Is it appropriate to model turbidity currents with the three‐equation model?

Abstract

The three-equation model (TEM) was developed in the 1980s to model turbidity currents (TCs) and has been widely used ever since. However, its physical justification was questioned because self-accelerating TCs simulated with the steady TEM seemed to violate the turbulent kinetic energy balance. This violation was considered as a result of very strong sediment erosion that consumes more turbulent kinetic energy than is produced. To confine bed erosion and thus remedy this issue, the four-equation model (FEM) was introduced by assuming a proportionality between the bed shear stress and the turbulent kinetic energy. Here we analytically proof that self-accelerating TCs simulated with the original steady TEM actually never violate the turbulent kinetic energy balance, provided that the bed drag coefficient is not unrealistically low. We find that stronger bed erosion, surprisingly, leads to more production of turbulent kinetic energy due to conversion of potential energy of eroded material into kinetic energy of the current. Furthermore, we analytically show that, for asymptotically supercritical flow conditions, the original steady TEM always produces self-accelerating TCs if the upstream boundary conditions ("ignition" values) are chosen appropriately, while it never does so for asymptotically subcritical flow conditions. We numerically show that our novel method to obtain the ignition values even works for Richardson numbers very near to unity. Our study also includes a comparison of the TEM and FEM closures for the bed shear stress to simulation data of a coupled Large Eddy and Discrete Element Model of sediment transport in water, which suggests that the TEM closure might be more realistic than the FEM closure.

Hydrodynamic analysis of fully developed turbidity currents over plane beds based on self‐preserving velocity and concentration distributions

“…[]. The present three‐equation model does not fail in predicting turbidity currents from ignition point and, in turn, does not violate the four‐equation TKE balance [ Hu et al ., ]. Besides, in view of the self‐preserving type of velocity and concentration laws that are validated by the experimental data and used to develop the gradually varied flow relationships, perhaps a more than a qualitative rationality can be claimed for the computed results.…”

“…However, Hu et al . [] found that the three‐equation model does not fail to simulate self‐accelerating turbidity currents, rendering unclear the need of using the four‐equation model. Felix [] proposed a two‐dimensional turbulence model to address the development of turbidity current.…”

“…Further, Pratson et al [2000] solved the four-equation model developed by Parker et al [1986] using numerical techniques. However, Hu et al [2015] found that the three-equation model does not fail to simulate selfaccelerating turbidity currents, rendering unclear the need of using the four-equation model. Felix [2001] proposed a two-dimensional turbulence model to address the development of turbidity current.…”

Investigations of dynamic behaviors of lock-exchange turbidity currents down a slope based on direct numerical simulation

“…A turbidity current is a type of subaqueous sediment-laden gravity current [1][2][3][4], which has been studied using both a lock-exchange configuration (i.e., fixed volume) and a continuous-flux configuration. This study focuses on lock-exchange depositional turbidity currents, which represent those triggered by abrupt landslides and short-lived floods [5].…”

“…For those experimental studies within the lock-exchange configuration, most results [7,8,14] are presented in the form of the bulk values, i.e., the duration-averaged entrainment ratio. Hallworth et al [15] determined an average entrainment ratio of the lock-exchange gravity current on a flat bed as 0.063 ± 0.003. , 3 in which the entrainment ratio is defined as the ratio of the volumes of ambient and original fluid in the head. Recently, both the experimental and large eddy simulation results [16][17][18] indicate that the entrainment ratio of a lock-exchange gravity current is closely related to the initial density difference and the aspect ratio of the initial water depth to the lock length.…”