Sheared stratified turbulence driven by Kolmogorov flow

Abstract

We investigate three-dimensional turbulence in a stably stratified fluid driven by a vertically sheared Kolmogorov flow using direct numerical simulations of the Boussinesq equations. As stratification increases, mean profiles evolve toward piecewise-linear shapes while layered density structures emerge, with sharp interfaces separating well-mixed bulk layers. These highly stable interfaces form in the low-shear regions of the mean velocity profile and tend to promote flow relaminarisation, while shear-generated turbulence persists in the bulk layers. We analyse turbulent fluctuations, buoyancy transport and its spatial organisation, and flow stability via profiles of the gradient Richardson number $Ri_g$. The Richardson number in the bulk layers remains of order unity or less, $Ri_g \lesssim 1$, so that efficient turbulent shear production can take place there. Mixing efficiency analysis shows that the Nusselt number scales with the buoyancy Reynolds number $Re_b$ as $Nu = 1 + ΓRe_b$ (with $Γ= ε_p / ε$), with the data collapsing onto a robust master curve and roughly following a power-law $Nu \sim Re_b^{0.8}$. Further increase of stratification leads to a temporally intermittent turbulent regime, characterised by quasi-periodic bursts. We propose that the transition from stationary turbulence to this temporally intermittent regime is controlled by the buoyancy Reynolds number and highlight the mechanisms disrupting the turbulence and layered structures.

Figures (14)

• Figure 1: Schematic of system under study. The numerical domain of the main DNS runs is a cube of size $L=2\pi$, on which the external force $\bm{f}(z)$ and the linear background density profile $\rho_0-\gamma z$ are shown. The external force is steady in time and will generate a mean flow $\overline{\bm{u}}(z)$ oriented in the same direction (the $x$-direction) and with a similar modulation along the $z$-direction.
• Figure 2: Section visualisations of density perturbation $\theta$ (upper row) and kinetic energy dissipation rate $\varepsilon$ (lower row) in the plane $(x,z)$ and $y=L/2$. Resolution $M=512$.
• Figure 3: Vertical profiles of mean flow: longitudinal velocity $\overline{u}(z)$ (upper left panel), spanwise velocity $\overline{v}(z)$ (upper right), density perturbation $\overline{\theta}(z)$ (lower left) and total density $\overline{\theta}(z)-z$ (lower right) across the runs at resolution $M=512$.
• Figure 4: Left panel: Evolution of the mean-flow amplitude $U$ (red circles) and input power $P_{\rm in}$ (blue squares) as a function of $1/Fr_F$ for all DNS runs. The data are shown in dimensionless form, with linear fits included. Right panel: Density amplitude $\Theta$ (normalized by $K = 1$) as a function of $1/Fr_F$. At low $1/Fr_F$ (weak stratification), $\Theta$ follows a power law, $\Theta \propto Fr_F^\beta$, with $\beta = 1.2$ (dashed black). At high $1/Fr_F$ (strong stratification), separate power-law curves are observed for the two resolutions: $M=512$ with $\beta = 0.3$ (solid black) and $M=256$ with $\beta = 0.4$ (dotted gray). Other fit parameters are omitted for clarity.
• Figure 5: Left panel: mean flow amplitude, $U/U_F$, and RMS turbulent velocity, $|\bm{u}'|_{\rm rms}/U_F$, versus Reynolds number, $Re=U/\nu K$. The empty symbols correspond to the data of musacchio2014, which was converted to the present form using the values given in Table 1 of musacchio2014. Right panel: RMS turbulent velocity components, normalized by $|\bm{u}'|_{\rm rms}$, versus $Re$. Isotropy of the turbulent velocity components would give a value of 1 for these normalized quantities.