Axial flow in laminar trailing vortices

Moore, D. W.; Saffman, P. G.

doi:10.1098/rspa.1973.0075

Cited by 141 publications

(33 citation statements)

References 13 publications

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“…Equation (66) for W is the equation of a 2D passive scalar following this flow. Many solutions of the 2D Euler equations are catalogued in Saffman's book [4] and also in [27][28][29][30][31]. Hence we look here at only two examples.…”

Section: Results For the Euler Equationsmentioning

confidence: 99%

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Dynamically stretched vortices as solutions of the 3D Navier–Stokes equations

Gibbon

Fokas

Doering

1999

Physica D: Nonlinear Phenomena

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A well known limitation with stretched vortex solutions of the 3D Navier-Stokes (and Euler) equations, such as those of Burgers type, is that they possess uni-directional vorticity which is stretched by a strain field that is decoupled from them. It is shown here that these drawbacks can be partially circumvented by considering a class of velocity fields of the type u u u = (u 1 (x, y, t), u 2 (x, y, t), γ (x, y, t)z + W (x, y, t)) where u 1 , u 2 , γ and W are functions of x, y and t but not z. It turns out that the equations for the third component of vorticity ω 3 and W decouple. More specifically, solutions of Burgers type can be constructed by introducing a strain field into u u u such that u u u = (−(γ /2)x − (γ /2)y, γ z) + −ψ y , ψ x , W . The strain rate, γ (t), is solely a function of time and is related to the pressure via a Riccati equationγ + γ 2 + p zz (t) = 0. A constraint on p zz (t) is that it must be spatially uniform. The decoupling of ω 3 and W allows the equation for ω 3 to be mapped to the usual general 2D problem through the use of Lundgren's transformation, while that for W can be mapped to the equation of a 2D passive scalar. When ω 3 stretches then W compresses and vice versa. Various solutions for W are discussed and some 2π -periodic θ -dependent solutions for W are presented which take the form of a convergent power series in a similarity variable. Hence the vorticity ω ω ω = r −1 W θ , −W r , ω 3 has nonzero components in the azimuthal and radial as well as the axial directions. For the Euler problem, the equation for W can sustain a vortex sheet type of solution where jumps in W occur when θ passes through multiples of 2π .

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Section: Results For the Euler Equationsmentioning

confidence: 99%

“…(29) shows that the sign of γ determines whether ω 3 and W grow or decay. The strain rate γ satisfies an equation of Riccati type which can be linearized.…”

Section: A Riccati Equation For γmentioning

confidence: 99%

Dynamically stretched vortices as solutions of the 3D Navier–Stokes equations

Gibbon

Fokas

Doering

1999

Physica D: Nonlinear Phenomena

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“…This account of the stretched-spiral vortex model proposed by Lundgren (1982) complements and extends an earlier review by Pullin & Saffman (1995). Approximate solutions to the unsteady Navier-Stokes equations corresponding to strictly twodimensional spiral vortices have been known since the 1930s and have found application to the structure of the starting vortex (see Saffman 1992, chapter 8) and the laminar trailing vortex (Moore & Saffman 1973). Lundgren (1982) adapted this structure to model the fine scales of turbulence, replacing the steady Burgers' vortices in the Townsend ensemble described in Section 2 by unsteady stretched-spiral vortices.…”

Section: The Lundgren Stretched-spiral Vortexmentioning

confidence: 99%

Vortex Dynamics in Turbulence

Pullin

Saffman

1998

Annu. Rev. Fluid Mech.

130

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“…Simplifying assumptions associated with the above method, 37,38 are well known. Nevertheless, when applying the method to a flow control problem, the limitations become less important when comparing changes, e.g.…”

Section: A Control Predictions Using Inviscid Rollup Relationsmentioning

confidence: 99%

Managing Flap Vortices via Separation Control

Greenblatt

2006

AIAA Journal

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A pilot study was conducted on a flapped semi-span model to investigate the concept and viability of near-wake vortex management by means of boundary layer separation control.Passive control was achieved using a simple fairing and active control was achieved via zero mass-flux blowing slots. Vortex sheet strength, estimated by integrating surface pressures, was used to predict vortex characteristics based on inviscid rollup relations and vortices trailing the flaps were mapped using a seven-hole probe. Separation control was found to have a marked effect on vortex location, strength, tangential velocity, axial velocity and size over a wide range of angles of attack and control conditions. In general, the vortex trends were well predicted by the inviscid rollup relations. Manipulation of the separated flow near the flap edges exerted significant control over either outboard or inboard edge vortices while producing small lift and moment excursions. Unsteady surface pressures indicated that dynamic separation and attachment control can be exploited to perturb vortices at wavelengths shorter than a typical wingspan. In summary, separation control has the potential for application to time-independent or time-dependent wake alleviation schemes, where the latter can be deployed to minimize adverse effects on ride-quality and dynamic structural loading.

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Axial flow in laminar trailing vortices

Cited by 141 publications

References 13 publications

Dynamically stretched vortices as solutions of the 3D Navier–Stokes equations

Dynamically stretched vortices as solutions of the 3D Navier–Stokes equations

Vortex Dynamics in Turbulence

Managing Flap Vortices via Separation Control

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