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), 199-218.

[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), 21-53.

[3]

Blumen, W., Stability of non-planar shear flow of a stratified fluid, J. Fluid Mech., 68 (1975), 177-189.

[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), 297-305.

[5]

Drazin,, P. G. and Howard, L. N., Hydrodynamic stability of parallel flow of inviscici fluid, Advan. Appl. Mech., 9 (1966), 1-89.

[6]

Ivanov, Y. A., and Y.G.Morozov, Deformation of internal gravity waves by a stream with horizontal shear, Okeanologie 14 (1974), 467-461.

[7]

Jackson, T.L. and C.E. Grosch, Inviscid spatial stability of a compressible mixing layer, J. Fluid Mech., 208 (1989), 609-637.

[8]

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

[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), 23-26.

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), 199-218.

[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), 21-53.

[3]

Blumen, W., Stability of non-planar shear flow of a stratified fluid, J. Fluid Mech., 68 (1975), 177-189.

[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), 297-305.

[5]

Drazin,, P. G. and Howard, L. N., Hydrodynamic stability of parallel flow of inviscici fluid, Advan. Appl. Mech., 9 (1966), 1-89.

[6]

Ivanov, Y. A., and Y.G.Morozov, Deformation of internal gravity waves by a stream with horizontal shear, Okeanologie 14 (1974), 467-461.

[7]

Jackson, T.L. and C.E. Grosch, Inviscid spatial stability of a compressible mixing layer, J. Fluid Mech., 208 (1989), 609-637.

[8]

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

[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), 23-26.

[1]

Raphael Stuhlmeier. Effects of shear flow on KdV balance - applications to tsunami. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1549-1561. doi: 10.3934/cpaa.2012.11.1549
[2]

Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong, Qiang Du. Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow. Mathematical Biosciences & Engineering, 2006, 3 (4) : 683-696. doi: 10.3934/mbe.2006.3.683
[3]

Patrick Martinez, Jean-Michel Roquejoffre. The rate of attraction of super-critical waves in a Fisher-KPP type model with shear flow. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2445-2472. doi: 10.3934/cpaa.2012.11.2445
[4]

Muhammad Mansha Ghalib, Azhar Ali Zafar, Zakia Hammouch, Muhammad Bilal Riaz, Khurram Shabbir. Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 683-693. doi: 10.3934/dcdss.2020037
[5]

P. Kaplický, Dalibor Pražák. Lyapunov exponents and the dimension of the attractor for 2d shear-thinning incompressible flow. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 961-974. doi: 10.3934/dcds.2008.20.961
[6]

Scott Gordon. Nonuniformity of deformation preceding shear band formation in a two-dimensional model for Granular flow. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1361-1374. doi: 10.3934/cpaa.2008.7.1361
[7]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Saeed M. Ali. New stability result for a Bresse system with one infinite memory in the shear angle equation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 995-1014. doi: 10.3934/dcdss.2021086
[8]

Hong Zhou, M. Gregory Forest, Qi Wang. Anchoring-induced texture & shear banding of nematic polymers in shear cells. Discrete and Continuous Dynamical Systems - B, 2007, 8 (3) : 707-733. doi: 10.3934/dcdsb.2007.8.707
[9]

Eric S. Wright. Macrotransport in nonlinear, reactive, shear flows. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2125-2146. doi: 10.3934/cpaa.2012.11.2125
[10]

Julie Lee, J. C. Song. Spatial decay bounds in a linearized magnetohydrodynamic channel flow. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1349-1361. doi: 10.3934/cpaa.2013.12.1349
[11]

Aniello Raffaele Patrone, Otmar Scherzer. On a spatial-temporal decomposition of optical flow. Inverse Problems and Imaging, 2017, 11 (4) : 761-781. doi: 10.3934/ipi.2017036
[12]

Anna Geyer, Ronald Quirchmayr. Weakly nonlinear waves in stratified shear flows. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2309-2325. doi: 10.3934/cpaa.2022061
[13]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control and Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1
[14]

V. Torri. Numerical and dynamical analysis of undulation instability under shear stress. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 423-460. doi: 10.3934/dcdsb.2005.5.423
[15]

Yukun Song, Yang Chen, Jun Yan, Shuai Chen. The existence of solutions for a shear thinning compressible non-Newtonian models. Electronic Research Archive, 2020, 28 (1) : 47-66. doi: 10.3934/era.2020004
[16]

Guillermo H. Goldsztein. Bound on the yield set of fiber reinforced composites subjected to antiplane shear. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 391-400. doi: 10.3934/dcdsb.2011.15.391
[17]

Zhenlu Cui, Qi Wang. Permeation flows in cholesteric liquid crystal polymers under oscillatory shear. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 45-60. doi: 10.3934/dcdsb.2011.15.45
[18]

Alberto Gambaruto, João Janela, Alexandra Moura, Adélia Sequeira. Shear-thinning effects of hemodynamics in patient-specific cerebral aneurysms. Mathematical Biosciences & Engineering, 2013, 10 (3) : 649-665. doi: 10.3934/mbe.2013.10.649
[19]

Alex Mahalov, Mohamed Moustaoui, Basil Nicolaenko. Three-dimensional instabilities in non-parallel shear stratified flows. Kinetic and Related Models, 2009, 2 (1) : 215-229. doi: 10.3934/krm.2009.2.215
[20]

Sergey A. Suslov. Two-equation model of mean flow resonances in subcritical flow systems. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 165-176. doi: 10.3934/dcdss.2008.1.165

 Impact Factor: 

Tools

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy