On the mixing models for stratified flows subjected to concomitant stable and unstable stratifications

ABSTRACT

The concurrence of stable and unstable stratification in stratified flows leads to dramatically different features of turbulent mixing. This unique flow is experimentally studied by introducing a horizontal jet of dense fluid into a pool of light fluid. The buoyancy flux from simultaneous velocity–density measurements is an indicator for competition between a stabilising mechanism and another destabilising mechanism. The difference of mixing efficiency, quantified by flux Richardson number Ri_f, between the (un)stable stratification is mainly contributed by the large-scale mixing. The behaviour of Ri_f can be modelled by the gradient Richardson number Ri_g linearly in the low-Ri case and nonlinearly in the high-Ri case (especially in a region where the counter-gradient flux emerges). The turbulent diapycnal diffusivity, quantifying the combined effect of reversible and irreversible mixing processes, increases as the buoyancy Reynolds number Re_b increases only when Re_b is large. The irreversible mixing diffusivity, which quantifies the sole irreversible mixing process, increases linearly as the turbulent Péclet number with the data points from the (un)stable stratification overlapped. The turbulent Prandtl number approaches 0.75 as Ri_g approaches zero, but does not show clear dependence on Ri_g in the examined regime.

KEYWORDS:

• Turbulent mixing
• mixing modelling

Acknowledgment

The authors would like to thank the anonymous reviewers for their constructive comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. The computational cost of implementing DNS of the same stratified jet is estimated by the procedure in Section 9.1.2 of Ref. [42], where the spatial resolution of DNS is set to the Batchelor length scale (Sc ≃ 600 in this study). The total number of computation nodes (N) along one spatial direction is estimated by N ∼ 1.6k^3/2/(εη_b) ≃ 0.4R^3/2_λSc^1/2. The total number of time steps is M = 120k^3/2/(πεη_b) ≃ 9.2 R^3/2_λSc^1/2. Thus the total floating-point operations are N³M ∼ 0.55 R⁶_λSc² in a box domain with N³ nodes. Using a computer of 1 gigaflop, the computation time to implement the DNS is thus T_G ≃ (R_λ/70)⁶Sc². For the low-Ri case in the present study (R_λ ≃ 100), T_G ≃ 4, 680, 000 days.