Long wave propagation: Difference between revisions

From CIRPwiki
Jump to navigation Jump to search
No edit summary
 
(7 intermediate revisions by the same user not shown)
Line 1: Line 1:
= Analytical Solution =
= Problem =
The purpose of this verification test is to assess the model performance in simulating long wave propagation. The case is also useful for testing the model performance and symmetry for a non-rectangular domain. The test case consists of a quarter annulus with a linear bed elevation in the radial direction. The inner radius is 60.96 km (200,000 ft), and the outer radius is 152.4 km (500,000 ft). The analytical is given by Lynch and Gray (1978).  
Lynch and Gray (1978) presented the analytical solution for depth-averaged long-wave propagation in am annular domain. Here the case tested with a linearly sloping bed, and without bottom friction, Coriolis, or horizontal mixing. The offshore boundary consists of an single tidal constituent. Table 1 summarizes the important model settings used for this test.  


Table 1. General Settings for Flow over a Bump
'''Table 1. General Settings for quarter annulus test'''
{|border="1"
{|border="1"
|'''Parameter''' ||'''Value'''
|'''Parameter''' ||'''Value'''
|-
|-
|Deepwater Tidal Amplitude|| 0.3048 m
|Deepwater tidal amplitude|| 0.3048 m
|-
|-
|Tidal Period || 12.42 hrs (M2)
|Tidal period || 12.42 hrs (M2)
|-
|Inner radius || 60.96 km
|-
|Outer radius || 152.4 km
|-
|Depth at inner radius || 10.02 m
|-
|Depth at outher || 25.05 m
|-
|-
|Bottom  friction || None
|Bottom  friction || None
|-
|-
|Horizontal eddy viscosity || None
|Horizontal eddy viscosity || 0.0
|-
|Corilis || 0.0
|}
|}


= Model Setup =
= Model Setup =
A Cartesian grid is used with two levels of refinement from 8 km spacing to 2 km spacing.  
[[Image:QA_Depth_8km.png|thumb|right|600px| Figure 1. Quarter Annulus]]
The computational domain (Figure 1) consists of a three-level telescoping Cartesian grid with a grid resolution of 4, 2, and 1 km for each level. Higher resolution is specified near the inner and outer boundaries in order to reduce errors associated with the representation of the curved boundaries with squares. The computational grid is shown in Figure 1. The grid has 1,160 active ocean cells. The important model settings are shown in Table 2.


Table 1. General Settings for Flow over a Bump
'''Table 2. General settings for the quarter annulus test'''
{|border="1"
{|border="1"
|'''Parameter''' ||'''Value'''
|'''Parameter''' ||'''Value'''
|-
|-
|Time step|| 600 sec (10 min)
| Solution scheme || Implicit
|-
|Time step || 600 sec (10 min)
|-
|Simulation duration || 120 hrs
|-
|-
|Ramp period  || 24 hrs
|Ramp period  || 24 hrs
|-
|-
|Simulation time  || 120 hrs
|Advection terms || Off
|-
|Mixing terms || Off
|-
|-
|Grid resolution || 1 - 8 km
|Wall friction || Off
|-
|-
|Number of active ocean cells || 1160
|Coriolis || Off
|}
|}
[[Image:QA_Depth_8km.png|thumb|right|600px| Figure 1. Quarter Annulus]]
*Horizontal eddy viscosity: :0 m^2/sec
*Run duration: 5 days
*Ramp period: 2 days
*Time step: 10 min
*Wall Friction: Off
*Grid resolution: 1000 x 1000 m (constant)
*Number of cells: 23409


<br style="clear:both" />
<br style="clear:both" />


= Results =
= Results =
The figure below shows a time series of water levels at the inner edge of the model domain. The goodness of fit statistics are listed in the below. The model accurately predicts the wave phase but slightly overestimates the amplitude by approximately 0.01 m.  
[[Image:QA_Water_Level_8km.png|thumb|right|600px| Figure 2. Comparison of analytical (solid black) and calculated (red dots) water surface elevation at the center of the inner radius. ]]
Figure 2 shows a time series of water levels at the inner edge of the model domain. The goodness of fit statistics are listed in Table 3. The model accurately predicts the wave phase but slightly overestimates the amplitude by approximately 0.01 m. No significant numerical dissipation is observed or numerical instabilities. The simulation takes about 52 seconds on a PC and a single processor.


Table 2. Goodness of fit statistics for the water elevation
'''Table 3. Goodness of fit statistics for the water elevation'''
{|border="1"
{|border="1"
|'''Statistic'''
|'''Statistic''' ||'''Value'''
|'''Value'''
|-
|-
|RMSE 
|NRMSE || 3.3 %
| 0.014 m
|-
|-
|RMAE
|NRMAE || 2.7 %
| 0.028
|-
|-
|R^2   
|R^2  || 0.999
| 0.999
|-
|-
|Bias  
|Bias || 0.002 m
| 0.003 m
|}
|}
[[Image:QA_Water_Level_8km.png]]


<br style="clear:both" />
<br style="clear:both" />
Line 74: Line 75:


----
----
[[Test_Cases]]
[[Test_Cases]]

Latest revision as of 23:15, 1 June 2011

Problem

Lynch and Gray (1978) presented the analytical solution for depth-averaged long-wave propagation in am annular domain. Here the case tested with a linearly sloping bed, and without bottom friction, Coriolis, or horizontal mixing. The offshore boundary consists of an single tidal constituent. Table 1 summarizes the important model settings used for this test.

Table 1. General Settings for quarter annulus test

Parameter Value
Deepwater tidal amplitude 0.3048 m
Tidal period 12.42 hrs (M2)
Inner radius 60.96 km
Outer radius 152.4 km
Depth at inner radius 10.02 m
Depth at outher 25.05 m
Bottom friction None
Horizontal eddy viscosity 0.0
Corilis 0.0

Model Setup

Figure 1. Quarter Annulus

The computational domain (Figure 1) consists of a three-level telescoping Cartesian grid with a grid resolution of 4, 2, and 1 km for each level. Higher resolution is specified near the inner and outer boundaries in order to reduce errors associated with the representation of the curved boundaries with squares. The computational grid is shown in Figure 1. The grid has 1,160 active ocean cells. The important model settings are shown in Table 2.

Table 2. General settings for the quarter annulus test

Parameter Value
Solution scheme Implicit
Time step 600 sec (10 min)
Simulation duration 120 hrs
Ramp period 24 hrs
Advection terms Off
Mixing terms Off
Wall friction Off
Coriolis Off


Results

Figure 2. Comparison of analytical (solid black) and calculated (red dots) water surface elevation at the center of the inner radius.

Figure 2 shows a time series of water levels at the inner edge of the model domain. The goodness of fit statistics are listed in Table 3. The model accurately predicts the wave phase but slightly overestimates the amplitude by approximately 0.01 m. No significant numerical dissipation is observed or numerical instabilities. The simulation takes about 52 seconds on a PC and a single processor.

Table 3. Goodness of fit statistics for the water elevation

Statistic Value
NRMSE 3.3 %
NRMAE 2.7 %
R^2 0.999
Bias 0.002 m


References

  • Lynch, D.R., and Gray, W.G. (1978). "Analytical solutions for computer flow model testing," J. Hydraulics Division, 104, 1409-28.

Test_Cases

Retrieved from "https://cirpwiki.info/index.php?title=Long_wave_propagation&oldid=6932"