Energetics and mixing efficiency of lock-exchange gravity currents using simultaneous velocity and density fields

A series of laboratory experiments on energy-conserving gravity currents in a lock-exchange facility are conducted for a range of Reynolds numbers, $\text{Re}=\frac{U_{F}h}{\nu }=485–12270$, where $U_F$ is the front velocity of the current, $h$ the current depth, and $\nu$ the kinematic viscosity of the fluid. The velocity and density fields are captured simultaneously using a particle image velocimetry–planar laser induced fluorescence system. A moving average method is employed to compute the mean field and a host of turbulence statistics, namely, turbulent kinetic energy ($K$), shear production ($P$), buoyancy flux ($B$), and energy dissipation ($\epsilon$) during the slumping phase of the current. The subsequent findings are used to ascertain the quantitative values of mixing efficiency, ${\text{Ri}}_f$, Ozmidov length scale ($L_o$), Kolmogorov length scale ($L_{\kappa }$), and eddy diffusivities of momentum (${\kappa }_m$) and scalar (${\kappa }_{\rho }$). Two different forms of ${\text{Ri}}_f$ are characterized in this study, denoted by ${\text{Ri}}_f^I=\frac{B}{P}$ and ${\text{Ri}}_f^{II}=\frac{B}{B+\epsilon }$. The results cover the entire diffusive regime (3 $<{\text{Re}}_b<$ 10) and a portion of the intermediate regime (10 $<{\text{Re}}_b<$ 50), where ${\text{Re}}_b=\frac{\epsilon }{\nu N^2}$ is the buoyancy Reynolds number that measures the level of turbulence in a shear-stratified flow, with $N$ being the Brunt-Väisälä frequency. The variation of turbulence quantities, $\overline{P}\left(z\right)$, $\overline{B}\left(z\right)$, and $\overline{\epsilon }\left(z\right)$, along the depth of the current shows a marked increase at the interface of the ambient and current fluids, owing to the development of a shear-driven mixed layer. Based on the changes in the turbulence statistics and the length scales, it is inferred that the turbulence decays along the length of the current. The mixing efficiency monotonically increases in the diffusive regime (${\text{Re}}_b<10$) and is found to be $\overline{{\text{Ri}}_f^I}\approx$ 0.15 and $\overline{{\text{Ri}}_f^{II}}\approx$ 0.2 in the intermediate regime. Using the value of $\overline{{\text{Ri}}_f^{II}}$, the normalized eddy diffusivity of momentum is parameterized as $\frac{{\kappa }_m}{\nu .{\text{Ri}}_g}=1.2{\text{Re}}_b$, where ${\text{Ri}}_g$ is the gradient Richardson number, and the normalized eddy diffusivity of scalar is parameterized as $\frac{{\kappa }_{\rho }}{\nu }=0.2{\text{Re}}_b$.

Figure 1

Schematic of the experimental setup (not to scale). Isometric view of the setup, where 1 is the tank, 2 is the lock-exchange gate, 3 is the laser source illuminating the central portion of the tank, 4 and 5 are PIV and PLIF cameras, and 6 and 7 are high-pass and low-pass filters.

Figure 2

Qualitative image of a lock-exchange gravity current, moving from right to left. The image is obtained by collating snapshots recorded at different time instances. Blue represents ambient fluid (lighter), and red is the source fluid (denser). Intermediate colors are a result of mixing between the two fluids. Dashed white line shows the region where turbulence and mixing statistics are computed in the present study.

Figure 3

Density contours sampled from the body of the current for (a) $\text{Re}=485$, (b) $\text{Re}=4270$, and (c) $\text{Re}=12270$ qualitatively capturing different energetic regimes. The figure also shows qualitatively how the thickness of the mixing zone varies with $\text{Re}$. The color bar indicates the densities across the height of the current in $\text{g}{\text{cm}}^{-3}$.

Figure 4

Evolution of turbulent kinetic energy, $\overline{K}\left(t\right)$, in the body of the current for (a) $\text{Re}=485$, (b) $\text{Re}=4270$, and (c) $\text{Re}=12270$. The images of the current are placed underneath to visualize the relative location with respect to the head.

Figure 5

Evolution of turbulence statistics $\overline{P}\left(t\right)$, $\overline{B}\left(t\right)$ and $\overline{\epsilon }\left(t\right)$ for (a) $\text{Re}=485$, (b) $\text{Re}=4270$, and (c) $\text{Re}=12270$. Refer to Fig. 4 to visualize the relative location with respect to the head.

Figure 6

Average turbulence statistics, $\overline{P}\left(z\right)$, $\overline{B}\left(z\right)$, and $\overline{\epsilon }\left(z\right)$ across the depth of the current (a) $\text{Re}=485$, (b) $\text{Re}=4270$, and (c) $\text{Re}=12270$.

Figure 7

Evolution of Ozmidov scale, $\overline{L_O}\left(t\right)$, and Kolmogorov scale, $\overline{L_{\kappa }}\left(t\right)$, for (a) $\text{Re}=485$, (b) $\text{Re}=4270$, and (c) $\text{Re}=12270$. Refer to Fig. 4 to visualize the relative location with respect to the head.

Figure 8

Mixing efficiency ($\overline{{\text{Ri}}_f^I}$ and $\overline{{\text{Ri}}_f^{II}}$) as a function of buoyancy Reynolds number, ${\text{Re}}_b$. The magenta lines account for all the uncertainties that could arise in the calculations.

Figure 9

(a) Normalized eddy diffusivity of momentum (${\kappa }_m$), and the dashed line (- - -) represents 1.25 ${\text{Re}}_b$ [45]. (b) Normalized eddy diffusivity of scalar (${\kappa }_{\rho }$), and the dashed line (- - -) represents 0.2 ${\text{Re}}_b$ [25]. The eddy diffusivities are parameterized using the second definition of flux Richardson number, $\overline{{\text{Ri}}_f^{II}}$. The magenta line shows deviation ($\pm$ 10 $\%$) from Crawford's and Osborn's parametrizations.