Dimension reduction for compressible pipe flows including friction

Abstract

We present the full derivation of a one dimensional “Saint–Venant” like equations for barotropic compressible pipe flows including friction. The one dimensional hyperbolic system is called γ-pressurized model where γ is the adiabatic constant. It is obtained through the three dimensional barotropic Navier–Stokes equations under “thin layer” assumptions as a first order approximation. Prescribing suitable boundary conditions, one can introduce a general friction law and then explicitly show its geometrical (w.r.t. the hydraulic radius) and hydrodynamical (w.r.t. the Oser number) dependencies in the reduced model. In particular, for linear pressure law ( $\mathit{\gamma }=1$ ), we justify the one dimensional pressurized model (called P-model) introduced by the author in the context of unsteady mixed flows in closed water pipes. For non linear pressure law ( $\mathit{\gamma }\ne 1$ ), the γ-pressurized model describes the evolution of a compressible (almost) gravityless flow.