transom-stern ventilation were repeatedin a more accurate manner, through the use of specially de-signed “wetting probes” mounted around the transom stern.We now possess much more reliable data for this purposeand have, in the process of the investigation, verified that thecritical value of the transom-stern Froude number to obtainfull ventilation is around 2.5 — at least in the case of a tran-som stern with a rectangular cross section at model scale.

• As a special task inspired by the model experiments to bedescribed here, a further procedure was incorporated in thecomputer code in order to take into account the viscous in-teraction between the subhulls. This simplistic proceduremakes an allowance that the subhulls are in the “shadow”of each other to some degree, as seen in a profile or a sideview. This shadow will be greatest when the subhulls areabreast of each other and will be particularly strong whenthey are of similar size. In the present theory, the local gapbetween the subhulls below the loaded waterplane was com-puted. From this, an estimate of the increase in local waterspeed was obtained. This increase in speed was applied tothe usual formula for frictional resistance, thus leading to anapproximate technique for determining the frictional formfactor.

Of course, it is clearly understood that a number of physicalphenomena is neglected here. Neglected effects include thede-formed shape of the free surface and the fact that the flow wouldtend to skirt around the outside of the vessel, in cases wheretheintersubhull gap is very small. However, we wished to preservethe characteristically simple approach of the software as well asits fast computational behavior.

The reader is referred to the work of Maki, Doctors, Rhee,Wilson, Beck, and Troesch (2007), for our first publication onthis research and development project.

Seakeeping Using Strip Theory The computer programKORVIN consists, broadly speaking, of two main modules.The purpose of the first main module is to compute the two-dimensional (sectional) hydrodynamiccoefficients. The approachutilized here is a direct enhancement of that developed by Doc-tors (1986) for the case of an infinite fluid domain, without a freesurface. In the current case, we are concerned with a ship sectionoscillation at or near the free surface; the mathematical analysiswas presented in detail by Doctors (1988). It is effected through

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the use of the boundary-element method (BEM), also called thepanel method.

The surface (more precisely, the line boundary) of the sec-tion is subdivided into a series of approximately equal straightsegments, typically 20 to 40 in number. A source strength, pul-sating harmonically in time, is distributed uniformly along eachline segment. The influence of the source panel on the field pointis derived analytically. The resulting simultaneous equations forthe source strengths are solved in the usual way, thus givingthesource strengths.

The source strengths are complex with respect to the time.One can then compute the velocity field and, in particular, theinduced pressures on the field panels. The in-phase pressuresare next integrated around the section to yield the added mass.The integrals of the out-of-phase pressures are used to obtain thedamping coefficients. Two particularly effective enhancementsare also employed in the computer program. The first enhance-ment is to integrate the induced velocities over the field panel aswell as the source panel, when computing the matrix of influ-ence coefficients. This approach is sometimes referred to astheGalerkin technique. The Galerkin technique provides more accu-rate results. That is, fewer panels are required in order to obtaina prescribed level of accuracy, compared to the original work ofFrank (1967).

It should be mentioned here that other methods exist for en-hancing the convergence rate of the basic panel method. These in-clude the use of higher-order singularities by Hess (1975),curvedpanels by Bristow (1977 and 1978), and a spline fit by Inoue,Kuroumaru, and Yamaguchi (1977).

We also claim that our approach is simpler to apply than theseapproaches. Moreover, it is more elegant and robust than that ofBarnell (1984), for example, who effected the field-panel inte-gration numerically. It is clearly better to do this algebraically iffeasible.

The second enhancement in the software is to utilize a “lid”over the internal free surface of the section. This feature pos-sesses the considerable advantage of eliminating the problem ofthe “irregular frequencies”, without any other special treatment.The irregular-frequency problem was described, for example, byAdachi and Ohmatsu (1979), and Haraguchi and Ohmatsu (1983).

The second main module of the software is based on the striptheory of Salvesen, Tuck and Faltinsen (1970). This paper hasbecome the core reference for the strip theory, in which the sec-tional hydrodynamic properties of the ship are integrated alongthe length of the ship in order to obtain an approximation to thethree-dimensional hydrodynamic coefficients of the ship.

In the case of a multihull vessel, as in the present study ofthe motions of a trimaran, the program can be executed using twodifferent methods. In the first method, the abovementioned two-dimensional sections are taken right through all through subhulls.It is considered that this should be the appropriate form of anal-ysis for a stationary or low-speed vessel. That is, the sectionsof one subhull would interact hydrodynamically with the corre-

sponding sections of the other subhull and this must be takeninto account in the theory. In the second method, the subhullsare assumed to behave independently, from the hydrodynamicviewpoint. This second approach is assumed to be more appro-priate for the vessel when traveling at a high forward speed andthe waves generated by one subhull are swept downstream beforethey can interact with the other subhull.

These ideas were studied by Doctors and Scrace (2004) on atrimaran hull model similar to theRV TritonTrimaran Demonstra-tor. It was shown that, at typical forward speeds, correspondingto Froude numbers up to 0.3112, using the independent-subhullanalysis yielded superior predictions.

Further confirming this speculation, a second and more re-cent application of the software to a trimaran by Hebblewhite,Sahoo, and Doctors (2007) demonstrated that, indeed, bettercorrelation with theory appeared to be achieved through theindependent-subhull analysis for both low and high speeds.

This satisfactory outcome was confirmed even more power-fully by Thomas, Doctors, Couser, and Hackett (2007). This thirdexample was for a catamaranat zero speed. It is true that gener-ally very good agreement with experimental measurements forthe heave, yaw, and pitch motion responses was achieved withboth approaches, even in this very extreme limiting case. Never-theless, the analysis assuming hydrodynamic interaction betweenthe two demihulls was markedly more accurate, as anticipated.

We note that there is very recent work that focuses preciselyon improving conventional strip theory for multihull vessel sea-keeping. The Ph.D. thesis of Belknap (2008) developed a theorythat includes hull-interaction effects between hulls for the lowership speeds for which the effects are likely. In his work, a far-field approximation is invoked, whereby the radiated waves fromone demihull section appear as incident waves to another demi-hull section. The interacting sections are determined by the vec-tor combination of the group velocity of demihull radiated wavesand forward speed of the ship. For very high speeds, all the ra-diated waves are swept downstream and there is no interaction.Comparisons of the theory of Belknap to model test data and 3Dcomputations have shown that it captures the correct trendsand isan improvement over conventional strip theories.

In the present investigation, however, the first two theorieswill be plotted and compared with the experimental data in orderto test this hypothesis of hull-interaction.

Resistance Using RANS Computational tools continue to im-prove in both speed and accuracy, and in particular, Reynolds-Averaged Navier-Stokes (RANS) methods have demonstrated in-creased capabilities in predicting the flow around ships andsub-marines in recent years. Much of this is due to recent advancesin computing technologies, particularly parallel processing tech-niques, and the application of complex unstructured meshing al-gorithms that can more efficiently produce suitable computationalgrids, even for complicated geometries. What is still required,however, is to continue to study predictions to more clearlyun-derstand how these expensive simulations can be best utilized in

9

the design and analysis of marine vehicles. A review of applyingRANS techniques to a variety of surface ship configurations wasgiven in Gorski (2004). An assessment of different computationaltools was given in Wilson, Fu, Fullerton, and Gorski (2006).

The commercially available software FLUENT is a generalpurpose Reynolds-Averaged Navier-Stokes (RANS) solver. It uti-lizes a volume-of-fluid (VOF) method to predict the locationandevolution of the free surface. In the VOF approach, it is assumedthat the discrete volume elements that comprise the domain arefilled with a combination of separate fluids, in this case air andwater, and the sum of the volume fractions of each fluid in a con-trol volume equals unity.According to this constraint, theadvec-tion of only a single phase must be computed.

FLUENT uses a cell-centered finite-volume method and canutilize discrete volume elements that are constructed of anarbi-trary number of sides, appropriately referred to as polyhedra. Theadvantage of allowing an arbitrary number of sides per cell is thatdifferent standard element types such as tetrahedra, hexahedra,pyramid, and prismatic, can be used to generate the mesh. In thelatest version of the software, tetrahedral zones can be convertedinto polyhedra using an agglomeration process where the tetrahe-dral nodes become the polyhedral cell-centers. This conversionhas several advantages, the most prominent are the reduced cellcount and greatly improved mesh quality through reduced skew-ness. The reduced cell count is an obvious advantage, and skew-ness degrades accuracy and impedes solution convergence. Herewe use a cell skewness metric that assesses the deviation of thecell face angles from those of a face belonging to an equiangularcell. A value of 0 indicates a best-case equiangular cell, and avalue of 1 indicates a completely degenerate cell. In the presentsimulations, the maximum skewness for the near-field regionwasreduced from approximately 0.8 when discretized using tetrahe-dral elements to 0.4 after converting to polyhedra. By reducingboth the cell count and the skewness, it is possible to reduceso-lution times and enhance accuracy.

The computational grids used in the present study exploitthe advantage of using different element types in the differentregions of the domain where they excel, while also attemptingto reduce the difficulties and time associated with generating asuitable mesh. The near-body mesh consists of boundary-layerprisms that have a structured nature in the wall normal direction.The use of prisms allows for a fewer number surface cells to coverthe body while providing the necessary spacing to resolve the vis-cous layer appropriately. The farfield domain consists of hexahe-dral elements which are aligned with the direction of the flowandclustered near the free surface. The intermediate area between theboundary-layer-prism cap and the farfield domain is connectedusing tetrahedral elements which are converted into polyhedra us-ing the previously explained procedure.

Figure 7 shows the surface mesh on the centerhull and thevessel symmetry plane, and demonstrates the use of each of theelement types. The hexahedral elements are effective in solvingthe undisturbed free surface locations due to the ability toallowsignificant stretching in the streamwise direction withoutcom-

promising accuracy. The polyhedra elements, on the other hand,are better utilized in resolving the near-body regions thatcontaincomplex geometry. Figure 7 also shows the predicted thicknessof the air-water interface, which is seen to occur over at mosttwo to three cells. While it is acknowledged that the use of thepolyhedra elements does compromise the interface resolution inthe near-body regions, it has been demonstrated that this does notsignificantly impact the accuracy of the force predictions (Makiand Wilson, 2008), which will be presented in the following.Thegrids used in this work had a total cell count of approximately1.6M cells, composed of 1.3M polyhedra and 0.3M hexahedra.

In the present study, the convective terms were discretizedusing the QUICK scheme (Leonard and Mokhtari, 1990). Thisscheme uses a weighted average of a second-order upwindmethod and a second-order central interpolation method, which istypically more accurate for structured grids aligned with the flowdirection. For unstructured regions of the domain, the scheme re-verts to the second-order upwind method. Therefore, this schemeis very appropriate for the present case, which involves a hybridmesh that contains both structured hexahedral elements which aremostly aligned with the general flow direction, as well as poly-hedra elements for flexibility in resolving the model geometryand near-body flow. The discretization of the volume fractionequation used a modified version of the high-resolution interfacecapturing (HRIC) method (Muzaferija et al, 1998). The HRICscheme has been shown to be more accurate than the QUICK orother second-order discretization schemes, and is less computa-tionally expensive than using a complete geometric reconstruc-tion of the interface, which is also available in FLUENT. Thetur-bulence closure was accomplished using a variant of thek−ωmodel as described in Wilcox (1998). The pressure-velocitycou-pling is based on a SIMPLE type segregated solution algorithm.The simulations were performed as fully unsteady in time, march-ing to an equilibrium solution. The time advancement was ac-complished using first-order Euler-implicit differencing. Alge-braic multi-grid (AMG) methods using pointwise Gauss-Seideliteration were used to aid in the solution convergence.

While the capabilities exist within FLUENT to perform pre-dictions of the motions of the ship, the simulations in this effortutilized experimental measurements for the sinkage and trim tofix the attitude and position of the model. This provided a sig-nificant reduction in the computational expense to perform thecalculations, and reduces the number of variables in comparingthe simulated and experimental drag forces. An effort is currentlyunderway to assess the accuracy and computational expense as-sociated with dynamically predicting the sinkage and trim as partof the solution process.

Results for Calm-Water ResistanceWe turn to Figure 8, whose six parts show the experimental

and computed results for the abovementioned six configurations.The symbols on these plots are listed in Table 5.

The curves for the total resistance (or the resistance compo-nents) are all represented in a dimensionless form known as thespecific resistance. This is the ratio of the total resistance (or the

10

(a) Configuration 1,∆ = 17,760 t,L = 168.6 m,B = 56 m (d) Configuration 4,∆ = 12,340 t,L = 168.6 m,B = 40 m

(b) Configuration 2,∆ = 17,760 t,L = 210.6 m,B = 56 m (e) Configuration 5,∆ = 12,340 t,L = 210.6 m,B = 40 m

(c) Configuration 3,∆ = 17,760 t,L = 253.1 m,B = 56 m (f) Configuration 6,∆ = 17,760 t,L = 253.1 m,B = 40 m

Figure 8: Calm-Water Specific Resistance at Model Scale Plotted at Full-Scale Speed

11

Figure 7: CFD Grid

resistance components) to the vessel weightW. It is important toclarify that the calculations are all executed at model scale, in or-der to be directly comparable with the model experiments. How-ever, the results for resistance are plotted versus the prototypevelocity scale, namely the speedU in knots, but without any cor-rection to the resistance due to the fact that the full-scaleReynoldsnumber is substantially greater. Thus, for a realistic prototype cal-culation, one must perform the appropriate Froude extrapolation.This would substantially lower the specific frictional dragand,consequently, the specific total drag.

The plots also indicate the relevant values of the offsets2/2of the sidehull for the six configurations. This is measured be-tween the centerplane of the centerhull and the centerplaneof thesidehull. Additionally, the staggerr2 of the sidehull is shown.This is measured as the location of the sidehull transomaheadofthe centerhull transom. The value ofr2 was zero or negative forall the configurations, because we believe that sidehulls staggeredaft lead to a more practical design. Curiously, however, thetheo-retical resistance of the trimaran is lowered a similar amount forpositive and negative staggers. This fact can be demonstrated by

Table 5: Nomenclature for Plots of Resistance

Symbol MeaningU Speed of vesselW Displacement weightfs Surface-velocity limit factorr2 Stagger of sidehull

s2/2 Offset of sidehull∆ Displacement massRF Frictional resistanceRH Hydrostatic resistanceRT Total resistanceRW Wave resistanceRair Air resistance

a study of the resistance formula attributed to Michell (1898).

We first consider Configuration 1 in Figure 8(a). The waveresistanceRW possesses its typical maximum at the “hump” con-dition. This occurs for our vessel at a speed which varies accord-ing to the configuration. Thus, for Configuration 1, the humpspeed is about 43 knots. This is coincidentally the design opera-tional speed. The wave resistance is the resistance experienced bythe vessel due to the creation of the wave system associated withthe forward motion of the vessel through the water.

The hydrostatic resistanceRH is seen to be relatively small,indicating that little of the transom is immersed in the water. Thehydrostatic resistance is due to the presence of the transomstern.It is defined and computed on the basis of the lack of hydrostaticpressure on the face of the transom, due to the assumed cleanseparation of the flow of water past the hull at the transom station.

It is clear that the presence of a transom gives rise to anadditional and unwanted resistance component in contrast to amore traditional vessel with a streamlined stern. However,it isnot immediately obvious whether the total resistance increases asa result of the transom. This is because the wave resistance isalso affected by the presence of the transom. The reader is re-minded that the wave resistance is computed here assuming thatthe effective vessel moving through the water is the actual vesselcombined with the added hollow in the water behind the tran-som. Thus, Doctors (1999), in his extensive comparison betweendifferent multihull configurations, demonstrated theoretically thatthe presence of a modest transom resulted in a lowering of theto-tal resistance — compared to the resistance of a vessel either withno transom or a vessel with a substantial transom.

We should also add here that the reason for incorporating atransom in a marine vessel may, in fact, not be related at all tohydrodynamic considerations. The transom, because of its ge-ometry, lends itself to the most convenient location for siting thewaterjets, which are often used for the propulsion of such vessels.

The frictional resistanceRF is estimated from the 1957 ITTCline and is seen to be the major drag component, at least at modelscale, for the higher-speeds, namely the 50 to 60 knot range.

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The air resistanceRair is the fourth drag component plottedon the graph. We computed this drag component assuming thatthe drag coefficient was 1.000, based on the frontal area of themodel ship. This drag component is negligible for our design.

The simple sum of these four components is indicated by thefirst of the three curves for total resistanceRT . This predictedresult may be compared with the experimental dataas representedby the symbols.

It can be seen that there is a total of four curves for the totalresistanceRT . As indicated on the graphs, the first curve (indi-cated by circular symbols) represents the experimental data de-rived from the model tests. The other three curves forRT (eachwith a different value offs associated with it) were obtained fromthe theoretical work, as explained in the following text.

It can be seen that this first prediction of the total resistance,indicated by the symbolfs = 1, falls short of the experimentalresistance. This discrepancy increases with speed. It is believedthat most of this discrepancy can be traced back to the interactionsbetween the subhulls increasing the frictional drag, as noted ear-lier in this paper. The symbolfs is a surface-velocity limit factor,which we employed in our simple resistance-prediction software(not the CFD computations, which requires no such correctionfactors).

To emphasize this point, one has only to note that the sub-hulls each possess a maximum local beam of approximately 10 mat the waterline. With an overall subhull centerplane spacing s2

of 46 m, this leads to a minimum gap of 13 m. This suggests thatthe water has been channeled from an initial width of 23 m downto 13 m, implying a large increase in the local speed of the waterover the surface of the hull. On the other hand, it should be notedthat this effect will be much less on other parts of the subhull sur-face. The current estimate of the velocity increase does take thiseffect fully into account in the third theoretical prediction, indi-cated byfs = ∞.

In the computer program, the local “geometric channelwidth”, or gap, between the subhulls is computed. The ratio ofthe subhull centerplane spacing to this local gap, which is alwaysgreater than unity, provides an upper limit on the ratio of the lo-cal water speed to the speed of the vessel. This ratio is termedthe “subhull surface-velocity ratio” in our analysis. (This ratiois 23/13= 1.769, at the point where the subhull local beam isgreatest, in the above example).

Frictional resistance essentially depends on the square ofve-locity. But there is a weak Reynolds-number effect as well, lead-ing to a drop in the frictional resistance coefficient with speed.Thus, the computer subroutine calculates the average valueof thesubhull-surface velocity ratio to the 1.8 power (rather than thesecond power), over the surfaces of the individual subhulls.

This new factor is used to correct the frictional resistancebased on the standard 1957 ITTC formulation, for each subhull,on an individual basis. This simple correction procedure isa one-dimensional method. It no doubt ignores ameliorating effects dueto the true three-dimensional nature of the flow over the subhulls.

Thus, we also allow the possibility of moderating the intersubhullinfluence, as detailed in the following paragraph.

It is considered that the prescription explained here overes-timates the effect under study, as noted when the vessel speed Uexceeds 44 knots. Thus, yet another curve is shown, indicated byfs = 1.2. The meaning of this is that the computer coding placesan upper limit on the subhull surface velocity of 1.2 times thevessel speed.

The other five parts of Figure 8 pertain to the other five con-figurations. On the whole, it is confirmed that the surface-velocitylimit factor fs = 1.2 does seem to be a reasonable compromisefor achieving the best correlation between theory and experiment.One may note that the hump-like nature of the wave-resistancecurve for Configuration 1 is diminished for Configuration 4 and itis essentially non-existent for the remaining four configurations.This is to be expected because it is well known that longer vessels(Configurations 2, 3, 5, and 6) possess hump-free wave-resistancecurves.

Figure 8(d) and Figure 8(f) also show some calculationsbased on the FLUENT software. Because computational timesare much greater for these very extensive and detailed CFD stud-ies, a limited number of computed points only is depicted here.For this reason, we have chosen to use square symbols to indi-cate the FLUENT predictions, whereas it is conventional to use acurve for theoretical predictions. It is indeed most encouraging toobserve the excellent matching between the FLUENT data pointsand the experimental data. This statement is particularly true forConfiguration 6, the laterally-closely-spaced, fully-extended con-figuration, for which we are most interested in achieving high-fidelity predictions.

The average relative difference between prediction and ex-periment, when using the speed range of 20 to 50 knots in Con-figuration 6, is 9.9% and 1.8% for the thin-ship theory and CFDresults respectively. For the less operationally important, and hy-drodynamically more challenging Configuration 4, the averagedifferences were 16% and 8% respectively. When comparing thetwo computational methods, we must recall the extra expensethatis associated with achieving the increased accuracy. The time re-quired to complete a converged drag result for a single forwardspeed is approximately 24 to 48 hours using 24 processors fortheRANS method and only about 1 second on a single processor forthe thin-ship theory code.

The RANS computation is also useful for the prediction ofthe free surface elevation. For the adjustable-length trimaran ge-ometry, the designer is concerned with wetness of the fore andunderdecks, the water level near the water jet intakes, and theinterference between the center- and sidehulls. A sample pre-diction of free-surface elevation from the RANS computationsfor an equivalent forward speed of 40 knots is shown in Fig-ure 9. A closer examination of this figure also points out howthe contour of the wave elevation appears discontinuous alongthe non-conformal grid interface separating the near-bodyregionfrom the outer domain. At the interface, the vertical dimensionof the polyhedral cells is much larger than that of the stretched-

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(a)

(b)

(c)

Figure 9: Free-Surface Contours

Table 6: Nomenclature for Plots of Ship Motions

Symbol MeaningL Length of vesselU Speed of vesselg Acceleration due to gravity

ω0 Sea-wave radian frequencyγ Heading of vesselA0 Wave amplitudeA2 Sway amplitudeA3 Heave amplitudeA4 Roll amplitudeA5 Pitch amplitudeA6 Yaw amplitudeε2 Sway phaseε3 Heave phaseε4 Roll phaseε5 Pitch phaseε6 Yaw phase

hexagonal neighbors. Subsequently, the interpolation performedduring post-processing is more refined in the region of cellswithsmaller spacing, thus leaving a visible discontinuity in the free-surface contour. The consequence of this is only a reminder thatthe method is solving an assumed continuous process discretely.Because the focus of this effort was to examine the ability ofthetools to predict the model resistance, the rapid change in cellspacing across the interface was deemed an acceptable way torelax the mesh resolution requirements, while not significantlyimpacting the resistance computation, as evidenced by the resultsin Figures 8(d) and 8(f). Of course, if one were interested inac-curately predicting the wave behavior further away from thehull,then commensurate mesh resolution would be required. Wilson,Fu, Fullerton, and Gorski (2006) have shown that increased gridresolution enables a more accurate prediction of the wave eleva-tion, but the effect on resistance prediction is still unclear and casedependent since the wave field contributes to only a portion of thetotal force.

It is conjectured that the RANS method more accurately rep-resents the sidehull interference. The inter-hull wave field can beseen in Figure 9(c), where the camera is placed in between thecenter and starboard hull and faces aft. As pointed out earlier, thesubhull interaction increases with increasing forward speed andhull closeness; hence, the discrepancy between the two computa-tional methods grows as these parameters increase.

Results for SeakeepingThe five parts of Figure 10 present the results corresponding

to the second row in Table 4, namely a heading of 90 degrees(beam seas). Figure 10(a) is a plot of the response-amplitude op-erator (RAO) for sway, as a function of the dimensionless waveradian frequency. The nomenclature used in these plots of shipmotions are presented in Table 6.

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Four sets of data are shown on each plot. The first set of data,indicated as symbols (and denoted by “Exp reg”), show the exper-imental results from the regular-wave tests. The second data (de-noted by “Exp irreg”) is derived from the irregular-wave tests andwas obtained through a fast-Fourier transform (FFT); it is plot-ted as a short-dashed curve. In this example, there were, in fact,three separate irregular-wave runs. So, these are graphed as threeseparate curves, joining the analysis points with straightline seg-ments, without any smoothing. One can observe relatively closerepeatability for these three irregular-wave tests. Perfect repeata-bility is not expected, even if the instrumentation were perfect,because the irregular-wave system would have been different onthe three occasions, thus altering the outcome of the FFT analysis.

There is also close agreement between the irregular-wavedata (as a group) and the regular-wave data.

Finally, the last two curves show the multihull strip-theorypredictions, using either the combined-subhull concept (denotedby “Theory comb”) or the separate-subhull concept (denotedby“Theory sep”).

By way of clarification of the theoretical analysis, we pointout that we have tested two different approaches for handling thecritical matter of the wave interactions between the subhulls. Itis clear that these interactions will influence the motion behaviorof the vessel. In a more sophisticated strip-theory analysis, onecould account for waves created by one subhull impinging on itsneighbor or neighbors.

Utilizing the standard strip-theory simplification, one shouldargue that the interaction due to the wave from a subhull sectionmust occur at a downstream location on the neighbor-subhullsec-tion. The interaction would be further downstream for a greaterseparation between the two subhulls under consideration and alsofor greater forward speeds of the vessel. This was amplified uponby Lloyd (1989), where he published a relevant formula for thisrelative downstream location.

In strip theory used for the present calculations, thecombined-subhull concept assumes simply that these inter-subhull interactions take place at the same ship section. Thus,one can use the same ship-motion analysis, provided that thehy-drodynamic coefficients are computed on the basis of a multihullcross section. The method of Doctors (1988), which is effec-tively an enhanced version of the original approach developed byFrank (1967), was employed. It is certainly understood thatthisapproach will be suitable at low speeds and would be most appro-priate at zero forward speed.

It should be pointed out that the problem with the combinedsubhull concept is that because strip theory has divided theprob-lem into a series of two-dimensional problems it is susceptible tocreating a “trapped wave” system between the subhulls. In two-dimensions there are certain resonant frequencies at whichwavescan build-up between the hulls. The build-up leads to unrealisti-cally large hydrodynamic forces and in turn motion predictions.For actual three-dimensional bodies, the energy build up betweenthe hulls does not happen because the energy can escape in the

longitudinal direction, and results in much smaller wave build-up. This issue can be clearly seen in figures 10 and 11 where thecombined subhull predictions become very large and have wildoscillations for certain frequencies.

On the other hand, the separate-subhull concept assumes thatthe waves generated by a subhull section are swept downstreambefore they can impinge on a neighbor-subhull section. Thismethod is more suitable for higher speeds. Indeed, it would beentirely appropriate at speeds beyond the critical speed, when thegenerated waves fail entirely to impinge on the neighbor subhull.

For the low-frequency part of the range, the predictions aresimilar but this is not the case for the high-frequency part of therange. For this wave-heading angle, the combined-subhull ap-proach is perhaps slightly more faithful.

Heave motion for the 90-degree heading case is presented inFigure 10(b). The correlation between the regular-wave tests andirregular-wave tests is seen to be better for heave. We also notethat the separate-subhull predictions are superior to those from thecombined-subhull analysis. Roll motion is shown in Figure 10(c).Similar comments can be made for this mode of motion, with theexception that there are some differences between the experimen-tal data towards the low-frequency end of the curve. We may dis-cern undesired oscillations in the combined-theory calculationsat the higher frequency. It is assumed that these oscillations area result of the trapped-mode phenomena alluded to above. Pitchis shown in Figure 10(d). All the data is of low magnitude be-cause, as is well known for beam seas, there is little excitation ofpitch motion for most marine vessels. Finally, yaw is presentedin Figure 10(e). The quality of the results is high as for the pre-vious subfigure. In both cases, it appears that the analysis of theirregular data breaks down, to some extent, for low frequencies.

We now turn to the case of the 150-degree heading in thefive parts of Figure 11. Once again, the five parts of the figurecorrespond to the standard modes of motion, numbered from 2through 6. While there are general similarities for the results withthe case of the 90-degree heading, there are also some notabledifferences. In Figure 11 for the sway, there are some unnatu-ral responses at the low frequencies, not replicated in the muchsmoother separate-subhull theory. Also, at a dimensionless wavefrequency of 2.50, there is a relatively large variation in the re-sponse from the repeated regular-wave experiments. Heave inFigure 11 is well predicted. Disturbingly for roll, in Figure 11(c),we observe similar misbehavior for the experimental data that wealready referred to for the sway; it is even more exaggeratedhere.Pitch in Figure 11(d) is well predicted. However, the experiments,again, appear to be a little misbehaved, because the proper low-frequency limit for the RAO of cos(30◦) = 0.8660 does not occur.Lastly, the yaw in Figure 11(e), again, seems high for the experi-ments.

In summary for the motion responses, it can be stated that inalmost all cases, the separate-subhull theory provides a smootherand more realistic prediction of the experimental responses. Thiswas anticipated in the introductory comments on the theory,basedon our previous experience.

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(a) Sway (b) Heave

(c) Roll (d) Pitch

(e) Yaw

Figure 10: Seakeeping RAO, Beam Seas (γ = 90 deg), Configuration 6,U = 43 kn

16

(a) Sway (b) Heave

(c) Roll (d) Pitch

(e) Yaw

Figure 11: Seakeeping RAO, Bow-Quartering Seas (γ = 150 deg), Configuration 6,U = 43 kn

17

(a) Sway (b) Heave

(c) Roll (d) Pitch

(e) Yaw

Figure 12: Seakeeping RAO Phase, Beam Seas (γ = 90 deg), Configuration 6,U = 43 kn

18

(a) Sway (b) Heave

(c) Roll (d) Pitch

(e) Yaw

Figure 13: Seakeeping RAO Phase, Bow-Quartering Seas (γ = 150 deg), Configuration 6,U = 43 kn

19

We shall now discuss the phase angles corresponding to theamplitudes in the previous two figures. The phases for the 90-degree heading case are plotted in the five parts of Figure 12.Onthe whole, the agreement between the separate-subhull calcula-tions (definitely not the combined-subhull calculations) and theregular-wave experimental data is seen to be very good. It shouldbe borne in mind, with plots showing angles, that an angle of 180◦

is equivalent to an angle of−180◦. Hence, some apparent grossdiscrepancies (note the sole experimental point in Figure 12(d))are actually areas of very close correlation.

Finally, the case of phase angles for the 150-degree head-ing are shown in Figure 13. There is reasonable correlation forthe predictions emanating from the separate-subhull analysis. Onthe other hand, the combined-subhull analysis is seen to containconsiderable oscillations due to undesired and unphysical(non-existing) hydrodynamic interactions between sections of the dif-ferent subhulls at the same station.

CONCLUSIONSIn this paper a novel design for a high-speed sealift vessel

has been described. Initially several candidates that utilized thehigh-performance technologies of air-cushion assistance, multi-hulls, and adaptive geometry were considered. The demandingspeed and range requirement of the vessel eliminated all of theoriginal prototypes. Subsequently, a standard trimaran necessar-ily evolved into an adjustable-length trimaran that achieved dras-tically reduced wave resistance by extending the centerhull.

The hydrodynamic performance of the adjustable-length tri-maran was studied using three different numerical techniques to-gether with a large model test program. Calm-water resistancewas predicted using a code based on thin-ship theory and the CFDsoftware FLUENT. Seakeeping motions were predicted using aprogram based on classical strip theory. The powering experi-ments tested the trimaran in six different configurations ofvary-ing overall length, beam, and draft. The seakeeping model testswere done in a wide range of sea states, heading angles, and twoforward speeds, with the model possessing the lowest-resistancelarge-length narrow-beam configuration.

The adjustable-length trimaran geometry has subhulls thatare located relatively closely to each other, and the subhulls areheavily loaded, particularly upon departure. These attributes chal-lenge the viscous prediction that is used for calculating total dragwith the thin-ship theory code. Consequently, the viscous predic-tion was sensitized to the closeness of the hulls by estimating theincrease in velocity realized by the intersubhull fluid.

The accuracy of thin-ship theory resistance predictions wassatisfactory when compared to the experiments; the averagenor-malized difference was approximately 10%. The most distin-guishing attribute of this code in the present endeavor is the com-putational efficiency of generating predictions. The efficiency ofthis code enabled the user to investigate new designs of greateroverall length in an effort to reduce fuel consumption. Further-more, the ability of the code to accurately predict the effect ofchange in length and beam on the resistance of the six different

hull forms was demonstrated in the comparison with the experi-ments.

The CFD predictions were conducted on computational gridsthat utilize polyhedral elements near the body, and hexagonal fi-nite volumes clustered near the free surface in the far field.Twoof the six hull configurations were simulated, and the comparisonto experiment was encouraging. The average difference overthetested speed range was approximately 5%. The difference relativeto experiment of the CFD results was generally smaller than thatof the thin-ship theory. It is important to emphasize the differencein computational expense between the two methods: the CFD runtime is about one million times longer than the thin-ship theoryfor each simulated forward speed.

Strip theory was used to predict the motion RAOs for the tri-maran. The added mass and damping were determined in oneof two ways, either solving the boundary-value problem withallsubhull sections present at each station, or with a proper summa-tion of the contributions to the forces of each individual subhull.The comparison to experiment is encouraging, especially with thepredictions utilizing the separate subhull analysis. The designermust accurately predict loads on the centerhull in order to designthe structure and articulation mechanism, and further analysis isnecessary.

The efficient velocity-potential based codes enabled the au-thors to explore design tradeoffs, and even examine geometricalvariants that fell outside the originally defined parameterbound-aries. In fact it was this capability that led to the novel adjustable-length trimaran. Regardless of the advancement in the speedandpower of parallel computers, the velocity-potential basedmeth-ods will always be orders of magnitude (in this study six) morerapid. Conversely, for applications that can afford the extra ex-pense, CFD is proving to deliver on its promise of higher accu-racy. Furthermore, CFD provides additional details of the entireflow domain while robustly handling breaking waves and viscouseffects.

ACKNOWLEDGMENTSThe authors would like to gratefully acknowledge the sup-

port of the US Office of Naval Research through the High-SpeedSealift (HSSL) Program, under the project title, “ONR BAA 05-007: Architectural Concepts and Hydrodynamic Technologies forHigh Speed Sealift to Austere Ports: Subtopic B: ComputationalApproach and Hydrodynamic Tools”.

The authors would also particularly like to express their ap-preciation to Dr Ir Frans van Walree, from the Maritime ResearchInstitute Netherlands (MARIN) in Wageningen, Netherlands, forhis considerable assistance with the execution of the seakeepingtests and the subsequent analysis of the experimental data.

The CFD computations have been performed using resourcesavailable at the Ship Engineering and Analysis TECHnology(SEATECH) center, located at the Naval Surface Warfare CenterCarderock Division (NSWCCD) and the Air Force Research Lab-oratory (AFRL) Major Shared Resource Center (MSRC) HighPerformance Computing (HPC) facility.

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