Optimizing lift-to-drag ratio of slender, flat-top wing of given planform in hypersonic flow by using variational calculus method

Hull, D. G.

Miele, A.

English

NASA Technical Reports Server

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AERO-ASTRONAUTtCS REPORT NO. 27 
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THREE-DIMENSIONAL WINGS 
OF MAXIMUM LIFT-TO-DRAG RATIO IN HYPERSONIC FLOW 
by 
A N G E L 0  MlELE and DAViD G.  HULL 
N 66 34797 
w -  
I ( T H R U I  (ACCESSION H L i h ? a E R )  
1966 
https://ntrs.nasa.gov/search.jsp?R=19660025507 2020-03-16T20:11:19+00:00Z
1 AAR-27 
- THREE- DIMENSIONAL WINGS 
(* 1 OF MAXIMUM LIFI'-TO-DRAG RATIO IN HYPERSONIC FLOW 
(***I ANGEL0 MIELE(**) and DAVID G. HULL I 
1 ,  , SUMMARY 
The problem of maximizing the lift-to-drag ratio of a slender, three-dimensional, 
flat-top wing of given planform in hypersonic flow is considered under the assumptions 
that the pressure coefficient is modified Newtonian and the skin-friction coefficient is 
constant. The indirect methods of the calculus of variations in two independent variables 
a r e  employed, and the maximum lift-to-drag ratio problem is solved for (a) unconstrained 
I 
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volume and (b) given volume. 
If the volume is unconstrained, the optimum wing surface is unique; it has a con- 
~ stant chordwise slope and a trailing edge thickness distribution similar to the chord 
I distribution. 
(*I This research was supported by the Langley Research Center of the National 
Aeronautics and Space Administration under Grant No. NGR-44-006-045. 
(* *) Professor of A S ~ ~ O M U ~ ~ C S  and Director of the Aero-Astronautics Group, 
Department of Mechanical and Aerospace Engineering and Materials Science, Rice 
University, Houston, Texas. 
**+ ( Research Associate in Astronautics, Department of Mechanical and 
Aerospace Engineering and Materials Science, Rice University, Houston, Texas. 
- 1  
1 AAR-27 
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2 
If the volume is given, the chordwise slope of the optimum wing is constant in 
the spanwise sense but not in the chordwise sense. A one-parameter family of extrema1 
solutions exists, depending on the value of the volume parameter: this parameter is 
directly proportional to the volume and inversely proportional to the span and the root 
chord squared. If the volume parameter exceeds a certain critical value, the optimum 
wing is convex. If the volume parameter is equal to the critical value, the optimum 
wing is identical with that of case (a). Finally, if the volume parameter is smaller 
than the critical value, the optimum wing is slightly concave. 
3 AAR- 27 
1. INTRODUCTION 
For three-dimensional, hypersonic wings, two extrema1 problems are of 
interest, that is, to  minimize the drag for a given lift and to  maximize the lift-to-drag 
ratio for unconstrained lift. Since problems of the former type were considered in 
Ref. 1, the object of this paper is to  consider problems of the latter type in connection 
with wings whose planform shape is given, whose thickness distribution on the trailing 
edge is free, and whose volume is either free o r  arbitrarily given. The following 
hypotheses are employed: (a) a plane of symmetry exists between the left-hand and 
right-hand parts of the wing; (b) the upper surface is flat; (c) the free-stream velocity 
is parallel to the line of intersection of the plane of symmetry and the flat top; (d) the 
wing is slender in both the chordwise and spanwise senses, that is, the squares of 
both the chordwise and spanwise slopes a re  small with respect to one; (e) the pressure 
coefficient is proportional to the cosine squared of the angle formed by the free-stream 
velocity and the normal to  each surface elementxf) the skin-friction coefficient is 
constant and equal to some suitably chosen average value; (g) the base drag is neglected; 
and (h) the contrihtion of the tangential forces to  the lift is negligible with respect to  
the contribution of the normal forces. 
4 
2. FORMULATION OF THE VARIATIONAL PROBLEM 
In order to relate the lift-to-drag ratio and the volume of a three-dimensional, 
-I AAR- 27 
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flat-top wing to its geometry, the following reference system xyz is used: x is a 
chordwise coordinate measured from the leading edge and positive toward the base; 
y is a spanwise coordinate measured from the plane of symmetry and positive toward 
the wing tip; and z is a coordinate perpendicular to the flat-top plane, measured from 
the flat top, and positive downward. In this coordinate system, the boundary of the 
wing is described by the equations (Ref. 1) 
Leading edge 
Trailing edge 
x = o  , 2 3 . 0  
x =  c(y) , z = t(y) 
where the chord distribution c(y) is given and the trailing edge thickness distribution 
t(y) is free. Furthermore, if z(x, y) denotes the lower surface, p = az/ax the chord- 
wise slope, and q = &/ay the spanwise slope, the lift-to-drag ratio E = L/D is given by 
(Ref. 1) 
where 
K = C f / n  
and where b is the prescribed span, C the skin-friction coefficient, and n a factor 
f 
modifying the Newtonian pressure distribution (*I . This lift-to-drag ratio is to be 
(*) The pressure coefficient employed in Eq. (2) is C = 2np 2 . 
P 
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5 AAR-27 
maximized subjected to the isoperimetric constraint of given volume 
and the inequality constraint 
P 20 
which expresses the limit of validity of the Newtonian pressure law. 
The previous problem consists of extremizing a ratio of integrals and is governed 
by the theory of Ref. 2 .  Therefore, we study the maximization of the functional 
(see Chapter 3 of Ref. 3) 
subject to the boundary conditions (1) and the isoperimetric constraint (4) with the 
understanding that the fundamental function is defined as  (* 1 
(7) 
F = p 2 -  E(p 3 + K ) + X z  
The unknown maximum lift-to-drag ratio E and the undetermined Lagrange multiplier 
X are held constant during the extremization process. The fundamental function (7) is 
characterized by the first partial derivatives 
(") For simplicity, Ineq. (5) has not been accounted for in the definition (7). 
However, it can be verified a posteriori that it is satisfied by all of the extremal 
surfaces calculated here. 
6 
and the second partial derivatives 
F = 2(1 - 3Ep) , 
PP 
F = O  ? 
zz 
F = O  
99 
F = O  
ZP 
F = O  
Pq 
F = O  
zq 
~~ 
7 AAR- 27 
I .  
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3. NECESSARY CONDITIONS 
The function z(x, y) atremizing the functional (6) must be a solution of the 
Euler equation 
aF /ax - F~ = o 
P I 
~ which, in the light of Eqs . (8) can be rewritten as 
Therefore, upon integrating Eq. (11) in the x-direction, we see that the following first 
integral is valid: 
2 
3Ep - 2p + hx = f(y) 
where f(y) is an arbitrary function of the spanwise coordinate. 
The boundary conditions for the Euler equation are partly of the prescribed type 
and partly of the natural type. The latter are to  be determined from the transversality 
condition 
( F  - p F  ) ( 6 ~  - XSy) 3- F ( 6 2  - k6y) = 0 
P P 
(i = dx/dy and 5 = dz/dy denote total derivatives evaluated on the boundary) which must 
be satisfied for every set of variations consistent with the conditions imposed on the 
planform shape and the thickness distribution on the periphery of the planform. For 
the leading edge, the following relations hold: 
6x-X6y=O , 6 2 - i 6 y = O  
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1 AAR - 2'7 1 
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so  that Eq. (13) is satisfied. For the trailing edge, the variations 6x and 6y satisfy 
the relationship 
6x - i 6 y  = 0 
and Eq. (13) is satisfied providing 
F = O  
P 
that is, because of Eq. (8-1), providing 
- Trailing edge 2 3Ep - 2p = O  
Once the solution of the Euler equation is obtained, it is necessary to verlfy that 
it maximizes the functional (6). In this connection, the Legendre condition 
F S O  
PP 
must be satisfied and ensures a relative maximum with respect to weak variations. 
Because of Eq. (9-1), its explicit form is 
If strong variations of the slope are considered, the Legendre condition is to be 
replaced by the Weierstrass condition 
where z and p a r e  the ordinate and the slope of the extremal surface and p is the slope 
C 
of the comparison surface. The explicit form of this inequality 
holds for every comparison slope consistent with the constraint (5) providing 
I p 2 1/2E 
Clearly, this inequality is more restrictive than Ineq . (19). 
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4. NONDIMENSIONAL QUANTITIES 
In order to s i m p l e  the representation of the results for the particular cases, 
it is convenient to introduce the nondimensional coordinates 
s = x/c(y) 7 q =  2Y/b 9 5 =  Z/tOl) 
the nondimensional chord distribution 
Y = C(Y)/C(O) 
and the thickness ratio of the generic airfoil 
7 = t(Y)/C(Y) 
Also, we define the thickness ratio, lift-to-drag ratio, and volume parameters 
7, = T / $ Z  
3- E, = E ,,/K 
v, = V/bC2(O) $K 
_ _ ~  
5. VOLUME FREE 
11 AAR-27 
If the volume is free, the relationship X = 0 holds so that the extremal surface 
is described by the first iaegral 
2 3Ep - 2p =f(y) 
and the boundary conditions (1) in which the thickness distribution t(y) is to be 
determined s o  that the natural boundary condition 
Trailing edge 2 3Ep - 2p = O  
is satisfied. Because of the Weierstrass condition (22), this equation admits the 
solution 
Trailing edge p = 2/3E 
indicating that the chordwise slope is constant over the trailing edge. By combining 
Eqs. (27) and (28), we see that 
f(Y) = 0 
meaning that the chordwise slope is also constant over the entFre extremal surface. As 
a consequence, the optimum wing is described by the partial differential equation 
p =2/3E 
which, in the light of the conditions (l- l) ,  admits the particular integral 
z = (2/3E)x 
12 AAR-27 
Next, the conditions (1-2) are applied to obtain the relationship 
implying that the trailing edge thickness distribution is similar to the chord distrihtion. 
Then, by forming the ratio of the above equations and introducing the dimensionless 
coordinates (23), we conclude that the optimum wing surface is given by 
s = 5  (34) 
The final step consists of combining Eqs. (2), (4), (26), (3l),and (32) to ok&ain the 
following values of the thickness ratio , lift-to-drag ratio, and volume parameters: 
3- 
7, = 412 (3 5) 
Equation (35-2) represents the highest lift-to-drag ratio which can be obtained with a 
three-dimensional, flat-top wing subjected to a flow parallel to the flat top. Should a 
wing of given planform be required to  have a volume other than that given by Eq. (35-3), 
a loss in the lift-to-drag ratio would occur with respect to that predicted by Eq. (35-2). 
13 AAR-27 
6. VOLUME GIVEN 
If the volume is given, the extremal surface is described by the first integral 
and the boundary conditions (1) in which the thickness distribution t(y) is to be 
determined so  that the natural boundary condition 
2 Trailing edge 3Ep - 2p = O  
is satisfied. Because of the Weierstrass condition (22), this equation admits the solution 
Trailing edge p = 2/3E (38) 
indicating that the chordwise slope is constant over the trailing edge. 
Lf the first integral (36) is applied at the trailing edge and is combined with 
Eq. (37), it is seen that 
and that 
3Ep 2 - 2p - X(C - X) = O  
This is an algebraic equation of the second degree in  p which--in the light of the 
Legendre condition (19)--admits the solution 
1/2 3Ep = 1 + [ 1 + 3EX(c - x)] 
14 AAR- 27 
If this partial differential equation is integrated in x-direction and the conditions (1- 1) 
are imposed, we obtain the relationship 
3Ez = x + (2/9EX) {[ 1 + ~ E X C ] ~ ' ~  - [l + 3EX(c - x)I3l2} (42) 
which, at the trailing edge, becomes 
3Et = c + (2/9EX) [(l + 3Ehc) 3/2 - 1) (43) 
Therefore, if the following definitions are introduced : 
a = 3EXc(O) 
and 
(44) 
Eqs . (41) and (42) can be rewritten as 
2 
(27E X)t  = G(l, q, a)  
Consequently, the optimum shape is described in nondimensional form by 
15 AAR-27 
The next step is to relate the quantity a to the prescribed value of the volume 
as well as to determine the optimum dimensions and the lift-to-drag ratio. To do this, 
we combine Eqs. (2), (4), (43), and (47) to obtain the relations 
where 
1 
The final step consists of eliminating the quantity a between Eqs. (47) and (48) 
to obtain the functional relationships 
5 = f,(% "-I, V,) 
7, - f2(% V,) 
E, = f3(V*) 
The corresponding extremal surface is convex for 
16 AAR-29 
and concave for 
where a and a a re  defined by the relations 
1 2 
B(al) = 0 , a2 +3/4 = O  (53) 
For  values of V, smaller than the lower bound in Ineq. (52), the optimum wing is 
composed of two subsurfaces, one of which is a flat-plate starting at the leading 
edge. However, these solutions a re  not analyzed here in  view of their limited technical 
interest. 
With the aid of Eqs . (47) through (53), the proposed problem can be solved 
numerically o r  analytically for each volume parameter V, and chord distribution 
y(a. In this connection, the results for a constant chord (Ref. 4), that is, for 
Y(rl) = 1 
are plotted in Figs. 1 through 3. Since the limiting values of a a r e  
u =5.42 , = - 3/4 1 2 
the extrema1 solution is convex for 
0.630 5 V, 
and concave for 
0.543 cvyA 5 0.630 
(54) 
(55) 
(56) 
(57) 
17 AAR-27 
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REFERENCES 
1. MIELE, A. and HULL, D.G., Three-Dimensional, Lifting Wings of Minimum 
Drag in Hypersonic Flow, Rice University, Aero-Astronautics Report No. 26, 
1966. 
2. MIELE, A. , The Extremization of Products of Powers of Functionals and Its 
Application to  Aerodynamics, Astronautica Acta, Vol. 12, No. 1, 1966. 
3 .  MIELE, A. , Editor, Theory of Optimum Aerodynamic Shapes, Academic Press, 
New York, 1965. 
4 .  HULL, D.G., Two-Dimensional Wings of Maximum Lift-to-Drag Ratio in 
Hypersonic Flow, Rice University, Aero-Astronautics Report No. 25, 1966. 
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LIST OF CAPTIONS 
AAR-27 
Fig. 1 Optimum shape. 
Fig. 2 Optimum thickness ratio. 
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6 
4 
7* 
2 
0 -
0.5 
0.6 
0.4 
E* 
0.2 
/ 
1 .o 1.5 2.0 2.5 
v* 
Fig. 2 Optimum thickness ratio. 
3.0 
5 1 .o 
Fig. 3 
1.5 2.0 
v* 
2.5 
Maximum lift- to- drag ratio. 
.O 


Three-dimensional Wings of Maximum Lift-to- Drag Ratio in Hypersonic Flow

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