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Prandtl number dependence of stratified turbulence

Published online by Cambridge University Press:  21 September 2020

Jesse D. Legaspi Open the ORCID record for Jesse D. Legaspi [Opens in a new window]  and
Michael L. Waite
Jesse D. Legaspi*
Affiliation:
Department of Applied Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, OntarioN2L 3G1, Canada
Michael L. Waite
Affiliation:
Department of Applied Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, OntarioN2L 3G1, Canada
*
†Email address for correspondence: jlegaspi@uwaterloo.ca
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Abstract

Stratified turbulence has a horizontally layered structure with quasi-two-dimensional vortices due to buoyancy forces that suppress vertical motion. The Prandtl number $\textit {Pr}$ quantifies the relative strengths of viscosity and buoyancy diffusivity, which damp small-scale velocity and buoyancy fluctuations at different microscales. Direct numerical simulations (DNS) require high resolution to resolve the smallest flow features for large $\textit {Pr}$. To reduce computational demand, $\textit {Pr}$ is often set to 1. In this paper, we explore how varying $\textit {Pr}$ affects stratified turbulence. DNS of homogeneous forced stratified turbulence with $0.7 \le \textit {Pr} \le 8$ are performed for four stratification strengths and buoyancy Reynolds numbers $\textit {Re}_b$ between 0.5 and 60. Energy spectra, buoyancy flux spectra, spectral energy flux and physical space fields are compared for scale-specific $\textit {Pr}$-sensitivity. For $\textit {Re}_b \gtrsim 10$, $\textit {Pr}$-dependence in the kinetic energy is mainly found at scales around and below the Kolmogorov scale. The potential energy and flux exhibit more prominent $\textit {Pr}$-sensitivity. As $\textit {Re}_b$ decreases, this $\textit {Pr}$-dependence extends upscale. With increasing $\textit {Pr}$, the spectra suggest eventual convergence to a limiting spectrum shape at large, finite $\textit {Pr}$, at least at scales at and above the Ozmidov scale. The $\textit {Pr}$-sensitivity of the spectra in the most strongly stratified $\textit {Re}_b<1$ case differed from the rest, since large horizontal scales are affected by viscosity and diffusion. These findings suggest that $\textit {Pr}=1$ DNS reasonably approximate $\textit {Pr} > 1$ DNS with large $\textit {Re}_b$, as long as the focus is on kinetic energy at scales much larger than the Kolmogorov scale, but otherwise stray from $\textit {Pr} > 1$ spectra around and below the Kolmogorov scale, and even upscale when $\textit {Re}_b \lesssim 1$.

JFM classification

Turbulent Flows: Stratified turbulence Turbulent Flows: Turbulence simulation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , A12
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Almalkie, S. & de Bruyn Kops, S. M. 2012 Kinetic energy dynamics in forced, homogeneous, and axisymmetric stably stratified turbulence. J. Turbul. 13 (29), 1–32.CrossRefGoogle Scholar
Bartello, P. & Tobias, S. M. 2013 Sensitivity of stratified turbulence to the buoyancy Reynolds number. J. Fluid Mech. 725, 1–22.CrossRefGoogle Scholar
Batchelor, G. K. 1959 Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5 (1), 113–133.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13 (6), 1645–1651.CrossRefGoogle Scholar
Bouruet-Aubertot, P., Sommeria, J. & Staquet, C. 1996 Stratified turbulence produced by internal wave breaking: two-dimensional numerical experiments. Dyn. Atmos. Oceans 23 (1–4), 357–369.CrossRefGoogle 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
Burgers, J. M. 1948 A mathematical model illustrating the theory of turbulence. In Advances in Applied Mechanics (ed. von Mises, R. & von Kármán, T.), vol. 1, pp. 171–199. Elsevier.Google Scholar
Carnevale, G. F., Briscolini, M. & Orlandi, P. 2001 Buoyancy- to inertial-range transition in forced stratified turbulence. J. Fluid Mech. 427, 205–239.CrossRefGoogle Scholar
Caulfield, C. P. & Peltier, W. R. 2000 The anatomy of the mixing transition in homogeneous and stratified free shear layers. J. Fluid Mech. 413, 1–47.CrossRefGoogle Scholar
Corrsin, S. 1951 On the spectrum of isotropic temperature fluctuations in an isotropic turbulence. J. Appl. Phys. 22 (4), 469–473.CrossRefGoogle Scholar
Davidson, P. A. 2013 Turbulence in Rotating, Stratified and Electrically Conducting Fluids. Cambridge University Press.CrossRefGoogle Scholar
Davidson, P. A. 2015 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.CrossRefGoogle Scholar
Dewan, E. M. 1997 Saturated-cascade similitude theory of gravity wave spectra. J. Geophys. Res.: Atmos. 102 (D25), 29799–29817.CrossRefGoogle Scholar
Durran, D. R. 2010 Numerical Methods for Fluid Dynamics: With Applications to Geophysics, Texts in Applied Mathematics, vol. 32. Springer.CrossRefGoogle Scholar
Frigo, M. & Johnson, S. G. 2005 The design and implementation of FFTW3. Proc. IEEE 93 (2), 216–231.CrossRefGoogle Scholar
Gerz, T., Schumann, U. & Elghobashi, S. E. 1989 Direct numerical simulation of stratified homogeneous turbulent shear flows. J. Fluid Mech. 200, 563–594.CrossRefGoogle Scholar
Gotoh, T. & Yeung, P. K. 2013 Passive scalar transport in turbulence: a computational perspective. In Ten Chapters in Turbulence (ed. Davidson, P. A., Kaneda, Y. & Sreenivasan, K. R.), pp. 87–131. Cambridge University Press.Google Scholar
Herring, J. R. & Métais, O. 1989 Numerical experiments in forced stably stratified turbulence. J. Fluid Mech. 202, 97–115.CrossRefGoogle Scholar
Holloway, G. 1988 The buoyancy flux from internal gravity wave breaking. Dyn. Atmos. Oceans 12 (2), 107–125.CrossRefGoogle Scholar
Howland, C. J., Taylor, J. R. & Caulfield, C. P. 2020 Mixing in forced stratified turbulence and its dependence on large-scale forcing. J. Fluid Mech. 898, A7.CrossRefGoogle Scholar
Ishihara, T., Kaneda, Y. & Hunt, J. C. R. 2013 Thin shear layers in high Reynolds number turbulence DNS results. Flow Turbul. Combust. 91 (4), 895–929.CrossRefGoogle Scholar
Kimura, Y. & Herring, J. R. 2012 Energy spectra of stably stratified turbulence. J. Fluid Mech. 698, 19–50.CrossRefGoogle Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk. SSSR 30, 301–305.Google Scholar
Kundu, P. K., Cohen, I. M. & Dowling, D. R. 2012 Fluid Mechanics, 5th edn. Academic.Google Scholar
Lang, C. J. & Waite, M. L. 2019 Scale-dependent anisotropy in forced stratified turbulence. Phys. Rev. Fluids 4 (4), 044801.CrossRefGoogle Scholar
Laval, J.-P., McWilliams, J. C. & Dubrulle, B. 2003 Forced stratified turbulence: successive transitions with Reynolds number. Phys. Rev. E 68 (3), 036308.CrossRefGoogle ScholarPubMed
Lilly, D. K. 1983 Stratified turbulence and the mesoscale variability of the atmosphere. J. Atmos. Sci. 40 (3), 749–761.2.0.CO;2>CrossRefGoogle Scholar
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207–242.CrossRefGoogle Scholar
Lindborg, E. & Brethouwer, G. 2007 Stratified turbulence forced in rotational and divergent modes. J. Fluid Mech. 586, 83–108.CrossRefGoogle Scholar
Lucas, D., Caulfield, C. C. P. & Kerswell, R. R. 2017 Layer formation in horizontally forced stratified turbulence: connecting exact coherent structures to linear instabilities. J. Fluid Mech. 832, 409–437.CrossRefGoogle Scholar
Maffioli, A. 2017 Vertical spectra of stratified turbulence at large horizontal scales. Phys. Rev. Fluids 2 (10), 104802.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
Okino, S. & Hanazaki, H. 2017 Turbulence in a fluid stratified by a high Prandtl-number scalar. In Sustained Simulation Performance 2017 (ed. Resch, M., Bez, W., Focht, E., Gienger, M. & Kobayashi, H.), pp. 113–121. Springer.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
Okino, S. & Hanazaki, H. 2020 Direct numerical simulation of turbulence in a salt-stratified fluid. J. Fluid Mech. 891, A19.CrossRefGoogle Scholar
Ozmidov, R. V. 1965 On the turbulent exchange in a stably stratified ocean. Izv. Akad. Nauk SSSR 1, 861–871.Google Scholar
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35 (1), 135–167.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
Riley, J. J. & Lelong, M.-P. 2000 Fluid motions in the presence of strong stable stratification. Annu. Rev. Fluid Mech. 32 (1), 613–657.CrossRefGoogle Scholar
Riley, J. J. & Lindborg, E. 2013 Recent progress in stratified turbulence. In Ten Chapters in Turbulence (ed. Davidson, P. A., Kaneda, Y. & Sreenivasan, K. R.), pp. 269–317. Cambridge University Press.Google Scholar
Rose, H. A. & Sulem, P. L. 1978 Fully developed turbulence and statistical mechanics. J. Phys. 39 (5), 441–484.CrossRefGoogle Scholar
Salehipour, H., Caulfield, C. P. & Peltier, W. R. 2016 Turbulent mixing due to the Holmboe wave instability at high Reynolds number. J. Fluid Mech. 803, 591–621.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
Siggia, E. D. 1981 Numerical study of small-scale intermittency in three-dimensional turbulence. J. Fluid Mech. 107, 375–406.CrossRefGoogle Scholar
Smith, L. M. & Waleffe, F. 2002 Generation of slow large scales in forced rotating stratified turbulence. J. Fluid Mech. 451, 145–168.CrossRefGoogle Scholar
Smyth, W. D. & Moum, J. N. 2000 Anisotropy of turbulence in stably stratified mixing layers. Phys. Fluids 12 (6), 1343–1362.CrossRefGoogle Scholar
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, 1969–1992.2.0.CO;2>CrossRefGoogle Scholar
de Stadler, M. B., Sarkar, S. & Brucker, K. A. 2010 Effect of the Prandtl number on a stratified turbulent wake. Phys. Fluids 22 (9), 095102.CrossRefGoogle Scholar
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: from instabilities to turbulence. Annu. Rev. Fluid Mech. 34 (1), 559–593.CrossRefGoogle Scholar
Stretch, D. D., Rottman, J. W., Venayagamoorthy, S. K., Nomura, K. K. & Rehmann, C. R. 2010 Mixing efficiency in decaying stably stratified turbulence. Dyn. Atmos. Oceans 49, 25–36.CrossRefGoogle Scholar
Taylor, G. I. 1935 Statistical theory of turbulence. Proc. R. Soc. Lond. A 151 (873), 421–444.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT.CrossRefGoogle Scholar
Waite, M. L. 2011 Stratified turbulence at the buoyancy scale. Phys. Fluids 23 (6), 066602.CrossRefGoogle Scholar
Waite, M. L. 2013 Potential enstrophy in stratified turbulence. J. Fluid Mech. 722, R4.CrossRefGoogle Scholar
Waite, M. L. 2014 Direct numerical simulations of laboratory-scale stratified turbulence. In Modeling Atmospheric and Oceanic Flows: Insights from Laboratory Experiments and Numerical Simulations, chap. 8, pp. 159–175. American Geophysical Union.Google Scholar
Waite, M. L. 2017 Random forcing of geostrophic motion in rotating stratified turbulence. Phys. Fluids 29 (12), 126602.CrossRefGoogle Scholar
Waite, M. L. & Bartello, P. 2004 Stratified turbulence dominated by vortical motion. J. Fluid Mech. 517, 281–308.CrossRefGoogle Scholar
Wyngaard, J. C. 2010 Turbulence in the Atmosphere. Cambridge University Press.CrossRefGoogle Scholar