Computer codes for the design and analysis of transonic airfoils are considered. The design code relies on the method of complex characteristics in the hodograph plane to construct shockless airfoil. The analysis code uses artificial viscosity to calculate flows with weak shock waves at off-design conditions. Comparisons with experiments show that an excellent simulation of two dimensional wind tunnel tests is obtained. The codes have been widely adopted by the aircraft industry as a tool for the development of supercritical wing technology

Garabedian, P. R.

English

NASA Technical Reports Server

47
i N79-20034
4
TRANSONIC AIRFOIL CODES*
P.R. Garabedian
New York University
SUMMARY
Three books on supercritical wing sections have been pub-
lished recently that document and list codes for the design and
analysis of transonic airfoils. The codes have had a significant
i impact on the development of supercritical wing technologu. This
paper is devoted to several new contributions to the theory on
_i which the codes are based. It has become possible to prescribe
• the pressure distribution within a reasonabl_ tolerance even over
the supersonic portion o£ a shockless airfoil. The purely sub-
sonic problem of modeling the flow near the trailing edge has
been handled in a way that eliminates any appreciable loss of
lift in practice. Boundary layer effects are taken into account
in an empirically satisfactory fashion. The work has been ex-
tended to the case of cascades of airfoils appropriate for the
design of compressor and turbine blades. These methods of com-
putational fluid dynamics have produced a family of airfoils that
deliver outstanding performance over a wide range of conditions.
INTRODUCTION
In the last few years three books have appeared that list
computer codes for the design and analysis of transonic airfoils
(refs. 1,2,3). The design code relies on the method of complex
_haracteristics in the hodograph plane to construct shockless
airfoils. The analysis code uses artifical viscosity to calcu-
late flows with weak shock waves at off-design conditions. Com-
parisons with experiments show that an excellent simulation of
two-dimensional wind tunnel tests is obtained. The codes have
been widely adopted by the aircraft inoustry as a tool for the
development of supercritical wing technology.
SYMBOLS
f complex analytic function
h real function
WWork supported by NASA under Grants NGR-33-016-167 and NGR-
33-016-201 and by the U.S. Dept. of Energy under Contract
EY-76-C-02-3077 with N.Y.U.
45
r
L......._, ............................................1979011uouuo"'-"-"I
https://ntrs.nasa.gov/search.jsp?R=19790011863 2020-03-20T18:37:52+00:00Z
i square root of -i
k real constant
M Mach number
q speed
u function of x and y
uI even function of y
_2 even function of y
x ca**onical coordinate
y canonical coordinate
I
characteristic coordinate
n characteristic coordinate
O flow angle
p density
velocity potential
I _ stream function
I ANALYSIS AND DESIGN
For analysis, differencing of the partial differential
equations of gas dynamics that do not adhere to strict con-
servation form turns out to give the best representation of
boundary layer-shock wave interaction over a broad range of
conditions (ref. 2). An improved formula for the wave drag
compensates for errors in the conservation of mass across shocks
that are of the third order in the shock strength. Modern tech-
niques of conformal mapping and fast Fourier transform have led
to an upgraded analysis code that has unusual speed and accuracy
(ref. 3).
Corrections for the displacement thickness of the turbulent
boundary layer have been made in both the design and the analysis
codes. Even more important is an adequate model of the flow near
46
.
1979011859-052
.°
the trailing edge of the airfoil. It suffices to represent the
wake by a pair of parallel streamlines across which the pressure
balances. Large favorable pressure gradients can be tolerated on
the lower surface to provide for heavy aft loading. However, a
Stratford distribution should be used at the rear of the upper
surface to limit the adverse pressure gradient there so as to
avoid separation. Airfoils designed with this in mind perform
well over a wide range of conditions.
Shockless airfoils serve as an acceptable mathematical
model for the design of supercritical wing sections. Drag creep
can be reduced by restricting the size of the supersonic zone in
the shockless flow. The design method has been extended to in-
clude cascades of airfoils such as occur in compressors and tur-
bines. Present codes can handle gap-to-chord ratios down to
unity (see figures 1 and 2). The concept of a supercritical com-
pressor blade has been tested successfully by Harry Stephens of
Pratt and Whitney Aircraft in a cascade wind tunnel of the DFVLR
in Germany (see figure 3).
The latest version of the design co_e enables one to assign
the pressure distribution with a certain tolerance and still ob-
tain shockless flow when it exists (ref. 3). This is achieved
by formulating a new boundary value problem in the unit circle
of the complex plane of one of the characteristic coordinates for
the gas dynamics equations. It is worthwhile to review briefly
the theory underlying the new code, which seems to be quite suc-
cessful in practice.
In terms of characteristic coordinates _ and n , the partial
differential equations for the velocity potential _ and stream
function _ of plane compressible flow can be expressed in the form
where M is the local Mach number and p is the density. Through
analytic continuation the equations remain valid in the complex
domain. The speed q and angle 8 of the flow are related to
and q by the formulas
log f(_) = q
log f(_) = j-- q
where f is an analytic function mapping the flow onto a region
that can be taken as the unit circle I_I _ i. Analytic contin-
uation around the sonic line can be performed along paths on
47
1979011859-053
0°
J
which I-M 2 does not vanish.
The problem of finding an airfoil on which the pressure is a
prescribed function of the arc length is equivalent to the prob-
lem of finding a profile on which the speed is assigned as a
function q =q($) of the velocity potential. In the unit circle
I_I < 1 this reduces to the question of determining the map func
tion f and the stream function _ from boundary conditions of the
form
R_ log f(_)} = h(q) ,
R_(6,[) -ik _ (_,_)) = 0 ,
where h is known in terms of q and k is a given real constant.
This nonlinear boundary value problem can be solved iteratively
by first guessing _ so that f can be calculated and then com-
puting _ and $ so that the process can be repeated. For an ap-
propriate choice of h and k the iterations converge even in the
transonic case to a shockless solution that yields the prescribed
pressure distribution, except for minor deviations that must be
expected in the supersonic zone. (See figures 4 and 5.)
Nu:aerical computations suggest that the boundary value
problem that has been formulated in the complex domain is well
posed. This has been proved in a very special case by Sanz
(ref. 4). He considers the Euler-Poisson-Darboux equation
1
u +U +--u =0
XX yy 3y y
obtained by bringing the Tricomi equation into canonical form.
The problem becomes to find a solution u in the unit circle with
prescribed values of Re{u} on the boundary. Sanz introduces a
decomposition
u(x,y) = y2/3Ul(X,y2) + u2(x,y2)
of u into two solutions that are easily reflected across the x-
axis. He is led to boundary value problems for Ul and u2 in the
upper unit semicircle that can be solved in closed form.
I Numerous examples of shockless airfoils have been calculated
i by the method of complex characteristics using the code with pre-
k scribed q($). These include symmetric airfoils with two super-
sonic zones (figs. 4 and 5), compressor airfoils suitable for
stators (fig. i), and turbine a_rfoils with large turning angles
(fig. 4). The procedure is largely automatic, and it is rela-
48
tively easy to implement new ideas about design that depend on
the pressure distribution. However, further conformal trans-
formation of the characteristic coordinates will be required if
gap-to-chord ratios significantly lower than unity art desired.
It would be of interest to generalize the method of designing
shockless airfoils based on giving a pressure distribution to the
case of three-dimensional transonic flow past a swept wing. This
might be achieved by modifying the analysis code appropriately
and introducing an artifical viscosity that smears shocks. To
start with it would be helpful to have a fast STAR version of the
swept wing code published recently by Jameson and Caughey (refs.
2 and 5). Work on these proposals is in progress at the Courant
Mathematics and Computing Laboratory.
REFERENCES
i. Bauer, F.; Garabedian, P.; and Korn, D.: Supercritical
Wing Sections, Lecture Notes in Economics and Mathematical
Systems, vol. 66, Springer-Verlag, New York, 1972.
2. Bauer, F.; Garabedian, P.; Korn, D.; and Jameson, A.:
Supercritical Wing Sections II, Lecture Notes in Economics
and Mathematical Systems, vol. 108, Springer-Verlag,
New York, 1975.
3. Bauer, F.;Garabedian, P.; and Korn, D.: Supercritical
Wing Sections III, Lecture Notes in Economics and
Mathematical Systems, vol. 150, Springer-Verlag, New York,
1977.
4. Sanz, J.: A Well Posed Boundary Value Problem in Transonic
Gas Dynamics, OER Research and Development Report C00-3077-
149, Courant Math. and Computing Lab., New York Univ.,
February 1978.
5. Jameson, A.; and Caughey, D.: Numerical Calculation of
the Transonic Flow Past a Swept Wing, E_DA Research and
Development Report C00-3077-140, Courant Inst. Math. Sci. ,
New York Univ., June 1977.
!
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1979011859-055
	DINPUT
-l.2 CP _" + OUTPUT
-.8
-.4
0.0
•4 __
:.21
Hl=.?O? H2=.534 DEL TH= R5.O0 G/C= .99
Figure i.- Stephens stator blade.
5O
1979011859-056
1.33
M
-- iNPUT
l.O0 + OUTPUT
M1:.363 M2:.762 OEL TH= 99.26 G/C=I.IO
Figure 2.- Sanz turbine blade.
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1979011859-057
I!_ _"_, ;'!
/ ,, :,-
;t
.2_
k
0.0
i
i'
PRRTTRND WHITNEYCOMPRE888RRIRFOIL R=I.I MILLION
-- THEORY MI=.?80 M2=.480 DEf.TH=25.O
EXPERIMENT MI=.775 M2=.544 DEL TH=2S.S L08S=.0196
, Figure 3.- Supercritical compressor test data.
i 52i
'_
f,
1979011859-058
INPUT
1-25 M + OUTPUT
M=.831 CL= .000 DX=-.O00 DY= .021 TIC-.110
Figure 4.- Symmetric sho,:kless airfoil.
1979011859-059
1.20
I
-1.20 40 .80 1.20
-1.20
M:.831 CL: .000 OX:-.OOd BY: .021 T/C:.110
Figure 5.- Complex hodoEraph plane.
!
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1979011859-060


Transonic airfoil codes

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