contrails and aircraft downwash by scorer davenport

8/18/2019 Contrails and Aircraft Downwash by SCORER DAVENPORT

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J . FEuid Mech. (1970), vol. 43 part 3, p p . 451 464

Printed in Great Britain45 1

Contrails and aircraft downwash

By R. S. SCORER A N D L. J. DAVENPORTImperial College, London

(Received 11 December 1969)

Aircraft downwash consists initially of a vortex pair descending with its accom-panying fluid through th e atmosphere. Condensation rails are formed in exhaustemitted in to the accompanying fluid and the shapes of them and their evolutiondepend on th e positions of the engines in relation to the wing tip vortices.

The atmosphere is stably stratified and so the descending accompanying fluidacquires upward buoyancy. Consequently vorticity is generated at the outsideof the accompanying fluid and the flow pattern in the vortex pair is altered so as toproduce detrainment of its exterior par t. So long as any air which is a mixture ofaccompanying fluid and exterior air is detrained, the vortices remain stable,bu t t he width of th e pair decreases and its downward velocity increases withtime as a result of the buoyancy. Eventually the upper stagnation point in themotion relative to the vortices begins to move upwards relative to the vortices sotha t some mixed fluid is entrained in to the circulation and the vortices immedi-

ately become unstable, mixing occurs, the pressure in the core rises, and anyvortex core trails th at may exist appear to burst.The motion produces downward-thrust blobs in trails from centrally placed

engines, which correspond to the holes sometimes seen in cloud when distrails areformed.

1. IntroductionContrails are clouds of water droplets or ice particles formed in the exhaust of

aircraft. The appearance of these trails is determined by the physics of thecondensation, freezing, and evaporation mechanisms and by the motion towhich they are subjected. I n order to interpret their shapes in terms of this motionwhich is the aircraft downwash, it is necessary t o understand the physics, and asummary of this is given first.

I n 1955 Scorer described the appearance of contrails in terms of the motion ofthe vortex pair which is the aircraft downwash. It was then suggested th at theloops which appear were a manifestation of the instability inherent in any vortexpair in which the nearness of the vortices is slightly varying, but no quantitivetheory was given. In 1957 Turner described the effect of buoyancy on vortex

rings, and th e implications of this were included in Scorer’s (1958, pp. 73-4 andplate 2) description of contrail forms. It was appreciated that if, anywhere, thecirculation round a vortex decreases outwards the motion is unstable and thebuoyancy forces in the accompanying fluid obviously operate t o produce this

29-2


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452 R. S . Xcorer and L . J . avenport

effect because the circulation in a cylindrical thermal would be opposite to t hat ina vortex pair moving downwards. The effect of the buoyancy is to producedetrainment of accompanying fluid, because it decreases the distance betweenthe vortices according to Turner s theory. The result according to the presentinvestigation is that t o begin with all the opposite vorticity generated is de-trained and so no instability is produced in the vortices, but after a time whichdepends almost entirely on the static stab ility of the air, instability will occurand the vortices will suddenly burst.

The consequences of these developments are clearly visible in contrails anddistrails (clear lanes in cloud made by the downwash), and a comprehensivecollection of pictures of them has been made by Scorer (1970). The results of thepresent paper are taken from Davenport (1967).

2. Contrail physicsI n figures 1-3 the curves give the saturation ratio in g kg-l of water vapour

to air as a function of temperature for three different pressures. The arrowmarked exhaus t indicates th e direction of the point representing typical

550 mb 5600 m

FIGURE . Water and ice saturation mixing ratio curves for air a t 550 mb. If the pointrepresenting the ambient air lies in t he shaded region a contrail will form as the exhaust,represented by a point in the direction indicated, is diluted isobarically. If tho ambientair point is in the doubly shaded region and th e trail is glaciated it will persist. It will beglaciated immediately if it reaches a temperature below - 0 C.

aircraft exhaust. As the exhaust mixes with the air, the mixture is represented bya point on the straight line joining the points representing th e ambient air and

the exhaust. As the mixing proceeds cloud will appear if water saturation isreached, which means that for a contrail t o occur in previously clear air theambient air point must lie within the shaded area below the water saturationline and above the tangent t o it from the exhaust point.


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Contrails and aircraft downwash 453

If the cloud freezes it will not evaporate if the ambient air point lies in thedoubly shaded area which lies above the ice saturation mixing ratio curve. Ifthe ambient air is at a temperature below - 40 C the cloud will certainly freezeand in the above situation persistent contrails will occur. The shaded areas thusgive where the ambient point must lie for the occurrence and persistence oftrails respectively, if they are formed by mixing a t constant pressure.

Figure 4 summarizes these conclusions and shows where in th e atmospherecontrails of different kinds are likely to be found. The Mintra ' is the boundary

2.0

400 mb 7200 m

-

-

- 0 - 0 - 0

FIGURE . The same as figure 1 for 400 mb.

250 mb 10,400 m

@ /

// / /

- 0FIGURE . The same as figure 1 for 250 mb.


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454 R. . Scorer and L. J . Davenport

FIGURE . The conditions in the atmosphere required for condensation tra il formation areshown on this T 5 diagram in which the two orthogonal co-ordinates are temperatu re (in “C)andlog (potential temp eratur e), he potential te mpe rature being indicated in O K . The isobarsare slightly curved an d the diagram is drawn so th at the y are almost horizontal. Pressureand temperature are then used to plot a sounding of the atmosphere on the diagram: andthe line marked ICAO represents the ICAO standard atmosphere, with a tropopause atabout 230 mb.

The shaded areas ar e where trai ls are possible from jet aircraft bu rning kerosene : heyare bounded b y the ‘M intr a’ which indicates the pressure, at each temperature, below th ealtitude of which trails cannot be formed. The ‘Mintra dry’ indicates the limit for anabsolutely d ry atmosphere.

(For a dcscription of the T diagram see, for example, Scorer (1958, pp. 256-60).)


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between where no trails are possible, and where they are possible. In th e latterregion they may remain unfrozen if th e temperature is above - 40 C, and whetherthey persist or not if they are frozen depends on the humidity of the environment.Generally unfrozen trails are non-persistent because dilution always producesevaporation ultimately. They are called semi-persistent if they are frozen but theenvironment is not saturated for ice, because ice clouds usually take much longerto evaporate. I n the stratosphere they are never persistent because of th e lowhumidity, and so th e upper boundary of the region where persistent trails wouldoccur if t he air were saturated with respect to ice is rather variable, and dependson where the tropopause is. This is indicated by th e wavy boundary in th e regionof 250 mb. It is usually around 240 mb in the middle latitudes but can be quite alot higher or lower.

The ICAO standard atmosphere is shown, to indicate th e rough whereabouts

of the points of a typical sounding: obviously an actual sounding can depart verymuch from this, but it indicates th at th e region where trails are most likely t o bepersistent is below the tropopause and above about 300mb (9000m veryroughly).

The position of the Mintra ' line depends on the nature of the aircraft exhaust,and to show its variability i ts position is given for a typical jet engine burningkerosene when the air is saturated for water and when i t is absolutely dry . Themanner in which its position might vary in saturated air is shown by examples forother hydrocarbon fuels which produce different amounts of heat and watervapour.

Vortex trails are not indicated in the diagram. They occur when exhaustenters the core of the wingtip vortices, where the pressure may be 50-100mblower, and they can occur when contrails are not formed by mixing, but theengine must usually be rather near to the wingtip. If vortex trails freeze theywill persist if the ambient air is supersaturated for ice.

3. Motion due to a vortex pairThe variation along the length of an aircraft wake and the longitudinal

component of velocity are so small tha t any section can be adequately representedas the same as that of an infinitely long two-dimensional vortex pair, and in th enext section our analysis is like tha t of Turner (1960) who described the accelera-tions and changes in size of vortex rings and vortex pairs possessing buoyancy.

The streamlines relative to the vortices are shown in figure 5 for the case whereth e vorticity is concentrated into two thin lines. I n practice it is diffused througha central core, but when the vorticity is not concentrated the velocity field out-side th e core, and more particularly th e shape of the boundary of th e accompany-ing fluid, is virtually unaltered.

Air circulates within this boundary and travels along with the vortices. The

exterior air passes round the outside as the accompanying fluid descends with aspeed which is equal to the speed of each vortex in t he field of motion of theother.

In th e right half we show the tracks along which narrow exhaust trails would


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456 R. S . Scorer and L. J . Davenport

circulate round t he vortices. A tra.il from a centrally placed engine would simplydescend vertically. In the left half we show the positions of three particles,originally placed along the line of the wing, at three successive times, and we seethat the line of particles joining them becomes wound up. This means that

FIGURE . The streamlines in the motion due to a vortex pair located at the point,s x .The trajectories of air parcels and lines of marked fluid particles in this motion pattern areindicated.

Z= Rb

S Boundary of accompanyingfluid

sR 3 R

1 * x= R[- R 4

Z

FIGURE . The streamlines in the motion due to a vortex pair located at the point,s x .The trajectories of air parcels and lines of marked fluid particles in this motion pattern areindicated.

fluid

.x= R[

rFIGURE . Co-ordinate system for a vortex pair.


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Contrails and aircraft downwash 45 7

although a trail near the wingtip circulates around the vortex rapidly, a trailnearer the centre of the wing will appear from a distance to circulate around theouter one. This is evident in the first photograph in figure 10, plate 1.

4. The effect of atmospheric stratification

with circulations 5 K , heir vertical velocity is given byIf 2R is the distance between two horizontal line vortices a t the same height Z,

W = dZ/d t = -K/4nB. (1)The impulse required to establish the motion of unit length of the vortex pair influid density p (see, for example, Lamb 1932, art. 152, equation 6, or Davenport1967) is

The front and rear stagnation points L nd S in figure 7 ) form equilateral tri -angles of side 2R with the vortices. The accompanying fluid, whose boundarypasses through these points, is composed of air from the original level of thevortices, which is the flight level of the aircraft. If the environmental potentialtemperature is r-l, and has value T i 1 a t th e flight level, and

Q = - pKR. (2)

p = - - -1 a7ax )

th e potential temperature anomaly of th e accompanying fluid is given by

(3)

AT / T,= PZ, (4)

when the flight level is a t 2 = 0. I n the atmosphere r plays the same role as thedensity in an incompressible stratified fluid.

We may equate the rate of change of the impulse of the vortex pair at anymoment with the total buoyancy force which results from the vertical displace-ment of th e accompanying fluid in the stratified environment. Thus

dQldt = gmR2 A p , 5 )

where A p is the density anomaly and is equal to p A r / r l , and mR2 s the cross-section area of the accompanying fluid. This leads to

2KdRldt = gmR2@Z, ( 6 )

which by ( 1 ) gives

from which we obtain

dR/dZ = - 2ngmp/K2) ?Z,

R = R,( 1 +R2,ZZ/h4)-*,w ~ q i~;22 /h4)4 ,

where h = K2j2nmg/3.

When expressed as functions of time th e values of the height, size, and verticalspeed of the vortex pair are given by

h2 KtR, 4nh2’

= inh


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45s R. . Scorer and L. J . Davenport

KtR = Rl ech77h2

Kt

4nh2W = Hi cash

( 1 3 )

It is important to note later that the coefficient of t , namely K/4nh2 oes notactually depend on K .

5. Vorticity generation and detrainmentWhen the accompanying fluid is displaced downwards there is a density dis-

continuity at its boundary. We assume this discontinuity to have th e same magni-tude all over and th e motion in the stratified environment to be approximately

the same as when there is no stratification. This means tha t we assume the vorti-city t o be generated only at the boundary and not a t all in the exterior fluid as aresult of displacement in it. Meanwhile the vortices in th e cores are assumed toretain a constant circulation because they are surrounded by fluid of uniformdensity in irrotational motion. These assumptions are justified when p, thestratification of the environment, is small and the density anomaly is changingslowly.

The rate of generation of vorticity a t th e boundary is given by

A rg sin 31-fsin 01

Dtr

where and 8 re the angles made by gravity and th e fluid acceleration vectorswith the normal to the boundary, which is a streamline relative to the vortices.7 is the vorticity density in the boundary and f is the magnitude of the fluidacceleration. This equation is simply the appropriate component of the vorticityequation applied to the thin layer of fluid which constitutes the boundary.

From equation (14) it is a routine matter t o obtain the value of the vortexsheet strength 7 a t all points on the boundary assuming it is zero a t the lowerstagnation point, for, with the co-ordinates of figure 7

and the velocity components u and w are assumed to be those due t o the vortexpair. The details are described in Davenport (1 967) . It emerges in the computationthat the contribution due to the term in s negligible except in circumstancesmentioned below, and can usually be ignored in the case of the atmosphere.

When the vorticity has been calculated the velocity field may be recalculated.I n particular the position of the upper stagnation point may be determined. It isnow found to be inside the original boundary when the buoyancy is upward.

Consequently when it is a t 8 n figure 7 , for example, th e fluid between the boun-dary and the streamline arriving at S, is detrained as a narrow strip of fluid leftbehind the system in the region bounded by the heavy line on the right side of thefigure.


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We suppose th a t the vortex layer on the boundary is dynamically unstable tothe extent of producing corrugations or even mixing. The region pervaded by thedensity fluctuations will be confined to some region such as th a t shaded in figure 7.I n the right half, when stagnation occurs a t S, all the mixed fluid is detrained;but in the left half it is seen th at , when stagnation is close t o the original point,mixed fluid may be entrained between the vortices. In that case the motion

FIGURE . Streamlines when t he accompanying fluid has buoyancy. The arrows on theboundary indicate the sense and m agnitude of th e vorticity generated by buoyancy. Theshaded area represents the mixed region, and in the right half the upper stagnation point isa t 8 and th e mixed fluid is detrained, whereas in th e left half th e p attern is shown whereth e stagnation point is in the mixed fluid, some of which is then entrained in to the accom-panying fluid.

inside the accompanying fluid would become dynamically unstable because thevorticity entrained would produce a decrease of circulation across the streamlinesoutwards from the vortices (without an alteration of the sign of the curvature).This is presumed to lead to a breakdown of the vortices.


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460 R. S Scorer and L. J . Davenport

Earlier theories of the mechanism of breakdown exist. Thus Squire (1954)supposed that there was an eddy viscosity proportional to the circulation, forthis has th e right dimensions. It s magnitude has t o be determined from observa-tion and Rose & Dee (1965) found th at an explanation of the decay of the vortexcores in the early stages could be obtained in this way but not the subsequentvery rapid breakdown. Owen (1964) proposed alternatively that the eddyviscosity might be proportional t o the square root of the circulation. On eitherSquire’s or Owen’s theories the time of decay depends on the square root orfourth root of K , but the observations of Kerr & Dee (1960) suggested that thetime was independent of the circulation, although this conclusion could not bedrawn firmly in view of the inaccuracies inevitable in th e measurements.

To us there seems no evidence of an eddy viscosity operating and some contrailsin vortex cores remain very sharp for long periods. The motion is dynamically

stable in the absence of an axial component of velocity, and i t is very unsatis-factory t o have to determine th e eddy coefficient from th e observations beingexplained.

6. The motion of the upper stagnation pointThe vorticity on the boundary is given by

7 = 2nR2gpz/K)A (16)

awt = W / U w c = 11 (17 )

x =CRY x = CR, 4 2 = u2 w2, (18)

where q is th e non-dimensional vorticity computed from

which are derived from (14) and (15 ) by writing

and ignoring the term in f . Table 1 gives some of the values of 4 obtained.

5 tFI - 5 & 50 1.73 0 5.420.42 1.70 0 .37 5.050-83 1.59 0.75 4.661-25 1-39 1.17 4.251.67 1.05 1.66 3.761.92 0.69 2.05 3.362.07 0.23 2.50 2.92

TABLE . Values of th e dimensionless vorticity at selectedpoints on the boundary of the accompanying fluid

According to (16) the actual vorticity varies inversely as K , because the largerthe circulation the more rapidly the fluid passes around the boundary, and so less

vorticity is generated during its passage. The buoyancy represented by PZincreases downwards, but a t the same time R decreases downwards accordingto (7 ) . The consequence of these variations is th at the vorticity at the boundaryreaches a maximum relative to K and then decreases again. The stagnation


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point therefore descends from S as the system descends from its s tarting pointof zero buoyancy, and then rises asymptotically towards S again.

If w s s th e vertical velocity produced by the non-dimensional vorticity q onthe boundary a t th e position 6 = 0, 5 = s, the value of s a t a stagnation point isgiven by K

wR(l+i?) K 4nR '

K -- 2 9 P

this being the condition th a t the velocity there is the same as the velocity of thevortices. This can be written

20)1 +s w - K

The right-hand side is negative and changes from zero to a maximum magnitudeand then decreases exponentially with t i m e in a manner determined by (1 11, and

3-S I 4rR3g/3Z

- - - - -1.7

1.65

1.6

1.55

50 20 40 60 80 100 120 140 160 180 200

t seconds

FIGURE . Changes in s with time for a typical atmospheric stratification. The stagnationpoint reaches its lowest position relative to the vortices after 66 see and within anotherminute or two vortex core bursting can be expected.

(12). This is why s has a minimum a t a certain time, which is independent of K ,and is equal to

The value of m is 11.5 and so if g = lo3cm s e r 2 we obtaint* = 0.66(mgp/8n)-b. (21)

t* = 98 @ x lo7)- ec (22)if p is measured in cm-l. With a typical value of for th e atmosphere in th e range10-6 t o 10-7 cm-1 we see that t* would be between about 30 and lOOsec, and isindependent of K and R.

The value of s at its minimum is not so readily calculated because it depends onall th e parameters of the problem. Its variation with time is shown in figure 8 forthe following values f i 1 = 13 metres, K = 190 m2 set-1

which corresponds roughly t o a Comet aircraft, and/3 = 2-2 x 10-7 cm-1, g = 103cm s e r 2

for a typical atmosphere.


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462 R. . Scorer and L. J . Davenport

Although the stagnation point returns towards its initial position ratherslowly the intensity of the boundary vorticity is increasing and the dimensionsof the accompanying fluid are decreasing. Consequently it is inevitable that fluidmixed at the boundary will become entrained into the accompanying fluideventually, and that the vortex cores will then become unstable and rapidlymixed with a consequent rapid rise in central pressure and disappearance of anytip vortex trails th at might have existed.

In th e above calculation the term in f n (14) was ignored. This is justified in theexample taken for t < 3t*. This condition is not satisfied when th e dimension Rbecomes so small th at th e fluid acceleration becomes comparable with gravity,with constant K . The effect is merely to retard the return of the stagnation pointtowards its original position.

7. The form of the instability of the vortex pairThe vortex cores can be seen to ‘burst ’ when tip contrails are present. Ac-

cording to 11) or (13 ) the vortex pair is seen to be carried downwards with aconstant velocity at first, but later with an exponentially increasing velocity.This means that in the later stages, though not at first, irregularities in theoriginal vortex pair will grow exponentially and the vortices will assume anundulated configuration as shown in figure 9 with the width of the pair decreased

FIGURE . Perspcetive diagram of the relative positions of the cloud blobs and the burstparts of the tip vortices in th e airc raft wake.

a t the troughs as compared with the crests. Furthermore, because of the exponen-tial growt,h, the configuration will be in a series of arches without significantregions of upward curvature. The tip trails will burst a t the bottoms of the troughs.When bursting begins it will spread along the ti p vortices as they descend.

If there is any cloud from a nearly centrally placed engine it will be detraineda t the top if it has had time to pass round the vortices leaving a straight upperedge, while the bottom of i t will be caused to descend in a series of blobs (figure 9)which correspond to where the vortices are closer together a t the troughs.

8. Interpretation of contrailsThe viewpoint of the observer is important. I n figure lO(a) , plate 1 , the two

trails from the s tarboard engines appear t o separate, while those from the portengines seem to merge. This is simply because the inner trails are rotating aroundthe outer ones. Soon after, a little further behind the aircraft, the fuzziness ofth e


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outside of th e trails vanishes because the trail is unfrozen and mixing causesevaporation; but the tubular trails in the lower pressure regions of th e vortexcores remain, and the vortex motion, being stable, becomes laminar away fromthe turbulence close to the aircraft. Evidently in this case exhaust did not pene-trate into the vortex centres. In figures 10 c) and d ) he irregularities grow andfinally mixing occurs and t he cloud disappears.

By contrast figure 11, plate 2 shows the configuration of the trail of centrallyplaced engines. The curtain of detrained cloud has the vortex pair below it whichproduces a row of blobs in the cloud as i t breaks up. This trail is frozen andpersistent, while in figure 12, plate 2, the detrained cloud is scarcely visible andwe see loops in the vortices rather than blobs in th e cloud.

I n figure 13, plate 3, we see artificial smoke trails emitted at 10,OOOft. from thewingtips of a Comet aircraft showing th e same evolution with time, photographed

from directly below. They are much narrower because they are emitted a t thewingtips. The linking of the two ends where breaks occur is fairly typical also incontrails when the cloud is dense enough, but i t is beyond the scope of thispaper t o discuss it.

Finally, the downwash of aircraft sometimes makes a clear lane (distrail) in athin layer of cloud. Very occasionally the height of the aircraft above the cloud issuch tha t a row of holes rather than a clear lane is made. This is illustrated infigure 14, plate 4.

The times and configurations of contrails are thus seen to fit the explanationsuggested here fairly well bu t we have not predicted any particular space for theblobs which are produced. Irregularities which disturb both vortices togetherwould grow more than those which only disturb one vortex. Consequently i t is tobe expected that irregularities with a space rather larger than the distancebetween the vortices would probably appear. If their spacing were too largethere could well be another disturbance growing between two widely spacedirregularities. The pictures show that the spacing is typically three or four timesthe distance between the vortices, but th at there is no clearly dominant wave-length. Probably it would be necessary t o have an absolutely calm atmospherefor a dominant wavelength to appear as such.

We are indebted to Mr J.B.Holliday, of Rolls Royce Limited, Derby, forinformation about engine exhaust used in figure 4, to Mr F. W. Dee for thephotographs in figure 13 which were first shown in a paper by Dee & Nicholas(1969), and t o the Meteorological Office for their sponsorship of the work.

REFERENCES

DAVENPORT, . J. 1967 Vortex motion in a stratified fluid. M.A. Thesis, University ofLondon.

DEE, F. W. & NICHOLAS, . P. 1969 Flight measurements of wingtip vortex motion nearthe ground. Aer o Res . Counc. Current Paper 1065.

KERR, T . H. DEE, F. W. 1960 A flight investigation into the persistence of trailingvortices behind large aircraft. Aero. Res. Counc. Current Paper489.

LAMB, H. 1932 Hydrodynamics. Cambridge University Press.OWEN,P. R. 1964 The decay of a turbulent trailing vortex. Aero. Res. Cournc.25818.


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464 R. S. Scorer and L. J . Davenport

ROSE, R. & DEE, F. W. 1965 Aircraft vortex wakes and thcir effects on aircraft. A e r o .

SCORER, . . 1955 Condensation trails. Weather, X, no. 9, 281.SCORER, . . 1968 Natura l Aerodynamics . Oxford: Pergarnori.

SCORER, . S. 1970 Colour Encyclopaedia of Clouds , (chaptor 11 ) . Oxford: Pergamon.(I n press.)

SQUIRE, H. B. 1954 The growth of a vortex in turbule nt flow. Aero. Res. Counc. 16666.TURNER, . S. 1957 Buo yan t vortex rings. Proc. Roy. SOC. A 239 61.TURNER, . S. 1960 A comparison between buoyant vortex rings and vortex pairs.

Res. Counc . Current Paper 695.

J . Fluid Mech. 7, 419.


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Journal of Fluid Mechanics, Vol. 43, art 3

3

R . S. SCORER AND L. . DAVENPORT

Plate 1

(Faacing p . 464)


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Journal of Fluid M echanics , Vol .43, art 3

dEH+

.fHt

m

f

r j

d

8+

mB3

P

Plate 2

5E

P6-

m

5d

f

W

f

R . S. SCORER AND L. J. DAVENPORT


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Journal of Fluid Nechanics, Vol. 43 art 3 Plate 3

FIGURE 3. A series of ten pictures of smoke trails produced a t the wingtips of a Cometaircraft at 10,000 ft., and photographed from below. Th e times are indic ated in seconds.

R. . SCORER A N D L . J. DAVENPORT


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Journal o Fluid Mechanics, Vol. 43, art 3 Plate 4

FIGURE 4. Four successive pictu res of th e effect produced by an aircraft descending int o athin layer o f cloud whilr: making a turn. In ( a ) he prominent feature is the dense trailwhore the aircraft is in an ice cloud. The distrail is ju st beginning t o show.

I n b ) , c) and d ) we see th e subseq uent development of the distrail, partly as a row ofholes, corresponding to th c tr ail blobs.

R. S. SCORER AND L. J. DAVENPORT

contrails and aircraft downwash by scorer davenport

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