Laminar-turbulent cycles in inclined lock-exchange flows

Yukie Tanino, Frédéric Moisy, and Jean-Pierre Hulin

Phys. Rev. E 85, 066308 – Published 11 June 2012

Abstract

We consider strongly confined, stably stratified shear flows generated as a lock exchange in a tube inclined at an angle of $θ = 45^{\circ}$ . This paper focuses on a transitional regime, in which the flow alternates between two distinct states: laminar, parallel shear flow and intense transverse motion characteristic of turbulence. Laminar-turbulent cycles were captured at Atwood numbers $At \equiv (ρ_{2} - ρ_{1}) / (ρ_{1} + ρ_{2})$ ranging from $2.45 \times 10^{- 3}$ to $4.0 \times 10^{- 3}$ , where $(ρ_{1}, ρ_{2})$ are the initial densities of the two fluids, with multiple cycles observed at $At = 2.55 \times 10^{- 3}$ . The evolution of the density and velocity fields in these flows was measured simultaneously using laser-induced fluorescence and particle image velocimetry. During each laminar-turbulent cycle, the axial velocity exhibits a distinctive ramp-cliff pattern, indicating that the flow accelerates as it relaminarizes, then decelerates rapidly as the Kelvin-Helmholtz billows break down. Within the range of experimental conditions, transverse stratification does not directly determine the onset of instability. Instead, the data suggest that a necessary criterion for the onset of instability is for the local Reynolds number to exceed 2200, with only a weak dependence on the Richardson number.

Received 13 August 2011

DOI:https://doi.org/10.1103/PhysRevE.85.066308

Authors & Affiliations

Yukie Tanino^*, Frédéric Moisy^†, and Jean-Pierre Hulin^‡

Université Pierre et Marie Curie-Paris 6, Université Paris-Sud, CNRS, Laboratoire FAST, Bât. 502, Campus Universitaire, Orsay F-91405, France

^*ytanino@alum.mit.edu; currently in the Department of Earth Science and Engineering, Imperial College London.
^†moisy@fast.u-psud.fr
^‡hulin@fast.u-psud.fr

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Vol. 85, Iss. 6 — June 2012

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Figure 1
Central section of the experimental setup. Not to scale.Reuse & Permissions
Figure 2
The temporal evolution of $u$ [cm s $^{- 1}$ ] at selected $x$ (a)–(d) and ${〈 w^{2} 〉}_{x}^{1 / 2}$ [cm s $^{- 1}$ ] (e)–(h). From top to bottom, the figures show runs [(a), (e)] A, [(b), (f)] C, [(c), (g)] D, and [(d), (h)] E in order of increasing $At$ . The horizontal axes each extend from $t = 10$ to 260 s. The same color map is used for (a)–(d) and (e)–(h), which is displayed to the right of (a) and (e).Reuse & Permissions
Figure 3
Turbulent fraction $γ$ [Eq. (2), $w_{c} = 0.12$ cm s $^{- 1}$ ] as a function of $At$ . The temporal average of $I (t)$ was taken over $50 < t < 200$ s. The dashed line is intended only as a guide to the eye. The upper and lower limits of the vertical bars correspond to $w_{c} = 1.08$ and 1.32 cm s $^{- 1}$ , respectively. Where bars are not visible, the variation is smaller than the marker. The two data points at $At = 1.19 \times 10^{- 2}$ coincide exactly and cannot be distinguished in the figure.Reuse & Permissions
Figure 4
Temporal evolution of (a) $Δ U$ , (b) ${〈 w^{2} 〉}_{x}^{1 / 2} (0, t)$ , (c) the transverse separation between the extrema of ${〈 u 〉}_{x}$ , (d) $- {\partial {〈 \tilde{ρ} 〉}_{x} / \partial \tilde{z}|}_{0}$ , and (e) the instantaneous Richardson number, $Ri$ [Eq. (14)], in run C. The three ramp-cliff cycles (and the beginning of the fourth) are numbered at the top. ${\tilde{t}}_{a}^{i}$ $(i = 1, 2, 3, . . .)$ corresponds to the arrival of relatively unmixed fluid that initiates the ramp phase (solid line); ${\tilde{t}}_{b}^{i}$ , ${\tilde{t}}_{c}^{i}$ , and ${\tilde{t}}_{d}^{i}$ correspond to the onset of instability (dash-dotted), the onset of the breakdown of billows (dashed), and the local minima in $Δ U$ (dotted). ${\tilde{t}}_{d}^{1}$ coincides with ${\tilde{t}}_{a}^{2}$ . The oblique, solid line after ${\tilde{t}}_{a}^{i}$ in (a) depicts the theoretical acceleration of a “free-fall” [Sec. 3b3, Eq. (9)]. The horizontal line in (b) depicts the threshold $w_{c} = 0.12$ cm s $^{- 1}$ [Eq. (2)]. Horizontal lines in (c) depict ${\tilde{z}}^{*}$ in laminar flows $(2 {\tilde{z}}^{*} = 0.60)$ and in uniformly turbulent flows ( $2 {\tilde{z}}^{*} = 0.76$ ).Reuse & Permissions
Figure 5
Transverse profiles of ${〈 u 〉}_{x} (z, t) / Δ U (t)$ and ${〈 \tilde{ρ} 〉}_{x} (z, t) - {〈 \tilde{ρ} 〉}_{x} (0, t)$ at the end of a relaminarization phase [(a), (c)] and during the subsequent turbulence decay phase [(b), (d)] in run C. $\tilde{t} = 28.1$ [solid blue line (a), (c)], $30.9$ [solid blue line (b), (d)], $39.2$ [dotted red line (a), (c)], $43.0$ [dotted red line (b), (d)]. Thick dashed lines represent the corresponding profiles in permanently laminar [run A (a), (c)] and uniformly turbulent flows [run E (b), (d)]; ${〈 u 〉}_{x}$ was temporally averaged over $120 \leq t \leq 199$ s (run A) and $25 \leq t < 107$ s (run E); $t = 199$ s (c) and $t = 106$ s (d).Reuse & Permissions
Figure 6
$(Re, Ri)$ at the onset of instability in runs B ( $▴$ ), C ( $•$ ), and D ( $▪$ ). The thick red line is the least-squares best-fit power function to all five points. Thin solid lines depict the evolution of $Re$ and $Ri$ during each relaminarization phase. The thick dashed line depicts run A ( $t = 120 - 199$ s), in which the flow remained laminar. Open markers depict $({Re}_{i}, {Ri}_{i})$ in inclined rectangular tubes at $At = 2.5 \times 10^{- 3}$ ( $\circ$ ) and $7.5 \times 10^{- 3}$ ( $⋄$ ) reported by Defina et al. (Ref. [17], $θ = 80^{\circ}$ to $88 . 4^{\circ}$ ), where ${Ri}_{i}$ was calculated taking $\sin θ = 0.99$ and ${Re}_{c}$ was estimated as $2 {Re}_{i}^{DLS}$ , where ${Re}_{i}^{DLS}$ is the Reynolds number as defined in Ref. [17]. Thick gray lines are the corresponding least-squares best-fit power functions.Reuse & Permissions

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