 # Depth-averaged shallow water equations

The Reynolds equations that were derived in "Turbulent flows" describe the motion processes of a flow in all three dimensions. This elaborate approach is appropriate for example for the mathematical calculation of flows close to constructs or in strongly meandering rivers, so basically everywhere where the flow is dominated by three-dimensional effects. Since the numerical three-dimensional calculation is still very costly, it makes sense to reduce the Reynolds equations for calculations with simpler flow conditions. The depth-averaged two-dimensional flow equations, also called shallow water equations, provide an example. The shallow water equations are obtained, as the name suggests, by averaging the Reynolds equations over the depth. The following conditions have to be met in order for the shallow water equations to be applicable:

• the vertical momentum exchange is negligible and the vertical velocity component w is a lot smaller than the horizontal components u and v:

w << u and w << v.

• the pressure gain is linear with the depth (parallel flow lines => hydrostatic pressure distribution): with z being the depth measured from the water surface and These assumptions make it possible to reduce the basic system of equations to only three equations: the continuity equation (as usual) and the motion equations in direction of the x- and y-axis, as follows:

x-component:

Prior to integrating the momentum equations over the depth, we will work on the continuity equation, as a small practical example to warm up. An important tool for the depth-integration are the so-called kinematical boundary conditions that give information about the change of the water surface over time.  