Optimal perturbations for controlling the growth of a Rayleigh-Taylor instability. (English) Zbl 1419.76224
Summary: 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.
MSC:
| 76E17 | Interfacial stability and instability in hydrodynamic stability |
| 76M30 | Variational methods applied to problems in fluid mechanics |
| 76F65 | Direct numerical and large eddy simulation of turbulence |
References:
| [1] | Aamo, O. M., Krstić, M. & Bewley, T. R.2003Control of mixing by boundary feedback in 2D channel flow. Automatica39 (9), 1597-1606.10.1016/S0005-1098(03)00140-7 · Zbl 1055.93039 |
| [2] | Alon, U., Hecht, J., Mukamel, D. & Shvarts, D.1994Scale invariant mixing rates of hydrodynamically unstable interfaces. Phys. Rev. Lett.72 (18), 2867-2870.10.1103/PhysRevLett.72.2867 |
| [3] | Anuchina, N. N., Kucherenko, Y. A., Neuvazhaev, V. E., Ogibina, V. N., Shibarshov, L. I. & Yakovlev, V. G.1978Turbulent mixing at an accelerating interface between liquids of different density. Fluid Dyn.13 (6), 916-920. |
| [4] | Babchin, A. J., Frenkel, A. L., Levich, B. G. & Sivashinsky, G. I.1983Nonlinear saturation of Rayleigh-Taylor instability in thin films. Phys. Fluids26 (11), 3159-3161.10.1063/1.864083 · Zbl 0521.76046 |
| [5] | Baker, L. & Freeman, J. R.1981Heuristic model of the nonlinear Rayleigh-Taylor instability. J. Appl. Phys.52 (2), 655-663.10.1063/1.328793 |
| [6] | Banerjee, A. & Andrews, M. J.20093D simulations to investigate initial condition effects on the growth of Rayleigh-Taylor mixing. Intl J. Heat Mass Transfer52 (17), 3906-3917.10.1016/j.ijheatmasstransfer.2009.03.032 · Zbl 1167.76337 |
| [7] | Bodony, D. J.2010Accuracy of the simultaneous-approximation-term boundary condition for time-dependent problems. J. Sci. Comput.43 (1), 118-133.10.1007/s10915-010-9347-4 · Zbl 1203.76043 |
| [8] | Braman, K., Oliver, T. A. & Raman, V.2015Adjoint-based sensitivity analysis of flames. Combust. Theor. Model.19 (1), 29-56.10.1080/13647830.2014.976274 · Zbl 1519.80030 |
| [9] | Cabot, W.2006Comparison of two-and three-dimensional simulations of miscible Rayleigh-Taylor instability. Phys. Fluids18 (4), 045101.10.1063/1.2191856 |
| [10] | Cabot, W. H. & Cook, A. W.2006Reynolds number effects on Rayleigh-Taylor instability with possible implications for type Ia supernovae. Nat. Phys.2 (8), 562-568.10.1038/nphys361 |
| [11] | Capecelatro, J., Bodony, D. J. & Freund, J. B.2017Adjoint-based sensitivity analysis of ignition in a turbulent reactive shear layer. In 55th AIAA Aerospace Sciences Meeting, AIAA 2017-0846. AIAA. |
| [12] | Capecelatro, J., Bodony, D. J. & Freund, J. B.2018Adjoint-based sensitivity and ignition threshold mapping in a turbulent mixing layer. Combust. Theor. Model.147-179. · Zbl 1519.76063 |
| [13] | Capecelatro, J., Vishnampet, R., Bodony, D. J. & Freund, J. B.2016Adjoint-based sensitivity analysis of localized ignition in a non-premixed hydrogen-air mixing layer. In 54th AIAA Aerospace Sciences Meeting, AIAA 2016-2153. AIAA. |
| [14] | Carles, P., Huang, Z., Carbone, G. & Rosenblatt, C.2006Rayleigh-Taylor instability for immiscible fluids of arbitrary viscosities: a magnetic levitation investigation and theoretical model. Phys. Rev. Lett.96 (10), 104501.10.1103/PhysRevLett.96.104501 |
| [15] | Carnarius, A., Thiele, F., Oezkaya, E. & Gauger, N. R.2010 Adjoint approaches for optimal flow control. AIAA Paper 2010-5088.10.2514/6.2010-5088 |
| [16] | Carpenter, M. H., Gottlieb, D. & Abarbanel, S.1994Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes. J. Comput. Phys.111 (2), 220-236.10.1006/jcph.1994.1057 · Zbl 0832.65098 |
| [17] | Chandrasekhar, S.1961Hydrodynamic and hydromagnetic stability. In International Series of Monographs on Physics. Clarendon Press. · Zbl 0142.44103 |
| [18] | Cimpeanu, R., Papageorgiou, D. T. & Petropoulos, P. G.2014On the control and suppression of the Rayleigh-Taylor instability using electric fields. Phys. Fluids26 (2), 022105.10.1063/1.4865674 · Zbl 1321.76029 |
| [19] | Cook, A. W.2009Enthalpy diffusion in multicomponent flows. Phys. Fluids21 (5), 055109.10.1063/1.3139305 · Zbl 1183.76157 |
| [20] | Cook, A. W. & Dimotakis, P. E.2001Transition stages of Rayleigh-Taylor instability between miscible fluids. J. Fluid Mech.443, 69-99.10.1017/S0022112001005377 · Zbl 1015.76037 |
| [21] | Dimonte, G.2004Dependence of turbulent Rayleigh-Taylor instability on initial perturbations. Phys. Rev. E69 (5), 056305. |
| [22] | Dimonte, G., Youngs, D. L., Dimits, A., Weber, S., Marinak, M., Wunsch, S., Garasi, C., Robinson, A., Andrews, M. J., Ramaprabhu, P.2004A comparative study of the turbulent Rayleigh-Taylor instability using high-resolution three-dimensional numerical simulations: the Alpha-Group collaboration. Phys. Fluids16 (5), 1668-1693.10.1063/1.1688328 · Zbl 1186.76143 |
| [23] | Duff, R. E., Harlow, F. H. & Hirt, C. W.1962Effects of diffusion on interface instability between gases. Phys. Fluids5 (4), 417-425.10.1063/1.1706634 · Zbl 0111.23601 |
| [24] | Foures, D. P. G., Caulfield, C. P. & Schmid, P. J.2014Optimal mixing in two-dimensional plane Poiseuille flow at finite Péclet number. J. Fluid Mech.748, 241-277.10.1017/jfm.2014.182 · Zbl 1416.76036 |
| [25] | Freund, J. B.1997Proposed inflow/outflow boundary condition for direct computation of aerodynamic sound. AIAA J.35 (4), 740-742.10.2514/2.167 · Zbl 0903.76081 |
| [26] | Fujimoto, M. Y.1993The evolution of accreting stars with turbulent mixing. Astrophys. J.419, 768-775.10.1086/173528 |
| [27] | Goncharov, V. N.2002Analytical model of nonlinear, single-mode, classical Rayleigh-Taylor instability at arbitrary Atwood numbers. Phys. Rev. Lett.88 (13), 134502.10.1103/PhysRevLett.88.134502 |
| [28] | Haan, S. W.1989Onset of nonlinear saturation for Rayleigh-Taylor growth in the presence of a full spectrum of modes. Phys. Rev. A39 (11), 5812-5825.10.1103/PhysRevA.39.5812 |
| [29] | Halpern, D. & Frenkel, A. L.2001Saturated Rayleigh-Taylor instability of an oscillating Couette film flow. J. Fluid Mech.446, 67-93. · Zbl 1029.76020 |
| [30] | Hecht, J., Alon, U. & Shvarts, D.1994Potential flow models of Rayleigh-Taylor and Richtmyer-Meshkov bubble fronts. Phys. Fluids6 (12), 4019-4030.10.1063/1.868391 · Zbl 0922.76169 |
| [31] | Huang, Z., De Luca, A., Atherton, T. J., Bird, M., Rosenblatt, C. & Carlès, P.2007Rayleigh-Taylor instability experiments with precise and arbitrary control of the initial interface shape. Phys. Rev. Lett.99 (20), 204502.10.1103/PhysRevLett.99.204502 |
| [32] | Jameson, A.1989Aerodynamic design via control theory. In Recent Advances in Computational Fluid Dynamics, pp. 377-401. Springer.10.1007/978-3-642-83733-3_14 |
| [33] | Jameson, A. & Martinelli, L.2000Aerodynamic shape optimization techniques based on control theory. In Computational Mathematics Driven by Industrial Problems, pp. 151-221. Springer.10.1007/BFb0103920 |
| [34] | Janka, H.-T. & Müller, E.1996Neutrino heating, convection, and the mechanism of type-II supernova explosions. Astron. Astrophys.306, 167-198. |
| [35] | Jones, E., Oliphant, T. & Peterson, P.2001 SciPy: Open source scientific tools for Python. http://www.scipy.org/. |
| [36] | Kraft, D.1988A software package for sequential quadratic programming. Forschungsbericht- Deutsche Forschungs- und Versuchsanstalt fur Luft- und Raumfahrt. · Zbl 0646.90065 |
| [37] | Kreiss, H.-O. & Scherer, G.1974Finite element and finite difference methods for hyperbolic partial differential equations. In Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 195-212. Elsevier.10.1016/B978-0-12-208350-1.50012-1 · Zbl 0355.65085 |
| [38] | Layzer, D.1955On the instability of superposed fluids in a gravitational field. Astrophys. J.122, 1-12.10.1086/146048 |
| [39] | Lea, D. J., Allen, M. R. & Haine, T. W. N.2000Sensitivity analysis of the climate of a chaotic system. Tellus A52 (5), 523-532.10.3402/tellusa.v52i5.12283 |
| [40] | Lindl, J.1995Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition and gain. Phys. Plasmas2 (11), 3933-4024.10.1063/1.871025 |
| [41] | Liu, W.2006Mixing enhancement via flow optimization. In Decision and Control, 2006 45th IEEE Conference on, pp. 5323-5328. IEEE. |
| [42] | Livescu, D.2004Compressibility effects on the Rayleigh-Taylor instability growth between immiscible fluids. Phys. Fluids16 (1), 118-127.10.1063/1.1630800 · Zbl 1186.76332 |
| [43] | Livescu, D.2013Numerical simulations of two-fluid turbulent mixing at large density ratios and applications to the Rayleigh-Taylor instability. Phil. Trans. R. Soc. Lond. A371 (2003), 20120185. · Zbl 1353.76028 |
| [44] | Livescu, D., Wei, T. & Petersen, M. R.2011Direct numerical simulations of Rayleigh-Taylor instability. In J. Phys: Conference Series, vol. 318, p. 082007. IOP Publishing. |
| [45] | Lopez-Zazueta, A., Fontane, J. & Joly, L.2016Optimal perturbations in time-dependent variable-density Kelvin-Helmholtz billows. J. Fluid Mech.803, 466-501.10.1017/jfm.2016.509 · Zbl 1462.76071 |
| [46] | Martins, J. R. R. A., Alonso, J. J. & Reuther, J. J.2004High-fidelity aerostructural design optimization of a supersonic business jet. J. Aircraft41 (3), 523-530.10.2514/1.11478 |
| [47] | Mattsson, K., Svärd, M. & Nordström, J.2004Stable and accurate artificial dissipation. J. Sci. Comput.21 (1), 57-79.10.1023/B:JOMP.0000027955.75872.3f · Zbl 1085.76050 |
| [48] | Miles, C. J.2018 Optimal control of the advection-diffusion equation for effective fluid mixing. PhD thesis, University of Michigan, Ann Arbor, MI. |
| [49] | Millman, K. J. & Aivazis, M.2011Python for scientists and engineers. Comput. Sci. & Engng13 (2), 9-12.10.1109/MCSE.2011.36 |
| [50] | Movahed, P.2014 High-fidelity numerical simulations of compressible turbulence and mixing generated by hydrodynamic instabilities. PhD thesis, University of Michigan, Ann Arbor, MI. |
| [51] | Nadarajah, S. & Jameson, A.2000. A comparison of the continuous and discrete adjoint approach to automatic aerodynamic optimization. AIAA Paper 2000-0667.10.2514/6.2000-667 |
| [52] | Nakai, S. & Takabe, H.1996Principles of inertial confinement fusion-physics of implosion and the concept of inertial fusion energy. Rep. Prog. Phys.59 (9), 1071-1131.10.1088/0034-4885/59/9/002 |
| [53] | Oliphant, T. E.2007Python for scientific computing. Comput. Sci. Engng9 (3), 10-20.10.1109/MCSE.2007.58 |
| [54] | Oron, D., Arazi, L., Kartoon, D., Rikanati, A., Alon, U. & Shvarts, D.2001Dimensionality dependence of the Rayleigh-Taylor and Richtmyer-Meshkov instability late-time scaling laws. Phys. Plasmas8 (6), 2883-2889.10.1063/1.1362529 |
| [55] | Peters, N.2000Turbulent Combustion. Cambridge University Press.10.1017/CBO9780511612701 · Zbl 0955.76002 |
| [56] | Piro, A. L. & Bildsten, L.2007Turbulent mixing in the surface layers of accreting neutron stars. Astrophys. J.663 (2), 1252-1268.10.1086/518687 |
| [57] | Pitsch, H.2006Large-eddy simulation of turbulent combustion. Annu. Rev. Fluid Mech.38, 453-482.10.1146/annurev.fluid.38.050304.092133 · Zbl 1097.76042 |
| [58] | Qadri, U. A., Chandler, G. J. & Juniper, M. P.2015Self-sustained hydrodynamic oscillations in lifted jet diffusion flames: origin and control. J. Fluid Mech.775, 201-222.10.1017/jfm.2015.297 · Zbl 1403.76014 |
| [59] | Ramaprabhu, P., Dimonte, G. & Andrews, M. J.2005A numerical study of the influence of initial perturbations on the turbulent Rayleigh-Taylor instability. J. Fluid Mech.536, 285-319.10.1017/S002211200500488X · Zbl 1122.76035 |
| [60] | Read, K. I.1984Experimental investigation of turbulent mixing by Rayleigh-Taylor instability. Physica D12 (1-3), 45-58.10.1016/0167-2789(84)90513-X |
| [61] | Reckinger, S. J., Livescu, D. & Vasilyev, O. V.2016Comprehensive numerical methodology for direct numerical simulations of compressible Rayleigh-Taylor instability. J. Comput. Phys.313, 181-208.10.1016/j.jcp.2015.11.002 · Zbl 1349.76138 |
| [62] | Ristorcelli, J. R. & Clark, T. T.2004Rayleigh-Taylor turbulence: self-similar analysis and direct numerical simulations. J. Fluid Mech.507, 213-253.10.1017/S0022112004008286 · Zbl 1134.76364 |
| [63] | Roberts, M. S. & Jacobs, J. W.2016The effects of forced small-wavelength, finite-bandwidth initial perturbations and miscibility on the turbulent Rayleigh-Taylor instability. J. Fluid Mech.787, 50-83.10.1017/jfm.2015.599 · Zbl 1359.76119 |
| [64] | Sharp, D. H.1984An overview of Rayleigh-Taylor instability. Physica D12 (1), 3-18.10.1016/0167-2789(84)90510-4 · Zbl 0577.76047 |
| [65] | Strand, B.1994Summation by parts for finite difference approximations for d/dx. J. Comput. Phys.110 (1), 47-67.10.1006/jcph.1994.1005 · Zbl 0792.65011 |
| [66] | Svärd, M. & Nordström, J.2008A stable high-order finite difference scheme for the compressible Navier-Stokes equations: no-slip wall boundary conditions. J. Comput. Phys.227 (10), 4805-4824.10.1016/j.jcp.2007.12.028 · Zbl 1260.76021 |
| [67] | Thiffeault, J.-L.2012Using multiscale norms to quantify mixing and transport. Nonlinearity25 (2), R1.10.1088/0951-7715/25/2/R1 · Zbl 1325.37007 |
| [68] | Trevisan, A. & Legnani, R.1995Transient error growth and local predictability: a study in the Lorenz system. Tellus A47 (1), 103-117.10.3402/tellusa.v47i1.11496 |
| [69] | Vermach, L. & Caulfield, C. P.2018Optimal mixing in three-dimensional plane Poiseuille flow at high Péclet number. J. Fluid Mech.850, 875-923.10.1017/jfm.2018.388 · Zbl 1415.76304 |
| [70] | Vikhansky, A.2002Enhancement of laminar mixing by optimal control methods. Chem. Engng Sci.57 (14), 2719-2725.10.1016/S0009-2509(02)00122-7 · Zbl 1185.76380 |
| [71] | Vishnampet, R., Bodony, D. J. & Freund, J. B.2015A practical discrete-adjoint method for high-fidelity compressible turbulence simulations. J. Comput. Phys.285, 173-192.10.1016/j.jcp.2015.01.009 · Zbl 1351.76190 |
| [72] | 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. |
| [73] | Wang, Q. & Gao, J.-H.2013The drag-adjoint field of a circular cylinder wake at Reynolds numbers 20, 100 and 500. J. Fluid Mech.730, 145-161.10.1017/jfm.2013.323 · Zbl 1291.76117 |
| [74] | Wei, M. & Freund, J. B.2006A noise-controlled free shear flow. J. Fluid Mech.546, 123-152.10.1017/S0022112005007093 · Zbl 1222.76084 |
| [75] | Wei, T. & Livescu, D.2012Late-time quadratic growth in single-mode Rayleigh-Taylor instability. Phys. Rev. E86 (4), 046405. |
| [76] | Woosley, S. E. & Weaver, T. A.1986The physics of supernova explosions. Annu. Rev. Astron. Astrophys.24 (1), 205-253.10.1146/annurev.aa.24.090186.001225 |
| [77] | Xie, C. Y., Tao, J. J., Sun, Z. L. & Li, J.2017Retarding viscous Rayleigh-Taylor mixing by an optimized additional mode. Phys. Rev. E95 (2), 023109. |
| [78] | Youngs, D. L.1984Numerical simulation of turbulent mixing by Rayleigh-Taylor instability. Physica D12 (1-3), 32-44.10.1016/0167-2789(84)90512-8 |
| [79] | Youngs, D. L.2003 Application of miles to Rayleigh-Taylor and Richtmyer-Meshkov mixing. AIAA Paper 2003-4102.10.2514/6.2003-4102 |
| [80] | Youngs, D. L.2009Application of monotone integrated large eddy simulation to Rayleigh-Taylor mixing. Phil. Trans. R. Soc. Lond. A367 (1899), 2971-2983.10.1098/rsta.2008.0303 · Zbl 1185.76734 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
