Difference between revisions of "Surface model (Water Overlay)"

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(Surface flow scheme)
 
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In the model, imbalances in the water surface elevation across the grid drive the flow of water until a state of equilibrium is reached in terms of ''w'' and flux. Behavior of the flow is described by a second-order semi-discrete central-upwind scheme produced by Kurganov and Petrova (2007)<ref name="Kurganov2"/>, which is based on the 2-D Saint-Venant equations (a.k.a. shallow water equations):
 
In the model, imbalances in the water surface elevation across the grid drive the flow of water until a state of equilibrium is reached in terms of ''w'' and flux. Behavior of the flow is described by a second-order semi-discrete central-upwind scheme produced by Kurganov and Petrova (2007)<ref name="Kurganov2"/>, which is based on the 2-D Saint-Venant equations (a.k.a. shallow water equations):
  
h<sub>t</sub> + (hu)<sub>x</sub> + (hv)<sub>y</sub> = 0,<br>
+
:<math>
(hu)<sub>t</sub> + (hu<sup>2</sup> + 0.5gh<sup>2</sup>)<sub>x</sub> + (huv)<sub>y</sub> = -ghB<sub>x</sub> - gh n<sup>2</sup>u (u<sup>2</sup>+v<sup>2</sup>)<sup>0.5</sup> h<sup>-4/3</sup> <br>
+
\begin{align}
(hv)<sub>t</sub> + (huv)<sub>x</sub> + (hv<sup>2</sup> + 0.5gh<sup>2</sup>)<sub>y</sub> = -ghB<sub>y</sub> - gh n<sup>2</sup>u (u<sup>2</sup>+v<sup>2</sup>)<sup>0.5</sup> h<sup>-4/3</sup> <br>
+
\frac{\partial h}{\partial t} &+ \frac{\partial (hu)}{\partial x} + \frac{\partial (hv)}{\partial y} = 0,\\[3pt]
 +
\frac{\partial (hu)}{\partial t} &+ \frac{\partial}{\partial x} \left( hu^2 + \frac{1}{2} gh^2 \right) + \frac{\partial (huv)}{\partial y} = -gh \frac{\partial B}{\partial x} - ghn^2u \sqrt{u^2 + v^2} h^{-\frac{4}{3}},\\[3pt]
 +
\frac{\partial (hv)}{\partial t} &+ \frac{\partial (huv)}{\partial x} + \frac{\partial}{\partial y} \left( hv^2 + \frac{1}{2} gh^2 \right) = -gh \frac{\partial B}{\partial y} - ghn^2u \sqrt{u^2 + v^2} h^{-\frac{4}{3}},
 +
\end{align}
 +
</math>
  
 
where
 
where
: h<sub>t</sub> = water depth at time t
+
:{|
: u<sub>t</sub> = velocity in the x-direction at time t
+
|- valign="top"
: v<sub>t</sub> = velocity in the y-direction at time t
+
| ''u'' || is the velocity in the ''x'' direction
: B<sub>x</sub> = slope in the x-direction
+
|- valign="top"
: B<sub>y</sub> = slope in the y-direction
+
| ''v'' ||  is the velocity in the ''y'' direction
: g = gravitational constant
+
|- valign="top"
: n = manning roughness value
+
| ''h'' || is the water depth
{{clear|left}}
+
|- valign="top"
 +
| ''B'' || is the bottom elevation
 +
|- valign="top"
 +
| ''g'' || is the acceleration due to gravity
 +
|-  valign="top"
 +
| ''n'' || is the Gauckler–Manning coefficient
 +
|}
  
 
This scheme deviates from the original Saint-Venant equations in that it approaches the system in terms of water surface elevation (''w = h + B'') and flux (''hu'' and ''hv''), instead of just the water depth (''h''). The conceptual image in Fig. 1 aims to clarify on the various terms in the model equations. This method relies on a continuous piecewise linear approximation of the surface, which is further explained on [[Elevation model (Water Overlay)|this]] page.<!--More technical information on the {{software}}'s implementation of the ''Well-Balanced Positivity Preserving Central-Upwind Scheme'' can be found [[Surface_flow_formula_(Water_Overlay)|here]].-->
 
This scheme deviates from the original Saint-Venant equations in that it approaches the system in terms of water surface elevation (''w = h + B'') and flux (''hu'' and ''hv''), instead of just the water depth (''h''). The conceptual image in Fig. 1 aims to clarify on the various terms in the model equations. This method relies on a continuous piecewise linear approximation of the surface, which is further explained on [[Elevation model (Water Overlay)|this]] page.<!--More technical information on the {{software}}'s implementation of the ''Well-Balanced Positivity Preserving Central-Upwind Scheme'' can be found [[Surface_flow_formula_(Water_Overlay)|here]].-->

Latest revision as of 11:30, 14 August 2019

Computation

Fig. 1. For each grid cell:
B = bottom elevation
h = water depth
w = water surface elevation
Fig. 2. [1]
Fig. 3. [1]

The Water Module's primary function is simulating two-dimensional flow of water across the surface. In order to do this, the project area is first discretized into x by y cells depending on the configured grid cell size. Secondly, water is initialized in the model. Finally, a set of rules is required that describes the behavior of the flow.

Water level initialization

In theory, each grid cell can have a unique bottom elevation and accompanying water depth, yielding a certain water surface elevation (or water level). However, in practice, water levels are often initialized for large groups of cells rather than individual cells, assuming that each (water level) area in a project has been assigned a water level. During the initialization phase a distinction is made between two types:

For all water terrains in a water level area, the volume of water per grid cell is such that the resulting water level at those locations conforms the value of the WATER_LEVEL attribute as provided by the corresponding water level area.

In contrast to water level areas, which generate water on water terrains, inundation areas generate water over all grid cells, regardless of its terrain type. Similarly, the volume of water per grid cell is determined by the INUNDATION_LEVEL attribute as provided by the corresponding inundation area.

If a grid cell is neither part of any water level area nor inundation area, or the assigned water level is lower than its bottom elevation, the water depth is assumed to be zero and the water level becomes equal to the bottom elevation. In turn, this bottom (or surface) elevation is equal to the local terrain height of the project area, though it may be altered by the presence of a construction or the BREACH_HEIGHT attribute of a breach.

Notes

Surface flow scheme

In the model, imbalances in the water surface elevation across the grid drive the flow of water until a state of equilibrium is reached in terms of w and flux. Behavior of the flow is described by a second-order semi-discrete central-upwind scheme produced by Kurganov and Petrova (2007)[2], which is based on the 2-D Saint-Venant equations (a.k.a. shallow water equations):

where

u is the velocity in the x direction
v is the velocity in the y direction
h is the water depth
B is the bottom elevation
g is the acceleration due to gravity
n is the Gauckler–Manning coefficient

This scheme deviates from the original Saint-Venant equations in that it approaches the system in terms of water surface elevation (w = h + B) and flux (hu and hv), instead of just the water depth (h). The conceptual image in Fig. 1 aims to clarify on the various terms in the model equations. This method relies on a continuous piecewise linear approximation of the surface, which is further explained on this page.

This scheme is further extended with the shoreline reconstruction method by Bollerman et al (2014), which ensures better numerical stability at the wetting and drying fronts of a flood wave[3]. An elaborate explanation is provided by Horváth et al. (2014)[1].

General sequence of computational steps:

  1. Each cell's elevation value (B in Fig. 3) is set equal to the value at its center and the interface midpoints.
  2. Slopes of conserved variables are reconstructed (continuity and momentum in x- and y-direction).
  3. Values of conserved variables at the cell interface midpoints are compared with the left-sided and right-sided values with respect to the cell's center.
  4. Slopes of partially dry cells are modified to prevent negative depth values and numerical instability.
  5. Fluxes are computed at each cell interface to determine the values of the conserved variable at the cell centers for the next time step.
  6. The largest allowed time step is calculated.
  7. Time is incremented with the calculated time step and changes in water level and fluxes are subsequently applied.

References

  1. 1.0 1.1 1.2 Zsolt Horváth, Jürgen Waser, Rui A. P. Perdigão, Artem Konev and Günter Blöschl (2014) ∙ A two-dimensional numerical scheme of dry/wet fronts for the Saint-Venant system of shallow water equations ∙ found at: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.700.7977&rep=rep1&type=pdf and http://visdom.at/media/pdf/publications/Poster.pdf (last visited 2018-06-29)
  2. Kurganov A, Petrova G (2007) ∙ A Second-Order Well-Balanced Positivity Preserving Central-Upwind Scheme for the Saint-Venant System ∙ found at: http://www.math.tamu.edu/~gpetrova/KPSV.pdf (last visited 2019-04-11)
  3. Bollermann, Andreas & Chen, Guoxian & Kurganov, Alexander & Noelle, Sebastian. (2014). A Well-balanced Reconstruction for Wetting, Drying Fronts. IGPM Report. 313. ∙ found at: https://www.researchgate.net/publication/269417532_A_Well-balanced_Reconstruction_for_Wetting_Drying_Fronts (last visited: 2019-07-24)