2. Conservation of mass

To capture or predict flow under varying conditions, one is often forced to use the computational power of computers. Since the introduction of computers various methods have been introduced and improved. Some aspects are true for all types of methods. Here, we will limit ourselves to the methods used in the computational core of 3Di.

3Di is a two-layer, subgrid based hydrodynamical model, where a surface and a subsurface layer can be defined in combination with a 1D network. The computations of flow in each domain are based on two fundamental laws of physics; i) Conservation of mass and ii) Conservation of momentum. In this section we will describe how we deal with conservation of mass.

Conservation of mass states that mass cannot disappear or appear in a certain domain without clear source. For a defined domain, when all fluxes (discharges) in and out of that domain are known, the change in mass can be computed. This can be described mathematically as:

(1)\frac{\Delta \rho V}{\Delta t}=\sum_i^{in} \rho_i Q_i -\sum_k^{out} \rho_k Q_k + \sum_i \rho_j S_j

In which:
\rho is the density,
V_\Omega is the volume,
Q is a discharge and
S is a sink or source term.

virtual_conservation_box

A virtual box for conservation of mass.

The counters i, j, k count over all existing discharges, sink and source terms. In 3Di we do not account for density variations, so the density \rho is assumed uniform and constant. This simplifies the equation for conservation of mass, to the following equation for conservation of volume:

(2)\frac{\Delta V}{\Delta t}=\sum_i^{in} Q_i -\sum_k^{out} Q_k + \sum_i S_j

It is important to define the domain for which this is true. In the finite volume approach, used in 3Di, a volume domain equals a computational cell, i.e. the water level domain. For such a domain, as shown in the Figure, all discharges (in blue) sources and sink terms (in yellow) entering and leaving the box are to be defined or computed. The discharges are computed based on the momentum equations. The Sections 2D Surface Flow, 1D Network, Interflow and Groundwater elaborate about the computation of flow in the specific domains. Sources and sink terms are generally terms for water that is added or extracted in a domain. Examples of such are Simple Infiltration and Rainfall. In 3Di, the conservation of volume is where all flow phenomena are combined. This is independent whether flow originates from the 1D, 2D surface or subsurface domain. So, if the discharges are defined, the system can be solved.