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OCR for page 950
24th Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
Tracking Vortices Over Large Distances
Using Vorticity Confinement
RainalLd LJohner i, Chi Yang ~ and Robert Roger2
(i George Mason University, 2 Johns Hopkins University)
ABSTRACT
This paper discusses the use of the vorticity confine-
ment method on unstructured grids to simulate vortex
dominated flows. A general vorticity confinement term
has been derived using dimensional analysis. The re-
sulting vorticity confinement is a function of the lo-
cal vorticity-based Reynolds-number, the local element
size, the vorticity and the gradient of the absolute value
of the vorticity. The vorticity confinement term disap-
pears for vanishing mesh size, and is applicable to un-
structured grids with large element size disparity. The
new term has been found to be successful for a num-
ber of test cases, allowing better definition of vortices
without any deleterious effects on the flow field.
INTRODUCTION
The requirement to accurately track vortices over large
distances is common to many areas of engineering, e.g.
rotating helicopter blades (Caradonna, 2000; Strawn
2000; Wenren et al 2001) and vortices shed by sub-
marines. Due to the inherent dissipation built into nu-
merical flux functions in order to avoid numerical in-
stabilities and unphysical solutions, any current Euler
or RANS field solver will tend to dissipate these vor-
tices too fast. In order to obtain a rough estimate for the
grid sizes required to capture accurately typical trailing
edges vortices, consider a helicopter blade with radius
r = 5m and a vortex of size ~ = 10cm. The element
size required to describe such a vortex will be smaller
than h = lcm (10 elements across the vortex zone).
The volume occupied by the vortex per blade rotation
Is approximately
Vie, = 1r 102cm2 Or 5 102cm ~ 106cm3
Given that the vortex is moving, and possibly interact-
ing with blades and other vortices, an isotropic grid
seems prudent. Such a grid would require at least 106
points per blade rotation. This should be viewed as a
lower limit, as the only free parameter is the mesh size,
which, in all likelihood, was estimated too large. Simi-
larly, estimates for the number of grid points required to
track accurately one or more vortices for one submarine
length easily approach billions of points, making such
calculations impractical for the next decade (if Moore's
law continues). In order to avoid the rapid dissipation
of vortices, Steinhoff and co-workers have introduced
the concept of vorticity confinement (Steinhoff et al,
1992; Steinhoff et al,1994; Steinhoff et al, 1999; Hu et
al, 2000), refining it over the last decade and applying
it successfully to a number of relevant flows (Moulton
et al, 2000; Wenren et al, 2001; Dietz et al, 20011. The
basic technique consists of adding a force-term to the
momentum equations, resulting in:
pa, + pvVv + Vp = V,uVv—con x A, (1)
where p, v, p, ,u and ~ denote, respectively, the density,
velocity, pressure, viscosity and vorticity of the fluid,
~ is a user-defined number and n is a 'normal' vec-
tor. Note that this additional force acts in the direction
normal to the vorticity and n, thus convecting vortic-
ity back towards the centroid as it diffuses away. It can
also be noted that this term is limited to the vertical re-
gions and does not affect nonvertical regions. A typical
choice for n is:
Vows
~V~ ~
(2)
Steinhoff and co-workers (Steinhoff et al, 1992; Stein-
hoff et al,1994; Steinhoff et al, 1999; Hu et al, 2000;
Moulton et al, 2000; Wenren et al, 2001; Dietz et al,
2001) were able to demonstrate that vortices could be
captured and maintained for long distances without dis-
sipation. Steinhoff and co-workers have worked mainly
on uniform Cartesian grids, for which ~ could be kept
constant. Murayama (Murayama et al, 2001 ) attempted
to use this type of vorticity confinement on an unstruc-
tured grid, i.e. leaving ~ constant. The results were
OCR for page 951
mixed. For some values of c, an improvement of re-
sults was observed. Other values of ~ lead to unphysi-
cal results, e.g. premature vortex burst on a delta wing.
These results make it clear that for non-uniform grids a
general solution has to be found. Vorticity confinement
has also been used recently for the visual simulation of
smoke (Fedkiw et al, 2001~. These animations were
performed on Cartesian grids using the incompress-
ible Navier-Stokes equations, and included (for the first
time, to the author's knowledge) an explicit, linear de-
pendence on the mesh size h.
DIMENSIONAL ANALYSIS
From dimensional analysis, one can see that ~ must
have the dimension of a velocity. One could either use
Eve, hack or h2 ~V~ I. Considering Eqn.~2), the last
form is particularly appealing, leading to:
pv,~ + pvVv + Vp = V]UVV—CV ph2 V ~w ~ x ~ , (3)
where cv is now a true constant, regardless of the grid.
One can see immediately that the vorticity confinement
term is of the form of an anti-diffusion, and that it will
disappear as the grid gets finer and finer (h ~ O).
PROPER LENGTH SCALE h
A crucial ingredient in the vorticity confinement given
by Eqn.~3) is the length scale h. It was found that for
isotropic grids, most of the possible forms: average of
edge-lengths surrounding a point, volume to surface ra-
tio of elements surrounding a point, etc., gave similar
results. As expected, the situation is markedly differ-
ent for highly stretched grids. Here, is was found that
taking h as the characteristic length in the direction of
V ~w ~ was the proper choice. The determination of char-
acteristic lengths in the x, y, z directions is performed
by observing that the gradient, computed as:
Ok = Mi · ~ Ck (Ui + uj) , (4'
. .
for point i, direction k and edges i, j surrounding point
i, will be of dimension ffu]/:h]~. Therefore, an approx-
imate estimate for the characteristic element length in
direction k may be obtained from:
(ink) = 2M~: · ~ ark ~ . (5)
Denoting by h the characteristic element lengths com-
puted, the final form of h is given by:
h=h IVI 11 . (6)
TREATMENT OF BOUNDARY LAYERS
The primary function of vorticity confinement is to en-
hance the capture of relevant physics in regions where
mesh density is insufficient. This is not the case in
well resolved boundary layers close to solid surfaces.
In fact, it was found that switching on vorticity con-
finement in these highly resolved regions could lead to
numerical instabilities. Therefore, an explicit switch,
based on the local Reynolds-number Reh, was at-
tempted:
he = min(`l'Reh\) h; Reh = ~ · (~7)
This form did not prove satisfactory. A more universal
form, that is tied to the terms used for vorticity con-
finement, is a local Reynolds-number defined with the
vorticity. From dimensional analysis, one may observe
that the following three candidates could be used:
p~Av~h
Rew,h= .. ;
Red h = Path
7 ~
plVlwI ~h3
Reich = I,,.
(8)
The final form of the vorticity confinement force then
takes the form:
f = g(Re~,7h)cvph2VI~l x ~
g=max O,min 1, R 1 Re0 (9)
[ [ Cash Lo he ]
NUMERICAL RESULTS
The vorticity confinement described above was imple-
mented into FEFLO, a general adaptive unstructured fi-
nite element flow solver (Lohner, 2002~. The results
shown were all obtained by using a projection-type in-
compressible flow solver (Lohner et al, 1998) that em-
ploys a second-order upwind advection operator and a
fourth-order pressure damping for the divergence con-
straint (Lohner et al, 19991.
OCR for page 952
NACA0012 (Euler)
The first case considered is that of a finite width
NACA0012 wing, characteristic of control surfaces.
The upstream boundary is located at 5 chord lengths
ahead of the leading edge and the downstream bound-
ary is located at 10 chord lengths behind the trailing
edge. The incompressible Euler equations are solved
for an angle of attack of or = 15°. Figure la shows
the surface mesh employed, as well as a cut normal to
the x-direction at 15% chord length downstream of the
trailing edge. Note that a line-source was specified in
the approximate position of the vortex in order to ob-
tain a finer grid. The mesh has approximately 120,000
points, with 12,000 points on the boundary.
Figure lb shows the results of the surface pressure
contours and vorticity contours in four cut planes with
the vorticity confinement terms switched on. These
four cut planes are normal to the x-direction and are
located at 5%, 1, 4 and 8 chord lengths downstream of
the trailing edge. The vorticity confinement coefficient
is cat, = 0.1 . The vortex core visualization in this fig-
ure shows how well the vortex is captured till 8 chord
lengths downstream of the trailing edge.
Figures lc,d compare the vorticity and helicity (v
w) in the same four cut planes downstream of the wing.
The effect of vorticity confinement is clearly visible.
Without vorticity confinement, the vortex is dissipated
after only one chord length of the airfoil. With vorticity
confinement, the vortex is still maintained very well up
to eight chord lengths of the airfoil.
Delta Wing (Laminar NS)
The second case considered is that of the delta-wing
measured by Hummel (Hummer et al, 1967) and com-
puted by Murayama (Murayama et al, 2001~. This
is a laminar case, making it ideally suited for bench-
marking. The angle of attack is al = 20.5°, and the
Reynolds-number based on the length of the wing is
Re = 106. The grid, shown in Figure 2a, is typical
of RANS calculations. In the proximity of the wall,
the elements are highly anisotropic with extremely fine
spacing normal to the wall. Away from the wall the
mesh coarsens rapidly and becomes isotropic. The pri-
mary vortex generated by the delta wing rapidly enters
regions of low mesh density. Figure 2b shows the vor-
ticity and pressure contours for planes located at 30%,
50% and 70% root chord length for the case with vor-
ticity confinement terms switched on and the vorticity
confinement coefficient cat,. = 0.1 . The vortex strength
for the 50% chord plane are compared in Figure 2c. As
before, vorticity confinement has a marked effect on the
strength of the detached vortex.
Figure 2d compares the measured and computed
cp distribution at the plane x = 0.7 . One can see that
the results with vorticity confinement deviate from the
experimental measurements at the wing tip.
2-D Cylinder (Laminar NS)
The third case considered is a 2-D circular cylinder.
Although the case is two-dimensional, it was modeled
as three dimensional, with two parallel walls in the
z-direction. The simulations are performed for two
Reynolds numbers, Re = 110 and Re = 190, re-
spectively. The Reynolds number is based on the di-
ameter of the circular cylinder. The incompressible
laminar Navier-Stokes equations are solved since the
flow is laminar for the Reynolds numbers considered
here. The mesh consists of 306,160 tetrahedral ele-
ments, 60,291 points and 17,332 boundary points. The
grid, shown in Figure 3a, is typical of RANS calcula-
tions. In the proximity of the cylinder, the elements
are highly anisotropic with extremely fine spacing nor-
mal to the cylinder. Away from the cylinder, the mesh
coarsens rapidly and becomes isotropic. The vortex
generated by the circular cylinder rapidly enters regions
of low mesh density.
Figure 3b shows comparison of the time history of
the lift coefficient for Re = 110 between present results
obtained without vorticity confinement and the numer-
ical results predicted by Walhorn (Walhorn, 2001) us-
ing a space finite element method. It can be seen from
Figure 3b that both predictions agrees fairly well ex-
cept slightly different frequency with which vortices are
shed in a Karman vortex street behind a circular cylin-
der. The dimensional frequency of vortex shedding ob-
tained from the present numerical prediction is com-
pared with the measurements performed by Roshko
(Roshko, 1954) in the table 1. Our numerical predic-
lion shows slightly high Strouhal number. The slight
high Strouhal number in our prediction in comparison
with both Roshko (Roshko, 1954) and Walhorn (Wal-
horn, 2001) is due to constrain of channel in our simu-
lation model.
Table 1: Strouhal Number
I 11 Re=110 1 Re=190 1
l
Roshko 0.171 0.188
Present results cv—0 0.185 0.205
Present results cv 76 0
0.188 0.207
OCR for page 953
NACA 0012: Surface Grid (nboun= 12,087, npoin=121,314) NACA 0012: Cut for Plane x=1.15
Figure la Finite NACA0012 Wing: Surface Mesh (left) and Cut Plane x = 1.15 (right)
A..
/
Figure lb Finite NACA0012 Wing: Surface Pressure, Vorticity in 4 Cut Planes and Vortex Core
(4 Cut Planes: x = 1.05,2,5,9; cat. = 0.1 )
OCR for page 954
Figure to Finite NACA0012 Wing: Comparison of Vorticity in 4 Cut Planes
4 Cut Planes: x = 1.05, 2, 5, 9
(left: cv = 0; right: cv = 0.1)
~1
-
Figure ld Finite NACA0012 Wing: Comparison of Helicity in 4 Cut Planes
(4CutPlanes: x = 1.05,2,5,9)
(left: cv = 0; right: cat. = 0.1)
OCR for page 955
Figure 2a Delta Wing: Surface Mesh (left), Detailed Surface Mesh and Mesh in Cut Plane (right)
Figure 2b Delta Wing: Vorticity (left) and Pressure (right) in Planes x = 0.3, 0.5, 0.7
Figure 2c Delta Wing: Comparison of Vorticity for Plane x = 0.5
OCR for page 956
0.8
0.6
Q
O 0.4
0.2
O
-0.2
0.8
0.6
0.4
J 0.2
c'
o
-0.2
-0.4
-0.6
-0.8
Experiment
Cv=O. x
Cv=0.05 0
Cv=0.1 ~
6<
- 11
~ ,.~.- _. _ ~ .,_.~ ~-'e'
0 0.2 0.4 0.6 0.8 1
X
Figure 2d Delta Wing: Comparison of Cp
for Plane x = 0.7
0.6
0.4
c' 0.2
o
-0.2
-0.4
-0.e
Present Results, Cv=O.O
Walhorn's Results --~
135
140
1 4t5
150
160
Figure 3b 2-D Cylinder: Comparison of C~,
for Re = 110
Figure 3a 2-D Cylinder: Surface Mesh
0.8
0.6
0.4
0.2
o
-0.2
-0.4
-0.6
-0.8
-1 -
150 155 160 90
. . · . · .
Re=110, Cv=O.O
Re=1 10. Cv=0.25 --- ----- -
130 135 140 145
Re=190, Cv=O.O
- Re=190, Cv=0.2
Figure 3c 2-D Cylinder: Comparison of CL, Computed with and without Vorticity Confinement
(left: Re = 110; right: Re = 190)
OCR for page 957
Re= 110 ,t= 146
Re=llO,t=146
Re= 110 ,t= 148
Re = 110 , t = 148
Re= 110 ,t= 150
Re = 110, t = 150
Re=llO,t=152
Re = 110, t = 152
Re = 110 , t = 154
Figure 3d 2-D Cylinder: Pressure Contours
without Vorticity Confinement (cv = 0)
Re = 110 , t = 154
Figure Be 2-D Cylinder: Pressure Contours
with Vorticity Confinement (cv = 0.25)
OCR for page 958
Re = 190, t = 108
Re = 190, t = 108
Re= 190 ,t= 110
Re=l90,t=110
Re= 190 ,t= 112
Re=l90,t=112
Re = 190 , t = 114
Re = 190, t = 114
Re= 190 ,t= 116
Figure 3f 2-D Cylinder: Pressure Contours
without Vorticity Confinement act, = 0)
Re=l90,t=116
Figure 3g 2-D Cylinder: Pressure Contours
with Vorticity Confinement act, = 0.2)
OCR for page 959
The time histories of the lift coefficient computed
with and without vorticity confinement are compared
one another in Figure 3c for both Reynolds numbers.
The vorticity confinement coefficients are taken as cv =
0.25 for Re = 110 and cat, = 0.2 for Re = 190. It can
be seen from Figure 3c that the vorticity confinement
has a relatively minor effect on the lift coefficient and
the frequency with which vortices are shed in a Karman
vortex street behind a circular cylinder.
The pressure contours at four different times are
shown in Figures 3d-g for the Re = 110 and Re = 190.
It can be seen from Figure 3d and Figure 3f that the
Karman vortex street is dissipated quickly for both
Reynolds numbers without vorticity confinement. It
can also be seen from Figure 3e and Figure 3g that the
Karman vortex street has been preserved very well for
both Reynolds numbers with the vorticity confinement
terms switched on.
Submarine (RANS + Baldwin-Lomax turbulence
model)
The fourth case considered is a submarine with a 42-
foot diameter. The speed is 7 knots and the Reynolds
number per foot is Re = 1. l X 106. The front half of the
submarine is modeled to study the vortex shed by the
sail. The mesh has approximately 6,000,000 tetrahedral
elements. The incompressible RANS equations with
Baldwin-Lomax turbulence model are solved. Fig-
ures 4a,b show the RANS grid used in the simulation.
Figure 4a Submarine: Surface Mesh
Figure 4b Submarine: Detailed Surface Mesh
Figure 4c shows the preliminary results of the pres-
sure contours on the submarine and the absolute ve-
locity contours in two cut planes. A tip vortex can be
seen from this figure. Note that the present preliminary
results are computed without vorticity confinement.
Figure 4c Submarine: Surface Pressure
Contours and Absolute Velocity
Contours at Two Cut Planes
CONCLUSIONS AND OUTLOOK
A general vorticity confinement term for unstructured
grids has been derived, implemented and found to be
successful for some cases. The vorticity confinement
terms are of the form:
f = g(Re`~,h~c~,ph2V~ x
OCR for page 960
where cat, = 0~1), h is a characteristic element size
and 9 depends on the local vorticity-based Reynolds-
number Rew,h.
We are currently investigating in more depth the
theoretical aspects associated with vorticity confine-
ment. In particular:
- One can see that the vorticity confinement as given
by Eqn.~3) is in the form of a body force. As such,
these terms may add or subtract axial and/or tan-
gential moment from the surrounding flow. We
are attempting to derive hard estimates/ proofs for
the axial and tangential moment attributed to vor-
ticity confinement. Our present conjecture of that
such estimates can be obtained by making use of
Stokes' theorem for vorticity.
It is possible that the vorticity confinement terms
introduce errors in the flow field. For benchmark
problems (e.g. delta wing), more tests are required
to determine the source of errors and to quantify
them.
Finally, other formulations for vorticity are cer-
tainly possible.
We will also continue the submarine run with vor-
ticity confinement terms switched on to track the vor-
tices shed by the sail over a large distance.
ACKNOWLEDGEMENTS
This research was partially supported by ONR, with Dr.
Patrick Purtell as the technical monitor.
REFERENCES
Caradonna, F., "Developments and Challenges in Ro-
torcraft Aerodynamics," AIAA-00-0109, 2000.
Dietz, W., Fan, M., Steinhoff, J. and Wenren, Y., "Ap-
plication of Vorticity Confinement to the Prediction
of the Flow Over Complex Bodies," AIAA-01-2642,
2001.
Fedkiw, R., Stam, J. and Jensen, H. W., "Visual Simula-
tion of Smoke," Proceedings of SIGGRAPH, Los An-
geles, CA, 2001.
Hummel, D. and Srinavasan, P. S., "Vortex Break-
down Effects on the Low-Speed Aerodynamic Charac-
teristics of Slender Delta Wings in Symmetrical Flow,"
Royal Aeronautical Society Journal, vol. 71, 1967, pp.
319-322.
Hu, G., Grossman, B. and Steinhoff, J., "A Numer-
ical Method for Vortex Confinement in Compressible
Flow," AIAA-00-0281, 2000.
Lohner, R., Yang, C. and Onate, E., "Viscous
Free Surface Hydrodynamics Using Unstructured
Grids," Proceedings of the 22nd Symposium on Naval
Hydrodynamics, Washington, D.C., 1998.
Lohner, R., "FEFLO User's Manual," GMU-CSVCFD-
01-01, 2002.
Moulton, M., and Steinhoff, J., "A Technique for the
Simulation of Stall with Coarse-Grid CFD," AIAA-00-
0277, 2000.
Murayama, M., Nakahashi K. and Obayashi S., "Nu-
merical Simulation of Vortical Flows Using Vorticity
Confinement Coupled With Unstructured Grid," AIAA-
01-0606, 2001.
Roshko, A., "On the Development of Turbulent Wakes
from Vortex Streets," NACA Report 1191, 1954.
Steinhoff, J., Yonghu, W., Mersch, T. and Senge, H.,
"Computational Vorticity Capturing: Application to
Helicopter Rotor Flow," AMA-92, 1992.
Steinhoff, J., "Vorticity Confinement: A New Tech-
nique for Computing Vortex Dominated Flows,"
Frontiers of Computational Fluid Dynamics, D.A.
Caughey and M.M. Hafez eds., J. Wiley & Sons, 1994.
Steinhoff, J., Yonghu, W. and Lesong, W., "Efficient
Computation of Separating High Reynolds Number
Incompressible Flows Using Vorticity Confinement,"
AIAA-99-3316-CP, 1999.
Strawn, R., "Computational Modeling of Hovering Ro-
tors and Wakes," AIAA-00-0110, 2000.
Walhorn, E., "Ein ganzheitliches Berechnungsmodell
fur Fluid-Struktur-Wechselwirkungen," Ph.D. thesis,
Der Technischen Universitat Carolo-Wilhelmina zu
Braunschweig, 2001.
Wenren, Y., Fan, M., Dietz, W., Hu, G., Braun, C.,
Steinhoff, J. and Grossman, B., "Efficient Eulerian
Computation of Realistic Rotorcraft Flows Using Vor-
ticity Confinement," AIAA-01-0996, 2001.
OCR for page 961
DISCUSSION
John Steinhoff
University of Tennessee Space Institute, USA
I believe that Dr. Loehner et al. have done
excellent, much needed research in method for
computing vertical flows. The authors have
added an important capability to their
unstructured grid methodology. Two points
should be made, however: First, earlier work has
been done on varying the Vorticity Confinement
parameter so that the effects vanish when the
grid is sufficiently refined. Second, we believe
that it is important that a "biased" finite
difference method should be used so that no
Vorticity Confinement corrections are made
outside the vortex core. Otherwise, they could
erroneously effect the computed velocities. This
could be the reason for the deviation from
experiment of the reported results for computed
delta wing surface pressures. Comments,
paraphrased from Ref. t1], are
"Vortices convecting past airfoils and wings
(blade - vortex interactions) were treated in
Ref. A. In this early study, unlike in our
current studies, near the surface a surface-
fitted grid was used for the wing with
surface grid refinement to resolve the actual
Navier-Stokes equations, since only a low
Reynolds number, laminar, case was treated.
To accommodate this grid refinement with
Vorticity Confinement, the parameter
specifying the strength of the Vorticity
Confinement term ~ ~ ~ was made to be
proportional to grid size so that it
automatically vanished in the fine-grid
boundary layer region, but was able to
confine the convecting vortex in the
external, coarse-grid region.
When using unstructured grids, which have
rapidly changing cell sizes, care must be taken
not only that £ varies properly with cell size,
but also that the confinement correction does not
extend beyond the vortex core due to numerical
artifacts of the implementation. This property is
true in the continuum limit, and should be
preserved in the discretization. If the correction
does extend beyond the vortex, then it could
erroneously affect surface pressure if a vortex is
passing near a surface. This could be important,
for example, for delta wings and similar cases,
where vortices convect near surfaces. In fact, for
delta wings, it is well known that there is a
feeding sheet from the leading edge causing the
vortex to grow in strength as it convects and
causing the characteristics to point towards it. In
such cases, for a reasonable grid, confinement is
not really needed (until the vortex convects past
the trailing edge). If confinement is used
correctly, however, it should not change the
nearby pressure on the surface even in these
cases, for high Reynolds number flow."
References
1. Fan, M., Wenren, Y., Dietz, W., Xiao, M.,
Steinhoff, J., "Computing Blunt Body Flows On
Coarse Grids Using Vorticity Confinement", to
appear in Journal of Fluids Engineering,
December, 2002.
2. Steinhoff, J., and Raviprakash, G., "Navier-
Stokes Computation of Blade-Vortex Interaction
Using Vorticity Confinement", AIAA-95-0161.
AUTHORS' REPLY
Thank you very much for your interest in our
paper and for providing information about your
own work on the vorticity confinement. We
would like to address your comments as follows.
Point 1: The effect of Vorticity confinement
should vanish as a) Antidiffusion: One can
develop V|CA X ~ with CO = V X v . One of the
terms appearing in the double vector product is
of the form V2v, i.e. an antidiffusion. The
problem is that one cannot get rid of the other
terms, making the analysis murky. Clearly, the
antidiffusion is conservative.
b) Different Limiting: One can also submit that
the diffusion one is trying to avoid is the one that
is trying to spread the vortex. If one decomposes
locally the velocity vector with the CO, n, v
system, one can apply different limiters in the
different directions. For example, one could
apply a steepening limiter like super-B in the
direction of n. We tried this. The results of the
vorticity confinement were still far superior to
those obtained with this procedure.
OCR for page 962
DISCUSSION
Luigi Martinelli
Princeton University, USA
I would like to congratulate the authors for their
interesting contribution and for a well written
paper. Tracking vortices over large distances
using CFD methods based on the Euler and
RANS equations is indeed a challenging and
important task. The authors have chosen to
implement a vorticity confinement method,
which is shown to improve the resolution of
vortices propagating in the far field.
In the paper, the authors correctly note that the
vorticity confinement term acts as a body force.
Thus, it alters the conservation of momentum,
and therefore the overall dynamics of the flow.
In light of this, I was surprised by the little
theoretical work done to justify this approach. I
believe that a theoretical investigation of the
vorticity confinement method should precede the
application of it to the very complex flows
discussed in the paper. So I am wondering why
the authors have decided to compute very
difficult flows first, and eventually carry out the
theoretical analysis at a later time. Also, I expect
that a more accurate formulation of their method
will require a confinement term consistent with
the discretization error of the numerical scheme.
Have the authors looked at this alternative
approach?
AUTHORS' REPLY
Thank you very much for your interest in our
paper and for the good comment. The point of
carrying out first some theoretical analysis
before proceeding to large-scale problems is well
taken. We have tried to bridge the gap between
theory and practice, but have not been successful
to date. As anecdotal note, among the things we
tried the following two seemed the most
. .
promising:
a) Antidiffusion: One can develop V|Cd XCO
with ~ = V X v . One of the terms appearing in
the double vector product is of the form V2v,
i.e. an antidiffusion. The problem is that one
cannot get rid of the other terms, making the
analysis murky. Clearly, the antidiffusion is
conservative.
b) Different Limitina: One can also submit that
the diffusion one is trying to avoid is the one that
is trying to spread the vortex. If one decomposes
locally the velocity vector with the do, n, v
system, one can apply different limiters in the
different directions. For example, one could
apply a steepening limiter like super-B in the
direction of n. We tried this. The results of the
vorticity confinement were still far superior to
those obtained with this procedure.
DISCUSSION
Ernie O. Tuck
University of Adelaide, Australia
This is a worrying procedure, and I am pleased
to see (in the 1St two data points of the written
"conclusions and outlook" section) that the
authors are hoping to address these worries in
future work.
For those of us who doubt methods like RANS,
precisely because they sometimes seem to over-
dissipate, the "black magic" of adding a non-
physical additional body force that somehow
reverses that over-dissipation is specifically
attractive.
However, there is a worry that what it might be
doing is allowing generation of vortices by the
code, but then propagating them into the far field
via a non-physical and perhaps even incorrect
mechanism. The quality of the present results
suggests otherwise, and I wish the authors luck
in further study of this interesting technique.
AUTHORS' REPLY
Thank you very much for your interest in our
paper. We agree that there is the worry of
generating vorticity due to too much
antidiffusion. We, like so many others, also wish
for the perfect non-dissipative scheme. But after
waiting for 50 years, we decided we could wait
no longer. If we were ever going to translate
vortices in our lifetime, this technique looked
very good. We furthermore believe that although
not perfect, it has certainly improved the quality
of the results obtained. Its biggest contribution
may be a renewed thinking along the
antidiffusive theme.
Representative terms from entire chapter:
pressure contours
