PHYSICS OF FLUIDS 24, 025104 (2012)
Vortex bursting and tracer transport of a counter-rotating
vortex pair
T. Misaka,
1,a)
F. H o l z
¨
apfel,
1
I. Hennemann,
1
T. Ger z ,
1
M. Manhart,
2
and F. Schwertfirm
3
1
Deutsches Zentrum f
¨
ur Luft- und Raumfahrt (DLR), Institut f
¨
ur Physik der Atmosph
¨
are,
82234 Oberpfaffenhofen, Germany
2
Technische Universit
¨
at M
¨
unchen, Fachgebiet Hydromechanik, 80333 M
¨
unchen, Germany
3
Kreuzinger+Manhart Turbulenz GmbH, 81765 M
¨
unchen, Germany
(Received 19 April 2011; accepted 21 December 2011; published online 27 February 2012)
Large-eddy simulations of a coherent counter-rotating vortex pair in different envi-
ronments are performed. The environmental background is characterized by varying
turbulence intensities and stable temperature stratifications. Turbulent exchange pro-
cesses between the vortices, the vortex oval, and the environment, as well as the
material redistribution processes along the vortex tubes are investigated employing
passive tracers that are superimposed to the initial vortex flow field. It is revealed that
the vortex bursting phenomenon, known from photos of aircraft contrails or smoke
visualization, is caused by collisions of secondary vortical structures traveling along
the vortex tube which expel material from the vortex but do not result in a sudden
decay of circulation or an abrupt change of vortex core structure. In neutrally strat-
ified and weakly turbulent conditions, vortex reconnection triggers traveling helical
vorticity structures which is followed by their collision. A long-lived vortex ring
links once again establishing stable double rings. Key phenomena observed in the
simulations are supported by photographs of contrails. The vertical and lateral extents
of the detrained passive tracer strongly depend on environmental conditions where
the sensitivity of detrainment rates on initial tracer distributions appears to be low.
C
2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.3684990]
I. INTRODUCTION
The dynamics of a pair of counter-rotating vortices in various background conditions has
been a subject of numerous studies over decades. The incentive to study this particular flow has
been motivated by the potential risk of the wake vortex pairs generated behind flying aircraft
posed to following aircraft. Comprehensive and detailed understandings of wake vortex behavior
under different atmospheric conditions are crucial for the development of wake vortex prediction
systems that aim to increase aviation safety and airport capacity.
1
But this type of flow is also of
fundamental interest to fluid dynamicists as aircraft wake vortices constitute a relatively simple
configuration of coherent vortex structures which are embedded in incoherent, i.e., turbulent and
stratified environments. The study of such flows—experimentally as well as numerically—yields
improved understanding of elementary vortex interaction that is also relevant in more complex
transitional and turbulent flows. On the other hand, turbulent mixing processes and detrainment of
the ice crystals generated from the exhaust jets during wake vortex descent may affect the generation
of condensation trails (contrails). The contrails in turn may trigger the formation of long-lived cirrus
clouds (contrail cirrus) that have been suspected to contribute to global warming by modifying the
radiation budget of the atmosphere.
2, 3
The relevance of these topics might become increasingly large
due to the expected increase of air traffic in the coming decades.
4
a)
Electronic mail: Takashi.Misaka@dlr.de.
1070-6631/2012/24(2)/025104/21/$30.00
C
2012 American Institute of Physics24, 025104-1
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025104-2 Misaka
et al.
Phys. Fluids 24, 025104 (2012)
The wake vortex phenomenon in the atmosphere is related to a wide range of flows with scales
ranging from few millimeters to several tens of kilometers. In particular, the flows around the aircraft’s
main wings, slats, flaps, and their interactions may affect wake vortex roll-up in high-lift conditions.
5
On the other hand, wake vortices generated by cruising aircraft may remain along several tens of
kilometers.
2
The transport and mixing of aircraft emissions can be divided into four regimes:
6
(1) jet regime, (2) vortex regime, (3) dissipation regime, and (4) diffusion regime. Numerical
simulation setups are typically limited to flow scales of one or two of these individual regimes.
Dynamics of wake vortices in the vortex and dissipation regime have been studied mainly by large-
eddy simulation (LES) or direct numerical simulation. In these simulations, the detailed temporal
evolution of a vortex pair with an initially longitudinally constant velocity profile is investigated
allowing for the formation of cooperative instabilities, e.g., short-wave (elliptic) instability
7–10
and
Crow instability.
11–14
Various atmospheric conditions of turbulence, stability, and wind shear have
been considered in order to assess the influence of these factors on wake vortex evolution and
decay.
15, 16
However, little is known about the further evolution of the vortex rings formed as a
consequence of the Crow instability in particular under different environmental conditions.
The vortex bursting phenomenon has been reported from flight experiments visualized by
smoke
17, 18
or contrails,
19
in the towing tank
20
and numerical simulations.
21
The details of the
descriptions differ but they are always connected with an abrupt change of the diameter of the marker
around the vortex core. It has been observed that the bursts travel along the vortex tube in either
direction. Sometimes two bursts would travel toward each other, eventually colliding and leaving
behind an intensely marked disk-like parcel of tracer,
18
which is also termed puff or pancake.
22
In the
vortex core region, funnel-shaped features have been observed surrounded by the pancake-shaped
structures.
19
No explanation of the causes and structure of the bursts has been offered until Moet
et al.
21
suggested that vortex bursting might be caused by the collision of helical vorticity structures
connected with pressure waves emanating from the location of vortex reconnection. We confirm
that the colliding helical vorticity structures indeed temporarily and locally reduce circulation, but
do not cause substantial radial spreading of the tracer. Instead, we suggest that vortex bursting
associated with radial spreading of tracer before the onset of vortex linking is caused by the collision
of propagating secondary vortical structures which do not cause a change of vortex core structure.
For numerical simulations of contrails, microphysics of the ice particles have been studied along
with wake vortex dynamics.
23–26
The consideration of microphysics is crucial for obtaining realistic
optical depths and coverage of contrails which is required to estimate radiation effects. On the other
hand, our focus on passive tracers enables to investigate turbulent transport and mixing processes
precluding uncertainties regarding the performance of the microphysics package and adulteration
caused by sublimation or sedimentation of ice particles. Nevertheless, it is still possible to compare
the simulated passive tracer distributions with wake vortices visualized by smoke experiments
17, 20, 27
or contrails under certain conditions.
With this LES study, we aim to explain some details of the decay of coherent vortices in stably
stratified and weakly turbulent background flows. The parameter setting represents conditions of
the atmosphere where large civil aviation aircraft typically cruise. By this we also shed light on
tracer distributions as observed along the vortices and tracer detrainment into the atmosphere. The
impact of coherent–incoherent and coherent–coherent vortex interactions on vortex decay and tracer
distribution and detrainment is discussed. In the following Secs. II, III, and IV, applied numerical
methodologies, the initialization of wake vortices with various cross-sectional distributions of the
passive tracers, and the generation of stably stratified ambient turbulence fields are respectively
described. In Sec. V, the results on characteristics of a vortex pair in the different environments:
secondary vortical structures, tracer redistribution, and detrainment mechanisms are presented and
discussed. Sec. VI concludes our study.
II. EQUATIONS AND NUMERICAL METHODS
The LES are conducted using MGLET, which is a finite volume solver for the incompressible
Navier–Stokes equations.
28
To enable wake vortex simulations in stably stratified atmospheric
conditions featuring buoyancy effects, an additional equation is solved for potential temperature,
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025104-3 Vortex bursting and tracer transport Phys. Fluids 24, 025104 (2012)
employing the Boussinesq approximation
29
∂u
i
∂t
+
∂(u
i
u
j
)
∂x
j
=−
1
ρ
0
∂p
∂x
i
+ (ν + ν
t
)
∂
2
u
i
∂x
2
j
+ g
θ
θ
0
δ
i3
, (1)
∂θ
∂t
+
∂(u
j
θ
)
∂x
j
= (κ + κ
t
)
∂
2
θ
∂x
2
j
+ u
3
dθ
s
dx
3
, (2)
∂u
j
∂x
j
= 0 . (3)
In Eqs. (1)–(3), u
i
, p
, and θ
represent the cell averaged velocity components in three spatial directions
(i, j = 1, 2, or 3), the cell averaged pressure and potential temperature fluctuations, respectively.
The summation convention is used for the velocity components u
i
and δ
ij
denotes Kronecker’s delta.
The primes for pressure and potential temperature indicate that those are defined by the deviation
from the reference states: p = p
0
+ p
, θ = θ
0
+ θ
. In this study, typical values of air density and
potential temperature at cruise altitude are employed: ρ
0
= 0.35 kg/m
3
and θ
0
= 332.1 K.
30, 31
The
background potential temperature θ
s
in Eq. (2) is used to specify the vertical gradient of potential
temperature and is set to constant in a neutrally stratified case. In the Boussinesq approximation,
potential temperature and the momentum equations are coupled via the vertical velocity component
u
3
. The kinematic viscosity in Eq. (1) is defined by the sum of molecular viscosity and eddy viscosity
obtained by a subgrid-scale model. The corresponding diffusion coefficients in Eq. (2) are obtained
by assuming constant molecular and turbulent Prandtl numbers of 0.7 and 0.9, respectively. In
addition, equations for passive tracers are employed considering advection induced by local velocity
and diffusion by molecular and turbulent viscosity.
The above equations are solved by a compact fourth-order finite volume scheme.
32, 33
A split-
interface algorithm is used for the parallelization of tri-diagonal systems of the compact scheme,
which realizes smaller overhead time and scalability in parallel environments.
34
The pressure field
is obtained by the velocity-pressure iteration method by Hirt and Cook.
35
Iterations are performed
until the divergence of the velocity field becomes smaller than a threshold value of 1.0×10
−5
s
−1
.
The third-order Runge–Kutta method is used for time integration.
36
The Lagrangian dynamic model is employed to define eddy viscosity ν
t
in Eq. (1) (Ref. 37).
The use of the standard Smagorinsky model results in excessive eddy viscosity in the vortex core,
hence, a correction procedure is usually used together with the standard Smagorinsky model.
38, 39
An alternative way to handle vortex flows is the use of a dynamic-type subgrid scale model. The
Lagrangian dynamic model does not require directions of statistical homogeneity for the averaging
process of subgrid model coefficients but calculates the required averages along path-lines. This
enables the Lagrangian dynamic model to distinguish between the centrifugally stable vortex core
regions and the external turbulent flow.
III. INITIAL CONDITIONS
The coherent wake vortices are initialized as a pair of counter-rotating Lamb–Oseen vortices
which represents fully rolled-up trailing vortices of a cruising large aircraft. The vortices possess a
circulation of
0
= 530.0m
2
/s, a vortex core radius of r
c
= 3.0 m, and a vortex spacing of b
0
= 47.1 m.
The corresponding reference length, velocity, and time scales used for normalization of the results are
defined by the initial vortex spacing b
0
, the initial vortex descent speed w
0
=
0
/(2πb
0
) = 1.79 m/s,
and the time for descending one vortex spacing t
0
= b
0
/w
0
= 26.3 s, respectively. Turbulent fluctua-
tions induced by the aircraft’s jets, wings, and fuselage are modeled by white noise with a weighting
of a Gaussian distribution possessing a maximum variance of 1 m/s at the vortex core radius.
40
Molecular kinematic viscosity is set to ν = 4.0 × 10
−5
m
2
/s such that the vortex circulation based
Reynolds number amounts to Re =
0
/ν ≈ 10
7
. The dimensions of the computational domain are
L
x
= 400, L
y
= 384, and L
z
= 512 m where x, y, and z denote flight, spanwise, and vertical directions,
respectively (see Fig. 1). The domain length covers a theoretical wave length of the Crow instability.
13
A uniform mesh spacing of 1 m is employed for all three directions unless otherwise stated.
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025104-4 Misaka
et al.
Phys. Fluids 24, 025104 (2012)
b
0
L
x
L
z
1.2b
0
L
y
O
y
z
x
0.16
0.12
0.08
0.04
0.00
Velocity
Magnitude
[m/s]
FIG. 1. (Color online) Schematic of computational domain and initial vortex position represented by thick black lines.
Velocity magnitude fields established after the pre-run of a weakly turbulent and moderately stable environment are shown
along the domain boundaries.
We consider three initial cross-sectional distributions of passive tracer superimposed to the wake
vortices. In the two extreme cases, the tracer is trapped either in the vortex core or in the vortex
oval envelope. In the more realistic case, the tracer is situated within the streamlines covering half
of the vortex oval’s cross-sectional area (see Fig. 2). The half-oval tracer distribution was motivated
by observations of contrails after completion of the vortex roll-up. Cross-sectional areas covered by
these distributions are 113, 3150, and 6300 m
2
for vortex core, half-oval, and full-oval, respectively.
Based on a fit obtained from observations of exhaust plume areas,
41, 42
the cross-sectional area of the
half-oval of 3150 m
2
corresponds to 38 seconds of plume age which is on the order of one vortex
reference time t
0
. Hence, the assumption of a fully rolled-up vortex pair and the initialized tracer
distribution are consistent.
The vortex core and full-oval initializations have an internal tracer value of one and the back-
ground value of zero where a smooth transition is realized by a hyperbolic tangent function. The
FIG. 2. Initial distributions of passive tracer: (a) cross-sectional distribution, (b) the distribution along section A-A.
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025104-5 Vortex bursting and tracer transport Phys. Fluids 24, 025104 (2012)
vortex core tracer distribution is given by
c
v
(y, z) = 0.5
1.0−tanh
2.5
r
1
r
c
−
r
c
r
1
+ 0.5
1.0−tanh
2.5
r
2
r
c
−
r
c
r
2
, (4)
where r
1
= [(y − y
s
)
2
+ (z − z
s
)
2
]
1/2
and r
2
= [(y − y
p
)
2
+ (z − z
p
)
2
]
1/2
are distances from the centers
of the starboard (y
s
, z
s
) and port vortex (y
p
, z
p
), respectively. Similarly, a profile for the full-oval is
defined as
c
f
(y, z) = 0.5
1.0−tanh
2.5
r
r
o
−
r
o
r
, (5)
where r
o
=[(1.0 − ( A
s
/A
l
)
2
)y
2
+ A
2
s
]
1/2
, −A
l
≤ y ≤ A
l
defines the vortex oval and r = [(y − y
o
)
2
+ (z − z
o
)
2
]
1/2
is the distance from the center of the oval (y
o
, z
o
). Here, A
s
and A
l
are short and long
axes of the oval, respectively.
The tracer concentration within the half-oval corresponds to the magnitude of the modified
stream function of a vortex pair c
h
(y, z) =|(y, z)|,
(y, z) =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
0.761
y
b
0
+ln
z
2
+(y −b
0
/2)
2
z
2
+(y +b
0
/2)
2
−0.128
1
2
(y > 0,<−0.128 or y <0,>0.128),
0 (otherwise),
(6)
where the constant 0.761 is a factor to normalize the maximum concentration value to one, while the
constant 0.128 adjusts the area covered by the tracer to half of the full-oval’s cross-sectional area.
The visualizations of vortex structures and tracer distributions in this paper are produced only
from every second grid point of the computational mesh. However, the full resolution of the simu-
lations is used for the evaluations of, e.g., vortex positions and circulation.
IV. AMBIENT TURBULENCE FIELD
The ambient atmospheric conditions are characterized by different eddy dissipation rates ε,
representing the strength of ambient turbulence and Brunt-V
¨
ais
¨
al
¨
a frequencies N = (g/θ
0
dθ/dz)
1/2
,
representing the stability of the temperature stratification. The analysis of in situ measurements of
the Falcon research aircraft
31
at altitudes between 9 and 11 km indicates dissipation rates between
10
−8
and 2 × 10
−7
m
2
/s
3
and Brunt-V
¨
ais
¨
al
¨
a frequencies typically ranging from 0.011 s
−1
to
0.023 s
−1
. The combinations of ε* and N* investigated in this study are listed in Table I in which
ε and N are non-dimensionalized based on wake vortex parameters according to ε* = (εb
0
)
1/3
/w
0
and N* = Nt
0
, respectively. Table I also provides the normalized integral turbulence length-scales,
L
∗
t
= L
t
/b
0
, reached at the end of the respective pre-runs for generating the ambient turbulence
field. It has been pointed out that the turbulence length-scale, in addition to eddy dissipation rate
TABLE I. Investigated combinations of normalized eddy dissipation rates ε*, Brunt-V
¨
ais
¨
al
¨
a frequencies N*, and respective
integral turbulence length scales L
∗
t
.
N* = 0.0 0.35 1.0
ε* = 0.0 – L
∗
t
=∞ –
0.01 L
∗
t
= 0.85 L
∗
t
= 0.96 L
∗
t
= 0.79
0.23 – L
∗
t
= 0.72 –
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