Surface Wave Contours Associated with the Forebody Wake of Stepped ...

Surface Wave Contours Associated with the Forebody Wake ofStepped Planing HullsAuthor Names: Daniel Savitsky 1 (Member) and Michael Morabito 2 (Associate Member)1 Professor Emeritus, Davidson Laboratory, Stevens Institute of Technology2 Research Engineer, Davidson Laboratory, Stevens Institute of TechnologyResults of an extensive series of model tests that define the longitudinal surface wake profiles aft of prismatic hullshaving deadrise angles of 10 º, 20 º and 30 º are presented. Empirical equations are developed that quantitativelydefine these profiles and are in a form that can be easily applied by designers of stepped planing hulls. Theseequations are applicable for an expected range of variations in trim angle, speed coefficient, and loading coefficienttypical for these hulls. A brief introduction to the concept and to the hydrodynamic advantages of stepped planinghulls is presented to orient the reader as to the importance of wake data in their design. Examples are presented thatillustrate the application of these wake data for stepped planing hulls with wetted forebody chine to achieve maximumhydrodynamic lift/drag ratios. Finally experimental results are presented that illustrate the potential resistancepenalty associated with the operation of chines dry forebodies where the stagnation line crosses the step.INTRODUCTIONThe increasing availability of light-weight, high-horsepowerengines has motivated the designers of planing craft to designfor increases in maximum speed. It was soon realized howeverthat the ubiquitous hard chine planing monohull configurationhas inherent hydrodynamic characteristics that result inexponentially increasing resistance as the maximum designspeed increases and thus quickly consumes the availablehorsepower.An alternative high speed planing hull configuration is thestepped hull. It enables the designer to substantially reduce thehydrodynamic resistance at high speed (compared to anequivalent monohull) thus resulting in a more efficientutilization of the installed power. As will be described in detailin the subsequent section of this paper the stepped hull has atransverse discontinuity (step) located somewhat aft of midship.This results in a forebody that planes on the oncomingfree water surface and the flow separates from the bottom atthe step. The afterbody interacts with the wake of theforebody. The combination of the forebody and afterbodyforces must provide for vertical equilibrium of the craft. Whilethe force on the forebody can be readily calculated usingpublished methods, the afterbody force is not readily calculatedsince it is dependent upon the shape of the forebody wake thatmay intersect the afterbody bottom. Unfortunately, there hasbeen a dearth of published data that describes the forebodysurface wake geometry.This study reviewed surface wave data obtained from earlystudies of water-based aircraft (all have stepped hulls) andconcludes that their parametric variations were not generallyapplicable to stepped planing craft. Consequently, it wasnecessary to conduct additional experimental studies of planinghull wakes at the Davidson Laboratory for test conditions moreappropriate to planing craft. The results are presented in a formthat is easily applied by the designers of stepped planing craftto select optimum step depth and afterbody trim orientationrelative to the forebody and to estimate afterbody wetted areas.Prior to the presentation of the wake results, and to providefurther background for the use of stepped hull geometries, briefsummaries are provided of the lift and drag characteristics ofplaning hulls; the limitations of the monohull at high speed;Presented at the March 10, 2009 Meeting of the New York Metropolitan Section of the Society of Naval Architects and MarineEngineers.

and finally how the stepped hull overcomes these limitations.A hull configuration having a length of 32 ft (9.8m), adisplacement of 10,000 lbs (4546 kg) and operating in thechines wetted condition is used to illustrate the results. Futurestudies of the chines dry operating condition, such as iscommon in multi-step high speed racing craft, are planned.It is believed that this background may be useful inunderstanding the necessity for defining the forebody surfacewake geometries in the design of stepped hulls.IMPORTANCE OF TRIM ANGLE ON THEHYDRODYNAMIC RESISTANCE OF PLANINGCRAFTOne of the most significant parameters that influence theperformance of a planing craft is the equilibrium trim anglewhich varies with speed. It has a major effect on resistance,seakeeping, porpoising, and lateral stability. Each of theseperformance characteristics is worthy of a separate discussion.However, for the purposes of this paper only the effect of trimangle on hydrodynamic resistance is considered. It is shownthat the particular relationship between trim and resistancenaturally leads to adaptation of the stepped hull form at highplaning speeds.To best illustrate the dependence of resistance on trim angle,use is made of the planing lift and resistance equations forprismatic hulls from Savitsky (1964). These are repeatedbelow:Lift Coefficient of Zero Deadrise Surface:Cl o = Δ/ 0.50 ρ V 2 B 2= τ 1.1 ( 0.120 λ ½ + 0.0055 λ 5/2 / C V 2 ) (1)Lift Coefficient of Deadrise Surface:Cl β = Cl o –0.0065 β Cl o 0.60 (2)Hydrodynamic Resistance (exclusive of whisker-spray drag):R T = Δ tan τ + 0.50ρ V 2 λ B 2 C f (3)For the purpose of this illustration, consider a planing craft thathas the following geometry; loading and speed:LOA = 32 ft (9.8 m)Chine Beam (B) = 7.8 ft (2.4 m)Deadrise angle (β) = 12.5ºDisplacement (Δ) = 10,000 lbs (4546 kg)Velocity (Vk) = 40 knotsCl β = 0.036Roughness allowance = 0Now consider the planing surface free to heave but runningover a range of fixed trim angles. Using the above equations,the wetted area (λ B 2 ) and the total resistance/ displacementratio (R T /Δ) can be calculated as a function of trim angle. Theresults are plotted on Fig. 1. (Similar calculations can be madefor other deadrise angles; width of beam; displacements; andspeeds.) Although the absolute values of the results will bedifferent, the general conclusions on the effects of trim angleon wetted area and total resistance/displacement ratio will beessentially similar to that shown on Fig.1.Fig. 1: Calculated Wetted Area and R T /Δ vs. Trim AngleVariation of Wetted Area with Trim AngleReferring to Fig. 1 it is to be noted that the wetted surface areavaries inversely as an exponential power of trim angle. Forexample, decreasing the trim angle from 4º to 2º nearlyquadruples the wetted surface for a prismatic hull. This issignificant since the viscous resistance component increaseslinearly with the size of the wetted area (neglecting therelatively small change in C f as the wetted length increases).Further, for trim angles less than 5º there is a rapid increase inwetted area as the trim is decreased, while for larger trimangles there is only a moderate decrease in wetted surface asthe trim increases.Variation of R T /Δ with Trim AngleAs is well known to designers of planing craft, the totalhydrodynamic resistance (exclusive of whisker spray drag) isthe sum of the pressure drag (Δ tan τ ) that increases withincreasing trim angle and the viscous resistance developed bythe wetted bottom area which decreases with increasing trimangle. For the current example, it is seen from Fig.1 that thesum of these two components attains a minimum value in theproximity of 4º and increases for smaller and larger trimangles. The trim angle for minimum resistance increasesslightly as the deadrise angle is increased. For the present 12.5ºdeadrise angle it is seen that decreasing the trim angle from 4ºto 2º will nearly double the hydrodynamic resistance. Practicalhull forms have a longitudinally convex bow curvature that, ifwetted at low trim angles, will add additional form resistance.Designers are certainly aware of this trim effect on resistance

and attempt to locate the LCG to achieve optimum trim anglesand hence minimum resistance/displacement ratios. Of coursethis is not always achievable, particularly with high-speedplaning monohulls. Some designers will introduce a “rocker”geometry at the transom in order to increase the running trimangle-but this alternation is usually not completely effective.LIMITATIONS OF PLANING MONOHULLS ATHIGH SPEEDUsing the planing monohull hull geometry and dimensionspreviously listed, but now locating the LCG at 14.4 ft forwardof the transom and allowing the craft to have freedom in heaveand trim, the equilibrium values of trim and resistance, asfunction of speed, can be calculated using the method fromSavitsky (1964). These results are shown on Fig. 2. For thepurpose of presenting a simple illustration, the added resistancedue to the whisker spray (Savitsky, 2007) is omitted. This willnot change the conclusions of the present discussion.Trim Variation with SpeedNote from Fig.2 that, as is usual, the equilibrium trim angledecreases with increasing speed. At a planing speed of 20knots, the trim angle is approximately 3º and, as shown onFig.1, this results in an essentially minimum value ofresistance/displacement ratio. As speed increases, theequilibrium trim decreases and attains a value ofapproximately 1º at 60 knots. (well below the optimum trimangle) The calculation method (Savitsky, 1964) will also showthat the wetted keel length will exceed the LOA of this hull at60 knots.—this will cause some bow penetration into theoncoming fluid and hence result in an added form resistance.This is an undesirable operating condition. To avoid bowwetting the wetted keel length should be less thanapproximately 0.90 LWL. The calculation method (Savitsky,1964) indicates that this will occur at a speed of 40 knots. Atthis speed the trim angle is 2º—unfortunately, as shown onFig. 1 this is substantially less than the optimum trim angle ofapproximately 4º.Resistance/Displacement Ratio Variation withSpeedAs shown on Fig. 2 the resistance/displacement ratio increasesrapidly with increasing speed and the associated decrease intrim. At 20 knots, R T /Δ = 0.10—a most acceptable value. At40 knots, R T /Δ = 0.20—an unavoidable value that may have tobe accepted by default. At 60 knots, R T /Δ = 0.35—acompletely unacceptably value.Fig. 2 Calculated Typical Trim R T /Δ and vs. Speed -MonohullSummarySince the equilibrium trim angle of planing monohullsnaturally decrease with increasing speed, they quickly attainvalues that are well below the optimum trim angle required forminimum resistance. This may be partially alleviated bydesigning with a further aft located LCG or incorporating some“rocker” at the transom. If these are not feasible or totallyeffective, an alternative option is the use of a stepped planinghull that will allow for high speed operation at a more optimumtrim angle. The next section discusses the geometric featuresand resistance characteristics of the stepped hull and theimportance of providing forebody surface wake profiles forproper total design of these hulls.STEPPED PLANING HULLS DESIGNEDTOACHIEVE MAXIMUM LIFT/DRAG RATIOAT HIGH SPEEDThis brief discussion is not intended to provide a detaileddesign procedure for stepped hulls. It merely presents anoverview of the hydrodynamic influences that must beconsidered when designing these hulls for minimumhydrodynamic resistance at high speed. This may be in contrastwith the multi-step racing hulls that are designed for maximum

speed and stability. It is expected that at some future date theperformance of these multi-step racing stepped hulls will bebetter understood and their hydrodynamic lift/drag ratiosdocumented.The previous discussions have shown that the high speedplaning monohull has relatively large hydrodynamic resistancedue to its inherent characteristic of running at trim angles wellbelow the optimum trim angle required for minimumresistance at high planing speeds. The stepped hull overcomesthis deficiency in the following manner:The hull is split transversally at a position somewhat aft of midship creating a forebody and afterbody. The afterbody keel isslightly raised above the forebody keel creating a “step” in theprofile view (see Fig.3) hence the name “stepped hull”. Inaddition, the keel of the afterbody may be oriented at anappropriate angle relative to forebody keel to achieve a desiredintersection with the forebody wake. The water flow from theforebody separates from the bottom at the step location andclears most of the afterbody bottom. The contour of thisforebody wake governs the step height and angle of theafterbody. It is this forebody wake shape that is the primarysubject of this report.Typically approximately 90% of the total weight of a steppedhull is supported by the forebody since its lift/drag ratio isdesigned to be as large as possible. The remainder of theweight is supported by the afterbody or a stern appendage thatpenetrates the forebody wake. This appendage can be madeadjustable to control the trim angle as speed is varied. It shouldalso provide pitch damping to avoid porpoising. The LCGposition is in the forebody region and slightly forward of thestep. This limits the extent of the planing area on the forebody.Thus, to support 90% of the total load on a relatively smallbottom area, the trim angle of the forebody must be larger thanthat of a monohull which uses both the forebody and afterbodyplaning areas. This is the key to attractiveness of the steppedhull—at high speed it can be designed to operate at the trimangle for minimum hydrodynamic resistance. To furtherincrease the lift/drag ratio of the forebody, its trailing edge (atthe step) can either be cambered longitudinally (Clement andKoelbel, 1992) or fitted with an adjustable angle trailing edgeflap. The effectiveness of this later modification is discussedbelow.Estimate of the Performance of a Stepped HullTo illustrate the advantage of a stepped planing hull at highspeed, the monohull shown in Fig. 1 is reconfigured into anequivalent stepped hull as shown in Fig. 3. The totaldisplacement, 10,000 lbs (4546 kg), the LOA, 32 ft (9.8 m), thechine beam, 7.8 ft (2.4 m) and deadrise angle, 12.5º, remain thesame. The afterbody length is taken to be 13.5 ft (4.1 m). Theforebody load (Δ f ) is 0.90 x 10,000 = 9,000 lbs (4091 kg). It isassumed that the remaining 1,000 lbs (455 kg) is carried by theafterbody or a submerged or surface piercing hydrofoil that isattached to the transom and penetrates the forebody wake.Fig. 3 Dimensions of Equivalent Stepped HullB = 7.8 ft, β = 12.5°For the purposes of this illustration, the afterbody resistance isassumed to be small compared to the forebody so thecalculated results will be for the forebody which supports aload of 9,000 lbs (4091 kg). The afterbody load is taken to belocated approximately 1 ft (0.3 m) forward of the transom—this will be demonstrated in the illustrative example providedin a latter section of this paper. For pitch equilibrium thelongitudinal location (Lo) of the 9,000 lb (4091 kg) load on theforebody relative to the load on the afterbody is obtained bytaking moments about the resultant afterbody load location.Thus:10,000 x 14.4 = 9,000 x LoThus: Lo = 16 ft (4.9 m)Hence, the center of pressure of the 9,000 lb (4091 kg) load is16.0 - 12.5 = 3.5 ft (1.1 m) forward of the step. This is animportant dimension in the design of the stepped hull since thestagnation line on the forebody should intersect the chineahead of the step. Given the 9,000 lbs (4091 kg) acting 3.5 ft(1.1 m) forward of the step, the simplified format equations inSavitsky (1964) can be used to calculate the orientation of thestagnation line relative to the keel and also the keel and chinewetted lengths relative to the step. If the stagnation lineintersects the step, the heavy main spray, that originates at thatpoint will impinge upon the bottom of the afterbody and resultin a substantial increase in resistance.Performance Without Flaps Using the performanceestimating equations for the forebody without flaps (Savitsky,1964) the calculated equilibrium trim angle and resistance/displacement ratio are plotted on Fig. 4. (solid symbols) andare compared with the monohull results (curves). A transversestep was used in this illustration since it was shown in Savitskyand Brown (1976) that while a reentrant vee-transom, such assuggested by Clement and Koelbel (1992), can attain higheraspect ratios, the lift-drag ratio of the reentrant step issomewhat smaller than that of a transverse step that has thesame wetted area.

The calculations for the stepped hull are limited to speeds lessthan 40 knots (F ∇ =5.2) and greater than 30 knots (F ∇ = 3.9). Atspeeds greater than 40 knots, the trim angle will decrease andthe stagnation line will cross the step. The main spray will thusimpinge on the afterbody bottom with an un-quantifiedassociated increase in resistance. This is an undesirableoperating condition, however it can be alleviated by anadjustable hydrofoil attached to the stern than will control thetrim. At speeds less than 30 knots, the flow from the forebodymay not completely ventilate at the step. This will result in anincreased form drag of the forebody which, at the present time,cannot be easily calculated. It is likely that, at speeds less than30 knots, the resistance of the stepped hull will be greater thatthat of the equivalent monohull.From Fig. 4 it is seen that the running trim angles of thestepped hull are larger than those of the monohull and wellwithin the favorable range of trim angles as shown on Fig. 1.At an operating speed of 40 knots, the trim angle is 4.3º andthe R T /Δ for the stepped hull is nearly ½ that of the monohull.This is a significant improvement in performance that justifiesconsideration of the stepped hull for high speed operation. At30 knots, (F ∇ = 3.9) the R T /Δ for both hull forms are nearlyequal, so that the more simply constructed monohull would bethe desired hull form.Performance With Controlled Forebody Transom FlapsClement and Koelbel (1992) recommended that the afterbottom of the forebody have longitudinal camber that isdefined by a Johnson 3 term profile (Johnson, 1961).Since the primary curvature of the cambered section is along asmall area just forward of the trailing edge, it was speculatedthat a deflected flap located at the trailing edge of the forebodymay have similar beneficial effects on improving the lift/dragratio. Studies of the effectiveness of transom flaps installed onplaning craft are reported in Savitsky and Brown (1976). Itwas shown that for a fixed trim, load and speed the drag/liftratio was reduced as the deflected flap area was increased.Savitsky and Brown (1976) also presents methods forcalculating the flap effectiveness on a planing craft. Usingthese methods, a flap was selected for the stepped hull justdescribed. The dimensions and deflection of the flap whichwas attached to the hull bottom were:Full span = 7.8 ft (2.4 m)Chord = 0.40 ft (0.12 m)Deflection: 2.5º @ 40 knots10.0º @ 30 knotsThe flap deflection angles were selected to assure that thestagnation line on the forebody intersected the chine at eachtest speed.It is important to understand that the objective of this briefintroduction to stepped planing hulls is not to compare theperformance of a transverse step with the re-entrant andlongitudinally cambered step described by Clement andKoelbel. Each has their advantages and disadvantages. Rather,since the primary objective of this study is to supply wakedata, the transverse step was used as a simple illustration toemphasize the need for such data when designing steppedhulls.Using the methods of Savitsky (1964) and Savitsky and Brown(1976), the equilibrium trim angles and R T /Δ values werecalculated and are presented on Fig. 4. As expected, the flapreduced the running trim and, in addition, reduced R T /Δ at 30and 40 knots At these speeds, the R T /Δ was 0.10, a mostsignificant improvement compared to the planing monohull. Itis important to note that, while the deflected flap reduced thetrim angle at 40 knots, this lower trim angle was still with inthe acceptable trim range as shown on Fig. 1. It is suggestedthat a stepped planing hull may be designed without active trimcontrol if the LCG is located so that the craft operates atoptimum trim at the design high speed. For modest changes intrim, due to changes in speed, the R T /Δ varies very little.Fig. 4: Calculated Trim and R T /Δ vs. Speed – Monohull vs.Stepped HullsComparison of Wetted Areas and Performanceof a Stepped Hull vs. a MonohullFigures 5(a-c) present this comparison for a 40 knot planingspeed. The results are self explanatory. Note that the afterbody

forces are not considered. Although not discussed in this paperit is likely that, without an afterbody contribution to lift andespecially pitch damping, the forebody will porpoise. This canbe demonstrated by comparing the tabulated equilibrium trimangles with the porpoising limit criteria given in Savitsky(1964).Fig. 5(c) Comparison of Trim and Wetted AreasStepped Hull With Flap on Transom of Forebody(Forebody Load = 9,000 lb; V K = 40; F V = 5.2)Fig. 5(a) Comparison of Trim and Wetted AreasMonohull(Δ = 10,000 lb; V K = 40; F V = 5.1)SummaryThe stepped planing hull configuration offers significantreductions in total resistance at F ∇ greater than approximately5.0 as compared to a similar planing monohull.Unfortunately, there has been a dearth of data on the shape ofthe forebody wake which interacts with the afterbody. Thisinhibits the complete development of an analytical predictionmethod. The subsequent section of this paper presents suchdata.SURFACE WAVE CONTOURS IN THE WAKEOF PLANING SURFACESFig. 5(b) Comparison of Trim and Wetted AreasStepped Hull(Δ = 0.9 x 10,000 = 9,000 lb; V K = 40; F V = 5.1)Previous Experimental StudiesFor the reader who may not be familiar with the geometricform of the surface wake aft of a planing craft, Fig. 6, takenfrom Korvin-Kroukovsky, Savitsky and Lehman (1948b),presents a sketch of typical transverse and longitudinal wavecontours aft of a 10º deadrise hull that has a ventilated transom.As shown, the defined surface geometry extends 6 beams aft ofthe transom and 3 beams on either side of the longitudinalcenterline. Two characteristic geometric features areimmediately apparent. Along the aft extension of thelongitudinal centerline of the hull, the separated flow from thetransom is initially deflected below the level waterline. As thedistance aft of the transom increases, this depressed flowslowly rises and crosses the level surface to form the familiar“rooster tail” associated with planing craft. The second readilyvisible geometric features are the two longitudinally directedwave crests that are initiated at the chines. Their transversedistance from the centerline increases with distance aft of the

transom. Further, these crests maintain a nearly constant heightfor a distance of approximately 6 beams aft of the transom.Korvin-Kroukovsky, Savitsky and Lehman (1948a, 1948b,1949) contain many such plots for a range of planingconditions (trim angle, loading, and speed coefficients)appropriate to seaplane hulls. Specifically, the summary reportof these three references (Korvin-Kroukovsky, Savitsky andLehman, 1949) questions the reliability of these data at trimangles less than 6º. This is precisely the trim range appropriateto stepped planing hulls.Fig. 6 Experimentally Obtained Surface Wave Contours Aft of Transom of 10 Degree Deadrise Hull (Korvin-Kroukovsky,Savitsky and Lehman, 1948b)The wake data presented in Van Dyck (1960) are mainlyapplicable to seaplanes using hydro skis as landing and take offdevices. For that application, the test loading, speed, and trimconditions are substantially out of range with those associatedwith stepped planing hulls. Of particular significance in thatstudy however was the observation that the flow in anytransverse section aft of the transom is essentially twodimensional.That is, the trough developed by the hull fills infrom the sides and bottom. Hence the time history of the surfaceoscillation in the transverse plane can be combined with theforward speed to develop the longitudinal profiles of the surfacewake. It was also shown, that for a given depth of keel anddownwash velocity (V sin τ ), the resultant wake shape remainsthe same regardless of the separate values of V and τ.One of the earliest publications on wake shapes was a paper bySottorf, a German scientist who, in 1932, conducted model testsrelated to water-based aircraft. His work, which wassubsequently translated by NACA, (Sottorf, 1934) presentsseveral examples of the longitudinal centerline profile of thewake. Unfortunately, his test conditions were limited to onespeed and one displacement for several deadrise hulls and henceare insufficient for the general design of stepped planing craft.The original intent of the present study was to extract wake datafrom these references that would be suitable for the steppedplaning hulls. It soon became apparent that most of thepublished data would not be applicable. Hence, a model testprogram was undertaken to provide wake data for planingconditions appropriate to the stepped hull. A subsequent section

of this report describes this test program and presents the resultsin a format that can be directly used by the naval architect todefine the step height and afterbody angle to properly orient theafterbody relative to the forebody wake. An example of thisdesign procedure is also included in the present report.Analytical Efforts to Define Surface WakeGeometryMacPhail and Tye In 1944, MacPhail and Tye of the RoyalAircraft Establishment, Farnborough, England published atheoretical analysis of the wake geometry of planing hulls(MacPhail and Tye, 1944). The motivation for this study was“high speed porpoising of flying boat models seem to beseriously aggravated by inopportune striking of the rear step onthe surface of the through behind the main step. It would clearlybe useful to know the shape of the trough in order to predict thelikelihood of such an occurrence”In their analysis, the authors examine a planing surface having afixed trim angle and constant forward speed passing through atransverse plane that is fixed in space and normal to the levelwater surface. At planing speeds the flow separates from thebottom of the hull and the transom and is completely ventilatedto the atmosphere. At the instant of passage of the transomthough this plane, a trough conforming to the shape of the hullbottom and having a slight local “wave rise “ just outboard ofthe chines appears. This trough has imposed on it a verticalvelocity equal to the planing velocity times the sine of the trimangle. It is further assumed that the trough fills in from the sidesand bottom under the influence of gravity and that the entireprocess is two dimensional in transverse planes. Thisassumption was essentially confirmed by model tests reportedby Van Dyck (1960). Using potential flow theory the authorscompute the time history of the vertical position of the freesurfaceas the though fills in. Multiplying time by the forwardvelocity of the hull translates into a longitudinal position aft ofthe transom. Thus the longitudinal profile of the wake can bedefined.MacPhail and Tye compare their analytical results with thelimited amount of model data that was available in 1944. Theyconclude that, even with the introduction of empirical factorsinto the analytical model that are required to obtain a reasonableagreement with the overall geometric features, there is stillfurther development required for a reliable quantitativeprediction of the wake shape. They suggest possible additions totheir analytical approach that are expected to improve theaccuracy of their method. Interested readers are encouraged tostudy this reference. The authors did not further pursue thisstudy because of the pressure of more urgent problems duringWorld War 2.Shoemaker Zero Deadrise Wake Profile In 1934 J.M.Shoemaker published an analytical method for calculating thewake behind a flat plate planing at a fixed trim angle(Shoemaker, 1934). He assumed that the wake shape crosssectionwas essentially a rectangular trough. Neglecting staticlift, and using the principle of conservation of energy, hecalculated the height of the wake profile aft of the plate. Theseresults compared favorably with data collected by Sottorf (1934)in tests of a flat plate.Computational Fluid Dynamics (CFD) Computational FluidDynamics methods can be a useful tool for predicting thesurface wake profiles behind a planing hull. There are manycommercially available codes that may be applied. As with allCFD developments, to assure accurate results, it is essential toselect a mesh density that is appropriate to the complexity of thegeometry of the body. This usually requires experimental data toguide the iterative process of selecting the proper mesh size andto also judge the reality of the final analytical predictions. Thepresent report provides experimental data that can be used byresearchers interested in developing a CFD solution for theplaning hull wake geometry.It is believed that, even if CFD software becomes available, itwill not be at the level where it can be easily used by small craftdesigners who may have a limited basic knowledge of theunderlying numerics. Consequently, the present study provides awide base of experimental data which are presented in a formatthat is directly and easily used by the designer of steppedplaning hulls. These data will also be quite useful to potentialCFD analysts who may be interested in this problem. Theauthors encourage the development of CFD since such a toolmay be useful in extending the range of parametric variables atmodest cost.Present Experimental StudiesScope of Model Tests The wake region surveyed in this testprogram was limited to the surface area located ½ beam oneither side of the aft extended longitudinal centerline, and 3beams aft of the transom of the test models. This is in contrast tothe large surface area shown on Fig. 6. The smaller area isjustified because the length of afterbodies are typically between2 and 3 beams so that the collected data are well within thepractical design range for stepped hulls. This limited range ofstudy reduced the number of test runs and hence expedited thecompletion of this study.Test Models The test models were 9 inch beam, 10º, 20º, and30º prismatic hulls. They were constructed of white pine and thechines and transom were sharpened to assure complete flowseparation from these edges. Fig. 7 shows the lines of the 20ºdeadrise parent model.

Fig. 7 Lines of 20 Degree, 9” Beam Parent ModelMeasurement of Surface Wake Contours Several methods formeasuring the wake contours were considered. These includedoverhead stereo photography; a traversing sonic probe mountedabove the water surface; and a system of vertical probes thatcould be adjusted vertically until they just touched the watersurface. It was concluded that the simplest method was toattach a thin vertical plate, marked with a grid, to the transomof the model. The bottom of the plate was aligned with thebottom of the model at the transom. It extended 3 beams aft ofthe transom (see photographs in Fig. 8). The longitudinal wakeprofile was readily identified against this grid.Fig. 8(b) Typical Wake Profile Looking into Trough fromCamera on CeilingFig. 8(a) Typical Stern Quartering View From Camera onTank SillFig. 8(c) Typical Under Water Photo Showing WettedLengths

Load Coefficient = C Δ = Δ/wB 3 : After a review of thedimensions and displacements of many planing hulls itappeared that loading coefficients varied between 0.40 and0.80. The smaller values are more nearly representative ofrecreational craft while the tendency to larger values is usuallyassociated with heavily loaded military craft. Of course, it is thedesigners’ decision to specify the loading on the craft. Theseobservations were used primarily to select a range of testloadings that would be realistic and also result in a reasonablysized test matrix. Thus, the test loading coefficients were:C Δ = 0.40, 0.60, and 0.80Trim Angle (τ) Range: The previous discussions of theoperating conditions of the stepped hull concluded that itsfavorable resistance characteristics are a result of running atoptimum trim angles as shown on Fig.1. It is seen thatresistance/weight ratio is at its minimum with only a smallvariation for trim angles between 3º and 5º. The test range oftrim angles were somewhat extended to include a trim angle of2º. Thus:τ =2, 3, 4, 5 º.Fig. 9 Cross Section of Towing Tank Showing CameraLocations Used During TestsAs shown on the sketch in Fig. 9, four cameras were used torecord the test results. One camera was mounted on the ceilingof the tank and provided a view directly into the wake through.A second camera was mounted high on the moving test carriageand also provided a view directly into the wake trough. Wakeprofiles were recorded by both these cameras. A third camerawas mounted on the tank sill and provided data on the sprayheight. The fourth camera was located in a water tight box andprovided underwater photographs of the wetted bottom area ofthe model. The camera on the test carriage photographed thegrid 4 times during each run. Photographs were taken of thelongitudinal wake contours along the extended hull centerline;¼ beam from the centerline; and along the chine.The underwater photographs provided the wetted keel andchine lengths. When combined with the trim angle, a measureof the keel draft at the transom was obtained.Range of Model Test Conditions A test matrix was developedfor a range of loading, wetted lengths, and speeds that wereconsidered to be representative of stepped hull designs and theiroperating conditions.Speed Coefficient = C v = V/ √ gB It was previously stated thatthe advantages of a stepped hull are primarily at high speedwhere the equivalent monohull is likely running at unfavorablysmall trim angles. It was also suggested the stepped hull is bestused at volume Froude numbers greater than 5.0. Using thedefinition of speed coefficient, C v , that is more commonly usedby planing hull hydrodynamic researchers, it appears that a C v =4 may represent a lower limit of speed for the stepped hull.Hence, the test values of C v were selected to be:C v = 4, 6, 8Test Procedure: The models were fixed in trim and were free toheave. The sequence of tests runs was as follows:For each deadrise hull the model was fixed at specified trimangle and loaded to correspond to a prescribed value of C Δ .Tests were then conducted over a speed range corresponding tothe range of C v values identified above. For the centerline andquarter buttock of the 20º hull, the surface wave profile relativeto the lower edge of the grid was photographed four timesduring the length of the run. The heights of the wave weremeasured at 9 evenly spaced (1/3 beam) intervals aft of thetransom. For the 10º and 20º hulls, data were obtained at 3evenly spaced (1.0 beams) intervals aft of the transom. Theresults from the different photographs were combinedarithmetically to obtain average values. In addition, anunderwater photograph of the wetted area was taken to identifythe wetted keel and chine lengths. For those tests conditionswhere the stagnation line crossed the transom (chines drycondition) the wetted width of the transom was alsophotographed. This procedure was repeated for each of thedeadrise models over the same range of trim angles, C Δ , and C v

identified above. There were nearly 2,500 data points collectedduring what was planned to be a limited test program to definethe surface wake profile over a relatively small area of thewake. It would be impractical to tabulate these data in thepresent report. They are in storage at the Davidson Laboratoryand can be made available to interested researchers.Presentation of Test ResultsAlthough the number of experimental data points appear to beoverwhelming, it was found that the longitudinal wake profilescould be presented simply in a graphical form easily used bystepped hull designers. The sketches of Fig. 10 are diagrams ofthe reference axis used to identify the surface wake profiles.Figures 11, 12, and 13 present these results for hull deadriseangles of 10º, 20º and 30º respectively. Each figure presents thelongitudinal centerline wake profile on the left and thecorresponding ¼ buttock line wake profile on the right. Eachcontain three vertically arranged plots that provide results forL k /B values that assure chines wet operating conditions andthese are a function of trim and deadrise angle. Fig. 14 presentsa tabulation of the minimum values of L k /B. These values weredeveloped using an assumed minimum value of 0.10B forminimum L c and the quantity (L k – L c ) as given in Savitsky(1964).Within each plot the ordinate is the wake height in beams(relative to the bottom edge of the grid) and the abscissa is thedistance aft of the transom in beams. Note that the vertical scaleis 4 times the horizontal scale so that the actual slopes of thewake contours are exaggerated. The curves themselves are forcombinations of trim angle = 3º and 5º and C v = 4 and 8.Although similar curves are available for a trim angle of 4º andC v of 6.0, they were not included on these figures since theywould crowd the plots and make it somewhat morecumbersome to use. For trim = 4º and C v = 6.0 the designer caneasily interpolate between the plotted curves.Development of Surface Wake Profiles on Figures 11,12 and13 While the wake profiles are shown to plot smoothly and tovary systematically as the operational parameters are varied, thedata itself had scatter (+/- .03 beams), and anomalies. As isusual in analyzing large data collections, a fairing procedurewas developed to provide faired curves through the data. Ratherthan use computer based methods to arbitrarily fair the data, itwas decided to develop empirical equations that are based uponphysical phenomena that can be associated with thedevelopment of the wake. Specifically, the following physicalphenomena were to be represented in the empirical equations.1) The depth of the keel at the transom relative to the levelwater-line. For a given deadrise this defines the intitialcavity just behind the planing surface This depth is equal toL k sinτ. For the relatively small trim angles associated withminimum lift/drag ratios, this can be written as L k τ / 57.3.2) The downwash velocity at the transom = f (planingvelocity and trim angle). In the empirical developments, itappeared that using √ τ best represented the data. In thiscase τ is in degrees.3) Combining (1) and (2), it was found that the combination(L k τ 1.5 /57.3) was an important parameter when collapsingthe data.4) The longitudinal profiles of the surface wake appear to berepresented by sine curves whose origins are at theintersection of the submerged keel and the transom or theintersection of the quarter beam line and the transom.5) For a given draft and trim angle the height of the wakeprofile at a given distance aft of the hull transom isinversely proportional to the speed coefficient C v . Thisfollows from the conclusions of Van Dyck (1960) where itis shown that, given fixed trim and draft, the formation ofthe three-dimensional wake geometry is essentially theresult of two-dimensional flow in a vertical plane. It wasshown that the time history of the two-dimensional flow(relative to the level waster surface) was essentially thesame for any combination of trim and speed as long asVsin τ is constant. For the relatively small trim angles,shallow drafts and longitudinal positions ≤ 3B aft of thestep that are associated with stepped planing hulls, there isonly a small variation in the two dimensional time historieswith V sin τ. This time history is converted to longitudinalpositions along the three-dimensional wake by multiplyingtime by the speed of the craft. Thus: T = X/V ≈ X/ C v Thevertical displacement of the wake increases with time, T (atleast for X ≤ 3B). Hence, for a given X, T decreases as 1/C v , so that the wake displacement decreases as a functionof 1/ C v .Fig. 10 Diagram of Reference Axis for Wake ProfileEquations (Top) Centerline Profile (Bottom) Quarter BeamProfile

Fig. 11 Longitudinal Surface Wave Contours Along Wake of Prismatic Planing Surface β = 10°

Fig. 12 Longitudinal Surface Wave Contours Along Wake of Prismatic Planing Surface β = 20°

Fig. 13 Longitudinal Surface Wave Contours Along Wake of Prismatic Planing Surface β = 30°

Using the above physical phenomena inputs as guidance,empirical equations were developed which best represented theexperimental data. This required the selection of correlationfactors which are necessary to quantify the equations and toobtain agreement with the data. The resulting formulations aresummarized as shown:Equations for Longitudinal Surface Wake ProfilesCenterline Profile:β = 10 ºH = 0.17 [1.5 + 0.03 L k τ 1.5 ⎡]πSin ⎢⎢⎣Cvβ = 20 ºH = 0.17 [2.0 + 0.03 L k τ 1.5 ⎡]πSin ⎢⎢⎣Cvβ = 30 ºH = 0.17 [2.0 + 0.03 L k τ 1.5 ⎡]πSin ⎢⎢⎣Cv1/4 Beam Buttockβ = 10 ºH = 0.17 [0.75 + 0.03 L k τ 1.5 ] Sin1 .5⎡ π ⎛ X ⎞ ⎤⎢ ⎜ ⎟ ⎥⎢⎣Cv ⎝ 3 ⎠ ⎥⎦(7)β= 20 ºH = 0.17 [0.75 + 0.03 L k τ 1.5 ] Sin1 .5⎡ π ⎛ X ⎞ ⎤⎢ ⎜ ⎟ ⎥ (8)⎢⎣Cv ⎝ 3 ⎠ ⎥⎦β = 30 ºH = 0.17 [0.75 + 0.03 L k τ 1.5 ] Sin1 .5⎡ π ⎛ X ⎞ ⎤⎢ ⎜ ⎟ ⎥ (9)⎢⎣Cv ⎝ 3 ⎠ ⎥⎦where:H = height of wake profile above extended keel or ¼buttock, beamsL k = wetted keel length, beamsτ = trim angle, degreesX = distance aft of transom, beamsB = beamC v = speed coefficient, V/√ gBLimits of Application It is essential that the application of anydata based equations be limited to the range and combinationof parameters used in the test program. For this study:⎛⎜⎝⎛⎜⎝⎛⎜⎝X3X3X3⎞⎟⎠⎞⎟⎠⎞⎟⎠10º ≤ β ≤ 30º3º ≤ τ ≤ 5ºL k ≥ 0.10 + tan β /π tan τ0.017L k τ 1.5 ≥ 0.18L k < 3.5B for 20° and 30° deadrise and < 2.5B for 10°4.0 ≤ C v ≤ 8.0X ≤ 3B1 .51 .51 .5⎤⎥⎥⎦⎤⎥⎥⎦⎤⎥⎥⎦(4)(5)(6)The reason for the lower limit on L k follows from therequirement that the stagnation line should cross the chinerather than the transom of the forebody. In fact, it isrecommended that the wetted chine length be at least 0.10beams. If the stagnation line does cross the transom (this iscalled the chines dry planing condition) intense aft flowingspray jets will be developed at these points. These jets willimpact the afterbody of a stepped boat and result in a largedrag increment and possible longitudinal instability. Asubsequent section of this paper illustrates the severity of thesejets.The quantity (L k τ 1.5 /57.3) ≥ 0.18 represents the lower limit ofthis parameter as tested in the present study.Fig. 14 Minimum Values of L k /B to Avoid Chines DryConditionDiscussion of Wake ProfilesComparisons of the above empirical equations with the testdata are shown on Fig. 15. There is good agreement betweenthe equations and test data. The curves shown on Figures 11,12 and 13 are developed from these equations and provide aclear graphical representation of the appearance of theforebody wakes. Note that the parameters defining the curvesare varied in even increments. A designer using these figurescan either interpolate between the curves for odd values ofthese parameters or can use the equations to obtain wakeprofiles for specific combinations of L k , τ, and C v . Theequations can be easily evaluated.

CALCULATED WAKE HEIGHT(BEAMS)CALCULATED WAKE HEIGHT(BEAMS)CALCULATED WAKE HEIGHT(BEAMS)CHINES WET CENTERLINE WAKE HEIGHTSMEASURED IN TESTS VS. EQUATION0.450.400.350.300.250.200.150.100.050.000.00 0.10 0.20 0.30 0.40 0.500.300.250.200.150.100.05MEASURED WAKE HEIGHT (BEAMS)CHINES WET QUARTER BEAM WAKE HEIGHTSMEASURED IN TESTS VS. EQUATION0.000.00 0.05 0.10 0.15 0.20 0.25 0.300.300.250.200.150.100.05MEASURED WAKE HEIGHT (BEAMS)CHINES DRY CENTERLINE WAKE HEIGHTSMEASURED IN TESTS VS. EQUATION0.000.00 0.05 0.10 0.15 0.20 0.25 0.30MEASURED WAKE HEIGHT (BEAMS)Fig. 15 Plots Showing Goodness of Fit of Wake EquationsProfile Heights at ½ Beam Outboard of Keel It will be notedthat while measurements were also made of the longitudinalprofiles at a transverse distance of ½ beam outboard of the keel(chine location), they are not presented in this paper. As will beshown in a subsequent section of this paper, for deadrise hulls,where the forebody carries 90% of the total load, the afterbodyintersection with the wake occurs primarily along the afterbodykeel.Effect of Deadrise on Wake Profiles It was found that thedeadrise angle has a small effect on the longitudinal wakeprofiles. As shown in the empirical equations, the center-lineprofiles for the 20º and 30º deadrise hulls were the same(within the scatter of the experimental data). The centerlineprofiles for the 10º deadrise hull were slightly smaller thanthose for the larger deadrise hulls. The wake profiles for the ¼beam buttock reference position were the same for all deadriseangles and were smaller than the center-line profiles. Theseobservations are in general agreement with the conclusionsstated in the seaplane wake paper (Korvin-Kroukovsky,Savitsky and Lehman, 1949).Effect of Wetted Keel Length on Wake Profiles As shown onFigures 9, 10, and 11, for a fixed trim angle and speed, theheight of the wake profiles increases with increasing wettedkeel length. Since the draft of the keel = L k sinτ , it follows thatan increase in L k increases the penetration of the transomrelative to the level water line, This in turn increases thehydrostatic pressure at the transom and the volume of theinitial cavity generated by the hull. These increased initialdisturbances result in larger wake profiles. The reader shouldremember that the reference lines for the wake profiles shownin this paper are the extension of the keel or ¼ buttock linesfrom the transom into the wake cavity.Effect of Trim Angle on Wake Profile As shown on Figures 9,10, and 11, for a fixed L k and speed, the height of the wakeprofiles increases with increasing trim angle. This is a result ofthe increase in keel draft (as discussed above) and also to anincrease in downwash velocity at the transom due to theincreasing trim angle. More energy is thus initially imparted tothe wake and, as a consequence, the height of the followingwake is increased.For the present test range of trim angles and speed, the wakewas observed to be tangent to the hull bottom as it separatedfrom the hull.Effect of 0.017L k τ 1.5 on Wake Profile In developing theempirical equations for wake profile it was found the L k and τeffects could be well represented by the quantity 0.017 L k τ 1.5 .The test parameters were such that the minimum value of thisquantity was 0.18. Hence the application of the equations isrestricted to 0.017 L k τ 1.5 values equal to or greater than 0.18.Effect of Speed Coefficient (C v ) on Wake Profile Forotherwise identical planing conditions, increasing C v“stretches” the longitudinal wake profiles. Thus, at anyposition X, the wake height decreases with increasing speedcoefficient, C v .

Application of Wake Profiles in Design ofStepped Planing HullsTo illustrate possible applications of the wake profiles to thedesign of stepped hulls, the configuration shown on Fig. 3 isused as an example. The principal features of this hull aresummarized below:Total displacement = 10,000 lbs (4546 kg)LOA = 32ft (9.8m)Load supported by forebody = 9,000 lbs (4091 kg)Location of forebody load = 3.5 ft (1.1 m) forwardof stepDeadrise angle of forebody = 12.5ºLength of afterbody = 13.5 ft (4.1 m)Load on afterbody = 1,000 lbs (455 kg)Deadrise angle of afterbody = 12.5ºBeam of forebody & afterbody = 7.8 ft (2.4 m)Speed = 40 knotsSpeed Coefficient, C v = 4.3The objective of this illustration is to define the depth of stepand the trim angle of the afterbody relative to the forebody fortwo possible operating conditions:1. The aft end of the afterbody intersects the wake anddevelops the afterbody load and center of pressureposition required to provide for vertical equilibrium. Inthe present illustration the required afterbody load is1,000 lbs.2. The afterbody is clear of the wake. In this case theafterbody load will be developed by submerged orsurface piercing adjustable lifting hydrofoil that isattached to the stern.Using published planing hull performance prediction methods,such as shown in Savitsky (1964), the equilibrium trim angleand wetted keel length of the forebody are readily calculatedbased on the loading and speed defined above. The results are:L k = 1.1 beamsL c = 0.12 beamsτ = 4.3ºC v = 4.3Using these inputs and the wake equations developed in thispaper, the longitudinal centerline wake profile can be definedand is plotted on Figures 16 and 17. Note that the vertical scaleis substantially larger than the horizontal scale. This was doneto more clearly demonstrate the longitudinal curvature of thewake profiles.The two illustrative cases are discussed as follows:Afterbody Intersecting Wake Profile (Fig. 16) The procedureconsists in first locating the forward keel end of the afterbodyat a sufficient distance above the forebody keel (called the stepdepth) to allow the atmospheric air to proceed downward fromthe chine to the keel. This will assure ventilation of the flow asit separates from the forebody. Based on many experimentalstudies of stepped seaplane hulls it was found that a step depthof approximately 5% of the beam would be sufficient to assurenatural separation of the flow. Artificial aeration injectedthrough the hull bottom just aft of the step should also beconsidered since it can promote flow separation. This willreduce the required step height and, as a consequence, reducethe hull drag at non-planing speeds.

Fig. 16 Stepped Hull (See Fig. 5b) Afterbody Orientation Relative to Wake Profile to Provide 1,000 lb (455 kg) of LiftThe next step is to orient the trim angle of the afterbody relativeto the forebody so that its after end intersects the wake andestablishes a small lifting area. The extent of the lifting area isrelated to the afterbody trim angle and thus, defines the load onthe afterbody. The quantitative relation between afterbody trimangle and afterbody wetted area is established by superposingthe profile of the afterbody on to the wake profile (as shown inFig. 16) and measuring the wetted keel length for different trimangles. These inputs are used to calculate the afterbody load.Unfortunately, at this time there is no verified analytical methodfor calculating the lift force on a hull planing on the surface ofthe wake profile. Until further studies are carried out, it issuggested that established planing lift equations be used with theadded condition that the vertical velocity of local free surfacewake profile be included when defining the effective trim angleof the afterbody. This vertical velocity is directly related to theslope of the wake contour at its intersection with the afterbody.(a) For the case when the afterbody wetted area includes chinewetting, the following lift equations can be used:C Lo = τ 1.1 (0.0120 λ ½ + 0.0055 λ 5/2 / C 2 v ) (1)C Lβ = Clo - 0.0065 β Clo 0.60 (2)(b) For the case when the afterbody chines are not wetted,The planform of the wetted bottom area has a triangular shapesuch as shown on Fig.16. This makes it amenable to simple twodimensional analytical treatments such as developed byMilwitzky (1948). While that study deals with the impact ofchines-dry prismatic hulls on a level water surface, the resultsare readily adaptable to the steady planing process byeliminating the impact acceleration terms. The followingequation for the planing lift of a chines-dry prismatic hull is thusdevelopedC L = W / ½ ρ V 2 2L k (10)C L = π( π / 2β - 1) 2 sin 3 τ [(1 – tan τ / 2 tanβ)] (11)The term just ahead of the brackets represents the results of thetwo dimensional flow analysis. The term in the brackets is thecorrection for end flow losses due to the finite aspect ratio of thetriangular plan form. Milwitzky states that these functions arein good agreement with the experimental data for deadriseangles of 22.5, 30 and 40 degrees and “may not be too far inerror for angles as low as 15 deg.”For the present illustrative example, it was found that theafterbody intersection with the wake resulted in a chines-drywetted bottom area. Also an afterbody trim angle of 0.50º.

elative to the forebody resulted in a wetted keel length of 2.8 ft.This is shown on Fig. 16. The hydrodynamic trim angle of thisarea is a combination of forebody trim, afterbody trim relative tothe forebody and the local vertical velocity of the rising surfaceprofile of the wake in the wetted bottom region. This verticalvelocity is of course related to the surface wave slope. For thepresent case, as can be seen in Fig. 16, the local slope of thewake contour is essentially parallel to the free water surface sothat there in no vertical component of velocity due to the wake.These components result in an effective local hydrodynamictrim angle of the afterbody of 3.8º Substituting these values intothe chines-dry lift equation results in an afterbody load ofapproximately 1080 lbs. Assume that the center of pressure ofthis load is 0.40 L K forward of the transom. (1.1 ft. forward ofthe stern). It will be recalled that an afterbody load of 1,000 lbsand a longitudinal position of its center of pressure 1 ft forwardof the transom was required for vertical equilibrium. Furtheriteration of the trim angle of the afterbody will result in a keellength that provides the required values of load and center ofpressure. Unfortunately, the present experimental set-up did notprovide means for measuring the afterbody load, hencecomparison with equation 11 was not possible.The relatively small calculated triangular shaped wetted bottomarea at the stern of the deadrise afterbody is limited to the keelregion. Hence the longitudinal wake profile along the forebodycenterline is of primary importance in defining the afterbodyinterference with the wake. Fig. 17 is an underwater photographof a typical stepped planing hull. The small triangular shapedafterbody wetted bottom is clearly evident.Fig. 17 Wetted Areas for Typical Stepped HullAfterbody Clear of Wake (Fig. 18) It may well be that, in theabove illustrative example, the attempt to obtain verticalequilibrium by using the afterbody load may not be a stablesolution. The uncertainty is associated with the relatively smallafterbody wetted area and the small wetted keel length, L k .Since the load varies as the square of L k , small perturbations inwetted length may produce significant perturbations in this loadand thus result in longitudinal instability. Further, as the speedvaries, the extent of the afterbody wetted area and its lift forcewill also vary. It may well be that longitudinal equilibrium maynot be attainable over the operating speed range. This possibilityshould be studied.A possible solution to this uncertainty is to trim the afterbody sothat it is entirely clear of the wake. The vertical force requiredfor equilibrium can be provided by a small, adjustable,submerged or surface piercing hydrofoil that is attached to thestern of the afterbody. The angle of attack of this foil can beadjusted to provide the lift force required for verticalequilibrium at each speed. Its pitch moment arm relative to theLCG is independent of the wake slope. In addition, the hydrofoilwill also provide pitch damping to attenuate any porpoisingtendency. Fig. 18 shows such an arrangement. The afterbody isnow oriented at a trim angle of 2.0º relative to the forebody.Designers of stepped planing hulls may find alternate ways toincorporate these wake profiles into their design. They areencouraged to do so. What is presented herein is just onepossible application that is intended to illustrate a possiblemethodology.Spray Pattern When the Stagnation Line Crossesthe StepThis paper emphasized the importance of designing a steppedplaning hull so that the forebody chines are always wetted(stagnation line crosses the chine). To accomplish this, it hasbeen recommended that the wetted chine length be at least 0.10beams. If the stagnation lines do cross the step, large, highvelocity, spray sheets originate at these intersection points andimpact against the bottom of the afterbody. This will result in alarge increase in total resistance and possibly initiatelongitudinal instability. Water-based aircraft commonly observethese effects during take-off since the stagnation line must crossthe step just prior to the aircraft becoming airborne.In some designs it may not be feasible to achieve chine wettingat high speed. Hence several runs were deliberately made withthe stagnation lines crossing the transom of the test model. Fig.19 is a photograph of the spray pattern associated with thisplaning condition. Although measurements were not made ofthe spray height or intensity, the photograph does clearlydemonstrate the severity of this spray. Measurements were madeof the longitudinal centerline surface wave profile in the wake.It was found that the data are also well represented by empiricalequations similar to those for the chines wetted case. These areequations 12-14.

Fig. 18 Stepped Hull (see Fig. 5b) Afterbody Orientation Relative to Centerline Wake Profile to Avoid Afterbody WettingCenterline Profile:β = 10 ºH = 0.17 [1.5 + 0.03 L k τ 1.5 ] Sinβ = 20 ºH = 0.17 [1.5 + 0.03 L k τ 1.5 ] Sinβ = 30 ºH = 0.17 [1.0 + 0.03 L k τ 1.5 ] Sin⎡ π⎢⎢⎣Cv⎡ π⎢⎢⎣Cv⎡ π⎢⎢⎣Cv⎛⎜⎝⎛⎜⎝⎛⎜⎝X3X3X3⎞⎟⎠⎞⎟⎠⎞⎟⎠1 .51 .51 .5⎤⎥⎥⎦⎤⎥⎥⎦⎤⎥⎥⎦(13)(12)(14)20º with Wetted Beam = 61% Chine Beam* In this chines dry case B = actual wetted beam width incontrast to the geometric beam used in the chines wettedequations. It is used to non-dimensionalize L k , C v , and X.Fig. 20 illustrates the additional afterbody wetting due to theforebody stagnation line intersecting the step. The wetting dueto the intersection with the forebody wake was calculated and isidentified separately on Fig. 20. The additional wetting due tothe spray cones originating at the stagnation lines with the stepis also shown and was obtained from an underwater photographof a stepped hull. Note the large increase in afterbody wettedarea due to the spray wetting. This emphasizes the need todesign a chine wetted forebody for stepped hulls to avoid largeincreases in hydrodynamic resistance.30º with Wetted Beam = 58% Chine BeamFig. 19 Spray Patterns Associated with Stagnation LineIntersecting StepNote: The height highlighted on the photos is the heightfrom the undisturbed water surface to the top of thespray

forebody upon which the afterbody planes. Unfortunately, therehas been a dearth of published data defining these surface wakecontours. The present study fills this void by presenting theresults of an extensive series of model tests to measure thesurface wake profiles for several prismatic planing hulls over arange of test parameters typical for high speed stepped hulls.Empirical equations are developed that represent thelongitudinal surface profiles of the wake at various lateraldistances from the model centerline. These equations can beeasily evaluated for specified operating conditions of a steppedplaning hull.Sketch of Wetted Areas Due to Solid Wake and Due to SprayConeTwo illustrative examples are presented that demonstrate amethodology for combining the wake form with establishedplaning lift equations to properly orient the afterbody relative tothe forebody and its wake profile so that both longitudinalequilibrium and maximum hydrodynamic lift/drag ratios areattained. It is demonstrated that this single stepped hull has ahydrodynamic drag approximately ½ that of an equivalentmonohull at high speeds. This presentation is intended todemonstrate to designers the ease and simplicity of using thesewake data to complete their designs.Although five longitudinal surface wake profiles were obtained,it is shown that the most relevant for deadrise hulls is thelongitudinal profile aft of the forebody keel centerline.The report also demonstrates the substantial increase inafterbody wetted area associated with the forebody stagnationline intersecting the step (the so-called chines dry planingcondition).ACKNOWLEDGMENTThe authors are indebted to the T&R Steering Committee of theSociety of Naval Architects and Marine Engineers for theirencouragement to undertake this study and for providing partialfinancial support. We also thank Stevens Institute ofTechnology for the use of the Davidson Laboratory towing tankand undergraduate students, Maggie Hayes, Eric Giesberg andKyle Manning, who enthusiastically assisted in all phases of thisstudy.REFERENCESUnder Water Photograph of Chines Dry Run, and samephotograph with areas highlightedFig. 20: Additional Afterbody Wetting Due to Chines DryForebody SprayCONCLUSIONSThe ubiquitous monohull configuration of planing hulls hasinherently large drag/lift ratios at very high planing speeds. It isdemonstrated that this deficiency can be overcome by the use ofstepped hulls. A complete design of stepped hulls howeverrequires a knowledge of the wake form developed by theCLEMENT, E.P. and KOELBEL, J.G., Jr. “OptimizedDesigns for Stepped Planing Monohulls andCatamarans.” HPMV-92, Intersociety HighPerformance Marine Vehicles and Exhibit,Washington D.C., June, 1992JOHNSON, V.E., Jr. “Theoretical and ExperimentalInvestigation of Supercavtating Hydrofoils OperatingNear the Free Water Surface.” NASA TechnicalReport R-93, 1961.

KORVIN-KROUKOVSKY, B.V., SAVITSKY,DANIEL and LEHMAN, W.F. “Wave Contours inthe Wake of a 20-degree Deadrise Planing Surface.”Davidson Laboratory, Stevens Institute ofTechnology Report R-337, June, 1948a.KORVIN-KROUKOVSKY, B.V., SAVITSKY,DANIEL and LEHMAN, W.F. “Wave Contours inthe Wake of a 10-degree Deadrise Planing Surface.”Davidson Laboratory, Stevens Institute ofTechnology Report-R-344.November, 1948bKORVIN-KROUKOVSKY, B.V., SAVITSKY,DANIEL and LEHMAN, W.F. “Wave Profile of aVEE Planing Surface, Including Test Data on a 30-degree Deadrise Surface.” Davidson Laboratory,Stevens Institute of Technology Report R-339, April,1949.MACPHAIL, D.C. and TYE, W.D. “The Waves CloseBehind a Planing Hull.” Royal AircraftEstablishment, Farnborough Report No. Aero, 1992.November, 1944.MILWITZKY, B. “A General Theoretical andExperimental Investigation of Motions andHydrodynamic Loads Experienced by V-BottomSeaplanes during Step-Landing Conditions.” NACATechnical Note No. 1516. Washington DC, February1948NOMENCLATUREC Lo = lift coefficient, zero deadriseΔ / 0.50 ρ V 2 B 2C Lβ = lift coefficient, deadrise surfaceΔ / 0.50 ρ V 2 B 2C v = speed coefficientV / (gB) 1/2C Δ = load coefficient= Δ / w B 3C f = friction coefficient= Rf / 0.50 ρ V 2 SL k = wetted keel length , beamsL c = wetted chine length , beamsλ = mean wetted length/beam ratio( L k + L c ) /2Δ = displacement, lbsSAVITSKY, DANIEL “Hydrodynamic Design ofPlaning Hulls.” SNAME–MARINE TECHNOLGYVol.1, No.1, Oct. 1964, pp 71-95SAVITSKY, DANIEL, AND BROWN, P.W.“Procedures for Hydrodynamic Evaluation of PlaningHulls in Smooth and Rough Water” SNAME MarineTechnology, Vol.13, No. 4, Oct. 1976, pp 381-400SAVITSKY, DANIEL, DELORME, M. F. andDATLA, R., “Inclusion of Whisker Spray inPerformance Prediction Method for High SpeedPlaning Hulls.” SNAME Marine Technology, Vol.44,No.1, Jan. 2007, pp 35-5SHOEMAKER, J. “Tank Tests of Flat and V-bottomPlaning Surfaces.” NACA Technical Note No.509. 1934SOTTORF, W. “Experiments with Planing Surfaces.”NACA Technical Memorandum No. 739, March1934.VAN DYCK, R. “Final Engineering Report On Wake-Shapes of Planing Forms Associated With High-Speed Water Based Aircraft” Davidson Laboratory,Stevens Institute of Technology Report R-768,October, 1960.Δ f = load on forebody, lbsW = load on afterbody, lbsB = chine beam, ftS = wetted area , ft 2τ = trim angle, degreesβ = deadrise angle, degreesX = distance aft of step, beamsH = height of wake profile above extendedforebody bottom, beamsT = time, sec.R T = total resistance, lbsR f = viscous resistance, lbsV = velocity, ft/secσ = flap deflection, degrees∇ = displaced volume, ft 3F ∇ = Volumetric Froude NumberV / (g∇ 1/3 ) 1/2