Gravity current propagating against constant and pulsating counter flows

TL;DR

The paper addresses gravity currents propagating against a pulsating counterflow by performing 2D DNS in a lock-exchange configuration to isolate the effects of a mean opposing flow ($Fr_m$) and an oscillatory component ($Fr_o$, with $KC_b$). It reveals that Kelvin-Helmholtz billows arise at the interface for low to moderate $Fr_m$, driving vertical density transport and coherent advection of heavy-fluid patches, while Rayleigh-Taylor-like instabilities emerge under certain lifting conditions during the oscillation cycle, leading to strong vertical mixing. The front dynamics broadly follow traditional propagation phases, with the mean flow mainly reducing front speed and the current height, and the pulsating component enhancing vertical mixing and tailward mass transport, though it barely shifts the cycle-averaged front position. Collectively, non-hydrostatic processes substantially alter horizontal density transport on large scales, underscoring the need to incorporate KH and RT-like dynamics in estuarine and riverine salt-intrusion models.

Abstract

This paper describes the evolution of two-dimensional (2D) gravity currents that flow against a horizontally uniform laminar pulsating flow. We study the effect of opposing mean flow amplitude and the oscillatory velocity amplitude on the evolution of the gravity current, the emergence of instabilities due to shear at the interface of heavy and light fluid and unstable density stratification near the bottom wall, and the associated density redistributions. The velocity amplitudes and the oscillation frequency are reminiscent of tidal estuarine flows. This study revealed two key processes affecting the horizontal density transport of the heavy fluid, in addition to the buoyancy-driven propagation of the gravity current. The first process concerns the presence of shear-driven Kelvin-Helmholtz (KH) billows, depending on the strength of the opposing mean flow and the thickness of the gravity current. These KH billows are generated in the inertial phase of gravity current propagation and are responsible for coherent advective transport of heavy-fluid patches away from the gravity current head. The second process is related to the lifting of the gravity current head due to differential advection near the bottom wall when the propagation direction of the gravity current and the oscillating externally imposed flow are in the same direction. It generates a layer of light fluid below the heavy fluid of the gravity current head and becomes unstable when the ambient flow opposes the gravity current propagation, generating Rayleigh-Taylor-like (RT-like) instabilities. This results in a strong vertical redistribution of light and heavy fluid. Non-hydrostatic effects, such as the presence of KH billows and RT-like instabilities, with associated vertical density transport, have significant implications for large-scale horizontal density transport and modeling of salt intrusions in rivers and estuaries.

Figures (16)

• Figure 1: (a) Side view of an estuarine flow, where denser saltwater intrudes beneath lighter freshwater, forming a characteristic salt wedge. (b) Schematic of the gravity current resulting from the lock-exchange setup used in this study, where heavy fluid ($\rho_1$) and light fluid ($\rho_0$) are initially separated by a gate. The configuration assumes a simplified geometry with a horizontal, flat bottom to isolate the essential physics. Upon removal of the gate, the density difference drives a gravity current, which resembles the salt wedge dynamics illustrated in panel (a). The oscillatory velocity profile corresponding to the tidal phases with maximum velocity in both directions ($\phi = 90^{\circ}$ and $\phi = 270^{\circ}$) are shown in blue, while the laminar mean velocity profile is shown in red. The expressions for the velocity at the surface for the oscillating and mean flows are indicated above the corresponding profiles.
• Figure 2: Schematic of the lock-exchange set-up, with heavy ($\rho=1$) and light ($\rho=0$) fluid separated by a gate. The line shows the gravity current an instant after removal of the gate.
• Figure 3: Dimensionless density fields for different velocity amplitudes of the imposed ambient flow at $t=50$ ($L_{AR}\approx 1.3$): (a) $Fr_m=0$, (b) $Fr_m=0.1$, (c) $Fr_m=0.2$, (d) $Fr_m=0.3$, and (e) $Fr_m=0.4$. Panel (f) illustrates a comparison of the current height $h(\xi,t=50)$ for these cases. The red lines and arrows in (a)-(e) represent the horizontal velocity $u(y,t=50)$ at $x=X_{fr}-0.5$, with $X_{fr}$ the front position, and the horizontal velocity $u_m(y)$ far upstream. The value of the density (with $0 \le \rho \le 1$) is indicated by the color bar.
• Figure 4: The spatial distribution of $N^2$, in panels (a-e), and $S^2$, in panels (f-j) at $t=50$ for $Fr_m=0,$ 0.1, 0.2, 0.3, and 0.4, respectively ($L_{AR}\approx 2.2$). The solid red curves represent the horizontal velocity $u(y)$ at the head of the gravity current at $x=X_{fr}-0.5$ (indicated by the red vertical dotted lines). The value of the squared buoyancy frequency (with $-20 \le N^2 \le 20$) and squared shear (with $0 \le S^2 \le 200$) is indicated by the color bar.
• Figure 5: Gradient Richardson number, $Ri_g$, presented for $Fr_m=0$ panel (a), $Fr_m=0.1$ (b), $Fr_m=0.2$ (c), $Fr_m=0.3$ (d), and $Fr_m=0.4$ (e) at $t=50$ ($L_{AR}\approx 1.6$). The red solid curves represent the horizontal velocity $u(y)$ at the head of the gravity current for different panels ($x=X_{fr}-0.5$; indicated by the vertical red dotted line). The horizontal velocity $u(y)$, squared buoyancy frequency $N^2$ and squared shear $S^2$, are plotted on the right side of each plot at $x=X_{fr}-0.5$. The horizontal black dotted lines in the panels for $u(y)$, $N^2$ and $S^2$ indicate the position where $u(y)=0$. The value of the gradient Richardson number (with $-0.5 \le Ri_g \le 1.0$) is indicated by the color bar.