Depthaveraging

Depthaveraging is a modeling approach used in fluid dynamics to simplify three-dimensional flows by integrating variables over the vertical coordinate. The result is a two-dimensional, depth-averaged description of the flow. The primary quantities are the water depth h(x,y,t) and the depth-averaged horizontal velocity components u(x,y,t) and v(x,y,t). Depth-averaged models are widely used for rivers, floods, coastal flows, and other shallow-water situations where horizontal scales dominate vertical structure.

The governing equations are obtained by integrating the Navier–Stokes equations over depth under the hydrostatic pressure assumption and neglecting vertical accelerations. The standard form yields the Saint-Venant (shallow water) equations in conservative form:

- Mass conservation: ∂h/∂t + ∂(hu)/∂x + ∂(hv)/∂y = 0.

- Momentum in x: ∂(hu)/∂t + ∂(hu^2 + 0.5 g h^2)/∂x + ∂(huv)/∂y = -g h ∂b/∂x - τbx.

- Momentum in y: ∂(hv)/∂t + ∂(huv)/∂x + ∂(hv^2 + 0.5 g h^2)/∂y = -g h ∂b/∂y - τby.

Here g is gravitational acceleration, b(x,y) is the bed elevation, and τb denotes bed shear stresses (friction). In practice, friction is often modeled with empirical laws (e.g., Manning or Darcy–Weisbach).

Extensions include multi-layer depth-averaged models that resolve vertical structure with several stacked layers, and various numerical schemes (finite volume, finite element) for solving the equations on complex topography. Depth-averaged models are best suited for flows with gentle vertical variation and may be inadequate for strongly stratified, rapidly changing, or highly three-dimensional phenomena.