The mixing layer: deterministic models of a turbulent flow. Part 1. Introduction and the two-dimensional flow

Corcos, G. M.; Sherman, Frederick S.

doi:10.1017/s0022112084000252

Cited by 223 publications

(121 citation statements)

References 24 publications

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“…Here, we show that a geometric phase contribution arises during the "vortex pairing" stage of nonlinear shear layer evolution in a simple two-dimensional model for the process thought to be fundamental for the generation of small-scale motion and enhanced mixing in a wide range of more complicated real flows [35], [36]. In the model, an infinite row of evenly spaced, equal strength vortices is given a subharmonic perturbation so that neighboring vortices pair up and undergo periodic motion as shown in Figure 4.…”

Section: Problem 3: Pairing In An Infinite Row Of Point Vorticesmentioning

confidence: 92%

Vortex Motion and the Geometric Phase. Part I. Basic Configurations and Asymptotics

Shashikanth

Newton

1998

J. Nonlinear Sci.

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Summary.The geometric, or Hannay-Berry, phase is calculated for three canonical point vortex configurations in the plane. The simplest configuration is the three-vortex problem with arbitrary (like signed) circulations, where two of the vortices are near each other compared to the distance between them and a third vortex. We show that the third (distant) vortex induces a geometric phase in the relative angle variable between the two nearby vortices. The second configuration is a particle and vortex in a circular domain. In this problem, the geometric phase is induced on the particle by the presence of the boundary. The third configuration is an infinite row of point vortices undergoing subharmonic pairing. In this case, a geometric phase is induced on a particle orbiting one of the vortices as the vortex pairs orbit each other. In each case we derive the formula for the geometric phase using an asymptotic procedure, then we give it a geometric interpretation. For the asymptotic derivation, we show how the geometric phase can be interpreted as the product of two terms, one of which goes to zero, the other to infinity. Because they go at rates that balance each other, there is a residual O(1) term in the limit → 0. In this way, we can see that the → 0 problem is fundamentally different from the = 0 problem. For the geometric interpretation, we introduce a 1-form γ defined on the plane and show that the phase θ g in the appropriate angle variable can be constructed by taking the contour integral of this 1-form over the closed vortex path. In each case this gives the simple formula θ g = γ .

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Section: Problem 3: Pairing In An Infinite Row Of Point Vorticesmentioning

confidence: 92%

Vortex Motion and the Geometric Phase. Part I. Basic Configurations and Asymptotics

Shashikanth

Newton

1998

J. Nonlinear Sci.

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“…On the other hand, the inviscid calculations are more difficult to compare with experiments since they represent an idealised limit. The work of Corcos and Sherman [8] goes further in identifying clearly the two distinct stages of the nonlinear evolution of a 2-D shear layer. The first stage is the roll-up of the interface around a local vorticity maximum.…”

Section: Introductionmentioning

confidence: 98%

“…In Corcos and Sherman [8] and in Pozrikidis and Higdon [9], one goal is to determine the growth rate of interfacial area between two separated fluid regions. This quantity is of great interest to chemical engineers who study reacting streams.…”

Section: Introductionmentioning

confidence: 99%

“…One of the conclusions of this work is that the growth rate of interfacial area approaches a constant value for all shear layers they consider, a fact consistent with the experiments of Breidenthal [10], who found constant reaction rates as long as the mixing layer maintains its two-dimensional structure. This paper should be read in conjunction with that of Corcos and Sherman [8], who perform a finite difference calculation on the corresponding viscous problem. In their relatively low Reynolds number simulation, there is a rapid diffusion of vorticity in contrast to the inviscid calculation of Pozrikidis and Higdon [9].…”

Section: Introductionmentioning

confidence: 99%

“…The interface evolves into a spiral formation where the marker particles migrate inward along the arms and accumulate near their center. The second stage is the by now well-documented [11] vortex pairing process in which, due to a dominant subharmonic instability, neighbouring vortices pair and orbit each other as the spiral arms continue to wrap locally and evolve globally (see, for example, Figure 4 of Corcos and Sherman [8]). In general, the process is complex and well studied-see, for example, the review of Ho and Huerre [12].…”

Section: Introductionmentioning

confidence: 99%

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Vortex Motion and the Geometric Phase. Part II. Slowly Varying Spiral Structures

Shashikanth

Newton

1999

J. Nonlinear Sci.

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Summary.We derive formulas for the long time evolution of passive interfaces in three "canonical" incompressible, inviscid, two-dimensional flow models. The point vortex models, introduced in Part 1 [1] are (i) a "restricted" three-vortex problem, (ii) a vortex and a particle in a closed circular domain, and (iii) a particle in the flowfield of a mixing layer model undergoing a vortex pairing instability. In each configuration, it was shown in Part 1 that the passive particle exhibits a geometric or Hannay-Berry phase over long time periods induced by the slowly varying periodic background field. In this paper we show how the formula for the evolution of a passive interface driven by the dynamics of the vortices inherits this geometric phase effect. The interface wraps into a spiral formation around the "parent" vortex, with a slowly varying component induced by the farfield vorticity. The length formula for the long time growth of the slowly rotating spiral decomposes into a "dynamic" part and a "geometric" part. The dynamic part is the length in the "unperturbed" system-i.e., in the absence of the background field-and the geometric part is the contribution of the geometric phase θ g for a passive particle in the flow. We derive the following simple formula for an interface along a smooth curve joining two arbitrary particles labelled A and B. Define L(T ) as the difference in interface lengths between the "unperturbed" system and the "perturbed" slowly varying system at the end of the long time period T . In each case,where ξ is the radial coordinate parametrizing the interface at t = 0. * Current address: Control and Dynamical Systems,

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Bibliography

2014

Transport and Coherent Structures in Wall Turbulence

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The mixing layer: deterministic models of a turbulent flow. Part 1. Introduction and the two-dimensional flow

Cited by 223 publications

References 24 publications

Vortex Motion and the Geometric Phase. Part I. Basic Configurations and Asymptotics

Vortex Motion and the Geometric Phase. Part I. Basic Configurations and Asymptotics

Vortex Motion and the Geometric Phase. Part II. Slowly Varying Spiral Structures

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