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The decay of stably stratified grid turbulence in a viscosity-affected stratified flow regime

Published online by Cambridge University Press:  08 August 2022

Tomoaki Watanabe Open the ORCID record for Tomoaki Watanabe [Opens in a new window] ,
Yulin Zheng Open the ORCID record for Yulin Zheng [Opens in a new window]  and
Koji Nagata Open the ORCID record for Koji Nagata [Opens in a new window]
Tomoaki Watanabe*
Affiliation:
Education and Research Center for Flight Engineering, Nagoya University, Furo-cho, Chikusa, Nagoya 464-8603, Japan
Yulin Zheng
Affiliation:
Department of Aerospace Engineering, Nagoya University, Nagoya 464-8603, Japan
Koji Nagata
Affiliation:
Department of Aerospace Engineering, Nagoya University, Nagoya 464-8603, Japan
*
†Email address for correspondence: [email protected]
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Abstract

The decay of stably stratified turbulence generated by a towed rake of vertical plates is investigated by direct numerical simulations (DNS) of temporally evolving grid turbulence in a linearly stratified fluid. The Reynolds number $Re_M=U_0M/\nu$ is 5000 or 10 000 while the Froude number $Fr_M=U_0/MN$ is between 0.1 and 6 ($U_0$: towing speed; $M$: mesh size; $\nu$: kinematic viscosity; $N$: Brunt–Väisälä frequency). The DNS results are compared with the theory of stably stratified axisymmetric Saffman turbulence. Here, the theory is extended to a viscosity-affected stratified flow regime with low buoyancy Reynolds number $Re_b$, and power laws are derived for the temporal variations of the horizontal velocity scale ($U_H$) and the horizontal and vertical integral length scales ($L_H$ and $L_V$). Temporal grid turbulence initialized with the mean velocity deficit of wakes exhibits a $k^{2}$ energy spectrum at a low-wavenumber range and invariance of $U_H^2L_H^2L_V$, which are the signatures of axisymmetric Saffman turbulence. The decay of various quantities follows the power laws predicted for low-$Re_b$ Saffman turbulence when $Fr_M$ is sufficiently small. However, the decay of $U_H^2$ at $Fr_M=6$ is no longer expressed by a power law with a constant exponent. This behaviour is related to the scaling of kinetic energy dissipation rate $\varepsilon$, for which $\alpha =\varepsilon /(U_H^3/L_H)$ is constant during the decay for $Fr_M\leq 1$ while it varies with time for $Fr_M=6$. We also examine the experimental data of towed-grid experiments by Praud et al. (J. Fluid Mech., vol. 522, 2005, pp. 1–33), which is shown to agree with the theory of low-$Re_b$ Saffman turbulence.

JFM classification

Turbulent Flows: Stratified turbulence Wakes/Jets: Wakes Turbulent Flows: Homogeneous turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 946 , 10 September 2022 , A29
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© The Author(s), 2022. Published by Cambridge University Press

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References

Anas, M., Joshi, P. & Verma, M.K. 2020 Freely decaying turbulence in a finite domain at finite Reynolds number. Phys. Fluids 32 (9), 095109.CrossRefGoogle Scholar
Antonia, R.A., Lee, S.K., Djenidi, L., Lavoie, P. & Danaila, L. 2013 Invariants for slightly heated decaying grid turbulence. J. Fluid Mech. 727, 379–406.CrossRefGoogle Scholar
Batchelor, G.K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Batchelor, G.K. & Proudman, I. 1956 The large-scale structure of homogeneous turbulence. Phil. Trans. R. Soc. A 248, 369–405.Google Scholar
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J.-M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343–368.CrossRefGoogle Scholar
Britter, R.E., Hunt, J.C.R., Marsh, G.L. & Snyder, W.H. 1983 The effects of stable stratification on turbulent diffusion and the decay of grid turbulence. J. Fluid Mech. 127, 27–44.CrossRefGoogle Scholar
de Bruyn Kops, S.M. & Riley, J.J. 2019 The effects of stable stratification on the decay of initially isotropic homogeneous turbulence. J. Fluid Mech. 860, 787–821.CrossRefGoogle Scholar
Chongsiripinyo, K. & Sarkar, S. 2020 Decay of turbulent wakes behind a disk in homogeneous and stratified fluids. J. Fluid Mech. 885, A31.CrossRefGoogle Scholar
Comte-Bellot, G. & Corrsin, S. 1966 The use of a contraction to improve the isotropy of grid-generated turbulence. J. Fluid Mech. 25 (4), 657–682.CrossRefGoogle Scholar
Davidson, P.A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
Davidson, P.A. 2009 The role of angular momentum conservation in homogeneous turbulence. J. Fluid Mech. 632, 329–358.CrossRefGoogle Scholar
Davidson, P.A. 2010 On the decay of Saffman turbulence subject to rotation, stratification or an imposed magnetic field. J. Fluid Mech. 663, 268–292.CrossRefGoogle Scholar
Davidson, P.A. 2013 Turbulence in Rotating, Stratified and Electrically Conducting Fluids. Cambridge University Press.CrossRefGoogle Scholar
Diamessis, P.J., Spedding, G.R. & Domaradzki, J.A. 2011 Similarity scaling and vorticity structure in high-Reynolds-number stably stratified turbulent wakes. J. Fluid Mech. 671, 52–95.CrossRefGoogle Scholar
Djenidi, L., Kamruzzaman, M. & Antonia, R.A. 2015 Power-law exponent in the transition period of decay in grid turbulence. J. Fluid Mech. 779, 544–555.CrossRefGoogle Scholar
Espa, S., Avallone, G. & Cenedese, A. 2018 Decaying grid turbulence experiments in a stratified fluid: flow measurements and statistics. Stoch. Environ. Res. Risk Assess. 32 (8), 2325–2336.CrossRefGoogle Scholar
Fincham, A.M., Maxworthy, T. & Spedding, G.R. 1996 Energy dissipation and vortex structure in freely decaying, stratified grid turbulence. Dyn. Atmos. Oceans 23 (1–4), 155–169.CrossRefGoogle Scholar
Gampert, M., Boschung, J., Hennig, F., Gauding, M. & Peters, N. 2014 The vorticity versus the scalar criterion for the detection of the turbulent/non-turbulent interface. J. Fluid Mech. 750, 578–596.CrossRefGoogle Scholar
Godoy-Diana, R., Chomaz, J.-M. & Billant, P. 2004 Vertical length scale selection for pancake vortices in strongly stratified viscous fluids. J. Fluid Mech. 504, 229–238.CrossRefGoogle Scholar
Goto, S & Vassilicos, J.C. 2016 Unsteady turbulence cascades. Phys. Rev. E 94 (5), 053108.CrossRefGoogle ScholarPubMed
Gregg, M.C., D'Asaro, E.A., Riley, J.J. & Kunze, E. 2018 Mixing efficiency in the ocean. Annu. Rev. Mar. Sci. 10, 443–473.CrossRefGoogle ScholarPubMed
Hayashi, M., Watanabe, T. & Nagata, K. 2021 Characteristics of small-scale shear layers in a temporally evolving turbulent planar jet. J. Fluid Mech. 920, A38.CrossRefGoogle Scholar
Huq, P. & Britter, R.E. 1995 Turbulence evolution and mixing in a two-layer stably stratified fluid. J. Fluid Mech. 285, 41–67.CrossRefGoogle Scholar
Isaza, J.C., Salazar, R. & Warhaft, Z. 2014 On grid-generated turbulence in the near-and far field regions. J. Fluid Mech. 753, 402–426.CrossRefGoogle Scholar
Ishida, T., Davidson, P.A. & Kaneda, Y. 2006 On the decay of isotropic turbulence. J. Fluid Mech. 564, 455–475.CrossRefGoogle Scholar
Ishihara, T., Gotoh, T. & Kaneda, Y. 2009 Study of high–Reynolds number isotropic turbulence by direct numerical simulation. Annu. Rev. Fluid Mech. 41, 165–180.CrossRefGoogle Scholar
Itsweire, E.C., Helland, K.N. & Van Atta, C.W. 1986 The evolution of grid-generated turbulence in a stably stratified fluid. J. Fluid Mech. 162, 299–338.CrossRefGoogle Scholar
Kitamura, T., Nagata, K., Sakai, Y., Sasoh, A., Terashima, O., Saito, H. & Harasaki, T. 2014 On invariants in grid turbulence at moderate Reynolds numbers. J. Fluid Mech. 738, 378–406.CrossRefGoogle Scholar
Komori, S. & Nagata, K. 1996 Effects of molecular diffusivities on counter-gradient scalar and momentum transfer in strongly stable stratification. J. Fluid Mech. 326, 205–237.CrossRefGoogle Scholar
Komori, S., Nagata, K., Kanzaki, T. & Murakami, Y. 1993 Measurements of mass flux in a turbulent liquid flow with a chemical reaction. AIChE J. 39 (10), 1611–1620.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
Krogstad, P.-Å. & Davidson, P.A. 2010 Is grid turbulence Saffman turbulence? J. Fluid Mech. 642, 373–394.CrossRefGoogle Scholar
Krogstad, P.-Å. & Davidson, P.A. 2012 Near-field investigation of turbulence produced by multi-scale grids. Phys. Fluids 24 (3), 035103.CrossRefGoogle Scholar
Lavoie, P., Djenidi, L. & Antonia, R.A. 2007 Effects of initial conditions in decaying turbulence generated by passive grids. J. Fluid Mech. 585, 395–420.CrossRefGoogle Scholar
Lienhard, J.H. & van Atta, C.W. 1990 The decay of turbulence in thermally stratified flow. J. Fluid Mech. 210, 57–112.CrossRefGoogle Scholar
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207–242.CrossRefGoogle Scholar
Liu, H.-T. 1995 Energetics of grid turbulence in a stably stratified fluid. J. Fluid Mech. 296, 127–157.CrossRefGoogle Scholar
Maffioli, A., Brethouwer, G. & Lindborg, E. 2016 Mixing efficiency in stratified turbulence. J. Fluid Mech. 794, R3.CrossRefGoogle Scholar
Maffioli, A. & Davidson, P.A. 2016 Dynamics of stratified turbulence decaying from a high buoyancy Reynolds number. J. Fluid Mech. 786, 210–233.CrossRefGoogle Scholar
Mahrt, L. 1999 Stratified atmospheric boundary layers. Boundary-Layer Meteorol. 90 (3), 375–396.CrossRefGoogle Scholar
Mazellier, N. & Vassilicos, J.C. 2008 The turbulence dissipation constant is not universal because of its universal dependence on large-scale flow topology. Phys. Fluids 20 (1), 015101.CrossRefGoogle Scholar
Melina, G., Bruce, P.J.K. & Vassilicos, J.C. 2016 Vortex shedding effects in grid-generated turbulence. Phys. Rev. Fluids 1 (4), 044402.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
Mora, D.O., Pladellorens, E.M., Turró, P.R., Lagauzere, M. & Obligado, M. 2019 Energy cascades in active-grid-generated turbulent flows. Phys. Rev. Fluids 4 (10), 104601.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
Nagata, K., Saiki, T., Sakai, Y., Ito, Y. & Iwano, K. 2017 Effects of grid geometry on non-equilibrium dissipation in grid turbulence. Phys. Fluids 29 (1), 015102.CrossRefGoogle Scholar
Nagata, K., Sakai, Y., Inaba, T., Suzuki, H., Terashima, O. & Suzuki, H. 2013 Turbulence structure and turbulence kinetic energy transport in multiscale/fractal-generated turbulence. Phys. Fluids 25 (6), 065102.CrossRefGoogle Scholar
Okino, S. & Hanazaki, H. 2019 Decaying turbulence in a stratified fluid of high Prandtl number. J. Fluid Mech. 874, 821–855.CrossRefGoogle Scholar
Panchapakesan, N.R. & Lumley, J.L. 1993 Turbulence measurements in axisymmetric jets of air and helium. Part 1. Air jet. J. Fluid Mech. 246, 197–223.CrossRefGoogle Scholar
Pham, H.T., Sarkar, S. & Brucker, K.A. 2009 Dynamics of a stratified shear layer above a region of uniform stratification. J. Fluid Mech. 630, 191–223.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Praud, O., Fincham, A.M. & Sommeria, J. 2005 Decaying grid turbulence in a strongly stratified fluid. J. Fluid Mech. 522, 1–33.CrossRefGoogle Scholar
Rehmann, C.R. & Koseff, J.R. 2004 Mean potential energy change in stratified grid turbulence. Dyn. Atmos. Oceans 37 (4), 271–294.CrossRefGoogle Scholar
Riley, J.J. & de Bruyn Kops, S.M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15 (7), 2047–2059.CrossRefGoogle Scholar
Saffman, P.G. 1967 The large-scale structure of homogeneous turbulence. J. Fluid Mech. 27, 581–593.CrossRefGoogle Scholar
Salehipour, H, Peltier, W.R. & Mashayek, A 2015 Turbulent diapycnal mixing in stratified shear flows: the influence of Prandtl number on mixing efficiency and transition at high Reynolds number. J. Fluid Mech. 773, 178–223.CrossRefGoogle Scholar
Sarpkaya, T. 2006 Structures of separation on a circular cylinder in periodic flow. J. Fluid Mech. 567, 281–297.CrossRefGoogle Scholar
Seoud, R.E. & Vassilicos, J.C. 2007 Dissipation and decay of fractal-generated turbulence. Phys. Fluids 19 (10), 105108.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
Simmonsr, L.F.G. & Salter, C. 1934 Experimental investigation and analysis of the velocity variations in turbulent flow. Proc. R. Soc. Lond. A 145 (854), 212–234.Google Scholar
Sinhuber, M., Bodenschatz, E. & Bewley, G.P. 2015 Decay of turbulence at high Reynolds numbers. Phys. Rev. Lett. 114 (3), 034501.CrossRefGoogle ScholarPubMed
Smyth, W.D., Moum, J.N. & Caldwell, D.R. 2001 The efficiency of mixing in turbulent patches: inferences from direct simulations and microstructure observations. J. Phys. Oceanogr. 31 (8), 1969–1992.2.0.CO;2>CrossRefGoogle Scholar
Spedding, G.R. 1997 The evolution of initially turbulent bluff-body wakes at high internal Froude number. J. Fluid Mech. 337, 283–301.CrossRefGoogle Scholar
Stillinger, D.C., Helland, K.N. & Van Atta, C.W. 1983 Experiments on the transition of homogeneous turbulence to internal waves in a stratified fluid. J. Fluid Mech. 131, 91–122.CrossRefGoogle Scholar
Takamure, K., Sakai, Y., Ito, Y., Iwano, K. & Hayase, T. 2019 Dissipation scaling in the transition region of turbulent mixing layer. Intl J. Heat Fluid Flow 75, 77–85.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
Teitelbaum, T. & Mininni, P.D. 2012 Decay of Batchelor and Saffman rotating turbulence. Phys. Rev. E 86 (6), 066320.CrossRefGoogle ScholarPubMed
Thorpe, S.A. 1978 The near-surface ocean mixing layer in stable heating conditions. J. Geophys. Res. 83 (C6), 2875–2885.CrossRefGoogle Scholar
Uberoi, M.S. & Wallis, S. 1967 Effect of grid geometry on turbulence decay. Phys. Fluids 10, 1216–1224.CrossRefGoogle Scholar
Valente, P.C., da Silva, C.B. & Pinho, F.T. 2016 Energy spectra in elasto-inertial turbulence. Phys. Fluids 28 (7), 075108.CrossRefGoogle Scholar
Valente, P.C. & Vassilicos, J.C. 2011 The decay of turbulence generated by a class of multiscale grids. J. Fluid Mech. 687, 300–340.CrossRefGoogle Scholar
Valente, P.C. & Vassilicos, J.C. 2015 The energy cascade in grid-generated non-equilibrium decaying turbulence. Phys. Fluids 27 (4), 045103.CrossRefGoogle Scholar
VanDine, A., Pham, H.T. & Sarkar, S. 2021 Turbulent shear layers in a uniformly stratified background: DNS at high Reynolds number. J. Fluid Mech. 916, A42.CrossRefGoogle Scholar
Vassilicos, J.C. 2015 Dissipation in turbulent flows. Annu. Rev. Fluid Mech. 47, 95–114.CrossRefGoogle Scholar
Watanabe, T. & Nagata, K. 2018 Integral invariants and decay of temporally developing grid turbulence. Phys. Fluids 30 (10), 105111.CrossRefGoogle Scholar
Watanabe, T. & Nagata, K. 2021 Large-scale characteristics of a stably stratified turbulent shear layer. J. Fluid Mech. 927, A27.CrossRefGoogle Scholar
Watanabe, T., Riley, J.J., Nagata, K., Matsuda, K. & Onishi, R. 2019 a Hairpin vortices and highly elongated flow structures in a stably stratified shear layer. J. Fluid Mech. 878, 37–61.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. 2016 Large eddy simulation study of turbulent kinetic energy and scalar variance budgets and turbulent/non-turbulent interface in planar jets. Fluid Dyn. Res. 48 (2), 021407.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., Tanaka, K. & Nagata, K. 2020 Characteristics of shearing motions in incompressible isotropic turbulence. Phys. Rev. Fluids 5 (7), 072601.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
Yap, C.T. & Van Atta, C.W. 1993 Experimental studies of the development of quasi-two-dimensional turbulence in stably stratified fluid. Dyn. Atmos. Oceans 19 (1–4), 289–323.CrossRefGoogle Scholar
Yoon, K. & Warhaft, Z. 1990 The evolution of grid-generated turbulence under conditions of stable thermal stratification. J. Fluid Mech. 215, 601–638.CrossRefGoogle Scholar
Yoshimatsu, K. & Kaneda, Y. 2018 Large-scale structure of velocity and passive scalar fields in freely decaying homogeneous anisotropic turbulence. Phys. Rev. Fluids 3 (10), 104601.CrossRefGoogle Scholar
Zecchetto, M. & da Silva, C.B. 2021 Universality of small-scale motions within the turbulent/non-turbulent interface layer. J. Fluid Mech. 916, A9.CrossRefGoogle Scholar
Zheng, Y., Nagata, K. & Watanabe, T. 2021 a Energy dissipation and enstrophy production/destruction at very low Reynolds numbers in the final stage of the transition period of decay in grid turbulence. Phys. Fluids 33 (3), 035147.CrossRefGoogle Scholar
Zheng, Y., Nagata, K. & Watanabe, T. 2021 b Turbulent characteristics and energy transfer in the far field of active-grid turbulence. Phys. Fluids 33 (11), 115119.CrossRefGoogle Scholar
Zhou, Q. & Diamessis, P.J. 2019 Large-scale characteristics of stratified wake turbulence at varying Reynolds number. Phys. Rev. Fluids 4 (8), 084802.CrossRefGoogle Scholar