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Journal Article•DOI•

The two- and three-dimensional instabilities of a spatially periodic shear layer

Raymond T. Pierrehumbert1, Sheila E. Widnall1•
Massachusetts Institute of Technology1
01 Jan 1982-Journal of Fluid Mechanics (Cambridge University Press)-Vol. 114, Iss: -1, pp 59-82
TL;DR: In this paper, the two-dimensional stability properties of coherent shear-layer vortices discovered by Stuart are investigated, and the stability problem is formulated as a non-separable eigenvalue problem in two independent variables, and solved numerically using spectral methods.
Abstract: The two- and three-dimensional stability properties of the family of coherent shear-layer vortices discovered by Stuart are investigated. The stability problem is formulated as a non-separable eigenvalue problem in two independent variables, and solved numerically using spectral methods. It is found that there are two main classes of instabilities. The first class is subharmonic, and corresponds to pairing or localized pairing of vortex tubes; the pairing instability is most unstable in the two-dimensional limit, in which the perturbation has no spanwise variations. The second class repeats in the streamwise direction with the same periodicity as the basic flow. This mode is most unstable for spanwise wavelengths approximately 2/3 of the space between vortex centres, and can lead to the generation of streamwise vorticity and coherent ridges of upwelling. Comparison is made between the calculated instabilities and the observed pairing, helical pairing, and streak transitions. The theoretical and experimental results are found to be in reasonable agreement.
Citations
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Journal Article•DOI•
Vortex Dynamics in the Cylinder Wake

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Charles H. K. Williamson
01 Jan 1996-Annual Review of Fluid Mechanics
TL;DR: A review of wake vortex dynamics can be found in this article, with a focus on the three-dimensional aspects of nominally two-dimensional wake flows, as well as the discovery of several new phenomena in wakes.
Abstract: Since the review of periodic flow phenomena by Berger & Wille (1972) in this journal, over twenty years ago, there has been a surge of activity regarding bluff body wakes. Many of the questions regarding wake vortex dynamics from the earlier review have now been answered in the literature, and perhaps an essential key to our new understandings (and indeed to new questions) has been the recent focus, over the past eight years, on the three-dimensional aspects of nominally two-dimensional wake flows. New techniques in experiment, using laser-induced fluorescence and PIV (Particle-Image-Velocimetry), are vigorously being applied to wakes, but interestingly, several of the new discoveries have come from careful use of classical methods. There is no question that strides forward in understanding of the wake problem are being made possible by ongoing three- dimensional direct numerical simulations, as well as by the surprisingly successful use of analytical modeling in these flows, and by secondary stability analyses. These new developments, and the discoveries of several new phenomena in wakes, are presented in this review.

3,206 citations

Journal Article•DOI•
Turbulence in plant canopies

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John Finnigan
01 Jan 2000-Annual Review of Fluid Mechanics
TL;DR: In this article, the mean velocity profile is inflected, second moments are strongly inhomogeneous with height, skewnesses are large, and second-moment budgets are far from local equilibrium.
Abstract: ▪ Abstract The single-point statistics of turbulence in the ‘roughness sub-layer’ occupied by the plant canopy and the air layer just above it differ significantly from those in the surface layer. The mean velocity profile is inflected, second moments are strongly inhomogeneous with height, skewnesses are large, and second-moment budgets are far from local equilibrium. Velocity moments scale with single length and time scales throughout the layer rather than depending on height. Large coherent structures control turbulence dynamics. Sweeps rather than ejections dominate eddy fluxes and a typical large eddy consists of a pair of counter-rotating streamwise vortices, the downdraft between the vortex pair generating the sweep. Comparison with the statistics and instability modes of the plane mixing layer shows that the latter rather than the boundary layer is the appropriate model for canopy flow and that the dominant large eddies are the result of an inviscid instability of the inflected mean velocity profi...

1,484 citations

Book Chapter•DOI•
Coherent eddies and turbulence in vegetation canopies: The mixing-layer analogy

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Michael R. Raupach1, John Finnigan1, Y. Brunet1•
Commonwealth Scientific and Industrial Research Organisation1
01 Mar 1996-Boundary-Layer Meteorology
TL;DR: In this paper, the authors argue that the active turbulence and coherent motions near the top of a vegetation canopy are patterned on a plane mixing layer, because of instabilities associated with the characteristic strong inflection in the mean velocity profile.
Abstract: This paper argues that the active turbulence and coherent motions near the top of a vegetation canopy are patterned on a plane mixing layer, because of instabilities associated with the characteristic strong inflection in the mean velocity profile. Mixing-layer turbulence, formed around the inflectional mean velocity profile which develops between two coflowing streams of different velocities, differs in several ways from turbulence in a surface layer. Through these differences, the mixing-layer analogy provides an explanation for many of the observed distinctive features of canopy turbulence. These include: (a) ratios between components of the Reynolds stress tensor; (b) the ratio K H /K M of the eddy diffusivities for heat and momentum; (c) the relative roles of ejections and sweeps; (d) the behaviour of the turbulent energy balance, particularly the major role of turbulent transport; and (e) the behaviour of the turbulent length scales of the active coherent motions (the dominant eddies responsible for vertical transfer near the top of the canopy). It is predicted that these length scales are controlled by the shear length scale L s = U(h)/U′(h) (where h is canopy height, U(z) is mean velocity as a function of height z, and U′ = dU/dz). In particular, the streamwise spacing of the dominant canopy eddies Λ x = mL s , with m = 8.1. These predictions are tested against many sets of field and wind-tunnel data. We propose a picture of canopy turbulence in which eddies associated with inflectional instabilities are modulated by larger-scale, inactive turbulence, which is quasi-horizontal on the scale of the canopy.

1,094 citations

Cites background from "The two- and three-dimensional inst..."

  • ...Next, three-dimensional instabilities lead rapidly to the development of longitudinal vorticity, in the form of “braid” or “rib” vortices in the highly-strained braid regions (Pierrehumbert and Widnall, 1982)....

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Journal Article•DOI•
Subharmonics and vortex merging in mixing layers

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Chih-Ming Ho1, Lein-Saing Huang1•
University of Southern California1
01 Jun 1982-Journal of Fluid Mechanics
TL;DR: In this paper, it was shown that the spreading rate of a mixing layer can be greatly manipulated at very low forcing level if the mixing layer is perturbed near a subharmonic of the most-amplified frequency.
Abstract: In the present study, it is shown that the spreading rate of a mixing layer can be greatly manipulated at very low forcing level if the mixing layer is perturbed near a subharmonic of the most-amplified frequency. The subharmonic forcing technique is able to make several vortices merge simultaneously and hence increases the spreading rate dramatically. A new mechanism, ‘collective interaction’, was found which can bypass the sequential stages of vortex merging and make a large number of vortices (ten or more) coalesce.A deeper physical insight into the evolution of the coherent structures is revealed through the investigation of a forced mixing layer. The stability and the forcing function play important roles in determining the initial formation of the vortices. The subharmonic starts to amplify at the location where the phase speed of the subharmonic matches that of the fundamental. The position where vortices are seen to align vertically coincides with the position where the measured subharmonic reaches its peak. This location is defined as the merging location, and it can be determined from the feedback equation (Ho & Nosseir 1981).The spreading rate and the velocity profiles of the forced mixing layer are distinctly different from the unforced case. The data show that the initial condition has a longlasting effect on the development of the mixing layer.

808 citations

Journal Article•DOI•
GLOBAL INSTABILITIES IN SPATIALLY DEVELOPING FLOWS: Non-Normality and Nonlinearity

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Jean-Marc Chomaz
12 Jan 2005-Annual Review of Fluid Mechanics
TL;DR: In this paper, the dynamics of open flows are considered as a superposition of linear or nonlinear instability waves that behave at each streamwise station as if the flow were homogeneous in the streamwise direction.
Abstract: The objective of this review is to critically assess the different approaches developed in recent years to understand the dynamics of open flows such as mixing layers, jets, wakes, separation bubbles, boundary layers, and so on. These complex flows develop in extended domains in which fluid particles are continuously advected downstream. They behave either as noise amplifiers or as oscillators, both of which exhibit strong nonlinearities (Huerre & Monkewitz 1990). The local approach is inherently weakly nonparallel and it assumes that the basic flow varies on a long length scale compared to the wavelength of the instability waves. The dynamics of the flow is then considered as a superposition of linear or nonlinear instability waves that, at leading order, behave at each streamwise station as if the flow were homogeneous in the streamwise direction. In the fully global context, the basic flow and the instabilities do not have to be characterized by widely separated length scales, and the dynamics is then viewed as the result of the interactions between Global modes living in the entire physical domain with the streamwise direction as an eigendirection. This second approach is more and more resorted to as a result of increased computational capability. The earlier review of Huerre & Monkewitz (1990) emphasized how local linear theory can account for the noise amplifier behavior as well as for the onset of a Global mode. The present survey first adopts the opposite point of view by demonstrating how fully global theory accounts for the noise amplifier behavior of open flows. From such a perspective, there is strong emphasis on the very peculiar nonorthogonality of linear Global modes, which in turn allows a novel interpretation of recent numerical simulations and experimental observations. The nonorthogonality of linear Global modes also imposes severe constraints on the extension of linear global theory to the fully nonlinear regime. When the flow is weakly nonparallel, this limitation is so severe that the linear Global mode theory is of little help. It is then much more appropriate to develop a fully nonlinear formulation involving the presence of a front separating the base state region from the bifurcated state region.

725 citations

Cites methods from "The two- and three-dimensional inst..."

  • ...The temporal stability of the Stuart vortex street was studied by Pierrehumbert & Widnall (1982) and more recently by Potylittsin & Peltier (1999)....

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References
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Journal Article•DOI•
On density effects and large structure in turbulent mixing layers

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Garry L. Brown1, Anatol Roshko2•
University of Adelaide1, California Institute of Technology2
24 Jul 1974-Journal of Fluid Mechanics
TL;DR: In this article, Spark shadow pictures and measurements of density fluctuations suggest that turbulent mixing and entrainment is a process of entanglement on the scale of the large structures; some statistical properties of the latter are used to obtain an estimate of entrainedment rates, and large changes of the density ratio across the mixing layer were found to have a relatively small effect on the spreading angle.
Abstract: Plane turbulent mixing between two streams of different gases (especially nitrogen and helium) was studied in a novel apparatus Spark shadow pictures showed that, for all ratios of densities in the two streams, the mixing layer is dominated by large coherent structures High-speed movies showed that these convect at nearly constant speed, and increase their size and spacing discontinuously by amalgamation with neighbouring ones The pictures and measurements of density fluctuations suggest that turbulent mixing and entrainment is a process of entanglement on the scale of the large structures; some statistical properties of the latter are used to obtain an estimate of entrainment rates Large changes of the density ratio across the mixing layer were found to have a relatively small effect on the spreading angle; it is concluded that the strong effects, which are observed when one stream is supersonic, are due to compressibility effects, not density effects, as has been generally supposed

3,339 citations

Journal Article•DOI•
Vortex pairing : the mechanism of turbulent mixing-layer growth at moderate Reynolds number

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C. D. Winant1, F. K. Browand1•
University of Southern California1
03 Apr 1974-Journal of Fluid Mechanics
TL;DR: A mixing layer is formed by bringing two streams of water, moving at different velocities, together in a lucite-walled channel as mentioned in this paper, where dye is injected between the two streams just before they are brought together, marking the vorticitycarrying fluid.
Abstract: A mixing layer is formed by bringing two streams of water, moving at different velocities, together in a lucite-walled channel. The Reynolds number, based on the velocity difference and the thickness of the shear layer, varies from about 45, where the shear layer originates, to about 850 at a distance of 50 cm. Dye is injected between the two streams just before they are brought together, marking the vorticity-carrying fluid. Unstable waves grow, and fluid is observed to roll up into discrete two-dimensional vortical structures. These turbulent vortices interact by rolling around each other, and a single vortical structure, with approximately twice the spacing of the former vortices, is formed. This pairing process is observed to occur repeatedly, controlling the growth of the mixing layer. A simple model of the mixing layer contains, as the important elements controlling growth, the degree of non-uniformity in the vortex train and the ‘lumpiness’ of the vorticity field.

1,335 citations

Additional excerpts

  • ...For p = 0-25, the spanwise wavelength associated with the translative instability is about half that characteristic of helical pairing. This scale is smaller than the scale implied by the spanwise correlation experiments reviewed in the previous section, Moreover, the broad wavenumber range in which the translative instabilit, -7 occurs does not reflect the definite scaling of observed correlation length with local vorticity thickness as well as does the helical pairing mode. There is a degree of arbitrariness in the comparison of instability scales with correlation lengths, and it is therefore difficuit to say with certainty the degree to which the translative instability is connected with the observed large-scale three-dimensional structures as reflected in the correlation measurements. However, there is a phenomenon occurring in the shear layer which seems to correspond to the translative instability. This phenomenon is the streaky structure observed by Breidenthal (1978) and by Bernal et al. (1979). The streaks set in with a spanwise spacing somewhat less than the wavelength of the initial Kelvin-Helmholtz instability; this is consistent with the properties of the translative instability, as the translative instability has a growth rate comparable to that of the pairing instability, so that it should set in simultaneously with the first pairing. This picture is also supported by the earlier work of Miksad (1972), who reported that the region of subharmonic growth following equilibration of the initial KelvinHelmholtz instability is coincident with the region of initial generation of threedimensionality. Like the translative instability, the streak pattern survives and grows through several pairings with little change in scale. The following description, from Bernal et al. (1979), underscores the resemblance to the translative instability: ‘The streaks sometirries originate from a sinusoidal pattern that develops in the spanwise vortices which emerge from the Kelvin-Helmholz instability....

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  • ...For p = 0-25, the spanwise wavelength associated with the translative instability is about half that characteristic of helical pairing. This scale is smaller than the scale implied by the spanwise correlation experiments reviewed in the previous section, Moreover, the broad wavenumber range in which the translative instabilit, -7 occurs does not reflect the definite scaling of observed correlation length with local vorticity thickness as well as does the helical pairing mode. There is a degree of arbitrariness in the comparison of instability scales with correlation lengths, and it is therefore difficuit to say with certainty the degree to which the translative instability is connected with the observed large-scale three-dimensional structures as reflected in the correlation measurements. However, there is a phenomenon occurring in the shear layer which seems to correspond to the translative instability. This phenomenon is the streaky structure observed by Breidenthal (1978) and by Bernal et al. (1979). The streaks set in with a spanwise spacing somewhat less than the wavelength of the initial Kelvin-Helmholtz instability; this is consistent with the properties of the translative instability, as the translative instability has a growth rate comparable to that of the pairing instability, so that it should set in simultaneously with the first pairing....

    [...]

  • ...For p = 0-25, the spanwise wavelength associated with the translative instability is about half that characteristic of helical pairing. This scale is smaller than the scale implied by the spanwise correlation experiments reviewed in the previous section, Moreover, the broad wavenumber range in which the translative instabilit, -7 occurs does not reflect the definite scaling of observed correlation length with local vorticity thickness as well as does the helical pairing mode. There is a degree of arbitrariness in the comparison of instability scales with correlation lengths, and it is therefore difficuit to say with certainty the degree to which the translative instability is connected with the observed large-scale three-dimensional structures as reflected in the correlation measurements. However, there is a phenomenon occurring in the shear layer which seems to correspond to the translative instability. This phenomenon is the streaky structure observed by Breidenthal (1978) and by Bernal et al. (1979). The streaks set in with a spanwise spacing somewhat less than the wavelength of the initial Kelvin-Helmholtz instability; this is consistent with the properties of the translative instability, as the translative instability has a growth rate comparable to that of the pairing instability, so that it should set in simultaneously with the first pairing. This picture is also supported by the earlier work of Miksad (1972), who reported that the region of subharmonic growth following equilibration of the initial KelvinHelmholtz instability is coincident with the region of initial generation of threedimensionality....

    [...]

  • ...For p = 0-25, the spanwise wavelength associated with the translative instability is about half that characteristic of helical pairing. This scale is smaller than the scale implied by the spanwise correlation experiments reviewed in the previous section, Moreover, the broad wavenumber range in which the translative instabilit, -7 occurs does not reflect the definite scaling of observed correlation length with local vorticity thickness as well as does the helical pairing mode. There is a degree of arbitrariness in the comparison of instability scales with correlation lengths, and it is therefore difficuit to say with certainty the degree to which the translative instability is connected with the observed large-scale three-dimensional structures as reflected in the correlation measurements. However, there is a phenomenon occurring in the shear layer which seems to correspond to the translative instability. This phenomenon is the streaky structure observed by Breidenthal (1978) and by Bernal et al. (1979). The streaks set in with a spanwise spacing somewhat less than the wavelength of the initial Kelvin-Helmholtz instability; this is consistent with the properties of the translative instability, as the translative instability has a growth rate comparable to that of the pairing instability, so that it should set in simultaneously with the first pairing. This picture is also supported by the earlier work of Miksad (1972), who reported that the region of subharmonic growth following equilibration of the initial KelvinHelmholtz instability is coincident with the region of initial generation of threedimensionality. Like the translative instability, the streak pattern survives and grows through several pairings with little change in scale. The following description, from Bernal et al. (1979), underscores the resemblance to the translative instability: ‘The streaks sometirries originate from a sinusoidal pattern that develops in the spanwise vortices which emerge from the Kelvin-Helmholz instability. Amplification of the pattern’s amplitude, possibly due to straining, as it convects downstream with the vortices tends to stretch out the streamwise oriented segments of the pattern; the streaks apparently result from this stretching.’ In figure 11 of Breidenthal (1978) a possible instance of this development is shown....

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Journal Article•DOI•
On finite amplitude oscillations in laminar mixing layers

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J. T. Stuart1•
Imperial College London1
06 Sep 1967-Journal of Fluid Mechanics
TL;DR: In this paper, a mixing layer of tanh y form is considered, and twodimensional solutions of the non-linear inviscid equations are found representing periodic perturbations from the neutral wave of linearized stability theory.
Abstract: In the first part of the paper, a mixing layer of tanh y form is considered, and twodimensional solutions of the non-linear inviscid equations are found representing periodic perturbations from the neutral wave of linearized stability theory. To second order in amplitude the solutions are equivalent to the equilibrium state calculated by Schade (1964), who studied the development of perturbations in time and found an evolution towards that equilibrium state. The present calculation, however, is taken to fourth-order in amplitude. It is noted that the solutions presented in this paper are regular, even though viscosity is ignored; and the relationships to the singular (if inviscid) time-dependent solutions of Schade are explained. Such regular, inviscid solutions have been found only for odd velocity profiles, such as tanh y.Although the details of the velocity distributions are not of the form observed experimentally, it is shown that the amplitude ratios of fundamental and first harmonic, for a given absolute amplitude, are comparable with those observed.In part 2 some exact non-linear solutions are presented of the inviscid, incompressible equations of fluid flow in two or three spatial dimensions. They illustrate the flows of part 1, since they are periodic in one co-ordinate (x), have a shear in another (y) and are independent of the third. Included, as two-dimensional cases, are (i) the tanh y velocity distribution for a flow wholly in the x-direction, (ii) the well-known solution for the flow due to a set of point vortices equi-spaced on the axis, and (iii) an example of linearized hydrodynamic (Orr-Sommerfeld) stability theory. The flows may involve concentrations of vorticity. In three-dimensional cases the z component of velocity is even in y, whereas the x component is odd. Consequently, the class of flows represents, in general, small or large periodic perturbations from a skewed shear layer. Time-dependent solutions, representing waves travelling in the x direction may be obtained by translation of axes.

373 citations

Journal Article•DOI•
The instability of a straight vortex filament in a strain field

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Derek William Moore1, Philip Geoffrey Saffman2•
Imperial College London1, California Institute of Technology2
04 Nov 1975-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences
TL;DR: In this article, a straight infinite vortex of finite cross section is deformed by the action of weak irrotational plane strain, and it is shown that the deformed vortex is unstable to disturbances whose axial wavelengths lie in a narrow band, whose width is proportional to the imposed strain.
Abstract: A straight infinite vortex of finite cross section is deformed by the action of weak irrotational plane strain. The deformed vortex is shown, in the absence of axial flow, to be unstable to disturbances whose axial wavelengths lie in a narrow band, whose width is proportional to the imposed strain. The band is centred on the wavelength of the helical wave which does not propagate on the unstrained circular vortex. Thus support is given to the instability mechanism proposed recently by Widnall, Bliss & Tsai (1974). The argument depends, however, on the mirror image of the helical wave also being a possible non-propagating disturbance on the unstrained vortex.

275 citations

Journal Article•DOI•
The stability of short waves on a straight vortex filament in a weak externally imposed strain field

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Chon-Yin Tsai1, Sheila E. Widnall1•
Massachusetts Institute of Technology1
24 Feb 1976-Journal of Fluid Mechanics
TL;DR: In this paper, the stability of short-wave displacement perturbations on a vortex filament of constant vorticity in a weak externally imposed strain field is considered and the growth rate is calculated by linear stability theory.
Abstract: The stability of short-wave displacement perturbations on a vortex filament of constant vorticity in a weak externally imposed strain field is considered. The circular cross-section of the vortex filament in this straining flow field becomes elliptical. It is found that instability of short waves on this strained vortex can occur only for wavelengths and frequencies at the intersection points of the dispersion curves for an isolated vortex. Numerical results show that the vortex is stable at some of these points and unstable at others. The vortex is unstable at wavelengths for which ω = 0, thus giving some support to the instability mechanism for the vortex ring proposed recently by Widnall, Bliss & Tsai (1974). The growth rate is calculated by linear stability theory. The previous work of Crow (1970) and Moore & Saffman (1971) dealing with long-wave instabilities is discussed as is the very recent work of Moore & Saffman (1975).

268 citations

Related Papers (5)
On density effects and large structure in turbulent mixing layers

[...]

24 Jul 1974-Journal of Fluid Mechanics
Garry L. Brown, Anatol Roshko
Streamwise vortex structure in plane mixing layers

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01 Sep 1986-Journal of Fluid Mechanics
Luis P. Bernal, Anatol Roshko
Secondary instability of a temporally growing mixing layer

[...]

01 Nov 1987-Journal of Fluid Mechanics
Ralph W. Metcalfe, Steven A. Orszag, Marc Brachet, Suresh Menon, James J. Riley 
Three-dimensional instability of a plane free shear layer: an experimental study of the formation and evolution of streamwise vortices

[...]

01 Apr 1988-Journal of Fluid Mechanics
Juan C. Lasheras, H. Choi
Perturbed Free Shear Layers

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01 Jan 1984-Annual Review of Fluid Mechanics
Chih-Ming Ho, Patrick Huerre