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Energy cascade and scaling in supersonic isothermal turbulence

Published online by Cambridge University Press:  22 July 2013

Alexei G. Kritsuk ,
Rick Wagner  and
Michael L. Norman
Alexei G. Kritsuk*
Affiliation:
Department of Physics and Center for Astrophysics and Space Sciences, University of California, San Diego, MC 0424, 9500 Gilman Drive, La Jolla, CA 92093-0424, USA
Rick Wagner
Affiliation:
San Diego Supercomputer Center, University of California, San Diego, MC 0505, 10100 Hopkins Drive, La Jolla, CA 92093-0505, USA
Michael L. Norman
Affiliation:
Department of Physics and Center for Astrophysics and Space Sciences, University of California, San Diego, MC 0424, 9500 Gilman Drive, La Jolla, CA 92093-0424, USA San Diego Supercomputer Center, University of California, San Diego, MC 0505, 10100 Hopkins Drive, La Jolla, CA 92093-0505, USA
*
†Email address for correspondence: [email protected]
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Abstract

Supersonic turbulence plays an important role in a number of extreme astrophysical and terrestrial environments, yet its understanding remains rudimentary. We use data from a three-dimensional simulation of supersonic isothermal turbulence to reconstruct an exact fourth-order relation derived analytically from the Navier–Stokes equations (Galtier & Banerjee, Phys. Rev. Lett., vol. 107, 2011, p. 134501). Our analysis supports a Kolmogorov-like inertial energy cascade in supersonic turbulence previously discussed on a phenomenological level. We show that two compressible analogues of the four-fifths law exist describing fifth- and fourth-order correlations, but only the fourth-order relation remains ‘universal’ in a wide range of Mach numbers from incompressible to highly compressible regimes. A new approximate relation valid in the strongly supersonic regime is derived and verified. We also briefly discuss the origin of bottleneck bumps in simulations of compressible turbulence.

JFM classification

Turbulent Flows: Compressible turbulence Turbulent Flows: Turbulence simulation Turbulent Flows: Turbulence theory
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 729 , 25 August 2013 , R1
Copyright
©2013 Cambridge University Press 

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References

Aluie, H. 2011 Compressible turbulence: the cascade and its locality. Phys. Rev. Lett. 106 (17), 174502.CrossRefGoogle ScholarPubMed
Aluie, H. 2013 Scale decomposition in compressible turbulence. Physica D 247 (1), 54–65.CrossRefGoogle Scholar
Aluie, H., Li, S. & Li, H. 2012 Conservative cascade of kinetic energy in compressible turbulence. Astrophys. J. Lett. 751, L29.CrossRefGoogle Scholar
Banerjee, S. & Galtier, S. 2013 Exact relation with two-point correlation functions and phenomenological approach for compressible magnetohydrodynamic turbulence. Phys. Rev. E 87, 013019.CrossRefGoogle ScholarPubMed
Colella, P. & Woodward, P. R. 1984 The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys. 54, 174–201.CrossRefGoogle Scholar
Davidson, P. A. & Pearson, B. R. 2005 Identifying turbulent energy distributions in real, rather than Fourier, space. Phys. Rev. Lett. 95 (21), 214501.CrossRefGoogle ScholarPubMed
Dobler, W., Haugen, N. E., Yousef, T. A. & Brandenburg, A. 2003 Bottleneck effect in three-dimensional turbulence simulations. Phys. Rev. E 68 (2), 026304.CrossRefGoogle ScholarPubMed
Falkovich, G. 1994 Bottleneck phenomenon in developed turbulence. Phys. Fluids 6, 1411–1414.CrossRefGoogle Scholar
Falkovich, G., Fouxon, I. & Oz, Y. 2010 New relations for correlation functions in Navier–Stokes turbulence. J. Fluid Mech. 644, 465–472.CrossRefGoogle Scholar
Federrath, C., Roman-Duval, J., Klessen, R. S., Schmidt, W. & Mac Low, M.-M. 2010 Comparing the statistics of interstellar turbulence in simulations and observations: solenoidal versus compressive turbulence forcing. Astron. Astrophys. 512, A81.CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence: the Legacy of A. N. Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
Galtier, S. & Banerjee, S. 2011 Exact relation for correlation functions in compressible isothermal turbulence. Phys. Rev. Lett. 107 (13), 134501.CrossRefGoogle ScholarPubMed
Grinstein, F. F., Margolin, L. G. & Rider, W. J. (Eds) 2007 Implicit Large Eddy Simulation: Computing Turbulent Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Hennebelle, P. & Falgarone, E. 2012 Turbulent molecular clouds. Astron. Astrophys. Rev. 20:55, 1–58.Google Scholar
Hobbs, A., Nayakshin, S., Power, C. & King, A. 2011 Feeding supermassive black holes through supersonic turbulence and ballistic accretion. Mon. Not. R. Astron. Soc. 413, 2633–2650.CrossRefGoogle Scholar
Ingenito, A. & Bruno, C. 2010 Physics and regimes of supersonic combustion. AIAA J. 48, 515–525.CrossRefGoogle Scholar
Kim, J. & Ryu, D. 2005 Density power spectrum of compressible hydrodynamic turbulent flows. Astrophys. J. Lett. 630, L45–L48.CrossRefGoogle Scholar
Kolmogorov, A. N. 1941 Dissipation of energy in the locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 19–21.Google Scholar
Kowal, G. & Lazarian, A. 2007 Scaling relations of compressible MHD turbulence. Astrophys. J. Lett. 666, L69–L72.CrossRefGoogle Scholar
Kritsuk, A. G., Norman, M. L., Padoan, P. & Wagner, R. 2007a The statistics of supersonic isothermal turbulence. Astrophys. J. 665, 416–431.CrossRefGoogle Scholar
Kritsuk, A. G., Padoan, P., Wagner, R. & Norman, M. L. 2007b Scaling laws and intermittency in highly compressible turbulence. In Turbulence and Nonlinear Processes in Astrophysical Plasmas (ed. Shaikh, D. & Zank, G. P.), AIP Conference Proceedings, vol. 932, pp. 393–399.Google Scholar
Kritsuk, A. G., Ustyugov, S., Norman, M. L. & Padoan, P. 2010 Self-organization in turbulent molecular clouds: compressional versus solenoidal modes. In Numerical Modelling of Space Plasma Flows (ed. Pogorelov, N., Audit, E. & Zank, G.), ASP Conference Series, vol. 429, pp. 15–21.Google Scholar
Moffat, A. F. J. & Robert, C. 1994 Clumping and mass loss in hot star winds. Astrophys. J. 421, 310–313.CrossRefGoogle Scholar
Ogden, D. E., Glatzmaier, G. A. & Wohletz, K. H. 2008 Effects of vent overpressure on buoyant eruption columns: implications for plume stability. Earth Planet. Sci. Lett. 268, 283–292.CrossRefGoogle Scholar
O’Shea, B. W., Bryan, G., Bordner, J., Norman, M. L., Abel, T., Harkness, R. & Kritsuk, A. 2004 Introducing Enzo, an AMR cosmology application. ArXiv(astro-ph/0403044).Google Scholar
Pan, L., Padoan, P. & Kritsuk, A. G. 2009 Dissipative structures in supersonic turbulence. Phys. Rev. Lett. 102, 034501.CrossRefGoogle ScholarPubMed
Porter, D. H., Pouquet, A. & Woodward, P. R. 1992 A numerical study of supersonic turbulence. Theor. Comput. Fluid Dyn. 4, 13–49.CrossRefGoogle Scholar
Porter, D. H., Pouquet, A. & Woodward, P. R. 1994 Kolmogorov-like spectra in decaying three-dimensional supersonic flows. Phys. Fluids 6, 2133–2142.CrossRefGoogle Scholar
Porter, D. H. & Woodward, P. R. 1994 High-resolution simulations of compressible convection using the piecewise-parabolic method. Astrophys. J. Suppl. 93, 309–349.CrossRefGoogle Scholar
Price, D. J. & Federrath, C. 2010 A comparison between grid and particle methods on the statistics of driven, supersonic, isothermal turbulence. Mon. Not. R. Astron. Soc. 406, 1659–1674.Google Scholar
Robertson, B. & Goldreich, P. 2012 Adiabatic heating of contracting turbulent fluids. Astrophys. J. Lett. 750, L31.CrossRefGoogle Scholar
Schmidt, W., Federrath, C. & Klessen, R. 2008 Is the scaling of supersonic turbulence universal? Phys. Rev. Lett. 101 (19), 194505.CrossRefGoogle ScholarPubMed
Schwarz, C., Beetz, C., Dreher, J. & Grauer, R. 2010 Lyapunov exponents and information dimension of the mass distribution in turbulent compressible flows. Phys. Lett. A 374, 1039–1042.CrossRefGoogle Scholar
Wagner, R., Falkovich, G., Kritsuk, A. G. & Norman, M. L. 2012 Flux correlations in supersonic isothermal turbulence. J. Fluid Mech. 713, 482–490.CrossRefGoogle Scholar
Wang, J., Shi, Y., Wang, L.-P., Xiao, Z., He, X. & Chen, S. 2012 Effect of compressibility on the small-scale structures in isotropic turbulence. J. Fluid Mech. 713, 588–631.CrossRefGoogle Scholar