Some general properties of plane-parallel viscous flows

Golubkin, V. N.; Sizykh, G. B.

doi:10.1007/bf01051932

Cited by 17 publications

(9 citation statements)

References 2 publications

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“…Let us recall that, with the correspondence of magnetic field to vorticity, the force-free motion of charged particles in magnetic field has mathematical similarity to Beltrami flow. The idea in [25,26] parallels the expectation in the Newtonian fluid dynamics that an increase in the Reynolds number tends to align velocity and vorticity [27,28]. Later, the importance of the maximization of helicity [21] and the minimization of magnetic energy due to dissipation in the evolution of magnetic field of nontrivial topology has been pointed out in [29,30].…”

Section: Introductionmentioning

confidence: 86%

“…Note that v(q, k, t) has a Gaussian damping factor with respect to q and k. At the same time, it exhibits a temporally exponential decay. Spectrum function f (q, k) is the remaining factor of v. The v(q, k, 0) will be determined from the Fourier-Hankel transformation of v θ obtained (numerically) by solving (27) or (28) for an unknown t 0 and by extracting the Gaussian damping factor from the solution at large q or k. Finally, the time dependence is recovered by substituting the damping factor exp(−ν(q 2 + k 2 )(t + t 0 )) as in (37).…”

Section: Time Dependencementioning

confidence: 99%

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Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations

Takahashi

2023

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This paper is aimed at eliciting consistency conditions for the existence of unsteady incompressible axisymmetric swirling viscous Beltrami vortices and explicitly constructing solutions that obey the conditions as well as the Navier–Stokes equations. By Beltrami flow, it is meant that vorticity, i.e., the curl of velocity, is proportional to velocity at any local point in space and time. The consistency conditions are derived for the proportionality coefficient, the velocity field and external force. The coefficient, whose dimension is of [length−1], is either constant or nonconstant. In the former case, the well-known exact nondivergent three-dimensional unsteady vortex solutions are obtained by solving the evolution equations for the stream function directly. In the latter case, the consistency conditions are given by nonlinear equations of the stream function, one of which corresponds to the Bragg–Hawthorne equation for steady inviscid flow. Solutions of a novel type are found by numerically solving the nonlinear constraint equation at a fixed time. Time dependence is recovered by taking advantage of the linearity of the evolution equation of the stream function. The proportionality coefficient is found to depend on space and time. A phenomenon of partial restoration of the broken scaling invariance is observed at short distances.

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Section: Introductionmentioning

confidence: 86%

Section: Time Dependencementioning

confidence: 99%

Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations

Takahashi

2023

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“…The second typethe plane-parallel and axisymmetric nontwisted flows of viscous incompressible fluid, for which according to Refs. [6,7]…”

Section: Introductionmentioning

confidence: 99%

On the vorticity behind 3-D detached bow shock wave

Golubkin

Sizykh

2019

Adv. Aerodyn.

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For the general case of a spatial isoenergetic flow of ideal gas, Helmholtz's theorems are generalized and the speed with which vortex tubes move is found, keeping the intensity. It is shown that along the streamline without stagnation point, vorticity either is equal to zero everywhere, or it is non zero at all. The pattern of vortex lines behind the three-dimensional detached bow shock wave is specified.

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“…Golubkin & Sizykh (1987) showed that R is directed along lines u 2 /2+p = const. It can be easily shown that the vorticity flux is equal to the difference between the total pressures at the arc ends (Chernyshenko 1998):…”

Section: Problem Formulation and Some Properties Of The Solutionmentioning

confidence: 99%

On the uniqueness of steady flow past a rotating cylinder with suction

2000

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The subject of this study is a steady two-dimensional incompressible flow past a rapidly rotating cylinder with suction. The rotation velocity is assumed to be large enough compared with the cross-flow velocity at infinity to ensure that there is no separation. High-Reynolds-number asymptotic analysis of incompressible NavierStokes equations is performed. Prandtl's classical approach of subdividing the flow field into two regions, the outer inviscid region and the boundary layer, was used earlier by Glauert (1957) for analysis of a similar flow without suction. Glauert found that the periodicity of the boundary layer allows the velocity circulation around the cylinder to be found uniquely. In the present study it is shown that the periodicity condition does not give a unique solution for suction velocity much greater than 1/Re. It is found that these non-unique solutions correspond to different exponentially small upstream vorticity levels, which cannot be distinguished from zero when considering terms of only a few powers in a large Reynolds number asymptotic expansion. Unique solutions are constructed for suction of order unity, 1/Re, and 1/ √ Re. In the last case an explicit analysis of the distribution of exponentially small vorticity outside the boundary layer was carried out.

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Some general properties of plane-parallel viscous flows

Cited by 17 publications

References 2 publications

Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations

Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations

On the vorticity behind 3-D detached bow shock wave

On the uniqueness of steady flow past a rotating cylinder with suction

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