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

2018 ◽  
Vol 68 (8) ◽  
pp. 1001-1012 ◽  
Author(s):  
Maxim Kalashnik ◽  
Michael Kurgansky
Keyword(s):  
Nonlinear Dynamics ◽  
Reynolds Numbers ◽  
Kolmogorov Flow ◽  
Long Wave

scholarly journals NONLINEAR DYNAMICS OF LONG-WAVE PERTURBATIONS OF THE INVISCID KOLMOGOROV FLOW

2019 ◽  
Vol 47 (1) ◽  
pp. 64-65
Author(s):  
M.V. Kalashnik ◽  
M.V. Kurgansky
Keyword(s):  
Nonlinear Dynamics ◽  
Nonlinear Oscillations ◽  
Initial Perturbation ◽  
Analytical Formula ◽  
Basic Research ◽  
Elliptic Functions ◽  
Structural Features ◽  
Kolmogorov Flow ◽  
Long Wave ◽  
The Galerkin Method

The nonlinear dynamics of long-wave perturbations of the inviscid Kolmogorov flow, which models periodically varying in the horizontal direction oceanic currents, is studied. To describe this dynamics, the Galerkin method with basis functions representing the first three terms in the expansion of spatially periodic perturbations in the trigonometric series is used. The orthogonality conditions for these functions formulate a nonlinear system of partial differential equations for the expansion coefficients (Kalashnik, Kurgansky, 2018). Based on the asymptotic solutions of this system, a linear, quasilinear and nonlinear stage of perturbation dynamics are identified. It is shown that the time-dependent growth of perturbations during the first two stages is succeeded by the stage of stable nonlinear oscillations. The corresponding oscillations are described by the oscillator equation containing a cubic nonlinearity, which is integrated in terms of elliptic functions. An analytical formula for the period of oscillations is obtained, which determines its dependence on the amplitude of the initial perturbation. Structural features of the field of the stream function of the perturbed flow are described, associated with the formation of closed vortex cells and meandering flow between them. The research was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications» and by the Russian Foundation for Basic Research (Projects 18-05-00414, 18-05-00831).

On stability of parallel flow of an incompressible fluid of variable density and viscosity

1962 ◽  
Vol 58 (4) ◽  
pp. 646-661 ◽  
Author(s):  
P. G. Drazin
Keyword(s):  
Incompressible Fluid ◽  
Water Waves ◽  
Salt Water ◽  
Reynolds Numbers ◽  
Variable Density ◽  
Stability Equation ◽  
Wave Disturbances ◽  
Long Wave ◽  
Vertical Variations ◽  
The Stability

ABSTRACTSome aspects of generation of water waves by wind and of turbulence in a heterogeneous fluid may be described by the theory of hydrodynamic stability. The technical difficulties of these problems of instability have led to obscurities in the literature, some of which are elucidated in this paper. The stability equation for a basic steady parallel horizontal flow under the influence of gravity is derived carefully, the undisturbed fluid having vertical variations of density and viscosity. Methods of solution of the equation for large Reynolds numbers and for long-wave disturbances are described. These methods are applied to simple models of wind blowing over water and of fresh water flowing over salt water.

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

1998 ◽  
Vol 67 (5) ◽  
pp. 1597-1602 ◽  
Author(s):  
Hiroaki Fukuta ◽  
Youichi Murakami
Keyword(s):  
Side Wall ◽  
Wall Effect ◽  
Wave Instability ◽  
Kolmogorov Flow ◽  
Long Wave

Instability of a bed of particles sheared by a viscous flow

2002 ◽  
Vol 452 ◽  
pp. 303-323 ◽  
Author(s):  
FRANÇOIS CHARRU ◽  
HÉLÈNE MOUILLERON-ARNOULD
Keyword(s):  
Shear Stress ◽  
Viscous Flow ◽  
Fluid Layer ◽  
Mass Conservation ◽  
Reynolds Numbers ◽  
Bed Shear Stress ◽  
Fluid Inertia ◽  
Wave Instability ◽  
Long Wave ◽  
Small Shear

The instability of a bed of particles sheared by a viscous fluid is investigated theoretically. The viscous flow over the wavy bed is first calculated, and the bed shear stress is derived. The particle transport rate induced by this bed shear stress is calculated from the viscous resuspension theory of Leighton & Acrivos (1986). Mass conservation of the particles then gives explicit expressions for the wave velocity and growth rate, which depend on four dimensionless parameters: the wavenumber, the fluid thickness, a viscous length and the shear stress. The mechanism of the instability is given. It appears that for high enough fluid-layer thickness, long-wave instability arises as soon as grains move, while short waves are stabilized by gravity. For smaller fluid thickness, the destabilizing effect of fluid inertia is reduced, so that the moving at bed is stable for small shear stress, and unstable for high shear stress. The most amplified wavelength scales with the viscous length, in agreement with the few available experiments for small particle Reynolds numbers. The results are also compared with related studies for turbulent flow.

scholarly journals Study of the Reynolds Number Effect on the Process of Instability Transition Into the Turbulent Stage

2014 ◽  
Vol 136 (9) ◽  
Author(s):  
N. V. Nevmerzhitskiy ◽  
E. A. Sotskov ◽  
E. D. Sen'kovskiy ◽  
O. L. Krivonos ◽  
A. A. Polovnikov ◽  
...  
Keyword(s):  
Reynolds Number ◽  
Turbulent Mixing ◽  
High Speed ◽  
Reynolds Numbers ◽  
Short Wave ◽  
Solid Particles ◽  
Reynolds Number Effect ◽  
Long Wave ◽  
Compressed Gas ◽  
Perturbation Growth

The results of the experimental study of the Reynolds number effect on the process of the Rayleigh–Taylor (R-T) instability transition into the turbulent stage are presented. The experimental liquid layer was accelerated by compressed gas. Solid particles were scattered on the layer free surface to specify the initial perturbations in some experiments. The process was recorded with the use of a high-speed motion picture camera. The following results were obtained in experiments: (1) Long-wave perturbation is developed at the interface at the Reynolds numbers Re < 104. If such perturbation growth is limited by a hard wall, the jet directed in gas is developed. If there is no such limitation, this perturbation is resolved into the short-wave ones with time, and their growth results in gas-liquid mixing. (2) Short-wave perturbations specified at the interface significantly reduce the Reynolds number Re for instability to pass into the turbulent mixing stage.

Experimental study of interfacial long waves in a two-layer shear flow

1995 ◽  
Vol 303 ◽  
pp. 23-53 ◽  
Author(s):  
Pierre Barthelet ◽  
François Charru ◽  
Jean Fabre
Keyword(s):  
Nonlinear Dynamics ◽  
Second Harmonic ◽  
Interfacial Stability ◽  
Amplitude Equations ◽  
Wave Instability ◽  
Frequency Locking ◽  
Flow Parameters ◽  
Plate Velocity ◽  
Long Wave ◽  
Even Harmonics

Interfacial stability of two-layer Couette flow was investigated experimentally in a channel bent into an annular ring. This paper is focused on the supercritical long-wave instability which arises for a broad range of flow parameters. Above the critical upper plate velocity, a slowly growing long wave appears with wavelength equal to the perimeter of the channel. Transients of this wave were studied within the theoretical frame of amplitude equations obtained from the long-wave interface equation. Near the onset of instability, the unstable fundamental harmonic is described by the Landau–Stuart equation, and the nonlinear dynamics of the harmonics closely follows the central and slaved modes analysis. For the higher upper plate velocity, harmonics gain some autonomy but they eventually are enslaved by the fundamental, through remarkable collapses of amplitudes and phase jumps leading to wave velocity and frequency locking. Dispersive effects play a crucial role in the nonlinear dynamics. Far from the threshold, the second harmonic becomes unstable and bistability appears: the saturated wave is dominated either by the fundamental harmonic, or by the even harmonics, after periodic energy exchange.

scholarly journals Cluster-based network modeling—From snapshots to complex dynamical systems

2021 ◽  
Vol 7 (25) ◽  
pp. eabf5006
Author(s):  
Daniel Fernex ◽  
Bernd R. Noack ◽  
Richard Semaan
Keyword(s):  
Nonlinear Dynamics ◽  
Statistical Physics ◽  
Network Modeling ◽  
Network Connectivity ◽  
Dimensional Subspace ◽  
Data Driven ◽  
Lorenz Attractor ◽  
Universal Method ◽  
Time Resolved ◽  
Kolmogorov Flow

We propose a universal method for data-driven modeling of complex nonlinear dynamics from time-resolved snapshot data without prior knowledge. Complex nonlinear dynamics govern many fields of science and engineering. Data-driven dynamic modeling often assumes a low-dimensional subspace or manifold for the state. We liberate ourselves from this assumption by proposing cluster-based network modeling (CNM) bridging machine learning, network science, and statistical physics. CNM describes short- and long-term behavior and is fully automatable, as it does not rely on application-specific knowledge. CNM is demonstrated for the Lorenz attractor, ECG heartbeat signals, Kolmogorov flow, and a high-dimensional actuated turbulent boundary layer. Even the notoriously difficult modeling benchmark of rare events in the Kolmogorov flow is solved. This automatable universal data-driven representation of complex nonlinear dynamics complements and expands network connectivity science and promises new fast-track avenues to understand, estimate, predict, and control complex systems in all scientific fields.

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