submerged)obstacles)and)linear)wave)propagaon)submerged)obstacles)and)linear)wave)propagaon)...
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Submerged Obstacles and Linear Wave Propaga8on Megan Golbek1, Yan Sheng2, Ravi Shankar3, Tucker Hartland3
Introduc8on
Submerged obstacles are constructed to create waves for recrea8onal surfing, harbor protec8on, and as a line of defense from tsunami waves. Accurate numerical models can approximate how obstacles affect waves. This study was part of a larger project that included linear and non-‐linear numerical methods to approximate the wave’s transmission coefficients aRer propaga8ng over obstacles. My team and I constructed the linear numerical method. ARer construc8ng our model, we checked our model’s validity by comparing our numerical solu8ons with experimental data (figure 1 & 2) and analy8cal solu8ons (figure 4) and the effec8veness of the one and two obstacle, shelf, and stairs scenarios.
Methods
Results Figure 1: Single obstacle configura8on Figure 2: Shelf configura8on
Figure 3: Plot of numerical solu8on two obstacles Figure 4: Plot of numerical solu8on shelf configura8on
Discussion • Our numerical solu8ons
• were accurate when compared to our Analy8cal solu8ons for the shelf and one obstacle configura8on.
• accurately model the physical behavior of this system.
• overes8mates when non-‐linear effects begin and this is caused by our model being linear and neglec8ng non-‐linear effects.
• When a wave travels over an obstacle, a reflected wave is sent backwards.
• This explains why the wave looses energy as it travels over each barrier.
• We found that the two obstacle configura8on dampened the wave the greatest
because of the amount of reflected waves (figure 3).
• We can predict that mul8ple obstacles could cause an isolated wave to dissolve into smaller waves.
Future work The simple linear model was accurate un8l non-‐linear effects of the solitary wave took effect For future work, • analy8cal solu8on for the two-‐obstacle configura8on must be found. • a new non-‐linear model must be constructed. • new model would be more accurate for larger rela8ve wave heights. • numerical method for predic8ng how mul8ple barriers effect an isolated wave. • mul8ple barriers can be tested in a wave flume.
Cita8ons • Bryant, E. (2008). Tsunami: the underrated hazard. Springer. • Chang, H. K., & Liou, J. C. (2007). Long wave reflec8on from submerged trapezoidal breakwaters. Ocean Engineering, 34(1), 185-‐191. • Cho, Y. S., & Lee, C. (2000). Resonant reflec8on of waves over sinusoidally varying topographies. Journal of Coastal Research, 870-‐876. • Shankar, R., Sheng, Y., Golbek, M., Hartland, T., Gerrodege, P., Fomin, S., & Chugunov, V. (2015). Linear long wave propaga8on over
discon8nuous submerged shallow water topography. Applied Mathema-cs and Computa-on, 252, 27–44. doi:10.1016/j.amc.2014.11.034
Acknowledgements The authors are supported in part by NSF award DMS-‐0648764, and the Undergraduate Research Opportuni8es Center of California State University, Monterey Bay.
Calculate values at ar8ficial boundaries
• Backward Finite differencing scheme for leR region and Forward Finite Differencing scheme for right region.
Sa8sfy given boundary condi8ons
• First 8me with upwind scheme.
• Second 8me with Beam-‐Warming.
Calculate intermediate
values
• Lax-‐Wendroff method with center differencing.
-10 -5 5 10x
-1.5
-1.0
-0.5
0.5
1.0h
h1
h0
xL xRx=0
l1
h0
l0
d0h2
2 01xM
Ini8al wave profiles for the one and two obstacle configura8on. *All approxima8ons were calculated using Mathema8ca 9.
-10 -5 5 10
-1.5
-1.0
-0.5
0.5
1.0
Riemann
Difference
Analytic
Conclusion • We developed simple and adaptable numerical method to approximate linear
long waves traversing over submerged topography • Our model
• is successful in modeling a variety of ocean floor topography • accurately simulates varying water depths for waves with small
amplitudes rela8ve to water depth • We solved the one-‐dimensional linear shallow water equa8ons over submerged
topography • We verified our model against
• data found in literature • against analy8cal solu8ons for the shelf and obstacle configura8on
• The two obstacle configura8on was shown to be the most effec8ve • This study was published in the journal: Applied Mathema-cs and Computa-ons
Transmitted WaveAmplitude HcmL
Incident WaveAmplitude HcmL
2 3 4 5 6
3
4
5
6
7
8
Experimental
Numerical
H=22.2cm
Plots of transmiged wave amplitudes against incident wave amplitudes, in cen8meters.
Plots of transmiged wave amplitudes against incident wave amplitudes, in cen8meters. The obstacle found in literature was a triangle so we approximated with one region (experimental line) and with 5 regions(5-‐regions line)
Plots of our numerical solu8ons with analy8c solu8ons at 8me t=5. The green line is the ocean floor. Please note that the three lines are on top of each other so they may not all be visible.
Plot of our numerical solu8on in Riemann variables for two-‐obstacle configura8on at 8me, t=11. Note the rippling due to mul8ple transmissions and reflec8ons.
1-‐California State University, Monterey Bay, Seaside, California 2-‐Emory University, Atlanta Georgia 3-‐California State University, Chico, Chico, California
Transmitted WaveAmplitude HcmL
Incident WaveAmplitude HcmL
7 8 9 10 11 12
7
8
9
10
11
12
13
Experimental
Numerical
5-Regions
H=25.0cm
h1
h0
xL xRx=0
l1
h0
l0
d0
h2h3h4
4 0123
xB1xB2 xM
l2
Equa8ons • We used , the Linear Shallow Water equa8ons
to model our waves where, g is the accelera8on due to gravity and
models the ocean floor. • For our ini8al profile, we used the Gaussian func8on,
Where is centered at (the ini8al distance from, , the wave amplitude to the boundary) with ini8al condi8ons: , where is the characteris8c wave amplitude and is the characteris8c fluid velocity. • ARer crea8ng our ini8al profile, we iterated our code to approximate each 8me
step using the Linear Shallow Water equa8ons and schemes described below.
! ! = !!!(!!!!)! !
Solu8ons In a previous Research Experience for Undergraduates, Ravi Shankar found the analy8cal solu8ons for the shelf and one obstacle configura8on. Using these analy8cal solu8ons, my team and I compared our numerical solu8ons to determine the validity of our model (figure 4). We then compared our model with experimental data found in literature (figure 1 & 2). Using the experimental data, we determined a benchmark for our method and the benchmarks for the shelf ranged from .8%(for H=18.1 cm) to 13.8% (H=30.0 cm) of water depth and the obstacle configura8on ranged from 9.7-‐18.8% of H, Where H is the depth of water in front of and behind the shelf/obstacle.