Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-30T02:09:28.463Z Has data issue: false hasContentIssue false
  • English
  • Français

New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence

Published online by Cambridge University Press:  20 July 2020

Hamed Sadeghi Open the ORCID record for Hamed Sadeghi [Opens in a new window]  and
Martin Oberlack Open the ORCID record for Martin Oberlack [Opens in a new window]
Hamed Sadeghi
Affiliation:
Department of Mechanical Engineering, University of Ottawa, Ottawa, ON K1N 6N5, Canada Chair of Fluid Dynamics, TU Darmstadt, Otto-Bernd-Strasse 2, 64287Darmstadt, Germany
Martin Oberlack*
Affiliation:
Chair of Fluid Dynamics, TU Darmstadt, Otto-Bernd-Strasse 2, 64287Darmstadt, Germany Centre for Computational Engineering, TU Darmstadt, Dolivostrasse 15, 64293Darmstadt, Germany
*
†Email address for correspondence: oberlack@fdy.tu-darmstadt.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We use the Lie symmetry theory to derive new scaling laws for passive scalar dynamics under the influence of a constant mean gradient of the scalar in decaying homogeneous isotropic turbulence. For this purpose, we apply symmetry analysis to the equations for two-point correlation of the scalar and velocity fluctuations. It is shown that, in contrast to the classical self-similarity approach, the general invariant solutions, respectively scaling laws, of the two-point functions are constructed using the symmetry approach, without requiring an a priori set of similarity scales to carry on the analysis. In the context of the current analysis also, scaling laws for one-point quantities of the scalar variance $\overline{\unicode[STIX]{x1D703}^{2}}$, the transverse scalar flux $\overline{u_{2}\unicode[STIX]{x1D703}}$ and the variance of the turbulent velocity fluctuations $\overline{u^{2}}$ are established, which are essentially related to the scaling symmetries. A key step to derive the scaling laws is the symmetry breaking induced by the constant mean scalar gradient. We use the results of a highly resolved direct numerical simulation of Gauding et al. (Comput. Fluids, vol. 180, 2019, pp. 206–217) to verify the scaling laws and the self-similarity of the two-point correlation functions. It is shown that the general symmetry solutions obtained from symmetry results provide a very good similarity to these functions.

JFM classification

Turbulent Flows: Turbulence theory Turbulent Flows: Isotropic turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 899 , 25 September 2020 , A10
Copyright
© The Author(s), 2020. Published by 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

Antonia, R. A. & Orlandi, P. 2003 Effect of Schmidt number on small-scale passive scalar turbulence. Appl. Mech. Rev. 56, 615–632.CrossRefGoogle Scholar
Bahri, C., Arwatz, G., George, W. K., Mueller, M. E. & Hultmark, M. 2015 Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient. J. Fluid Mech. 780, 215–225.CrossRefGoogle Scholar
Batchelor, G. K. & Townsend, A. A. 1948 Decay of isotropic turbulence in the initial period. Proc. R. Soc. Lond. A 193 (1035), 539–558.Google Scholar
Batchelor, G. K. & Proudman, I. 1956 The large-scale structure of homogeneous turbulence. Phil. Trans. R. Soc. Lond. A 248, 369–405.Google Scholar
Bluman, G. W., Cheviakov, A. F. & Anco, S. C. 2010 Applications of Symmetry Methods to Partial Differential Equations. Springer.CrossRefGoogle Scholar
Corrsin, S. 1951a The decay of isotropic temperature fluctuations in an isotropic turbulence. J. Aeronaut. Sci. 18, 417–423.Google Scholar
Corrsin, S. 1951b On the spectrum of isotropic temperature fluctuations in an isotropic turbulence. J. Appl. Phys. 22, 469–473.CrossRefGoogle Scholar
Gauding, M., Danaila, L. & Varea, E. 2017 High-order structure functions for passive scalar fed by a mean gradient. Intl J. Heat Fluid Flow 67, 86–93.CrossRefGoogle Scholar
Gauding, M., Goebbert, J. H., Hasse, C. & Peters, N. 2015 Line segments in homogeneous scalar turbulence. Phys. Fluids 27 (9), 095102.CrossRefGoogle Scholar
Gauding, M., Wang, L., Goebbert, J. H., Boded, M., Danaila, L. & Vareaa, E. 2019 On the self-similarity of line segments in decaying homogeneous isotropic turbulence. Comput. Fluids 180, 206–217.CrossRefGoogle Scholar
George, W. K. 1989 The self-preservation of turbulent flows and its relation to initial conditions and coherent structures. In Advances in Turbulence (ed. George, W. K. & Arndt, R.). Springer.Google Scholar
George, W. K. 1992a The decay of homogeneous isotropic turbulence. Phys. Fluids 4 (7), 1492–1509.CrossRefGoogle Scholar
George, W. K. 1992b Self-preservation of Temperature Fluctuations in Isotropic Turbulence, Studies in Turbulence, pp. 514–528. Springer.Google Scholar
Hinze, O. J. 1959a Turbulence, An Introduction to its Mechanism and Theory. McGraw-Hill.Google Scholar
Hydon, P. E. 2000a Symmetry Methods for Differential Equations: A Beginner’s Guide. Cambridge University Press.CrossRefGoogle Scholar
Ishida, T., Davidson, P. & Kaneda, Y. 2006 On the decay of isotropic turbulence. J. Fluid Mech. 564, 455–475.CrossRefGoogle Scholar
von Kármán, T. & Howarth, L. 1938 On the statistical theory of isotropic turbulence. Proc. R. Soc. Lond. A (164), 192–215.CrossRefGoogle Scholar
Loitsianskii, L. G.1939 Some basic laws of isotropic turbulent flow. Cent. Aero. Hydrodyn. Inst. Moscow, Rep. 440 (translation in NACA Tech. Memo. 1079).Google Scholar
Lundgren, T. S. 2003 Kolmogorov turbulence by matched asymptotic expansions. Phys. Fluids 15, 1074–1081.CrossRefGoogle Scholar
Mohamed, M. S. & Larue, J. C. 1990 The decay power law in grid-generated turbulence. J. Fluid Mech. 219, 195–214.CrossRefGoogle Scholar
Mydlarski, L. 2003 Mixed velocity-passive scalar statistics in high-Reynolds-number turbulence. J. Fluid Mech. 475, 173–203.CrossRefGoogle Scholar
Oberlack, M. 2000a Asymptotic expansion, symmetry groups, and invariant solutions of laminar and turbulent wall-bounded flows. Z. Angew. Math. Mech. 80, 791–800.3.0.CO;2-5>CrossRefGoogle Scholar
Oberlack, M.2000b Symmetrie, Invarianz und Selbstähnlichkeit in der Turbulenz. Habilitation thesis, RWTH Aachen.Google Scholar
Oberlack, M. 2001 A unified approach for symmetries in plane parallel turbulent shear flows. J. Fluid Mech. 427, 299–328.CrossRefGoogle Scholar
Oberlack, M. 2002 On the decay exponent of isotropic turbulence. Proc. Appl. Maths. Mech. 1, 294–297.3.0.CO;2-W>CrossRefGoogle Scholar
Oberlack, M. & Peters, N. 1993 Closure of the two-point correlation equation as a basis for Reynolds stress models. Appl. Sci. Res. 51, 533–539.CrossRefGoogle Scholar
Oberlack, M. & Rosteck, A. 2010 New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws. Discrete Continuous Dyn. Syst. 3, 451–471.CrossRefGoogle Scholar
Oberlack, M. & Rosteck, A. 2011 Applications of the new symmetries of the multi-point correlation equations. J. Phys.: Conf. Ser. 318, 042011.Google Scholar
Oberlack, M., Waclawczyk, M., Rosteck, A. & Avsarkisov, V. 2015 Symmetries and their importance for statistical turbulence theory. Mech. Engng. Rev. 2, 1–72.Google Scholar
Oberlack, M. & Zieleniewicz, M. 2013 Statistical symmetries and its impact on new decay modes and integral invariants of decaying turbulence. J. Turbul. 14, 4–22.CrossRefGoogle Scholar
Rotta, J. C. 1972 Turbulente Strömungen. Teubner.CrossRefGoogle Scholar
Sadeghi, H., Oberlack, M. & Gauding, M. 2018 On new scaling laws in a temporally evolving turbulent plane jet using Lie symmetry analysis and direct numerical simulation. J. Fluid Mech. 854, 233–260.CrossRefGoogle Scholar
Sadeghi, H., Oberlack, M. & Gauding, M. 2020 On new scaling laws in a temporally evolving turbulent plane jet using Lie symmetry analysis and direct numerical simulation – corrigendum. J. Fluid Mech. 885, E1.CrossRefGoogle Scholar
Saffman, P. G. 1967 The large scale structure of homogeneous turbulence. J. Fluid Mech. 27, 581–593.CrossRefGoogle Scholar
Sreenivasan, K. R. 2019 Turbulent mixing: a perspective. Proc. Natl Acad. Sci. USA 116, 18175–18183.CrossRefGoogle ScholarPubMed
Townsend, A. A. 1956 The Structure of Turbulent Shear Flows, 1st edn. Cambridge University Press.Google Scholar
Waclawczyk, M., Staffolani, N., Oberlack, M., Rosteck, A., Wilczek, M. & Friedrich, R. 2014 Statistical symmetries of the Lundgren–Monin–Novikov hierarchy. Phys. Rev. E 90, 013022.Google ScholarPubMed
Warhaft, S. & Lumley, J. L. 1978 An experimental study of decay of temperature fluctuations in grid-generated turbulence. J. Fluid Mech. 88, 659–685.CrossRefGoogle Scholar
Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32, 203–240.CrossRefGoogle Scholar
Watanabe, T. & Gotoh, T. 2004 Statistics of a passive scalar in homogeneous turbulence. New J. Phys. 6, 40.CrossRefGoogle Scholar