On the merging and splitting processes in the lobe-and-cleft structure at a gravity current head

Abstract

High-resolution simulations are performed for gravity currents propagating on a no-slip boundary to study the merging and splitting processes in the lobe-and-cleft structure at a gravity current head. The simulations reproduce the morphological features observed in the laboratory and provide more detailed flow information to elucidate the merging and splitting processes. Our mean lobe width $\tilde {b}$ and mean maximum lobe width $\tilde {b}_{max}$ satisfy the empirical relationships $\tilde {b}/\tilde {d}=7.4 Re^{-0.39}_f$ and $\tilde {b}_{max}/\tilde {d}=12.6 Re^{-0.38}_f$ , respectively, over the front Reynolds number in the range $383 \le Re_f \le 3267$ , where the front Reynolds number is defined as $Re_f= \tilde {u}_f \tilde {d} / \tilde {\nu }, \tilde {u}_f$ is the front velocity, $\tilde {d}$ is the height of the gravity current head and $\tilde {\nu }$ is the fluid kinematic viscosity. When measured in terms of the viscous length scale $\tilde {\delta }_\nu = \tilde {\nu }/\tilde {u}^*$ , where $\tilde {u}^*$ is the shear velocity at the gravity current head, the mean lobe width and the mean maximum lobe width increase with increasing front Reynolds number and asymptotically approach $126 \tilde {\delta }_\nu$ and $230 \tilde {\delta }_\nu$ at $Re_f=3267$ , respectively. The vortical structure inside a lobe has an elongated tooth-like shape and a pair of counter-rotating streamwise vortices are positioned on the left- and right-hand sides of each cleft. For the merging process, it requires the interaction of three tooth-like vortices and the middle tooth-like vortex breaks up and reconnects with the two neighbouring tooth-like vortices. Therefore, a cleft may continually merge with another neighbouring cleft but may never disappear. For the splitting process, even before the new cleft appears, a new born streamwise vortex is created by the parent vortex of opposite orientation and the parent vortex can be either the left part or the right part of the existing tooth-like vortex inside the splitting lobe. The new born streamwise vortex then induces the other counter-rotating streamwise vortex as the new cleft develops. The initiation of the splitting process can be attributed to the Brooke–Hanratty mechanism reinforced by the baroclinic production of vorticity. Depending on the orientation of the parent vortex, the resulting new cleft after the splitting process can shift laterally in the positive or negative spanwise direction along the leading edge of the gravity currents as the lobe-and-cleft structure moves forward in the streamwise direction. For gravity currents propagating on a no-slip boundary, the lobe-and-cleft structure is self-sustaining and the manifestations of the merging and splitting processes are in accord with reported laboratory observations.

Energy balances for the collision of gravity currents of equal strengths

“…Our results indicate that the occurrence of an array of vertical vortices developing along the interface between the two colliding gravity currents is due to the stretching of pre-existing x 3 vorticity inside the lobes. Our simulations show that, consistent with previously published reports (Cantero et al 2007b;Espath et al 2015;Dai & Huang 2022), the x 3 vorticity pre-exists inside the lobes of the two colliding gravity currents even before the two gravity currents enter the H-H domain. Inside a lobe moving in the streamwise direction, positive x 3 vorticity exists in the left portion of the lobe, while negative x 3 vorticity exists in the right portion of the lobe.…”

“…The initial density field was seeded with minute, uniform, random disturbances following Härtel, Michaud & Stein (1997) and Cantero et al (2006). The de-aliased pseudo-spectral code has been employed in previous high-resolution simulations for lock-exchange flows (Cantero et al 2007b;Dai 2015;Dai & Wu 2016;Dai & Huang 2022).…”

“…The front Reynolds number is defined as Re f = ũf d/ν and is related to the Reynolds number via Re f = u f d Re, where u f is the dimensionless front velocity, and d is the dimensionless height of the head. To provide sufficient resolution for the simulations of collision of gravity currents, the grid spacing must be of the order of O(Re Sc) −1/2 (Härtel et al 1997(Härtel et al , 2000Birman, Martin & Meiburg 2005), and the grids employed in the three-dimensional simulations for the preceding four Reynolds numbers follow Dai & Huang (2022) and are listed in table 1. The time step was chosen such that the Courant number remained less than 0.5.…”

Turbidity currents propagating down an inclined slope: particle auto-suspension

“…During the evolution of TCs, typical streamwise and transverse coherent vortical structures are exhibited (Dai & Huang 2022), the qualitative cognition of which is independent of the Reynolds number (Nasr-Azadani & Meiburg 2014; Nasr-Azadani, Meiburg & Kneller 2018). The coherent structure can also reflect the lobe-and-cleft structures of the TC (Espath et al.…”

“…Note that the increasing slope angle thickens the lower region (figure 5d) where the particle bed-normal velocity is negative, and it simultaneously inhibits the upward movement of the upper particles and the settling of the lower particles, implying the decrease in the separation rate of the upper and lower particles. During the evolution of TCs, typical streamwise and transverse coherent vortical structures are exhibited (Dai & Huang 2022), the qualitative cognition of which is independent of the Reynolds number (Nasr-Azadani & Meiburg 2014; Nasr-Azadani, Meiburg & Kneller 2018). The coherent structure can also reflect the lobe-and-cleft structures of the TC (Espath et al 2015;Francisco, Espath & Silvestrini 2017), which are related to the mixing dynamics.…”