Mixed (Transcritical) Flow

One of the challenging features in the unsteady flows in a sewer or storm water drainage system is the interchanging or moving interface of different flow regimes between subcritical and supercritical flows. This is largely due to the fact that an urban hydraulic system can experience a large range of slopes of conduits and it is common to have significant slope changes at many pipe junctions. A good numerical model for sewer and storm water system has to 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) 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 Inertia modification (LPI) technique is used in the dynamic sewer model. In the LPI technique, the momentum equation, Equation 11.2, is modified by a numerical filter, σ, 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 σ 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 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.

The LPI technique was developed by Dr. Ming Jin and Dr. Danny Fread and this technique has been adapted by Federal dynamic models such as NWS Fldwav model, USACE HEC-RAS unsteady flow model and EPA-SWMM model.