Dynamic equilibrium evaluation for planing hulls with arbitrary geometry and variable deadrise angles – The Virtual Prismatic Hulls Method

Ocean Engineering 115 (2016) 67–92

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Dynamic equilibrium evaluation for planing hulls with arbitrary geometry and variable deadrise angles – The Virtual Prismatic Hulls Method R.D. Schachter a,n, H.J.C. Ribeiro b, C.A.L. da Conceição c a

Department of Naval and Ocean Engineering, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil Federal Institute of Education and Technology of Bahia, Bahia, Brazil c Program of Ocean Engineering, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil b

art ic l e i nf o

a b s t r a c t

Article history: Received 4 March 2015 Accepted 29 December 2015 Available online 18 February 2016

This paper describes the development of a semi-empirical computational method to solve the dynamic equilibrium problem for planing hulls with arbitrary geometry, including variable deadrise angles along the length. The method is based on Savitsky's method (1964) and it is named ‘Virtual Prismatic Hulls Method’, allowing for the determination of the intensity of the lift force and the resulting center of pressure, in order to determine the dynamic equilibrium position and the drag force, at the fully developed planing regime, for any arbitrary, non-prismatic hard chine hull form. Results produced by the method are presented and investigated and their applicability is established and compared with experimental results of well-known systematic series of planing boats. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Planing hulls Dynamic equilibrium Variable deadrise Warped hulls Concept design

1. Introduction In 1964, Daniel Savitsky presented a publication (Savitsky, 1964) that updated 40 years of research in planing surfaces (flat plates and prismatic surfaces), providing the maritime community with the first practical method of determining with reasonable precision, the lift, the drag and the dynamic equilibrium of planing hulls. This method has proved to be greatly successful and instantly became a landmark, the main reference for dynamic equilibrium of planing hulls. Ever since, the method had many evolutionary contributions, but in essence it remains very much the same, as it is applied by planing boat designers even nowadays. Some of the developments were carried out in several publications by authors such as Hadler (1966), Savitsky and Brown (1976), Blount and Fox (1976), and later Almeter (1993), Serter (1992), Royce (1996), among others. These works include contributions such as the addition of propulsion, taking into consideration the inclination of the propeller shafts, the integration of all forces, including the spray effect, influences of the behavior in waves, inclusion of trim flaps, comparative analyses with Series 62 (Clement and Blount, 1963) and 65, dynamic instability considerations, among many other contributions. In 2007 the now n

Corresponding author. E-mail address: [email protected] (R.D. Schachter).

http://dx.doi.org/10.1016/j.oceaneng.2015.12.053 0029-8018/& 2016 Elsevier Ltd. All rights reserved.

Professor Emeritus of Stevens Institute of Technology, Savitsky et al. (2007) made an improved contribution introducing a new component of resistance, the Whisker Spray, that made his method even better for the estimates. In 2010, once again Prof. Savitsky innovated proposing a solution for a problem that designers could only solve or dimension experimentally: he provided the Marine Community with a method of determining the Dynamic Equilibrium of Stepped Hulls (Savitsky and Morabito, 2010). Nonetheless, it is interesting to notice that until recently these formulations were still for prismatic hull forms, with no deadrise variation along the length or even keel height variations, which is not the case of most planing boats in operation or being designed. As very well argued in Savitsky (1985, 2012) and Begovic and Bertorello (2012), planing boats in the past were dominantly of constant deadrise, although many of them had variable beams along the length. As time passed, improvements were made for planing hull forms, the most common being the adoption of warped forms. This is because low deadrise angles produce less resistance, thus demanding less power, but are very bad for motions, producing very high bow and CG heave accelerations in waves. A logical warp form would be one that had low deadrise angles at the stern in order to minimize the installed power and consequently the weight. Since this part of the boat would supposedly always be submerged, it would not bounce on waves. For the mid-ship section, higher deadrise angles should be used to provide much better motions in waves and more ahead, onto the

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Nomenclature AP AR, BR As b(X) b ðxÞ BP BPA BPT BPX CAP Cf CfS CLβ CL(x) CL0 CLR CLS(x) CLSP(x) CPD CpH Cv D Df DT Error % f Fn FWD g h hC hP k1, k2 KF KG KI LCG LCP LD LF LH Lk Lm LP

projected area of bottom surface auxiliary constants surface wetted by spray (prismatic) local breadth local non-dimensional breadth breadth of prismatic hull associated average chine width: AP /LP chine width at transom maximum chine width longitudinal position of the center of projected area as a fraction of projected length of chines flat plate frictional coefficient frictional coefficient in spray resistance calculations lift coefficient, deadrise surface describes the height of central buttock line lift coefficient, flat plate (zero deadrise) sectional dynamic lift coefficient at trailing edge local sectional lift coefficient for the boat local sectional lift coefficient for the prismatic hull longitudinal position of dynamic lift longitudinal position of hydrostatic lift force speed coefficient depth drag due to friction abbreviation for dynamic trim angle percentage error distance between Df and CG Froude number forward abbreviation acceleration of gravity stern keel immersion, experimental stern keel immersion. stern keel immersion – calculated stern keel immersion for a prismatic hull non-dimensional constants vertical position of the point of application of frictional drag vertical position of the center of gravity abbreviation for keel immersion longitudinal position of center of gravity longitudinal position of center of total pressure dynamic lift force longitudinal position of the center of application of frictional drag hydrostatic lift force wetted keel length mean wetted length projected length of chines

bow, even bigger deadrise angles could be used to better pierce the incoming waves and minimize heave and pitch motions. To achieve such behaviors, a number of hull form variations are used, such as forms with chines that decline to stern, others with keels that lift towards the stern, called by some ‘rocker shapes’ – ‘keel negative trim, chine horizontal or chine positive trim’ (Savitsky, 2012), that are not so fast, but provide a better fit for propellers, or even ‘jet boats’ like forms, where the maximum beam is astern. In order to take into account the deadrise variation (warped or non-monohedral planing hull forms) along the length, Bertorello and Oliviero (2007) developed a method using conformal mapping

Ls waterline length of the boat LT total lift produced by the flow LOA length overall Lw wave length MS abbreviation for midship section NS number of sections Q Taylor number R1, R2(h,τ) remain functions Rex local Reynolds number Rp viscous pressure resistance Rs spray resistance Rt total resistance – experimental RT abbreviation for total resistance RtC total resistance – calculated SR surface wetted by spray (variable deadrise) SW wetted surface t dynamic trim – experimental T propeller thrust tC dynamic trim – calculated U incident flow speed, speed of boat, constant Vk speed in knots V(x) local speed W weight of the boat X dimensional longitudinal coordinate x non-dimensional longitudinal coordinate xc non-dimensional length of leading edge, nondimensional longitudinal coordinate of chine immersion point. Xc longitudinal coordinate of chine immersion point XchFW longitudinal coordinate of beginning of chine line XFW dimensional longitudinal coordinate where the keel begins to rise forward Xt dimensional longitudinal coordinate of the point of application of thrust force αCR rear angle of chines αE half angle of entrance of design water line. β deadrise angle β(x) local deadrise for the boat βP deadrise of the prismatic hull δCf Schoenherr´s roughness correction δF frictional drag force finite increment δX longitudinal coordinate finite increment ε Inclination of thrust line relative to keel λ mean wetted length-beam ratio λc1, λc wetted chine length ratio λc2 side wetness length ratio τ dynamic trim angle τ0 wall tension Δ displacement ∇ volume of displacement

(Finite Prismatic Hull Method) to evaluate the contribution of each section. They show towing tank tests with two models (warped and prismatic) compared with calculations of their method and Savitsky's (1964) method, the latter using deadrises at: the transom, 1/4 of Lpp and the LCG. This new method showed more accuracy for the non-monohedral model. Savitsky (2012) addresses this contribution, showing that the prismatic original method can provide reasonable results for warped rates at the order of 3°/ beam in a 10–30° deadrise (from transom, forward) hull form tested, and if the equivalent deadrise angle is well chosen, as being the one at the mean wetted length (Lm), provided it is assumed

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that the LCG and the center of pressure are superimposed. One other acceptable choice for the prismatic equivalent deadrise angle could be the one at the LCG, resulting in slightly lower resistance values. It should be stated that methods such as the one presented in this work and Bertorello and Oliviero (2007) ones could not exist if it was not for the method of Savitsky (1964), on which they are based. This work provides a different approach to solve the same problem. It is an attempt to determine iteratively the equilibrium position, the lift force and its center of pressure. After solving dynamic equilibrium, the method also determines boat's resistance. It was developed for use at the fully developed planing regime, for any arbitrary, non-prismatic (warped), variable deadrise, beam and keel height, single hard chine hull form, by means of a method named Virtual Prismatic Hulls method (VPH). The intention of this development was to try to provide a small improvement contribution to the landmark method of Savitsky (1964), to test hull form variations that could go beyond the boundaries a prismatic method could cover. In order to determine the total dynamic lift force and its center of pressure at each iteration, this method integrates the values of a sectional dynamic lift (dynamic lift force per unit of length) which, in its turn, is determined through a sectional dynamic lift coefficient. This dynamic lift coefficient associated to each section of the bottom of the hull is accessed through an interpolating function. Each value of sectional lift is integrated along the length, finding the center of pressure in the process, by summation of moments of all different sectional lift forces generated for each section, with its deadrise, beam and keel height.

2. Dynamic equilibrium of planing hulls Planing hulls are the simplest and most classical type of High Performance Marine Vessel (HPMV). One way of defining the need and the motivation to create conceptions of High Performance Marine Vessels is to overcome speed limits that exist for displacement ships and boats, and for most of them, to maintain high speeds in rough water, trying to maximize their loading capacity, safety and operational ability, that become strongly compromised with high speed and high seastates, associated to their usually smaller size. It can be said that the basic motivation that lead to the creative conception of HPMVs was the need to lift the craft out of the water, which is 815 times denser than the air, trying to avoid the free surface and its effects, as an attempt to maximize the lift in order to minimize the drag. Therefore, these concepts were created to allow for higher speeds by breaking away from the traditional displacement mode, a consequence of the displacement hull forms, which limits their maximum speed. Therefore, dynamic lift is generated and it will balance weight. This is not only for planing boats; it is also applied for the pre-take-off of Hydrofoil Boats, Wing in Ground Effect and for operations of Surface Effect Ship. 2.1. Speed length regimes As shown by many authors and mentioned by Savitsky (1985), ‘length is the main dimension used to define speed–size relationships at low speeds because the resistance of the hull to motion through the water is specially dependent upon the formation of waves which, of course, move at the same speed of the hull’. Displacement hulls must be designed to minimize flow separation near the stern, in order to reduce the drag due to the effect of boundary layer separation. On the other hand, the pressure field perturbation generated by the hull moving at the free surface generates the wave system responsible for

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the wave making resistance, being the bow and stern the main regions of wave formation along the hull, producing its components of ‘wave breaking’ (divergent waves) and ‘wave making’ (transverse waves), respectively. It is precisely the growth of these two components that becomes a barrier for higher speeds for most displacement hulls. The following discussion is based on potential theory, due to its simplicity. It was found that despite the fact that a real fluid flow is under consideration, the essence of the phenomena remains unchanged. Applying the dispersion equation at the condition of deep water to the divergent wave created by the bow, for instance, the following expression relating the waterline length with the length of the wave can be found: Fn2 ¼

1 LW 2π LS

ð2:1Þ

As explained in Savitsky (1985), at low speeds (below Q¼ Vk/ √L ¼1) marine craft span two or more wavelengths along the hull from their own bow train. At lower speeds, under Fn ¼ 0.23 or Q¼0.77, for instance, the train of waves spanned by the bow has a length equal to less than half of the craft's water line length. In this condition the power requirements are rather low and the resistance is predominantly frictional. For higher speeds the interference between the bow and stern waves becomes increasingly important, since the wave resistance becomes a more intensified barrier for the speed development, as will be shown ahead. In order to simplify the explanation, assume that the wave systems generated by the bow and stern initiate as crests, and have same lengths, as the celerities are the same. As a coincidence of two hollows or a hollow with a crest occur at the stern, maxima and minima peaks of wave resistance result due to this interference of both systems. Fig. 2.1 illustrates three particular situations of this interference between systems, for Fn of 0.28, 0.33 and 0.40 (Q of 0.94, 1.11 and 1.34, respectively). For the situations where Fn are 0.28 and 0.33, a maximum and a minimum occur, respectively. As the Froude number increases, such barrier becomes more intense, since at a limit situation of very high values – if it were possible – the interferences are formed with greater waves, preventing higher speeds to be achieved. Theoretically, for most displacement ships, these situations of constructive and destructive interferences would happen up to a limit condition, variable according to the hull form where the occurrences of constructive interferences at a certain high speed would cause increasingly energy dissipation with the wave formation, making higher speeds unfeasible. It is to be noted that the celerity C on the crest of a gravity wave in deep water, is expressed as: C ¼√(gLw/2π). If these last equations are expressed in terms of the Taylor number (Q¼Vk/√L, with Vk in knots and L in feet), which is usually adopted for fast craft, substituting the constants by their values and C converted from ft/ s to knots, C ¼1.34√Lw. This helps to explain the fixed relation between the speed (celerity) and the length of gravity deep water waves and why one wave length is spanned along the hull when it reaches a speed where Q¼1.34 or Fn ¼1.34/3.355 ¼0.399. At a certain point, as speed increases, besides the wave systems interference at high speed, it also starts to occur at a boundary layer separation at the stern, decreasing the pressure at that area. On the other hand, there is a surface elevation at the bow, due to its proximity to the stagnation point. For fuller forms, typical of displacement hulls, this causes the elevation of the bow. The combined effect results in a trim elevation with speed, followed by a progressive sinkage of the stern, that increases the wetted surface, the frictional and viscous pressure resistances, lowering the speed limit situation. In order to overcome this type of problem, it becomes necessary to abandon smooth buttock lines and round streamlined

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Fig. 2.2. Situation of a planing hull at Fn ¼ 0.89.

wetted length of side

Fig. 2.1. Wave systems interference (Fn ¼ 0.28, 0.33 and 0.4). c2

sterns, recommended for displacement vessels. Strait buttock lines, ending abruptly at a transom stern, reinforce the problem once avoided for displacement hulls. At the same time that they increase the viscous pressure resistance, the strait buttocks and transom force the flow separation out of the hull at higher speeds. This can be easily explained if it is noticed that because of the straight buttock lines of planing vessels, the flow separates at the transom, creating a low pressure region, below atmospheric pressure in this vicinity. It is exactly this low pressure region that is responsible for lowering the water level and drying the transom. When the water level is lowered, the atmospheric pressure acts all along the edge of the transom, making the bottom a high pressure area with no more separation. Therefore, the low pressure generated modifies the stern wave profile form initiating with a crest to a hollow. This wave system modification at the stern is now essential to cause destructive interferences at high speeds (see Fig. 2.2). It should be noted that from Froude number above 0.89 or Taylor number above 3.0, where the wave length is around five times the waterline length, the interferences will always be destructive, minimizing more and more the wave resistance. As the ‘wave making’ resistance becomes negligible at these speeds, the major component of resistance becomes once more frictional, together with the spray resistance. This component starts to be considerable from the moment the boat achieves the fully planing regime, as it represents the energy given to the fluid to maintain the stagnation line and keep the additional bottom surface wet by spray.

c1

Fig. 2.3. Side wetness prior to the fully planing regime.

the side at a certain position (see Fig. 2.3). Using Savitsky and Brown's (1976) nomenclature, let λc2 be the ratio defined by the wetted length of side and λc1 the ratio defined by the wetted length of the chine. Therefore, according to the reference:

λc1  λc2 ¼ 3C 2V sin τ Since λc1 ffi λ 

1 tan β 2 π tan τ

And making C 2V ¼

λ

ð2:3Þ ð2:4Þ

λc2 ¼ 0, the condition of dry sides is determined:

β  0:16 tan tan τ

3 sin τ

ð2:5Þ

Therefore, when this speed is achieved, the fully planing condition is characterized; with the dryness of the transom and sides, as well as the lack of separation at the bottom of the hull. Notice that, according to these expressions (2.3)–(2.5), a planing boat may be moving at a Froude number over 0.89, but have no dry sides, and thus not to be fully planing. From Eq. (2.5) it can also be noticed that longer boats have greater difficulty to reach the fully planing condition, as well as those whose hydrostatic trim is ‘nose down’. They tend to develop speed at lower trims, which implies in less ability to plane.

2.2. Fully planing regime

3. The Virtual Prismatic Hulls Method (VPH)

In most cases the fully planing regime is achieved at Froude numbers above 0.89. According to Yeh (1965) the fully planing condition occurs at about this Froude number, but the fully planing regime is also characterized by the condition where there is no separation of the flow at the bottom and the transom and sides (flare) are dry. The dry transom condition is strongly connected to the buttock lines, typical of planing boats, in conjunction to low transom angles. According to observations made by Savitsky and Brown (1976), the dryness of the transom occurs at a speed coefficient of about 0.5. This coefficient is evaluated as a function of the chine width astern, by the transom.

The Virtual Prismatic Hulls Method has been created during one of the authors' M.Sc. course (Ribeiro, 2002). The ideas that allowed for its creation were developed from the intention to carry out towing tank tests with models with combinations of deadrise variations, in order to compare results and complement Savitsky's Curves. In solving the dynamic equilibrium problem, it is needed to solve iteratively a system of equations that, in fact, imposes equilibrium of forces and moments when the attitude (dynamic trim and keel immersion aft) of the boat is steady, with constant speed. Frequently, this is done just as it is in this paper, that is: assuming an initial attitude and searching for equilibrium iteratively. As all the forces involved depend on the attitude of the boat, it is necessary to calculate, the more precisely possible, the values of these forces. Recognizing that the most difficult of these tasks is to determine total lift force in each iteration, the method presented herein consists of calculating this force breaking it into its two

U C V ¼ pffiffiffiffiffiffiffiffiffiffi gBPT

ð2:2Þ

For the condition related to the deadrise angle, the flare and the dynamic trim angle, the dryness occurs from forward to stern, from the stagnation point. But for a certain speed, the flow wets

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components. The hydrostatic component is easily calculated by direct integration of the pressure field on the bottom. The dynamic component is calculated using concepts of Savitsky's (1964) method, by integrating a sectional force. This sectional dynamic lift force is accessed through a sectional dynamic lift coefficient, which will be associated to each section of the boat. Once calculated all the sectional values of the sectional dynamic lift force, it can be integrated along the wetted length to compose the total dynamic lift and its center of pressure. Therefore, in order to determine the value of each sectional force, this coefficient is associated to each section of the boat. Each of these coefficients are determined by making it the same that acts in a given section of a virtual prismatic hull, that is, in its turn, associated to each section of the boat. Then, each of these sections of the boat will have a virtual prismatic hull associated to it. Also, each section of the boat has a value of the sectional coefficient to be interpolated through the variation of the coefficient in the prismatic hull associated. Assuming that each virtual prismatic hull has a known variation of the sectional coefficient along its length, the value of the sectional coefficient for the boat can be interpolated. Once the number of sections is defined for the application of the integration procedure, the process described above is repeated as many times as there are sections, in order to calculate the dynamic lift force value through integration. In the next subsections it will be explained that the whole process for determining the magnitude of the dynamic lift force starts with the division of the submerged bottom of the hull of the boat in sections. At the same time, a virtual prismatic hull is associated to each section. Each section has a dynamic lift coefficient interpolated through the longitudinal variation of the same coefficient for a prismatic hull. In imposing conditions to the function that represents the longitudinal variation of the sectional dynamic lift coefficient, some of these conditions use results obtained by the regression made by Savitsky (1964) for the dynamic lift coefficient for a deadrise surface. Each of these sectional dynamic lift coefficients becomes a sectional dynamic lift, that will be integrated to result in the magnitude of the total dynamic lift of the boat, to be inserted in each of the iterations to solve the dynamic equilibrium problem. It should be observed that since it is an iterative method, there will be a number of iterations until the steady state of the attitude of the boat is reached. For each of these iterations, the boat will be divided into a number of sections in order to determine hydrostatic and dynamic lift force by integration. Finally, each of these sections will have a virtual prismatic hull associated to it, in order to interpolate the sectional dynamic lift coefficient. This method should have the same limits of application as Savitsky's (1964) method, since some of the formulations used herein are the same, for being based on the method. Nonetheless, the present method has been applied in the forward region of most hulls of Section 5, with deadrise angles of up to 45 þ degrees and greater Lm/b (λ) values. So, the applicable limits are: λ r9; 2° r τ r15°; β r45°; 0.5 rCV r10.

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associated to each section of the boat, is interpolated from the variation of the same coefficient in the virtual prismatic hull associated to the boat's section. This subsection describes the process of defining the parameters of a prismatic hull to be associated to the boat's section. The definition of these parameters is necessary because it is through them that the interpolating function is fitted to each prismatic hull. Then, for associating a prismatic hull to a section, it is only needed to define the prismatic hull's attitude and geometry. Following Professor Savitsky's (1964) definitions, the attitude of this hull is defined by the mean wetted length ratio and dynamic trim. The geometry of the hull is defined by the deadrise angle and chine width. In this work, the mean wetted length ratio (λ) is substituted by the keel immersion aft (hP), which also represents the boat's attitude. Eq. (3.1) shows the relation of keel immersion with mean wetted length ratio. hP

λ ¼ tan

BP tan βP τ  2 π tan τ

ð3:1Þ

BP

3.1.1. Keel immersion aft (hP) Fig. 3.1 shows a boat with a certain attitude. The curve that describes the stem, or keel elevation astern, in relation to the base line, was named CL(X), as shown in Fig. 3.1. The keel immersion of the boat aft is h. So, the keel immersion of the prismatic hull associated to a section located at the longitudinal coordinate X on the center line of the boat is hP ¼ h  CLðXÞ

ð3:2Þ

3.1.2. Dynamic trim angle (τ) The dynamic trim angle of the prismatic hull associated to any of the boat's sections is the same of the boat for the actual iteration, as this parameter is one for the entire hull. This means that any of the boat's sections faces the flow direction with the same angle of incidence. Once in another iteration, when it is assumed another dynamic trim angle, this parameter changes to this new value assumed. 3.1.3. Breadth (BP) For the determination of this parameter two situations are possible. The section may be situated in the chine dry region or behind the chine immersion point, XC. For a section located at the dry chine region, the breadth of the associated prismatic hull is made equal to the beam at the chine immersion point. For the association to a section located at the submerged chine region, the breadth of the prismatic hull is taken as the sectional beam at the considered position. Therefore: Bp ¼ bðX C Þfor X oX C

ð3:3Þ

Bp ¼ bðX Þ for X Z X C

ð3:4Þ

3.1. Associating a prismatic hull to each section The present method consists of the determination of the dynamic lift force after integrating values of sectional dynamic lift forces. The value of the dynamic component and its center of pressure will be composed with the hydrostatic force and its point of application. The total force is inserted in the iterative process of solution of the dynamic equilibrium problem. Values of the sectional dynamic lift force are accessed through the values of a sectional dynamic lift coefficient. This coefficient,

CL(X) Water line

h CL(X) CL(X)=0

CL(X)=0

Fig. 3.1. Determination of the parameters of Eq. (3.2) in order to calculate the keel immersion aft of the prismatic hull associated.

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3.1.4. Deadrise angle (βP) The deadrise angle of the associated prismatic hull will be the same as the deadrise angle of the section under consideration. At this point, the virtual prismatic hull associated to each section is defined with a stern immersion of the keel, a dynamic trim angle, a deadrise angle and a breadth. Other parameters needed for the interpolation of the sectional dynamic lift coefficient can be calculated from these ones. 3.2. Determination of the sectional dynamic lift coefficient To allow the interpolation of each value of the sectional dynamic lift coefficient for the boat, it is essential to know how this sectional coefficient varies in the prismatic hull associated to each section. Once the longitudinal variation of the coefficient is known for the prismatic hull, the value for the boat will be interpolated in the same rational longitudinal position, relative to the wetted length of keel, LK. The value of the sectional coefficient will be transformed into a value of the sectional dynamic lift force, that integrated, will calculate the value of the total dynamic lift force. For convenience, a dimensionless longitudinal coordinate was created with its origin at the stagnation point, oriented backwards, parallel to the keel line, as shown by Fig. 3.2. If a dimensionless longitudinal coordinate is defined by x¼

X Lk

ð3:5Þ

And a dimensionless local beam as (x bðXÞ xC for x o xC ; bðxÞ ¼ bðxÞ ¼ BP 1 for x Z xC

C LSP ðxÞ ¼ a0 þ a12 x

þ a13 x

1=3

þ a1 x

ð3:7Þ

where a0, a12, a13 and a1 are constants to be determined through some conditions to be imposed to this function: iÞ iiÞ

At the stagnation point : C LSP ð0Þ ¼ 1

ð3:8Þ

At the trailing edgeðtransomÞ : C LSP ð1Þ ¼ C LR Z

1

iiiÞ 0

Z

1

ivÞ 0

xC ¼

BP tan β Lk  LC ¼ π Lk tan τ Lk

ð3:9Þ

C LSP ðxÞbðxÞdx ¼ k1

ð3:10Þ

C LSP ðxÞbðxÞ x dx ¼ k2

ð3:11Þ

ð3:12Þ

In Eq. (3.9), it is to be noted that at the trailing edge (transom) the disturbed free surface leaves the bottom tangentially. So, if the total pressure equals zero, then dynamic pressure equals the negative of hydrostatic pressure:   2g h B4P tan β ð3:13Þ C LR ¼  U2 Also, the sectional dynamic lift coefficient is here defined as C LSP ðX Þ ¼

dLD =dX 1 2

ð3:14Þ

ρU 2 bðXÞ

Then the equations for dynamic lift (3.15) and dynamic centre of pressure (3.16): Z Lk 1 1 C LSP ðX Þ ρU 2 bðX ÞdX ¼ ρU 2 B2P C LβD ð3:15Þ 2 2 0 Z

Lk

0

1 1 C LSP ðX Þ ρU 2 bðX ÞXdX ¼ ρU 2 B2P C LβD C PD 2 2

k1 ¼ C LβD

BP Lk

ð3:17Þ

k2 ¼ C LβD

BP C PD Lk Lk

ð3:18Þ

In the above equations CLβD represents the dynamic lift coefficient of the prismatic hull, according to Savitsky (1964). The longitudinal position of the dynamic center of pressure is determined by C PD ¼ 0:842119 þ 0:09981λ þ 0:02314τ  0:01299β  0:06183C V Lk ð3:19Þ This was developed to account for the determination of the hydrostatic pressure separately. Savitsky (1964) proposed an equation for both dynamic and hydrostatic lift components added, allowing for the separation of them, but for the center of pressure expression proposed, these components could not be separated. The polynomial above was generated from the regression of 117 runs of Fridsma's (1969) model tests.

Undisturbed free surface

Chine line

X XC Stagnation point

X

BP

ð3:16Þ

Yield to Eqs. (3.10) and (3.11), respectively, when shifting to the dimensionless coordinate system. Constants k1 and k2 are

ð3:6Þ

A function that describes the longitudinal variation of the sectional dynamic lift coefficient is 1=2

In Eq. (3.6) the non-dimensional length of leading edge is defined by

Leading Edge

Lk

Fig. 3.2. Coordinate system, leading edge and bottom surface of a prismatic hull.

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Finally, the imposition of Eqs. (3.8)–(3.11) allows the determination of the values of the unknown constants that define function CLSP(x), the sectional dynamic lift coefficient. Once this function is known for the prismatic hull associated to the section of the boat, the interpolation process of a value for the sectional lift in the boat's section is shown. Let CLS(x) be the sectional dynamic lift coefficient for the boat's section. Let also xi be the dimensionless position of the section of the boat, where it is desired to determine sectional dynamic lift coefficient's value. If xi ¼

Xi Lk

ð3:20Þ

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diagram describing the equilibrium position. Once the equilibrium position is obtained, the resistance can be determined. Also, the performance of the boat can be analyzed for the weight and longitudinal position of the center of gravity in the design process. Basically, the iterative process is followed through until all the forces reach the balance. Some forces are dependent on the boat's attitude and others are not. The lift and drag forces are dependent on the attitude, while thrust and the weight of the boat are not. So, the equilibrium will occur when the equilibrium attitude is reached. That is, at this attitude all the forces meet the balance. Briefly, the real unknowns are the dynamic trim angle and the keel immersion aft of the boat.

Then, the value of sectional dynamic lift coefficient for the boat's section can be interpolated as

4.1. Equilibrium position determination

C LS ðxi Þ ¼ C LSP ðxi Þ

The determination of the equilibrium position is directly connected to the solution of the system of equations that describes the balance of forces and moments. The free body diagram that describes the equilibrium position is shown in Fig. 4.1. From the diagram above the balance of the horizontal forces is found to be:

ð3:21Þ

3.2.1. Determination of the total lift force The total lift acting on the bottom of the hull is the summation of the dynamic and hydrostatic components. LT ¼ LD þ LH

ð3:22Þ

Let, then, LD be the dynamic lift force and dLD/dX the sectional dynamic lift force, for the boat's section. Through the sectional dynamic lift coefficient definition, the dynamic lift force can be determined by dLD 1 ¼ ρU 2 bðX ÞC LS ðXÞ dX 2 Z And :LD ¼

Lk

0

ð3:23Þ

dLD dX dX

ð3:24Þ

The hydrostatic force and its center of pressure can be determined through direct integration of the hydrostatic pressure field acting on the bottom surface: Z   ρ g z nz dS ð3:25Þ LH ¼ 

T cos ε cos τ  Df ¼ 0 The vertical balance of forces is given by LT þ T sin ε cos τ  W ¼ 0 And the balance of moments: T cos εKG  T sin εðLCG  X T Þ  LT ðLCG  LCP Þ þ LT KG sin τ  Df ðKG  KF Þ cos τ þDf sin τðLF LCG Þ ¼ 0

ð4:1Þ

Eliminating the total lift (LT) and the propeller thrust (T) from the equations above and substituting in the third equation, it gets to:   Df AR  W  Df tan ε BR  Df f ¼ 0 cos τ

ð4:2Þ

where AR ¼ KG  ðLCG  xT Þ sin ε

ð4:3Þ

BR ¼ ðLCG  LCP Þ cos τ  KG sin τ

ð4:4Þ

LT ¼ W  Df tan ε

ð4:5Þ

3.3. Determination of the center of pressure of the total lift

f ¼ ðKG  KF Þ cos τ  ðLF LCG Þ sin τ

ð4:6Þ

Analogous to the lift calculation, the longitudinal position of the center of pressure can be obtained through the summation of moments of the dynamic and hydrostatic forces. Therefore: Z 1 Lk dLD C PD ¼ XdX ð3:27Þ LD 0 dX

It was observed above that the true unknowns of this system of equations are the variables that define the attitude of the boat, that is, the keel immersion aft and dynamic trim angle. So, finally, two equations are needed in order to determine the two unknowns. Therefore, at the equilibrium position, certainly Eq. (4.7) is satisfied, as one of the conditions for the equilibrium:

Bottom

C PH ¼ 

LCP ¼

1 LH

Z

 Bottom



ρ g z x nz dS

LD C PD þ LH C PH LT

ð3:26Þ

ð3:28Þ

LT  W þDf tan ε ¼ 0

4. Dynamic equilibrium solution In the previous section the Virtual Prismatic Hulls method was presented, developed in order to determine the magnitude of the dynamic lift force in a planing boat of arbitrary geometry along the length. This value is to be inserted in the iterative process of the dynamic equilibrium solution. The present section presents the methodology in order to solve the dynamic equilibrium problem. At steady state, the dynamic equilibrium can be solved from the balance of forces and moments, carried out from the free body

Fig. 4.1. Free body diagram for the craft at planing steady state.

ð4:7Þ

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R.D. Schachter et al. / Ocean Engineering 115 (2016) 67–92

Also, at the equilibrium position Eq. (4.2) is satisfied. Due to such, the solution process will be composed by the numerical solution of a system of two transcendental non-linear equations, with two unknowns (4.8):

along the length to provide the total friction drag at the bottom:

LT  W þ Df tan ε ¼ 0

Note that the coordinates of the point of application of the drag force can be easily determined composing the vertical and longitudinal moments of each sectional contribution (Eq. (4.15)).

  Df AR  W  Df tan ε BR  Df f ¼ 0 cos τ

ð4:8Þ

All of the variables involved depend on both keel immersion and dynamic trim angle. Then, the system (4.8) can be written as a pair of remaining functions: R1 ðh; τ Þ ¼ 0 R2 ðh; τ Þ ¼ 0

ð4:9Þ

The system (4.9) can then be solved interactively by any numerical method, as wished, to determine the equilibrium position where the steady state occurs. In this case, the Newton–Raphson method was used. 4.2. Resistance calculation The resistance may also be divided into parts, and its calculation will be the result of the summation of its components due to friction, spray formation and viscous pressure. 4.2.1. Frictional resistance Once the dynamic equilibrium is found, the frictional part will be estimated integrating the tension over the average skin wall of each section of the bottom along the length. The use of a local Reynolds number, as a function of a local speed field, intends to improve the precision of resistance calculation. The following equations define a wall tension as a function of the local Reynolds number: Rex ¼

V ðxÞx

ð4:10Þ

υ

1 2

τ 0 ¼ ρU 2 C f

ð4:11Þ

The frictional coefficient Cf may be determined after solving Schoenherr's(ATTC) equation:   1 pffiffiffiffiffi ¼ 4:13log 10 Rex C f Cf

C LSP ðX Þ ¼ 1 2

ρU 2 bðXÞ

ð3:140 Þ

It is to be noted that the sectional dynamic lift used in (3.14) may be calculated using mean sectional speed V(x) by dLD ¼2 dX

Z

bðXÞ 2

0

   1  2 1 ρ U  V 2 ðXÞ dY ¼ ρbðXÞ U 2  VðXÞ2 2 2

NS X

ð4:13Þ

δF

ð4:16Þ

i¼1

4.2.2. Spray resistance Supposing that the energy received by the fluid at the spray root is dissipated in contact with the surface wetted by the spray, it is possible to determine the resistance due to spray:   1 RS ¼ ρU 2 SR C fS þ δC fS 2

4.2.2.1. Spray wetted surface. The actual hull spray wetted surface of an arbitrary form is calculated by an analogous process to the one used in the VPH method (Section 3). The leading edge is decomposed in transverse stripes and the spray wetted surface is determined by arithmetic average, as detailed ahead. Eq. (4.18) is presented at Savitsky's work (1964) and estimates the surface of a prismatic hull wetted by spray.   B2 tan ðβÞ 1 AS ¼ P  ð4:18Þ 2 π tan ðτÞ 4 tan ðΦÞ cos ðβÞ The angle Φ, according to Savitsky (1964), is the angle between the keel and spray edge, measured at the bottom plane. The averaging of this surface for the actual hull is made according to the sequence below: i) Divide the leading edge into NS sections; ii) In the section under consideration, identify the deadrise angle and the actual beam at the chine waterplane (X¼XC); iii) Compute the spray wetted surface of the equivalent prismatic hull (B, β, τ), (Eq. (4.1)); iv) If each computed surface of the later item is designated ASi, then the actual hull surface, SR can be approximated by NS P

SR ¼

i¼1

ASi

NS

RP ¼ LT sin τ

Once the sectional frictional coefficient is calculated (4.12), the elementary drag force at a station will be:

5. Applications

1 2

bðX ÞδX C cos βðX Þ f

ð4:15Þ

where δF represents the elementary drag force at a transverse station of the hull. This elementary force can then be integrated

ð4:19Þ

4.2.3. Viscous pressure resistance The condition of dry transom and sides, where the fully planing regime occurs, implies in the absence of separation along the whole bottom. This indicates that the total pressure is not negative, what allows for the possibility of decomposing the pressure integral in the opposite direction of the boat's motion and to determine the pressure resistance component. The total lift force can then be decomposed in the opposite direction of the motion to obtain:

Therefore, substituting (4.13) in (3.14), the average speed at the section under consideration will be pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ðxÞ ¼ U 1  C LS ðxÞ ð4:14Þ

δF ¼ ρU 2

ð4:17Þ

where CfS is the frictional coefficient from the ATTC line (ATTC, 1956), δCfs is the roughness correction, proposed by Schoenherr and SR is the spray wetted surface.

ð4:12Þ

where the value of speed required for computing Reynolds number in (4.10) and (4.12) is approximated through the sectional lift coefficient, as defined in Eq. (3.14). dLD =dX

Df ¼

ð4:20Þ

In order to validate the VPH model proposed here, some experimental results were simulated with the computational method developed in the previous sections. Initially the method was programmed and run in Mathcad (Ribeiro, 2002). The recent simulations were made using the recently developed computer

R.D. Schachter et al. / Ocean Engineering 115 (2016) 67–92

program in Delphi (Castelli, 2015), that also includes VPH, as part of a System under development. In Ribeiro (2002) the VPH method was applied to Series 62 (Clement and Blount, 1963) and Keuning and Gerritsma (1982) Series, in 50 simulations of the program. In this work only results of Series 62 were selected, in order to give space for two more modern Series, Begovic and Bertorello (2012) and Kowalyshyn and Metcalf (2006), to better demonstrate the method. Series 62, a classical series, was left not only to show part of the original work, but also because of the forward deadrise angles, that go up to the vicinity of 45°, although having constant deadrise angles in the aft sections of 12.5°. These variable angles in the forward sections of the models are taken into consideration in the calculations, since the VPH method deals with the wetted keel length, LK, from transom to the stagnation point. In fact, it is in the region enclosed by the leading edge, near the stagnation point, that the pressures are greater. Table 5.1 Geometrical parameters of the models presented from Series 62. Note: 1 ft2 ¼ 0.30482 m2 (SI); 1 ft ¼0.3048 m (SI); 1 dm ¼0.1 m. Geometry (as presented)

The models tested by Begovic and Bertorello (2012) were also simulated because of their very large deadrise variation and Kowalyshyn and Metcalf's (2006) for being a modern, successful real life concept. The non-monohedral model of Bertorello and Oliviero (2006) and the model tested by Savitsky (2012) could not be simulated due to lack of enough information to allow for comparisons of the methods. 5.1. Characteristics of the models Table 5.1 shows the main characteristics of the models of Series 62 used to validate the numerical results produced by the method. Fig. 5.1 presents brief lines drawings of each Series 62 model used. Table 5.2 shows the main characteristics of the models of Begovic and Bertorello (2012) used to validate the numerical results produced by the method.

Table 5.2 Characteristics of Begovic and Bertorello (2012)'s models (the parameters shown are those used by the method, taken from the publication).

Series 62 Geometric parameters

Ap, ft2(dm2) LP, ft (dm) BPA, ft (dm) BPX, ft (dm) BPT, ft (dm) LP /BPA LP /BPX BPX /BPA BPT /BPX CAP (%LP) ε [deg] xT αCR [deg] αE [deg] βMS [deg] βFWD [deg]

75

4665

4667-1

4669

6.469 3.912 1.654 1.956 1.565 2.365 2.00 1.18 0.80 47.5 19.41  0.6698 5.0 58 12.5 50

12.800 8.000 1.600 1.956 1.250 5.00 4.09 1.22 0.64 48.8 10.00  0.7150 5.0 46 12.5 50

7.479 8.000 0.935 1.143 0.734 8.56 7.00 1.22 0.64 48.8 5.75  0.7247 2.9 37 12.5 50

Displacement [t] LOA [m] BPX [m] BPT [m] D [m] LCG [m] KG [m] βMS [deg] βAft [deg] βFW [deg] Transom Rake [deg] XFW [m] XchFW [m] ε [deg] xT [m]

Models MONO

WARP1

WARP2

WARP3

0.033 1.900 0.424 0.424 0.2 0.697 0.143 16.7 16.7 16.7 0.001 1.500 1.900 0 0

0.033 1.900 0.424 0.424 0.2 0.660 0.152 20.3 14.31 23.75 0.001 1.500 1.900 0 0

0.033 1.900 0.424 0.424 0.2 0.609 0.155 23.3 11.59 30.11 0.001 1.500 1.900 0 0

0.032 1.900 0.424 0.424 0.2 0.586 0.156 26.0 9.09 35.75 0.001 1.500 1.900 0 0

Fig. 5.1. Lines of Series 62 models (Clement and Blount, 1963) used. See Table 5.1 for dimensions.

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Fig. 5.2. Sketch of Begovic and Bartorello's (2012) Models. Table 5.3 Characteristics of the USCG models (the parameters shown are those used by the method, taken from the publication). Geometric parameters

Displacement [lb] LOA [ft] BPX [ft] BPT [ft] D [ft] LCG [ft] KG [ft] βMS [deg] βAft [deg] βFW [deg] Transom Rake [deg] XFW [ft] XchFW [ft] ε [deg] xT [ft]

Models Parent 5628

Variant 1 5629

Variant 3 5631

375 10.900 3.258 3.258 2.0 3.8 1.5 22.8 16.61 35 0.001 9.000 10.500 10 0

375 10.900 2.643 2.643 2.0 3.8 1.5 22.8 16.61 35 0.001 9.000 10.500 10 0

375 10.900 2.364 2.364 2.0 3.8 1.5 25.4 20 35 0.001 9.000 10.500 10 0

The four models used are shown in Fig. 5.2, taken from the reference. Table 5.3 shows the main characteristics of the models of Kowalyshyn and Metcalf (2006) – USCG used to validate the numerical results produced by the method. Fig. 5.3 presents body plans of each model used, taken from the reference. 5.1.1. Models selected to simulate When simulating Series 62 models, Fig. 5.1, a selection of models 4665, 4667-1 and 4669 was found to be representative and to show attributes and limitations of the method. They show the simulations made by the mathematical model and they present results of keel immersion aft, dynamic trim angle after equilibrium, and resistance. It is to be noted that these two first results show the attitude of the hull at dynamic equilibrium. Consequently, it shows the capacity of this mathematical model to solve this problem. It is also to be noted that these two variables define

the region of the bottom exposed to the flow, which allows other characteristics to be calculated, such as lift, center of pressure, total resistance and wetted surface. All Begovic and Bertorello models were tested, Mono, Warp 1, Warp 2 and Warp 3 (Fig. 5.2) and for Kowalyshyn and Metcalf's (USCG) Series, models 5628, 5629 and 5631, Fig. 5.3, for one displacement (375 lb) and one LCG (3.8 ft), were selected for simulation. For all models tested, Savitsky's (1964) method was also simulated for comparison. For Series 62, Hadler's spray calculations were done and for Begovic and Bertorello's and for Kowalyshyn and Metcalf's Series, the Whisker Spray (Savitsky et al., 2007) calculation was used. All Savitsky's method simulations were done with the deadrise angle at the LCG: for Series 62, 12.5°; for Begovic and Bartorello, 16.7° for the Mono, 18.5° for Warp 1, 19.1° for Warp 2 and 19.5° for Warp 3 models; for the USCG, 19.3° for models 5628 and 5629 and 22° for model 5631. The Begovic and Bertorello and the USCG's Model were simulated in the computer program mentioned, using Savitsky (1964) and the VPH method (Ribeiro, 2002). The program is still in an initial version, Castelli (2015), as part of a System to be published, as can be seen in Fig. 5.4. 5.2. Simulation results Next, the simulations made are presented in graphs of Keel Immersion (KI), Dynamic Trim (DT) and Total Resistance (RT), comparing experimental results with the application of the VPH and Savitsky's Methods. In order to convert the rise of CG of the USCG Series tests and the sinkage of Bergovic and Bertorello, the following formulations were used: δKG ¼ ðKG  h2 þ LCG tan t 2 Þ cos t 2  ðKG  h1 þ LCG tan t 1 Þ cos t 1 h2 ¼ 

δKG þ ðKG  h1 þ LCG tan t 1 Þ cos t 1  ðKG þ LCG tan t 2 Þ cos t 2 cos t 2

ð5:1Þ ð5:2Þ

where h1 is the keel immersion aft at rest, h2 is the keel immersion aft after equilibrium, t1 is the trim at rest, t2 is the dynamic trim angle after equilibrium and δKG is the rise of CG after equilibrium.

R.D. Schachter et al. / Ocean Engineering 115 (2016) 67–92

77

Fig. 5.3. USCG's Models Selected to simulate (body plans).

Fig. 5.4. Interface of part of the program (Castelli, 2015), showing Resistance vs. Speed for the simulation of model W3 of Begovic and Bartorello, using the VPH Method.

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Table 5.4 Model 4665 – Tested Condition 1:

AP : ∇2=3

7.1/LCG located 6% aft of center of projected area.

VPH Error %: V(kt)

KI

DT

RT

8.14

1.2

13.2

10.0

8.98

4.0

21.5

12.9

9.72

7.4

27.3

12.3

10.56

3.0

30.9

10.7

11.42

5.0

34.0

8.8

12.22

1.7

33.2

7.1

13.11

1.8

35.0

4.8

14.74

0.0

33.2

0.6

16.38

0.0

33.0

0.9

18.02

2.3

28.0

1.2

19.64

4.7

22.0

2.9

Table 5.5 Model 4665 – Tested Condition 11:

AP : ∇2=3

7.1/LCG located 4% aft of center of projected area.

VPH Error %: V(kt) 6.58 7.34 8.14 8.94 9.78 10.62 11.42 12.22 13.00 14.70 16.42 18.04 19.66

KI 1.0 3.3 8.9 5.3 1.4 1.5 3.2 3.2 1.7 1.9 2.0 6.1 34.8

DT 14.3 1.1 12.0 20.9 24.7 31.7 34.8 32.4 35.2 36.4 30.4 26.7 18.5

RT 4.7 6.0 12.5 18.3 15.5 14.8 13.3 12.8 7.6 3.9 2.5 5.1 50.1

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Table 5.6 Model 4665 – Tested Condition 12:

AP : ∇2=3

7.1/LCG located 8% aft of center of projected area.

VPH Error %: V(kt) 6.52 7.36 8.10 8.96 9.78 10.62 11.42 12.22 13.10 14.74 16.38 18.04 19.64

KI 8.4 2.2 0.0 0.0 1.4 1.5 4.6 4.8 8.3 9.3 8.2 12.8 13.3

DT 2.4 10.0 19.1 26.3 30.8 33.1 35.4 36.8 36.5 37.8 35.4 30.5 21.9

RT 0.8 8.6 12.9 15.4 15.1 13.4 13.0 10.8 8.7 3.8 0.7 1.4 0.1

KI 1.3 4.3 3.1 1.6 0.0 1.9 3.7 7.8 4.3 59.1

DT 9.5 18.9 25.3 26.9 34.7 36.6 35.6 32.4 28.3 4.2

RT 17.8 21.0 23.3 18.4 9.9 7.7 2.8 0.9 3.4 77.7

Table 5.7 Model 4665 – Tested Condition 15:

AP : ∇2=3

8.5/LCG located 4% aft of center of projected area.

VPH Error %: V(kt) 7.86 8.62 9.42 10.20 11.00 11.76 12.58 14.19 15.78 17.40

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Table 5.8 Model 4667 – Tested Condition 1:

AP : ∇2=3

7.0/LCG located 6% aft of center of projected area.

VPH Error %: V(kt) 9.76 10.74 11.72 12.70 13.68 14.70 15.70 17.64 19.62 21.60 23.56

A table showing the error percentage of the VPH method in relation to the experimental results is also presented. The discussion of the results is presented soon after. 5.2.1. Series 62 results See Tables 5.4–5.16. 5.2.2. Begovic and Bartorello's results See Tables 5.17–5.20. 5.2.3. USCG's results See Tables 5.21–5.23.

6. Discussion The present model was developed for predicting behavior of planing hulls with variable form, like sectional beam, deadrise angle and central buttock line. It considers that the boat has already developed fully planing regime in order to use Savitsky's equations (1964), together with an interpolating function (3.7). Then, the quality of results produced by the virtual prismatic hulls method (VPH), also depends on the applicability of Savitsky's equations. When validating the method, the mathematical model has been tested in some of the models of Series 62, mainly the parent hull – model 4667-1 – and the forward sections reached deadrise angles of up to 45°. As the results produced showed small

KI 8.2 8.6 9.0 9.8 8.8 9.0 7.1 7.7 8.6 9.6 8.8

DT 8.2 1.3 1.3 4.1 5.6 5.3 9.1 11.8 10.6 8.2 11.4

RT 4.6 2.4 1.1 0.3 0.1 0.2 0.7 1.2 0.7 1.1 1.4

percentages of error (see Tables 5.4–5.16), it was concluded that the limits proposed by Savitsky (1964) for the deadrise angles could be extended to 45°. There are also good results obtained by the method when simulating runs of model 4669, which has a mean wetted length ratio of about 9 (Tables 5.13–5.16). So the applicable limits of the method can be said to be:

λ r 9;

2 3 r τ r 15 3 ;

β r 45 3 ;

0:5 rC V r 10:

On the other hand, a kind of function was chosen for the interpolating function (3.7), which has to be monotonically decreasing, what is naturally expected, as the pressure at the bottom comes from stagnation to atmospheric, from bow to stern. As this characteristic is not formally imposed, this is another source of some diverging results, like the ones at Tables 5.5, 5.7 and 5.12, all at top speeds. This is also motivating the continuation of the research, in order to impose a monotonically decreasing interpolating function. Nonetheless, the present interpolating function seems to play its role quite reasonably. The cubic root function has a minus infinite derivative at the stagnation point, what is desired in order to simulate a quick decreasing of the stagnation pressure in the vicinity of the stagnation point. Though shorter hulls are under the limits of applicability of Savitsky's equations, because of the surrounding ideas of strip theory, the model tends to produce less precise results when slenderness ratio is below 2.0, which is typical of the 4665 model results. But it is to be noted that even these results agree satisfactorily with experimental data.

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81

Table 5.9 Model 4667 – Tested Condition 9:

AP : ∇2=3

5.5/LCG located 12% aft of center of projected area.

VPH Errors: V(kt) 9.26 10.30 11.40 12.48 13.50 14.40 15.64 16.68

A last observation must be made in order to point another motivation to continue the research to improve this promising computational tool. It is important to impose formally that the interpolation of the value of the sectional lift coefficient be made correctly when the boat's section is inside the leading edge. It is inside the leading edge that the bigger dynamic pressures appear. So, it is very important that the sectional dynamic lift coefficient interpolated in the prismatic hull's bottom be inside its leading edge also. This is not also imposed formally. Although these errors cannot be observed, they may be present and also deserve to be fixed in the future. Once these previous observations were understood, it is possible to analyze the results from other points of view, as this discussion is divided into two sections. One for discussing the equilibrium position, and another to discuss the results produced for total resistance. 6.1. Equilibrium position As it was noted in Section 4.1 and in the system of Eq. (4.9), hydrodynamic results depend on the quality of the results produced for the equilibrium position. For example, for calculating frictional resistance one needs to know, unlike Savitsky's method, that uses b², the area of wetted surface of the hull, which in turn depends on the final equilibrium attitude of the hull. Firstly, direct examination of Tables 5.4–5.23 shows that these results produced by the model are qualitatively following a consistent tendency. It is also to be observed that when predicting the equilibrium position the model does it better for keel immersion aft than for the dynamic trim angle, in most cases. But it is easy to see that

KI 9.26 10.30 11.40 12.48 13.50 14.40 15.64 16.68

DT 3.4 0.9 1.1 4.0 6.8 5.8 8.2 9.1

RT 5.2 2.5 0.1 2.0 3.3 4.4 5.3 5.8

when slenderness of the bottom area increases, both predicted values improve. This is easy to explain, since the model is much more sensitive to a little error in the prediction of the position of center of pressure than to an error in the value of the lift force, depending on the magnitude of the latter. Also, when disconnecting the hydrostatic force from dynamic force, it was observed that the regression on longitudinal position of dynamic force (Eq. (3.19)) here used needs improvement. In order to improve the precision, in this work the effects of dynamic and hydrostatic forces have been separated. What can be seen is that Eq. (6.1) includes both the effects of dynamic and hydrostatic forces, and each contribution, built in the equation, cannot be separated. It is also to be noted that there is no need to estimate the hydrostatic force, as it can be determined exactly by direct integration, improving precision. On the other hand, the first part of Eq. (6.2) represents the contribution of the dynamic force, while the second part represents the contribution of hydrostatic force. What has been done here was to get the first part, and let the second to be determined by direct integration of pressure field on the bottom of the hull. CP ¼

LCP

λ BP

¼ 0:75 

1

ð6:1Þ

C 2V

5:21 2 þ2:39 λ

C L0 ¼ τ1:1 0:012λ

1=2

þ 0:0055

λ5=2 C 2V

! ð6:2Þ

So, an alternative formulation was proposed for the prediction of the dynamic contribution on the longitudinal position of the center of dynamic pressure, which resulted in a simple regression of numerical data, collected from the work presented by Fridsma

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Table 5.10 Model 4667 – Tested Condition 12:

AP : ∇2=3

7.0/LCG located 8% aft of center of projected area.

VPH Error %: V(kt) 9.80 10.78 11.78 12.82 13.74 14.74 15.74 17.70 19.62 21.62 23.58

KI 10.0 10.5 12.0 10.2 5.4 4.8 4.0 5.6 8.5 9.2 8.5

DT 7.6 4.1 2.5 1.7 3.3 5.5 8.4 7.5 7.0 9.1 11.5

RT 7.4 4.8 3.3 1.0 1.2 1.4 1.3 0.3 0.4 0.5 2.0

Table 5.11 Model 4667 – Tested Condition 13:

AP : ∇2=3

7.0/LCG located 12% aft of center of projected area.

VPH Errors: V(kt) 7.81 8.80 9.72 10.78 11.74 12.72 13.64 14.68 15.72 17.52 19.64 21.62 23.60

KI 14.0 13.2 12.5 11.1 8.7 6.7 4.5 5.7 3.0 5.6 8.7 9.6 13.2

DT 12.5 10.7 6.0 1.1 3.3 1.2 7.6 7.2 6.7 10.4 8.4 3.3 0.8

RT 12.2 7.0 3.2 0.1 1.2 3.1 4.4 3.5 4.4 2.6 1.1 2.1 3.7

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83

Table 5.12 Model 4667 – Tested Condition 15:

AP : ∇2=3

8.5/LCG located 4% aft of center of projected area.

VPH Errors: V(kt) 7.42 8.34 9.24 10.22 11.15 12.12 13.28 13.98 14.98 16.74 18.72 20.52 21.74 22.42

KI 1.8 0.9 0.9 1.7 3.5 1.9 2.0 2.1 3.3 12.0 2.6 1.4 5.3 20.2

DT 22.7 11.5 3.0 6.6 4.3 5.0 10.5 12.5 15.4 17.1 18.0 18.3 16.2 18.4

RT 19.4 13.3 6.3 0.9 5.6 1.6 4.0 1.7 3.5 4.6 2.6 1.1 4.5 19.4

KI 20.0 22.0 12.0 17.3 14.5 12.5

DT 2.1 1.8 3.6 6.1 9.4 12.6

RT 15.3 15.6 15.2 11.8 9.0 2.9

Table 5.13 Model 4669 – Tested Condition 1:

AP : ∇2=3

7.0/LCG located 6% aft of center of projected area.

VPH Error %: V(kt) 12.80 13.64 15.35 17.10 18.80 20.50

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Table 5.14 Model 4669 – Tested Condition 7:

AP : ∇2=3

5.5/LCG located 4% aft of center of projected area.

VPH Error %: V(kt) 12.70 13.60 14.55 16.38 18.20 20.00 21.80

KI 14.2 11.7 14.0 11.7 10.8 7.1 2.5

DT 4.8 1.4 0.5 4.5 8.8 6.5 7.9

RT 17.9 14.7 14.8 12.8 11.8 10.7 7.9

KI 19.9 17.8 18.2 14.8 11.7 9.7 3.6 10.4

DT 0.8 2.8 3.9 6.3 5.3 7.0 13.0 11.1

RT 15.8 13.2 13.3 13.0 12.7 11.2 8.1 5.1

Table 5.15 Model 4669 – Tested Condition 8:

AP : ∇2=3

5.5/LCG located 8% aft of center of projected area.

VPH Error %: V(kt) 11.78 12.70 13.68 14.55 16.40 18.10 20.00 21.84

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Table 5.16 Model 4669 – Tested Condition 11:

AP : ∇2=3

7.0/LCG located 4% aft of center of projected area.

VPH Error %: V(kt) 12.80 13.60 15.40 17.10 18.84 20.55

(1969). This regression was developed by Ribeiro (2002), as mentioned in Section 3.2: C PD ¼ 0:842119 þ 0:09981λ þ 0:02314τ  0:01299β  0:06183C V Lk ð3:19Þ As shown in Section 3.2, this formulation has substituted the dynamic contribution in the longitudinal position of the center of pressure, but since just about 70% of data has fitted reasonably to the regression, this formulation also needs improvement. 6.2. Total resistance After the equilibrium position is determined, the computational tool provides the calculation of the total resistance, considering that at fully planing regime, wave resistance can be neglected, and that there is no more separation at the bottom of the hull. Under these circumstances the total resistance can be computed as a summation of components due to friction, viscous pressure and spray, as shown in Section 4, above. Precise calculation of the frictional resistance is certainly linked to the precision obtained at the solution of equilibrium position, as this integration is taken at the wetted surface of the bottom. But the comparison of the results obtained by the mathematical model with experimental showed how the method created here was an effective choice. The use of a local Reynolds number (Eq. (4.10)) together with a local speed (Eq. (4.14)) showed to be reasonable contributions of the work. This explains why the vast majority of the results have very good precision, as it can be seen in the low percentages of error found in the result Tables of Section 5.

KI 21.1 17.6 16.9 15.8 12.9 9.2

DT 7.3 3,6 1.7 5.3 7.8 7.6

RT 20.5 17.2 13.7 11.5 10.0 7.8

7. Conclusions A method, named Virtual Prismatic Hulls, and its model, have been presented, with the feature to be applicable to the hydrodynamic design of planing boats with single hard chine, allowing for a totally variable geometry, from sectional beams to deadrise angles. From the results shown, it can be seen that the model has produced quite successful figures, as compared with model testing results. As mentioned, there are some improvements that need to be made, but the quality and consistency of the results indicate a promising direction on how to achieve this. It could also be seen that Savitsky's method still produces good results for new warped forms. When solving the dynamic equilibrium problem it is desirable to determine the attitude of the hull and the boat's resistance, mainly. It was shown that the results produced for keel immersion and resistance are very precise. The dynamic trim results are generally less precise but, as can be seen from the graphs and errors, the model produces also very good results for dynamic trim when L/B or λ increases, which is a positive characteristic of this method. As these results also agree very well with experimental data, it can be said that the model has indeed succeeded. Since one of the main concerns in the hydrodynamic design of planing boats is to get the resistance right, after finding its equilibrium position, this could become a good tool for designing and propelling these boats, since resistance precision showed to be a reasonable feature of this method. Despite the successful results, it was noted that the model needs improvement. One detected as necessary is to search for a monotonically decreasing interpolating function and also to

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Table 5.17 Monohedral hull.

V(kt) 3.32 4.45 5.44 6.61 7.78 8.96 9.89 11.18 12.29 13.41 14.5

Error % - VPH KI DT 5% 60% 9% 9% 6% 9% 1% 20% 4% 36% 1% 33% 5% 27% 10% 23% 9% 23% 10% 24% 11% 24%

RT 45% 2% 1% 5% 10% 9% 8% 6% 2% 1% 1%

Error % - Savitsky V(kt) KI DT 3.32 19% 3% 4.45 31% 36% 5.44 3% 3% 6.61 10% 16% 7.78 13% 23% 8.96 10% 19% 9.89 6% 14% 11.18 1% 11% 12.29 2% 11% 13.41 1% 11% 14.5 1% 10%

RT 33% 28% 5% 6% 9% 8% 9% 8% 6% 4% 3%

Table 5.18 Warp 1 hull.

Error % - VPH

Error % - Savitsky

V(kt)

KI

DT

RT

V(kt)

KI

DT

RT

3.32

1%

59%

28%

3.32

45%

41%

27%

4.45

11%

10%

27%

4.45

29%

28%

27%

5.44

6%

11%

2%

5.44

0%

9%

2%

6.61

2%

24%

8%

6.61

15%

33%

10%

7.78

6%

45%

9%

7.78

17%

41%

11%

8.96

2%

49%

8%

8.96

11%

37%

10%

9.89

1%

48%

9%

9.89

9%

33%

11%

11.18

2%

50%

6%

11.18

5%

33%

8%

12.29

2%

54%

1%

12.29

4%

35%

3%

13.41

0%

58%

1%

13.41

5%

37%

1%

14.50

2%

62%

4%

14.50

7%

39%

2%

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Table 5.19 Warp 2 hull.

V(kt) 3.32 4.45 5.44 6.61 7.78 8.96 9.89 11.18 12.29 13.41 14.50

Error % KI 0% 8% 3% 7% 12% 11% 10% 11% 14% 13% 16%

- VPH DT 60% 13% 18% 39% 63% 70% 80% 86% 104% 113% 135%

RT 28% 12% 9% 1% 8% 10% 9% 4% 0% 5% 10%

V(kt) 3.32 4.45 5.44 6.61 7.78 8.96 9.89 11.18 12.29 13.41 14.50

Error % - Savitsky KT DT 47% 40% 22% 14% 9% 33% 22% 58% 22% 59% 19% 54% 17% 59% 15% 66% 18% 75% 15% 81% 17% 97%

RT 38% 27% 1% 10% 9% 8% 7% 3% 1% 5% 10%

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Table 5.20 Warp 3 hull.

V (kt) 3.34 4.47 5.44 6.61 7.78 8.94 9.91 11.17 12.28 13.40 14.50

Error % KI 2% 7% 0% 11% 20% 25% 23% 29% 30% 38% 35%

- VPH DT 60% 13% 22% 46% 82% 107% 124% 148% 174% 209% 260%

RT 19% 21% 17% 7% 6% 10% 7% 1% 5% 9% 16%

V (kt) 3.32 4.45 5.44 6.61 7.78 8.96 9.89 11.18 12.29 13.41 14.50

Error % - Savitsky KT DT 39% 26% 17% 5% 18% 51% 28% 73% 29% 74% 32% 80% 29% 90% 33% 107% 31% 127% 37% 153% 33% 192%

RT 33% 29% 2% 7% 7% 5% 3% 2% 8% 11% 17%

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Table 5.21 Parent hull – 5628.

V (kt) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Error % - VPH KI DT 3% 49% 7% 17% 4% 20% 2% 17% 2% 19% 2% 23% 3% 25% 3% 28% 2% 32% 0% 35% 1% 36% 1% 35% 1% 33% 3% 31% 5% 28% 4% 27% -

RT 83% 21% 3% 1% 2% 5% 6% 6% 4% 3% 1% 0% 1% 2% 2% 2% 2% 8% 9% 9% 11% 10% 14%

V (kt) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Error % - Savitsky KI DT 8% 52% 3% 15% 7% 22% 8% 18% 6% 9% 5% 1% 4% 4% 5% 5% 7% 6% 10% 6% 11% 7% 12% 9% 12% 11% 10% 14% 8% 17% 10% 19% -

RT 61% 25% 8% 4% 3% 4% 4% 5% 7% 9% 12% 15% 19% 23% 28% 33% 39% 35% 39% 44% 80% 84% 86%

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Table 5.22 Variant 1 hull – 5629.

V (kt) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Error % - VPH KT DT 0% 93% 7% 31% 4% 30% 1% 27% 0% 26% 0% 32% 2% 38% 1% 38% 0% 38% 2% 37% 3% 37% 4% 38% 4% 40% 4% 44% 4% 49% 5% 54% 5% 60% 5% 65% 4% 69% 4% 69% 3% 65% 2% 57% -

RT 82% 21% 0% 6% 4% 0% 3% 2% 1% 1% 2% 3% 3% 3% 3% 2% 1% 10% 10% 11% 4% 4% 4%

V (kt) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Error % - Savitsky KT DT 6% 59% 13% 5% 6% 32% 10% 33% 9% 28% 7% 22% 5% 17% 6% 12% 8% 8% 10% 6% 12% 4% 13% 3% 13% 3% 14% 5% 14% 7% 15% 9% 16% 12% 16% 15% 16% 16% 16% 15% 15% 11% 15% 5% -

RT 25% 8% 5% 2% 1% 1% 1% 1% 0% 1% 2% 4% 7% 10% 14% 18% 23% 16% 19% 21% 45% 49% 53%

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Table 5.23 Variant 3 hull – 5631.

V (kt) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

improve the prediction of the position of the longitudinal position of dynamic center of pressure, since Eq. (3.19) is just a first estimation, using the simplest linear regression form.

Acknowledgment The authors would like to acknowledge Lucas Castelli for his great help on running the Begovic and Bertorello and the USCG's test results and the Methods and also for producing the graphs and error tables.

References Almeter, J.M., 1993. Resistance prediction of planing hulls: state of the art. SNAME Mar. Technol. 30 (4), 297–307.

Error % - VPH KI DT 6% 166% 3% 52% 0% 55% 5% 56% 9% 49% 12% 58% 10% 71% 5% 65% 2% 45% 2% 31% 4% 23% 5% 19% 6% 19% 6% 19% 6% 20% 5% 22% 5% 23% 6% 24% 7% 23% 6% 20% 10% 17% 13% 12% -

RT 133% 30% 1% 5% 8% 10% 9% 7% 5% 4% 3% 3% 3% 3% 3% 4% -

V (kt) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Error % - Savitsky KI DT 3% 105% 12% 14% 9% 53% 15% 58% 18% 48% 19% 42% 17% 43% 12% 32% 10% 13% 6% 0% 4% 8% 3% 11% 3% 13% 3% 14% 4% 14% 4% 14% 5% 14% 5% 14% 5% 16% 5% 18% 2% 21% 1% 25% -

RT 54% 11% 4% 10% 12% 10% 7% 5% 5% 5% 7% 10% 13% 17% 21% 25% -

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