LPI technique

One of the challenging features of unsteady flows in a sewer or storm water drainage system is the interchanges or moving interfaces of different flow regimes between subcritical and supercritical flows. This is largely due to the fact that a piped urban drainage system can have a great variation in the range of slopes of the conduits and it is common to have significant slope changes at pipe junctions. A good numerical model for sewer and storm water system must be able to handle the mixed flow regimes and interchanges with great robustness.

When modeling unsteady flows, the dynamic routing technique, using the four-point implicit numerical scheme, tends to be less numerically stable than the diffusion (zero inertia) routing technique for certain mixed flows, especially in the near critical range of the Froude number (Fr ~ 1) or mixed flows with moving supercritical/subcritical interfaces. It has been observed that the diffusion technique, which eliminates the two inertial terms in the momentum equation, produces stable numerical solutions for flows where Fr is in the range of critical flow (Fr = 1.0) and for supercritical flows. To take advantage of the diffusion method's stability and retain the accuracy of the fully dynamic method, the Local Partial Inertial modification (LPI) technique is used. In the LPI technique, the momentum equation is dynamically modified by a numerical filter, ?, so that the inertial terms are partially or totally omitted based on the time-dependent local hydraulic conditions. The modified equation and numerical filter are:

in which is a numerical modifier and its value for every finite-difference box (between xi and xi+1) will be determined at each time step by the following equation:

in which m is a user specified constant (LPI coefficient in the calculation option) and m 1.0. It is found that smaller values of m tend to stabilize the solution in some cases while larger values of m provide more accuracy.

Virtual Flow

Dry flow condition is common in the storm and sewer modeling. A zero or near zero flow can cause instability in the implicit numerical model, in order to overcome this small flow instability a virtual flow method is used. The method adds a small base flow to every conduit which has a depth < the virtual depth so that the model is stable for low flow conditions. The virtual flow is determined by a pre-scribed (as a calculation option) virtual depth, the virtual depth is typically between 0.0 to 0.04 ft and the default value of 0.04 is proven to be stable for almost all conditions. After the calculation is finished for a time step a filtering algorithm is used to filter out the virtual flow and depth so that the final results are more accurately presented.