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NUMERICAL SIMULATION OF VORTICAL
FLOWS USING VORTICITY CONFINEMENT
COUPLED WITH UNSTRUCTURED GRID
AIAA-2001-0606
Mitsuhiro Murayama and Kazuhiro Nakahashi
Department of Aeronautics and Space Engineering
Tohoku University, Sendai, JAPAN
Shigeru Obayashi
Institute of Fluid Science
Tohoku University, Sendai, JAPAN
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American Institute of Aeronautics and Astronautics 1
AIAA-2001-0606
NUMERICAL SIMULATION OF VORTICAL FLOWS USING
VORTICITY CONFINEMENT COUPLED WITH UNSTRUCTURED GRID
Mitsuhiro Murayama*, Kazuhiro Nakahashi
†
and Shigeru Obayashi
‡
Tohoku University, Sendai 980-8579 Aoba-yama 01, JAPAN
ABSTRACT
This paper discusses the use of the vorticity
confinement method coupled with the unstructuredgrid approach to simulate vortical flows. The method
is evaluated by several vortical flow computations of
the leading-edge separation vortices on delta wings
and the wing tip vortices of NACA0012 wing. It is
shown that the vorticity confinement can keep the
vorticity away from the numerical diffusion
effectively. Although further study to reduce the
dependency of the confinement coefficient on thegrid density is required, the present results indicate
the possibility of accurate vortical flow computations
by the vorticity confinement method coupled with
unstructured and adaptive refinement grids.
1. INTRODUCTION
With the recent progress, the computational fluiddynamics (CFD) is very close to its matured stage in
the computation of flows around airplanes at
designed conditions. However, it is still difficult to
deal with complex flows at off-design conditions
where flow separates and vortices characterize theflow features. These vortical flows are often
encountered at various important engineering
problems. For example, flows around a delta wing at
high incidence are characterized by the leading-edgeseparation. This separation vortex generates a large
non-linear lift increment called vortex-induced lift at
moderate angles of attack. At higher angles of attack,
however, this vortex is to burst, resulting in a suddendecrease of the lift. The BVI (Blade-Vortex
Interaction) problems and vibratory loading problems
are also caused by impingements of vortices on
helicopter rotor and aircraft fuselage. Wing-tipvortices of airplanes during the take-off and landing
are another serious problem to deal with the
congestion at airports accompanied with the rapid
growth of the aviation.
* Graduate student† Professor, Department of aeronautics and space engineering,Associate Fellow AIAA‡ Associate professor, Institute of Fluid Science, Senior Member Copyright © 2001 by American Institute of Aeronautics andAstronautics, Inc. All rights reserved.
Accuracy of the vortical flow simulations,
however, is still not good enough. Generally, the
computational grid becomes rapidly coarser as it
becomes far away from the body surface. There, the
vortices are highly diffused due to the numericaldiscretization error. By the use of a highly dense grid,
numerical dissipation of the vortices may be
minimized. However, flow computations around 3-D
complex bodies with large-scale separations andvortices are difficult within the realistic number of
grid points.
One approach to crisply capture the vortex is to
use the adaptive grid method. We proposed an
effective and efficient adaptive grid refinement
method for the improvement of grid resolution
around a vortex center using the vortex center identification method. The resulting method was
applied to flows around a delta wing and showed
significant improvements in resolution of the
leading-edge separation vortices [1]. For further improvements, however, not only the way to reduce
dissipation by the refinement of the grid but also the
model to keep the vortices from diffusion will be
needed.
“Vorticity confinement method” has been
proposed to reduce the diffusive property of the
vortical flow simulations [2-4]. In the method, thesource term added to the Navier-Stokes equations
works as it convects the discretization error back into
the vortex center and thus confine the vortex. The
method is applied to flows around helicopter rotors
with fuselage, and shows reasonable improvements
in the vortex resolution [3]. However, the method is
still needed to be tuned for the confinement
parameter at a particular grid. In addition, the
vorticity confinement has been used on Cartesian
grids or structured surface conforming grids, not on
unstructured grids. Unstructured grids will be more
suitable for the numerical simulation around 3-Dhighly complex bodies.
The objective of this paper is to develop the
vorticity confinement method coupled with the
unstructured grid method. The capability of theresulting method is evaluated by the computations of
the following four flow fields with vortices,
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whereijS∆ is a segment area of the control volume
boundary associated with the edge connecting points
i and j . The term h is an inviscid numerical flux
vector normal to the control volume boundary, and±
ijQ are values on both sides of the control volume
boundary. The subscript of summation, )(i j , refersto all node points connected to node i .
The Harten-Lax-van Leer-Einfeldt-Wada Riemannsolver (HLLEW) [9] is used for the numerical flux
computations. Second-order spatial accuracy is
realized by a liner reconstruction of the primitive
variables T pwvu ],,,,[ ρ =q inside the control volume,
viz.,
)(),,( iiii z y x rrqqq −⋅∇Ψ+= (6)
where r is a vector pointing to point ),,( z y x ; and
i is the node number. The gradients associated with
the control volume centroids are volume-averagedgradients computed using the value in the
surrounding grid cells. A limiter, Ψ , is used to make
the scheme monotone. Here Venkatakrishnan’slimiter [10] is used because of its superior
convergence properties.
To compute viscous stress and the heat flux terms
in )(QG , spatial derivatives of the primitive
variables at each control volume face are evaluated
directly at the edges.
A one-equation turbulence model by Goldberg-
Ramakrishnan (G-R)[11] was implemented to treatturbulent flows. This model does not involve wall
distance explicitly so that it is of great benefit to
unstructured grid method.
The lower-upper-symmetric Gauss-Seidel
(LU-SGS) implicit method originally developed for
structured grid is applied in order to compute thehigh Reynolds number flows efficiently. The
LU-SGS method on unstructured grid can be derived
by splitting node points )(i j into two groups,
)(i L j∈ , and )(iU j∈ for the first summation in
LHS of Eq. (5). With nnQQQ ∆−∆=∆
+1 , the final
form of the LU-SGS method for the unstructured grid
becomes the following two sweeps:
Forward sweep:
])(5.0[)(
¦∈
∗∗−∗∆−∆∆−=∆
i L j
j A jiji S QhRDQ1 ρ (7a)
Backward sweep:
¦∈
−∗∆−∆∆−∆=∆
)(
)(5.0iU j
A jijii S j1
QhDQQ ρ (7b)
where )()( QhQQhh −∆+=∆ , and
¦ ∆+=∆ )(
)5.0(i j Aijt
V S j ID ρ , (8)
ii
i j
n
ijij
i j
ijijiV SS SGhR +∆+∆−= ¦¦ ∗
)()(
(9)
The term D is diagonalized by the Jameson-Turkel
approximation [12] of the Jacobian as
)(5.0 IAA A ρ ±=± , where A ρ is a spectral radius of
Jacobian A .
The lower/upper splitting of Eq. (7) for the
unstructured grid is done by a grid reordering
technique [13] that was developed to improve theconvergence and the vectorization. For unsteady flow,
the time accuracy of the LU-SGS solution algorithm
is recovered by the Newton iteration using
Crank-Nicolson method.
5. RESULTS
5.1 CASE1: Single Vortex in Freestream
For the validation of the present method, a 2Dsingle vortex in freestream was computed first. The
formulation of the initial vortex suggested in Ref. [4,14] is written as follow,
( )¯®-
≤≤+
<=
0,
,
Rr Rr B Ar
Rr Rr U r u
c
cccθ (10)
220 c
cc
R R
RU A
−
−= (11)
220
20
c
cc
R R
R RU B
−
= (12)
( ) 00 =θ u , ( ) cc U Ru =θ , ( ) 00 = Ruθ (13)
where 0 R is an outer radius and Rc is a core
radius of the vortex. cU is a maximum core velocity.This initial vortex conditions are added to freestreamconditions.
Solutions were obtained at a freestream Mach
number of 5.0=∞ M . The Reynolds number is
1.2×106. The initial vortex conditions were set to
∞=U U c , 05.0=c R , c R R ×=100 . The outer
boundary is a square whose edges have 1.0 length
and periodic boundary conditions are applied. Two
types of the unstructured computational grid wereused as shown in Fig. 2. Grid 1 which has 14,094
node points has approximately uniform cells. Grid 2
which has 18,024 node points is constructed by the
division of the Grid 1 and has relatively dense region
in the middle of the grid and non-uniform grid
density.
The variations of the maximum vorticity
magnitude around vortex center and vorticity
contours with/without the vorticity confinement are
shown in Figs. 3 and 4, respectively. The centered
vortex moves downstream and passes through thedownstream boundary and reappears from the other
upstream side and moves to the center of the grid
again. This process is defined as one cycle in these
computations.
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(a) Grid 1 (14,094 nodes)
(b) Grid2 (18,024 nodes)
Fig. 2. Computational grid
0
10
20
30
40
50
60
70
0 10 20 30 40 50
Original ResultEPS0.001EPS0.003EPS0.005EPS0.01
V o r t i c i t y M a g n i t u d e
Cycle (a) Grid 1
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30
Original ResultEPS0.001
EPS0.003
EPS0.005
EPS0.01
V o r t i c i t y M a g n
i t u d e
Cycle
(b) Grid 2
Fig. 3. The variations of the maximum vorticitymagnitude around vortex center with/without the
vorticity confinement (EPS: the value of the vorticityconfinement parameter, ε )
(a) (b) (c)Without confinement
(a) (b) (c)
With confinement (ε=0.003)
Fig. 4. The vorticity contours on Grid 1 with/without
the vorticity confinement, (a) after one cycle, (b)
after ten cycles, (c) after fifteen cycles
In the case without the vorticity confinement, thevortex rapidly diffused. In the case with the vorticity
confinement with moderate values of the
confinement parameter ε, it can be seen that the
degrees of the dissipation of the vortex are decreased
by the confinement method and the strength of the
vorticity is preserved even after 50 cycles.
However, this effect is influenced by the values of
the confinement parameter ε. Too large values of the
parameter may lead to the non-physical results.
Moreover, in the case using the non-uniform grid,Grid 2, the results are fluctuant, especially when therelatively large value of the parameter ε is employed.
These results suggest that the effect of the
confinement highly depends on the grid density and
the confinement parameter.
5.2 CASE2: Vortical Flows Around a Double Delta
Wing
The method was applied to vortical flows on delta
wings with high incidence. The geometry used in the
present study is a double delta wing shown in Fig. 5.The leading-edge sweep angle is 80 degrees at the
strake and 60 degrees at the main wing. The
thickness is 0.6% of the root chord length and the
leading edge is rounded. The first point above the
wing surface is located at a distance of 8.0×10-5
of
the root chord length. The outer boundaries are
located 10-root chord length away from the body
surface. The total number of the initial grid points is676,541. The computational grids in the cross flow
plane at 62.5% chord length are shown in Fig. 6.
Solutions were obtained at a freestream Mach
number of 3.0=∞
M and angle of attack of 12º. The
Reynolds number based on the root chord is 1.3×106.
In this study, laminar flow was assumed.
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Fig. 5. Surface grid of the double-delta wing
Fig. 6. Cut view of the grid at x/c=0.625
First, the results without using vorticity
confinement are discussed. Total pressure contours
computed on the initial grid and the adaptive gridrefined twice are shown in Fig. 7. The adaptive grid
has 1,042,292 grid points. In this flow field, two
leading-edge separation vortices originating from the
strake and main wing interact with each other in the
main wing region. The strength of the strake vortex
becomes weaker at the downstream of the kink
because the energy feeding to the vortex from the
strake leading edge will decrease, while the strengthof the main wing vortex increase as the distance from
the kink increases downstream. The weak strake
vortex is moved outward influenced by the velocity
field induced by the relatively strong wing vortex andmerges with the wing vortex eventually. Compared
with the results on the initial grid, the primary and
secondary vortices are more clearly captured by the
adaptive grid refinement method.
For the validation, the surface pressure coefficients
at different axial locations were compared with
experimental data by the Brennenstuhl and Hummel[15] in Fig. 8. Two pressure peaks by the inner strake
vortex and outer wing vortex can be seen in this
experimental data. In the computational results on the
initial grid, however, the inner peak at x/c=0.625 bythe strake vortex is much lower and it can be seen
that the strake vortex has already been weaken by the
numerical diffusion due to the lack of the grid density.
By using the adaptive grid, the diffusion of the strake
vortex is suppressed as shown in Fig. 7. However,
the improvement of the surface pressure prediction is
very small as shown in Fig. 8. The main reason of
this discrepancy may be caused by the fact thatlaminar flow was assumed in the computation
although the laminar/turbulent transition was probably observed in the experiment, as discussed in
Ref. 16.
(a) Initial Grid
(b) Adaptive grid
Fig. 7 The total pressure contours
( 6100.1Re,3.0,0.12 ×=== M α )
-0.5
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1
ExperimentInitial gridAdaptive grid
- C p
Spanwise location (a) Surface pressure coefficients at x/c=0.625
-0.5
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1
ExperimentInitial gridAdaptive grid
- C p
Spanwise location (b) Surface pressure coefficients at x/c=0.75
Fig. 8. Surface pressure coefficients at differential
axial locations without the vorticity confinement( 6100.1Re,3.0,0.12 ×=== M
α )
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The results using vorticity confinement are
compared in Fig. 9. The confinement was applied to
the regions without a boundary layer. Computations
were performed with varying the value of the
confinement parameter ε on the initial grid. With
moderate values of the confinement parameter, the
resolution of vorticity is effectively improved.However, the method creates extra correction in the
case that the values of ε are too large.
The total pressure contours with/without vorticity
confinement at x/c=0.625 and 0.75 are shown in Fig.
10. From these results, it can be seen that the vortex
core position is not affected by the confinement and
the vorticity is concentrated to the vortex core
reasonably. The confinement is more effective than
the adaptive grid refinement method even with much
less grid points.
-0.5
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1
ExperimentInitial gridEPS0.005EPS0.01EPS0.05
- C p
Spanwise location (a) Surface pressure coefficients at x/c=0.625
-0.5
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1
ExperimentInitial gridEPS0.005EPS0.01EPS0.05
- C p
Spanwise location
(b) Surface pressure coefficients at x/c=0.75
Fig. 9. Surface pressure coefficients at differentialaxial locations with the vorticity confinement
( 6100.1Re,3.0,0.12 ×=== M α ); EPS is a value of the
confinement parameter ε
(i) x/c=0.625 (ii) x/c=0.75
(a) Initial grid
(i) x/c=0.625 (ii) x/c=0.75
(b) Adaptive grid
(i) x/c=0.625 (ii) x/c=0.75
(c) Initial grid with the vorticity confinement,ε
=0.01
Fig. 10. The total pressure contours with/without
vorticity confinement
5.3 CASE3: Vortical Flows with Vortex
Breakdown Around a Delta Wing
The confinement method was tested for the vortex
breakdown simulations shown in Fig. 11. The
geometry used in this computation is a slender delta
wing of aspect ratio of unity and a sweep angle of 76
degrees. The freestream Mach number is 0.3, angleof attack is 32º, and the Reynolds number based on
the root chord is 6100.1 × . Laminar flow was
assumed. This flow field was numerically simulated
by the present unstructured grid method [1] and the
predicted positions of the vortex breakdown showed
good agreements with experiments of Hummel andSrinivasan [17].
In Fig. 11, the streamlines starting from the wing
apex show the vortex breakdown pattern near the
trailing edge of the delta wing. The results using the
vorticity confinement computed on the same grid and
conditions are shown in Fig. 12. With larger valuesof ε, the vortex breakdown does not appear.
Fig. 11. The computed flow fields around a delta
wing with vortex breakdown
(6
100.1Re,3.0,0.32 ×=== M
α )
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(a) ε=0.001 (b) ε=0.005
(c) ε=0.01
Fig. 12. The computed flow fields around a delta
wing with vortex breakdown using the vorticity
confinement ( 6100.1Re,3.0,0.32 ×=== M α )
Figures 13 and 14 show the results with large-scale
vortex breakdown at higher angle of attack of 40º. By
the introduction of the vorticity confinement with
large values of ε, the starting points of the vortex
breakdown and its pattern are considerably
influenced. The vorticity confinement gives rotation
components to the vortex and confines the vortex.
However, with larger values of confinement parameter, excessive rotational components are
added and may destroy the flow physics.
Fig. 13. The computed flow fields around a delta
wing with vortex breakdown
( 6100.1Re,3.0,0.40 ×=== M α )
a) ε=0.001 (b) ε=0.005
(c) ε=0.01
Fig. 14. The computed flow fields around a delta
wing with vortex breakdown using the vorticity
confinement ( 6100.1Re,3.0,0.40 ×=== M α )
5.4 CASE4: Wing Tip Vortex of NACA0012 Wing
Finally, the present method is applied to the
computations of a wing tip vortex of a NACA0012
wing. The geometry used in the present study is a
NACA0012 rectangular wing of aspect ratio 3 as
shown in Fig. 15. The initial grid has nearlyhomogeneous grid density in the wake region and the
total number of the grid points is 701,037. The axial
direction coincides with the wing chord direction.
The outer boundary is a sphere whose radius is
15-root chord length. Solutions were obtained at a
freestream Mach number, M ∞=0.12. The Reynolds
number based on the root chord is 0.9×106
and angle
of attack is 10.0°.
(a) Surface grid (b) Close-up view
(c) Cut view of the fine grid at 95% semi-span
Fig. 15. Computational grid of NACA0012
The vortex centerlines and vorticity contours
obtained on the initial grid is shown in Figs. 16 and
17, respectively. The wing tip vortex far away from
the trailing edge is highly diffused although the
vortex centerlines are identified clearly.
Fig. 16. The vortex centerlines at the initial grid
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Fig. 17. Contours of the vorticity magnitude at theinitial grid
Therefore, adaptive grid refinement method using
the vortex center for the concentration of the grid
points around the vortex core to decrease the
diffusion of the vorticity is applied to the initial grid.The grid is refined twice and the resulting number of grid points is 815,262. The results are shown in Figs.
18-20. These results show the improvement about the
vorticity magnitude by the adaptive grid refinement.
More grid refinement may improve the results, but
from the limitation of the computational resources,
the physical model to keep the vortices from
diffusion may be required.
(a) Initial Grid (b) Adapted grid refined twice
Fig. 18. Cut views of the grid at x/c=5.0
Fig. 19. Contours of the vorticity magnitude at the
adaptive grid refined twice
0
2
4
6
8
10
12
14
2 4 6 8 10
Initial GridAdaptive Grid 1Adaptve Grid 2Adaptive Grid 3Adaptive Grid 4
V o r t i c i t y
Distance From Trailing Edge (x/c)
Fig. 20. The variation of the maximum vorticity
magnitude around vortex center
The results using vorticity confinement are shown
in Fig. 21. The confinement was applied to the wakeregions. The vorticity contours with vorticity
confinement on the initial grid are shown in Fig. 22.In the case of the initial grid, the results show some
improvements although the overall level of the
vorticity magnitude is still low due to the poor grid
density of the grid in the wake region near thetrailing edge. However, larger values of the
confinement parameter ε lead to extra effects as
shown in Fig. 22(b).
The results on the adapted grid refined twice with
the vorticity confinement are shown in Figs. 21(b)
and 23. It is apparent that the vorticity was confinedand the method successfully decreased the numerical
diffusion of the vorticity by the coupling of the
adaptive grid refinement method and the vorticity
confinement. By the use of the confinement, the
diffusion of the vortex is suppressed at the stationabout ten chord length away from the trailing edge.
The vorticity contours on a cut view at x/c=5.0 are
shown in Fig. 24. In these figures, it can be seen that
the vortex core position is not affected by theconfinement and the vorticity is concentrated to the
vortex core.
However, the effect highly depends on the value of the confinement constant coefficient, ε and grid
density. Moderate ε improves the results, while the
large ε works extra correction. And in Fig. 21(b),
the computed vorticity magnitude using the vorticityconfinement is fluctuant and not smooth at both
value of ε . This is because the grid size on
unstructured grid is not regular although the adaptive
grid refinement was applied and the grid density
becomes approximately homogenous around vortex
center. The confinement effects may depend on the
grid. From these results, it can be seen that optimal
ε which is different by the place and grid may berequired again.
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0
2
4
6
8
10
12
14
2 4 6 8 10
Initial GridEPS0.005EPS0.01
V o
r t i c i t y
Distance From Trailing Edge (x/c)
(a) Initial grid
0
2
4
6
8
10
12
14
2 4 6 8 10
Adaptive Grid 2EPS0.01EPS0.02
V o r t i c i t y
Distance From Trailing Edge (x/c) (b) Adaptive grid refined twice
Fig. 21. The variation of the maximum vorticity
magnitude around vortex center with vorticity
confinement (EPS: the value of the vorticity
confinement parameter ε )
(a) ε =0.01
(b) ε =0.05Fig. 22. Contours of the vorticity magnitude at theinitial grid with vorticity confinement
ε =0.02
Fig. 23. Contours of the vorticity magnitude at theadaptive grid refined twice with vorticity
confinement
(a) Without confinement (b) With confinement
ε =0.02
Fig. 24. Vorticity contours in the cross flow plane at
x/c=5.0 of the adaptive grid refined twice
6. CONCLUSION
The vorticity confinement method coupled with
unstructured grid has been applied to the numerical
simulations of four vortical flows. In the case of a
single vortex in freestream, the effect of the vorticityconfinement on unstructured was validated and
problems about the grid dependency were pointed
out. In the case of vortical flows around delta wings,
the confinement was found more effective than the
adaptive grid refinement method. In the vortex
breakdown case, it was demonstrated that the use of
excessive confinement parameters could suppress the
breakdown and thus destroy the flow physics. Finally,in the case of a wing tip vortex of NACA0012 wing,
it was shown that the confinement method could
suppress the numerical diffusion of the vortex far
away from the trailing edge by the coupling of theadaptive grid refinement method.
The results obtained in this study show that,
although further study to reduce the dependency of
the confinement coefficient, ε on the grid density
is required, the vortex confinement method coupled
with unstructured and adaptive refinement grids has
the possibility of accurate simulations of vortical
flows.
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REFERENCES
[1] Murayama, M., Nakahashi, K., and Sawada,
K., “Numerical Simulation of Vortex Breakdown
Using Adaptive Grid Refinement with
Vortex-Center Identification,” AIAA Paper
2000-0806, 2000.
[2] Steinhoff, J. and Underhill, D., “Modificationof the Euler Equations for “Vorticity
Confinement”: Application to the computation of
interacting vortex rings,” Physics of Fluids, Vol.
6, 1994, pp.2738-2744.
[3] Steinhoff, J., Yonghu, W., and Lesong, W.,
“Efficient Computation of Separating High
Reynolds Number Incompressible Flows Using
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