Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T18:03:03.777Z Has data issue: false hasContentIssue false
  • English
  • Français

A mechanism for the amplification of interface distortions on liquid jets

Published online by Cambridge University Press:  01 February 2021

Hanul Hwang Open the ORCID record for Hanul Hwang [Opens in a new window] ,
Parviz Moin Open the ORCID record for Parviz Moin [Opens in a new window]  and
M.J. Philipp Hack Open the ORCID record for M.J. Philipp Hack [Opens in a new window]
Hanul Hwang*
Affiliation:
Center for Turbulence Research, Stanford University, Stanford, CA94305, USA
Parviz Moin
Affiliation:
Center for Turbulence Research, Stanford University, Stanford, CA94305, USA
M.J. Philipp Hack
Affiliation:
Center for Turbulence Research, Stanford University, Stanford, CA94305, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A novel mechanism for the amplification of distortions to the material interface of liquid jets is identified. The mechanism is independent of the exponential instability of the flow and can intensify small perturbations to the material interface by several orders of magnitude. Depending on the parameters, it can amplify interfacial distortions at a faster pace than modal mechanisms such as the Kelvin–Helmholtz instability. The study is based on spatial linear stability theory in a two-fluid formulation that accounts for the effects of both viscosity and surface tension. The analysis of the mechanism is cast into an optimization problem in the surface tension energy of the interface distortion and discounts the trivial redistribution of perturbation kinetic energy. The identified mechanism is related to the Orr mechanism, and amplifies distortions to the material interface via a reorientation of perturbations by the mean shear. Analyses of the linearized energy budgets show that energy is extracted from the mean shear by the production term of the streamwise perturbation velocity component and subsequently transferred to the radial perturbation velocity component, where it is absorbed by the surface tension potential of the interface. The gain in surface tension energy attributable to the mechanism is shown to scale linearly with the Reynolds number. A critical Weber number is identified as a lower bound beyond which the mechanism becomes active, and a power-law relation to the Reynolds number is established. Nonlinear simulations based on the full two-fluid Navier–Stokes equations substantiate the observability and realizability of the mechanism.

JFM classification

Wakes/Jets: Jets
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 911 , 25 March 2021 , A51
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bague, A., Fuster, D., Popinet, S., Scardovelli, R. & Zaleski, S. 2010 Instability growth rate of two-phase mixing layers from a linear eigenvalue problem and an initial-value problem. Phys. Fluids 22 (9), 092104.CrossRefGoogle Scholar
Boronin, S.A., Healey, J.J. & Sazhin, S.S. 2013 Non-modal stability of round viscous jets. J. Fluid Mech. 716 (11), 96–119.CrossRefGoogle Scholar
Butler, K. & Farrell, B.F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4 (8), 1637–1650.CrossRefGoogle Scholar
Chigier, N. & Eroglu, H. 1989 Atomization of liquid jets from injection element in liquid rocket combustion chamber. Second Quarterly Report to NASA.Google Scholar
Cruz-Mazo, F., Herrada, M.A., Gañán-Calvo, A.M. & Montanero, J.M. 2017 Global stability of axisymmetric flow focusing. J. Fluid Mech. 832, 329–344.CrossRefGoogle Scholar
Duda, J.L. & Vrentas, J.S. 1967 Fluid mechanics of laminar liquid jets. Chem. Engng Sci. 22 (6), 855–869.CrossRefGoogle Scholar
Farrell, B. 1987 Developing disturbances in shear. J. Atmos. Sci. 44 (16), 2191–2199.2.0.CO;2>CrossRefGoogle Scholar
Foures, D.P.G., Caulfield, C.P. & Schmid, P.J. 2012 Variational framework for flow optimization using seminorm constraints. Phys. Rev. 86, 026306.Google ScholarPubMed
Fuster, D., Matas, J.-P., Marty, S., Popinet, S., Hoepffner, J., Cartellier, A. & Zaleski, S. 2013 Instability regimes in the primary breakup region of planar coflowing sheets. J. Fluid Mech. 736, 150–176.CrossRefGoogle Scholar
González-Mendizabal, D., Olivera-Fuentes, C. & Guzmán, J.M. 1987 Hydrodynamics of laminar liquid jets, experimental study and comparison with two models. Chem. Engng Commun. 56, 117–137.CrossRefGoogle Scholar
Gustavsson, L.H. 1991 Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224 (1), 241–260.CrossRefGoogle Scholar
Hack, M.J.P. & Moin, P. 2017 Algebraic disturbance growth by interaction of Orr and lift-up mechanisms. J. Fluid Mech. 829, 112–126.CrossRefGoogle Scholar
Hack, M.J.P. & Zaki, T.A. 2015 Modal and non-modal stability of boundary layers forced by spanwise wall oscillations. J. Fluid Mech. 778, 389–427.CrossRefGoogle Scholar
Hinch, E.J. 1984 A note on the mechanism of the instability at the interface between two shearing fluids. J. Fluid Mech. 144, 463–465.CrossRefGoogle Scholar
Hoyt, J.W. & Taylor, J.J. 1977 Waves on water jets. J. Fluid Mech. 83 (1), 119–127.CrossRefGoogle Scholar
Khorrami, M.R., Malik, M.R. & Ash, R.L. 1989 Application of spectral collocation techniques to the stability of swirling flows. J. Comput. Phys. 81 (1), 206–229.CrossRefGoogle Scholar
Landahl, M.T. 1975 Wave breakdown and turbulence. SIAM J. Appl. Maths 28 (4), 735–756.CrossRefGoogle Scholar
Landahl, M.T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98 (2), 243–251.CrossRefGoogle Scholar
Lin, S.P. & Chen, J.N. 1998 Role played by the interfacial shear in the instability mechanism of a viscous liquid jet surrounded by a viscous gas in a pipe. J. Fluid Mech. 376, 37–51.CrossRefGoogle Scholar
Lin, S.P. & Ibrahim, E.A. 1990 Instability of a viscous liquid jet surrounded by a viscous gas in a vertical pipe. J. Fluid Mech. 218, 641–658.CrossRefGoogle Scholar
Lozano, A., Barreras, F., Hauke, G. & Dopazo, C. 2001 Longitudinal instabilities in an air-blasted liquid sheet. J. Fluid Mech. 437, 143–173.CrossRefGoogle Scholar
Malik, S.V. & Hooper, A.P. 2007 Three-dimensional disturbances in channel flows. Phys. Fluids 19 (5), 052102.CrossRefGoogle Scholar
Marmottant, P. & Villermaux, E. 2004 On spray formation. J. Fluid Mech. 498, 73–111.CrossRefGoogle Scholar
Orr, W.M'F. 1907 The stability or instability of the stead motions of a perfect liquid and of a viscous liquid. Part II: a viscous liquid. Proc. R. Irish Acad. 27, 9–68.Google Scholar
Popinet, S. 2015 A quadtree-adaptive multigrid solver for the Serre-Green-Naghdi equations. J. Comput. Phys. 302, 336–358.CrossRefGoogle Scholar
Popinet, S. 2018 Numerical models of surface tension. Annu. Rev. Fluid Mech. 50, 49–75.CrossRefGoogle Scholar
Preziosi, L., Chen, K. & Joseph, D.D. 1989 Lubricated pipelining: stability of core-annular flow. J. Fluid Mech. 201, 323–356.CrossRefGoogle Scholar
Reddy, S.C. & Henningson, D.S. 1993 Energy growth in viscous channel flow. J. Fluid Mech. 252, 209–238.CrossRefGoogle Scholar
Renardy, Y. 1987 The thin-layer effect and interfacial stability in a two-layer couette flow with similar liquids. Phys. Fluids 30 (6), 1627–1637.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 1994 Optimal energy density growth in Hagen-Poiseuille flow. J. Fluid Mech. 277, 197–225.CrossRefGoogle Scholar
Shkadov, V.Ya. & Sisoev, G.M. 1996 Instability of a two-layer capillary jet. Intl J. Multiphase Flow 22 (2), 363–377.CrossRefGoogle Scholar
Söderberg, L.D. 2003 Absolute and convective instability of a relaxational plane liquid jet. J. Fluid Mech. 493, 89–119.CrossRefGoogle Scholar
Söderberg, L.D. & Alfredsson, P.H. 1998 Experimental and theoretical investigations of plane liquid jets. Eur. J. Mech. B/Fluids 17 (5), 689–737.CrossRefGoogle Scholar
South, M.J. & Hooper, A.P. 1999 Linear growth in two-fluid plane Poiseuille flow. J. Fluid Mech. 381, 121–139.CrossRefGoogle Scholar
Trefethen, L.N., Trefethen, A.E., Reddy, S.C. & Driscoll, T.A. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578–584.CrossRefGoogle ScholarPubMed
van Noorden, T.L., Boomkamp, P.A.M., Knaap, M.C. & Verheggen, T.M.M. 1998 Transient growth in parallel two-phase flow: analogies and differences with single-phase flow. Phys. Fluids 10 (8), 2099–2101.CrossRefGoogle Scholar
Yecko, P. & Zaleski, S. 2005 Transient growth in two-phase mixing layers. J. Fluid Mech. 528, 43–52.CrossRefGoogle Scholar
Yecko, P., Zaleski, S. & Fullana, J.M. 2002 Viscous modes in two-phase mixing layers. Phys. Fluids 14 (12), 4115–4122.CrossRefGoogle Scholar
Yih, C.S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27 (2), 337–352.CrossRefGoogle Scholar