Long wavelength instability of the ABC-flows
Authors:
A. Libin and G. Sivashinsky
Journal:
Quart. Appl. Math. 48 (1990), 611-623
MSC:
Primary 76E99; Secondary 76D99
DOI:
https://doi.org/10.1090/qam/1079909
MathSciNet review:
MR1079909
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Abstract: Multiple-scale and spectral (Galerkin) techniques are employed to analyze the long-wave stability of a simple periodic helical flow at large Reynolds numbers. For a certain class of long-wave perturbations the corresponding mathematical problem admits an exact asymptotic solution, showing that the helical flow is always unstable.
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J. S. A. Green, Two-dimensional turbulence near the viscous limit, J. Fluid Mech. 62, 273–287 (1974)
A. A. Nepomniashchii, On stability of secondary flows of a viscous fluid in unbounded space, J. Appl. Math. Mech. 40, 836–841 (1976)
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K. Gotoh and M. Yamada, Instability of a cellular flow, J. Phys. Soc. Japan 53, 3395–3398 (1984)
G. Sivashinsky and V. Yakhot, Negative viscosity phenomena in three-dimensional flows, Phys. Review A 35 (2), 810–820 (1987)
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H. K. Moffatt, Magnetostatic equilibria and analogous Euler flows of arbitrary complex topology. Part 2. Stability considerations, J. Fluid Mech. 166, 359–387 (1986)
V. Yakhot and G. Sivashinsky, Negative viscosity phenomena in three-dimensional flows, Phys. Review A 35 (2), 815–820 (1987)
L. Shtilman and G. Sivashinsky, Negative viscosity effect in three-dimensional flows, J. de Physique 47, 1137–1140 (1986)
B. J. Bayly and V. Yakhot, Positive- and negative-effective-viscosity phenomena in isotropic and anisotropic Beltrami flows, Phys. Rev. A 34 (1), 381–391 (1986)
K. Gotoh and M. Yamada, The instability of rhombic cell flows, Fluid Dynamics Res. 1, 165–176 (1986)
D. Galloway and U. Frisch, A note on the stability of a family of space-periodic Beltrami flows, J. Fluid Mech. 180, 557–564 (1987)
- Zhen-Su She, Metastability and vortex pairing in the Kolmogorov flow, Phys. Lett. A 124 (1987), no. 3, 161–164. MR 910886, DOI https://doi.org/10.1016/0375-9601%2887%2990244-1
U. Frisch, Z. S. She, and P.-L. Sulem, Large-scale flow driven by the anisotropic kinetic alpha effect, Physica D 28, 382–392 (1987)
A. Libin, G. I. Sivashinsky, and E. Levich, Long-wave instability of periodic flows at large Reynolds numbers, Phys. Fluids 30, 2984–2986 (1987)
A. Libin and G. I. Sivashinsky, Long wavelength instability of triangular eddies at large Reynolds numbers, Phys. Lett. A 144, 172–178 (1990)
L. D. Meshalkin and Ya. G. Sinaǐ, Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous fluid, J. Appl. Math. Mech. 25, 1700–1705 (1961)
J. S. A. Green, Two-dimensional turbulence near the viscous limit, J. Fluid Mech. 62, 273–287 (1974)
A. A. Nepomniashchii, On stability of secondary flows of a viscous fluid in unbounded space, J. Appl. Math. Mech. 40, 836–841 (1976)
D. N. Beaumont, The stability of spatially periodic flows, J. Fluid Mech. 108, 461–474 (1981)
K. Gotoh, M. Yamada, and J. Mizushima, The theory of stability of spatially periodic parallel flows, J. Fluid Mech. 127, 45–58 (1983)
K. Gotoh and M. Yamada, Instability of a cellular flow, J. Phys. Soc. Japan 53, 3395–3398 (1984)
G. Sivashinsky and V. Yakhot, Negative viscosity phenomena in three-dimensional flows, Phys. Review A 35 (2), 810–820 (1987)
G. Sivashinsky, Weak turbulence in periodic flows, Physica D 17, 243–255 (1985)
H. K. Moffatt, Magnetostatic equilibria and analogous Euler flows of arbitrary complex topology. Part 2. Stability considerations, J. Fluid Mech. 166, 359–387 (1986)
V. Yakhot and G. Sivashinsky, Negative viscosity phenomena in three-dimensional flows, Phys. Review A 35 (2), 815–820 (1987)
L. Shtilman and G. Sivashinsky, Negative viscosity effect in three-dimensional flows, J. de Physique 47, 1137–1140 (1986)
B. J. Bayly and V. Yakhot, Positive- and negative-effective-viscosity phenomena in isotropic and anisotropic Beltrami flows, Phys. Rev. A 34 (1), 381–391 (1986)
K. Gotoh and M. Yamada, The instability of rhombic cell flows, Fluid Dynamics Res. 1, 165–176 (1986)
D. Galloway and U. Frisch, A note on the stability of a family of space-periodic Beltrami flows, J. Fluid Mech. 180, 557–564 (1987)
Z. S. She, Metastability and vortex pairing in the Kolmogorov flow, Phys. Lett. A 124, 161–164 (1987)
U. Frisch, Z. S. She, and P.-L. Sulem, Large-scale flow driven by the anisotropic kinetic alpha effect, Physica D 28, 382–392 (1987)
A. Libin, G. I. Sivashinsky, and E. Levich, Long-wave instability of periodic flows at large Reynolds numbers, Phys. Fluids 30, 2984–2986 (1987)
A. Libin and G. I. Sivashinsky, Long wavelength instability of triangular eddies at large Reynolds numbers, Phys. Lett. A 144, 172–178 (1990)
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Article copyright:
© Copyright 1990
American Mathematical Society
