Hydrodynamic computations in river hydraulics use the **Shallow Water Equations (SWE)** to model open channel flow. The SWE are partial differential equations, being derived from the 3D Navier-Stokes equations (3D-NSE).

**Remember**: partial differential equations (PDEs) involve **at least two variables**

**in space** (boundary value problems)

**or time** (initial value problems).

The SWEs are commonly used **to approximate the water depth (h)** and the spatial pattern of the velocity field (u, v) and most commonly describe more than one time instance, but an entire hydrograph.

This approximation within the SWEs is a simplification, introducing time averaging and depth integration. The 2D model theory in river hydraulics assumes that vertical velocity components are negligible. Consequently, a hydrostatic pressure distribution is presumed. This simplification comes along with a reduction in computational costs afforded by the SWE approach. This is particularly important for ocean and estuary simulations, usually involving huge domains and a serious time span for a hydrodynamic simulation period, resulting in a large number of time steps. Calculations of complex flow situations such as caused by interactions with building structures or in urban areas show differences between 3D-NSE and SWE. They raise vertical velocity components that are generally not negligible.