https://cirpwiki.info/index.php?title=Quarter_Annulus&feed=atom&action=historyQuarter Annulus - Revision history2021-01-20T14:29:16ZRevision history for this page on the wikiMediaWiki 1.35.0https://cirpwiki.info/index.php?title=Quarter_Annulus&diff=10502&oldid=prevU4hcsdaw: Created page with "Test C1-Ex3: Tidal Propagation in a Quarter Annulus ''Purpose'' The purpose of this verification test is to assess the model performance in simulating long wave propagation...."2014-04-15T22:08:09Z<p>Created page with "Test C1-Ex3: Tidal Propagation in a Quarter Annulus ''Purpose'' The purpose of this verification test is to assess the model performance in simulating long wave propagation...."</p>
<p><b>New page</b></p><div>Test C1-Ex3: Tidal Propagation in a Quarter Annulus<br />
<br />
''Purpose''<br />
<br />
The purpose of this verification test is to assess the model<br />
performance in simulating long wave propagation. The case is useful<br />
for testing the model performance and symmetry for a non-rectangular<br />
domain with a tidal forcing specified on one of the curved<br />
boundaries. Because there is no bottom friction or mixing, the test<br />
case is also useful for looking at numerical dissipation.<br />
<br />
''Problem''<br />
<br />
Lynch and Gray (1978) presented the analytical solution for<br />
depth-averaged long-wave propagation in an annular domain. The case<br />
was for a linearly sloping bed, and without bottom friction, Coriolis<br />
force, or horizontal mixing. The offshore boundary consisted of a<br />
single tidal constituent (see Figure7). Table7 summarizes the<br />
important model settings used for this test.<br />
{| class="wikitable" border="1"<br />
|+ Table 7. Quarter annulus setup parameters<br />
!Parameter !! Value<br />
|-<br />
|Deepwater tidal amplitude || 0.3048 m (1 ft)<br />
|-<br />
|Tidal period || 12.42 hr (M2 tide)<br />
|-<br />
|Inner radius || 60.96 km<br />
|-<br />
|Outer radius || 152.4 km<br />
|-<br />
|Inner water depth || 10.02 m<br />
|-<br />
|Outer water depth || 25.05 m<br />
|-<br />
|Bathymetry profile || Linear<br />
|-<br />
|Bottom friction || None<br />
|-<br />
|Mixing terms || Off<br />
|-<br />
|Coriolis force || Off<br />
|}<br />
<br />
[[File:tidal_quarter_annulus.png || leftthumb|400px|alt=framework]]<br />
Figure 7. Computational domain for tidal propagation in a quarter<br />
annulus.<br />
<br />
''Model Setup''<br />
<br />
The computational grid (Figure 8) consists of a three-level<br />
telescoping Cartesian grid with resolution of 4, 2, and 1 km for the<br />
three levels. Higher resolution is specified near the inner and outer<br />
boundaries in order to reduce errors associated with the<br />
representation of the curved boundaries with squares. The grid has<br />
1,160 active ocean cells. Model settings are shown in Table 8.<br />
<br />
{| class="wikitable" border="1"<br />
|+ Table 8. CMS-Flow setup parameters for the quarter annulus test case.<br />
!Parameter !!Value<br />
|-<br />
|Solution scheme || Implicit<br />
|-<br />
|Time step || 10 min<br />
|-<br />
|Simulation duration || 120 hr<br />
|-<br />
|Ramp duration || 24 hr<br />
|-<br />
|Mixing terms || Off<br />
|-<br />
|Advection terms || Off<br />
|-<br />
|Wall friction || Off<br />
|-<br />
|Coriolis force || Off<br />
|}<br />
<br />
[[File:tidal_quarter_annulus_grid.png || leftthumb|400px|alt=framework]]<br />
Figure 8. Computation grid used for tidal propagation in a quarter annulus.<br />
<br />
''Results and Discussion''<br />
<br />
Figure 9 shows a time series of water levels at the inner edge of the<br />
simulation domain. The goodness-of-fit statistics are listed in<br />
Table 9. The model accurately predicts the wave phase but slightly<br />
overestimates the amplitude by approximately 0.01 m. No significant<br />
numerical dissipation is observed or numerical instability. The<br />
simulation takes about 1 min on a Windows PC on a single 2.67 GHz<br />
processor.<br />
<br />
[[File:wl_timeSeries.png || leftthumb|400px|alt=framework]]<br />
Figure 9. Comparison of analytical (solid black) and calculated (red<br />
dots) water surface elevations at the center of the inner radius.<br />
<br />
<br />
{| class="wikitable" border="1"<br />
|+Table 9. Water level goodness-of-fit statistics for the quarter<br />
annulus test case<br />
!Statistic !!Value<br />
|-<br />
|NRMSE, % ||3.3<br />
|-<br />
|NMAE, % ||2.7<br />
|-<br />
|<math>R^2</math> ||0.999<br />
|-<br />
|Bias, m ||0.002 m<br />
|}<br />
<br />
<br />
An example of the simulated water level and current velocity magnitude<br />
fields is shown in Figure 10. The water level contours are very smooth<br />
and do not show any significant instabilities. However, the current<br />
magnitude shows some errors at the offshore boundary. This is due to<br />
staircase representation of the curved open boundary. This problem<br />
would be fixed by specifying the analytical current velocities at the<br />
offshore boundary, but it was not done in this example. Sensitivity<br />
tests with smaller time steps showed that the problem persists for<br />
smaller time steps. For practical applications all model forcing is<br />
specified on straight boundaries and this problem does not occur as<br />
demonstrated in subsequent test cases.<br />
<br />
[[File:quarter_annulus_water_level.png || leftthumb|400px|alt=framework]]<br />
[[File:quarter_annulus_current.png || leftthumb|400px|alt=framework]]<br />
<br />
Figure 10. Snap shot of water levels at 62 hr (a) and current<br />
magnitude at 65.5 hr (b).<br />
<br />
''Conclusions and Recommendations''<br />
<br />
The CMS-Flow can accurately simulate linear long-wave propagation in a<br />
quarter annulus with a linear bed and zero bottom friction and<br />
Coriolis. The water level NRMSE, NMAE, and R2 were 3.3%, 2.7%, and<br />
0.999, respetively. For practical applications, it is recommended to<br />
specify water level boundary conditions on straight boundaries. If a<br />
curved forcing boundary is necessary, then it is recommended to<br />
specify both water levels and current velocities.</div>U4hcsdaw