The three-dimensional interaction of a vortex pair with a wall
J. Alan Luton and Saad A. Ragab
Citation: Physics of Fluids (1994-present) 9, 2967 (1997); doi: 10.1063/1.869408
View online: http://dx.doi.org/10.1063/1.869408
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The three-dimensional interaction of a vortex pair with a wall
J. Alan Luton and Saad A. Ragab
a)
Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University,
Blacksburg, Virginia 24061-0219
~Received 4 September 1996; accepted 19 June 1997!
The interaction of vortices passing near a solid surface has been examined using direct numerical
simulation. The configuration studied is a counter-rotating vortex pair approaching a wall in an
otherwise quiescent fluid. The focus of these simulations is on the three-dimensional effects, of
which little is known. To the authors’ knowledge, this is the first three-dimensional simulation that
lends support to the short-wavelength instability of the secondary vortex. It has been shown how this
Crow-type instability leads to three dimensionality after the rebound of a vortex pair. The growth of
the instability of the secondary vortex in the presence of the stronger primary vortex leads to the
turning and intense stretching of the secondary vortex. As the instability grows the secondary vortex
is bent, stretched, and wrapped around the stronger primary. During this process reconnection was
observed between the two secondary vortices. Reconnection also begins between the primary and
secondary vortices but the weaker secondary vortex dissipates before the primary, leaving
reconnection incomplete. Evidence is presented for a new type of energy cascade based on the
short-wavelength instability and the formation of continual smaller vortices at the wall. Ultimately
the secondary vortex is destroyed by stretching and dissipation leaving the primary vortex with a
permanently distorted shape but relatively unaffected strength compared to an isolated vortex.
© 1997 American Institute of Physics. @S1070-6631~97!03110-3#
I. INTRODUCTION
The interaction of vortices with surfaces is an important
phenomenon in many engineering applications. For instance,
trailing vortices from aircraft which interact with the ground
can present a danger to following aircraft.
1
In assessing the
danger it is important to understand the motion, structure,
and decay of the vortices. These characteristics could be
strongly influenced by three-dimensional effects. Vortices
shed from the aircraft body and leading edge extension of
delta wings pass over the wing thereby affecting the lift,
drag, and possible control of the aircraft. Similarly in sepa-
rating flows, vortical structures can form which later interact
with the solid surface downstream of reattachment.
2,3
These
vortical structures can induce secondary separation and pres-
sure fluctuations along the solid surface. Also, in helicopter
aerodynamics the wakes from upstream blades can interact
with blades downstream, possibly leading to undesirable
vibrations.
4,5
There are many studies which focus on either vortex
rings or pairs impinging on a no-slip wall in an otherwise
stagnant fluid. A recent review of vortex/wall interactions
can be found in Doligalski et al.
6
A vortex moving toward a
wall is often observed to reverse course and move away from
the wall. This movement is known as vortex rebound and has
been observed in free-flight studies for wing tip vortices near
the ground ~see, for instance, Dee and Nicholas
7
!. An expla-
nation for the rebound phenomenon was given by Harvey
and Perry
8
who experimentally studied the motion of a single
wing tip vortex near a moving wall. The primary vortex cre-
ates a layer of vorticity of opposite sense next to the wall.
This vortex sheet becomes unstable, separates, and rolls up
to form a secondary vortex. The secondary vortex induces an
upward motion to the primary vortex in an inviscid-like fash-
ion. Boldes and Ferreri
9
examined a vortex ring approaching
a wall. The ring was created by a drop of colored water
impacting the free surface of a quiescent body of water. They
reported the rebound of the vortex from the wall, and in
some cases, multiple rebounds. In their two-dimensional
Navier–Stokes calculations, Peace and Riley
10
demonstrated
the rebounding phenomenon of a vortex pair from a no-slip
boundary. Their simulations were for low Reynolds number
~up to Re
G
5 G/
n
5 150). Flow separation was not observed.
In their detailed experimental study of vortex rings, Walker
et al.
11
~also see Cerra and Smith
12
for related earlier work!
created vortex rings by the sudden ejection of fluid through a
sharp edged orifice. As the ring approached a wall, the for-
mation of secondary and tertiary vortex rings was observed.
The secondary ring moves around the primary ring and into
its center. The secondary ring not only causes the primary
ring to rebound, but arrests the radial expansion of the ring.
For a sufficiently strong vortex, the diameter of the primary
ring will shrink, a process known as reversal. As the second-
ary ring moves into the center of the primary ring, azimuthal
waves develop on the secondary ring, but not the primary
ring. Walker et al. showed that these fluctuations are associ-
ated with the compression of the secondary ring. They also
observed the ejection of the secondary vortex ring from the
center of the primary ring. Orlandi,
13
in his two-dimensional
Navier–Stokes simulations, observed the formation of sec-
ondary and tertiary vortices for a vortex pair impacting a
no-slip wall. His simulations were for Reynolds number up
to Re
G
5 3200. He also observed the formation of a second-
ary vortex pair that moved far from the wall, similar to the
secondary ring ejection in the experiments of Walker et al.
a!
Corresponding author. Telephone: ~540! 231-5950; Electronic mail:
ragab@ragab1.esm.vt.edu
2967Phys. Fluids 9 (10), October 1997 1070-6631/97/9(10)/2967/14/$10.00 © 1997 American Institute of Physics
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Copyright by the AIP Publishing. Luton, J. A.; Ragab, S. A., "the three-dimensional interaction of a vortex pair with a wall,"
Phys. Fluids 9, 2967 (1997); http://dx.doi.org/10.1063/1.869408
In a subsequent study Orlandi and Verzicco
14
observed the
multiple formation of vortices for the case of a vortex ring.
They concluded that vortex pairing was the mechanism for
the ejection of the secondary ring from the center of the
primary ring. The results also showed the growth of azi-
muthal instabilities in the secondary ring, similar to the ex-
perimental results of Walker et al. The stability of the sec-
ondary vortex ring was investigated by Swearingen et al.
15
using numerical simulations and a localized stability analy-
sis. In the simulations the secondary vortex ring developed
azimuthal perturbations similar to those seen by Walker
et al.,
11
Orlandi and Verzicco,
14
and others. The stability
analysis followed the approach of Widnall and Sullivan
16
who showed that an isolated vortex ring is stable to long-
wavelength disturbances ~but unstable to short-wavelength
ones!. However, the analysis of Swearingen et al. showed
that the presence of another vortex ring causes the long-
wavelength disturbances to grow. Their analysis gave results
in good agreement with the simulations in the early stages of
development of the instability.
Most numerical studies of vortex pairs near a wall have
been two-dimensional. However, in three dimensions the
pair is subject to instabilities, in particular, those named after
Crow. Crow
17
modeled a pair of trailing vortices as sinusoi-
dally perturbed vortex filaments. The mode shape is a sinu-
soidal perturbation of each filament that is confined to a
plane inclined at approximately 45° to the plane in which the
undisturbed vortices lie. The configuration is symmetric
about a plane that bisects the distance between the vortices.
The analysis predicts two groups of unstable waves with one
set having long wavelengths and the other having short
wavelengths ~comparable to the core diameter!. Crow gives
the maximum dimensionless growth rate (
a
¯
5 2
p
b
2
a
/G)as
0.8 for the long waves and 1.0 for the short waves. Here b is
the distance between the vortices and G is the circulation of
each vortex. Widnall et al.
18
showed that the short-
wavelength instability found by Crow is spurious since its
predicted wavelength violates the assumptions of the model.
The analysis of Widnall et al. is valid for slender vortices
(a/b! 1, where a is the core diameter! and ka5O(1),
where k is the wavenumber. This analysis is therefore appro-
priate for short-wavelength disturbances, as opposed to the
analysis of Crow which requires ka!1. Although the short-
wavelength instability predicted by Crow does not exist,
Widnall et al. showed that for vortices of finite core size
there exists higher radial bending modes which are unstable.
Recently Thomas and Auerbach
19
observed the formation of
both the long- and short-wavelength instabilities for a vortex
pair. The pair was composed of the starting and stopping
vortices that formed on the edge of a rotated plate. Both long
and short-wavelength instabilities have been seen in the cur-
rent simulations.
The disturbances grow until portions of each vortex
come into close contact with one another. At these points
vortex reconnection occurs which transforms the pair into a
series of rings. While vortex reconnection has received much
attention, especially in recent years, it is still a poorly under-
stood phenomenon. A central concept is the dissipation ~or
‘‘cancellation’’! of vorticity in regions where antiparallel
vorticity lines are close together. The vortex lines then un-
dergo a cross linking, or bridging, process which connects
the two vortices. The cross linked vorticity is amplified by
vorticity stretching. Common configurations for the study of
vortex reconnection are the collision of two vortex rings
20,21
and a sinusoidally perturbed vortex pair.
22,23
An extensive
discussion of the reconnection process is given by Kida and
Takaoka.
24
Recently Dommermuth presented three-dimensional
simulations of vortex pairs interacting with free-slip and no-
slip walls
25
as well as free surfaces.
26
The primary motiva-
tion for this work was to understand the formation of free
surface features such as striations and scars that can form
when a rising vortex pair approaches a free surface. These
features were observed in the experiments of Sarpkaya and
Suthon
27
and Sarpkaya.
28
Since Dommermuth was con-
cerned with the reconnection process and not the mechanism
by which the primary vortex is deformed, a sinusoidal per-
turbation was initially imposed on the core position of the
vortex. Sheets of helical vorticity were observed spiraling off
of the vortex before interaction with the surface. The author
suggests that the vorticity sheets originated from an inviscid
instability caused by large changes in the curvature of the
vortex along its axis. The helical vorticity sheets evolved
into ‘‘beads’’ of cross-axis vorticity as they revolved around
the primary vortex. For no-slip wall interactions, the helical
vorticity sheets merged with the secondary vorticity sheet at
the wall to form U-shaped vortices wrapped around the pri-
mary vortex. For free surface interactions, the cross-axis vor-
ticity reconnected at the free surface. The cross-axis vortices
appear to be responsible for the formation of the striations
and scars observed in the experiments.
In Dommermuth’s work the axial length of the compu-
tational domain was chosen to be too short to permit the
long-wavelength Crow instability to form. The short-
wavelength Crow–Widnall instability was not addressed. In
the current work the Crow–Widnall instability and the inter-
action between the primary and secondary vortices are key
features of the flowfield. Instead of an initial sinusoidal dis-
turbance of the primary vortex core a random disturbance is
used which allows any instabilities present to develop natu-
rally. Because of this form of the initial disturbance the he-
lical sheets of vorticity which played a fundamental role in
the vorticity dynamics of Dommermuth’s simulations are not
present. The instabilities seen in the current work occur after
the primary vortex interacts with the wall and the subsequent
formation of a secondary vortex.
In the current study we consider the case of a counter-
rotating vortex pair approaching a solid surface. Our object
is to reveal the three-dimensional characteristics of the inter-
action of the pair of vortices with the wall. The full three-
dimensional, unsteady, incompressible Navier–Stokes equa-
tions are solved. In Section II the numerical scheme is
outlined as well as initial and boundary conditions. The re-
sults are presented in Section III while the conclusions are
discussed in Section IV.
2968 Phys. Fluids, Vol. 9, No. 10, October 1997 J. A. Luton and S. A. Ragab
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II. METHOD OF SOLUTION
A. Numerical scheme
The incompressible Navier–Stokes equations are given
by
]
u
i
]
t
1
]
]
x
j
u
i
u
j
52
]
p
]
x
i
1
1
Re
]
]
x
j
]
]
x
j
u
i
, ~1!
]
u
i
]
x
i
5 0, ~2!
where all variables have been nondimensionalized by a char-
acteristic velocity and length. The equations are solved by a
scheme first proposed by Kim and Moin
29
and later modified
by Le and Moin.
30
The scheme is described in detail in these
references so a mere overview is given herein. The scheme
consists of a fractional-step ~time-splitting! method com-
bined with the approximate factorization technique. The mo-
mentum equation is advanced in time in two steps, first ap-
plying the convection and diffusion operators and then the
pressure operator. Finding the pressure consists of solving
Poisson’s equation, which is equivalent to satisfying the con-
tinuity equation. Solving Poisson’s equation is by far the
most computationally expensive step. Le and Moin proposed
a modification to the Kim and Moin method to increase the
CFL while reducing the computational effort. Their method
employs a three-stage Runge–Kutta scheme in which the
convective terms are advanced explicitly and the viscous
terms implicitly. The stability limit is CFL5
A
3 for the one-
dimensional convection-diffusion equation. The Kim and
Moin time splitting is applied at each stage, yet the Le and
Moin modification permits Poisson’s equation to be solved
only at the end of the time step, rather than at each of the
three stages. This results in a substantial reduction in the
CPU time. Le and Moin estimated the CPU time savings to
be 68% over the Kim and Moin scheme for their simulations
of flow over a backward facing step. In the current work a
multigrid method is used to solve Poisson’s equation. This
implementation of the multigrid method uses Gauss–Seidel
line relaxation and semi-coarsening on uniform or stretched
meshes. The scheme provides a very efficient solution to
Poisson’s equation which is critical to reducing the compu-
tational time. The accuracy of the code has been thoroughly
examined by Luton.
31
B. Initial and boundary conditions
Now we consider the initial and boundary conditions for
a vortex pair approaching a wall. The computational domain
is shown in Fig. 1. The two primary vortices are initially
parallel to the z-axis and extend from z52` to z51`.
The x5 0 plane bisects the distance between the vortices and
is assumed to be a plane of symmetry throughout the simu-
lation. For the boundary conditions, which are only needed
for the velocity, the no-slip and no-penetration conditions are
applied at the wall (y5 0 plane!. Due to symmetry only one
vortex of the pair is simulated. Thus symmetry boundary
conditions are applied along the center (x5 0) plane of the
vortex pair. Symmetry conditions are also specified on the
boundaries opposite the wall ~top! and opposite the center
plane ~right!. These boundaries are much further away and
do not affect the flowfield substantially as will be discussed
in Section III D ~also see Luton
31
!. Periodic conditions are
used in the axial (z) direction. The velocity field is initialized
using a Lamb–Oseen vortex which has a velocity profile
given by
v
u
5
A
v
c
r/r
c
@
12 e
2 Br
2
/r
c
2
#
, ~3!
where
v
u
is the tangential velocity,
v
c
is the maximum tan-
gential velocity, r is the distance in the radial direction, r
c
is
the core radius, and A and B are constants given by A
5 1.39795 and B5 1.25643.
The vortex images across the center plane and wall are
included as well. The images across the other two symmetry
boundaries are also taken into account, but since they are
further away they have little effect. While these initial con-
ditions are divergence free and satisfy the no-penetration
boundary condition on the wall, they violate the no-slip con-
dition. However, the vortex is sufficiently far away from the
wall that the resulting disturbance is negligible.
In addition to this basic state, a random disturbance is
added to the initial velocity field at all points in the compu-
tation domain. All possible modes are excited randomly to
ensure that all, if any, instabilities of the flow are affected.
The disturbance is subject to certain restrictions. The first is
that the disturbance must have a periodicity of L
z
~the length
of the domain in the axial direction! in order to be compat-
ible with the boundary conditions. Thus the form of
x-component of the disturbance velocity is taken to be
u
8
~
x,y,z
!
5 Real
F
(
m51
N
z
/2
u
ˆ
m
~
x,y
!
e
ikz
G
, ~4!
where u
ˆ
m
is the complex amplitude, k5 2
p
m/L
z
is the wave
number, N
z
is the number of points in the z-direction, m is
the mode number, and i5
A
2 1. The other velocity compo-
nents have a similar form. Since all incompressible flows
must have a divergence free velocity field the amplitudes
FIG. 1. The computational domain.
2969Phys. Fluids, Vol. 9, No. 10, October 1997 J. A. Luton and S. A. Ragab
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cannot be specified independently. If u
ˆ
m
and
v
ˆ
m
are given
then w
ˆ
m
must be found from the discrete form of the conti-
nuity equation. Two methods have been used to determine
u
ˆ
m
and
v
ˆ
m
. Both rely on calls to a random number genera-
tor. The first method, denoted here as Rand, assigns random
numbers between 2 1/2 and 1/2 to the real and imaginary
parts of the complex amplitude. This is done at every point
in the cross (xy) plane and for every mode number m. Thus
the only structure to the disturbance field is that it is periodic
in z. The second method, denoted by RandB, assigns random
numbers in the range of 0 to 1. Consequently the velocity
field has some structure in the cross plane, while the Rand
disturbance does not. The RandB disturbance is more repre-
sentative of a flow that is being forced. For instance, let us
consider the trailing vortex of a wing. Time varying pertur-
bations could be introduced at the wing tip. These distur-
bances would be wrapped up into the vortex and create varia-
tions along its axis. This has been done experimentally
32
and
during flight
33
to excite the Crow instability between trailing
vortices. The current simulations, however, are applicable to
instabilities that occur between a trailing vortex and the sec-
ondary vortex that is created as the trailing vortex interacts
with the ground. The RandB disturbance could give one an
idea as to the potential of affecting the long time character-
istics of the flow by forcing.
A variation of the RandB disturbance, denoted by
RandC, was also used. This disturbance is identical to RandB
except that u
ˆ
m
5
v
ˆ
m
. Thus the cross plane velocity lies in the
directions defined by the line that is 45° from the positive x-
and y-axes. This was used in order to more strongly excite
the Crow instability.
III. RESULTS
A. The long-wavelength Crow instability
As an introduction to some of the three-dimensional ef-
fects of a vortex pair impinging on a wall we shall briefly
examine one type of instability exhibited by an isolated vor-
tex pair. The instability is named after Crow who laid the
theoretical foundation.
17
The initial and boundary conditions
are the same as those in Section II B with the exception that
the wall is replaced with symmetry conditions. Thus the
boundaries parallel to the vortex axis are in conflict with the
configuration as given by Crow for an isolated vortex pair.
The boundary conditions were selected so that a small do-
main could be used. Thus the vortex follows a circuit along
the inside edges of the domain instead of traveling long dis-
tances in a straight line ~as an isolated pair would!.Inany
case, we are not concerned here with a detailed analysis of
the Crow instability but merely a demonstration of its prop-
erties.
Let us first consider a vortex initially located at x5 2,
y55 with a Reynolds number based on the circulation of
Re
G
5 8784. In the cross plane the domain extends from x
5 0tox510 and y5 0toy510. The length of the domain in
the z-direction is 28. All quantities have been nondimension-
alized by the initial core radius, r
c
, and maximum tangential
velocity,
v
c
. The grid is uniform with a size of 643 64
3 32. The vortex has positive vorticity and therefore initially
moves in the negative y-direction due to the influence of its
image across the symmetry plane at x5 0. A RandB distur-
bance of magnitude U
rms
8
5 0.04
v
c
was added to the initial
velocity field. Only the second mode (l
z
5 14) was excited.
A surface of constant vorticity at time t5 167 reveals the
undulations of the vortex that have arisen due to the long-
wavelength Crow instability @Fig. 2~a!#. The disturbance
continues to grow until parts of the vortex approach a sym-
metry plane, that is, the image vortex. The process of recon-
nection then begins which transforms the vortex pair into a
series of rings. In Fig. 2~b! the vortex is beginning to recon-
nect as evident by the formation of half-ring structures—the
other halves are the images. After this point in time the cor-
ner of the domain begins to significantly alter the flow. Nev-
ertheless, the simulation illustrates the nature of the Crow
instability. The characteristic 45° angle of the Crow instabil-
ity can be seen by projecting the isovorticity surface onto an
xy-plane. This has been done at time t5167 ~Fig. 3!. The
vortex is moving in the clockwise direction. The slight bend
at one end is most likely due to the interaction at the corners.
The wavelength of the instability seen in the previous
simulation is 14r
c
. However, for this vortex pair Crow pre-
dicts that the most unstable wave has a wavelength of 21r
c
.
In order to compare with the theory another simulation was
run with a 643 643 8 grid and L
z
5 21. Only the first mode
(l
z
5 L
z
) was excited with a RandC disturbance of U
rms
8
5 0.20
v
c
. The growth rate was found by taking a fast Fou-
rier transform in the z-direction of the total kinetic energy.
After an initial decay the energy in the first mode grows
exponentially at a rate of 0.071. When nondimensionalized
as defined by Crow the growth rate is
a
¯
5 0.81. The growth
rate predicted by Crow for this mode is
a
¯
5 0.80. This com-
pares very well, especially considering the very coarse grid
in the z-direction. Later in time the corners of the domain
cause interactions that alter the growth rate. A systematic
FIG. 2. Surfaces of constant vorticity V5 1.0. ~a! t5 167, ~b! t5 196.
2970 Phys. Fluids, Vol. 9, No. 10, October 1997 J. A. Luton and S. A. Ragab
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