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Reynolds number dependence of the turbulent/non-turbulent interface in temporally developing turbulent boundary layers

Published online by Cambridge University Press:  26 May 2023

Xinxian Zhang Open the ORCID record for Xinxian Zhang [Opens in a new window] ,
Tomoaki Watanabe Open the ORCID record for Tomoaki Watanabe [Opens in a new window]  and
Koji Nagata Open the ORCID record for Koji Nagata [Opens in a new window]
Xinxian Zhang*
Affiliation:
School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, PR China Department of Aerospace Engineering, Nagoya University, Nagoya 464-8603, Japan
Tomoaki Watanabe
Affiliation:
Education and Research Center for Flight Engineering, Nagoya University, Nagoya 464-8603, Japan
Koji Nagata
Affiliation:
Department of Mechanical Engineering and Science, Kyoto University, Kyoto 615-8540, Japan
*
†Email address for correspondence: zhangxinxian@buaa.edu.cn
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Abstract

Direct numerical simulations (DNS) of temporally developing turbulent boundary layers are performed with a wide range of Reynolds numbers based on the momentum thickness $Re_{\theta } = 2000$–$13\,000$ for investigating the Reynolds number dependence of the turbulent/non-turbulent interface (TNTI) layer. The grid spacing in the DNS is determined carefully such that small-scale turbulent motions near the TNTI are well resolved. The outer edge of the TNTI layer, called the irrotational boundary, is detected with vorticity magnitude. The mean thicknesses of the TNTI layer, $\delta _{TNTI}$, turbulent sublayer, $\delta _{TSL}$, and viscous superlayer, $\delta _{VSL}$, are found to be approximately $15\eta _{TI}$, $10\eta _{TI}$ and $5\eta _{TI}$, respectively, where $\eta _{TI}$ is the Kolmogorov scale taken in the turbulent region near the TNTI layer. The mean curvature of the irrotational boundary is also characterized by $\eta _{TI}$. The shear parameter and the shear-to-vorticity ratio show that the mean shear effects near the TNTI layer are not significant for both large and small scales. The anisotropy tensors of Reynolds stress and vorticity suggest that the turbulence under the TNTI layer tends to be isotropic at high $Re_{\theta }$, for which $\eta _{TI}/\delta \sim Re_{\theta }^{-3/4}$ is valid with the boundary layer thickness $\delta$. The surface area of the irrotational boundary is consistent with the fractal analysis of the interface, where the fractal dimension $D_f$ is found to be $2.14$–$2.20$. The present results suggest that the mean entrainment rate per unit horizontal area normalized by the friction velocity varies slowly as $Re_{\theta }^{(3/4)(D_f-2)}$ for $Re_{\theta } \geq 4000$.

JFM classification

Turbulent Flows: Turbulent boundary layers Turbulent Flows: Turbulence simulation Turbulent Flows: Intermittency
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 964 , 10 June 2023 , A8
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Abe, H., Antonia, R.A. & Toh, S. 2018 Large-scale structures in a turbulent channel flow with a minimal streamwise flow unit. J. Fluid Mech. 850, 733–768.CrossRefGoogle Scholar
Adrian, R.J., Meinhart, C.D. & Tomkins, C.D. 2000 Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54.CrossRefGoogle Scholar
del Álamo, J.C., Jiménez, J., Zandonade, P. & Moser, R.D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135–144.CrossRefGoogle Scholar
Attili, A., Cristancho, J.C. & Bisetti, F. 2014 Statistics of the turbulent/non-turbulent interface in a spatially developing mixing layer. J. Turbul. 15 (9), 555–568.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. & Jiménez, J. 2016 Properties of the turbulent/non-turbulent interface in boundary layers. J. Fluid Mech. 801, 554–596.CrossRefGoogle Scholar
Borrell, G., Sillero, J.A. & Jiménez, J. 2013 A code for direct numerical simulation of turbulent boundary layers at high Reynolds numbers in BG/P supercomputers. Comput. Fluids 80, 37–43.CrossRefGoogle Scholar
Carlier, J. & Stanislas, M. 2005 Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry. J. Fluid Mech. 535, 143–188.CrossRefGoogle Scholar
Chauhan, K., Philip, J., de Silva, C.M., Hutchins, N. & Marusic, I. 2014 b The turbulent/non-turbulent interface and entrainment in a boundary layer. J. Fluid Mech. 742, 119–151.CrossRefGoogle Scholar
Chauhan, K., Philip, J. & Marusic, I. 2014 a Scaling of the turbulent/non-turbulent interface in boundary layers. J. Fluid Mech. 751, 298–328.CrossRefGoogle Scholar
Cimarelli, A. & Boga, G. 2021 Numerical experiments on turbulent entrainment and mixing of scalars. J. Fluid Mech. 927, A34.CrossRefGoogle Scholar
Corrsin, S. 1958 Local isotropy in turbulent shear flow. NACA Tech. Rep. 58B11.Google Scholar
Corrsin, S. & Kistler, A.L. 1955 Free-stream boundaries of turbulent flows. NACA Tech. Rep. 1244.Google Scholar
Crow, S. 1968 Turbulent Rayleigh shear flow. J. Fluid Mech. 32, 113–130.CrossRefGoogle Scholar
De Graaff, D.B. & Eaton, J.K. 2000 Reynolds-number scaling of the flat-plate turbulent boundary layer. J. Fluid Mech. 422, 319–346.CrossRefGoogle Scholar
Eisma, J., Westerweel, J., Ooms, G. & Elsinga, G.E. 2015 Interfaces and internal layers in a turbulent boundary layer. Phys. Fluids 27 (5), 055103.CrossRefGoogle Scholar
Erm, L.P. & Joubert, P.N. 1991 Low-Reynolds-number turbulent boundary layers. J. Fluid Mech. 230, 1–44.CrossRefGoogle Scholar
Guarini, S.E., Moser, R.D., Shariff, K. & Wray, A. 2000 Direct numerical simulation of a supersonic turbulent boundary layer at Mach 2.5. J. Fluid Mech. 414, 1–33.CrossRefGoogle Scholar
Hawkes, E.R., Sankaran, R., Sutherland, J.C. & Chen, J.H. 2007 Scalar mixing in direct numerical simulations of temporally evolving plane jet flames with skeletal CO/H$_2$ kinetics. Proc. Combust. Inst. 31, 1633–1640.CrossRefGoogle Scholar
Holzner, M. & Lüthi, B. 2011 Laminar superlayer at the turbulence boundary. Phys. Rev. Lett. 106 (13), 134503.CrossRefGoogle ScholarPubMed
Holzner, M., Liberzon, A., Nikitin, N., Kinzelbach, W. & Tsinober, A. 2007 Small-scale aspects of flows in proximity of the turbulent/nonturbulent interface. Phys. Fluids 19 (7), 071702.CrossRefGoogle Scholar
Hunt, J.C.R., Eames, I., Westerweel, J., Davidson, P.A., Voropayev, S., Fernando, J. & Braza, M. 2010 Thin shear layers – the key to turbulence structure? J. Hydro-Environ. Res. 4 (2), 75–82.CrossRefGoogle Scholar
Ishihara, T., Ogasawara, H. & Hunt, J.C.R. 2015 Analysis of conditional statistics obtained near the turbulent/non-turbulent interface of turbulent boundary layers. J. Fluids Struct. 53, 50–57.CrossRefGoogle Scholar
Jahanbakhshi, R. 2021 Mechanisms of entrainment in a turbulent boundary layer. Phys. Fluids 33 (3), 035105.CrossRefGoogle Scholar
Jahanbakhshi, R., Vaghefi, N.S. & Madnia, C.K. 2015 Baroclinic vorticity generation near the turbulent/non-turbulent interface in a compressible shear layer. Phys. Fluids 27 (10), 105105.CrossRefGoogle Scholar
Jiménez, J. 2013 Near-wall turbulence. Phys. Fluids 25 (10), 101302.CrossRefGoogle Scholar
Jiménez, J., Wray, A.A., Saffman, P.G. & Rogallo, R.S. 1993 The structure of intense vorticity in isotropic turbulence. J. Fluid Mech. 255, 65–90.CrossRefGoogle Scholar
Kang, S.J., Tanahashi, M. & Miyauchi, T. 2007 Dynamics of fine scale eddy clusters in turbulent channel flows. J. Turbul. 8, N52.CrossRefGoogle Scholar
Kennedy, C.A. & Carpenter, M.H. 1994 Several new numerical methods for compressible shear-layer simulations. Appl. Numer. Maths 14 (4), 397–433.CrossRefGoogle Scholar
Kozul, M., Chung, D. & Monty, J.P. 2016 Direct numerical simulation of the incompressible temporally developing turbulent boundary layer. J. Fluid Mech. 796, 437–472.CrossRefGoogle Scholar
Lalescu, C.C., Meneveau, C. & Eyink, G.L. 2013 Synchronization of chaos in fully developed turbulence. Phys. Rev. Lett. 110 (8), 084102.CrossRefGoogle ScholarPubMed
Lee, J., Sung, H.J. & Zaki, T.A. 2017 Signature of large-scale motions on turbulent/non-turbulent interface in boundary layers. J. Fluid Mech. 819, 165–187.CrossRefGoogle Scholar
Long, Y., Wang, J. & Pan, C. 2022 Universal modulations of large-scale motions on entrainment of turbulent boundary layers. J. Fluid Mech. 941, A68.CrossRefGoogle Scholar
Lozano-Durán, A. & Jiménez, J. 2014 Effect of the computational domain on direct simulations of turbulent channels up to $Re_\tau = 4200$. Phys. Fluids 26 (1), 011702.CrossRefGoogle Scholar
Martín, M.P. 2007 Direct numerical simulation of hypersonic turbulent boundary layers. Part 1. Initialization and comparison with experiments. J. Fluid Mech. 570, 347.CrossRefGoogle Scholar
Marusic, I., Mathis, R. & Hutchins, N. 2010 High Reynolds number effects in wall turbulence. Intl J. Heat Fluid Flow 31 (3), 418–428.CrossRefGoogle Scholar
Marusic, I., Monty, J.P., Hultmark, M. & Smits, A.J 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.CrossRefGoogle Scholar
Meneveau, C. & Sreenivasan, K.R. 1990 Interface dimension in intermittent turbulence. Phys. Rev. A 41 (4), 2246.CrossRefGoogle ScholarPubMed
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. 820, 690–725.CrossRefGoogle Scholar
Morinishi, Y., Lund, T.S., Vasilyev, O.V. & Moin, P. 1998 Fully conservative higher order finite difference schemes for incompressible flow. J. Comput. Phys. 143 (1), 90–124.CrossRefGoogle Scholar
Moser, R.D., Kim, J. & Mansour, N.N. 1999 Direct numerical simulation of turbulent channel flow up to $Re_\tau = 590$. Phys. Fluids 11 (4), 943–945.CrossRefGoogle Scholar
Mouri, H., Hori, A. & Kawashima, Y. 2007 Laboratory experiments for intense vortical structures in turbulence velocity fields. Phys. Fluids 19 (5), 055101.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
Neamtu-Halic, M.M., Krug, D., Mollicone, J.P., Van Reeuwijk, M., Haller, G. & Holzner, M. 2020 Connecting the time evolution of the turbulence interface to coherent structures. J. Fluid Mech. 898, A30.CrossRefGoogle Scholar
Osaka, H., Kameda, T. & Mochizuki, S. 1998 Re-examination of the Reynolds-number-effect on the mean flow quantities in a smooth wall turbulent boundary layer. JSME Intl J. B 41 (1), 123–129.CrossRefGoogle Scholar
Philip, J., Meneveau, C., de Silva, C.M. & Marusic, I. 2014 Multiscale analysis of fluxes at the turbulent/non-turbulent interface in high Reynolds number boundary layers. Phys. Fluids 26 (1), 015105.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Prandtl, L. 1928 Motion of fluids with very little viscosity. Tech. Memo. Natn. Advis. Comm. Aeronaut. Wash. No. 452, pp. 484–491.Google Scholar
Prasad, R.R. & Sreenivasan, K.R. 1996 Scalar interfaces in digital images of turbulent flows. Exp. Fluids 7, 259–264.CrossRefGoogle Scholar
van Reeuwijk, M. & Holzner, M. 2014 The turbulence boundary of a temporal jet. J. Fluid Mech. 739, 254–275.CrossRefGoogle Scholar
van Reeuwijk, M., Vassilicos, J.C. & Craske, J. 2021 Unified description of turbulent entrainment. J. Fluid Mech. 908, A12.CrossRefGoogle Scholar
Schlatter, P. & Örlü, R. 2010 Assessment of direct numerical simulation data of turbulent boundary layers. J. Fluid Mech. 659, 116–126.CrossRefGoogle Scholar
Schlatter, P. & Örlü, R. 2012 Turbulent boundary layers at moderate Reynolds numbers: inflow length and tripping effects. J. Fluid Mech. 710, 5–34.CrossRefGoogle Scholar
Sillero, J.A., Jiménez, J. & Moser, R.D. 2013 One-point statistics for turbulent wall-bounded flows at Reynolds numbers up to $\delta ^+ \approx 2000$. Phys. Fluids 25 (10), 105102.CrossRefGoogle Scholar
Sillero, J.A., Jiménez, J. & Moser, R.D. 2014 Two-point statistics for turbulent boundary layers and channels at Reynolds numbers up to $\delta ^+ \approx$ 2000. Phys. Fluids 26 (10), 105109.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 in a jet. J. Fluid Mech. 685, 165–190.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, 567–590.CrossRefGoogle Scholar
da Silva, C.B. & Pereira, J.C.F. 2008 Invariants of the velocity-gradient, rate-of-strain, and rate-of-rotation tensors across the turbulent/nonturbulent interface in jets. Phys. Fluids 20 (5), 055101.CrossRefGoogle Scholar
de Silva, C.M., Philip, J., Chauhan, K., Meneveau, C. & Marusic, I. 2013 Multiscale geometry and scaling of the turbulent-nonturbulent interface in high Reynolds number boundary layers. Phys. Rev. Lett. 111 (4), 044501.CrossRefGoogle ScholarPubMed
de Silva, C.M., Philip, J., Hutchins, N. & Marusic, I. 2017 Interfaces of uniform momentum zones in turbulent boundary layers. J. Fluid Mech. 820, 451–478.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
Smits, A.J., McKeon, B.J. & Marusic, I. 2011 High-Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43, 353–375.CrossRefGoogle Scholar
Spalart, P.R 1988 Direct simulation of a turbulent boundary layer up to $R_\theta = 1410$. J. Fluid Mech. 187, 61–98.CrossRefGoogle Scholar
Sreenivasan, K.R., Ramshankar, R. & Meneveau, C.H. 1989 Mixing, entrainment and fractal dimensions of surfaces in turbulent flows. Proc. R. Soc. Lond. A 421, 79–108.Google Scholar
Tanaka, S., Watanabe, T. & Nagata, K. 2019 Multi-particle model of coarse-grained scalar dissipation rate with volumetric tensor in turbulence. J. Comput. Phys. 389, 128–146.CrossRefGoogle 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 (2), 021702.CrossRefGoogle Scholar
Taveira, R.R., Diogo, J.S., Lopes, D.C. & da Silva, C.B. 2013 Lagrangian statistics across the turbulent–nonturbulent interface in a turbulent plane jet. Phys. Rev. E 88 (4), 043001.CrossRefGoogle Scholar
Van der Vorst, H.A. 1992 Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13 (2), 631–644.CrossRefGoogle Scholar
Vreman, A.W., Sandham, N.D. & Luo, K.H. 1996 Compressible mixing layer growth rate and turbulence characteristics. J. Fluid Mech. 320, 235–258.CrossRefGoogle Scholar
Watanabe, T., da Silva, C.B. & Nagata, K. 2019 a 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., da Silva, C.B., Nagata, K. & Sakai, Y. 2017 c Geometrical aspects of turbulent/non-turbulent interfaces with and without mean shear. Phys. Fluids 29 (8), 085105.CrossRefGoogle Scholar
Watanabe, T., Jaulino, R., Taveira, R.R., da Silva, C.B., Nagata, K. & Sakai, Y. 2017 a Role of an isolated eddy near the turbulent/non-turbulent interface layer. Phys. Rev. Fluids 2 (9), 094607.CrossRefGoogle Scholar
Watanabe, T., Riley, J.J. & Nagata, K. 2017 b Turbulent entrainment across turbulent–nonturbulent interfaces in stably stratified mixing layers. Phys. Rev. Fluids 2 (10), 104803.CrossRefGoogle Scholar
Watanabe, T., Riley, J.J., Nagata, K., Onishi, R. & Matsuda, K. 2018 a A localized turbulent mixing layer in a uniformly stratified environment. J. Fluid Mech. 849, 245–276.CrossRefGoogle Scholar
Watanabe, T., Sakai, Y., Nagata, K., Ito, Y. & Hayase, T. 2014 Wavelet analysis of coherent vorticity near the turbulent/non-turbulent interface in a turbulent planar jet. Phys. Fluids 26 (9), 095105.CrossRefGoogle Scholar
Watanabe, T., Sakai, Y., Nagata, K., Ito, Y. & Hayase, T. 2015 Turbulent mixing of passive scalar near turbulent and non-turbulent interface in mixing layers. Phys. Fluids 27 (8), 085109.CrossRefGoogle Scholar
Watanabe, T., Zhang, X. & Nagata, K. 2018 b Turbulent/non-turbulent interfaces detected in DNS of incompressible turbulent boundary layers. Phys. Fluids 30 (3), 035102.CrossRefGoogle Scholar
Watanabe, T., Zhang, X. & Nagata, K. 2019 b Direct numerical simulation of incompressible turbulent boundary layers and planar jets at high Reynolds numbers initialized with implicit large eddy simulation. Comput. Fluids 194, 104314.CrossRefGoogle Scholar
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
Wu, D., Wang, J., Cui, G. & Pan, C. 2020 Effects of surface shapes on properties of turbulent/non-turbulent interface in turbulent boundary layers. Sci. China Technol. Sci. 63 (2), 214–222.CrossRefGoogle Scholar
Zhang, X., Watanabe, T. & Nagata, K. 2018 Turbulent/nonturbulent interfaces in high-resolution direct numerical simulation of temporally evolving compressible turbulent boundary layers. Phys. Rev. Fluids 3 (9), 094605.CrossRefGoogle Scholar
Zhuang, Y., Tan, H., Huang, H., Liu, Y. & Zhang, Y. 2018 Fractal characteristics of turbulent-non-turbulent interface in supersonic turbulent boundary layers. J. Fluid Mech. 843, R2.CrossRefGoogle Scholar