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NASA Technical Memorandum 4506

Computational Prediction of IsolatedPerformance of an Axisymmetric Nozzle atMach Number 0.90John R. Carlson

February 1994

NASA Technical Memorandum 4506

Computational Prediction of IsolatedPerformance of an Axisymmetric Nozzle atMach Number 0.90John R. CarlsonLangley Research Center � Hampton, Virginia

National Aeronautics and Space AdministrationLangley Research Center � Hampton, Virginia 23681-0001

February 1994

National Aeronautics andSpace AdministrationCode JTTWashington, D.C.20546-0001

Official BusinessPenalty for Private Use, $300

B U L K R A T E

POSTAGE & FEES PAIDNASA Permit No. G-27

Postmaster: If undeliverable (Section 158 Postal Manual) Do Not Return

Abstract

An improved ability to predict external propulsive performance

has been incorporated into the three-dimensional Navier-Stokes code

PAB3D. The improvements are the ability to account for skin fric-

tion and external pressure forces. Performance parameters for two axi-

symmetric supersonic cruise nozzle con�gurations were calculated to

test the improved methodology. Internal and external ow-�eld re-

gions were computed using a two-equation k-� turbulent viscous-stress

model. The computed nozzle discharge coe�cient matched the experi-

mental data within 0.5 percent in both level and trend. The calculated

thrust-minus-drag ratios were within 1 percent of the absolute level of

experimental data, and the trends of data were predicted accurately. The

predicted trend of integrated nozzle pressure drag matched the trend of

the integrated experimental pressure drag over a range of nozzle pressure

ratios, but absolute drag levels were not accurately predicted.

Introduction

Highly maneuverable aircraft operate over a widerange of power settings and Mach numbers that re-quire a propulsion system with variable geometry forobtaining e�cient performance at di�erent ight con-ditions. Understanding the e�ects of various nozzlegeometries on the internal ow region and the sur-rounding boattail-nozzle region is vital for design-ing an e�cient afterbody for these aircraft. Thedevelopment and utilization of advanced computa-tional methods will play a vital role in developing thisunderstanding. Several on-going research activitiescurrently exist at the Langley Research Center whichare directed at establishing an experimental databasefor new nozzle concepts. Subsequent improvementsto the computational methods are guided by thesedata.

A nozzle internal performance module (ref. 1) hasbeen modi�ed to include external aerodynamic ef-fects. The module was incorporated into a Navier-Stokes solver PAB3D (ref. 2), which provides the ow-�eld solution. The nozzle performance moduleuses the control volume concept to calculate the bodyforces resulting from the uid ow (ref. 3). Forcesand moments are calculated from the integration ofthe momentum uxes through the control volumefaces. Skin friction on solid walls is calculated byusing the gradient of the local velocity in the direc-tion normal to the wall and the local viscosity. Thesecalculations can be performed at intermediate stepsthroughout the solution procedure to provide an inte-grated ow quantity convergence in addition to mon-itoring the computational residuals in the ow solveralgorithm.

This paper evaluates the module by using thedata from the axisymmetric single-engine test bodywith jet-plume simulation reported in reference 4.Computed discharge coe�cient, thrust-minus-dragratio (measured axial force nondimensionalized byideal isentropic thrust), and internal and externalsurface static pressures were compared with exper-imental data for a �xed free-stream Mach numberof 0.90 at several nozzle pressure ratio settings fortwo of the nozzle concepts in reference 4.

Symbols

Ae nozzle exit area, cm2

Ai;j local rotation matrix

Am maximum cross-sectional area of

model, 182.415 cm2

At nozzle throat area, cm2

�A incremental cross-sectional

area, cm2

CD;� boattail pressure-drag coe�cient

Cd discharge coe�cient,wp

wi

CF;i aerodynamic ideal thrust coe�cient,Fi

q1Am

Cp;� boattail pressure coe�cient

Dn nozzle total drag, Pa

de nozzle exit diameter, cm

dm model maximum diameter,15.240 cm

dt nozzle geometric throatdiameter, cm

eij rate-of-deformation tensor

F gross thrust along body axis, N

F total force vector, N

Fi ideal isentropic gross thrust alongbody axis, N

i; j; k indices, when associated withcomputational grid planes, denotingorientation in lateral, streamwise,and normal directions, respectively

l axial length of boattail, cm

lC axial length of nozzle convergentsection, cm

lD axial length of nozzle divergentsection, cm

M Mach number

M total moment vector, N �m

n̂ local surface normal vector in globalsystem

NPR nozzle pressure ratio,pt;jp1

NPRdes design nozzle pressure ratio

p static pressure, Pa

pt;j jet total pressure, Pa

q dynamic pressure, Pa

R gas constant ( = 1.4),287.3 J/kg �K

R moment arm vector, cm

T static temperature, K

Tt;j jet total temperature, K

u total velocity vector, m/sec

Ui local Cartesian velocity, m/sec

wi ideal mass- ow rate, kg/sec

wp actual mass- ow rate, kg/sec

x axial distance, positive down-stream, cm

y+ nondimensional distance from wallin turbulent boundary layer

� angle of attack, deg

� nozzle boattail angle, deg

ratio of speci�c heats, 1.4 for air

� nozzle divergence angle downstreamof throat, deg

� nozzle convergence angle upstreamof throat, deg

� viscosity, m2/sec

�i local Cartesian coordinate systemon surface

� density, kg/m3

�ij viscous-stress tensor, Pa

�i local Cartesian skin friction forcevector, N

Subscripts:

1 free-stream conditions

ux ux and pressure forces ( ow-through face)

pres pressure forces (solid face)

fric viscous forces (solid face)

ref reference conditions

local local conditions

Abbreviations:

A/B afterburning

C-D convergent-divergent

CFL Courant-Friedrichs-Lewy

C2 con�guration 2 of reference 4

C4 con�guration 4 of reference 4

2-D two-dimensional

3-D three-dimensional

Experimental Model Con�guration

General Model Description

The experimental results selected for this com-putational study are reported by Carson and Leein reference 4. A photograph of the model in theLangley 16-Foot Transonic Tunnel and a sketch ofa representative test model and support system arepresented in �gures 1 and 2. The facility, which is asingle-return, continuous- ow atmospheric wind tun-nel with an octagonal slotted-throat test section, hasa continuously variable Mach number range from 0.20to 1.30. A detailed description of this wind tunnel isgiven in reference 5.

The models of the axisymmetric, convergent-divergent (C-D) nozzles (�gs. 3 and 4) had a circular-arc throat contour and conical divergent sections.

2

The nozzle geometries simulated a variable-geometryaxisymmetric nozzle designed for a variable-cycle en-gine that was proposed for a supersonic cruise air-craft. Experimental data were obtained for �ve noz-zle con�gurations that represented the geometries fordi�erent ight conditions and power settings. Thesegeometries ranged from a subsonic cruise, dry powercon�guration, which would have a low expansion ra-tio and a high nozzle boattail angle, to a super-sonic acceleration, maximum afterburning con�gu-ration, which would have a high expansion ratioand a low nozzle boattail angle. Tests were con-ducted at nozzle pressure ratios (NPR's) from jeto� to approximately 10 at free-stream Mach num-bers from 0.60 to 1.30. The nozzles were attachedto an axisymmetric single-engine host body mountedon a sting-strut system and had a maximum cross-sectional area of 182.415 cm2 (28.274 in2). The ogive-shaped nose had an apex angle of 14� and a radiusof curvature of 128.37 cm (50.54 in.). The nonmetricportion of the host body was 67.31 cm (56.50 in.)long. The model was cylindrical from the metricbreak to the nozzle connect station and was 69.85 cm(27.50 in.) long.

The jet was simulated by high-pressure air ex-iting the nozzle into the external free stream. Anair system at the wind tunnel provides a continuous ow of clean, dry air at a stagnation temperatureof nominally 300 K (540�R). The nozzle dischargecoe�cient was determined from experimentally mea-sured jet total temperature, jet total pressure, andmeasured mass- ow rate. The thrust ratio was deter-mined from the measured balance axial force that wasnondimensionalized by the ideal thrust (determinedfrom the measured nozzle mass ow) and correctedfor internal pressure tare forces. Static pressure ori-�ces were located on the external boattail and theinternal nozzle surfaces.

Test Case Details

Two nozzle con�gurations from reference 4 wereselected as test cases for the computational method,and they are described below.

Con�guration 2. Con�guration 2 of reference 4is a high expansion ratio, axisymmetric, C-D nozzle.The external geometry is a circular-arc and straight-line type with a terminal boattail angle of 3:82�. Theinternal geometry has a circular-arc throat and a di-vergent section angle of 13:18� with an expansion ra-tio of 3.000 that results in a design NPR of 21.23.This geometry was chosen because the experimentaldata generally indicate no external ow separationat M1 = 0:90 for the NPR settings tested. In spite

of the absence of data around the design NPR atM1 = 0:90 because of limitations in the attainablemass- ow rate of the experimental apparatus, it re-mains an interesting case for evaluation of a compu-tational method. The internal ow does separate atlower NPR settings because of the high expansionratio.

Con�guration 4. Con�guration 4 of reference 4is a low expansion ratio, axisymmetric, C-D nozzle.The external geometry is a circular-arc and straight-line type with a terminal boattail angle of 8:28�. Theinternal geometry has a circular-arc throat and astraight-line divergent section angle of 4:78� with anexpansion ratio of 1.500 and a design NPR of 6.23.This geometry was chosen because the experimentaldata were obtained around the design NPR andnozzle internal ow appears to remain attached at thelower NPR conditions because of the low expansionratio.

Computational Procedure

Flow-Field Calculation

A three-dimensional Navier-Stokes code PAB3Dwas developed to predict the e�ects of the jet exhaustplume on nozzle-afterbody con�gurations. The thin-layer Navier-Stokes formulation (ref. 6) was modi�edto simulate jet mixing problems (ref. 2). The codeallows for the discretization of the ow-�eld domaininto multiblock grids and can utilize several numeri-cal schemes to solve the governing equations and tur-bulence models as discussed in reference 7. The Roeupwind scheme with third-order accuracy is used toevaluate the explicit part of the governing numericalequations for relaxation in the streamwise direction,typically along the i index. The van-Leer scheme,with its faster convergence rate, is used to constructthe implicit operator in the cross plane; typically thisplane is a grid along the j and k indices. (Normallythis code utilizes the Roe scheme to sweep streamwisethrough the computational domain and the van-Leerscheme for the solution of the cross plane.) For thepresent study, the streamwise grid plane is orientedalong the j and k indices to take advantage of the im-plicit scheme for e�ciency and accuracy. Solutionswere developed using two versions of a two-equationk-� turbulent viscous-stress model. A low Reynoldsnumber k-� model was used for the ow near solidsurfaces while a high Reynolds number k-�model wasused in the regions of free-shear ow.

A user-written control �le determines the com-munication between blocks and the type of bound-ary condition to be used at each face. Di�erent grid

3

topologies for neighboring blocks and mixed bound-ary conditions on a block face are permitted, withsome restrictions on grid matching at block bound-aries. The code permits di�erent numerical schemesto be selectively applied to each block.

Grid De�nition

The block arrangement and dimensions of thegrid are shown in �gure 5(a). A sectional view ofthe overall grid in the j and k planes is shown in�gure 5(b), and a sectional view of the two blocksde�ning the internal ow path is shown in �gure 5(c).A 2-D wedge grid was used, and the ow was assumedto be axisymmetric. A single-cell-width cylindricalwedge grid was generated with a wedge angle of 5:00�,which represents 1/72 of an axisymmetric geometry.

The nozzle ow �eld was modeled by using �vecomputational blocks. Two blocks represented thenozzle internal grid, and an additional two blocksrepresented the external grid region surrounding thenozzle. The �fth block modeled the external re-gion downstream of the nozzle exit. The dimen-sions of the nozzle internal blocks were 2 � 6 � 49and 2� 78� 49. The nozzle internal boundary-layergrid region contained approximately 24 points withthe �rst grid point speci�ed by y+ = 2:5. The ex-ternal blocks surrounding the nozzle grid were dimen-sioned 2 � 6 � 87 and 2 � 64 � 87. Approximately40 streamwise grid points were used in the region ofthe nozzle boattail. The external downstream region,modeled by block 5, was dimensioned 2 � 51 � 135.The interfaces matched point for point between theblocks upstream and downstream of the nozzle exitplane. The grid was extended to a distance of 15 noz-zle exit radii away from the outer boundary. Theout ow boundary was 5 exit radii downstream of thenozzle exit plane.

Boundary Conditions

The PAB3D code allows di�erent boundary con-ditions to be applied to a given block face. Solid wallsare treated as no-slip adiabatic surfaces. The in owboundary conditions used for the internal nozzle owpath are the total pressure pt;j and the total temper-ature Tt;j. This particular in ow boundary conditionassumes a uid ow angle normal to the in ow face.The operating NPR of the nozzle and the free-streamstatic pressure p1 determines the jet total pressurept;j = (NPR)(p1).

A jet total temperature of 300 K was used for allthe calculations, and it was the nominal temperatureset during the experimental investigation. The in- ow Mach number, ow angle, total pressure, and to-tal temperature were �xed along the external in ow

boundary. A boundary condition for Riemann invari-ants (ref. 2) along the characteristics was speci�ed forthe lateral free-stream boundary of the ow domain.An extrapolation boundary condition was applied onthe downstream out ow face where both the exter-nal free stream and the nozzle plume stream exit thecomputational domain. The wedge-angle boundarycondition determines the angle of the wedge lateralfaces and the proper boundary values to apply forsatisfying the symmetry assumptions. Flow proper-ties were transferred between blocks through cell-by-cell conservation of the mass and momentum uxesacross the block interface with third-order continu-ity. However, some restrictions on the multiblockgrid were expected by the code, such as an integermultiple to one correspondence between cells at theblock interface and matched cell sizes in directionsnormal to the block interface.

Performance Calculation

Nozzle performance is obtained through the ap-plication of the momentum theorem to a controlvolume surrounding the nozzle (ref. 3). Cheatham,Walker, and Gridley calculated both 2-D and 3-D in-viscid nozzle performance in reference 8, and Carlsoncalculated 3-D viscous nozzle performance in refer-ence 1 by using the control volume method. Inte-gration of the ow quantities is typically performedacross the nozzle exit. The method integrates themass and momentum uxes and the pressure forcesover each cell by using equations (1) and (2) for ow-through sections of the control volume:

wp =X

(�u � n̂) �A (1)

F ux =X

[�u(u � n̂) + (p� p1)n̂] �A (2)

where �A is the incremental cross-sectional area ofthe cell face and R is the moment arm vector fromthe reference center to the cell face.

Ideal mass- ow rate and thrust are determinedfrom the isentropic ow equations (3) and (4), re-spectively (ref. 9), and they are used to normalizethe calculated mass- ow rate and thrust for compar-isons with the experimental data:

wi =

r

R

�2

+ 1

� +1

2( �1)At

pt;jpTt;j

(3)

Fi =

s2 R

� 1wp

vuuuutTt;j

2641�

p1

pt;j

! �1

375 (4)

4

Skin friction and pressure forces are calculated forsolid wall sections of the control volume. The solidsurface pressure force is calculated by extrapolatingthe cell centered static pressure to the surface walland multiplying this pressure by the cell face area:

Fpres =X

[(p� p1)n̂] �A (5)

The viscous-stress tensor used for determining theskin friction force is calculated by using only the ve-locity derivatives normal to the surface. The velocitygradients are determined by a two-point di�erence.The �rst velocity is a zero magnitude vector posi-tioned on the surface. The second velocity is thevelocity at the cell center as sketched in �gure 6.Equations (6a) and (6b) are two of the nonzero com-ponents of the rate-of-deformation tensor calculated(ref. 10):

e13 =@U1

@�3(6a)

e23 =@U2

@�3(6b)

where Ui = Aijuj. The remaining velocity deriva-tives are assumed to be zero. The set of local velocitycomponents Ui are in a local surface Cartesian coor-dinate system �i (where �3 is the direction normal tothe surface), which is determined from an orthogonaltransformation of the global velocity components by

the rotation matrix Aij. In general eij =@Ui@�j

, where

i 6= j.

The local shear-stress tensor �ij, where i 6= j,was the local viscosity multiplied by the velocity gra-dients eij, where i 6= j. The viscosity was deter-mined from Sutherland's formula (ref. 11) by usingthe static temperature at the local cell center:

�local = �ref

�(Tref + 110:33)

(Tlocal+ 110:33)

��Tlocal

Tref

�1:50

(7)

�ij = �localeij (i 6= j) (8)

The local skin friction force components �i were

�i = (�ijni) �A (9)

where ni is the ith component of the unit normalvector n̂ and �i were transformed back to a forcevector expressed in global coordinates Ffric. Thesolid surface forces are then added to F ux for thetotal volume forces:

(Ffric)j = �iATij (10)

F = F ux+Fpres+Ffric (11)

The performance package is incorporated intoPAB3D to permit monitoring of various performanceparameters as the solution convergences.

Discussion of Results

Experimentally measured external-surface static-pressure coe�cient distributions and the internal-surface pressure ratio distributions are comparedwith calculations in �gures 7 to 12. Various measuredforce parameters are compared with calculations in�gures 13 to 17. Speci�c comparisons are presentedin the following �gures:

FigureExternal and internal surface distributionsof con�guration 2, NPR = 4, M1 = 0:90 . . . 7

External and internal pressure distributionsof con�guration 2, NPR = 5, M1 = 0:90 . . . 8

External and internal pressure distributionsof con�guration 2, NPR = 6, M1 = 0:90 . . . 9

External and internal pressure distributionsof con�guration 4, NPR = 5, M1 = 0:90 . . 10



Nozzle discharge coe�cients . . . . . . . . 13Variation in nozzle ideal thrust coe�cientwith NPR . . . . . . . . . . . . . . . 14

Variation in aeropropulsive performance withNPR . . . . . . . . . . . . . . . . . 15

Variation of increment in aeropropulsiveperformance with nozzle con�guration . . . 16

Variation in integrated nozzle pressure dragwith NPR . . . . . . . . . . . . . . . 17

Discussion

Solutions were obtained for two nozzle con�gu-rations by using the PAB3D ow solver. The solu-tions were computed by using a single-cell-width 2-Dmesh oriented so that the entire ow �eld was solvedimplicitly with each iteration. All solutions weredeveloped by using a two-equation, k-� turbulentviscous-stress model. Discharge coe�cient, idealthrust coe�cient, thrust-minus-drag ratio, and noz-zle drag coe�cient were calculated for di�erent noz-zle operating conditions, which exhausted into aM1 = 0:90 free-stream air ow at an angle of attackof 0�. The calculations are summarized as follows:

5

NPR

Parameter 4 5 6 7

Con�guration 2

Iteration . . . . . . 8000 6800 2000 2500

Computer time, hr . 3.3 2.8 0.8 1.0

Cd

. . . . . . . . 0.963 0.963 0.963 0.963

(F �Dn)=Fi . . . . 0.789 0.787 0.849 0.887

Con�guration 4

Iteration . . . . . . 2500 2500 1500

Computer time, hr . 1.0 1.0 0.6

Cd

. . . . . . . . 0.966 0.966 0.966

(F �Dn)=Fi . . . . 0.969 0.979 0.984

The solutions started with a CFL number of 1.The CFL number for subsequent iterations duringa run were adjusted automatically within the codefor convergence acceleration. Flow solution residuals ,discharge coe�cient, thrust ratio, and nozzle dragwere obtained at regular intervals during the solutiondevelopment. Typical solution convergence criteriaare less than 0.05 percent change in discharge coef-�cient, less than 0.1 percent change in thrust ratio,and less than 0.001 change in nozzle drag coe�cientover the previous 1000 iterations. Solutions with owseparation typically required an additional 3000 to4000 iterations beyond the average 1500 to 3500 iter-ations to establish an acceptable level of performanceconvergence. Situations of ow instability because ofeither real ow physics or numerical modeling di�-culties occasionally caused some performance param-eters to oscillate about a median value. The solutionsfor this study met the general convergence criteriawith little or no occurrence of oscillations in perfor-mance numbers.

Comparisons of predicted internal and exter-nal static pressure distributions with experimentaldata are made for con�guration 2 at NPR = 4, 5,and 6. Predicted static pressure distributions arecompared with experimental data for con�guration 4at NPR = 5, 6, and 7. Predicted performance pa-rameters are compared with experimental data.

Static-Pressure Comparisons

Computed external and internal static-pressuredistributions are compared with experimental datafor con�guration 2 in �gures 7 to 9. In general, theexternal static-pressure coe�cients are matched wellfor all three NPR conditions tested (�gs. 7(a), 8(a),and 9(a)). The corner ow expansion near x

dm= 0

is slightly overpredicted and the trailing-edge ow

expansion is slightly low, which is possibly a result ofgrid density in those locations. The ow separationover the internal divergent section near x

dm= 0:7 is

fairly closely matched (�g. 7(b)) at NPR= 4. Neitherthe apparent internal ow separation at NPR = 5(�g. 8(b)) at x

dm= 0:92 nor the trailing-edge ow

compression at NPR = 6 (�g. 9(b)) at x

dm= 0:995

was predicted. Extremely �ne grid in the vicinity ofthe nozzle trailing edge in the external and internalregions would probably be required to resolve suchdetailed local ow structure. The slight mismatch ofthe computed pressures and the experimental dataupstream of x

dm= 0:20 is probably because of a

geometry documentation di�culty in reference 4.Analytic results performed in reference 4 and thepresent method showed very similar trends in theinternal static-pressure distributions in the regionupstream of the throat.

Computed external and internal static-pressuredistributions for con�guration 4 are compared withexperimental data in �gures 10 to 12. The predictedexternal pressure-coe�cient distributions match wellwith experimental data except for the magnitude ofthe ow expansion around the circular-arc closure of

the boattail�0 < x

dm< 0:2

�in �gures 10(a), 11(a),

and 12(a). The overprediction of this ow accelera-tion is, again, possibly because of grid density in thatregion of the model. The internal ow for this nozzleremained fully attached for the range of NPR ex-amined. The internal static-pressure ratio distribu-tion downstream of the throat typically changed littlefor choked nozzle ow that �lled the internal nozzlevolume completely. The solutions matched the ex-perimental internal pressure distributions closely forthe three NPR computed and shown in �gures 10(b),11(b), and 12(b) (NPR = 5, 6, and 7, respectively).

Performance Comparisons

Figures 13 to 16 show comparisons of com-puted performance parameters with experimentaldata from reference 4. All data were at a free-streamMach number of 0.90 and a free-stream angle of at-tack of 0�. Additional experimental data were plot-ted to show general trends that surrounded the con-ditions chosen for the analytic study.

Discharge coe�cient. A comparison of pre-dicted discharge coe�cients with experimentallydetermined discharge coe�cients is shown in �g-ure 13. The experimental data, shown by the circleand square symbols, were obtained at static condi-tions. Theoretically, external ow has no e�ect on

6

discharge coe�cients once the internal ow is choked(NPR > 1:89). Thus, a comparison of computed dis-charge coe�cients at M1 = 0:90 with experimentaldischarge coe�cients at M1 = 0:0 is valid. The pre-dicted discharge coe�cients for con�guration 2 arewithin 0.5 percent of the experimental data. The pre-dicted discharge coe�cients for con�guration 4 arewithin 0.1 percent.

Ideal thrust coe�cient. Predicted ideal thrustcoe�cients for both con�gurations are comparedwith experimental data in �gure 14. The idealthrust coe�cient is the ideal nozzle thrust non-dimensionalized by using the free-stream dynamicpressure q1, a reference area Am, and the physicalmass ow wp. This coe�cient is used as a meansof converting aerodynamic coe�cients normalized byq1Am to propulsive-force ratios based on Fi. Forthese con�gurations, the reference area is the maxi-mum cross-sectional area of the model. The ability topredict this quantity should be similar to the abilityto predict the discharge coe�cient. In general, thelevel and slope of the computed ideal thrust coe�-cient and the experimental data are closely matched,which indicates that the trends in the mass ow withNPR were predicted properly.

Aeropropulsive performance. Figure 15shows a comparison of the predicted aeropropulsiveperformance (thrust-minus-drag ratio) with the ex-perimental data for both con�gurations forM1 = 0:90. The ow that exhausted from thehigh expansion ratio nozzle (con�guration 2) is highlyoverexpanded and exhibits some internal ow sepa-ration at NPR = 4, which causes the discontinuity inthe trend between NPR = 4 and 5 (square symbols,�g. 15). The method predicted the change in perfor-mance as the region of nozzle internal ow separationdiminished. The predicted thrust-minus-drag ratiowas within 0.010 of the experimental data. Consid-erably higher performance was attained by con�gu-ration 4 as the design NPR was bracketed by theNPR range from 5 to 7. The level and trend of thepredicted thrust-minus-drag ratio for con�guration 4was within 0.004 of the experimental data in thrustratio, when the circle symbols are compared with thesolid line.

The di�erence in aeropropulsive performance be-tween con�gurations 2 and 4 was predicted within0.005 in thrust ratio for the three NPR settingsshown in �gure 16. Limitations in calculating theinternal ow separation of the turbulence model forcon�guration 2 likely contributed to the disagree-ment with the experimental data at NPR = 5. The

e�ect on performance of any ow separation, whichmay be present at NPR = 6 and 7, appeared to beless signi�cant than at NPR = 5. The comparisonof calculated with experimentally determined incre-ments in performance was within 0.002 in thrust ratiofor NPR = 6 and 7.

Nozzle pressure-drag coe�cient. The inte-grated nozzle pressure-drag coe�cient is shown in�gure 17. The trend in pressure-drag coe�cientswith NPR is closely predicted for both con�gura-tions, although the absolute level of drag is notwell matched. Predicted pressure-drag coe�cientsfor con�guration 2 are on the average 0.006 belowthe experimentally determined pressure-drag coe�-cients. The opposite condition occurred for con�g-uration 4 where the predicted pressure drag is typi-cally 0.004 above the experimentally determined dragcoe�cients for the three NPR conditions examined.The lower than predicted drag of con�guration 2 waspossibly caused by the underprediction of the owexpansion near the nozzle trailing edge. Similarly,the higher than predicted drag of con�guration 4 waspossibly caused by the overprediction of the expan-sion on the nozzle boattail shoulder. Accurate pres-sure drag numbers are sometimes di�cult to obtainfrom wind-tunnel models through the pressure-areaintegration method. The ability to predict trends,however, is very useful. The prediction of accuratelevels of performance may occur with improvementsin both testing and computational techniques.

Conclusions

A nozzle internal performance module, which ispart of the three-dimensional Navier-Stokes PAB3Dmethod, has been modi�ed to include external aero-dynamic ow e�ects. The ow quantities calcu-lated were external and internal surface static pres-sure, discharge coe�cient, ideal thrust coe�cient,thrust-minus-drag ratio, and pressure-drag coe�-cient. Comparisons of the results of the Navier-Stokes PAB3D method were made with a selection ofexperimental data from a model of an axisymmetricsupersonic cruise nozzle investigation performed inthe Langley 16-Foot Transonic Tunnel. Calculationswere performed for a single transonic Mach numberat several nozzle pressure ratio settings. Solutions foroperating conditions close to design and below designnozzle pressure ratios were obtained. The results aresummarized as follows:

1. Internal and external surface static-pressure dis-tributions were closely predicted for the selectedcon�gurations. In particular, the location of the

7

internal separation of ow from the divergent sec-tion of the high expansion ratio con�guration at

a nozzle pressure ratio of 4 was predicted closely.

2. Nozzle discharge coe�cient was predicted to

within 0.5 percent of experimental data for nozzle

pressure ratios at design and overexpanded nozzle ow conditions. The trends and the levels of ideal

thrust coe�cients matched the experimental data

well.

3. Thrust-minus-drag ratios were predicted to with-

in 0.010 in thrust ratio. The predicted incrementin thrust-minus-drag ratio between the two noz-

zles for nozzle pressure ratios of 5, 6, and 7 was

within 0.005 of the experimental data.

4. Integrated nozzle pressure-drag coe�cient trends

with nozzle pressure ratio matched fairly closely ;however, absolute levels of drag coe�cient were

not well predicted.

NASA Langley Research CenterHampton, VA 23681-0001

November 12, 1993

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Supersonic Cruise Nozzle Geometry at Mach Numbers

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7. Abdol-Hamid, Khaled S.; and Compton, William B., III:Supersonic Navier-StokesSimulations of TurbulentAfter-

body Flows. A Collection of Technical Papers|AIAA

7th Applied Aerodynamics Conference, July{Aug. 1989,pp. 268{277. (Available as AIAA-89-2194-CP.)

8. Cheatham, P. L.; Walker, S. H.; and Gridley, M. C.:Computation of Vectoring Nozzle Performance. AIAA-

90-2752, July 1990.

9. Shapiro, AscherH.: The Dynamicsand Thermodynamicsof CompressibleFluid Flow, Volume I. Ronald Press Co.,

1953.

10. Schlichting, Hermann (J. Kestin, transl.): Boundary-

Layer Theory, Seventh ed. McGraw-HillBook Co., 1979.

11. Ames Research Sta�: Equations, Tables, and Charts for

CompressibleFlow. NACA Rep. 1135, 1953. (SupersedesNACA TN 1428.)

8

5-percent thickness ratio parallel tomodel centerline; 50.80 chord

Airflow8 equally spacednozzles exitingradially

2.54 rad.

15.24

5.0845°

Balance

30 percentof chord

Sta. 0

Sta. 67.31Total-temperatureprobe (rotated45° for clarity)

Total-pressure rake

14°

40 percentof chord

30 percentof chord

High-pressureplenum Sta. 137.16.15 gap

Flexible seal(metal bellows)

AirflowScreens withsupport vanes

Seals

55.88

CL

CL

45°

Tunneldm = 15.24

128.37 rad.

Sta. 15.24

Figure 2. General arrangement of model and support system. All linear dimensions are in centimeters. (From ref. 4.)

10

1

2

3

4

5

Subsonic cruise

Supersonic cruise

Subsonic accel.

Transonic accel.

Supersonic accel.

Dry

Dry

Max. dry

Partial A/B

Max. A/B

4.25

21.23

5.03

6.23

10.64

1.250

3.000

1.350

1.500

2.000

0.250

.250

.300

.350

.420

0.312

.750

.405

.525

.840

0.500

.500

.548

.592

.648

0.286

.286

.299

.309

.319

0.800

.779

.799

.797

.789

42.35

42.35

33.58

26.90

19.58

2.12

13.18

3.18

4.78

9.70

1.000

.979

1.012

1.020

1.022

0.559

.866

.636

.725

.917

15.05

3.82

11.63

8.28

2.12

Configuration Flight

segment Powersetting NPRdes Ae/At At/Am Ae/Am dt/dm ιC/dm ιD/dm θ, deg δ, deg ι/dm de/dm β, deg

Design dimensions

Sta. 137.16

1.303

lC

1.270 rad.

lD

δ

dmx

θ

.076 (basethickness)

de

dm = 15.240

2.540 rad.

22.860 rad.

12.863

l

β

dt

Figure 4. Geometric details of test nozzle con�gurations. Absolute linear dimensions are in centimeters.

11

k

j

3

2 × 6 × 87

H - O

4

2 × 64 × 87

H - O

5

2 × 51 × 135

H - O

1H - O

2 × 6 × 492 H - O2 × 78 × 49

(a) Blocking arrangement and grid dimensions.

Figure 5. Sketch of computational mesh.

12

(b) Complete ow �eld.

(c) Close-up of nozzle ow path.

Figure 5. Concluded.

13

Grid lines

ξ3

ξ1

u

∂U1∂ξ3

Cellcenter

Figure 6. Sketch of cell-centered velocity gradient normal to surface.

14

Experiment (ref. 4)PAB3D (ref. 2)Base pressure

.4

.2

0

-.2

-.4

-.6

-.8-.2 0 .2 .4 .6 .8 1.0

Cp,β

x/dm

(a) External surface.

1.0

.8

.6

.4

.2

0-.2 0 .2 .4 .6 .8 1.0

p/pt,j

x/dm


(b) Internal surface.

Figure 7. Comparison of calculation with experimental data for surface-pressure distributions of nozzlecon�guration 2 with NPR = 4 and M1 = 0.90.

15


.4

.2

0

-.2

-.4

-.6

-.8-.2 0 .2 .4 .6 .8 1.0

Cp,β

x/dm


1.0

.8

.6

.4

.2

0-.2 0 .2 .4 .6 .8 1.0

p/pt,j

x/dm



Figure 8. Comparison of calculation with experimental data for surface-static pressure distributions of nozzlecon�guration 2 with NPR = 5 and M1 = 0.90.

16


.4

.2

0

-.2

-.4

-.6

-.8-.2 0 .2 .4 .6 .8 1.0

Cp,β

x/dm


1.0

.8

.6

.4

.2

0-.2 0 .2 .4 .6 .8 1.0

p/pt,j

x/dm




17


.4

.2

0

-.2

-.4

-.6

-.8-.2 0 .2 .4 .6 .8 1.0

Cp,β

x/dm


Experiment (ref. 4)PAB3D (ref. 2)

1.0

.8

.6

.4

.2

0-.2 0 .2 .4 .6 .8 1.0

p/pt,j

x/dm



18


.4

.2

0

-.2

-.4

-.6

-.8-.2 0 .2 .4 .6 .8 1.0

Cp,β

x/dm



1.0

.8

.6

.4

.2

0-.2 0 .2 .4 .6 .8 1.0

p/pt,j

x/dm



19


.4

.2

0

-.2

-.4

-.6

-.8-.2 0 .2 .4 .6 .8 1.0

Cp,β

x/dm



1.0

.8

.6

.4

.2

0-.2 0 .2 .4 .6 .8 1.0

p/pt,j

x/dm



20

.90

.92

.94

.96

.98

1.00

3 4 5 6 7 8

Cd

Configuration

4

2

NPR

ExperimentM = 0.90 (ref. 4)

PAB3DM = 0 (ref. 2)

Figure 13. Comparison of calculation with experimental data for nozzle discharge coe�cients.

0

1

2

3

4

5

6

2 3 4 5 6 7 8

CF,i

NPR

Configuration

4

2

Experiment(ref. 4)

PAB3D(ref. 2)

Figure 14. Comparison of calculation with experimental data for variation in nozzle ideal thrust coe�cient

with NPR with M1 = 0.90 and � = 0�.

21

.70

.75

.80

.85

.90

.95

1.00

2 3 4 5 6 7 8

NPR

Configuration

4

2

Experiment(ref. 4)

PAB3D(ref. 2)

(F - Dn)/Fi

Figure 15. Comparison of calculation with experimental data for variation in aeropropulsive performance with

NPR with M1 = 0.90 and � = 0�.

0

.05

.10

.15

.20

5 6 7

Experiment (ref. 4)

PAB3D (ref. 2)

NPR

(F - Dn)Fi C4–

(F - Dn)Fi C2

Figure 16. Comparison of calculation with experimental data for variation of increment in aeropropulsive

performance with nozzle con�guration.

22

0

.01

.02

.03

.04

.05

2 3 4 5 6 7 8NPR

ConfigurationExperiment(ref. 4)

PAB3D(ref. 2)

CD,β

42

Figure 17. Comparison of calculation with experimental data for variation of integrated nozzle pressure drag

with NPR with M1 = 0.90 and � = 0�.

23

Figure 1. Photograph showing installation of a typical nozzle in test section of Langley 16-Foot TransonicTunnel.

Figure 3. Photograph showing �ve nozzle con�gurations tested (ref. 4).

L-78-5575

L-78-5750.1

REPORT DOCUMENTATION PAGEForm Approved

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources,gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of thiscollection of information, including suggestions for reducing this burden, toWashington Headquarters Services, Directorate for Information Operations and Reports, 1215 Je�ersonDavis Highway, Suite 1204, Arlington, VA 22202-4302, and to the O�ce of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503.

1. AGENCY USE ONLY(Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

February 1994 Technical Memorandum

4. TITLE AND SUBTITLE

Computational Prediction of Isolated Performance of an Axi-

symmetric Nozzle at Mach Number 0.90

6. AUTHOR(S)

John R. Carlson

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-0001

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

5. FUNDING NUMBERS

WU 505-62-30-01

8. PERFORMING ORGANIZATION

REPORT NUMBER

L-17248

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM-4506

11. SUPPLEMENTARY NOTES

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Unclassi�ed{Unlimited

Subject Category 02

13. ABSTRACT (Maximum 200 words)

An improved ability to predict external propulsive performance has been incorporated into the three-dimensional Navier-Stokes code PAB3D. The improvements are the ability to account for skin friction andexternal pressure forces. Performance parameters for two axisymmetric supersonic cruise nozzle con�gurationswere calculated to test the improved methodology. Internal and external ow-�eld regions were computedusing a two-equation k-� turbulent viscous-stress model. The computed nozzle discharge coe�cient matchedthe experimental data within 0.5 percent in both level and trend. The calculated thrust-minus-drag ratios werewithin 1 percent of the absolute level of experimental data and the trends of data were predicted accurately. Thepredicted trend of integrated nozzle pressure drag matched the trend of the integrated experimental pressuredrag over a range of nozzle pressure ratios, but absolute drag levels were not accurately predicted.

14. SUBJECT TERMS 15. NUMBER OF PAGES

Nozzle; Navier-Stokes; Computational uid dynamics; Transonic ow; Supersonic

cruise; Isolated performance; Performance prediction25

16. PRICE CODE

A0317. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION

OF REPORT OF THIS PAGE OF ABSTRACT OF ABSTRACT

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