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Nonlinear effects in buoyancy-driven variable-density turbulence

  • P. Rao (a1), C. P. Caulfield (a2) (a3) and J. D. Gibbon (a4)

We consider the time dependence of a hierarchy of scaled $L^{2m}$ -norms $D_{m,\unicode[STIX]{x1D714}}$ and $D_{m,\unicode[STIX]{x1D703}}$ of the vorticity $\unicode[STIX]{x1D74E}=\unicode[STIX]{x1D735}\times \boldsymbol{u}$ and the density gradient $\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$ , where $\unicode[STIX]{x1D703}=\log (\unicode[STIX]{x1D70C}^{\ast }/\unicode[STIX]{x1D70C}_{0}^{\ast })$ , in a buoyancy-driven turbulent flow as simulated by Livescu & Ristorcelli (J. Fluid Mech., vol. 591, 2007, pp. 43–71). Here, $\unicode[STIX]{x1D70C}^{\ast }(\boldsymbol{x},t)$ is the composition density of a mixture of two incompressible miscible fluids with fluid densities $\unicode[STIX]{x1D70C}_{2}^{\ast }>\unicode[STIX]{x1D70C}_{1}^{\ast }$ , and $\unicode[STIX]{x1D70C}_{0}^{\ast }$ is a reference normalization density. Using data from the publicly available Johns Hopkins turbulence database, we present evidence that the $L^{2}$ -spatial average of the density gradient $\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$ can reach extremely large values at intermediate times, even in flows with low Atwood number $At=(\unicode[STIX]{x1D70C}_{2}^{\ast }-\unicode[STIX]{x1D70C}_{1}^{\ast })/(\unicode[STIX]{x1D70C}_{2}^{\ast }+\unicode[STIX]{x1D70C}_{1}^{\ast })=0.05$ , implying that very strong mixing of the density field at small scales can arise in buoyancy-driven turbulence. This large growth raises the possibility that the density gradient $\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$ might blow up in a finite time.

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Andrews, M. J. & Dalziel, S. B. 2010 Small Atwood number Rayleigh–Taylor experiments. Phil. Trans. R. Soc. Lond. A 368 (1916), 16631679.
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), 562568.
Cook, A. W. & Dimotakis, P. E. 2001 Transition stages of Rayleigh–Taylor instability between miscible fluids. J. Fluid Mech. 443, 6999.
Davies-Wykes, M. S. & Dalziel, S. B. 2014 Efficient mixing in stratified flows: experimental study of a Rayleigh–Taylor unstable interface within an otherwise stable stratification. J. Fluid Mech. 756, 10271057.
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, 16681693.
Dimotakis, P. E. 2005 Turbulent mixing. Annu. Rev. Fluid Mech. 37, 329356.
Donzis, D. A., Gibbon, J. D., Kerr, R. M., Gupta, A., Pandit, R. & Vincenzi, D. 2013 Vorticity moments in four numerical simulations of the 3D Navier–Stokes equations. J. Fluid Mech. 732, 316331.
Gibbon, J. D. 2015 High–low frequency slaving and regularity issues in the 3D Navier–Stokes equations. IMA J. Appl. Maths 81, 308320.
Gibbon, J. D., Donzis, D. A., Kerr, R. M., Gupta, A., Pandit, R. & Vincenzi, D. 2014 Regimes of nonlinear depletion and regularity in the 3D Navier–Stokes equations. Nonlinearity 27, 119.
Glimm, J., Grove, J. W., Li, X. L., Oh, W. & Sharp, D. H. 2001 A critical analysis of Rayleigh–Taylor growth rates. J. Comput. Phys. 169 (2), 652677.
Hyunsun, L., Hyeonseong, J., Yan, Y. & Glimm, J. 2008 On validation of turbulent mixing simulations for Rayleigh–Taylor instability. Phys. Fluids 20, 012102.
Lawrie, A. G. W. & Dalziel, S. B. 2011 Rayleigh–Taylor mixing in an otherwise stable stratification. J. Fluid Mech. 688, 507527.
Lee, H., Jin, H., Yu, Y. & Glimm, J. 2008 On validation of turbulent mixing simulations for Rayleigh–Taylor instability. Phys. Fluids 20, 18.
Livescu, D. 2013 Numerical simulations of two-fluid mixing at large density ratios and applications to the Rayleigh–Taylor instability. Phil. Trans. R. Soc. Lond. A 371, 20120185.
Livescu, D., Canada, C., Kanov, K., Burns, R. & Pulido, J.2014 Homogeneous buoyancy driven turbulence data set. LA-UR-14-20669.
Livescu, D., Mohd-Yusof, J., Petersen, M. R. & Grove, J. W.2009 A computer code for direct numerical simulation of turbulent flows. Tech. Rep. LA-CC-09-100. Los Alamos National Laboratory.
Livescu, D. & Ristorcelli, J. R. 2007 Buoyancy-driven variable-density turbulence. J. Fluid Mech. 591, 4371.
Livescu, D. & Ristorcelli, J. R. 2008 Variable-density mixing in buoyancy-driven turbulence. J. Fluid Mech. 605, 145180.
Luo, G. & Hou, T. 2014a Potentially sinugular solutions of the 3D axisymmetric Euler equations. Proc. Natl Acad. Sci. USA 111, 1296812973.
Luo, G. & Hou, T. 2014b Toward the finite time blow-up of the 3D incompressible Euler equations: a numerical investigation. Multiscale Model. Simul. 12, 17221776.
Petrasso, R. D. 1994 Rayleigh’s challenge endures. Nat. Phys. 367 (6460), 217218.
Rayleigh, Lord 1900 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. In Scientific Papers, vol. 2, p. 598. Cambridge University Press.
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Phys. D 12D, 318.
Tailleux, R. 2013 Available potential energy and exergy in stratified fluids. Annu. Rev. Fluid Mech. 45, 3558.
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond A 201 (1065), 192196.
Youngs, D. L. 1984 Numerical simulation of turbulent mixing by Rayleigh–Taylor instability. Phys. D 12 (1–3), 3244.
Youngs, D. L. 1989 Modelling turbulent mixing by Rayleigh–Taylor instability. Phys. D 37, 270287.
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Journal of Fluid Mechanics
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