NASA Contractor Report 3991 

Transonic Airfoil Analysis 

and Design in Nonuniform Flow 

Jen-Fu Chang and C. Edward Lan 

GRANT NAGI-308 

JUNE 1986 

NI\S/\ 

LIBRARY COpy 

' ....... 

111111111111111111111111111111111111111111111 1 

NF01990 
NASA Contractor Report 3991 

Transonic Airfoil Analysis 

and Design in Nonuniform Flow 

Jen-Fu Chang and C. Edward Lan 

Flight Research Laboratory 

The University of Kansas Center for Research, Inc. 

Lawrence, Kansas 

Prepared for 

Langley Research Center 

under Grant NAGl-308 

NI\SI\ 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Branch 

1986 
SUMMARY 

A nonuniform transonic airfoil code is developed for 

applications in analysis, inverse design and direct optimization 

involving an airfoil immersed in propfan slipstream. Problems 

concerning the numerical stability, convergence, divergence and 

solution oscillations are discussed. The code is validated by 

comparing with some known results in incompressible flow. A 

parametric investigation indicates that the airfoil lift-drag ratio 

can be increased by decreasing the thickness ratio. A better 

performance can be achieved if the airfoil is located below the 

slipstream center. Airfoil characteristics designed by the inverse 

method and a direct optimization are compared. The airfoil designed 

with the method of direct optimization exhibits better 

characteristics and achieves a gain of 22 percent in lift-drag ratio 

with a reduction of 4 percent in thickness. 

i 
TABLE OF CONTENTS 

SUMMARy ••••••••••••••••••••••••••••••••••••••••••••••••••••••• i 

TABLE OF CONTENTS •••••••••••••••••••••••••••••••••••••••••••• ii 

LIST OF SYMBOLS •••••••••••••••••••••••••••••••••••••••••••••• 1 V 

1 • INTRODUCTION •••••••••••••••••••••••••••••••••••••••••••••••••• 1 

2. THEORETICAL DEVELOPMENT ••••••••••••••••••••••••••••••••••••••• 7 

2.1 Equations for Two-Dimensional Flow ••••••••••••••••••••••• 7 

2.2 Boundary Conditions ••••••••••••••••••••••••••••••••••••• 10 

2.3 Coordinate Transformation ••••••••••••••••••••••••••••••• 12 

2.4 solution Procedures for Analysis and Inverse 

Design •••••••••••••••••••••••••••••••••••••••••••••••••• 18 

2.5 Formulation for Direct Optimization ••••••••••••••••••••• 19 

3. CONVERGENCE PROBLEMS AND SOLUTIONS ••••••••••••••••••••••••••• 22 

3.1 Grid Halving •••••••••••••••••••••••••••••••••••••••••••• 22 

3.2 Supersonic Damping Factor ••••••••••••••••••••••••••••••• 23 

3.3 Over-Relaxation Factor •••••••••••••••••••••••••••••••••• 24 

3.4 Convergence Criterion ••••••••••••••••••••••••••••••••••• 24 

3.5 Under-Relaxation Factor for Airfoil Shape 

in Inverse Design ••••••••••••••••••••••••••••••••••••••• 25 

3.6 Under-Relaxation Factor for Boundary Layer 

Displacement Thickness •••••••••••••••••••••••••••••••••• 26 

4. NUMERICAL RESULTS AND DISCUSSIONS •••••••••••••••••••••••••••• 27 

4.1 Program Validation •••••••••••••••••••••••••••••••••••••• 27 

4.2 A Parametric Study in Transonic Nonuniform 

Flow •••••••••••••••••••.•••••••••••••••..••••••••••••.•. 28 

ii 
4.3 

4.4 

TABLE OF CONTENTS, continued 

Inverse Design ...•..••.••..•.....•............••..•..... 30 

Direct Optimization ••••••••••••••••••••••••••••••••••••• 32 

5. CONCLUSIONS •••••••••••••••••••••••••••••••••••••••••••••••••• 34 

6 • REFERENCES ••••••••••••••••••••••••••••••••••••••••••••••••••• 35 

iii 
English 

Symbols 

a 

b 

F 

f 

G 

g 

ILE 

ITE 

IMAX 

LIST OF SYMBOLS 

Definition 

--+e • 

Constant in y-stretching function controlling the 

grid size near the airfoil 

Constant in x-stretching function controlling the 

grid size near the interface 

Constant in x-stretching function controlling the 

far-field grid size. 

Speed of sound, or 

Coefficient of coordinate transformation in airfoil 

region. See Equation (21). 

Fourier coefficients in Equation (39) 

Coefficient in coordinate transformation. See 

Equation (21). 

Sectional lift coefficient 

Sectional drag coefficient 

Sectional pitching moment coefficient relative to the 

quarter chord 

Pressure coefficient 

Total derivative, or 

Drag 

Rotation function, or 

Objective function 

Transformation in the x direction 

Constraint function 

Transformation in 'the y direction 

x-direction index at the leading edge 

x-direction index at the trailing edge 

x-direction index in the positive far field 

iv 
English 

.Symbol 

JMAX 

JB 

LID 

M 

N 

NIC 

p 

P 

+ 

q 

t 

u 

v 

x 

y 

LIST OF SYMBOLS, continued 

Definition 

y-direction index in the positive far field 

Index for solid boundary points 

Unit vector in the z-direction 

Lift 

Lift-to-drag ratio 

Mach number 

Number of interpolation points 

Number of constraints 

Pressure 

Some quantity 

Veloci ty vector 

Radius of the propfan slipstream 

Allowable trailing edge thickness 

Time 

Velocity component in the x direction 

Velocity component in the y direction 

Axis pointing downstream 

x-coordinate after which airfoil shape is allowed to 

change 

x-coordinate where two coordinate transformations 

meet 

Axis normal to the x axis and pointing upwards, or 

Airfoil ordinate 

v 
Greek 

Symbols 

y 

r 

n 

8 

p 

w 

~y 

Mlmax 

LIST OF SYMBOLS, continu~d 

Definition 

Angle of attack 

Prandtl-Glauert compressibility factor 

Ratio of specific heats 

Circulation 

Transformed coordinate in the x direction 

~ coordinate where two coordinate transformations 

meet 

Transformed coordinate in the y direction 

Polar angle 

Density 

Reference density 

Disturbance potential 

Velocity function 

Stream function 

Vorticity 

Del operator 

Spacing in the x coordinate 

Spacing in the y coordinate 

Spacing in the transformed x coordinate 

Maximum difference among ~9 relative to 

previous iteration 

vi 
Superscripts 

and subscrip~~ 

o 

00 

I I 

b 

n 

ref 

s 

sw 

te 

u 

LIST OF SYMBOLS, continued 

Definition . 

Vector 

Nondimensional quantity 

Stagnation quantity, or 

Quantity on the axis, or 

Reference quantity 

Freestream quantity 

Absolute value 

Boundary quantity 

Lower limit 

Normal direction 

Reference quantity 

Streamline direction 

swirl quantity 

Trailing edge 

Upper limit 

vii 
LIST OF SYMBOLS, continued 

Abbreviations Definition 

NASA National Aeronautics and Space Administration 

ACEE Aircraft Energy Efficiency program 

QVLM Quasi-Vortex Lattice Method (Ref. 24) 

FL028 A conservative transonic code (Ref. 28) 

FL022NM A nonconservative transonic code with Nash-McDonald 

boundary layer method (Ref. 29) 

CPU Central Processing Unit 

TRANDES TRANsonic DESign (Ref. 42) 

CONMIN CONstrained MINimization (Ref. 35) 

viii 
1. INTRODUCTION 

Due to the energy crisis in 1973, NASA and industry beqan to 

search for ways to reduce aircraft fuel consumption and established 

the Aircraft Energy Efficiency (ACEE) program. It was found that 

turboprop engines have the potential to significantly reduce fuel 

consumption for a qiven mission relative to advanced turbofans used 

today by extending their hiqh propulsive efficiency to about Mach 

0.8. To do this, Hamilton Standard proposed the Propfan (Ref. 1) 

concept, using thin, short, mUlti-swept blades and area-ruled 

spinner and nacelle to reduce the wave drag. 

Under the sponsorship of ACEE, NASA and industry (Refs. 2-4) 

performed extensive studies and concluded that in the absence of 

excessive aerodynamic interference, propfan propulsion systems are 

18 percent more efficient in the net thrust specific fuel 

consumption than high by-pass-ratio turbofans at Mach 0.8. They are 

even hetter for lower Mach numbers. Such potential benefit led to 

the establishment of the NASA high-speed turboprop program (Ref. 

5). Windtunnel tests (Refs. 6-15) were conducted to provide data 

bases for verification of analytical and computational techniques, 

assessment of the maqnitude of each factor, and installation aspects 

of the system. Welge and Crowder (Ref. 6) used an ejector-nacelle 

simulator to assess the magnitude of the aerodynamic interference of 

a simulated propeller stream on a 32°-swept supercritical wingbody. The interference drag amounts to an increase of ten drag 

counts (one drag count = .0001 of drag coefficient) for normal swirl 
angle. Up-inboard (posi ti ve) swirl generally was found to have less 

drag than up-outboard. These results were found to be independent 

of airplane lift coefficient or free-stream Mach number. For a 30 0 -

swept supercritical wing-body, Boctor, et ale (Ref. 8) used a 

compressible panel method to perform the analysis and design. Test 

results for the configuration at a Mach number of 0.7 showed the 

potential for recovering up to 50% of the thrust lost due to 

swirl. However, at higher Mach numbers, the occurrence of shock 

wave and flow separation could offset the thrust recovery. Welge 

(Ref. 9) also used a compressible panel method to determine the flow 

field. He also studied the propfan integration for a OC-9 Super 80 

with design Mach number of 0.8 based on the work in References 7 and 

10. The propeller and forward nacelle are canted inboard 0.5 degree 

for alignment with local streamlines, and the region of the nacelle 

over the upper surface is aligned with the average surface 

streamline angle. The nacelle also had a 4.50 downtilt angle with 

respect to the propfan axis. He claimed that after offsetting 

unfavorable increment for weight, a 27 percent advantage in cruise 

specific fuel consumption over the current JT80-209 was achieved. 

Smith and Levin (Refs. 11-12, 14) confirmed high drag for a nacelle 

installation in wind-tunnel testing. Inboard fillets and an 

outboard strake similar to that recommended in Reference 8 were 

tested and verified to reduce drag. Bartlett (Ref. 15) conducted 

wind-tunnel testing for a propfan installed on an upswept 

supercritical wing including overwing and underwing 

2 
configurations. He found that the swirl is recoverable as thrust by 

the wing, and the propfan slipstream causes increase in wing lift. 

From a theoretical point of view, when the free stream is 

nonuniform, the flow becomes rotational so that the potential flow 

equation is not valid. The main type of flow nonuniformity 

considered here arises from velocity variation inside the propfan 

slipstream and from different velocity magnitudes inside and outside 

the slipstream. When a wing section is immersed in the slipstream, 

its aerodynamic characteristics can be substantially different from 

those under a uniform flow condition with a free-stream velocity 

equal to an average value inside the slipstream or outside it. To 

predict correctly the aerodynamic characteristics, any theoretical 

method must be capable of properly accounting for the effect of this 

velocity nonuniformity. 

Historically, Koning (Ref. 16) pioneered the formulation of the 

incompressible interaction problem involving a slipstream. Ferrari 

(Ref. 17) treated the compressible counterpart. A lifting surface 

theory for a circular jet was used by Rethorst (Ref. 18). Chow, et 

ale (Ref. 19), used the vorticity equation to solve the twodimensional nonuniform incompressible flow by a finite difference 

method. Jameson (Ref. 20) combined the lifting line theory (for 

wing) and vortex sheets (for slipstream) to solve problems involving 

rectangular and elliptical slipstream. Ting, et ale (Ref. 21), used 

the lifting line theory and asymptotic expansion techniques to solve 

the interference of wings and multi-propellers. Using flow 

3 
singularities, Shollenberger (Ref. 22) investigated wing/jet 

interaction without Mach number nonuniformity. Lan (Ref. 23) 

included compressibility effects by using the Quasi-Vortex-Lattice 

Method (QVLM, Ref. 24) plus two vortex sheets to represent the 

slipstream effects. Chin (Ref. 25) solved the inverse problem in 

two-dimensional nonuniform incompressible flow by using stream 

function formulation and the thin airfoil theory. 

More recently, transonic effect on slipstream-wing interaction 

associated with propfan installation has received great attention. 

various theoretical flow models and numerical schemes have been 

proposed. Rizk (Ref. 26) used two separate sets of 'transonic smalldisturbance equations for inside and outside the slipstream and 

then imposed the interface conditions. He also included the 

assumption that the undisturbed velocity in the slipstream is nearly 

uniform. Samant, et ale (Ref. 27), applied a full potential code 

called FL028 (Ref. 28) based on a conservative finite volume 

approach to a wing-body configuration. The slipstream effect is 

simulated by the transpiration condition on the solid surface within 

the slipstream instead of the usual no-flow-through condition. 

Narain (Ref. 29) incorporated in the FL022NM code (Ref. 30) the 

slipstream effect by superimposing the rotational slipstream 

velocities into the coefficients of the steady nonconservative full 

potential equation by using either a rigid cylindrical semi-infinite 

slipstream model or a rigid converging one. The magnitude of these 

rotational velocities was calculated from propeller experimental 

4 
data. Whitfield and Jameson (Ref. 31) used the three-dimensional, 

time-dependent Euler equation with force term (jump in total 

pressure) to simulate the propeller. They also incorporated a twodimensional compressible turbulent inverse integral boundary layer 

code (Ref. 32) for the viscous displacement thickness effect. Shock 

can be captured. 

Since supercritical wings have blunt noses, the smalldisturbance assumption is not accurate near that critical region. 

Cylindrical modeling of the slipstream is not adequate for the 

lifting cases because the boundary will deform. The transpiration 

boundary condition, although accounting for the velocity increment 

of the propeller, causes absorption of energy from outside the 

cylindrical surface because the energy is conserved. The Euler code 

is theoretically ideal for the inviscid flow, but it requires 

comparatively long computer time. In addition, in all these 

theoretical methods no design of airfoils or wings has been 

considered. 

In the present investigation, design of supercritical airfoils 

in transonic nonuniform flow will be emphasized. The formulation is 

to allow existing full-potential transonic codes to be modified for 

nonuniform flow through the introduction of a rotation function. 

The idea of using a rotation function in the full-potential equation 

was originated by Brown (Ref. 23) in solving a transonic nozzle 

flowfield. It is similar to the dual-potential approach adopted by 

Chaderjian and Steger (Ref. 34) to solve the transonic rotational 

5 
flow. The effects of airfoil thickness, camber, swirl and airfoil 

location in and out of the slipstream will be examined. The design 

of airfoils will be done by both an inverse method and a direct 

optimization through an optimizer (Ref. 35). 

6 
2. THEORETICAL DEVELOPMENT 

2.1 Equations for Two-Dimensional Nonuniform Flow 

In the absence of viscosity, heat conduction and body forces, 

steady compressible flow is described by Euler's equation, 

+ + 

(q • V)q 

1 - - Vp P 

and the continuity equation, 

+ + + V • (pq) = q • Vp + p(V • q) 

= .Qe. + p(V • q) = 0 Dt 

+ where q is the velocity vector, p is the density, and p is the 

(1) 

(2) 

static pressure. If the two-dimensional stream function is defined 

as 

1/1 = f .L udy Po 

(3) 

where Po is the reference density, the Euler's equation along a 

streamline can be shown to be 

Dp = _ e. DlCil 2 

Dt 2 Dt (4) 

Under isentropic assumption along a streamline, the substitution of 

Dp _ 2 Qe. into Equation (4) gives Dt - a Dt 

Dp = _ --.E....- Dlql2 (5) Dt 2a2 Dt 

where a is the local speed of sound. For a two-dimensional flow, 

Equation (5) can be expressed in the following form: 

7 
(6) 

According to Brown (Ref. 33), nondimensionalizing Equation (6) with 

respect to the stagnation speed of sound gives 

2 -2 au -- av au -2 -2 av (a - u ) - - uv(- + -) + (a - v ) - = 0 (7) ax ax ay ay 

which is seen to be of the same form as Equation (6). For 

simplicity, the bar notation will be neglected from now on. Define 

a velocity function ~ and a rotation function F such that 

u ~ + F 

x 

v = ~ 

y 

Substituting Equation (8) into Equation (7) gives 

or 

(8) 

2 2 2 2 2 2 (a - u)~ - 2uv~ + (a - v)~ = uv F - (a - u )F xx xy yy y x 

where subscripts denote partial differentiation. 

The two-dimensional vorticity vector t 

+ + w=Vxq=K---) 7( av au 

ax ay 

is given by 

(9) 

(10) 

where k is the unit vector in the z-dir~ction. The magnitude of the 

vorticity vector. using Equation (8) is given by 

w = It I = v - u = ~ - (¢ + F ) = -F 

x Y xy xy Y Y (11 ) 

It follows that 

F = u - v 

y y x 

8 
1 2 1 elP 0 

= (1 - y; Mo (~)J yuP (~) ayo 

(12) 

where Po is the stagnation pressure on a streamline, y is the ratio 

of specific heats, and F, Mo and u are all referenced to the 

stagnation speed of sound (ao) on a streamline. The F values are 

obtained by integrating Equation (12). 

Local density is needed to calculate the stream function (see 

Equation 3). Along a streamline, density can be related to its 

stagnation quantity as 

p = po(~)(1 - Y2- 1 Mo 2(~)JY~I (13) 

00 

Since both po(~) and Mo (~) vary with streamline, they have to be 00 

calculated iteratively as follows: 

(1) Assume the initial density distribution to be that of the 

undisturbed one. 

(2) Calculate the stream function at a constant x according to 

Equation (3). 

(3) Determine p (~) and M (~) as a function of ~. o 0 00 

(4) Update p according to Equation (13). 

(5) Check the convergence and decide if steps 2 to 5 should be 

repeated. 

The iterative process will converge in several iterations. Note 

that the calculation of the rotation function F is updated according 

to Equation (12) as soon as the stream function is known at a given 

field point (step 5). 

9 
Along a streamline, the following form of .energy equation for a 

perfect gas can be used: 

2 2 a 2(1/1) 1 2 1 2 a~ (1/1) a 0 zq + = - q (1/1) + = (14) 

y - 1 2 ~ y - 1 y - 1 

If the entropy is assumed to be nearly constant along a streamline, 

i.e. only weak shocks are present, the pressure coefficient can be 

derived from Equation (14) as 

C 

P 

p - p~ 

= -=----~-- 1 2 "2 p~ q 

ref ~ ref 

2 

y;' co ref 

2.2 Boundary Conditions for Two-Dimensional Flow 

Assume p u to be a function of y only in the free stream. ~ ~ 

Using the continuity equation for a two-dimensional flow, 

a(pu) + a(pv) = 0 

ax ay 

it can be shown that p v is a function of x only. Thus in the ~ ~ 

(15) 

(16) 

free stream, if v~ vanishes, then p~ may vary with y. On the other 

hand. if v F 0 • then both p and v~ must be constant. ~ ~ 

The free-stream rotation function is constructed from the 

difference in the undisturbed nonuniform velocity and the uniform 

part. 

10 
Since the free stream static pressure must be constant (Ref. 

36), one of the following five parameters must be specified or 

assumed to be constant: stagnation pressure, temperature, 

stagnation temperature, density and stagnation density. For 

example, the experimental stagnation pressure is specified to 

simulate the power effect. Once one of the parameters is specified, 

the rest can be determined through the state equation and/or the 

isentropic compressible relations along a streamline. 

According to References 37 and 38, the far-field boundary 

condition for ~ can be expressed as 

r -1 ~ = q=(~)(x COSa + y sina) + 2TI tan [Stan(a - 8)] (17a) 

where q (~) is the free-stream velocity, a is the angle of attack, r 

= 

is the circulation, e is the compressibility factor and 8 is the 

corresponding polar angle. If the measured swirl angles (asw) are 

included, the following equation is used: 

r -1 ~ = q (~)(x cos a + y sina ) + ~ tan [Stan(a - 8)] = sw SW L.TI SW 

(17b) 

Since ~ approaches infinity (Equation 17) at the far field, the farfield boundary condition is satisfied at midpoints of the two 

outermost transformed grid points. 

The surface tangency condition is written as 

(dy) = (~) dx bub 

where b represents the boundary values. This equation, in 

(18) 

combination with Equation (15), can be used for the inverse design: 

11 
{I -

(y -

1 

q co {--v 

ref 1 + (~)2 

ub 

2.3 Coordinate Transformation 

(19) 

Following Carlson (Ref. 38), the flow field is divided into 

three regions as shown in Figure 1. The stretching is symmetrical 

about the origin and is given by 

in regions I and III and by 

2 

x = ~(a + bi; ) 

in region II. The constants a and b are determined by the 

requirements that 

and 

(21) 

(22a) 

(22b) 

The constant A2 controls the grid spacing in the vicinity of x4' 

usually near the leading and trailing edges; while A3 determines the 

physical location of the grid line adjacent to the grid edges. 

In the y-direction, the stretching function is 

(23) 

12 
where Al controls the grid size near the airfoil. 

Notice that these stretchings map the infinite x,y plane 

-00 " x .. 00 

(24) 

into the finite computation plane 

-1 .. n .. 1 (25) 

where ~4 determines the amount of the computational plane confined 

to the vicinity of the airfoil. 

A typical grid system is shown in Figure 2. 

Let P(x,y) be a function of x and y, and f and g be the 

transformations used in the y directions, respectively; i.e., 

f = (26) 

where ~ and n are the transformed coordinates of x and y. The 

following relation can be derived: 

P P~~x fP ~ x 

P = P n gP Y n Y n 

P 

xx (fp ) ~ 

~ ~ x f(fP ~\ 

P 

xy (fP ) n = ~ n Y fgP ~n 

P (gP ) n = g(gP ) (27) 

yy n n Y n n 

Thus Equations (8) and (9) can be transformed into 

13 
2 2 

= uvgF n - (a - u )fF~ 

u = F + u~ 

v = g4> n 

Since far-field boundary conditions of 4> are satisfied at the 

outermost half grid points instead of the edge points, using 

(28) 

(29) 

analytic expresions of f and g will make the computation of u and v 

very inaccurate, particularly in the far field. Therefore, the 

following first order approximation of f is used in regions I and 

III; and that of g is used for the outer half regions. 

f = ~ '" lls. 

ax llx 

g =~ ay 

lln "'-

lly 

(30) 

(31) 

Similar problems occurred in calculating the stream function 

w. The stream function values are obtained by integrating from the 

far field in the lower half plane. In the far field. the y 

intervals are so large that any slight inaccuracy in u can cause 

large difference in Wand thus the stagnation quantities on the 

airfoil streamline. Therefore, the reference point from which 

Equation (12) is integrated is shifted to the lower second grid 

point, instead of the exact far -field. 

14 
Note that the value of stream function, and hence the rotation 

function, on the airfoil lower surface should be equal to that on 

the upper surface. However, because of numerical inaccuracy 

associated with integration between grid points and airfoil surface 

boundary points, their values may be different. In this case, their 

values are averaged for the purpose of identifying the dividing 

streamline. 

The transformed Equation (28) is then reduced to a finitedifference form. To obtain the correct zone of dependence of ~ 

values in the supersonic region, the rotated difference scheme (Ref. 

39) is used. Note that the latter is based on streamline 

coordinates sand n. Derivatives in streamline coordinates s,n and 

x,y coordinates are related as follows: 

Ps 

u v 0 0 0 Px -

q q 

Pn 

v u 0 0 0 Py - - q q 

2 2uv 2 

Pss 0 0 u v Pxx = (32) 2" -2- 2 

q q q 

2 2 

Psn 0 0 uv u - v uv P -2 2 2 xy 

q q q 

2 2uv 2 

Pnn 0 0 v u P 2 yy - -2- 2 

q q q 

or, conversely, 

15 
o o u v -

q q 

v u - - 0 0 

q q 

Pxx 0 0 

2 

u 2uv 

2 - -2-

q q 

2 

o o uv u - v 

Z 2 

q q 

o 

2 

v 2uv 

2 o -2-

q q 

Now Equation (28) can be written as 

where 

cp 

ss 

.0 Ps 

0 Pn 

2 

v 

2 Pss (33) 

q 

2 

uv -Z Psn 

q 

2 

u 

2 Pnn 

q 

(34) 

(35) 

Equation (28) or (34) is seen to be quite similar to those used 

by Carlson (Refs. 38, and 40-42), except for the nonhomogeneous 

terms on the right side of the equation. Therefore, Carlson's 

transonic potential flow code, TRANDES, is modified to solve the 

present problem. Carlson's code has the following features: 

16 
(a) the coordinate system used was the Cartesian system. 

(b) A rotated finite difference scheme was used. 

(c) A damping term was added explicitly to the difference 

equations for numerical stability, instead of using 

Jameson's implicit damping formulation. 

(d) Airfoil boundary conditions in both analysis and inverse 

design were satisfied by introducing dummy points inside 

the airfoil. 

(e) The resulting finite-difference equations were of the 

tridiagonal type and were solved iteratively by column 

relaxation. 

(f) The Nash-McDonald method is used for boundary layer 

calculation. 

Main differences between the present formulation and Carlson's 

are as follows: 

(a) A velocity function ~ is used in the present formulation. 

instead of Carlson's disturbance velocity potential. The 

latter approaches zero in the far field, but not the 

former (see Equation 17a). This behavior makes the farfield boundary conditions more difficult to satisfy in the 

present formulation. The ~ formulation is necessary in 

the present problem to result in the same form of the 

governing equation. 

(b) The present equations are nondimensionalized with the 

stagnation speed of sound which varies from streamlines to 

17 
streamlines. Therefore, the streamline with which a field 

point is associated must be identified in the relaxation 

solution. 

2.4 Solution Procedures for Analysis and Inverse Design 

The solution procedure for the transonic nonuniform flow 

consists of the following steps: 

(1) Input options and parameters. 

(2) Set up grid coordinates and transformations. 

(3) Initiate or input variables. 

(4) Input airfoil shape (analysis) or pressure distribution 

(design). 

(5) Set up far-field values for ~. 

(6) Calculate stream function at the far field and set up 

interpolation coefficients for p , p , etc. 

o 0 

(7) Solve the flow field in front of the airfoil. 

(8) Solve the flow field on the airfoil depending on analysis 

or design mode. 

(9) Solve the wake region. 

(10) Calculate circulation from Kutta conditions. 

(11) Update the far-field boundary conditions. 

(12) Periodically update the stream function, density, 

stagnation pressure and stagnation pressure gradient to 

obtain the new rotation function. 

18 
(13) In the design mode, periodically update the airfoil 

shape. 

(14) In the analysis mode with viscous interaction option, 

periodically update the boundary layer thickness 

distribution. 

(15) Repeat steps 7-14 until ~ converges or the number of 

iterations exceeds a limit. 

(16) Calculate the last values of rotation funtion, shapes and 

boundary layer if applicable. 

(17) Compute pressure distribution. 

(18) Plot Mach plot. 

(19) Compute airfoil characteristics. 

(20) Plot pressure distribution. 

(21) Halve the grid if applicable. 

(22) Repeat steps 2-21 if applicable. 

(23) In the design mode, subtract the boundary layer to obtain 

the actual airfoil shape. 

2.5 Formulation for Direct Optimization 

In a direct optimization to design an airfoil, the analysis 

method described above is coupled with an optimizer, CONMIN (Ref. 

35). The design problem can be stated as follows: 

Minimize 

(36) 

19 
Subject to 

Gi ' 0, i = 1, ••• , NIC (37) 

where cd is the sectional drag coefficient including wave drag and 

friction 'drag. Gi represents the constraints. Since the 

constraints work best when magnitudes of their gradients are of the 

same order of magnitude, a scale factor may be applied to 

constraints whose gradients are too small or too large. 

The following constraint formulations have been used in the 

present optimization problem. 

1. Lift constraint: 

(38) 

x 10 

where ct(u) and ct(t) are the prescribed upper and lower 

bounds of lift coefficient, respectively. This type of 

constraints is necessary because an exact numerical 

constraint on c~ can not be achieved in a numerical 

optimization. 

2. Trailing-edge closure constraint: 

G = (Y - Yn ) /THTE - 1.0 te u ~ te (39) 

where Y and Yare the nondimensiona1 upper and lower 

Ute R. te 

surface coordinates at the trailing edge, respectively, 

and THTE is the allowable trailing edge thickness. 

20 
To reduce the number of design variables, airfoil coordinates 

are expressed in a Fourier cosine series, i.e. 

N 

Y = I 

j=I 

a. cos(j - 1) e 

J 

(40) 

where y may be the upper or lower surface coordinates. The Fourier 

coefficients are given by 

1 

2 

n 

J y de 

o 

n 

J y cos(j - I)ede 

o 

e

k = (2k - I)n/2N 

(4Ia) 

(4Ib) 

(42a) 

(42b) 

where xl is the x-station after which the airfoil shape is to be 

modified. These Fourier coefficients are used as the design 

variables. Equation (40) is used only away from the nose region. 

This means that the nose region, within 5% say, remains unchanged 

during optimization. If the nose shape is to be changed, an 

expression different from Equation (40) should be used. The flow 

chart for the optimization is presented in Figure 3. 

21 
3. CONVERGENCE PROBLEMS AND SOLUTIONS 

3.1 Grid Hal vin~ 

Multigrid calculation is frequently used to accelerate the 

convergence. When the coarse grids are halved, medium grids are 

obtained with the number of grid points doubled. The starting 

values of ~ in the latter are normally obtained by interpolating the 

values of ~ in the coarse grids. In this process, the new starting 

~ values in the far field may change so greatly from the values in 

the coarse grids as to cause divergence. A solution to this problem 

is to set the values next to the new far-field boundary to the old 

far-field values, i.e. 

~i,jmax-l(new) = ~i,jmax-1/2 (old) 

~i,2(new) ~i,1+ 1/2 (old) 

(42) 

~ imax-I ,j (new) ~ imax-1f2 ,j (old) 

~2,j(new) = ~1+1/2 ,j(old) 

where "new" indicates values in the medium grids and "old" means 

values in the coarse grids. Note that jmax is different in both 

grids. The new far-field ~ values (at the outermost half grid 

points) are calculated with Equation (17a). 

Interpolation of old ~ values for the new grids is made only on 

the difference between the total ~ values and the undisturbed 

ones. The same process is used again when grids are further halved. 

22 
For ~ values across the Kutta-Joukowsky cut downstream of the 

trailing edge, the ~ values just below the x-axis are obtained by 

extrapolating those from the lower flow region. 

3.2 Supersonic Damping Factor 

Supersonic damping is needed for the stability of the 

relaxation solution involving supersonic flow region. In the 

present ~-formulation, instead of the formulation with the 

disturbance velocity potential, a larger value of damping factor 

appears to be always needed. At the beginning of calculation in 

each grid system, a value of up to 5.0 is used in a transonic 

flow. During the iterative solution process, the damping factor 

will be increased by 0.1 if the cumulative ~~max increases by a 

certain critical value, except that in the last grid calculation 

where an increment of 0.01 is used after 200 iterations. On the 

other hand, if the cumulative ~~max decreases by the same critical 

value, the damping factor will be decreased by 0.1. In addition, 

the damping factor is also restricted to be within a minimum and a 

maximum value. This method of changing the damping factor during 

iterations appears to work well in the transonic nonuniform flow. 

In direct optimization, the starting damping factor usually 

should be less because the initial ~ values are from the converged 

solution of a previous analysis run. Typically, a value equal to 

half of that used in the initial analysis run would be adequate. 

23 
3.3 Over-Relaxation Factor 

An over-relaxation factor is used only in the subonic region. 

If ~~max changes sign frequently, the over-relaxation factor needs 

to be reduced. On the other hand, if ~~max does not change sign, 

the factor should be increased. However, oscillatory convergence is 

also possible. The criteria for changing the over-relaxation factor 

built into the present code are that if the number of sign changes 

is greater than five and more than three of which are with ~~max 

greater than twice the convergence criterion, then the factor will 

be decreased by 0.05. On the other hand, if the number of sign 

changes is less than three, it will be increased by 0.05. A maximum 

value of 1.7 is built into the program, and a minimum value is to be 

input. 

3.4 Convergence Criterion 

In the nonuniform flow, the equation has been shown to be 

nondimensionalized with respect to the stagnation speed of sound. 

Therefore, the convergence criterion should be more stringent for 

the nonuniform flow. However, if a too stringent convergence 

criterion is used in the coarse grids, divergence may occur when the 

grids are halved. Therefore, a relaxed convergence criterion is 

usually needed for coarse and medium grids. 

On the other hand, at the beginning of iterations for each grid 

system, ~ values are usually far from being correct. It tends to 

24 
work better if the convergence criterion is relaxed initially until 

the convergence reaches a certain stage (called an acceptable region 

in the code). Then the criterion is tightened until the convergence 

reaches the next stage (called a stable region in the code). After 

the stable region has been reached, inverse design, viscous 

interaction, update of rotation function and airfoil shape in the 

inverse design can be started. 

In direct optimization, the original convergence criterion is 

multiplied by 2.5 to reduce the computing time. The final 

convergence is achieved by a separate run in the analysis mode using 

the converged ~ values in the last run. 

All of the aforementioned ideas have been built into the code. 

3.5 Under-Relaxation Factor for Airfoil Shape in Inverse Design 

Subroutine SHAPE is used to update the airfoil shape in the 

inverse design. Relaxation solution may diverge if the shape is 

changed too much during the first few updates. An under-relaxation 

factor can take care of divergence. A maximum value of 1.0 and a 

minimum value of 0.4 are built into the code. If the shape change 

increases by more than 1.0 x 10-4 , the relaxation factor is 

decreased by 0.05. On the other hand, if the shape change decreases 

by more than 1.0 x 10-4 , the factor is increased by 0.05. 

25 
3.6 Under-Relaxation Factor for Boundary Layer Displacement 

Thickness 

This factor was found to be important to the final 

convergence. This factor should be small, such as 0.20, to avoid 

oscillation in the modified airfoil 'shape and thus the values of ~. 

In direct optimization, this factor is set to one half of its 

input value. 

26 
4. NUMERICAL RESULTS AND DISCUSSIONS 

4.1 Program Validation 

A Joukowsky airfoil in incompressible uniform flow was used for 

comparison with the exact solution, Carlson's code (Ref. 42), and 

Chow's results (Ref. 19). Note that Carlson's code is based on 

disturbance velocity potential. The resulting pressure distribution 

is plotted in Figure 4. The present theory agrees well with 

Carlson's formulation with a slight discrepancy near the leading and 

trailing edges. The difference in the present results from those in 

Reference 19 may be caused by compressibility effect and different 

numerical techniques. The calculated lift coefficients at a = 0 are 

compared in the following table. 

Method 

(cR,) a=O 

Exact 

0.6943 

Present 

0.7117 

Carlson 

0.7147 

Chow, et al. 

0.7233 

It can be seen that the present result is consistent with others. 

For a nonuniform flow case, the pressure distribution is 

compared in Figure 5. The present results show more negative Cp 

nearly everywhere than those of Reference 19, again perhaps because 

of compressibility effect. The corresponding predicted lift 

coefficients based on maximum free-stream dynamic pressure are as 

follows: 

Present 

0.6547 

27 

Chow, et al. 

0.6188 
4.2 A Parametric Study in Transonic Nonuniform. Flow 

The following results are based on the velocity profile 

measured at Lewis Research Center and the location of the airfoil 

relative to the propfan slipstream center taken to be that of a test 

configuration used at Langley Research Center as shown in Figure 

6. If the airfoil is decomposed into a thickness part and a camber 

part, the thickness of a NASA supercritical airfoil can be varied to 

see its influence on the airfoil characteristics. This study is 

needed in choosing proper pressure distributions for airfoil 

design. Since results are more realistic to include boundary layer 

in the computation, both cases are run and compared. Cases with 

boundary layer interaction will be called the interaction cases, and 

those without are called inviscid cases here. 

In Figures 7(a)-(d), the effect of thickness variation within 

±25 percent for both inviscid and viscous cases are plotted. The 

angle of attack is zero. It is seen that the lift coefficients 

increase almost linearly with decreasing thickness. The drag 

coefficients increase with increasing thickness, and the pitching 

moment coefficients become more negative with increasing 

thickness. The lift-to-drag ratios increase with decrease in 

thickness. Inviscid results indicate higher lift-drag ratios, much 

higher lift coefficients and higher drag coefficients. It also 

means more negative pitching moment coefficients. These 

characteristics can be understood better by plotting the 

corresponding pressure distributions in Figures 8(a)-(d). From 

28 
Figures 8(a) and 8(b), it is seen that as the thickness is 

decreased, the shock strength is reduced and the loading tends to 

shift forward to make em more positive [Figure 7(c)]. The inviscid 

calculation shows higher and sharper trailing-edge pressure peaks 

and hence higher shock strength [Figures 8(c) and 8(d)]. 

The effect of camber is plotted in Figures 9(a)-(d). The 

general trend is that both lift and drag coefficients increase with 

increasing camber. However, there are large changes in aerodynamic 

characteristics for camber changes of 10 to 15%. In an inviscid 

flow, a large change occurs when the camber is decreased. On the 

other hand, in the viscous flow increasing camber tends to cause the 

sharp change. Examination of pressure distribution [Figures 10(a)-

(d)] reveals that these abrupt changes are caused by shifting of 

shock positions and strengths. 

The airfoil location relative to the slipstream is an important 

issue in propulsion integration. Since the swirl effect has been 

found to be important in three-dimensional experiments, it is of 

interest to examine this effect in the present two-dimensional 

flow. Note that the swirl effect is accounted for by an incremental 

angle of attack inside the slipstream. The locations of airfoil in 

the slipstream to be examined are illustrated in Figure 11. The 

results are presented in Figures 12(a)-(d). It is seen that the 

swirl effect is minor in two-dimensional cases. In Reference 9, the 

swirl velocity was shown to be converted into side velocity as the 

wing is approached. The nacelle used in the experiment also induced 

29 
side velocity and thus the three-dimensional boundary layer 

effect. Since the side velocity does not exist in a two-dimensional 

model, the airfoil will not experience the resulting threedimensional boundary layer effect. It follows that the swirl effect 

is decreased. 

When the airfoil is above the slipstream center, both lift and 

drag decrease; but the drag decreases faster. Thus the lift-drag 

ratio increases greatly. Pitching moment coefficients become more 

positive. The pressure distribution of some selected vertical 

locations is plotten in Figures 13(a) and (b). It is seen that the 

pressure distribution is closer to that in a uniform flow when the 

airfoil moves away from the slipstream center in the lower half 

plane. From Figure 13(a), it is seen that as y/R becomes more 

negative, the shock strength on the upper surface tends to be 

decreased. On the other hand, if y/R becomes more positive, not 

only does the upper-surface shock remain strong, but also a lowersurface shock tends to be generated. This is expected because the 

velocity on the airfoil lower surface will be increased by the 

slipstream. 

4.3 Inverse Design 

The inverse design process can be described as follows: 

(1) Specify a pressure distribution based on the backward 

fini te difference (to be called the "backward" pressure) 

and an initial airfoil shape. Note that the design 

30 
process in the Carlson's code is formulated in terms of 

the "backward" pressure. 

(2) The final "backward" pressure and the pressure based on 

the central difference (to be called the "central" 

pressure) are obtained together with a shape which 

includes a boundary layer. This shape will be called the 

initial design shape (i.e. outer shape) here. The 

aerodynamic characteristics based on this outer shape are 

designated as "design" in figures 14-16. 

(3) The design shape is obtained by subtracting a boundary 

layer from the initial design shape. 

(4) Input the design shape in the interaction analysis mode to 

obtain a more reliable boundary layer, and hence the 

pressure distribution. The design shape plus this 

boundary layer forms the "modified" shape. Its 

corresponding "central" pressure distribution will be 

called the "viscous" pressure. The aerodynamic 

characteristics based on interaction analysis are 

designated as "analysis" in Figure 15. 

Different pressure distributions for inverse design are plotted 

in Figure 14. For an ideal situation, the specified pressure should 

be identical to the "backward" pressure. In this inverse design 

this is not satisfied near the trailing edge due to the large 

pressure gradient caused by the shock. There is some discrepancy 

near the nose. Usually, the "central" pressure distributions are 

31 
close to the "backward" pressure values for smooth pressure 

distributions. The difference between "central" pressure and 

"viscous" pressure results from the different boundary layers. The 

resulting design shown in Figures 15 and 16 indicates that a 7.5 

percent decrease in maximum thickness results in a 16.5 percent 

increase in lift-drag ratio. However, the trailing-edge thickness 

is considered too thick to be desirable. However, it is very 

difficult to specify a better pressure distribution to realize a 

better performance, and at the same time control the trailing-edge 

thickness. Therefore, an approach based on direct optimization will 

be employed in the following. 

4.4 Direct Optimization 

Direct optimization is achieved by integrating the analysis 

code with an optimizer. The problem formulation was described in 

section 2.5. Although it takes many iterations to obtain a 

converged solution, it does not need an experienced person to do the 

job. 

For the flow condition shown in Figure 6, a NASA supercritical 

airfoil is used as an input shape. Seven Fourier coefficients for 

both upper and lower surfaces are chosen (see Equation 39). That 

is, there are 14 design variables. The front 5% of the nose region 

is kept unchanged in the optimization. The results show that a 4 

percent decrease in thickness results in a gain of 22 percent in 

lift-drag ratio (Figure 17). This should be compared with a direct 

32 
thickness reduction from which the same gain in LID can be achieved 

only by a reduction of thickness of about 17.5% [Figure 7(d)]. The 

improvement appears to lie in a small thickness increase in the nose 

region to accelerate the flow and a flatter mid upper surface to 

reduce the shock strength. On the lower surface, two weak 

supersonic regions appear. 

Based on extensive numerical experience in designing an airfoil 

in transonic nonuniform flow, the method of direct optimization is 

found to be much esier to use and has a better chance to design a 

good airfoil in comparison with the inverse method. However, it has 

the disadvantage of requiring long computer CPU time. On the other 

hand, the CPU time to determine an airfoil shape in the inverse 

method is relatively small. The main disadvantage in the inverse 

method lies in the difficulty of specifying a good pressure 

distribution for low drag at a given lift coefficient and at the 

same time controlling the thickness of the trailing edge. 

33 
5. CONCLUSIONS 

A nonuniform transonic airfoil code capable of performing 

analysis, inverse design and direct optimization was 

developed. Problems involving numerical stability, convergence, 

divergence and solution oscillations were discussed. Application of 

the code to an airfoil immersed in a transonic propfan slipstream 

indicated that decrease in thickness would increase the lift and 

lift-drag ratio and make the pitching moment more negative. 

Increase in camber, however, tended to decrease the lift-drag ratio 

in a nonlinear manner. The nonlinearity arose from the rapid shift 

in shock locations when camber was changed. Swirl effect was found 

to be insignificant in the present two-dimensional case. The 

airfoil performance tended to be better if it was located below the 

propfan slipstream center. 

Both inverse design and direct optimization of an airfoil were 

conducted. It was found that the airfoil designed through the 

direct optimization offered a better performance resulting in a gain 

of 22 percent in lift-drag ratio for a reduction of 4 percent in 

thickness. The airfoil shape could also be better controlled than 

that based on the inverse design method. 

34 
6. REFERENCES 

1. Rohrback, C., "A Report on the Aerodynamic Design and 

Windtunnel Test of a Propfan Model." AIM paper 76-667, July 

1976. 

2. Kraus, E. F., and van Abkoude, J. C., "Cost/Benefit Tradeoffs 

for Reducing the Energy Consumption of the Commercial Air 

Transportation System." Douglas Aircraft Company, NASA CR137925, June 1976. 

3. Hopkins, J. P., and Wharton, H. E., "study of the Cost/Benefit 

Tradeoffs for Reducing the Energy Consumption of the Commercial 

Air Transportation System." Lockheed California Company, NASA 

CR-137927, August 1976. 

4. "Energy Consumption Characteristics of Transports Using the 

Propfan Concept." The Boeing Commercial Airplane Company, NASA 

CR-137938, November 1976. 

5. Dugan, J. F., Miller, B. A., Graber, E. J., and Sagerser, 

D. A., "The NASA High-Speed Turboprop Program." SAE paper 

801120. 

6. Welge, H. R., and Crowder, J. P., "simulated Propeller 

Slipstream Effects on a Supercritical Wing." NASA CR-152138, 

June 1978. 

7. Bencze, D. P., Smith, R. C., Welge, H. R., and Crowder, J. P., 

"Propeller Slipstream Wing Interaction at Mach 0.8." SAE paper 

780997. 

35 
8. Boctor, M. L., Clay, C. W., and Watson, C. F., "An Analysis of 

propfan/Airframe Aerodynamic Integration." NASA CR-152186, 

October 1978. 

9. Welge, H. R., "propfan Integration at Cruise Speeds." AGARD 

CP-301, paper 33, May 1981. 

10. Welge, H. R., Neuhart, D. H., and Dahlin, J. A., "Analysis of 

Mach 0.8 Turboprop Slipstream Wing/Nacelle Interaction." NASA 

CR-166214, Douglas Aircraft Company, Long Beach, California, 

August 1981. 

11. Smith, R. C.; and Levin, A. D.: "Propfan Installation Aerodynamics of 

a Supercritical Swept Wing Transport Configuration." AIAA/SAE/ASME 

17th Joint Propulsion Conference, July 27-29, 1981, Colorado Springs, 

Colorado, AIAA paper 81-1563. 

12. Levin, A. D.; and Smith, R. C.: "Prop fan Propulsion Integration 

Test Techniques." AIAA paper 82-0577, 1982. 

13. Samant, S. S.; Hultman, D.; Lines, T. R.; Yu, N. J., and Rubbert, P. E.: 

"Propfan Nacelle Installations at Transonic Speeds." NASA CR-166376, 

October 1982. 

14. Levin, A. D., and Smith, R. C., "Installed Nacelle DragImprovement Tests of a M = 0.8 Turboprop Transport 

Configuration." NASA TM-84302, January 1983. 

15. Bartlett, G. R., "An Experimental Investigation of Propfan 

Installations on an Unswept Supercritical Wing." M.S. Thesis, 

George Washington University, August 1983. 

36 
16. Koning, C., "Influence of the Propeller on Other Parts of the 

Airplane." Aerodynamic Theory, Vol. IV, Springer-Verlag, 

Berlin, 1935. 

17. Ferrari, C., "propeller and Wing Interaction at Subsonic 

Speeds," in Aerodynamic Components of Aircraft at High Speed. 

Edited by A. F. Donovan and H. R. Lawrence, Sec. C, Princeton 

University press, Princeton, New Jersey, 1957, Chapter 3. 

18. Rethorst, S., "Aerodynamics of Nonuniform Flow as Related to an 

Airfoil Extending Through a Circular Jet." Journal of the 

Aeronautical Science, January 1958, pp. 11-28. 

19. Chow, F., Krause, E., Liu, C. H., and Mao, J., "Numerical 

Investigations of an Airfoil in a Nonuniform Stream." Journal 

of Aircraft, Vol. 7, No.6, November-December 1970, pp. 531-

537. 

20. Jameson, A., "Analysis of Wing Slipstream Flow Interaction." 

NASA CR-1632, August 1970. 

21. Ting, L., Liu, C. H., and Kleinstein, G., "Interference of Wing 

and Multipropeller." AIM Journal, Vol. 10, No.7, July 1972, 

pp. 906-914. 

22. Shollenberger, C. A., "Three-Dimensional Wing/Jet Interaction 

Analysis Including Jet Distortion Influences." Journal of 

Aircraft, Vol. 12, No.9, September 1975, pp. 706-713. 

23. Lan, C. E., "Wing-Slipstream Interaction with Mach Number 

Nonuniformity." Journal of Aircraft, Vol. 12, No. 10, October 

1975, pp. 759-760. 

37 
24. Lan, C. E., "A Quasi-Vortex-Lattice Method in Thin Wing 

Theory." Journal of Aircraft, Vol. 11, No.9, September 1974, 

pp. 518-527. 

25. Chin, W. C., "Thin Airfoil Theory for planar Inviscid Shear 

Flow." Journal of Applied Mechanics, ASME, Vol. 51, March 

1984, pp. 19-26. 

26. Rizk, M. H., "Propeller Slipstream/Wing Interaction in the 

Transonic Regime." Journal of Aircraft, Vol. 18, No.3, March 

1981, pp. 184-191. 

27. Samant, S. S., YU, N. J., and Rubbert, P. E., "Transonic Flow 

Simulation of Propfan Configurations." AIM paper 83-0187, 

January 1983. 

28. Jameson, A., and Caughey, D. A., "A Finite Volume Hethod for 

Transonic Potential Flow Calculations." AIM 3rd Computational 

Fluid Dynamics Conference, Albuquerque, New Mexico, June 1977. 

29. Narain, J. P., "A Transonic Analysis of Propfan Slipstream 

Effects on a Supercri tical t'1ing." AIM paper 83-0186, January 

1983. 

30. Jameson, A., and Caughey, D. A., "Numerical Calculation of the 

Transonic Flow Past a Swept Wing." Courant Institute of 

Mathematical Sciences, New York University, COO-3077-140, June 

1977. (Available as NASA CR-153297.) 

31. Whitfield, D. L., and Jameson, A., "Three-Dimensional Euler 

Equation Simulation of propeller-Wing Interaction in Transonic 

Flow." AIM paper 83-0236, January 1983. 

38 
32. Whitfield, D. L., Swafford, T. W., and Jacocks, J. L., 

"Calculation of Turbulent Boundary Layers with Separated and 

Viscous-Inviscid Interaction." AIAA Journal; Vol. 19, No. 10, 

october 1980, pp. 1315-1322. 

33. Brown, E. F., Brecht, T. J. F., and Walsh, K. E., "A Relaxation 

Solution of Transonic Nozzle Flows Including Rotational Flow 

Effects." Journal of Aircraft, Vol. 14, No. 10, October 1977, 

pp. 944-951. 

34. Chaderj ian, N. M., and Steger, J. L., "A Zonal Approach for the 

Steady Transonic Simulation of Inviscid Rotational Flow." AIM 

paper 82-1927, January 1983. 

35. Vanderplaats, G. N., "CONMIN-A Fortran Program for Constrained 

Function Optimization." NASA TM X-62282, August 1973. 

36. Schouten, G., "Static Pressure in the Slipstream of a 

propeller." Journal of Aircraft, Vol. 19, No.3, March 1982, 

pp. 251-252. 

37. Ludford, G. S. S., "The Behavior at Infinity of the Potential 

Function of a TWo-Dimensional Subsonic Compressible Flow." 

Journal of Mathematical Physics, vol. 30, October 1951, pp. 

117-130. 

38. Carlson, L. A., "Transonic Airfoil Flowfield Analysis Using 

Cartesian Coordinates." NASA CR-2577, August 1975. 

39 
39. Jameson, A., "Iterative solution of Transonic Flow over 

Airfoils and wings, Including Flows at Mach One." 

Communication of Pure and Applied Mathematics, Vol. 27, 1974, 

pp. 283-309. 

40. Carlson, L. A., "Transonic Airfoil Design Using Cartesian 

Coordinates." NASA CR-2578, April 1976. 

41. Carlson, L. A., "Transonic Airfoil Analysis and Design Using 

Cartesian Coordinates." Journal of Aircraft, Vol. 13, No.5, 

May 1976. 

42. Carlson, L. A., "TRANDES: A Fortran Program for Transonic 

Airfoil Analysis or Design." NASA CR-2821, June 1977. 

40 
~ 

I-' 

I = 

J = JMAX 

,~y, n 

I - Regio 1 II 

-.- III -. 

V ILE ~ r---r--- ITE 

j ~ 

...... 0 ':: X ,E;, 

1 I = IMAX 

S4 J = 1 

4 

Figure 1. Flow Field Subdivision for Coordinate Stretching 
j~~ 

>< 

~ 

1 

I 

I 

42 
L 

I 

1-

(Start) 

+ I 

/Read Case Description and Option/ ,.-

.J. , 

IOPT ....... analysis 

'" optimization 

jRead Optimization Parameters 7 - - -- - -- -- - - - I jRead Analysis Parameters! I 

v I 

Coarse and Medium Grid 

Calculation I 

~. I 

Fine Grid Calculation/ I 

- -- ...J 

LITRANDES-NFi - - ~ ____ J 

__ 1._ -- --- -----

!calculate gradients for Objective 

Function and Constraints 

i 

{Stop 

-, 

I' 

I 

I 

Isearch for Optimal Design variablesl I 

_1 . -r-------- ----

LrCO~~HNi - - -

L- :...:..-..:..J. 

Save Design Variablesl 

~ converged Check Convergence 

J. not converged limit exceeded Check Iteration Limit not 

limit exceeded 

, 

Save q,~ 

-.". I 

~ 

Figure 3. Flow Chart of Integrating CONMIN and TRANDESNF 

43 
.p.. 

.p.. 

C 

P 

-1.0· 

-0.8 

-0.6 

o 

0.4 

Present, M..,=O.l 

Chow, et a1., tl",,=O.(Ref. 19) 

--- .... -. ... , Carlson, Moo=O.~Ref. 38) 

----

Figure 4. Pressure Distribution on a Joukowsky Airfoil in Uniform Flow 

at zero Angle of Attack 
..,... 

VI 

c 

p 

Present, M (uniform) =0.1 00 

------ Chow, et al. ,M =O.(Ref. 19) co 

-0.8 

Yl 

c==:::--. --+ x 

N22 u =U {I + exp(-y /d )} 00 00 

d=c/l. 81 

-0.2' . 

o 

0.3 "" .............. ~.4 O.S 0.6 0.7 ................ ...... ~ x/c -_. --------

0.1 

---

Figure 5. Pressure Distribution of a Joukowsky Airfoil in a Nonuniform Flow 

at Zero Angle of Attack. C Based on the Freestream Dynamic Pressure 

p 

on the Airfoil Surface Streamline 
+:--

'" 

M .. 0.8 ~ A 

Slipstream ~ 

V- ~ 

Airfoil 

r.. 1 r= =::::----

'I-' .12575 ,- ""'""'"""" 

--- ~ T 

.2604 /C=.5208S-

. 

.~ .I 

J 

.I 

'7 

... A 

A-A 

Figure 6. Mach Number Profile of a Propfan at M = 0.8 and Relative Airfoil Location 
1.1 ~------------------------------------~ 

<:J 

C!) 

1.0 C!) 

r- (!) 

r- C!) 

C!) 

rrt--=! 

t.) r-

+J (~ 

s::O.9 r-

(l) 

'r-! 

C) r-

.r-! 

'H 

'H t-

(l) 

0 

t.) ~ .... +-l 

'H 

'r-! 

t--=!O.8 rt- .... 

trO.7~ Inviscid 

.... with Boundary Layer Displacement Effect 

O.6~--~--~--~--~--~--~--~--~----~~ I I I I I I I I • 

- 25 - 20 -1 5 -10 -5 0 5 10 1 5 20 25 

Percent Thickness Variation Relative to the Original Airfoil 

Figure 7a. Thickness Effect of a NASA Airfoil on the Lift 

Coefficients in a Nonuniform Flow at Zero Angle 

of Attack and Mach Number 0.8 

47 
0.09r-------------------------------------~ 

• with Boundary Layer Displacement Effect 

CD Inviscid CD 

CD 

0.081-

CD 

"'0 

U CD 

-+J 

c: 

CD l-

.-

0 e!) 

.-

~ 0.07 ICD 0 

U 

0'1 

0 

La ... CD 

le!) 

I0.06 ~ 

... 

lI-

.~ 

... 

0.05 I I I I I I I I I 

-25. -20. -15. -1 O. -5. O. 5. 10. 15. 20. 25. 

Percent Thickness Variation Relative to the Original Airfoil 

Figure 7b. Thickness Effect of a NASA Airfoil on the Drag 

Coefficients in a Nonuniform Flow at Zero Angle 

of Attack and Mach Number 0.8 

48 
-0.30 

-0.31 ~ 

.~ 

... -0.32 ~ 

-0.33 ~ 

C) 

IJ.l 

~-0.34 ~ 

E 

u -0.35 ~ 

-+J 

C 

Q) 

t-

.2 -0.36 ~ 

~ CD 

0 u -0.37 ~ 

-+J 

C E -0.38 ~ 

0 

:E -0.39

Ol c ~ C 

.- CD .c 0-0.40 ~ 

-+J 

ta.. CD 

-0.41 ~ 

-0.42 ~ CD 

CD 

-0.43 ~ C!) Inviscid CD 

CD 

-0.44 f- ~ with Boundary Layer Displacement Effect 

-0.45 I I 1 I I 

-25. -20. -15. -1 O. -5. O. 

1 

5. 

I L I 

10. 15. 20. 25. 

Percent Thickness Variation Relative to the Original AirfoiJ 

Figure 7c~Thick~ess Effect of a NASA Airfoil on the Pitching 

Moment Coefficients in a Nonuniform Flow at Zero 

Angle of Attack and Mach Number 0.8 

49 
0 

~ 

18.0 

f17.0. f0-

b 

I16.0· ~ CD 

f15.0 fo-

~ 

14.0 fo-

~ 

13.0 ~ 

~ 

12.0 ~ 

~ 

11.0 ff10.0 ~ 

I9.0 fCD 

CD 

CD 

CD 

CD 

CD 

CD Inviscid 

A with Boundary· Layer Displacement Effect II> .... 

. 

8.0 J I I I I J 

5. 

J I J 

-25.-20.-15.-10. -5. O. 10. 15. 20. 25. 

Percent Thickness Variation Relative to the Original Airfoil 

Figure 7d. Thickness Effect of a NASA Airfoil on the LiftDrag Ratios in a Nonuniform Flow at Zero Angle 

of Attack and Mach NumbeT 0.8 

50 
-1.5~---------------------------------------- Percent Thickness Variation 

Original 

-570 . . . . . . . . . . . -10" 

- - -- - - -- - -- -1 5% 

G 0 -20% ~.-.--. -25% 

-1.0 

0.0 

0.5 

1.0~--~--~--~--~--~---L--~--~--~--~ 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.B 0.9 1.0 

X/C 

Figure Sa. Thickness Effect of a NASA Airfoil on Pressure 

Distributions in a Nonuniform Flow at Zero Angle 

of Attack and Mach Number o.s Including Boundary 

Layer Displacement Effect 

51 
Q. 

u 

-1.5--------------------------------------~ Percent Thickness Variation 

-1.0 

Original 

+5% 

. . . . . . . . . . . + 10% 

----------- + 1 5% 

o 0 +20% ~.-.---+ +25% 

~-0.5 

a> 

.-

o 

.-'+- '+-

a> 

o 

(,) 

a> 

~ 

:J 

en 

en 

Q) 

.~ 

a.. 

0.0 

0.5 

1.0L---~~L-~--~--~--~--~--~--~--~ 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0 

X/C 

Figure Sb. Thickness Effect of a NASA Airfoil on Pressure 

Distributions in a Nonuniform Flow at Zero Angle 

of Attack and Mach Number O.S Including Boundary 

Layer Displacement Effect 

52 
0-

U 

-1.5--------------------------------------------, Percent Thickness Variation 

-1.0 

Original 

-5% . . . . . . . . . . . -10% 

- - - - - - - - - -- -1 5% 

o 0 -20% ~.-.---+ -25% 

1: -0.5 

CD 

.-

o 

t;: 

\tCD 

o 

u 

CD 

L-

:J 

U) 

en 

CD 

La. 

0.0 

0.5 

1.0L.---~--~--~--~--~--~--~--~--~--~ 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

X/C 

Figure Sc. Thickness Effect of a NASA Airfoil on Pressure 

Distributions in an Inviscid Nonunifonn Flow at 

Zero Angle of Attack and Mach Number 0.8 

53 
a. 

(.) 

-1.5----------------------------------------~ Percent Thickness Variation 

-1.0 

Original 

+5% 

. . . . . . . . . . . + 10% 

----------- + 1 5% 

G 0 +20% ~.-.----+ +25% 

1: -0.5 

Cl) 

0-

o 

t;: 

'+-

Cl) 

o 

u 

CD 

L-

::J 

en 

en 

Cl) 

La.. 

0.0 

0.5 

1.0~---~--~--~--~--~--~--~--~--~--~ 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 . 1.0 

X/C 

Figure 8d. Thickness Effect of a NASA Airfoil on Pressure 

Distributions in an Inviscid Nonuniform Flow at 

Zero Angle of Attack and Mach Number 0.8 

54 
o 

1.1------------------------------------------

ff- CD Inviscid 

ICD 1.0 f- CD 

CD 

I- CD 

.. CD 

.. 

0.9 -

- CD 

CD 

-

C) 

-

- ... 0.8-

I- .... .... 

f- .... 

I-

.... 0.7 I I I I I I 

5. 

I I I 

-25. -20. -15. -1 O. -5. O. 10. 15. 20. 25. 

Percent Camber Variation Relative to the Original Airfoil 

Figure ga. Camber Effect of a NASA Airfoil on the Lift 

Coefficients in a Nonuniform Flow at Zero Angle 

of Attack and Mach Number 0.8 

55 
~ 

() 

of.J 

C 

a> 

.-0 

.-

~ CD 

0 

() 

0'1 

0 

LCl 

0.09--------------------------------------~ 

• 0ith Boundary Layer Displacement Effect ~ 

CD Inviscid 

~ 

0.08 l-

~ 

~ 

~ CD 

CD 

~ 

CD 

0.07 ~ 

~ 

~ 

.... 

.... "- CD 

0.06 I- CD 

cb .... . ~ 

f0.05 J I I I I 

-25. -20. -15. -1 O. -5. o. 

CD 

.... 

I 

5. 

ll> 

CD 

CD 

I I I 

10. 15. 20. 25. 

Percent Camber Variation Relative to the Original Airfoil 

Figure 9b. Camber Effect of a NASA Airfoil on the Drag 

Coefficients in a Nonuniform Flow at Zero -Angle 

of Attack and Mach Number 0.8 

56 
-0.30r------------------------------

E 

o -0.35 ..... 

c 

CD 

o 

..... 

C 

<V 

E 

o 

2 

en 

(!) 

I 

I 

! 

I 

I 

I 

I 

~ -0.40~ ,6 

t (!) 

(!) 

(!) 

(!) Inviscid CD 

t!) 

-0.45 l __ "--_--J.

A

_

W

._ 1_' t_hL-.B_O_U--LD _d_a,_r_Y...lo-L_a_y_e_rJ-D._i_s-p...,JILa_c_e_m_eJ..D_t_E.f ~ ec t I 

-25. -20. -15. -10. -5. O. 5. 10. 15. 20. 25. 

Percent Camber Variation Relative to the Original Airfoil 

Figure 9c. Camber Effect of a NASA Airfoil on the Pitching 

Moment Coefficients in a Nonuniform Flow at Zero 

Angle of Attack and Mach Number 0.8 

57 
15----------------------------------------~ 

CD 

CD 

CD 

CD 

CD 

CD Inviscid 

.. 1 1 • with Boundary Layer Displacement Effect 

10~--~--~--~--~--~--~--~--~--~~~ 

-25 -20 -15 -10 -5 o 5 10 15 20 25 

Percent 

Figure 9d. 

Camber Variation Relative to the Original Airfoil 

Camber Effect of a NASA Airfoil on the Lift-Drag 

Ratios in a Nonuniform Flow at Zero Angle of 

Attack and Mach Number 0.8 

58 
0-

U 

-1.5r--p-e-r-c-e-n-t--C-a-m-b-e-r--v-a-r-i-a-t-io-n------------------------~ 

Original 

-5~ . . . . . . . . . . . -10% 

----- -. --. - -1 570 

G 0 -2070 ~.-.---+ -2570 

-1.0 

1: -0.5 

CD 

.-

o 

t;: 

\tCD 

o 

u 

~ :::J 

~ 0.0 

CD 

L.. 

0-

0.5 

1.0~--~--~--~--~--~--~----~--~--~--~ 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

X/C 

Figure lOa. Camber Effect of a NASA Airfoil on Pressure 

Distributions in a Nonuniform Flow at Zero Angle 

of Attack and Mach Number 0.8 Including Boundary 

Layer Displacement Effect 

59 
a. 

() 

-1.5~----------------------------------------~ Percent Camber Variation 

-1.0 

Original 

+5% ........... +10% 

. -. --. --. . . + 1 5% 

o 0 +20%. ~.-.~ +25% 

1: -0.5 

a> 

'0 

.-'+-

\fa> 

o 

() 

a> 

L. 

:J 

~ 0.0 

CD 

L. 

.0. 

0.5 

1.0~--~--~--~--~--~--~--~--~----~~ 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

X/C 

Figure lOb. Camber Effect of a NASA Airfoil on Pressure 

Distributions in a Nonuniform Flow at Zero Angle 

of Attack and Mach Number 0.8 Including Boundary 

Layer Displacement Effect 

60 
CO 

- 1.5 ---P-e-r-c-e-n-t-C-a-m-b-e-r-V-a-r-i-a-t-io-n------------~ 

-1.0 

Original 

-5~ 

. . . . . . . . . . . -10% 

- - - - - -- - . - - ~ 1 5% 

o 0 -20% ~.-.--+ -25% 

~-0.5 

CD 

.-

o 

tt= 

...... 

CD 

o 

o 

CD 

'-

:J 

(IJ 

(IJ 

~ Q.. 

0.0 

0.5 

1.0~--~--~--~--~--~---L--~--~--~--~ 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

X/C 

Figure lOco Camber Effect of a NASA Airfoil on Pressure 

Distributions in an Inviscid Nonuniform Flow at 

Zero Angle of Attack and Mach Number 0.8 

61 
CO 

-1.5--------------------------------------~~ 

-1.0 

Percent Camber Variation 

Original 

+5% . . . . . . . . . . . + 10% 

----------- + 1 5% 

o 0 +20% ~.-.--+ +25% 

1:: -0.5 

CD 0-

o 

tt=. 

\fCD 

o 

o 

0.5 

1.0~--L---~--~--~--~--~--~--~----~--~ 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

X/C 

Figure 10d. Camber Effect of a NASA Airfoil on Pressure 

Distributions in an Inviscid Nonuniform Flow at 

Zero Angle of Attack and Mach Number 0.8 

62 
0\ 

W 

swirl angle , de g • 

o. 

o 

positive swirl o 

y/R 

1.2 

.015 

-1.01 

-1.2 

negative swirl 

x/R 

Figure 1~. Airfoil Locations Relative to the Propfan Slipstream to Be Examined 
1.0~------------------------------------------~ 

-

- A Positive Swirl 

CD Without Swirl 

0.9 f- • Negative Swirl 

• 

- • o 0.8-

.f.J 

c: 

Q) 

. -

o 

!E 

Q) 

o 

o 

-

• 

0.61-

-

-

• 

• 

A 

• 

5 o. ~~~~~~~~~~~~~~~~~~~~~ t t t J I I Itt I I t I Itt I I 

-1 .2 -0.8 -0.4 -0.0 0.4 0.8 

y/R 

Figure 12a. Lift Coefficient Variations Due to Relative 

Airfoil-Slipstream Locations in a Nonuniform 

Flow at Zero Angle of Attack and Mach Number 

0.8 

64 

1.2 
~ 

c 

Q) 

.-

o 

.-

o.,o----------------------------------------~ 

0.09 

0.08 

.. Positive Swirl 

CD Without Swirl • 

• Negative Swirl 

0.07 • 

• 

• • 

I • 

• 

~ 0.05 • 

o 

() 

• 

0.03 • 

0.01 

O.OO~~~~~~~~~~~~~~~~~~~~~ 

-1.2 -0.8 -0.4 -0.0 

y/R 

0.4 0.8 

Figure 12b. Drag Coefficient Variations Due to Relative 

Airfoil-Slipstream Locations in a Nonuniform 

Flow at Zero Angle of Attack and Mach Number 

0.8 

65 

1.2 
-o.,----------------------------------------~ 

.. Positive Swirl 

l- G) Without Swirl 

Ii"' • Negative Swirl 

I:) rIf) 

C\1 .~ 

E rU -0.2 I-

-+J 

C r- Q) • -0 l-

.- • :::: !- QJ 

0 r- U • ...... .. C 

QJ - E 

0 - ~ - Ol 

C • - .-

.c 0-0.3 - • +' .-a. -

-

-

III-

• .. • • 

• 

-0.4~~~~~~~~~~~~~~~~~~~~~ I I I I I I t I I I I t I I I • • I I t 

-1.2 -0.8 -0.4 -0.0 

y/R 

0.4 0.8 

Figure 12c. Pitching Moment Coefficient Variations Due to 

Rerative Airfoil-Slipstream Locations in a 

Nonuniform Flow at Zero Angle of Attack and 

Mach Number 0.8 

66 

1.2 
0 

J 

0 .--+oJ 

0 

0:: 

Ol 

e 

0 

I 

-tJ 

..... 

~ 

35.----------------------------------------, 

30. .. Positive Swirl 

CD Without Swirl 

• Negative Swirl 

25. 

• 

20. .. 

III 

15. 

• 

a • • • • 

.. 

10 .. 

-1.2 -0.8 -0.4 -0.0 0.4 0.8 

Figure 12d. 

y/R 

Lift-Drag Ratio Variations Due to Relative 

Airfoil-Slipstream Locations in a Nonuniform 

Flow at Zero Angle of Attack and Mach Number 

0.8 

67 

1.2 
0. 

() 

-1.5r-------------------·--------------------~ 

-1.0 

C -0.5 

(I) 

·u .-'+-

'+-

(I) 

o 

() 

Q} 

L-

:J 

............... .. ' .:.:---

~ 0.0 

(I) 

La... z/R=-0.051 

. .......... , z/R=-0.433 

-----. z/R=-0.817 

------------ z/R=-1.2 

0.5· 

1 0 I I . ~--~--~---~----~--~--~---~----~--~--~ 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

X/C 

Figure 13a. Pressure Distribution Variations Due to Relative 

Airfoil-Slipstream Location in a Nonuniform Flow 

at Zero Angle of Attack and Mach Number 0.8 with 

Positive Swirl 

68 
a. 

U 

-1 .5 r---------

-1.0 , , 

,:/.\: 

/. 

I 

C -0.5 

-:: '.'. . . . . . .\\ • I 

I 

I 

\\ 

IV \\ .-o .-l+-

I+-

IV 

o 

U 

IV 

"-

::J 

UJ 

rn 

IV 

"-

a.. 

r ,:.': 

0.0 It 

f I 

~ 

j 

----.- y /R=0.051 

............. y /R=.433 

y /R=O.817 

- - - - - - - - - - - - y /R= 1.2 

• I 

I 

I 

I \ 

I \ 

I • 

I 

1 0 I I I I • __ ~ __ ~ _____ ~ _____ ~ _____ ~_~ ____ ~ ________ L ___ -L-. ____ ~ 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

X/C 

Figure 13b. Pressure Distribution Variations Due to Relative 

Airfoil-Slipstream Location in a Nonuniform Flow 

at Zero Angle of Attack and Mach Number 0.8 with 

Positive Swirl 

69 
Q. 

(,) 

.-'+-

'+-

eD 

o 

(,) 

CD 

-2.0~--------------------------------------~ 

------ Central 

G G a Specified 

----------- Backward 

---_. Viscous 

Modified Shape 

5-0.0 

(I) 

(I) 

CD 

LQ.. 

0.5 _ 1>esis.."- _Anglysis _ 

Ct · .9827 .935 • 

Cd · .0704 .0662 • 

1.0 Cm · -.3851 -.362 

· 

LID 13.96 14.12 

1.5~--~--~------~---L--~--~--~~--~~ 

0.0 0.1 0.2· 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

X/c 

Figure 14. Comparison of Pressure Terminology for Inverse 

Airfoil Design in a Nonuniform Flow at Mach 

Number 0.8 

70 
-1.5----------------------------------------~ 

0. 

o 

-1.0 

1: -0.5 

CD 

1; 

ti: 

"fCD 

o 

o 

e 

::s 

~ 0.0 

~ 0.. 

Original 

. - - . - - -... - Design 

.. .. .. ... .. .. 

.. .-

.. .. . . _. ... .. .. .. .. 

Onginal Design 

.. .. .. .. .. .. .. 

.......... 

, 

, 

, 

, 

, 

, 

.. 

, 

, 

• • 

, 

• • ,

• • • , 

, 

• 

• ------- ----- • 

0.5 . Ct · .7804 . 935 • 

• 

Cd · .0645 .0662 • 

<;" • -.3347 -.362 • 

LID • 12.11 14.12 • 

1.0~------~--~--~------~--~--~------~ 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 .. 9 1.0 

X/C 

Figure 15. Comparison of Original and Inverse-Design Shapes 

and Pressure Distributions in a Nonuniform Flow· 

at Mach Number 0.8 

71 
-.....J 

N 

"'" CIl 

CD 

~ 

o 

c: 

~ o 

o 

0.2. ------------------------------~ 

0.1 

Design 

__ 0- Start 

- - - - - - - - - -- Modified 

o 0 0 0 0 0 0 ••• 0 Outer 

() -0.0 

-·0 

't: 

~ 

o 

~ -0.1 

-0.2 1 I 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

X/C 

Figure 16. Comparison of Shape Terminology for Inverse Airfoil Design 

and Original Airfoil in a Nonuniform Flow at Zero Angle of 

Attack and Mach Number 0.8 
-1.5----------------------------------------, 

-1.0 

0.0 

0.5 

-------10ri gina I 

----------- Direct Design 

, 

, 

, ' .. 

, , 

" 

, 

, 

, -

, 

, , 

Original 

, , 

------

Ct · .7804 • 

Cd · .0645 · 

Cm · -.3347 • 

LID 12. 11 

, , , 

..... , 

Design 

----- .7747 

.0524 

-.3167 

14.79 

1.0L---~--~--~--~--~--~--~--~--~~~ 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

Xjc 

Figure 17. Comparison of Original and Direct-Design Shapes 

and Pressure Distributions in a Nonuniform Flow 

at Mach Number 0.8 

73 
1. Report No. I 2. Government Accession No. 3. Recipient's Catalog No. 

NASA CR-399l 

4. Title and Subtitle 5. Report Date 

Transonic 

June 1986 

Airfoil Analysis and Design in Nonuniform 

6. Performing Organization Code 

Flow 

7. Author(sl 8. Performing Organization Report No. 

Jen-Fu Chang and C. Edward Lan CRINC-FRL-602-2 

10. Work Unit No. 

9. Performing Organization Name and Address 

Flight Research Laboratory 11. Contract or Grant No. The University of Kansas Center for Research, Inc. 

Lawrence, KS 66045 NAGI-308 

13. Type of Report and Period Covered 

12. Sponsoring Agency Name and Address Contractor Report 

National Aeronautics and Space Administration 14. Sponsoring Agency Code 

Washington, DC 20546 

535-03-01-01 

15. Supplementary Notes 

Langley Technical Monitor: William P. Henderson 

16. Abstract 

A nonuniform transonic airfoil code is developed for applications in 

analysis. inverse design and direct optimization involving an airfoil innnersed 

in propfan slipstream. Problems concerning the numberical stability, convergence, divergence and solution oscillations are discussed. The code is 

validated by comparing with some known results in incompressible flow. A 

parametric investigation indicates that the airfoil lift-drag ratio can be 

increased by decreasing the thickness ratio. A better performance can be 

achieved if the airfoil is located below the slipstream center. Airfoil 

characteristics designed by the inverse method and a direct optimization are 

compared. The airfoil designed with the method of direction optimization 

exhibits better characteristics and achieves a gain of 22 percent in lift-drag 

ratio with a reduction of 4 percent in thickness. 

17. Key Words (Suggested by Author(s)) 18. Distribution Statement 

Airframe/Propulsion Integration Unclassified-Unlimited 

PropFan 

Transonic Flows Subject Category 02 

19. Security Oassif. (of this report) 20. Security Classif. (of this pagel 21. No. of Pages 22. Price 

Unclassified Unclassified 84 AD5 

For sale by the National Technical Information Service, Springfield, Virginia 22161 

NASA-Langley, 1986 
End of Document