Numerical-analytical solutions to mud mechanical responses to the waves in shallow water regions: From cohesive sediment to fluid mud

 Abstract:

A comprehensive overview to the mud liquefaction and fluid mud mass transport induced by waves and currents are proposed in the shallow water regions. In this regard, the Shallow Water Equations (SWEs) must be solved for the upper water layer and the mud mechanical response to the free surface must be investigated. The aim of the present study, thus, is two folded: 1) To numerically investigate the newly developed energy-stable skew-symmetric form of the linear and nonlinear shallow water equations (SWE) using high-order numerical schemes with the so-called summation by parts (SBP) property; 2) analytical solutions to the interactions between waves, currents, and the muddy bed layer and compare the results for different constant, linear, and second-order current profiles.

The nonlinear stable boundary treatments with penalty-like simultaneous approximation terms (SAT), have been applied to mimic the lifting approach of continuous characteristic boundary conditions. In order to test the skew-symmetric form of SWE with the new variables, a manufactured solution (MS) is deployed, and the scheme is shown to be robust in the domain and at the boundary sides. The free parameters in the new form of the equations slightly change the convergence.

The effects of mean shear stress and its variations on wave dispersion relations as well as mud (particle and mass transport) velocities are investigated. It is found that the second-order profile presents the maximum effects on the wave field (wave dissipation and mud mass transport velocities) compared to the constant and linear current profiles. However, assuming the constant current profile, frequently applied in the literature models, results in the minimum effects. A local peak exists in the mud mean discharge over the current profile curvature. The mud velocity induced by the linear current profile presents the closest value to the particle velocity for the no-current case. Additionally, the second-order current profile provides slightly better results for the mud mass transport velocity rather than the constant current profile when comparing the results with the laboratory data.

There is a rather huge gap between the existing agreed mechanisms in the literature for non-cohesive and cohesive sediment which is addressed. Also, the lack of experimental and theoretical results for the mud liquefaction mechanism is pointed out. Open questions in the field and potential topics for further research are presented.

Keywords:

Shallow water equations, Mud mass transport, Wave-current-mud interaction, Summation by parts, Stable boundary conditions, Mud liquefaction and transport