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.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The authors gratefully acknowledge the support of the U.S. Bureau of Reclamation.
References
Aggarwal, S. K., and Peng, F. (1995). “A review of droplet dynamics and vaporization modeling for engineering calculations.” J. Eng. Gas Turbines Power, 117(3), 453–461.
Arya, S. P. (1988). Introduction to micrometeorology, Academic Press, New York.
Carrion, P., Tarjuelo, J. M., Montero, J. (2001). “SIRIAS: A simulation model for sprinkler irrigation. I: Description.” Irrig. Sci., 20(2), 73–84.
Fukui, Y., Nakanishi, K., and Okamura, S. (1980). “Computer simulation of sprinkler irrigation uniformity.” Irrig. Sci., 2(1), 23–32.
Granger, R. A. (1995). Fluid mechanics, Dover Publications, New York.
Jourdan, G., Houas, L., Igra, O., Estivalezes, J. L., Devals, C., and Meshkov, E. (2007). “Drag coefficient of a sphere in a non-stationary flow: new result.” Proc. R. Soc. A, 463, 3323–3345.
Karanfilian, S. K., and Kotas, T. J. (1978). “Drag on a sphere in unsteady motion in a liquid at rest.” J. Fluid. Mech., 87(1), 85–96.
Kim, I., Elghobashi, S., and Sirignano, W. A. (1998). “On the equation for spherical-particle motion: Effect of Reynolds number and acceleration number.” J. Fluid Mech., 367, 221–253.
Kincaid, D. C., and Longley, T. S. (1989). “A water droplet evaporation and temperature model.” Trans. ASAE, 32(2), 457–462.
Kincaid, D. C., Solomon, K. H., and Oliphant, J. C. (1996). “Drop size distribution for irrigation sprinklers.” Trans. ASAE, 39(3), 839–845.
Kohl, R. A. (1974). “Drop size distribution from medium-sized agricultural sprinklers.” Trans ASAE, 17(4), 690–693.
Laws, J. O. (1941). “Measurement of the fall velocity of water-drops and raindrops.” Trans. Am. Geophys. Union, 22(3), 709–721.
Odar, F., and Hamilton, W. S. (1964). “Forces on a sphere accelerating in a viscous fluid.” J. Fluid Mech., 18(2), 302–314.
Playan, E., et al. (2009). “Mathematical problems and solutions in sprinkler irrigation.” 〈http://www.unizar.es/acz/05Publicaciones/Monografias/MonografiasPublicadas/Monografia31/153.pdf〉 (Sep. 2015).
Salvador, R., Bautista-Capetillo, C., Burguete, J., Zapata, N., Serreta, A., and Playan, E. (2009). “A photographic method for drop characterization in agricultural sprinklers.” Irrig. Sci., 27(4), 307–317.
Seginer, I., Nir, D., and von Bernuth, R. D. (1991). “Simulation of wind distorted sprinkler patterns.” J. Irrig. Drain. Eng., 285–306.
Shames, I. H. (1966). “Engineering mechanics.” Dynamics, Vol. II, Prentice Hall, Upper Saddle River, NJ.
Silva, W., and James, L. G. (1988). “Modeling evaporation and microclimate changes in sprinkle irrigation: I. Model formulation and calibration.” Trans. ASAE, 31(5), 1481–1486.
Sirignano, W. A. (2010). Fluid dynamics and transport of droplets and sprays, Cambridge University Press, New York.
Soutas-Little, R. W., and Inman, D. J. (1999). Engineering mechanics, dynamics, Prentice Hall, Upper Saddle River, NJ.
Temkin, S., and Kim, S. S. (1980). “Droplet motion induced by weak shock waves.” J. Fluid Mech., 96(1), 133–157.
Temkin, S., and Mehta, H. K. (1982). “Droplet drag in accelerating and decelerating flow.” J. Fluid Mech., 116(1), 297–313.
Thompson, A. L., Gilley, J. R., and Norman, J. M. (1993). “A sprinkler water droplet evaporation and plant canopy model. I: Model development.” Trans. ASAE, 36(3), 735–741.
Vennard, J. K. (1940). Elementary fluid mechanics, Wiley, New York.
von Bernuth, R. D., and Gilley, J. R. (1984). “Sprinkler droplet size distribution estimation from single leg test data.” Trans. ASAE, 27(5), 1435–1441.
Vories, E. D., von Bernuth, R. D., and Michelson, R. H. (1987). “Simulating sprinkler performance in wind.” J. Irrig. Drain. Eng., 119–130.
Zerihun, D., and Sanchez, C. A. (2014). “Sprinkler irrigation precipitation pattern simulation model: Development and evaluation.” USBR.
Zerihun, D., and Sanchez, C. A., and Warrick, A. W. (2015). “Sprinkler irrigation droplet dynamics. II: Numerical solution and model evaluation.” J. Irrig. Drain. Eng., 04016008.
Information & Authors
Information
Published In
Copyright
© 2016 American Society of Civil Engineers.
History
Received: May 13, 2015
Accepted: Oct 27, 2015
Published online: Feb 1, 2016
Published in print: May 1, 2016
Discussion open until: Jul 1, 2016
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.