Side-Wall Effect on the Long-Wave Instability in Kolmogorov Flow

Fukuta, Hiroaki; Murakami, Youichi

doi:10.1143/jpsj.67.1597

Cited by 5 publications

(5 citation statements)

References 7 publications

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“…As in figure 2, the triangles and cross-hatched line mark the critical Reynolds numbers in the bounded and unbounded incompressible problems, respectively. At first we observe that the lower left part of the neutral curve approaches the incompressible limit in an unbounded domain with increasing n, in accordance with the result of Fukuta & Murakami (1998). This is accompanied by a decrease in the critical wavenumber (not presented here), which, in the unbounded problem, obtains the limit k cr = 0 (infinitely long waves).…”

Section: The Case Nsupporting

confidence: 88%

“…The corresponding DSMC results, denoted by circles, crosses and squares, mark parameter combinations where the reference state is found to be stable, unstable and marginally stable, respectively. The triangle depicts the critical Reynolds number Re cr ≈ 2.65 for 38 A. Manela and J. Zhang the onset of instability according to incompressible analysis in a bounded domain for n = 4 (Fukuta & Murakami 1998), and the cross-hatched line marks the critical value Re cr = √ 2 in the counterpart unbounded set-up (Meshalkin & Sinai 1961).…”

Section: Stability Analysismentioning

confidence: 99%

“…The triangle depicts the value of k cr in the incompressible limit (Fukuta & Murakami 1998) and the dashed lines mark two of the wavenumbers (k/π = 4, 5) included in the discrete spectrum of DSMC calculation (see § 3). Again we note that our solution coincides with the incompressible solution for Kn → 0.…”

Section: Stability Analysismentioning

confidence: 99%

“…Later on, Fukuta & Murakami (1998) found that the effect of bounding surfaces at finite n is to stabilize the system by increasing the value of Re cr . However, in the limit n → ∞ it was shown that the impact of the walls vanishes and the critical Reynolds number obtains its unbounded limit.…”

Section: The Case Nmentioning

confidence: 99%

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The effect of compressibility on the stability of wall-bounded Kolmogorov flow

Manela

Zhang

2012

J. Fluid Mech.

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We extend the stability analysis of incompressible Kolmogorov flow, induced by a spatially periodic external force in an unbounded domain, to a compressible hardsphere gas confined between two parallel isothermal walls. The two-dimensional problem is studied by means of temporal stability analysis of a 'slip flow' continuumlimit model and the direct simulation Monte Carlo (DSMC) method. The neutral curve is obtained in terms of the Reynolds (Re) and Knudsen (Kn) numbers, for a given non-dimensional wavenumber (2πn) of the external force. In the incompressible limit (Kn, Kn Re → 0), the problem is governed only by the Reynolds number, and our neutral curve coincides with the critical Reynolds number (Re cr ) calculated in previous incompressible analyses. Fluid compressibility (Kn, Kn Re = 0) affects the flow field through the generation of viscous dissipation, coupling flow shear rates with irreversible heat production, and resulting in elevated bulk-fluid temperatures. This mechanism has a stabilizing effect on the system, thus increasing Re cr (compared to its incompressible value) with increasing Kn. When compressibility effects become strong enough, transition to instability changes type from 'exchange of stabilities' to 'overstability', and perturbations are dominated by fluctuations in the thermodynamic fields. Most remarkably, compressibility confines the instability to small (O(10 −3 )) Knudsen numbers, above which the Kolmogorov flow is stable for all Re. Good agreement is found between 'slip flow' and DSMC analyses, suggesting the former as a useful alternative in studying the effects of various parameters on the onset of instability, particularly in the context of small Knudsen numbers considered.

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Section: The Case Nsupporting

confidence: 88%

Section: Stability Analysismentioning

confidence: 99%

Section: Stability Analysismentioning

confidence: 99%

Section: The Case Nmentioning

confidence: 99%

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The effect of compressibility on the stability of wall-bounded Kolmogorov flow

Manela

Zhang

2012

J. Fluid Mech.

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“…She (1988); Platt et al (1991); Armbruster et al (1996)), forcing form (e.g. Gotoh & Yamada (1987); Kim & Okamoto (2003); Rollin et al (2011); Gallet & Young (2013)), boundary conditions (Fukuta & Murakami (1998); Thess (1992); Gallet & Young (2013)), and 3-dimensionalisation (e.g. Borue & Orszag (2006); Shebalin & Woodruff (1997); Sarris et al (2007); Musacchio & Boffetta (2014)).…”

Section: Introductionmentioning

confidence: 99%

Spatiotemporal dynamics in two-dimensional Kolmogorov flow over large domains

Lucas

Kerswell

2014

J. Fluid Mech.

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Kolmogorov flow in two dimensions -the two-dimensional (2D) Navier-Stokes equations with a sinusoidal body force -is considered over extended periodic domains to reveal localised spatiotemporal complexity. The flow response mimics the forcing at small forcing amplitudes but beyond a critical value develops a long wavelength instability. The ensuing state is described by a Cahn-Hilliard-type equation and as a result coarsening dynamics is observed for random initial data. After further bifurcations, this regime gives way to multiple attractors, some of which possess spatially localised time dependence. Co-existence of such attractors in a large domain gives rise to interesting collisional dynamics which is captured by a system of 5 (1-space and 1-time) partial differential equations (PDEs) based on a long wavelength limit. The coarsening regime reinstates itself at yet higher forcing amplitudes in the sense that only longest-wavelength solutions remain attractors. Eventually, there is one global longest-wavelength attractor which possesses two localised chaotic regions -a kink and antikink -which connect two steady one-dimensional (1D) flow regions of essentially half the domain width each. The wealth of spatiotemporal complexity uncovered presents a bountiful arena in which to study the existence of simple invariant localised solutions which presumably underpin all of the observed behaviour.

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Nonlinear dynamics of long-wave perturbations of the Kolmogorov flow for large Reynolds numbers

Калашник

Kurgansky

2018

Ocean Dynamics

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Side-Wall Effect on the Long-Wave Instability in Kolmogorov Flow

Cited by 5 publications

References 7 publications

The effect of compressibility on the stability of wall-bounded Kolmogorov flow

The effect of compressibility on the stability of wall-bounded Kolmogorov flow

Spatiotemporal dynamics in two-dimensional Kolmogorov flow over large domains

Nonlinear dynamics of long-wave perturbations of the Kolmogorov flow for large Reynolds numbers

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