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Length scales and the turbulent/non-turbulent interface of a temporally developing turbulent jet

Published online by Cambridge University Press:  04 September 2023

S. Er Open the ORCID record for S. Er [Opens in a new window] ,
J.-P. Laval Open the ORCID record for J.-P. Laval [Opens in a new window]  and
J.C. Vassilicos Open the ORCID record for J.C. Vassilicos [Opens in a new window]
S. Er*
Affiliation:
Univ. Lille, CNRS, ONERA, Arts et Metiers Institute of Technology, Centrale Lille, UMR 9014 – LMFL – Laboratoire de Mécanique des Fluides de Lille – Kampé de Fériet, F-59000 Lille, France
J.-P. Laval
Affiliation:
Univ. Lille, CNRS, ONERA, Arts et Metiers Institute of Technology, Centrale Lille, UMR 9014 – LMFL – Laboratoire de Mécanique des Fluides de Lille – Kampé de Fériet, F-59000 Lille, France
J.C. Vassilicos*
Affiliation:
Univ. Lille, CNRS, ONERA, Arts et Metiers Institute of Technology, Centrale Lille, UMR 9014 – LMFL – Laboratoire de Mécanique des Fluides de Lille – Kampé de Fériet, F-59000 Lille, France
*
†Email addresses for correspondence: sarp.er.etu@univ-lille.fr, john-christos.vassilicos@centralelille.fr
†Email addresses for correspondence: sarp.er.etu@univ-lille.fr, john-christos.vassilicos@centralelille.fr
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Abstract

The temporally developing self-similar turbulent jet is fundamentally different from its spatially developing namesake because the former conserves volume flux and has zero cross-stream mean flow velocity whereas the latter conserves momentum flux and does not have zero cross-stream mean flow velocity. It follows that, irrespective of the turbulent dissipation's power-law scalings, the time-local Reynolds number remains constant, and the jet half-width $\delta$, the Kolmogorov length $\eta$ and the Taylor length $\lambda$ grow identically as the square root of time during the temporally developing self-similar planar jet's evolution. We predict theoretically and confirm numerically by direct numerical simulations that the mean centreline velocity, the Kolmogorov velocity and the mean propagation speed of the turbulent/non-turbulent interface (TNTI) of this planar jet decay identically as the inverse square root of time. The TNTI has an inner structure over a wide range of closely spatially packed iso-enstrophy surfaces with fractal dimensions that are well defined over a range of scales between $\lambda$ and $\delta$, and that decrease with decreasing iso-enstrophy towards values close to $2$ at the viscous superlayer. The smallest scale on these isosurfaces is approximately $\eta$, and the length scales between $\eta$ and $\lambda$ contribute significantly to the surface area of the iso-enstrophy surfaces without being characterised by a well-defined fractal dimension. A simple model is sketched for the mean propagation speeds of the iso-enstrophy surfaces within the TNTI of temporally developing self-similar turbulent planar jets. This model is based on a generalised Corrsin length, on the multiscale geometrical properties of the TNTI, and on a proportionality between the turbulent jet volume's growth rate and the growth rate of $\delta$. A prediction of this model is that the mean propagation speed at the outer edge of the viscous superlayer is proportional to the Kolmogorov velocity multiplied by the $1/4$th power of the global Reynolds number.

JFM classification

Wakes/Jets: Jets Turbulent Flows: Turbulence simulation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 970 , 10 September 2023 , A33
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Attili, A., Cristancho, J.C. & Bisetti, F. 2014 Statistics of the turbulent/non-turbulent interface in a spatially developing mixing layer. J. Turbul. 15, 555–568.CrossRefGoogle Scholar
Balamurugan, G., Rodda, A., Philip, J. & Mandal, A.C. 2020 Characteristics of the turbulent non-turbulent interface in a spatially evolving turbulent mixing layer. J. Fluid Mech. 894, A4.CrossRefGoogle Scholar
Bisset, D.K., Hunt, J.C.R. & Rogers, M.M. 2002 The turbulent/non-turbulent interface bounding a far wake. J. Fluid Mech. 451, 383–410.CrossRefGoogle Scholar
Borrell, G. & Jimenez, J. 2016 Properties of the turbulent/non-turbulent interface in boundary layers. J. Fluid Mech. 801, 554–596.CrossRefGoogle Scholar
Cafiero, G. & Vassilicos, J.C. 2019 Non-equilibrium turbulence scalings and self-similarity in turbulent planar jets. Proc. R. Soc. A 475 (2225), 20190038.CrossRefGoogle ScholarPubMed
Cafiero, G. & Vassilicos, J.C. 2020 Non-equilibrium scaling of the turbulent–nonturbulent interface speed in planar jets. Phys. Rev. Lett. 125 (17), 174501.CrossRefGoogle Scholar
Catrakis, H.J. & Dimotakis, P.E. 1999 Scale-dependent fractal geometry. In Mixing: Chaos and Turbulence (ed. H. Chaté, E. Villermaux & J.-M. Chomaz), pp. 145–162. Springer.CrossRefGoogle Scholar
Corrsin, S. & Kistler, A.L. 1955 Free-stream boundaries of turbulent flows. NACA Tech. Rep. TR-1244.Google Scholar
Dairay, T., Obligado, M. & Vassilicos, J.C. 2015 Non-equilibrium scaling laws in axisymmetric turbulent wakes. J. Fluid Mech. 781, 166–195.CrossRefGoogle Scholar
Dimotakis, P.E. & Catrakis, H.J. 1999 Turbulence, fractals, and mixing. In Mixing: Chaos and Turbulence (ed. H. Chaté, E. Villermaux & J.-M. Chomaz), pp. 59–143. Springer.CrossRefGoogle Scholar
Flohr, P. & Olivari, D. 1994 Fractal and multifractal characteristics of a scalar dispersed in a turbulent jet. Physica D 76 (1–3), 278–290.CrossRefGoogle Scholar
Gauding, M., Bode, M., Brahami, Y., Varea, E. & Danaila, L. 2021 Self-similarity of turbulent jet flows with internal and external intermittency. J. Fluid Mech. 919, A41.CrossRefGoogle Scholar
George, W.K. 1989 The self-preservation of turbulent flows and its relation to the initial conditions and coherent structures. In Advances in Turbulence (ed. W.K. George & R. Arndt), pp. 39–73. Hemisphere.Google Scholar
Goto, S. & Vassilicos, J.C. 2016 Unsteady turbulence cascades. Phys. Rev. E 94 (5), 053108.CrossRefGoogle ScholarPubMed
Gutmark, E. & Wygnanski, I. 1976 The planar turbulent jet. J. Fluid Mech. 73, 465–495.CrossRefGoogle Scholar
Krug, D., Chung, D., Philip, J. & Marusic, I. 2017 Global and local aspects of entrainment in temporal plumes. J. Fluid Mech. 812, 222–250.CrossRefGoogle Scholar
Lane-Serff, G.F. 1993 Investigation of the fractal structure of jets and plumes. J. Fluid Mech. 249, 521–534.CrossRefGoogle Scholar
Mandelbrot, B.B. 1982 The Fractal Geometry of Nature. WH Freeman & Co.Google Scholar
Miller, P.L. & Dimotakis, P.E. 1991 Stochastic geometric properties of scalar interfaces in turbulent jets. Phys. Fluids A 3 (1), 168–177.CrossRefGoogle Scholar
Mistry, D., Dawson, J.R. & Kerstein, A.R. 2018 The multi-scale geometry of the near field in an axisymmetric jet. J. Fluid Mech. 838, 501–515.CrossRefGoogle Scholar
Mistry, D., Philip, J., Dawson, J.R. & Marusic, I. 2016 Entrainment at multi-scales across the turbulent/non-turbulent interface in an axisymmetric jet. J. Fluid Mech. 802, 690–725.CrossRefGoogle Scholar
Nagata, R., Watanabe, T. & Nagata, K. 2018 Turbulent/non-turbulent interfaces in temporally evolving compressible planar jets. Phys. Fluids 30 (10), 105109.CrossRefGoogle Scholar
Nedić, J. 2013 Fractal-generated wakes. PhD thesis, Imperial College London.Google Scholar
Nedić, J., Vassilicos, J.C. & Ganapathisubramani, B. 2013 Axisymmetric turbulent wakes with new nonequilibrium similarity scalings. Phys. Rev. Lett. 111 (14), 144503.CrossRefGoogle ScholarPubMed
Obligado, M., Dairay, T. & Vassilicos, J.C. 2016 Nonequilibrium scalings of turbulent wakes. Phys. Rev. Fluids 1 (4), 044409.CrossRefGoogle Scholar
Ortiz-Tarin, J.L., Nidhan, S. & Sarkar, S. 2021 High-Reynolds-number wake of a slender body. J. Fluid Mech. 918, A30.CrossRefGoogle Scholar
Prasad, R.R. & Sreenivasan, K.R. 1990 The measurement and interpretation of fractal dimensions of the scalar interface in turbulent flows. Phys. Fluids A 2 (5), 792–807.CrossRefGoogle Scholar
Ramaprian, B.R. & Chandrasekhara, M.S. 1985 LDA measurements in plane turbulent jets. Trans. ASME J. Fluids Engng 107, 264–271.CrossRefGoogle Scholar
da Silva, C.B., Hunt, J.C.R., Eames, I. & Westerweel, J. 2014 Interfacial layers between regions of different turbulence intensity. Annu. Rev. Fluid Mech. 46 (1), 567–590.CrossRefGoogle Scholar
da Silva, C.B. & Pereira, J.C. 2008 Invariants of the velocity-gradient, rate-of-strain, and rate-of-rotation tensors across the turbulent/nonturbulent interface in jets. Phys. Fluids 20 (055101), 1–18.CrossRefGoogle Scholar
da Silva, C.B. & Taveira, R.R. 2010 The thickness of the turbulent/nonturbulent interface is equal to the radius of the large vorticity structures near the edge of the shear layer. Phys. Fluids 22, 121702.CrossRefGoogle Scholar
Silva, T.S., Zecchetto, M. & da Silva, C.B. 2018 The scaling of the turbulent/non-turbulent interface at high Reynolds numbers. J. Fluid Mech. 843, 156–179.CrossRefGoogle Scholar
Sreenivasan, K.R. 1991 Fractals and multifractals in fluid turbulence. Annu. Rev. Fluid Mech. 23 (1), 539–604.CrossRefGoogle Scholar
Sreenivasan, K.R., Ramshankar, R. & Meneveau, C. 1989 Mixing, entrainment and fractal dimensions of surfaces in turbulent flows. Proc. R. Soc. A 421 (1860), 79–108.Google Scholar
Taveira, R.R. & da Silva, C.B. 2014 Characteristics of the viscous superlayer in shear free turbulence and in planar turbulent jets. Phys. Fluids 26, 021702.CrossRefGoogle Scholar
Tennekes, H. & Lumley, J.L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Townsend, A.A. 1949 The fully developed turbulent wake of a circular cylinder. Aust. J. Sci. Res. 2 (4), 451–468.Google Scholar
Townsend, A.A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.Google Scholar
Tritton, D.J. 1988 Physical Fluid Dynamics. Oxford University Press.Google Scholar
Van Reeuwijk, M. & Holzner, M. 2013 The turbulence boundary of a temporal jet. J. Fluid Mech. 739, 254–275.CrossRefGoogle Scholar
Vassilicos, J.C. 2015 Dissipation in turbulent flows. Annu. Rev. Fluid Mech. 47 (1), 95–114.CrossRefGoogle Scholar
Virtanen, P., et al. 2020 SciPy 1.0: fundamental algorithms for scientific computing in Python. Nat. Meth. 17, 261–272.CrossRefGoogle ScholarPubMed
Watanabe, T., da Silva, C.B. & Nagata, K. 2019 Non-dimensional energy dissipation rate near the turbulent/non-turbulent interfacial layer in free shear flows and shear free turbulence. J. Fluid Mech. 875, 321–344.CrossRefGoogle Scholar
Watanabe, T., Riley, J.J., De Bruyn Kops, S.M., Diamessis, P.J. & Zhou, Q. 2016 Turbulent/non-turbulent interfaces in wakes in stably stratified fluids. J. Fluid Mech. 797, R1.CrossRefGoogle Scholar
Zhou, Y. & Vassilicos, J.C. 2017 Related self-similar statistics of the turbulent/non-turbulent interface and the turbulence dissipation. J. Fluid Mech. 821, 440–457.CrossRefGoogle Scholar