Stability of Kolmogorov flow in a channel with rigid walls

Oparina, E. I.; Troshkin, O. V.

doi:10.1134/1.1815419

Cited by 12 publications

(6 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The results of this analysis were a good match with those of Thess [29] and Kolesnikov [45]. Oparina and Troshkin [46] performed an analytical study to determine the stability of the k-flow in a channel with rigid walls. Their study displays that k-flows with short periods will remain stable inside a channel with rigid walls.…”

Section: Stability Analysis and Drag Reductionsupporting

confidence: 67%

Kolmogorov Flow: Seven Decades of History

Fylladitakis¹

2018

JAMP

View full text Add to dashboard Cite

The Kolmogorov flow (k-flow) is generated by a stationary sinusoidal force that varies in space. This flow is rather academic since generating such a periodic forcing in an unbounded flow is difficult to appear in nature. Nevertheless, it allows for simple experimental measurements and for a fairly detailed analytical treatment. Although simple, the k-flow makes a good test case for investigating simultaneously inhomogeneous, sheared, and anisotropic features in a flow, and several studies concerning the stability, transition, and turbulence of the k-flow have been published. The present article reviews the most important published works incorporating the k-flow as a test-bed for studying fluid mechanics, testing numerical or experimental methods, or even studying the properties of the k-flow itself.

show abstract

Section: Stability Analysis and Drag Reductionsupporting

confidence: 67%

Kolmogorov Flow: Seven Decades of History

Fylladitakis¹

2018

JAMP

View full text Add to dashboard Cite

show abstract

“…Theorem 2.In weakly compact algebra M with sub ordinated and positive inertia at ν ≥ 0 and exact and strong dissipation at ν > 0, enstrophy norm ||Aϕ|| of variation ϕ of normal free rotation(9) for any perturba tion χ = from(7)satisfies the estimates Indeed, at JOINT MODES OF INERTIA AND DISSIPATION Theorem 3. The presence of the following system of joint modes ζ orthogonal in scalar product to inertia (ξ, η) A = (ξ, Aη) and complete in its norm ||ξ|| A = , with discrete (as having no finite limiting points) moments μ > 0, for positive inertia A and exact dissipation B…”

mentioning

confidence: 84%

“…They were already useful in obtaining the conditions of nonlinear stabil ity of basic flows on smooth two dimensional varieties with compact closing and the Kolmogorov flow in a channel at the no slip conditions [7][8][9].…”

Section: Basic Flows and Rotations Of The Topmentioning

confidence: 99%

A dissipative top in a weakly compact lie algebra and stability of basic flows in a plane channel

Troshkin

2012

Dokl. Phys.

Self Cite

View full text Add to dashboard Cite

Based on the equation of an abstract dissipative top, it is shown that the linear and parabolic velocity profiles are stable with respect to arbitrary smooth per turbations of the initial data in a flat periodic channel filled with an ideal or viscous incompressible medium, under the impermeability OR no slip boundary condi tions, respectively.The stability of the Couette flow with a linear velocity profile and the instability of the Poiseuille flow with a parabolic profile for a viscous incompressible fluid in a plane channel had been proved at restricted three dimensional (the less the molecular viscosity, the less they are) [1] and indefinitely small two dimensional perturbations of the velocity field [2], respectively.Here, we consider arbitrary two dimensional per turbations of the initial data in a periodic cell V = {0 ≤ x ≤ l, 0 ≤ y ≤ h} of the ideal or viscous incompressible medium with impermeability or no slip conditions at the cell walls y = 0, h, respectively. Allowing for equiv alency of weak and strong solutions of the Euler or Navier equations for plane parallel flows in a bounded region and the impossibility of increasing the smooth ness of the flow due to the boundary and initial condi tions [3-5], we below show the nonlinear stability of the mentioned basic flows with respect to two dimen sional smooth perturbations at any norm of the initial vorticity of the velocity field in the space of quadrati cally integrable functions L 2 (V). The functional algebraic constructions used can be considered as an alternative continuation of the known group theory analogy between differomor phisms of the flow region conserving the element of the volume and rotations of the top (an absolutely rigid body with one fastened point) [6]. They were already useful in obtaining the conditions of nonlinear stabil ity of basic flows on smooth two dimensional varieties with compact closing and the Kolmogorov flow in a channel at the no slip conditions [7-9].In contrast with the latter, the flows under study are free of the rotor of the field of external body forces and correspond to normal (in the direct and indirect sense) rotation of a typical top: the "kinetic moment" (a vor tex, or vorticity of the velocity field) of the basic flow of a continuous top generated by hydrodynamic equa tions is normal, i.e., orthogonal (in the sense of the scalar product of space L 2 (V)) to all its gyroscopic moments as "vector products" (the Poisson brackets) of "angular velocities" (functions of the flow) and their "kinetic moments" (vorticities).For example, rotation of the whirligig is normal. This simplest gyroscope has two equal moments of inertia μ 2 = μ 3 = μ (the top is symmetric), the residual moment μ 1 ≠ μ unequal to them (the top is nontrivial), and a constant vector of angular velocity ψ = (ψ 1 , 0, 0), ψ 1 ≠ 0, the kinetic moment of which Aψ = μ 1 ψ is directed along the axis of the isolated basic moment (the top is normal).The gyroscopic moment [χ, Aχ], which is deter mined by the angular velocity χ = (χ 1 , …), by the k...

show abstract

“…The latter enabled us to conduct some qualitative analysis of the set of solutions of each subsystem (with respect to the compatibility, finiteness or infiniteness of the set of the subsystems' solutions, etc.) and hence to obtain information about the whole set of system's (11) solutions (respectively, (10) and (9)) and find out some groups of solutions.…”

Section: Obtaining Stationary Solutionsmentioning

confidence: 99%

“…Such analogies allow one to conduct, for example, analysis of stability in problems of above type by classical methods of rigid body dynamics, and to suggest clear interpretation of results obtained [10].…”

Section: Introductionmentioning

confidence: 99%

On One Approach to Investigation of Mechanical Systems

Irtegov¹

2006

SIGMA

View full text Add to dashboard Cite

Abstract. The paper presents some results of qualitative analysis of Kirchhoff's differential equations describing motion of a rigid body in ideal fluid in Sokolov's case. The research methods are based on Lyapunov's classical results. Methods of computer algebra implemented in the computer algebra system (CAS) "Mathematica" were also used. Combination of these methods allowed us to obtain rather detailed information on qualitative properties for some classes of solutions of the equations.

show abstract

Stability of Kolmogorov flow in a channel with rigid walls

Cited by 12 publications

References 2 publications

Kolmogorov Flow: Seven Decades of History

Kolmogorov Flow: Seven Decades of History

A dissipative top in a weakly compact lie algebra and stability of basic flows in a plane channel

On One Approach to Investigation of Mechanical Systems

Contact Info

Product

Resources

About