Flow over a bump: Difference between revisions

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<font color=red>'''UNDER  CONSTRUCTION'''</font>
<font color=red>'''UNDER  CONSTRUCTION'''</font>
== Setup ==
== Setup ==
The spatial domain consists of a rectangular  
The spatial domain consists of a rectangular domain with the bed elevation <math>z_b</math> given by
    {{Equation| <math> z_b = \begin{pmatrix} 0 & \forall x<8m \\ 0.2-0.05(x-10)^2 & \forall 8 \leq x \leq 12 \\ 0 & \forall x>12 \end{pmatrix} </math> |2=1}}
\begin{equation} \tag{1}
  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}
\end{equation}


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.
where <math>x</math> is the horizontal distance.
 
'''Table 1. General Settings for Flow over a Bump'''
{|border="1"
|'''Parameter'''
|'''Value'''
|-
|Discharge
| 0.18 m^3/s
|-
|Downstream water level
| 0.33 m
|-
|Bottom friction  
| None
|}


== 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.  
[[Image:Bump_Grid.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 0.1-1 s is applied. The computational time on a single processor is approximately 1 min.
 
<br style="clear:both" />
 
== Results ==
[[Image:Bump_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.  
 


==  Results ==
Table 2. Goodness of fit statistics for the water elevation
[[Image:Wind_Setup_Dir0_WSE.png|thumb|left|600px|   Figure 1. Computed water surface elevation for the irregular domain with constant water depth.]]
{|border="1"
|'''Statistic''' ||'''Value'''
|-
|RMSE || 0.0074 m
|-
|NMAE || 0.0068
|-
|R^2 || 0.991
|-
|Bias || 0.0017 m
|}
* 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" />


<br    style="clear:both" />
== References ==
*Caleffi, V., Valiani, A., and Zanni, A. (2003). "Finite volume method for simulating extreme flood events in natural channels," Journal of Hydraulic Research, 41(2), 167-177.


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----

Latest revision as of 14:36, 16 July 2012

UNDER CONSTRUCTION

Setup

The spatial domain consists of a rectangular domain with the bed elevation given by \begin{equation} \tag{1} 

 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}

\end{equation}

where is the horizontal distance.

Table 1. General Settings for Flow over a Bump

Parameter Value
Discharge 0.18 m^3/s
Downstream water level 0.33 m
Bottom friction None

Model Setup

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 0.1-1 s is applied. The computational time on a single processor is approximately 1 min.


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, 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

Statistic Value
RMSE 0.0074 m
NMAE 0.0068
R^2 0.991
Bias 0.0017 m


References

  • Caleffi, V., Valiani, A., and Zanni, A. (2003). "Finite volume method for simulating extreme flood events in natural channels," Journal of Hydraulic Research, 41(2), 167-177.

Test Cases

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