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Paul A Palo.

The 1980 CEL mooring dynamics seminar online

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Online LibraryPaul A PaloThe 1980 CEL mooring dynamics seminar → online text (page 5 of 11)
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64




Figure 6
Heaving and Pitching Amplitudes for Fr = 0-35



65



Another point of interest regarding the effect of forward motion is
about the value of the parameter y —

If Y> 0.25, then the waves generated by the ship oscillation travel
more slowly than the vessel and hence are confined in the sector behind
the ship, whereas if y< 0.25 the waves travel faster than and ahead of
the vessel. As indicated by [36] and [37] and recently [38] this feature is
not a theoretical anomaly because experimental measurements around the para-
meter value Y =0.25 show some irregularities and scatter.

The computation of the wave excitation is carried out either a) from
the Froude-Krylov theory by using the defined relative motion between the
ship and the wave, as given in [ 8 ], [15], [ 1 ] or 2) from the diffraction
theory by using Has kind -Newman relationship [23], [37], [26],

Consideration of the forward speed in the coefficients of equations of
motion is another source of difference between various strip theories.
Wave Exciting Forces and Moments

There are at present two methods for the calculation of wave exciting
forces and moments, namely:

1) Korvin-Kroukovsky type of approach

2) Use of Haskind-Newman formulae

The first method makes use of the relative motion concept and in a
way employs Froude-Krylov theory combined with this relative motion definition.
Consequently, this approximate method is valid only for the medium range of
frequencies; for the short waves the Froude-Krylov hypothesis is not valid
whereas for the very long waves the strip theory fails.

Recently [18] experimentally obtained the magnitude and the distribution
of wave exciting forces on a segmented tanker model and showed this difference
between the theoretical predictions of this type and the experimental measure-
ments for the short wavelengths as illustrated in Figs. 7 and 8. The good

66



1



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STRIP THEORY




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Figure 7
Sectional VJaves Forces W _ i qq



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67



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agreement is due to the fact that the longitudinal ship motion amplitudes at
high and very low frequency ranges are insignificant. But in case of spring-
ing and mooring problems, where the high and very low frequencies are involved
respectively, this kind of engineering approach may not be satisfactory.

Use of the Haskind-Newman relationship in calculating the wave exciting
forces is a useful method so far as avoiding the solution of the diffraction
problem, while calculating the forces and moments created by the diffraction
of waves. So the approach is, in a way, equivalent to the solution of the
wave diffraction problem. The main difference is due to the evaluation of
the Haskind-Newman relationship. The original approach requires that in the
evaluation of diffraction force (moment) the perturbation potential <^ is the
three-dimensional potential satisfying the same state equations and radiation
condition as the diffraction potential, whereas in "strip theory" only the
two-dimensional potential is available which satisfied different state equa-
tions and radiation condition.

Newman [42], however, proved that for the high-frequency range this
difference does not cause any significant error.

For longer waves McCreight [43] recently developed a relationship similar
to that of Haskind-Newman for the computation of wave exciting forces.

As the numerical evaluation of the wave excitation by the Haskind-
Newman relationship is not difficult, this approach should be preferred in-
stead of the previous approach as it eliminates the somewhat arbitrary choice
on the relative motion between the ship and the waves.

It should, however, be mentioned that this approach also fails in very
long waves because of the breakdown of the strip theory. For such long waves
the approach adopted by [11] is preferrable as it includes the effect of wave
deformation in an approximate way.



69



Approximate Calculation of the Wave Force

In the strip method, when calculating the diffraction force, one uses
approximations in which the orbital- velocity of the regular wave is represented
by the value in the mean draft. As to the circular cylinder subjected to trans-
verse waves, there is exact solution of [31] and [19] compared them with the
approximate solutions with reference to the force Z which is proportional to
the orbital acceleration of the wave and the force Z. which is proportional
to the orbital velocity. Fig. 9 shows the comparison whereas Fig. 10 shows
the similar calculation for a circular cylinder subjected to longitudinal waves.

In case of transverse waves, there are considerable differences in the
regions of high frequency but in case of longitudinal waves there is no
noticeable difference. However, in case of longitudinal waves, final conclus-
ion cannot be drawn at present since it includes the problem of the three-
dimensional effect.
Critical Revi ew of Strip Theory

It must be mentioned that the usefulness of the strip theory approach,
especially for longitudinal motions and associated predictions, has surpassed
the imagination of many theorists and engineers.

Predictions for the transverse motions and the associated effects, how-
ever, were not so good because of the difficulties arising from the modeling
and computation of the roll associated parameters. But recent efforts of
[28] have been proven to be very successful.

The difficulties in connection with the transom stem (or more generally
blunt-ended) ships have been removed by the inclusion of end effects.

In utilizing the results of strip theory one should always remember
that this approach is valid as long as:

1) The vessel is slender, smooth and the geometrical variations in
the longitudinal direction are gradual, and not abrupt.

70



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71



2) Frequencies are high

3) Forward speed (or Froude number) is low.

If these conditions are not satisfied, experimentally-determined transfer
functions should be used for the prediction of ship motions in an irregular
seaway.

The accuracy of the strip method is to be investigated with regard to
longitudinal ship motions, namely, heaving and pitching. The items which
should be studied are listed as follows:

1) Three-dimensional effect

2) Non-linear effect

3) Approximation by Lewis form

4) Viscous effect

5) Approximate calculation of wave forces

6) Displacement effect

Thus, from the mathematical point of view the limitations and inaccuracies
of the classical methods for the ship wave problems are for the following as-
sumptions :

1) Viscous/wave interactions

The interactions between the viscous effects and the gravity waves
are assumed to be small so that potential flow theory can be used for pre-
dicting ship motions.

In case of longitudinal motions (i,e. heaving and pitching) the level
of wavemaking is higher than in case of rolling motion and it can be justify-
ably said that the viscous effect caused by the bilge-keel etc. occur seldom
except for the region in which the frequency is high [33] .

Considering viscous effect another important effect is that of dynamic
lift [30], [40].



72



In case of longitudinal motions there are two cases which can be
considered as dynamic lift, namely,

a) One is that proportional to the pitching angle

b) The other is that proportional to the ratio of the velocity of
heaving to the advance speed.

The latter may contribute considerably to the damping force.
2 ) Linearization of free-surface conditions

It is assumed that the wave slopes of the incoming as well as ship
generated waves are sufficiently small so that the non-linear free-surface
boundary conditions can be replaced by the linearized condition.

Ref [4 5] dealt with hull-shape non-linearity and showed that the
calculated amplitude of motion differs considerably from that of the linear
calculation. Ref [44], by calculating the second-order approximation of the
diffraction problem regarding the two-dimensional body, showed that when the
period of motion is short there are considerable differences from the first-
order approximation. Also Ref. [30] discussed about the hydrodynamic force
which is proportional to the product of the perturbed velocity due to the
forward velocity of the vessel and that due to ship motions and calculated
its effect on the motions as shown in Fig. 11 and IZ

Ref. [29] concludes that the ship motion calculations must take into
account the non-linearity effect which is extremely important for slamming and
deck wetness.

In the strip method, all the perturbed potential which are more
than the square are neglected.

In the slender body theory, aside from the effect already stated in
the non-linear effect as discussed above, the existence of the effect of
the hydrodynamic force which is proportional to the product of the perturbed



73




•H 2>



74



velocity of the forward motion, and the oscillatory displacement are considered.

Ref, [29] call this as "displacement effect" and calculated the effect on the

motions.

Figs. Hand 12 show the three-dimensional effect, non-linear effect

and displacement effect for the motion of series 60, C = 0.70. The respec-

B

tive effects are significant, but the agreement with the experimental value
in the totally corrected calculation is still unsatisfactory.
3) Small amplit u de ship motions

Here it is assumed that the unsteady body displacements are small so
that the hull boundary condition can be satisfied at the mean position of the
ship.

Large-Amplitude Ship Motion

In linear ship-motion theories, it is assumed not only that the free-
surface conditions can be linearized, but also that the ship displacements are
small relative to the ship dimensions . The exact body boundary condition then
can be approximated by satisfying it at the mean position of the hull. However,
ship motions are not always small. In fact, they can be on the order of magni-
tude of the ship dimensions even in typically moderate sea conditions.

So a method should be developed for predicting large amplitude ship
motions. This is a difficult non-linear problem both for the boundary con-
ditions at the hull or at the free-surface. Non-linearities resulting from
the large amplitude rolling motion influence both the hydrodynamic problem and
the equations of motion.

In the hydrodynamic problem, the use of average wetted surface is no
longer justified as the geometry of the wetted surface changes significantly
during one cycle of motion. This means that the added inertia is a function
of the angular position and systematic experiments conducted by [37 ] indicate
that the added inertia of rolling mota n varies with the amplitude of motion.

75



Recently, [^6] used a quasi^steady treatment and calculated the
hydrodynamic properties at different angles of heel. His treatment may be
useful at very low frequency range.

In the dynamic problem, two additional complications arise:

1) The effect of non-linearity of rolling motion is not confined
to the equation of motion of this mode alone, but also makes the coupled sway-
roll-yaw equations non-linear.

2) The existence of the position-dependent added inertia gives
rise to the existence of additional velocity-dependent terms which may take
both positive and negative values. Some of these problems have been considered
already by [24], [21].

However, if it is assumed that the frequency of ship motions is suf-
ficiently small (which means that the slope of the body generated waves will
also be small) and that the slope of the Incident waves is fairly small, then
it may be valid to linearize the free-surface conditions even for large body
displacements. There are some occasions when the oscillation frequency is
low, e.g., ship motions in following and quartering seas, roll motions in
beam seas, pitching and heaving in long head waves.

Chapman [5] is developing such a method (JSR vol. 23, No. 1
also) .

[ 3 ] has developed a three dimensional numerical method for predict-
ing ship motions which solves the complete three-dimensional hydrodynamics
problem and satisfies correctly all forward speed effects.

The hydrodynamics problem is solved by distributing three-dimensional
oscillating (Kelvin) sources (which satisfy the linearized free-surface boundary
condtion) on the wetted hull surface. The strength of these singularities is
obtained by solving the hull boundary condition. It is assumed that ship
motions are small enou gh that the hulP boundary condition can be satisfied at

the mean position of the hull,

76



30



2.0



Fn =0



Heave



o(3-dimension1
a^2-clinDension]



Ratios of Kim ond HavekxkN
are corrected so that they
become unit when ar,-»<x) /



Pitch

A(3-dimensionol)

A(2-dimensional)




Hovelock( Sphere)



caf=-^L



' g

Figure 13
Three-dimensional Effect (Added Mass)



77



Fn=0



Heave
b(3-dimensional)



Piteh
B(3-dirnensional)



Takogi (thin ship)
(L/B-7,B/(j -25)




Hovelock
(spheroid. L/B« 8)



Vossers



Figure 14



Three-dimensional Effect (Damping Coefficient)



78



Fig. 15a and I5b show some results of added mass and damping co-
efficients which, next to the exciting forces, are the most important hydro-
dynamic ingredients needed in predicting ship motions and wave induced loads.

It is seen that Chang's predictions agree well with the experimental
results throughout the frequency range whereas the strip-theory results only
agree well with the experimental values in the high frequency range.

A complete evaluation of the ship motions by Chang's method is now
in progress at DTNSRDC.

Chapman [ ^] has shown that by applying slender body theory, the
three-dimensional problem of a ship oscillating in the lateral modes of
motion (sway and yaw) can be reduced to a series of transient unsteady two-
dimensional flow problems in the transverse plane.

Fig. 16 shows some of his results. In some cases Chapman's results
are even more accurate than Chang's because Chapman takes into account some
non-linear free-surface effects.
^) Hull form approximation

Exact hull boundary condition is replaced by some approximate con-
dition and so the theories are called, thin-ship theory, strip theory, slender
body theory, etc.

Three-dimensional effect

In the strip method, the three-dimensional ship hydrodynamics problem
is replaced by a summation of two-dimensional sectional problems and the for-
ward-speed effects are only satisfied approximately. The strip theory provides
good results for heaving, pitching motions in moderate seas and moderate ship
speeds for most conventional hull forms; however, the method gives inadequate
results for low frequencies, higher ship speeds, local pressure distributions
and for sway and yaw motions. The forward speed limitations is the most severe
restriction for naval applications.

79



STRIP

D MEASURED (WHOLE MODEL)

O MEASURED (SUM OF SECTIONAL MODEL)

• NUMERICAL METHOD (CHANG. NP 1977)





FREQUENCY, GJ (sec )



FREQUENCY, OJ(sec"



a)-



Pitch Added-Mass and Damping Coefficients at F =0





12 3 4 5 6

FREQUENCY. a.(sec M FREQUENCY, u,(sec^)

b) - Yaw Added-Mass Coefficients
Figure 15
Added Mass and Damping Coefficients for Series 60
(B = 0.70)



80



In case of advancing ship [29] found three-dimensional correction
factor for each coefficient of the equations of motion of the ordinary strip
method by the thin ship theory of [ ] , In addition they corrected the co-
efficients of the equations of motions using the assumption of the slender
body to which normal ships are subjected. The calculated results of the
said three-dimensional effect, together with the results as mentioned above,
are illustrated in Fig. 1 and 2 which show that the three-dimensional effect
is rather significant. Thus, it is necessary to consider this effect in
ship motion calculations.

Also one should be careful in obtaining the longitudinal derivations
of the two-dimensional (sectional) added mass and damping coefficients as
these may cause significant errors in the calculated force distribution for
a ship with forward speed. One should therefore adopt a smoothing procedure
before the numerical derivation.

For critical reviews of strip theories one should refer to [30 ] and
[48].

In order to overcome some conceptual and practical shortcomings of
the strip theory various attempts have been made to include the effects of
three-dimensionality .

However, calculations have shown that these corrections did not pro-
vide an improved accuracy. In fact, in most of the cases the predictions
become worse when the three-dimensional corrections were applied. Only the
technique proposed by [10] may be acceptable. This method proposed an interest-
ing quasi-three-dimensional method which, however, did not receive wide ac-
ceptance because of the more complicated calculations needed.

After investigations of all the topics as mentioned above, [29] made the
following observations:



81



EXPERIMENTAL
DATA



NUMERICAL
METHOD




FREQUENCY

Figure 16



Flat Plate Survey Added Mass Coefficients



82



1) The study of the three-dimensional correction is fotind to be
rather significant. Therefore practical correction factors must be developed
for prediction of ship motions.

2) The effects of the dynamic lift on the hull should be examined
experimentally and theoretically.

3) Investigations should be performed with regard to non-linear effect
including the displacement effect, etc. Both theoretical and experimental
studies should be the basis of this investigation.

Further Investigations

1) Combined action of steady and unsteady excitation:

The equations of motion, which are now in use, are valid in the
frequency domain, and therefore, if there is also a steady force, for example,
wind, rudder and drift forces, acting on the ship, these equations are no
more useful.

2) Low frequency motions:

As it is well-known, even for the heaving motion, the results for
low frequencies may not be realistic. A knowledge in the low frequency range
is generally very important, especially for the prediction of lateral motions
(i.e. sway, roll and yaw) in following waves, because of intact and course-
keeping stability of ships. From its basic assumption it is clear that the
strip theory may not be suitable for this purpose as t hree-dimensional effects
as well as viscous effects will be significant.

3) Impact pressures:

Determination of pressures during slamming is important in avoiding
bow damage and in determining the hull bending moment and springing. The theo-
retical results are still not satisfactory because of various simplifications
made in the problem formulation.

Since' a solution which should consider the effects of compressibility,

83



viscosity and three-dimensionality is very difficult one may split the problem
in a number of stages such as contact, immersion and cavity formation (i.e.
formation of inner free surface and spraying) and then considering each stage
with different assumptions.

4) Sea loads on a vibrating ship:

As it is known, the presently available seakeeping theories are
valid for rigid body motions . Therefore, as the frequency increases, the
wave damping vanishes and the natural frequencies obtained by using so-deter-
mined added masses does provide correct results,

5) Wave forces on discontinuous structures

In the present theories it is assumed that the change in the body
geometry is gradual. If, however, there are abrupt ends as in the case of
a barge, the flow around the ends will be different from a potential flow
due to vortex shedding. As a result, the forces exerted by the fluid on the
body may differ considerably from the results obtained from potential theory.
For these types of forms also the effects of viscosity should be included in
the calculations.

6) Interaction problems:

When there is more than one body and each is in close proximity to
the other, the flow field around each will differ from the case where the
other bodies are not present. Present methods of super-position of the flow
fields can provide reasonable approximation, provided the distance between
the bodies is large compared to the characteristic dimension of the largest
body. For configurations where the bodies are close, interaction effects
ought to be considered more carefully.



84



1 Betts, Bishop, Price, Trans. RINA, 1977

2 Beukelman, W. , "Pitch and Heave Characteristics of a Destroyer" ISP, 1970

3 Chang, Poc. 2nd Numerical Hydrodynamics Conference, 1977


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