HYDRODYNAMIC LIFT IN A TIME-DOMAIN PANEL METHOD FOR THESEAKEEPING OF FAST SHIPS

Pepijn de Jong, Delft University of Technology, The NetherlandsFrans van Walree, Maritime Research Institute, The Netherlands

SUMMARY

A method is presented for the seakeeping of high speed craft with transom stern flow. The method consists of a time domainboundary element method utilizing a free surface Green function. For the solution a combined source-doublet formulationis applied, while satisfying two boundary conditions explicitly. Firstly, a zero normal flow on the body condition andsecondly a condition at the transom stern based on the unsteady Bernoulli equation to model transom stern flow. Thesolution is done in two steps. First a source system is solved in absence of the transom condition and subsequently thedoublet strength is solved incorporating the previously solved source strengths and the transom condition. Although theformulations enable a non-linear treatment of the submerged hull form, partial linearization is employed for computationalefficiency.

The fundamentals are elaborated and subsequently the method is applied to wedge shapes with constant forward speedin calm water. The results are compared with the outcome of Savitsky’s empirical model for planing wedges and with anumber of alternative formulations with encouraging results. Although the method is capable of dealing with unsteadyseakeeping problems in the present paper it will only be applied to steady cases, as development is ongoing.

NOMENCLATURE

1+ k Form factorβ Deadrise angleη Free surface vertical locationλ Wetted length/beam ratioµ Doublet strengthω Wave frequencyΦ Velocity potentialΦd Disturbance velocity potentialΦw Wave velocity potentialψ Wave directionρ Density of waterσ Source strengthτ Past time or trim angleV Rigid body velocityξ,η,ζ Location source pointζa Wave amplitudeB BeamCD Cross-flow drag coefficientC f ITTC friction coefficientCL0 Lift coefficient flat plateCLβ

Lift coefficient deadrise planing surfaceCv Beam Froude numberFzv Cross-flow drag forceFn Froude numberG Green functiong Gravity constantG0 Rankine part of Green functionG f Free surface (memory) part of Green functionJ0 Bessel function of order zerok Wave numberLc Wetted length chine

Lk Wetted length keeln Normal to surfacep,q Field en source pointpa Atmospheric pressureRv Viscous resistanceRn Reynolds numberS Wetted surfacet TimeUvel Constant forward speedVN Projection normal velocity on free surfaceVn Normal velocityx0,y0,z0 Earth fixed coordinateszT Vertical location transom

1 INTRODUCTION

The continuous demand for high speed operation whilefulfilling existing and extended operational and mission re-quirements has become a constant challenge for the navalarchitect. There is a perpetual competition in the industryto develop innovative methods of reducing resistance andexpanding maximum speeds in a seaway.

Evaluation of advanced and/or high speed concepts re-quires advanced numerical tools that can deal with the hy-drodynamic issues involved on a first principles basis. In-vestigations should not be limited to issues like motion in-duced accelerations in the vertical plane, but need to ad-dress course keeping and dynamic stability as well.

The research presented in this paper is aimed at devel-oping a practical numerical model for the evaluation ofthe seakeeping behavior of high speed vessels in terms ofmotions, acceleration levels, loads and dynamic behavior.

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The formulation of the numerical model is based on thework of Lin and Yue [6] and further developed by VanWalree [10, 11] and Pinkster [7]. The formulation orig-inally adopted by Van Walree employs unsteady impul-sive sources on the hull with combined source-doublet-elements to represent submerged lifting control surfaces.The free surface boundary conditions are linearized to theundisturbed free surface, while it is possible to retain thebody boundary condition on the actual submerged geome-try. Practically, it is necessary to linearize the body bound-ary condition as well, to reduce the computational burdenof the method, enabling the seakeeping analysis to run ona normal desktop computer.

The numerical model is capable of dealing with signifi-cant forward speeds and arbitrary three-dimensional (largeamplitude) motions due to the transient Green function, asshown by for example King et al. [5]. The free surfacelinearization in the numerical model is a disadvantage, es-pecially for high speed cases, where significant nonlinearfree surface effects can occur. The recent implementationof pressure stretching based on the calculation of the freesurface deformation as presented by De Jong et al. [2]provide a means to partly overcome this disadvantage.

In the current paper this method is further extended for ap-plication to high speed vessels. For these vessels, mostlyfitted with a transom stern, the flow is characterized byhigh pressure values in the stagnation regions along thewaterline in the fore part and smooth separation from thestern at moderate and high speeds. The flow around thebody develops significant hydrodynamic lift, while thetransom typically is left dry.

In the existing code the high pressure regions near thebow are well predicited, however the flow leaving at thestern is not modeled very well. The flow leaving the sterncan be modeled in two ways in the existing code:

• By applying a dummy segment elongating the shipat the stern. This ensures that the streamlines re-main attached at the stern location instead of devel-oping very large velocities around the transom edge,although at the same time the total pressure at thetransom edge does not equal atmospheric pressure,violating the Bernoulli equation.

• By empirically post-process the pressure distribu-tion near the transom with a function that decreasesthe total pressure over a certain length to the atmo-spheric pressure at the transom edge, as proposed byGarme [1]. Although the pressure distribution nowis more in agreement with experimental experiencethis does not have any influence on the solution it-self.

Both approaches largely ignore dynamic effects that areimportant when considering for instance the forward speedmotion damping in waves. The latter is especially impor-tant for the damping of pitching motions at high forwardspeed. Emperical evidence suggests that flow leaving thetransom plays an important role in this.

Another solution is the applicition of a combinedsource-doublet distribution on the hull coupled with a trail-ing edge condition and wake sheet equivalent to the oneused for foils. This condition can be formulated in suchway that both the flow separates tangentially at the transomand that the dynamic pressure and the hydrostatic pressureat the transom edge are equal to the atmospheric pressure.Reed et al. [8] proposed such condition making use ofthe steady linearized Bernoulli equation applied just foreand aft of the transom stern. By employing this condi-tion the flow at the transom will smoothly separate at thestern while satisfying the atmospheric pressure expectedwith a dry transom, while at the same time the doublet ele-ments introduce the possibility of circulation lift, possiblyenhancing the prediction of trim and rise. To allow for dy-namic effects the unsteady Bernoulli equation is used inpresent model.

Besides the implementation of a transom condition, the so-lution process is modified as well. As pointed out by Reedet al. [8] presetting of the source strength and subsequentsolution for the doublet strength often yields instable re-sults. For this reason a solution in two steps has beenimplemented, where first the source strength is solvedwithout the transom condition and secondly the doubletstrength is solved using the known source strengths andthe transom condition.

The new method is applied to wedge shapes traveling ina fixed reference position with constant forward speed incalm water and the resulting vertical force is comparedwith the results of the semi-empirical model by Savit-sky [9]. Different versions of the code are compared.Although development is still ongoing, the comparisonshows that the new method with a two step solution pro-cess and with a trailing edge condition based on the workof Reed et al. [8] shows the best agreement.

The second section will describe the numerical back-ground of the model and will detail the transom condi-tion and solution process. The next section will presentthe comparison of the different versions of the code withthe Savitsky empirical model. The final section will sum-marize the conclusions and recommendations that followfrom the research presented in this paper.

2 NUMERICAL BACKGROUND

The numerical method presented in this paper is an exten-sion of the work presented by Lin and Yue [6], Pinkster [7]and Van Walree [11]. The code containing the numericalmethod is termed PANSHIP.

2.1 TIME DOMAIN GREEN FUNCTION METHOD

Potential flow is assumed based on the following simplifi-cations of the fluid:

• The fluid is homogeneous

• The fluid is incompressible

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• The fluid is without surface tension

• The fluid is inviscid and irrotational

The medium of interest is water, while there is an interfacewith air. The ambient pressure is assumed to equal zero.The water depth is infinite and waves from arbitrary direc-tions are present. Under all these assumptions it can beshown that the Laplace equation, resulting from conserva-tion of mass, is valid in the interior of the fluid:

The following definitions are used to describe the do-main:

• V (t) is the fluid volume, bounded by:

• SF (t) the free surface of the fluid,

• SH (t) the submerged part of the hull of the ship,

• SW (t) wake sheets and

• S∞ (t) the surface bounding the fluid infinitely farfrom the body.

Assuming linearity, the total potential can be split into twoparts, the wave potential and the disturbance potential

Φ = Φw +Φ

d (1)

The wave potential is given by:

Φw =

ζagω

ekz0 sin(k (x0 cosψ+ y0 sinψ)−ωt) (2)

The subscript 0 refers to earth fixed coordinates. At thefree surface two conditions are imposed. First, a kinematiccondition assuring that the velocity of a particle at the freesurface is equal to the velocity of the free surface itself.

∂η

∂t+∇Φ ·∇η− ∂z0

∂t= 0 ∀ x0 ∈ SF (3)

Second, a dynamic condition assuring that the pressure atthe free surface is equal to the ambient pressure. For thiscondition use is made of the unsteady Bernoulli equationin a translating coordinate system.

∂Φ

∂t+gη+

12

(∇Φ)2 = 0 ∀ x0 ∈ SF (4)

Both can be combined and linearized around the still waterfree surface, yielding:

∂2Φ

∂t2 +g∂z0

∂t= 0 at z0 = 0 (5)

On the instantaneous body surface a zero normal flow con-dition is imposed be setting the instantaneous normal ve-locity of the body equal to:

Vn =∂Φd

∂n+

∂Φw

∂n∀ x0 ∈ SH (6)

At a large distance from the body (at S∞) the influence ofthe disturbance is required to vanish.

Φd → 0

∂Φd

∂t→ 0 (7)

At the start of the process, apart from the incoming waves,the fluid is at rest, as is reflected in the initial condition.

Φd∣∣∣t=0

=∂Φd

∂t

∣∣∣∣t=0

= 0 (8)

In this time-domain potential code the Green functiongiven in will be used. This Green function specifies theinfluence of a singularity with impulsive strength (sub-merged source or doublet) located at singularity pointq(ξ,η,ζ) on the potential at field point p(x0,y0,z0).

G(p, t,q,τ) = G0 +G f =1R− 1

R0+

2Z

∞

0

[1− cos

(√gk (t− τ)

)]ek(z0+ζ)J0 (kr)dk

for p 6= q , t ≥ τ (9)

It has been shown, by for example Pinkster [7], that theGreen function satisfies both the Laplace equation and theboundary conditions, making it a valid solution for theboundary value problem stated above. Using the above, itis possible to derive a boundary integral formulation. Thefirst step is to apply Greens second identity to:

Φd(

ξ, t)

and∂G∂τ

(x0,ξ, t,τ

)(10)

Subsequently the resulting volume integral is equal to zeroby using the Laplace equation. Integrating in time yieldsfor the surface integral:

Z t

0

ZSFHW (τ)

(Φ

dGτn−GτΦdn

)dSdτ = 0 (11)

Next, the free surface integral is eliminated by virtue ofthe Green function. Finally, a general formulation of thenonlinear integral equation is obtained for any field point:

4πT Φd (p, t) =−

ZSHW (t)

(Φ

dG0n−G0

Φdn

)dS+Z t

0

ZSHW (τ)

(Φ

dGτn−GτΦdn

)dSdτ+

1g

Z t

0

ZLw(τ)

(Φ

dGττ−GτΦdτ

)VNdLdτ (12)

VN is the projection of the normal velocity at the curve inthe plane of the free surface, for example G0

n = ∂G0

∂n , and Tis defined as:

T (p) =

1 p ∈V (t)1/2 p ∈ SH (t)0 otherwise

(13)

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Now the choice of surface singularity elements can bemade. The current version of the code is able to deal withsource-only distributions and combined source-doubletdistributions or any mix of the both. For the purposesof this paper a combined source-doublet distribution dis-tributed on the body surface will be elaborated. The sourcestrength is set equal to the jump in the normal derivativeof the potential between the inner (-) and outer (+) sides ofthe surface, while the doublet strength is set equal to thejump of the potential across the inner and outer surfaces.This results in:

Φd+−Φd− =−µ∂Φd+

∂n− ∂Φd−

∂n= σ

∀ q ∈ SH (14)

For the infinite thin wake sheets there is no jump in thenormal derivative of the potential:

Φd+−Φd− =−µ∂Φd+

∂n=

∂Φd−

∂n

∀ q ∈ SW (15)

Substituting equations 13, 14 and 15 in eq. 12, taking thenormal derivative for a field point lying on the outer faceof the hull and applying the body boundary condition eq.6 results in an expression for the normal velocity at fieldpoint p(x0, t) in terms of integrals over time and sourcepoints q(x0, t)

4π

(Vnp −

∂Φw

∂np

)= 2πσ(p, t)+Z

SH (t)σ(q, t)

∂G0

∂npdS +

ZSHW (t)

µ(q, t)∂2G0

∂np∂nqdS−Z t

0

ZSH (τ)

σ(q,τ)∂2G f

∂np∂τdSdτ−Z t

0

ZSHW (τ)

µ(q,τ)∂3G f

∂np∂nq∂τdSdτ−

1g

Z t

0

ZLw(τ)

σ(q,τ)∂2G f

∂np∂τVNVndLdτ−

1g

Z t

0

ZLw(τ)

µ(q,τ)∂3G f

∂np∂τ2 VNdLdτ (16)

Equation 16 is the principal equation to be solved to obtainthe unknown singularity strengths. Two steps have yet tobe taken:

1. The definition of a Kutta or trailing edge conditionto formulate the problem as such that an unique so-lution can be obtained.

2. To chose an appriopiate solution scheme to ob-tain an equal amount of equations and unknowns,as for now there are roughly double the numberof unknowns (one source strength and one doubletstrength per panel) per equation (one normal veloc-ity condition per panel);

The first step will be elaborated in section 2.3 and the latterin section 2.4.

2.2 LINEARIZATION

Especially the evaluation of the free surface memory termof the Greens function requires a large amount of compu-tational time. These terms need to be evaluated for eachcontrol point for the entire time history at each time step.To decrease this computational burden, the evaluation ofthe memory term has been simplified. For near time his-tory use is made of interpolation of predetermined tabularvalues for the memory term derivatives, while for largervalues further away in history polynomials and asymp-totic expansion are used to approximate the Green func-tion derivatives.

Moreover, the position of the hull relative to the pasttime panels is not constant due to the unsteady motions,making recalculation of the influence of past time panelsnecessary for the entire time history. This recalculation re-sults in a computational burden requiring the use of a su-percomputer. To avoid this burden, the unsteady positionof hull is linearized to the average position (moving withthe constant forward speed). Now the memory integral canbe calculated a priori for use at each time step during thesimulation.

The prescription of the wake sheets in this linear ap-proach leads to a flat wake sheet behind the hull. Again aconstant distance exist to the past time wake panels. Onlythe influence coefficients of the first row of wake elementsneed to be calculated at each time step, until the maxi-mum wake sheet length is reached. For all other rows theinduced velocity can be obtained by multiplying the influ-ence by their actual circulation.

2.3 WAKE MODEL

The wake model is necessary for an unique solution of thepotential problem set up in terms of a mixed source soubletformulation. The wake model relates the dipole strengthat the trailing edge of lifting surfaces to the location andshape of a wake sheet, by using the unsteady linearizedBernoulli equation in the body fixed axis system, as pro-posed by Reed et al. [8] for steady cases.

gzT = Uvel

(∂Φd

∂x+

∂Φw

∂x

)−

(∂Φd

∂t+

∂Φw

∂t

)(17)

This condition will be appoximately satisfied at the tran-som edge. In fact, it will not be satisfied exactly at the tran-som edge due to numerical problems arising when evalu-ating influence functions on panel edges. Instead, the con-dition will be satisfied at the collocation points of the lasthull panel row in front of the transom edge.

The wave influence can be calculated by taking the ap-propiate derivatives of eq. 2. The tangential induced ve-locities of all singularities at source points q

(ξ, t

)at the

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transom edge panels w(x0, t) are given by:

4π∂Φd

∂x(w, t) =−2π

∂µ(w, t)∂x

+ZSH (t)

σ(q, t)∂G0

∂xdS +

ZSHW (t)

µ(q, t)∂2G0

∂x∂nqdS−Z t

0

ZSH (τ)

σ(q,τ)∂2G f

∂x∂τdSdτ−Z t

0

ZSHW (τ)

µ(q,τ)∂3G f

∂x∂nq∂τdSdτ−

1g

Z t

0

ZLw(τ)

σ(q,τ)∂2G f

∂x∂τVNVndLdτ−

1g

Z t

0

ZLw(τ)

µ(q,τ)∂3G f

∂x∂τ2 VNdLdτ (18)

Because constant strength singularities are used, it is notpossible to directly obtain the x-derivative of µ. The solu-tion is to estimate this derivative at the transom edge panelby using the value of µ at the panel just in front of thispanel and at the panel just behind the transom edge panel,the first wake sheet panel and dividing over the length.i + 1 refers to the panel directly upstream and i− 1 refersto first wake panel downstream of the transom panel asindicated in figure 1.

∂µ(w, t)∂x

≈ µi+1−µi−1

2Lpan(19)

The disturbance part of the second term of eq. 17 can beevaluated as follows:

4π∂Φd

∂t(w, t) =−2π

∂µ∂t

(p, t)+ZSH (t)

∂σ

∂t(q, t)G0dS +

ZSHLW (t)

∂µ∂t

(q, t)∂G0

∂nqdS−Z t

0

ZSH (τ)

σ(q,τ)∂2G f

∂t∂τdSdτ−Z t

0

ZSHW (τ)

µ(q,τ)∂3G f

∂t∂τ∂nqdSdτ−

1g

Z t

0

ZLw(τ)

σ(q,τ)∂2G f

∂t∂τVNVndLdτ−

1g

Z t

0

ZLw(τ)

µ(q,τ)∂3G f

∂t∂τ2 VNdLdτ+ZLw(t)

µ(q, t)∂G0

∂nqVNdL (20)

The final term appears due to time derivation of the dou-blet waterline integral and the fact that the time integrationborder of this integral is dependent on time. This term issimplified using the free surface boundary condition andthe definition of the Green’s function.

The implementation of eq. 20 is slightly more com-plicated, as the doublet G0-terms and the wake terms areestimated by a simple first order backward scheme. The

other terms are calculated analytically as the approxima-tion method is unsuitable for these terms.

The wake sheet position and shape is prescribed to reducethe computational effort. This prescription is that a wakeelement remains stationary once shed. This eliminates theeffort needed to calculate the exact position of each wakeelement at each time step. This violates the requirement ofa force free wake sheet. However, for practical purposesthis does not have significant influence as shown by VanWalree [11] and Katz and Plotkin [3].

Per time step only the first wake row, consisting ofthe elements attached to the transom edge, is treated asunknown. Once shed these wake elements keep theirstrength. The number of extra equations by the above con-dition is equal to the number of wake panels in the firstwake row.

Body

Wake sheet

i

2 L

pan

i−1

i+1

Figure 1: Panel identification for local dµ/dx

2.4 SOLUTION

Equation 16 and equations 17-20 are discretized in termsof a combined source-doublet element distribution on thehull and an equivalent vortex ring elements on the wakesurface. In the current method constant strength quadrilat-eral source and doublet panels are used. This results in asystem that is over-determined as both a source strengthand a doublet strength are defined for each hull panel. Ontop of this there are unknown doublet strengths in the firstwake row. To resolve this a number of possibilities exist.Two of these are:

1. To set the source strength equal to the undisturbednormal velocity at each body panel. In this, thememory integrals of the past time influences of thesources and doublets could be included.

2. To solve the system in two steps. Step one is to solvefor the source strength without wake influences andwithout the G0-influences of the doublet panels. Thesecond step consists of a solution for the doubletstrengths and first wake row strengths including thewake influences and with the G0 influences of thesource strengths determined in the first step in theright hand side.

The second method is chosen as it gives the best results, asshown in the next section.

Figure 2 illustrates the system that is solved for the com-bined sourcedoublet system, with known source strengths.

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The latter can be obtained by any of the two methods. Firstthe different parts of the influence matrix A:

A1 The normal G0-influence terms of the doublet singu-larities of the body on the other panels and them-selves.

A2 The normal G0-influence terms of the first wake rowsingularities on the body panels.

A3 The tangential G0-influences of the body singularitieson the u-velocity on the pressure condition (appliedon the last hull panel row at the transom edge) aswell as the term used to construct the local doubletx-derivative in equation 19. Additionally the esti-mated terms for the time derivative at the transomcondition that are dependent on the current doubletstrength.

A4 The tangential G0-influences of the first wake row sin-gularities on the other first wake row and themselvesas well as their contribution to the local doublet x-derivative in equation 19 and their contribution tothe time derivative in the transom pressure condi-tion.

A3

A1 A2

A4 x2

x1 b1

b2

=

Figure 2: Setup of solution of combined source-doubletsystem

The solution vector b contains in the b1-part the unknowndoublet strength on the body and in part b2 the unknowndoublet strengths of the first wake row. The RHS vec-tor part x1 houses the normal velocity contributions of allmemory integrals and known G0-integrals on each bodypanel along with the local wave and rigid body normal ve-locities. The x2-part of the RHS vector x holds all mem-ory and known G0-term contributions to the u-velocity anddΦ/dt at the transom panels along with the wave velocityin x-direction.

At the start of the simulation the body is impulsively setinto motion. At each subsequent time step the body is ad-vanced to a new position with an instantaneous velocity.Both position and velocity are known from the solution ofthe equation of motion. The singularity strengths are ob-tained by solving the systems following from either of theboth methods.

2.5 FORCE EVALUATION

Forces can be obtained from integration of the pressure ateach collocation point over the body. The pressures canbe obtained by using the unsteady Bernoulli equation (in abody fixed axis system):

pa− pρ

=12

{(∂Φ

∂x

)2

+(

∂Φ

∂y

)2

+(

∂Φ

∂z

)2}

+

∂Φ

∂t−V ·∇Φ (21)

In eq. 21 V is the total velocity vector at the collocationpoint of the rigid body, including rotations.

The spatial derivatives of the potential in eq. 21 followstraight from the solution. The only difficulty remainingis to obtain the time derivative. For the contribution of thewake and the Rankine part of the doublet panels this canbe done by utilizing a straightforward backward differencescheme. However, this gives unstable results when usedfor the contribution of the source panels and the memorypart of the doublet panels to the time derivative. This canbe resolved by calculating the time derivative of these con-tributions analytically from the Green function derivatives.

This means that additional Green function derivativesneed to be obtained, besides the derivatives needed forthe solution itself. Furthermore, the time derivative of thesource strength is needed. One solution is to derive thisderivative directly from the solution itself:

σ = A−1σ x

ddt

σ = A−1σ

ddt

x(22)

In this equation Aσ is the matrix relating the sourcestrengths via the Rankine influences to the RHS. The vec-tor x is the RHS vector of the solution, containing all in-fluences due to incident wave, free surface memory effectsand rigid body motions in terms of normal velocity in thecollocation points. To obtain the time derivative of thefree surface memory part of this vector, again extra Greenfunction derivatives need to be obtained. The time deriva-tive of the wave contributions can be obtained analytically.The time derivative of the rigid body velocity is the rigidbody acceleration. This acceleration is multiplied by theinverse of the Rankine influence matrix that equals theadded mass. This contribution can be transferred to themass times acceleration part of the equation of motion.

2.6 VISCOUS RESISTANCE

With respect to the viscous resistance Rv, empirical formu-lations are applied to each part separately (hull, outriggers,lifting surfaces). The formulations used can be generalisedas follows:

Rv =12

ρU2velS (1+ k)C f

C f =0.075

(log10 (Rn)−2)2

(23)

100

where U is the ship speed, S is the wetted surface area, kis a suitable form factor, and Rn is the Reynolds numberof the body part considered.

2.7 VISCOUS DAMPING

Especially for high speed vessels, having only slight po-tential damping, viscous damping can play an importantrole. This is especially true around the peak of verticalmotions. Then forces that arise due to separation in thebilge region due to vertical motions can be of significance.The magnitude of these forces depends on oscillation fre-quency, Froude number and section shape. In the currentmodel a cross flow analogy is used to account for theseforces. The viscous damping coefficient only depends onsection shape, other influences are neglected. The follow-ing formulation is used in a strip wise manner:

Fzv =12

ρVr |Vr|SCD (24)

Vr is the vertical velocity of the section relative to the lo-cal flow velocity, while S is the horizontal projection of thesection area. The cross-flow drag coefficient CD has valuesin-between 0.25 and 0.80.

3 RESULTS

Figure 3: Typical geometry seen from below, includingwake sheet

In this section results of a number of calculations areshown with wedges traveling with constant foward speedthrough calm water fixed in a reference position. Thelength over beam ratio is 4.3, the deadrise is 15 degreesand trim either 3 or 6 dgrees. Figure 3 shows a typical

wedge paneling, including wake sheet. The results aremeant as a preliminary investigation of the applicabilityof the method and the improvement of adjusted methodover previous versions. Currently the results are limited tosteady cases, in a later stage the method will be applied tounstaedy cases.

3.1 GRID STUDY

To investigate the influence of the number of elements onthe predicted vertical force calculations have been per-formed with a wedge shape with 15 degrees deadrise, 6degrees trim for three different Froude numbers for a gridwith respectively 248, 444 and 828 elements. Figure 4shows the ratio of the total vertical force with the displace-ment for these calculations.

Figure 4: Grid study

It shows that the results are quite independent of the num-ber of elements, although for the highest Froude numberthe calculation with 828 elements shows a slight deviationfrom the calculations with less elements. Especially thehigh pressure regions along the waterline in the fore partcould be responsible for this deviation. The pressure var-ries rapidly over this region while the currently used dis-cretization is possibly not fine enough at that location toresolve that gradient properly. Although the influence onthe total force is only slight, the sufficient resolution of thehigh pressure gradients in the fore part requires ongoingattention.

3.2 COMPARISON OF METHODS

Figures 5, 6 and 7 show three-dimensional representationsof the total pressure calculated with 3 different version ofthe code, respectively:

1. A source-only formulation with an empirical tran-som pressure modification based on the work ofGarme [1].

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2. A combined source-doublet formulation with aKutta condition based on that for two-dimensionalfoils. The source strength is fixed by the incom-ing flow plus memory effects, the doublet strengthis solved for.

3. A combined source-doublet formulation with thetransom condition based on the Bernoulli equationpresented in this paper. Both the source strength andthe doublet strength are solved for in two steps aspresented in this paper.

Figure 5: Total pressure plotted on the xy-grid for a source-only formulation with an empirical transom pressure cor-rection

The empirical transom pressure correction accordingGarme [1]:

aBCv

= 0.35

fred = tanh(

2.5a

x1

) (25)

Where the correction length a is determined in the firstequation (Garme uses a factor of 0.34 for his model), withB the width of the transom and Cv the beam Froude num-ber. The second equation determines the pressure reduc-tion factor fred , with x1 the distance in front of the tran-som. The reduction factor becomes unity at a distance a infront of the transom and is zero at the transom. The result-ing total pressure for a source-only formulation using thiscorrection is shown in figure 5.

The Kutta condition for a finite angle trailing edge fora two-dimensional foil as presented by Katz and Plotkin[3] is that the wake doublet strength becomes equal to thedifference in doublet strength of the upper and lower foilsides at the trailing edge. For a ship hull with transom sternthe upper foil side is absent and one could do by transfer-ing the doublet strength of the last hull panels before thetransom edge to the first wake row, ensuring velocity con-tinuity. The resulting total pressure is shown in figure 6.

Figure 6: Total pressure plotted on the xy-grid for a source-doublet formulation with a Kutta condition derived fromfoils

Figure 7: Total pressure plotted on the xy-grid for a source-doublet formulation with Bernoulli transom condition

Figure 7 shows the resulting total pressure for the methodpresented in this paper with the Bernoulli transom condi-tion and two step solution. When comparing the three fig-ures it shows that the pressure distribution in the fore partis not affected very much by the choice of element dis-tribution or transom condition/correction. For the source-only formulation the pressure in the fore part is marginallylarger. The differences show mostly at the transom.

The empirical formulation does result in zero pressure atthe transom (figure 5), but its region of influence is con-fined to a region close to the transom. The Kutta conditionbased on two-dimensional foils with a combined source-doublet system (figure 6) does reduce the pressure near

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the transom somewhat. However, the pressure is not re-duced to zero at the transom edge, voilating the Bernoulliequation there. The formulation with the Bernoulli tran-som condition (figure 7) does reduce the pressure to zeroat the transom edge. Not exactly though, most probablydue to the fact that the transom condition is satisfied at thecollocation point of the last panel row before the transominstead of at the transom itself, as pointed out in the previ-ous section. The influence of the transom flow condition ison larger region around the transom, when compared withthe other two methods and the predicted vertical force willbe less.

All three methods calculate somewhat unbelievablepressures on the submerged part of the body above thechines, especialy at the point where the chines cross thewater surface. In real life these parts of the body would beat least partly dry, something that is ignored by the freesurface linearization. Remarkable are the near transompressures in this part in the last figure. Due to the Bernoullicondition one expects the total pressure to approach zerohere. Closer inspection is necessary here.

Figures 8 and 9 show a comparison of the ratio of the to-tal vertical force (lift) with the displacement for differentcode versions with the outcome with Savitsky’s empiri-cal model for the vertical lift, Savitsky [9]. In Savitsky’smodel first the lift for a flat plate at trim angle τ is predictedby:

CL0 = τ1.1

[0.0120λ

1/2 +0.0055λ5/2

C2v

](26)

With λ:

λ =Lk +Lc

2BLk−Lc =

Bπ

tanβ

tanτ(27)

β is deadrise angle, τ the trim angle, Lk the wetted lengthof the keel and Lc the wetted length of the chine taking intoaccount the actual wetted width (due to wave rise). The liftof a deadrise planing surface is then calculated by:

CLβ= CL0 −0.0065βC0

L0.60 (28)

The code versions that are compared in figures 8 and 9 are:

• Source-only formulation with near transom empiri-cal pressure correction (source pc)

• Source-doublet formulation with Bernoulli transomcondition and two step solution (dbl transom)

• Source-doublet formulation with Kutta conditionand fixed source strength (dbl kutta/no pc)

• Source-doublet formulation with Kutta conditionand fixed source strength and near transom empir-ical pressure correction (dbl kutta pc)

• Source-doublet formulation with Bernoulli transomcondition and fixed source strength (dbl transom srcfixed)

Figure 8 shows results for a wedge with 15 degrees dead-rise and 3 degrees trim and figure 8 shows results for awedge with 15 degrees deadrise and 6 degrees trim.

Figure 8: Comparison of the lift/displacement ratio for 3degrees trim

Figure 9: Comparison of the lift/displacement ratio for 6degrees trim

The method outlined in the paper with the Bernoulli tran-som condition and two step solution (first source systemand subsequently the doublet system) clearly produces re-sults closest to Savitsky’s empirical model. Using thesame method, but setting the source strength a priori equalto the incoming flow and just solving for the doublet sys-tem performs only slightly worse.

The combined source-doublet system with the Kuttacondition derived from foils with a finite trailing edge an-gle clearly produces too much lift relative to the other

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methods. Introducing the empirical near transom pressurecorrection obviously reduces the total vertical force anddrastically improves this method. However, a source-onlyformulation with the same empirical pressure correctionperforms much better. Of course it is possible to enhancethis method further by fitting the empirical pressure cor-rection better for the case under consideration, this, how-ever, is only possible for cases where one has this oppor-tunity.

The comparison for the two trim angles is very simi-lar, the computational methods performing slightly betterat 6 degrees trim. However, the emperical model predicts alower lift in all cases. There could be a number of reasonsfor this. Among these:

• Panship is not aimed for the high speeds Savitsk’smodel is aimed for. The predictions here are in thelower speed regime for the empirical model with Cvranging from 1.9 tot 2.7, while the speeds are quitelarge for the numerical model with the Froude nu-mer over the length ranging from 0.9 tot 1.3.

• Both methods include buoyancy in the verticalforce. Panship includes the hydrostatics based onthe calm water wetted geometry. The the empiricalmodel includes in the lift the buoyancy of a flat plateat least partly, but when corrected for deadrise thebuoyancy is also implicitly corrected with measureddata. Two-dimensional empirical models based onwedges impacting the water surface often include abuoyancy correction factor reducing the hydrostaticforce at high forward speeds. This is related to thefact that part of the geometry is dry when saling athigh speeds, refer to for instance Keuning [4].

• Panship is a potential method without viscosity. Ab-sence of viscosity generally leads to overpredictionof the lift. The model inlcudes empirical formula-tions for viscous effects (refer to sections 2.6 and2.7). The influence of the tweaking of the coeffi-cients in these formulations on the lift needs to bestudied.

• Panship does not include wave rise and dry chines,free surface effects that are ignored by the lineariza-tion of the free surface.

Also a larger number of panels and a better resolution ofthe large pressure gradients in the bow area could reducethe predicted lift somewhat as indicated by the grid study.

4 CONCLUSION AND FUTURE WORK

A transom flow condition has been incorporated into atime-domain potential flow panel method for the seakeep-ing of high speed ships using a combined source-doubletformulation on the hull with a wake sheet extending fromthe transom. The method makes use of the unsteady lin-earized Bernoulli equation to ensure that the pressure atthe transom becomes zero. The potential method makes

use of a transient Green function with a linearized freesurface condition. Although it is possible to solve on theactual submerged body surface below the calm waterline,also the body boundary condition is linearized to reducethe computational effort. The source and doublet strengthsare obtained by solving per time step two systems:

• A source system without the presence of the wakesheet and the influence of the current time step dou-blet elements

• A doublet system extended with the transom con-dition and wake sheet. The source strength deter-mined in the previous step are treated as knowns inthis system.

As a preliminary validation study the method has been ap-plied to the lift generated on wedges moving with con-stant forward speed through calm water. The predicted to-tal vertical force has been compared with the outcome ofan empirical model by Savitsky for planing deadrise sur-faces and to the outcome of alternative formulations us-ing only source elements or combined source-doublet el-ements with an alternative Kutta condition with a wakesheet.

Although development is still ongoing, it has beenshowed that the new method using the transom conditionperforms best, and offers the advantage over an empiri-cal pressure correction that the physical properties of theflow are better incorporated into the solution. Of coursethe empirical pressure correction could be modified to im-prove its predictions, but still the flow properties at thetransom would not be properly solved for. Especially forseakeeping cases where one is interested in for instancepitch damping due to the accelerated flow leaving the tran-som this is important.

Still, it is evident that the current numerical modeloverpredicts the lift in comparison with the empiricalmodel. This could have a number of reasons, one beingthe absence of viscosity in the numerical model, anotherthe use of the full calm water hydrostatics in the numericalmodel. The latter is in contrast with for instance semi-empirical models based on the two-dimensional wedgeimpact for high speed planing often use buoyancy correc-tion factors. The first can only be addressed by tweakingthe viscous coefficients of the model, the latter needs to beinvestigated. It should be noted that the comparison herehas been carried out at the minimum speed range of theempirical model and at the maximum speed range of thenumerical model.

Future work includes the further implementation and vali-dation of the unsteady transom condition and to study theinfluence of this condition on ship motions in seaways.Also some details of the current implementation need at-tention, specifically the pressure on the above chine wettedregions near the transom, the number of elements used topredict the pressure peak along the waterline in the forepart and the influence of satifying the transom pressurecondition on the centers of the last hull panels instead ofexactly at the transom.

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ACKNOWLEDGEMENTS

The authors gratefully acknowledge the permission of theparticipants of the FAST project: Royal Netherlands Navy,Damen Shipyards at Gorinchem (NL), Royal ScheldeGroup at Vlissingen (NL), Marin at Wageningen (NL), andthe TUDelft (NL) to use the results of the FAST project.

References

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[3] J. Katz and A. Plotkin. Low-speed aerodynamics.Cambridge University Press, second edition, 2001.

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[8] A. Reed, J. Telste, and C. Scragg. Analysis of tran-som stern flows. In Eighteenth Symposium on NavalHydrodynamics, pages 207–219, 1991.

[9] D. Savitsky. Hydrodynamic design of planing hulls.Marine Technology19, 1(1):71–95, 1964.

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