Exploring the Limits of Shock-free Transonic Airfoil Design

17 Pages·2012·1.4 MB·English

43rdAerospaceSciencesMeetingandExhibit,January10{13,2005,Reno,Nevada
Exploring the Limits of Shock-free Transonic Airfoil
Design
Michael Harbeck(cid:3) and Antony Jamesony
Stanford University, Stanford, CA 94305-4035,USA
In this study the limits of shock-free transonic airfoil design are explored via computa-
tional (cid:13)uid dynamics and shape optimization. Using an inviscid (cid:13)ow solver coupled with
the adjoint method for shape optimization, the best possible Mach number and lift coef-
(cid:12)cient combinations are determined for various airfoil sections. In addition, the e(cid:11)ects of
airfoil thickness and a trailing edge wedge are examined. Although shock-free solutions
only occur for a singular Mach number and lift coe(cid:14)cient for a given airfoil shape, it is
useful to know the best possible combination that can be attained. We have found that
there is a band of attainable shock free solutions between a lift coe(cid:14)cient of 1.0 at Mach
0.72 and a lift coe(cid:14)cient of 0.3 at Mach 0.83. Also, we are able to shift the curve of
attainable solutions outward by making the airfoil thinner and by adding a trailing edge
wedge. Once a shock-free solution is found, the von Ka(cid:19)rma(cid:19)n transonic similarity rule can
be used to determine shock-free solutions by changing airfoil thickness. In addition, the
o(cid:11)-design performance of the shape optimized shock-free airfoils is examined, and the in-
viscid results are veri(cid:12)ed by using a viscous (cid:13)ow solver coupled with the adjoint method
for shape optimization.
I. Introduction
In the late 1960s and early 1970s there was considerable work done on shock-free transonic airfoil design.
Morawetz determined in the late 1950s that shock-free solutions are isolated points, and a small pertur-
bation from the point would result in a shock.1 Also, in the sixties Whitcomb developed his well known
supercriticalairfoilsfortransonic(cid:13)ow,2 butthemostsuccessfulmethodforcreatingshock-freeairfoilshapes
isthe method of complexcharacteristicsin the hodographplane, developedunderthe leadershipof Garabe-
dian.3,4 Using hodograph methods, Boerstoel and Nieuwland published reports on transonic (cid:13)ows around
quasi-elliptical airfoils sections in 1967.5,6
While these methods were successful in producing a variety of shock-free airfoils, the di(cid:14)culty of re-
covering realizable physical shapes from the solution in the hodograph plane made them di(cid:14)cult and time
consuming to use. Moreover, no boundaries were established for the lift coe(cid:14)cient and Mach number com-
binations achievable in shock-free transonic (cid:13)ow. This question has remained open to the present day.
The last decade has seen the development of e(cid:11)ective shape optimization methods based on control
theory.7,8,9,10,11 These methods use the adjoint equation to derive the gradient of a cost function with
respect to the shape, and then approach the optimum by a preconditioned descent procedure. When the
drag coe(cid:14)cient is minimized at a (cid:12)xed lift, the result is typically a shock-free solution, which corresponds
to zero drag in inviscid (cid:13)ow. Optimum shapes can be found in minutes on a laptop computer.
The purpose of this paper is to exploit the speed and e(cid:14)ciency of adjoint based optimization methods
to explore the attainable limits of shock-free design. For this purpose we use a variety existing airfoils
as starting points for optimization, such as airfoils designed using the hodograph method, Whitcomb’s
supercritical airfoils, and other transonic airfoils. The results establish a boundary of the attainable lift
(cid:3)HughH.SkillingStanfordGraduateFellow,DoctoralCandidate,AIAAMember
yThomasV.JonesProfessorofEngineering,DepartmentofAeronauticsandAstronautics,AIAAMember
Copyright(cid:13)c 2005bytheAmericanInstituteofAeronauticsandAstronautics,Inc. TheU.S.Governmenthasaroyalty-free
licensetoexerciseallrightsunderthecopyrightclaimedhereinforGovernmentalpurposes. Allotherrightsarereservedbythe
copyrightowner.
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coe(cid:14)cient versus Mach number for airfoils of a given thickness. The boundary can be shifted upward by
reducingthethickness,andinsomecasesbyaddingawedgeatthetrailingedge. Whenthedesignispushed
to too extreme a limit there is a risk that the performance may degrade very rapidly away from the design
point. Therefore, we also examine the behavior of the optimized designs over a range of Mach numbers in
order to establish performance boundaries attainable by "good" airfoils, which might be usable in practical
designs.
II. MATHEMATICAL FORMULATION
In this work the equations of steady inviscid (cid:13)ow are
@
f (w)=0 (1)
i
@x
i
where w is the solution vector,and f (w) is the inviscid (cid:13)ux vectoralongthe x axis, which areapplied in a
i i
(cid:12)xed computational domain, with coordinates (cid:24) , so that
i
@ @
R(w;S)= F (w)= S f (w)=0 (2)
i ij j
@(cid:24) @(cid:24)
i i
where S are the coe(cid:14)cients of the Jacobian matrix of the transformation. Then geometry changes are
ij
represented by changes (cid:14)S in the metric coe(cid:14)cients. Suppose one wishes to minimize the cost function of
ij
a boundary integral
I = M(w;S)dB + N(w;S)dB (3)
(cid:24) (cid:24)
Z Z
B B
wherethe integralof M(w;S) could be an aerodynamiccost function, e.g. dragcoe(cid:14)cient, and the integral
of N(w;S) could be a structural cost function, e.g. wing weight. Then one can augment the cost function
through Lagrangemultiplier as
I = M(w;S)dB + TR(w;S)dD + N(w;S)dB (4)
(cid:24) (cid:24) (cid:24)
Z Z Z
B D B
A shape variation (cid:14)S causes a variation
@
(cid:14)I = (cid:14)MdB + T (cid:14)F dD + (cid:14)N dB ; (5)
(cid:24) i (cid:24) (cid:24)
Z Z @(cid:24) Z
i
B D B
The second term can be integrated by parts to give
@ T
n T(cid:14)F dB (cid:0) (cid:14)F dD : (6)
i i (cid:24) i (cid:24)
ZB ZD @(cid:24)i
Now, choosing to satisfy the adjoint equation
@f @
(S j)T =0 (7)
ij
@w @(cid:24)
i
with appropriate boundary conditions
n TS f =M +N ; (8)
i ij jw w w
where
M = @M, and
w @w
N = @N,
w @w
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the explicit dependence on (cid:14)w is eliminated, allowingthe costvariationsto be expressedin terms of (cid:14)S and
the adjoint solution, and hence (cid:12)nally in terms of the change (cid:14)F in a function F((cid:24)) de(cid:12)ning the shape.
Thus, one obtains
(cid:14)I = G(cid:14)Fd(cid:24) =hG;(cid:14)Fi (9)
Z
where G is the in(cid:12)nite dimensional gradient (Frechet derivative) at the cost of one (cid:13)ow and one adjoint
solution. Then one can make an improvement by setting
(cid:14)F =(cid:0)(cid:21)G (10)
In fact the gradient G is generally of a lower smoothness class than the shape F. Hence, it is important
torestorethe smoothness. This maybe e(cid:11)ected bypassingtoaweighted Sobolev innerproduct of the form
@u@v
hu;vi= (uv+(cid:15) )d(cid:24) (11)
Z @(cid:24) @(cid:24)
This is equivalent to replacing G by G(cid:22), where in one dimension
@ @G(cid:22)
G(cid:22)(cid:0) G(cid:22) =G; G(cid:22)=zero at end points (12)
@(cid:24) @(cid:24)
and making a shape change (cid:14)F =(cid:0)(cid:21)G(cid:22).
In addition, in this work a comparisonof viscousand inviscid shock-freepro(cid:12)les is made. The equations
of steady viscous (cid:13)ow are
@ @
f (w)= f (w) (13)
@x i @x vi
i i
where w, and f (w) are de(cid:12)ned as above, and f (w) is the viscous (cid:13)ux vector.
i vi
Contained in the viscous (cid:13)ux vector is viscous stress tensor, which for Newtonian (cid:13)uids is
@u @u 2 @u
i j k
(cid:28) =(cid:22) + (cid:0) (cid:22) (cid:14) (14)
ij ij
(cid:20)@x @x (cid:21) 3 (cid:20)@x (cid:21)
j i k
where u is velocity, (cid:22) is the coe(cid:14)cient of viscosity, and (cid:14) is the Kronecker delta function. The coe(cid:14)cient
ij
of viscosity is determined by the Sutherland equation:
3
T2
(cid:22)=C (15)
1
T +C
2
where T is temperature and C and C are constants for a given gas. For air C =1:458x10(cid:0)6kg=(msK)
1 2 1
and C =110:4K. The viscous adjoint formulation follows the steps outlined above for inviscid (cid:13)ow.
2
III. PROCEDURE
Drag coe(cid:14)cient (C ) is the cost function for shape optimization in this study. An existing transonic
D
airfoil is used as the starting point for shape optimization at a particular lift coe(cid:14)cient (C ), and Mach
L
number (M) is varied until the best shock-freesolution is found, if oneexists. The procedure is repeated for
multiple lift coe(cid:14)cients and airfoils forming a curve of attainable shock-free solutions for each airfoil.
IV. DETERMINING SHOCK-FREE PROFILES
The di(cid:11)erence between a small shock wave and a steep pressure recovery can be negligible. Therefore,
we take a moment to de(cid:12)ne our criteriafor a shock-freesolution. In many casesit is helpful to examine the
Machcontoursinadditiontothepressuredistributiontodetermineifaparticularcaseisshock-free. Asmall
shock wave consists of a nearly vertical segment in the pressure recoveryregion of the pressure distribution
with Mach contours that coalesce. In contrast, a shock-free pro(cid:12)le consists of a smooth pressure recovery
with spread out Mach contours.
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V. RESULTS
A. Grid
-.2E+01 -.2E+01
-.2E+01 -.2E+01
Cp 0.1E+010.8E+000.4E+00-.2E-15-.4E+00-.8E+00-.1E+01 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Cp 0.1E+010.8E+000.4E+00-.2E-15-.4E+00-.8E+00-.1E+01 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
KORN AIRFOIL KORN AIRFOIL
MACH 0.780 ALPHA 0.369 MACH 0.780 ALPHA -0.100
CL 0.5934 CD 0.0033 CM -0.1268 CL 0.6006 CD 0.0018 CM -0.1465
GRID 192X32 NDES 20 RES0.366E-02 GMAX 0.286E-02 GRID 384X64 NDES 20 RES0.388E-02 GMAX 0.181E-02
(a)192x32 gridnotcapturingtheshock (b)384x64gridcapturingtheshock
Figure 1. Comparison of grids
In order to determine if an airfoil is shock-free the
grid (cid:12)neness must be reasonably small in order capture
the shock wave. A 192x32 grid does not capture the
shock (Figure 1a), but a 384x64 grid does (Figure 1b).
Hence, the shock-free conditions were determined using
a 384x64 grid, which is shown for the Korn airfoil in
Figure 2.
B. The e(cid:11)ects of airfoil shape
Airfoil shape is the most important factor in produc-
ing shock-freepro(cid:12)les. A minute changein airfoilshape
can cause an airfoil to become shock-free for a particu-
lar Mach number and lift coe(cid:14)cient. This is where the
adjoint method for shape optimization comes in. Fig-
ures 3 and 4 show the di(cid:11)erence a subtle shape change
can make for the Korn and DLBA-243 airfoils respec-
tively. Figure 5 shows the shape change necessary to
makethe Korn airfoilshock-freeat Mach0.77and alift Figure 2. 384x32 grid for the Korn airfoil
coe(cid:14)cient of 0.6.
Variousairfoilsareoptimized foreachMachnumber
andliftcoe(cid:14)cientcombinationgivingacurveofthebest
possibleshock-freesolutions,whichisshowninFigure 6. Thereisbandof feasibleshock-freeMachnumbers
and lift coe(cid:14)cients between C = 1.0 at Mach 0.72 and C = 0.3 at Mach 0.83. The G78-06-10, W110,
L L
and G79-06-10 airfoils are the best starting airfoils for low lift coe(cid:14)cients and high Mach numbers. The
DLBA-243 is the best for the middle to high CL range, and thKeORGN7 A0I-R1F0O-I1L3 a i r f o i l r e a c h e s t h e h i g h e s t l i f t
coe(cid:14)cients. Most of the airfoilshavearounda 10% thicknesstoGRcIhD o 3r84d X r 6a4tio,but the G70-10-13is is slightly
thicker at 13%.
We have seen that the starting airfoil shape is very important. Moreover, if we start with an airfoil not
designed for transonic (cid:13)ow, like the GAW airfoil, the appropriate shape changes cannot be made to get the
airfoil to be shock-free at high Mach numbers. Thus, the GAW airfoil is well inside the band of attainable
shock-free solutions (Figure 6).
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C. The e(cid:11)ects of airfoil thickness
If the airfoil section is made thinner than the original airfoil, it is possible to push the attainable shock-
free band outward. We de(cid:12)ne the thickness ratio ((cid:28)) as the airfoil thickness divided by the original airfoil
thickness. Figure7showsthatbymaking(cid:28) =0.9,theRAE2822airfoilisshock-freeataMachnumber0.01
higher than the original airfoil. As thickness ratio decreases the band continues to shift outward. When (cid:28)
= 0.7, the airfoil is shock-free at Mach 0.03-0.04 higher than the original, but this trend does not continue
inde(cid:12)nitely. If we decrease (cid:28) to 0.6, the airfoil is no longer shock-free for any Mach number in the range
of interest (greater than Mach 0.70). Thus, the curve can be shifted outward by decreasing thickness, but
there are limits to how thin the airfoil can be and still be shock-free for given C .
L
D. The e(cid:11)ects of a trailing edge wedge
The addition of a small wedge at the trailing edge can also shift the attainable shock-free curve outward.
Figure 8 shows that the addition of a 2% trailing edge wedge (see Figure 9) to the Korn airfoil extends the
curve to higher lift coe(cid:14)cients. The airfoil with the wedge has a lift coe(cid:14)cient 0.1 higher than the airfoil
without a wedge for all Mach numbers. In addition, the maximum shock-free C is 0.9 with the wedge
L
compared to 0.7 without the wedge. Figure 10 shows that a trailing edge wedge also shifts the attainable
shock-freecurveupwardfor the RAE 2822airfoil. The maximum C forthe RAE 2822airfoilwith a wedge
L
is 0.8 compared to 0.7 for the standard airfoil. The only Mach number for which C does not increase with
L
the wedge is 0.79.
E. von K(cid:19)arm(cid:19)an transonic similarity rule
Once a shock-free solution is found, shock-free solutions at di(cid:11)erent Mach numbers can also be found, but
lift coe(cid:14)cientand thicknessmust be scaledaccordingto the vonKa(cid:19)rma(cid:19)ntransonicsimilarityrule. The von
Ka(cid:19)rma(cid:19)n transonic similarity rule, in a form to to Sprieter is
1(cid:0)M2 1(cid:0)M2
1 = 2 (16)
[t ((cid:13) +1)M2]2=3 [t ((cid:13) +1)M2]2=3
1 1 1 2 2 2
and
C [((cid:13) +1)M2]1=3 C [((cid:13) +1)M2]1=3
L;1 1 1 = L;2 2 2 (17)
t2=3 t2=3
1 2
where t is the airfoil thickness and (cid:13) is the ratio of speci(cid:12)c heats. Since we are using the similarity rule to
compare similar airfoils in the same (cid:13)uid, (cid:13) = (cid:13) , and the formulas can be simpli(cid:12)ed as follows.
1 2
1(cid:0)M2 3=2 M2
(cid:28) = 2 1 (18)
(cid:18)1(cid:0)M2(cid:19) M2
1 2
and
2=3
M t
1 2
C =C (19)
L;2 L;1
(cid:18)M t (cid:19)
2 1
Figure11showshowC andMachnumbervarywiththicknessratioaccordingtothetransonicsimilarity
L
rule for an airfoil that is shock-free at C = 0.6 and Mach 0.79. Using the transonic similarity rule we are
L
abletoobtainshock-freesolutionsalongthe C versusMcurveforvariousthicknessesshowninFigure 12a.
L
Figure 12 also shows the pressure distributions and Mach contours for similar solutions along the transonic
similarity curve from C = 1.0 at Mach 0.7 in Figure 12b to C = 0.19 at Mach 0.92 in Figure 12f for the
L L
J78-06-10airfoil.
F. The o(cid:11)-design performance
When an airfoil is shape optimized for a particular C and M combination, the performance at conditions
L
other than the design point can be less satisfactory than the original airfoil performance. Figure 13 shows
that the low speed behavior of the optimized Korn and RAE 2822airfoilsis worsethan the originalairfoils,
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but the airfoils have better performance at high speeds. Figure 13 is the C versus Mach plot for both
D
airfoils optimized at a lift coe(cid:14)cient of 0.6 and Mach 0.77. There is a dip in C at the design point, and
D
at Mach numbers below the design point there is an increase in drag. Pressure distributions are shown in
Figure14forthedesignMachnumberandMach0.1aboveandbelowthedesignpoint. Theoptimizedairfoil
hasadoubleshockatMach0.76resultinginthe increaseddrag(Figure 14a-b). Figure15showsthe original
and optimized pressuredistributions for the RAE 2822airfoilat the same three Mach numbers asthe Korn
airfoil. Again there is a double shock on the optimized airfoil for Mach 0.76 that was not present on the
originalairfoil(Figure15a-b). Inaddition,thereisashockwaveonthelowersurfaceofthe optimizedairfoil
at Mach 0.78 (Figure 15), but the optimized airfoil still has a lower drag. Overall, the performance of the
optimized airfoils on the outer limit of the shock-free band degrades rapidly away from the design point.
G. Viscous Veri(cid:12)cation
The preceding discussion was on results obtained using a 2-D inviscid (cid:13)ow solver coupled with the adjoint
method for shape optimization. We now take a moment to verify the above results for the attainable
shock-freesolutionsusinga2-Dviscous(cid:13)owsolvercoupledwith the adjointmethod forshapeoptimization.
Figure 16 shows the comparison between attainable inviscid and viscous shock-free solutions. The overall
band of shock-free solutions appears to be roughly the same for the viscous computations with only minor
variations.
VI. CONCLUSION
These studies suggest that shock-free airfoils can be designed for a band extending from C = 0.3 at
L
Mach 0.83 to C = 1.0 at Mach 0.72 with a thickness to chord ratio around 10%. However, airfoils on the
L
limit de(cid:12)ned by this band may exhibit severe degradation of performance away from the design point. The
starting airfoil played a role in determining the attainable shock-free band. The G70-10-13 airfoil reached
the highest lift coe(cid:14)cients; the DLBA-243 airfoil was best for the middle to high lift coe(cid:14)cients, and the
G78-06-10,W110,andtheG79-06-10airfoilswerethebestshock-freestartingairfoilsforlowliftcoe(cid:14)cients.
By examining the Korn and RAE 2822airfoils, we showed that a trailing edge wedge may be introduced to
shift this bandupwardwith lift coe(cid:14)cients 0.2greaterthan werepossible with the standardairfoil. Making
theairfoilthinnermayalsobeabletoshifttheshock-freecurveoutward. Wehaveshownthatbymakingthe
RAE 2822 airfoil 70% of the original thickness shock-free solutions occurred at Mach numbers 0.04 greater
than the original. In addition, we used the transonic similarity rule to (cid:12)nd shock-free solutions at various
Machnumberscorrespondingtodi(cid:11)erentliftcoe(cid:14)cientsandthicknessratios. Finally,bycomparinginviscid
and viscous shock-free solutions, we veri(cid:12)ed that the results for viscous (cid:13)ow are essentially similar.
VII. ACKNOWLEDGMENT
This material is based upon work supported under a Stanford Graduate Fellowship, and this work has
bene(cid:12)ted greatlyfrom the support of the Air Force O(cid:14)ce of Science Researchunder grant No. AF F49620-
98-1-2005.
References
1Morawetz, C., \On the Non-existence of Continuous Transonic Flows Past Pro(cid:12)les," Comm. Pure Appl. Math, Vol. 9,
1956,pp.45{68.
2Whitcomb, R., \Review of NASA Supercritical Airfoils," Haifa paper 74-10, 9th ICAS Congress, Alburguerque, NM,
June1974.
3Bauer,F.,Garabedian,P.,andKorn,D.,Supercritical WingSections,SpringerVerlag,NewYork,1972.
4Bauer,F.,Garabedian,P.,Korn,D.,andJameson,A.,SupercriticalWingSectionsII,SpringerVerlag,NewYork,1975.
5Boerstoel, J., \A Survey of Symmetrical Transonic Potential Flows around Quasi-elliptical Aerofoil Sections," N.L.R.
ReportTR.T179,1967.
6Nieuwland,G.,\TransonicPotentialFlowaroundaFamilyofQuasi-ellipticalAerofoilSections,"N.L.R.ReportTR.T172,
1967.
7Jameson,A.,Alonso,J.,Reuther,J.,andMarinelli,L.,\AerodynamicShapeOptimizationTechniquesBasedonControl
Theory,"AIAA paper 98-2538,29th AIAAFluidDynamicsConference,Alburguerque,NM,June1998.
8Jameson,A.,\AerodynamicDesignviaControlTheory,"Journal of Scienti(cid:12)cComputing,Vol.3,1988,pp.233{260.
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AmericanInstituteofAeronauticsandAstronautics Paper2005-1041
9Jameson,A.,\ComputationalAerodynamicsforAircraftDesign,"Science,Vol.245,1989,pp.361{371.
10Jameson, A., \Optimum Transonic Wing Design Using Contril Theory," Symposium Transsonicum IV, International
UnionofTheoretical andAppliedMechanics,Gottingen, Germany,September 2002.
11Jameson,A.,\AerodynamicShapeOptimizationUsingtheAdjointMethod,"LectureSeriesattheVonKarmanInstitute,
Brussels,Belgium,Febuary2003.
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-.2E+01
-.2E+01
Cp 0.4E+00-.2E-15-.4E+00-.8E+00-.1E+01 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
++
0.8E+00 +++++
++
0.1E+01 ++++++++
KORN AIRFOIL
MACH 0.770 ALPHA -0.301
CL 0.6061 CD 0.0047 CM -0.1686
GRID 384X64 NDES 0 RES0.563E-02 GMAX 0.100E-05
(a)Beforeoptimization
-.2E+01
-.2E+01
Cp 0.4E+00-.2E-15-.4E+00-.8E+00-.1E+01 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
0.8E+00 +++++++
++
0.1E+01 +++++++++
KORN AIRFOIL
MACH 0.770 ALPHA -0.066
CL 0.6005 CD 0.0004 CM -0.1410
GRID 384X64 NDES 20 RES0.186E-02 GMAX 0.719E-03
(b)Afteroptimization
Figure 3. Pressure distribution and Mach contours for the Korn airfoil
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-.2E+01
-.2E+01
Cp 0.8E+000.4E+00-.2E-15-.4E+00-.8E+00-.1E+01 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
0.1E+01 +++++++++
DLBA-243 DRAG REDUCTION
MACH 0.770 ALPHA -1.752
CL 0.7021 CD 0.0139 CM -0.3026
GRID 384X64 NDES 0 RES0.593E-03 GMAX 0.100E-05
(a)Beforeoptimization
-.2E+01
-.2E+01
-.1E+01
Cp 0.8E+000.4E+00-.2E-15-.4E+00-.8E+00 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
++
0.1E+01 +++++++
DLBA-243 DRAG REDUCTION
MACH 0.770 ALPHA -0.966
CL 0.7008 CD 0.0013 CM -0.2196
GRID 384X64 NDES 20 RES0.288E-02 GMAX 0.123E-02
(b)Afteroptimization
Figure 4. Pressure distribution and Mach contours for the DLBA-243 airfoil
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0.08
Original
0.06 Optimized
0.04
0.02
0
−0.02
−0.04
−0.06
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
Figure 5. Shape change required to make Korn airfoilshock-free at M=0.77 and CL=0.6
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