Turbulent entrainment into natural gravity-driven flows

Abstract

Observations of entrainment into natural gravity-driven flows on sloping surfaces are described. It is shown that the laboratory-based entrainment law of Ellison & Turner (1959), which is often used for modelling of atmospheric and oceanic flows, underestimates the entrainment rates substantially, arguably due to the fact that the laboratory flows have been conducted at Reynolds numbers (Re . 10 3 ) below what is required for mixing transition (Re ∼ 10 3 -10 4 ) whereas natural flows occur at much higher Reynolds numbers (Re ∼ 10 7 ). A new entrainment law of the form E ∼ Ri −3/4 is proposed for the atmospheric Richardson number range 0.15 < Ri < 1.5. In contrast to the laboratory observation that entrainment ceases at Ri = 0.8, field observations show continuous entrainment over the entire Richardson number range.

Mixing in a density-driven current flowing down a slope in a rotating fluid

“…The increase in entrainment parameter with increasing Froude number shown in figure 7 is a robust result previously found in laboratory experiments of both rotating (CWAO; Wells 2007) and non-rotating dense currents (Ellison & Turner 1959;Wells & Wettlaufer 2005), and in theoretical (Turner 1986) as well as observational studies (Price & Barringer 1994;Dallimore, Imberger & Ishikawa 2001;Mauritzen et al 2005;Princevac, Fernando & Whiteman 2005;Arneborg et al 2007) (figure 12).…”

Dynamics of the head of gravity currents

“…In Fig. 4, entrainment parameter obtained in katabatic flows [52], with Re ∼ 10 7 , is always higher than values obtained in continuously fed gravity currents in laboratory by Ellison and Turner [20] with Re ∼ 10 3 . It is important to note that entrainment occurs under subcritical conditions (Fr < 1), which is supported by oceanic observations, by our experimental results and from Cenedese et al [17] (cf.…”

“…Previous experimental results herein presented concern continuously fed gravity currents developing over smooth sloping beds, in both rotating [15,17,62] and non rotating [4,20] experimental apparatus. Field data was obtained in Lake Ogawara [18], Mediterrean [51], Denmark Strait [24], Faroe Bank Channel [39], Baltic Sea [8] and in atmospheric katabatic flows [52]. Figure 4 also presents well-known entrainment parameterizations proposed by Ellison and Turner [20], for conservative gravity currents, and by Parker et al [48], based on both conservative and non-conservative currents.…”

“…They predict the entrainment parameter to be negligible for high Richardson numbers (Ri > 0.8). Their work motivated the development of further research on mixing in gravity currents ( [17,16,22,26,48,52] among many others), being the parameterization of mixing in dense currents still an area of active research nowadays. The influence of Richardson number, or densimetric Froude number, on the entrainment parameter, as well as the slope of the bed, have been considered since the early works.…”

“…The influence of Richardson number, or densimetric Froude number, on the entrainment parameter, as well as the slope of the bed, have been considered since the early works. Only recently, the dependence on Reynolds number was also investigated and seen to be fundamental in the prediction of entrainment [15,21,52], especially in natural flows with high Reynolds numbers (Re ∼ 10 7 ) where results deviate substantially from the laboratory-based entrainment laws, usually obtained for Re ∼ 1, 000. In Adduce et al [2] a two-layer shallow water model with entrainment was developed and tested by laboratory experiments with lock-exchange gravity currents in which both initial density and initial saline volume in the lock were varied.…”