Effect of water depth on Kelvin–Helmholtz instability in a shallow water flow

Abstract

It is well known that the interface between two regions of an incompressible ideal fluid flow moving in a relative motion is necessarily destabilized, regardless of the velocity difference’s strength. This phenomenon is the so-called Kelvin–Helmholtz instability. However, a large number of works demonstrated a surprising result that the instability is suppressed for shallow water flows; the interface is stabilized if the Froude number, defined by the velocity difference’s ratio to the gravity wave’s speed, is sufficiently large. In a limited way, these authors used the shallow water equations without the higher-order effect of the dispersive terms. Thus, this investigation aims at examining these higher-order dispersive effects to analyze the interface stability problem of tangential-velocity discontinuity in shallow water flows. In particular, we use the Green–Naghdi equations to introduce the dispersive terms related to the depth and the depth-averaged horizontal velocities of the fluid. We show that the interface stability depends on the Froude number (i.e., the velocity difference’s strength) and the water depth. A critical value of the Froude number to stabilize the interface is smaller than the case of no dispersive terms, and the flow in a deeper region is more stable than in a shallower one. We also consider the distribution of kinetic and potential energy to clarify a feature characteristic of a large class of instabilities in shallow water flow. The instability of flows is caused by the decrease in the kinetic energy during the perturbation of waves. This phenomenon is known as negative energy modes and plays a vital role in applying the model to industrial equipment. A conclusion is that the equipartition of energies occurs if and only if the velocity difference is zero and the water depth is shallow enough to ignore the dispersive terms.