Fully nonlinear free-surface computations for arbitrary and complex hull forms

ABSTRACT

In this paper is described the successful extension to realistic hull forms of a desingularized boundary integral method that computes fully nonlinear, inviscid water waves in the time domain and which had been limited thus far to applications involving mathematical body shapes. Results are presented for the Series 60, C8=OE6 advancing in calm water for two different Froude numbers. The calculated wave profile and wave cuts

are shown to be

in good agreement with the experimental dr . The method is also successfully demonstrated for a ship with a bulbous bow as a prelude to the extension of the method for a naval combatant. Lastly, ways in which recent work on two-dimensional inviscid transom stern flow could be extended to compute the flow characteristics of three-dimensional transom sterns e discussed.

INTRODUCTION

Concomitant with the improvements in the

techniques for the design of ships

and offshore structures has been an increase in the demand for improved analysis tools. This is especially so in the area of marine hydrodynamics, as indicated by the degree to which the computational mtthods are being

extended for use in the testing of newer naval

combatant designs.

The very use of NurnericaF Hydrodynamics tools in the design process is aided considerably by the availability of many, diverse solution methodologies, exh with its own different strengths and drawbncks. The wide choice in methodologies probably owes most to two factors: the essential difficulty of marine hydmdynamic computations ail the relentless increases in computational power over the last few decades. The latter is recognized widely;

to elaborate on the former even the reduction to

To be presented at the

Twenty-Second Symposium on Naval Hydrodynamics. Washington D. C., August 9-14, 1998 Anil K. Subramani', Robert F. Beck' and Stephen M. Scorpio2

(' University of Michigan. Ann Arbor. U.S.A.) (2 Design Research Engineering, Novi, U.S.A.)

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-potential flow leaves significant challenges because of the nonlinear nature of the free surface boundary conditions and the complex geometric features of typical marine vehicles. This paper will discuss recent advances made with a panicular potential flow solver - in particular, the solving of fully nonlinear hydrodynamic problems for arbitrary and complex hull forms.

Fully nonlinear potential flow computations can be performed in many ways. For steady fcrwad speed problems, fully nonlinear computations may be performed by using an iterative technique to solve a series of linearized boundary value problems that an based on the solution to the previous iteration ail which are satisfied on the deformed free surface ail

the exact wetted surface of the body.

Iterative techniques have been used successfully by, among others, Jensen et al. (j), Kim and Lucas (2) CID, Raven (4.), and, more recently, Scullen and Tuck () or Scullen (e.). These techniques have the likely advantage of converging to the fully nonlinear solution in less computer time than a time-stepped

method, but they cannot be extended easily

to unsteady scakeeping problems.

A time-domain solution of the fully nonlinear water wave problem may be obtained using the Euler-Lagrange method, which was introduced by Longuet-Higgins and Cokelet (Z) in their study of two-dimensional water wave problems. This time-stepping method consists of two major tasks at eath time step: In the Lagrange phase, individual nodes on the free surface are tracked by integrating the nonlinear free surface conditions with respect to time, while in the Euler phase a mixed boundary value problem is solved to obtain the fluid velocities that in turn are neukd in the integration of the free surface boundary conditions.

Variations of the Euler-Lagrange method have been applied to a wide variety of two- and

ttiee-s

Fully Nonlinear Free-Surface Computations for Arbitrary and Complex

Hull

dimensional problems. However, the variations anse

primarily from differences in the time-integration schemes used and in the techniques used to solve the resulting mixed boundary value problem at each time step. The vast majority of results have been obtained for applications limited to mathematical bcdy

geometries such as cylinders, spheroids, and the

Wigley hull, or for water wave problems with no

body at all.

At the University of Michigan, a multipole-accelerated desingularized boundary integral

method-denominated UM-DELTA, for University of

Michigan Desingularized Euler-Lagrange Time-Domain Approach---has been developed over the past

several years to solve fully nonlinear water-wave problems including wave loads on offshore structures and nonlinear ship motions. The development of the UM-DELTA method was recently chronicled by Beck (J. As previously indicated, in this method the fluid is assumed incompressible, inviscid and started from

rest so that the resultant flow is irrotational

i

governed by the Laplace equation for the velocity potential.

The velocity potential is constructed by

singularities distributed on auaiuiary surfaces separated from the problem boundaries (hence, 'desingularized') and outside the flow domain--above the fr e-surface and inside' the body surface, arid 'outside' the appropriate boundaries at infinity. The strengths of the singulanties are determined so that the boundary conditions are satisfied at chosen collocation points or nodes. [Figure 1. (a) provides a perspective view of the typical grid distribution on the body and the free surface, while in figure 1. (b) are sketched the relative

locations, at a reesentative cross-section, of the

nodes and the dsingulazized sources.] To ensure the

convergence of the method, the desingiilarization

distance decreases as the computational grid becomes finer. Because of the desingularization, the resulting

kernel in the integral equation is nonsingu lar:

therefore, special e is not required to evaluate

integrals over the panels. Simple numerical

quadratures can be ed to reduce the computational

effort greatly. Further, simple isolated Rankine sources may be used, allowing the direct computation of the induced velocities in the fluid without further numerical integration or differentiation. The solution is advanced in time using the Euler-Lagrange time-stepping procedure xl the body position and surface

normal velocities c updated from the equatons of

motion of the vessel.

As discussed in Scorpio et al. _{), multipole} acceleration techniques have also been implemented

within UM-DELTA. The underlying idea of multipole acceleration is that the potential due to a

group of sources may be evaluated at distant points

by first accumulating the influences into a single

multipole expansion. By a comç4ementary

arrangement of the evaluation points into groups, the

accumulated multipole expansions may be

transformed into local expansions centered around each group. Using a tree-structured hierarchy of

source groups and evaluation point groups, the normally O(N5 process of evaluating the influence of N sources at N points is reduced to an 0(N) process with only an 0(N) storage requirement. This leads to substantial reductions in computational time tial storage, enabling the simulations to be performed on high-end workstations. However, because of the extensive requirements for logical branching, multipole acceleration disrupts the high degree of vectonzation inherent in the UM-DELTA code and is not, therefore, invoked on vector supercomputers such

as the CRAY C-90. On the other band, it also

allows for easier implementation of the method on scalable parallel computers and thus presents a useful option on more than one platform.

To date, the UM-DELTA method has been successfully applied to a wide variety of problems. These include: the generation of shallow water

solitons [Cao et al. (10)]; wave resistance, a±kd mass, and damping and exciting forces on a Wigley hull [Beck et al. (li)]; and wave loads on a verticai

cylinder [Scorpio and Beck (L2JJ. However, the

applications thus far have been limited to simplified body shapes - body shapes that could be represented mathematically.

An accurate and efficient modeling of the

geometry is not a trivial task within 11M-DELTA.

Unlike panel methods that use flat quadrilateral panels

to define the surface and the unit normal,

UM-DELTA needs to be able to determine the surface location and unit normals at arbitrazy node locations

on the body. Determining the surface location ariI

unit normal at arbitrary locations on a typical ship hull ha ndifficult, and ithas servedasa hurdle in

the'extension of the method to arbitrary hulls. Nevertheless, the first such extension was carried out byCelebi and Beck (U), who used a N1JRBS (Non-Uniform Rational B-SpUme) surface representation of the body surface and a variational adaptive curve grid generation method for node placement on the body. However, the use of this techniquewhen tested on the parabolic Wigky bull modified with the inclusion of bow and stern rakesresulted in a 40% increase in CPU time when compared to a calculation that used instead the mathematical representation of the hull. Considering that the computational bordee would only increase for ships with complex features such bulbous bows and skegs, which may requiie more than a single NURBS surface to represent the hull

surface effectively, the NIJRBS technique was not

-

Therefore, the development of a faster aid more robust hull-surface poxessor was initiated aid subsequently incorporated into the Up-DELTA C(Xk along with other improvements. This paper describes then the recent development of the method in its ongoing extension to arbitrary and complex hull for-ms. while also touching upon the inherent difficulties in the task.

In the following sections shall be presented a brief description of the geometry processor and a discussion of the recent efforts to handle wave breaking in the computations. Results of the calculations for arbitrary and complex hull shapes ae then presented. Specifically, calm water results ae

presented for the Series 60, C=0.6 hull,

which typifies arbitrary body shapes, at two different Froude numbers. The calculations of the wave profile on the bull arid wave cuts will be shown to compare well

with experimental data

The extension of the method for complex hulls such as naval combatants with bulbous bows and transom sterns was divided into two stages

-seeking to mn

capable the handling of, first, bulbous bows and, then, transom sterns. extensions for transom sterns have not been completed at the time of this writing. Therefore, the method is

herein demonstrated for a ship with a

bulbous bow alone (as far as complex geometric features are concerned). Additionally, the strategy to be adopted for handling the inviscid flow behind three-dimensional transom sterns is described by presenting some two-dimensional results. First, however, the theoretical background of the method is provided for the sake of completeness.

THEORETICAL BACKGROUND

Problem Formulation

('isider a vessel floating on a free surface aid

translating with speed 130(t) in the negative

x-direction with respect to a right-handed se-fixed

coordinate system. For the (x,y,z) system attached to the body, the z4) plane defines the calm waL level, with +z directed upward and the x-z plane coincident with the centerplane of the vessel. The total velocity potential describing the fluid motion is then

4(x,y,4t)= U0(t)x +$(x,y,z,t)

where U0(t)x is the potential due to the free-stream velocity and (x,y,z,t) is the perturbation potential. In the fluid domain, both potentials and satisfy Laplace's equation,

V24=O

(2)

In order to define a well-posed problem, boundary conditions must be specified on all surfaces surrounding the fluid domain. On the body wetted surface (SH) and on the bottom (SB), the boundary conditions are:

=U0n1 +n.VH (3)

aid

= + 'B - on S8 (4)

where ñ=(n1,n2,n3) istheunit normal out of the fluid and ''H is the velocity of a point on the body due to unsteady ship motion and '' is the velocity

of the bottom relative to the

O(x,y,z) system. Because we are solving an initial value problem, the boundary condition on the surface at infinity (S,.,) is

as

2+y2_co

(5)

Other far field boundary conditions are possible; for example, incident waves may be prescribed at the upstream boundary (SE). The boundazy condition on an outer wall () aligned with the free-stream direction will have the form:

SWM1 (6)

Downstream, an absorbing beh may be used; foe forrd speed problems, an open bounday condition is often used. The kinematic and dynamic free swface boundary conditions are prescribed on the free suiface

(SF), although they are numerically implemented in

a special form for an improve4;descripuon of the fr-surface as the solution evolv [see Beck et al. (fl) for details]. The kinematic condition is

z=T(X.y.t) is the free-surface elevation,

i

= (U(t). V(t). &nJ&) is the velocity with which the ree-surfax nodes are prescribed to move along

pre-ordained tracks; thedynamicconditionis

where p

is the fluid density, g, the gravitational acceleration, and _a the _{atmosphenc pressure.} fl

initial values of the potential and free surface must also be specified such that

= O t O in fluid domain (IO)

i=O tO

--

I

-IicxsI

(9)

As mentioned earlier, a solution to the above nonlinear initial boundary value problem is obtained using the Euler-Lagrange time-stepping procedure. The linear mixed boundary value problem for the perturbation potential is solved at each time step (using a desingularized boundary integral method) ai the nonlinear free surface boundary conditions integrated in time. The body position and surface normal velocities are updated from the equations of motion of the vessel.

In the desingularized method, Rankine sources ax distributed on an integration surface (Ci) that is offset from the physical problem boundary by a small distance. The potential anywhere in the fluid

domain can then

be calculated, in an indirect application of Green's theorem, by integrating the influences of all the sources:

$(ic)

=_{JJ(is)G(is;is)dcz} Q

where i

_{isapointin thefluiddomain,}

_{x is a}

point on the integration surface,

_a(is)

is the strength of the source located al i,

_{aix! G(;i5)}

is the Rankme source Green function. The Oreen function for three-dimensional problems is

(12)

An application of the appi

opi jale

Dirichiet ud

Ncumann boundary conditions to (11) yields the

integral equations that must be solved for the unknown source strength

_a(is)

at each time step:

without the need for numerical derivatives.

Finally, the free surface boundary conditions, (7) and (9), are integrated in time using the fourth-order Runge-Kutta method to obtain new values of

N

1.i=1

to N

(15)

Numerical Method

_i=I

XCErD (13)

Q

is

Er (14)

where is the given potential at ,

r)

is

boundary on which

_{is known, x}

_{is the given} normal velocity on the solid boundaries, and rN ae

surfaces on which is known. For isolated sources, the above integral equations become simple summations; when discretized over a total cf N collocation points

_(is)

,

they result in an NxN

linear system of the form

where: the influence matrix A11 = ₌

fr1

-I

for collocation points on the free surface ud A _{=iii.VIG!j for collocation points on solid} boundaries; a is the vector of unknown source strengths; and b1 is the vector of boundary conditions - b1 ₌4,, for boundaries on which the

Dirichiet condition is prescribed (e.g., the free

surface), b1=_{U0n1 + ñI.VH on the body surface,}

and b1=O on a flat bottom. A variety of options have been exercised for the solution of this system of equations. These are the use of LU decomposition for the smallest problems (mostly two-dimensional); the use of the iterative solver, GMRES [Saad and Schultz

(I4)J for larger (three-dimensional, especially)

problems; and, more recently, the incorporation of mulupole acceleration techniques [Scorpio et al.

_()J

to reduce the computational burden to 0(N).

Once the integral equations are solved foc , hence the fluid velocities can be found using

=VG

(16)

and rì() , and the cycle is repeated. A

steady-state solution is obtsincd by accelerating the vessel from rest up to steady forward speed.

GEOMETRY PROCESSOR

The geometry processor was developed to enable UM-DELTA to handle arbitrary and complex hull

forms easily and efficiently. Christened HULLGEO,

the geometry processor can handle in addition to

conventional raked bows arid cruiser sterns, bulbous

bows, sterns with deadwocd, and transom stems. It

can also be extended easily to multihulls or tunnel

hulls.

The underlying idea is of using precomputed

cubic spline fits of the waterlines and stations to

return values of the unit normals at the desired noie locations. Figure 2 shows the unit normals returned

by HULLGEO for the DDGS4 15 naval combatant

model. The unit normals have been evaluated at test nodes distributed along the stations that fonned the

input to the process-. As can be seen clearly, the vectors closely represent the complex geometric

features of the combatani

Given that the normals axe evaluated based

on cubic spline fits, some amount of noise in the

results is inevitable, especially if the body surface is coarsely discretize*l. The noise may be minimized by providing as input to HULLGEO a discretized surface

of fine resolution; then, at the interface between

HIJLLGEO and UM-DELTA, a select subset of the input stations may be retained as the stations on

which collocation points are actually distributed.

UM-DELTA's calculations using HULLGEO have been verified by comparisons of the computed waves for the mathematical Wigley hull. In the results section are presented the newer results

obtained for arbitrary and complex hulls by

UM-DELTA.

BREAKING WAVE DAMPER

A greater challenge than the accurate and efficient

modeling of the geometry to UM-DELTA has been

finding ways to cope with breaking waves and spray

in the fully nonlinear computations. In linear or

higher-order expansion methods, the wave amplitudes

can get unrealistically large without causing

numerical difficulties. In fully nonlinear

computations, however, wave breaking or spray formation ai the bodyfree-irface intersection will

cause the computations to stop promptly even though

the difficulty may be confined to a small ea such as

the breaking bow wave. To overcome this difficulty we have been working on the local suppression of wave breaking and spray by absorbing energy using the free surface boundary conditions. Recently, on the basis of studies in a two-dimensional numerical wave tank, Subramani et al. (15)proposedatechnique by which to absorb energy locally from waves that

are about to break, thereby suppressing wave breaking.

The "local absorbing patch" model may be activated in two steps: 1) detecting the likely

occurrence of the wave breaking, and 2) determining the appropriate amount of local damping so as to render reasonably realistic waves in the post-breaking regime. The likely incidence of wave breaking was detected by monitoring the local free-surface curvature; when the curvature exceeded a suitably prescribed threshold value, a local wave damper was activated in the vicinity of the wave crest where the threshold was exceede*i The actual damping was provided by applying an artificial pressure on the fiee surface in such a manner that the pressure would

always take energy out of the waves. As

demonstrated in (15), the technique has been successfully applied to the suppression of spray tid wave breaking in the case of two-dimensional water waves. However, we have yet to determine the right form of the damper for three-dimensional calculations with forward speed.

RESULTS

Arbitrary Hulls

The Series 60, CB=O.Ó parent hull form, for which extensive experimental data is available, was selected as the realistic, arbitrary hull on which to test first

the extended UM-DELTA code. Calculations we

performed for a ship advancing straight ahend in calm water at two Froude numbers (Fr), 0.25 and 0.316,

and using two body discretizationsa coarse grid

consisting of 26 stations and a finer grid consisting of 51 stations. The node density along a station was prescribed to be 10 nodes along a girth erjuai to the ship draft The total number of nodes on the body was344forthecoarsegrid hull and 680 for the finer one.

The domain of the numerical wave tank for either case was bounded by an upstream end 05 Limes the ship length, L, ahead of the bow, with the tank

extending for 0.75 L downstream of the stern id

measuring 0.6 L across. The free-surface

discrecizaiion corresponding to the coarse body grid

was 58 ns in the longitudinal direction and li

case, it was 114x22. For the coarse body grid, the number of free-surface nodes ahead of the body, across the body, and downstream of the body was, respectively, 13, 26, and 19; and for the fine grid, the corresponding distribution was 25, 51, and 38. The time-step size chosen was z=O.J, sufficiently small to ensure a stable time-integration.

Figure 3 shows a perspective view of the free-surface at steady state from a Fr=0.316

calculation on the finer grid.

Note that due to

symmetry, the calculations are performed in the half-domain, Y>O. The entire domain is depicted in figure 3 for illustrative purposes. The computed wave profile along the hull at Fr=O.25 for the two different discretizations considered is peesented in figure 4 aal compared with the experimental measurements of Toda et al. (J,). The differences in the results for the two different grids are small and the calculations show reasonably good agreement with the data However. clear differences exist, with an underprediction of the bodyfree-surface intersection, a shift in the bow wave crest, and a slightly overpredicted stern shoulder wave crest. The underprediction of the water-level ax the bow is probably because of the thin sheet of fluid that shoots upward ax the bow in reality and which is noi easily captured in numerical methods. To some extent, the differences may also be due to the

simplification of the Series 60, C80.6's bow to a line bow of zero radius of curvature in the numerical method. _{Given the capability of HULLGEO to} handle raked bows, the calculations could be repeated without the simplifications, in a more detailed study on the computed bow flow.

The computed wave profile along the hull at F-0.3l6 is presented next, in figureS. In addition to the calculations for the two spatial discretizations considered, a fine grid calculation performed with a smaller time-step size of 0.05 is also presented and all the results are compared again with the data. The calculations show somewhat better agreement with the data, although the differences at the bow and at the stern shoulder wave crest persist. The differences in the results due to the different temporal discretizations are almost negligible; however, clear differences exist between the coarse and fine grid calculations cbx likely to inadequate resolution on the coarse grid. Owing to constraints on the available time, calculations with a still finer grid could not be maie and shall be undertaken in the near future to complete the grid convergence study.

Figure 6, which presents a comparison of longitudinal wave-cuts with the available dataat YIL=O.0755, 0.108, and 0.14108, for Fr=0.316-provides an indication of the accuracy of the computed wave field - the inviscid calculations are in v good

agreement with the data over the length of the ship,

except for a

slight overprediction of the waves

amidships.

The computational requirements for these calm-water calculations (steady state was attained at about t=25.O, accelerating gradually from rest) is as follows: the fine grid calculation required about 23 MW of memory and 2.5 hours on the CRAY C-90, while the coarse grid calculation attained steady-state in about 30 minutes of computer time on the same machine. Note that the high degree of vectonz.ation within the UM-DELTA has it operating consistently over 500 MFLOPS on the C-90.

Finally, given that the UM-DELTA is a time-domain method for solving unsteady seakeeping problems, a preliminary calculation for the Series 60, C8=O.6 advancing in head seas is presented in figure 7. The calculation was performed on a coarse gud as described above (with 26 stations). The incident waves have an amplitude of about 0.006 L and a wavelength of about 1.0 L and they were turned on in the computation after the ship had accelerated from rest to steady forward speed, at Fr=025. Eventually, computations of the radiation and the exciting forces

shall be presented for a range of Froude numbers.

Complex Hulls

One of the stated objectives for the current project is the calculation of the flow around complex hull forms such as naval combatants. lie development of the HULLGEO module has made possible improvements to the UM-DELTA code that go precisely towards the meeting of this objective: enabling, for instance, an accurate resolution of the flow around bulbous bows by allowing a node to track the stemfree-surface junction as the water level rises oc falls al the raked stem; allowing for nodes to be droed or a±kd in a robust manner and even for an entire station to be inserted at a new location, as necessary, depending on the change in the wetted portion of the hull.

However, additional extensions aid modifications remain to be made to the code before results can be obtained foc, e.g, the DDGS4I5 naval combatant model. This work is expected to be completed in the coming months.

For the time

being, we present bere a calculation to demonstrate the capability of UM-DELTA for bulbous bows. The calculation was performed for a hull with a bulbous bow and a croiser stem. The hull shape was creased by fusing the fcrward half of the DDG54IS model with a cruiser stern whose lines were extrxted from

the Series 60, C).6 model and which was also

closed in with deadwood.

Calm water results were obtained for this hull at Fr=0.25 using two different body discretizations -- a coarse grid with 32 stations and a fine grid with 52 stations. The number of nodes on the body was 302 for the coarse grid and 492 for the finer grid.

The free-surfe grid, in a domain of

identical proportions to that discussed for the Senes 60, Ca=O.6 calculation, had a distribution 67x13 for the coarse grid run and 108x21 for the fine grid run. The calculated wave profile along the hull, for the forward portion of the hull---the portion of interest--is presented in figure 8. The differences between the calculations on the two grids are considerable ni further calculations need to be performed to ensure grid convergence. However, this calculation was intended only to demonstrate the successful handling of a bulbous bow, before the initiating of calculations for a naval combatant model with a transom stern too A technique for handling the flow characteristics

of transom sterns has been devised from a

two-dimensional study which is described in the next section.

2D TRANSOM STERN FLOW

To provide a starting point foc the extension of the method to unsteady, fully nonlinear, three-dimensional transom stern flows, Scorpio and Beck (17) recently investigated, using the 11M-DELTA method, the unsteady flow past a two-dimensional, semi-infinite body with a transom stern (unsteady in the sense that the problem was started from rest l accelerated to steady forward speed). To aid in the present discussion, the relevant material is discussed again here in some detail.

The cases studied by them cOEresponded to those in Broeck and Tuck (18) and Vanden-Broeck (19). wherein nonlinear waves behind a transom stern were computed using a series expansion in the Fronde number. In

(JI)

and (19). the problem was solved in an inverse manner using the x and y coordinates the dependent variables and the velocity potential and stream function as the independent variables.

The series expansions in x and y we

everywhere divergent but could be swnmed by

axidanl methods. Integro-differential equations with nonlinear boundary conditions nere solved in the inverse space to obtain the expansion coefficients.

Problem Formulation

Figure 9 shows the problem configuration. The x-y coordinate system is translating with speed Ub i negative x-direction. Laplace's equation governs in

the

fluid domain and the

velocity potential is (I) = Ubx + The surfaces bounding the fluid are:

SF = Free Surface; SH = Body Surface;

Sj = Upstream Truncation Surface;

SD = Downstream Truncation Surface; and

SB = Bottom Surface.

The boundary conditions are:

Dt ESH

= gT +

= Ubnj

X E SH en

as y9oo

ES8 Here, - =

+ V$.V is the material or Lagrangian derivative,

i=(x1,x2,x3)

are the coordinates of control points on the problem boundary,

=(n1,n,n3) is

the

unit normal on the body

pointing out of the fluid, g is the acceleration of gravity, r is the free surface elevation, and is the perturbation potential. The boundaries S and SD are unspecified. Cases have been run with

SD prescribed to satisfy continuity and very little differences seen in the results as long as St,, and SD are far enough up- and down-stream, respectively. The truncation boundaries were placed about 12

wavelengths away from the transom for these calculations.

Results

Vanden-Broeck (19) sugestcd that two realistic

solutions exist for the steady flow behind a transom stern. For small values of the Fronde number, the flow rises up the transom to a stagnation point The free surface separates from the transom at the stagnation point, creating waves downstream that increase in steepness with increasing Fronde number. This solution will be refared to as "A" hereafter.

Solution A is

physically unreasonable for large Fronde numbers because the ratio of stagnation height to transom depth goes to infinity as Fronde number goes to infinity. For large Fronde numbers a second, more physically realizable solution exists where the

flow separates cleanly from the bottom of the

transom. _{This solution will be referred to as "B"} hereafter. Solution B reduces to the uniform stream as Froude number tends to infinity and the downstream waves steepen as Froude number becomes small. In _{fact, Vanden-Broeck found a} minimum Froude number (= 2.26) below which the downstream waves exceed the theoretical breaking wave steepness limit (2aIX=O.141)

The problem is started from rest and the hull is accelerated up to steady fonvard speed. Using the UM-DELTA method, the inviscid solution always

frpj

yypjj

solution A as the hull reaches steady

forward speed, regardless of the Froude number. In a viscous fluid, we know that the flow behaves like solution A at low Fronde numbers and transitions to solution B as Froude number increases. As the hull speed mcreases from rest, the flow separating from

the bottom of the transom becomes

_turbulent, resulting in the "dead water" region commonly observed behind transom sterns. _{Conseuendy, the} pressure behind the transom is lowered. Eventually the falling pressure causes the free surface to drop to the bottom of the transom resulting in the solution B flow. Once the free surface_{separases cleanly from the} transom, the turbulence is confined to a thin boundary layer (foc high speeds) and the viscous wake. Using an invscid flow model, it appears to be impossible to proceed from transom wetted _{to transom dry bei'ci'} of the lack of turbulence. However, _{two techniques} were found by which to transition artificially the solution from A to B.

In the first method, the problem is started at steady fccward speed with the transom out of the water. The hull _{is then slowly lowered into the} water. As the hull is _{lowered, the free surface}

remains separated from the bottom of the transom

i

solution B results. However, this technique will not work for a problem starting from

_{rest with the}

transom immersed. To obtain solution B for the problem starting from rest, a second techniquewas tried, attempting to mimic the effect of the dealwater region by artificially lowering the stagnationpressure on the transom This pressure drop can be modeled in the inviscid flow code by modifying the boundary condition on the transom. The condition,

s

(21)

causes the stagnation pressure. The technique consists of reducing the stagnation pressure by modifying the transom boundary condition to;

(2e2

_{- 1)}

(22) =

This modified forni of the transom boundary condition was chosen arbitrarily. There is no special significance in the exponential decay other than it provides a smooth transition for the bc*indazy condition. As the hull accelerates up to speed, the pressure on the transom drops until the free surface drops to the bottom of the transom. V'/hen the hull reaches steady speed, solution B is recovered.

The general numerical details are similar to those outlined for the three dimensional problem. There is a double ncxie where the free surface _meets

the transom in the solution A flow.

_{One node} satisfies the body boundary condition while the other satisfies the free surface boundary condition. Treating the intersection in this manner has consistently worked well in the desingularized method. _{There is} one additional constraint (or Kutta condition) at the bottom of the transom in the solution B flow. The free surface nodes move downstreani with the fluid velocity during the intermediate time steps (4th cider Runge-Kutta). At the end of a major time step the free surface nodes are regridded to their original positions by interpolating elevations and potentials. The Kutta condition is imposed by regridding the first free surface node back to the bottom of the transom. The potential at this point is computed from the source strengths.

Figure 10 shows the waves generated by the

transom stern at FH =Ub/.Jji:Ï=6.3.

_{Figure II}

shows wave steepness versus _{Froude number for the}

waves downstream of the transom. The lines on the graph represent the relationship between wave steepness and Fronde number derived by Vanden-Broeck (12). The Fronde number can be related to the mean potential and mean kinetic energy in the wave train by integrating the momentum equation. _T mean potential and kinetic energy are in turn functions of the wave steepness. The points represent results from the UM-DELTA method for various Fronde numbers. _{Encellent agreement with the} theoretical wave steepness is shown over a lxoa4.

range of Froude numbers.

-CONCLUDING REMARKS

A multipole-acceleraLed desingularized boundary integral method, 15M-DELTA, has been &vekped over the past several years to solve fully nonlinear water-wave problems including wave loads on offshore structures and nonlinear ship motions. _To date, the UM-DELTA method has been successfully applied to a wide variety of problems. However, the applications thus far have been limited to simplified body shapes such as spberoids, cylinders, and the

mathematical Wigley hull. An efficient and robust geometry processor was therefore developed to enable the handling of arbitrary and complex hull forms. The extensions of the method to realistic hull forms, with the incorporation of this geometry processor, is described herein.

Results are presented for the Series 60,

CB=O.6 advancing in calm water for two different Froude numbers. The calculated wave profile nJ wave cuts are shown to be in good agreement with the experimental data The method is also successfully demonstrated for a ship with a bulbous bow (modeled after the DDGS4I5 naval combatant model's bulbous bow). However, work is still ongoing on extensions of the method to ships with transom sterns.

To aid in

these extensions, a numerical investigation of unsteady, two-dimensional inviscid transom stern flow was recently carried out. Tr numerical results were shown to agree well with theory. Further, a technique for effecting the transition from transom wetted to transom diy at high Froude number, by artificially modifying the transom boundary condition, was successfully implemented. Additionally, it was seen that a transom dry solution could be maintained by regndding the first free-surface node back to the bo«om of the transom at each step, with no other requirements.

Besides the continued extensions 3rd

improvements to the method, future work involves firstly the carrying out of additional, fine grid simulations to demonstrate better grid convergence; and equally importantly, the devising of ways in which to suppress the disruptive effects of spray *i wave breaking in the fully nonlinear ship wave computations.

ACKNOWLEDGMENTS

This research was funded by the Office of Naval Research and the University of Michigan I Sea Grant I Industry Consortium on Offshore Engineering. The computations were made in part using an allocation of high performance computing resources through the National Partnership for Advanced Computational Infrastructure and through the Department of Defense High Performance Computing Modernization Program.

REFERENCES

U.) Jensen, G., Bertram, V., and Soding, H, "Ship Wave-Resistance Computations," Proceedings.

'

--5 International Conference on Numerical Ship Hydrodynamics, Hiroshima. Japan, 1989. pp. 593.6(J6

(2) Kim. Y.-H. and Lucas, T., "Nonlinear Ship Waves," Proceedings, l8 Symposium on Naval Hydrodynamics, Ann Arbor, Michigan, 1990, pp. 439-452.

Q) Kim, Y.-H. and Lucas, T., "Nonlinear Effects on High Block Ship at Low and Moderate Speed," Proceedings l9 Symposium on Naval

Hydrodynamics, Seoul, Korea, 1992, pp. 43-52. (4.) Raven, H.C., "A Practical Nonlinear Method for

Calculating Ship Wavemaking and Wave Resistance," Proceedings, l9 Symposium on Naval Hydrodynamics, 1992, pp. 349-370. () Scullen, D.C. and Tuck, E.O., "Nonlinear

Free-Surface Flow Computations for Submerged Cylinders," Journal of Ship Researçb, Vol. 39, No. 3, 1995, pp. 185-193.

() Scullen, D.C., MAccurate Computation of Steady Nonlinear Free-Surface Flows," Ph.D. Thesis, Department of Applied Mathematics, University of Adelaide, Feb.1998)

Q) Longuet-Higgins, M.S. and Cokelet, ED., "The Deformation of Steep Surface Waves on Water: I. A Numerical Method of Computation,"

edin's inyaLSociety of London," Vol. A350, 1976, pp. 1-26.

() Beck, R.F., "Fully Nonlinear Water Wave Computations Using a Desingularized Euler-Lagrange Time-Domain Approach." Nonlinear Water Wave Interactions Computational Mechanics Publishers, U.K., 1998 (to appear). (9) Scorpio, S.M., Beck. RF., and Korsmeyer,

F.T., Nonlinear Water Wave Computations Using a Multipole Accelerated, Desingularized Method," Proceedings. 21*Symposium on Naval Hydrodynamics, Trondheim, Norway, 1996, pp. 69-74.

(j.Q)Cao, Y., Beck, R.F., and Schultz, W.W., "Numerical Computations of Two-Dimensional Solitary Waves Generated by Moving

Disturbances," International Journal of Numerical Methods in fluids, Vol. 17, 1993, pp. 905-920.

(JjJ 8-eck, R.F., Cao, Y, Scorpio, S.M., and Schultz, WW., "Nonlinear Ship Motion Computations Using the Desingularized Method," ?roceeding, 20's Symposium on Navai Hydrodynamics, Santa Barbara, August 1994, pp. 227-246.

(12) Scorpio, S.M. and Beck, R.F., "A Multipole Accelerated Desingularized Method for

Computing Nonlinear Wave Forces On Bodies," Journal of Offshore Mechanics and Arctic Engineering, Jan. 1998.

(13)Celebi, MS. and Beck, R.F., "Geometric Modeling for Fully Nonlinear Ship-Wave Interactions" Journal of Ship Researçtì, Vol. 41, No. 1, 1997, pp. 17-25.

(14) Sand, Y. and Schultz, M.H., "GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems," _£L.M

Journal of_{Scientific arid Statistical Computing}

Vol. 7, 1986, pp. 856-869.

(12

Subra.mani, A.K., Beck, RF., and Schultz, "Suppression of Wave-Breaking in Nonlinear Water Wave Computations," Proceedings, 1 3 International Workshop and on Water Waves and Floating Bodies, Aiphen ann Rijn, Netherlands, 1998.

(th) Toda, Y., Stern F., and Longo, J., "Mean-Flow Measurements in the Boundary Layer and Wake and Wave Field of a Series 60, CB=0.6 Ship Model-- Part l:Froude Nurnbersø.16 and 0316," Journal of Ship Research, Vol. 36, No. 4, 1992, pp. 360-377.

(Ji) Scorpio, S.M. and Beck, RF., "Two-Dimensional Inviscid Transom Stern Flow," Pnxecsiings, 12th International Workshop on Water Waves and Floating Bodies, Carry-le-Rouet, France, 1997.

(].) Vanden-Broeck, J.-M., "Nonlinear Stern Waves," Journal of fluid Mechanics, Vol. 96, No. 3, 1980, pp. 603-611.

(1.2) Vanden-Broeck, J.-M. and Tuck, E.O., "Computation of Near-Bow oc Stern Flows, Using a Series Expansion in the Froude Number," Proceedings. 2 International Conference on Numciical Ship Hydrodynamics, B.erkeky, California, 1977, pp. 371-38 1.

Figure 1. (a) Perspective view of typical grid distribution and (b) cross-section at A-A showing the relative locations of the nodes and sources

Figure 2. Unit normals on the body as obtained by the geometry processoc (top) bow and (bottom) stern rtgions of the DDG54I5 combatant model

o-3 023 o NO25 4 a o A A A £ A S

-.-I Body & -0.3 Body -OE75 4 a Fre-swfe DX Frc'e-mrfe3-c -125 ¿ Oi i 1, 2 2i

i

N

Figure 3. Perspective view of the steady-state

free-surface for the Series 60, CB=O.6at Fr=0.316

(I 14x22 free-surface grid,680body nodes)

002

0_0

o

XJL

Figure 4. Wave profile along the hull for the Series60, CB=0.6at Fr=0.25 (.t=). 1)

Figure 5. Wave profile along the hull for the Series60,C1=0.6 at Fr=0.316

0.02

00)

-002

- ÇkL

(1UI Irid. 6& 05)

Etpcnnnt[Tothetd fl6)l

(b) Y/L=0.108

0.02

- C*ktu, (fue grid. O5)

E.çuni ÍTOÖa et iI (16(1

Figure 7. Perspective view of the Senes 60, C1=0.6 advancing into incident waves of amplitude, 0.006 L and wavelength, 1.0 L

CkJaitoit (fue pid. .M 05)

T4a et ¡1 (16)J

-

E rent

CI3x (fee grid)

C'JJ)crgnI

-)fu grid. S&.6) 05)

ht,o(fp,d..&.0.l) gr6! & I) C2JCUIaDOO cid

I_N -

-i-0.5 xii.. IS (c)Y/L = 0.14108

Figure6. Longitudinal wave cuts for the Series 60,

CB=0.6 at Frì316 o 0.25 0.5 0.75 XiL Io 0 05

_x.

(a) Y/L = 0.0755 o 025 05 0.75 0.02 ODI -00! -002 0.02 001 -0.01 -0.02 0.0) -0.0) .0.02

orn

oDI

-OE02

01 02 03 04 05

XJL

Figure 8. Wave profile along the hull (forward of midship) at Fr=O.25 for a ship with the bow of a DDG5415 combatant model

T

S,

SQ

Figure 9. Problem configuration for 2D transom stern flow

li

14 0.12 0.1 0.06 0.04

o.

o S. So

Figure 10. Waves generated by the 2D transom stem at a Froude number based on transom depth

of Ff6.3

M UW-.TA A L LtTA 0oM

V-I.o

(ITs So A

V-1

(IT) I 2 3 4 5 6 7 F1

Figure 11. Wave steepness versus Froude number, FH for waves downstream of the transom