Velocity profile development and friction in compressible micro-flows

From Poiseuille theory, it is known that incompressible laminar fully-developed flow of a Newtonian fluid in a constant cross-section channel is characterised by steady parabolic velocity profiles after a fully-developed flow condition is attained. In turbulent fully-developed flow the velocity profiles are non-parabolic and become more flat for higher Reynolds numbers. When the incompressible hypothesis does not hold, as in the case of high velocity ideal gas flow, the velocity profile becomes flatter, as if more turbulent, due to the superposition of compressibility and turbulence effects, if applicable. This is typical in micro-channel flows, where pressure gradients are high and the gas is rapidly accelerating, eventually up to the sound velocity. As the flow accelerates the effects of compressibility grow stronger and the velocity profile keeps changing shape. The radial velocity component does not zero as in fully-developed flow but reverses after the entrance effects have damped out and grows with the Mach number. A net mass transfer toward the walls is thus generated making the velocity profile more flat. This affects the friction factor which is no longer constant, being proportional to the normal-to-wall velocity gradient, and needs to be evaluated. In the present work, the compressible friction factor is numerically investigated and correlations are proposed based on the velocity profile shape evolution as a function of the Mach number. This, together with other considerations on the velocity profile shape change, is shown to enhance the predictive capability of the Fanno theory for compressible flows.