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An eddy viscosity model for two-dimensional breaking waves and its validation with laboratory experiments
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10.1063/1.3687508
/content/aip/journal/pof2/24/3/10.1063/1.3687508
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/3/10.1063/1.3687508

Figures

Image of FIG. 1.
FIG. 1.

Illustration of the two-dimensional wave tank (not to scale) and measurement devices.

Image of FIG. 2.
FIG. 2.

Example time series of measured surface elevation of (a) a focusing group and (b) a group subject to modulational instability. Both groups lead to wave breaking. An offset of 5 cm is applied in the ordinates to separate measurements from different stations. Locations of the wave station relative to the wavemaker are shown in (a), e.g., 1.83 and 2.53 (m), and correspond to those in (b) also.

Image of FIG. 3.
FIG. 3.

Example images of the surface profiles of steep wave crests due to (a) energy focusing and (b) modulational instability. Both steep crests propagate from left to right and develop into plunging breakers subsequently.

Image of FIG. 4.
FIG. 4.

Sketch of a steep wave crest and definitions of local wave parameters. The wave propagation direction is from left to right.

Image of FIG. 5.
FIG. 5.

Definition of the falling crest height, H br for (a) a spiller and (b) a plunger according to Drazen et al. 7

Image of FIG. 6.
FIG. 6.

Temporal evolution of the breaking wave crest height, (negative) height of the wave troughs of (a) a spiller and (b) a plunger due to modulational instability and (c) a plunger due to energy focusing. Triangles represent H C (t)/H C0 ; circles represent H t1 (t)/H C0 , and asterisks depict H t2 (t)/H C0 . Here H C0 is the wave crest height at wave breaking, i.e., at tt b = 0. The relative time, tt b , is non-dimensionalized with the local angular wave frequency, ω b , at wave breaking.

Image of FIG. 7.
FIG. 7.

Evolution of normalized L C as wave crests approach breaking. L C0 is L C at wave breaking, where tt b = 0. Note that the wavelength of the plunger due to energy focusing (solid triangles) increased abruptly after breaking. This sudden increase is corresponding to the disappearance of one zero-crossing point due to the trough in front rising above the mean water level, as shown in (c) of Figure 6. This also causes the horizontal crest asymmetry parameter, R b , to decrease to a very small value, as shown later in Figure 8. In this scenario, the wavelength should be redefined with the horizontal location of the wave trough in front rather than the next available zero-crossing point. The relative time, tt b , is non-dimensionalized with the local angular wave frequency, ω b , at wave breaking.

Image of FIG. 8.
FIG. 8.

Evolution of the wave asymmetry parameter, R b , as wave crests approach breaking. t b represents the time associated with wave breaking. The relative time, tt b , is non-dimensionalized with the local angular wave frequency, ω b , at wave breaking.

Image of FIG. 9.
FIG. 9.

Connection between pre- and post-breaking scales. (a) and (b) show k b L br and ω b T br as a function of S b , respectively; (c) can be used to estimate the breaking crest speed, C br = L br /T br , relative to the local phase speed at breaking onset, C b = ω b /k b . Open triangles indicate results of breaking waves due to energy focusing (essentially plunging breakers), including results of the experiments in Tian et al.;6 solid circles represent breaking waves due to modulational instability (most are spilling breakers). Solid lines in (a) and (b) represent linear least-squares fits and their equations are provided in Sec. IV B. The slope of the dash line in (c) is one, indicating C br = C b .

Image of FIG. 10.
FIG. 10.

Falling crest height, k b H br , as a function of (a) local steepness, S b , and (b) the crest asymmetry parameter, R b . Open symbols indicate results of breaking waves due to energy focusing (essentially plunging breakers), including results of the experiments in Tian et al.;6 circles represent breaking waves due to modulational instability (open circles for plungers and solid ones for spillers).The solid line in (b) represents a linear least-squares fit and its equation is provided in Sec. IV B.

Image of FIG. 11.
FIG. 11.

A comparison of the long time integration of the surface variance, , for the four energy focusing wave groups (Group EF 1 is a non-breaking one, the rest are breaking ones). Here, T = 40.95 s. According to linear wave theory, the integration is proportional to the total energy passing a wave station. Symbols are experimental measurements and solid lines are numerical results. Dash lines indicate the breaking regions of the major breaking events in experiments; Dash-dot lines are for those predicted in the simulations. Note that the shown breaking regions are for the major breaking events only; locations of secondary breaking prior and/or subsequent to the major ones are not indicated.

Image of FIG. 12.
FIG. 12.

(a)–(d) present the surface elevation at wave stations along the tank. Wave group EF 1 is non-breaking, the rest are breaking cases. Solid lines are measured and dash lines are predicted. Locations of the wave stations are provided in the figure, e.g., x = 1.83 m. For clarity, an offset of 7.5 cm is applied to the ordinate to separate the surface elevations at different stations.

Image of FIG. 13.
FIG. 13.

Comparison of the long time integration of the surface variance, , for the wave groups subject to modulational instability. Here, T = 50 s. According to linear wave theory, the integration is proportional to the total energy passing a wave station. Symbols are experimental measurements and solid lines are numerical results. Note that for the BFI 1712 comparison, once breaking ensued, it continued to the downstream extent of the tank. Regions in which wave breaking was observed in the experiments are highlighted with the dash lines. Note that the decrease of the integration (total energy) is mainly due to two factors. One is energy dissipation in breaking events; the other is that, since the integration is limited from 0 to 50 s, not all generated waves upstream have arrived at the downstream wave station yet (see Figure 14).

Image of FIG. 14.
FIG. 14.

(a) and (b) present the measured (solid lines) and the predicted (dash lines) surface elevation at different wave stations along the tank. Locations of wave stations are the same as those shown in Figure 12. For clarity, an offset of 5 cm is applied to the ordinate to separate the surface elevations at different stations.

Image of FIG. 15.
FIG. 15.

Evolution of long-crested irregular waves characterized by the JONSWAP spectrum with H s = 0.12 m, f p = 1 Hz, and γ = 3.3: (a) Comparison between the measured (solid lines) and the predicted (dashed lines) surface elevations at 20 different wave stations along the tank. For clarity, an offset of 10 cm is applied to the ordinate to separate the surface elevations at different stations; (b) Detailed comparison at the 1st, 10th, and 20th wave stations located at x/λ p = 0, 6.92, and 14.62 (from bottom to top).

Image of FIG. 16.
FIG. 16.

Comparison between the measured (solid lines) and the predicted (dashed lines) surface elevations at the 1st, 10th, and 20th wave stations wave stations (from bottom to top) along the tank for the evolution of long-crested irregular waves characterized by the JONSWAP spectrum with f p = 1 Hz. Other physical parameters are (a) H s = 0.14 m and γ = 3.3; (b) H s = 0.12 m and γ = 20; and (c) H s = 0.14 m and γ = 20.

Image of FIG. 17.
FIG. 17.

Determination of a critical local surface slope for wave breaking prediction in the numerical simulations. In the numerical simulations, initial conditions are generated with linear wave theory and surface elevations of five non-breaking, focusing wave groups (i.e., W1G1, W2G1, W3G1, W4G1, and W5G1). Characteristics of these five non-breaking wave groups and surface elevation measurements are discussed in detail in Tian et al. 6,22 Solid symbols: wave breaking cases (numerical failure); open symbols: non-breaking groups. Circles for W1G1; diamonds for W2G1; squares for W3G1; triangles for W4G1; and down-pointing triangles for W5G1. The abscissa, Gain, indicates the multiplier of the magnitude of the non-breaking wave groups.

Tables

Generic image for table
Table I.

Specified parameters for the energy focusing wave groups. Note that EF 1 is a non-breaking wave group while the remainder are breaking groups.

Generic image for table
Table II.

Specified parameters for the wave groups subject to modulational instability. Note that BFI 1710 is a non-breaking group while the rest are breaking groups.

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/content/aip/journal/pof2/24/3/10.1063/1.3687508
2012-03-07
2014-04-25
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: An eddy viscosity model for two-dimensional breaking waves and its validation with laboratory experiments
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/3/10.1063/1.3687508
10.1063/1.3687508
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