Review Article

Stably Stratified Wall-Bounded Turbulence

[+] Author and Article Information
Francesco Zonta

Institute of Fluid Mechanics and
Heat Transfer,
Vienna University of Technology,
Getreidemarkt 9,
Vienna 1060, Austria
e-mail: francesco.zonta@tuwien.ac.at

Alfredo Soldati

Institute of Fluid Mechanics and
Heat Transfer,
Vienna University of Technology,
Getreidemarkt 9,
Vienna 1060, Austria;
University of Udine,
Udine 33100, Italy
e-mail: alfredo.soldati@tuwien.ac.at

1Corresponding author.

Manuscript received March 16, 2018; final manuscript received July 10, 2018; published online August 9, 2018. Assoc. Editor: Jörg Schumacher.

Appl. Mech. Rev 70(4), 040801 (Aug 09, 2018) (17 pages) Paper No: AMR-18-1036; doi: 10.1115/1.4040838 History: Received March 16, 2018; Revised July 10, 2018

Stably stratified wall-bounded turbulence is commonly encountered in many industrial and environmental processes. The interaction between turbulence and stratification induces remarkable modifications on the entire flow field, which in turn influence the overall transfer rates of mass, momentum, and heat. Although a vast proportion of the parameter range of wall-bounded stably stratified turbulence is still unexplored (in particular when stratification is strong), numerical simulations and experiments have recently developed a fairly robust picture of the flow structure, also providing essential ground for addressing more complex problems of paramount technological, environmental and geophysical importance. In this paper, we review models used to describe the influence of stratification on turbulence, as well as numerical and experimental methods and flow configurations for studying the resulting dynamics. Conclusions with a view on current open issues will be also provided.

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Grahic Jump Location
Fig. 1

The structure of a weakly/moderately stratified wall-bounded turbulent flow. Close to the boundary, classical boundary layer turbulence is sustained. Farther from the wall, a buoyancy dominated region characterized by the presence of IGWs is observed. Contour maps of the temperature field are used for visualization purposes. Trajectories of Lagrangian tracers randomly released in the flow and colored by the magnitude of their turbulent kinetic energy are also shown. The flow direction is explicitly indicated.

Grahic Jump Location
Fig. 2

Panel (a) vector plots on a yz cross section of a stratified channel (only half of the channel, from the boundary up to the channel center, is shown); (b) Wall-normal behavior of the Ozmidov scale (Lo/h), Kolmogorov scale (9η/h), and distance from the boundary (Lz/h, taken here as representative for the largest turbulence scales). Results from Ref. [36].

Grahic Jump Location
Fig. 3

A parameter space (ΔT, h) of wall-bounded stratified turbulence with the different numerical approach that can be used for its description. Panel (a): air; panel (b): water. In this schematic, the solid lines indicate approximately the point at which the basic Oberbeck–Boussinesq model begins to fail, and more complex non-Oberbeck–Boussinesq models (both incompressible NOB or low-Mach) must be used. The dashed line indicates the point at which the thermodynamic Boussinesq model should be used. Specific indication of the parameter (ε) that describes each threshold line is also explicitly given. The label inside each region indicates the corresponding numerical approach according to the following color-code: Oberbeck–Boussinesq, low-Mach number, incompressible NOB, and thermodynamic Boussinesq. Some examples of the most suitable approaches to be used for flows of interest in environmental and industrial applications are provided in the following. In the nocturnal boundary layer, for example, h ≃ 102/103m and ΔT ≃ 5 °C. In this case, the thermodynamic Boussinesq model would be appropriate. However, if h > 103m, a low-Mach number approach would be recommended. In the deep ocean, h ≃ 103m and ΔT ≃ 2 °C, whereas in the upper ocean, h ≃ 102m and ΔT ≃ 10  °C [62]. In both cases, an incompressible NOB approach is required. In industrial heat transfer processes, typical sizes are h ≃ 1 m, whereas ΔT are usually larger than in environmental applications. For air, temperature gradients can easily be ΔT = 10/100 °C, while for water ΔT can achieve few tens, in particular for high heat flux cooling technologies [63]. In these latter cases, a low Mach number approach (air) and an incompressible NOB approach (water) is recommended.

Grahic Jump Location
Fig. 4

Structure of the (Reτ − Riτ) space diagram for wall bounded stratified flows. Circles represent the critical Riτ (neutral curve) obtained from the linear stability analysis [68] and recast to fit with the present parameter space. The fitting of the neutral curve proposed here has the following form: log(Riτ,cr)=m·[log(Reτ)]b+n·[log(Reτ)−d]a+c, where the value of the parameters is a = −0.1843, b = 1.047, c = 1.914, d = 1.927, m = 1.651, and n = −2.204. The shaded area in the parameter space corresponds to strongly stratified turbulent flow conditions (appearance of laminar patches in the near wall region). The strongly stratified laminar and the weakly stratified turbulent regions are also explicitly indicated. Symbols below the neutral curve represent a collection of numerical simulations of wall-bounded stratified turbulence [6,36,45,6971]. Note that, in the context of stably stratified turbulence, the term subcritical (resp. supercritical) condition refers to a flow characterized by a lower-than-critical (resp. larger-than-critical) value of the shear Richardsons number Riτ, i.e., falling below (resp. above) the neutral curve.

Grahic Jump Location
Fig. 5

Time measurements of the temperature as a function of the distance from the bottom boundary (h.a.b) during an experimental campaign focused on the bottom oceanic boundary layer. (Reproduced with permission from Cimatoribus and van Haren [125]. Copyright 2015 by Cambridge University.)

Grahic Jump Location
Fig. 6

Contour map of the streamwise velocity fluctuations on a xy plane in the near wall region (z+ ≃ 15) for Reτ = 550. Panel (a): unstratified flow; Panel (b): stratified case at Riτ = 480. (Reproduced with permission from García-Villalba and Álamo [45]. Copyright 2011 by AIP Publising.)

Grahic Jump Location
Fig. 7

Contour map of density (panel a) and wall normal velocity fluctuations on a horizontal plane at the center of the channel for Reτ = 550 and Riτ = 480. The black solid line indicates the trace ∂ρ/∂x = 0 obtained after smoothing the temperature field by 2D cutoff filter. (Reproduced with permission from García-Villalba and Álamo [45]. Copyright 2011 by AIP Publishing.)

Grahic Jump Location
Fig. 8

Behavior of the phase angle ϕ() between the vertical velocity w and the temperature fluctuations θ as a function of the normalized wavenumber in the streamwise direction kxδ. Measurements are taken at different locations from the wall (y+) for moderately stratified conditions. (Reproduced with permission from Iida et al. [75]. Copyright 2002 by Elsevier.)

Grahic Jump Location
Fig. 9

Contour map of the streamwise velocity ux on a horizontal parallel plane located at 10 wall units from the wall. Panels are as follows: panel (a) Reτ = 80 and Riτ = 0; panel (b) Reτ = 113 and Riτ = 38; panel (c) Reτ = 192 and Riτ = 273; panel (d) Reτ = 334 and Riτ = 985. (Reproduced with permission from Brethouwer et al. [70]. Copyright 2012 by Cambridge University.)

Grahic Jump Location
Fig. 10

Panel (a) behavior of the friction factor Cf as a function of the shear Richardson number Riτ. Panel (b) behavior of the Nusselt number, Nu, as a function of the shear Richardson number Riτ. Data are gathered both from numerical [6,35,44,45,116] (filled symbols) and experimental studies [119,142] (open symbols).

Grahic Jump Location
Fig. 11

Distribution of density ρ, viscosity μ and Prandtl number Pr of supercritical CO2 (sCO2) in a cross section of an annular pipe heated from the inner wall and cooled from the outer wall. Panel (a) refers to the forced convection case (no buoyancy); panel (b) refers to the mixed convection case (buoyancy is accounted). (Reproduced with permission from Peeters et al. [70]. Copyright 2016 by Cambridge University.)

Grahic Jump Location
Fig. 12

Summary points at a glance

Grahic Jump Location
Fig. 13

Future challenges and perspectives at a glance



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