The goals are to visualize, predict and understand detailed hydrodynamics at the fine scales to better understand the larg escale processes occurring on beaches.
Hydrodynamic modeling provides a means whereby we can visualize not only the processes occurring in the swash zone, but also estimate important physical parameters such as fluid velocities, acceleration, bed shear stress, vorticity, pressure gradients and turbulence. A variety of models exist for estimating hydrodynamics in the swash zone. In our studies we utilize the well known RBREAK2 model developed by Nobu Kobayashi and his colleagues (also from the University of Delaware). This model relies on the depth averaged non linear shallow water equations (DNLSWE), a simplification of the governing Navier-Stokes equations, and has been shown to qualitatively (and in some cases quantitatively) reproduce hydrodynamic features observed in the swash zone. The movie below shows a simulation on a 20 degree planar sloping beach using 0.8 s monochromatic waves with a height of 4 cm. The vectors denote the depth averaged cross-shore fluid velocity and for convenience have been placed a fixed distance above the bed in the movie. Note that as the wave approaches the still water line near x = 0.55 m, the wave form steepens and obtains a near vertical face but does not curl over or break. This is just one limitation from the shock capturing method in the DNLSWE.
Note: file is over 13 Mb so it may take a while to play.
Rather than using a depth averaged approach, more sophisticated models rely on the full Navier-Stokes equations. A model known as RIPPLE for free surface incompressible flows has been obtained from Los Alamos National Laboratory and modified for use in the swash zone. The RIPPLE model is solved using a volume of fluid (VOF) technique that does not explicitly track the free surface, but instead calculates the appropriate force balances in each control volume and the flux of water across each of the control volume surfaces. In not using the DNLSWE, the vertical fluid velocity does not scale out of the equations, pressure is not required to be hydrostatic, the model is not restricted to shallow water, and the flow field is depth dependent rather than depth averaged. Presently, model domains are typically several meters in the cross-shore with very small control volumes (3 millimeters in the vertical and 5 millimeters in the horizontal) and time steps of generally 1/100 of a second such that the detailed structure of the boundary layer, shear stresses and breaking processes can be calculated. The movie below shows a RIPPLE simulation using the same conditions for the RBREAK2 simulation above. Note now that as the wave approaches the still water line, the wave form curls over as a plunger and then rushes up the foreshore area. The vectors are plotted at every 20th in the horizontal and every 10th in the vertical such that much more detailed structure is calculated than is shown. It is clear from the simulation that the vertical velocity can be large especially during the breaking process suggesting that the RBREAK2 model over-simplifies this process.
Note: file is over 22 Mb so it may take a while to play.
Both models were validated using data from the Large scale Sediment Transport Facility (LSTF) at the USACE Coastal Hydraulics Laboratory in collaboration with Ernie Smith. Both monchromatic and irregular waves were run. FIGURE 1. shows a short time series form the monochromatic wave case moving from offshore (top) to onshore (bottom). Red is the in situ surface elevation measurement, black is from the 1D model and blue from the 2D model. As can be seen, both models perform quite well in predicting the free surface elevation at these locations.
FIGURE 1. Comparison for monochromatic waves between the 1D (RBREAK), 2D (RIPPLE) and actual data for surface elevation at three locations.
In addition, the RIPPLE model has been used to investigate swash processes using the data from the LSTF. FIGURE 2. shows horizontal swash velocities (outer swash - top; inner swash - bottom) for the monochromatic (left) and irregular (right) wave case. The data is shown in grey and the model output in black. In most cases, the model performs well in predicting hte measured swash velcoity. The measurements were collected using ADV's which may have some difficulty in the thin, potentially aerated, swash flow. Based on Correlations and signal to noise ratios, the data has been culled as per manufacturers recommendations. The missing data is represented by the gaps in the grey time series. Clearly, there is little validation data as the upper swash is approached.
FIGURE 2. Horizontal swash velocities for monochromatic (left) and irregular (right) waves from the outer (top) to inner (bottom) swash. Measurements are in grey and model predictions in black.
Mean flows for the simulations for both days are shown in FIGURE 3. The mean was taken by averaging a given cell only when it was occupied by fluid so the averaging is not biased by empty fluid volumes. In both instances, mean flow patterns show a dominance of shoreward motion in the upper water column near the propagating bores and small values of mean offshore flow below. It is also evident that an undertow is predicted by the model denoted by the 0.01 to 0.02 m thick region of offshore-directed flow landward of cross-shore locations x = 4.0 and x = 4.5 for July 2001 and June 2002 respectively. We find that the undertow feature extends up into the seaward portion of the swash zone and shows that the feature of uprush overriding backwash (observed in Figure 3) has some persistence.
FIGURE 3. Mean cross-shore velocity fields for monochromatic (top) and irregular (bottom) waves. Colorscale is velocity in m/s.
Local fluid accelerations are thought to alter the boundary layer structure due to the potential for streamline contraction [Nielsen, 1992] or dilation [Hanes, 1986]. In addition, recent work has suggested that local fluid acceleration parameterizations can improve sediment transport predictions (e.g. [Drake, 2001]). While we are not concerned with sediment transport in this study, we can still investigate the existence and magnitudes of fluid acceleration predicted by the model in these regions (based on second order differences). Note that care was taken in calculating the acceleration so that only one-sided differences were used at the first and last instance of a swash cycle. In this way, the acceleration is not biased by differencing with a zero velocity value when the location is unwetted. The cross-shore local, and convective accelerations for the June, 2002 simulation are shown in FIGURE 4. for locations at x = 3.95 m, x = 4.29 m and x = 4.53 m all at 4 mm above the bed. It is clear that local fluid accelerations (analogous to those that would be measured with a fixed instrument in the field) occur under broken waves. In addition, short-lived, large onshore-directed local fluid accelerations are also observed near the shoreline in the inner surf/swash zone where fluid collisions occur. In contrast, large shore-directed local accelerations are not predicted at the locations shown when they are landward of the initial start point for a swash cycle, consistent with ballistic motion theory for swash (e.g. [Hughes, 2004]). The model also predicts backwash local deceleration that has been observed in field data [Puleo, 2003].
While convective accelerations have been largely ignored in the inner surf and swash zones with the exception of [Jensen, 2003], it is evident from this model simulation (FIGURE 4) that they exist and can be the same order of magnitude as the local acceleration. Therefore, if acceleration is thought to affect sediment transport or is important to understanding of swash flows, both the local and convective (i.e. total) acceleration should be considered.
FIGURE 4. Horizontal velocity (top), local acceleration (middle) and convective acceleration (bottom) at three locations (upper swash to left) for the irregaular wave case.
Shallow water theory predicts that the pressure at any location in the fluid is obtained via the hydrostatic approximation and hence solely dependent on depth below the free surface. Modeled pressure and hydrostatic approximation data from 3 cross-shore locations for the irregular wave simulation show that this approximation is often valid (FIGURE 5) with greater than 94 %, 84 % and 69 % of the estimates having smaller than 20 % error at x = 3.95, x = 4.29 and x = 4.53 respectively. Yet, the fact that large deviations are occasionally predicted near the bore and collapse point suggests that variations from hydrostatic pressure cannot be totally ignored in the swash zone.
FIGURE 5. Modeled (black) and hydrostatic (gray) pressure (left panels). Modeled and hydrostatic pressure gradients (right panels; same color format).
For more information on this topic see Puleo et al. ICCE proceedings (2002), or Puleo et al. Coastal Engineering Journal (submitted).
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