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Large-scale characteristics of a stably stratified turbulent shear layer

Published online by Cambridge University Press:  28 September 2021

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]
Tomoaki Watanabe*
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: watanabe.tomoaki@c.nagoya-u.jp
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Abstract

Implicit large eddy simulation is performed to investigate large-scale characteristics of a temporally evolving, stably stratified turbulent shear layer arising from the Kelvin–Helmholtz instability. The shear layer at late time has two energy-containing length scales: the scale of the shear layer thickness, which characterizes large-scale motions (LSM) of the shear layer; and the larger streamwise scale of elongated large-scale structures (ELSS), which increases with time. The ELSS forms in the middle of the shear layer when the Richardson number is sufficiently large. The contribution of the ELSS to velocity and density variances becomes relatively important with time although the LSM dominate the momentum and density transport. The ELSS have a highly anisotropic Reynolds stress, to a degree similar to the near-wall region of turbulent boundary layers, while the Reynolds stress of the LSM is as anisotropic as in the outer region. Peaks in the spectral energy density associated with the ELSS emerge because of the slow decay of turbulence at very large scales. A forward interscale energy transfer from large to small scales occurs even at a small buoyancy Reynolds number. However, an inverse transfer also occurs for the energy of spanwise velocity. Negative production of streamwise velocity and density spectra, i.e. counter-gradient transport of momentum and density, is found at small scales. These behaviours are consistent with channel flows, indicating similar flow dynamics in the stratified shear layer and wall-bounded shear flows. The structure function exhibits a logarithmic law at large scales, implying a $k^{-1}$ scaling of energy spectra.

JFM classification

Turbulent Flows: Stratified turbulence Wakes/Jets: Shear layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 927 , 25 November 2021 , A27
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Balaras, E., Piomelli, U.G.O. & Wallace, J.M. 2001 Self-similar states in turbulent mixing layers. J. Fluid Mech. 446, 1–24.CrossRefGoogle Scholar
Berkooz, G., Holmes, P. & Lumley, J.L. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25, 539–575.CrossRefGoogle Scholar
Bernal, L.P. & Roshko, A. 1986 Streamwise vortex structure in plane mixing layers. J. Fluid Mech. 170, 499–525.CrossRefGoogle Scholar
Biancofiore, L. 2014 Crossover between two-and three-dimensional turbulence in spatial mixing layers. J. Fluid Mech. 745, 164–179.CrossRefGoogle Scholar
Bogey, C. & Bailly, C. 2009 Turbulence and energy budget in a self-preserving round jet: direct evaluation using large eddy simulation. J. Fluid Mech. 627, 129–160.CrossRefGoogle Scholar
Caulfield, C.P. 2021 Layering, instabilities, and mixing in turbulent stratified flows. Annu. Rev. Fluid Mech. 53, 113–145.CrossRefGoogle Scholar
Champagne, F.H., Pao, Y.H. & Wygnanski, I.J. 1976 On the two-dimensional mixing region. J. Fluid Mech. 74 (2), 209–250.CrossRefGoogle Scholar
Davidson, P.A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
Davidson, P.A., Nickels, T.B. & Krogstad, P.-Å. 2006 The logarithmic structure function law in wall-layer turbulence. J. Fluid Mech. 550, 51–60.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
Fritts, D.C., Arendt, S. & Andreassen, Ø. 1998 Vorticity dynamics in a breaking internal gravity wave. Part 2. Vortex interactions and transition to turbulence. J. Fluid Mech. 367, 47–65.CrossRefGoogle Scholar
Fritts, D.C., Wieland, S.A., Lund, T.S., Thorpe, S.A. & Hecht, J.H. 2021 Kelvin–Helmholtz billow interactions and instabilities in the mesosphere over the Andes Lidar Observatory: 2. Modeling and interpretation. J. Geophys. Res.: Atmos. 126 (1), e2020JD033412.CrossRefGoogle Scholar
Gargett, A.E. & Wells, J.R. 2007 Langmuir turbulence in shallow water. Part 1. Observations. J. Fluid Mech. 576, 27–61.CrossRefGoogle Scholar
Gerz, T. & Schumann, U. 1996 A possible explanation of countergradient fluxes in homogeneous turbulence. Theor. Comput. Fluid Dyn. 8 (3), 169–181.CrossRefGoogle Scholar
Holt, S.E., Koseff, J.R. & Ferziger, J.H. 1992 A numerical study of the evolution and structure of homogeneous stably stratified sheared turbulence. J. Fluid Mech. 237, 499–539.CrossRefGoogle Scholar
Hutchins, N. & Marusic, I. 2007 Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28.CrossRefGoogle Scholar
Iida, O. & Nagano, Y. 2007 Effect of stable-density stratification on counter gradient flux of a homogeneous shear flow. Intl J. Heat Mass Transfer 50 (1–2), 335–347.CrossRefGoogle Scholar
Jiménez, J. 2013 How linear is wall-bounded turbulence? Phys. Fluids 25 (11), 110814.CrossRefGoogle Scholar
Kaminski, A.K. & Smyth, W.D. 2019 Stratified shear instability in a field of pre-existing turbulence. J. Fluid Mech. 862, 639–658.CrossRefGoogle Scholar
Kempf, A., Klein, M. & Janicka, J. 2005 Efficient generation of initial-and inflow-conditions for transient turbulent flows in arbitrary geometries. Flow Turbul. Combust. 74 (1), 67–84.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
Kim, K.C. & Adrian, R.J. 1999 Very large-scale motion in the outer layer. Phys. Fluids 11 (2), 417–422.CrossRefGoogle Scholar
Klaassen, G.P. & Peltier, W.R. 1991 The influence of stratification on secondary instability in free shear layers. J. Fluid Mech. 227, 71–106.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
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
Lee, M. & Moser, R.D. 2019 Spectral analysis of the budget equation in turbulent channel flows at high Reynolds number. J. Fluid Mech. 860, 886–938.CrossRefGoogle Scholar
Lee, M.J., Kim, J. & Moin, P. 1990 Structure of turbulence at high shear rate. J. Fluid Mech. 216, 561–583.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
Mahrt, L. 1999 Stratified atmospheric boundary layers. Boundary-Layer Meteorol. 90 (3), 375–396.CrossRefGoogle Scholar
Mashayek, A. & Peltier, W.R. 2012 The ‘zoo’ of secondary instabilities precursory to stratified shear flow transition. Part 1. Shear aligned convection, pairing, and braid instabilities. J. Fluid Mech. 708, 5–44.CrossRefGoogle Scholar
Mashayek, A. & Peltier, W.R. 2013 Shear-induced mixing in geophysical flows: does the route to turbulence matter to its efficiency? J. Fluid Mech. 725, 216–261.CrossRefGoogle Scholar
McMullan, W.A. 2015 Spanwise domain effects on the evolution of the plane turbulent mixing layer. Intl J. Comput. Fluid Dyn. 29 (6–8), 333–345.CrossRefGoogle Scholar
Miles, J.W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10 (4), 496–508.CrossRefGoogle Scholar
Mizuno, Y. 2016 Spectra of energy transport in turbulent channel flows for moderate Reynolds numbers. J. Fluid Mech. 805, 171–187.CrossRefGoogle Scholar
Monty, J.P., Hutchins, N., Ng, H.C.H., Marusic, I. & Chong, M.S. 2009 A comparison of turbulent pipe, channel and boundary layer flows. J. Fluid Mech. 632, 431–442.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
Nickels, T.B., Marusic, I., Hafez, S. & Chong, M.S. 2005 Evidence of the $k_1^{-1}$ law in a high-Reynolds-number turbulent boundary layer. Phys. Rev. Lett. 95 (7), 074501.CrossRefGoogle Scholar
Peltier, W.R. & Caulfield, C.P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35 (1), 135–167.CrossRefGoogle Scholar
Perry, A.E. & Chong, M.S. 1982 On the mechanism of wall turbulence. J. Fluid Mech. 119, 173–217.CrossRefGoogle Scholar
Pham, H.T. & Sarkar, S. 2010 Transport and mixing of density in a continuously stratified shear layer. J. Turbul. 11, N24.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Rahmani, M., Lawrence, G.A. & Seymour, B.R. 2014 The effect of Reynolds number on mixing in Kelvin–Helmholtz billows. J. Fluid Mech. 759, 612–641.CrossRefGoogle Scholar
Redford, J.A., Lund, T.S. & Coleman, G.N. 2015 A numerical study of a weakly stratified turbulent wake. J. Fluid Mech. 776, 568–609.CrossRefGoogle Scholar
Riley, J.J. & Lindborg, E. 2008 Stratified turbulence: a possible interpretation of some geophysical turbulence measurements. J. Atmos. Sci. 65 (7), 2416–2424.CrossRefGoogle Scholar
Rogers, M.M. & Moser, R.D. 1994 Direct simulation of a self-similar turbulent mixing layer. Phys. Fluids 6 (2), 903–923.CrossRefGoogle Scholar
Salehipour, H. & Peltier, W.R. 2015 Diapycnal diffusivity, turbulent Prandtl number and mixing efficiency in Boussinesq stratified turbulence. J. Fluid Mech. 775, 464–500.CrossRefGoogle Scholar
Savill, A.M. 1987 Recent developments in rapid-distortion theory. Annu. Rev. Fluid Mech. 19 (1), 531–573.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
Smyth, W.D. & Moum, J.N. 2000 a Anisotropy of turbulence in stably stratified mixing layers. Phys. Fluids 12 (6), 1343–1362.CrossRefGoogle Scholar
Smyth, W.D. & Moum, J.N. 2000 b Length scales of turbulence in stably stratified mixing layers. Phys. Fluids 12 (6), 1327–1342.CrossRefGoogle Scholar
Smyth, W.D. & Moum, J.N. 2012 Ocean mixing by Kelvin–Helmholtz instability. Oceanography 25 (2), 140–149.CrossRefGoogle Scholar
Smyth, W.D. & Winters, K.B. 2003 Turbulence and mixing in Holmboe waves. J. Phys. Oceanogr. 33 (4), 694–711.2.0.CO;2>CrossRefGoogle Scholar
Strang, E.J. & Fernando, H.J.S. 2001 Entrainment and mixing in stratified shear flows. J. Fluid Mech. 428, 349–386.CrossRefGoogle Scholar
Tai, Y., Watanabe, T. & Nagata, K. 2020 Implicit large eddy simulation of passive scalar transfer in compressible planar jet. Intl J. Numer. Meth. Fluids 93 (4), 1–16.Google Scholar
Takahashi, M., Iwano, K., Sakai, Y. & Ito, Y. 2019 Three-dimensional visualization of destruction events of turbulent momentum transfer in a plane jet. Phys. Fluids 31 (10), 105114.CrossRefGoogle Scholar
Takamure, K., Ito, Y., Sakai, Y., Iwano, K. & Hayase, T. 2018 Momentum transport process in the quasi self-similar region of free shear mixing layer. Phys. Fluids 30 (1), 015109.CrossRefGoogle 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 (15), 128–146.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
Taylor, J.R., de Bruyn Kops, S.M., Caulfield, C.P. & Linden, P.F. 2019 Testing the assumptions underlying ocean mixing methodologies using direct numerical simulations. J. Phys. Oceanogr. 49 (11), 2761–2779.CrossRefGoogle Scholar
Thorpe, S.A. 1978 The near-surface ocean mixing layer in stable heating conditions. J. Geophys. Res. 83 (C6), 2875–2885.CrossRefGoogle Scholar
Thorpe, S.A. 2012 On the Kelvin–Helmholtz route to turbulence. J. Fluid Mech. 708, 1–4.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
Watanabe, T., Riley, J.J. & Nagata, K. 2016 a Effects of stable stratification on turbulent/nonturbulent interfaces in turbulent mixing layers. Phys. Rev. Fluids 1 (4), 044301.CrossRefGoogle Scholar
Watanabe, T., Riley, J.J. & Nagata, K. 2017 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., 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 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 b 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., da Silva, C.B. & Nagata, K. 2020 Scale-by-scale kinetic energy budget near the turbulent/nonturbulent interface. Phys. Rev. Fluids 5 (12), 124610.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
Wingstedt, E.M.M., Fossum, H.E., Pettersson Reif, B.A. & Werne, J. 2015 Anisotropy and shear-layer edge dynamics of statistically unsteady, stratified turbulence. Phys. Fluids 27 (6), 065106.CrossRefGoogle Scholar
Wu, X. & Moin, P. 2009 Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer. J. Fluid Mech. 630, 5–41.CrossRefGoogle Scholar