Sprinkler Irrigation Droplet Dynamics. I: Review and Theoretical Development

Abstract

Droplet dynamics simulations are key to predicting sprinkler irrigation precipitation patterns. This paper includes derivations of equations describing droplet motion through a steady, uniform horizontal airflow (wind). The assumptions on which sprinkler irrigation droplet dynamics is based are stated, and the limitations they entail are highlighted. The motion of droplets is treated as an impulsively started accelerated motion of rigid spheres, originating at the sprinkler nozzle with known initial conditions, and involving no interactions between themselves. The following steps are used in the derivation of pertinent equations. First, the forces that the ambient air exerts on a water droplet undergoing a steady or accelerated rectilinear relative motion are defined, and their significance in the context of sprinkler irrigation droplet dynamics is discussed, based on which relevant equations are derived. This is followed by a discussion on the dynamics of accelerated motion of a water droplet through a quiescent ambient air. Then, the more general case of the dynamics of the motion of a droplet undergoing unsteady three-dimensional curvilinear motion under wind is discussed, the major forces acting on a droplet are defined, the type of droplet motion they produce is described, and pertinent equations are derived. An important feature of these equations, distinct from earlier approaches, is the manner in which the effect of wind on droplet motion is taken into account. The wind-induced aerodynamic forces, acting on a droplet, are differentiated into tangential and normal drag. The normal drag is shown to be responsible for the curvilinear droplet motion produced by wind, an important component of the wind drift effects on droplet motion. Wind effects on the tangential drag force, on the other hand, are shown to be represented in terms of the damping or amplification effects that wind introduces to the attenuation of the droplet absolute velocity compared with an equivalent no-wind condition. A companion paper presents a numerical solution for the droplet dynamics equations presented here and results of model evaluation.