Concreto Asfáltico

Conventional strip theory (such as that of Salvesen, Tuck and Faltinsen [84]) developed initially from a need for analysing the seakeeping of conventional boats, and as such all authors agree that it is not valid for low frequencies or high speeds (although curiously many of the same authors list F r=O(1) as one of the starting assumptions). Typical comments include: “strip theory is invalid at low frequencies of encounter” [73], “This is the short wavelength theory”

[102], “The conventional strip theory is deficient not only for low frequencies, but also for high speeds” [73], “Strip theories have no justification for high speed” [26], “The strip theory as described does not have any three-dimensional effects representing interactions between sections nor any forward-speed effects on the free-surface condition. It is known to be incorrect in the low frequency limit” [102].

Often cited is the inconsistency in conventional strip theory of applying forward speed correc- tions on the hull boundary condition but not on the free surface boundary condition, although some versions (for example Ogilvie and Tuck [77]) do contain higher order terms on the free sur- face. Faltinsen [28] states “for high Froude numbers unsteady ‘divergent’ wave systems become important. This effect is neglected in [conventional] strip theory...”. By this he was probably referring in particular to the wedge shaped three-dimensional wave pattern characteristic of the caseτ =U ω

g > 14, as opposed to the infinite lateral propagation of waves of the steady state pe-

riodic solution for a two-dimensional strip. All the high speed theories presented below (sections 4.3.4, 4.3.5 and 4.3.6, as well as 4.4), by inclusion of forward speed terms in the free surface boundary condition, contain the divergent wave system that Faltinsen was referring to.

Newman [74] questions the use of forward speed corrections altogether in conventional strip theory. He gives an example where (at a Froude number of only 0.2) some of the hydrodynamic coefficients are predicted worse if the forward speed terms are omitted while an equal number are predicted better. The findings of the current research are that this is by no means an isolated occurrence. Results from the conventional strip theory programHYDROS presented in chapter 6 of this thesis suggest in some cases that motions at high speeds are only predicted accurately presumably because they resemble the zero speed result, but that incremental changes in the magnitude of resonant peaks as speed is increased are not necessarily in the right direction.

It is typically the coupling between heave and pitch motions that is predicted most poorly by strip theory (see for example [86], [102], [74], [11]), and given that coupling effects are introduced as a result of the hydrodynamic asymmetry caused by forward speed [32] the use of forward speed corrections must further come into question.

In spite of this Faltinsen [28] states that conventional strip theories, although theoretically questionable, are “still the most successful theories for wave-induced motions of conventional ships at forward speed.” This is perhaps as much a comment on conventional ships as it is on conventional strip theories.

In relation to the low frequency limitation, Newman [73] states that conventional strip theory makes “reasonable predictions” for long wavelength and zero speed, although “An explanation of this fortunate situation is the disparity between the natural frequencies and the wave fre- quencies where the exciting force and moment are significant.” In other words, if the encounter frequency is near or above resonance the accurate prediction of dynamic response characteristics is unimportant if the forcing is negligible, while significantly below resonance hydrostatic forces dominate and one can get away with poor estimates of added mass and damping as long as

the exciting force can be calculated with reasonable accuracy (which turns out to be possible since this is usually dominated by hydrostatic and Froude Krylov components: Gerritsma [32] states that, except for short incident waves, the diffraction force is small compared with the Froude-Krylov force, and “Even gross errors in the computed diffraction forces may be masked in the predicted total wave force.”). Newman [73] also shows, by considering typical relation- ships between displacement, waterplane area and length, that for conventional boats “resonance will occur in combination with significant wave excitation only if the frequency of encounter is substantially greater than the wave frequency”, which of course is the case only at high speeds. In the short wavelength regime the three-dimensional wave pattern in the near field of the hull is equivalent to the strip theory solution (except perhaps near the bow and stern), as can be shown by consideration of the high frequency limit in Newman’sunified theory (described in section 4.3.3), and strip theory in this case is entirely appropriate.

Actual estimates of limits of Froude number or dimensionless frequency for which strip the- ory is valid vary considerably from one author to another, and depend on the type of vessel, the information sought, and the criteria for judgement. For example Gerritsma [32] suggests F r <0.4 for reasonable predictions of damping for conventional ships, while in the same paper he cites a case where some motions were well predicted atF r= 1.14. Gerritsma also investigates wavelength effects by comparing the strip theory predictions of added mass and damping distri- butions at zero forward speed with predictions from a three-dimensional panel method and with experimental results. He shows some significant discrepancy at a frequency ofω

q L

g = 1.9, but claims very little difference at higher frequencies. Generally speaking, because of cancellation of opposing errors, overall motions tend to be better predicted than individual coefficients or local forces.

Effect of steady wave pattern

Inclusion of steady wave pattern effects is considered important both at high speed (“The in- teraction between the unsteady and the local steady flow is important at high forward speed.” [28]) and for full bodied hulls (“There is growing evidence that the influence of the steady-state velocity field is important, and the degree of completeness required to account for the steady field depends on the fullness of the ship.” [74]). It is in these two situations that significant waves are generated by the steady forward motion of the ship.

Opinion differs regarding the importance of the steady wave system. Newman’s [74] view is that “In the perturbation hierarchy any attempt to analyse these unsteady motions must be preceded by a solution of the basic steady-state flow”, while Yeung and Kim [102] say “In as much as the steady-state potential φs is not immediately amenable to reliable numerical description even with present state-of-the art computational techniques, and that there is already considerable amount of complexity in tackling just the homogeneous equation of (2.7) alone, it is a little premature to consider the inclusion of these interactive terms at this time”.

Interaction between steady and unsteady wave patterns should in particular be considered however in the calculation of added resistance in waves [28].

Steady wave effects fall into three main categories.

First is the question of linearisation of the free surface. This is only a problem in frequency domain theories, in which the steady and unsteady potentials must be solved separately. The free surface boundary condition with interaction includes cross products of the two potentials. Since the two potentials are not necessarily of the same order it is difficult to justify which, if any, of the terms should be discarded in a linerised form of the boundary condition. In a time domain theory the steady and unsteady problems can be solved simultaneously, and use of a linearised free surface boundary condition will automatically distinguish which is of higher order.

Second is the calculation of the so called ‘mj-terms’ in the body boundary condition. The boundary condition on the hull for the potential corresponding to thejth mode of motion (j= 3 for heave andj= 5 for pitch) can be written in the form ∂φj

∂n =iωnj+U mj, wherenjis thejth component of the unit normal,m3 =−∂

2_φ

s

∂z∂n, m5=−U n3−xm3, (and the othermjs similarly defined) and φs is the steady potential. In the simplest case, corresponding to the slender ship assumption of strip theory,φsis approximated as−U x(the free stream flow), and all interaction with the steady potential disappears. The next level of sophistication is to use the double body flow as an approximation to φs (as was done in the three-dimensional theory of Nakos and Sclavounos [70]), and finally the full calculation of φsmust take into account the steady wave system generated by the forward motion of the boat. A particular difficulty associated with calculating the steady interaction term ∂2_φ

s

∂z∂n is the fact that it is singular at sharp corners because the normal direction changes abruptly (for example at chines), and may also be very large at the intersection between the free surface and the hull where the latter is not vertical (see [26], [28]). Again the problem can be avoided in the time domain using a ‘body-nonlinear’ approach in which the body boundary condition is satisfied at the exact body position at each time step: “This approach obviates the need to consider the troublesome m-factors, and the ships motion may be prescribed arbitrarily, in principle” [74].

Third is the effect of sinkage and trim. Although this is indirect it may nevertheless be important. An example of significant interactions with motions, particularly as a result of trim, is the lifting effects associated with high speed vessels with ventillated transoms (to be discussed in section 6.5.1).

Other limitation

A limitation not only of strip theory, but of most unsteady motion theories, is the inability to adequately represent the tangential separation of flow at a transom stern in the downstream direction and the resulting atmospheric pressure. (At slower speeds there may be a head of water built up above the streamline separating tangentially from the transom, but if this is of

constant depth the unsteady component of the transom force is still zero.)

Although the theory of Salvesen, Tuck and Faltinsen [84] contains transom terms these are not to be confused with the issue in question here. The transom terms in strip theory give rise to the forces that would in theory be present if the hull were extrapolated rearward. The actual force that would be present with atmospheric pressure at the transom is of course zero, meaning that the added mass force per unit length should exactly cancel with the restoring force per unit length, and the damping should be zero. Faltinsen and Zhao [26] demonstrate that this is not the same as the values predicted by strip theory. However Faltinsen [28] concludes from actual force measurements “...that there must be a rapid change of pressure back to atmospheric pressure in a small neighbourhood of the transom stern.” This means that in terms of motion predictions the error is not critical, but it would make a difference in predicting loads very near the stern.

Conventional strip theory also represents the radiated and diffracted waves simplistically. They are assumed to propagate perpendicularly to the hull, extending an infinite distance later- ally, and none of the three-dimensional characteristics of the wave pattern due to a translating and pulsating source are present. In particular the two-dimensionality of the representation pre- cludes the presence of any waves with characteristics at all similar to the transverse waves in the three-dimensional solution. While these approximations may seem very severe they are actually nothing more than the physical manifestations of the high-frequency low-speed assumptions.

Finally, conventional strip theory assumes small motions and wave heights, and therefore a linear relationship between the two.