Large-Scale Instability of Generalized Oscillating Kolmogorov Flows

The stability of an incompressible unidirectional flow that depends periodically, but otherwise arbitrarily, on a transverse coordinate and on time is considered. An iterative solution of an infinite-dimensional eigenvalue problem is constructed by a rigorous perturbation method. The critical Reynol...

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Bibliographic Details
Published in:SIAM journal on applied mathematics 1998-04, Vol.58 (2), p.540-564
Main Authors: Zhang, Xiaojing, Frenkel, Alexander L.
Format: Article
Language:English
Subjects:
Algebra
Coordinate systems
Critical frequencies
Critical values
Eddy viscosity
Eigenvalues
Flow velocity
Matrices
Reynolds number
Scholarships & fellowships
Spacetime
Time dependence
Velocity
Viscosity
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Summary:The stability of an incompressible unidirectional flow that depends periodically, but otherwise arbitrarily, on a transverse coordinate and on time is considered. An iterative solution of an infinite-dimensional eigenvalue problem is constructed by a rigorous perturbation method. The critical Reynolds number Rc and the critical direction for which the large-scale "eddy viscosity" is minimum (and equal to zero) are determined by a system of two algebraic equations. For both time-independent and time-dependent cases, it turns out that the fastest-growing critical disturbances generally do not have the same transverse periodicity as that of basic flow. In the limit of large frequencies of oscillation, stability is essentially determined by the time-averaged flow. When the latter vanishes, the flow is absolutely stable for sufficiently large frequencies.
ISSN:0036-1399
1095-712X
DOI:10.1137/s003613999630527x