Absolute and global instability of plane submerged jets

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The family of velocity profiles of a submerged jet, which are absolutely unstable in the plane-parallel approximation, is considered. The profiles are specified by two parameters, the first of them is responsible for the location of the only inflection point in the velocity profile, and the second is responsible for the shear layer thickness. An algorithm for determining the length of the section of local absolute instability of the jet with a given input velocity profile, that is, the distance at which absolute instability gives way to convective instability, has been implemented. The dependence of this length on the parameters defining the input profile is obtained. A connection between the characteristics of local absolute instability calculated in the plane-parallel approximation and global instability of the jet evolving in space is analytically obtained. The input velocity profile that corresponds to sufficiently large length of the zone of local absolute instability, at which global instability of spatially developing jet occurs is demonstrated. Thus, the possibility of existence of global instability of plane submerged jets with special velocity distributions is demonstrated.

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作者简介

V. Vedeneev

Moscow State University, Institute of Mechanics

编辑信件的主要联系方式.
Email: vasily@vedeneev.ru
俄罗斯联邦, Moscow

L. Gareev

Moscow State University, Institute of Mechanics

Email: gareev@imec.msu.ru
俄罗斯联邦, Moscow

Ju. Zayko

Moscow State University, Institute of Mechanics

Email: zayko@imec.msu.ru
俄罗斯联邦, Moscow

N. Exter

Moscow State University, Institute of Mechanics

Email: exter@imec.msu.ru
俄罗斯联邦, Moscow

参考

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2. Fig. 1. Velocity profiles defined by a pair of parameters ξ, ζ: ξ ≤ 0.5, ζ = 1 (a), ξ = 0.5, ζ ≥ 1 (b), ξ = 0.25, ζ ≥ 1 (c). The red circle marks the inflection point.

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3. Fig. 2. Computational domain for the symmetrical half of a flat unidirectional submerged jet.

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4. Fig. 3. Velocity profiles (a) and the corresponding level lines Im α = 0 on the plane for a jet with parameters ξ = 0.25, ζ = 2 (b). Dashed curves indicate the velocity profiles and level lines for x = 1, dotted for x = 2, dashed-dotted for x = 4, and solid for x = 6.6.

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5. Fig. 4. Level lines Im ω = 0 on the plane for cases of absolute (a) and convective (b) instability for jet velocity profiles at different distances from the origin. The black dot denotes the saddle point. The signs “±” in the gray circles show the directions of decrease and increase of Im ω in the vicinity of the saddle point. The thin line with arrows shows the integration path. The type of curves corresponds to Fig. 3.

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6. Fig. 5. Dependence of the length of the local absolute instability section on the parameters ξ, ζ. The red dots indicate the velocity profiles considered.

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7. Fig. 6. Velocity profiles for x = 0.0, 0.8, 1.8, 3.4, 5.0, 7.2 with the initial profile corresponding to ξ = 0.1, ζ = 1.6, the circles show the position of the inflection point (a). Velocity values ​​at the profile inflection point depending on the longitudinal coordinate x (b).

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8. Fig. 7. Mapping of the line Im ω = Im ωs into the plane α for the jet profile corresponding to the distance x = 0 (a), 0.8 (b), 1.8 (c), 3.4 (d), 5.0 (d), 7.2 (e) from the initial cross-section. The calculation based on the Rayleigh equation is shown in black, the approximation with the parameters of Table 1 is shown in gray, the saddle point α = αs is shown by the circle.

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9. Fig. 8. Dependence of the increment of local absolute instability Im ωs on the longitudinal coordinate x.

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