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Spectral theory of passive scalar with mean scalar gradient

Published online by Cambridge University Press:  02 August 2021

Takuya Kitamura Open the ORCID record for Takuya Kitamura [Opens in a new window]
Takuya Kitamura*
Affiliation:
Graduate School of Engineering, Nagasaki University, Nagasaki852-8521, Japan
*
†Email address for correspondence: [email protected]
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Abstract

A single-time two-point spectral closure is developed by approximation of the Lagrangian direct interaction approximation (LDIA) for a passive scalar in the presence of a mean scalar gradient in homogeneous isotropic turbulence. In the derivation of a single-time two-point spectral closure, the two assumptions, Markovianisation and the exponential form of Lagrangian velocity response function, are made for the LDIA, and angle dependence of the passive-scalar field is expressed by the second-order truncation of Legendre polynomials, in which such a truncation is justified by the linear theory. The resulting closure equations are derived in a straightforward way except for the above assumptions and further simplifications. The closures studied agree qualitatively with direct numerical simulation for one- and two-point statistics of a passive-scalar field in the case of unity Schmidt number. For both direct numerical simulation and closures, we show that the dependence of one-point passive-scalar statistics on the Péclet number based on scalar Taylor microscales collapses properly compared with that based on velocity microscales. We also propose universal scaling laws for second-order scalar structure functions and demonstrate their validity.

JFM classification

Mixing: Turbulent mixing Turbulent Flows: Isotropic turbulence Turbulent Flows: Turbulence theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 923 , 25 September 2021 , A28
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Arfken, G.B. & Weber, H.J. 1999 Mathematical Methods for Physicists. Cambridge University Press.Google Scholar
Ariki, T. & Yoshida, K. 2020 Hessian-based Lagrangian closure theory for passive scalar turbulence. arXiv:1909.05482.Google Scholar
Ariki, T., Yoshida, K., Matsuda, K. & Yoshimatsu, K. 2018 Scale-similar clustering of heavy particles in the inertial range of turbulence. Phys. Rev. E 97, 033109.CrossRefGoogle ScholarPubMed
Batchelor, G.K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Bos, W.J.T. 2014 On the anisotropy of the turbulent passive scalar in the presence of a mean scalar gradient. J. Fluid Mech. 744, 38–64.CrossRefGoogle Scholar
Bos, W.J.T., Clark, T.T. & Rubinstein, R. 2007 Small scale response and modeling of periodically forced turbulence. Phys. Fluids 19, 055107.CrossRefGoogle Scholar
Bos, W.J.T., Touli, H. & Bertoglio, J.P. 2005 Reynolds number dependency of the scalar flux spectrum in isotropic turbulence with a uniform scalar gradient. Phys. Fluids 17, 125108.CrossRefGoogle Scholar
Briard, A., Gomez, T. & Cambon, C. 2016 Spectral modelling for passive scalar dynamics in homogeneous anisotropic turbulence. J. Fluid Mech. 799, 159–199.CrossRefGoogle Scholar
Briard, A., Gomez, T., Sagaut, P. & Memari, S. 2015 Passive scalar decay laws in isotropic turbulence: Prandtl number effects. J. Fluid Mech. 784, 274–303.CrossRefGoogle Scholar
Cambon, C. & Jacquin, L. 1989 Spectral approach to non-isotropic turbulence subjected to rotation. J. Fluid Mech. 202, 295–317.CrossRefGoogle Scholar
Cambon, C., Jeandel, D. & Mathieu, J. 1981 Spectral modelling of homogeneous non-isotropic turbulence. J. Fluid Mech. 104, 247–262.CrossRefGoogle Scholar
Cambon, C., Mansour, N.N. & Godeferd, F.S. 1997 Energy transfer in rotating turbulence. J. Fluid Mech. 337, 303–332.CrossRefGoogle Scholar
Corrsin, S. 1951 On the spectrum of isotropic temperature fluctuations in an isotropic turbulence. J. Appl. Phys. 22, 469–473.CrossRefGoogle Scholar
Doering, C.R. & Foias, C. 2002 Energy dissipation in body-forced turbulence. J. Fluid Mech. 467, 289–306.CrossRefGoogle Scholar
Donzis, D. & Sreenivasan, K.R. 2010 The bottleneck effect and the Kolmogorov constant in isotropic turbulence. J. Fluid Mech. 657, 171–188.CrossRefGoogle Scholar
Donzis, D., Sreenivasan, K.R. & Yeung, P.K. 2005 Scalar dissipation rate and dissipative anomaly in isotropic turbulence. J. Fluid Mech. 532, 199–216.CrossRefGoogle Scholar
Elperin, T., Kleeorin, N. & Rogachevskii, I. 1996 Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow. Phys. Rev. E 53, 3431–3441.CrossRefGoogle ScholarPubMed
Favier, B.F.N., Godeferd, F.S., Cambon, C., Delache, A. & Bos, W.J.T. 2011 Quasi-static magnetohydrodynamic turbulence at high Reynolds number. J. Fluid Mech. 681, 434–461.CrossRefGoogle Scholar
Goto, S. & Kida, S. 1999 Passive scalar spectrum in isotropic turbulence: prediction by the Lagrangian direct-interaction approximation. Phys. Fluids 11, 1936–1952.CrossRefGoogle Scholar
Gotoh, T., Kaneda, Y. & Bekki, N. 1988 Numerical integration of the Lagrangian renormalized approximation. J. Phys. Soc. Japan 57, 866–880.CrossRefGoogle Scholar
Gotoh, T., Nagaki, J. & Kaneda, Y. 2000 Passive scalar spectrum in the viscous-convective range in two-dimensional steady turbulence. Phys. Fluids 12, 155–168.CrossRefGoogle Scholar
Gotoh, T. & Watanabe, T. 2012 Scalar flux in a uniform mean scalar gradient in homogeneous isotropic steady turbulence. Physica D 241, 141–148.CrossRefGoogle Scholar
Gotoh, T., Watanabe, T. & Suzuki, Y. 2011 Universality and anisotropy in passive scalar fluctuations in turbulence with uniform mean gradient. J. Turbul. 12, 1–27.CrossRefGoogle Scholar
Gotoh, T & Yeung, P.K. 2012 Passive scalar transport in turbulence: a computational perspective. In Ten Chapters in Turbulence (ed. P.A. Davidson, Y. Kaneda & K.R. Sreenivasan), pp. 87–131. Cambridge University Press.CrossRefGoogle Scholar
Greene, P.R. 1989 A useful approximation to the error function: Applications to mass, momentum and energy transport in shear layers. J. Fluid Engng 111, 224–226.CrossRefGoogle Scholar
Herr, S., Wang, L. & Collins, L.R. 1996 EDQNM model of a passive scalar with a uniform mean gradient. Phys. Fluids 8, 1588–1608.CrossRefGoogle Scholar
Herring, J.R. 1974 Approach of axisymmetric turbulence to isotropy. Phys. Fluids 17, 859–872.CrossRefGoogle Scholar
Herring, J.R., Schertzer, D., Lesieur, M., Newman, G.R., Chollet, J.P. & Larcheveque, M. 1982 A comparative assessment of spectral closures as applied to passive scalar diffusion. J. Fluid Mech. 124, 411–437.CrossRefGoogle Scholar
Holzer, M. & Siggia, E. 1994 Turbulent mixing of a passive scalar. Phys. Fluids A 6, 1820–1837.CrossRefGoogle Scholar
Ishihara, T., Yoshida, K. & Kaneda, Y. 2002 Anisotropic velocity correlation spectrum at small scales in a homogeneous turbulent shear flow. Phys. Rev. Lett. 88, 154501.CrossRefGoogle Scholar
Ito, Y., Watanabe, T., Nagata, K. & Sakai, Y. 2016 Turbulent mixing of a passive scalar in grid turbulence. Phys. Scr. 91, 074002.CrossRefGoogle Scholar
Kaneda, Y. 1981 Renormalized expansions in the theory of turbulence with the use of Lagrangian position function. J. Fluid Mech. 107, 131–145.CrossRefGoogle Scholar
Kaneda, Y. 1986 Inertial range structure of turbulent velocity and scalar fields in a Lagrangian renormalized approximation. Phys. Fluids 29, 701–708.CrossRefGoogle Scholar
Kaneda, Y. 1993 Lagrangian and Eulerian time correlations in turbulence. Phys. Fluids 5, 2835–2845.CrossRefGoogle Scholar
Kaneda, Y. & Yoshida, K. 2004 Small-scale anisotropy in stably stratified turbulence. New J. Phys. 34, 16.Google Scholar
Kida, S. & Goto, S. 1997 A lagrangian direct-interaction approximation for homogeneous isotropic turbulence. J. Fluid Mech. 345, 307–345.CrossRefGoogle Scholar
Kitamura, T. 2020 Single-time Markovianized spectral closure in fluid turbulence. J. Fluid Mech. 898, A8.CrossRefGoogle Scholar
Kitamura, T. 2021 Constant–energetics control-based forcing methods in isotropic helical turbulence. Phys. Rev. Fluids 6, 044608.CrossRefGoogle Scholar
Kolmogorov, A.N. 1941 a Dissipation of energy in the locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 15–17.Google Scholar
Kolmogorov, A.N. 1941 b The local structure of turbulence in imcompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk. SSSR 31, 301–305.Google Scholar
Kraichnan, R.H. 1966 Dispersion of particle pairs in homogeneous turbulence. Phys. Fluids 9, 1937–1943.CrossRefGoogle Scholar
Kraichnan, R.H. 1968 Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11, 945–953.CrossRefGoogle Scholar
Kraichnan, R.H. 1994 Anomalous scaling of a randomly advected passive scalar. Phys. Rev. Lett. 72, 1016–1019.CrossRefGoogle ScholarPubMed
Leslie, D.C. 1973 Developments in the Theory of Turbulence. Oxford University Press.Google Scholar
Lumley, J.L. 1964 The spectrum of nearly inertial turbulence in a stably stratified turbulence. J. Atmos. Sci. 21, 99–102.2.0.CO;2>CrossRefGoogle Scholar
Lumley, J.L. 1967 Similarity and the turbulent energy spectrum. Phys. Fluids 10, 855–858.CrossRefGoogle Scholar
McComb, W.D., Filipiak, M.J. & Shanmugasundaram, V. 1992 Rederivation and further assessment of the LET theory of isotropic turbulence, as applied to passive scalar convection. J. Fluid Mech. 245, 279–300.CrossRefGoogle Scholar
Mydlarski, L. & Warhaft, Z. 1998 Passive scalar statistics in high-Péclet-number grid turbulence. J. Fluid Mech. 358, 135–175.CrossRefGoogle Scholar
Nakayama, K. 2002 Lagrangian statistical theory of anisotropic MHD turbulence. Publ. Astron. Soc. Japan 54, 1065–1078.CrossRefGoogle Scholar
Newman, G.R. & Herring, J.R. 1979 A test field model study of a passive scalar in isotropic turbulence. J. Fluid Mech. 94, 163–194.CrossRefGoogle Scholar
Novikov, E.A. 1965 Functionals and the random-force method in turbulence theory. Sov. Phys. JETP 20, 1290–1294.Google Scholar
Obukhov, A.M. 1949 Structure of the temperature field in turbulent flows. Izv. Akad. Nauk SSSR. Geogr. Geofiz 13, 58–69.Google Scholar
O'gorman, P.A. & Pullin, D.I. 2005 Effect of Schmidt number on the velocity-scalar cospectrum in isotropic turbulence with a mean scalar gradient. J. Fluid Mech. 532, 111–140.CrossRefGoogle Scholar
Overholt, M.R. & Pope, S.B. 1996 Direct numerical simulation of a passive scalar with imposed mean gradient in isotropic turbulence. Phys. Fluids 8, 3128–3148.CrossRefGoogle Scholar
Pumir, A. 1994 A numerical study of the mixing of a passive scalar in three dimensions in the presence of a mean gradient. Phys. Fluids 6, 2118–2132.CrossRefGoogle Scholar
Pumir, A., Shraiman, B.I. & Siggia, E.D. 1997 Perturbation theory for the $\delta$-correlated model of passive scalar advection near the batchelor limit. Phys. Rev. E 55, R1263.CrossRefGoogle Scholar
Sagaut, P. & Cambon, C. 2008 Homogeneous Turbulence Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Shraiman, B.I. & Siggia, E.D. 2000 Scalar turbulence. Nature 405, 639–646.CrossRefGoogle ScholarPubMed
Sirivat, A. & Warhaft, Z. 1983 The effect of a passive cross-stream temperature gradient on the evolution of temperature variance and heat flux in grid turbulence. J. Fluid Mech. 128, 323–346.CrossRefGoogle Scholar
Sreenivasan, K.R. 2018 Turbulent mixing: A perspective. Proc. Natl Acad. Sci. USA 116, 18175–18183.CrossRefGoogle ScholarPubMed
Staquet, C. & Godeferd, F.S. 1998 Statistical modelling and direct numerical simulations of decaying stably stratified turbulence. Part 1. Flow energetics. J. Fluid Mech. 360, 295–340.CrossRefGoogle Scholar
Suzuki, H., Nagata, K., Sakai, Y. & Ukai, R. 2010 High-Schmidt-number scalar transfer in regular and fractal grid turbulence. Phys. Scr. 2010, 014069.CrossRefGoogle Scholar
Tavoularis, S. & Corrsin, S. 1981 a Experiments in nearly homogeneous turbulent shear flow with a uniform mean temperature gradient. Part 2. The fine structure. J. Fluid Mech. 104, 349–367.CrossRefGoogle Scholar
Tavoularis, S. & Corrsin, S. 1981 b Experiments in nearly homogenous turbulent shear flow with a uniform mean temperature gradient. Part 1. J. Fluid Mech. 104, 311–347.CrossRefGoogle Scholar
Wang, L.P., Chen, S. & Brasseur, J.G. 1999 Examination of hypotheses in the Kolmogorov refined turbulence theory through high-resolution simulations. Part 2. Passive scalar field. J. Fluid Mech. 400, 163–197.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
Yasuda, T., Gotoh, T., Watanabe, T. & Saito, I. 2020 Péclet-number dependence of small-scale anisotropy of passive scalar fluctuations under a uniform mean gradient in isotropic turbulence. J. Fluid Mech. 898, A4.CrossRefGoogle Scholar
Yeung, P.K., Xu, S. & Sreenivasan, K.R. 2002 Schmidt number effects on turbulent transport with uniform mean scalar gradient. Phys. Fluids 14, 4178–4191.CrossRefGoogle Scholar
Yoshida, K., Ishihara, T. & Kaneda, Y. 2003 Anisotropic spectrum of homogeneous turbulent shear flow in a Lagrangian renormalized approximation. Phys. Fluids 15, 2385–2397.CrossRefGoogle Scholar