Depthaveraged

Depthaveraged refers to the reduction of a three‑dimensional flow problem to two horizontal dimensions by averaging quantities over the vertical depth of the fluid. In shallow water contexts, this technique yields the depth-averaged variables, most commonly the water depth h(x,y,t) and the depth-averaged velocity Ū(x,y,t). The depth-averaged velocity is defined as the vertical integral of the local velocity divided by the depth: Ū = (1/h) ∫0^h u(z) dz. This approach relies on the hydrostatic pressure assumption and on relatively small vertical accelerations, allowing a two-dimensional description of horizontal flow.

The resulting depth-averaged equations are often called the Saint-Venant or shallow water equations. They consist of a continuity equation, expressing mass conservation, and a momentum equation, expressing horizontal momentum balance. In compact form: ∂h/∂t + ∇·(h Ū) = 0 for continuity, and ∂(h Ū)/∂t + ∇·(h Ū ⊗ Ū) + g h ∇h + bed-friction terms = 0 for momentum. Bed friction is typically modeled with empirical relations such as Manning’s n or Chezy coefficients. These equations describe the propagation of floods, tides, dam-break waves, and other long, shallow flows where vertical structure is not resolved.

Extensions include multilayer depth-averaged models, which divide the vertical column into several layers, each with its own depth and velocity to capture some vertical shear. Limitations of depth-averaged models include the neglect of vertical accelerations and stratification, making them less suitable for strongly three-dimensional or non-hydrostatic flows.