Evolution Equations for Finite Amplitude Waves in Parallel Shear Flows

A novel method is presented for obtaining rigorous upper bounds on the finite-amplitude growth of instabilities to parallel shear flows on the beta-plane. The method relies on the existence of finite-amplitude Liapunov (normed) stability theorems, due to Arnol'd, which are nonlinear generalizations of the classical stability theorems of Rayleigh and Fjørtoft. Briefly, the idea is to use the finite-amplitude stability theorems to constrain the evolution of unstable flows in terms of their proximity to a stable flow. Two classes of general bounds are derived, and various examples are considered. It is also shown that, for a certain kind of forced-dissipative problem with dissipation proportional to vorticity, the finite-amplitude stability theorems (which were originally derived for inviscid, unforced flow) remain valid (though they are no longer strictly Liapunov); the saturation bounds therefore continue to hold under these conditions.

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On the Nonlinear Stability of Inviscid Homogeneous Shear Flows in Sea Straits of Arbitrary Cross Sections

Mapana Journal of Sciences ◽

10.12723/mjs.22.2 ◽

2012 ◽

Vol 11 (3) ◽

pp. 31-50

Author(s):

V Ramakrishnareddy ◽

M Subbiah

Keyword(s):

Nonlinear Stability ◽

Cross Sections ◽

Shear Flows ◽

Stability Result ◽

Finite Amplitude ◽

Steady Flows ◽

Stability Theorems ◽

Sea Straits ◽

Parallel Shear Flows ◽

Special Case

In this paper we study the nonlinear stability of steady flows of inviscid homogeneous fluids in sea straits of arbitrary cross sections. We use the method of Arnol'd [1] to obtain two general stability theorems for steady basic flows with respect to finite amplitude disturbances. For the special case of plane parallel shear flows we find a finite amplitude extension of the linear stability result of Deng et al [2]. We also present some examples of basic flows which are stable to finite amplitude disturbances.

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Finite-Amplitude Waves in a Rotating Cylindrical Layer of Inhomogeneous and Homogeneous Fluids

International Journal of Fluid Mechanics Research ◽

10.1615/interjfluidmechres.v23.i1-2.140 ◽

1996 ◽

Vol 23 (1-2) ◽

pp. 122-126

Author(s):

P. A. Shestopal ◽

N. V. Saltanov

Keyword(s):

Finite Amplitude ◽

Cylindrical Layer ◽

Finite Amplitude Waves

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Finite-amplitude steady waves in plane viscous shear flows

Journal of Fluid Mechanics ◽

10.1017/s0022112085003482 ◽

1985 ◽

Vol 160 ◽

pp. 281-295 ◽

Cited By ~ 34

Author(s):

F. A. Milinazzo ◽

P. G. Saffman

Keyword(s):

Boundary Layer ◽

Reynolds Number ◽

Stokes Equations ◽

Reynolds Numbers ◽

Finite Amplitude ◽

Linear Instability ◽

Navier Stokes ◽

Unidirectional Flow ◽

Viscous Shear ◽

Finite Amplitude Waves

Computations of two-dimensional solutions of the Navier–Stokes equations are carried out for finite-amplitude waves on steady unidirectional flow. Several cases are considered. The numerical method employs pseudospectral techniques in the streamwise direction and finite differences on a stretched grid in the transverse direction, with matching to asymptotic solutions when unbounded. Earlier results for Poiseuille flow in a channel are re-obtained, except that attention is drawn to the dependence of the minimum Reynolds number on the physical constraint of constant flux or constant pressure gradient. Attempts to calculate waves in Couette flow by continuation in the velocity of a channel wall fail. The asymptotic suction boundary layer is shown to possess finite-amplitude waves at Reynolds numbers orders of magnitude less than the critical Reynolds number for linear instability. Waves in the Blasius boundary layer and unsteady Rayleigh profile are calculated by employing the artifice of adding a body force to cancel the spatial or temporal growth. The results are verified by comparison with perturbation analysis in the vicinity of the linear-instability critical Reynolds numbers.

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