Linear spatial stability analysis of particle-laden stratified shear layers

Abstract

Hydrodynamic instabilities at the interface of stratified shear layers could occur in various modes. These instabilities have an important role in the mixing process. In this work, the linear stability analysis in spatial framework is used to study the stability characteristics of a particle-laden stratified two-layer flow. The effect of parameters such as velocity-to-density thickness ratio, bed slope, viscosity as well as particle size on the stability is considered. A simple iterative method applying the pseudospectral collocation method that employed Chebyshev polynomials is used to solve two coupled eigenvalue equations. Based on the results, the flow becomes stable for Richardson number larger than 0.25 (same as the result of temporal stability analysis); the stability is not affected by spatial wavenumber. The increase in bed slope makes the current more unstable as does in temporal framework. For 1% bed slope, the spatial growth rate increases by 70% in J\u2009=\u20090.23. For R\u2009=\u20095 (velocity-to-density thickness ratio) and zero bed slope, there are four zones: (a) two Kelvin–Helmholtz modes (0\u2009<\u2009J\u2009<\u20090.09, J is local Richardson number), (b) two Holmboe modes (0.09\u2009<\u2009J\u2009<\u20090.65), (c) no unstable mode (0.65\u2009<\u2009J\u2009<\u20092.5) and (d) two Holmboe modes (2.5\u2009<\u2009J\u2009<\u20094 where the second type of Holmboe modes appears). The second type of Holmboe modes does not appear in temporal framework in this condition. In spatial analysis, for nonzero bed slope there is no stable region. Also, just one type of Holmboe modes and two types of Kelvin–Helmholtz modes appear. Th existence of particles changes the instability characteristics of the flow. Particles increase the spatial growth rate. In temporal analysis, particles larger than a certain size (e.g., kaolin particles larger than 20 micron in Stokes’ law settling velocity) make the flow unstable, but in spatial framework particles with any size do this. As expected, the viscosity makes the current more stable like temporal analysis.