Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T11:27:08.043Z Has data issue: false hasContentIssue false
  • English
  • Français

Convective/absolute instability in miscible core-annular flow. Part 2. Numerical simulations and nonlinear global modes

Published online by Cambridge University Press:  10 January 2009

B. SELVAM ,
L. TALON ,
L. LESSHAFFT  and
E. MEIBURG
B. SELVAM
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
L. TALON
Affiliation:
FAST, Universités Paris VI et Paris XI, CNRS (UMR 7608) Bâtiment 502, Campus Universitaire, 91405 Orsay Cedex, France
L. LESSHAFFT
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
E. MEIBURG*
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
*
†Email address for correspondence: meiburg@engineering.ucsb.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The convective/absolute nature of the instability of miscible core-annular flow with variable viscosity is investigated via linear stability analysis and nonlinear simulations. From linear analysis, it is found that miscible core-annular flows with the more viscous fluid in the core are at most convectively unstable. On the other hand, flows with the less viscous fluid in the core exhibit absolute instability at high viscosity ratios, over a limited range of core radii. Nonlinear direct numerical simulations in a semi-infinite domain display self-excited intrinsic oscillations if and only if the underlying base flow exhibits absolute instability. This oscillator-type flow behaviour is demonstrated to be associated with the presence of a nonlinear global mode. Both the parameter range of global instability and the intrinsically selected frequency of nonlinear oscillations, as observed in the simulation, are accurately predicted from linear criteria. In convectively unstable situations, the flow is shown to respond to external forcing over an unstable range of frequencies, in quantitative agreement with linear theory. As discussed in part 1 of this study (d'Olce, Martin, Rakotomalala, Salin and Talon, J. Fluid Mech., vol. 618, 2008, pp. 305–322), self-excited synchronized oscillations were also observed experimentally. An interpretation of these experiments is attempted on the basis of the numerical results presented here.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 618 , 10 January 2009 , pp. 323 - 348
Copyright
Copyright © Cambridge University Press 2008

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

Bai, R., Chen, K. P. & Joseph, D. D. 1992 Lubricated pipelining: stability of core-annular flow. Part 5. Experiments and comparison with theory. J. Fluid Mech. 240, 97–132.CrossRefGoogle Scholar
Balasubramaniam, R., Rashidnia, N., Maxworthy, T. & Kuang, J. 2005 Instability of miscible interfaces in a cylindrical tube. Phys. Fluids 17, 052103.CrossRefGoogle Scholar
Boomkamp, P. A. M. & Miesen, R. H. M. 1992 Nonaxisymmetric waves in core-annular flow with a small viscosity ratio. Phys. Fluids 240, 97.Google Scholar
Briggs, R. J. 1964 Electron-Stream Interaction with Plasmas. MIT-Press.CrossRefGoogle Scholar
Büchel, P., Lücke, M., Roth, D. & Schmitz, R. 1996 Pattern selection in the absolutely unstable regime as a nonlinear eigenvalue problem: Taylor vortices in axial flow. Phys. Rev. E 53 (5), 4764.Google ScholarPubMed
Buell, J. C. & Huerre, P. 1988 Onflow/outflow boundary conditions and global dynamics of spatial mixing layers. Rep. CTR-S88. Center for Turbulence Research, Stanford University pp. 19–27.Google Scholar
Chen, C.-Y. & Meiburg, E. 1996 Miscible displacement in capillary tubes. Part 2. Numerical simulations. J. Fluid Mech. 326, 57.CrossRefGoogle Scholar
Chen, K., Bai, R. & Joseph, D. D. 1990 Lubricated pipelining. Part 3. Stability of core-annular flow in vertical pipes. J. Fluid Mech. 214, 251.CrossRefGoogle Scholar
Chomaz, J. M. 2003 Fully nonlinear dynamics of parallel wakes. J. Fluid. Mech. 495, 57.CrossRefGoogle Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357.CrossRefGoogle Scholar
Chomaz, J. M., Huerre, P. & Redekopp, L. G. 1988 Bifurcations to local and global modes in spatially developing flows. Phys. Rev. Lett. 60 (1), 25–28.CrossRefGoogle ScholarPubMed
Chomaz, J. M., Huerre, P. & Redekopp, L. G. 1991 A frequency selection criterion in spatially developing flows. Stud. Appl. Maths 84, 119–144.CrossRefGoogle Scholar
Couairon, A. & Chomaz, J. M. 1997 a Absolute and convective instabilities, front velocities and global modes in nonlinear systems. Phys. D 108 (3), 236.Google Scholar
Couairon, A. & Chomaz, J. M. 1997 b Pattern selection in the presence of a cross flow. Phys. Rev. Lett. 79 (14), 2666–2669.CrossRefGoogle Scholar
Couairon, A. & Chomaz, J. M. 1999 Fully nonlinear global modes in slowly varying flows. Phys. Fluids 11 (12), 3688.CrossRefGoogle Scholar
Deissler, R. J. 1987 The convective nature of instability in plane Poiseuille flow. Phys. Fluids 30 (8), 2303.CrossRefGoogle Scholar
Delbende, I. & Chomaz, J. M. 1998 Non-linear convective/absolute instabilities in parallel two-dimensional wakes. Phys. Fluids 10 (11), 2724.CrossRefGoogle Scholar
Govindarajan, R. 2004 Effect of miscibility on the linear instability of two-fluid channel flow. Intl J. Multiphase Flow 30, 1177.CrossRefGoogle Scholar
Goyal, N. & Meiburg, E. 2004 Unstable density stratification of miscible fluids in a vertical Hele--Shaw cell: influence of variable viscosity on the linear stability. J. Fluid Mech. 516, 211–238.CrossRefGoogle Scholar
Goyal, N. & Meiburg, E. 2006 Miscible displacements in Hele–Shaw cells: two-dimensional base states and their linear stability. J. Fluid Mech. 558, 329.CrossRefGoogle Scholar
Hallberg, M. P. & Strykowski, P. J. 2008 Open-loop control of fully nonlinear self-excited oscillations. Phys. Fluids 20, 041703.CrossRefGoogle Scholar
Hickox, C. E. 1971 Instability due to viscosity and density stratification in axisymmetric pipe flow. Phys. Fluids 14, 251.CrossRefGoogle Scholar
Hu, H. H. & Joseph, D. D. 1989 Lubricated pipelining: stability of core-annular flow. Part 2. J. Fluid Mech. 205, 359.CrossRefGoogle Scholar
Hu, H. H., Lundgren, T. S. & Joseph, D. D. 1990 Stability of core-annular flow with a small viscosity ratio. Phys. Fluids A 2 (11), 1945.CrossRefGoogle Scholar
Hu, H. H. & Patankar, N. 1995 Non-axisymmetric instability of core-annular flow. J. Fluid Mech. 290, 213.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473.CrossRefGoogle Scholar
Huerre, P. & Rossi, M. 1998 Hydrodynamics and Nonlinear Instabilities, Chapter 2. Cambridge University Press.Google Scholar
Joseph, D. D., Bai, R., Chen, K. P. & Renardy, Y. Y. 1997 Core-annular flows. Annu. Rev. Fluid Mech. 29, 65.CrossRefGoogle Scholar
Joseph, D. D., Renardy, Y. & Renardy, M. 1984 Instability of the flow of immiscible liquids with different viscosities in a pipe. J. Fluid Mech. 141, 319.CrossRefGoogle Scholar
Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308.CrossRefGoogle Scholar
Kouris, C. & Tsamopoulos, J. 2001 Dynamics of axisymmetric core-annular flow in a straight tube. I. The more viscous fluid in the core, bamboo waves. Phys. Fluids 13 (4), 841.CrossRefGoogle Scholar
Kouris, C. & Tsamopoulos, J. 2002 Dynamics of axisymmetric core-annular flow in a straight tube. II. The less viscous fluid in the core, saw tooth waves. Phys. Fluids 14 (3), 1011.CrossRefGoogle Scholar
Kuang, J., Maxworthy, T. & Petitjeans, P. 2003 Miscible displacements between silicone oils in capillary tubes. Eur. J. Mech. 22, 271.CrossRefGoogle Scholar
Lele, S. K. 1992 Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16.CrossRefGoogle Scholar
Lesshafft, L. & Huerre, P. 2007 Linear impulse response in hot round jets. Phys. Fluids 19, 024102.CrossRefGoogle Scholar
Lesshafft, L., Huerre, P. & Sagaut, P. 2007 Frequency selection in globally unstable round jets. Phys. Fluids 19, 054108.CrossRefGoogle Scholar
Lesshafft, L., Huerre, P., Sagaut, P. & Terracol, M. 2006 Nonlinear global modes in hot jets. J. fluid Mech. 554, 393.CrossRefGoogle Scholar
Li, J. & Renardy, Y. Y. 1999 Direct simulation of unsteady axisymmetric core-annular flow with high viscosity ratio. J. Fluid Mech. 391, 123.CrossRefGoogle Scholar
Monkewitz, P. A., Huerre, P. & Chomaz, J. M. 1993 Global linear stability analysis of weakly non-parallel shear flows. J. Fluid Mech. Digital Arch. 251, 1–20.CrossRefGoogle Scholar
Müller, H. W., Lücke, M. & Kamps, M. 1992 Transversal convection patterns in horizontal shear flow. Phys. Rev. A 45 (6), 3714.CrossRefGoogle ScholarPubMed
d'Olce, M., Martin, J., Rakotomalala, N., Salin, D. & Talon, L. 2009 Convective/absolute instability in miscible core-annular flow. Part 1. Experiments. J. Fluid. Mech. 618, 305–322.CrossRefGoogle Scholar
Osher, S. & Fedkiw, R. 2003 Level Set Methods and Dynamic Implicit Surfaces. Springer.CrossRefGoogle Scholar
Panton, R. L. 1984 Incompressible Flow. John Wiley & Sons.Google Scholar
Payr, M., Vanaparthy, S. H. & Meiburg, E. 2005 Influence of variable viscosity on density-driven instabilities in capillary tubes. J. Fluid Mech. 525, 333–353.CrossRefGoogle Scholar
Petitjeans, P. & Maxworthy, T. 1996 Miscible displacements in capillary tubes. Part 1. Experiments. J. Fluid Mech. 326, 37.CrossRefGoogle Scholar
Pier, B. 2003 Open-loop control of absolutely unstable domains. Roy. Soc. Lond. Proc. Series A 459 (2033), 1105–1115.CrossRefGoogle Scholar
Pier, B. & Huerre, P. 2001 Nonlinear self-sustained structures and fronts in spatially developing wake flows. J. Fluid Mech. 435, 145.CrossRefGoogle Scholar
Pier, B., Huerre, P. & Chomaz, J. M. 2001 Bifurcation to fully nonlinear synchronized structures in slowly varying media. Phys. D 148 (1), 49–96.Google Scholar
Preziosi, L., Chen, K. & Joseph, D. D. 1989 Lubricated pipelining: stability of core-annular flow. J. Fluid Mech. 201, 323.CrossRefGoogle Scholar
Rai, M. M. & Moin, P. 1991 Direct simulations of turbulent flow using finite-difference schemes. J. Comput. Phys. 96, 15.Google Scholar
Ruith, M. R., Chen, P. & Meiburg, E. 2004 Development of boundary conditions for direct numerical simulations of three-dimensional vortex breakdown phenomena in semi-infinite domains. Comput. Fluids 33, 1225.CrossRefGoogle Scholar
van Saarloos, W. 1989 Front propagation into unstable states. II. Linear versus nonlinear marginal stability and rate of convergence. Phys. Rev. A 39 (12), 6367.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Scoffoni, J., Lajeunesse, E. & Homsy, G. M. 2001 Interface instabilities during displacements of two miscible fluids in a vertical pipe. Phys. Fluids 13 (3), 553.CrossRefGoogle Scholar
Selvam, B., Merk, S., Govindarajan, R. & Meiburg, E. 2007 Stability of miscible core-annular flow with viscosity stratification. J. Fluid Mech. 592, 23.CrossRefGoogle Scholar
Tan, C. T. & Homsy, G. M. 1986 Stability of miscible displacements: rectilinear flow. Phys. Fluids 29, 3549.CrossRefGoogle Scholar
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123, 402.CrossRefGoogle Scholar
Yin, X. Y., Sun, D. J., Wei, M. J. & Wu, J. Z. 2000 Absolute and convective instability character of slender viscous vortices. Phys. Fluids 12 (5), 1062.CrossRefGoogle Scholar