Hypersonic Airfoils for Maximum Lift to Drag

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 5, No.6, 1970

Hypersonic Airfoils of Maximum lift-to-Drag Ratio!

R. A. THOMPSON2 AND D. G. HULL3

Communicated by A. Miele

Abstract. The problem of determining the slender, hypersonic airfoil shape which produces the maximum lift-to-drag ratio for a given profile area, chord, and free-stream conditions is considered. For the estimation of the lift and the drag, the pressure distribution on a surface which sees the flow is approximated by the tangent-wedge relation. On the other hand, for surfaces which do not see the flow, the Prandtl-Meyer relation is used. Finally, base drag is neglected, while the skin-friction coefficient is assumed to be a constant, average value. The method used to determine the optimum upper and lower surfaces is the calculus of variations. Depending on the value of the governing parameter, the optimum airfoil shapes are found to be of three types. For low values of the governing parameter, the optimum shape is a flat plate at an angle of attack followed by slightly concave upper and lower surfaces. The next type of solution has a finite thickness over the entire chord with the upper surface inclined so that the flow is an expansion. Finally, for the last type of solution, the upper surface begins with a portion which sees the flow and is followed by an inclined portion similar to that above. For all of these solutions, the lower surface sees the flow. Results are presented for the optimum dimensionless airfoil shape, its dimensions, and the maximum lift-to-drag ratio. To calculate an actual airfoil shape requires an iteration procedure due to the assumption on the skin-friction coefficient. However, simple results can be obtained by assuming an ap­proximate value for the skin-friction coefficient.

1 Paper received November 26, 1969. This research was supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, U.S. Air Force, under AFOSR Grant No. 69-1744.

2 Graduate Student of Aerospace Engineering, University of Texas, Austin, Texas. 3 Assistant Professor of Aerospace Engineering, University of Texas, Austin, Texas.

432

]OTA: VOL. 5, NO.6, 1970 433

1. Introduction

Previous analyses of optimum hypersonic airfoil shapes have been con­cerned with nonlifting, symmetric airfoils of minimum drag or lifting, flat-top airfoils of minimum drag for a given lift or of maximum lift-to-drag ratio. The former are summarized in Ref. 1, while the latter are contained in Refs. 2-7. In this paper, lifting airfoils are optimized, but the only restriction imposed on the upper and lower airfoil surfaces is that their slopes relative to the free-stream direction are small, implying that the airfoils are slender and are at small angles of attack.

If such conditions as extreme heat-transfer rates, use of the wing as a storage volume, or the necessity to maintain structural strength are not con­sidered, the lift-to-drag ratio is the primary indicator of wing performance. However, it is easily shown that a wing designed solely to maximize the lift-to­drag ratio is too thin for practical purposes, at least from the standpoint of internal volume or structural strength (for example, see Ref. 6, where thickness ratios on the order of 2 % are predicted). Therefore, it is desired to determine the greatest performance that can be achieved when the prescribed characteris­tics of the wing are considerably different from those of the unconstrained optimum wing (Ref. 6) when both are operating at a specified flight condition, that is, altitude and velocity.

In view of the above statement, the problem of determining the airfoil shape which maximizes the lift-to-drag ratio for a given profile area and chord is considered here. It is assumed that the pressure distribution on surfaces which see the flow can be approximated by the tangent-wedge relation. On surfaces which do not see the flow, the Prandtl-Meyer pressure relation for small disturbances in hypersonic flow is employed. With regard to skin friction, it is assumed that the surface-averaged skin-friction coefficient can be held constant during the optimization procedure. It has been shown in Ref. 7 that this assumption is valid and that it simplifies the extremization problem considerably. Finally, the base drag is neglected with respect to the forewing drag. The effect of including the base drag in optimum hypersonic airfoil studies is negligible for moderate to high hypersonic Mach numbers (Ref. 7).

2. Lift-to-Drag Ratio and Profile Area

In order to determine an optimum airfoil shape, it is necessary to predict the appearance of the optimum airfoil shape. Now, the family of optimum

434 lOTA: VOL. 5, NO.6, 1970

airfoil shapes obtained by specifying the profile area and the chord must contain the family of optimum airfoil shapes obtained by specifying only the chord. Next, the optimum airfoil shape for a given chord is a wedge at such an angle of attack that the upper surface is located in an expansion region of the flow field (Ref. 6). Therefore, to begin the solution of the present problem, it is assumed that the optimum airfoil shape has the form shown on the left in Fig. 1.

If it is assumed that the airfoil is slender and is operating at a small angle of attack, that the effect of skin-friction forces on the lift is negligible with respect to pressure forces, that the base drag is negligible with respect to the forebody drag, and that the surface-averaged skin-friction coefficient C fa is constant during the optimization procedure, the expressions for the lift L and the drag D can be written as

Ljq = f: (Cp! - Cpu) dg (1)

Djq = 5: (Cp!TJ!' - CpuTJu' + 2Cta) dg

where q is the free-stream dynamic pressure, Cp is the pressure coefficient, u and I refer to the upper and lower surfaces, respectively, and r/ denotes the derivative dry/dg. It should be noted that c, the length being held constant, is the chord under the assumptions of slender airfoils at small angles of attack. In these expressions, the pressure coefficient on the upper surface is given by the Prandtl-Meyer expansion formula, while the pressure coefficient on the lower surface is obtained from the tangent-wedge approximation (Ref. 8), that is,

c ~ ! x

T T t MT

1 1 7J y,z

Fig. 1. Coordinate systems.

}OTA: VOL. 5, NO.6, 1970 435

where y is the ratio of specific heats, M is the free-stream Mach number, and ki = (y - 1)/2, k2 = 2y/(y - 1), k3 = (y + 1)/4. Finally, the profile area A is given by

(3)

At this point, the following dimensionless variables and parameters are introduced (Fig. I-right):

x = ~/c,

E* =L/MD,

as are the functions

y = M'1]u/c,

A* = MA/c2,

Then, the lift-to-drag ratio can be written as

where

while the profile area becomes 1

A*=fo(z-y)dX

3. Maximum Lift-to-Drag Ratio Problem

T = tic

M* = M {leta (4)

(5)

(6)

(7)

(8)

The problem of maximizing the lift-to-drag ratio for a given profile area and chord is equivalent to that of maximizing the ratio of integrals (6) subject to the isoperimetric constraint (8) and certain prescribed boundary conditions. This is not a standard problem of the calculus of variations, but it can be transformed into one (Ref. 9). In other words, it can easily be shown that the problem of maximizing the ratio of functionals E * = I L/I D is equivalent to that of maximizing the modified functional E' = IL - E*ID . However, it is known a priori that the maximum value of E' is zero, that is, E* = IL/ID , and that E* is actually the maximum value being sought and, hence, is held constant during the extremization procedure. To constrain the profile area,

436 JOTA: VOL. 5, NO.6, 1970

Eq. (8) is adjoined to the modified functional E' by an undetermined, constant Lagrange multiplier A. The proof of these statements can be conducted by reformulating the above problem as a Mayer problem or by variational differen­tiation.

The result of the above discussion is that the problem of finding the airfoil shape of given profile area and chord which maximizes the lift-to-drag ratio is equivalent to that of maximizing the standard-form functional

XI

1= J F(y, z,y, i, E*, A) dx Xi

(9)

with

F = (yQ - P) - E*(yiQ - yP + yM*3) + A(Z - Y) (10)

subject to the prescribed boundary conditions

Xi = 0, Yi = 0,

YI free,

Zi = 0

ZI = free (11)

XI = 1,

and the integral relations (6) and (8) which are used to determine E* and A as functions of the profile-area parameter A * .

4. Necessary Conditions and Solution Process

It is known that the functions y(x) and z(x) which extremize the functional (9) must be solutions of the Euler equations (see Ref. 1, for example)

Fy - dFy/dx = 0, Fz - dFi/dx = 0 (12)

which, since Fy = -A and Fz = A from Eq. (10), admit the first integrals

F 11 = -Ax + C1 , (13)

The constants of integration C1 and C2 can be determined by applying the transversality. condition

[(F - yFy - iFi ) ox +Fy oY +Ft oz]r = 0 (14)

which must be consistent with the prescribed boundary conditions. Since 8y j *- 0 and 8z j *- 0 while the remaining variations at the end points vanish, the transversality condition requires that (Fy)j = (Fi)j = 0, so that C1 = A

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and C2 = -,\ from Eq. (13). Finally, taking the required derivatives of the fundamental function (10) leads to the following equations for determining the optimum upper and lower airfoil surfaces:

P' - E*(P + yP') = -A(1 - x), yQ' - E*(yQ + yzQ') = -A(1 - x) (15)

where P' = dPjdy and Q' dQjdz. The procedure employed to obtain the functions y(x) and z(x) which

maximize Eq. (9) is an iterative one. Values are chosen for y and M*, and values are guessed for the constants E* and ,\. Then, Eqs. (15) are solved for y(x) and z(x) by incrementing x in the interval 0 :( x :( 1. These values of the derivatives are then used in Eq. (6) to calculate a new value of E* . This value of E* differs from the guessed value, and the difference can be denoted by ,1E*. What remains is to iterate on the guessed E* until ,1E* = 0 (,1E* :( 10-7, in this case), and this is done by the Newton-Raphson method. Once the correct value (that is, the maximum value of E*) is determined, the final set of slopes y(x), z(x) is integrated by Simpson's rule to obtain the optimum airfoil shape y(x), z(x). Finally, the thickness ratio parameter T* = Zj - Yj can be determined, and the integration indicated in Eq. (8)

2.0 r---.::::-r---..:---.,----...,----..,..---.,----...,----..,..,,-----,

o

-1.0 ~-----t------,k:_--__:;;;.f'_--_+~-""'=---t__--_t_--=""'_I.;::---__1

-2D 1---+-----1-- --- -- -

-3.0~~ ___ ~ __ ~ __ ~ ___ ~ __ ~ __ ~ __ ~ -0.1 o 0.1 .. 0_2

Y,z 0.3

Fig. 2. Euler equations.

0.4 0.5 0.6

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can be carried out to find the value of A* corresponding to the prescribed value of A.

A word as to how E* and A are guessed is in order. Hypersonic airfoils can be shown to develop lift-to-drag ratios on the order of five. Hence, at a Mach number of five, E* would have a value of unity, and one might make the first guess E* = 1. With regard to possible choices of A, Eqs. (15) are plotted in Fig. 2 for y = 1.4 and various values of E* , and only that part of the graph where y, z are positive pertains to the discussion. At the origin, where x = 0, the ordinate of Fig. 2 becomes just A. Hence, for E* = 1, one could choose any value of A in the range -0.02 ::::;;; A ::::;;; 1.4 and obtain a solution for which Zi ~ Yi and Yi ~ O. By working around these values, the complete set of solutions for which Y, Z are positive can be found.

In carrying out these calculations, the ratio of specific heats is chosen to be 'Y = 1.4, and the parameter M * is varied in the range 0.5 ::::;;; M * ::::;;; 1. 5. The lower limit defines the beginning of hypersonic flow, while the upper limit defines that part of hypersonic flow where viscous interactions become important. The optimum shapes are shown in Figs. 3-5 with A as a parameter. The relation between A and A* (the parameter containing all the prescribed quantities) is shown in Fig. 6. Finally, the approximate thickness-ratio para-

-1.0

A =6 y

o --- 3 1.4

~ -----= -CD ------ 1.4

~ --3 1.0

M.=0.5 '~ ~ ~

z

2.0

3.0 o 0.2 0.4 0.6 0.8 1.0

x

Fig. 3. Optimum airfoil shapes, M* = 0.5.

JOTA: VOL. 5, NO.6, 1970 439

-1.0

Y A=

3 0 1.4

0 -CD

1.0 t-----t----=~-t_----=:-I"___=_---+__-=,..,..,.._J

z

2.0 I-----+----+------i---=::,~-__I_---__I

3.0 0 0.2 0.4 0.6 0.8 1.0

x Fig. 4. Optimum airfoil shapes, M* = 1.0.

-1.0

Y A=6

3 0

z

2.0t-----r-----+----~~----+_---~~~

3.00 ':::-----=-=------:~---__::~---__,_J----:::......J

0.6 0.8 1.0 x

Fig. 5. Optimum airfoil shapes, M* = 1.5.

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8 r---------,---------,---------,---------,--------,

6 r---------T---------r---------r-------~~~~--~

2

o

-2 o 0.4 0.8 1.2 1.6 2.0

Fig. 6. Lagrange multiplier.

~Or----------r----------.---------,----------.----------,

4.01--------+--------1------+------+--------1

Aft Fig. 7. Thickness-ratio parameter.

IOTA: VOL. 5, NO.6, 1970 441

meter 'T * is shown in Fig. 7, while the maximum lift-to-drag ratio parameter E* is plotted in Fig. 8.

Theoretically, all the arcs obtained so far are extremal arcs. To ascertain that these extremal arcs are indeed maximal arcs, one must show that the Legendre condition

(16)

is satisfied at each point of each extremal arc. Because of the nature of the problem, FYi = 0, and the Legendre condition is satisfied if

F~iJ ~ 0, (17)

at each point. The explicit forms of these inequalities are obtained by using Eq. (10) and are given by

_pH + E*(2P' + yP") ~ 0, Q" - E*(2Q' + zQ") ~ ° (18)

and are plotted in Fig. 9 for several values of E * . Since the Legendre condition is satisfied, the extremal solutions are maxima.

The solutions obtained in this section are only valid for a finite range of

22

14

I\~ ! ~ /Yj=O I

! 1 :~ i i \. VYj=i j i \

\~ ~~O5 ~ Y - ". 1.0 t'-..... \

\ '-. ""-

~ \ ' ...... --

1.8

10

0.6

1.5 r--._

0.2 o 0.4 0.8 1.2

A.

Fig. 8. Lift-to-drag ratio parameter.

1.6 2.0

442 JOTA: VOL. 5, NO.6, 1970

Or-------~------._------_rr_----~------_,------_,

-5 ~---4_----7+-r-+.~-+-~--~ __ --_+---~

F" yy

-10 ---.- --+---+---f--+-----\----+----'\---+--'''<----+-.---''~-___{

-20L-~ ____ ~ ______ L-______ ~ ______ ~~ ____ ~ ____ ~

-2.0 -1.0 o Y,Z 1.0 2.0 3.0 4.0

Fig. 9. Legendre condition.

values of ,\. One limiting value, ,\ = y, occurs when Yi = 0; for ,\ > y, the initial slope of the upper surface becomes negative, creating a compression rather than an expansion for which the pressure law has been chosen. However, the expansion formula holds for the compression provided Y is very small (Ref. to). At any rate, for ,\ > y, the problem is reformulated to include the compression on the upper surface, and this is discussed in the following section. The other limiting value occurs when Yi = Zi ; however, the value of ,\ at which this occurs varies with M * . For values of ,\ lower than this limiting value, Eqs. (15) predict that Yi > Zi' which is not physically possible. To find these solutions, it is necessary to add whatever constraints are necessary to ensure that the two surfaces do not intersect. It is observed that this class of solutions must contain that for A* = 0 (Fig. 5), which is a flat plate (zero profile area) at the optimum angle of attack. Finally, it is observed that the optimum shapes for the values of ,\ being considered are composed of a flat plate followed by concave upper and lower surfaces which give the airfoil thickness near the trailing edge. These shapes have not been computed in their entirety because they are not practical. However, the correct trends are shown in the figures by dashed lines. The points for A * = 0 (,\ = - (0) have been calculated.

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5. Necessary Conditions and Solution Process, A> Y

From the trend of the solutions for'\ ~ y, it is apparent that the solutions for ,\ > y have a similar lower surface, but the upper surface begins with a negative-slope portion (tangent-wedge pressure distribution) and switches to a positive-slope portion (Prandtl-Meyer pressure distribution) at some abscissa Xc to be determined during the optimization procedure. For these shapes, the lift-to-drag ratio is given by

(19)

where

IL = {C- (yQ - yR) dx + f (yQ - P)dx o Xc+

(20)

ID = {C- (yzQ - yjR + yM*3) dx + f (yzQ - yP + yM*3) dx o Xc+

and where

(21)

In view of the discussion at the beginning of the previous section, the problem of maximizing the lift-to-drag ratio (19) for a given profile area (8) and chord is equivalent to the problem of maximizing the functional

with

f"'C- fl I = G(y, z, y, Z, E* , ..\) dx + F(y, z, y, Z, E* , ..\) dx

o Xc+

G = (yQ - yR) - E*(yzQ - yjR + yM*3) + ..\(z - y)

F = (yQ - P) - EAyzQ - yP + yM*3) + ..\(z - y)

subject to the prescribed boundary conditions

XI = 1,

Yi = 0,

YI = free,

Zi = 0

Zl == free

(22)

(23)

(24)

and the integral relations (8) and (19) which are used to determine E* and A as functions of the area parameter A * .

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The functional (22) is not one of the standard functionals for which necessary conditions have been derived. It is therefore necessary to develop the necessary conditions through variational differentiation. Hence, after taking the first variation of Eq. (22), performing the usual integration by parts, and relating variation 8 taken at constant x to arbitrary variations S (see Ref.1, for example), one obtains the following relation:

OJ = (C- [(Gy - dG,iJ/dx) 8y + (Gz - dGz/dx) 8z] dx "'.

+ [(G - yGy - zGz) 8x + Gy 8y + Gi 8z]f-

+ (' [(Fy - dFiI/dx) 8y + (Fz - dFt/dx) 8z] dx "'c+

(25)

This relation does not allow for a corner except at the point Xc ; however, if a corner exists on either the upper or lower surface between Xi and Xc or Xc

and xf ' then the standard corner conditions apply. The vanishing of the first variation (25) leads to the Euler equations

Gy - dGy/dx = 0,

Fy - d}~/dx = 0,

Gz - dGi/dx = 0,

Fz - dFi/dx = 0, (26)

the natural boundary conditions (SXi = SYi = SZi = SXf = 0, SYt * 0, SZf * 0)

(27)

(G - yGy - zGi)c_ = (F - yFy - zFt)c+

(Gy)c_ = (Fy)c+ , (Gz)c- = {Fz)c+ (28)

In applying these conditions, there is no reason to suspect a corner on the lower surface. On the other hand, the corner conditions must be applied to the upper surface to determine the conditions under which the compression and expansion segments are to be joined. In this connection, if zc_ = Z'c+ , then the last corner condition is satisfied since Gz = Fi from Eqs. (23). With this information, the remaining corner conditions reduce to

(29)

F· Y

6

4

2

o

-2

20

16

12

8

4

JOTA: VOL. 5, NO.6, 1970

1\ \ \

\ \\ ( Fyle_

'0~ ~

I ~ ~.=O r-----I-----\: r- 1.0

2.0

-3 -2 -I 'Ie -

o 2 Ye+

Fig. 10. Corner condition.

\E.=2.0

\ V G-yGy

\.

'" '--Z F-yFy o

-1.2

~ ~ -0.8 -0.4 o 0.4 0.8

Yc- Yc+ Fig. 11. Corner condition.

445

(Fyle+

3 4

r2.0

Jf'O E.=O

1.2 1.6

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and, after the use of Eqs. (23), are plotted in Figs. 10-11. These figures show that the only way the compression and expansion subarcs can be joined so that E * is the same for each subarc is if

Yc- = Yc+ = 0 (30)

Next, if it is observed that Gy = -A, Gz = A, Fy = -A, and Fz = A from Eqs. (23), the Euler equations (26) can be integrated to obtain

Gy = -Ax + CI ,

Fy = -Ax + C3 ,

(31)

Hence, the last two corner conditions in Eq. (28) require that C1 = Ca and C2 = C4 , while the natural boundary conditions (27) show that Ca = A and C4 = -A. Combining these results with the first integrals (31) and the fundamental functions (23) leads to the following equations for determining the optimum airfoil shape:

yR' - E*(yR + yjR') = -A(1 - x)l yQ' - E*(yQ + yzQ') = -A(l - x)

P' - E*(P + yP') = -A(l - x)l yQ' - E*(yQ + yzQ') = -A(l - x)!

(32)

Finally, the abscissa Xc where the upper surface subarcs are joined can be determined from (Gy)c_ = -A(l - xc), Yc- = 0, so that (Gy)c_ = Y; therefore,

Xc = 1 - yfA (33)

To show that the extremal arcs obtained above are maximal arcs, It IS

necessary to show that the second variation is nonpositive. For fixed end points, the second variation becomes

'iN = {C- [Gyy(&y)2 + GU (&Z)2] dx + {f [Fyy(&y)2 + Fu(8z)2] dx ~ 0 (34) :Ilj :Ilc+

This condition is satisfied if

Gyy ~ 0,

Fyy :< 0,

Gu~O,

Fu~O,

Xi ~ X < Xc

where, from Eqs. (23), it is seen that GJi = FJi .

(35)

JOTA: VOL. 5, NO.6, 1970 447

The procedure to be followed in the calculation of the airfoil shapes is nearly identical with that used previously. The difference is that the calculation is started using the first two of Eqs. (32). When the upper-surface slope y becomes zero, the equations must be changed to the last two of Eqs. (32). The selection of the values for E * is essentially determined by where the previous solutions left off. However, the choice of values for A is limited to the values A ~ y for which Yi ~ 0 (Fig. 2). The optimum shapes are shown in Figs. 3-5; the quantities A, T*, E* are plotted versus A* in Figs. 5-8; and the satisfaction of the Legendre condition is displayed in Fig. 9.

6. Discussion and Conclusions

The problem of determining the airfoil shape which produces the maximum lift-to-drag ratio for a given profile area and chord has been consid­ered. It is found that the airfoil shapes fall into three categories, depending on the value of the area parameter A* . For low values of A* , the optimum shape is composed of a flat plate at an angle of attack followed by concave upper and lower surfaces which give the airfoil thickness near the trailing edge. For A* = 0, the optimum airfoil shape is a flat plate at the optimum angle of attack. These airfoil shapes are considered impractical because of the low thickness ratios. The airfoil shapes for moderate values of A* have a finite thickness all along the chord and include the optimum shape for the case where the profile area is free (A = 0), that is, the wedge. These airfoil shapes have the highest maximum lift-to-drag ratios, but the thickness ratios may be too low (around 2 %, see Ref. 6) for practical purposes. For the larger values of A* , the optimum airfoil shapes have a lower surface which always sees the flow, while the forward portion of the upper surface sees the flow and the rearward portion does not see the flow. For these shapes, the thickness-ratio parameter increases with A* and the lift-to-drag ratio parameter decreases.

In order to determine actual airfoil shapes, it is necessary to employ an iteration procedure. This procedure begins by specifying the free-stream conditions and guessing a value for C ta . An optimum shape is then determined, and the value of Cta for this shape is calculated and is used as the guess for the next iteration. The procedure is repeated until the guessed and calculated values agree. It is possible to obtain some approximate information by assuming that the boundary layer is laminar and that C ta is for a flat plate at zero angle of attack. If the flight conditions are characterized by a free-stream Mach number of M = 10 and a free-stream Reynolds number of Re = 106, the value of Cta given in Ref. 11 is Cta = 10-3• Using these numbers, one finds

448 ]OTA: VOL. 5, NO.6, 1970

that M * = 1, so that, if Ajc2 = 0.05, the following results are obtained: T = 0.084 andLjD = 6.7.

While this analysis has omitted the base drag as far as the optimization problem is concerned, the calculation of the final lift-to-drag ratio must include the base drag. Hence, the above value ofLjD decreases somewhat.

Finally, it is noted that the optimum airfoil shapes obtained here have sharp leading edges and, hence, are subject to heating rates excessive for present-day materials. The unavoidable blunting of the leading edge causes a further decrease in the lift-to-drag ratio.

References

1. MIELE, A., Editor, Theory of Optimum Aerodynamic Shapes, Academic Press, New York, 1965.

2. DONALDSON, C. DUP., and GRAY, D. E., Optimization of Airfoils for Hypersonic Flight, Aerospace Engineering, Vol. 20, No.1, 1961.

3. LEWELLEN, W. S., and MIRELS, H., Optimum Lifting Bodies in Hypersonic Viscous Flow, AlAA Journal, Vol. 4, No. 10, 1966.

4. MIELE, A., and LUSTY, A. H., Jr., On Optimum Wedges and Semicones in Hyper­sonic Viscous Flow, AIAA Journal, Vol. 5, No.1, 1967.

5. HULL, D. G., Two-Dimensional, Lifting Wings of Minimum Drag in Hypersonic Flow, Rice University, Aero-Astronautics Report No. 24, 1966.

6. HULL, D. G., Two-Dimensional, Hypersonic Wings of Maximum Lift-to-Drag Ratio, Journal of the Astronautical Sciences, Vol. 14, No.2, 1967.

7. WEEKS, D. E., Jr., and HULL, D. G., Optimum Flat-Top Airfoils in Viscous Hypersonic Flow, Journal of Optimization Theory and Applications, Vol. 5, No.1, 1970.

8. Cox, R. N., and CRABTREE, L. F., Elements of Hypersonic Aerodynamics, Academic Press, New York, 1965.

9. MIELE, A., On the Minimization of the Product of Powers of Several Integrals, Journal of Optimization Theory and Applications, Vol. 1, No.1, 1967.

10. DORRANCE, W. H., Two-Dimensional Airfoils at Moderate Hypersonic Velocities, Journal of the Aeronautical Sciences, Vol. 19, No.9, 1952.

11. SCHLICHTING, H., Boundary Layer Theory, Pergamon Press, New York, 1955.