Basic 2D Hydraulic Theory
The 2D Engine used in InfoDrainage is based on the procedures described in Alcrudo and Mulet-Marti (2005). This is the same 2D engine that is used in InfoWorks ICM.
The shallow water equations (SWE), that is, the depth-average version of the Navier-Stokes equations, are used for the mathematical representation of the 2D flow. The SWE assume that the flow is predominantly horizontal and that the variation of the velocity over the vertical coordinate can be neglected. The conservative formulation of the SWE used in InfoDrainage is described below:
The effect of turbulence is considered to be included in the energy loss due to the bed resistance and modelled via the Manning's n parameter specified in the Deluge options.
The conservative formulation of the SWE is essential in order to preserve the basic fundamental quantities of mass and momentum. This type of formulation allows the representation of flow discontinuities and changes between gradually and rapidly varied flow. The conservative SWE are discretized using a first-order finite volume explicit scheme. Finite volume schemes use control volumes to represent the area of interest. With finite volume methods the modelling domain is divided into geometric shapes over which the SWE are integrated to give equations in terms of fluxes through the control volume boundaries. The scheme that is used to solve the SWE is based upon the Gudunov numerical scheme, with the numerical fluxes through the boundaries of the control volumes computed using the standard Roe’s approximate Riemann solver. Finite volume methods are generally considered to have a number of advantages in terms of conservativeness, geometric flexibility and conceptual simplicity. As the scheme is an explicit solution it does not require iteration to achieve stability within defined tolerances like the 1D scheme. Instead, for each element, the required timestep is calculated using the Courant-Friedrichs-Lewy condition in order to achieve stability, where the Courant-Friedrichs-Lewy condition is:
The management of cell wetting and drying is performed using a threshold depth as a criteria to determine whether a cell is wet and the velocity is set to zero if the depth is below the threshold value of 0.001m. This avoids the formation of artificially high velocities in wetting/drying areas. InfoDrainage uses an unstructured mesh to represent the InfoDrainage surface and this together with the scheme used allow robust simulation of rapidly varying flows (shock capturing) as well as super-critical and transcritical flows. During a Deluge simulation, for each mesh element, final depth and direction of flow at maximum depth*velocity will be calculated.
2D Mesh Generation Methodology
When creating 2D meshes, the 2D mesh is generated using the Shewchuk Triangle meshing functionality. Heights at the vertices of the generated mesh elements are calculated by interpolation from the InfoDrainage surface.
The triangles are recursively split until all triangles are smaller than the maximum element area, and certain geometrical constraints are satisfied. The maximum element area considered is four times the specified minimum area size in the Deluge properties.
A single mesh element may be made up of more than one triangle, if a triangle has an area less than the Minimum element area specified in the Deluge properties. Triangles will be aggregated with adjacent triangles until the minimum area is met.
The ground elevation for a mesh element is calculated by sampling the surface within the 2D triangles making up the element and then taking the average of the sample point elevations. The number of sample points for each triangle is determined by subdividing the triangle until the minimum element area is reached. The sample points are the centroids of the resulting triangles. If a triangle is smaller than the minimum element area, the centroid of the triangle will be the only point sampled.