Analysis of wake vortex decay mechanisms in the atmosphere … · 2003-07-16 · F. Holzäpfel et al. / Aerospace Science and Technology 7 (2003) 263–275 265 Table 1 Survey of cases,

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Aerospace Science and Technology 7 (2003) 263–275www.elsevier.com/locate/aesc

Analysis of wake vortex decay mechanisms in the atmosphere

Analyse der Zerfallsmechanismen von Wirbelschleppenin der Atmosphäre

Frank Holzäpfela,∗, Thomas Hofbauera, Denis Darracqb, Henri Moetb,François Garnierc, Cecile Ferreira Gagoc

a Institut für Physik der Atmosphäre, DLR Oberpfaffenhofen, 82234 Weßling, Germanyb CERFACS, 31057 Toulouse, France

c Département de Physique Unité d’Environnement Atmosphérique, ONERA, 92320 Châtillon, France

Received 10 October 2001; received in revised form 13 December 2002; accepted 26 February 2003

Abstract

Results of high-resolution numerical simulations of aircraft wake vortex evolution and decay in different regimes and atmconditions are presented. The different cases comprise (i) the near field interaction of a trailing vortex with an exhaust jet, (ii) the evsingle vortices and counter-rotating vortex pairs in homogeneous isotropic turbulence, as well as (iii) the decay of wake vortices in astably stratified atmosphere, and (iv) in a weakly turbulent sheared environment. The different cases are used to analyse commovortex dynamics and decay mechanisms. In all scenarios the formation of coherent secondary vorticity structures that enclose tvortices is observed. These secondary vorticity structures deform and weaken the primary vortices and in some cases lead to rdecay. It is shown that the mean swirling flow effectively rearranges and intensifies any secondary vorticity by tilting and stretchsecondary vorticity may either originate from the turbine jet, ambient turbulence or may be produced baroclinically. Based on thephenomena, eleven postulates are established that pinpoint fundamental aspects of the observed decay mechanisms. 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

Zusammenfassung

Mittels hochauflösender numerischer Simulationen wird die Entwicklung und der Zerfall von Flugzeug-Wirbelschleppen unschiedenen atmosphärischen Bedingungen untersucht. Die betrachteten Szenarien umfassen (i) die Wechselwirkung eines Triebmit einem Nachlaufwirbel, (ii) die Entwicklung von Einzelwirbeln und gegensinnig rotierenden Wirbelpaaren in homogener isTurbulenz, (iii) den Wirbelschleppenzerfall in einer turbulenten stabil geschichteten Atmosphäre und (iv) in einer schwachten Umgebung mit überlagerter Windscherung. Die Untersuchungen konzentrieren sich auf den verschiedenen Fällen gemeipekte der Wirbeldynamik und des Zerfalls. Stets werden kohärente Sekundärwirbelstrukturen beobachtet, die die Primärwschließen. Die Sekundärwirbel deformieren und schwächen die Wirbelschleppe und können ihren schnellen Zerfall auslösengezeigt, dass die mittlere Wirbelströmung die zunächst inkohärente sekundäre Wirbelstärke in effektiver Weise durch Kippganisiert und mittels Streckung verstärkt. Die sekundäre Wirbelstärke kann durch den Triebwerksstrahl, die Umgebungsturbubarokline Produktion eingebracht werden. Es werden elf Postulate aufgestellt, die wesentliche Aspekte der beobachteten Zerfnismen beschreiben. 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

Keywords: Wake vortices; Numerical simulation; Exhaust jet; Turbulence; Stable stratification; Shear; Secondary vorticity structures

Schlüsselwörter: Wirbelschleppe; Numerische Simulation; Triebwerksstrahl; Turbulenz; Stabile Schichtung; Scherung; Sekundärwirbelstrukturen

This article was presented at ODAS 2001.* Corresponding author.

E-mail address: [email protected] (F. Holzäpfel).

1270-9638/03/$ – see front matter 2003 Éditions scientifiques et médicalesdoi:10.1016/S1270-9638(03)00026-9

Elsevier SAS. All rights reserved.

264 F. Holzäpfel et al. / Aerospace Science and Technology 7 (2003) 263–275

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1. Introduction

The lift exerted on aircraft wings produces long-livtrailing vortices. Especially during an aircraft’s critical laning phase these can endanger following aircraft. To avwake vortex encounters, follower aircraft must maintaisafe distance to a landing aircraft up ahead of them. Inder to increase airport capacities whilst at least mainting safety levels, the knowledge of wake vortex behaviunder varying meteorological conditions achieves consiable significance. Also, the influence of aircraft designthe strength of the shed vortices and the related potentmore rapid decay is of crucial importance.

The mostly prevailing decay mechanism in cruise atude, the so-called Crow instability [2], can frequently beserved being visualized by contrails. Minor atmosphericbulence is sufficient to trigger sinusoidal oscillations ofvortices that lead to vortex reconnection and, finally, themation of descending vortex rings. Although the Crow insbility is well understood in principle, the longevity of vorterings and their effects on encountering aircraft are discucontroversly.

Another controversy applies to the temporal evolutionvortex intensity and decay. Two radically different conceare debated, namely the concept of gradual and predicdecay and the concept of stochastic collapse [30]. In theter concept a phase of minor gradual decay is followeda rapid decay phase [9] which coincides with a transitto fully turbulent vortices [15,26]. The controversy canpart be resolved by the insight that the observed circulaevolution largely depends on its definition. On small raa gradual decay is observed which reflects diffusion invortex core region, whereas larger radii are not affectedthis diffusion and rapid circulation decay sets only in whinstability mechanisms have developed. On the other hrapid decay could well be masked in lidar1 measurementdue to several reasons: (i) gradual spatial deformation oftices will apparently decrease gradually swirl velocitiesthe measurement plane, (ii) the advection of deformedtices through the measurement plane will cause substascatter. If that scatter shall be reduced by averaging ofculation data from several flights, characteristic featurecirculation evolution are smeared out, (iii) the late phasepotentially rapid circulation decay is not accessible by liprobably because vortices decay turbulently [15] or canbe distinguished from their turbulent environment. Somerical approaches have to be employed to contributecomprehensive understanding of vortex decay characttics. The above mentioned and further implications ofculation evaluation from lidar data are discussed in dein [16].

In-situ measurements in the wakes of various airchave shown that the exhaust jets contribute dominantl

1 LIght Detection And Ranging, remote sensing technique to meathe line-of-sight velocity along a Laser beam that scans wake vortices.

,

l

-

aircraft-induced wake turbulence compared to other soulike boundary layer separation [14]. So the turbineconstitutes a top candidate to affect wake evolutionaircraft design. Turbulence, stratification and shear, onother hand, are the primary influences on vortex evoluin the atmosphere. All the mentioned scenarios are tacin this manuscript. Merely, the effect of ground proximitynot considered.

In the current article, a condensed survey is givenfour wake vortex calculations that were in part alreadescribed in previous publications. In each of the four cadifferent numerical approaches and parameters are ap(cf. Table 1). A detailed description of the approachwould go beyond the scope of this paper and is gielsewhere [6,11,14,24]. However, the dissimilar pecularof the described approaches on one hand, and the sitopologies of the resulting vortex evolutions on the othand, indicate a relative independency of results frspecific methodological aspects. This finding encouraus to consider the extracted common aspects of wvortex decay as generic and universally valid results.analysed phenomena are related to the formation of cohsecondary vorticity structures and their impact on vordecay. We argue that the observed effects, though extredifficult to verify experimentally, generally occur in wakvortex evolution and play a crucial role for vortex decBased on the current results, eleven postulates are dethat characterize prominent mechanisms of wake vodecay. The interrelation of azimuthal vorticity structuand wake vortex decay was first pointed out by Riet al. [27] and is typical for three-dimensional turbuleninitialization [19].

2. Vortex jet interaction

In the near field of the aircraft wake the exhaust jets ofturbines are entrained into the two counter-rotating wingvortices which at the same time roll up from the sheevorticity induced by the wings [7]. To investigate the highcomplex entrainment and mixing process of the turbujet into a trailing vortex it is convenient to identify twoverlapping regimes [5,22]: the jet regime and the deflecregime. In the current numerical approach these two regare modeled sequentially. First the turbulent jet evoltemporarily and uninfluenced by the wake flow. Thenthe deflection regime the interaction of jet and wake flis modeled. The characteristic dimensions of the numeset-up are provided by a wind tunnel experiment [1] andbased on similarity parameters of a large transport aircwith two engine jets. The experimental results show thatvortex flow does not affect the engine jet behaviour ua downstream distance of 0.5–1 wing spans. For molarge transport aircraft, the characteristic size of theregime is of the order of 1–50 diameters of the nozzle ewhile the deflection regime, scaled to the wingspan, exte

F. Holzäpfel et al. / Aerospace Science and Technology 7 (2003) 263–275 265

ty;

Table 1Survey of cases, numerical codes, methods, vortex Reynolds numbers,Re = Γ/ν, boundary conditions in flight direction,x, lateral direction,y,and vertical direction,z (per. stands for periodic and nr. for non-reflecting), ratio of domain size,L, to vortex core radius,rc , and number of gridpoints across core radius

Case Jet (Section 2) HIT (Section 3) Strat (Section 4) Shear (Section 5)

Code PEGASE [6] NTMIX3D [31] LESTUF [18] MESOSCOP [28]Method DNS LES LES DNS

Re 5000 (jet: 1000) 100· 106 ∞ 2200Boundary conditions per.× nr. × nr. periodic periodic per.× per.× free slip

Lx/rc ×Ly/rc ×Lz/rc 12× 66.9× 66.9 28.23;28.2× 56.42(SV5.5) 102× 64× 135 68.8× 128× 85.3rc/x × rc/y × rc/z 7.5× 10× 10 2.33 0.63× 4 × 4 0.93× 3 × 3

Fig. 1. Normalized axial vorticity contours in a vertical lateral cross-section atx/B = 0.5 (left) andx/B = 1 (right). Dashed lines denote negative vorticicontour range from−1 to 4 in steps of 0.5.

–10

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tytwo

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ndsx.and

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downstream of the airplane to a distance of about 1wingspans.

The simulations are performed with a temporal DNSthe three-dimensional Navier–Stokes equations. Key pmeters of all simulations are listed in Table 1. For detailsthe numerical method we refer to [6]. The axial lengththe domain of 6 nozzle radii corresponds to twice the walength of the maximum growth rate of the first azimuthinstability of a spatially evolving jet [23]. An unstable nozle outlet velocity profile [23] is prescribed which is supimposed by three-dimensional Gaussian shape randomturbations. When the jet simulation has reached an agecorresponds to a downstream distance of 0.5 wing sp(x/B = 0.5), the cross-section of the domain is enlargeda Lamb–Oseen vortex is superimposed on the flow fielddistance of 14 vortex core radii from the jet center (Fig. 1

The tangential velocity profile,vθ (r), of the Lamb–Oseevortex is given by

vθ (r)= 1.4v0rc

r

(1− exp

−1.256r2

r2c

), (1)

wherev0 denotes the maximum tangential velocity atcore radiusrc . The ratio of the jet radius at the nozzle eand the initial vortex core radius amounts torjet/rc = 1.3and the ratio of the maximum jet and tangential velociat x/B = 0.5 to ujet/v0 = 2.72. In all cases dimensionletime, t∗ = t/t0, is normalized by the vortex time scalt0 = 2rc/v0. Here it is set zero at the beginning of the seco

-t

simulation when the jet has reached an age oft∗ = 9. Theconversion of time to downstream distance is establisemploying the Taylor assumption with the experimental fstream velocity ofu0/v0 = 2.2.

Fig. 1 depicts cross-sections of axial vorticity atx/B =0.5 and x/B = 1. The undisturbed concentric vorticicontours represent the trailing vortex whereas theopposite-signed regions of vorticity atx/B = 0.5 reflect themost unstable mode of the jet instability. These unstastructures burst subsequently and the jet reaches aturbulent state atx/B = 1. Fig. 2 shows axial and azimuthvorticity contours in a perspective view at different instaof time. At t∗ = 90 the fully turbulent jet is deflected anentrained by the vortex-induced velocity field and stato wrap around the primary vortex. During that procethe jet’s vorticity is progressively stretched and rearranto coherent – but now spiral-shaped – secondary vortstructures (SVS) of opposite signs (t∗ = 110).

Fig. 3 sketches how a passive tracer that initially exteradially between 4rc and 6rc is stretched in a Rankine vorteAt 32t0 the tracer completely encloses the vortex centeris stretched by a factor of 5π . In the long-term limit thepassive tracer would be evenly distributed between 4rc and6rc. A similar stretch-rate of approximately one revolutiin 30t0 takes place fromt∗ = 90 to 120 (see Fig. 2).

Fig. 2 shows, furthermore, that the counter-rotating sondary vortex rings approach each other driven by a sinduced propagation velocity. At the same time they incre


Fig. 2. Subset of the computational domain showing vorticity contours in a perspective view att∗ = 90,110,120,130,150, and 210 (from left to right). Blacksurface denotes axial component (ω∗

x = 3), azimuthal components (ω∗θ = 1 (−1)) are plotted dark grey (light grey).

4e,

ntatiofor

the

heddi-

in.

vely

icsat

ional

Fig. 3. Schematic of a passive tracer that initially extends betweenrcand 6rc and is stretched in a Rankine vortex. Different instants in timt = n115.2π2r2c /Γ , are plotted at which the tracer encloses thenthpart of the vortex wheren = 1/8,1/4,1/2, and 1. Axisymmetric andinstantaneous local Cartesian coordinate systems as well as represeof tangential velocity,Vθ , in both coordinate systems are illustratedusage in Section 7.

in radius due to mutual induction which again stretchesSVS. Finally, the counter-rotating SVS connect (t∗ = 150)and decay by direct turbulent interaction (t∗ = 210).

During that process the primary vortex has been stretcand deformed in turn by SVS driven induction. Fig. 4 in

n

Fig. 4. Tangential velocity profiles of wake vortex at different instantstime. Velocity profiles are averaged over axial and azimuthal directions

cates that maximum tangential velocities are progressireduced but circulation is essentially unmodified atr = 4rc(see Fig. 5).

3. Vortex evolution in homogeneous isotropicturbulence

The effects of ambient turbulence on vortex dynamare studied using the numerical code NTMIX3D [31] thsolves the Navier–Stokes equations for a three-dimens


Fig. 5. Temporal evolution of axially averaged normalized circulation for all the cases. Circulations are integrated over locally centered circleswith radii of4rc . For vortex pair with small vortex separation (VPb0/rc=4) circulation is determined onr = b0/2 = 2rc .

Fig. 6. Sequence of perspective views of iso-surfaces ofλ2 for vortices withIa = 5.5% att∗ = 25,50,75,100. (a) Single vortex, (b) vortex pair withb0/rc = 8.

iong tocale

hat

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unsteady flow. The code is applied in its LES verswhere the subgrid-scale fluxes are modeled accordinthe Boussinesq gradient approach and the subgrid-sviscosity is estimated from the kinetic energy at cut-off tis assessed by the filtered structure function [4].

A field of homogeneous isotropic turbulence (HIT)generated from a given spectrum with random phaseseach mode which is subsequently allowed to adjustpreparatory run. The resulting turbulence is characterwith respect to wake vortex parameters by the ratio of romean-square velocity and maximum tangential velocitythe vortex,Ia = u′/v0, and the ratio of integral length scato vortex core radiusΛ/rc = 5.8. Since the linear superpostion of two Lamb–Oseen vortices may introduce additioperturbances to the flow [29], the wake vortices are allow

to form quasi-steady dipoles in a two-dimensional viscpre-simulation. For further details regarding the numerapproach and turbulence initialization see Table 1 and [2

Figs. 6, 9 display the temporal evolution of a singvortex (SV) and two vortex pairs (VP) with different vorteseparations,b0/rc, for the turbulence intensityIa = 5.5%in a sequence of perspective views of iso-surfaces ofλ2.2

Different values forλ2 are chosen to visualize the primavortices and the SVS. In Fig. 6 eddies of the ambiturbulence field become visible when they are intensi

2 The second eigenvalueλ2 of the symmetric tensorS2 + Ω2 is ameasure for coherent vortex structures [17], whereS andΩ denote thesymmetric and antisymmetric parts of the velocity gradient tensor.coherent vortex structures, the resultingλ2 iso-surfaces largely corresponto iso-surfaces of the vorticity norm.


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Fig. 7. Iso-lines of vorticity and streamfunction in a reference frathat descends with the vortices. Additionally, the baroclinical vorticproduction in a stably stratified atmosphere is illustrated.

by vortex stretching which is induced by the primavortices. The amount, length and intensity of the Sincrease with time. The numerous coherent SVS deformprimary vortices and lead initially to a similar decay rain the single vortex case and the case with large voseparation (see Fig. 5). Att∗ = 60 the decay rates ovortex pair and single vortex start to deviate from eaother and, eventually, att∗ = 100 the decay of the singlvortex levels off. It should be noted that for the singvortex case a domain of transversally doubled sizeused to assure the independence of results from the sproduced in between the single vortex and its periocounterparts. For vortex pairs this shear is much weabecause neighbouring vortices induce velocities of the sdirection and magnitude in the vicinity of the periodboundaries.

Additionally to the mechanisms that produce SVS aroa single vortex, the topology of vortex pairs provides a fther mechanism that can be explained by considering strelines in a framework that descends with the vortices (Fig. 7). Along the streamlines that connect the two disthyperbolic stagnation points, the flow is accelerated andscelerated which exerts both stretching and squeezing obedded eddies. Fig. 6 illustrates that while for the singletex all SVS tend to be aligned azimuthally, the vortex pproduces also vertical vorticity streaks midway betweenvortices (stretching by vertical gradients of the verticallocity ∂w/∂z in the vicinity of the upper stagnation poinand oval-shaped streaks (stretching by lateral gradienthe lateral velocity∂v/∂y in the vicinity of the lower stagnation point). These additional SVS contribute to decayturbulent diffusion and, in particular, by exchange of vortity in between the counter-rotating vortices. The compariof decay rates for SV and VP in Fig. 5 indicates that the lamechanisms advance decay effectively.

r

-

-

f

Fig. 8. Temporal evolution of the extrema of each vorticity component osingle vortex and vortex pair cases withIa = 5.5% andb0/rc = 8. Vorticityextrema are averaged over the axial direction,x, and are normalized by thinitial maximum of axial vorticity,ωx,0.

Fig. 8 depicts the evolution of the extrema of eavorticity component normalized by the initial maximumaxial vorticity, ωx,0 for the cases shown in Fig. 6. Threpresentation allows to quantitatively assess the efficieof vortex stretching mechanisms. Due to diffusion procesthe axial vorticity maxima decrease with time. However,lateral and vertical vorticity maxima,ωy,max, ωz,max, whichat t∗ = 0 represent eddies of the background turbulenincrease with time, driven by vortex stretching, and rougtriplicate initial values aroundt∗ = 125.ωy,max andωz,maxare of similar magnitude throughout the whole evolutibecause the SVS appear alternately in both componSince the vortex pair exerts stretching both across (cf. Figand along (cf. Fig. 7) streamlines, its vorticity maxima grfaster and maintain higher values than in the single vocase. Finally, all vorticity components of the vortex pair hadecayed to a similar level att∗ = 200, when circulation isreduced to 20%. The vorticity extrema of the case SV sto deviate from those of case VP at approximatelyt∗ = 60,a time that correlates well to the deviation of the respeccirculation evolutions (see Fig. 5). Att∗ = 100, when thecirculation of case SV remains almost constant, the deof ωx is reduced and the SVS are not further stretchi.e. vorticity valuesωy,max, ωz,max merely fluctuate arounconstant values.

When vortex separation is reduced, the evolution ofcirculation shows a very different decay behaviour. Afteshort phase of gradual decay governed by turbulent dsion, the elliptic/short-wave instability [20,21] is developiweakly (linear phase) (see Fig. 9). Subsequently, fort∗ > 15the Crow/long-wave instability is largely amplified and tcirculation starts to decay rapidly (which in part is duethe fact that circulation is only determined from the vortic


Fig. 9. As Fig. 6 for vortex pair withb0/rc = 4 at t∗ = 15,25,50.

Fig. 10. Sequence of perspective views of iso-surfaces ofλ2 for (a) a single vortex and (b) a vortex pair withIa = 24% andb0/rc = 8 at t∗ = 0,30,60.

n-

ingth.

allyienticespairatedo

esan

sim-ti-

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ft

ee-ur-e

at-he

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calcesd. In

kecee

inententsly

component aligned with the flight direction). After reconection of the two primary vortices att∗ = 25 a vortex ringis formed att∗ = 50. This ring remains coherent for a certatime but deforms under its own induction and loses stren

In the highly turbulent case (Ia = 24%) depicted inFig. 10 the primary vortices are too weak to essentistretch the turbulent eddies. Here it is the strong ambturbulence that directly deforms and destroys the vortrapidly. No differences between the decay of the vortexand the single vortex are seen (see also Fig. 5). This indicthat interactions between the vortices of the vortex pairnot contribute to decay in strong turbulence.

4. Vortex decay in a turbulent stably stratifiedatmosphere

The large-eddy simulation code LESTUF which solvthe Boussinesq-approximated Navier–Stokes equationsuses the classical Smagorinsky closure is employed toulate wake vortex evolution in a thermally stably strafied atmosphere with superimposed moderate, anisotrand decaying turbulence. The turbulence, which isscribed in detail in [8], obeys prescribed spectra withIa =3.6%, andΛ/rc = 12. Turbulence induced by the aircra

s

d

,

is superimposed on the vortices by adding initially thrdimensional random perturbations with a maximum tbulence intensity ofIv = 12% at the core radius. Thprescribed mean potential temperature gradient of themosphere,dΘ/dz, is constant in each calculation and tcorresponding normalized Brunt–Väisälä frequenciesN∗ =(g/Θ0 · dΘ/dz)1/2(2πb2

0/Γ0) are set to 0, 0.35, and 1The pair of superimposed Lamb–Oseen vortices is iniized with a ratio ofb0/rc = 12 for which a self-adaptionphase is not necessary because the resulting distortionweak [29]. For further details regarding the numerical setwe refer to [14] and Table 1.

Fig. 11 depicts iso-contours of the lateral and vertivorticity components that are induced by the wake vorti(represented by tubularλ2-contours) in a stably stratifieturbulent atmosphere for two degrees of stratificationthe caseN∗ = 0.35, the iso-lines of the lateral velocity,v,illustrate the converging flow that is induced by the wavortices. This converging flow is deformed by turbulensuch that the iso-linev = 0 (bold) meanders along thsymmetry plane between the vortices. As describedquantitative detail in [14], such a superposition of turbuland wake vortex induced velocities produces axial gradiof the lateral velocity,∂v/∂x, and, equivalently, verticavorticity, ωz ∼ ∂v/∂x in a volume above and midwa


are
Fig. 11. Wake vortex evolutions in a turbulent and weakly stratified (N∗ = 0.35) and in a turbulent and strongly stratified (N∗ = 1) atmosphere att∗ = 120.Iso-surfaces of lateral and vertical vorticity components (ω∗
y = ω∗z = 0.125 (−0.125) plotted dark grey (light grey)) in a perspective view. Wake vortices

represented by black tubularλ2-contours. ForN∗ = 0.35, iso-lines for lateral velocity,v, are plotted in a horizontal plane above the vortices.

ks,tiones.

ary

ongd-icalc-

amnds

-di-

fiessetedat-

onract

between the vortices. The resulting vertical vorticity streaωz, are amplified by vortex stretching due to the acceleraof the downwards directed flow between the main vorticThen, the vorticity is tilted, and wraps around the primvortices.

The intense SVS in Fig. 11 below demonstrate that strstable stratification (N∗ = 1) intensifies these effects consierably. The source of this intensification is the baroclinvorticity, ωx , which is produced by the baroclinic torque acording to

Dωx

Dt∼ 1

ρ2∇ρ × ∇p (2)

along the oval-shaped interface (see Fig. 7) between thebient flow and the adiabatically heated flow that desce

-

with the vortices [13]. This baroclinical vorticity additionally induces lateral velocities above the vortices that arerected towards the symmetry plane. Herewith it intensithe axial gradients of the lateral velocity which finally cauthe vertical and azimuthal vorticity structures and the reladecay. Fig. 5 quantifies the impact of a stably stratifiedmosphere on the longevity of wake vortices.

5. Wake vortices in a turbulent environment withconstant shear

Shear flows exhibit vigorous and variform influenceswake vortex transport and decay. Wake vortices that inte


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Fig. 12. Axial view of wake vortices in a turbulent environment wconstant positive background shear att∗ = 47. Iso-surfaces of lateraand vertical vorticity components (ω∗

y = ω∗z = 0.63 (−0.63) plotted dark

grey (light grey)); wake vortices represented by black tubularλ2-contours.Streamlines in a reference system that moves with the vortices in thex = Lx/2.

with a shear layer may experience vortex tilting, separaand subsequent rebound of vortices, whereupon the vowith opposite signed vorticity to the shear layer showsstronger tendency to rebound [25]. It is again the vortex wopposite-signed vorticity that decays distinctly faster whthe vortex pair immerges into a turbulent shear-layer [1Also the spectacular rebound observed at London-HeathInt’l Airport [10] can be at least in part attributed to shelayer effects [15]. However, these effects are not observelaminar flows with constant shear [11].

Here, we report a DNS of vortex evolution in a constansheared and turbulent environment. For this purpose theMESOSCOP [28] is employed which solves the Boussineapproximated Navier–Stokes equations. The pair of LamOseen vortices with a vortex spacing ofb0/rc = 8 isperturbed sinusoidally with an amplitude of 0.01b0 atthe wavelength of 8.6b0 to trigger the Crow instabilitywhich allows to study the influence of vortex spacingdecay. The weak homogenous and initially isotropic ambturbulence is characterized byIa = 0.7%, andΛ/rc ≈ 8.The prescribed constant background strain just compenthe strain that the left vortex would induce on the right vorcenter. Employing the common normalization for vorpairs the shear corresponds to(∂v/∂z)(2πb2

0/Γ0)= 1.Fig. 12 shows the wake vortices represented by b

tubular λ2-contours in an axial view. The correspondistreamlines indicate an intrinsic asymmetry of the vorflow, as known from analytical investigations of wavortices subjected to background shear [3] (compare Ffor the unsheared case). The formation of SVS (greysurfaces of lateral and vertical vorticity) is in strikincorrelation with the streamline that separates the fluidand outside the vortex oval. The right vortex, which

s

Fig. 13. Wake vortices in turbulent sheared environment in a perspeview. Iso-surfaces as in Fig. 12.

composed of opposite signed vorticity compared tobackground vorticity of the shear, is closely encompassethe SVS. A vertical streak evolves above the vortex pairlags progressively behind the descending vortices becit develops partially outside the oval that descends wthe vortices. This structure is similar as in unsheared c(cf. Fig. 11) but here it is deflected to the right due toadvection by the background shear flow.

The perspective view for two different instants, depicin Fig. 13, gives a spatial impression of the evolutionthe flow structures. Att∗ = 47 the SVS indicate that vortestretching of ambient turbulence is most effective nearright vortex at the axial position where the vorticesclosest. At this particular position the area around the rvortex which is enclosed by the bold separating streamachieves the smallest size (cf. Fig. 12). ConsequentlySVS structures that extend from the stagnation pointshave to cover a small distance to completely encompasvortex and (ii) the SVS experience strong accelerationhence, strong stretching in the vicinity of the right vorteFluid, almost at rest at the stagnation point, followsstreamlines to locations of high circumferential velocityclose proximity to the right vortex core, resulting in stroacceleration. Contrary to non-sheared cases, the disbetween left vortex (same signed vorticity as the backgroshear) and vortex oval boundary is increased, which gan explanation for unbalanced decay rates and prolonglifetimes of the vortices [11].

At t∗ = 87 SVS have formed all along the right vortand have initiated a rapid decay phase (see Fig. 5). A


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ofceshethe

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ede,

on-of

itythis

dichthe

result of the initial perturbation the vortices link at abot∗ = 100 which initiates rapid decay also for the left vorteThis is combined with tilting of the closely spaced vortsegments caused by mutual induction of the unequstrong vortices.

6. Comparison of decay characteristics

So far we employed normalized time based on the votime scale,t0 = 2rc/v0, to allow for the comparison of decacharacteristics of both single vortex and vortex pair ca(see Fig. 5). We found in Section 3 that a reductionnormalized vortex spacing by a factor of two may acceledecay by a factor of four. Alternatively, we now emplthe normalization of time that is commonly used for vorpairs and is based on the time scalet0 = 2πb2

0/Γ0. Fig. 14indicates that now the decay rates of the two casesIa = 5.5% with a variation of vortex separation by a factortwo almost coincide. This means that the time normalizafor vortex pairs considers the effects of vortex separacorrectly even when the underlying decay mechanismsdiffer considerably (cf. Figs. 6, 9). Another consequencthat increasing vortex core radii for a given vortex separareduces lifetime only slightly, whereas a reduction ofb0 maybe very beneficial for an accelerated vortex decay.

A comparison of all cases in Fig. 14 indicates that thespan of the vortices is basically correlated to the intenof ambient turbulence. However, the specific pecularitiethe different cases, like additional superposition of aircrinduced turbulence in the stratified cases or initial pertution of the vortex spacing in the sheared case, modifyranking according to background turbulence. Imposing

ditionally a stable temperature stratification withN∗ = 1may reduce wake vortex lifetime by roughly two time unwhereas an increase of ambient turbulence fromIa = 5.5%by a factor of four may shorten wake vortex life by four timunits. In contrast, dimensional analysis tells us that thespan of wake vortices in a given environment may be haby a reduction of vortex spacing by a factor of

√2.

7. Formation of secondary vorticity and decaymechanisms

As shown in the previous sections the formationcoherent secondary vorticity around the primary vortiis a prominent feature of wake vortex evolution in tatmosphere. In the following we list common aspects ofrelated mechanisms.

(i) Incoherent vorticity of background turbulence achiecoherence by tilting and stretching. It is the primary vorthat alignes (tilting) and reinforces (stretching) random vticity such that SVS are produced.

(ii) However, a single vortex filament that is stretchwill conserve its circulation (Helmholtz theorem). Hencthe vorticity of the vortex filament may be increased csiderably by stretching, but its far field effect in termsvelocity induction is not modified.

(iii) Although the merger of several coherent vorticfilaments of equal sign may generate larger SVS,mechanism likewise does not amplify far field effects.

(iv) Far field velocity induction (circulation) is generateand increased when vorticity of arbitrary orientation whhas no far field effect is aligned to coherent structures bytilting process.

mined
Fig. 14. Temporal evolution of normalized circulation where the common normalization of time for vortex pairs is employed. Circulations deterasin Fig. 5.


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(v) The tilting mechanism is effective for any vorticistructure that at least contains some radial vorticity comnent and (vi) tilting provokes simultaneous stretching. Tcan be illustrated by considering vortex tilting and streting in a background flow given by a single plane potenvortex,Vθ(r)= Γ/(2πr). Viscous and baroclinic effects aneglected.

For convenience, the secondary vorticity,ωs , is givenin an instantaneous local Cartesian coordinate systemdirections along the spiral-shaped vortex line, andn awayfrom the center of curvature (cf. Fig. 3). The tilting astretching terms read

Dωn

Dt= ωs ∂Un

∂s,

Dωs

Dt= ωs ∂Us

∂s, (3)

where the velocity gradients along the vortex line,∂Un/∂s,∂Us/∂s, are prescribed by the background flowVθ(r). Forthe tilting term it follows

Dωn

Dt= ωs

∂Un

∂r

∂r

∂s= ωs ∂(Vθ sinϕ)

∂r

∂r

∂s

= ωsΓ

2πr2sin2ϕ, (4)

whereϕ denotes the angle betweenVθ andUs . In analogythe stretching term yields

Dωs

Dt= ωs

∂Us

∂r

∂r

∂s= ωs ∂(Vθ cosϕ)

∂r

∂r

∂s

= ωs cosϕΓ

2πr2sinϕ. (5)

Eqs. (4) and (5) specify the limits of the processes: Tiltis strongest for radially oriented vorticity filaments beca∂r/∂s = sinϕ = 1 and tilting ceases for azimuthally orientstructures because in azimuthal direction sinϕ = 0. Radiallyaligned filaments are not yet stretched because cosϕ = 0,whereas azimuthally oriented structures can not be furstretched (sinϕ = 0). In summary, radially oriented vorticitis first tilted and then tilted and stretched simultaneouuntil both processes decay when in the long-term limitSVS tend to be aligned azimuthally.

(vii) SVS can deform primary vortices. In particulacounter-rotating SVS can stretch (squeeze) the primvortices when approaching (departing from) each otherFig. 2, t∗ = 210).

(viii) In a reference frame that descends with the vorpair the vortices are engulved by a oval-shaped streamwith stagnation points below and above the vortices (Fig. 7). Due to the low velocities at the stagnation poialready minor disturbances are sufficient to displacestagnation points which enables entrainment of ambvorticity into the oval (see Fig. 11). Entrained vorticitysubsequently intensified according to the right equatio(3) by vortex stretching along streamlines in the accelerflow midway between the vortices or along the half moshaped streamlines around the vortices.

(ix) Counter-rotating vorticity streaks produced midwbetween the vortices can be effective in exchanging fl

Fig. 15. Sketch to illustrate the increase of rotational energy of a stretRankine vortex.

across the symmetry line. This effect is prerequisitedirect cancelation of primary vorticity [3,14].

(x) A primary vortex that stretches SVS as depictedFig. 3 performes work on the SVS. During the stretchprocess the SVS gain rotational energy, whereas the primvortex, in turn, loses rotational energy. This can be deeated phenomenologically by considering a vortex tube wa Rankine tangential velocity distribution that is stretchby a factor of 2 (see Fig. 15). The increase in length cosponds to a contraction by a factor of 1/

√2 (mass conser

vation). During stretching angular momentum is conser(rvθ = const.), which means that the potential vortex rgion is not affected by stretching. But the angular velity of the contracted vortex core is doubled accordingΩ2 = r2

1/r22Ω1 becauser2 = r1/

√2. The work performed

on the vortex core,W1,2, which corresponds to its gain orotational energy,E, amounts to

W1,2 = E2 −E1 = 1

2

(mr22

2Ω2

2 −mr21

2Ω2

1

)

= 1

2

(mr21

44Ω2

1 −mr21

2Ω2

1

)=E1. (6)

Hence, a doubling of length corresponds directly todoubling of rotational energy of the flow in the vortex coof the SVS. The described mechanism is in direct anato the energy transfer from large eddies to small eddieturbulent flows [32]. In our cases it is most obvious in tjet vortex interaction where rotational energy of the primvortex decreases only on smaller radii (Fig. 4) and in the lturbulence cases of Section 3 (Figs. 5, 6, 8). In particulathe single vortex case HIT SV5.5% circulation decay ends aa time oft∗ = 100 when the SVS are not further stretched

(xi) The formation of SVS may trigger the formationcooperative instabilities as the short-wave instability [and the Crow [2] instability.

With the exception of cooperative instabilities the dscribed mechanisms are independent from the ratio oftex core radius and vortex separation. We therefore assthat the relatively large vortex cores that we used herethat are commonly employed in numerical simulationsnot derogate the current results.


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8. Conclusions

The presented numerical simulations of wake vortex elution in the atmosphere indicate that stretching, tilting amerger of ambient vorticity caused by the primary vorticare the prominent mechanisms in vortex evolution andcay. In the cases with low to moderate turbulence posscombined with thermal stable stratification and/or consshear as well as with jet–vortex interactions these meanisms create spatially and temporarily extended cohesecondary vortices out of the ambient incoherent flow.first, the creation of these secondary vorticity structures csumes energy of the primary vortices. Secondly, theondary vorticity structures, in turn, have a far field impaand thus, deform the primary wake vortices. It is shownthese mechanisms lead to an initially gradual and subseqrapid decay. Whereas continous decay can be attributediffusion and the work that the primary vortices performthe secondary vorticity structures, rapid decay is associwith the interaction of secondary vorticity with the primavortex pair which causes an exchange and mutual annition of primary vorticity, also in combination with instabiity mechanisms and the transition to fully turbulent vorticFor single vortices, longitudinal stretching of secondary vticity and the mutual exchange of primary vorticity betwecounter-rotating vortex pairs does not apply. Therefore,decay of single vortices is strongly reduced when the entransfer from primary to secondary vortices ceases becthe secondary vortices are no longer stretched substanwhen they tend to be aligned azimuthally. In a highly turlent atmosphere, the primary vortices are too weak andtime scales of vortex decay are too short to essentially stratmospheric eddies. As a consequence, the wake vorticerapidly disrupted directly by ambient turbulence.

It should be noted that numerical simulations of wavortices in the atmosphere unavoidably suffer from limiresolution of the vortex core region. Whereas DNS oreach small Reynolds numbers it is the type of subgrid-sclosure that controls vortex core evolution in LES [15]. Boapproaches do not meet the complex interaction of tulence and rotation at highRe flows together with the peculiarities of specific aircraft configurations and environmtal conditions. This limitation will persist despite the enmous increase of computational power and the developmof smart numerical methods. Full-scale experiments willmain both challenging and mandatory to ensure the valiof the conclusions drawn from numerical simulations.

References

[1] S. Brunet, L. Jacquin, P. Geffroy, Experiment on heated jets/wvortex interaction, ONERA RT 15/2496 DAFE/Y, 1999.

[2] S.C. Crow, Stability theory for a pair of trailing vortices, AIAA J.(1970) 2172–2179.

[3] C.P. Donaldson, A.J. Bilanin, Vortex wakes of conventional aircrAGARDograph 204, 1975.

t

t

e

t

[4] F. Ducros, P. Comte, M. Lesieur, Large-eddy simulation of spatigrowing boundary layer over an adiabatic flat plate, Internat. J. HFluid Flow 16 (1995) 341–348.

[5] F. Garnier, F.C. Beaudoin, P. Woods, N. Louisnard, Engine emisalteration in the near field of an aircraft, Atmosph. Environ. 31 (191767–1781.

[6] C. Ferreira Gago, S. Brunet, F. Garnier, Numerical investigationturbulent mixing in a jet/wake vortex interaction, AIAA J. 40 (200276–284.

[7] T. Gerz, T. Ehret, Wingtip vortices and exhaust jets during theregime of aircraft wakes, Aerosp. Sci. Technol. 1 (1997) 463–474

[8] T. Gerz, F. Holzäpfel, Wingtip vortices, turbulence, and the distributof emissions, AIAA J. 37 (1999) 1270–1276.

[9] T. Gerz, F. Holzäpfel, D. Darracq, Commercial aircraft wake vorticProg. Aerosp. Sci. 38 (2002) 181–208.

[10] J.S. Greenwood, J.M. Vaughan, Measurements of aircraft waketices at Heathrow by laser doppler velocimetry, Air Traffic ContQuarterly 6 (1997) 179–203.

[11] T. Hofbauer, Numerische Untersuchungen zum Einfluss von Wscherung und Turbulenz auf Flugzeugwirbelschleppen, Ph.D. Ditation, DLR, Oberpfaffenhofen, 2002.

[12] T. Hofbauer, T. Gerz, Shear-layer effects on the dynamics of a courotating vortex pair, AIAA Paper 2000-0758, 2000.

[13] F. Holzäpfel, T. Gerz, Two-dimensional wake vortex physics instably stratified atmosphere, Aerosp. Sci. Technol. 3 (1999) 261–

[14] F. Holzäpfel, T. Gerz, R. Baumann, The turbulent decay of traivortex pairs in stably stratified environments, Aerosp. Sci. Techno(2001) 95–108.

[15] F. Holzäpfel, T. Hofbauer, T. Gerz, U. Schumann, Aircraft wavortex evolution and decay in idealized and real environmemethodologies, benefits and limitations, in: R. Friedrich, W. R(Eds.), Advances in LES of Complex Flows, Kluwer, Dordrecht, 202002, pp. 293–309.

[16] F. Holzäpfel, T. Gerz, F. Köpp E. Stumpf, M. Harris, R.I. YounA. Dolfi-Bouteyre, Strategies for circulation evaluation of aircrwake vortices measured by lidar, J. Atmos. Ocean. Tech., acceptepublication.

[17] F. Jeong, J. Hussain, On the identification of a vortex, J. FMech. 285 (1995) 69–94.

[18] H.-J. Kaltenbach, T. Gerz, U. Schumann, Large-eddy simulatiohomogeneous turbulence and diffusion in stably stratified shearJ. Fluid Mech. 280 (1994) 1–40.

[19] L. Kleiser, U. Schumann, Spectral simulations of the laminar-turbutransition process in plane poiseuille flow, in: Spectral MethodsPartial Differential Equations, SIAM, Philadelphia, 1984, pp. 14163.

[20] F. Laporte, D. Darracq, A. Corjon, On the vortex-turbulence intertion: DNS of elliptic instability of a vortex pair, AIAA Paper 99-34171999.

[21] T. Leweke, C.H.K. Williamson, Cooperative elliptice instability ofvortex pair, J. Fluid Mech. 360 (1998) 85–119.

[22] R.C. Miake-Lye, M. Martinez-Sanchez, R.C. Brown, C.E. KoPlume and wake dynamics, mixing and chemistry behind a high scivil transport, J. Aircraft 30 (1993) 467–479.

[23] A. Michalke, G. Hermann, On the inviscid instability of a circularwith external flow, J. Fluid Mech. 114 (1982) 343–359.

[24] H. Moet, D. Darracq, F. Laporte, A. Corjon, Investigation of ambiturbulence effects on vortex evolution using LES, AIAA Paper 200756, 2000.

[25] F.H. Proctor, D.A. Hinton, J. Han, D.G. Schowalter, Y.-L. Lin, Twdimensional wake vortex simulations in the atmosphere: Preliminsensitivity studies, AIAA Paper 97-0056, 1997.

[26] F.H. Proctor, G.F. Switzer, Numerical simulation of aircraft trailivortices, in: 9th Conf. on Aviation, Range and Aerospace Meteorol7.12, 2000, pp. 511–516.

[27] F. Risso, A. Corjon, A. Stoessel, Direct numerical simulations of wvortices in intense homogeneous turbulence, AIAA J. 35 (1997) 101040.


aleun-. 60

n in000)

30

ion95

ss,

[28] U. Schumann, T. Hauf, H. Höller, H. Schmidt, H. Volkert, A mesoscmodel for the simulation of turbulence, clouds and flow over motains: Formulation and validation examples, Beitr. Phys. Atmosph(1987) 413–446.

[29] D. Sipp, L. Jacquin, C. Cossu, Self-adaption and viscous selectioconcentrated two-dimensional vortex dipoles, Phys. Fluids 12 (2245–248.

[30] P.R. Spalart, Airplane trailing vortices, Annu. Rev. Fluid Mech.(1998) 107–138.

[31] A. Stoessel, An efficient tool for the study of 3d turbulent combustphenomena on MPP computers, in: Proceedings of the HPCNConference, Springer-Verlag, 1995, pp. 306–311.

[32] H. Tennekes, J.L. Lumley, A First Course in Turbulence, MIT PreCambridge, MA, 1972.

Documents

Analysis of wake vortex decay mechanisms in the atmosphere … · 2003-07-16 · F. Holzäpfel et al. / Aerospace Science and Technology 7 (2003) 263–275 265 Table 1 Survey of cases,