Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-12-01T03:44:18.180Z Has data issue: false hasContentIssue false
  • English
  • Français

Self-similar mixing in stratified plane Couette flow for varying Prandtl number

Published online by Cambridge University Press:  04 May 2017

Qi Zhou Open the ORCID record for Qi Zhou [Opens in a new window] ,
John R. Taylor Open the ORCID record for John R. Taylor [Opens in a new window]  and
C. P. Caulfield Open the ORCID record for C. P. Caulfield [Opens in a new window]
Qi Zhou*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
John R. Taylor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
C. P. Caulfield
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate fully developed turbulence in stratified plane Couette flows using direct numerical simulations similar to those reported by Deusebio et al. (J. Fluid Mech., vol. 781, 2015, pp. 298–329) expanding the range of Prandtl number $Pr$ examined by two orders of magnitude from 0.7 up to 70. Significant effects of $Pr$ on the heat and momentum fluxes across the channel gap and on the mean temperature and velocity profile are observed. These effects can be described through a mixing length model coupling Monin–Obukhov (M–O) similarity theory and van Driest damping functions. We then employ M–O theory to formulate similarity scalings for various flow diagnostics for the stratified turbulence in the gap interior. The midchannel gap gradient Richardson number $Ri_{g}$ is determined by the length scale ratio $h/L$, where $h$ is the half-channel gap depth and $L$ is the Obukhov length scale. As $h/L$ approaches very large values, $Ri_{g}$ asymptotes to a maximum characteristic value of approximately 0.2. The buoyancy Reynolds number $Re_{b}\equiv \unicode[STIX]{x1D700}/(\unicode[STIX]{x1D708}N^{2})$, where $\unicode[STIX]{x1D700}$ is the dissipation, $\unicode[STIX]{x1D708}$ is the kinematic viscosity and $N$ is the buoyancy frequency defined in terms of the local mean density gradient, scales linearly with the length scale ratio $L^{+}\equiv L/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$, where $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ is the near-wall viscous scale. The flux Richardson number $Ri_{f}\equiv -B/P$, where $B$ is the buoyancy flux and $P$ is the shear production, is found to be proportional to $Ri_{g}$. This then leads to a turbulent Prandtl number $Pr_{t}\equiv \unicode[STIX]{x1D708}_{t}/\unicode[STIX]{x1D705}_{t}$ of order unity, where $\unicode[STIX]{x1D708}_{t}$ and $\unicode[STIX]{x1D705}_{t}$ are the turbulent viscosity and diffusivity respectively, which is consistent with Reynolds analogy. The turbulent Froude number $Fr_{h}\equiv \unicode[STIX]{x1D700}/(NU^{\prime 2})$, where $U^{\prime }$ is a turbulent horizontal velocity scale, is found to vary like $Ri_{g}^{-1/2}$. All these scalings are consistent with our numerical data and appear to be independent of $Pr$. The classical Osborn model based on turbulent kinetic energy balance in statistically stationary stratified sheared turbulence (Osborn, J. Phys. Oceanogr., vol. 10, 1980, pp. 83–89), together with M–O scalings, results in a parameterization of $\unicode[STIX]{x1D705}_{t}/\unicode[STIX]{x1D708}\sim \unicode[STIX]{x1D708}_{t}/\unicode[STIX]{x1D708}\sim Re_{b}Ri_{g}/(1-Ri_{g})$. With this parameterization validated through direct numerical simulation data, we provide physical interpretations of these results in the context of M–O similarity theory. These results are also discussed and rationalized with respect to other parameterizations in the literature. This paper demonstrates the role of M–O similarity in setting the mixing efficiency of equilibrated constant-flux layers, and the effects of Prandtl number on mixing in wall-bounded stratified turbulent flows.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Mixing: Mixing Turbulent Flows: Stratified turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 820 , 10 June 2017 , pp. 86 - 120
Check for updates
Copyright
© 2017 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

Armenio, V. & Sarkar, S. 2002 An investigation of stably stratified turbulent channel flow using large-eddy simulation. J. Fluid Mech. 459, 1–42.CrossRefGoogle Scholar
Barry, M. E., Ivey, G. N., Winters, K. B. & Imberger, J. 2001 Measurements of diapycnal diffusivities in stratified fluids. J. Fluid Mech. 442, 267–291.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, 113–133.CrossRefGoogle Scholar
Bewley, T. R. 2010 Numerical Renaissance: Simulation, Optimization, and Control. Renaissance Press (available at http://numerical-renaissance.com).Google Scholar
Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13, 1645–1651.CrossRefGoogle Scholar
Bouffard, D. & Boegman, L. 2013 A diapycnal diffusivity model for stratified environmental flows. Dyn. Atmos. Oceans 61–62, 14–34.CrossRefGoogle Scholar
Bradshaw, P. & Huang, G. P. 1995 The law of the wall in turbulent flow. Proc. R. Soc. Lond. A 451, 165–188.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.1974 An experiment on turbulence in a density stratified fluid. PhD thesis, Monash University, Victoria, Australia.Google Scholar
de Bruyn Kops, S. M. 2015 Classical scaling and intermittency in strongly stratified boussinesq turbulence. J. Fluid Mech. 775, 436–463.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
Caulfield, C. P., Tang, W. & Plasting, S. C. 2004 Reynolds number dependence of an upper bound for the long-time-averaged buoyancy flux in plane stratified Couette flow. J. Fluid Mech. 498, 315–332.CrossRefGoogle Scholar
Chung, D. & Matheou, G. 2012 Direct numerical simulation of stationary homogeneous stratified sheared turbulence. J. Fluid Mech. 696, 434–467.CrossRefGoogle Scholar
Davidson, P. A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
Davis, K. A. & Monismith, S. G. 2011 The modification of bottom boundary layer turbulence and mixing by internal waves shoaling on a barrier reef. J. Phys. Oceanogr. 41, 2223–2241.CrossRefGoogle Scholar
Deusebio, E., Caulfield, C. P. & Taylor, J. R. 2015 The intermittency boundary in stratified plane Couette flow. J. Fluid Mech. 781, 298–329; referred to in the text as DCT.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
van Driest, E. R. 1956 On turbulent flow near a wall. J. Aero. Sci. 23, 1007–1011.CrossRefGoogle Scholar
Eaves, T. S. & Caulfield, C. P. 2015 Disruption of SSP/VWI states by a stable stratification. J. Fluid Mech. 784, 548–564.CrossRefGoogle Scholar
Ellison, T. H. 1957 Turbulent transport of heat and momentum from an infinite rough plane. J. Fluid Mech. 2, 456–466.CrossRefGoogle Scholar
Fernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23, 455–493.CrossRefGoogle Scholar
Flores, O. & Riley, J. J. 2011 Analysis of turbulence collapse in stably stratified surface layers using direct numerical simulation. Boundary-Layer Meteorol. 139, 241–259.CrossRefGoogle Scholar
Foken, T. 2006 50 years of the Monin–Obukhov similarity theory. Boundary-Layer Meteorol. 119, 431–447.CrossRefGoogle Scholar
Galperin, B., Sukoriansky, S. & Anderson, P. S. 2007 On the critical Richardson number in stably stratified turbulence. Atmos. Sci. Lett. 8, 65–69.CrossRefGoogle Scholar
García-Villalba, M. & del Álamo, J. C. 2011 Turbulence modification by stable stratification in channel flow. Phys. Fluids 23, 045104.CrossRefGoogle Scholar
García-Villalba, M., Azagra, E. & Uhlmann, M. 2011a A numerical study of turbulent stably-stratified plane Couette flow. In High Performance Computing in Science and Engineering ’10 (ed. Nagel, W. E. et al. ), pp. 251–261. Springer.Google Scholar
García-Villalba, M., Azagra, E. & Uhlmann, M. 2011b Mixing efficiency in stably-stratified plane Couette flow. In Proceedings of the 7th International Symposium on Stratified Flows, Rome, Italy, August 2011. IAHR.Google Scholar
Holford, J. M. & Linden, P. F. 1999 Turbulent mixing in a stratified fluid. Dyn. Atmos. Oceans 30, 173–198.CrossRefGoogle Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512.CrossRefGoogle Scholar
Ivey, G. N., Winters, K. B. & Koseff, J. R. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40, 169–184.CrossRefGoogle Scholar
Karimpour, F. & Venayagamoorthy, S. K. 2014 A simple turbulence model for stably stratified wall-bounded flows. J. Geophys. Res. 119, 870–880.CrossRefGoogle Scholar
Karimpour, F. & Venayagamoorthy, S. K. 2015 On turbulent mixing in stably stratified wall-bounded flows. Phys. Fluids 27, 046603.CrossRefGoogle Scholar
Lilly, D. K. 1983 Stratified turbulence and the mesoscale variability of the atmosphere. J. Atmos. Sci. 40, 749–761.2.0.CO;2>CrossRefGoogle Scholar
Linden, P. F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn. 13, 3–23.CrossRefGoogle Scholar
Linden, P. F. 1980 Mixing across a density interface produced by grid turbulence. J. Fluid Mech. 100, 691–703.CrossRefGoogle Scholar
Lozovatsky, I. D. & Fernando, H. J. S. 2013 Mixing efficiency in natural flows. Phil. Trans. R. Soc. Lond. A 371, 20120213.Google ScholarPubMed
Maffioli, A., Brethouwer, G. & Lindborg, E. 2016 Mixing efficiency in stratified turbulence. J. Fluid Mech. 794, R3.CrossRefGoogle Scholar
Maffioli, A. & Davidson, P. A. 2015 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, 375–396.CrossRefGoogle Scholar
Mahrt, L. 2014 Stably stratified atmospheric boundary layers. Annu. Rev. Fluid Mech. 46, 23–45.CrossRefGoogle Scholar
Mashayek, A., Caulfield, C. P. & Peltier, W. R. 2013 Time-dependent, non-monotonic mixing in stratified turbulent shear flows: implications for oceanographic estimates of buoyancy flux. J. Fluid Mech. 736, 570–593.CrossRefGoogle Scholar
Mater, B. D. & Venayagamoorthy, S. K. 2014 The quest for an unambiguous parameterization of mixing efficiency in stably stratified geophysical flows. Geophys. Res. Lett. 41, 4646–4653.CrossRefGoogle Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496–508.CrossRefGoogle Scholar
Oglethorpe, R. L. F., Caulfield, C. P. & Woods, A. W. 2013 Spontaneous layering in stratified turbulent Taylor–Couette flow. J. Fluid Mech. 721, R3.CrossRefGoogle Scholar
Osborn, T. R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 83–89.2.0.CO;2>CrossRefGoogle Scholar
Park, Y. G., Whitehead, J. A. & Gnanadeskian, A. 1994 Turbulent mixing in stratified fluids: layer formation and energetics. J. Fluid Mech. 279, 279–311.CrossRefGoogle Scholar
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.CrossRefGoogle Scholar
Pham, H. T., Sarkar, S. & Winters, K. B. 2013 Large-eddy simulation of deep-cycle turbulence in an Equatorial Undercurrent model. J. Phys. Oceanogr. 43, 2490–2502.CrossRefGoogle Scholar
Phillips, O. M. 1972 Turbulence in a strongly stratified fluid – is it unstable? Deep-Sea Res. 19, 79–81.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
van Reeuwijk, M. & Hadžiabdić, M. 2015 Modelling high schmidt number turbulent mass transfer. Intl J. Heat Fluid Flow 51, 42–49.CrossRefGoogle Scholar
Riley, J. J. & de Bruyn Kops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15, 2047.CrossRefGoogle Scholar
Riley, J. J. & Lindborg, E. 2012 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.CrossRefGoogle Scholar
Rohr, J. & Van Atta, C. 1987 Mixing efficiency in stably stratified growing turbulence. J. Geophys. Res. 92, 5481–5488.Google 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. 2015 Diapycnal diffusivity, turbulent Prandtl number and mixing efficiency in Boussinesq stratified turbulence. J. Fluid Mech. 775, 464–500.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
Schlichting, H. & Gersten, K. 2003 Boundary-Layer Theory. Springer.Google Scholar
Scotti, A. 2015 Biases in Thorpe-scale estimates of turbulence dissipation. Part II: energetics arguments and turbulence simulations. J. Geophys. Res. 45, 2522–2543.Google Scholar
Scotti, A. & White, B. 2016 The mixing efficiency of stratified turbulent boundary layers. J. Phys. Oceanogr. 46, 3181–3191.CrossRefGoogle Scholar
Shih, L. H., Koseff, J. R., Ferziger, J. H. & Rehmann, C. R 2000 Scaling and parameterization of stratified homogeneous turbulent shear flow. J. Fluid Mech. 412, 1–20.CrossRefGoogle Scholar
Shih, L. H., Koseff, J. R., Ivey, G. N. & Ferziger, J. H. 2005 Parameterization of turbulent fluxes and scales using homogeneous sheared stably stratified turbulence simulations. J. Fluid Mech. 525, 193–214.CrossRefGoogle Scholar
Smyth, W. D. & Moum, J. N. 2013 Marginal instability and deep cycle turbulence in the eastern equatorial Pacific Ocean. Geophys. Res. Lett. 40, 6181–6185.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
Sukoriansky, S. & Galperin, B. 2013 An analytical theory of the buoyancy–Kolmogorov subrange transition in turbulent flows with stable stratification. Phil. Trans. R. Soc. Lond. A 371, 20120212.Google ScholarPubMed
Tang, W., Caulfield, C. P. & Kerswell, R. R. 2009 A prediction for the optimal stratification for turbulent mixing. J. Fluid Mech. 634, 487–497.CrossRefGoogle Scholar
Tastula, E.-M., Galperin, B., Sukoriansky, S., Luhar, A. & Anderson, P. 2015 The importance of surface layer parameterization in modeling of stable atmospheric boundary layers. Atmos. Sci. Lett. 16, 83–88.CrossRefGoogle Scholar
Taylor, J. R.2008 Numerical simulations of the stratified oceanic bottom boundary layer. PhD thesis, University of California, San Diego.Google Scholar
Taylor, J. R., Sarkar, S. & Armenio, V. 2005 Large eddy simulation of stably stratified open channel flow. Phys. Fluids 17, 116602.CrossRefGoogle Scholar
Thorpe, S. A. & Liu, Z. 2009 Marginal instability? J. Phys. Oceanogr. 39, 2373–2381.CrossRefGoogle Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.CrossRefGoogle Scholar
Venayagamoorthy, S. K. & Stretch, D. D. 2010 On the turbulent Prandtl number in homogeneous stably stratified turbulence. J. Fluid Mech. 644, 359–369.CrossRefGoogle Scholar
Walter, R. K., Squibb, M. E., Woodson, C. B., Koseff, J. R. & Monismith, S. G. 2014 Stratified turbulence in the nearshore coastal ocean: dynamics and evolution in the presence of internal bores. J. Phys. Oceanogr. 119, 8709–8730.Google Scholar
Wells, M., Cenedese, C. & Caulfield, C. P. 2010 The relationship between flux coefficient and entrainment ratio in density currents. J. Phys. Oceanogr. 40, 2713–2727.CrossRefGoogle Scholar
Wilson, J. M. & Venayagamoorthy, S. K. 2015 A shear-based parameterization of turbulent mixing in the stable atmospheric boundary layer. J. Atmos. Sci. 72, 1713–1726.CrossRefGoogle Scholar
Wyngaard, J. C. 2010 Turbulence in the Atmosphere. Cambridge University Press.CrossRefGoogle Scholar
Zhou, Q.2015 Far-field evolution of turbulence-emitted internal waves and Reynolds number effects on a localized stratified turbulent flow. PhD thesis, Cornell University, Ithaca, New York.Google Scholar