Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-07-02T22:43:58.110Z Has data issue: false hasContentIssue false
  • English
  • Français

Transition to stably stratified states in open channel flow with radiative surface heating

Published online by Cambridge University Press:  09 February 2015

N. Williamson ,
S. W. Armfield ,
M. P. Kirkpatrick  and
S. E. Norris
N. Williamson*
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, NSW 2006, Australia
S. W. Armfield
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, NSW 2006, Australia
M. P. Kirkpatrick
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, NSW 2006, Australia
S. E. Norris
Affiliation:
Department of Mechanical Engineering, University of Auckland, Auckland, New Zealand
*
†Email address for correspondence: nicholas.williamson@sydney.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

Direct numerical simulations (DNS) of turbulent stratified flow in an open channel with an internal heat source following the Beer–Lambert law from the surface are used to investigate the transition from neutral to strongly stable flow. Our buoyancy bulk parameter is defined through the ratio of the domain height ${\it\delta}$ to $\mathscr{L}$ , a bulk Obukhov length scale for the flow. We cover the range ${\it\lambda}={\it\delta}/\mathscr{L}=0{-}2.0$ , from neutral conditions to the onset of the stable regime, with the Reynolds number range $Re_{{\it\tau}}=200{-}800$ , at a Prandtl number of 0.71. The result is a boundary layer flow where the effects of stratification are weak in the wall region but progressively stronger in the outer layer up to the free surface. At ${\it\lambda}\simeq 1$ the turbulent kinetic energy (TKE) budget is in local equilibrium over a region extending from the near-wall region to a free-surface affected region a distance $l_{{\it\nu}}$ from the surface, with $l_{{\it\nu}}/{\it\delta}\sim Re^{-1/2}$ . In this equilibrium region the flow can be characterised by the flux Richardson number $R_{f}$ and the local Obukhov length scale ${\it\Lambda}$ . At higher ${\it\lambda}$ local mixing limit conditions are observed over an extended region. At ${\it\lambda}=2$ the flux Richardson number approaches critical limit values of $R_{f,c}\simeq 0.18$ and gradient Richardson number $Ri_{c}\simeq 0.2$ . At high ${\it\lambda}$ , we obtain a flow field where buoyancy interacts with the smallest scales of motion and the turbulent shear stress and buoyancy flux are suppressed to molecular levels. We find that this regime can be identified in terms of the parameter $Re_{\mathscr{L},c}=\mathscr{L}u_{{\it\tau}}/{\it\nu}\lesssim 200{-}400$ (where $u_{{\it\tau}}$ is the friction velocity and ${\it\nu}$ the kinematic viscosity), which is related to the $L_{\ast }$ parameter of Flores and Riley (Boundary-Layer Meteorol., vol. 139 (2), 2011, pp. 241–259) and buoyancy Reynolds number $\mathscr{R}$ . With energetic equilibrium attained, the local buoyancy Reynolds number, $Re_{{\it\Lambda}}={\it\Lambda}\langle u^{\prime }w^{\prime }\rangle ^{1/2}/{\it\nu}$ , is directly related to the separation of the Ozmidov ( $l_{O}$ ) and Kolmogorov ( ${\it\eta}$ ) length scales in the outer boundary layer by $Re_{{\it\Lambda}}\simeq \mathscr{R}\equiv (l_{O}/{\it\eta})^{4/3}$ . The inner wall region has the behaviour $\mathscr{R}\sim Re_{\mathscr{L}}Re_{{\it\tau}}$ , in contrast to stratified boundary layer flows where the buoyancy flux is non-zero at the wall and $\mathscr{R}\sim Re_{\mathscr{L}}$ .

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Turbulent Flows: Stratified turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 766 , 10 March 2015 , pp. 528 - 555
Check for updates
Copyright
© 2015 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

Abe, H., Kawamura, H. & Matsuo, Y. 2001 Direct numerical simulation of a fully developed turbulent channel flow with respect to Reynolds number dependence. Trans. ASME: J. Fluids Engng 123, 382–393.Google Scholar
Armenio, V. & Sarkar, S. 2002 An investigation of stably stratified turbulent channel flow using large-eddy simulation. J. Fluid Mech. 459, 1–42.Google Scholar
Armfield, S. W., Norris, S. E., Morgan, P. & Street, R.2002 A parallel non-staggered Navier–Stokes solver implemented on a workstation cluster. In Proceedings of the Second International Conference on Computational Fluid Dynamics (ed. S. Armfield, P. Morgan & K. Srinivas), pp. 30–45. Springer.Google 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.Google Scholar
Bormans, M. & Webster, I. T. 1997 A mixing criterion for turbid rivers. Environ. Model. Softw. Environ. Data News 12 (4), 329–333.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
Businger, J. A., Wyngaard, J. C., Izumi, Y. & Bradley, E. F. 1971 Flux-profile relationships in the atmospheric surface layer. J. Atmos. Sci. 28, 181–189.Google Scholar
Calmet, I. & Magnaudet, J. 2003 Statistical structure of high-Reynolds-number turbulence close to the free surface of an open-channel flow. J. Fluid Mech. 474, 355–378.CrossRefGoogle Scholar
Chung, D. & Matheou, G. 2012 Direct numerical simulation of stationary homogeneous stratified sheared turbulence. J. Fluid Mech. 696, 434–467.CrossRefGoogle Scholar
Deusebio, E., Schlatter, P., Brethouwer, G. & Lindborg, E. 2011 Direct numerical simulations of stratified open channel flows. J. Phys. Conf. Ser. 318, 022009.CrossRefGoogle Scholar
Dillon, T. M. & Caldwell, D. R. 1980 The Batchelor spectrum and dissipation in the upper ocean. J. Geophys. Res. 85 (C4), 1910–1916.Google Scholar
Ellison, T. H. 1957 Turbulent transport of heat and momentum from an infinite rough plane. J. Fluid Mech. 2, 456–466.CrossRefGoogle Scholar
Flores, O. & Riley, J. J. 2011 Analysis of turbulence collapse in the stably stratified surface layer using direct numerical simulation. Boundary-Layer Meteorol. 139 (2), 241–259.CrossRefGoogle Scholar
Garcia-Villalba, M. & del Alamo, J. C. 2011 Turbulence modification by stable stratification in channel flow. Phys. Fluids 23 (4), 045104.Google Scholar
Garg, R. P., Ferziger, J. H., Monismith, S. G. & Koseff, J. R. 2000 Stably stratified turbulent channel flows. I. Stratification regimes and turbulence suppression mechanism. Phys. Fluids 12 (10), 2569–2594.CrossRefGoogle Scholar
Gargett, A. E., Osborn, T. R. & Nasmyth, P. W. 1984 Local isotropy and the decay of turbulence in a stratified fluid. J. Fluid Mech. 144, 231–280.Google Scholar
Garrett, C. J. R., Keeley, J. R. & Greenberg, D. A. 1978 Tidal mixing versus thermal stratification in the Bay of Fundy and Gulf of Maine. Atmos.-Ocean 16 (4), 403–423.Google 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
Gonzalez-Juez, E. D., Kerstein, A. R. & Shih, L. H. 2011 Vertical mixing in homogeneous sheared stratified turbulence: a one-dimensional-turbulence study. Phys. Fluids 23 (5), 055106.Google Scholar
Grachev, A. A., Andreas, E. L., Fairall, C. W., Guest, P. S. & Persson, P. O. G. 2013 The critical Richardson number and limits of applicability of local similarity theory in the stable boundary layer. Boundary-Layer Meteorol. 147 (1), 51–82.CrossRefGoogle Scholar
Grachev, A. A., Fairall, C. W., Persson, P. O. G., Andreas, E. L. & Guest, P. S. 2005 Stable boundary-layer scaling regimes: the SHEBA data. Boundary-Layer Meteorol. 116 (2), 201–235.CrossRefGoogle Scholar
Handler, R. A., Saylor, J. R., Leighton, R. I. & Rovelstad, A. L. 1999 Transport of a passive scalar at a shear-free boundary in fully developed turbulent open channel flow. Phys. Fluids 11, 2607–2625.Google Scholar
Handler, R. A., Swean, T. F., Leighton, R. I. & Swearingen, J. D. 1993 Length scales and the energy balance for turbulence near a free surface. AIAA J. 31 (11), 1998–2007.Google Scholar
Hearn, C. J. 1985 On the value of the mixing efficiency in the Simpson-Hunter $h/u^{3}$ criterion. Dtsch. Hydrogr. Z. 38 (3), 133–145.Google Scholar
Holloway, P. E. 1980 A criterion for thermal stratification in a wind-mixed system. J. Phys. Oceanogr. 10, 861–869.2.0.CO;2>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.Google Scholar
Hunt, J. C. R. 1984 Turbulence structure and turbulent diffusion near gas–liquid interfaces. In Gas Transfer at Water Surfaces (ed. Brutseart, W. & Jirka, G. H.), Water Science and Technology Library, vol. 2, pp. 67–82. Springer.Google Scholar
Hunt, J. C. R. & Graham, J. M. R. 1978 Free-stream turbulence near plane boundaries. J. Fluid Mech. 84 (2), 209–235.Google Scholar
Itsweire, E. C., Koseff, J. R., Briggs, D. A. & Ferziger, J. H. 1993 Turbulence in stratified shear flows: implications for interpreting shear-induced mixing in the ocean. J. Phys. Oceanogr. 23 (7), 1508–1522.2.0.CO;2>CrossRefGoogle Scholar
Ivey, G. N. & Imberger, J. 1991 On the nature of turbulence in a stratified fluid. Part I: the energetics of mixing. J. Phys. Oceanogr. 21, 650–658.2.0.CO;2>CrossRefGoogle Scholar
Ivey, G. N., Winters, K. B. & Koseff, J. R. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40 (1), 169–184.Google Scholar
Komori, S., Nagaosa, R., Murakami, Y., Chiba, S., Ishii, K. & Kuwahara, K. 1993 Direct numerical simulation of three-dimensional open-channel flow with zero-shear gas–liquid interface. Phys. Fluids A 5, 115–125.Google Scholar
Komori, S., Ueda, H., Ogino, F. & Mizushina, T. 1983 Turbulence structure in stably stratified open-channel flow. J. Fluid Mech. 130, 13–26.CrossRefGoogle Scholar
Kullenberg, G. E. B. 1976 On vertical mixing and the energy transfer from the wind to the water. Tellus 28 (2), 159–165.Google Scholar
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207–242.Google Scholar
Monin, A. S. 1970 The atmospheric boundary layer. Annu. Rev. Fluid Mech. 2, 225–250.CrossRefGoogle Scholar
Moser, R. D., Kim, J. & Mansour, N. N. 1999 Direct numerical simulation of turbulent channel flow up to $Re=590$ . Phys. Fluids 11, 943–945.CrossRefGoogle Scholar
Nagaosa, R. & Handler, R. A. 2003 Statistical analysis of coherent vortices near a free surface in a fully developed turbulence. Phys. Fluids 15 (2), 375–394.Google Scholar
Nezu, I. & Rodi, W. 1986 Open-channel flow measurements with a laser Doppler anemometer. J. Hydraul. Engng 112 (5), 335–355.Google Scholar
Nieuwstadt, F. T. M. 1984 The turbulent structure of the stable, nocturnal boundary layer. J. Atmos. Sci. 41 (14), 2202–2216.2.0.CO;2>CrossRefGoogle Scholar
Nieuwstadt, F. T. M. 2005 Direct numerical simulation of stable channel flow at large stability. Boundary-Layer Meteorol. 116 (2), 277–299.CrossRefGoogle Scholar
Osborn, T. R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 83–89.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google 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.Google Scholar
Simpson, J. H., Allen, C. M. & Morris, N. C. G. 1978 Fronts on the continental shelf. J. Geophys. Res. 83 (C9), 4607–4614.Google Scholar
Simpson, J. H. & Hunter, J. R. 1974 Fronts in the Irish sea. Nature 250, 404–406.Google Scholar
Smyth, W. D. & Moum, J. N. 2000 Length scales of turbulence in stably stratified mixing layers. Phys. Fluids 12, 1327–1342.Google Scholar
Sorbjan, Z. 1986 On similarity in the atmospheric boundary layer. Boundary-Layer Meteorol. 34 (4), 377–397.Google Scholar
Sorbjan, Z. & Grachev, A. A. 2010 An evaluation of the flux-gradient relationship in the stable boundary layer. Boundary-Layer Meteorol. 135 (3), 385–405.Google Scholar
Taylor, J. R., Sarkar, S. & Armenio, V. 2005 Large eddy simulation of stably stratified open channel flow. Phys. Fluids 17 (11), 116602.Google Scholar
Wang, L. & Lu, X.-Y. 2005 Large eddy simulation of stably stratified turbulent open channel flows with low-to high-Prandtl number. Intl J. Heat Mass Transfer 48 (10), 1883–1897.Google Scholar
Wiel, B. J. H., Moene, A. F., Ronde, W. H. & Jonker, H. J. J. 2008 Local similarity in the stable boundary layer and mixing-length approaches: consistency of concepts. Boundary-Layer Meteorol. 128 (1), 103–116.Google Scholar
Zonta, F., Onorato, M. & Soldati, A. 2012 Turbulence and internal waves in stably-stratified channel flow with temperature-dependent fluid properties. J. Fluid Mech. 697, 175–203.CrossRefGoogle Scholar