On Boussinesq and Coriolis coefficients and their derivatives in transient pipe flows

A rigorous analytical justification is developed for a simplification that is widely used in one-dimensional simulations of steady and unsteady flows in pipes, namely treating the flow as a 'plug flow' in which cross-sectional variations in axial velocity are neglected except for consequential shear forces on pipe walls. The proof is obtained without assuming that the flow is nearly incompressible flow and, indeed, it is found that the plug flow approximation remains good even for compressible flows at moderate, subsonic speeds. In more general analyses, explicit allowance is sometimes made for the influence of (i) the Boussinesq coefficient β and (ii) its axial rate of change ∂β/∂x. Typically, such analyses discard the terms in ∂β/∂x in the basic equations and proceed using β alone. This is done on the grounds that no method of evaluating ∂β/∂x is available. It is shown in this paper that discarding ∂β/∂x is not only unnecessary, but that it actually leads to less accurate outcomes than simply assuming plug flow. The process used to derive the analytical justification has a spin-off benefit of shedding light on alternative methods of integrating source terms over the pipe cross-section. Although the primary purpose of the paper is to demonstrate that the use of plug flow approximations can be justified rigorously for most flows, brief attention is also paid to cases where it has potential to mislead.