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On new stability modes of plane canonical shear flows using symmetry classification. (English) Zbl 1327.76080

Summary: In the work of the first two authors [Phys. Fluids 25, No. 10, Paper No. 104101, 18 p. (2013; Zbl 1327.76079)], it was shown that three different instability modes of the linear stability analysis perturbing a linear shear flow can be derived in the common framework of Lie symmetry methods. These modes include the normal-mode, the Kelvin mode, and a new mode not reported before. As this was limited to linear shear, we now present a full symmetry classification for the linearised Navier-Stokes equations which are employed to study the stability of an arbitrary plane shear flow. If viscous effects for the perturbations are neglected, then we obtain additional symmetries and new Ansatz functions for a linear, an algebraic, an exponential, and a logarithmic base shear flow. If viscous effects are included in the formulation, then the linear and a quotient-type base flow allow for additional symmetries. The symmetry invariant solutions derived from the new and classical generic symmetries for all different flow types naturally lead to algebraic growth and decay for all cases except for two linear base flow cases. In turn this leads to the formulation of a novel eigenvalue problem in the analysis of the transition to turbulence for the respective flows, all of which are very distinct from the classical Orr-Sommerfeld eigenvalue problems.{
©2015 American Institute of Physics}

MSC:

76F10 Shear flows and turbulence
76F06 Transition to turbulence
76D05 Navier-Stokes equations for incompressible viscous fluids

Citations:

Zbl 1327.76079

Software:

GeM
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Full Text: DOI Link

References:

[1] Orr, W. M., The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II: A viscous liquid, Proc. R. Ir. Acad., Sect. A, 27, 9 (1907)
[2] Lundbladh, A.; Johansson, A. V., Direct simulation of turbulent spots in plane Couette flow, J. Fluid Mech., 229, 499 (1991) · Zbl 0850.76256 · doi:10.1017/S0022112091003130
[3] Tillmark, N.; Alfredsson, P. H., Experiments on transition in plane Couette flow, J. Fluid Mech., 235, 89 (1992) · doi:10.1017/S0022112092001046
[4] Kelvin, L. W. T., Stability of fluid motion: Rectilinear motion of viscous fluid between two parallel plates, Philos. Mag., 24, 188 (1887) · doi:10.1080/14786448708628078
[5] Rosen, G., General solution for perturbed plane Couette flow, Phys. Fluids, 14, 2767 (1971) · Zbl 0255.76048 · doi:10.1063/1.1693404
[6] Case, K. M., Stability of inviscid plane Couette flow, Phys. Fluids, 3, 143 (1960) · Zbl 0213.54306 · doi:10.1063/1.1706010
[7] Trefethen, L. N.; Trefethen, A. E.; Reddy, S. C.; Driscoll, T. A., Hydrodynamic stability without eigenvalues, Science, 261, 578 (1993) · Zbl 1226.76013 · doi:10.1126/science.261.5121.578
[8] Butler, K.; Farrell, B., Three-dimensional optimal perturbations in viscous shear flow, Phys. Fluids A, 4, 1637 (1992) · doi:10.1063/1.858386
[9] Reddy, S. C.; Schmid, P. J.; Henningson, D. S., Pseudospectra of the Orr-Sommerfeld operator, SIAM J. Appl. Math., 53, 15 (1993) · Zbl 0778.34060 · doi:10.1137/0153002
[10] Gustavsson, L. H., Energy growth of three-dimensional disturbances in plane Poiseuille flow, J. Fluid Mech., 224, 241 (1991) · Zbl 0717.76044 · doi:10.1017/S002211209100174X
[11] Grossmann, S., The onset of shear flow turbulence, Rev. Mod. Phys., 72, 603 (2000) · doi:10.1103/RevModPhys.72.603
[12] Schmid, P. J., Nonmodal stability theory, Annu. Rev. Fluid Mech., 39, 129 (2006) · Zbl 1296.76055 · doi:10.1146/annurev.fluid.38.050304.092139
[13] Horton, W.; Kim, J. H.; Chagelishvili, G. D.; Bowman, J. C.; Lominadze, J. G., Angular redistribution of nonlinear perturbations: A universal feature of nonuniform flows, Phys. Rev. E, 81, 066304 (2010) · doi:10.1103/PhysRevE.81.066304
[14] Garnaud, X.; Lesshafft, L.; Schmid, P.; Huerre, P., Modal and transient dynamics of jet flows, Phys. Fluids, 25, 044103 (2013) · Zbl 1284.76149 · doi:10.1063/1.4801751
[15] Nold, A.; Oberlack, M., Symmetry analysis in linear hydrodynamic stability theory: Classical and new modes in linear shear, Phys. Fluids, 25, 104101 (2013) · Zbl 1327.76079 · doi:10.1063/1.4823508
[16] Boisvert, R. E.; Ames, W. F.; Srivastava, U. N., Group properties and new solutions of Navier-Stokes equations, J. Eng. Math., 17, 203 (1983) · Zbl 0546.35056 · doi:10.1007/BF00036717
[17] Simonsen, V.; Meyer-ter Vehn, J., Self-similar solutions in gas dynamics with exponential time dependence, Phys. Fluids, 9, 1462 (1997) · Zbl 1185.76861 · doi:10.1063/1.869258
[18] Oberlack, M.; Wenzel, H.; Peters, N., On symmetries and averaging of the G-equation for premixed combustion, Combust. Theory Modell., 5, 363 (2001) · Zbl 1114.80306 · doi:10.1088/1364-7830/5/3/307
[19] Oberlack, M., A unified approach for symmetries in plane parallel turbulent shear flows, J. Fluid Mech., 427, 299 (2001) · Zbl 1007.76067 · doi:10.1017/S0022112000002408
[20] Avramenko, A. A.; Blinov, D. G.; Shevchuk, I. V.; Kuznetsov, A. V., Symmetry analysis and self-similar forms of fluid flow and heat-mass transfer in turbulent boundary layer flow of a nanofluid, Phys. Fluids, 24, 092003 (2012) · doi:10.1063/1.4753945
[21] Barenblatt, G.; Galerkina, N.; Luneva, M., Evolution of a turbulent burst, J. Eng. Phys. Thermophys., 53, 1246 (1987) · doi:10.1007/BF00871083
[22] Grebenev, V., On a certain system of degenerate parabolic equations which arises in hydrodynamics, Sib. Math. J., 35, 670 (1994) · Zbl 0861.35079 · doi:10.1007/BF02106610
[23] Bluman, G. W.; Cheviakov, A. F.; Anco, S. C., Applications of Symmetry Methods to Partial Differential Equations, 168 (2010) · Zbl 1223.35001
[24] Bluman, G.; Kumei, S., Symmetries and Differential Equations (1989) · Zbl 0698.35001
[25] Bluman, G.; Anco, S., Symmetry and Integration Methods for Differential Equations, 154 (2002) · Zbl 1013.34004
[26] Cantwell, B., Introduction to Symmetry Analysis, 29 (2002) · Zbl 1082.34001
[27] Steeb, W.-H., Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra (2007) · Zbl 0916.34001
[28] Cheviakov, A., GeM software package for computation of symmetries and conservation laws of differential equations, Comput. Phys. Commun., 176, 48 (2007) · Zbl 1196.34045 · doi:10.1016/j.cpc.2006.08.001
[29] Carminati, J.; Vu, K., Symbolic computation and differential equations: Lie symmetries, J. Symbolic Comput., 29, 95 (2000) · Zbl 0958.68543 · doi:10.1006/jsco.1999.0299
[30] Akhatov, I.; Gazizov, R.; Ibragimov, N., Group classification of the equations of nonlinear filtration, Soviet Math. Dokl., 35, 384-386 (1987) · Zbl 0632.76004
[31] Drazin, P.; Reid, W., Hydrodynamic Stability (2004) · Zbl 1055.76001
[32] Schmid, P.; Henningson, D., Stability and Transition in Shear Flows, 142 (2001) · Zbl 0966.76003
[33] Pope, S. B., Turbulent Flows (2000) · Zbl 0966.76002
[34] Wygnanski, I., On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug, J. Fluid Mech., 59, 281 (1973) · doi:10.1017/S0022112073001576
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