Origin of lobe and cleft at the gravity current front

In this Rapid Communication, the Rayleigh-Taylor instability (RTI) along the density interfaces of gravity current fronts is analyzed. Both the location and the spanwise wave number of the most unstable mode determined by the local dispersion relation agree with those of the strongest perturbation obtained from numerical simulations, suggesting that the original formation mechanism of lobes and clefts at the current front is RTI. Furthermore, the predictions of the semi-infinite RTI model, i.e., the original dominating spanwise wave number of the Boussinesq current substantially depends on the Prandtl number and has a 1/3 scaling law with the Grashof number, are confirmed by the three-dimensional numerical simulations.

Figure 1

(a) The dimensionless temperature $T$ at the current front. The solid curve represents $T=0.5$. (b) The front velocity, the velocity of the forefront with $T=0.5$ in the $x$ direction, as a function of time for $\text{Pr}=1$ and $\mathrm{Gr}={10}^6$ without introducing artificial disturbances.

Figure 2

The isosurfaces of the dimensionless temperature $T=0.5$ at $t=8$ and 10 are shown in (a) and (b), respectively. The temperature perturbation amplitude $T_{\text{max}}^{*}$ at $t=8$ is shown in (c), where $T^{*}{(x,y,t)}_{\text{max}}={[T(x,y,z,t)-\overline{T}(x,y,t)]}_{\text{max}}$ and the spanwise mean temperature $\overline{T}=\frac{1}{Lz}{\int }_0^{Lz}T(x,y,z,t)dz$. The black solid line indicates $\overline{T}=0.5$ and the open square, diamond, and triangle symbols represent the positions of the maximum $T_{\text{max}}^{*}$, the stagnation point of the spanwise mean flow, and the maximum curvature of the interface with $\overline{T}=0.5$, respectively. Some streamlines are drawn as well to show the mean flow field in the translational frame moving with the front. (d) The energy spectral density $E_T$ of the temperature perturbation $T^{*}$ along the spanwise line where $T_{\text{max}}^{*}$ reaches its maximum at the front region. The vertical dashed line indicates the dominating spanwise wave number, where the spectral density reaches the maximum. In the simulation, $\mathrm{Gr}={10}^6, \text{Pr}=1$, and the random disturbances are introduced at $t=5$.

Figure 3

(a) The maximum growth rate ${\gamma }_m$ and (b) the corresponding wave number $k_m$ calculated from the local dispersion relation as functions of $\theta$ for $\text{Pr}=1$ and $\mathrm{Gr}={10}^6$. The short solid and dashed lines in (a) represent the positions of the stagnation point and the maximum curvature, respectively. The symbol $★$ in (b) indicates the value of the strongest perturbation obtained from simulations.

Figure 4

The original dominating wave number obtained in simulations as a function of (a) the Grashof number and (b) the Prandtl number. The theoretical curves of the semi-infinite RTI model [Eq. (9)] are shown in (a) and (b) as well for comparison. The solid square in (a) represents the estimated mean wave number [12] based on Simpson's saltwater and freshwater experiments [10], where the equivalent of Pr (Schmidt number) is about 650.