Local Friction Factor of Compressible Laminar or Turbulent Flow in Micro-Tubes

Laminar/turbulent flows of compressible fluid in microtubes were simulated numerically to obtain the effect of compressibility on the local pipe friction factors. For gaseous flows, the effect of compressibility had not been clarified except for laminar flow whose Mach number is less than 0.45, so the present work extended this to handle higher speed flows including choked ones and turbulent flows. The numerical procedure based on arbitrary-Lagrangian-Eulerian method solves two-dimensional compressible momentum and energy equations. The Lam-Bremhorst Low-Reynolds number turbulence model was adopted to calculate eddy viscosity coefficient and turbulence energy. The physical domain of simulation with the back region downstream from the outlet of the micro-tube was used to be able to calculate the case of under-expansion flow in the tube. The orthogonal curvilinear grid was used for the computational mesh to obtain accurate results. The computations were performed for a wide range of Reynolds number and Mach number including laminar/turbulent choked flows. It was found that in laminar regimes the ratio of the Darcy friction factor to its conventional (incompressible flow’s) value is a function of Mach number and the same goes for the Fanning friction factor. On the other hand, in turbulent regimes, the ratio is still a function of Mach number for the Darcy friction factor but is equal to about unity for the Fanning friction factor. Namely, the Fanning friction factor of gaseous flow in micro-tubes coincides with Blasius formula, even when Mach number is not small and compressibility effect appears. This fact can be seen in choked flow.