1INSTABILITY AND UNSTEADINESS OF AIRCRAFT WAKE VORTICES

L. Jacquin(1), D. Fabre(1), D. Sipp(1), V. Theofilis(2) & H. Vollmers(2)

(1) Dept of Fundamental and Experimental AerodynamicsONERA, 8 rue des Vertugadins, 92190 Meudon - France

(2) Institute of Aerodynamics and Flow TechnologyDLR, Bunsenstrasse 10, D-37073 Goettingen - Germany

SUMMARY

This paper presents a review of theoretical and experimental results on stability and other unsteady properties ofaircraft wakes. Theory mainly concern the dynamics of two-dimensional vortices. The basic mechanismsresponsible for the propagation and the amplification of perturbation along vortices, namely the Kelvin wavesand the co-operative instabilities, are first detailed. These two generic unsteady mechanisms are described byconsidering asymptotic linear stability analysis of model flows such as vortex filaments or Lamb-Oseen vortices.Extension of the linear analysis to more representative flows, using a global stability approach, is also described.Experimental results obtained using LDV, hot wire and PIV in wind tunnels are presented and they arecommented in the light of theory .

Key words : Wake Vortex - Stability theory Nonparallel global stability theory Unsteady flow Turbulence Laser velocimetry - Hot wire velocimetry Particle Image Velocimetry .

1. INTRODUCTIONTwo strategies may be considered to produce lessharmful wakes behind aircraft, see e.g. Gerz et al.(2001), Jacquin et al. (2001). The first is to increasethe characteristic radii of the final vortices so as todecrease the rolling momentum of a followingaircraft during encounters. This may be achieved byintroducing "as much turbulence as possible" intothe vortex system. The second strategy is to promotethe co-operative instabilities which develop in asystem of several (at least two) vortices and whichleads to destructive interactions between the twohalves of the wake. Theoretical analysis and specific measurementsare both needed to investigate this topic. This paperaims at presenting the status of our understanding ofunsteadiness in wake vortices which has been gainedboth from theoretical analysis of model vortex flowsand from experimental investigations ofrepresentative trailing wakes. As it will be shown below, useful theoreticalresults are provided by linear stability analysis ofgeneric flows. This enable the introduction ofimportant physical mechanisms which lead tounsteadiness. As for the experiments, wind tunneltests give access to measurements withindownstream distances limited to typically ten modelspans. In this regime, the wake unsteadinessamounts to small amplitude displacements of thevortex cores. Due to the presence of very sharp

velocity gradients within the cores, this"meandering" of the vortices leads to energeticvelocity temporal fluctuations which can bemeasured both by LDV and hot wire techniques.PIV-measurements also show the fluctuation of tipand flap tip vortices. Examples will be described insection 4. These wind tunnel tests can be prolongedby using a catapult facility, as described in Stuff etal. (2001). The paper is organized as follows. Averageproperties of a vortex wake are described in section2. In section 3, basic linear mechanisms whichparticipate to vortex wake unsteadiness andinstability are reviewed. This comprises the Kelvinwaves and the co-operative instabilities thatdevelop on long- or short-wavelengths withinmulti-polar arrangements of vortices. In section 4,some experimental observations are presented andare discussed in the light of theory. Open questionsand perspectives are listed in Section 5.

2. THE MEAN FIELD AND LENGTH-SCALES

The trailing wake produced by an aircraft amountsto an antisymmetric distribution of vorticity ofcirculation :

=0

dzdy (1)

2where denotes the x-component of the vorticity.Convention are those of figure 1. Let consider awing with lift coefficient LC , span b, airfoil surfaceS and aspect ratio SbAR 2= which is placed in aflow with velocity

V . From conservation of

vertical momentum, the lift, SVL2C21

, is equalto the flux of the wake vertical momentum b~V

where :

= dzdyyb~ 1 (2)

is the separation between the vorticity centroids (seee.g. Saffman, 1992). It follows that b~V

= Lift .

In the case of an elliptic loading, ( )bb~ 4= , thecirculation is therefore given by ARbVCL = 2 .For a landing transport aircraft one typically has :

2LC , 7AR .

Figure 1 -Sketch of far-field flow downstream of a wing

The half-wake wake circulation can onlydecrease through diffusion and cancellation ofcirculation through the plane that separates the twovortices. In an aircraft vortex wake, this is a veryslow process due to the high value of the Reynoldsnumber. A decrease of is compensated by anincrease of b~ in order to maintain the wake verticalimpulse b~V

.

In order to characterize the internal structure of thevortices, several vortex core length-scales may bedefined (see Jacquin et al., 2001). In particular, anaircraft vortex may be assumed to consist of an"internal core" (or "viscous core") of radius 1r ,rotating as a solid body, and of an "external core" (or"inviscid core") of radius 2r , which characterizesthe region containing vorticity surrounding theinternal core (see Spalart, 1998). The internal coreradius 1r is usually defined as the position ofmaximum azimuthal velocity and the external coreradius 2r is the position where the total circulationof the vortex is almost attained. The need of such a two-scale model to describe theinternal structure of a wake vortex has beenillustrated by LDV measurements performed in thewake of a scale 1:100 scale generic A300 model, see

Jacquin et al. (2001). In this experiment, the modelhad a wing span mmb 448= and it was set in twoconfigurations, a cruise configuration (noted "cleancase") and a high lift configuration (noted "highlift"), with wings fitted with single block flaps. Thelift coefficients were respectively 70.CL and

71. . Tests have been performed for a free-streamvelocity 150

= msV . The Reynolds number

based on the aerodynamic chord mmc 66 was000220cRe . The Reynolds number based on the

circulation was 000230=Re in the high

lift case ( sm. 2463 ) and 00095Re in theclean case ( sm. 2421 ). Values of areobtained using the elliptic expression given above. Figures 2 and 3 show the tangential velocity Vafter cylindrical averaging of the LDV mean valuesobtained in three vertical sections, =bx 3,5,9,behind the model in the high lift and clean casesrespectively. In both cases, the velocitydistributions remain almost unchanged between

3=bx and 9 ; difference in the maximum of Vat 9=bx results from change in the probing meshsize. In the high lift case, see figure 2, the tangentialvelocity is characterized by the presence of aplateau where V remains around its maximumvalue, that is approximately 23% of

V . Such a

velocity plateau was also observed by Devenport etal. (1996) ; as suggested by these authors, this couldcorrespond to a remnant of the initial conditionsdue to a merging of several cores. In the high liftcase, the merger between the wing tip and the flaptip vortices occurs around 2=bx . After thisplateau region, the decrease of V seems to follow

a power law r with around 0.5 (the exactvalue of the slope is indicated in figure 2(b)). Thisis in accordance both with the classical modelproposed by Betz (1932), which leads to 50.=for a wing with an elliptic load, and with the self-similar solution of the roll-up given by Kaden (seeSaffman, 1992). Departure from these laws occursaround 102 ,br .

In the clean case, see figure 3, no plateau isobserved. A r law with slightly larger than0.5, is particularly well adapted here. Vortices areless intense (the maximum of V is worthapproximately 15% of

V ) and more concentrated.

A two scale model of the type :

ArV:rr = 1

,BrV:rrr 2121

= (3)1

2

= CrV:rr

x

yzb

b~

1

2

3has been proposed to model the flow, as sketched infigure 4 . The values of the constants are :( )2112 rrrA = , ( )22 rB = , = 2C .Our results suggests : ( )21 10= Ob~r ,( )12 10= Ob~r . A fourth region, corresponding tothe plateau observed in figure 2, could also be addedfor the high lift case.

(a)

0 0.1 0.2 0.3 0.4 0.50.00

0.05

0.10

0.15

0.20

0.25

x/b=3x/b=5x/b=9

(b)

10-2 10-10.05

0.10

0.15

0.20

0.25

x/b=3x/b=5x/b=9

Figure 2 - Tangential velocity VV versus the radius

centred on the vortex - high lift configuration : (a) lin-linplot, (b) log-log plot (from Jacquin et al., 2001).

In both the clean and high lift cases, the inner coreradius 1r is very small and the mesh size used( mm.52 ) is not sufficient to discriminate the innerregion 1rr . As shown by Jacquin et al. (2001), inthis experiment 1r is close to mm1 , which means

0101 .br < .

Another useful lengthscale is the "dispersionradius", defined as :

( ){ }

+

=

0

222 1 )zz()yy(dydzr cc (4)

where cy and cz are the coordinates of the vorticitycentroid in half a plane. This provides an evaluationof the vorticity field dispersion in the plane. If the

evolution of the wake is purely two-dimensional, rcan only increase by viscous diffusion (seeSaffman, 1992). Other definitions for vortex corelength-scales are discussed in Jacquin et al. (2001).

(a)

0 0.1 0.2 0.3 0.4 0.50.00

0.05

0.10

0.15

0.20

x/b = 3

x/b = 5

x/b = 9

(b)

10-2 10-1

0.05

0.10

0.15

0.20

x/b = 3

x/b = 5

x/b = 9

Figure 3 - Tangential velocity VV versus the radius

centred on the vortex clean configuration : (a) lin-linplot, (b) log-log plot (from Jacquin et al., 2001).

Figure 4 A vortex model.

The available experimental and numerical resultsshow a large discrepancy on the internal core radius

1r . On the other hand, the external radius and thedispersion radius are generally found to be of theorder ( )b.O 10 .

br

r

1r

b~.r 102 b~

.r 0101

VV

r

br

br

br

470.

570.

VV

VV

VV

VV

43. STABILITY OF WAKE VORTICES

3.1. Long-wave instabilities

The wake issuing from a high lift designed wing isusually compounded of several vortex filamentswhose evolution may be computed using a 3Dvortex method. In general, such a vortexarrangement is unstable with respect to 3Dperturbations due to mutual straining of the vortices.Following Crow (1970) and Crouch (1997), asystem of stability equations may be derived byconsidering a set of parallel vortex filaments withslight sinusoidal perturbations of their respectivepositions. We suppose that the centerline position( ))t(Z),t(Y nn of the vortex filament labeled n isdisplaced with an amplitude proportional to ikxewhere x is the coordinate in the axial direction :

( ) ( )( )( )( ) zikxnn

yikx

nnxn

eetz)t(Zeety)t(Yext,xX

++

++= (5)

At the leading order, the linearization of the Biot-Savart law leads to the system :

2

2

2

2

mn

mn

nm

mn

mn

mn

nm

mn

RYY

dtdZ

RZZ

dtdY

=

=

(6)

where ( ) ( )222 nmnmmn ZZYYR += . This systemdescribes the evolution of an ensemble of N pointvortices. At the following order, the equations whichdescribes the evolution of the perturbation amplitudevector ( ) ( )NN z,y,...,z,ytX 11= may be put underthe following form :

( ) ( ) ( )tXtLdt

tXd= (7)

The developed expressions of this linear system aregiven in Crow (1970) for the case of a pair ofcounter rotating vortex filaments, in Crouch (1997)for the case of 22 co-rotating filaments and inFabre & Jacquin (2000) for 22 contra-rotatingfilaments. The right hand side of the system (7)amounts to a superposition of three effects : (i) thestraining experienced by filament n when displacedfrom its mean position in the velocity field of theunperturbed filaments m, (ii) the self inducedrotation of the filament n and (iii) the velocity fieldinduced on the filament n by the other vortices whensinusoidally displaced from their mean positions.

Mechanism (i) may be easily understood whenconsidering the 2D flow corresponding to a pair ofstraight counter-rotating vortex filaments separated

by a distance b~h = , as in figure 1. In a co-ordinatesystem moving downward with a speed( )b~dtdZ = 2 , linearization of (6) around thecenter of the vortex labeled 2, i.e. 1

5merged wing and flap tip vortices and merged innerflap and horizontal tail vortices. Donaldson &Bilanin (1975) showed that the evolution of thissystem is either divergent (the internal and externalpairs separate) or periodic (the vortices orbit aroundthe center of vorticity in each half-plane). As shownby Rennich & Lele (1999), a stationaryconfiguration, such as CtedtdZ,dtdY nn == 0 ,see (6), is possible if :

Figure 5 - Four-vortex arrangement

0331

2

1

22

1

2

1

23

1

2=

++

+

bb

bb

bb

(11)

A temporal analysis may be then conducted.Solutions of (7) are seek under the form( ) teXtX = 0 . The results of the three dimensional

stability analysis by Fabre & Jacquin (2000) isdescribed in figure 6 which sketches theamplification rate of the unstable eigenmodes forthe base flow corresponding to the set of parameters

4012 .= , 14012 .bb = , ,.ba 1011 = ,.ba 05012 =a choice suggested by Rennich & Lele (1999). Notethat, according to the above discussion, the methodrequires the core radii of the vortices to be smallwith respect to their separations, i.e. :

( ) 2212121 bb,b,ba,a

6Figure 7 - Long-wave modes 801 .kb = ,( )211 289 b. = : (a) symmetric 1S - mode , (b)antisymmetric A - mode, (c) most amplified 1S - mode :( )2111 22187 b.,kb == . These modes are identifiedby a circle in figure 6 (from Fabre & Jacquin, 2000).

The case of a basic flow corresponding to a Rankinevortex has been extensively described in theliterature (see e.g; Saffman, 1992, Rossi , 2000). Weconsider here the case of a Lamb-Oseen vortex, withangular velocity :

( ) ( )

==

221

2 2are

rr

rVr (13)

which has been described by Sipp et al. (1999), seealso Sipp (2000), Fabre et al. (2001). We linearizethe incompressible Euler equations around thesebasic flow and introduce the following smallperturbations :

( ) ( )( ) ( )tmkxirx erP,H,iG,Fp,v,v,v + = (14)where r

x v,v,v are the axial, radial and azimuthalcomponents of the velocity and p the pressure.

ir i+= denotes a complex frequency.Following Howard & Gupta (1962), Lessen, Singh& Paillet (1974), we are led to the following secondorder equation for the variable ( ) ( )rrrGZ = ,( ) ( ) = rmr :

0= ZBZAZ (15)

with Ar

m k rm k r

=

+

1 2 2 22 2 2

,

2kB = +2

2

r

m

r

m

2 - ( )222

24

rkmmk+

2 22

k

, where ( ) r =( )( )drrVdr 1 is the basic flow axial vorticity.

This equation is valid whatever ( )r . With theboundary conditions ( ) =0Z ( ) 0=Z , this equationconstitutes an eigenvalue problem for . Thisproblem admits a countable infinity of eigenvaluesindexed as ( )kn,m where k and m are the axialand azimuthal wavenumbers and where the secondindex n is related to the number of zeros of theeigenmode (the higher the label, the more radialoscillations the mode contains). Resolution isachieved using a shooting method. Results areshown in figure 8 for the axisymmetric modes

0=m and the helical modes 1=m . The results arevery similar to those obtained with a Rankinevortex except for the modes that possess a criticallayer. Critical layers occur for non-axisymmetricmodes ( 0m ) whenever the angular phase speedof the perturbation, mr , coincides with theangular velocity of the vortex ( )cr at some radius

cr , i.e. ( ){ } 0= crRe . Modes which possess acritical layer are necessarily damped, i.e. 0

7vortices, energy of the perturbations is thusconvected downstream. This prevents the possibilityof perturbation energy traveling upstream at a higherspeed than that of the flow, a necessary condition foroccurrence of a "hydraulic jump" leading to vortexbursting.

(a)

k a

/

0

0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5

0,2

0,1

0,-1

0,-2

0,-30,-4

0,-5

0,50,4

0,3

Con

tinu

ou

ssp

ectru

m

k a

/

0

0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5

0,2

0,1

0,-1

0,-2

0,-30,-4

0,-5

0,50,4

0,3

(b)

k a0 1 2 3 4 5-1

-0.5

0

0.5

1

1.5

2

1,2

1,1

1,51,41,3

1,-2

1,-1

1,0

/

0

0 1 2 3 4 5-1

-0.5

0

0.5

1

1.5

2

Figure 8 - Frequencies r of the Kelvin waves in aLamb-Oseen vortex : (a) axisymmetric mode 0=m , (b)helical mode 1=m . 20 2 a= (from Fabre et al.2001).

- Let consider now the case of helical modes1=m . According to the symmetries of the base

flow, one has n,n, 11 = . This means that thehelical waves must be considered by pairs, the left-handed modes ( 1=m ) propagating along the vortexaxis in opposite directions than their right-handedcounterpart ( 1=m ). As shown by Widnall et al.(1971), the asymptotic behavior for 0ka of thebranch 01, is :

( )

+

= Cteka

lnkk,

14

02

01 (17)

This branch is called the "slow branch" since boththe frequency and the phase velocity kc r=(and also the group velocity dkd r ) tend towardszero when k goes to zero. The corresponding mode,which has the least complex spatial structure(eigenfunctions with no zero), is responsible for themain part of the vortex core displacement. Thismode corresponds to the self-induced oscillationmode of a filament vortex introduced in 3.1, seerelations (9)-(10). - The oscillations which correspond to thebranches n,1 , n=1, 2, in figure 8(b), verify :

( ) ( )( )( )

+++

nn

kaa

kn, 2121

20

2

21, (18)

see Leibovich et al. (1986). The angular velocity( ) mm,k 0 of these modes exactly matches

that of the vortex axis ( )0= r . They are called"fast branches" because their phase velocity tendsto as 0k (note however that, according to(18), the group velocity of these "fast waves" tendsto zero with k). Calculation of their spatial structureshows that in the limit 0k , these modes areconcentrated in a very small region near the vortexaxis and that, consequently, they are likelyeliminated by viscosity, see Fabre et al. (2001). - It exists wavenumbers where 0=r , for whichthe helical perturbations 1=m are steady. For aLamb-Oseen vortex, see figure 8(b), thesewavenumbers are found to be 262.ka , 963. ,

615. , etc. The superposition of these steady waveswith their counterparts 1=m leads to steadyuntwisted perturbations. As an example, the vortexdeformation resulting from the superposition of thetwo branches 11 , is visualized in figure 9 for thecase of a Rankine vortex (in a Rankine vortex, thewave number for which 0=r is 52.ka , avalue slightly higher than that of the Lamb-Oseenwhich is 262.ka ). The vorticity field associatedto this wave is shown in figure 10 for the case of aLamb-Oseen vortex. An interesting mechanism is now a possibleexponential amplification of these steady Kelvinwaves by straining imposed by other vortices,which means instability.

3.4. Core instabilities

As seen in 3.1, the presence of a second vortexleads to imposition of a strain field on the currentvortex. Such a field has been described in (8) for acounter-rotating vortex pair ; it corresponds to astrain of rate ( )22 b~ with axes oriented 45and its is responsible for the displacement of thetwo vortex filaments in the Crow instabilitymechanism.

0

8Figure 9 - Wave motion produced by the combination oftwo steady Kelvin waves 11 , and 11 , for 52.ka on a Rankine vortex. The vortex core boundary isdisplayed with an arbitrary amplitude (from Eloy, 2000).

Now, considering the flow within thecore, the strainfield amplifies, by stretching, any vorticityperturbations aligned with the strain axis, such asthat in figure 10(a). This mechanism, called Widnallinstability, may be quantified by means of a multipletime scale analysis, see Moore & Saffman (1975),Tsai & Widnall (1976). Considering the flowevolution on the characteristic length and time scales

aL = and = 22 aT , the strain amounts to a

perturbation at order ( )2b~a= . The assumption :1

2

9for akka c= in which case 22 b~R = . As an

example, figure 11 shows the amplification rate ofthe instabilities due to resonance of the strainingfield with the helical waves ( 0=r ,

1=m , 615963262 .,.,.ka ) for a lamb-Oseendipole of aspect ratio 20.b~a = . The results areplotted here versus kb~ . The first lobe, close to theorigin 0=b~k , corresponds to the Crow instability.The short-wave instabilities concerns wave numbers

12b~k . The status about the co-operativeinstabilities is the following : - Amplification rate of these short wave co-operative instabilities is ( )2241 b~. = , seefigure 11. It is slightly larger than the amplificationrate of the Crow instability, ( )2280 b~.Crow .The time scale 1= of the two instabilities arethus equivalent and of the order ( ) 22 b~O .

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 11 - Amplification rate of the co-operativeinstabilities due to resonance of the straining field withthe steady helical waves ( )k

, 11 such that 0=r in a

Lamb-Oseen vortex pair ( 20.b~a = ).

- Compared with the viscous time scale

2a , we have ( ) Reb~a 2 . Therefore,

if we assume that ( )10.Ob~a = , for 100>>Re , theco-operative instabilities are free from viscousdamping.- Dependence of co-operative instabilities on thevortex model is a question of importance. Onlysmall differences are found when comparing twomodels based on a single scale, such as the Rankinevortex and the Lamb-Oseen vortex for instance : themost amplified wave numbers ck and theamplification rates are comparable. The maindifference lies in the suppression of resonances(such as 20 21 == m,m ) due to critical layers. Notethat, given any of the one scale vortex models

considered so far, one may replace the core scale aby the dispersion radius r , see (4), these two scalesbeing of the same order. - Things become different when considering morerepresentative models such as that suggested by ourexperimental results, see figure 2 to 4. Figure 12shows the first amplification lobe of the short waveinstabilities obtained when considering threesimplified vortex models. The first one is theLamb-Oseen vortex pair, such that 20.b~a = , thesecond one is a "plateau" vortex pair with aconstant tangential velocity region from

0201 .b~

r = to 20.b~r = and the third is a vortex

0 25 50 75 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 12 - Amplification rate of the first Widnallinstability in pairs of vortices corresponding to differentmodels : Lamb-Oseen ( 20.b~a = ), "plateau" and Betz-like, with 0201 .b

~

r = , 202 .b~

r = . For the Betz-likemodel which is defined by equation (3), different valuesof the exponent have been considered.

pair corresponding to the model (3), with internalradius 0201 .b

~

r = and external radius 202 .b~

r = .

In accordance with figure 11, the short-waveinstabilities in the Lamb-Oseen dipole starts around

12b~k . Compared to this reference case, it is seenthat the two scale models leads to a shift towardssmaller wavelengths of the instability regions. The"plateau" model leads to occurrence of the firstWidnall instability at a slightly higher wavenumber( 20b~k ). In the case of the Betz-vortex dipole,the higher , the smaller are the unstable

b~.a 20Oseen-Lamb

=

V

V1

"plateau"r

1r

V

b~.r 0201 = b~

.r 202 =

40.=450.=

50.=550.=

22 b~

b~k

b~k

22 b~

b~.r 0201 = b~

.r 202 =

Equation (3)

10

wavelengths. For instance, using 50.= , it is foundthat the first shortwave co-operative instability is

85b~kc . This corresponds to 711 .rkc whichmeans that the instability scales with the inner coreradius. Note that, interestingly, the wave length andthe width of the instability band are extremelysensitive to the value of within the range ofvalues observed in the experiments, see section 2.This topic is the object of ongoing work. - A last interesting theoretical result is thatobtained by Sipp (2000) using a weakly non-linearanalysis of the Widnall instabilities of a strainedLamb-Oseen vortex. The result is that short-waveinstabilities saturate very quickly. The mechanismresponsible for this saturation is a sudden selfinduced rotation of the plane waves that occurswhen their amplitude become large. Detailed resultsshows that the maximum vortex core distortionsinduced by the Widnall instability is then limited tovalues smaller that b~.010 before non-linearinteractions occurs, see Sipp (2000). In conclusion, the above results suggest that, inrepresentative vortex wakes, the shortwave co-operative instabilities are likely confined in the verycenter region of the vortex cores and that theycannot lead to significant core distortions beforethey saturate.

3.5. Global instability analysis

The analysis of long- and short-wave perturbationsconducted above can be merged using a moregeneral approach, by relaxing the assumptions (12)and (19). Any flow quantity ( )tp,w,v,uq = is againconsidered as :

( ) ( ) ( )t,z,y,xq~z,yqt,z,y,xq += , (22)with a small-amplitude perturbation of order andform :

( ) ( ) ( ) .c.cey,xqt,z,y,xq~ tkxi += (23)where { }= Rer is related with the frequency of aglobal eigenmode q~ while the imaginary part,

{ }= Imi is its growth/damping rate. Theobjective of the analysis is still identification ofunstable eigenvalues and associated eigenvectoramplitude functions q for a given basic state qdescribing the wake-vortex system. The system forthe determination of and the associatedeigenfunctions q in its most general form can bewritten as the complex nonsymmetric generalisedEVP :

( ) ( )( )[ ] ( )

( ) ( )[ ]

=+

=

=

=++

wipDwwDLvwD

vipDwvDvvDL

uipikwuDvuDuLwDvDuik

zzy

yzy

zy

zy 0

(24)

where the linear operator is

( )( ) zyzy DwDvuikDDkReL ++= 2221 (25)and yDy = , yDy 222 = , Significantcomments regarding the global instability analysisare, first, that the two-dimensional eigenvalueproblem (24) permits considering wake-vortexsystems having a velocity component u in thedirection of the aircraft motion, x, in addition tothose defined on the Oyz plane, v and w ; theonly assumptions of the analysis are

0== xqtq (26)

the first of which may be relaxed in case of a time-periodic basic state by employing Floquet theory(Herbert, 1983, Barkley & Henderson, 1996). Itshould also be noted here that solution of one of thealternative simplified forms of the partial derivativeeigenvalue problem (24) valid in case of a singlevelocity component (Tatsumi & Yoshimura, 1990)or, additionally, in the inviscid limit (Hall &Horseman, 1991) is not permissible in the wake-vortex stability problem. Successful applications of global instabilityanalysis based on (24) are the studies of the sweptattachment-line boundary layer, Lin & Malik(1996), who exploited the symmetries of thatproblem to reduce the computing effort, that ofTheofilis (1998) who solved (24) for the sameconfiguration without resort to symmetries as wellas the analyses of Theofilis (2000) in open and lid-driven cavity flows and that of Theofilis et al.(2000) in boundary layer-flow which encompassesa closed recirculation bubble. Compared with thelatter applications, in which a matrix eigenvalueproblem of leading dimension in excess of 410 wassolved using state-of-the-art numerical algorithmsfor the discretisation of (24) and efficientalgorithms for the eigenspectrum, the wake-vortexsystem presents the additional challenge of yethigher resolution being necessary for the adequatedescription of the wake-vortex system basic flowitself and, consequently, the sought globalinstabilities. One simplification which halves the storagerequirements for the solution of (24) that aretypically of the order of several gigabytes, is thecase in which the basic flow velocity component uis absent in the wake-vortex system. In conjunctionof the redefinitions = i~ , wiw~ = , this results in

11

a real partial-derivative eigenvalue problemenabling storage of real arrays alone, as opposed tothe complex arrays appearing in (24). Freeing half ofthe necessary storage results in the ability to addressflow instability at substantially higher resolutionsand/or higher Reynolds numbers compared with ananalysis based on solution of (24). However, from aphysical point of view, neglecting the axial velocitycomponent u in the basic flow restricts the classesof flows that can be addressed by a global instabilityanalysis. With this consideration in mind, thepotentiality of the method will be illustrated byconsidering the case of a system composed ofBatchelor vortices, each of which is characterizedby :

( ) 2rez,yu =( ) recosqz,yv r =

21 (27)

( ) resinqz,yw r = 2

1

where ( ) ( ) nnn azzyyr 22 += , ( )nn z,ydenotes the centre and na the radius of vortex n.Here the centreline axial velocity of the vortices hasbeen taken equal to unity. In constructing a basicflow composed of several Batchelor vorticessatisfying (27), the additional freedom exists in thechoice of the relative circulation, radius and locationof the vortices. The basic flow analysed here wasconstructed along the lines of those discussed in3.2. It consists in two pairs of co-/counter-rotatingvortices. A first case was constructed by taking

11 =q and 502 .q = and assuming a stationaryconfiguration fixed by the Rennich & Lele condition(11). The first vortex was placed at the outmoststarboard location ( ) ( )0711 ,y,x = , lengths beingmade nondimensional using the radius parameter ofthe outer vortices. The radii were chosenconsistently with the definition (27), 11 =a ,

502 .a = . The Rennich-Lele condition delivers thelocation of the second vortex ( ) ( )271022 .,y,x = inthis case. The locations and swirl of these twovortices were mirrored with respect to the centreline

0=x to construct the fields which are illustrated infigure 13 (a) through the distribution of the verticalvelocity ( )z,yw . A second case was constructedusing the same vortex locations and 11 =q ,

502 .q = . Such a configuration is not stationary. A Reynolds number can be defined as

= 11 qRe . The global instability analysis was

performed at 31 10=Re and several axial periodicitywavenumbers k, of which results at a short wavenumber, 32=k i.e. 492 .kLx = , are

(a)

(b)

Figure 13 Vertical velocity ( )z,yw of the basic flowconsidered : (a) 02 >q (co-rotating), (b) 02 qq and 021 qq can be easier

12

(a)

(b)

Figure 14 Global eigenspectra at 310=Re ,32=k : (a) 02 >q (co-rotating), (b) 02

13

vortices. This is put in evidence by considering thespectral contain of the perturbations. Figure 17shows the energy density of the axial component ofthe velocity measured using a standard single wireDISA P11 probe placed in the center of the high liftand clean configuration vortices at 5=bx . Some

0 2 4 6 8 100

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2k

max(HL)/V

kmax

(C)/V

Figure 16 - Downstream evolution of the peak value ofthe turbulence rate

Vk measured with a 3D-LDV

system in the centre of the wing tip vortex of a A300model in the high lift and clean configurations (in thehigh lift case, merging with the flap vortex occurs around

2=bx ; from Jacquin et al., 2001).

characteristic slopes are indicated. It is seen that thevortex meandering corresponds to a broadbandspectrum which exhibits a sharp energy excess forfrequencies smaller than, say Hzf 1000 (redcurves). Searching for possible relationships withinstabilities lead to identification of energyovershoots in the spectra. In the high lift wakevortex, three energy accumulations may beidentified (see arrows in figure 17). They are locatedat, approximately, Hzf 15 , Hz55 and Hz400 .These energy peaks are not found in the clean caseshown in figure 17(b), except an energy bumparound the first of these frequencies. It is also veryclear from these figures that the small scales of theclean case vortex are much less energetic than in thehigh lift case. Thus, one may wonder if these features are relatedto linear mechanisms. Some answers were proposedby Jacquin et al. (2001). They are summed-upbellow with some additional remarks that can bemade in the light of the new theoretical resultspresented in figure 12 concerning short-waveinstabilities. - As seen in 3.2, the theoretical wavelength forthe Crow instability which may develop in a dipoleor in a four-vortex arrangement is 80.b~k whichmeans b~Crow 8 . At 5=b

~

x , one has

mmb~ 340 which gives =Crowf CrowV Hz18 , a value close to Hz15 . Consequently, the

(a)

100 101 102 103 10410-6

10-5

10-4

10-3

10-2

10-1

(b)

100 101 102 103 10410-7

10-6

10-5

10-4

10-3

10-2

10-1

Figure 17 Spectral densities of the axial component ofvelocity measured with a hot-wire probe in the center ofthe vortex at 5=bx . (a) high lift case, (b) clean case.The red and black curves correspond to 20kHz and 2kHzsampling, respectively (from Jacquin et al., 2001)

peak located around Hzf 15 may correspond tothe emergence of a long-wave co-operativeinstability of this type. As mentioned in 3.2, the"signature" of a four vortex instability was detectedin two-point correlations measured in the high liftcase. This may contribute to differences in the lowfrequency energy contain between figures 17(a) and17(b).- The amplification rate of the Crow instability is( )2280 b~.Crow . As seen in 3.2, it mayreach a value ten times higher when co-operationwith an inner counter-rotating vortex pair isaccounted for. Development of such long-waveinstabilities takes place on a characteristic distance

bx

high liftclean

( )fSuu

( )Hzf

-4/3

-3

-5/3

-3

( )Hzf

1010.1 100 b~k

1010.1 100 b~k

14

=

Vx . Using an elliptic model for evaluating and b~ (see section 2), the characteristic distancescorresponding to an 1e amplification of the Crowinstability is found to be 30bx in the high liftcase ( 71.CL ) and 70bx in the clean case( 70.CL ). These distances become ten timessmaller if a four vortex system is considered. Inconclusion, only the very first stagesof the development of the Crow instability can befelt at 5=bx . But the vortices are veryconcentrated and slight vortex displacements maygenerate strong fluctuations on a fixed hot wireprobe. So, there is no impossibility to establish arelationship between the first energy peak observedin the spectrum in figure 17 and a long-wave co-operative instability, even if the latter only emerges. - Let consider now the short-wave instabilities. Inthe clean case, we saw that a Betz-like vortex with

550. fits correctly the velocity profile of theclean case vortex at 5=bx , see figure 3(b). In thiscase, from figure 12 it is found that the firstshortwave co-operative instability occurs around

95b~kc that is Hzfc 2200 using mmb~ 340= .

In figure 17(b), such high frequencies correspond tovery low levels of energy compared with those ofthe low frequency part. The conclusion is that, assuggested by theory, short wave instabilitycontributions are almost undetectable in the cleancase.- Things are different in the high lift case. First, thepresence of a plateau region shown in figure 2 letsroom to occurrence of short-wave instabilities atsmaller frequencies than in the clean case. This issuggested by figure 12. The third energy peak foundat Hzf 400= in figure 17(a) correspond to awavelength m.1250 . This correspond to

17b~k which is not incompatible with thetendencies depicted by the plateau model in figure12. The much higher energy contained in theseintermediate frequencies in figure 17(a) comparedwith figure 17(b) does suggest that short waveinstabilities could be more active there. This is alsoillustrated in figure 18 which shows the product

( )fSf uu obtained when the hot wire is movedalong a vertical line crossing the vortex core up to itscenter. Each curve correspond to a displacement of

mmz 1= . Integral under these curves correspondsto the signal energy. It is seen that energy of theperturbations undergo a dramatic increase within adistance of mm21 when approaching the vortexaxis. The energy increase essentially comes fromshort-wave contributions.- Note finally that neutral Kelvin waves, initiated byperturbations emanating from the model (turbulence,separations..) and propagating downstream along thevortex', see 3.3, may also contribute to the

broadband spectra shown in both figures 17(a) and(b).

100 101 102 103 104

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Figure 18 - Lin-log plot of ( )fSf uu obtained bytranslation of the hot wire in the vortex of the high liftcase at 5=bx , along a vertical line crossing the vortexcore. The highest energy is obtained on the vortex axis.Each curve corresponds to a displacement of mmz 1=with respect to the vortex axis.

PIV measurementsPIV measurements enable to characterize vortexunsteadiness in a different way. One considers herethe wake generated by a 1:13.6 A320 half-modelwith a semi-span equal to 1.25m. The experimentwas conducted in DNW-LLF. For three differentconfigurations of the wing, samples of one to twohundred particle images were taken and the vectorfields evaluated. Besides some exceptions whichshowed three vortices, the topology of the field inthe frame of reference of the camera was formed bytwo vortices in each half plane, as shown in figure19. In order to estimate the position and strength ofthe vortices, the velocity field was approximated bytwo Lamb-Oseen vortices. The distributions ofcirculation and radius a, cf. equation (13), wereinvestigated. The mean value and the standarddeviation of both quantities are given in Table 1.Though this overlay of velocity fields is notconsistent with Navier-Stokes equations theapproximation is acceptable as shown in figure19(b), see also Vollmers (2001). In figure 20,variation of the vortex center points for threedifferent model configurations have been plotted.Both the vortex displacements depicted in figure 20and the variations in the model parameters intable 1define measures of the instationarity of thevortices.Figure 20 gives an indication about meanderingamplitude whose typical value is found to be

m.050 , that is 020.b . This is much higherthan the value which were inferred from LDVmeasurements in the experiment presented above. It

( )fSf uu

( )Hzf

1010.1 100b~kb~k

15

(a)

(b)

Figure 19 (a) the velocity field with two vorticesgenerated by flap and tip of a model wing isapproximated by superposing the velocity fields of twoLamb-Oseen vortices, (b) comparison of v-component ofvelocity in cuts through the approximated vortices(dashed lines) given in figure (a). PIV data in solid lines.Circles represent location and radius of Lamb-Oseenvortices. The respective lines of zero values indicate thecenter of vortices as determined by the approximationwith two Lamb-Oseen vortices.

is seen that the flap vortex is subjected to largerexcursions than the wing tip and that its movementsadmit a preferential orientation which suggest a co-operative instability. This could confirm that multi-polar wakes are much more unstable than dipolarones. From figure 20, it can be also inferred than the"blue" and "green" model planforms produce moreunsteady wakes than the "red" one. At last, thefluctuations of circulation and vortex core radiuswhich are reported in table 1 show that theconfiguration corresponding to case m05 is moreunsteady than the others.

5. CONCLUSIONS

The characterization of the unsteady properties ofthe wake of a given aircraft configuration may betested in wind tunnels using conventional unsteadymeasurement techniques, i.e. LDV, hot wire andPIV. Sample results have been presented.

Figure 20 - Distribution of center points from theapproximation by two Lamb-Oseen vortices. Thedifferent colors refer to different configurations of thesame model (95 samples in blue, 298 in green, 100 inred). The different rectangles refer to the same region ofthe flow.

( )12 1

m s ( )22 1

m s ( )a

m

1( )a

m

2

Case m06Mean 12.3 5.1 0.0501 0.0251 0.32 0.258 0.00197 0.00245% 2.6% 5.1% 3.9% 9.7%

Case m04Mean 10.4 5.6 0.0504 0.0345 1.08 0.711 0.00789 0.00869% 10.4% 12.7% 15.6% 25.2%

Case m05Mean 10.4 5.6 0.0506 0.0352 1.00 1.02 0.00737 0.0136% 9.6% 18.1% 14.5% 38.6%

Table 1 - Mean values and standard deviation ofapproximations of velocity fields by two Lamb-Oseenvortices for different configurations of a half model.

Asymptotic linear stability theory leads to thecharacterization of important physical mechanismsthat promote unsteadiness in wake vortices. Thistheory provides useful reference guidelines tounderstand some of the experimental observations.Application of such quasi-analytical methods tocharacterize representative wake flows is underway. A more complete and predictive approach forrealistic flows, based on a global stability analysis,is also in progress. Results of comparisons withasymptotic results and experiments will bepresented in due course.

AcknowledgmentsThe authors gratefully acknowledge the EuropeanCommunity and the French-Service des ProgrammesAronautiques (SPA) for their support.

16

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