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iii 

TABLE OF CONTENTS 

NOMENCLATURE ............................................................................................................................. v 

SUMMARY .......................................................................................................................................... 1 

I. INTRODUCTION ............................................................................................................................. 1 

II. OVERVIEW OF THRUST-BASED INITIATION PROCESS ...................................................... 3

A. Negligible Engine Trust ............................................................................................................... 3

B. Robust Engine Thrust ................................................................................................................... 3 

III. OBSERVATIONS AND PHOTOGRAPHS OF THRUST-BASED INITIATION ...................... 6

A. Overview ...................................................................................................................................... 6

B. Shear-Layer Vortex Mesh Around Exhaust Streams of Jet Engines ........................................... 7

C. From Roll-Up Through Linking of Vortex Pair ........................................................................... 9 

IV. TIME-TO-LINKING ESTIMATE OF LINKING ....................................................................... 12 

V. CONCLUSIONS............................................................................................................................ 15 

APPENDIX: CONFIRMATION OF THEORY BASED ON AMBIENT TURBULENCE ............. 16

A. Overview .................................................................................................................................... 16

B. Parameters Used ......................................................................................................................... 16

C. Vortex-Linking Experiments in Water Tow Tanks ................................................................... 17 

REFERENCES ................................................................................................................................... 18
iv 

LIST OF FIGURES 

Figure 1. Qualitative computation of assimilation of engine flow-through streams into rolled-up 

vortex wake at various distances behind generating aircraft (engine thrust is negligible) 

(ref. 42). .............................................................................................................................. 4 

Figure 2. Qualitative computation of formation of shield composed of engine-exhaust products 

(open circles) around the vortex wake (filled circles) when engine thrust is robust 

(ref. 42). .............................................................................................................................. 5 

Figure 3. Diagonal view of condensation trail displayed by jet-engineaexhaust streams of 

two-engine aircraft. Small amounts of condensate visualize shear-layer vortex arrays 

around both exhaust streams in part (a), and their merger in part (c) as ice crystals 

sublime. ............................................................................................................................... 8 

Figure 4. View of condensation trail from aircraft with 4 jet engines as the two streams from 

engines on each side of the aircraft rotate about each other because of the action of the 

vortex wake and then merge; the view is about 0 to 10 bg. Merger of port and starboard 

jet streams does not occur until about 100 bg after aircraft passage. Lift-generated vortices 

are not visible. ..................................................................................................................... 8 

Figure 5. Photographs of progress of structure of condensation trail from when first shed by 

aircraft to shortly after linking and loop formation of lift-generated vortices. ................. 10 

Figure 6. Comparison of spacings between shear-layer vortices with spacings between loops 

of vortex filaments that are in the process of diffusing. ................................................... 11 

Figure 7. Minimum increase in time to linking when long-wave instability is initiated by 

shear-layer vortex array. ................................................................................................... 12
v 

NOMENCLATURE 

AR b2

/S 

b wingspan, ft (m) 

bg spanwise distance between vortex centers, ft (m) 

CL L/qS 

L ρU|Γo| bg, lb (N) 

q ρU

2

/2, lb/ft2 (N/m2) 

S wing planform area, ft2 (m2) 

t time, s

y, z distance in lateral and vertical directions, ft (m) 

u, v, w velocity components in x, y, and z directions, ft/s (m/s) 

U velocity of wake-generating aircraft, ft/s (m/s) 

Γ circulation bound in wing, ft2/sec (m2/s) 

λ wavelength, ft (m) 

ρ air density, slugs/ft3 (kg/m3) 

(ε bg)1/3/wpr Crow-Bate turbulence parameter 

τ tLinkwpr/bg 

Subscripts 

fil vortex filament 

g wake-generating aircraft 

Link time to vortex linking 

o reference quantity 

pr vortex pair 

 free-stream condition 
vi 
1 

INITIATION OF LONG-WAVE INSTABILITY OF VORTEX PAIRS 

AT CRUISE ALTITUDES 

Vernon J. Rossow1

Ames Research Center 

SUMMARY 

Previous studies have usually attributed the initiation of the long-wave instability of a vortex pair to 

turbulence in the atmosphere or in the wake of the aircraft. The purpose here is to show by use of 

observations and photographs of condensation trails shed by aircraft at cruise altitudes that another 

initiating mechanism is not only possible but is usually the mechanism that initiates the long-wave 

instability at cruise altitudes. The alternate initiating mechanism comes about when engine thrust is 

robust enough to form an array of circumferential vortices around each jet-engineaexhaust stream. In 

those cases, initiation begins when the vortex sheet shed by the wing has rolled up into a vortex pair 

and descended to the vicinity of the inside bottom of the combined shear-layer vortex arrays. It is the 

in-and-out (up and down) velocity field between sequential circumferential vortices near the bottom 

of the array that then impresses disturbance waves on the lift-generated vortex pair that initiate the 

long-wave instability. A time adjustment to the Crow and Bate estimate for vortex linking is then 

derived for cases when thrust-based linking occurs. 

I. INTRODUCTION 

The mutually induced or long-wave instability of a vortex pair (refs. 1a3) is generally recognized as 

the aerodynamic mechanism that most rapidly decomposes the organized flow field of lift-generated 

vortex wakes. Unfortunately, the instability is not self-starting but requires that a sequence of wavelike in-phase disturbances be impressed along the pair of lift-generated vortex filaments to initiate 

the process. When the dynamics of the instability was first derived, it was assumed that the 

instability was initiated by ambient turbulence in the atmosphere. Analysis showed that more intense 

levels of turbulence in the atmosphere would produce more rapid growth in the amplitude of the 

disturbance waves and thereby require less time to complete the decomposition of lift-generated 

vortex wakes. An overview presented in the appendix summarizes experimental studies carried out 

in water tow tanks that confirm the analysis of Crow (ref. 1) and Crow and Bate (ref. 3) when 

the instability is initiated by ambient turbulence or by artificially generated disturbance waves 

(refs. 3a11). 

 

1

 Aviation Systems Division, Ames Research Center, Moffett Field, CA 94035-1000. 
2 

The discovery and analysis of the long-wave instability was of immediate interest to airport 

managers and to the operators of subsonic transport aircraft because persistence of the wakes of 

preceding aircraft was (and still is) the primary limiting impediment to airport capacity. Research 

that followed was directed at improved understanding of the instability and at finding other, more 

rapid ways to initiate the instability. It was hoped that a mechanism could be found that would 

decrease the time needed for the instability to do its work of decomposition, or that would decrease 

the hazard posed by the vortex wakes shed by larger aircraft in the transport fleet (refs. 4a28). 

Studies were also conducted on how best to measure the turbulent structure of the atmosphere and 

on how best to relate the measurements to the parameters that control the initiation and growth of the 

long-wave instability (refs. 29a37). 

The foregoing studies were informative but did not find a way to use the instability so that it would 

solve the airport capacity problem, largely because the research did not find a satisfactory initiating 

solution. The effort did, however, provide much information about the characteristics of the 

instability. For example, the research efforts showed that, in addition to atmospheric turbulence, the 

long-wave instability could also be initiated by pitching motions (refs. 1a3 and 6), by roll-oscillation 

motions (refs. 10 and 11), or by special flap arrangements on the wake-generating aircraft 

(refs. 5a16). It was also found that a given wake-alleviation procedure could be successful on some 

aircraft but not on others. For example, roll oscillations were very effective in initiating the longwave instability to thereby quickly reduce the wake hazard posed by the Boeing B-747 (refs. 10, 11, 

and 16). However, the same roll-oscillation maneuver did not induce the instability in the wake of 

the Lockheed L-1011, but substantially increased the magnitude and the duration of the hazard 

posed by the wake. 

The research described herein was developed as part of a continuing effort by the author to use 

condensation trails behind aircraft at cruise altitudes to observe and study the time-dependent 

structure of lift-generated vortex wakes (ref. 17). The effort is directed at a search for useable 

methods for rapid reduction in the hazard level posed by vortex wakes and for improved prediction 

methods for the time-dependent size of the hazardous region (refs. 37a41). This paper reports on 

observations of condensation trails of aircraft at cruise altitudes to determine how vortex wakes 

decompose and spread as a function of time when engine thrust is robust. The observations 

described indicate that robust thrust levels used by aircraft during cruise cause the long-wave 

instability to be initiated by a thrust-based mechanism. Supporting computations show that the 

exhaust streams from the jet engines are redistributed from just below the generating aircraft and 

near its centerline to the regions around the sides and on top of the lift-generated vortex pair 

(ref. 42). In this way, the exhaust streams appear to produce an insulating layer of exhaust products 

between the lift-generated vortex pair and any ambient turbulence that may be in the atmosphere. 

That is, when engine thrust is robust, it is theorized that ambient turbulence must be very severe in 

amplitude if it is to initiate waves on the lift-generated vortex pair. Therefore, it appears likely that 

the products of robust engine thrust may move such that they shield the vortex pair from 

atmospheric turbulence. If the long-wave instability is to be initiated, it needs to be exposed to some 

other form of disturbance waves. It is important to note that, like other alternative initiation methods, 

once initiated, the remainder of the time-dependent dynamics of the long-wave instability is the 

same as postulated by Crow (ref. 1). 
3 

This paper first briefly describes the thrust-based initiation process for the long-wave instability 

when jet-engine thrust is robust. It then uses observations and photographs of condensation trails to 

describe the development of the long-wave instability from just after aircraft passage to when wakes 

become benign. It provides an estimate for the extra time needed for the thrust-based initiation 

waves to develop and bring about linking between the two vortices in the lift-generated pair. An 

appendix then presents an overview of water tow-tank experiments that confirm the reliability of the 

Crow-Bate prediction method when the long-wave instability is initiated by ambient turbulence in 

the atmosphere (refs. 1a3 and 19a28]. The two wave-initiation processes discussed are distinguished 

from each other using the names kambient-turbulence initiating mechanisml for the original process 

and kthrust-based initiating mechanisml for the process identified herein. 

II. OVERVIEW OF THRUST-BASED INITIATION PROCESS 

A. Negligible Engine Trust 

When engine thrust is negligible, the lift-generated wake shed by the wing of the aircraft rolls up 

into a vortex pair that can be approximated by two-dimensional (2-D) inviscid theory (ref. 16). After 

the wake has rolled up, three-dimensional (3-D) time-dependent deformations of the lift-generated 

vortex pair come about by motions of the aircraft or by disturbances from the atmosphere. As 

indicated in figure 1, when engine thrust is negligible the flow-through exhaust streams of the 

engines become a part of the rolled-up vortex wake. Any turbulence in the atmosphere is then 

immediately effective in bringing about wave-like distortions in the vortex filaments because they 

are in intimate contact with any non-uniformities in the atmosphere near the flight path of the 

aircraft. The figure is taken from an earlier paper (ref. 42) to display qualitative computations of the 

roll-up process when the exhaust streams from the engines have about the same velocity as that of 

the free stream; e.g., during approach to an airport runway. The boundaries of the streams through 

each of the two engines are each marked by 20 small circles in order to identify how the streams 

from engines at idle are assimilated into the structure of the vortex pair. The span loading on the 

wing is designed for minimum weight and lift-induced drag for a specified span of the wing. 

 

The reason for presentation of the computations in figure 1 is to call attention to the fact that, when 

engines have negligible thrust, the rolled-up vortex wake (round filled symbols) is in immediate and 

intimate contact with any nearby turbulence eddies present in the atmosphere. This situation is 

assumed in the ambient-turbulenceabased theory for the long-wave instability (refs. 1a3). It was this 

theory that was simulated and confirmed by experiments conducted in water tow tanks (refs. 19a28). 

B. Robust Engine Thrust 

When engine thrust is robust, the hot and energetic exhaust streams slow and spread by entrainment 

of air because of mixing with the cold stationary air of the atmosphere. The enlargement of the wake 

by entrainment and slowing of the exhaust streams is simulated in figure 2 using active point-source 

distributions, shown as open circles. The vortex wake (shown by filled circles) now rolls up inside a 

shroud composed of fluid elements marked by open circles and by fluid emitted from the sources 

and not marked. The lift-generated, or vortex part, of the wake is now shielded by exhaust gases 
4 

from any ambient turbulence in the vicinity of the wake. As a consequence, distortion of the vortex 

filaments by any turbulence in the atmosphere is cushioned, and not likely to initiate the long-wave 

instability. If the long-wave instability is to be initiated, it must be initiated by some other 

mechanism. In each of the figures, the descending vortex pair is formed by the two largest 

accumulations of point vortex markers (filled circles). At x/bo = 100, the grouping of point vortices 

near the bottom of the exhaust wake makes up most of the circulation in the filament pair. The other 

point vortices form the outer layers of the vortex pair. The purpose of figure 2 is to show that, at 

x/bo = 25 and beyond, exhaust gases form a shield around the vortex part of the wake to reduce the 

likelihood that ambient turbulence will initiate the long-wave instability. 

Figure 1. Qualitative computation of assimilation of engine flow-through streams into rolled-up 

vortex wake at various distances behind generating aircraft (engine thrust is negligible) (ref. 42). 
5 

It is suggested here that when jet-engineathrust levels are robust like those used during takeoff, 

climb, and cruise, and possibly also during short periods of time where intermittent thrust is used to 

adjust the velocity or altitude of an aircraft on approach to a runway (refs. 38a40), a thrust-based 

mechanism initiates the instability. Observations indicate that the process is more complicated than 

the mechanism based on ambient turbulence. That is, the thrust-based initiation mechanism begins 

with the development of an array of circumferential vortex rings in the shear layer between each 

engine-exhaust stream and the surrounding stationary atmosphere. The individual exhaust streams 

and their shear-layer vortex arrays then merge over time to form a single exhaust stream with a 

single array of irregularly shaped vortex loops at the interface between the energetic part of the 

exhaust stream and the atmosphereni.e., in the region just outside of the region occupied by open 

circles illustrated in figure 2. 

Figure 2. Qualitative computation of formation of shield composed of engine-exhaust products 

(open circles) around the vortex wake (filled circles) when engine thrust is robust (ref. 42). 
6 

Observations at cruise altitudes show that the vortex filaments remain straight and without waves 

along their length as they descend. Unfortunately, the computational method used to produce 

figures 1 and 2 is not able to simulate the effect of atmospheric turbulence on the vortex filaments, 

nor how exhaust products might shield the vortex pair from the turbulence. However, when the pair 

of vortex filaments has descended far enough to reach the vicinity of the bottom of the shear-layer 

vortex array, the in and out velocity fields induced by the array of circumferential shear-layer 

vortices around the merged and distorted jet-engineaexhaust streams generate a series of sinusoidal 

waves along the vortex pair. Because the wavelengths of the waves (or loops) formed are too short 

for use by the long-wave instability, the relatively uniform set of short waves induced by the shearlayer vortex array is re-formed into a set of longer wavelengths that then lead to linking of the vortex 

pair to form closed loops of vortex filaments that decompose the organized structure of the wake. 

The re-formation process from short to longer wavelengths may be about the same as when vortex 

wakes encounter ambient turbulence of smaller wavelengths in the atmosphere (as derived by Crow 

and Bate, ref. 3), or it may be a formation process whereby short loops first form and then merge to 

form loops of longer wavelength that are acceptable for the linking process. Thereafter, either 

process appears to be able to cause the re-formed waves to grow in amplitude until the two filaments 

link to form large irregularly shaped vortex loops that decompose the entire lift-generated wake shed 

by the aircraft. 

III. OBSERVATIONS AND PHOTOGRAPHS OF THRUST-BASED INITIATION 

A. Overview 

At cruise altitudes, the atmosphere is often at a temperature below a40° F (a40° C) with a humidity 

level above 40%. When these conditions are present, the water vapor in jet-engineaexhaust streams 

freezes into small ice crystals as the hot, moist exhaust gases mix with the cold, ambient atmosphere 

to form what is referred to as a condensation trail. The light scattering characteristics of the ice 

crystals make it possible to observe much of the dynamics, decomposition, and decay of liftgenerated wakes for an improved understanding of how lift-generated vortex wakes redistribute 

themselves at cruise altitudes as a function of time (refs. 12a17). 

Over the years the author has taken numerous photographs of condensation trails (ref. 17). Good 

candidate photographs were taken as they appeared in the field of view of the observer/photographer 

even though the conditions for viewing and the orientation of condensation trails at cruise altitudes 

were often not ideal. Not all wakes produce a condensation trail, even though the temperature of the 

atmosphere at cruise altitudes is usually well below the required temperature of a40° F (a40° C). 

When the humidity of the atmosphere is below 40%, the ice crystals formed by the mixing of 

exhaust products with cold, ambient air evaporate or sublime and do not persist. When ice crystals 

form in marginal numbers, only the outer parts of the jet-exhaust streams are visualized. When the 

humidity at cruise altitudes is well above 40%, large quantities of ice crystals form to visualize both 

the shear-layer vortex system and the lift-generated vortex wake for an hour or more. Unfortunately, 

when ice crystals form in large numbers, many of the internal features of the wake are obscured. A 

satisfactory history of wake dynamics can then be obtained only by making a large number of 

observations that include those with a bare minimum of condensate to those where there is too much 
7 

condensate. In addition, the viewing angle of a condensation trail is often not the one desired 

because the observations and photographs are made from the ground and not from a nearby aircraft. 

An attempt was made to take photographs when the viewing angle was around 45° or less. Not only 

was it hard to keep such a rule, but the rule was aggravated by the fact that the viewing angle 

changes as the aircraft passes the viewing station. 

The photographs were taken as they appeared in the skynneither plan nor side viewsnand an 

estimate of the viewing angle was not recorded because the author was most interested in getting 

good pictures. Uncertainty exists, therefore, as to the viewing angle, the relative locations of wake 

components, and processes in addition to the aircraft type. As a result, the authorgs interpretation of 

some of the details in the pictures presented may later be shown to be incorrect when improved 

photographs are obtained by use of aircraft flying near the wakes being explored. However, it is 

believed that the process described herein for the initiation of the long-wave instability when engine 

thrust is robust is basically correct. Because the photographs were taken under dynamic conditions, 

very little information is available about the circumstances for a given photograph. 

In spite of data-taking difficulties, it is believed that the photographs presented support the thrustbased initiation process described herein. The photographs were chosen to provide pictures of the 

condensation trails that were related to the time history of the shapes of the exhaust streams and of 

the vortex pair because they were the best available, even though not ideal. Most of the pictures 

presented have had their contrast enhanced a small amount to improve identification of various flowfield components. The flight direction exhibited in all of the figures has the aircraft flying from the 

right to the left for ease of interpretation. 

B. Shear-Layer Vortex Mesh Around Exhaust Streams of Jet Engines 

As a reminder, two vortex systems exist behind aircraft flying at cruise altitudes (figure 3). The first 

is the lift-generated vortex pair that trails from the wing of the aircraft and moves downward with 

time in response to the self-induced velocity field between the two equal and opposite vortices in the 

pair. The second is the shear-layer vortex system that begins just behind the aircraft to form at the 

interface between each jet-engineaexhaust stream and the nearly stationary atmosphere. The shearlayer vortex system is first illustrated in this section by using photographs of several condensation 

trails at a time when conditions are marginal for formation of ice crystals. Therefore, only the shearlayer vortices are visualized, and the remainder of the exhaust system is transparent. In the 

photographs, the lift-generated vortex pair is not visualized by ice crystals, because it is believed that 

the two vortices are surrounded by exhaust gases and not yet by a mixture of atmosphere and 

exhaust gases. 

Condensation trails begin almost immediately behind the wake-generating aircraft as exhaust 

streams exit from the jet engines that propel the aircraft. At first, only the outer parts of the exhaust 

streams are visible, but more becomes visible as the exhaust mixes with the ambient atmosphere for 

formation of ice crystals that serve as the fluid markers. For this reason, photographs are first 

presented in figure 3 for a two-engine aircraft, and in figure 4 for a four-engine aircraft to show the 

shear-layer vortex structures that form around the jet-engineaexhaust streams as they trail behind 

aircraft. In both cases the jet-engineaexhaust streams slow relative to the atmosphere and then merge 
8 

so that their radii grow to eventually become much larger than the wingspan of the wake-generating 

aircraft. 

Figure 3 shows that the shear-layer vortex arrays or meshes that surround the two engine-exhaust 

streams first appear to spread at a total angle of about 4°, and then decrease to a spreading angle of 

about 1° or less. The reason for changes in the apparent rate of spreading of the jet streams is that the 

visible ice crystals do not mark the exteriors of the exhaust streams, but instead mark where ice 

crystals have survived. That is, ice crystals first form at the interface between each exhaust stream 

and the atmosphere and then gather around each segment of the vortex-filament mesh as the swirling 

flow of each vortex segment mixes hot, moist air from the inside edge of each jet stream outward 

into the cold, ambient atmosphere where ice crystals form. If conditions for condensation are 

marginal, the ice crystals outside of the exhaust streams and in the shear-layer vortex arrays sublime 

and disappear, rather than remaining visible. Therefore, as the jet streams age their cross-sectional 

size first appears to grow as it mixes with the atmosphere, and then appears to shrink in crosssectional size because the ambient air has a low humidity that causes the ice crystals to sublime. The 

Figure 3. Diagonal view of condensation trail displayed by jet-engineEexhaust streams of twoengine aircraft. Small amounts of condensate visualize shear-layer vortex arrays around 

both exhaust streams in part (a), and their merger in part (c) as ice crystals sublime. 

Figure 4. View of condensation trail from aircraft with 4 jet engines as the two streams from 

engines on each side of the aircraft rotate about each other because of the action of the 

vortex wake and then merge; the view is about 0 to 10 bg. Merger of port and starboard jet 

streams does not occur until about 100 bg after aircraft passage. Lift-generated vortices 

are not visible. 
9 

last picture in figure 3 shows the exhaust streams as shrinking in size, but not yet completely gone. 

As a consequence, the outside of the vortex filaments that visualize the exhaust trails shown in 

figures 3 and 4 do not include all of the gases that contain exhaust products, so they do not 

necessarily represent the outside surfaces of all of the mixture of engine exhaust and ambient air. 

Although the shear-layer vortex arrays are sometimes observed as a neat sequence of coaxial vortex 

rings around each exhaust stream, the patterns most often observed consist of a somewhat random 

diagonal system of vortex segments that are tied together like a network of stretched strings around a 

long sausage, as photographed in figures 3 and 4. As distance behind the aircraft increases, the 

multiple exhaust streams mix with the atmosphere and with each other to form a single exhaust 

stream with a single shear-layer vortex array around its exterior. The reduced velocity in the merged 

exhaust stream causes the shear-layer vortex mesh to be compressed in the streamwise direction and 

to become enlarged in cross-sectional size. 

The segments of irregularly shaped vortex rings around the combined exhaust streams then become 

more evenly spaced, and more nearly perpendicular to the axis of the combined jet stream, so that 

the re-formed mesh pattern more closely approximates coaxial vortex rings (fig. 3c). Although the 

overall structure is still complex, the circumferential shear-layer vortex lines appear to become more 

evenly spaced and nearly transverse to the flight direction. As a result, the shear-layer vortex system 

has developed the capability to initiate regularly spaced waves along the lift-generated vortexfilament pair to provide the disturbances needed to carry out the initiation of the long-wave 

instability. Overlap of exhaust streams as they spread and as the viewing angle changes makes it 

difficult to interpret some features in the pictures. Of importance is the fact that the individual jet 

streams grow in cross-sectional size, and they appear to overlap and then merge with distance behind 

the wake-generating aircraft. 

C. From Roll-Up Through Linking of Vortex Pair 

As illustrated in figures 3, 4, and 5a, a shear-layer vortex array first forms around each jet-enginea

exhaust stream almost immediately after aircraft passage. In figures 5b and 5c, the shear-layer vortex 

arrays are visible in the photographs as diffuse, nearly vertical striations in the middle and upper part 

of the wake. It should be noted that the upper part of the wake has about doubled in depth between 

figures 5b and 5c. Also shown in both of the figures as thicker, nearly straight white lines is the liftgenerated vortex pair first near the bottom of the combined jet-engineaexhaust streams and their 

shear-layer vortex array and then at the bottom (fig. 5c). In figure 5d, the vortex pair is at the inside 

bottom of the flow field just inside of the vortex array where the vortex pair has become distorted 

into a sequence of in-phase waves that appear as segmented humps. That is, when the pair of vortex 

cores or filaments approaches the inside of the bottom of the shear-layer vortex array that surrounds 

the exhaust condensate, waves of nearly the same wavelength form along the filament pair (figs. 5c 

and 5d). It is the up- and downward velocity field between the shear-layer vortices at the bottom of 

the shear-layer vortex array that induces these waves on the vortex filaments. The amplitude of these 

short waves grows rapidly at first and then stops growing because of the limited depth of the vertical 

velocity field inside the shear-layer vortex array. 
10 

Figure 5. Photographs of progress of structure of condensation trail from when first shed by 

aircraft to shortly after linking and loop formation of lift-generated vortices. 

Figure 5e presents a view of the closed loops of decomposing and diffusing lift-generated vortex 

segments that have formed into irregularly shaped vortex loops by the long-wave instability. The 

lower lobes of diffuse condensate indicate where the loops of vortex filaments have their greatest 

downward extension. The upper parts of the shear-layer vortex array are indicated by a closely 

spaced sequence of white-filled ellipses. The lower part of the wake consists of the diffuse, 
11 

irregularly shaped vortex loops that were formed by linking of lift-generated vortices brought about 

by the long-wave instability. The vortex loops have lost their crisp appearance by becoming diffuse, 

so they appear as long puffs of condensate loops rather than small-diameter filaments (fig. 5e). In the 

cases observed, the ratio of the spacing between shear-layer vortices and the loops formed by the 

long-wave instability is consistently somewhere between 3 and 4 (figs. 5e and 6b). 

The spacing between shear-layer vortices is apparent in both figures 5 and 6, but is most clearly 

defined in figure 6b. The vortex loops that have formed by vortex linking have had their structure 

changed from straight-line filaments to filaments with waves, and from continuous filaments to 

closed loops of vortex filaments that then become diffused, smeared out, or destroyed by actions of 

the vortex loops formed by the long-wave instability. Figures 5e and 6b also indicate the 

approximate shapes and spacing between the vortex loops that are formed after the two vortices in 

the lift-generated vortex pair have linked. The ratio of distance between successive shear-layer 

vortices and vortex-loop formations consistently appear to be between three and four shear-layer 

vortex spacings for each single loop of the long-wave instability. All of the condensation trails 

observed exhibited the same spacing rule. 

Also shown in figures 5e and 6b is a curious characteristic where the loops formed by the long-wave 

instability slope to the right. It is believed that part of the reason for the slanting shape is that the 

loops are moving downward as they age. Therefore, because the older part (the left part) had a head 

start at descending motion, it enters the stationary part of the atmosphere well before the upper part. 

Another reason for the slanted form is that as the jet-engineaexhaust streams have merged into a 

single stream that broadens as their velocity slows relative to the surrounding atmosphere, fluid 

Figure 6. Comparison of spacings between shear-layer vortices with spacings between loops 

of vortex filaments that are in the process of diffusing. 
12 

elements within the center part of the wake and inside or next to the rearward-moving exhaust 

stream continue to move rearward. In contrast, fluid elements (e.g., loops of vortex filaments) that 

are near the outside of the exhaust stream are almost stationary relative to the atmosphere. As a 

consequence, those parts of the exhaust streams move the flow field rearward, causing the vortex 

loops to slant (figs. 5 and 6). 

IV. TIME-TO-LINKING ESTIMATE OF LINKING 

The Crow-Bate theory assumes that the formation of disturbance waves for initiation of the longwave instability begins almost immediately after aircraft passage. The assumption is a good one 

because the pair of vortex filaments shed by an aircraft is in immediate contact with the atmosphere 

and its turbulent eddies when engine-thrust levels are negligible (fig. 1). This is not the case when 

disturbance waves are initiated on vortex filaments by the shear-layer vortex arrays generated by 

robust jet-engineaexhaust streams. In those cases, as shown in the foregoing figures, the vortex pair 

must first travel from about the vertical center of the wake (or condensation trail) down to the 

bottom of the wake where the bottom of the array of shear-layer vortices resides. Thereafter, the 

time interval required for amplification and reassembly of disturbance waves into those that progress 

into the linking and loop formation stages of the instability should be about the same as the CrowBate estimate for vortex linking (ref. 3). Therefore, the Crow-Bate estimate (fig. 7) for a time to 

linking after wake generation by disturbance waves that consist of ambient turbulence is used as a 

basis for a time-to-linking estimate that takes place when the disturbance waves are generated by 

jet-engineaexhaust shear-layer vortices. 

Figure 7. Minimum increase in time to linking when long-wave instability is initiated by 

shear-layer vortex array. 
13 

The estimate for the downward travel time of the vortex pair from the original location just behind 

the generating aircraft down to the vicinity of the bottom of the shear-layer vortex array is based on 

two parameters. The first is an estimate for the maximum travel velocity of the vortex pair, and the 

second is an estimate for the shortest likely travel distance from where the vortex sheet is generated 

down to near contact level at the bottom of the inner part of the flow field of the shear-layer vortex 

array. Those values yield an estimate for the minimum amount of time needed for the vortex pair to 

move from the elevation where the vortex pair was generated down to roughly where the bottom of 

the wake is located at the time that the vortex wake arrives. 

The derivation begins with an estimate for the maximum self-induced downward travel velocity of a 

vortex pair as given by 

 wpr = a|Γo|/2πbg (1) 

where it is assumed that the vortex strength, |Γo|, and the spanwise distance between the centers of 

the vortices in the pair, bg, are the same as when the wake was first generated by the aircraft. The 

estimate made by equation (1) yields the maximum self-induced downward velocity because the 

strength of each vortex filament is assumed to be the same for the entire downward trip as when first 

generated by the roll-up of the vortex sheet shed by the wing. Those values were chosen even though 

some of the circulation has been stripped away from around the vortex centers and has been moved 

to the top of the wake. The reduction of the circulation content of each vortex is made by removal of 

some of the layers of vorticity in the outer layers of the vortices to the top of the wake by the 

swirling action of the opposite vortex in the pair. Turbulence in the exhaust streams of the jet 

engines also tends to spread the vortices. These processes tend to reduce the circulation content and 

thereby increase the downward travel time of the pair of vortex filaments or cores. It is also assumed 

that the separation distance, bg, between the two vortices in the pair is unchanged by any of the 

foregoing processes from its initial value of about bg § πbg/4. 

Because the cross-sectional size of the wake is increasing with time as the vortex pair descends, the 

travel distance of the vortex pair is also uncertain. For the purposes of the present approximate 

estimate of travel time, the travel distance (one-half of the wake diameter) is estimated as 

somewhere between 2 bg and 3 bg. It is argued that the jet-engineaexhaust streams expand from their 

original diameter to about 3 bg, or perhaps more. Observations of wake sizes at the time of arrival of 

the vortex pair at the bottom of the shear-layer vortex array appear sometimes to be as much as or 

more than 4 bg in diameter. Because the smallest value for travel time is desired, the downward 

travel distance of the pair is taken as 1.5 bg. An estimate for the minimum travel time of the vortex 

pair is then given by: 

∆tDwn = 1. 5 bg/wpr = 1. 5 bg/(|Γo| /2πbg) (2) 
14 

The assumed circulation content of each vortex in the pair can be converted to aircraft parameters by 

using two equations for the lift on the wing of the wake-generating aircraft. Because aircraft flying 

at cruise altitudes are in steady level flight, the weight of the aircraft is the same as the lift produced 

by the wing of the aircraft, or 

 Wt = L = ρU|Γo|bF (3) 

The lift is also given by 

 L = CLgSρU

2

/2 (4) 

The dimensionless circulation may then be written as 

 |Γo|/boU = 2CLg/πARg (5) 

so that the self-induced downward velocity of the lift-generated vortex pair may be written as 

 wpr /U = Q4CLg/π

3

ARg § Q0.0184 (6) 

for all similar aircraft whose lift coefficient is about 1.0 and wing aspect ratio is about 7. An estimate 

for the time required for the vortex pair to descend from the center of the wake down to the bottom 

of the wake where disturbance waves are impressed on the filaments is then given by 

∆tDwn = 1. 5 bg/wpr § 14 s (7) 

when the wake-generating aircraft has a wing span of 200 ft and is flying at about 800 ft/s. In 

comparison, observed times are difficult to determine visually from the ground, but appear to vary 

from about 20 s to 100 s. 

 

In terms of the Crow-Bate time-to-linking parameter, the downward travel time may be written as 

∆τDwn = ∆tDwnwpr/bF = 1. 5 bg/bF = 1.91§ 2 (8) 

When the downward travel time of the vortex pair is added to the Crow-Bate estimate of 6 units (for 

when the ambient turbulence is small), the sum yields a total value of around 8 units. A short 

horizontal line is shown in figure 7 for shear-layer thrust initiation, which is in fair agreement with 

the data point for the B-727. 
15 

V. CONCLUSIONS 

It is well known that the long-wave instability of a vortex pair is the aerodynamic mechanism that 

decomposes and spreads lift-generated vortex wakes more rapidly than any other mechanism. It has 

also usually been assumed that turbulence in the atmosphere is the initiation mechanism for the 

long-wave instability. Theoretical and experimental studies of the growth of the long-wave 

instability have been confirmed by experiments in water tow tanks when engine thrust is absent or at 

low levels. 

However, when engine-thrust levels are robust as during takeoff, climb, and cruise, and during 

periods of flight-path adjustments during approach to a landing, a thrust-based mechanism initiates 

the long-wave instability in the lift-generated vortex pair. Robust engine thrust first generates an 

array of vortices in each of the shear layers between the engine-exhaust streams and the atmosphere. 

Shortly after generation, the self-induced velocity field of the lift-generated vortex pair moves the 

exhaust gases and their vortex arrays from near the bottom of the wing around to the sides and the 

top of the wake. The vortex pair may then be shielded from any turbulence in the ambient 

atmosphere near the aircraft wake. During this time, the lift-generated vortex pair moves downward 

to the bottom of the wake, where it encounters the up and down velocity field of the combined 

shear-layer vortex arrays, which initiates the long-wave instability. 

The photographs provided illustrate and support the thrust-based initiation process described herein. 

Unfortunately, the photographs of condensation trails at cruise altitudes were taken from the ground 

at a variety of different viewing angles and condensation conditions. As a consequence, the 

photographs are somewhere between side views and views from directly below the condensation 

trail, making it difficult to discern the exact detailed dynamics within the wake of the aircraft. 

Although the uncertainty in the interpretation of the dynamics within the wake is bothersome, the 

evidence is clear that the long-wave instability is initiated by the thrust-based mechanism described 

here and not by ambient turbulence. It is also obvious that the velocity field at the bottom of the 

shear-layer vortex array that surrounds the unified streams of the engines does provide the waveinitiation process. Because some of the details of the thrust-based initiation process are vague, it is 

recommended that further experimental evidence be obtained by means of photographs taken of the 

process from an aircraft flying beside, from beneath, and from above condensation trails to more 

fully describe the thrust-based initiation process and to provide a more exact description of its 

details. An improved theoretical analysis is also needed for the process to improve the understanding 

of and accuracy for the estimate of the time for the process. 
16 

APPENDIX: 

CONFIRMATION OF THEORY BASED ON AMBIENT TURBULENCE 

A. Overview 

This appendix presents an overview of the experimental studies that were carried out to confirm the 

ambient-turbulence model developed by Crow and Bate for the initiation and dynamics of the longwave instability in a vortex pair. It is found that the experiments conducted in water tow tanks on the 

basics of the long-wave instability have a small amount of scatter. That fact helps to demonstrate the 

accuracy of the Crow-Bate model and the fidelity of the experimental work on matching the 

ambient-turbulence initiation process assumed in the theory. The larger amount of scatter in the 

flight data presented in the Crow-Bate paper is not discussed because, as explained in the paper, it is 

now believed that the long-wave instability is not initiated at cruise altitudes by the ambientturbulence initiation process but by a thrust-based initiation process. Therefore, the discussion 

presented here treats only the confirmation of the ambient-turbulenceabased model derived by Crow 

and Bate. Confirmation of the theoretical model for the long-wave instability when initiated by 

ambient turbulence is important for an understanding of the dynamics of the long-wave instability 

and for an estimate of the time needed for its execution when initiated by the shear-layer vortex 

arrays produced by robust engine thrust. 

The theoretical representations derived by Crow and Bate indicate that the preferred wavelengths for 

vortex linking by the ambient-turbulence process range from about 4 bg to about 10 bg (bg § πbg/4). 

After initiation, the planes of oscillation of the growing waves on the two filaments in the vortex pair 

each rotate in a direction opposite to that of the swirl direction in the vortex itself as the wave 

amplitude grows (refs. 10 and 11). The increase in wave amplitude and the rotation of the two wave 

planes combine to move the wave troughs inboard towards each other until the troughs are close 

enough to each other to link across the center plane of the flow field. The linking process causes the 

vortex filaments to form irregularly shaped loops of vortex filaments that continuously and rapidly 

reform their shape to bring about large-scale fluid motions that decompose the originally wellorganized flow field of the vortex pair. The instability has then changed the hazard posed by the 

wake to penetrating aircraft from a coherent over-powering rolling moment to an incoherent random 

bumpy ride (ref. 16). 

B. Parameters Used 

The parameters used to describe the characteristics of the long-wave instability (refs. 1a3) include 

the eddy-dissipation rate, ε, of the turbulence in the fluid where the lift-generated wake is embedded, 

and the time after aircraft passage, tLink, at which linking occurs across the centerline of the wake by 

the two vortex filaments. These two parameters are made dimensionless by combining them with bg, 

the initial spanwise distance between the centers of the two vortices in the pair, and wpr, the initial 

estimated self-induced downward velocity of the vortex pair. The quantity bg is assumed to be 

related to the span of the wake-generating aircraft by bg § (π/4)bg, and wpr is taken as Γo/2πbg, 

where Γo is the circulation bound in the wake-generating wing at its centerline. Two dimensionless 
17 

parameters, a time-to-linking parameter, [tLnkwpr/b8], and a turbulence parameter, [(ε b8)1/3/wpr], are 

then used to compare the Crow-Bate theory for time to linking with measured data (fig. 7). 

C. Vortex-Linking Experiments in Water Tow Tanks 

Experimental studies of the characteristics of the long-wave instability of a vortex pair were carried 

out in water tow-tank test facilities (refs. 19J28) to confirm the theoretical analyses (refs. 1J3) and 

to better understand the relationship between the structure of the turbulence in the ambient fluid and 

the time to linking of the two vortices in the pair, tLink. The water tow tank is the ideal instrument for 

testing the analysis of Crow and Bate because the characteristic parameters, ε, b8, wpr, and tLnk, can 

all be readily generated and measured reliably in a water tow tank. A broad range of turbulence 

levels were generated in the ambient fluid by using different grid sizes that were towed through the 

water before each test run with a lifting wing to generate various levels and scales of turbulence. The 

tow tank does have the disadvantages of being at small scale and small Reynolds numbers, but the 

cost of many laboratory measurements is small compared with even a small number of flight tests, 

where control and measurement of important parameters are difficultQif not impossibleQto obtain. 

The large number of data points obtained in water tow tanks are not shown in figure 7 because their 

large number obscures the two theoretical curves presented. Instead, only the theoretical curves 

developed by Crow and Bate (ref. 3) and by Sarpkaya (ref. 23) are shown because the experimental 

data are distributed closely around the theoretical curves to provide a convenient average of all of 

the tow-tank data taken by Sarpkaya (refs. 19 and 22J24) and by the staff of the Northwest Research 

Associates (refs. 25J28). The averaged tow-tank data and Sarpkaya8s recommended curve are noted 

to have about the same shape as and to be just above the curve predicted by the theory of Crow and 

Bate (ref. 3). The results obtained by Liu (refs. 20 and 21) confirm the results of the Crow-Bate 

analysis and provide flow-field visualization and other information on the decomposition of vortex 

wakes that are brought about by ambient turbulence, stratification, and the proximity of the ground 

plane. 

Even though the tow-tank data are not in exact agreement with the theoretical predictions of Crow 

and Bate, the data confirm the predicted trends and indicate that proper attention to the details of the 

flow-field structure in the development of the long-wave instability does produce data that have a 

small amount of scatter and are in good agreement with theory. A negative aspect of the data taken 

in water tow tanks is that, because of its small scale and the confinement by the walls of the test 

tank, it may have a systematic change in the level of linking times from what might be found if 

larger sizes of water tow tanks and turbulence-generating grids had been used in the experiments. 

Such a conjecture is suggested by the observation that the largest grid size used in the various tow 

tanks is only large enough to produce turbulence eddies of wavelength sizes up to about 4 b8, 

whereas the optimum wavelength for stimulation and support of the instability is 8.6 b8. Therefore, if 

larger grid sizes had been used (a larger facility would have been required) to generate turbulence, 

the measured linking times might have been in complete agreement with the theoretical estimates 

made by Crow and Bate (ref. 3). None of the experiments in water tow tanks simulated the action of 

jet-exhaust streams for a simulation of vortex wakes at cruise altitudes. 
18 

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