The numerical towing tank - fact of fiction

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Scheepsbouwkunde Technische Hogeschool

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The Numerical Towing Tank - Fact or Fiction?

(T.he Eleventh Georg Weinbium Memorial Lecture)

John Nicholas Newman

Ocean Engineering Department,, MIT

(Manuscript submitted for publication in Schiffstechnik) September 1989

ABSTRACT

A personal view is offered for the present and future status of numerical ship hydrody-namics. As in the use of conventional towing tanks, the discussion is focussed on the two complementary problems of wave resistance in calm water and ship motions in waves Emphasis is given to the development of programs based on rational theories which are sufficiently complete and reliable to be used in place of towing-tank experiments. Appro-priate steps are outlined to validate these programs, and a distinction is made between validation of the theory, generally by means of experiments, and validation of the compu-tational model The linear analysis of wave interactions with stationary vessels has reached a relatively mature state, by comparison to the analysis of wave resistance and seakeeping for ships moving at a substantial forward velocity. The cOntinual improvements in com-putational hardware will assist future progress in this field, but even greater importance is associated with new developments in numerical mèthods and software.

Methods to predict the hydrodynarnic performance of ships from computations, based on

rational theoretical models, have been a continuing objective since the time of Froude and Michell. Georg Weinblum was a leading advocate of these developments. He contributed particularly to the theories of wave resistance and seakeeping, and provided unique leader-ship to the international community in the broad field of leader-ship theory. It is most appropriate that his name is associated with these lectures, an4 Jam honored by the invitation to present the Eleventh Georg Weinbium Memorial Lectures in Berlin and Washington.

1. Introduction

As with every field of engineering, the growth in capacity and use of digital computers has greatly accelerated work in ship hydrodynamics during the past 20-30 years. Numerical methods are used routinely by modern towing tanks to assist and complement experimental work The speed and economy of relatively simple numerical modelling facilitate the plannrng, monitormg and analysis of experiments If the numerical models are sufficiently

reliable they also can be used to interpolate and extrapolate experimental data, thus

reducing the level of experimental effort or eçpanding the validity of the results.

Ultimately, if their reliability is sufficient, theoretical and computational models can be used directly to make engineering predictions, without the need to perform routine

This implies more accurate and reliable predictions than those associated with planning, interpolating, or even extrapolating experimental data; to be acceptable as a substitute for experiments the numerical model must be at least as reliable, if not more só.

The very concept of a numerical towing tank is provocative, and can lead to polarization and extreme views, on óne hand defending the perpetual need fOr experiments and on

the other proclaiming the ultimate power of the computer. The latter view might be

considered an inevitable coróllary of the exponential growth in computing facilities, and their continuing decline in cost relative to experiments. According to this view it is only a matter of time until any physical system of finite complexity is reduced to anumerical

model with the required degree of accuracy. On the other hand, the concept of finite

complexity may be strained by the analysis of external flows at high Reynolds numbers, in the presence of free-surface waves.

My lecture is intended to offer one personal view of the present status of the numer-ical towing tank, specifnumer-ically in respect to the types of engineermg predictions generally assòciated with the conventional experimental facility. Thti I shall discuss only the two complementary problerris of wave resistance in calm water and ship motions in waves, or the generic problem of a floating body in an extern3l flow bounded by a free surface. Only irrotational potential flows without vorticity or circulation are considered. As with the experimental towing tank itself, this idealization proscribes the consideration of vis-cous effects at the correct Reynolds number, so that boundary-layer analysis and other assumptions appropriate to unseparated flows must be invoked to complete the required engineering predictions. Substantial efforts are being devoted to efficient Navier-Stokes solvers and to the modelling of turbulence and separation, but these efforts are considered to be complementary to the representation of free-surface effects and it is premature to

onsider both viscous and free-surface effects simultaneously for practical problems. Only two special areas exist in ship hydrodynamics where numerical methods are com-parable in value to experiments: (1) the analysis of ship propellers by lifting surface theory, and (2) the prediction of wave effects on vessels without substantial mean forward velocity. In both cases there are shortcomings in the numerical model, for example cavitation incep-tion near the leading edge of a propeller, or nonlinear wave effects, but other limitaincep-tions of comparable importance exist in the corresponding experimental procedures.

The restricted scope of my lecture rules out further consideration of propeller problems, where some of the greatest successes of numerical ship hydrodynamics have been achieved. On the other hand the problem of zero-forward-speed seakeeping is specifically included, since it falls within the province of conventional towing tanks and is particularly important in the design of offshore platforms; moreover, the achievements in this limited area provide

useful guidelines for future accomplishments where the forward velocity ¡s nonzero. Figure 1 shows the problems to be considered in a three-parameter space with coor-dinates corresponding to the forward speed, unsteady parameter such as the frequèñcy of oscillation, and the degree of nonlinearity. Various established or developing problems are considered within this space including the linearized steady wave-resistance problem, unsteady problem at zero forward velocity, and the general case of unsteady motions in the

presence of forward speed. The relatively mature computational model for zero-forward-speed linear theory is indicated by a solid circle; other well-known theories, still in the research phase, are identified by open circles.

The analytical description of these problems is that of classical potential-flow theory, with Laplace's equation applied in the fluid domain. Neumann boun4ary conditions are imposed on the submerged surface of the body and on the bottom of the fluid, coupled kine-matic and dynamic boundary conditions are imposed on the free surface, and a radiation condition or initial condition specifies the field far from the body.

2. Numerical Approaches

.

Three principal approaches have been follówed to describe this class of prob1err

corn-putationally: (A) discretization of the body surface alone, ai4 use of a 'free-surface' Green function (source potential) which satisfies the complete linearized problem except for the boundary condition on the body; (B) discretjzation of both the body and free surface, and use of a simpler 'Rankine' Green function; and (C) discretization of the entire fluid domain and use of still simpler basis functions such as polynomials which separately do not satisfy any part of the boundary-value problem. Approaches (A) and (B) are boundary-integral-equation-methods based on Green's theorem, which can be interpreted more physically as the representation of the flow field in terms of suitable source and dipole distributions on the boundary surface. Approach (C) is usually pursued as an application of the finite-element-method.

Figure 2 illustrates these three approaches in two dimensions, for the flow exterior to a ship's cross-section. Figure 3 indicates the three-dimensional discretization of the ship's surface and the free surface, the latter being included in

approach (B) but not in (A)

Figure 4 shows the corresponding volume discretization, i.e. approach (C), for the flow

exterior to the ship hull. Suitable closures are required in the far field, except in approach (A) where the analytic radiation condition is invoked for the Green function. Thus in (B) and (C) the horizontal extent of the free surface is artificially restricted, and a suitable

algorithm is used to correspond in a discretized manner to the radiation condition; in

approach (C) the vertical domain of the fluid is restrictedsimilarly. In the latter respect,

at least, approach (C) is most effective for shallow depths of water, and vice versa.

The transition from (A) to (C) involves the use of simpler mathematical elements to build up the solution, at the expense of an increased computational domain and larger number of unknowns. Similarly, one might say that the

transition from (A) to (C)

in-volves a greater emphasis on computational methodology a.ùd less utilization of analytical developments such as the free-surface Green functions.

In linear' water-wave theory, for a fluid of unbounded horizontal extent and constant

finite or infinite depth, the Green functions which satisfy the free-surface boundary

condi-tion are well known. (Wehausen & Laitone, 1960, §13). Thus approach (A) can be used

when the free-surface condition is linearized, but the analytic complexity of the appropriate Green functions makes this approach non-trivial. Typically these functiOns are expressed

evalu-ation is to integrate numerically in the Fourier domain. This is an expedient approach, which minimizes the investment in numerical analysis, but at the expense of substantial computing time and/or loss of accuracy. The alternative is to develop special algorithms based on analytic expansions and systematic appraximations, as is done more commonly for transcendental functións of a single variable.

Approaches (B) and (C) have been. advocated to avoid the usé of these complicated Green functions, but recent years have seen a renaissance of approach (A) resulting from intensive efforts to develop effective algorithms and subroutines for the evaluation of the free-surface Green fünctions. To the extent that these efforts are successful, approach (A) is likely to prevail, at least for problems where the linear free surface condition is applicable, but one cannot discount the possibility of further cycles in the development of these different approaches, as in a game of 'leap-frog'.

For nonlinear computations the free-surface Green functions are not directly useful, and either approach (B) or (C) must be utilized at least in the domain where the nonlinear features are significant. A successful 'hybrid' thatching, between a nonlinear inner solütion described by Rankine singularities on both the body and free surface, and a linear outer solution represented by free-surface Green functions, is demonstrated for axisymmetric bodies by Dommermuth and Yue (1986).

Strictly from the computational standpoint, an important practical consideration is the total number of unknowns, or computational elements, assoçiated with each of the

approaches (1-3). From the hydrodynamic standpoint a closely-related issue isthe efficient

design of the discretized grid. This involves consideration of both geometrical and flow gradients. The former is associated with the complexity of the body surface; the latter with the characteristic wavelength and with local singularities resulting from corners and local rapid changes along the body surface. (In nonlinear work one also must consider the geometrical and hydrodynamic fluctuations along the free surface.) In linear applications

of the two-dimensional situations depicted in Figure 2, typical numbers of unknownswould

be 16 for approach (A), 32 for approach (B), and 256 for approach (C). In three dimensions these numbers are increased substantially, e.g. 500, 1000, and 20,000. For a given approach the determination of these numbers is related to the desired degree of accuracy in the final

results. The most reliable approach is to perform a convergence test, decreasing the scale

of geometric discretization systematically and increasing the number Of unknowns until the results no longer change within a specified tolerance. Such tests may be difficult or impossible to perform, if the program has insufficient speed or capacity to accommodate the larger number of unknowns; in such cases the program cannot be considered properly validated.

3. Validation

The validation of computer programs is a topic of special importance, which is receiving growing recognition. It is vital to possess confidence in the correctness of a program, if it is to be used for practical engineering analysis, but the manner in which this confidence

is established,, i.e. the validation of the program, is less obvious. It is common to cite

underlying theory upon which the program is based. The program itself must be validated from the standpoint that it correctly represents the theory, to someclearly defined degree of accuracy.

The distinction between validation of atheory and of a program is illustrated by the strip theory of ship motions. Here there are ad hoc approximations in the theory, based on

\the assumption of two-dimensional flow in each transverse plane; this basic assumption can

be tested by comparisons with more exact three-dimensional theories or with experiments.

On the other hand, a program based on the strip theory must be validated to ensure

that it is an accurate and reliáble numerical model for the theory itself. The use of an

experimental comparison to simultaneously serve both purposes is an appealing sho±tcut

from the practical standpoint, but uncertainties in the experimental results and limitations of the theory mask the detection of numerical errors and inaccuracies.

Two or three decimals accuracy in the final results of a program is generally sufficient

for engineering predictions, particularly where the underlying theory is based on idealized

assumptions. However, careful programmers are taught to test software to substantially higher tolerances. Assuming the use of 32-bit single-precision floating-point arithmetic,

and alkwing for the loss of one or two decimals accuracy due to truncation and round-off

errors, a robust program should be accurate to between four and six decimals. Only by testing to this tolerance can one be reasonably confident that the program is free of errors

and that it correctly reflects the underlying theory. This type of checking should be done

for several test cases including difficult, as well as simple geometries, frequencies, or Froude numbers. A variety of complementary checks should be made including (i) comparisons with established benchmarks such as analytic results for simple geometries and limiting values of the parameters; (2) verification of reciprocity and other analytic relations, such as the Haskind relations for the wave exciting forces; and (3) tests to confirm that the

results converge tó the estimated internal tolerance of the program as the number

of

panels, or other discretization elements, is increased. Comparisons between two or more independent programs are especially important, and should' be approached with the same target of tolerance for error.

In practice these complementary validation tests are rarely performed to such an ideal degree. Even for the 'model' problem discussed below, where a unique degree of validation has been established and tests have been made with up to 12,000 panels on the body, the degree of convergence is on the order of three decimals rather than four. A related mathe-matica.l issue is that the rate of convergence of panel methods has not been established. In the most widely-used formulations where the ource strength or velocity potential is

con-stant on each panel, empirical evidence suggests a rate of convergence between

1/N and

1/N2, with increasing number N of panels, depending on the geometrical complexity of the body and its discretization. Special hydrodynam.ic or numerical complications such as proximity to an irregular frequency, or imposition of a Kutta condition, generally impede the convergence rate.

4. A Model Problem (U=0)

motion at a prèscribed frequency w, with the body in a fixed mean position. Not surpris-ingly, this problem has reached the most advanced state in its re4uction to a computa-tionally efficient and reliable form. Thus it is of interest, not only for direct application to vessels without significant forward velocity, but also as a model to provide guidance for the more. difficult cases involving steady or unsteady forward motion.

A substantial number of computer programs have been developed to solve this problem for arbitrary three.dimensipnal floating (or submerged) bodies, in water which is either infinitely deep or of constant finite depth Most of these programs are extensions of the panel method originally developed by Hess and Smith (1962) to analyze the potential flow external to a body in the absence of a free surface; the principal modification is to utilize the free-surface Green function. A flow chart for this approach is shown in Figure 5. Thé remaining prograis are based either on the use of simpler Rankine singularities, distributed on the free surface as well as the body, or on finite-element discretization of

the fluid volume. .

These computer programs have found wide practical use, particularly in the design of offshore platforms for the petroleum industry, but limitations of the 'first-generatiqn' programs were apparent when complicated structures such as tension-leg platforms were analyzed. Figure 6, re-plotted from Eatock Taylor and Jefferys (1986), shows the surge added-mass coefficient for a generic TLP, as predicted by 17 different organizations. The substantial scatter of these results has been a source of great concern. We now know that only four of these programs are within ten-percent of the correct results.

To improve upon this situation a 'second-generation' panel programhas been developed

with various numerical refinements These include fast algorithms for evaluating the free-surface Green function with controlled accuracy, and the use of an iterative solver for the linear system of algebraic equations leading to the value of the unknown potential on each panel. The use of this program has made it possible to increase the maximum number of panels from a few hundred to several thousand, and to establish convergence of the solution as this number is increased. These new results are shown for comparison in Figure 6. Further details are given by. Korsmeyer et aI. (1988) and by Nèwman and

Sclavounos (1988).

One of the most important validations for the second-generation program was based on an independent time-domain analysis by Korsmeyer (1988). In accordance with linear-system theory the Fourier transforms of the impulse-response function are equivalent to the

added-mass and damping coefficients, and the comparison shown in Figure 7confirms this

relationship in the numerical context. A similar comparison has been made subsequently by Ferrant (1988), with practically identical results.

No satisfactory explanation has béen given for the wide discrepancies

in the other

results of Figure 6. The possibilities range from the use of unsuitable body discretizations

to errors in the. programs 'themselves. The most generous explanation is that insìfficient

numbers of panels have been used, but the rapid convergence of the new results in Figure 6 implies that an accuracy within 10% can be achieved with only 512 panels.

Aside from its use as a validating tool for the more common frequency-domain pro grams, the time-domain formulation offers several possibilities for direct use in the analysis of wave-body interactions. These incde couplingwith nonlinear structural-analysis pro-grams, and of considering semi-nonlinear formulations where the body boundary condition

is exact and the free-surface condition is linear Of more mterest for ship motions is the

possibility investigated by King et aL (1988), of considering forward velocity via a time-stepping procedure with the zero-forward-speed Green function. Several fast and efficient algorithns have been used successfully to evaluate the time-domain free-surface Green function in infinitely deep water; the ftnite-depth eictension is substantially more difficult and no results have yet been reported.

5. The Forward-Speed Steady Problem

The most important problems in ship hydrodynamics involvesteady or unsteady

mo-tion with a constant forward velocity U. The resulting complicamo-tions of the theory are profound, and have frustrated many years of dedicated research. These complications es-sentially fall intO two categories: (i) the analytical formulation, and (2) numerical problems associated with developing the corresponding solutions.

Uncertainties regarding the análytical formulation are associated

with the issue of

linearization. If the geometrkal shape of a body and its forward velocity are completely general the resulting fluid motion will be finite and there is no basis for linearization of the free-surface disturbance. Broadly speaking three possibilities exist: (1) to linearize the problem in a consistent manner, by imposing appropriate restrictions on the body

geometry or the forward velocity; (2) to linearizethe mathematical statement of the free.

surface boundary condition on an ad hoc basis and to solve the resulting problem without other restrictions; and (3) to attack the nonlinear problem directly.

The literature of ship-wave theory includes various examples of consistent lineariza-tion, based on geometrical restrictions including thin, slender, or flat hull shapes, and the consideration of submerged bodies where the depth of submergence is sufficiently large to justify the assumption of a small free-surface disturbance. In these circumstances the an-alytical formulation is similar to the zero-forward-velocity case; the principal modification is to replace time-derivatives in the fixed reference frame by Lorentz transforms, which add derivatives in the. direction of forward velocity to the time. derivatives.

Well known integral expressions exist for the free-surface Creen function, with forward velocity included. However these functions, and presumably also the exact solution of the linearized problem, contain subtle and very complicated mathematical singularities.

From the physical viewpoint, if the source is located in the

free surface the resulting

Kelvin wave field is infinitely short and steep along the track downstream of the source.

(For submerged bodies this problem does not occur since the source point isalways a finite

distance below the free surface.) This short-wavelength singularity was first emphasized by

Ursell (1960). lt has received more detailed examinations by UrseIl (1988,1989), Newman

(1988), and Clarisse (.1989) with the objective to provide suitable algorithms fornumerical

purposes. It remains unclear what the

integrated effect of this singularity is, or how it

solution . '

An expedient approach is simply to ignore the short diverging waves, assumiiig that they are cancelled by destructive interférence if the sources (or normal dipoles) are dis-tributed in a continuous manner over the submerged surface of the hull Such filtering can be achieved simply by truncating the integral representation- for the free-surface Green function in approach (A) , or by finite discretization of the free surfacé in approach (B).

However this leaves open several questions: (1) the numerical consistency and convergence

of these ad hoc approaches; (2) local effects from domainswhere the actual hull surface

is not mathematically continuous, e.g. the waterline, bow and stern; (3) the effects of

piecewise discrétization of the hull surface, which destroys the above continiity in a more

global sense; and (4) the likelihood that the short-wavelength singularity will have a relá-tively weak effect on global parameters such as the wave resistaitce, whereas computation of the far-field waves may be fundamentally more difficult. Consideráble support for the last conjecture comes from the results reported by Ratcilife et al. (1989).

The mathematical singularity whiéh extends downstream from the intersection of the ship hull and free surface leads to an interesting physical question: if a towing-tank ex-periment could be conducted in an inviscid fluid, without surface tension, would singular short waves occur in the same region? If so, is viscosity or surface tension the more impo tant constraint which prevents this phenomenon from being observed in nature? Limited computational evidence, based on the results shown in Figure 8, suggests that no singular short waves are observed for a continuous distribution of sources on a thin ship.

Ad hoc linearization of the free-surface condition, withOut restricting the geometry or Froude number, is known as the Neumann-Kelvin approach in wave-resistance theory.

Several computer programs have been written on this basis where the short-wave

compo-nent of the spectrum is excluded in evaluating the Green function. There is insufficient evidence of numerical convergence to accept these programs for routine use at the present time. This situation could change if fast algorithms are used for the Green function, and if effective numerical filtering of the short-wavelength singularity can be implemented in a rational manner.

The alternative approach (B), based on distributions of Rankine singularities on both the body and the free surface, was used in seminal papers by Gadd (1976) and Da'wson

(1977). Here a variety of free-surface boundary conditions can be introduced, including

(1) the conventional condition linearized about a uniform free stream, (2) the

'low-Froude-númber' boundary condition linearized about the double-body flow, and (3) the complete nonlinear condition. The efficacy of the low-Froude-number condition was established by Dawson, using a special discretized differential operator to impose the free-surface conditiofl along the streathlines of the double-body flow; the latter had the undesirable feature of giving a coarse discretization near the bow and stern stagnation points. More recent extensions have allowed for a general choice of the free-surface panelization. The work reported by Raven (1988) Sclavounos and Nakos (1988), and Nakos and Sclavounos (1989) has clarified the role of the differential operator with respect to numerical damping, dispersion, and stability. Iterative schemes have been used by Xia and Larssen (1986), and

also by 3ensen (1988) , to satisfy the nonlinear free-súrface condition. Raven (1989) observes that the choice of free-surface condition is relatively unimportant, and that differences in the results obtained by various programs are more likely due to the inherent difficulty of: obtaining sufficient numerical accuracy and robustness

As with the Neumann-Kelvin theory, current Rankine-type programs lack the capacity to perform thorough convergence tests. The principal means of validation is via compar-isons with experiments and with other numerical results Nevertheless, these programs have found substantial acceptance for practical use in predicting the wave resistance and other hydròdynamic force components, as well as local flow phenomena close to the hull.

Since the free surface is discretized and truncated, it is fundamentally impossible to use

this approach to evaluate far-field or short.wavelength effects, but a hybrid extension in

conjunction with free-surface Green functions may overcome the latter difficulty.

Parallel developments of both the Rankine and free-surface Green function approaches can provide not only the possibility of matching in a hybrid scheme, but also of comple-mentary checks for the purpose of validation. Figures 8 and 9 show such a comparison for a thin vertical strut of infinite draft and parabolic thickness. It is encouraging to see good agreement between these two very different numerical methods, but graphical comparisons such as these are limited in accuracy and are not suffiçient by themselves to satisfy the validation goals listed in Section 3.

6. The Forward-Speed Unsteady Problem

The same techniques used for the steady problem can be extended to unsteady ship motions in waves. Progress along these lines has been made with both the Neumann-Kelvin

and Rankine approaches. Nakos and Sclavounos (1989) demonstrate the promise of the

Rankine approach using similar algorithms for the far-field conditions as in the steady

forward-speed problem, provided the parameter r wU/g is sufficiently large so that all

of the radiated waves are convected downstream.

Parallel work on the Neumann-Kelvin approach has been inhibited by the lack of fast subroutines for the Green function, but recent progress in this direction has been reported by Ohkusu and Iwashita (1989).

One issue unique to this problem is the appropriate hull boundary condition to be satisfied; the so-called 'rn-terms' in this boundary condItion result from consistent

lin-earization of the body motions withm the steady velocity field The rn-terms are difficult

to implement, especially near the bow and stern, and most numerical solutionsdelete them for the sake of simplicity or robustness The practical consequence of this simplification is unclear.

Also unclear is the extent to which nonlinearities are important, for both the steady and unsteady problems with forward speed. It would seem on physical grounds that this question is a direct consequence of the fullness of the ship's ends.

For seakeeping work the strip theory approach continues to provide a simple

com-putational tool, but not one which can be used safely in the

absence of experimental

confirmation. Faltinsen, (1989) aptly states that "To make further improvements in ship motion predictions at moderate and high Froude number it is felt that one first has to study the steady wavepotential problem in more detail."

7. Future Prospects and Conclusions

Unlike other unsolved problems in computational fluid dynamics, such as the

descrip-tions of turbulence, separated flows, or cavitatkn inception, the potential flow past a

moving ship hull is described by a well-defined boundary-value problem and is free. of

micro-scale physical effects It seems remarkable that this problem, which is sufficiently

important to justify elaborate experimental facilities throughout the world, has persistently defied a suitable numerical description. Certainly this is not due to a lack of intellectual

in-vestment, or computing resources. And,while there is a perpetual desirefor morepowerful

computers, it is unclear what algorithms would be appropriate to use if we were suddenly presented with a twenty-first-century 'hyper-computer'.

Extrapolations of past progress in computational engineering are usually inferred from the exponential growth in computer power, e.g. the line 'H' in Figure 10 which indicates an 0(10) increase every 10 years in the speed of the fastest computers. Equally important, if not more so, is the parallel development of numerical methods, algorithms, and software. Two examples of the latter progress are also shown in Figure 10, and it is remarkable to observe that their rates of improvement have been just as great as those in hardware. The first example is for the solution of the Reynolds-averaged Navier Stokes equations, a problem of broad interest where a large number of scientists and engineers are active. (This curve, and that for hardware, are adapted from a report prepared by the Committee on Computational Aerodynamics of the National Research Council., Anon. 1983.) The second example shown in Figure 10 is the estimated rate of improvement in numerical methods for solving the 'Model Problem' of unsteady motions of a floating body, as described in Section 4. (This estimate is based on opinions submitted from eight organizations using four different panel programs and one hybrid finite-element program; for the decade 1979-1989 estimated improvement factors ranged between 4 and 50 with a geometric mean equal to 17.) Note that the. indicated rates of improvement for both examples are substantially independent of compuìer speed.

In the small community of ship hydrodynamics the development of numerical methods may seem relatively slow, but occasionally as work on a particular topic matures quantum jumps in numerical efficiency can be anticipated. An example is given by the unsteady

problem described in Section 4, where an order-of-magnitude gain in computational effi-ciency has been achieved in the. past ten years due solely to advances in numerical methods. Not only has this facilitated the routine use of numerical predictions by practicing engi-neers, but even more important has been the possibility to make extensive convergence tests and to validate the programs in a comprehensive manner. Corresponding advances must be made with the forward-speed problem, if it is to achieve a similar computational status.

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Ship

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6' 2nd-Order ___ Waves

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Platform Motions Figure 1 - The scope of three-dimensional ship-wave

theories, intermediate _{between the}

linear and exact levels are several partially nonlinear _{methods. 'MEL' denotes}

the mixed

Eulerian-Lagrg ¡an

_{technique used to describe}

some simple cases of fully _{nonlinear body} motions (Dommerrnuth and Yue, 1986).

Body-nonlinear Dawson

Ship Motions

Figure 2 - Discretization schemes for the two-dimensional flow exterior to a ship section: (A) free-surface singularities on the body boundary only; (B) Rankine singularities on the body and free surface; and (C) discretization of the fluid domain.

1111111111

¡liii

01111111

IIflhIflflhiiqui

IIlohIIIIIIIIlII

'IIIIIIIIIUIfluii

IlilIlliti

.

2fl1111111i'

Figure 3 - Boundary discretization of the three-dimensional flow exterior to a ship hull. In approach (A) only the ship hull is discretized; in (B) both the ship hull and free

surface

/ modes

freqUencies Froude numbers etc. V

Add Free-surface

influence functions

0(N2)

y"Discretization

of Body (+ Free-surface)

(Interactive)

V

Evaluate

Panel geometric data

V Setup

Rankine infltence functions

.0(N)

.4 Loop over frequencies, Froude numbers etc.

Post processors

(RAO's, Spectral Data,

Forces and Field data)

Solve 0(N2, F?)

Figure 5 - Flow chart for a free-su±face panel code.

80000-O

40O00

20000

-O e e

1pi.

s e e e s I L I ¿ j T T i I T T T T I t l r t

5

10

15

20

PERIOD (see)

Figure 6

Surge added mass of the ISSC tension-leg platform. The symbols (s)

de-note computations reported by Eatock Taylor and Jefferys (1986) based

on 17 different

programs

The resulta indicated by the two curves and symbols (x)

are based on the

second-generation program described by Newman and Sciavounos (1988) with three levels of discretization between 512 (broken line) and 4096 (x) panels

on the complete structure.

s e s e e e

t

e s s e

..

3 e e se e a s s e

:

-.

51S. s -; a * e

.

2 s e s s S

.

-3 s

₂₅

u u

z

Q

o

o

o

Cbo

_i.0

2.0 3.0

FREQUENCY w

4.0 5.0

Figure 7 Damping and added-mass coefficients of the ISSC tension-leg platform in surge, based on Fourier transform of the time-domain impulse-response function (solid line) and frequency-domain panel code (+). (From Korsmeyer et al., 1988.)

3.0 2.0

Figure 8 TWo computational solutions of the wave elevation behind a thin strut advancing

at Froude Number F = 0.4. The strut,

which has parabolic waterlines an4 infinite draft,

is situated in the rectangular domain outlined to the right. The lower figure is based

on the thin-ship solution with free-surface singularities distributed continuously over the strut mid-plane The upper figure is based on the Rankine-source method with discrettzed

-3.50

1.50 1.00 0.50 0.00

0.50

1.00

1.50

3.50

2.50

1.50

irr

t'

x/L

1/111))))

2.50

1.50

0.50

1 .50 1.00 0.50 0.00

0.50

1.00

1.50

0.50

Figure 9 - Contour plots of the wave elevations shown in Figure 8. Comparison between the continuous solution' (lower half) and the discrete solution (top half). (From Nakos and Sciavounos, 1989.)

I

i

I I

1970 1980 1990 2000.

YEAR

Figure 10 - Rates of improvement in hardware and numerical methods. The line 'H'

indicates the relative reduction in time required to perform standard floating-point com-putations on state-of-the-art hardware (mainframes and super-computers). The curve 'NS' indicates the number of óperations per grid point required forsolution of the Navier Stokes equations at a Reynolds number Re= 10v. The line 'W' indicates the relative reduction in computing time for solving the zero-forward-speed unsteady wave-interaction problem described in Section 4, due solely to numerical methods and independent of improvements in hardware. Dashed lines indicate future extrapolations or projections.