American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 611-618. doi: 10.3934/proc.2013.2013.611

Spatial stability of horizontally sheared flow

Iordanka N. Panayotova 1, , Pai Song 1, and  John P. McHugh 2,

1. 

Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, United States, United States
2. 

Department of Mechanical Engineering, University of New Hampshire, Durham, NH 03824, United States

Received  August 2012 Revised  January 2013 Published  November 2013

We investigate the stability of a shear flow in a stratified fluid. The flow is assumed to be inviscid and Boussinesq and the base state density gradient is vertical with constant Brunt-Vaisala frequency. The shear is taken as horizontal, where the base-state velocity has uniform direction and it's magnitude depends on the transverse horizontal coordinate, U(y). Unlike vertical shear flows, this combination of horizontal shear with vertical stratification is inherently three-dimensional and Squire's theorem is inapplicable. Spatial stability characteristics are obtained using the normal-mode approach and the Riccati transform. Sensitivity of the stability characteristics and their qualitative features are investigated by numerical methods for free-shear flow approximated by the hyperbolic tangent.
Keywords: Spatial stability, unstable shear flow., Riccati equation, horizontal shear flow.
Mathematics Subject Classification: Primary: 76F10; Secondary: 76E0.
Citation: Iordanka N. Panayotova, Pai Song, John P. McHugh. Spatial stability of horizontally sheared flow. Conference Publications, 2013, 2013 (special) : 611-618. doi: 10.3934/proc.2013.2013.611
References:
[1]

Badulin, S.I., V.I.Shrira, and L.Sh. Tsimring, The trapping and vertical focusing of internal waves in a pycnocline due to the horizontal inhomogeneities of density and currents,, J. Fluid Mech., 158 (1985), 199.   Google Scholar

[2]

Badulin, S.I. and V.I.Shrira, On the irreversibility of internal wave dynamics due to wave trapping by mean flow inhomogeneities, Part 1. Local analysis,, J. Fluid Mech., 251 (1993), 21.   Google Scholar

[3]

Blumen, W., Stability of non-planar shear flow of a stratified fluid,, J. Fluid Mech., 68 (1975), 177.   Google Scholar

[4]

Deloncle, A., J. M. Chomaz, and P. Billant, Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid,, J. Fluid Mech., 570 (2007), 297.   Google Scholar

[5]

Drazin,, P. G. and Howard, L. N., Hydrodynamic stability of parallel flow of inviscici fluid,, Advan. Appl. Mech., 9 (1966), 1.   Google Scholar

[6]

Ivanov, Y. A., and Y.G.Morozov, Deformation of internal gravity waves by a stream with horizontal shear,, Okeanologie 14 (1974), 14 (1974), 467.   Google Scholar

[7]

Jackson, T.L. and C.E. Grosch, Inviscid spatial stability of a compressible mixing layer,, J. Fluid Mech., 208 (1989), 609.   Google Scholar

[8]

Maslowe, S.A., Hydrodynamic instabilities and the transition to turbulence., (Springer-Verlag), (1981).   Google Scholar

[9]

Panayotova, I.N. and J.P. McHugh, On the stability of three-dimensional disturbances in staratified flow with lateral and vertical shear,, Open Atm. Sci. J., 5 (2011), 23.   Google Scholar

show all references

References:
[1]

Badulin, S.I., V.I.Shrira, and L.Sh. Tsimring, The trapping and vertical focusing of internal waves in a pycnocline due to the horizontal inhomogeneities of density and currents,, J. Fluid Mech., 158 (1985), 199.   Google Scholar

[2]

Badulin, S.I. and V.I.Shrira, On the irreversibility of internal wave dynamics due to wave trapping by mean flow inhomogeneities, Part 1. Local analysis,, J. Fluid Mech., 251 (1993), 21.   Google Scholar

[3]

Blumen, W., Stability of non-planar shear flow of a stratified fluid,, J. Fluid Mech., 68 (1975), 177.   Google Scholar

[4]

Deloncle, A., J. M. Chomaz, and P. Billant, Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid,, J. Fluid Mech., 570 (2007), 297.   Google Scholar

[5]

Drazin,, P. G. and Howard, L. N., Hydrodynamic stability of parallel flow of inviscici fluid,, Advan. Appl. Mech., 9 (1966), 1.   Google Scholar

[6]

Ivanov, Y. A., and Y.G.Morozov, Deformation of internal gravity waves by a stream with horizontal shear,, Okeanologie 14 (1974), 14 (1974), 467.   Google Scholar

[7]

Jackson, T.L. and C.E. Grosch, Inviscid spatial stability of a compressible mixing layer,, J. Fluid Mech., 208 (1989), 609.   Google Scholar

[8]

Maslowe, S.A., Hydrodynamic instabilities and the transition to turbulence., (Springer-Verlag), (1981).   Google Scholar

[9]

Panayotova, I.N. and J.P. McHugh, On the stability of three-dimensional disturbances in staratified flow with lateral and vertical shear,, Open Atm. Sci. J., 5 (2011), 23.   Google Scholar

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