Stability theory for a pair of trailing vortices
01 Dec 1970-AIAA Journal (American Institute of Aeronautics and Astronautics (AIAA))-Vol. 8, Iss: 12, pp 2172-2179
TL;DR: In this article, the authors considered the early stages of the formation of a train of vortex rings and found that their stability depends on the products of vortex separation 6 and cutoff distance d times the perturbation wavenumber.
Abstract: x(/3) Trailing vortices do not decay by simple diffusion. Usually they undergo a symmetric and nearly sinusoidal instability, until eventually they join at intervals to form a train of vortex rings. The present theory accounts for the instability during the early stages of its growth. The vortices are idealized as interacting lines; their core diameters are taken into account by a cutoff in the line integral representing self-induction. The equation relating induced velocity to vortex displacement gives rise to an eigenvalue problem for the growth rate of sinusoidal perturbations. Stability is found to depend on the products of vortex separation 6 and cutoff distance d times the perturbation wavenumber. Depending on those products, both symmetric and antisymmetric eigenmodes can be unstable, but only the symmetric mode involves strongly interacting long waves. An argument is presented that d/b = 0.063 for the vortices trailing from an elliptically loaded wing. In that case, the maximally unstable long wave has a length 8.66 and grows by a factor e in a time 9.4(^4#/CL)(6/F0), where AR is the aspect ratio, CL is the lift coefficient, and V0 is the speed of the aircraft. The vortex displacements are symmetric and are confined to fixed planes inclined at 48° to the horizontal.
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TL;DR: In this paper, the authors show that a large-scale orderly pattern may exist in the noiseproducing region of a round subsonic jet by observing the evolution of orderly flow with advancing Reynolds number.
Abstract: Past evidence suggests that a large-scale orderly pattern may exist in the noiseproducing region of a jet. Using several methods to visualize the flow of round subsonic jets, we watched the evolution of orderly flow with advancing Reynolds number. As the Reynolds number increases from order 102 to 103, the instability of the jet evolves from a sinusoid to a helix, and finally to a train of axisymmetric waves. At a Reynolds number around 104, the boundary layer of the jet is thin, and two kinds of axisymmetric structure can be discerned: surface ripples on the jet column, thoroughly studied by previous workers, and a more tenuous train of large-scale vortex puffs. The surface ripples scale on the boundary-layer thickness and shorten as the Reynolds number increases toward 105. The structure of the puffs, by contrast, remains much the same: they form at an average Strouhal number of about 0·3 based on frequency, exit speed, and diameter.To isolate the large-scale pattern at Reynolds numbers around 105, we destroyed the surface ripples by tripping the boundary layer inside the nozzle. We imposed a periodic surging of controllable frequency and amplitude at the jet exit, and studied the response downstream by hot-wire anemometry and schlieren photography. The forcing generates a fundamental wave, whose phase velocity accords with the linear theory of temporally growing instabilities. The fundamental grows in amplitude downstream until non-linearity generates a harmonic. The harmonic retards the growth of the fundamental, and the two attain saturation intensities roughly independent of forcing amplitude. The saturation amplitude depends on the Strouhal number of the imposed surging and reaches a maximum at a Strouhal number of 0·30. A root-mean-square sinusoidal surging only 2% of the mean exit speed brings the preferred mode to saturation four diameters downstream from the nozzle, at which point the entrained volume flow has increased 32% over the unforced case. When forced at a Strouhal number of 0·60, the jet seems to act as a compound amplifier, forming a violent 0·30 subharmonic and suffering a large increase of spreading angle. We conclude with the conjecture that the preferred mode having a Strouhal number of 0·30 is in some sense the most dispersive wave on a jet column, the wave least capable of generating a harmonic, and therefore the wave most capable of reaching a large amplitude before saturating.
2,108 citations
TL;DR: In this article, a general scheme for educing coherent structures in any transitional or fully turbulent flow is presented, based on smoothed vorticity maps in convenient flow planes, which recognizes patterns of the same mode and parameter size, and then phase-aligns and ensembles them to obtain coherent structure measures.
Abstract: This is a personal statement on the present state of understanding of coherent structures, in particular their spatial details and dynamical significance. The characteristic measures of coherent structures are discussed, emphasizing coherent vorticity as the crucial property. We present here a general scheme for educing structures in any transitional or fully turbulent flow. From smoothed vorticity maps in convenient flow planes, this scheme recognizes patterns of the same mode and parameter size, and then phase-aligns and ensemble-averages them to obtain coherent structure measures. The departure of individual realizations from the ensemble average denotes incoherent turbulence. This robust scheme has been used to educe structures from velocity data using a rake of hot wires as well as direct numerical simulations and can educe structures using newer measurement techniques such as digital image processing. Our recent studies of coherent structures in several free shear flows are briefly reviewed. Detailed data in circular and elliptic jets, mixing layers, and a plane wake reveal that incoherent turbulence is produced at the ‘saddles’ and then advected to the ‘centres’ of the structures. The mechanism of production of turbulence in shear layers is the stretching of longitudinal vortices or ‘ribs’ which connect the predominantly spanwise ‘rolls’; the ribs induce spanwise contortions of rolls and cause mixing and dissipation, mostly at points where they connect with rolls. We also briefly discuss the role of coherent structures in aerodynamic noise generation and argue that the structure breakdown process, rather than vortex pairing, is the dominant mechanism of noise generation. The ‘cut-and-connect’ interaction of coherent structures is proposed as a specific mechanism of aerodynamic noise generation, and a simple analytical model of it shows that it can provide acceptable predictions of jet noise. The coherent-structures approach to turbulence, apart from explaining flow physics, has also enabled turbulence management via control of structure evolution and interactions. We also discuss some new ideas under investigation: in particular, helicity as a characteristic property of coherent structures.
1,117 citations
TL;DR: A summary of physical insights gained into three-dimensional linear instability through solution of the two-dimensional partial-differential-equation-based nonsymmetric real or complex generalised eigenvalue problem is presented in this article.
485 citations
Cites background or methods from "Stability theory for a pair of trai..."
...The analogon of the Crow instability is denoted by the second symmetric mode S2, while it can be seen that the discovered modes S1 and A are respectively amplified by factors 10 and 5 stronger than S1; the spatial structure of the new eigenmodes is shown in Fig....
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...The first is akin to Crow instability and corresponds to wavelengths and growth rates that are, respectively, too large and too small to be interesting for the wake–vortex minimisation problem (top of figure)....
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...He employed the vortex-filament approach also used by Crow [44] and Jim!enez [164] and numerical solutions of the initial value problem resulting from matching the induced velocities that the Biot–Savart law delivers with kinematic conditions resulting from temporal differentiation of the position vectors of the vortex system....
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...Since the key objective of the current efforts is to provide recommendations leading to reduction of aircraftseparation limits, currently of the order of a few minutes, focus of instability analyses is on mechanisms other than the classic Crow [44] instability, the longwavelength nature of which excludes it from playing an active role in the sought process....
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...Prime examples of such modelling have been performed in the framework of the classic analysis of trailing vortex instability by Crow [44] and the recent efforts of de Bruin et al....
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TL;DR: A review of the formation, motion, and persistence of trailing vortices is presented in this article, which highlights findings or shifts made since Widnall's (1975) review in this series.
Abstract: ▪ Abstract This review surveys the formation, motion, and persistence of trailing vortices as relevant to the safety and productivity of air travel. It highlights findings or shifts made since Widnall's (1975) review in this series. This review also examines the predictability of the vortices (particularly in terms of lifespan), the durability of multiple vortex pairs, the controversy between expectations of vortex decay and of vortex collapse, the many types of turbulence that may influence the flow, the rich interplay between the rotational and the axial velocity fields in the vortex, the various atmospheric and ground-related factors that dominate the late behavior, a few instances of wakes rising back, and the still unexplained bursting of the vortices. The article also briefly covers prospects for detection and control.
484 citations
TL;DR: In this article, it was shown that this value is not only numerically predictable but also that it is expected to be a nonmonotonic function of the Richardson number that characterizes preturbulent stratification strength.
Abstract: ▪ Abstract The issue of the physical mechanism(s) that control the efficiency with which the density field in stably stratified fluid is mixed by turbulent processes has remained enigmatic. Similarly enigmatic has been an explanation of the numerical value of ∼0.2, which is observed to characterize this efficiency experimentally. We review recent work on the turbulence transition in stratified parallel flows that demonstrates that this value is not only numerically predictable but also that it is expected to be a nonmonotonic function of the Richardson number that characterizes preturbulent stratification strength. This value of the mixing efficiency appears to be characteristic of the late-time behavior of the turbulent flow that develops after an initially laminar shear flow has undergone the transition to turbulence through an intermediate instability of Kelvin-Helmholtz type.
416 citations
Cites background from "Stability theory for a pair of trai..."
...Arendt et al. (1997) and Fritts et al. (1998) also found that vortex-vortex interactions with longer wavelengths [associated with the cooperative Crow (1970) instability of neighboring elliptical vortices] could also induce turbulence, and it is not yet clear which of these competing mechanisms, if…...
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References
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TL;DR: In this paper, the authors show that a large-scale orderly pattern may exist in the noiseproducing region of a round subsonic jet by observing the evolution of orderly flow with advancing Reynolds number.
Abstract: Past evidence suggests that a large-scale orderly pattern may exist in the noiseproducing region of a jet. Using several methods to visualize the flow of round subsonic jets, we watched the evolution of orderly flow with advancing Reynolds number. As the Reynolds number increases from order 102 to 103, the instability of the jet evolves from a sinusoid to a helix, and finally to a train of axisymmetric waves. At a Reynolds number around 104, the boundary layer of the jet is thin, and two kinds of axisymmetric structure can be discerned: surface ripples on the jet column, thoroughly studied by previous workers, and a more tenuous train of large-scale vortex puffs. The surface ripples scale on the boundary-layer thickness and shorten as the Reynolds number increases toward 105. The structure of the puffs, by contrast, remains much the same: they form at an average Strouhal number of about 0·3 based on frequency, exit speed, and diameter.To isolate the large-scale pattern at Reynolds numbers around 105, we destroyed the surface ripples by tripping the boundary layer inside the nozzle. We imposed a periodic surging of controllable frequency and amplitude at the jet exit, and studied the response downstream by hot-wire anemometry and schlieren photography. The forcing generates a fundamental wave, whose phase velocity accords with the linear theory of temporally growing instabilities. The fundamental grows in amplitude downstream until non-linearity generates a harmonic. The harmonic retards the growth of the fundamental, and the two attain saturation intensities roughly independent of forcing amplitude. The saturation amplitude depends on the Strouhal number of the imposed surging and reaches a maximum at a Strouhal number of 0·30. A root-mean-square sinusoidal surging only 2% of the mean exit speed brings the preferred mode to saturation four diameters downstream from the nozzle, at which point the entrained volume flow has increased 32% over the unforced case. When forced at a Strouhal number of 0·60, the jet seems to act as a compound amplifier, forming a violent 0·30 subharmonic and suffering a large increase of spreading angle. We conclude with the conjecture that the preferred mode having a Strouhal number of 0·30 is in some sense the most dispersive wave on a jet column, the wave least capable of generating a harmonic, and therefore the wave most capable of reaching a large amplitude before saturating.
2,108 citations
TL;DR: In this paper, it was shown that the degree to which the vortices are rolled up depends upon the distance behind the wing and upon the lift coefficient, span loading, and aspect ratio of the wing.
Abstract: The motion of the trailing vortices associated with a lifting wing is investigated by theoretical and visual-flow methods for the purpose of determining the proper vortex distribution to be used for downwash calculations. Both subsonic and supersonic speeds are considered in the analysis. I t is found that the degree to which the vortices are rolled up depends upon the distance behind the wing and upon the lift coefficient, span loading, and aspect ratio of the wing. While the rolling up of the trailing vortices associated with high-aspectratio wings is of little practical importance, it is shown that, with low-aspect-ratio wings, the trailing vortex sheet may become essentially rolled up into two trailing vortex cores within a chord length of the trailing edge. The downwash fields associated with the two limiting cases of the flat vortex sheet and the fully rolled-up vortices are investigated in detail for both subsonic and supersonic speeds. The intermediate case in which the rolling-up process is only partially completed at the tail position is also discussed.
187 citations
TL;DR: In this paper, the authors investigated the effect of self-induction on the configuration of vortices in the wake behind a cylinder of an allowance for the thickness of the vortice.
Abstract: This paper is an attempt to investigate the effect on the configuration of vortices in the wake behind a cylinder of an allowance for the thickness of the vortices. The vortices themselves are assumed to be initially rectilinear and of equal circular section, and we assume also that they arrange themselves in an “unsymmetrical double row.” We therefore find a relationship between the “stability ratio” of the double row—that is, the ratio of the distance between the rows to the distance between consecutive vortices on the same row, in the stable configuration—and the diameters of the vortices. The problem in its initial stages can no longer be treated as one in two dimensions, for the “self-induction” of a vortex only enters when we deal with a three-dimensional disturbance, and it is the self-induction that produces the difference between this and the original treatment of the subject. By the “self-induction” of a vortex we mean the effect of the vortex on itself. The isolated rectilinear vortex is treated separately and the results obtained from it are extended to meet the case of the double row of rectilinear vortices. The three-dimensional stability of the Benard-Karman street has already been discussed, but the present treatment introduces various simplifications which, while not altering the general nature of the problem, make the expressions more amenable to treatment and yield results that appear to have been masked by the complexities of the algebra in the previous investigation. I would like to express my thanks to Dr. H. Jeffreys for many helpful criticisms which have had the effect of altering entirely certain sections of this paper.
138 citations
TL;DR: In this article, the effect of the viscosity in the inner core of the vortex ring is matched with the classical inviscid solution of the outer region by the boundary layer technique.
Abstract: In the classical inviscid theory of a vortex ring, the velocity at a point near the vortex ring becomes singular due to terms of r −1 and ln r where r is the shortest distance from the point to the vortex ring. Also the velocity of the vortex ring depends on the logarithm of the effective radius of the cross section of the vortex ring and is infinite for zero radius. The effect of the viscosity in the inner core of the vortex ring is now included and the inner viscous solution is matched with the classical inviscid solution of the outer region by the boundary layer technique. By means of the systematic matching, the singularities of r −1 and ln r in the classical inviscid theory is removed. By the requirement that the velocity at the center of the viscous core is finite, a unique and finite value is obtained for the velocity of the translation of the vortex ring which is decreasing with respect to time as − ln (ντ), where ν is the kinematic viscosity. From this analysis, the effective radius of the cross section of the vortex ring can be identified as 2(ντ)½. The variable τ is transformed from the time variable t by the relationship τ = ∫0 t R(t′)dt′/R(t), where R(t) is the radius of the ring.
85 citations