On Boussinesq and Coriolis coefficients and implications for the Joukowsky equation

The purpose of this paper is to show how derivatives of the Boussinesq and Coriolis- coefficients, β and α, can be handled formally in 1-D analyses of unsteady flow. In the case of low Mach number flows typical of liquid flows in many pipes, it is usual to
disregard differences between these coefficients and unity, thereby simplifying expressions such as the Joukowsky equation. When this is deemed to be unacceptable – e.g. in moderate and high Mach number flows – a different approach is usually followed,
namely allowing for the actual values of the coefficients, but disregarding derivatives of them. It is shown herein that this approach is not only unnecessary, but is actually less accurate than disregarding the coefficients altogether (i.e. using plug-flow
approximations). Mathematically, the new result is obtained by deriving expressions that relate derivatives of β and α to derivatives of the principal flow parameters (pressure p, density ρ and mean velocity U). Because these relationships involve derivatives, they do
not enable actual values of β and α to be deduced. However, it is shown rigorously that inertial waves do not change the product ρ^2 U^2 (β -1) and so, if β is known a priori before a wave-induced velocity change, its value after the change can be deduced.