Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-20T04:47:20.324Z Has data issue: false hasContentIssue false
  • English
  • Français

Measurements of molecular mixing in a high-Schmidt-number Rayleigh–Taylor mixing layer

Published online by Cambridge University Press:  27 July 2009

NICHOLAS J. MUESCHKE ,
OLEG SCHILLING ,
DAVID L. YOUNGS  and
MALCOLM J. ANDREWS
NICHOLAS J. MUESCHKE
Affiliation:
Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA
OLEG SCHILLING
Affiliation:
Lawrence Livermore National Laboratory, Livermore, CA 94551, USA
DAVID L. YOUNGS
Affiliation:
Atomic Weapons Establishment, Aldermaston, Reading, Berkshire RG7 4PR, UK
MALCOLM J. ANDREWS*
Affiliation:
Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA Los Alamos National Laboratory, Los Alamos, NM 87545, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Molecular mixing measurements are reported for a high-Schmidt-number (Sc ~ 103), small-Atwood-number (A ≈ 7.5 × 10−4) buoyancy-driven turbulent Rayleigh–Taylor (RT) mixing layer in a water channel facility. Salt was added to the top water stream to create the desired density difference. The degree of molecular mixing was measured as a function of time by monitoring a diffusion-limited chemical reaction between the two fluid streams. The pH of each stream was modified by the addition of acid or alkali such that a local neutralization reaction occurred as the two fluids molecularly mixed. The progress of this neutralization reaction was tracked by the addition of phenolphthalein – a pH-sensitive chemical indicator – to the acidic stream. Accurately calibrated backlit optical techniques were used to measure the average concentration of the coloured chemical indicator. Comparisons of chemical product formation for pre-transitional buoyancy- and shear-driven mixing layers are given. It is also shown that experiments performed at different equivalence ratios (acid/alkali concentrations) can be combined to obtain a mathematical relationship between the coloured product formed and the density variance. This relationship was used to obtain high-fidelity quantitative measures of the degree of molecular mixing which are independent of probe resolution constraints. The dependence of molecular mixing on the Schmidt and Reynolds numbers is examined by comparing the current Sc ~ 103 measurements with previous Sc = 0.7 gas-phase and Pr = 7 (where Pr is the Prandtl number) liquid-phase measurements. This comparison indicates that the Schmidt number has a large effect on the quantity of mixed fluid at small Reynolds numbers Reh < 103. At larger Reynolds numbers, corresponding to later times in this experiment, all mixing parameters indicated a greater degree of molecular mixing and a decreased Schmidt number dependence. Implications for the development and quantitative assessment of turbulent transport and mixing models appropriate for RT instability-induced mixing are discussed.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 632 , 10 August 2009 , pp. 17 - 48
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

Anuchina, N. N., Kucherenko, Y. A., Neuvazhaev, V. E., Ogibina, V. N., Shibarshov, L. I. & Yakovlev, V. G. 1978 Turbulent mixing at an accelerating interface between liquids of different densities. Izv. Akad Nauk. SSSR, Mekh. Zhidk. Gaza 6, 157–160.Google Scholar
Atzeni, S. & Meyer-ter-Vehn, J. 2004 The Physics of Inertial Fusion: Beam Plasma Interaction, Hydrodynamics, Hot Dense Matter, International Series of Monographs on Physics, vol. 125. Oxford University Press.CrossRefGoogle Scholar
Benedict, L. H. & Gould, R. D. 1996 Towards better uncertainty estimates for turbulence statistics. Exp. Fluids 22, 129–136.CrossRefGoogle Scholar
Betti, R., Umansky, M., Lobatchev, V., Goncharov, V. N. & McCrory, R. L. 2001 Hot-spot dynamics and deceleration-phase Rayleigh–Taylor instability of imploding inertial confinement fusion capsules. Phys. Plasmas 8, 5257–5267.CrossRefGoogle Scholar
Bishop, E. (Ed.) 1972 Indicators, International Series of Monographs in Analytical Chemistry. Pergamon.Google Scholar
Breidenthal, R. 1979 A chemically reacting turbulent shear layer. PhD dissertation, California Institute of Technology.CrossRefGoogle Scholar
Breidenthal, R. 1981 Structure in turbulent mixing layers and wakes using a chemical reaction. J. Fluid Mech. 109, 1–24.CrossRefGoogle Scholar
Browand, F. K. & Weidman, P. D. 1976 Large scales in the developing mixing layer. J. Fluid Mech. 76, 127–144.CrossRefGoogle Scholar
Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816.CrossRefGoogle Scholar
Burton, G. C. 2008 The nonlinear large-eddy simulation method applied to Sc ≈ 1 and Sc ≫ 1 passive-scalar mixing. Phys. Fluids 20, 035103-1–035103-14.Google Scholar
Cabot, W. H. & Cook, A. W. 2006 Reynolds number effects on Rayleigh–Taylor instability with possible implications for type-Ia supernovae. Nat. Phys. 2, 562–568.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Chassaing, P., Antonia, R. A., Anselmet, F., Joly, L. & Sarkar, S. 2002 Variable Density Fluid Turbulence. In Fluid Mechanics and Its Applications, vol. 69. Kluwer Academic.Google Scholar
Cook, A. W., Cabot, W. & Miller, P. L. 2004 The mixing transition in Rayleigh–Taylor instability. J. Fluid Mech. 511, 333–362.CrossRefGoogle Scholar
Cook, A. W. & Dimotakis, P. E. 2001 Transition stages of Rayleigh–Taylor instability between miscible fluids. J. Fluid Mech. 443, 69–99.CrossRefGoogle Scholar
Cook, A. W. & Dimotakis, P. E. 2002 Corrigendum. J. Fluid Mech. 457, 410–411.CrossRefGoogle Scholar
Cui, A. Q. & Street, R. L. 2004 Large-eddy simulation of coastal upwelling flow. Environ. Fluid Mech. 4, 197–223.CrossRefGoogle Scholar
Dalziel, S. B., Linden, P. F. & Youngs, D. L. 1999 Self-similarity and internal structure of turbulence induced by Rayleigh–Taylor instability. J. Fluid Mech. 399, 1–48.CrossRefGoogle Scholar
Danckwerts, P. V. 1952 The definition and measurement of some characteristics of mixtures. Appl. Sci. Res. A 3, 279–296.CrossRefGoogle Scholar
Desai, M. A. & Vadgama, P. 1991 Estimation of effective diffusion coefficients of model solutes through gastric mucus: assessment of a diffusion chamber technique based on spectrophotometric analysis. Analyst 116, 1113–1116.CrossRefGoogle ScholarPubMed
Dimotakis, P. E. 2005 Turbulent mixing. Annu. Rev. Fluid Mech. 37, 329–356.CrossRefGoogle Scholar
Fox, R. O. 2003 Computational Models for Turbulent Reacting Flows. Cambridge University Press.CrossRefGoogle Scholar
Green, F. J. 1990 The Sigma-Aldrich Handbook of Stains, Dyes, and Indicators. Aldrich Chemical Co.Google Scholar
Haan, S. W. 1989 Onset of nonlinear saturation for Rayleigh–Taylor growth in the presence of a full spectrum of modes. Phys. Rev. A 39, 5812–5825.CrossRefGoogle ScholarPubMed
Harris, D. C. 2003 Quantitative Chemical Analysis, 6th ed. W. H. Freeman.Google Scholar
Hecht, E. 2002 Optics, 4th ed. Addison Wesley.Google Scholar
Kolthoff, I. M. 1937 Acid–Base Indicators. Macmillan.Google Scholar
Konrad, J. H. 1977 An experimental investigation of mixing in two-dimensional turbulent shear flows with applications to diffusion-limited chemical reactions. PhD dissertation, California Institute of Technology.Google Scholar
Koochesfahani, M. M. & Dimotakis, P. E. 1986 Mixing and chemical reactions in a turbulent liquid mixing layer. J. Fluid Mech. 170, 83–112.CrossRefGoogle Scholar
Koop, G. K. 1976 Instability and turbulence in a stratified shear layer. PhD dissertation, University of Southern California.Google Scholar
Kraft, W., Banerjee, A. & Andrews, M. J. 2009 On hot-wire diagnostics in Rayleigh–Taylor mixing layers. Exp. Fluids. doi:10.1007/S00348-009-0636-3.CrossRefGoogle Scholar
Kukulka, D. J. 1981 Thermodynamic and transport properties of pure and saline water. MS thesis, State University of New York at Buffalo.Google Scholar
Lide, D. R. (Ed.) 2006 CRC Handbook of Chemistry and Physics, 87th ed. CRC Press.Google Scholar
Linden, P. F. & Redondo, J. M. 1991 Molecular mixing in Rayleigh–Taylor instability. Part I. Global mixing. Phys. Fluids A 3, 1269–1277.CrossRefGoogle Scholar
Linden, P. F., Redondo, J. M. & Youngs, D. L. 1994 Molecular mixing in Rayleigh–Taylor instability. J. Fluid Mech. 265, 97–124.CrossRefGoogle Scholar
Lindl, J. D. 1998 Inertial Confinement Fusion: The Quest for Ignition and Energy Gain Using Indirect Drive. Springer.Google Scholar
Liu, Y. & Fox, R. O. 2006 CFD predictions for chemical processing in a confined impinging-jets reactor. AIChE J. 52, 731–744.CrossRefGoogle Scholar
Marmottant, P. H. & Villermaux, E. 2004 On spray formation. J. Fluid Mech. 498, 73–111.CrossRefGoogle Scholar
Molchanov, O. A. 2004 On the origin of low- and middler-latitude ionospheric turbulence. Phys. Chem. Earth 29, 559–567.CrossRefGoogle Scholar
Mueschke, N. J. & Andrews, M. J. 2005 a Investigation of scalar measurement error in diffusion and mixing processes. Exp. Fluids 40, 165–175.CrossRefGoogle Scholar
Mueschke, N. J. & Andrews, M. J. 2005 b Erratum. Exp. Fluids 40, 176.CrossRefGoogle Scholar
Mueschke, N. J., Andrews, M. J. & Schilling, O. 2006 Experimental characterization of initial conditions and spatio-temporal evolution of a small-Atwood-number Rayleigh–Taylor mixing layer. J. Fluid Mech. 567, 27–63.CrossRefGoogle Scholar
Mungal, M. G. & Dimotakis, P. E. 1984 Mixing and combustion with low heat release in a turbulent shear layer. J. Fluid Mech. 148, 349–382.CrossRefGoogle Scholar
Poinsot, T. & Veynante, D. 2005 Theoretical and Numerical Combustion, 2nd ed. R. T. Edwards.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Pullin, D. I. 2000 A vortex-based model for the subgrid flux of a passive scalar. Phys. Fluids 12, 2311–2319.CrossRefGoogle Scholar
Ramaprabhu, P. & Andrews, M. J. 2003 Simultaneous measurements of velocity and density in buoyancy-driven mixing. Exp. Fluids, 34, 98–106.CrossRefGoogle Scholar
Ramaprabhu, P. & Andrews, M. J. 2004 Experimental investigation of Rayleigh–Taylor mixing at small Atwood numbers. J. Fluid Mech. 502, 233–271.CrossRefGoogle Scholar
Ramaprabhu, P., Dimonte, G. & Andrews, M. J. 2005 A numerical study of the influence of initial perturbations on the turbulent Rayleigh–Taylor instability. J. Fluid Mech. 536, 285–319.CrossRefGoogle Scholar
Rayleigh, J. W. 1884 Investigation of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14, 170–177.Google Scholar
Ristorcelli, J. R. & Clark, T. T. 2004 Rayleigh–Taylor turbulence: self-similar analysis and direct numerical simulations. J. Fluid Mech. 507, 213–253.CrossRefGoogle Scholar
Shea, J. R. 1977 A chemical reaction in a turbulent jet. J. Fluid Mech. 81, 317–333.CrossRefGoogle Scholar
Smarr, L., Wilson, J. R., Barton, R. T. & Bowers, R. L. 1981 Rayleigh–Taylor overturn in super nova core collapse. Astrophys. J. 246, 515–525.CrossRefGoogle Scholar
Snider, D. M. & Andrews, M. J. 1994 Rayleigh–Taylor and shear driven mixing with an unstable thermal stratification. Phys. Fluids A 6, 3324–3334.CrossRefGoogle Scholar
Stillinger, D. C., Head, M. J., Helland, K. N. & Van Atta, C. W. 1983 A closed loop gravity-driven water channel for density stratified shear flow. J. Fluid Mech. 131, 73–89.CrossRefGoogle Scholar
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proc. R. Soc. Lond. 201, 192–196.Google Scholar
Thomas, G. O. 2003 The aerodynamic breakup of ligaments. Atom. Sprays 13, 117–129.CrossRefGoogle Scholar
Veynante, D & Vervisch, L. 2002 Turbulent combustion modeling. Prog. Energy Combust. Sci. 28, 193–266.CrossRefGoogle Scholar
Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32, 203–240.CrossRefGoogle Scholar
Wilson, P. N. & Andrews, M. J. 2002 Spectral measurements of Rayleigh–Taylor mixing at low-Atwood number. Phys. Fluids A 14, 938–945.CrossRefGoogle Scholar
Youngs, D. L. 1984 Numerical simulations of turbulent mixing by Rayleigh–Taylor instability. Physica D 12, 32–44.CrossRefGoogle Scholar
Youngs, D. L. 1994 Numerical simulation of mixing by Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Laser Part. Beams 12, 725–750.CrossRefGoogle Scholar
Zhang, S. & Schneider, S. P. 1995 Quantitative molecular-mixing measurements in a round jet with tabs. Phys. Fluids 7, 1063–1070.CrossRefGoogle Scholar
Zhang, S., Schneider, S. P. & Collicott, S. H. 1995 Quantitative molecular-mixing measurements using digital processing of absorption images. Exp. Fluids 19, 319–327.CrossRefGoogle Scholar