On the flow field induced by a hovering rotor or a static jet

Abstract

The flow in the far field of an isolated static momentum source is considered, taking into account the entrainment of fluid by the turbulent jet which develops far downstream irrespective of the type of device. The result is a simple analytical model for the irrotational region, which depends only on the thrust applied. This equation is implied by Stewart (J. Fluid Mech., vol. 1, 1956, pp. 593–606) for a jet. For a rotor, the model is radically different from the classical one derived from an actuator disk without turbulence or mixing in the wake, which led to a sink flow in the far field. The velocities decay like rather than , where is the radius, and are everywhere directed in the direction opposite to the thrust, rather than pointing towards the origin. The momentum source drives a co-flow which converges towards the turbulent region, thus supplying the entrained fluid. This flow pattern supports the assumption that the fluid surrounding the turbulent region is irrotational, better than the sink-flow model would. The model depends only on one empirical constant, a measure of the entrainment in a fully developed jet, for which a range of values is determined from the experimental literature. If the rotor is climbing, the sink flow is recovered; however, the limit of that equation as the climb velocity tends to zero, leading to hover, is singular. For both jets and rotors, this model used in a boundary condition should eliminate extraneous parameters and reduce the computational cost of numerical simulations, and may guide the design of chambers used for experiments, following Ricou & Spalding (J. Fluid Mech., vol. 11, 1960, pp. 21–32).