Plunging Flow at Pipe Outfalls

Plunging flow occurs when the downstream water level is below the invert of the pipe exit. In these cases, a stage boundary is applied to the end of the pipe.

For steeply sloped pipes, where the normal depth is less than the critical depth, a normal depth boundary condition is applied at the end of the pipe with the friction slope equal to the slope of the pipe.

For more gradually sloped pipes, the water depth is assumed to pass through critical depth near the exit. The work of Rouse (1936) showed the depth of the free outfall to be less than the critical depth, and critical depth occurred at a length four times the critical depth into the pipe. However since these effects are confined to the area very near the end of the pipe, the outfall stage boundary in HEC-RAS is simply set to the critical depth.

Plunging Flow into Junction Boxes

Plunging flow may also occur internally in the pipe system when a perched pipe enters a junction box. In these cases, two modifications are made to the discretization of the momentum equation at the terminal computational face of the perched pipe.

First, the barotropic pressure gradient term is modified to account for the presence of the free outfall. To accomplish this, a term is added to the calculated water surface slope between the cells on either side of the drop.

1) \left(g\frac{\partial H}{\partial x}\right)_j\approx g\frac{H_{i+1}^{n+\theta} - H_i^{n+\theta} + \Delta z_{\textrm{drop}}}{\Delta x_j}\newline
2) \Delta z_{\textrm{drop}} = \textrm{max}(\frac{2}{3} E_i - H_{i+1}^n, 0)

where  

E_i = H_i + \frac{V_{j-1}^2}{2g} is the upstream specific energy.

In effect this term limits the water surface slope at the plunging face so that the downstream water level is not considered. When the downstream water level does rise to the level where it would effect the upstream cell, the maximum function in 2) negates any modification of the slope, and the pressure gradient term is calculated as normal.

Second, modifications are made to the Lagrangian backtracking method for momentum advection (see Numerical Methods (1D FV)) so that velocities are not traced back through the plunging flow. This prevents advection of momentum through the plunging face. When flow reconnects as downstream water levels rise above the plunging face invert, advection backtracking is reinstated.