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Instability of forced planar liquid jets: mean field analysis and nonlinear simulation

Published online by Cambridge University Press:  25 November 2019

S. Schmidt Open the ORCID record for S. Schmidt [Opens in a new window]  and
K. Oberleithner
S. Schmidt*
Affiliation:
Laboratory for Flow Instability and Dynamics, Technische Universität Berlin, 10623 Berlin, Germany
K. Oberleithner
Affiliation:
Laboratory for Flow Instability and Dynamics, Technische Universität Berlin, 10623 Berlin, Germany
*
†Email address for correspondence: s.schmidt@fdx.de
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Abstract

The stability of forced planar liquid jets in a still gaseous environment is explored using nonlinear simulation and spatial linear stability analysis. Harmonic modulation of the transverse component of the inlet velocity leads to an excitation of sinuous modes in the jet. Two forcing amplitudes, 1 % and 5 %, are investigated. While for 1 % forcing, the interfacial disturbance retains a sinuous shape throughout the domain, for 5 % forcing, an increasing downstream deviation from the sinuous wave shape is found. Both forcings lead to sufficient mean flow correction to render linear stability analysis on a base flow unfeasible. Hence, an analysis on the time-averaged mean flow is performed. A correction scheme is introduced, to account for the spreading of the interface position in the mean flow. Comparison of eigenfunctions and growth rates with their counterparts from the nonlinear simulation shows an excellent agreement for 1 % forcing. For 5 % forcing, agreement of the eigenfunctions deteriorates significantly and growth rates are falsely predicted, resulting in a breakdown of the stability model. Subsequent analysis reveals a strong interaction of the fundamental wave with the second higher harmonic wave for 5 % forcing and a reversed energy flow from the coherent motion to the mean flow. These findings provide an explanation for the failure of the linear stability model for large forcing amplitudes. The study extends the applicability of mean flow stability analysis to convectively unstable planar liquid/gas jets and supports previous findings on the limits of mean flow stability, involving pronounced influence of higher harmonic modes.

JFM classification

Instability: Absolute/convective instability Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 883 , 25 January 2020 , A7
Copyright
© 2019 Cambridge University Press

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References

Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75 (5), 750–756.CrossRefGoogle Scholar
Beneddine, S., Sipp, D., Arnault, A., Dandois, J. & Lesshafft, L. 2016 Conditions for validity of mean flow stability analysis. J. Fluid Mech. 798, 485–504.CrossRefGoogle Scholar
Bobusch, B. C., Woszidlo, R., Bergada, J. M., Nayeri, C. N. & Paschereit, C. O. 2013 Experimental study of the internal flow structures inside a fluidic oscillator. Exp. Fluids 54 (6), 1559.CrossRefGoogle Scholar
Boeck, T. & Zaleski, S. 2005 Viscous versus inviscid instability of two-phase mixing layers with continuous velocity profile. Phys. Fluids 17 (3), 032106.CrossRefGoogle Scholar
Boomkamp, P. A. M. & Miesen, R. H. M. 1996 Classification of instabilities in parallel two-phase flow. Intl J. Multiphase Flow 22, 67–88.CrossRefGoogle Scholar
Boujo, E., Bauerheim, M. & Noiray, N. 2018 Saturation of a turbulent mixing layer over a cavity: response to harmonic forcing around mean flows. J. Fluid Mech. 853, 386–418.CrossRefGoogle Scholar
Brackbill, J. U., Kothe, D. B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100 (2), 335–354.CrossRefGoogle Scholar
Cheung, L. C. & Zaki, T. A. 2010 Linear and nonlinear instability waves in spatially developing two-phase mixing layers. Phys. Fluids 22 (5), 052103.CrossRefGoogle Scholar
Clark, C. J. & Dombrowski, N. 1972 Aerodynamic instability and disintegration of inviscid liquid sheets. Proc. R. Soc. Lond. A 329 (1579), 467–478.CrossRefGoogle Scholar
Cohen, J. & Wygnanski, I. 1987 The evolution of instabilities in the axisymmetric jet. Part 1. The linear growth of disturbances near the nozzle. J. Fluid Mech. 176, 191–219.CrossRefGoogle Scholar
Crighton, D. G. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77 (2), 397–413.CrossRefGoogle Scholar
Cummins, S. J., Francois, M. M. & Kothe, D. B. 2005 Estimating curvature from volume fractions. Comput. Struct. 83 (6), 425–434.CrossRefGoogle Scholar
De Luca, L. 2001 Non-modal growth of disturbances in free-surface flows. In Proceedings of International Conference RDAMM-2001, vol. 6.Google Scholar
Delbende, I., Chomaz, J. & Huerre, P. 1998 Absolute/convective instabilities in the batchelor vortex: a numerical study of the linear impulse response. J. Fluid Mech. 355, 229–254.CrossRefGoogle Scholar
Dongarra, J. J., Straughan, B. & Walker, D. W. 1996 Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems. Appl. Numer. Maths 22 (4), 399–434.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge Mathematical Library 1. Cambridge University Press.CrossRefGoogle Scholar
Emerson, B., Lieuwen, T. & Juniper, M. P. 2016 Local stability analysis and eigenvalue sensitivity of reacting bluff-body wakes. J. Fluid Mech. 788, 549–575.CrossRefGoogle Scholar
Francois, M. M., Cummins, S. J., Dendy, E. D., Kothe, D. B., Sicilian, J. M. & Williams, M. W. 2006 A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework. J. Comput. Phys. 213 (1), 141–173.CrossRefGoogle Scholar
Gordillo, J. M. & Pérez-Saborid, M. 2005 Aerodynamic effects in the break-up of liquid jets: on the first wind-induced break-up regime. J. Fluid Mech. 541, 1–20.CrossRefGoogle Scholar
Hagerty, W. & Shea, J. F. 1955 A study of the stability of moving liquid film. ASME J. Appl. Mech. 22, 509–514.Google Scholar
Harvie, D. J. E., Davidson, M. R. & Rudman, M. 2006 An analysis of parasitic current generation in volume of fluid simulations. Appl. Math. Model. 30 (10), 1056–1066.CrossRefGoogle Scholar
Herbert, T. 1997 Parabolized stability equations. Annu. Rev. Fluid Mech. 29 (1), 245–283.CrossRefGoogle Scholar
Hirt, C. W. & Nichols, B. D. 1981 Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39 (1), 201–225.CrossRefGoogle Scholar
van Hooft, J. A., Popinet, S., van Heerwaarden, C. C., van der Linden, S. J. A., de Roode, S. R. & van de Wiel, B. J. H. 2018 Towards adaptive grids for atmospheric boundary-layer simulations. Boundary-Layer Meteorol. 167 (3), 421–443.CrossRefGoogle ScholarPubMed
Jarrahbashi, D. & Sirignano, W. A. 2014 Vorticity dynamics for transient high-pressure liquid injection. Phys. Fluids 26 (10), 101304.CrossRefGoogle Scholar
Krüger, O., Bobusch, B. C., Woszidlo, R. & Paschereit, C. O. 2013 Numerical modeling and validation of the flow in a fluidic oscillator. In 21st AIAA Computational Fluid Dynamics Conference, AIAA Paper 2013-3087.Google Scholar
Li, X. 1993 Spatial instability of plane liquid sheets. Chem. Engng Sci. 48 (16), 2973–2981.CrossRefGoogle Scholar
Li, X. & Tankin, R. S. 1991 On the temporal instability of a two-dimensional viscous liquid sheet. J. Fluid Mech. 226, 425–443.CrossRefGoogle Scholar
Lin, S. P., Lian, Z. W. & Creighton, B. J. 1990 Absolute and convective instability of a liquid sheet. J. Fluid Mech. 220, 673–689.CrossRefGoogle Scholar
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2014 Self-consistent mean flow description of the nonlinear saturation of the vortex shedding in the cylinder wake. Phys. Rev. Lett. 113 (8), 084501.CrossRefGoogle ScholarPubMed
Marmottant, P. & Villermaux, E. 2004 On spray formation. J. Fluid Mech. 498, 73–111.CrossRefGoogle Scholar
Mehring, C. & Sirignano, W. A. 1999 Nonlinear capillary wave distortion and disintegration of thin planar liquid sheets. J. Fluid Mech. 388, 69–113.CrossRefGoogle Scholar
Noack, B. R., Afanasiev, K., Morzyński, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.CrossRefGoogle Scholar
Oberleithner, K., Paschereit, C. O. & Wygnanski, I. 2014a On the impact of swirl on the growth of coherent structures. J. Fluid Mech. 741, 156–199.CrossRefGoogle Scholar
Oberleithner, K., Rukes, L. & Soria, J. 2014b Mean flow stability analysis of oscillating jet experiments. J. Fluid Mech. 757, 1–32.CrossRefGoogle Scholar
Orszag, S. A. & Crow, S. C. 1970 Instability of a vortex sheet leaving a semi-infinite plate. Stud. Appl. Maths 49 (2), 167–181.CrossRefGoogle Scholar
Otto, T.2012 Spatio-temporal stability analysis in two-phase mixing layers. PhD thesis, Technische Universität Ilmenau.Google Scholar
Pier, B. 2002 On the frequency selection of finite-amplitude vortex shedding in the cylinder wake. J. Fluid Mech. 458, 407–417.CrossRefGoogle Scholar
Popinet, S. 2003 Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190 (2), 572–600.CrossRefGoogle Scholar
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228 (16), 5838–5866.CrossRefGoogle Scholar
Rayleigh, Lord 1878 On the instability of jets. Proc. Lond. Math. Soc. s1‐10 (1), 4–13.CrossRefGoogle Scholar
Reynolds, W. C. & Hussain, A. K. M. F. 1972 The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments. J. Fluid Mech. 54 (2), 263–288.CrossRefGoogle Scholar
Scardovelli, R. & Zaleski, S. 1999 Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 31 (1), 567–603.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2006 Grenzschicht-theorie. Springer.Google Scholar
Schmidt, S., Krüger, O., Göckeler, K. & Paschereit, C. O. 2018 Numerical investigation of the breakup behavior of an oscillating two-phase jet. Phys. Fluids 30 (7), 072101.CrossRefGoogle Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358.CrossRefGoogle Scholar
Söderberg, L. D. 2003 Absolute and convective instability of a relaxational plane liquid jet. J. Fluid Mech. 493, 89–119.CrossRefGoogle Scholar
Squire, H. B. 1953 Investigation of the instability of a moving liquid film. British J. Appl. Phys. 4 (6), 167–169.CrossRefGoogle Scholar
Tammisola, O., Lundell, F. & Söderberg, L. D. 2012 Surface tension-induced global instability of planar jets and wakes. J. Fluid Mech. 713, 632–658.CrossRefGoogle Scholar
Tammisola, O., Sasaki, A., Lundell, F., Matsubara, M. & Söderberg, L. D. 2011 Stabilizing effect of surrounding gas flow on a plane liquid sheet. J. Fluid Mech. 672, 5–32.CrossRefGoogle Scholar
Terhaar, S., Oberleithner, K. & Paschereit, C. O. 2015 Key parameters governing the precessing vortex core in reacting flows: an experimental and analytical study. Proc. Combust. Inst. 35 (3), 3347–3354.CrossRefGoogle Scholar
Torrey, M. D., Cloutman, L. D., Mjolsness, R. C. & Hirt, C. W.1985 NASA-VOF2D: A computer program for incompressible flows with free surfaces. Tech. Rep. Los Alamos National Lab.Google Scholar
Tryggvason, G., Scardovelli, R. & Zaleski, S. 2011 Direct Numerical Simulations of Gas–Liquid Multiphase Flows. Cambridge University Press.CrossRefGoogle Scholar
Turton, S. E., Tuckerman, L. S. & Barkley, D. 2015 Prediction of frequencies in thermosolutal convection from mean flows. Phys. Rev. E 91 (4), 043009.Google ScholarPubMed
Yecko, P. & Zaleski, S. 2005 Transient growth in two-phase mixing layers. J. Fluid Mech. 528, 43–52.CrossRefGoogle Scholar
Yih, C. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27 (2), 337–352.CrossRefGoogle Scholar
Zandian, A., Sirignano, W. A. & Hussain, F. 2018 Understanding liquid-jet atomization cascades via vortex dynamics. J. Fluid Mech. 843, 293–354.CrossRefGoogle Scholar