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 = 0.23. For R = 5 (velocity-to-density thickness ratio) and zero bed slope, there are four zones: (a) two Kelvin–Helmholtz modes (0 < J < 0.09, J is local Richardson number), (b) two Holmboe modes (0.09 < J < 0.65), (c) no unstable mode (0.65 < J < 2.5) and (d) two Holmboe modes (2.5 < J < 4 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.
Similar content being viewed by others
References
Khavasi E, Afshin H, Firoozabadi B (2012) Effect of selected parameters on the depositional behavior of turbidity currents. J Hydr Res 50(1):60–69
Ortiz S, Chomaz JM, Loiseleux Th (2002) Spatial Holmboe instability. Phys Fluids 14(8):2585–2597
Zhu DZ, Lawrence GA (2001) Holmboe’s instability in exchange flows. J Fluid Mech 429:391–409
Balmforth NJ, Roy A, Caulfield CP (2012) Dynamics of vorticity defects in stratified shear flow. J Fluid Mech 694:292–331
Gelfgat AY, Kit E (2006) Spatial versus temporal instabilities in a parametrically forced stratified mixing layer. J Fluid Mech 552:189–227
Hajesfandiari A, Forliti DJ (2014) On the influence of internal density variations on the linear stability characteristics of planar shear layers. Phys Fluids 26:054102
Smyth WD, Winters KB (2003) Turbulence and mixing in Holmboe waves. J Phys Oceanogr 33:694–711
Carpenter JR, Tedford EW, Rahmani M, Lawrence GA (2010) Holmboe wave fields in simulation and experiment. J Fluid Mech 648:205–223
Pawlak G, Armi L (1998) Vortex dynamics in a spatially accelerating shear layer. J Fluid Mech 376:1–35
Panayotova IN, Song P, Mchgh JP (2013) Spatial stability of horizontally sheared flow. Discrete Continuous Dyn Syst Supplement:611–618
Amini P, Khavasi E, Asadizanjani N (2017) Linear stability analysis of two-way coupled particle-laden density current. Can J Phys 95(3):291–296
Barmak I, Yu. Gelfgat A, Ullmann A, Brauner N (2017) On the Squire’s transformation for stratified two-phase flows in inclined channels. Int J Multiph Flow 88:142–151
Negretti ME, Socolofsky SA, Jirka GH (2008) Linear stability analysis of inclined two-layer stratified flows. Phys Fluids 20(9):104–111
Khavasi E, Firoozabadi B, Afshin BH (2014) Linear analysis of the stability of particle-laden stratified shear layers. Can J Phys 92(2):103–115
Alba K, Taghavi SM, Frigaard IA (2013) Miscible density-unstable displacement flows in inclined tube. Phys Fluids 25(6):067101
Barros R, Choi W (2011) Holmboe instability in non-Boussinesq fluids. Phys Fluids 23:124103
Barros R, Choi W (2014) Elementary stratified flows with stability at low Richardson number. Phys Fluids 26:124107
Hazel P (1972) Numerical studies of the stability of inviscid stratified shear flows. J Fluid Mech 51(1):39–61
Alexakis A (2009) Stratified shear flow instabilities at large Richardson numbers. Phys Fluids 21(5):054108
Author information
Authors and Affiliations
Corresponding author
Additional information
Technical Editor: Jader Barbosa Jr., Ph.D.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Khavasi, E., Firoozabadi, B. Linear spatial stability analysis of particle-laden stratified shear layers. J Braz. Soc. Mech. Sci. Eng. 41, 246 (2019). https://doi.org/10.1007/s40430-019-1745-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40430-019-1745-4