A vortex in an infinite viscous fluid

Long, Robert R.

doi:10.1017/s0022112061000767

Cited by 117 publications

(95 citation statements)

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“…Its computation requires consideration of the jet development region for x ϳO(1), giving a value that depends on S and on the shape functions u i (r) and ⌫ i (r). Note that the SchlichtingGörtler-Loitsianskii asymptotic solution is fundamentally different from the asymptotic solutions given by Long 15 and Fernández-Feria et al 16 for jets with outer circulation, in that the axial and azimuthal motions remain intimately coupled in the latter solutions.…”

Section: ͑4͒mentioning

confidence: 76%

The quasi-cylindrical description of submerged laminar swirling jets

Revuelta

Sánchez

Liñán³

2004

Physics of Fluids

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The quasi-cylindrical approximation is used to describe numerically the structure of a submerged swirling jet for subcritical values of the swirl ratio SрS c . The emerging flow structure is affected by the swirling motion, which enhances the entrainment rate of the jet and induces an adverse pressure gradient that reduces its momentum flux. The effect is more pronounced as the swirl ratio S is increased, yielding for sufficiently large values of S a jet with an annular structure. The integration describes the smooth transition towards the far-field self-similar solution for all values of S smaller than a critical value SϭS c , at which the numerical integration fails to converge at a given downstream location. The comparisons with previous experimental results confirm the correspondence between the onset of vortex breakdown and the failure of the quasi-cylindrical approximation. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1645850͔ This Brief Communication investigates the submerged swirling jet that forms when an incompressible fluid of density and kinematic viscosity v discharges with both forward and swirling motion through a circular orifice of radius a into a stagnant region of the same fluid, a flow configuration recently studied experimentally by Billant et al. 1 The initial fluxes of momentum J and angular momentum L of the jet can be used to define the swirl ratio SϭL/(Ja) and the jet Reynolds number R j ϭ͓J/( )͔ 1/2 /v as the main parameters characterizing the flow structure. The description below corresponds to moderately large values of R j , for which the laminar jet remains steady, and to values of S of order unity, a distinguished regime that allows us to explore the effect of the swirl on the flow structure and the onset of vortex breakdown. [2][3][4][5] Near the orifice, the jet is separated from the outer stagnant fluid by an annular mixing layer that thickens downstream, so that the effect of viscosity starts reducing significantly the velocity at the axis at distances of the order of R j times the jet radius. The swirling motion causes the pressure near the axis to be smaller than the ambient value, which in turn induces an adverse axial pressure gradient that is largest at the axis. 3 For SϳO(1), the pressure differences induced are of the order of the dynamic pressure, so that the swirling motion results in a significant flow deceleration in the jet development region, additional to that associated with viscous stresses. The magnitude of the azimuthal flow velocity is seen to decay more rapidly that the axial velocity, so that in the far-field self-similar solution that arises at distances from the orifice much larger than R j a the pressure gradient induced no longer affects the forward motion at leading order. 6 For swirl ratios below a critical value of order unity, the jet remains slender, and can be therefore described with relative errors of order R j Ϫ2 with the boundary-layer, or quasi-cylindrical ͑QC͒ approximation. 7 The resulting set of parabolic equations is integrate...

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Section: ͑4͒mentioning

confidence: 76%

The quasi-cylindrical description of submerged laminar swirling jets

Revuelta

Sánchez

Liñán³

2004

Physics of Fluids

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“…Since there is no source of momentum outside the half-line vortex, the surface can be chosen rather arbitrarily. For example, the plane, z = z 0 > 0, is an appropriate choice which Long (1961) applied to the near-axis boundary layer. The corresponding dimensionless parameters are the swirl Reynolds number Re s = Γ b (−1) and J 0 = J /(2πρν 2 ) or Long's parameter M = 2πJ 0 /Re 2 s .…”

Section: Stability Of Swirling Jetsmentioning

confidence: 99%

“…Conical flows include swirl-free round jets (Schlichting 1933;Landau 1944;Squire 1952), swirling jets (Long 1961), and many other flows (e.g. see Shtern & Hussain 1998, referred to herein as SH98).…”

Section: Introductionmentioning

confidence: 99%

Effect of deceleration on jet instability

Shtern

Hussain

2003

J. Fluid Mech.

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A non-parallel analysis of time-oscillatory instability of conical jets reveals important features not found in prior studies. Flow deceleration significantly enhances the shearlayer instability for both swirl-free and swirling jets. In swirl-free jets, flow deceleration causes the axisymmetric instability (absent in the parallel approximation). The critical Reynolds number Re a for this instability is an order of magnitude smaller than the critical Re a predicted before for the helical instability (where Re a = rv a /ν, r is the distance from the jet source, v a is the jet maximum velocity at a given r, and ν is the viscosity). Swirl, intensifying the divergence of streamlines, induces an additional, divergent instability (which occurs even in shear-free flows). For the swirl Reynolds number Re s (circulation to viscosity ratio) exceeding 3, the critical Re a for the single-helix counter-rotating mode becomes smaller than those for axisymmetric and multi-helix modes. Since the critical Re s is less than 10 for the near-axis jets, the boundary-layer approximation (used before) is invalid, as is Long's Type II boundary-layer solution (whose stability has been extensively studied). Thus, the nonparallel character of jets strongly affects their stability. Our results, obtained in a far-field approximation allowing reduction of the linear stability problem to ordinary differential equations, are more valid for short wavelengths.

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“…Long [20] obtained a solution for a line vortex in an infinite viscous fluid. Similarity arguments led to a reduction of the Navier-Stokes equations to a set of ordinary differential equations, which can be simplified by a boundary-layer approximation.…”

Section: Introductionmentioning

confidence: 99%

“…Similarity arguments led to a reduction of the Navier-Stokes equations to a set of ordinary differential equations, which can be simplified by a boundary-layer approximation. The simplified equations were integrated numerically by Long [20]. The inviscid stability of Long's vortex was studied by Foster and Duck [21] and by Foster and Smith [22], the latter included many references on stability problems involving a variety of vortices.…”

Section: Introductionmentioning

confidence: 99%

The swirling round laminar jet

Hwang

Chwang

1992

J Eng Math

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The swirling round laminar jet in an unbounded viscous fluid is investigated in this paper. The axisymmetric laminar jet with a swirling velocity is simulated by a linear-momentum source and an angularmomentum source, both located at the origin. The first-order and the second-order solutions in the far field have been obtained by solving the complete Navier-Stokes equations. It is found that the first-order solution is the well-known round-laminar-jet solution without the swirling velocity obtained by Landau [2] and Squire [3]. The second-order solution represents a pure rotating flow. The swirling velocity predicted by the present solution is compared with that obtained by Loitsyanskii [15] and G6rtler [16], who solved the corresponding boundary-layer equations. It is found that the swirling velocity predicted by the present theory is smaller than that obtained from the boundary-layer equations.

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A vortex in an infinite viscous fluid

Cited by 117 publications

References 3 publications

The quasi-cylindrical description of submerged laminar swirling jets

The quasi-cylindrical description of submerged laminar swirling jets

Effect of deceleration on jet instability

The swirling round laminar jet

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