Sudden Expansion: Difference between revisions

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(Created page with "<big> __NOTOC__ <font color=red>'''UNDER CONSTRUCTION'''</font> == Setup == The spatial domain consists of a rectangular {{Equation| <math> z_b = \begin{cases} 0, & \mb...")
 
 
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__NOTOC__
__NOTOC__
<font color=red>'''UNDER    CONSTRUCTION'''</font>
<font color=red>'''UNDER    CONSTRUCTION'''</font>
== Setup ==
== Experimental Setup ==
The spatial domain  consists of a rectangular  
The flume experiment consisted of a rectangular flume 18 m long with an inflow section 0.6 m wide which expands suddenly into a wider section 1.2 m wide. The experiment parameters are shown in Table 1.
  {{Equation| <math> z_b =
\begin{cases} 
0, & \mbox{if } x<8  \\
0.2-0.05(x-10)^2, &  \mbox{if } 8 \leq x \leq 12 \\
0, & \mbox{if } x>8  \\
\end{cases}  </math> |2=1}}


where  <math>z_b</math> is the bed elevation and  <math>x</math> is the horizontal distance.
Table  1. General Settings for Sudden Expansion Case
 
Table  1. General Settings for Flow over a Bump
{|border="1"
{|border="1"
|'''Parameter'''
|'''Parameter'''
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|-
|-
|Discharge
|Discharge
| 0.18 m^3/s
| 0.0385 m^3/s
|-
|-
|Downstream water level
|Downstream water level
| 0.33 m
| 0.0 m
|-
|-
|Bottom  friction 
|Manning's coefficient
| None
| 0.015
|}
|}


== Model Setup ==
== Model Setup ==
[[Image:Bump_Grid.png|thumb|right|600px|  Figure 1. Computational grid.]]
[[Image:Sudden_Expansion_Grid_V2.png|thumb|right|600px|  Figure 1. Computational grid.]]
The computational domain is 25 m  long and has a constant grid resolution of 0.1 m. A flux boundary  condition is specified at the inflow boundary and a constant water level  boundary condition is applied to the downstream boundary. An adaptive time between 0.1-10 seconds is applied.  
The streched 3-level quadtree CMS-Flow grid is shown in Figure 1. The grid has a resolution between 0.03 and 0.45 m. A flux boundary  condition is specified at the inflow boundary and a constant water level  boundary condition is applied to the downstream boundary. The model setup parameters are shown in Table 2.
 
Table 2. Model
{|border="1"
|'''Parameter'''
|'''Value'''
|-
|Numerical scheme
|Implicit
|-
|Time step
| 30 s
|-
|Simulation Duration
| 1 hr
|-
|Ramp time
| 0.5 hr
|}


<br  style="clear:both" />
<br  style="clear:both" />
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== Results ==
== Results ==
[[Image:Sudden_Expansion_Results.png|thumb|right|600px| Figure 2. Comparison of analytical and calculated water surface  elevations and bed elevations.]]
[[Image:Sudden_Expansion_Results.png|thumb|right|600px| Figure 2. Comparison of analytical and calculated water surface  elevations and bed elevations.]]
A comparison between the  analytical solution for water levels is compared to the calculated water  levels in Figure 2. As shown in the goodness of fit statistics, the  model results agree well with the analytical solution. The minimum water  level is captured well, however there is a small shift in the location  of the water level drop over the bump toward the downstream direction.  
A comparison between the  analytical solution for water levels is compared to the calculated water  levels in Figure 2. As shown in the goodness of fit statistics in Table 3, the  model results agree well with the analytical solution. The minimum water  level is captured well, however there is a small shift in the location  of the water level drop over the bump toward the downstream direction.  




Table  2. Goodness of fit statistics for the water elevation
Table  3. Goodness of fit statistics for the current velocities
{|border="1"
{|border="1"
|'''Statistic'''
|'''Statistic''' ||'''x = 0 m''' || '''x = 1 m''' || '''x = 2 m''' || '''x = 3 m''' || '''x = 4 m''' || '''x = 5 m'''
|'''Value'''
|-
|-
|RMSE
|RMSE, m/s || 0.0105 || 0.0260 || 0.0294 || 0.0386 || 0.0775 || 0.0584
| 0.0074 m
|-
|-
|RMAE
|NMAE, % || 1.28 || 0.72 || 0.56 || 3.19 || 4.33 || 0.45
| 0.0068 m
|-
|-
|R^2   
|R^2  || 0.789 || 0.995 || 0.990 || 0.989 || 0.936 || 0.980
| 0.991
|-
|-
|Bias  
|Bias, m/s || -0.0083 || -0.0046 || 0.0035 || 0.0187 || -0.0017 || 0.0022
| 0.0017 m
|}
|}
* For a definition of the goodness of fit statistics see [[Statistics |  Goodness of fit statistics]].


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


== References ==
== References ==
*Caleffi, V., Valiani, A., and Zanni, A. (2003). "Finite volume method for  simulating extreme flood events in naturalchannels," Journal of Hydraulic Research, 41(2), 167-177.
*Xie, B.L. (1996). "Experiment on flow in a sudden-expanded channel," Technical Report, Wuhan University of Hydraulic and Electric Engineering, China.


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----

Latest revision as of 17:35, 1 June 2011

UNDER CONSTRUCTION

Experimental Setup

The flume experiment consisted of a rectangular flume 18 m long with an inflow section 0.6 m wide which expands suddenly into a wider section 1.2 m wide. The experiment parameters are shown in Table 1.

Table 1. General Settings for Sudden Expansion Case

Parameter Value
Discharge 0.0385 m^3/s
Downstream water level 0.0 m
Manning's coefficient 0.015

Model Setup

Figure 1. Computational grid.

The streched 3-level quadtree CMS-Flow grid is shown in Figure 1. The grid has a resolution between 0.03 and 0.45 m. A flux boundary condition is specified at the inflow boundary and a constant water level boundary condition is applied to the downstream boundary. The model setup parameters are shown in Table 2.

Table 2. Model

Parameter Value
Numerical scheme Implicit
Time step 30 s
Simulation Duration 1 hr
Ramp time 0.5 hr


Results

Figure 2. Comparison of analytical and calculated water surface elevations and bed elevations.

A comparison between the analytical solution for water levels is compared to the calculated water levels in Figure 2. As shown in the goodness of fit statistics in Table 3, the model results agree well with the analytical solution. The minimum water level is captured well, however there is a small shift in the location of the water level drop over the bump toward the downstream direction.


Table 3. Goodness of fit statistics for the current velocities

Statistic x = 0 m x = 1 m x = 2 m x = 3 m x = 4 m x = 5 m
RMSE, m/s 0.0105 0.0260 0.0294 0.0386 0.0775 0.0584
NMAE, % 1.28 0.72 0.56 3.19 4.33 0.45
R^2 0.789 0.995 0.990 0.989 0.936 0.980
Bias, m/s -0.0083 -0.0046 0.0035 0.0187 -0.0017 0.0022


References

  • Xie, B.L. (1996). "Experiment on flow in a sudden-expanded channel," Technical Report, Wuhan University of Hydraulic and Electric Engineering, China.

Test Cases

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