Skip to main content
Log in

Linear spatial stability analysis of particle-laden stratified shear layers

  • Technical Paper
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript


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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others


  1. 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

    Article  Google Scholar 

  2. Ortiz S, Chomaz JM, Loiseleux Th (2002) Spatial Holmboe instability. Phys Fluids 14(8):2585–2597

    Article  MathSciNet  Google Scholar 

  3. Zhu DZ, Lawrence GA (2001) Holmboe’s instability in exchange flows. J Fluid Mech 429:391–409

    Article  Google Scholar 

  4. Balmforth NJ, Roy A, Caulfield CP (2012) Dynamics of vorticity defects in stratified shear flow. J Fluid Mech 694:292–331

    Article  Google Scholar 

  5. Gelfgat AY, Kit E (2006) Spatial versus temporal instabilities in a parametrically forced stratified mixing layer. J Fluid Mech 552:189–227

    Article  MathSciNet  Google Scholar 

  6. 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

    Article  Google Scholar 

  7. Smyth WD, Winters KB (2003) Turbulence and mixing in Holmboe waves. J Phys Oceanogr 33:694–711

    Article  MathSciNet  Google Scholar 

  8. Carpenter JR, Tedford EW, Rahmani M, Lawrence GA (2010) Holmboe wave fields in simulation and experiment. J Fluid Mech 648:205–223

    Article  Google Scholar 

  9. Pawlak G, Armi L (1998) Vortex dynamics in a spatially accelerating shear layer. J Fluid Mech 376:1–35

    Article  MathSciNet  Google Scholar 

  10. Panayotova IN, Song P, Mchgh JP (2013) Spatial stability of horizontally sheared flow. Discrete Continuous Dyn Syst Supplement:611–618

    MathSciNet  MATH  Google Scholar 

  11. 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

    Article  Google Scholar 

  12. 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

    Article  MathSciNet  Google Scholar 

  13. Negretti ME, Socolofsky SA, Jirka GH (2008) Linear stability analysis of inclined two-layer stratified flows. Phys Fluids 20(9):104–111

    Article  Google Scholar 

  14. 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

    Article  Google Scholar 

  15. Alba K, Taghavi SM, Frigaard IA (2013) Miscible density-unstable displacement flows in inclined tube. Phys Fluids 25(6):067101

    Article  Google Scholar 

  16. Barros R, Choi W (2011) Holmboe instability in non-Boussinesq fluids. Phys Fluids 23:124103

    Article  Google Scholar 

  17. Barros R, Choi W (2014) Elementary stratified flows with stability at low Richardson number. Phys Fluids 26:124107

    Article  Google Scholar 

  18. Hazel P (1972) Numerical studies of the stability of inviscid stratified shear flows. J Fluid Mech 51(1):39–61

    Article  Google Scholar 

  19. Alexakis A (2009) Stratified shear flow instabilities at large Richardson numbers. Phys Fluids 21(5):054108

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Ehsan Khavasi.

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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: