Long-wave Runup: Difference between revisions

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<font color=red>'''UNDER  CONSTRUCTION'''</font>
<font color=red>'''UNDER  CONSTRUCTION'''</font>
== Analytical Solution ==
== Overview ==
The goal of this verification test to asses the model performance in simulating nonlinear runup/rundown over a plane slopt. Carrier et al. (2003) presented an analytical solution to the nonlinear shallow water equations over a plane slope for several initial wave forms. Here the analytical solution for a the initial wave form given by leading-depression N-wave is used.


where <math>\eta</math> is the water surface elevation, <math>\rho</math> is the water density, <math>\rho_a</math> is the air density, <math>g</math> is the gravitational acceleration, <math>W</math> is the wind speed, <math>h</math> is the water depth, and <math>C</math> is a constant of integration.
== Initial Condition ==
The bed has a constant slope of 1/10 with the initial shoreline located at x=0. Figure 1 shows the initial water level (is given by a leading depression N-wave (characteristic of the waves caused by submarine landslides). The initial current velocity is equal to zero everywhere.
 
[[Image:Long-wave_Runup_Initial_Water_Level.png|thumb|left|600px|  Figure 1. Initial water level]]
 
<br  style="clear:both" />


== Model Setup ==
== Model Setup ==
A computational grid with constant water depth of 5 m and irregular boundaries is used in order to test the model performance. The computational grid has 60 columns and 70 rows and a constant resolution of 500 m.  
The computational grid has a 3 m resolution for x<300 and increases to 10 m with an aspect ratio of 1.05. The general model parameters used in the simulation are shown in Table 1.
 
Table  1. Model Parameters
{|border="1"
!'''Parameter''' !! '''Value'''
|-
| Time step  || 0.1 s
|-
| Ramp period || 0.0 s
|-
| Drying depth || 0.01 m
|-
| Wall friction || Off
|-
| Mixing terms  || Off
|-
| Manning's  coefficient || 0.0
|}


== Results ==
== Results ==
[[Image:Wind_Setup_Dir0_WSE.png|thumb|left|600px| Figure 1. Computed water surface elevation for the irregular domain with constant water depth.]]
Figure 2 shows a comparison of computed and analytical water surface elevations near the shoreline at 4 different time steps. The goodness of fit statistics are shown in Table 2.
[[Image:Long-wave_Runup_Water_Level.png|thumb|none|600px| Figure 2. Comparison of calculated and analytical water levels for different time steps. ]]
 
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'''Table 2. Goodness of Fit Statistics'''
{| border="1"
!  Time, s !! RRMSE, % !! RMAE, %  !!  R^2 !!  Bias, m
|-
|  160 ||  3.7 ||  3.8 ||  0.999 ||  -0.012 
|-
|  175 ||  6.5 ||  5.9 ||  0.997 ||  -0.113 
|-
| 220 ||  4.6 ||  5.4  ||  1.000 ||  -0.066
|}
* For a definition of the goodness of fit statistics see [[Statistics |  Goodness of fit statistics]].


[[Image:Wind_Setup_Dir0_V2.png|thumb|none|600px| Figure 2. Comparison of computed water surface elevation to the analytical solution for an irregular basin with constant depth.]]
<br  style="clear:both" />
 
A comparison of the calculated and analytical shoreline position are shown in Figure 3.
 
[[Image:Long-wave_Shoreline.png|thumb|none|600px| Figure 3. Time series comparison of calculated and analytical shoreline position.]]


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<br  style="clear:both" />


== References ==
== References ==
 
* Carrier, G.,  Wu, T.T., and Yeh, H. (2003). "Tsunami runup and draw-down on a plane beach", Journal of Fluid Mechanics, 475, 79-99.
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Latest revision as of 22:06, 11 May 2011

UNDER CONSTRUCTION

Overview

The goal of this verification test to asses the model performance in simulating nonlinear runup/rundown over a plane slopt. Carrier et al. (2003) presented an analytical solution to the nonlinear shallow water equations over a plane slope for several initial wave forms. Here the analytical solution for a the initial wave form given by leading-depression N-wave is used.

Initial Condition

The bed has a constant slope of 1/10 with the initial shoreline located at x=0. Figure 1 shows the initial water level (is given by a leading depression N-wave (characteristic of the waves caused by submarine landslides). The initial current velocity is equal to zero everywhere.

Figure 1. Initial water level


Model Setup

The computational grid has a 3 m resolution for x<300 and increases to 10 m with an aspect ratio of 1.05. The general model parameters used in the simulation are shown in Table 1.

Table 1. Model Parameters

Parameter Value
Time step 0.1 s
Ramp period 0.0 s
Drying depth 0.01 m
Wall friction Off
Mixing terms Off
Manning's coefficient 0.0

Results

Figure 2 shows a comparison of computed and analytical water surface elevations near the shoreline at 4 different time steps. The goodness of fit statistics are shown in Table 2.

Figure 2. Comparison of calculated and analytical water levels for different time steps.


Table 2. Goodness of Fit Statistics

Time, s RRMSE, % RMAE, % R^2 Bias, m
160 3.7 3.8 0.999 -0.012
175 6.5 5.9 0.997 -0.113
220 4.6 5.4 1.000 -0.066


A comparison of the calculated and analytical shoreline position are shown in Figure 3.

Figure 3. Time series comparison of calculated and analytical shoreline position.


References

  • Carrier, G., Wu, T.T., and Yeh, H. (2003). "Tsunami runup and draw-down on a plane beach", Journal of Fluid Mechanics, 475, 79-99.

Test Cases

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