Hostname: page-component-7bb8b95d7b-pwrkn Total loading time: 0 Render date: 2024-09-19T09:26:42.071Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal perturbations for controlling the growth of a Rayleigh–Taylor instability

Published online by Cambridge University Press:  31 July 2019

Ali Kord Open the ORCID record for Ali Kord [Opens in a new window]  and
Jesse Capecelatro Open the ORCID record for Jesse Capecelatro [Opens in a new window]
Ali Kord*
Affiliation:
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125, USA
Jesse Capecelatro
Affiliation:
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125, USA
*
†Email address for correspondence: akord@umich.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A discrete adjoint-based method is employed to control multi-mode Rayleigh–Taylor (RT) instabilities via strategic manipulation of the initial interfacial perturbations. We seek to find to what extent mixing and growth can be enhanced at late stages of the instability and which modes are targeted to achieve this. Three objective functions are defined to quantify RT mixing and growth: (i) variance of mole fraction, (ii) a kinetic energy norm based on the vertical velocity and (iii) variations of mole fraction with respect to the unperturbed initial state. The sensitivity of these objective functions to individual amplitudes of the initial perturbations are studied at various stages of the RT instability. The most sensitive wavenumber during the early stages of the instability closely matches the most unstable wavenumber predicted by linear stability theory. It is also shown that randomly changing the initial perturbations has little effect at early stages, but results in large variations in both RT growth and its sensitivity at later times. The sensitivity obtained from the adjoint solution is employed in gradient-based optimization to both suppress and enhance the objective functions. The adjoint-based optimization framework was capable of completely suppressing mixing by shifting all of the interfacial perturbation energy to the highest modes such that diffusion dominates. The optimal initial perturbations for enhancing the objective functions were found to have a broadband spectrum resulting in non-trivial coupling between modes and depends on the particular objective function being optimized. The objective functions were increased by as much as a factor of nine in the self-similar late-stage growth regime compared to an interface with a uniform distribution of modes, corresponding to a 32% increase in the bubble growth parameter and 54% increase in the mixing width. It was also found that the interfacial perturbations optimized at early stages of the instability are unable to predict enhanced mixing at later times, and thus optimizing late-time multi-mode RT instabilities requires late-time sensitivity information. Finally, it was found that the optimized distribution of interfacial perturbations obtained from two-dimensional simulations was capable of enhancing the objective functions in three-dimensional flows. As much as 51% and 99% enhancement in the bubble growth parameter and mixing width, respectively, was achieved, even greater than what was reached in two dimensions.

JFM classification

Flow Control: Instability control Mathematical Foundations: Variational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 876 , 10 October 2019 , pp. 150 - 185
Copyright
© 2019 Cambridge University Press 

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

Aamo, O. M., Krstić, M. & Bewley, T. R. 2003 Control of mixing by boundary feedback in 2D channel flow. Automatica 39 (9), 1597–1606.Google Scholar
Alon, U., Hecht, J., Mukamel, D. & Shvarts, D. 1994 Scale invariant mixing rates of hydrodynamically unstable interfaces. Phys. Rev. Lett. 72 (18), 2867–2870.Google Scholar
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 density. Fluid Dyn. 13 (6), 916–920.Google Scholar
Babchin, A. J., Frenkel, A. L., Levich, B. G. & Sivashinsky, G. I. 1983 Nonlinear saturation of Rayleigh–Taylor instability in thin films. Phys. Fluids 26 (11), 3159–3161.Google Scholar
Baker, L. & Freeman, J. R. 1981 Heuristic model of the nonlinear Rayleigh–Taylor instability. J. Appl. Phys. 52 (2), 655–663.Google Scholar
Banerjee, A. & Andrews, M. J. 2009 3D simulations to investigate initial condition effects on the growth of Rayleigh–Taylor mixing. Intl J. Heat Mass Transfer 52 (17), 3906–3917.Google Scholar
Bodony, D. J. 2010 Accuracy of the simultaneous-approximation-term boundary condition for time-dependent problems. J. Sci. Comput. 43 (1), 118–133.Google Scholar
Braman, K., Oliver, T. A. & Raman, V. 2015 Adjoint-based sensitivity analysis of flames. Combust. Theor. Model. 19 (1), 29–56.Google Scholar
Cabot, W. 2006 Comparison of two-and three-dimensional simulations of miscible Rayleigh–Taylor instability. Phys. Fluids 18 (4), 045101.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 (8), 562–568.Google Scholar
Capecelatro, J., Bodony, D. J. & Freund, J. B. 2017 Adjoint-based sensitivity analysis of ignition in a turbulent reactive shear layer. In 55th AIAA Aerospace Sciences Meeting, AIAA 2017-0846. AIAA.Google Scholar
Capecelatro, J., Bodony, D. J. & Freund, J. B. 2018 Adjoint-based sensitivity and ignition threshold mapping in a turbulent mixing layer. Combust. Theor. Model. 147–179.Google Scholar
Capecelatro, J., Vishnampet, R., Bodony, D. J. & Freund, J. B. 2016 Adjoint-based sensitivity analysis of localized ignition in a non-premixed hydrogen-air mixing layer. In 54th AIAA Aerospace Sciences Meeting, AIAA 2016-2153. AIAA.Google Scholar
Carles, P., Huang, Z., Carbone, G. & Rosenblatt, C. 2006 Rayleigh–Taylor instability for immiscible fluids of arbitrary viscosities: a magnetic levitation investigation and theoretical model. Phys. Rev. Lett. 96 (10), 104501.Google Scholar
Carnarius, A., Thiele, F., Oezkaya, E. & Gauger, N. R.2010 Adjoint approaches for optimal flow control. AIAA Paper 2010-5088.Google Scholar
Carpenter, M. H., Gottlieb, D. & Abarbanel, S. 1994 Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes. J. Comput. Phys. 111 (2), 220–236.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and hydromagnetic stability. In International Series of Monographs on Physics. Clarendon Press.Google Scholar
Cimpeanu, R., Papageorgiou, D. T. & Petropoulos, P. G. 2014 On the control and suppression of the Rayleigh–Taylor instability using electric fields. Phys. Fluids 26 (2), 022105.Google Scholar
Cook, A. W. 2009 Enthalpy diffusion in multicomponent flows. Phys. Fluids 21 (5), 055109.Google Scholar
Cook, A. W. & Dimotakis, P. E. 2001 Transition stages of Rayleigh–Taylor instability between miscible fluids. J. Fluid Mech. 443, 69–99.Google Scholar
Dimonte, G. 2004 Dependence of turbulent Rayleigh–Taylor instability on initial perturbations. Phys. Rev. E 69 (5), 056305.Google Scholar
Dimonte, G., Youngs, D. L., Dimits, A., Weber, S., Marinak, M., Wunsch, S., Garasi, C., Robinson, A., Andrews, M. J., Ramaprabhu, P. et al. 2004 A comparative study of the turbulent Rayleigh–Taylor instability using high-resolution three-dimensional numerical simulations: the Alpha-Group collaboration. Phys. Fluids 16 (5), 1668–1693.Google Scholar
Duff, R. E., Harlow, F. H. & Hirt, C. W. 1962 Effects of diffusion on interface instability between gases. Phys. Fluids 5 (4), 417–425.Google Scholar
Foures, D. P. G., Caulfield, C. P. & Schmid, P. J. 2014 Optimal mixing in two-dimensional plane Poiseuille flow at finite Péclet number. J. Fluid Mech. 748, 241–277.Google Scholar
Freund, J. B. 1997 Proposed inflow/outflow boundary condition for direct computation of aerodynamic sound. AIAA J. 35 (4), 740–742.Google Scholar
Fujimoto, M. Y. 1993 The evolution of accreting stars with turbulent mixing. Astrophys. J. 419, 768–775.Google Scholar
Goncharov, V. N. 2002 Analytical model of nonlinear, single-mode, classical Rayleigh–Taylor instability at arbitrary Atwood numbers. Phys. Rev. Lett. 88 (13), 134502.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 (11), 5812–5825.Google Scholar
Halpern, D. & Frenkel, A. L. 2001 Saturated Rayleigh–Taylor instability of an oscillating Couette film flow. J. Fluid Mech. 446, 67–93.Google Scholar
Hecht, J., Alon, U. & Shvarts, D. 1994 Potential flow models of Rayleigh–Taylor and Richtmyer–Meshkov bubble fronts. Phys. Fluids 6 (12), 4019–4030.Google Scholar
Huang, Z., De Luca, A., Atherton, T. J., Bird, M., Rosenblatt, C. & Carlès, P. 2007 Rayleigh–Taylor instability experiments with precise and arbitrary control of the initial interface shape. Phys. Rev. Lett. 99 (20), 204502.Google Scholar
Jameson, A. 1989 Aerodynamic design via control theory. In Recent Advances in Computational Fluid Dynamics, pp. 377–401. Springer.Google Scholar
Jameson, A. & Martinelli, L. 2000 Aerodynamic shape optimization techniques based on control theory. In Computational Mathematics Driven by Industrial Problems, pp. 151–221. Springer.Google Scholar
Janka, H.-T. & Müller, E. 1996 Neutrino heating, convection, and the mechanism of type-II supernova explosions. Astron. Astrophys. 306, 167–198.Google Scholar
Jones, E., Oliphant, T. & Peterson, P.2001 SciPy: Open source scientific tools for Python. http://www.scipy.org/.Google Scholar
Kraft, D. 1988 A software package for sequential quadratic programming. Forschungsbericht- Deutsche Forschungs- und Versuchsanstalt fur Luft- und Raumfahrt.Google Scholar
Kreiss, H.-O. & Scherer, G. 1974 Finite element and finite difference methods for hyperbolic partial differential equations. In Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 195–212. Elsevier.Google Scholar
Layzer, D. 1955 On the instability of superposed fluids in a gravitational field. Astrophys. J. 122, 1–12.Google Scholar
Lea, D. J., Allen, M. R. & Haine, T. W. N. 2000 Sensitivity analysis of the climate of a chaotic system. Tellus A 52 (5), 523–532.Google Scholar
Lindl, J. 1995 Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition and gain. Phys. Plasmas 2 (11), 3933–4024.Google Scholar
Liu, W. 2006 Mixing enhancement via flow optimization. In Decision and Control, 2006 45th IEEE Conference on, pp. 5323–5328. IEEE.Google Scholar
Livescu, D. 2004 Compressibility effects on the Rayleigh–Taylor instability growth between immiscible fluids. Phys. Fluids 16 (1), 118–127.Google Scholar
Livescu, D. 2013 Numerical simulations of two-fluid turbulent mixing at large density ratios and applications to the Rayleigh–Taylor instability. Phil. Trans. R. Soc. Lond. A 371 (2003), 20120185.Google Scholar
Livescu, D., Wei, T. & Petersen, M. R. 2011 Direct numerical simulations of Rayleigh–Taylor instability. In J. Phys: Conference Series, vol. 318, p. 082007. IOP Publishing.Google Scholar
Lopez-Zazueta, A., Fontane, J. & Joly, L. 2016 Optimal perturbations in time-dependent variable-density Kelvin–Helmholtz billows. J. Fluid Mech. 803, 466–501.Google Scholar
Martins, J. R. R. A., Alonso, J. J. & Reuther, J. J. 2004 High-fidelity aerostructural design optimization of a supersonic business jet. J. Aircraft 41 (3), 523–530.Google Scholar
Mattsson, K., Svärd, M. & Nordström, J. 2004 Stable and accurate artificial dissipation. J. Sci. Comput. 21 (1), 57–79.Google Scholar
Miles, C. J.2018 Optimal control of the advection-diffusion equation for effective fluid mixing. PhD thesis, University of Michigan, Ann Arbor, MI.Google Scholar
Millman, K. J. & Aivazis, M. 2011 Python for scientists and engineers. Comput. Sci. & Engng 13 (2), 9–12.Google Scholar
Movahed, P.2014 High-fidelity numerical simulations of compressible turbulence and mixing generated by hydrodynamic instabilities. PhD thesis, University of Michigan, Ann Arbor, MI.Google Scholar
Nadarajah, S. & Jameson, A.2000. A comparison of the continuous and discrete adjoint approach to automatic aerodynamic optimization. AIAA Paper 2000-0667.Google Scholar
Nakai, S. & Takabe, H. 1996 Principles of inertial confinement fusion-physics of implosion and the concept of inertial fusion energy. Rep. Prog. Phys. 59 (9), 1071–1131.Google Scholar
Oliphant, T. E. 2007 Python for scientific computing. Comput. Sci. Engng 9 (3), 10–20.Google Scholar
Oron, D., Arazi, L., Kartoon, D., Rikanati, A., Alon, U. & Shvarts, D. 2001 Dimensionality dependence of the Rayleigh–Taylor and Richtmyer–Meshkov instability late-time scaling laws. Phys. Plasmas 8 (6), 2883–2889.Google Scholar
Peters, N. 2000 Turbulent Combustion. Cambridge University Press.Google Scholar
Piro, A. L. & Bildsten, L. 2007 Turbulent mixing in the surface layers of accreting neutron stars. Astrophys. J. 663 (2), 1252–1268.Google Scholar
Pitsch, H. 2006 Large-eddy simulation of turbulent combustion. Annu. Rev. Fluid Mech. 38, 453–482.Google Scholar
Qadri, U. A., Chandler, G. J. & Juniper, M. P. 2015 Self-sustained hydrodynamic oscillations in lifted jet diffusion flames: origin and control. J. Fluid Mech. 775, 201–222.Google 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.Google Scholar
Read, K. I. 1984 Experimental investigation of turbulent mixing by Rayleigh–Taylor instability. Physica D 12 (1-3), 45–58.Google Scholar
Reckinger, S. J., Livescu, D. & Vasilyev, O. V. 2016 Comprehensive numerical methodology for direct numerical simulations of compressible Rayleigh–Taylor instability. J. Comput. Phys. 313, 181–208.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.Google Scholar
Roberts, M. S. & Jacobs, J. W. 2016 The effects of forced small-wavelength, finite-bandwidth initial perturbations and miscibility on the turbulent Rayleigh–Taylor instability. J. Fluid Mech. 787, 50–83.Google Scholar
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica D 12 (1), 3–18.Google Scholar
Strand, B. 1994 Summation by parts for finite difference approximations for d/dx . J. Comput. Phys. 110 (1), 47–67.Google Scholar
Svärd, M. & Nordström, J. 2008 A stable high-order finite difference scheme for the compressible Navier–Stokes equations: no-slip wall boundary conditions. J. Comput. Phys. 227 (10), 4805–4824.Google Scholar
Thiffeault, J.-L. 2012 Using multiscale norms to quantify mixing and transport. Nonlinearity 25 (2), R1.Google Scholar
Trevisan, A. & Legnani, R. 1995 Transient error growth and local predictability: a study in the Lorenz system. Tellus A 47 (1), 103–117.Google Scholar
Vermach, L. & Caulfield, C. P. 2018 Optimal mixing in three-dimensional plane Poiseuille flow at high Péclet number. J. Fluid Mech. 850, 875–923.Google Scholar
Vikhansky, A. 2002 Enhancement of laminar mixing by optimal control methods. Chem. Engng Sci. 57 (14), 2719–2725.Google Scholar
Vishnampet, R., Bodony, D. J. & Freund, J. B. 2015 A practical discrete-adjoint method for high-fidelity compressible turbulence simulations. J. Comput. Phys. 285, 173–192.Google Scholar
Vishnampet Ganapathi Subramanian, R.2015 An exact and consistent adjoint method for high-fidelity discretization of the compressible flow equations. PhD thesis, University of Illinois at Urbana-Champaign.Google Scholar
Wang, Q. & Gao, J.-H. 2013 The drag-adjoint field of a circular cylinder wake at Reynolds numbers 20, 100 and 500. J. Fluid Mech. 730, 145–161.Google Scholar
Wei, M. & Freund, J. B. 2006 A noise-controlled free shear flow. J. Fluid Mech. 546, 123–152.Google Scholar
Wei, T. & Livescu, D. 2012 Late-time quadratic growth in single-mode Rayleigh–Taylor instability. Phys. Rev. E 86 (4), 046405.Google Scholar
Woosley, S. E. & Weaver, T. A. 1986 The physics of supernova explosions. Annu. Rev. Astron. Astrophys. 24 (1), 205–253.Google Scholar
Xie, C. Y., Tao, J. J., Sun, Z. L. & Li, J. 2017 Retarding viscous Rayleigh–Taylor mixing by an optimized additional mode. Phys. Rev. E 95 (2), 023109.Google Scholar
Youngs, D. L. 1984 Numerical simulation of turbulent mixing by Rayleigh–Taylor instability. Physica D 12 (1–3), 32–44.Google Scholar
Youngs, D. L.2003 Application of miles to Rayleigh–Taylor and Richtmyer–Meshkov mixing. AIAA Paper 2003-4102.Google Scholar
Youngs, D. L. 2009 Application of monotone integrated large eddy simulation to Rayleigh–Taylor mixing. Phil. Trans. R. Soc. Lond. A 367 (1899), 2971–2983.Google Scholar