NASA Technical Reports Server (NTRS) 19960015539: Transonic Turbulent Flow Predictions With Two-Equation Turbulence Models

16 Pages·0.74 MB·English

by  NASA Technical Reports Server (NTRS)

/
NASA Contractor Report 198444
ICOMP-96--02; CMOTT-96-01
Transonic Turbulent Flow Predictions With New
Two-Equation Turbulence Models
William W. Liou and Tsan-Hsing Shih
Institute for Computational Mechanics in Propulsion and
Center for Modeling of Turbulence and Transition
Cleveland, Ohio
January 1996
Prepared for
Lewis Research Center
Under Cooperative Agreement NCC3-370
NationalAeronauticsand
Space Administration

TRANSONIC TURBULENT FLOW PREDICTIONS WITH NEW
TWO-EQUATION TURBULENCE MODELS
William W. Liou* and Tsan-Hsing Shih**
Center .far Modeling o/Turbulence and Tran4ition
ICOMP, NASA Lewis Research Center, Cleveland, Ohio
Abstract are closely related to these factors. Therefore, accu-
Solutions of the Favre-averaged Navier-Stokes rate predictions of transonic flows with embedded shock
equations for two well-documented transonic turbulent waves can help improve the efficiency of many impor-
flows are compared in detail with existing experimental tant fluid machines. The location and the strength of
data in this paper. While the boundary layer in the first the shock wave in a transonic flow is determined by the
case remains attached, a region of extensive flow separa- interaction between the shock waves and the boundary
tion has been observed in the second case. Two recently layer. With a sufficiently fine grid, modern numerical
developed k - _, two-equation, eddy-viscosity models schemes, such as the various TVD schemes and the new
are used to model the turbulence field. These mod- a-# scheme 1, etc., can capture the shock wave satisfac-
els satisfy the realizability constraints of the Reynolds torily. Therefore, a successful prediction of such flows
stresses. Comparisons with the measurements are made hinges largely upon the physical modeling of the inter-
for the wall pressure distribution, the mean streamwise action process. In this paper, two new eddy-viscosity
velocity profiles and turbulent quantities. Reasonably turbulence models are used to predict transonic turbu-
good agreement is obtained with the experimental data. lent separated flows. These two-equation models, one
a low-Reynolds number model and the other a high-
I. Introduction Reynolds number model, satisfy the realizability con-
In this paper, the accuracy and the range of appli- straints of the Reynolds stresses. The value of the coef-
cability of two new two-equation turbulence models for ficient for the eddy viscosity is determined by the mean
transonic shock-wave/boundary-layer interactions are flow deformation rate and the turbulence quantities.
assessed. Generally, when regions of subsonic Mach The models, therefore, can be particularly suitable for
number and regions of supersonic Mach number both flows with large or sudden flow deformation.
exist in a flow, the flow is said to be transonic. Tran- Two transonic flows with embedded shock waves are
sonic flow fields may be found in internal flows, for calculated in this paper. In the first, an internal tran-
example, in engine compressor passages and external sonic flow field is generated by using floor-mounted,
flows, for example, around airfoils. Other than a few two-dimensional bump models 2, Fig. 1, and the flow
exceptional cases, where fluid machines were specifi- is symmetric. The flow was observed to be incipiently
cally designed for shock-free compression, the transi- separated. The quasi-normal shock extends across the
tion from supersonic to subsonic flows invariably oc- tunnel and the flow is choked. The flow was iden-
curred through shock waves. The location and the tified as ONERA Bump A in the EUROVAL effort s,
strength of the shock wave impact significantly the a program devoted to the validation of computational
characteristics of a fluid machine involving the tran- fluid dynamics (CFD) codes with special attention di-
sonic flow. For example, the major part of the loss asso- rected toward the validation of turbulence models. In
ciated with a compressor blade passage and the loading the second configuration studied here 4, Fig. 2, a cir-
on an airfoil cular cylinder aligned with the flow direction is used
to generate the external transonic flow field. In this
*Senior Research Associate, Institute for Computa- case, the shock, which terminates before reaching the
tional Mechanics in Propulsion. Member AIAA tunnel wall, is sufficiently strong to induce an extensive
**Technical Leader, CMOTT, Institute for Comu- region of flow separation. The axisymmetric configara-
tational Mechanics in Propulsion. tion is relatively free from side-wall interference with
the turbulent boundary layer and is particularly suit-
able for the evaluation of model performance in the
prediction of strong viscous-inviscid interactions with
flowseparationT.his particular flow was also selected where
2
as a test case for transonic separated flows in the 1980
vii = 2(# + #t)S 0 - ]k6ij
AFSOR-HTTM-Stanford conference 5 and was identi-
fied as Case 8611. p,denotesthemean molecularviscosityand Siidenotes
These two cases, although relatively simple, contain themean strainratetensor,i.e.,
fundamentally essential characteristics that are impor-
tant in the design process of, for example, wings, en- &j= 1 + ui, -)1 6 (3)
gine compressors and turbines. Therefore, successful
predictions of these flows can have profound industrial
The turbulent Reynolds stresses are modeled via the
importance. turbulent eddy viscosity, St. In all of the models used
To provide an objective comparison of the model in this study, the turbulent eddy viscosity is determined
performance, care has been taken to obtain solutions by the turbulent kinetic energy, k, and the dissipation
on sufficiently refined computational grids so that the rate, _, i.e.,
observed difference in the results can be attributed di- k2
rectly to the turbulence models used in the calculations. = Cj.p T (4)
In the following sections, the turbulence models
k and e are obtained from the solution of their respec-
and the numerical platform used in this study are de-
tive model transport equations, f_ is the wall damping
scribed. The results of flow calculations are then com-
function for the eddy-viscosity. For reference, these
pared with those obtained by using two existing two-
models are described briefly in the following.
equation models, e.g., Chien's model 6 (CH) and the
standard k - _ model (SKE). Turbulence Models
II. Analysis In Shih et.alz,a generalconstitutiverelationbe-
tween theReynolds stressesand themean flowdefor-
Mean Flow Equations
mation ratewas derivedby usingtheinvarianceprin-
The flow properties are decomposed into two parts: cipleofLumley. The model satisfietshe realizability
amean value and afluctuation with respect to the mean constraints:forexample, the energy component
value. That is, shouldalwaysbe positive.Note thatthestandardk-
= p + p" (1.a) model withC_,= 0.09isan unrealizablemodel. Forex-
ample,_ becomes negativewhen
'Q,-i-Ui + u_ (1.b)
Snk 1
i_= p + p" (l.c) m > __ (5)
E 0.27"
_h= T + T' (1.d)
Therefore, the value of C_ should not be a constant for
= E + E' (1.e) a realizable model. In Shih et. alz, the realizability
constraints have led to,
where p,p, T, E, Ui denote Reynolds-averaged density
1
and pressure, mass-weighted-averaged temperature, to- (6)
tal energy, and velocity, respectively. It is customary to Ao+A.U(')
use both the Reynolds average and the mass-weighted
where
average in the decomposition process for compressible
turbulent flows to simplify the final form of the mean U(') = ¢SijSij + flijflij
flow equations. The governing equations for the mean
_ij = _i1 - 2eiikwk
flow may be obtained by substitution of flow properties
in the form of eq.(1) into the Navier-Stokes equations
followed by a Reynolds average of the equations. The
flij is the mean rotation rate viewed in a rotating ref-
mean flow equations become,
erence frame with the angular velocity wk. The param-
eter A8 is determined by
p,,+ = o (2.a)
1
A, ----Vr6cos¢, ¢ = -_arccos(vr6W)
(pUi)4 + (pUiUj + rij + p60),j = 0 (2.b) (7)
Sii Sj kSk
w- s3 s=
(pE),t + (pUiE + pUi + qT,i + T_jUj + qk.i),i = 0 (2.c)
The new formulation of C,, with an explicit depen- Kolmogorov eddies by using both direct numerical sim-
dence on the mean strain rate, cam be used to render ulation results and an asymptotic analysis of near wail
a model realizable. It is also in accord with the exper- turbulence. According to their analysis, this turbulent
imental observation that the value of C_ can be dif- limit point is located at
ferent for different flows. The value of A0 is set equal
6u
to 4.08 . It is determined by examining the log-law of Yn= -- (10)
U,r
the inertial sublayer. The corresponding value of C_ is
0.09. As was noted by Shih et. a_, the resulting for- At thislimitpoint,
mulation for C_, also worked very well for homogeneous
shear flows. The first model tested here is the Shih and kn= 0.25u_ and en= 0.251 u_ (Ii)
v
Lumley model 9, modified by including the new formu-
lation of C_ (Yang et. all°). The second model by Shih In practice, the boundary conditions are enforced at the
et. a_, on the other hand, has applied the variable C_, wall. With the application of eqns.(10) and (11), the
formulation during the development of the model. turbulent time scale near a wall, similar to the velocity
A: Shih and Lumley Model 9 - KE1 and length scales, is determined by the Kolmogorov
time scale. Therefore, there is no unphysical singularity
The model equations for k and e in KE1 model are, in the current model e equation.
pk, + = + - p U ,j - pe (S.a) B: Shill et. al Model s - KE2
A new form of model equation for the turbulent
dissipation rate was proposed by Shih et. ala. The
(s.b) equation for the mean-square vorticity fluctuation was
_2
first examined by using an order of magnitude anal-
- C2f_p-_ + vmU_,jkU_,_k
ysis. The truncated low-order equation is then mod-
where eled through physical reasoning. The modeled equa-
C1-1.44, C2=1.92, _re---1.3 tion for the mean-square vorticity fluctuation can be
transformed into an equation for the turbulent dissipa-
f: = 1- 0.22exp [- 2 Rt = -- tion rate in the limit of high Reynolds number. The
resulting model equation for g is,
The damping function is defined by
_2
(9) +puj , = ,J]a+C pS - C:p-k +
], = [1. - exp(-(alR_ H-a3Rk ÷ asRk))] ]
(12)
where where
O,1: 1.7 X 10-3, a3 = 10-9, as = 5 x 10-1°
C_ = max{0.43, 5--_}, (_ = 1.2, C: = 1.9
Rk = pvfky Sk
S= _, _l=_
#
Note that the value a_ has been modified due to the ap- In Shih et. aIa, the new modeled dissipation rate equa-
plication of the new formulation of variable C_ which tion has been coupled with the standard model equa-
is bounded by 0.09 in the current application. This tions for k, eq.(8.a), to form a two-equation model. Be-
modified Shih and Lumley model 9 has been shown to cause _ appears in the denominator of the sink term,
predict well a variety of flows in Yang et. aP°. The this new dissipation rate equation will not become sin-
near-wall boundary conditions for the turbulent quan- gular even if k vanishes. C_, is defined by eq.(6). Near
tities are determined by examining the Kolmogorov be- the wall, a compressible wail-function was applied.
havior of near-wall turbulence proposed by Shih and
u 1
Lumley °. They have shown that energetic eddies re- -- In(y +) Jr C (13)
duce to "Kolmogorov eddies" at a finite distance from
the wall and all the wall parameters are characterized
u is the Van Driest transformed velocity defined as,
by Kolmogorov microscales. Therefore, an estimate can
be obtained for the turbulent kinetic energy and its dis- A+U, A
sipation rate at the location where large eddies become u = vrB[arcsin(_).- /) - arcsin(-_)] (14)
where
extends from 0.5 chords upstream of the leading edge
(x = 0) of the bump to 0.128 m downstream of the
A = q--_-_B- 2cpT_ D = V:_ + B trailing edge of the bump. At the computational in-
Yw ' PTt ' let, the stagnation conditions were prescribed and the
mean velocity in the transverse direction was assumed
The heat flux near the wall is defined as,
zero. At the exit, the back pressure was prescribed.
Mass conservation was also ensured at the exit. For
q = q_ + U v (15)
Case 8611, the bump is 20.32 cm in length (=C). The
computational domain extends from -2C to 3.5C. At
The turbulent quantities are defined as,
the computational inlet, the known experimental stag-
nation conditions and the flow Mach number (=0.875)
k = r_lp (r,,,Ip) 3n (16) were prescribed. For both cases, low levels of turbulent
kinetic energy and dissipation rate were prescribed at
the inlet.
The value of y+(= u_.YlV_) for the first grid point The computational domains were covered with non-
away from the wall, where the wail-function is ap- orthogonal surface-fitted meshes, with grid clustering
plied,is about 30. Although the validity of the wall- at the shock location and near the wall. Several dif-
function boundary conditions in complex separated ferent meshes were used to ensure grid-independence,
flows is somewhat ambiguous, previous work seems to by varying both the number of grid nodes and the grid
show that it can provide reasonably accurate predic- clustering. For ONERA Bump A, it was found that
tions for a wide range of flows 11,12. the isentropic wall Mach number changed by less than
It should be noted that except for the mean flow vol- 0.2% for two grids (180x60 and 160×60). For low-
ume dilatation, no explicit compressibility effect models Reynolds number models, a mesh of 180x70 nodes was
have been included in any of the models in the calcula- found sufficient to provide a grid-independent solution.
tions performed in this study. Similarly, for Case 8611, 180x70 for the models em-
The results of calculations using these two realiz- ploying wall-function and 180x80 for the low-Reynolds
able models, which are presented in a later section, number models were found sufficient to ensure grid-
have been compared with those obtained by using the independence. About forty to fifty grid points are typi-
standard k - _ model (SKE) and Chien's (CH) low- cally located inside the boundary layer. In the following
Reynolds number k - _ model. The SKE and CH mod- section, the solutions obtained with the fine meshes are
els are representative of the high- and low-Reynolds presented.
number types of model, respectively. They are chosen
IV. Results and Discussions
here for comparison due to their simplicity and stabil-
ity. In the following, the numerical solution procedure In this section, the results obtained by using KE1
is described. and KE2 models are presented and compared with the
experiments.
III. Nmmerical Solutions
ONERA Bump A
The Favre-averaged Navier-Stokes equations and
the model transport equations have been solved nu- As was mentioned earlier, at the exit the back pres-
merically by using the COMTUR code developed by sure was imposed. With the experimental value of 0.641
Huang and Coakley la. Briefly, it uses a line-by-line (P/PT), it was found that the predicted shock locations
Gauss-Seidel algorithm and Roe's approximate Rie- were different. Consequently, the models' capability in
mann solver. Yee's MINMOD TVD scheme was applied predicting the flow structures in the interaction region
in all the computations. The mean and the turbulence can be obscured due to their difference in predicting the
equations are solved in a sequential manner. All the shock locations. To obtain a meaningful comparison of
calculations have been carried out with the same initial the model predictions, it has been suggested a that the
and boundary conditions, including those for k and _. experimental exit pressure be perturbed for the individ-
Since the model bump mounted on the top wall is uai model so that the calculated isentropic wall Mach
the same as the one mounted on the bottom wall, see number on the bump wall is 1.047 at x=0.158 m. This
Fig. 1, only the lower half of the channel is computed in allows a fair comparison of the model predictions of
ONERA Bump A. Symmetry conditions were imposed the interaction process without any influence resulting
along the centerline of the channel. The chord length from a mismatch of the shock location. The computed
of the floor bump is 0.2 m. The computational domain centerline Mach numbers have been compared with the
measuremeanntd are shown in Fig. 3. With the ad- interaction region, the turbulent Reynolds shear stress
justed exit pressure, the computed and measured shock were not well predicted. It should be noted, however,
locations agree well. The computed shocks are quite that experimental uncertainly can be higher for mea-
sharp, reconfirming the ability of the TVD scheme in suring stresses than for mean velocity. Also, the turbu-
lent structures are more sensitive to shocks than is the
capturing shock waves in the inviscid region. The two
mean flow.
high-Reynolds number models, SKE and KE2 mod-
els, predicted slightly higher post-shock expansion than Case 8611
the two low-Reynolds number models, CH and KE1
A sketch of the flow is shown in Fig. 2. The ex-
models, in the immediate post-shock region. All of
periment has shown a large region of flow separation
them, however, lie below the the measurement. This
immediate downstream of the shock due to the effects
appears to indicate three-dimensional effects related to
of shock/turbulent boundary-layer interaction and the
the thickening of the boundary layers on the side-wall
geometry of the floor model. As noted earlier, this flow
of the wind tunnel, causing the acceleration of the flow
configuration is particularly suitable for the study of
in the center of the channel.
transonic turbulent separated flows, since the axisym-
Figure 4 shows the comparison of the predicted
metric flow model provides a flow that is relatively free
and measured wall pressure distributions. KE2 model
from three-dimensional and tunnel wall effects. In fact,
shows the best agreement with the measured values in
the experiments have been repeated later in a larger
the post-shock region, followed by SKE, KE1, and CH
facility and the reported changes in the shock loca-
models. All the model predictions of the wall pressure tions are within 1% chord. The data were acquired
asymptote to avalue higher than the measurement. at a freestream Mach number of 0.875. At this Mach
Figure 5 shows the predicted and measured distri- number, the flow separates at x/C_ 0.7 and the reat-
butions of the boundary-layer displacement thickness tachment occurs at x/C_ 1.1. The griding of the com-
on the bump wall. A method proposed by Stock and putational domain is basically the same as those in the
Haase 14has been used in determining the displacement previous case. Again, the fine grid solutions are pre-
thickness. Overall, the low-Reynolds number CH and sented.
KE1 models return fairly good agreement with the mea- Figure 8 shows the Mach number contours using
surement. SKE and KE2 models predict higher peak the SKE model. The increment between the contour
values, suggesting a greater sensitivity to the shock. fines is a constant value of 0.05. A supersonic pocket is
The larger displacement of the boundary layer pre- formed above the bump surface and the shock has been
dicted by the SKE and KE2 models in the immedi- captured very well. The shock is curved, suggesting
ate post-shock region has led to the lower wall pres- that the computational mesh is sufficiently fine.
sure, Fig. 4, or higher isentropic wall Mach number. The measured and the computed surface pressure
In the recovery region, all the models show excessive distributions are compared in Fig. 9. The KE2 model's
boundary-layer displacement. This suggests that the prediction on a coarser grid (160×60) has also been
models would return higher centerline Mach numbers
shown for comparison. There is no significant difference
and lower wall pressures than the measurements, which between the fine- and coarse-grid solutions. The high-
are not observed in Figs. 3 and 4. Since the experi-
Reynolds number KE2 model gives the best prediction
mental flow is confined in the spanwise direction, this
of the wall pressure near the shock location. The shock
apparent inconsistency may also be attributed to the
predicted by the CH model is located downstream of
three-dimensional effects mentioned earlier.
the experimental shock. In the region of flow sepa-
Measured and computed streamwise mean veloc- ration, the CH model underestimates the level of the
ity profiles are shown in Fig. 6. The model predic- viscous/inviscid interaction and gives the highest pres-
tions agree well with the measured profiles upstream of sure rise among all the models tested. Compared with
the interaction region, x=0.15 m. The mean velocity the CH model, the KE1 model has shown a significant
profiles in the interaction region, say, x=0.155 m and improvement. The KE2 model predicts rather well the
x=0.16 m, were found very sensitive to the shock loca- wall pressure in this region. This model also shows a
tions. Downstream of this region, all the models give significant improvement over the SKE model. To quan-
reasonable predictions of the mean velocity profiles. tify the comparison, at the trailing edge of the bump,
In Fig. 7, the Reynolds shear stress predictions are x/C=l, the differences between the predicted and mea-
compared with the data. The shear stress profiles up- sured wall pressure are about 1.6%, -0.1%, 4.3%, and
stream and downstream of the interaction region have 2.8% for the KE1, KE2, CH, and SKE models, respec-
been reasonably well predicted by all the model. In the tively.
Comparisons of the computed and measured able models perform better than the selected existing
streamwise mean velocity profiles are shown in Fig. models of the same type. The performance differences,
10. The model predictions agree reasonably well with however, are not significant. For the case with strong
the measurements upstream of the interaction region, shock/turbulent boundary-layer interactions, the high-
x/C=0.563. In the region of flow separation, x/C=0.75, Reynolds number model of Shih et. a/6 has shown the
0.875, and 1.0, The KE2 model seems to predict a greatest sensitivity to the interaction process and given
boundary layer that is more sensitive to the effect of the a far better prediction for the wall pressure distribution
shock than the other models tested and the CH model and the shock location than the other models tested. In
gives slightly better predictions. The KE2 model has addition, the two high-Reynolds number models tested
predicted afaster flow recovery than the other models. here, i.e., the Shih et. a_ model and the standard k-
Figure 11 shows the computed and measured Reynolds model, have predicted better overall performance than
shear stress profiles. Near the shock wave, x/C=0.75, the two low-Reynolds number models.
the KE2 model has correctly predicted the peak value
Acknowledgements
of the Reynolds shear stress. The predicted location
of the peak, however, is slightly higher than the mea- The authors would like to thank Dr. P.G. Huang at
surement. Further downstream, the flow predicted by MCAT Inc. Mountain View, CA, for helpful discussions
the KE2 model relaxes rather quickly. At x/C=1.375, and comments.
the KE2 model has predicted a better profile distribu-
References
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els. The turbulent kinetic energy profiles are compared 1 Chang, S.-C., "The method of space-time conserva-
in Fig. 12. In the interaction region, the KE2 model tion element and solution element-A new approach
overpredicts the location of the peak turbulence energy. for solving the Navier-Stokes and Euler equations,"
Due to the relatively fast flow recovery predicted by To appear in J. Corny. Phys. (1995).
the KE2 model, however, the KE2 model predictions
2 Delery, J. and Reisz, J., "Analyse experimentale
of the turbulent kinetic energy agree far better than
d'une interaction choc-couche limite turbulente a
the other models downstream of the immediate region
Mach 1.3 (decollment naissant)" ONERA Rapport
of strong interaction. Figure 13 shows the distributions
Technique No. 42/7078 AY 014, Chatillon, Decem-
of the eddy-viscosity coefficient, C_, ahead, amid, and ber 1980.
behind the interaction region. For the CH and SKE
models, the value has a constant value of 0.09. For 3 EUROVAL-A European Initiative on Validation of
the KE1 and KE2 models, its value depends on the CFD Codes. Edt. W. Haase, F. Bradsma, E.
local mean and turbulent flow field. At x/C=0.563, Elsholz, M. Leschziner and D. Schwanborn, 1992.
the value of C_ given by KE2 model is different from 4 Bachalo, W. D. and Johnson, D. A., "Transonic,
0.09, indicating the fact that the flow has been affected turbulent boundary-layer separation generated on
by the bump. The distribution at x/C=0.75 shows an axisymmetric flow model," AIAA Journal, 24,
that the KE1 and KE2 models react rather differently 437-443 (1986).
to the shock/turbulent boundary-layer interaction pro-
s The 1980-81 AFSOR-tITTM-Standford Confer-
cess. Since there is no upper limit when the variable C_
formulation is used with the KE2 model, the maximum ence on Complex Turbulent Flows: Comparison of
Computalon and Experiment. Edited by S. J. Kline,
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BJ. J. CantweU, and G. M. Lilley (1981).
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V. Smnmary
7 Shih, T. H., Zhu, J., and Lumley, J. L., "A new
Two new realizable, k - e, eddy-viscosity models
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TM 106644 (1994). Accepted for publication in
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els, the same initial, boundary conditions and compu- and Zhu, J., "A new k - ¢ eddy viscosity model for
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10 Yang, Z., Georgiadis, N., Zhu, J., and Shih, T.-H., 13 Huang, P. G. and Coakley, T. J., "An implicit
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11 Huang, P. G. and Coakley, T. J., "Calculations of 14 Stock, H. W. and Haase, W., "Determination of
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on Engineering Turbulence Modeling and Measure- (1989).
l,lili,il,il,i,li_,. .,].ll III IIII III III I!
//---" ] M<I
//M>I]^
llllli¢llllcllll _illi(|lll(llllll lillll Ill II lilll |l I- "'t}i||l|tli|lllll|
Figure 2. A sketch for Case 8611.
Figure 1. A sketch for ONERA Bump A.
1.5
........i.............._.........-..........:............._.............._.............f.i.................
13
i i I _P i :;
Mo,. 1.1
O.9
_..__ ....i............i......._.._..o.._._..o......
i i
0.7
0.10 0.15 0.20 0.25 0.30
X
Figure 3. Variation of Mach number along the centerline. ONERA Bump A
0.70
__--ami__S__2_ -______:-___
0.60 _ _ _ ! _
P/PT o._ ....... ;........ _.... ............ I............ ........... -.......... _-.........
................_...............i.............i..............i................;............. ".............._..............
-iiii.;P
OAo .........._..............; ............i..............i..............._..............=..............i...............
0.3]
0.10 0,15 0.20 0.25 0.30
x(m)
Figure 4. Variation of isentropic wall Mach number. ONERA Bump A
O.OO4
0.0_ OQ i
6"(111)0111 ',, , _--.I-. _ -o _: _:_
......4..-..-.-.-..= ..........l..............i................+............._.............
o.%._o _ t ,,:.,
0.15 _ 0.25
X
Figure 5. Variation of boundary-layer displacement thickness. ONERA Bump A
O.OLOx---0.15 f 0"16 i_ 0.21 0.2,
0.008 I o
o._ ---- ,_+w+ i"
°+f+J j
0.002
y(m) =_ /
0.000 = I I J j . j m . I
0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0
U/U_
Figure 6. Streamwise mean velocity profiles. ONERA Bump A.
0.24
0.0100.008-0-.-0-0--6X-----S'0I"_1÷5w--F-CH_I,_2.w0.F1K55EI _Im+,_ 0.16 , .17 ..i.21
+
y(m) i _":,-,,
o.oo2 f"_ _' _
0
0.000 I ....... ,I ,.I ..,.I ,, I ..,,,,.,°1
0.000 O.OOS0.000 0.0050.000 0.0050.000 0.0050.000 0.0050.000 0.005
-uv/U_2
Figure 7. Comparison of Reynolds shear stress profiles. ONERA Bump A.
8