JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 12, No. 6, 1973

Continuous and Discontinuous Solutions for Optimum Thrust Nozzles of Given Length

K. G. GUDERLEY, 1 D. TABAK, e M. C. BREITER, ~ AND O. P. BHUTANI 4

Abs t rac t . In its first sections, the paper deals with optimum thrust nozzles of given length and exit radius for flows with swirl. The computation is based on a modification of methods familiar for flows without swirl. Rather extensive numerical results show that the swirl does not impair the specific impulse attainable at a given nozzle length. The analysis suggests that the assumption of isentropic continuous flows, on which this approach is based, may sometimes be too restrictive. A survey of plane nozzles shows, on the other hand, that discontinuities need to be admitted only if, besides the length, a rather large radius of the nozzle is prescribed. Discontinuous solutions have been thoroughly investigated by Shmyglevskiy. At least in principle, we use the same line of thought, but considerable simplifications are possible if one starts with the variational formula- tion of Rao. In its numerical discussion and also in some analytical details, the present paper goes beyond Shmyglevskiy's results. The problem is conveniently discussed in a state plane, which has the local state of the flow (flow direction and speed or Mach number) as independent variables. By taking into account second variations, one can determine the boundary of the region for which continuous solutions give the (local) maximum. This boundary coincides with the locus of points at which the solution in the physical plane would fold back into itself. Another limitation of the original approach emerges if one asks under which conditions the thrust can be increased by admitting along the control surface values of the

1 Senior Scientist, Applied Mathematics Research Laboratory, .Aerospace Research Laboratories (AFSC), Wright-Patterson AFB, Ohio. Visiting Scientist, Applied Mathematics Research Laboratory, Aerospace Research Laboratories (AFSC), Wright-Patterson AFB, Ohio.

3 Mathematician, Applied Mathematics Research Laboratory, Aerospace Research Laboratories (AFSC), \Vright-Patterson AFB, Ohio.

4 NRC-Resident Research Associate, Applied Mathematics Research Laboratory., Aerospace Research Laboratories (AFSC), Wright-Patterson AFB, Ohio.

588 © 1973 Plenum Publishing Corporation, 227 W'est 17th Street, New York, N.Y. 100t 1.

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entropy that are higher than those of the oncoming flow. The conditions for isentropic and nonisentropic jumps are formulated and evaluated next, and a survey of the discontinuities which satisfy conditions for isentropic and also for selected nonisentropic jumps is given. Up to this point, the analysis is concerned only with the state distribution along the control surface. Jumps of the state in the interior require the occurrence of centered compression waves. Sample computations show that, in most cases, flow fields of this character can be generated by the choice of the nozzle shape. In some cases, no nozzle contours exist which generate the op- timizing state distribution along the control surface as determined by the present analysis. It would then be necessary- to include from the very beginning conditions for the realizability of the flow field.

I. I n t r o d u c t i o n

This paper presents in its first sections the theory and some numeri- cal results about the optimization of thrust nozzles of prescribed length in a flow with swirl. More details can be found in internal reports of the authors (Refs. 1-2). The remainder of the report is devoted to a special question which arises in this context. Initially, it is assumed that the state distribution along a certain control surface is continuous. However, the character of the equations shows that one may encounter optimum flow fields for which this assumption is not satisfied. It is true that this difficulty will not appear if only the length of the nozzle is prescribed, but it may arise if length and exit radius are given and if the opening angle of the nozzle is rather large. At least from a theoretical point of view, one would like to know what kind of flow patterns to expect under these circumstances. The answer may be of use for prob- lems where constraints other than length and exit radius are prescribed This problem has been explored rather thoroughly by Shmyglevskiy. Nevertheless, the reader may find the presentation given here worthwhile. It is simpler, mainly because we use a simpler form of the variational formulation but also because of a more natural choice of the dependent variables. Moreover, a considerable amount of numerical details has been provided. The analysis is not quite complete; there exist nozzle configurations where the whole flow pattern, rather than merely the flow at the control surface, must be included in the analysis, even if only the length and the exit radius are prescribed. The nozzle shapes where this would happen are rather unrealistic and a discussion of such cases would hardly be justified unless it is required for a specific practical purpose.

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2. B a s i c E q u a t i o n s

Considered are axisymmetric flows; let the x-axis coincide with the axis of symmetry, and let y be the distance of a point of the flow field from this axis. In a meridian plane, the velocity components in the x- and y-directions are denoted, respectively, by u and v. Because of the presence of swirl, a velocity component normal to the meridian plane is admitted; it is denoted by v t . Let p be the pressure, p the density, T the temperature, i the enthalpy, and s the entropy. The outside pressure is denoted by p , . We define

( d p / d p ) s = c o n s t ~ a s.

Furthermore, we introduce

q~ = u Z + v ~, 0 =a rc tan (v/u), (1)

o r

u = q c o s 0 , v = q s i n 0 . (2)

Notice that the velocity component v t does not contribute to q. Le t

sin ~ = aI~ (3)

The equation of continuity reads

(a/ax)(ypu) ~- (~/ay)(ypv) - - O. (4)

Equation (4) can be integrated by introducing a stream function

~ / ~ , == ypu, O4~i~x = --ypv. (5)

~b, defined in this manner, represents the mass flow through an axisym- metric surface whose intersection with a meridian plane extends from the axis to a point x, y and which is bounded by the intersection with two meridian planes which form a dihedral angle 1. This definition avoids the occurrence of factors 2~r in some of the formulas.

The streamlines have the form of rather steep helices with varying diameter. A surface ~b ---- const in the three-dimensional space is naturally axisymmetric, a streamline that has one point in common with such a surface lies completely within it. All streamlines of one surface 4~ = const can be brought into coincidence by rotation around the axis of symmetry. Since all streamlines lying in a surface 4s = const are equivalent, one has, for an inviscid flow,

s ----- s(~b). (6)

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For short, we shall also call streamlines curves arising by the intersection of a surface ¢ = const with a meridian plane. Applying the law of moment of momentum with respect to the axis of symmetry to a particle, one fmds

yv, = ~ ( ¢ ) .

For later convenience, we set

(yvt) ~ = (~1(¢))~ = ga(¢). (7)

In an inviscid steady flow, Bernoulli's equation is valid with a constant that may differ from strearnline to streamline. Under the present con- ditions, all streamlines lying in a surface ~ - const are equivalent; therefore, one has

i + ½(u 2 + v e) q- ½vt ~ = g2(¢), (8)

or

i + ½q? + ½ [gl(¢)/y ~] - g~(¢) = 0. (9)

Equation (7) is an integrated form of Euler's equation for the direction normal to the meridian plane. Within the meridian plane, the Euter's equations have the form

(1/p)(~p/~x) + u(ou/~x) + v(~u/~y) = o, ( lo)

(1/p)(~p/~y) + u(Ov/~x) + v(~v/ey) -- v,Z/y = 0. (1t)

Aside from these equations, one will have relations describing the thermodynamic properties of the medium. The system of differential equations is hyperbolic for q > a.

3. O p t i m i z a t i o n P r o c e d m ' e

At first glance, it seems as if the solution of the variational problem must the computation of the flow field; one would then be confronted with a two-dimensional variational problem. But, for nozzles for which either the length or the position of the endpoint is prescribed, the number of independent variables is reduced to one if one chooses the characteristic surface that runs through the endpoint of the nozzle as control surface for thrust and mass flow. The flow equations are then compressed into the compatibility condition for this characteristic, and the endpoint of the controI surface is directly given by the formulation of the variational problem. This observation was made independently by Guderley and Hantsch (Ref. 3) and Nikol'skij (Ref. 4). This approach has

592 JOTA: VOL. 12, NO. 6, 1973

been thoroughly discussed in the literature. A variant, due to Rao (Ref. 5), does not stipulate explicitly that the control surface be a characteristic; instead, the postulate is imposed that variations of the shape of the control surface do not affect the thrust. Rao's formulation leads to great simplifications of the numerical work; in the case of isentropic isoenergetic flows, for instance, the solution appears in a closed form. The equivalence of the two formulations has been shown analytically in Rao's first publication. The underlying reason for the equivalence is shown in Ref. 6. It is based on the following observations. In either formulation, the control surface wilt have the characteristic direction. This is a consequence of the fact that quantities with special physical properties, namely mass flow and the momentum, are subject to variations (see, for instance, Ref. 7). A second condition is needed to ensure that the control curve can be embedded in a flow field with finite velocity gradients. In the theory of characteristics, this requirement leads directly to the compatibility conditions. The same is accomplished by Rao's postulate in an indirect manner. We consider a flow field in which the first variation of the thrust caused by a variation of the flow conditions vanishes while the mass flow is kept constant. In such a field, the thrust is insensitive against a variation of the shape of the control surface iff the embedding mentioned above can be carried out. The first part of this statement is trivial.

In the present paper, we shall use Rao's formulation without further discussion. Figure 1 shows schematically the flow field within a thrust nozzle. The variations of the contour are restricted to the portion IF; therefore, the flow field in region I is not influenced by the variations. We are concerned with the control surface as determined by the characteristic KF. Let 9(y) be the angle of the tangent to the control surface -with the x-axis. The contribution of the control curve KF to the thrust is given by

M = f l d y , (12) K

F

~ R t S T I C S A B Fig. 1. Th ru s t nozzle.

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with

f l = y[P - - P~ q- pqe sin(~0 -- 0) cos 0/sin ~o]. (t3)

In the definition of A//, as formerly in the definition of ~b, we have restricted ourselves to the wedge-shaped portion of the flow field which is bounded by two meridian planes with dihedral angle 1. From (5), one finds a differential equation for the stream function along the control curve. It is written in the form

g(¢, ¢, q, 0, y ) = ¢ - - ypq sin(5o -- O)/sin % (14)

where derivatives along the control curve with respect to y are denoted by a dot. Equation (14) is considered as one of the constraints of the problem. Point F lies on the same streamline as the throat contour; this determines the value of ~ at point F. At the boundary of the region I and II, that is, at point K, ~b is continuous. The contribution of the control curve to the length of the nozzle is written as

1" VF L = | .f~@, (15)

~K

with

f~ = cot % (16)

If L is prescribed, then (15) constitutes a constraint on the choice of ~. In Rao's formulation, one has the task of maximizing (1.2) under

observation of the constraints (14)-(15). Independent are the variations of q, 0, % We have additional equations which will be taken into account directly, namely,

s = s(¢), g , = g~(¢), g~ = g~(,/,). (17)

According to the second law of thermodynamics, one has

di = Tds -4- (l/p) alp.

Inserting this equation into the differentiated form of (9), one obtains

(l/p) dp -4- qdq q- (gl ' /2y ~ - - ga' q- Ts') de - - [gl(¢)/y ~1 dy ~= 0. (18)

Here, the derivatives of the functions g l , g2, s with respect to their argument ~b are denoted by a prime. In carrying out the variations, we consider p as a function of q, ~b, y. Then, from the last equation,

@t~q = - p q , @ l e e = p ( - -Ts ' + g2' - - g~'/2y2), @/~y == pg~/y.a. (19)

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p is considered as a function of p and s. Then,

dp = dt,/a + (ap/o,) (20)

We introduce a function h(y) as Lagrange multiplier for the constraint (14) and a constant multiplier A for (15). According to Rao's formulation, one must now postulate that

fyF 8 [fl + h(y)g + ,~f2] dy = 0, (21) UK

where the variations of q, 0, ~b, ~0 are considered as independent. Varying 0, one is led directly to

h = q cos(9 -- 20)/cos(q~ -- 0). (22)

In varying q, one expresses the variations o fp and p by means of (19)-(20); furthermore, we set a = q sin a. After some manipulations involving h, one finds that the variation of q vanishes if

~o - - 0 = ~ . ( 2 3 )

Now, h is rewrit ten as

h = q cos(0 - - )/cos (24)

The vanishing of the variation of q~ leads to the condition

h = --ypq= sin = 0 tan a. (25)

The variation 3~b must vanish at the points K and F. One is led to the equation

dh/dy ---- y[(Op/~) cot(0 + c 0 tan ~ -- s'(Op/&)q ~ sin 0 sin 2 a/sin (0 q- a) cos ~]. (26)

Inserting (23) into (14), one obtains

&b/dy = ypq sin a/sin(0 + a). (27)

the equations Equations (24)-(27), together with (3), (9), (17), are governing the problem. The essential variables are h, q, 0, ~b. I f the position of point K is considered as known, then q, 0, ~, h [because of (24)] are kno~,a at this point, i.e., one has initial conditions at point K. The integration is terminated at the value y, to be denoted by Ye, for which ~ assumes a value which is given by the mass flow through the throat. As one computes the flow conditions along the control curve,

JOTA: VOL. 12, NO. 6, 1973 595

one also computes its x-coordinates. The value of x for y = ye gives the length of the nozzle. The control curve determined in this manner satisfies all the conditions of the optimization problem, provided that the coordinates of point F obtained from the computation are the preassigned values xF and y~. In a survey of nozzle shapes, the values of xF and YF for which the examples are drawn do not matter. Therefore, it is sufficient if one simply carries out the integrations for different choices of the initial point K and considers the values of xF and YF as preassigned, although they arise only in the course of the computations. By varying the position of K in the x y - p l a n e , one obtains a two-dimen- sional family of control curves. Along these control curves, the flow conditions are known. For each of these control curves, one can find the nozzle contour which wouId generate along it the desired flow conditions.

If only the length is prescribed, then Y e is free to vary. Naturally, the condition that ~v has an assigned value must still be maintained. This leads to the following additional condition:

[sin(20)] p -- [cot o~(p - - p,)/~pq~] F. (28)

For a derivation, we refer to the literature, for instance ReL I, Eq. (4.18). The problem can be brought into a form which is more suitable for

numerical work. A starting point is given by the observation that the equations valid for the control surface imply that the compatibility condition for a left-going characteristic is satisfied. This can be de- monstrated by elimination of h from the system. The compatibility conditions can be taken as a substitute for (26); and the only equation in which h is still present, namely (24), can be disregarded for h is merely an auxiliary quantity. The resulting equation contains the derivatives of q and 0. One of these derivatives can be eliminated by using the differentiated form of (25). Finally, we consider ~ rather than y as independent variable. For details of these manipulations, see Ref. 1. One finally arrives at the following system:

(d log q/d~) sin(2c~ - - 0) / s in ~ ~ s i n 0

+ (1 /s in ~ cos c~)(dc~/d~) + [sin(0 - - a ) / s in ~](1/pqy2)(1 + gl/y ~ q2 s in S ~)

q- (1/q2)(gz ' - - g l ' /2y 2 - - Ts')(sin 0 -- 2sin e~ cos c~ cos 0)/sin 2 o~ sin 0

+ O!p)(aplas)s' = o , (29)

ypq2 sin 2 0 tan ~ = --3. = const, (30)

dy/d$ = cos(0 + oO/ypq sin % (31)

dx/d~ = sin(0 + @/ypq sin ~. (32)

8o9/~z,/6-5

596 JOTA: VOL. 12, NO. 6, 1973

Later, we shall refer to an alternate form of (29). I t arises from (6.7) of Ref. 1, if one uses c~ in the form

= ~(q, y, s, g l , &)-

is a function of q and the thermodynamic state, say s and i. By (9), i is expressed by q, y , g l , g2 • Then, one obtains, instead of (29),

(d log q/d¢)(1/sin ~ sin 0)[sin(2c~ -- 0)/sin a 4- (sin 0/cos a)(a~/a log q)]

+ (1/sin = cos c~)[(Oo~/~y)(dy/d¢) + ( ~ / 0 , ) , ' + (Oo@gt)g 1' + (~4Og2)gz']

+ [sin(0 -- ~)/sin ,]( t /y2pg)(1 4-g~/y2q2 sin 2 ~)

4- (1/q2)(g~ ' - - g~'/2y ~ - - Ts')(sin 0 -- 2sin a cos c~ cos 0)/sin ~ a sin 0

+ (1/9)(ep/&)s' = 0. (33)

4. E q u a t i o n s U s e d in the N u m e r i c a l E v a l u a t i o n

In the computations, the problem has been specialized to an ideal gas with a constant ratio of the specific heats. A reference state is charac- terized by a subscript zero. Let

One has

Let

with

H .... exp[--(s -- So)/R ].

p/p - RT,

y = c~/c~ = const,

i = c~T = [7/(~' - - 1)] R T ,

~2 = ~ , ( p / p ) = ( 7 - 1 ) i .

(34)

(35)

a S is expressed in terms of i by the last equation of (35), and i is expressed in terms of q and ¢ by Bernoulli 's equation (9). The flow equations appear in the form of characteristic conditions [see Eqs. (A.9) and (A. 10) of Ref. 1]. They will be used in the following form. Let d l j , j = t or 2, be respectively the line element of left-going or right-going char- acteristics. In the following formulas, the upper sign is always connected with j = 1, and the lower with j = 2.

p . = ( r / T o ) ' / ( , - ' ~ = ( i / io) . I , -~>.

P ---- po Hp,:~, (36)

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T h e s lope of the charac ter i s t ic l ines is

dy/dx = tan(0 ± c 0. (37)

T h e l eng th of a l ine l e m e n t is

dl, = dx/cos(O i a) .~ dy/sin(O ± a). (38)

T o avoid divis ion by small n u m b e r s , we a lways use the f o r m u l a wh ich has the la rger denomina to r . M o r e o v e r ,

d¢/dlj == IOoY Hpi~q sin ~. (39)

T h e t h r u s t M can be expres sed as a l ine in tegra l car r ied out a long the character is t ics . O n e has

aM/d~ = q cos 0 - - (1/r)a sin(0 ± ce). (40)

T h e compa t ib i l i t y cond i t ions fo r the charac ter i s t ic lines r ead

± cot a(1/q)(dq/dlj) ± dO/dt~ ± (sin 0 sin ~/y)(1 47 ga/y2a 2)

± (~ot (I/y2)(dgl/d¢) 47 d&/d¢](d¢/dl,) = O. ( 4 l )

T h e m e t h o d of charac ter i s t ics is ca r r ied out in the usual fo rm, i.e., by rep lac ing the der iva t ives b y finite differences and b y in t roduc ing on the r i g h t averages over the line e l e m e n t in ques t ion wi th one i te ra t ion s tep to i m p r o v e the averages .

T h e s y s t e m of differential equa t ions f r o m wh ich the flow condi t ions a long K F are c o m p u t e d n o w as sumes the f o r m

dy/d¢ = sin(0 + c~)/pe~pisy q sin a,

dx/d¢ = cos(0 47 oO/poHpi,y q sin a,

(42)

(43)

dM/d¢ = q cos 0 q- (1/y)a sin(0 + e~) - - Pa sin(0 47 a)/poHp,,q sin a, (44)

(1/q)(dq/d¢){2 sin a cos a ~ cos 0 - - sin 0(sin 4 a 47 cos 4 a) - - [(y - - 1)/2] sin 0}

47 (sin ~ sin O/poHp~sqy 2)

× {sin(0 - - ~) cos a ~(1 47 gl/y2a ~) 47 [(y - - 1)/2](gx/yZa ~) sin(0 47 ~)}

47 (1/q2)[d&/a4, - - (dg,/a¢)(1/2y=)]

× {cos ~ ~(sin 0 - - 2 sin ~ cos ~ cos 0) 47 [(7 - - 1)/2] sin 0}

q- (1/H)(dH/d~b) sin s ~ cos 2 a[sin 0 - - (2/9,) sin ~ cos a cos 0] = 0,

yHp id "~ sin 2 0 tan a = --2t/p o = const.

(45)

(46)

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The flow field in region I of Fig. 1 is not affected by the variational procedure; for the computation of the state along KF, it can be considered as known. At the present, we assume that there is a continuous transition from the flow conditions in region I to the flow conditions in region II. Therefore, the flow- conditions at point K are available, once its position has been chosen. The constant A is then determined by substituting the data valid for point K into (46). The differential equations for y and q [Eqs. (42) and (45)] and the algebraic equation (46) are coupled with each other. The thermodynamic state is determined by two quantities, that i s , / / a n d the enthalpy i. H, as well as gl and g2, is given as a function of the independent variable ~b; i is expressed in terms of q, g t , g~ by Bernoulli's equation (9), a is found from (35), and a is defined in (3); finally, 0 can be computed from (46). The x-coordinate is needed for the construction of the flow field. M serves for the computation of the thrust. Point F is the point of the characteristic KF for which ~b assumes the value which corresponds to the throat contour.

For each choice of the point K, one can determine the state along the curve KF. But, of course, not every curve so obtained corresponds to a nozzle which has maximum thrust for a given length. Nevertheless, the curves are of interest as the solution of a different variational problem, namely the problem of maximizing the thrust if not only the x-coordinate of point F is fixed (as in the problem of nozzles of given length) but also the y-coordinate. The nozzle which is best if only the length is prescribed is, of course, contained in this family. It is characterized by the conditions (28). For outside pressure zero, this simplifies to the condition

[sin(20)]F = (1/y)[sin(2~)]e. (47)

5. N u m e r i c a l Eva lua t ion

The determination of the flow field in the region I requires in principle the computation of the entire subsonic flow field. Actually, the influence of the subsonic portion of the nozzle contour on the supersonic part of the flow field is only weak; therefore, it should be sufficient if only an approximation to the flow in the throat area is sued. We have used for this purpose the results shown in Ref. 8. They yield the state of the flow along a noncharacteristic curve of the supersonic part of the nozzle. The flow in region I is now computed with the method of characteristics. The line IB is one of the characteristics of this flow field, but its position is not specified in advance. The field in region I is computed fairly far out, and then a number of its right-going charae-

JOTA: VOL. 12, NO. 6, 1973 599

teristics is chosen to take the place of IK. Along the axis of the nozzle, a quotient sin 0/3, is encountered in the characteristic conditions which must be evaluated by some suitable interpolation. The method which we have used is described in Ref. 2.

The flow field in region II can be computed from the flow conditions along the characteristic IK taken from region I and the conditions along the characteristic KF, which are to be determined by the optimization process. So far, we do not know which position K will assume if a certain value for the length of the nozzle is assigned. We choose some point of field I as point K, and then solve the system of optimization equations (42)-(46). The computation terminates at the value of ~b which corresponds to the wall streamline in the nozzle throat. The results will satisfy- all the necessary conditions of the problem, except (47). Solutions in which (47) is disregarded can be interpreted as solutions for nozzles for which the length and the exit radius of the nozzle are prescribed. The integration of (44) immediately gives the thrust. This value of the thrust can be compared with the maximum thrust that can be obtained for a nozzle of infinite length and infinite diameter (assuming that the outer pressure is zero). At the exit surface of such a nozzle, one would have i = 0 and for most of the outflowing mass y = Go. Then, from Bernoulli's equation (9),

q = [2g2(~b)]~,

and hence,

= f [2&(¢)]~ M de.

Let us call nozzle effectiveness the ratio between the actual thrust and the maximum attainable thrust. The result for arbitrarily chosen positions of K can be used to determine charts showing the nozzle effectiveness in dependence of the position point F. For a family of points K which lie on the same left-going characteristic, the corresponding points F will also lie on a smooth curve. Along curves of this kind, one can interpolate for different values of the nozzle effectiveness. From curves of constant nozzle effectiveness, one can determine the endpoints of the nozzles for which the thrust is a maximum if only the length is prescribed; they are given by the point where these curves have a vertical tangent. Actually, this fact is used merely as a check; a direct characterization of these optimal nozzles is given by (47). For the tentative curves KF described above, we have computed the difference of the left- and right-hand sides in (47). Then, it is possible to interpolate for those positions of point K where the criterion (47) is satisfied. For these position of K, the state along KF has also been computed.

600 JOTA: VOL. 12, NO. 6, 1973

These data are used to compute the flow field in region II. Of technical interest is the shape of the streamline IF. To find it, one determines points of the flow" field for which ~ assumes the value pertain- ing to the throat contour. This is done by interpolation along the characteristic lines.

6. Results

The evaluation of nozzle shapes has been carried out for flows in which the functions g l , g 2 , / / a r e given by

g~ = 452~b ~, & = ½(y + 1)/(y -- 1), / / = 1.

We have here y = 1.25 and/92 = 0,0.5573, 0.8063. The Mach number of the swirl motion at the intersection of the sonic line with the throat contour is given by M, = 0, 0.6781, 0.7822, and the values of ~b at the throat contour are given by 0.49525, 0.44317, 0.42237, respectively. The initial conditions for these flow fields are taken from Ref. 8,

From the data generated during the computation, it is easy to compute charts which give the maximum nozzle effectiveness for dif- ferent points of the flow field. They are shown in Fig. 2; best nozzles

--T- THROAT RADIUS

I.

L 2 = 0 2, CORRESPOND TO ~ =.55734

3. =.80627

THE CORRESPONDING ~ALUES d/ ARE .495 .445 .422

82%

il \ ° . ( % 2 " ~ - - - I

78 ',~ { ~ \ ~ " = " 5

84%

3

I I I I f I I . X ................. 2. 3~ 4. 5. 6. 7. e, 9, io.

Fig. 2. Curves of constant nozzle effectiveness.

J O T A : V O L . 12, N O , 6, 1973 601

J

1 I I ........................... ~ . . . . . . I . . . . I I 1 I. 2. 5, 4, 5. 6. Z 8. X

Fig, 3. Nozzle shapes for/32 = 0,

1

9. I0,

I t i --1 i . . . . . . I ............ I' I ~" :

Y

I I t , I , , I I I I t I. 2. 5. 4, 5. 6. 7. 8. X 9. I0,

Fig, 4, Nozzle shapes for/32 = 0.55734.

602 J O T A : V O L . 12, N O . 6, 1973

ii I I I l I I I I I

J

I 1 I I I I I I I I. 2. 3. 4. 5. 6. 7. 8. X 9. I0 .

Fig. 5. N o z z l e s h a p e s for f12 = 0 .80627.

of a given length are obtained for the points of curves of constant nozzle effectiveness which have a vertical tangent. The nozzle effectiveness increases slightly if swirl is present, at least for the initial conditions shown here. A similar observation, although in a different setting, was made by Shmyglevsky and Naumova (Ref. 9). Because of the reduction of the mass flow through the nozzle in the presence of swirl, the total thrust obtained is somewhat lower for nozzles with swirl than for nozzles without swirl. Thus, the presence of swirl is only advantageous with respect to fuel consumption, but not with respect to nozzle weight.

The actual shapes for nozzles whose outer radius is allowed to vary are shown in Figs. 3, 4, 5. The nozzles are very similar for different values of the swirl. If we had superimposed these three figures, the nozzle shapes would practically coincide.

7. Further Discuss ion of the Solut ions

The factor of dq/d~b in (45) can vanish, and then this derivative will be infinite. Of course, one may defer the discussion of such a singularity until it occurs in a problem of practical importance. But the phenomenon shows that the class of flow fields admitted so far is too

JOTA: VOL. 12, NO. 6, I973 603

limited to give a solution under all circumstances; for this reason, a further discussion appears justified. Similar phenomena can be expected if constraints other than given length are prescribed. This question has been thoroughly discussed by Shmyglevski (Ref. i0). The present treatment is somewhat simpter, partially because it is based on Rao's optimization procedure.

At the moment, we are primariIy interested in basic phenomena. The derivative dq/d~ can become infinite even in an axisymmetric potential flow. Then, h is constant, as one recognizes from (26) and the definition (19) for ?p/~b, and the solution can be found from algebraic equations. The constant A is rather unimportant; it merley determines the geometric scale of the flow field. With h given, one can compute 0 as a function of q or ~. Figure 6 shows some curves of this kind, calculated for an ideal gas with V = 1.25. The field of curves has a singular point S. At this point, cos ~ ~ = (V -k 1)/4 and 0 = ~. This point can be found in the following manner. From the expression for h [Eq. (24)], one determines d~/dO (assuming h to be constant). This gives a directional field in the n0-piane. Then, point S is a point where d,~/dO is undeter- mined. The curves h = const running through this point divide the c~0-plane into four different regions, which are shown more distinctly

_z

Fig. 6.

E

Curves of 0 versus c~ for the state along the control char- acteristic in axisymmetric isentropic flows without swM.

604 JOTA: VOL. 12, NO. 6, 1973

0

f

¢

Fig. 7.

B 3

>Y6 \,

, g

Y='0D 2 U

Subdivision of Fig. 6 into dif- ferent regions. The curves B 1 and Ba are the loci of points for which the solutions in the yq-plane have a vertical tangent (see Fig. 8).

in Fig. 7. T h e curve VPSU of Fig. 6 is the locus of points where the coefficient of dq/&b in (23) vanishes. We shall denote this curve as B 1 . For each of the curves in Fig. 6, 2~ is const [see (25)]. Choosing ;~ in some fashion, one can compute curves of q versus y. Th e value of A may differ f rom curve to curve; to obtain a figure in which the curves are arranged systematically, it is desirable to choose ;~ as a smooth funct ion of h. Some curves of q versus y are shown in Fig. 8. T h e four regions shown in Fig. 7 can also be found in Fig. 8. For a general orientation, we remark that q = 1 corresponds to ~ = w/2. T h e curves of region c start at a - - 7r/2 and end at ~ = rr/2; those of region b start at rr/2 and end at the line 0 = 0 at some value of ~ different f rom ~r/2. According to (25), the lines 0 ~- 0 and c~ = 0 correspond to y = oo. A closer discussion of the curves of q versus y shows that they have a vertical tangent along the curve B , and along the line 0 + c~ = rr of Fig. 6. For 0 + a = rr, the control characteristic has a horizontal tangent. Of course, those parts of the control surface have no technical interest, for 0 exceeds rr/2; but we mention this property, because it explains the character of the curves in the regions c and d. Fig. 8 also includes the map of the curve B 1 .

In the actual computation, we usually choose a point K in the region I (Fig. 1) and determine f rom its values of q and 0 the values of h and ;~ which characterize the integral curves; under the present assumptions, h and A are constants. With these data, one can then compute the state

JOTA: VOL. 12, NO. 6, 1973 605

. . # " a~

o

Fig. 8.

j/

Curves of q versus y tbr the state along the control characteristic in axisym- metric isentropic flows without swirl.

along the control characteristic. It is in the nature of this construction that one follows these curves in the direction of increasing values of y. This leads to a surprising consequence: depending upon whether the state at point K gives a point in the qy-plane which lies above or below the locus of the points for which the curves of q versus y have a vertical tangent, the solutions will be given either by the upper or the lower branches of these curves. This may happen even if the initial points K lie close together. It is rather unlikely that a slight shift of K in the physical plane will result in a radical change of the solution of the variational problem. A jump of the flow conditions at point K, which leads from one side of the locus mentioned above to the other side, would lead out of this dilemma. The occurrence of such a discontinuity is also suggested by the vertical tangent of the curve of q versus y. The

606 JOTA: VOL. 12, NO. 6, 1973

jump conditions will be studied in the next section. At the moment, we show that only certain portions of the solution curves in the a0-plane can be regarded as solutions of the variational problem.

With the simplification mentioned above, only the total mass flow (not the values of ~b at the individual points of the control curve) are needed. We denote by f3 the second term on the right-hand side of (14), namely,

A = y o q sii~(~ - O)/sin ~. (48)

Then, the augmented integral corresponding to (21) assumes the form

e j~e ( f l - h A + ),f2) dy. (49) YK

Assume that we have determined a solution which satisfies the necessary condition of the problem, i.e., that the first variations of the last expres- sion caused by variations of 8q, 80, 8~0 vanish. We consider now a second flow in the neighborhood of this flow. In the neighboring flow, we choose as control curve a line which has again the characteristic direction. Furthermore, we assume that the control surface in the neighboring flow begins and ends at the same points as the control surface of the original flow. We consider as variations the differences of q, 0, ~ between the control surfaces in the original and the neighboring flows taken for the same value of y. Because of this choice of the control surface, one has 3cp = 8a q- 80. In a physically existing neighboring flow the variations must also be such that the compatibility conditions for a characteristic are satisfied. At the moment, this condition is disregarded; the class of variations admitted here is wider than in reality; the results will then be less restrictive. First variations vanish because of the conditions which govern the basic flow. The decision of whether one has a maximum or only a stationary value of the thrust is given by second variation con- siderations. Denoting the integrand of (49) by F [see Eq. (86)], one must compute

28F -- FqqSq 2 q-FooSO 2 q-F~,~3q~ 2 q- 2Fqo6q60 + 2Fq~3qSqo

q- 2Fo~806q), 6q~ == 3o~ 4- 80.

The manipulations are carried out in Appendix A. One finds that

for

3 F < 0

tan 0 o < sin 2%/[cos 2% -- tan %(dc~/d log q)]. (50)

JOTA: VOL. 12, NO. 6, 1973 607

That is, a maximum of the thrust is guaranteed (in an isentropic flow) if the state ties underneath of the curve B 1 . If this condition is not satisfied, then the sign of SF is undetermined. One might ask whether the compatibility condition for a characteristic line in the neighboring flow is sufficiently restrictive to guarantee that a maximum of the thrust is obtained even if the condition (50). is not satisfied This is not the case, as can be shown by the following example. In plane flow, one has along a left-going characteristic

cot o~(3qfq) = 30. (5t)

But, for variations of this kind, F has actually a minimum, as shown in Appendix A.

Next, we ask whether the thrust could be increased if one allows the entropy to increase. This could perhaps be done by a suitable system of shock waves introduced upstream of the control surface. Assuming an ideal gas, we write, in analog), to (36),

with p = p o H p . , (52)

(53) p . = (iiio)~l<~-~>. Then,

c~F/~H = y{p~s + [sin(p -- 0)/sin ~o]pq0 If cos 0 -- e0 cos(00 -- %)/cos ~d}-

F can be increased if dF/dor < 0. For an ideal gas, one finds that

tan 0 > sin(2~)/[2y -- 1 -- cos(2c0]. (54)

This is the curve VPWQU in Fig. 6. We shall call it the curve B~. A true maximum of the thrust requires that the solution lie below the curve B~ and below the curve B 1 .

According to these discussions, the solutions obtained so far are useful only if they lie below the curve B 1 of Fig. 6. If point K in Fig. 1 should lie outside this region, then it is likely that the assumption of continuity of the data along the control characteristic is no longer tenable, and thus one is led to explore whether the conditions for an optimum nozzle can be satisfied by a control curve along which the flow conditions are discontinuous. But, before we carry out such a discussion, it appears desirable to define the limits of the previous approach, at least approxi- mately. For this purpose, a problem has been studied where the solution can be obtained with a minimum of work, namely best nozzles in two- dimensional isentropic flow. For nozzles with fixed endpoint, the solutions are simply given by nozzles which would ultimately produce a parallel supersonic jet, but chopped to whichever length is desirable

608 JOTA: VOL. 12, NO. 6, 1973

(Ref. 3). These nozzles form a one-parameter family; in contrast, the solution of the axisymmetric problem requires a two-parameter family of nozzle shapes. For the initial conditions in the nozzle throat, the simple assumption has been made that one has a parallel sonic jet. The expansion part of the nozzle is formed by the flow around a corner. The flow field has been determined numerically by the method of charac- teristics using the special simplifications which are possible in plane isentropic flows. Figure 9 shows computer-drawn nozzle shapes. By

D o

@

I

J

c~

> ~ -

Fig. 9.

~.oo ,1;o ~:oo ~'oo i~.~0 ~.oo :~.oo ~ 0 ~oo 2~.~o x

Optimum nozzle shapes for plane isentropic flows. The curves cut off at any point give optimum shapes for nozzles terminating at the cutoff point. Curve C: is the locus of nozzle endpoints which, for outside pressure zero, have maximum thrust if only the length (not the exit diameter) is assigned, Curve B1 is the left boundary of the region which has no discontinuities along the control charac- teristic.

JOTA: VOL. 12, NO. 6, 1973 609

interpolation within this field, we have determined the points for which the nozzles are best if only the length is assigned. For such points, the condition (28) is satisfied. This is the curve C 1 . Moreover, we have drawn a curve which corresponds to the curve B 1 of Fig. 6, again denoted by B I . This is the curve which bounds the region where the solutions correspond to a maximum of the thrust (provided that we do not admit changes of entropy), at least in the small. Any point of the portion of the xy-plane that is covered by these curves and ties to the right of B 1 can be taken as the endpoint of the nozzle, and the curve running through it gives the best nozzle which has this assigned endpoint. For a nozzle endpoint lying to the left of the curve B1, the continuous solution is not a maximum of the thrust, even in the small, and the previous procedure must be modified. For points to the right of this curve, the procedure holds, even if the nozzle contour intersects the curve B 1 .

One comes to the conclusion that opt imum nozzles for which the flow conditions along the control characteristics are discontinuous correspond to cases where the opening angle of the nozzle exceeds by far the angle which would be desirable for maximum thrust at the available length. Thus, the practical importance of the following discussions is limited. Nevertheless, it may be welcome if we clarify some of the basic questions.

8. l u m p C o n d i t i o n s

We assume now that, at some point K of the control curve A K F , a discontinuity occurs (see Fig. 10). The control curve of the unvaried flow is a characteristic. The portion A K in Fig. 10 lies in region I; the portion K F lies in region II. We assume that the flow fields I and II are defined in a neighborhood of the control curve, but we postpone the question of how a jump of a certain kind can be physically established.

F

.i K!

A 8

Fig. 10. Control surface in the original flow and in the varied flow for a nozzle in which a discontinuous change of state is admitted at point K.

610 JOTA: VOL. 12, NO. 6, 1973

We shall write K~ or KH in order to differentiate between the state in regions I or II. I f a variation is carried out, point K will move to point K' . Let

x,v" - - XK = 3X and YK" - - Y K == ~Y"

T h e control curve is then chosen as AK~K'KnF. We assume that the port ion AKI which lies in the flow field I is not subject to variations. For the port ion KHF of the control surface, the conditions for a continuous solution (that is, the equations derived previously) are applicable. At the moment , we do not assume that the en t ropy is continuous at point K, but in later discussions this assumption will be made. T h e functions gl and g2 are assumed to be continuous; for, within the nozzle, one has no practical means to control them. T h e mome n t u m in the x-direction is

fo M = K~Ady @ f~dy YK H

+- ~Y[(P -k oq z cos ~ O)K, -- (p -t- pq z COS 2 0)X,,]

- - 8x[(pq ~ cos 0 sin O)K~ -- (pq~ cos 0 sin O)Kr~].

One has now

~ K l l = ~Y[(oq COS O)K I - - (Dq c o s O)Kll ] - - ~ x [ ( ~ q sin 0)K, - - (pq sin O)KII ],

T h e augmented integral now contains additional terms due to Sx and Sy, namely,

YKI1

[fl + h(y)g ~- h.f2] dy + Sy[(p + pq2 cos 20)g~ - - (p ~, pq2 COS ~ 0)g~l]

- - ~X[(pq 2 COS 0 sin 0)/¢ i -- (pq~ cos 0 sin 0)~iI ].

I f 3x and 3y are taken to be zero, the procedure is the same as before. T h e integration by parts which is encountered in the t reatment of the constraint ( t4) gives a nonvanishing contr ibut ion at point KI I , namely a te rm --3~b~Hh(y%). Then , the requi rement that contr ibutions due to 3x and ~y vanish independent ly gives the following jump conditions:

(p + pq2 cos a 0)K~ _ (p + pp2 cos 2 0)z¢ix -- h[(pq cos 0)Kx -- (Pq cos 0)/¢u] =-- 0,

(55)

(pq~ cos 0 sin 0)~c~ -- (#qZ cos 0 sin 0)K H -- h[(pq sin 0)g I -- (pq sin 0)Kix] = 0. (56)

J O T A : V O L . 12, NO. 6, 1973 611

According to this derivation, h is determined by evaluating (24) in region II, that is,

h = [q cos(0 -- ~)/cos ~]K,.

These relations have been evaluated for a number of cases, in order to obtain an overview of the results. The evaluation is easier if one assumes the state at Kn to be known and computes the state K~ from which Kn can be reached by a discontinuity. From now on, subscripts I and I I , rather than K x and K l I , will be used. Rearranging (55)-(56) and substituting h, one obtains

COS Oipiqt(qi COS 01 - - h) = - - [ ( P l - - _DII) @- c cos 011], (57)

sin 0iplqi(qt cos 01 -- h) = --c sin 0iz, (58)

where

tience,

c = PIIq~ sin 011 tan ~xI.

t a n 0i = c s i n Oxi/(pi - - Pl i -:i-" c cos 01x ). (59)

Now 0 x is substituted into (58). With

A = [(/t -- pif) 2 + 2(pi -- pi1)c cos 01~ + c2] ~, (60)

one obtains the following condition:

B = Piqt{qt[(Pi - - PII) + c cos 0i, 1 - - h A } q- A 2 = 0. (61)

If one considers the state II as known, then, for isentropic discontinuities, this expression depends solely upon q~ ; therefore, one has the task of determining the zeros of the last expression. Discontinuities can be physically generated either by shock waves or by coalescence of com- pression waves; therefore, only cases where qx > qll are of physical interest. For nonisentropic discontinuities, one chooses the value of H in addition to the state II. Then, the expression (61) is again only a function of ql • After having found the value of q~ which gives the zero of (61), 01 is determined from (59). To obtain, a general survey about the location of the zeros of (61) and how they change with qH and 011, we first plotted B [Eq. (61)] versus qt for a number of values q H, 011. These curves depend continuously upon the parameters q~t and 011; therefore, they show dearly whether and how the number and the positions of the zeros wilt change. The details are somewhat involved, but one finds that all points below the c u r v e B I of Fig. 6 can be reached by" isentropic discontinuities. A survey of the results is shown in Fig. 11. In the region below the curve B1, a grid consisting of dashed

809[t2:6-6

612 JOTA: VOL. 12, NO. 6, 1973

0

5"0 V

2"0 t

] , O n

0 •

Fig. 11.

t l a L l *~

~iii,~i!lli

/:li;ll!rj l/ !lwr~ IL IJ';ll

/ / ! ' ! I l l tr

7 N

4: . / , :2

a l ' - ~ ! , i , ! !~ fl~I, i , L. i l i i q

I 2 5

Survey of states which are connected by jump conditions. Dashed curves above Bt are mapped by isentropic jumps into dashed lines below B~, solid curves are mapped into solid lines. Points of the right part of B 1 are mapped into themselves.

vertical lines and solid horizontal lines has been imposed. This grid is mapped by the jump conditions into another grid of the q0-plane. In Fig. 11, dashed or solid lines are mapped into dashed or solid lines, respectively. We refer to the portion of the curve B 1 which lies to the right of its maximum as its right part. Points of the right part of B 1 are mapped into themselves. In this manner, one can identify the dashed and solid curves with their maps. The lower points of the left part of B 1 map into rather distant points of the q0-plane. They can be identified by the symbols used for these points. In its upper portion, the left part of B 1 is mapped into itself. For details of this kind, the plotting of the curves (61) is very helpful. Figure 11 extends up to 0x = ~r; actually, 0, = =/2 is the outside limit for values that are of practical interest; the wider range has been included to give a survey of the whole field of curves. The dashed line of Fig. 14 in the field below the curve B 1 gives approximately the locus of the points that arise from a state I for 0i = ~r/2.

For nonisentropic discontinuities, we have chosen the state q , , 011 along the portion of the curve B~ which lies underneath of B 1 . If the jump would lead to points that lie below this curve, then the thrust could be improved if a smaller value of the entropy change would have been

JOTA: VOL. 12, NO. 6, 1973 6t3

chosen; if the points lie above this curve, a larger entropy change could be advantageous. There may arise specific technical problems where other possibilities should be admitted; but, for a first survey, the choice made here is probably sufficient. Plotting the expression (61) versus ql

.2 1 I

/ 0 .

Fig. 12.

[ 1.5 2 2.5 3

Survey for nonisentropic jumps for which state II lies at the point 1 through 10 of curve B2. Each point gives rise to two curves, depending upon the entropy change (here, characterized by H). One of these family of curves (numbered from 1 to I0, in correspondence with the points 1 to 10 along B~) is shown in detail. Also shown are curves H-const. C5 and C4 are the curves of the other family for which the jumps lead respectively to points t and 10. Tick marks along these curves give the values of H.

614 JOTA: VOL. 12, NO. 6, 1973

one recognizes that, for a given entropy change, each state qH, 0u chosen along B 2 can be reached from two different states q~, O,.

Figure 12 gives a survey of the results. By varying 17, one obtains, for each point qii, 0tl, two curves for the state q~, Ox from which this point can be reached. By varying the position of the point along B 2 , two families of curves are generated. Only one of them is shown in detail, because the curves of the other family overlap rather strongly. The cross-hatched region in Fig. 12 shows one of the regions from which suitable nonisentropic jumps lead to the curve B 2 . The boundary of the region for isentropic jumps is given by the curve B1. The cross-hatched region lies entirely within the region of isentropic jumps. For a given state qi, 0~, one can therefore find one solution for an optimum nozzle without entropy change and a second one with an entropy change. Of course, the endpoints of these nozzles will be different, but it is conceivable that, even for the same nozzle endpoint, two different solutions of the optimization problem (in the small) may exist. Challenging as such speculations may be, it seems to be best to defer a discussion until the problem arises in a more realistic technical setting.

9. Discon t inu i t i e s in the F low F ie ld

So far, our discussion has been restricted to the state along the control surface. The question arises of now state distributions which have jumps along the control surface can be generated in an actual flow field. Flow fields with discontinuities can exist only if qx > qH ; for, in the interior of a flow, sudden expansions are impossible. A sudden isentropic compression is brought about by centered compression waves. We ask whether such a configuration will generate the discontinuities required here, namely, a transition from a state qt, 0i to the state qu, OH that belongs to it. Our discussion is restricted to the immediate vicinity of the discontinuity; therefore, the results are the same for plane and axisymmetric flows and even for flows with swirl. The oncoming flow has the state I. A family of centered right-going compression waves coalesce at point K (Fig. 13) and form a shock downstream of K. The streamline that runs through K separates the streamlines that do not pass through the shock from those that pass through the shock and, therefore, experience an increase in entropy. The velocity direction and the pressure in the parts of the flow field adjacent to this separating streamline are the same (the velocity has a jump). From K, some left- going Mach waves (or a left-going shock) will emerge. The discontinuity

JOTA: VOL. 12, NO. 6, 1973 615

left golng pertu~ion

velocity vector do.stream centered eompression~ 4 of compression waves

K ~ 3 do~stre~ of the shock

velocity vector in t ~i. ~ ~ shock he oncom ng ~ *I X ~ I

5/

Fig. 13. Sketch of the flow field in the vicinity of point K.

required by the jump conditions can be realized if the state II is present in the field of left-going characteristics that start at K. Our computations have been carried out under the assumption that the left-going charac- teristics starting at K are expansion waves. For the subscripts used to differentiate between different states in the vicinity of K, see Fig. 13.

Let/~ be the angle of the streamlines upstream of the shock with the shock. For a left-going shock, one has ~z ~< --/~ ~< ~r/2. The equations will be derived for this case. The right-going shock, which we have in our application, is obtained by choosing either ~i ~< fi ~< 7r/2 or ~; ~ ~" - - /~ ~ 7r/2. The final results are the same. We shall vary ]~ in order to obtain the desired discontinuity. Let subscripts 1 and 2 characterize, respectively, the states upstream and downstream of the shock; let w~ denote the normal components of the velocity; let a, p, p , s

denote the velocity of sound, the density, the pressure, and the entropy; and let H = exp[(s 2 - -S l ) /R ]. We specialize the shock condition im- mediately for ideal gases with a constant ratio of the specific heats. The computat ions are carried out for y = 1.25. One has

w~l = qr sin/~. (62)

Let

~ 2 ~,~ = [2/(r + O]{(a? + [(r - U/2]wgO}.

Then, from the Prandtl relation w ~ l w ~ = a s ,

zv, 2 = [2/(y + 1)](1/w,~i){al 2 + [(y -- 1)/2]w~l)}.

The equation of conservation of mass gives

Pl/P2 = Wn2/Wnl "

Bernoulli 's equation gives

a2 z = at 2 -1- [(y -- 1)/21(w~1 -- wn~).2

(63)

(64)

(65)

(66)

616 JOTA: VOL. 12, NO. 6, 1973

For an ideal gas with constant ratio of the specific heats, one has

P2/Pl' = H(a2~/a12) 1/(~'-1)

Therefore,

Let

17 = (w,aiw,~2)(a12ta~2) la~-1). (67)

0a = 0 a -- 01. (68)

From the shock relations jus t derived and the further condition that the vdoc i ty component tangential to the shock remains unchanged, one obtains

( 6 9 ) qd = q? c°s~ fi + w,,2,

tan 0 z = (qi sin/~ cos fi - - wn2 cos fi)/(qx cos2/~ q- wn2 sin/~), (70)

0a = 0, q- 0a. (71)

These formulae allow us to compute the state 3, if state I and/~ are given. State 4 is connected to state 3 by the equations

G = 04, pa " -p~ .

The second condition leads to

a42 = H(y -- 1)/yaa 2. (72)

Then, using Bernoulli 's equation, we have

a42 = ( y q - 1 ) / 2 - - [ ( y - - 1)/21q42 = H ~'-1)t" { [ ( y ÷ 1 ) /2 ] - - [ 0 ' - - 1) /2]qa2}, (73)

and hence,

qa = {[(~' + 1 ) / ( y - 1)][1 --F/(,-1)/,] q-H(~-~)/,qa2}L (74)

State 2 and 4 lie on the same right-going characteristic. The character- istic conditions are

J(q) q= 0 = const, (75)

where the upper sign applies to left-going and the lower sign to right- going characteristics, and

J(q) = cot c~(q)(dq/q). (76)

JOTA: VOL. 12, NO. 6, 1973 617

For an ideal gas with constant ratio of the specific heats, one has

J(q) = [(7 + 1)/(v - 1)]=* arctan{[(y - 1)/(7 q- 1)]'~v} - arctan v, (77)

where

v = c o t ~ = [(q* - ~ ) / {1 - [ ( r - 1)/(r + 1 ) l q ~ } P . ( 7 8 )

Specifically, for y = 1.25,

v = [(qZ -- 1)/(1 -- q~/9)]} (79)

and

J = 3 arctan(v/3) -- arctan v. (80)

T h e fact that states II and 4 lie on the same right-going characteristic is then expressed by

F = J(q~) + 0~ - y(q,~) - 0H, F = 0. (81)

F can be evaluated as a funct ion of/~, and one must seek the value of fi for which F vanishes.

State 5 and state I lie on the same left-going characteristic, and state 5 and state II lie on the same right-going characteristic; hence,

J(qs) - oa = J(qr) - 0 , , J(qs) "@ 05 = J(qiI) -}- Oil"

Hence, 1 I 05 = ~[J(qzx) - J(ql) + 0ii ~- or]. (82)

Of course, one can also determine J(qs ) and qs. In order for the per- turbance which propagates f rom K to the left to be an expansion fan, one must have

05 - o4 > o . ( 8 3 )

In order for state I I to lie within this expansion fan, one must have

05 -- 01, >~ 0, (84)

e , , - o4 > 0 . ( 8 5 )

T h e j u m p discontinuity can be realized by centered compression waves if the last two conditions are satisfied.

These computat ions have been carried out for the points q~,, 0,z marked in Fig. 14. T h e y lie in the region bounded by B 1 and the locus of the points for which one would obtain 0, = rr/2 (Fig. t4 complements Fig. 11). For practical orientation, we have included the curve for the

618 JOTA: VOL. 12, NO. 6, 1973

0

0

Fig. 14. Realizability of isentropic jumps: © ~ can be realized by adiabatic jumps; ® ~ cannot be realized by adiabatic jumps; x --= can be nearly realized by adiabatic jumps.

flow conditions q, 0 which arise from a sonic parallel stream by a fan of right-going expansion waves (curve C8). All states which occur in the upper half of a Laval nozzle will lie to the right of C a . The point K of the control characteristic lies in this region. As long as K lies below B~, the solutions will be continuous along the control characteristics. If point K lies above this curve, then discontinuities will appear, and the flow" conditions at such a point give the state qt, 0i- The states qi1,01t which are reached from the upper part of C~ by isentropic jumps lie on the curve C 4 . In the usual LavaI nozzles, only the points qi1, 0lt lying to the right of C~ are of interest. At the points marked by the empty circles, the desired isentropic jumps can be generated by centered compression waves; that is, for these points, the conditions (83)-(85) are satisfied. Close to the curve for which 01 = 7r/2, the states I and iI which should be connected by a jump are so far remote from each other that there is no value of/~ for which the condition (81) can be satisfied. These are the points marked by B. A physical insight is obtained if one considers the states I and II as two separate parallel supersonic jets which impinge upon each other. The point where the boundaries meet is K. Downstream of K, the jets have a common boundary. They adjust to each other by a shock which propagates in region I and a rather weak disturbance which propagates in region II. It is easy to imagine that the jets impinge on each other with an angle which is so large that the deflection necessary to allow for a common boundary cannot be brought about by a shock alone. In this case, one

JOTA: VOL. 12, NO. 6, 1973 619

has a subsonic region downstream of the shock. The shock itself arises upstream of the ideal point of impingement, and part of the necessary turning of the streamlines is brought about in the subsonic field.

For points marked by m, a value of/~ can be found, but the con- ditions (84)-(89) are violated. In some of these cases, the left-going perturbation which starts at point K is a (weak) shock, and therefore no characteristic in which one encounters the state II exists. In other cases, there is a left-going fan of expansion waves, but it does not contain state II. But these states can occur only in flow fields of a rather un- common character. In principle, phenomena of this character indicate that the variational problem can no longer be solved by studying the state distribution along the control characteristic alone; the flow conditions the control characteristic are now limited by the requirement that a flow field generating these conditions must exist. Nevertheless, one would expect in many cases that the actual flow field is still rather well ap- proximated by the configurations considered so far, and that the modifi- cations which must be admitted are only minor. For instance, the right-going compression waves which are responsible for the jump might

I F

2

I

o l - 2 ~

Fig. 15. Computed example of a plane nozzle with discontinuity at point K.

620 JOTA: VOL. 12, NO. 6, 1973

not be completely centered; then, one would find, instead of the weak shock in the region II, a sequence of left-going compression waves, which do not coalesce within the flow field, and one of these compression waves might be suited as the control characteristic. Again, the authors believe that detailed discussions should be postponed until they occur in examples of actual technical interest.

An example for a nozzle with an isentropic discontinuity is shown in Fig. 15. The basic problem is the same as in Fig. 9. The line IF is one of the characteristics of the expansion fan that starts at point I; it re- presents the boundary of region I. The nozzle contour between I and H has been constructed in such a manner that it generates compression waves centered at point K. The state at KI is given by q~ = 2.108, 0~ = 0.6632. The state at KH is then computed using the formulas of Section 8; one finds qH = 2.0266, 0ix = 0.5645. For these values, the states 3, 4, 5 have been determined using the procedure of Section 9. Only the state 5 is important for nozzle shape. The left-going perturba- tion that starts at K is extremely weak, but the state at KII corresponds indeed to one of the expansion waves of this perturbation. The flow conditions are the same all along KF, since we have the special case of an isentropic plane flow. The nozzle shape has been found graphically by means of the method of characteristics, taking as starting points the state along the chosen characteristic IK, previously computed in con- nection with Fig. 9. If one would transfer this particular nozzle shape into Fig. 9, one would find that it ends at a point F slightly to the right of the curve B~, that is, in a region for which solutions which are continuous along the control characteristics are also possible. Thus, there are two candidates for the best nozzle. Of course, it is not difficult to decide, in a specific case, which of the two solutions gives the larger values of the thrust, but the example shows the difficulties which one would encounter in a general theory.

10. Appendix A: Second-Order Variations

We substitute h and A into (49). Under the assumptions made here, these quantities are constant; nevertheless, it is preferable to express them in terms of data along the solution curve of the variational problem. These data are characterized by a subscript 0. One then obtains

F --= p + (1/sin ~o){pq 2 sin(~o -- 0) cos 0 -- qo[COS(00 -- %)/cos %]pq sin(~o -- 0)}

-- cot ~opoqo 2 sin 2 00 tan ~o- (86)

jOTA: VOL. 12, NO. 6, 1973 62t

S i n c e

sin(50 - - 0)/sin 50 = cos 0 - - sin 0 cot 50,

o n e f inds t h e f o l l o w i n g a l t e r n a t e e x p r e s s i o n :

F = p + pq2 cos ~ 0 - - qo[cos(Oi - - %)/cos %] pq cos 0 + cot ~o A(q, 0), (87)

w h e r e

A = ~ (pq2 __ 00%2) sin 0 cos 0 + %[cos(0 o - - %)/cos o:d(pq - - 0o%) sin 0

+ poqo2[--sin 0(cos 0 - - cos 0o) + sin0 o tan %(sin 0 - - s in 0o) ]. (88)

T h e f o r m of A s h o w s i m m e d i a t e l y t h a t A ~ 0 for q = qo , 0 = 0 o ,

50 = 500 • O n e ha s

F~ == - - (1 / s i n 2 50)A, (89)

a n d h e n c e i m m e d i a t e l y

F,,(qo, O 0 , %) -= 0, F ~ ( q o , 0 o , %) = 0. (90)

I n c o m p u t i n g F~o , we set q -~ qo , P = Po •

F~o = --poqo 2 sin 0/sin(0 o + %) cos % . (91)

M o r e o v e r ,

/Tq = _ (1/sin 2 50)Aq = - - (1 / s i n s 50)

× {--2pq sin 0 cos 0 + qo[cos(0 o - - %)/cos %]p sin 0

-{- (pq/sin 2 c~) sin 0 cos 0 - - qo[cos(0 o - - %)/cos %](p/sin 2 s) s in 0}, (92-1)

F~q(qo, 0o, %) -~ Po% sin 0o/sin(0 o + %) sin % . (92-2)

D i f f e r e n t i a t i n g (86), o n e o b t a i n s

Fo = (1/sin ~)(--pq~ cos(~o - - 20) + pq qo[cos(0 o - - so)/cos %] cos(9 - - 0)}.

T h i s e x p r e s s i o n v a n i s h e s fo r q = qo , 0 = 0 o , 50 = 500, as o n e r e c o g n i z e s i m m e d i a t e l y . H e n c e ,

Foo(qo, 0o, 500) = (Ooqo2/sin%)[2 sin(0o - - %) + cos(0o - - %) tan %]. (93)

F o r t he c o m p u t a t i o n o f F q o , we se t i m m e d i a t e l y 0 = 0 o , 50 ----- 500 • T h e n ,

[Fo]o=oo -~ (pq/sin 50)[--q cos(0 o - - %) + qo cos(0o ~ %)].

622 JOTA: VOL. 12, NO. 6, 1973

Hence ,

Fqo(q o , 00 , %) = -- (p0qo/sin %) cos(0 o - - %). (94)

I n f o r m i n g derivat ives wi th respec t to q, we set immedia te ly 0 = 0 o . Fq can u l t imate ly be b r o u g h t into the f o r m

[Fq]o=oo = (1/sin %)[(pq sin 0o/COS n o sin ~ ~)(sin 2 % -- sin z ~) q~=qo 0

"t- p(% - - q) cos(0o - - %) tan % cot 2 ~].

I t is t h e n obvious tha t Fq = 0 for q = qo, 0 = 0 o , 9 = 9o • Hence ,

Fqq(qo, 0o, %) = (p0/sin %)[--2(sin 0o/sin % ) ( & / d log q) -- cot % cos(00 - - %)].

(95)

Next , we c o m p u t e

26F = F~q(3q) 2 + Foo(30) 2 + F~(3~) "~ + Fqo3q30 --r 2F~o6¢p30 + 2F~&p3q, (96)

wi th

89 = 6o~ + 60 = (do~/d log q)(~q/q) + 60. (97)

U s i n g the derivat ives c o m p u t e d above, one finds

26F = a(3q) ~ -j- 2b6q60 4:- c(60) ~

= - -co t % cos(0 o - - %)(6q/qo) 2

+ [2 cot % sin(0 o -- %) - - 2(sin 0o/COS %)(dc~/d log q)] (3q/qo) 60

- - tan % cos(0 o - - %)(30) z. (98)

a and b are negative. The re fo re , one will have a m a x i m u m of F if

ac - - b 2 > O,

or, m o r e specifically,

cos2(0o - - %) - - [cot % sin(0 o - - %) - - (sin 00/cos %)(d~/d log q)]2 > 0. (99)

T h i s express ion can be split in to two factors. One of the factors is given by

cos(0 o - - %) + cot % sin(0o - - %) - - (sin 0o/COS ao)(dc~/d log q)

= sin00[1/sin ~ - - (1/cos o~)(dcz/d log q)] > 0, (100)

for d a / d log q is negative. One has, for an ideal gas and potent ia l flow,

sin2~ = [(~, + 1)/2](q*/q) ~ - - (~, - - 1)/2, (101)

do(d log q = [cos(2~) - - 9,]/sin(2~). (102)

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The sign of (99) is therefore decided by the second factor, namely,

cos(0o -- %) -- cot % sin(00 -- %) + (sin 0o/cos %)(da/d tog q).

The first two terms in this expression can be combined, and we obtain as condition for 8F < 0

sin(2% -- 00)/sin % + (sin 00/cos ao)(do~/d log q) > 0, (103)

or

tan 00 < sin(2%)/[cos(2%) -- tan ao(do~/d log q)]. (104)

After substituting the relation (102) for an ideal gas, one obtains

tan 0 o < sin(2%)[1 + cos(2%)]/[7 + cos2(2%)]. (105)

If (104) is not satisfied, then ~F may be positive or negative, depending upon the choice of 8q and 80. In the main text, the question was raised of whether the compatibility conditions, which are also satisfied along the varied control surface, would impose a constraint, which caused SF to be negative even if (104) is not satisfied. Here, we offer the following comments. The relations obtained here are correct also for the plane problem. Then, )t does not depend upon y, and the state is constant along the entire control surface. For plane potential flow, the compatibility condition gives

cot o~(Sq/q) = 30.

Hence, by substitution into (98),

23F = --2 tan %[sin(2% -- 00)/sin % + (sin 00/cos %)(do~/d log q)](80) 2.

The term in the bracket agrees exactly with the left-hand side of (103). It follows that, for the special variation considered here, 3F will be positive outside of the region where (103) or (104) are satisfied.

A comparison with (33) shows that the boundary of the region of the ~0-plane in which (103) holds coincides exactly with the curve for which dq/d~b is infinite. The critical terms are

(d log q/d~) sin(2~ -- 0)/sin 2 ~ sin 0 + (1/sin ~ cos ~)(d~/d~b)

= (1/sin ~ sin 0)[sin(2~ - 0)/sin ~ + (sin 0/cos o~)(da/d log q)](d Iog q/d~b).

11. A p p e n d i x B: Spec ia l P r o p e r t i e s o f the Cond i t i ons fo r I s en t rop i c Discon t inu i t i e s

First, we collect some results from the variational treatment of continuous solutions. The equations which we shall use are direct

8o9/I2/6-7

624 JOTA: VOL. 12, NO. 6, 1973

consequences of the variational procedure; the specific form of the functions f t plays only a limited role. In this manner, it will become obvious that the properties of the jump condition derived here are a consequence of the conditions imposed by the variational procedure. In f l , we set immediately Pa = 0, for only pressure differences are of importance. Accordingly, from (13),

f~(q, O, % y , ~)

= y { p + pq2[sin(9 -- 0)/sin 9] cos 0), p = p(q, Y, ¢), P = p(q, Y, ~). (106)

The second term on the r ight-hand side of (14) is now written as

fa(q, 0, 9, Y, ~b) = ypq sin(9 -- 0)/sin ~o. (107)

The dependence of p and p on ~b and y occurs because of BernoulIi's equation and because of the dependence of entropy on ~b. In this section, we consider isentropic discontinuities; therefore, the H-dependence does not play a role. The variational procedure has led us to conditions

f:tq - - h fs~ = O, f lo - - hf~o = 0. (I08)

They are satisfied if

h(q, O, y , ¢) = q cos(O -- cO/cos c~, 9 = 0 + ~. (109)

All partial derivatives refer to the functions as they are written here. In particular, we do not assume that ~ in (108) has been replaced by 0 q- c~. We have, as a further consequence of the variational procedure,

We introduce

Here,

f l~ - - h f2~ - - )t/sin~ 9 = 0. (110)

f~q, - - hf2q, = --(y/sin e ~o)A. (111)

f4(q, O, y , ~b) = pq~ sin s 0 tan e. (112)

In (25), we found

- -Y f4 (q , O ,y , ~b) = h = const. (113)

Now, we derive a characterization of the curve B I . This curve gives the boundary of the region in which solutions satisfying the necessary conditions of the problem give a maximum of the thrust, provided that

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entropy changes are not admitted. At the same time, B 1 is the curve for which dy/dq = dy/dO = 0. For a solution of the variational equations, one obtains, from (26), (109), (113),

(ah/aq) dq + (Oh/aO) dO ÷ [ah/~y ÷ (ah/a40(d4,/dy)] dy = O,

(afJoq) dq + (af~laO) dO -p- [afgay + @f4ta~)(d$!dy)] dy = O.

For the curve B1, dy = 0. Therefore, if the rank of this system is 2,

l eh/aq ah/ao ~A/Oq of~/ao[ = o. (114)

This is the desired characterization. Incidentally, if one considers dq or dO as first-order terms and if dy is of second order, then the changes of g l , & , / 7 are of second order, too. Curves of 0 versus q (or 0 versus a) for solutions in which g l , g~, o r / 7 are not necessarily taken to be constant are therefore tangent along B 1 to the corresponding curves for potential flow. In more geometric form, the curve B 1 can be described in the following manner. Le t d~/dO be the slope of a curve h = const at its intersection with the curve B 1 ; dq~/dO is determined by the equation

(ahlae)(d41do) + ahlaO = o. (115)

Then, the following condition is satisfied, too:

(~f, Jeq)(d~/dO) + Of4/O0 = O. (116)

In preparation for the discussion of the jump conditions, we now form the derivatives of Eqs. (108) in the direction of a curve h = const for a point of B~. One finds, f rom the first equation,

Ea~f~feq~ - h(o~I.l&)](d4fao) + Ea~kfeeeO - h(a~Lleqeo)]

+ [a~fllOqe~ - h(a%/e¢~)](di~/dO) = O,

where d~/dO is the derivative of ~ along such a curve. The definition of f4 yields

O~fl/~q&p -- h(O~f3lOqgq)) = --(y/sin S ~o)(gf41Oq) ÷ (Oh!~q)(Ofsl&p)° (117)

Then, the last equation can be rewritten in the following form:

[ 02f l/ ~q ~ -- h( a~f d eq2) J( d4 /dO ) ÷ [ e~f l/ oqaO -- h( a2f ,/ ~q~O ) ]

+ [(y/s in2 cp)(ef4/Oq) -l- (eh/eq)(ef~/&p)](d~/dO) .= O. (118)

626 JOTA: VOL. 12, NO. 6, 1973

Similarly,

[8~fl/Oq80 - h(8~f~/~qSO)](d~/dO) -t- [(~fl/80 2 - - h(8~fa/802)]

+ [(y/sin z 50)(Of4/(;qO ) -~- (8h/80)(Ofs/O50)](dc~/30) = O.

One observes that

f t + sin 50 cos 50(Sfl/(%p) =: y ( p + pq2 cos e 0),

fz q- sin 50 cos q~(Sf3/850) -~ ypq cos 0.

The jump conditions, as derived in Section 8, can therefore be written in terms of the integrands occurring in the variational problem as follows:

[f~ -c sin 50 c o s 5 0 ( C g f l / ~ 5 0 ] ) K t I - - [f l + sin 50 cos 50(Sfx/850)]K ~

-- hxH{[fa + sin 50 cos 50(Ofa/84,)]x~ ,

-- [ f , + sin 50 cos 50(S f , /~)]K,} = 0, (119-1)

(8A/e50)Kix - - (SA/O50)r., - - h x u [ ( e f a / 8 5 0 ) x u - (Ofa/e50)x,] = 0. (119-2)

Actually, these conditions are independent of 50: in the first equation, the terms containing 50 cancel; in the second one, 50 occurs only in a common factor. Multiplying the second equation by sin 50 cos 50 and subtracting the result from the first equation, one obtains

(f~)x, , - - ( A ) x , - - hxn[( f3)Kn - - (f3)K,] -- 0. (120)

This equation holds for any choice of 50. In this appendix, the jump conditions are used in the forms (1 19) and (120).

In the following discussions, the state II is considered as given, and we ask for which values of qz and 01 Eqs. (119)-(120) are satisfied. The system (119)-(t20) has one trivial solution, namely, q~--~ qH and 0 t = OiZ. We shall show that, for this solution, these equations can be satisfied up to terms of the first order by a suitable choice of dq f fdOt . I f we had a single condition, this would correspond to a zero of the second order. In these discussion, 50 is arbitrary, it can be considered as constant, and h is constant, too, for we consider the state II as given. One then obtains, for the first-order terms in (120),

[(efl/Oq) - - h (SA/eq)]r ,n dq~ + [oS~t80 - h(eA/OO)]x , dO, = O,

but because of (t08) the coefficients of dql and dot vanish separately. Therefore, we are free to choose dqffd01 in such a manner that (119) is satisfied up to the first order. One has specifically

[8~'f1/Sq850 - - h( ~ f a/3qc~50)]xu( dq~/dO,) + [82f~/80850 - - h( O2f a/80850)]xu .... O.

JOTA: VOL. 12, NO. 6, 1973 627

For cases where state II corresponds to a point of the curve B 1 , one can use (117) to rewrite this equation as

(y/sin 2 q~)[(~fa/Oq)(dqt/dOz) -- ~f4/~O] + (Of3/~q~)i(~h/~q)(dqr/dO,) - Oh/~O] = O.

This equation is then satisfied if one chooses

dqI/dO, = d4/dO. (121)

Next, we s tudy second-order terms in (t20), assuming that the state II lies on B 1 and that (121) is satisfied:

[~,f~/~q -- h(~f~/Oq)](d2qr/dO± 2) 4- [~2]- /~q2 _ h(~f,~/~qZ)](dq/dO~)2

4- 2[e2fa/~q~O -- h(~fa/Oq~O)](dqz/dOz) + [e2fa/eO z -- h(~f~/e02)].

The first term vanishes because of (t08); after substitution of (117) and of a similar equation involving derivatives with respect to 0, one obtains

[--(y/sin z 9)(eA/eq) 4- (eh/eq)(eA/e~)](dqz/dO,)

-- (y/sin 2 ~)(af4/O0 ) ~- (~h/bO)(af~/aq~)

~ - : -- (ylsin 2 ~)[(ef,~tOq)(dq/dOs) + (Df~lO0)]

+ (Ofale~o)[(Oh/~q)(dqr/dO,) + ~h/~O].

These expressions vanish because of (119)-(116) on account of the choice of dql/dO~. Conditions (119) can now be satisfied up to terms of the second order by the choice of dZql/dO~, 2.

References

1. GUDERLEY, K. G., and TABAK, D., On the Determination of Optimum Supersonic Thrust Nozzles of a Given Length for a Flow with Swirl: Theoretical Part, Office of Aerospace Research, United States Air Force, Report No. ARL 66-0013, 1956.

2. GUDERLEY, K. G., and BREITER, ~ . C., On the Determination of Optimum Thrust Nozzles of a Given Length for a Flow with Swirl: Numerical Results, Office of Aerospace Research, United States Air Force, Report No. ARL- 70-0161, 1970.

3. GUDERLEY, K. G., and HANTSCH, ]~., Beste Formenfuer Achsensymmetrische Uberschallschubduesen, ZFW, Vol. 3, pp. 305-314, I955.

4. NIKOL'SKI¥, A. A., Concerning Bodies of Revolution with a Duct which Possess l]/[inimum Wave Drag in Supersonic Flow (in Russian), Collection of Theoretical Works on Aerodynamics, Oborongiz, Moscow, USSR, 1957.

628 JOTA: VOL. 17, NO. 6, 1973

5. RAO, G. V. R., Exhaust Nozzle Contour for Optimum Thrust, Theory of Optimum Aerodynamic Shapes, Edited by A. Miele, Academic Press, New York, New York, 1965.

6. GUDERLEY, K. G., The Rote of Rao' s Postulate in the Optimization of Thrust Nozzles of a Given Length, ZFW, Vol. 18, pp. 41-44, 1970.

7. G~'DERLEY, K. G., On Rao's 3lethod for the Computation of Exhaust Nozzles, ZFW, Vol. 7, pp. 345-350, 1955.

8. GUDERLEY, I~. G., and BREITER, M. C., Approximation for Swirl Flows in the Vidnity of the Throat of a Laval Nozzle, Office of Aerospace Research, United St,~tes Air Force, Report No. ARL-70-0009, 1970.

9. SHMYGLEVSKIY, Yu. D., Some Variational Problems in Gasdynamics (in Russian), Computing Center of the Academy of Sciences, Moscow, USSR, 1963.

10. NAUMOVA, I. N., and SIaMYGLEVSKIY, Yu. D., Increase in Nozzle Thrust by Flow Rotation, Izvestiia Mechanika Zhidkosti i Gaza, pp. 34-37, 1967,