Abstract
The aim of this work is to study the nonlinear temporal evolution of an alternating swirling streaming jet. According to laboratory observations, the viscous dissipation is simulated by means of a phenomenological damping coefficient included in the equation of motion (Jiang et al. in J Fluid Mech 369:273–299, 1998; Wright et al. in J Fluid Mech 402:1–32, 2000) and added to the Bernoulli equation or to the evolution equation. The multiple scales are used for finding out the evolution equations. The fixed points of the solutions have been determined. Really, the modulation concept is exploited in order to analyze the stability criteria in the possible cases of resonance. While in the non-resonant case, the nontrivial solutions are obtained numerically. Different numerical applications have been considered. The studied cases have showed that the Weber number, the phenomenological number and the streamwise circulation play a significant role in determining the dynamics of the developing interfacial patterns.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Jiang L., Ting C.L., Perlin M., Schultz W.W.: Period tripling and energy dissipation of breaking standing waves. J. Fluid Mech. 369, 273–299 (1998)
Wright J., Yon S., Pozrikidis C.: Numerical studies of two-dimensional Faraday oscillations of inviscid fluids. J. Fluid Mech. 402, 1–32 (2000)
Panda I., Mclaughlin D.K.: Experiments on the instabilities of a swirling jet. Phys. Fluids 6, 263–276 (1994)
Liao Y., Jeng S.M., Jog M.A., Benjamin M.A.: Instability of an annular liquid sheet surrounded by swirling airstreams. AIAA J. 38, 453–460 (2000)
Vanierschot M., van den Bulck E.: Hysteresis in flow patterns in annular swirling jets. Exp. Therm. Fluid Sci. 31, 513–524 (2007)
Weber C.: Zum zerfall eines flussigkeitsstrahles. Z. Angew. Math. Mech. 11, 138–145 (1931)
Drazin P.G., Reid W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (1981)
Martin J.E., Meiburg E.: On the stability of the swirling jet shear layer. Phys. Fluids 6, 424–426 (1994)
Martin J.E., Meiburg E.: The growth and nonlinear evolution of helical perturbations in a swirling jet model. Eur. J. Mech.-B/Fluids 17, 639–651 (1998)
Park H., Heister S.D.: Nonlinear simulation of free surfaces and atomization in pressure swirl atomizers. Phys. Fluids 18, 052–103 (2006)
Park H., Yoon S.S., Heister S.D.: On the nonlinear stability of a swirling liquid jet. In. J. Multi phase Flow 32, 1100–1109 (2006)
Zakaria K.: Nonlinear dynamics of magnetic fluids with a relative motion in the presence of an oblique magnetic field. Physica A 327, 221–248 (2003)
Zakaria K.: Wilton ripples between two uniform streaming magnetic fluids. Int. J. Non-Linear Mech. 39, 66–1051 (2004)
Zakaria K.: Standing waves between immiscible liquids inside an infinite boxed basin. J. Phys. A Math. Theor. 41, 175501–175521 (2008)
Sirwah M.: Non-linear temporal dynamics of two-mode interactions of magnetized flow. Int. J. Non-Linear Mech. 43, 416–436 (2008)
Zakaria K., Sirwah M., Alkharashi S.: Nonlinear behavior of creeping flow on the inclined permeable substrate plane subjected to an electric field. Int. J. Non-Linear Mech. 47(6), 577–598 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zakaria, K., Sirwah, M.A. & Adham, M. Nonlinear evolution of a column swirling jet. Z. Angew. Math. Phys. 64, 811–830 (2013). https://doi.org/10.1007/s00033-012-0266-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-012-0266-0