Explain GVF and spatially varied flow

Gradually varied flow (GVF)

GVF is steady non-uniform flow in which the depth changes slowly from section to section over a long distance. The streamlines are nearly parallel, so the pressure is hydrostatic and the friction loss can be found by applying a uniform-flow formula (Manning or Chezy) locally. The discharge is constant: $dQ/dx=0$.

Governing (dynamic) equation:

$$\frac{dy}{dx}=\frac{S_0-S_f}{1-Fr^2}$$

Example: the backwater curve upstream of a dam or weir (the depth rises gradually towards the weir); the drawdown curve in a mild channel approaching a free overfall.

    backwater (GVF)
  ~~~~~~~___-----~~~~|
  y_n               ||  weir
  ____________________||

Spatially varied flow (SVF)

SVF is steady flow in which the discharge varies along the channel length because water is added or removed along the way, so $dQ/dx\neq0$. For increasing discharge (lateral inflow), the momentum equation is used, because the mixing of the incoming water causes an unknown energy loss. For decreasing discharge (outflow over a side weir), the energy equation is usually used, because the outflow causes little energy loss.

Dynamic equation for lateral inflow (with negligible momentum of the inflow in the flow direction): $\dfrac{dy}{dx}=\dfrac{S_0-S_f-\dfrac{2Q}{gA^2}\dfrac{dQ}{dx}}{1-Fr^2}$.

Example: a side channel spillway or a roadside gutter (increasing Q); a side weir or a canal with several outlets (decreasing Q).