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OCR for page 633
Nav~er-Stokes Analysis of Turbulent Boundary Layer
and Wake for Tw - Dimensional Lifting Bodies
P. Nguyen, I. Gorski (David Taylor Research Center, USA)
ABSTRACT
Navier-Stokes calculations were per-
formed on two 2-D lifting foils which have
been tested in a wind tunnel. In the experi-
ment, the angles of attack for the two foils
were set up to yield approximately the same
lift at a Reynolds number of 2.~6xio6 ~ based
on chord). One foil has a thicker trailing
edge than the other, and has mild flow
separation on the last No chord of the suc-
tion side. The flow solver, called the Davic!
Taylor Navier-Stokes (DTNS) code, is for-
mulatec} with artificial compressibility and
upwind differencing. The Launder-Spalding
k-e turbulence mode! is used. Predictions
of the turbulent flow quantities of the
boundary layer and wake are compared
with the experimental data for both foils.
These predictions, including flow separation
location, agree reasonably well with the
data. After these validation predictions, the
Navier-Stokes analysis method and a design
technique baser] on conformal mapping are
combined to develop new 2-D foil sections.
Since the turbulent kinetic energy is the
dynamic pressure, and the ReynoIcis shear
stresses are related to the turbulence pro-
duction, these quantities are used to
develop new 2-D sections with desirable
turbulent boundary layer characteristics.
The characteristics of one new section are
presented as results of the new foil design
process.
INTRODUCTION
In this paper, with the aid 0 f a
Navier-Stokes ~ N-S) analysis method we
explore the potential of tailored blade
633
sections instead of stan(lar(1 NACA sections
for optimization of propeller performance.
Propeller designers normally use sections
with NACA 16 or NACA 66 thickness dis-
tributions and an a 0.S meanline. Due to
recent improvement in computational capa-
bility, it is now feasible to shape the section
to achieve a specific design goal, whether it
. ~ ~ ~ · · · · · -
~e maximizing e~nclency, mlnlmlzlng cavl-
tation, or boundary layer control. Also, for
some applications, it is desirable to maxim-
ize the section thickness without degrading
the propeller performance by massive flow
separation. The motivation for this N-S
analysis is due to the experimental results
of Gershfeld et al. A, and Huang et al.
{2i, which have shown that the pressure
spectra on the trailing edge are related to
the turbulent flow characteristics in the
near-wake region. The turbulent flow data
in Ref. t2] are used for validation of the
N-S analysis. This paper presents the vali-
dation results, and the calculated flow
characteristics for a new section developed
with the aid of the N-S analysis.
The mean momentum balance for
viscous flow at high Reynolds number
yields the time-averaged N-S equations.
The full N-S formulation is used here as
separate(1 flow is analyzecl. There are two
fundamental difficulties in using the N-S
equations to predict the flows: 1 ) numerical
instability clue to the convection terms, anti
2) mo(lelling of turbulence. The instability
problem has been attacked by various
numerical techniques such as 1 ) central
differencing with artificial damping ~ 34, and
2) upwind (1ifferencing with Total
OCR for page 634
Variational D iminishing ~ TVD ~ schemes
A. These techniques, however, only
address the mean flow. The nonlinearity of
the fluctuating flow yields turbulence,
which is a more challenging problem.
Little progress has been made in the
development of a general theory for com-
plex turbulent flows TV. Most of the funda-
mental understanding of turbulent flow has
been acquired through experimentation,
and just recently through direct numerical
simulation. In practice, turbulence models
have evolved from the simple m~xing-
length models, to the more physically real-
istic models such as Mean Vorticity and
Covariance ~ 6] . Most turbulence models
are based on the eddy-viscosity concept
which, although not very rigorous, has
been widely used since it was proposed.
The turbulence models used in this paper
are baser! on this approach.
OUTLINE OF ANALYTICAL METHOD
The objective is to maximize the
thickness of a 2-D lifting foil and to reduce
trailing edge turbulent kinetic energy
without incurring significant flow separa-
tion. This is achieved by control of the tur-
bulent boundary layer and wake charac-
teristics through careful shaping of the 2-D
section. The N-S analysis is used in the last
stage of a foil design process to calculate
the turbulent flow characteristics. The
design parameters are the turbulent kinetic
energy, and the turbulent shear stress.
There are two steps in obtaining a N- Rig I
S so lutio n: 1 ) geo m etry preparatio n, inclu(l
ing grid generation, and 2) flow calculation.
For this study, the grid generation is basest
on the work of Coleman Hi, which uses
partial differential equations to define a
body-fitted grid. Here, a multi-zone grid is
used for better control of the grid struc
ture, which is especially useful when com
bined with a multi-zone flow code such as
the D avid Taylor Navier-Stokes (D TNS)
code developed by Gorski Gil. For a high
Reynolds number flow ~ greater than io6)
the first grid point should be as close as
in- 5 chord length away from the body,
approximately y+ of 5, so that the sub-layer
can be reso Ived. Fig. ~ sh o ws a 3- z o n e grid
used for computing the flow over a foil
with Reynolds number of ~ x io6. The flow
634
calculation steps are described in the fol-
lowing sections.
Using the idea of artificial compressi-
bility developed by Chorin [9] allows the
N-S equations for an incompressible fluid,
in cartesian coordinates, to be written in
the following conservative form:
in + 0(f~+9~) + (~3+g2) ° (1)
where the subset
~q oft ~f2 _O (2)
constitutes the corresponding inviscid flow
equations. For 2-D flow, the dependent
variable q and the inviscid fluxes fir and f2
are given by:
q [I/], /} j:2+p~, /2-t UV ~
V US Lv2+p
where p is pressure, and u and v are the
the x and
y directions, respectively. The term p/,3 is
the pseudocompressibility which should
approach zero as the solution converges.
controls the convergence rate of the
scheme with a value of ~ being used for the
present calculations. The viscous "fluxes"
91 and 92 are given by:
Cartesian velocity components in
~ Bu
0~ ~ 92 -~ By
REV . ~ REV
. .
where Re is the Reynolds number and ,u is
the molecular viscosity. The equations (1)
and ( 2) form a hyperbolic system which
can be marched in time using implicit tech-
niques.
k- ~ Equations
The k- ~ mode} used here is
developed by Launder and Spaiding 0.
The mode! equations already have a time
derivative term and can be written in a
form similar to the N-S equations ( 1).
ark + Bf Jk l+9k 1) + Bt Jk 2+9k 2)
at ox By
(3)
OCR for page 635
where
qk t6 ~ ~ fk} ~6 ) ~ Jk2 = EVEN
1
ski R
e
ok
Ink 0~ 1
SE ~ 9k2=R
8z e
1 P- cRe
S ~ Re C1 kP - C2 k Re
and
ok
ilk By
~ By
. ~
lark (1~ + fitly) ~ Ice ~ (1] + lastly)
where k is turbulent kinetic energy, ~ is
turbulent dissipation, and ,u t is the eddy
viscosity. P represents the production of
kinetic energy and the following form of it
is used here:
P pt(Uy2+Vx2+2nyVx)
Here fk~ and fk2 are convective terms and
Ski and 9k2 are viscous diffusion terms. S is
a source term added to the equations which
moclels the production and dissipation of
turbulent kinetic energy.
The k- ~ model still employs the eddy
viscosity/diffusivity concept as it relates
eddy viscosity to the kinetic energy and dis-
sipation by
k2
~ t-Cp-R e
-This eddy viscosity is then used to create
all effective viscosity (,u +,u`) which
replaces ,u in the N-S equations ( 14. To
implement the above turbulence mode! the
following constants are specified as given in
Ref. ~ 104: ~ k 1.0, ~ ~ = 1.3, Ci-1.44,
c2 i.g2, and c~=o.os.
Because the N-S and k- ~ equations
are similar the same numerical technique
has been used for both sets of equations.
Solution Procedure
The N-S and the k-e equations con-
tain both first derivative convective terms
and second derivative viscous terms. The
viscous terms are numerically well-behaved
terms and central di~erencing is used. An
upwind differenced TVD scheme was used
for differencing the convective part of the
equations. This upwind differenced scheme
gives third-order accuracy without any
artificial dissipation terms being added to
the equations. Details of how this discreti
zation method is applied to the N-S equa
tions for incompressible flows can be found
in Gorski iS].
The equations are solved in an impli
cit coupled manner using approximate fac
torization. The implicit sides of the equa
tions (the sides in which the values at the
grid points are unknown) are discretized
with a firsLorder accurate upwind scheme
for the convective terms. This creates a
iagonally (lominant system which requires
the inversion of block tri-cliagonal matrices.
The implicit sides of the equations are only
firsLorder accurate but the final converged
solution has the higher order of accuracy of
the explicit sides of the equations (the sides
in which the values at the grid points are
known already).
An important quality of any scheme is
its convergence rate. The (liagonal domi
nance of the present method allows large
time steps to be used for fast convergence.
A spatially varying time step was also
implemented but not used in this case.
The solution starts with intial esti
mates for the kinetic energy and dissipation
fields. Here, a calculation with the
; 4y Baldwin-Lomax t Il] turbulence mode} pro
vides the estimates. The N-S equations and
the k- ~ mode} equations are iterated in
pseudo-time until convergence is obtained.
The N-S equations are solved to the wall
with proper no-sTip boundary conditions for
all cases. With the Baldwin-Lomax tur
bulence model, the Van D riest m~xing
length mode! takes care of the near-wall
region. The k- ~ model however needs
approximation as the flow physics in the
near-wall sublayer ( y+< 15) is not well
represented by the standard k- ~ equations.
This DINS code has a novel near-wall cal
culation technique 112] in which the N-S
equations are solved to the wall, but near
wall empirical algebraic equations are used
to calculate the kinetic energy and dissipa
tion in this region.
635
OCR for page 636
RESULTS AND DISCUSSION
The experimental data A] for two 2-
dimensional lifting foils are used as bench-
marks for the N-S analysis. Both foils have
an a 0.8 meanline and NACA 16 Type IT
thickness distribution. These foils are plot
lied in Figs. 2.a and 3.b. The thick foil (also
referred to as Ti) has acIditional thickness
from the midchord to the trailing edge as
compared to the thin foil (also referred to
as TNO). The wedge angle at the trailing
edge of the TNO foil is approximately 20
deg. That angle for the T! foil is about 4S
deg which results in a bevel shape on the
suction side. The geometrical details are
carefully documented in Ref. ~ 2] .
Precision of Validation Data and Accuracy of
Calculations
The data from Ref. [2] include: sur-
face pressure distribution, wake mean velo-
city profiles, and turbulence characteristics
such as Reynolds stresses, and power spec-
tral density. Surface pressures were meas-
ured with a scanning valve system and a
precision pressure transducer. Repeater!
measurements of the streamwise and nor-
mal velocity components yielded precision
within two of the measured free-stream
velocity at any position. An(l, the precision
of the measured turbulence intensities and
Reynolds shear stress was within No of the
maximum measured values of a given wake
profile. Wall shear stress measurement
across the span, and hot-fiIm measurement
across the wake indicated that the mean
flow approximates a 2-dimensional flow
field well. Since the acoustic measurements
of Ref. ~ I~ were also performed with
exactly the same foil models and setup,
the models had to be located with the aft
1/3 foil sticking outside of the test tunnel.
Later pressure measurements in free 2-D
jet agrees! with the earlier measurements at
most positions. The geometrical angle of
attack for both foils was set at 0.68 deg.
D ue to the effect of the 2-D jet
configuration on the foils, the corresponcI-
ing free-field angles of attack were calcu-
lated in Ref. t2] to be -1.01 cleg for foil
TNO, and -~.54 deg for T1 foil. To calcu-
late these angles, iterations were performed
on the free-field lift coefficient using the
free 2-D jet correction formula of Rae and
Pope t13] and the boundary layer program
of Cebeci et al. t 14~ . All experimental
measurements were reported without any
"correction". The above free-field angles of
attack were used only for analytical calcula-
tions to-compare with the data.
For the N-S calculations, several grid
sizes were used to establish computational
accuracy. First of all, the boundaries were
established by preliminary calculations to be
at least 9 to t0 chord lengths upstream and
normal to the foil so that free-stream con-
dition applies. And the downstream boun-
dary was set at 10 chords from the trailing
edge to assume negligible streamwise gra-
dients there. With the wake streamwise grid
fixed at 40 points;, different grid sizes were
use(l: 121x40, 121x60, ISIx60, 211x60, and
241x60. Convergence of the calculation
with respect to the grid was established
when the foil surface pressures changes
within 0.2~o of the free-stream pressure.
Validation Results
The bench-mark cases were simulated
as free-fiel(1 2-D flows with the following
conditions:
- Reynolds number 2.25x io6
- The angles of attack use(1 were the same
as the "corrected angles" use(1 for boun-
dary layer calculations in Ref. t2] (-~.01 deg
for foil TNO, en cl -~.54 deg for foil TI).
- A 3-zone C-gricl was used with 241
points; around the bo(ly and 60 points nor-
mal to the body. The upstream boundary is
14 chord lengths from the leading edge.
Top and bottom boundaries are 10 chords
from the bo(ly. The downstream boundary
is 10 chords from the trailing edge.
- The first grill point normal to body
locatecl at the trailing edge region is about
i.4xi0- 5 chord away which translates to
y+
OCR for page 637
calculated pressure distributions are in rea-
sonable agreement with the experimental
data of Ref. t2], more so for the thin
trailing edge foil than for the thick one.
Note that the predicted loading for foil T!
is higher than the data (see Fig. 2.b), espe-
cially in the aft region. Therefore, the 2-D
jet correction to the angles of attack may
not be adequate. Several different angles of
attack were tried for both foils and the
resulting pressure distributions were not
any better. One of the reasons for the
discrepancy between the calculations and
the data may be clue to the wall effect. The
foil chord length is 0.9144 m (3 fig, and the
walls are about 2.667 chords away from the
foil. And according to Rae Knot Pope t13],
the walI/chorc} ratio should be at least 4 so
that the measured lift coefficient is negligi-
bly different from the free-field value. How
the 2-D jet arrangement affects the foil
surface pressure distribution is unknown.
Also, recall in the previous section that the
foils were sebup to be 2/3 in the tunnel
test section and 1/3 outside. It is probable
that this arrangement changes the pressure
field of the foils more than the usual free
2-D jet arrangement. No rigorous reason
can be found at this time to explain the
above discrepancy. Other than the calcu-
latecI pressure, the calculated flow separa-
tion location for the thick foil agrees well
with the experimental value, which is
around 96~% chord on the suction side.
Figs. 3.a and 3.b show good match
between the computed and measured velo-
city vectors for foil TN0, and foil TI,
respectively. This match is relatively better
than that of the foil surface pressure distri-
bution. A possible explanation is that the
surface pressure is more sensitive than the
boundary layer flow, and the presumed wall
effect is not significant for the boundary
layer development. These velocity vectors
are in the near-wake region, from To to
10~o chord length downstream of the trail-
ing edge. The wake deficit for the thick foil
is larger than that for the thin foil. The
thick foil also shows larger normal velocity
component than the thin foil, which is clue
to the flow separation on the suction side of
the thick foil. This flow separation, even
though very mild, results in a recirculating
region which the N-S code does not calcu
late very well as seen in Fig. 3.b for the
x/C=1.02 station. A possible reason is that
in the turbulence models local isotropy is
assumed, i.e. all three velocity components
contribute equally to the turbulent kinetic
energy. This assumption is not met when
there is flow separation. Further down-
stream in the wake, however, the calcu-
late(1 velocity vectors agree well with the
data.
The N-S calculations of the turbulent
kinetic energy k also match the data rea-
sonably well. Figs. 4.a and 4.b show the
calculated and measured k for the TN0 foil,
and the T! foil, respectively. The data are
actually approximated because the z-
component was not measured en c! only the
x- and y- components were obtained from
Huang et al. in Ref. [2~. According to the
boundary layer data of Klebanoff t16], the
z-component can be assumed to be approx-
imately equal to the average of the x-
component and the y-component. For the
TN0 foil, the N-S calculation tends to
over-predict k on the suction side con-
sistently for all three wake stations. The
pressure side, which looks almost flat, has
better agreement. Therefore, this observa-
tion could signal that the turbulence mo(lels
do not work well with a highly curved wall
or large adverse pressure graclient. For the
thick T1 foil, the miTcl flow separation on
the suction sicle causes large over-
prediction of the (lata for the wake station
closest to the trailing edge, x/C 1.02.
Further downstream in the wake, however,
the calculate(1 k agrees relatively better with
the data. As discussed earlier in the previ-
ous paragraph for the velocity profiles, this
observation could mean that the turbulence
models do not work well even for mild flow
separation. Also, this observation illustrates
the elliptic nature of the N-S simulation,
i.e. errors upstream do not neccessarily pro-
pagate (lownstream anti increase as seen in
the boundary layer simulation in Ref. [24.
The N-S calculations of the Reynolds
shear stress uv also match the data reason-
ably well. Figs. 5.a and 5.b show the caTcu-
late(1 and measured uv for foil TNO, and
foil T1, respectively. The magnitude of uv
is slightly over-predicted on the suction side
for the TN0 foil. But the distribution shape
is well predicte(1 for both foils, i.e. the cal
637
OCR for page 638
culated locations of the two extreme of uv
agree well with the data. This could be
because the modelled uv term is propor-
tional to the local velocity gradient which is
predicted very well. Again, we observe
that the pressure side is predicted better
than the suction sicle because it is almost
flat. Also, the data for foil TI in the closest
wake station x/C-1.02 are over-predicted
due to the mild flow separation on the suc-
tion sicle. Both of these observations corre-
late with the previous ones for the velocity
vectors and the turbulent kinetic energy.
Application of N-S Analysis to Section Design
With this successful validation of the
N-S analysis, we can have confidence in
using such a too! to develop new section
shapes. Here, the design goal is to maxim-
ize the thickness of the section without too
much flow separation. The baseline sec-
tion, from an existing design, has an a-0.S
meanline an] a NACA 16 thickness distri-
bution with thickness of 17.16 To chord en cl
camber of 4.79 ~ chord. The design lift
coefficient is approximately 0.68. For this
study, the chosen Reynolds number is
5xio6 to match the conditions for 1/4-scale
tests of naval propellers. The new section,
shown in Fig. 6, is initially designed with
the conformal mapping technique of
Eppler-Somers t 17~ . The (resign approach
is: I) move the minimum pressure on the
suction side further upstream, 2) start
recovering the pressure with a steep gra-
dient because the boundary layer is still
strong after the minimum pressure point,
and 3) decrease the adverse pressure gra-
dient as the trailing edge is approached to
avoid flow separation. After the initial
design, a thin section is produced which has
the three features stated above. As the
desired pressure distribution is not input to
this Eppler-Somers code, a final design is
not easily obtained at this step. Also, the
conformal mapping technique is based on
the potential flow model and therefore can
not account for the thickness effect accu-
rately. The N-S analysis is used iteratively
to obtain the final design. Two parameters
are used in the iteration with the N-S
analysis: thickness, and angle of attack. For
simplicity, thin airfoil theory is used to cal-
culate the "camber-versus-lif~coefficient"
behavior of the new section. The criterion
for the final design is "no-separation" in
the ~ 4 (leg around the design angle of
attack with the thickness as high as possi-
ble. The ~ 4 deg range is normally the
fluctuation of angle of attack that a pro-
peller section sees in straighLahea`;1 opera-
tion.
After some iterations with the N-S
analysis, the final design is produced, with
the pressure distribution at design angle of
attack (2.5 deg) shown in Fig. 6. The lift
coefficient from this pressure distribution is
approximately 0.6S, the same as the base-
line. The desirable characteristics for the
boundary layer development are presented
in this pressure distribution. Maximum suc-
tion peak is around! 5097O chord on the suc-
tion side; steep pressure recovery follows
immediately, then the gradient becomes
milder to minimize flow separation as the
trailing edge is approached. Also, the pres-
sure si(le distribution is rather flat over
most of the surface; this should reduce the
turbulent kinetic energy. The velocity vec-
tor plot in Fig. 7 shows attached flow on
both the pressure side and suction side at
design angle of attack. This attached flow
field of the new section produces lower tur-
bulent kinetic energy as seen in Fig. 8.a,
and lower Reynolds shear stress as seen in
Fig. S.b when compared to the baseline sec-
tion. This trend is more pronounced for
the suction side than the pressure side. At
this design lift for the baseline foil, the N-S
analysis indicates some flow separation on
the suction side which accounts for the high
turbulence activity. And the pressure side
of the new section has low turbulence
activity because the pressure distribution
there is almost flat over the entire surface.
An undesirable feature of this new section
is the thin trailing edge. This could prove
harmful when structural analysis is per-
formed even though care is taken during
the designing to ensure minimum loading
in that region. More details of the design
process, and the comparison between the
new section and the baseline can be found
in Ref. t 181 .
CONCLUSIONS
In this paper, a N-S analysis is per-
formed on the turbulent boundary layer
and wake flows over lifting surfaces. This
638
OCR for page 639
analysis is performed as bench-mark calcu-
lations for 2 airfoils at high Reynolds
number for which turbulent flow data are
available. Overall agreement between data
and calculations is reasonably good. There
is a better match of the mean velocity than
the turbulence stresses. A possible reason
for this is the inability of the turbulence
mode! to simulate accurately flows with
strong adverse pressure gradient or flow
separation. Since the normal stress data
from Refs. t2,14] show that the streamwise
turbulence intensity uu is significantly larger
(up to a factor of 2) than the transverse
component vv, assumptions of local iso-
tropy should be reconsidered.
A new 2-dimensional airfoil section is
developed by combining a conformal maw
ping technique with an iterative N-S
analysis. Results show that the new section
has better boundary layer characteristics,
for the same lift coefficient, than the base-
line. Since the design goal is to maximize
thickness with minimum flow separation,
this new section is not recommended for
other applications in which high thickness is
not needed. This particular new section
will certainly have poor cavitation perfor-
mance. Nevertheless, this paper illustrates
that N-S analysis is very useful in guiding
2-D section design. The N-S analysis, how-
ever, can only give insight about the mag-
nitude and the spatial distribution of the
mean flow, and Reynolds stresses. The
spectral behavior is entirely unknown. Until
better turbulence models, numerical tech-
niques, and computers become more easily
accessible, the N-S analysis should only be
used for final design fine-tuning or off-
design predictions as done in this case.
From the results, further work is recom-
mended to: I) simulate the wall in the N-S
calculation of the same 2-dimensional foils
to establish the significance of the wall
effect; 2) develop a new section with
thicker trailing edge; 3) develop a series of
new sections with different locations of the
minimum pressure point on the suction
side and experimentally evaluate them in
the same manner as in Refs. tI,2~; and 4)
concentrate on the development of tur-
bulence models that can calculate more
accurately turbulent flows with strong
adverse pressure gradient, and even separa-
tion.
AC~NOWLE:DGMEN I
The authors would like to thank Drs.
Tommy Huang, Pat Purtell, and Yu-Tai
Lee (DTRC) for making available the vali-
dation dicta, and D r. Rod Coleman
(DTRC) for the mesh generation code. The
guidance and support of Dr. Frank Peter-
son (DTRC) for the New Section work is
much appreciated. And the New Section
work is supported by the Office of Naval
Technology under work unit number 1-
1506-060-34 for FY-90.
REVERENCES
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Congress, Cincinnati, Ohio, paper S8-3826-
CP, July 1988.
2. Huang, T.T., I,.P. Purtell and Y.T. Lee,
" Turbulence Characteristics of Trailing-
Edge Flows on Thick and Thin Hydro-
foils, " presented at the 4th Symposium of
Numerical and Physical Aspects of Aero-
ynamics Flows, CSU Long Beach, CA,
Jan 16-19, 1989.
3. Sung, C.-H., "An Explicit Runge-Kutta
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4. Harten, A., "High Resolution Schemes
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6. Bernard, P. S. and Berger B.S., "A
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7. Coleman, R.M., "Tnmesh: An lnterac-
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S. Gorski, Jot, "Solutions of the
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Scheme," lith Int. Conf. on Num.
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9. Chorin, A.~. "A Numerical Method for
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10. Launder, B.E. and Spalding, D.B.,
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269-89, 1974.
11. Baldwin, B.S. and Lomax, H., "Thin
Layer Approximation and Algebraic Mode!
for Separated Turbulent Flows, " AlAA
16th Aerospace Sciences Meeting, Hunts-
ville, Al., Jan. 1978.
12. Gorski, J.~., "A New Near-Wall For-
mulation for the k-e Equations of Tur-
buTence, " ALLA Paper 86-0556, 1986.
14. Cebeci, T., R.W. Clark, K.C. Chang,
N.D. Halsey and K. Lee, "Airfoils with
Separation and the Resulting Wakes," Jou.
Fluid Mech., Vol. 163, 1986, pp.323-47.
15. Mehta, U., K.C. Chang, and T. Cebeci,
"A Comparison of Interactive Boundary-
Layer and Thin Layer Navier-Stokes Pro-
ce(lures, " Num. and Phys. Aspects of
Aero. Flows T1:T, Chapter 11, p. 198, 1985.
3, pp. 16. Klebanoff, P., " Characteristics of
Turbulence in a Bounciary-Layer with Zero
Pressure Gradient, " NACA TN 3178,
1954.
17. Eppler, R. and Somers, D.M., allow
Speed Airfoil D esign and Analysis, "
Advanced Technology Airfoil Research
Conference, Langley Research Center,
NASA, Hamton, Va., Mar 1978.
18. Nguyen, P.N., "A Design Method for
Boun(lary-l,ayer Control of 2-D Lifting Sur
13. Rae, W.H., Jr., and A. Pope, Low- faces, " DTRC/SHD 1262-04, 1990 (in
Speed Wind Tunnel Testing, John Wiley & review).
Sons, New York, 1984, p.361.
640
OCR for page 641
0.6
0~4
0.2
0.0
-0.2
-0.4
-0.8
[~
_
I
r '=
_~
-0.1 0.1 0.3 0.5 0.7
X/C
Figure 1. C-type grid with 3-zone structure for a lifting surface
0.6
l
n 0.4
0.2
0~0
-0.2
0 -Cp data (Re=2.25E6, - 1.01 deg.)
-- - - -Cp calculation by DTNS
section geometry
A\ ,°'\6
opt\
P~ =~\
I I I I ~
0.0 0.2 0.4 0.6
X/C
0.8 1.0
Figure 2.a) Pressure distribution for the thin
section (data from Ref. 2)
0 -Cp data (Re=2.25E6, - 1.54 deg.)
-Cp calculation by DINS
section geometry
0.6
0.4
0.2
0.0
-0.2
o ,p _ of-o_
~ o >_ ~-o_ ~
~ o'er
9
I I I I I
0.0 0.2 0.4 0.6 0.8 1.0
X/C
\
Figure 2.b) Pressure distribution for the thick
section (data from Ref. 2)
64~
OCR for page 642
data from Ref. 2
calculation
data from Ref. 2
calculation
0.02
0.00
-0.02
-0.04
-0.06
_
~ _
_
,
_
r~ __
_ __.~,
~ _ ~
~_
_
a.
_ -
_ _ _
-_~
_
e
~ _ . -
_ _ _ _ _
__..~.
0.04
0.02
0.00
1.00 1.04 1.08 1 .12 1 .16
x/C
Figure 3.a) Velocity vector data and DTNS
calculations for the thin section
o k data at x/C=1.02
o x/C= 1 .04
x/C=1.10
-- -- calculation by DTNS
I S
; o~ ~
~Oo`' [o~ ta&~)
,,pa~ -; ~
I ~ I ~ I
1 0 1 0 1
100 k/U 2
0.05
, ~
_
-0.05
0.03
0.01
-0.01
-0.03
Figure 4.a) Profiles of turbulent kinetic energy
for the thin section (data from Ref. 2)
0.05
0.03
0.01
-0.01
-0.03
-0.05 0 5
I ~ I ~ I ~ I
8 ~ ~
t} ~ ~
~ -' ~'}
,_' , ~ , ~ ,
0 5 0 5 0 5
1000 uv/U.2
-0.02
-0.04
1.00 1.04 1.08
x/C
1.12 1.16
Figure 3.b) Velocity vector data arid DTNS
calculations for the thick section
o k data at x/C=1.02
o
0.05
0.03
0.0'1
-0.01
-0.03
-0.05
o
x/C= 1 .05
x/C=l .10
-- -- calculation by DTNS
1 ~ 1 ~ 1
]\ ~ ~ ~_
~1
i. I ~ I ~ I
1 0 1 0 1
100 k/U 2
Figure 4.b) Profiles of turbulent kinetic energy
for the thick section (data from ReT. 2)
0.05
0.03
0.01
-0.01
-0.03
-0.05
' ~ ' ~ ~ t ~
b ~ ~
~W
o ~
~ g ~ 8 ~ .
0 5 0 5 0 5 0
1000 uv/UO2
s
~'
Figure 5.a) Profiles of Reynolds shear stress forFigure 5.b) Profiles of ReYnolds shear stress for
the thin section (legend in Fig. 4.a)the thick section (legend in Fig. 4.b)
642
OCR for page 643
DTNS calc (Re=SE6,2.5 deg.)
section geometry
0.e
my, 0.4
l
._ 0.2
o
I I ~
,' \
r
-0.2 ~I I I I
0.0 0.2 0.4 0.6 0.8 1.0
X/C
Figure 6. Pressure distribution for the new
section
0.07
0.500
0.250
0.05
0.03
0.01
-0.01
o-new section calc, x/C=1.02
a-x/C= 1 .05
x/C= 1 .09
baseline section calc
I I I ~ I
>~
-0.03 .o 1.5 0.0 1.5 0.0 1.5 a
100 k/U 2
Figure 8.a) Profiles of turbulent kinetic energy
for the baseline and the new sectRon
643
0.000
-0.250
_- = = ~ _
o.oo 0.25 0.50 0.75 1.00
x/C
Figure 7. \/elocl~ vector calculations for the new
section
0.07
0.05
0.03
0.01
-0.01
-0.03
0 new section calc, x/C=1.02
0 x/C= 1 .05
&- x/C= 1 .09
-- -- baseline section calc
it'
5 0 5 0 5 0 5
1000 w/U 2
Figure 8.b) Profiles of Reynolds shear stress
for the baseline and the new section
OCR for page 644
DISCUSSION
Wolfgang Faller
Sulzer Escher Wyss, Germany
For your comparison between experiment and 2-D N-S calculation,
the computational domain used in infinite. Would a closer modelling
of the actual experimental configuration improve the correlation, e.g.,
Cp distribution and B.L. development?
AUTHORS' REPLY
Modelling of the foil and wall configuration would possibly lead to
better match of the bench-mark calculations to the hydrodynamics
data. Plan is underway to implement a grid structure necessary for
this study.
DISCUSSION
All H. Nayfeh
Virginia Polytechnic Institute and State University, USA
1. How sensitive is the pressure distribution of the new section to
variation in the angle of attack? 2. How sensitive is the designed
shape to the turbulence model? 3. How much is the drag reduced by
the new section?
AUTHORS' REPLY
1. The new section does not have flow separation in the + 4 deg.
range around the design angle of attack. 2. Preliminary study
indicates that the flow solution converges to approximately the same
pressure distribution for both the Baldwin-Lomax and the k-e
turbulence models. 3. Drag is not computed in this study as the
current project focuses on the turbulence activity. The drag of the
new section is included in the plan for future investigation.
DISCUSSION
Philippe Genoux
Bassin d'Essais des Cartnes, France
1. Is your model able to take in account turbulent levels of the
incoming flow? 2. What would become of the turbulent energy
levels when the profile is placed in incoming flow?
AUTHORS' REPLY
1. The code currently does not have a model for the free-stream
turbulence. 2. When placed in flow with free-stream turbulence, the
turbulent kinetic energy level of the new section would likely increase
(as compared to incoming flow without turbulence). The
redistribution and the magnitude of the increase in turbulence energy
would need to be calculated with a proper turbulence model.
DISCUSSION
Hyoung-Tae Kim
The University of Iowa, USA (Korea)
First, I want know is there any reason not to show the distribution of
the shear stress on the surface of the foil section? Secondly, I want
to point out that the Low turbulence activity simulated in the
computation doesn't necessarily mean the new foil section has a lower
drag than the baseline section.
AUTHORS' REPLY
The study focuses on the turbulence activity in the near-wake region
of lifting surfaces. The drag itself is, however, included in the plan
for future investigation. The authors agree with the discusser on his
second point. No claim is made about drag reduction in the paper.
The resistance of the new section to flow separation only provides for
higher lift at the same angle of attack as the baseline section. And
the low turbulence activity provides for lower fluctuating pressure on
the trailing edge of the section.
644
Representative terms from entire chapter:
turbulent kinetic
