aiaa-1996-3401-807

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Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, 1996, pp. 328-335A9635118, AIAA Paper 96-3401

Parametric investigation of idealized hypersonic cruise configurations

Rick GravesColorado Univ., Boulder

George EmanuelOklahoma Univ., Norman

AIAA Atmospheric Flight Mechanics Conference, San Diego, CA, July 29-31, 1996,

Technical Papers (A96-35084 09-08), Reston, VA, American Institute of Aeronautics and

Astronautics, 1996

The lift and drag of 2D hypersonic configurations are evaluated for steady, compressible, inviscid flow. The twoconfigurations differ in their engine design with one configuration exhibiting a shock wave followed by a downstreamexpansion wave, while the other has the waves in reverse order. The theoretical formulation is outlined, and results arepresented for a variety of parameters, including vehicle length, nozzle exit Mach number, and lift and drag. Thelift-to-drag ratio is shown to be a maximum when the size of the engine inlet opening is a maximum, with the lip of the

cowl still inside the shock layer. In addition, the utility of configurations possessing negative lift and configurationsfree of shock wave-boundary layer interaction are discussed. (Author)

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96*101

A96-35118

AIAA-96-3401-CP

PARAMETRIC INVESTIGATION OF IDEALIZED

HYPERSONIC CRUISE CONFIGURATIONS

Rick Graves*Department of Aerospace Engineering Sciences

Th e Universityof ColoradoBoulder, Colorado 80309-0429

George Emanuel*School of Aerospace and Mechanical Engineering

The University of OklahomaNorman, Oklahoma 73019-0601

ABSTRACT

Th e lift and drag of two-dimensional h ypersonicconfigurations are evaluated for steady, com pressible,

inviscid flow. The two configurations differ in theirengine design w ith one configuration exhibiting a shockwave followed by a downstream expansion wave whilethe other has the waves in reverse order. T he theoreticalformulation is outlined and results are presented for avariety of parameters, including vehicle length, nozzleexit Mach number, and lift and drag. Th e lift-to-dragratio is shown to be a maximum when the size ofthe engine inlet opening is a maximum, with the lipof the cowl still inside the shock layer. In addition,the utility of configurations possessing negative liftand configurations free of shock wave-boundary layerinteraction are discussed.

NOMENCLATURE

Cd,C{ = two-dimensional drag and lift coefficientsC5^= control surfaced, t= two-dimensional drag and lift forces$x,ey =Cartesian unit vectorsFx, F y = drag an d lift componentsLf = forebody lengthLSE, LES = cowl length parameterM = Mach numberH = unit normal vectorp = pressurer = radial lengths = arc length« T = flow velocityx,y = Cartesian distances in the flow and transverse

directionsac = cowl location param eterass, ( X E S = cowl rotation parameter

* Graduate Student, Member, AIAA.t P rofessor, Associate Fellow, AIA A.

a* = boundary between cases A and B (ES family)a* * = boundary between cases B and C (ES family)f t = shock angle

7 = ratio of specific heats6 = angle relative to t t f o o

O f = forebody angleO j = inlet angle7 7 = expansion fan anglep = density( J , (M ) = sin

-1(l/M) = Mach angle

= Prandtl-Meyer function( ) I _ B = point po sitions( )SE = shock-expansion family( )ES = expansion-shock family

( )i-vi = uniform flow regions( ]00= freestream quantity

INTRODUCTION

Ongoing interest in researching the performanceaspects of hypersonic cruise vehicles (HCVs) withrespect to scramjet inlet design has resulted in thepredominance of certain configurations. On e promisingfamily of configurations, the single expansion rampnozzle (SERN), has been investigated by severalresearchers.

1"

3This family of vehicles (Fig. 1) features

a single-sided expansion nozzle and high nozzle exitvelocities with minimal external engine drag. Theinternal flow exiting the combustor interacts with the

Copyright © 1996 by the American Institute ofAeronautics and Astronautics, Inc. All rights reserved. Figure 1. Generic HCV scramjet engine geometry.

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y

Lf

control surrace

. . . . \

bow shock

Prandtl-Meyer

expansion

Figure 2. Schematic of the shock-expansion (SE) configuration.

free stream through a shear layer, while the nozzle wallbounds the expanding gas on the other side.

Other researchers have shown that simplifiedmodels can be useful in fluid mechanics, since they areeasy to interpret and provide dominant trends. Thisis particularly important for supersonic and hypersonicflow problems. Fo r example, Shaw and Duclr studiedthe problem of stability in supersonic boundary-layerflows, where inviscid disturbances are more important

than viscous disturbances. Bonataki et al.

5

calculatedthe optimal shape of turbomachinery blade sectionsassuming compressible flow using an inviscid designmethod. Broadbent

6 emphasized scramjet combustionand the role of inlet shock waves. Our investigation issimilar to Broadbent's in that most of his assumptions areretained, including the two-dimensional, inviscid flowof a perfect gas. In contrast, the flowfield is allowedto be hypersonic, there is no combustion process, andthe configuration geometries and overall approach aredifferent. W e further assume that all shock waves are ofthe weak solution variety and the forebody is a wedge.

Th e constant slope nozzle wall characteristic ofSERN configurations is limiting when designing HCVs.Therefore, we generalize this family by allowing thenozzle wall to possess some degree of curvature.Furthermore, parameters are introduced which allowdirect control over the amount of air entering thevehicle's inlet and the characteristics of the enginecowl. Analytical models representing these extendedconfigurations are outlined, and their lift and dragcoefficients are computed. Our principal objective isto examine basic trends, such as the effects of Mx and

the cowl's shape and location on lift and drag. Th e nextsection discusses the formulation, with the remainingsections containing results and conclusions. For furtherdetails, Hsu

7and Graves

8should be consulted.

FORMULATION

Shock-expansion configuration

Figure 2 provides a schematic of the shock-expansion (SE) configuration, in which the nozzle inlet

duct contains an oblique shock wave followed by anexpansion wave. Wall turn angles are such that at a givenM O O , the oblique shock does not reflect from point 4 andthe expansion does not reflect from the upper surfaceof the cowl. The cowl extends from point 3 to point 8,confines the inlet flow, and between points 7 and 8 is thelower nozzle wall. The forebody is a wedge of lengthL f, with its upper surface parallel to W Q O - The lengthL SE of the upper surface of the inlet region is arbitrary.Between points 3 and 6, the cowl is parallel to wall 4-5,with both walls at an angle A O = O f - O j relative to W Q O -The angle O f is specified, whereas

6 '/ = aSEOf

determines O j. The specified parameter ag^ rangesfrom zero, when the inlet shock wave between points 3and 4 becomes a Mach wave, to unity when the wallsbetween points 4 and 5 and between 3 and 6 are parallelto tSao, and the point 5 expansion reduces to a Machline. Th e labeled regions (I-VI) represent uniform flows,

with region IV here limited to the trailing edge of theexpansion. Centered Prandtl-Meyer expansions emanate

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upper cowl

wall

lower cowl

wallm

Figure 3. Curved cowl schematic.

from points 3 and 5, and the cowl w all between points 6

and 7 has the curved shape of a streamline of the point 5expansion. On the underside of the cowl, between points6 and 7, is another Prandtl-Meyer expansion wave, whichis non-centered, since M/// > M//.

The inlet shock wave and downstream expansionwave operate in combination to turn the flow in regionI parallel to W O Q . Th e nozzle flow, which is boundedby points 5-7-8-9-5, isentropically expands the gas to auniform state and a velocity that is also parallel to W Q O •This expansion is achieved with a nozzle of minimumlength using a lens analogy.

9 In general, pv differs fromPoo and pvi- Thus, there is a shock wave, slipstream,and expansion wave that emanate from points 8 and9. The subsequent downstream flow has a downwashvelocity when the vehicle has a positive lift. Th e

point 3 expansion intersects the bow shock and reflectsas a compression wave. When the reflected wave isdownstream of p oint 8, the l i f t and drag results of the SEconfiguration are exact. Nevertheless, the strength of thereflected wave is quite weak.10 As in shock-expansiontheory, we neglect the reflected wave.

The dotted line in Fig. 2 represents the controlsurface (CS), where point 2 is the location where thestreamline that wets the cowl crosses the bow shock.To apply the momen tum theorem to the correspondingcontrol volume requires knowledge of the shape of theCS, the pressure on it, and the velocity on those sectionsof the CS that are not tangent to a wall. In addition,

the shape of the 6-7 wall and the pressure integralsthat provide the l i f t and drag contributions need to be

established.The momentum theorem is written as

f iJcs

[pw(w •n) + pn]ds = -de^ -

and each of the seven sections of the CS in Fig. 2 isevaluated. Thus, for section 1-2, we have

with the result

where ac = X 2 / X - ) . When ac approaches zero, the cowlapproaches the body, while when ac = 1, the lip of thecowl touches the bow shock. Hence, the parameter ac

determines th e amount of air that enters the inlet. Thefinal form for the l i f t and drag coefficients are based onthe vehicle's overall length, and are given by

-2

% 2 L/ d

X g pooLf

where

'— PooQ-cX^ +PVl(x& —£7) "~ PVV&(My —1 ) '

+P///[(a;6 - £3 )2+ ( 2 / 6 - J /3)

2]

1/ 2 cos(% - 6 / )

and

- ( 2 / 6 - 2 / 3 )2]

1 / 2s i n ( 6 > / - 0 / )

- Q c)a;3 tan*?/ +L fdl_-,}

Th e quantities Z g _ 7 and d < J _ 7 are associated with thecurved part of the cowl (Fig. 3), and are given by

i * 6 - - , = c rJrn

•+1\V

2

( 1 \ 'rj tan zn( 7 7 ) cos ? / > „

1/2

T O - cos il > n + f jJ tan 2//(7?) sin^dr j

where

= P / / / K X 6 -

T+

(1)

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control surface

1 • •

M

bow shock

/

forebody expansion

cowl

Prandtl-Meyer

expansion

Figure 4. Schematic of the expansion-shock (ES) configuration.

-1 + tan 2 / 7 ( 7 7 ) tan

tan i / ) ( n ) +

T J 6 = 7J7 =

tan 2/7(77)

-0f+0l

Equations (1) and (2) provide M(rj) along section 6-7,and the above integrals are numerically evaluated. The

geometry of the SE configuration, as well as the l i f t anddrag coefficients, depends only on 7, M ,, O f, LSE/L/,U S E * a n d a c .

Expansion- shock configuration

Figure 4 is a schematic of the expansion-shock (ES)configuration. An expansion wave followed by a shockwave now turns the inlet flow so that in region IV it isparallel to w<». The isentropic nozzle flow, boundedby points 10-11-12-13-10, is again based on the lensanalogy. 9

In contrast to the SE case, the cowl is a flatplate that is parallel to W O Q , and thus has no drag. Th estreamline that wets the cowl is 2-3-5-9, where 3-5 iscurved. In this case, ac is defined as 2/2/1/7. wherex^ = xs is on the trailing edge of the expansion; this ac

definition differs from the one used in the SE case. Whenac is small compared to unity, we have #7 < x?,, whichis denoted as case A. With an increase in ac, points 6and 7 coincide and we set ac = a*. For larger valuesof a . c, we obtain the configuration sketched in Fig. 4 ,denoted as case B, where x -j > x& , but X f, > X T . .

For a still larger ac value, £2 = #6 an d Q C = a * * .When x2 > X6, we have case C, point 5 is then closeto point 8, and point 9 is close to the bow shock.

Furthermore, point 3 no longer occurs and some of thegas that crosses the curved bow shock between points 6and 8 now enters the inlet. In contrast to cases A andB, an accurate mass flux calculation at the inlet wouldrequire inclusion of the reflected wave from the curvedpart of the bow shock. This could be accomplished witha method-of-characteristics comp utation; however, thisis beyond the scope of this study. Hence, only cases A

and B are considered (i. e., ac < a**) .As in the SE case, the CS consists of sevensections, which are individually evaluated. Althoughthe geometry depends on whether the configuration iscase A or case B, the lift and drag coefficient formulas

2 Lf I 1 Lf d

do not, where

+ Pl [ (x3 -ar2)2+(2/3 -2/2)

2]

1/ 2cos 6> / -

+pn[(x 9 - xs)2 + ( 2 / 9 - 2 /5) 2]1/ 2 cos(0/ -

and

x2f + ( 2 / 3 - 3/ 2)2] 1

- F

x sin(0/ - #/)- - F x

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In these relations, F x an d F y are force components

associated with section 3-5 in Fig. 4 (see Graves8

fortheir lengthy analytical expressions). Results depend on7, Moo, 6f, LEs/Lf, < X E S , and ac.

Limiting case

Hsu7

considered the limiting SE case when LSE = 0and asE = 1 - In this circumstance, points 4 and 5coincide (Fig. 2), the expansion from point 5 is reducedto a Mach line, and the cowl is a flat plate parallel toWK, . The CS now has five sections and the associatedlift an d drag of each section is evaluated. One importantconclusion is that a properly defined lift coefficient forthe forebody substantially exceeds that for the entireconfiguration. Much effort ha s gone into optimizingwaverider (W R) forebodies for performance parameterssuch as the lift-to-drag (1/d) ratio.

11"

13The wedge

forebody in our study can be viewed as a WR, albeit asimple one. Nevertheless, this result calls into questionthe usefulness of separately optimizing a forebody

configuration relative to some criteria and constraints.Moreover, the large difference (a multiplicative factor of5) between the two lift coefficients raises doubts aboutthe viability of a W R forebody. This configurationis typically very slender and has little useful volumecompared to its large surface area.

The SE formulation is singular when ac equalsunity.

7 In this circumstance, the lip of the cowl, point3 in Fig. 2, is on the bow shock. The shock wavemay continue below the cowl, as it does in the presentformulation, or terminate at the cowl's lip. In eithercase, there is still an expansion emanating from point3, although the strength differs, since the upstreamMach number also differs. As a consequence, the liftcoefficients differ, as will the drag coefficients when

( y 2 ) s

0.0

0.0 0.2 0.4 0.6 0.0

(b)

0.0 0.2 0.4 0.6 0.6 1.0

Figure 5. Mass flow comparison vs. ac when Me*, =1,6f = 10°,

and L SE = LES = 0.5Lf .

Figure 6. Vehicle length vs. ac when 6; = 10°, LSE = LES=

0.5Lf , and U SE = ass = 0.5; (a) SE configuration, (b) ESconfiguration.

RESULTS

A parametric study8

was performed with 7 = 1.4,

M O O = 4,7,10, 9 f = 5,10,15°,aSE =aEs =0,0.5,1,LsE/Lf = LES/Lf = 0,0.5,1, an d ac = 0.01 to 1for the SE configuration an d ac = 0.01 to a** for theES configuration. The definition of the cowl locationparameter ac differs for the SE and ES configurations.Therefore, results are provided in Fig. 5 which showa 2 /2 ratio that is proportional to the ratio of the inletmass flow rates. The ES inlet often has a substantiallylarger flow rate for the same ac value. The curvesare truncated when ac - a* * for the E S configuration.

Curve truncation occurs in later figures for the samereason.

Selected results are shown in Figs. 6-11 for the SEand ES configurations (see reference 8 for additionalresults). As is evident in Fig. 6, the ES designapproach produces configurations that are longer thanSE configurations fo r large values of ac. In addition, fo rthe parameters in Fig. 6, both configurations are seen tohave a minimum length of about 4Lf . However, furtherexamination of the global parameter space reveals thatboth configurations have a minimum length of about2L/.

8Figure 7 shows the nozzle exit Mach number,

which approaches M O Oat large ac values. Conversely,

M y approaches infinity when ac approaches zero.W hen Q C is large, the nozzle area ratio is small, whereas

when ac approaches zero, th e area ratio becomes infinite.This behavior also causes the nozzle exit pressure ratio inFig. 8 to increase steadily with Q C . Although not evident

in the figure, th e pressure ratio Pv/Poo can exceed unitywhen ac is large and USE = 1, or O ES = 0- In fact,for the parameter space considered, both configurationshave a maximum value of about 2.2 for pv/Poo- WhenPv/Poo is less than unity, a substantial truncation of the

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(a)

0.0 0.2 0,4 0.6 0.8 1.0

(b)

0.0 0,2 0.4 0.6 O.B 1.0 0.0 0.2 04 0.6 0,8 1.0

Figure 7. Nozzle exit Mach number (My) vs. «c when 9f = 10°,

LS£ = LES = O.SLf , and OSE = «ES = 0.5; (a) SE configuration,

(b) ES configuration.

Figure 9. Lift coefficient (Cj) vs. ac when O f = 10°, LSE =LES = O.SLf , and USB = < * E S = 0.5; (a) SE configuration, (b)

ES configuration.

Figure 8. Nozzle exit pressure ratio (PV/POO)VS. ac when 6f = 10°,LSE = LES = 0.5L/, an d O SE = C U E S - 0.5; (a) SE configuration,(b) ES configuration.

long upper nozzle wall results in an increase in lift anda modest loss of thrust. This point is further examinedfor viscous nozzle flow by Bae and Emanuel.

14Note

that pv/Poo is relatively insensitive to Mx, an d wouldbe large if a constant area combustion process occurs

upstream of the nozzle's inlet. In this circumstance,cycle analysis would clearly favor the higher pressurelevel associated with the SE configuration.

Figure 9 shows the lift coefficient, which has amaximum value when ac is near 0.1. Note that Cjbecomes negative when ac approaches zero. This is

caused by the low pressure, compare to P O O on the uppernozzle wall 5-9 (SE) or 10-13 (ES). Except at small ac

0.0 0,2 0.4 0.6 0.6 1.0

a.

(b)

a.

Figure 10. Drag coefficient (C<j)vs

-ac when ff f = 10", LSE -

LES = O.SLf , and < X S E - & ES - 0.5; (a) SEconfiguration, (b) ESconfiguration.

values, there is little effect of MO O on C\. Th e dragcoefficient is shown in Fig. 10, where it is observedthat minimum drag occurs at a large ac value. Althoughthe drag coefficient decreases with increasing M O O , thedrag itself has the opposite trend. Figure 11 shows

l /d, which becomes negative when ac approaches zeroan d is a maximum when ac is a maximum. As theanalysis in Graves

8shows, the main influence on l/d is

M O O an d ac,with increasing values for these parametersincreasing l/d. For the parameter space considered,the SE configuration has a maximum l/d value of 4.9,

whereas the ES configuration has a maximum value of6.1.

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o—o M= 4 (b)Q——EJ7

« — 0 1 0

0.0 0.2 0.4 0.6 0.8 1.0

Figure 11. Lift-to-drag ratio ( I / i t ) vs. a0 when 6f = 10°, L SE =L E S = 0.5Lf , an d asE = OES = 0.5; (a) SE configuration, (b) ES

configuration.

An important observation of this study is that dis a maximum when ac « 0.1, whereas l/ d is a

maximum when ac is a maximum, say, unity for theSE configuration. As shown by Fig. 9, C j is reducedfrom its maximum value by at least a factor of twowhen ac is a maximum. Thus, the large C\ valuerequired by HCVs during acceleration conflicts with therange maximization characteristics of a large l / d . Oneresolution, of course, is a moveable cowl.

Combinationvehicle

By adjusting the forebody length L f, it is possibleto combine two configurations, as illustrated in Fig. 12.Th e configurations can be designed using either the SE

or ES approach, have different ac and O f values, etc.All that is required is that Mx and the overall vehiclelength be the same. The lift an d drag coefficients for thecombined vehicle are

Ci = CIB - CIA, C d = CdB + C d A

Additional l i f t is thus obtainable if configuration A hasnegative lift.

Figure 12. Com bination vehicle.

bow shock expansion

Figure 13. Wind tunnel configuration.

Wind tunnel model

The ES configuration with a^s - 1 possesses

the unique characteristic of having no shock wave-boundary layer interaction. A scaled view of this typeof configuration is shown in Fig. 13 , where MOO = 4,O f - 30°, LES/Lf =0.2, UES = 1, andac =0.33. The

inlet shock wave is now a Mach wave and the Mach waveat the start of the nozz le is also shown (this is line 10-11 inFig. 4 ). If there is a sizeable pressure mismatch betweenPv and poo. boundary layer separation inside the nozzleor on the external surface may occur. Fo r turbulent

boundary layers, this may not happen for the illustratedcase, since pv/Poo = 0.80 andpv//p/// = 0.62.

CONCLUSIONS

A momentum theorem formulation is providedfor th e lift an d drag of two supersonic/hypersonicconfigurations. The relative amount of inlet ai r flow

rate is controlled by the parameter ac. This flow

rate decreases to zero when ac approaches zero, an dthe lift becomes negative. Th e l i f t coefficient is amaximum when ac « 0.1, whereas the lift-to-dragratio is a maximum when ac is a maximum. Boththe shock-expansion (SE) and expansion-shock (ES)configurations have similar aerodynamic properties.However, the SE configuration can be used with a ramjet

or scramjet engine, while the ES configuration may bemore advantageous for wind tunnel testing.

REFERENCES1

Ruffin, S. M ., Venkatapathy, E., Lee, S. H., Keener, E.R., and Spaid, F. W., 1992, "Single expansion rampnozzle simulations," A I A A Paper No. 92-0387.

2Canupp, P. W. and Candler, G. V., 1993, "Effects of

thermochemical nonequilibrium on scramjet nozzleperformance," A I A A Paper No. 93-2838.

3Spaid, F. W. and Keener, E. R., 1993, "Hypersonic

nozzle/afterbody CFD code validation part I: exper-imental measurements," A I A A Paper No. 93-0607.

4Shaw, S. J. and Duck, P. W ., 1992, "The inviscid

stability of supersonic flow past heated or cooledaxisymmetric bodies," Physics of Fluids A, Vol. 4 ,No. 7, pp . 1541-1557.

5Bonataki, E. , Chaviaropoulos, P., and Papailiou, K .

D. , 1993, "An inverse inviscid design method for thedesign of quasi-three-dimensional turbomachinerycascades," Journal of Fluids Engineering, Vol. 115,pp . 121-127.

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