Incompressible, viscous, swirling flow through a nozzle

The viscous core of an incompressible, swirling flow through the nozzle is treated by momentum-integral equations. These equations, together with the statement of the conservation of mass in the whole nozzle and the conservation of momentum along the nozzle axis, form a system of four ordinary, nonlinear differential equations. This system of equations is solved for the circumferential and axial velocities both inside and outside the core. This system has a singularity which occurs when the square of the ratio of the circumferential to axial velocities, at core boundary, reaches a certain value depending upon the axial velocity profile. This singularity divides the possible solutions into those in which the core is massflow dominated and those in which the core is swirl dominated. The massflow-dominated core occurs at low swirl and is characterized by high discharge coefficient, which is essentially unaffected by the swirl and by a certain maximum total pressure loss which is inversely dependent on nozzle contraction. The flows with the swirl-dominated core have their discharge strongly throttled by the swirl, have the capability to sustain very large total pressure loss, and usually have regions of reversed axial velocity. The increase of Reynolds number tends to decrease the discharge coefficient, whereas the change from laminar to turbulent flow tends to increase the discharge coefficient. Both of these effects are particularly pronounced in swirl-dominated flows.