Modelo de Sustitución de Importaciones

The linear dispersion relation predicts a wavelength o f 0.996m for a 0.8s period wave in 0.49m water depth. Such a wave lies on the cusp between a deep water conditions (d/X>0.5) and intermediate depth conditions (0.05<û?/à<0.5). Now, water waves are inherently unstable when kd>l . 363 (Benjamin, 1967). The instability can result in initially uniform waves breaking down into a highly modulated wave as energy is transferred to sideband frequencies about the primary wave frequency. In a water depth o f 0.49m , the inequality predicts that a wave will be unstable if À<2.305m or equivalently if r < 1 .3 0 s assuming the linear dispersion relation. Clearly a 0.8s wave will be susceptible to these sideband instabilities. However, small amplitude waves are less prone to growth in the sideband frequencies. Lake et al. (1977) observed no instabilities for deep water waves with ka < 0 .\, arguing that for these low energy waves viscous dissipation suppressed the growth o f energy in the sideband frequencies. Thus, to minimise the influence of sideband instabilities, attention is restricted to amplitudes that satisfy the condition ka<Q.l, which corresponds to amplitudes o f less than 0.016m for the 0.8s wave considered. Furthermore, the time-scale required for such modulations to manifest themselves is much greater than that available in the tests.

The aim was to generate wave conditions that were linear or at least near-linear. Figure 4.4 shows the ranges of validity o f several wave theories. From this figure, the wave amplitudes corresponding to

0.05

Stream function theory

0.01

Cokelet exact'

solution \ _{Stokes 2nd order}

0001

:noidol theorj^ Small - amplitude 1 st order

0-0001

0-0005

0-001

0.0002 0.01 d 0.1 0.2

Figure 4.4: The range of validity o f several com m on wave theories (taken from Le M éhauté (1976), figure 7).

the limits o f validity o f the various wave theories can be determined for any given wave period and water depth. For a 0.8s period wave in 0.49m water depth, Stokes first order (linear) theory is only applicable when o<0.003m while Stokes second order theory is applicable when 0.003m <a<0.022m . The generation of a wave in the linear regime was impractical as the very small am plitude wave involved would be easily masked by other effects in the basin. Restricting attention to am plitudes satisfying the wave stability criterion discussed above results in a second order wave, but only weakly second order. Even at the upper limit, when a=0.016m , the ratio of the second order com ponent of surface elevation am plitude to that o f the first order com ponent, given by

3 - idiwh^kd R = ka

4 tanhU r/ [4.3]

is only 5%.

4.1.3 Prelim inary experim ents

The properties o f the design wave condition chosen for the experim ent are sum m arised in table 4.1.

Active A bsorption A ssessm ent

Phase 1 of this study, was the first project to use the U K C R F after the active absorption capability had been brought on-line. This capability was considered beneficial to these experim ents as the shear

Water depth, d 0.49m Wave period, T 0.8s Wave length \ X 0.995m Phase speed, c 1.244m/s d i x 0.492 kd 3.091 0.078 Maximum wave amplitude, Stokes first order theory ^ 0.003m Maximum wave amplitude, Stokes second order theory ^ 0.022m

Maximum wave amplitude, such that ka = 0.\ 0.016m

Wave amplitude, a 0.016m

Wave slope, ka 0.10

Ratio of 2nd order wave amplitude to 1st order wave amplitude ^ 0.051

Ursell number, X^H ! d^ 0.269

^ Calculated from linearized dispersion relation, o)^ = gk tanh kd

^ Maximum wave amplitude that still satisfies conditions for Stokes wave theory (see figure 4.4) ^ Given by ka (3-tanh^ kd) ! (4 tanh^ kd)

Table 4.1 : Wave parameters for design wave.

layers o f the jet current were expected to reflect a proportion o f the incident wave. However, in preliminary tests the wave field was observed to be o f poorer quality with active absorption than without. The desired wave appeared to have short period multi-directional waves superimposed upon it with active absorption that were not present when the active absorption was turned off. A basic assessment was undertaken, generating a single wave condition at normal incidence first with and then without active absorption. The assessment was conducted in still water with no reflective stmctures present, apart from the spending beach, with surface elevation measured at the centre o f the test section on the basin centre-line. The time series in figure 4.5 confirmed the observations that the wave quality deteriorated when active absorption was used.

The source of this short wave energy appears to be the inability o f the absorption system installed in the UKCRF to determine the propagation direction of wave energy incident upon a paddle element of the wave generator. Thus, a paddle element will respond equally to a normally incident wave as to one propagating parallel to the paddle face. In the latter case no absorption is required, however, the response of paddle element generates a disturbance which propagates into the basin. Consequently the active absorption capability was not used in the further phases o f the study.

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