Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T05:01:51.822Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear interaction between a boundary layer and a liquid film

Published online by Cambridge University Press:  07 October 2009

M. VLACHOMITROU  and
N. PELEKASIS
M. VLACHOMITROU
Affiliation:
Department of Mechanical & Industrial Engineering, University of Thessaly, Leoforos Athinon, Pedion Areos, 38834 Volos, Greece
N. PELEKASIS*
Affiliation:
Department of Mechanical & Industrial Engineering, University of Thessaly, Leoforos Athinon, Pedion Areos, 38834 Volos, Greece
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The nonlinear stability of a laminar boundary layer that flows at high Reynolds number (Re) above a plane surface covered by a liquid film is investigated. The basic flow is considered to be nearly parallel and the simulations are based on triple deck theory. The overall interaction problem is solved using the finite element methodology with the two-dimensional B-cubic splines as basis functions for the unknowns in the boundary layer and the film and the one-dimensional B-cubic splines as basis functions for the location of the interface. The case of flow above an oscillating solid obstacle is studied and conditions for the onset of Tollmien–Schlichting (TS) waves are recovered in agreement with the literature. The convective and absolute nature of TS and interfacial waves is captured for gas-film interaction, and the results of linear theory are recovered. The evolution of nonlinear disturbances is also examined and the appearance of solitons, spikes and eddy formation is monitored on the interface, depending on the relative magnitude of Froude and Weber numbers (Fr, We), and the gas to film density and viscosity ratios (ρ/ρw, μ/μw). For viscous films TS waves grow on a much faster time scale than interfacial waves and their effect is essentially decoupled. The influence of interfacial disturbances on short-wave growth in the bulk of the boundary layer bypassing classical TS wave development is captured. For highly viscous films for which inertia effects can be neglected, e.g. aircraft anti-icing fluids, soliton formation is obtained with their height remaining bounded below a certain height. When water films are considered interfacial waves exhibit unlimited local growth that is associated with intense eddy formation and the appearance of finite time singularities in the pressure gradient.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 638 , 10 November 2009 , pp. 199 - 242
Copyright
Copyright © Cambridge University Press 2009

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

Bezos, G. M., Dunham, R. E., Gentry, G. L. Jr. & Melson, W. E. 1992 Wind tunnel aerodynamic characteristics of a transport-type airfoil in a simulated heavy rain environment. Tech Rep. TP-3184. NASA.Google Scholar
Bowles, R., Davies, C. & Smith, F. T. 2003 On the spiking stages in deep transition and unsteady separation. J. Engng Math. 45, 227–245.CrossRefGoogle Scholar
Brotherton-Ratcliffe, R. V. & Smith, F. T. 1987 Complete breakdown of an unsteady interactive boundary layer (over a surface distortion or in a liquid layer). Mathematica 34, 86–100.Google Scholar
Caponi, E. A., Fornberg, B., Knight, D. D., Mclean, J. W., Saffman, P. G. & Yuen, H. C. 1982 Calculations of laminar viscous flow over a moving wavy surface. J. Fluid Mech. 124, 347–362.CrossRefGoogle Scholar
Cassel, K. W., Ruban, A. I. & Walker, D. A. 1995 An instability in supersonic boundary-layer flow over a compression ramp. J. Fluid Mech. 300, 265–285.CrossRefGoogle Scholar
Cassel, K. W., Ruban, A. I. & Walker, D. A. 1996 The influence of wall cooling on hypersonic boundary-layer separation and stability. J. Fluid Mech. 321, 189–216.CrossRefGoogle Scholar
Chomaz, J. M., Huerre, P. & Redekopp, L. G. 1991 A frequency selection criterion in spatially developing flows. Stud. Appl. Math. 84, 119–144.CrossRefGoogle Scholar
Craik, A. D. D. 1966 Wind generated waves in thin liquid films. J. Fluid Mech. 26 (2), 369–392.CrossRefGoogle Scholar
De Boor, C. 1978 A Practical Guide to Splines. Springer.CrossRefGoogle Scholar
Duck, P. W. 1985 Laminar flow over unsteady humps: the formation of waves. J. Fluid Mech. 160, 465–498.CrossRefGoogle Scholar
Elliott, J. W., Cowley, S. J. & Smith, F. T. 1983 Breakdown of boundary layers: (i) on moving surfaces; (ii) in self-similar unsteady flow; (iii) in fully unsteady flow. Geophys. Astrophys. Fluid Dyn. 25, 77–138.CrossRefGoogle Scholar
Fletcher, A. J. P., Ruban, A. I. & Walker, J. D. A. 2004 Instabilities in supersonic compression ramp flow. J. Fluid Mech. 517, 309–330.CrossRefGoogle Scholar
Gresho, P. M. & Sani, R. L. 1998 Incompressible Flow and the Finite Element Method: Volume II. Wiley.Google Scholar
Hendrickson, G. S. & Hill, E. G.Effects of aircraft de-anti-icing fluids on airfoil characteristics, von Karman Inst. for Fluid Dynamics Lecture Series, Influence of Environmental factors on aircraft performance. von Karman Inst. for Fluid Dynamics, Brussels, Belgium, Feb. 16–19, 1987.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.CrossRefGoogle Scholar
Jenson, R., Burggraf, O. R. & Rizzetta, D. P. 1975 Asymptotic solution for supersonic viscous flow past a compression corner. Lecture Notes Phys. 35, 218–224.CrossRefGoogle Scholar
Lac, E., Barthes-Biesel, D., Pelekasis, N. & Tsamopoulos, J. 2004 Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and the onset of buckling. J. Fluid Mech. 516, 303–334.CrossRefGoogle Scholar
Lin, C. C. 1946 On the stability of two-dimensional parallel flows. Stability in viscous fluid. Quart. Appl. Math. 3 (4), 277–301.CrossRefGoogle Scholar
Ludwieg, H. & Hornung, H. 1989 The instability of a liquid film on a wall exposed to an airflow. J. Fluid Mech. 200, 217–233.CrossRefGoogle Scholar
Meza, C. E. & Balakotaiah, V. 2008 Modeling and experimental studies of large amplitude waves on vertically falling films. Chem. Engng Sci. 63, 4704–4734.CrossRefGoogle Scholar
Nelson, J. J., Alving, A. E. & Joseph, D. D. 1995 Boundary layer flow of air over water on a flat plate. J. Fluid Mech. 284, 159–169.CrossRefGoogle Scholar
Ozgen, S., Carbonaro, M. & Sarma, G. S. R. 2002 Experimental study of wave characteristics on a thin layer of de/anti-icing fluid. Phys. Fluids 14, 3391–3402.CrossRefGoogle Scholar
Ozgen, S., Degrez, G. & Sarma, G. S. R. 1998 Two-fluid boundary layer stability. Phys. Fluids 10 (11), 2746–2757.CrossRefGoogle Scholar
Pelekasis, N. A. & Tsamopoulos, J. A. 2001 Linear stability analysis of a gas boundary layer flowing past a thin liquid film over a flat plate. J. Fluid Mech. 436, 321–352.CrossRefGoogle Scholar
Pelekasis, N. A., Tsamopoulos, J. A. & Manolis, G. D. 1992 A hybrid finite-boundary element method for inviscid flows with free surface. J. Comput. Phys. 101 (2), 231–251.CrossRefGoogle Scholar
Peridier, V. J., Smith, F. T. & Walker, J. D. A. 1991 a Vortex-induced boundary-layer separation. Part 1. The unsteady limit problem Re → ∞. J. Fluid Mech. 232, 99–131.CrossRefGoogle Scholar
Peridier, V. J., Smith, F. T. & Walker, J. D. A. 1991 b Vortex-induced boundary-layer separation. Part 2. Unsteady interacting boundary layer theory. J. Fluid Mech. 232, 133–165.CrossRefGoogle Scholar
Prender, P. M. 1989 Splines and Variational Methods. John Wiley & Sons.Google Scholar
Purvis, R. & Smith, F. T. 2004 Air–water interactions near droplet impact. Eur. J. Appl. Math. 15, 853–871.CrossRefGoogle Scholar
Rothmayer, A. P., Matheis, B. D. & Timoshin, S. N. 2002 Thin liquid films flowing over external aerodynamic surfaces. J. Engng Math. 42, 341–357.CrossRefGoogle Scholar
Ruban, A. I. 1978 Numerical solution of the local asymptotic problem of the unsteady separation of a laminar boundary layer in supersonic flow Comput. Math. Math. Phys. 18 (5), 175–187.CrossRefGoogle Scholar
Ryzhov, O. S. & Terent'ev, E. D. 1986 On the transition mode characterizing the triggering of a vibrator in the subsonic boundary layer on a plate. Prikl. Mat. Mekh., J. Appl. Math. Mech. (Engl. Transl) 50 (6), 753–762.CrossRefGoogle Scholar
Saad, Y. & Schultz, M. H. 1986 GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869.CrossRefGoogle Scholar
Smith, F. T. 1976 a Flow through constricted or dilated pipes and channels. Part 1. Quart. J. Mech. Appl. Math 29 (3), 343–364.CrossRefGoogle Scholar
Smith, F. T. 1976 b Flow through constricted or dilated pipes and channels: Part 2. Quart. J. Mech. Appl. Math 29 (3), 365–376.CrossRefGoogle Scholar
Smith, F. T. 1979 a On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A 366, 91–109.Google Scholar
Smith, F. T. 1979 b Nonlinear stability of boundary layers for disturbances of various sizes. Proc. R. Soc. Lond. A 368, 573–589.Google Scholar
Smith, F. T. 1985 A structure for laminar-flow past a bluff body at high Reynolds-number. J. Fluid Mech. 155, 175–191.CrossRefGoogle Scholar
Smith, F. T. & Bodonyi, R. J. 1985 On short-scale inviscid instabilities in flow past surface-mounted obstacles and other non-parallel motions. Aeronaut. J. 89 (886), 205–212.CrossRefGoogle Scholar
Smith, F. T., Brighton, P. W. M., Jackson, P. S. & Hunt, J. C. R. 1981 On boundary-layer flow past two-dimensional obstacles. J. Fluid Mech. 113, 123–152.CrossRefGoogle Scholar
Smith, F. T. & Burggraf, O. R. 1985 On the development of large-sized short-scaled disturbances in boundary layers. Proc. R. Soc. Lond. A 399, 25–55.Google Scholar
Smith, F. T. 1986 Two-dimensional disturbance travel, growth and spreading in boundary layers. J. Fluid Mech. 169, 353–377.CrossRefGoogle Scholar
Smith, F. T. 1988 Finite time singularity can occur in any unsteady interacting boundary layer. Mathematica 35, 256–273.Google Scholar
Smith, F. T. 1995 On spikes and spots: strongly nonlinear theory and experimental comparisons. Phil. Trans. R. Soc. Lond. A. 352, 405–424.Google Scholar
Smith, F. T., Li, L. & Wu, G. X. 2003 Air cushioning with a lubrication/inviscid balance. J. Fluid Mech. 482, 291–318.CrossRefGoogle Scholar
Smyrnaios, D. N., Pelekasis, N. A. & Tsamopoulos, J. A. 2000 Boundary layer flow of air past solid surfaces in the presence of rainfall. J. Fluid Mech. 425, 79–110.CrossRefGoogle Scholar
Sychev, VL. V., Ruban, A. I., Sychev, VI. V. & Korolev, G. L. 1998 Asymptotic theory of separated flows. Cambridge University Press.CrossRefGoogle Scholar
Terent'ev, E. D. 1981 The linear problem of a vibrator in a subsonic boundary layer. J. Appl. Math. Mech. 45 (6), 791–795.CrossRefGoogle Scholar
Terent'ev, E. D. 1984 The linear problem of a vibrator performing harmonic oscillations at supercritical frequencies in a subsonic boundary layer. J. Appl. Math. Mech. 48 (2), 184–191.CrossRefGoogle Scholar
Thompson, J. F., Warsi, Z. U. A. & Mastin, C. W. 1995 Numerical Grid Generation: Foundations and Applications. North Holland.Google Scholar
Timoshin, S. N. 1997 Instabilities in a high-Reynolds-number boundary layer on a film-coated surface. J. Fluid Mech. 353, 163–195.CrossRefGoogle Scholar
Tsao, J. C., Rothmayer, A. P. & Ruban, A. 1997 Stability of air flow past thin liquid films on airfoils. Comput. Fluids 26 (5), 427–452.CrossRefGoogle Scholar
Tsiglifis, K. & Pelekasis, N. A. 2005 Nonlinear oscillations and collapse of elongated bubbles subject to weak viscous effects. Phys. Fluids 17, 102101.CrossRefGoogle Scholar
Tutty, O. R. & Cowley, S. J. 1986 On the stability and the numerical simulation of the unsteady interactive boundary-layer equation. J. Fluid Mech. 168, 431–456.CrossRefGoogle Scholar
Veldman, A. E. P. 1981 New quasi-simultaneous method to calculate interacting boundary layers. AIAA J. 19, 79–85.CrossRefGoogle Scholar
Yih, C. S. 1990 Wave formation on a liquid layer for de-icing airplane wings. J. Fluid Mech. 212, 41.CrossRefGoogle Scholar