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Weber2000.pdf
Journal of Fluids and Structures (2000) 14, 779}798
doi:10.1006/j#s.2000.0299, available online at http://www.idealibrary.com on
COMPUTATIONAL SIMULATION OF DYNAMIC STALL
ON THE NLR 7301 AIRFOIL
S. WEBERs
AND
M. F. PLATZER
Department of Aeronautics and Astronautics, Naval Postgraduate School
Monterey, CA 93943-5000, U.S.A.
(Received 28 February 1999, and in "nal form 20 March 2000)
The dynamic stall behavior of the supercritical NLR 7301 airfoil is analyzed with a 2-D
thin-layer Navier}Stokes code. The code solves the compressible Reynolds-averaged NavierStokes equations with an upwind biased numerical scheme in combination with the
Baldwin}Lomax or the Baldwin}Barth turbulence models. The e!ect of boundary layer
transition is incorporated using the transition length model of Gostelow et al. The transition
onset location is determined with Michel's formula or it can be speci"ed as an input parameter.
The two turbulence models yield signi"cantly di!erent steady-state lift coe$cients at incidences
greater than 83. The use of the one-equation Baldwin}Barth model together with the Gostelow
transition model is found to give substantially better agreement with the experimental data of
McCroskey et al. than the Baldwin}Lomax model. Also, the unsteady computations are
strongly a!ected by the choice of the turbulence model. The Baldwin}Barth model predicts
the lift hysteresis loops consistently better than the algebraic turbulence model. However, the
one-equation model improves the prediction of the moment hysteresis loops only for one test
case.
( 2000 Academic Press
1. INTRODUCTION
DYNAMIC STALL LIMITS THE HELICOPTER FLIGHT ENVELOPE and methods are required to predict
its occurrence. Dynamic stall also occurs on wind turbines, propellers, and turbomachinery
blades. At present, dynamic stall prediction method used by the helicopter, turbomachinery,
aircraft, and wind turbine industries are largely based on semi-empirical approaches. The
very complex dynamic stall phenomena which have recently been reviewed by Carr
& Chandrasekhara (1996) greatly impede the development of nonempirical prediction
methods. Nevertheless, recent progress in the numerical analysis of dynamic stall phenomena, as summarized by Ekaterinaris & Platzer (1997), suggests the possibility of
developing prediction methods which are based on the solution of the viscous #ow
equations.
The present paper is a continuation of systematic studies begun by Ekaterinaris et al.
(1995), Ekaterinaris & Platzer (1996) and Sanz & Platzer (1998) to explore the feasibility of
computing stall onset and dynamic stall using the Navier}Stokes equations in combination
with advanced turbulence and transition models. Encouraging results were obtained for the
NACA 0012 and Sikorsky SC 1095 airfoils. It is the objective of the present paper to report
similar computations for the NLR 7301 airfoil. This airfoil was chosen for the study because
detailed experimental data are available from McCroskey et al. (1982) and Schewe
& Deyhle (1996).
sDaimlerChrysler Aerospace, MTU MuK nchen, Munich, Germany.
0889}9746/00/080779#20 $35.00/0
( 2000 Academic Press
780
S. WEBER AND M. F. PLATZER
2. NUMERICAL METHOD
The unsteady, nonlinear and compressible Navier}Stokes algorithm solves the strong
conservation-law form of the two-dimensional, thin-layer Navier}Stokes equations in
a curvilinear coordinate system (m, g). The governing equations are given in vector form by
L Q) #L E) #L F) "Re~1L S) ,
t
m
g
g
)
where Q is the vector of conservative variables
(1)
o
GH
1 ou
Q) "
,
J ov
(2)
e
E) and F) are the inviscid #ux vectors
G
o;
o<
H G
H
ou;#m p
ou<#g p
1
x
x
, FK "
,
(3)
J
ov;#m p
ov<#g p
y
y
(e#p);!m p
(e#p)<!g p
t
t
and S) is the thin-layer approximation of the viscous #uxes in the g direction (normal to the
airfoil surface),
1
E) "
J
1
S) "
J
G
0
H
km u #(k/3)m g
1 g
2 x ,
km v #(k/3)m g
1 g
2 y
km m #(k/3)m m
1 3
2 4
(4)
in which
m "g2#g2 ,
y
x
1
m "g u #g v ,
2
x g
y g
m "(u2#v2)/2#(i!1)~1Pr~1L (a2),
3
g
m "g u#g v.
4
x
y
The terms ; and < are the contravariant velocity components given by
(5)
;"um #vm #m , <"ug #vg #g ,
x
y
t
x
y
t
and J is the metric Jacobian, where
(9)
(6)
(7)
(8)
J~1"x y !x y .
(10)
m g
g m
Pressure is related to the other variables through the equation of state for an ideal gas:
p"(i!1)[e!o(u2#v2)/2].
(11)
Equations (1}11) are nondimensionalized using c as the reference length, a as the reference
=
velocity, c/a as the reference time, o as the reference density and o a2 as the reference
=
=
= =
energy.
For Euler solutions, the viscous terms on the right-hand side are set to zero, and #ow
tangency boundary conditions are applied at the surface. For Navier}Stokes solutions the
SIMULATION OF DYNAMIC STALL ON THE NLR 7301 AIRFOIL
781
no-slip condition is applied. Density and pressure are extrapolated to the surface for both
Euler and Navier}Stokes solutions. For unsteady motions, the #ow-tangency and no-slip
conditions are modi"ed to include the local motion of the surface which also contributes to
the pressure on the surface. Therefore, the momentum equation normal to the surface
(g direction) is solved to predict the pressure for a viscous #ow more accurately:
C G H
D
x5 D
1
8!-- ' $g#L pD $m ' $g ,
oL
(12)
L pD "!
t
g 8!-g 8!-+ 2g
y5 D
8!-where x5 D
and y5 D
are the components of the blade velocity. Furthermore, it is assumed
8!-8!-that the grid is orthogonal at the surface, and therefore $m ' $g"0. If the blade does not
move, the normal pressure gradient is equal to zero according to boundary layer theory.
The time integration is performed using the third-order upwind biased, factorized,
iterative, implicit scheme of Rai & Chakravarthy (1988) tested by Ekaterinaris & Platzer
(1996) and given by
K )]p
[I#h (+ AK ` #D AK ~ )]p][I#h (+ BK ` #D BK ~ !Re~1d M
g g i,k
g i,k
m i,k
g i,k
m m i,k
)
!EK p
](QK p`1!QK p )"![(QK p !QK n )#h (EK p
[email protected],k
i,k
m i`[email protected],k
i,k
i,k
i,k
#h (FK p
!FK p
)!Re~1h (SK p
!SK p
)] .
g i,k`[email protected]
i,[email protected]
g i,k`[email protected]
i,[email protected]
(13)
In equation (13), h "Dt/Dm, etc., AK $"LEK /LQK , etc., are the #ux Jacobian matrices and +,
m
D and d are the forward, backward and central di!erence operators, respectively. The
are numerical #uxes. The superscript ( )n denotes
quantities EK
, FK
and SK
i`[email protected],k i,k`[email protected]
i,k`[email protected]
the time step, and the superscript ( )p refers to Newton subiterations within each time step.
The inviscid #uxes, EK and FK , are evaluated using Osher's third-order upwinding scheme
(Chakravarthy & Osher 1985). For the linearization of the left-hand side of equation (13) the
#ux Jacobian matrices, A and B, are evaluated by the Steger}Warming #ux-vector splitting
(Steger & Warming 1981). The viscous #uxes are computed with second-order central
di!erences. Time accuracy is improved by performing Newton subiterations to convergence
at each step. These subiterations minimize the linearization and factorization errors and
help drive the left-hand side of equation (13) to zero at each time step. The present authors
found that larger CFL numbers (i.e. a larger time step) could be used if the number of
Newton iterations was increased. The optimum seemed to depend on the grid topology and
#ow conditions, but the best computational performance seemed to occur with 2}3
sub-iterations for Navier}Stokes simulations. The Navier}Stokes solver has been tested
extensively in a variety of unsteady subsonic and transonic studies, such as Clarkson et al.
(1993), Grohsmeyer et al. (1991), Ekaterinaris et al. (1994), and Ekaterinaris & Platzer
(1996).
The turbulence modeling is based either on the standard algebraic model of Baldwin
& Lomax (1978) or the one equation models of Baldwin & Barth (1990). The eddy viscosity
obtained from the models is used for the computation of the fully turbulent region and
for the evaluation of an e!ective eddy viscosity in the transitional #ow region as is
explained next.
3. TRANSITION MODELING
The transition modeling for all turbulence models follows Sanz & Platzer (1998). In this
publication, the model of Gostelow et al. (1996) was introduced, which permits the
782
S. WEBER AND M. F. PLATZER
calculation of the transition length as a function of pressure gradient and free-stream
turbulence level. This method continuously adjusts the turbulent spot growth in response to
changes of the local pressure gradient.
The intermittency function in the transitional region is given by
C P
A BP
D
xi p dx xi
tan e dx ,
(14)
tan e ;
xt
xt
where the correlations for the variation of the spot propagation parameter p and the spot
spreading half-angle e as functions of the pressure gradient parameter j are
h
22)14
e"4#
(15)
0)79#2)72 exp(47)63j )
h
and
c(x)"1!exp !n
0)37
p"0)03#
.
(16)
0)48#3)0 exp(52)9j )
h
Here j "(h2/l)/(d;/dx) with the boundary-layer momentum thickness, h, and the
h
outer-edge velocity, ;. The spot generation rate, n, is inferred from the dimensionless
breakdown-rate parameter, N, where
and
N"npj3t /l,
h
N"0)86~3 exp[2)134j t ln(q )!59)23j t !0)564 ln(q )], for j t40
h
t
h
t
h
(17)
(18)
(19)
N"N(j "0)]exp(!10Jj t), for j t '0 ,
h
h
h
and where q denotes the free-stream turbulence.
t
The spot-propagation rate and the spot spreading half-angle asymptotically approach
a maximum value for high negative values of j , but n is allowed to increase to in"nity for
h
high negative values of j t , where j t is the pressure gradient at the transition onset location,
h
h
x . The value of the intermittency parameter, c(x), is zero for x4x , and increases
t
t
downstream from the transition point, asymptotically to a maximum value of unity, which
corresponds to fully turbulent #ow. An e!ective eddy-viscosity for the transitional region is
obtained by scaling the turbulent eddy-viscosity computed by c(x), i.e., k "c(x)k .
53!/4
563"
Sanz & Platzer (1998) have used the Gostelow model, originally developed for attached
#ow, for the prediction of laminar separation bubbles by using the spot-generation rate as
a second adjustable parameter along with the location of transition onset. They investigated
the in#uence of the spot-generation rate on the separation bubble by either limiting the
breakdown-rate parameter to 1)0, which forces instantaneous transition, or by assuming the
value for a zero pressure-gradient. In the present study, a break-down-rate parameter of 1)0
was chosen and the transition onset was either predicted by the Michel criterion (Cebeci
& Bradshaw 1977) or by speci"cation as an input parameter.
4. RESULTS AND DISCUSSION
All steady-state and unsteady computations for the NLR 7301 airfoil were performed on
a C-type Navier}Stokes grid with 221]91 grid points. The initial wall spacing used was
8]10~6 resulting in a y`(2 for the chosen Mach number of 0)3 and the Reynolds number
of 4]106. A total of 40 grid points were used in the wake and the far"eld boundary
extended 20 chord lengths from the surface. The grid is shown in Figure 1.
SIMULATION OF DYNAMIC STALL ON THE NLR 7301 AIRFOIL
783
Figure 1. Navier}Stokes C-type grid (221 x 91) for the NLR 7301 airfoil.
A detailed set of measurements for the steady #ow at di!erent angles of attack, Mach
numbers up to 0)3 and Reynolds numbers up to 4]106 have been performed at the NASA
Ames Research Center by McCroskey et al. (1982). They investigated the static and dynamic
characteristics of seven helicopter blades and a supercritical "xed wing airfoil. The supercritical airfoil was chosen for the present study because it is and will be experimentally
investigated at the DLR Goettingen to determine its dynamic stall behavior.
4.1. FULLY TURBULENT STEADY-STATE COMPUTATIONS
The steady-state computations as well as the measurements were obtained at "xed angles of
attack in a range from !53 up to 203. The Mach number for the experiments was
approximately 0)3 and the Reynolds number 4]106. In Figure 2 the computed steady-state
results are compared with the measured data.
Figure 2 shows the linear results obtained with the potential #ow panel code UPOT
(Teng 1987) and results for viscous computations using the Baldwin}Lomax (BL) and the
Baldwin}Barth (BB) turbulence model. All viscous computations were performed assuming
fully turbulent #ow. The BL- and BB-computed lift coe$cients are in close agreement with
the measured data for a range of the angle of attack of !5 to 83. Although the BL- and
BB-computed lift coe$cients are almost the same up to a"83, simulations with BB show
more #ow separation near the trailing edge on the upper surface than BL. For both models
this separation grows signi"cantly with increasing angle of attack but the #ow stays
attached longer for #ow simulations with BL due to BL computing higher viscosity. The
locations of the separation onset s/c versus the angle of attack for both turbulence models
can be seen in Figure 3. Once the #ow is separated, it does not reattach. For #ow angles of
0 and 53 both turbulence models show no separation. Steady-state solutions in terms of the
pressure coe$cient distributions versus the chord for a"14, 17 and 17)63 are given in
784
S. WEBER AND M. F. PLATZER
Figure 2. Comparison of the computed and measured lift coe$cient: r*r, BB fully turbulent; - - - -, panel code;
d*d, BL fully turbulent; h*h, experimental data by McCroskey et al. (1982).
Figures 4}6. These plots show that a higher suction peak is predicted by BL than by BB.
Furthermore, the BB model yields a much more constant static pressure in the separation
region, as can be seen from the pressure plateaus in front of the trailing edge. Both models
predict a nonlinear increase of the lift coe$cient for angles of attack higher than 83. The lift
coe$cients calculated with the BB model are in good agreement with the measurements up
to 143 while the BL model overpredicts the lift for angles of attack in excess of 83 due to the
higher suction peak and smaller #ow separation.
The BL model predicts the highest lift coe$cient at 173 angle of attack in agreement with
the experiments, but overpredicts the lift coe$cient by 11%. BB predicts the highest lift at
an angle of attack of 163. For angles of attack greater than 143, the BB model yields a lower
lift coe$cient than the measurements. It underpredicts the lift by 8%. For angles of attack
greater than 173 the convergence behavior of both models is quite di!erent. Although the
#ow, after reaching the stall angle of approximately 173, will no longer be steady and
therefore the measured and the computed data have to be regarded as averaged values,
the BL model converges to a constant lift coe$cient up to an angle of attack of 17)63. In
Figure 6 the BL- and BB-computed pressure distributions are compared with the experimental data.
It can be seen that the BB-computed #ow on the pressure side is in slightly better
agreement with the experiment. On the suction side, except near the suction peak, the BL
computed pressures agree with the experiment up to 70% chord. Further downstream the
pressure is too low. The BB computed suction peak almost yields the measured peak, but
the pressure coe$cient between 10 and 70% chord is underpredicted which explains the
underprediction of the lift coe$cient (Figure 2). The development of the boundary layer
along the airfoil for 17)63 is given in Figure 7. For #ow angles exceeding 17)63 the computed
lift coe$cients are no longer steady. This can be seen in Figure 8 where the BL- and the
BB-computed lift histories are plotted for the case of 203 incidence angle. It is shown that
the BL-computed lift is periodic but reaches values which are considerably higher than the
SIMULATION OF DYNAMIC STALL ON THE NLR 7301 AIRFOIL
785
Figure 3. Comparison of the computed separation onset location: d*d, BL fully turbulent; r*r, BB fully
turbulent.
Figure 4. Comparison of the computed pressure coe$cient for a"143: - - - -, BL fully turbulent; *, BB fully
turbulent.
786
S. WEBER AND M. F. PLATZER
Figure 5. Comparison of the computed pressure coe$cient for a"173: - - - -, BL fully turbulent; *, BB fully
turbulent.
Figure 6. Comparison of the compute and measured pressure coe$cient for a"17)63: - - - -, BL fully turbulent;
*, BB fully turbulent; h, experimental data by McCroskey et al. (1982); } } } } , BB with transition.
SIMULATION OF DYNAMIC STALL ON THE NLR 7301 AIRFOIL
787
Figure 7. Comparison of the computed boundary layer pro"les on the upper surface for a"17.63: - - - -, BL
fully turbulent; *, BB fully turbulent; } } } } , BB with transition.
Figure 8. Comparison of the computed time development of the lift coe$cient for a"203: ......., BL fully turbulent;
*, BB fully turbulent.
788
S. WEBER AND M. F. PLATZER
measured value. The BB-computed lift, on the other hand, approaches values considerably
closer to the measured value, but then drops dramatically. Figures 9 and 10 depict typical
time-instantaneous #ow "elds computed with the BL and the BB models. The BB model is
seen to produce massively separated #ow. It should be noted that all the computations were
performed time-accurately.
4.2. STEADY-STATE COMPUTATIONS INCLUDING TRANSITION
The fully turbulent #ow computations described in the previous section show that the BB
model is superior to the BL model. However, the sensitivity to transitional #ow e!ects is an
important aspect which needs to be studied. Therefore, the computations were repeated
with the BB model by assuming that transition starts at x/c"0)02 for angles of attack
between 14 and 17)63. The numerical results can be seen in Figure 11.
Figure 11 shows that transition becomes important as the incidence angle is increased.
Incorporation of transition substantially improves the agreement with the measured lift
values in the high angle of attack range. This improvement can also be seen in Figure 6.
On the pressure side, the computations with and without transition di!er only slightly,
but inclusion of transition yields better agreement with the experimental data on
the suction side. Only the suction peak is overpredicted. The development of the
boundary layer including transition is given in Figure 7. The separation onset on the
suction side was moved downstream from 48% chord without transition to 60% chord with
transition.
These numerical results for the NLR 7301 airfoil con"rm the "ndings of Ekaterinaris
& Platzer (1996) and Sanz & Platzer (1998) for the NACA 0012 and the Sikorsky SC 1095
airfoils which showed that the numerical prediction of the stall onset is signi"cantly
improved by the incorporation of boundary layer transition.
4.3. DYNAMIC STALL COMPUTATIONS
Unsteady computations were performed for the NLR 7301 airfoil oscillating in the pitching
mode described by the equation a(t)"a #a sin ut. As in the experiment, the pitch axis
0
1
was located at the quarter chord point. Two test cases were considered and compared with
the unsteady experimental data.
The "rst test case had a mean angle of a "11)963 and an amplitude of a "2)03. The
0
1
reduced frequency was k"0)4, the Mach number 0)293 and the Reynolds number
3)72]106. All unsteady numerical computations were performed time-accurately after
calculating a steady-state solution for the mean angle. The unsteady solutions were assumed
to be converged if consecutive hysteresis loops did not change. The computations were
converged after 3}4 periods.
The experiment was performed with free transition. Figure 12 shows the computed
hysteresis loops using the BL model with and without transition. The prediction of the
unsteady onset location of transition was obtained with the Michel criterion. It is apparent
that the BL model fails to predict the measured hysteresis loop. Repeating the calculations
with the BB model, with and without transition, yields much better agreement with the
measured hysteresis loop, (Figure 13). Because the unsteady onset locations of transition
were not measured, the Michel criterion was used for the calculations.
Additionally, the BB-computed unsteady pressure coe$cient distributions on the suction
side including transition are compared with the measured pressure distributions in
Figure 14 for 20 di!erent angles of attack during the oscillation of the airfoil. It is seen that
the computed pressure distributions agree well with the measured distributions.
SIMULATION OF DYNAMIC STALL ON THE NLR 7301 AIRFOIL
Figure 9. Baldwin}Lomax computed instantaneous Mach number distribution for a"203.
Figure 10. Baldwin}Barth computed instantaneous Mach number distribution for a"203.
789
790
S. WEBER AND M. F. PLATZER
Figure 11. Comparison of the computed and measured lift coe$cient with experimental data by McCroskey
et al. (1982): e*e, BB fully turbulent; m, BB with transition; h*h, experimental data; s*s, BL fully turbulent.
Figure 12. Comparison of the computed and measured unsteady lift coe$cient for a(t)"11)963#2)03 sin ut:
d*d, unsteady experimental data by McCroskey et al. (1982); *, BL fully turbulent; - - - -, BL with transition.
SIMULATION OF DYNAMIC STALL ON THE NLR 7301 AIRFOIL
791
Figure 13. Comparison of the computed and measured unsteady lift coe$cient for a(t)"11)963#2)03 sin ut
data by McCroskey et al. (1982): d*d, unsteady experimental data by McCroskey et al. (1982); *, BB fully
turbulent; - - - -, BB with transition.
Figure 14. Comparison of the computed and measured unsteady pressure coe$cient for a(t)"
11)963#2)03 sin ut including transition with experimental data by McCroskey et al. (1982).
792
S. WEBER AND M. F. PLATZER
From the stability point of view, one is interested whether the oscillation will be damped
or excited. This test con"guration resulted in a damped oscillation which can be seen from
the moment coe$cient versus the angle of attack distribution running counter clockwise.
The numerical results including transition in comparison with the measured data are given
in Figure 15. Both turbulence models predicted a damped oscillation and again the BB
result is in much better agreement with the experiment then the BL result.
A second series of computations was performed for the case a "9)983 and a "4)93. The
0
1
reduced frequency was k"0)098, the Mach number 0)299 and the Reynolds number
3)79]106. In Figures 16 and 17 one can see the numerically predicted lift coe$cient
distributions including transition for both turbulence models in comparison with the
experimental data. The unsteady onset locations of transition for both calculations were
predicted with Michel's criterion because of the unavailability of relevant experimental
Figure 15. Comparison of the computed and measured unsteady moment coe$cient for a(t)"
11)963#2)03 sin ut including transition: d*d, unsteady experimental data by McCroskey et al. (1982); - - - -,
unsteady BL with transition; *, unsteady BB with transition.
Figure 16. Comparison of the computed and measured unsteady lift coe$cient for a(t)"9)983#4)93 sin ut:
d*d, unsteady experimental data by McCroskey et al. (1982); * unsteady BL with transition.
SIMULATION OF DYNAMIC STALL ON THE NLR 7301 AIRFOIL
793
Figure 17. Comparison of the computed and measured unsteady lift coe$cient for a(t)"9)983#4)93 sin ut:
d*d, unsteady experimental data by McCroskey et al. (1982); * unsteady BB with transition.
transition data. Again, the BB model predicts the measured hysteresis loop much
better than the BL model. This is con"rmed by the comparison of the computed and
measured pressure distributions shown in Figure 18. The experimental data plotted in
Figures 16}19 were obtained with free transition. The corresponding moment coe$cient
distribution in comparison with the experimental data are shown in Figure 19. The
oscillation in the second test case investigated was found to be damped in the experiment which was predicted by both turbulence models. All loops of Figure 19 are
counter clockwise. In contrast, to the computed lift coe$cient distributions, the prediction of the moment coe$cient distribution with the BB model is not better then the
prediction with the BL model. Both models failed to compute the experimental loop which,
as can be seen by the small integrated area of the measured loop, was close to an unstable
oscillation.
Dramatically di!erent hysteresis loops were measured if transition was forced. This is
shown in Figures 20 and 21. Unfortunately, no information is available on the precise
location of the transition strip. The computations assuming fully turbulent #ow are shown
in Figures 18, 20 and 21. As the fully turbulent approach did not predict the measured lift
during the downstroke, it therefore failed to produce an unstable oscillation. In Figure 21
the right part of the experimental hysteresis is clockwise during the downstroke of the airfoil
while the BB-computed loop is counterclockwise during the whole oscillation. Clearly, the
inclusion of transition has a negligible in#uence if transition onset is predicted from
Michel's criterion.
Therefore, it is not surprising that such a computation fails to reproduce the experimental
loop of Figures 20 and 21. It is apparent from the measured pressure distributions shown in
Figure 22 that severe #ow separation occurs at the start of the downstroke. One would
expect that this #ow behavior is caused by the sudden burst of a laminar separation bubble.
Van Dyken et al. (1996) were able to compute the formation and bursting of the bubble on
a NACA 0012 airfoil. Therefore, a computation was performed varying the transition onset
794
S. WEBER AND M. F. PLATZER
Figure 18. Comparison of the computed and measured unsteady pressure coe$cient for a(t)"
9)983#4)93 sin ut; experimental data by McCroskey et al. (1982).
Figure 19. Comparison of the computed and measured unsteady moment coe$cient for a(t)"9)983#4)9 sin ut
including transition: d*d, unsteady experimental data by McCroskey et al. (1982); - - - -, unsteady BL with
transition; *, unsteady BB with transition.
SIMULATION OF DYNAMIC STALL ON THE NLR 7301 AIRFOIL
795
Figure 20. Comparison of the computed and measured unsteady lift coe$cient for a(t)"9)983#4)93 sin ut:
d*d, unsteady experimental data by McCroskey et al. (1982); *, unsteady BB fully turbulent.
Figure 21. Comparison of the computed and measured unsteady moment coe$cient for a(t)"
9)983#4)93 sin ut: d*d, unsteady experimental data by McCroskey et al. (1982); * unsteady BB fully turbulent.
796
S. WEBER AND M. F. PLATZER
Figure 22. Measured unsteady pressure coe$cient for a(t)"9)983#4)93 sin ut; experimental data by
McCroskey et al. (1982).
location periodically such that the mean position was at x/c"0)02 and the amplitude was
1% of the mean position.
However, this simple attempt to model the #ow physics was unsuccessful. Clearly, a more
detailed analysis is necessary to clarify the #ow mechanism which produces the measured
loop.
The average computational e!ort on steady and unsteady calculations can be summarized as follows. The steady-state computations were performed on SGI Octane workstations. Typical steady-state time-accurate computations using the Baldwin}Barth
turbulence model including transition needed between 10 000 and 25 000 time steps in the
stall region. The time-accurate unsteady computations were run on a C-90 and a T-932
where between 10 000 and 20 000 time steps per cycle with 3 Newton subiterations for every
time step were necessary.
5. CONCLUSIONS
A 2-D thin-layer Navier}Stokes code was used in conjunction with the Baldwin-Lomax and
Baldwin}Barth turbulence models to predict the steady and dynamic stall behavior of the
supercritical NLR 7301 airfoil. The in#uence of transition was investigated by the incorporation of the transition model of Gostelow et al.
The BB model was found to give consistently better agreement with the experimental
data than the BL model except for the predicted moment coe$cient loop for the second test
case (Figure 19). Furthermore, the incorporation of boundary layer transition yielded
additional improvements, especially for the steady-state analysis, provided that the &&correct'' transition onset location was chosen. Further work is clearly needed to develop
reliable criteria for transition onset under unsteady adverse pressure gradient conditions.
SIMULATION OF DYNAMIC STALL ON THE NLR 7301 AIRFOIL
797
Ultimately, the computations need to be extended to account for 3-D e!ects based on the
full Navier}Stokes equations.
ACKNOWLEDGEMENTS
The "rst author gratefully acknowledges the support of the Deutsche Forschungsgemeinschaft (DFG) for the Post-Doctoral Research stay and the Naval Postgraduate School.
Also, the authors are indebted to Dr J. Ekaterinaris, Senior Research Scientist at NIELSEN
Engineering and Research Inc., Dr W. Sanz at the Technical University Graz, Austria, and
Dr K. Jones at the Naval Postgraduate School for their advice. NAVO provided computing
time on the DoD High Performance Computing Systems.
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798
S. WEBER AND M. F. PLATZER
APPENDIX: NOMENCLATURE
a
=
c
C
l
C
p
e
f
k
M
=
Re
t
;
;
=
u, v
x
x
t
y`
h
i
k
l
l
t
o
o
=
())
D
8!--
free-stream speed of sound
chord length
lift coe$cient
pressure coe$cient
total energy per unit volume
frequency in Hz
reduced frequency, 2nfc/;
=
free-stream Mach number
Reynolds number, c;/l
nondimensional time, tL a /c
=
local free-stream velocity
free-stream velocity magnitude
velocity components
coordinate parallel to chord
transition onset location
nondimensional normal wall distance
boundary layer momentum thickness
ratio of speci"c heats
viscosity
kinematic viscosity
turbulent viscosity
density
free-stream density
di!erentiation with respect to t
quantity on the surface of the blade
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