Laminar swirling slender pipe flows

Swirling flows are extensively used to enhance mixing in chemical and physical processes. Although many fundamental studies involve decaying laminar swirls in circular ducts with constant area-for which several theoretical descriptions exist-converging and diverging pipe sections are often employed, demanding the extension of the analysis to swirling laminar flows in pipes with varying areas. This work analytically investigates axial swirl decay in laminar axisymmetric flows in converging and diverging pipes at moderate Reynolds numbers. The theoretical model extends the classical theory for laminar low swirling flows in straight pipes with impermeable walls to pipes with slowly varying areas. Inlet Reynolds number and converging/diverging angle effects are analyzed for incompressible axisymmetric laminar flows in slender pipes (lengths much greater than the radius). Results show that diverging pipes enhance the swirl decay rate, whereas converging systems have a weakening effect for flows with the same inlet Reynolds number. Conversely, increasing the inlet angle in converging flows decreases the decay rate, whereas larger diverging pipe inlet angles promote swirl decay. A simplified eigenvalue analysis shows that the relationship between the swirl decay lengths in diverging ( & ell;(d,div)), straight ( & ell;(d,st)), and converging ( & ell;(d,conv)) pipes is given by & ell;(d,div)<& ell;(d,st)<& ell;(d,conv) for flows with similar inlet Reynolds numbers. Comparisons with numerical simulations show that the model predictions are almost exact for nondimensional pipe lengths up to 5x10(-2) times the inlet Reynolds number. Approximating the axial velocity profile through a fully developed function representative of laminar flows with heat transfer through the walls shows that the narrower axial velocity profile in diverging flows increases viscous effects, enhancing the swirl decay rate. On the contrary, the flatter axial velocity profile in converging pipes further extends the decay length.