No CrossRef data available.
Article contents
On the role of unsteadiness in impulsive-flow-driven shear instabilities: precursors of fragmentation
Published online by Cambridge University Press: 23 October 2023
Abstract
Shear instabilities at the interface of two fluids, such as classical Kelvin–Helmholtz instability (KHI), is the precursor of interface destabilization, leading to fluid fragmentation critical in a wide range of applications. While many insights into such instabilities are derived for steady background forcing flow, unsteady impulse flows are ubiquitous in environmental and physiological processes. Yet, little is understood on how unsteadiness shapes the initial interface amplification necessary for the onset of its topological change enabling subsequent fragmentation. In this combined theoretical, numerical and experimental study, we focus on an air-on-liquid interface exposed to canonical unsteady shear flow profiles. Evolution of the perturbed interface is formulated theoretically as an impulse-driven initial value problem using both linearized potential flow and nonlinear boundary integral methods. We show that the unsteady airflow forcing can amplify the interface's inherent gravity–capillary wave, up to wave-breaking transition, even if the configuration is classically KH stable. For impulses much shorter than the gravity–capillary wave period, it is the cumulative action, akin to total energy, that determines amplification, independent of the details of the impulse profile. However, for longer impulses, the details of the impulse profile become important. In this limit, akin to a resonance, it is the entangled history of the interaction of the forcing, i.e. the impulse, that changes rapidly in amplitude, and the response of the oscillating interface that matters. The insights gained are discussed and experimentally illustrated in the context of interface distortion and destabilization relevant for upper respiratory mucosalivary fluid fragmentation in violent exhalations.
- Type
- JFM Papers
- Information
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press
References
Abramowitz, M. & Stegun, I.A. 1972 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55. Courier Corporation.Google Scholar
Baker, G.R. & Beale, J.T. 2004 Vortex blob methods applied to interfacial motion. J. Comput. Phys. 196 (1), 233–258.CrossRefGoogle Scholar
Baker, G.R., Meiron, D.I. & Orszag, S.A. 1982 Generalized vortex methods for free-surface flow problems. J. Fluid Mech. 123, 477–501.CrossRefGoogle Scholar
Baker, G. & Nachbin, A. 1998 Stable methods for vortex sheet motion in the presence of surface tension. SIAM J. Sci. Comput. 19 (5), 1737–1766.CrossRefGoogle Scholar
Baker, G.R. & Pham, L.D. 2006 A comparison of blob methods for vortex sheet roll-up. J. Fluid Mech. 547, 297–316.CrossRefGoogle Scholar
Basser, P.J., McMahon, T.A. & Griffith, P. 1989 The mechanism of mucus clearance in cough. J. Biomech. Engng 111 (4), 288–297.CrossRefGoogle ScholarPubMed
Bender, C.M. & Orszag, S. 1999 Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, vol. 1. Springer Science & Business Media.CrossRefGoogle Scholar
Berné, O., Marcelino, N. & Cernicharo, J. 2010 Waves on the surface of the orion molecular cloud. Nature 466 (7309), 947–949.CrossRefGoogle ScholarPubMed
Betchov, R. & Szewczyk, A. 1963 Stability of a shear layer between parallel streams. Phys. Fluids 6, 1391–1396.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
Bourouiba, L. 2020 Turbulent gas clouds and respiratory pathogen emissions: potential implications for reducing transmission of COVID-19. JAMA 323 (18), 1837–1838.Google ScholarPubMed
Bourouiba, L. 2021 a The fluid dynamics of disease transmission. Annu. Rev. Fluid Mech. 53, 473–508.CrossRefGoogle Scholar
Bourouiba, L. 2021 b Fluid dynamics of respiratory infectious diseases. Annu. Rev. Biomed. Engng 23 (1), 547–577.CrossRefGoogle ScholarPubMed
Bourouiba, L. 2022 Respiratory system simulator systems and methods. US Patent PCT/US2021/046393/ WO/2022/040247.Google Scholar
Bourouiba, L., Dehandschoewercker, E. & Bush, J.W.M. 2014 Violent expiratory events: on coughing and sneezing. J. Fluid Mech. 745, 537–563.CrossRefGoogle Scholar
Cowley, S.J., Baker, G.R. & Tanveer, S. 1999 On the formation of moore curvature singularities in vortex sheets. J. Fluid Mech. 378, 233–267.CrossRefGoogle Scholar
DeVoria, A.C. & Mohseni, K. 2018 Vortex sheet roll-up revisited. J. Fluid Mech. 855, 299–321.CrossRefGoogle Scholar
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.CrossRefGoogle Scholar
Fraser, A.E., Cresswell, I.G. & Garaud, P. 2022 Non-ideal instabilities in sinusoidal shear flows with a streamwise magnetic field. J. Fluid Mech. 949, A43.CrossRefGoogle Scholar
Funada, T. & Joseph, D.D. 2001 Viscous potential flow analysis of kelvin-helmholtz instability in a channel. J. Fluid Mech. 445, 261–283.CrossRefGoogle Scholar
Han, Z.Y., Weng, W.G. & Huang, Q.Y. 2013 Characterizations of particle size distribution of the droplets exhaled by sneeze. J. R. Soc. Interface 10 (88), 20130560.CrossRefGoogle ScholarPubMed
Helmholtz, H.V. 1868 On discontinuous fluid motions. Phil. Mag. 36 (4), 337–346.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
Hou, T.Y., Lowengrub, J.S. & Shelley, M.J. 1994 Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. 114 (2), 312–338.CrossRefGoogle Scholar
Hou, T.Y., Lowengrub, J.S. & Shelley, M.J. 1997 The long-time motion of vortex sheets with surface tension. Phys. Fluids 9 (7), 1933–1954.CrossRefGoogle Scholar
Jin, L., Le, T.T. & Fukumoto, Y. 2019 Frictional effect on stability of discontinuity interface in tangential velocity of a shallow-water flow. Phys. Lett. A 383 (26), 125839.CrossRefGoogle Scholar
Johnson, J.R., Wing, S. & Delamere, P.A. 2014 Kelvin Helmholtz instability in planetary magnetospheres. Space Sci. Rev. 184 (1), 1–31.CrossRefGoogle Scholar
Kant, P., Pairetti, C., Saade, Y., Popinet, S., Zaleski, S. & Lohse, D. 2022 Bags mediated film atomization in a cough machine. arXiv:2202.13949.CrossRefGoogle Scholar
Kelly, R.E. 1965 The stability of an unsteady Kelvin–Helmholtz flow. J. Fluid Mech. 22 (3), 547–560.CrossRefGoogle Scholar
Kelvin, L. 1871 On the motion of free solids through a liquid. Phil. Mag. 42 (281), 362–377.Google Scholar
Kerswell, R.R. 2018 Nonlinear nonmodal stability theory. Annu. Rev. Fluid Mech. 50, 319–345.CrossRefGoogle Scholar
Khenner, M.V., Lyubimov, D.V., Belozerova, T.S. & Roux, B. 1999 Stability of plane-parallel vibrational flow in a two-layer system. Eur. J. Mech. (B/Fluids) 18 (6), 1085–1101.CrossRefGoogle Scholar
King, M., Brock, G. & Lundell, C. 1985 Clearance of mucus by simulated cough. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 58 (6), 1776–1782.CrossRefGoogle ScholarPubMed
Kleinstreuer, C. & Zhang, Z. 2010 Airflow and particle transport in the human respiratory system. Annu. Rev. Fluid Mech. 42 (1), 301–334.CrossRefGoogle Scholar
Krasny, R. 1986 A study of singularity formation in a vortex sheet by the point-vortex approximation. J. Fluid Mech. 167, 65–93.CrossRefGoogle Scholar
Le, T.T. 2021 Effect of water depth on Kelvin–Helmholtz instability in a shallow water flow. J. Math. Phys. 62 (10), 103101.CrossRefGoogle Scholar
Lobanov, A.P. & Zensus, J.A. 2001 A cosmic double helix in the archetypical quasar 3c273. Science 294 (5540), 128–131.CrossRefGoogle ScholarPubMed
Lyubimov, D.V. & Cherepanov, A.A. 1986 Development of a steady relief at the interface of fluids in a vibrational field. Fluid Dyn. 21 (6), 849–854.CrossRefGoogle Scholar
MacKay, R.S. & Saffman, P.G. 1986 Stability of water waves. Proc. R. Soc. Lond. A. Math. Phys. Sci. 406 (1830), 115–125.Google Scholar
Marmottant, P. & Villermaux, E. 2004 On spray formation. J. Fluid Mech. 498, 73–111.CrossRefGoogle Scholar
Mishin, V.V. & Tomozov, V.M. 2016 Kelvin–Helmholtz instability in the solar atmosphere, solar wind and geomagnetosphere. Sol. Phys. 291 (11), 3165–3184.CrossRefGoogle Scholar
Moore, D.W. 1979 The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. R. Soc. Lond. A. Math. Phys. Sci. 365 (1720), 105–119.Google Scholar
Perlin, M., Choi, W. & Tian, Z. 2013 Breaking waves in deep and intermediate waters. Annu. Rev. Fluid Mech. 45, 115–145.CrossRefGoogle Scholar
Poulin, F.J., Flierl, G.R. & Pedlosky, J. 2003 Parametric instability in oscillatory shear flows. J. Fluid Mech. 481, 329–353.CrossRefGoogle Scholar
Pozrikidis, C. 2011 Introduction to Theoretical and Computational Fluid Dynamics, 2nd edn. Oxford University Press.Google Scholar
Pullin, D.I. 1982 Numerical studies of surface-tension effects in nonlinear Kelvin–Helmholtz and Rayleigh–Taylor instability. J. Fluid Mech. 119, 507–532.CrossRefGoogle Scholar
Rangel, R.H. & Sirignano, W.A. 1988 Nonlinear growth of Kelvin–Helmholtz instability: effect of surface tension and density ratio. Phys. Fluids 31 (7), 1845.CrossRefGoogle Scholar
Rayleigh, Lord 1879 On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. s1-11 (1), 57–72.CrossRefGoogle Scholar
Saffman, P.G. 1985 The superharmonic instability of finite-amplitude water waves. J. Fluid Mech. 159, 169–174.CrossRefGoogle Scholar
Sato, N. & Yamada, M. 2019 Superharmonic instability of nonlinear travelling wave solutions in Hamiltonian systems. J. Fluid Mech. 876, 896–911.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
Scharfman, B.E., Techet, A.H., Bush, J.W.M. & Bourouiba, L. 2016 Visualization of sneeze ejecta: steps of fluid fragmentation leading to respiratory droplets. Exp. Fluids 57 (2), 1–9.CrossRefGoogle ScholarPubMed
Scherer, P.W. & Burtz, L. 1978 Fluid mechanical experiments relevant to coughing. J. Biomech. 11 (4), 183–187.CrossRefGoogle ScholarPubMed
Schwartz, L.W. & Fenton, J.D. 1982 Strongly nonlinear waves. Annu. Rev. Fluid Mech. 14 (1), 39–60.CrossRefGoogle Scholar
Siegel, M. 1995 A study of singularity formation in the Kelvin–Helmholtz instability with surface tension. SIAM J. Appl. Maths 55 (4), 865–891.CrossRefGoogle Scholar
Smyth, W.D. & Moum, J.N. 2012 Ocean mixing by Kelvin–Helmholtz instability. Oceanography 25 (2), 140–149.CrossRefGoogle Scholar
Sohn, S.-I., Yoon, D. & Hwang, W. 2010 Long-time simulations of the Kelvin–Helmholtz instability using an adaptive vortex method. Phys. Rev. E 82 (4), 046711.CrossRefGoogle ScholarPubMed
Tanaka, M. 1983 The stability of steep gravity waves. J. Phys. Soc. Japan 52 (9), 3047–3055.CrossRefGoogle Scholar
Tanaka, M. 1985 The stability of steep gravity waves. Part 2. J. Fluid Mech. 156, 281–289.CrossRefGoogle Scholar
Villermaux, E. 1998 On the role of viscosity in shear instabilities. Phys. Fluids 10 (2), 368–373.CrossRefGoogle Scholar
Wang, Y. & Bourouiba, L. 2018 Unsteady sheet fragmentation: droplet sizes and speeds. J. Fluid Mech. 848, 946–967.CrossRefGoogle Scholar
Wang, Y., Dandekar, R., Bustos, N., Poulain, S. & Bourouiba, L. 2018 Universal rim thickness in unsteady sheet fragmentation. Phys. Rev. Lett. 120, 204503.CrossRefGoogle ScholarPubMed
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
Yoshikawa, H.N. & Wesfreid, J.E. 2011 Oscillatory Kelvin–Helmholtz instability. Part 1. A viscous theory. J. Fluid Mech. 675, 223–248.CrossRefGoogle Scholar
Shen and Bourouiba Supplementary Movie 1
Response of the water-air interface in a cylindrical geometry analogous to a human trachea, to the impulsive shearing airflow impulse profile A given in Figure 20 of the main text. The airflow direction is right-to-left. Transient growth and de-cay of period gravity-capillary surface waves are recorded. No fragmentation occurs throughout the impulse.
Video
854.4 KB
Shen and Bourouiba Supplementary Movie 2
Response of the water-air interface in a cylindrical geometry analogous to a human trachea to the impulsive shearing airflow impulse profile B given in Figure 20 of the main text. The airflow direction is right-to-left. Transient growth of period gravity-capillary surface waves leads to surface wave overturn-ing and onset of fragmentation. Onset of droplet formation is visible.
Video
514.9 KB