Individual Wave Analysis

1. 0 9 5 8 4 -0.5 6 3 7 z (m) -1 2 1 -1.5 -2 30 35 40 45 50 55 60 65 70 75 80 x (m) 0 0 0 10 10 10 E -2 -2 -2 P 10 10 10 -4 -4 -4 10 10 10 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 * * * D D D P P P 0.8 0.6 0.01 onshore 0.4 0.05 * 0.01 P 0.1 0.01 D 0.1 0.01 0.2 0.05 0.05 0.1 0.1 0.5 0.5 0.5 0 0.5 0.5 0.1 0.5 0.05 0.1 0.05 0.01 0.05 0.1 0.01 -0.2 offshore 0.1 0.01 * -0.4 P 0.05 D -0.6 0.01 -0.8 30 35 40 45 50 55 60 65 70 75 80 x (m) Table 1. Deep water and breaking wave conditions measured during the experiment. PRESSURE GRADIENTS OVER A BARRED BEACH Sungwon Shin (shinsu@engr.orst.edu), Takayuki Suzuki, William Boylston, and Daniel T. Cox O.H. Hinsdale Wave Research Laboratory, College of Engineering, Oregon State University http://wave.oregonstate.edu Exceedance Probability Madsen (1974) proposed that large horizontal pressure gradients can cause a “momentary failure” or instability of bed material. The paper suggested a critical value of the pressure gradient for this failure of 0.5, based on common assumptions for fully saturated sands. Skewness and Asymmetry of Pressure Gradients and Accelerations • Motivation/Objectives • Sandbars play an important role in beach morphology and wave climate. It has been proposed that onshore sandbar migration is affected by wave-induced acceleration and offshore migration occurs when undertow is dominant (Elgar et al, 2001). However, onshore sandbar migration is not well-understood. Recent model studies (Rakha et al., 1997; Drake and Calantoni, 2001; Karambas and Koutitas, 2002; Long and Kirby, 2003) include acceleration skewness for either the entire time series or on a wave-by-wave basis to simulate sediment transport. In the present study, cross-shore pressure gradients were observed in a large-scale wave flume over a barred beach and compared with the local wave induced fluid acceleration to get a better understanding of these processes. • The objectives of this study are: • To obtain a synoptic data set of the free surface elevation, pressure gradients, and fluid velocities on a barred beach for a random wave case. • To compare the pressure gradient skewness with the acceleration skewness at each cross-shore location for the entire time series and on a wave-by-wave basis to test sediment transport model assumptions. • To investigate the probability of the pressure gradient exceeding a critical value for “momentary failure” of the sea bed. • Experimental Setup • The experiment was carried out in the Large Wave Flume of O. H. Hinsdale Wave Research Laboratory at Oregon State University (Figure 1). The flume is 104 m long, 3.7 m wide, and 4.3 m deep and can generate a maximum wave height of 1.6 m. The bathymetry approximated the bar geometry observed on Oct 11, 1994, of the Duck94 field experiments at 1:3 scale. Since the local fluid acceleration outside the bottom boundary layer is related to the pressure gradient, many modelers have used fluid acceleration to predict sediment transport. In the present study, the dimensionless pressure gradient is defined as: (a) (a) (b) L 1 L 5 L 8 L5 L8 L1 The exceedance probability (PE) can be used to investigate the occurrence probability of “momentary failure” based on this critical value (Cox et al, 1991). It can be modeled by a Rayleigh distribution: (c) and is compared to the local acceleration measured at the same elevation. Skewness and asymmetry statistics are computed in Figure 3. (b) (c) (d) (d) (e) Figure 3 Cross shore variation of skewness and asymmetry of free surface, pressure, horizontal velocity, ∆P*, and acceleration (du/dt) for entire time series for L 1 – L 9. Arrows refer to details of Figure 4 & 5. Figure 3 (a) and (b) shows that skewness and asymmetry of water surface elevation, pressure, and near bottom horizontal velocities have good correlation. Also, horizontal velocity asymmetry, acceleration and pressure gradient skewness using entire time series were well correlated in most regions (Figure 3 (c) and (d)). (e) (f) L 5 Figure 6 PE of pressure gradients in both onshore (blue) and offshore (red) direction (b – d) and contour plot of PE for L1 – L9 (e) based on unfiltered data . PE for Rayleigh distribution (b – d) is plotted vs ΔP*. Figure 6 (b) – (d) show that extreme values (ΔP*>0.5) are greater in offshore direction than onshore direction at L 5 but typical values (ΔP*<0.5) are smaller in offshore direction than onshore direction. Contour plot in Figure 6 (e) shows that, in most cross-shore location except near the bar location, pressure gradients in onshore direction are larger than those in offshore direction. Individual Wave Analysis How well do these quantities compare on a wave-by-wave basis? Joint distribution is used to examine hydrodynamic quantities for the 354 individual waves (Figure 4 and 5). To consider larger waves which may dominate the total and net sediment transport, the analysis was repeated using only the 1/3 highest waves. Four pressure transducers were mounted in a thin plate oriented lengthwise along the center of the tank, 1 cm from the bottom. Two Sontek ADVs were collocated in the cross-shore direction and measured the velocity at 1 cm and 5 cm from the bottom. Figure 4 Joint distribution of horizontal velocity skewness vs. acceleration skewness (left and center panels). Filled circles are the 1/3 highest waves and red open circles are smaller waves than the 1/3 highest waves. Dashed lines are the best fit using all waves (left panels) and the 1/3 highest waves (center panels); Cross shore variation of r2 value and slope of best fit lines. Open circles are using all waves. Filled circles are the 1/3 highest waves. (right panels) Elgar, et al (2001) suggested that acceleration skewness and velocity asymmetry are related to onshore sediment movement. Figure 3 shows that these values are well correlated except at L 8. However, the joint distribution for individual wave in Figure 4 shows that correlation between the two quantities are low in both cases (all waves and 1/3 highest waves) near the bar and L8. Cross-shore variation of r2 values, in spite of using 1/3 highest waves, still do not improve. L 1 L 1 Figure 7 Time series of unfiltered extreme case indicated in Figure 6 at L 5. This kind of event occurred under broken waves. Magnified figure (left) of extreme event shows that the signal is not noise and several data points were involved in the event. The occurrence of extreme events corresponds with large horizontal velocity signal measured by ADV. This result shows that extreme event of “momentary failure” can be affected by intermittent turbulence. L 5 L 5 • Conclusions • Large-scale, synoptic data of the pressure gradient on a barred beach for random waves were acquired. • The skewness of the local fluid acceleration has a high correlation with that of the pressure gradient, when averaged over many waves. • The correlation of the local fluid acceleration skewness with pressure gradient skewness is low when considered on a wave-by-wave basis, even for the 1/3 highest waves. • Exceedance probability of ΔP * shows that extreme events occur over the bar and correspond to large velocities due to intermittent wave breaking. • Acknowledgements • This work was partially funded by the National Science Foundation under grants • No. CMS-0086571, EEC-0244205 and OCE-0351741. The success of this project • was due to the contributions of many, especially Terry Dibble and Chris Scott. • References • Cox. D. T., Kobayashi, N., and Mase, H. (1991). “Effects of fluid accelerations on sediment transport in surf zones.” Coastal Sediments 91, 447-461. • Drake, T. G., and Calantoni, J. (2001). “Discrete particle model for sheet flow sediment transport in the nearshore.” JGR-Oceans, 106 (C9), 19,859-19,868. • Elgar, S., Gallagher, E. L., and Guza, R. T. (2001). “Nearshore sandbar migration. ” JGR-Oceans, 106 (C9), 11,623-111,627. • Goring, D.G., and Nikora, V.I. (2002). “Despiking Acoustic Doppler Velocimeter Data.” J. Hydr. Engrg., 128(1), 117-126. • Hoefel, F., and Elgar, S. (2003). “Wave-induced sediment transport and sandbar migration.” Science, 299, 1,885-1,887. • Karambas, T. V., and Koutitas, C. (2002). “Surf and swash zone morphology evolution induced by nonlinear waves.” JWPCOE., 128 (3), ASCE, 102-113. • Long, W., and Kirby, J. T. (2003). “Cross-shore sediment transport model based on the Boussinesq equations and improved Bagnold formula.” Coastal Sediments 03. • Madsen, O. S. (1974). “Stability of a sand bed under breaking waves.” Proc., 14th Int. Conf. on Coastal Engineering (ICCE), ASCE, 776-794. • Rakha, K. A., Deigaard, R., and Broker, I. (1997). “A phase-resolving cross shore sediment transport model for beach profile evolution.” Coastal Eng., 31, 231-261. L 8 L 8 Figure 5 Joint distribution of horizontal pressure gradients skewness vs. acceleration skewness (left and center panels). Filled circles are the 1/3 highest waves and red open circles are smaller waves than the 1/3 highest waves. Dashed lines are the best fit using all waves (left panels) and the 1/3 highest waves (center panels); Cross shore variation of r2 value and slope of best fit lines. Open circles are using all waves. Filled circles are the 1/3 highest waves. (right panels) Figure 5 shows that horizontal pressure gradients have almost no correlation with acceleration skewness for individual wave. Therefore, the assumption that pressure gradients are related to the local fluid acceleration is questioned. A time dependent phase resolving model which relies on the local fluid acceleration skewness or asymmetry on a wave-by wave basis may not accurately reflect the basic physics. L 1 L 1 L 5 L 5 Figure 1. Experimental setup and instrumentation at the O. H. Hinsdale Wave Research Laboratory at Oregon State University Data Reduction The data were band passed filtered to remove frequencies higher than 4 times the spectral peak period. The velocity data were despiked using the method of Goring and Nikora (2002) and the pressure gradients were computed using the 4 sensors by L 8 L 8 L 5 u (cm/s) Figure 5 shows that pressure gradient skewness is not uniquely determined by the local fluid acceleration. Therefore, sediment transport models driven by the local fluid acceleration skewness on a wave-by-wave basis may not correctly represent the physical processes. du/dt (cm/s2) Figure 2. Example time series at L5 over the bar