Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-21T01:38:51.423Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic scaling laws for the irrotational motions bordering a turbulent region

Published online by Cambridge University Press:  05 May 2021

Ricardo P. Xavier Open the ORCID record for Ricardo P. Xavier [Opens in a new window] ,
Miguel A. C. Teixeira Open the ORCID record for Miguel A. C. Teixeira [Opens in a new window]  and
Carlos B. da Silva Open the ORCID record for Carlos B. da Silva [Opens in a new window]
Ricardo P. Xavier
Affiliation:
LAETA, IDMEC, Instituto Superior Técnico, Universidade de Lisboa, 1049-001Lisboa, Portugal
Miguel A. C. Teixeira
Affiliation:
Department of Meteorology, University of Reading, Meteorology Building, Whiteknights Road, Earley Gate, ReadingRG6 6ET, UK
Carlos B. da Silva*
Affiliation:
LAETA, IDMEC, Instituto Superior Técnico, Universidade de Lisboa, 1049-001Lisboa, Portugal
*
†Email address for correspondence: carlos.silva@ist.utl.pt
Get access Rights & Permissions [Opens in a new window]

Abstract

Turbulent flows are often bounded by regions of irrotational or non-turbulent flow, where the magnitude of the potential velocity fluctuations can be surprisingly high. This includes virtually all turbulent free-shear flows and also turbulent boundary layers, and is particularly true near the so-called turbulent/non-turbulent interface (TNTI) layer, which separates the regions of turbulent and non-turbulent fluid motion. In the present work, we show that in the non-turbulent region and for distances $x_2$ sufficiently far from the TNTI layer, the asymptotic variation laws for the variance of the velocity fluctuations $\langle u_i^{2} \rangle$ ($i=1,2,3$), Taylor micro-scale $\lambda$ and viscous dissipation rate $\varepsilon$ depend on the shape of the kinetic energy spectrum in the infrared region $E(k) \sim k^{n}$. Specifically, by using rapid distortion theory (RDT), we show that for Saffman turbulence ($E(k) \sim k^{2}$), we obtain the asymptotic laws $\langle u_i^{2} \rangle \sim x_2^{-3}$ ($i=1,2,3$), $\lambda \sim x_2$ and $\varepsilon \sim x_2^{-5}$. Additionally, we confirm the classical results obtained by Phillips (Proc. Camb. Phil. Soc., vol. 51, 1955, p. 220) for Batchelor turbulence ($E(k) \sim k^{4}$), with $\langle u_i^{2} \rangle \sim x_2^{-4}$ ($i=1,2,3$), $\lambda \sim x_2$ and $\varepsilon \sim x_2^{-6}$. The new theoretical results are confirmed by direct numerical simulations (DNS) of shear-free turbulence and are shown to be independent of the Reynolds number. Therefore, these results are expected to be valid in other flow configurations, such as in turbulent planar jets or wakes, provided the kinetic energy spectra in the turbulence region can be described by a Batchelor or a Saffman spectrum.

JFM classification

Turbulent Flows: Turbulence theory Turbulent Flows: Shear layer turbulence Turbulent Flows: Homogeneous turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 918 , 10 July 2021 , A3
Check for updates
Copyright
© The Author(s), 2021. 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

REFERENCES

Antonia, R.A., Shah, D.A. & Browne, L.W.B. 1987 The organized motion outside a turbulent wake. Phys. Fluids 30 (7), 2040–2045.CrossRefGoogle Scholar
Batchelor, G.K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Birkhoff, G. 1954 Fourier synthesis of homogeneous turbulence. Commun. Pure Appl. Maths 7 (1), 19–44.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
Bradbury, L.J.S. 1965 The structure of a self-preserving turbulent plane jet. J. Fluid Mech. 23 (1), 31–64.CrossRefGoogle Scholar
Bradshaw, P. 1967 Irrotational fluctuations near a turbulent boundary layer. J. Fluid Mech. 27 (2), 209–230.CrossRefGoogle Scholar
Canuto, C., Hussaini, M.Y., Quarteroni, A. & Zang, T.A. 1987 Spectral Methods in Fluid Dynamics. Springer-Verlag.Google Scholar
Carruthers, D.J. & Hunt, J.C.R. 1986 Velocity fluctuations near an interface between a turbulent region and a stably stratified layer. J. Fluid Mech. 165, 475–501.CrossRefGoogle Scholar
Cimarelli, A., Cocconi, G., Frohnapfel, B. & Angelis, E.D. 2015 Spectral enstrophy budget in a shear-less flow with turbulent/non-turbulent interface. Phys. Fluids 27, 125106.CrossRefGoogle Scholar
Davidson, P.A. 2004 Turbulence, An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
Eyink, G.L. & Thomson, D.J. 2000 Free decay of turbulence and breakdown of self-similarity. Phys. Fluids 12, 477–479.CrossRefGoogle Scholar
Fabris, G. 1979 Conditional sampling study of the turbulent wake of a cylinder. Part 1. J. Fluid Mech. 94 (4), 673–709.CrossRefGoogle Scholar
Ishida, T., Davidson, P.A. & Kaneda, Y. 2006 On the decay of isotropic turbulence. J. Fluid Mech. 564, 455–475.CrossRefGoogle Scholar
Kovasznay, L.S.G., Kibens, V. & Blackwelder, R.F. 1970 Large-scale motion in the intermittent region of a turbulent boundary layer. J. Fluid Mech. 41 (2), 283–325.CrossRefGoogle Scholar
Lesieur, M. 1997 Turbulence in Fluids. 3rd edn. Kluwer.CrossRefGoogle Scholar
Oberlack, M. & Zieleniewicz, A. 2013 Statistical symmetries and its impact on new decay modes and integral invariants of decaying turbulence. J. Turbul. 14 (2), 4–22.CrossRefGoogle Scholar
Perot, B. & Moin, P. 1995 Shear-free turbulent boundary layers. Part 1. Physical insights into near-wall turbulence. J. Fluid Mech. 295, 199–227.CrossRefGoogle Scholar
Phillips, O.M. 1955 The irrotational motion outside a free turbulent boundary. Proc. Camb. Phil. Soc. 51, 220.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Saffman, P.G. 1967 The large-scale structure of homogeneous turbulence. J. Fluid Mech. 27, 581–593.CrossRefGoogle Scholar
da Silva, C.B., Hunt, J.C.R., Eames, I. & Westerweel, J. 2014 Interfacial layers between regions of different turbulent intensity. Annu. Rev. Fluid Mech. 46, 567–590.CrossRefGoogle Scholar
da Silva, C.B., dos Reis, R.J.N. & Pereira, J.C.F. 2011 The intense vorticity structures near the turbulent/non-turbulent interface a jet. J. Fluid Mech. 685, 165–190.CrossRefGoogle Scholar
Silva, T.S. & da Silva, C.B. 2017 The behaviour of the scalar gradient across the turbulent/non-turbulent interface in jets. Phys. Fluids 29, 085106.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
Sunyach, M. & Mathieu, J. 1969 Zone de melange d'un jet plan fluctuations induites dans le cone a potentiel-intermittence. Intl J. Heat Mass Transfer 12 (12), 1679–1697.CrossRefGoogle Scholar
Taveira, R.R. & da Silva, C.B. 2013 Kinetic energy budgets near the turbulent/nonturbulent interface in jets. Phys. Fluids 25, 015114.CrossRefGoogle Scholar
Teixeira, M.A.C. & da Silva, C.B. 2012 Turbulence dynamics near a turbulent/non-turbulent interface. J. Fluid Mech. 695, 257–287.CrossRefGoogle Scholar
Thomas, R.M. 1973 Conditional sampling and other measurements in a plane turbulent wake. J. Fluid Mech. 57 (3), 549–582.CrossRefGoogle Scholar
Vassilicos, J.C. 2011 An infinity of possible invariants for decaying homogeneous turbulence. Phys. Lett. A 375 (6), 1010–1013.CrossRefGoogle Scholar
Watanabe, T., Jaulino, R., Taveira, R., da Silva, C.B., Nagata, K. & Sakai, Y. 2017 Role of an isolated eddy near the turbulent/non-turbulent interface layer. Phys. Rev. Fluids 2, 094607.CrossRefGoogle Scholar
Watanabe, T., Sakai, Y., Nagata, K., Ito, Y. & Hayase, T. 2014 Enstrophy and passive scalar transport near the turbulent/non-turbulent interface in a turbulent planar jet flow. Phys. Fluids 26, 105103.CrossRefGoogle Scholar
Westerweel, J., Fukushima, C., Pedersen, J.M. & Hunt, J.C.R. 2005 Mechanics of the turbulent-nonturbulent interface of a jet. Phys. Rev. Lett. 95, 174501.CrossRefGoogle ScholarPubMed
Westerweel, J., Fukushima, C., Pedersen, J.M. & Hunt, J.C.R. 2009 Momentum and scalar transport at the turbulent/non-turbulent interface of a jet. J. Fluid Mech. 631, 199–230.CrossRefGoogle Scholar
Williamson, J.H. 1980 Low-storage Runge–Kutta schemes. J. Comput. Phys. 35, 48–56.CrossRefGoogle Scholar
Wygnanski, I. & Fiedler, H.E. 1970 The two-dimensional mixing region. J. Fluid Mech. 41 (2), 327–361.CrossRefGoogle Scholar