February 1999

NASA/TP-1999-209093

Effects of Convoluted Divergent FlapContouring on the Performance of aFixed-Geometry NonaxisymmetricExhaust Nozzle

Scott C. Asbury and Craig A. HunterLangley Research Center, Hampton, Virginia

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February 1999

NASA/TP-1999-209093

Effects of Convoluted Divergent FlapContouring on the Performance of aFixed-Geometry NonaxisymmetricExhaust Nozzle

Scott C. Asbury and Craig A. HunterLangley Research Center, Hampton, Virginia

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NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS)7121 Standard Drive 5285 Port Royal RoadHanover, MD 21076-1320 Springfield, VA 22161-2171(301) 621-0390 (703) 605-6000

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Summary

An investigation was conducted in the modelpreparation area of the Langley 16-FootTransonic Tunnel to determine the effects ofconvoluted divergent-flap contouring on theinternal performance of a fixed-geometry exhaustnozzle. Testing was conducted at staticconditions using a sub-scale, nonaxisymmetric,convergent-divergent nozzle model designed withinterchangeable divergent flap inserts. Force,moment, and pressure measurements were takenand internal focusing schlieren flow visualizationwas obtained for one baseline and four convolutedconfigurations. All tests were conducted with noexternal flow at nozzle pressure ratios from 1.25to approximately 9.50.

Results indicate that baseline nozzleperformance was dominated by unstable,shock-induced, boundary-layer separation atoverexpanded (below the design nozzle pressureratio) conditions, which came about through thenatural tendency of overexpanded exhaust flow tosatisfy conservation requirements by detachingfrom the nozzle divergent flaps. Convolutedconfigurations were found to significantlyreduce, and in some cases totally alleviate,shock-induced, boundary-layer separation atoverexpanded conditions. This result wasattributed to the ability of convoluted contouringto energize and improve the condition of thenozzle boundary layer. Separation alleviationresulted in off-design nozzle thrust ratio penaltiesthat ranged from 3.6% to 6.4% below the fullyseparated baseline configuration; thus, imposing atradeoff between separation alleviation and nozzlethrust ratio which may be acceptable in someapplications. Separation alleviation offerspotential for installed nozzle aeropropulsive(thrust-minus-drag) performance benefits byreducing drag at forward flight speeds, eventhough this may reduce nozzle thrust ratio atoff-design conditions. At on-design conditions,nozzle thrust ratio for the convolutedconfigurations ranged from 1% to 2.9% below thebaseline configuration; this was a result ofincreased skin friction and oblique shock lossesinside the nozzle.

Introduction

Supersonic cruise transport aircraft andmodern military aircraft with supersonic cruise ordash capabilities utilize variable-geometryexhaust nozzles to ensure efficient aeropropulsive(thrust-minus-drag) performance across a widespeed range. A variable-geometry nozzlefunctions by adjusting throat area and expansionratio to provide the optimum nozzle configurationfor each engine throttle setting and flightcondition. Independent throat area (At) control isnecessary to satisfy engine afterburningrequirements, and separate control of the exit area(Ae) provides the proper nozzle expansion ratio(Ae/At) at each flight condition (ref. 1). Forexample, a typical fighter aircraft might have alow nozzle pressure ratio of about 3.0 at takeoff,requiring a nozzle expansion ratio of about 1.1 foroptimum nozzle performance. During asupersonic dash to Mach 2.0, nozzle pressure ratioincreases to approximately 10.0, and a nozzleexpansion ratio of 1.9 is required for optimumnozzle performance. Figure 1 illustrates a typicalvariable-geometry nozzle at several operatingconditions.

Nozzle geometry variation is achieved usingactuators and movable nozzle flaps as shown infigure 2. While effective, these systems can beheavy, mechanically complex, and prone tofatigue through thermal, aerodynamic, andaeroacoustic loading. In addition, variable-geometry mechanisms are inherently difficult tointegrate into fighter aircraft afterbodies and canbe a primary cause of afterbody drag. Additionalrequirements such as multiaxis thrust vectoring(ref. 2), thrust reversing (ref. 3), low observability(ref. 4), and noise suppression (ref. 5) furthercomplicate the propulsion-airframe integration ofvariable-geometry nozzle systems.

The capabilities of future high performancemilitary aircraft will be critically dependent on thedevelopment of simple, lightweight exhaustsystems that are aerodynamically efficient,compact, and low observable. Supersonictransport aircraft will rely heavily on efficientnozzle performance for extended cruise at high

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supersonic speeds where the ratio of lift to drag islow and fuel consumption is high. There istremendous incentive to improve both militaryand transport aircraft performance by reducing thecomplexity of exhaust nozzles.

The desire for reduced weight and complexityin exhaust systems has led designers to considerreducing, or even eliminating, the need forvariable-geometry mechanisms in exhaustnozzles. The fundamental problem with thissolution is that a fixed-geometry nozzle will onlyoperate efficiently at the flight condition forwhich it is designed. When operated away fromthe design point (which may be common if asupersonic aircraft is expected to cruisesubsonically, loiter, or divert to alternate airports),a fixed-geometry nozzle suffers large off-designperformance penalties. For example, if the fighteraircraft mentioned previously were to operatewith a fixed-geometry, 1.9 expansion ratio nozzleat the takeoff condition, a 20-percent loss in thrustratio would result from nozzle overexpansioneffects (ref. 1). Large performance penalties suchas this would be unacceptable in mostapplications.

The successful utilization of fixed-geometrynozzles in most aircraft applications will requireimprovements in off-design performance. Athighly overexpanded conditions, exhaust flowseparation results from the natural tendency ofoverexpanded exhaust flow to satisfyconservation requirements by separating from thenozzle divergent flaps. This increases off-designperformance by allowing the nozzle to effectively“adjust” to a shorter nozzle with a lowerexpansion ratio. At forward flight speeds,however, external flow can aspirate the separatedportion of the divergent flaps, causing increaseddrag (ref. 6). In some instances, separationalleviation may be necessary to ensure efficientaeropropulsive performance, even if losses inthrust ratio result from increased exhaust flowoverexpansion. A detailed study would berequired to determine the conditions at whichseparation alleviation is beneficial to nozzleaeropropulsive performance.

Numerous research programs have shown thatthree-dimensional convoluted contouring canenhance multistream mixing and reduce subsonicboundary-layer separation in various applications(refs. 7 through 10). The objective of the researchdescribed in this report was to determine theeffects of convoluted divergent-flap contouring onthe internal performance of a fixed-geometryexhaust nozzle. Testing was conducted at staticconditions in the model preparation area of theLangley 16-Foot Transonic Tunnel using a sub-scale, nonaxisymmetric, convergent-divergentnozzle model designed with interchangeabledivergent flap inserts. The nozzle model had anexpansion ratio of 1.797 and a design nozzlepressure ratio of 8.78. Force, moment, andpressure measurements were taken and internalfocusing schlieren flow visualization wasobtained for one baseline and four convolutedconfigurations. All tests were conducted with noexternal flow and high-pressure air was used tosimulate jet-exhaust flow at nozzle pressure ratiosranging from 1.25 to approximately 9.50.

Symbols

All forces and moments are referred to themodel centerline (body axis). The model(balance) moment reference center was located atstation 29.39. A discussion of the data reductionprocedure, definitions of force and moment termsand propulsion relationships used herein can befound in reference 11. All pressures presented areabsolute unless otherwise noted.

Ae nozzle exit area, 7.758 in2

At nozzle throat area, 4.317 in2

F measured thrust along body axis,positive in forward direction, lbf

Fi ideal isentropic thrust, lbf

F/Fi nozzle thrust ratio

(F/Fi)peak peak nozzle thrust ratio

3

g acceleration due to gravity, 32.174ft/sec2

M Mach number

NPR nozzle pressure ratio, pt,j/p

a

NPRd design nozzle pressure ratio (NPR forfully expanded flow at the nozzle exit)

p local static pressure, psi

pa ambient pressure, psi

pt,j average jet-total pressure, psi

Rj gas constant (for γ=1.3997), 1716ft2/sec2-°R

Tt,j average jet-total temperature, °R

wp measured weight-flow rate, lbf/sec

x linear dimension measured along modelcenterline from nozzle connect station(Sta. 41.13), positive downstream (seefigs. 8, 9, and 13), in.

xt distance between nozzle connect station(Sta. 41.13) and nozzle throat, measuredalong model centerline, positivedownstream (see fig. 13), 2.275 in.

y vertical distance measured from modelcenterline, positive upwards (see figs. 8and 9), in.

z lateral distance measured from modelcenterline, positive to right whenlooking upstream (see figs. 8 and 13),in.

∆(F/Fi)f skin friction thrust ratio penalty

α nozzle divergence half angle, 11.01 deg

β oblique shock-wave inclination angle,measured from upstream flow direction,deg

γ ratio of specific heats, 1.3997 for air

θ angle of flow direction across anoblique shock wave, measured fromupstream flow direction, deg

Subscripts:

1 conditions just upstream of a shockwave

2 conditions just downstream of a shockwave

Abbreviations:

C-D convergent-divergent

Hz Hertz

NPAC Nozzle Performance Analysis Code

R radius, in.

Sta. model station, in.

Apparatus and Procedures

Test Facility

This investigation was conducted in the modelpreparation area of the Langley 16-FootTransonic Tunnel. Although this facility isnormally used for setup and calibration of wind-tunnel models, it can also be used for nozzleinternal performance testing at static (no externalflow) conditions. Testing is conducted in a 10 x29-foot chamber where a cold-flow (Tt,j≈540°R)jet from a single-engine propulsion simulationsystem exhausts to the atmosphere through anacoustically treated exhaust passage. A controlroom is adjacent to the test chamber, and offersaccess through a sound-proof door andobservation window. The model preparation areashares a high-pressure air system with the 16-FootTransonic Tunnel that includes valving, filters,and a heat exchanger to provide a continuous flowof clean, dry air to the propulsion simulation

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system for jet-exhaust simulation. A completedescription of the test facility is provided inreference 12.

Single-Engine Propulsion SimulationSystem

The single-engine propulsion simulationsystem used in this investigation is shown indetail in figure 3. High-pressure air supplied tothe propulsion simulation system was varied fromatmosphere up to about 140 psi total pressure inthe instrumentation section at a constantstagnation temperature of approximately 540°R.As shown in figure 3(b), the high-pressure air wasdelivered by six air lines through a support strutinto a annular high-pressure plenum. The air wasthen discharged radially into a low-pressureplenum through eight equally spaced, multiholedsonic nozzles. This flow transfer system wasdesigned to minimize any forces imposed by thetransfer of axial momentum as the air passed fromthe non-metric high-pressure plenum to the metric(attached to the balance) low-pressure plenum.Two flexible metal bellows functioned as sealsbetween the non-metric and metric portions of themodel and compensated for axial forces caused bypressurization. The air then passed through acircular-to-rectangular transition section, arectangular choke plate (primarily used for flowstraightening), a rectangular instrumentationsection, and then through the nozzle, whichexhausted to atmospheric back pressure. Theinstrumentation section had a ratio of flow pathwidth to height of 1.437 and was identical ingeometry to the nozzle airflow entrance (nozzleconnect station). All nozzle configurations testedwere attached to the downstream end ofinstrumentation section at model station 41.13.

Nozzle Concept

A fixed-geometry, nonaxisymmetric, C-Dnozzle was designed with symmetric pairs (upperand lower) of convergent and divergent flaps andflat (internally) sidewalls to contain the exhaustflow in the lateral direction. The nozzle wasbased on a previous design described in reference13. In an effort to improve off-design

performance, the nozzle divergent flap surfaceswere modified with three-dimensional convolutedcontouring.

Convoluted contouring. Numerous researchprograms have shown that three-dimensionalconvoluted contouring can enhance multistreammixing and reduce subsonic boundary-layerseparation in various applications (refs. 7 through10). The most familiar application of convolutedcontouring is the turbofan forced mixer shown infigure 4, which efficiently mixes engine-core andfan exhaust flow in mixed-flow, long-duct,turbofan nacelles by generating streamwisevorticity as shown in figure 5 (ref. 14). Inaddition, convoluted contouring has provensuccessful in alleviating boundary-layerseparation on fighter aircraft afterbodies (ref. 9)and airfoil trailing edges (ref. 7). Part of thisseparation alleviation is due to energization of theboundary layer from vorticity generated by theconvoluted contours, but research has shown thatthe contouring can also delay separation in theconvoluted section itself (refs. 7 to 10).

The "bump" type convoluted contouring usedin this investigation is depicted in figure 6. Thiscontouring generates multi-dimensional pressuregradients and inviscid secondary flows in anormally two-dimensional onset flow, providingthree-dimensional relief for the onset boundarylayer as it approaches an adverse pressuregradient. As boundary-layer flow nears theconvolutions, it receives additional freedom ofmovement in the lateral direction, which reducesthe tendency for two-dimensional separation tooccur. In addition, the convolutions generatesecondary flows in the form of horseshoe vorticesdue to the inviscid turning and stretching ofvortex filaments as they pass over the contours.These vortical secondary flows can traildownstream 5 to 10 convolution heights beforebreaking down, continuously energizing theboundary layer in that region (ref. 9).

Nozzle Models

The model used in this investigation was asub-scale, nonaxisymmetric, C-D nozzle with an

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expansion ratio Ae/At of 1.797 (NPRd=8.78), anominal throat area At of 4.317 in2, and a constantflow path width of 3.990 in. The model wascomposed of upper and lower nozzle flapassemblies, (each equipped with interchangeabledivergent flap inserts) and two sidewallassemblies (each equipped with optical qualityboro-silicate crown glass windows to permitinternal focusing schlieren flow visualization). Aphotograph, sketch, and geometric details of thenozzle model with baseline (no convolutions)divergent flap inserts installed are presented infigures 7, 8, and 9, respectively. The fourconvoluted geometries investigated consisted of afine configuration (fig. 10(a)), a mediumconfiguration (fig. 10(b)), a medium-longconfiguration (fig. 10(c)), and a coarseconfiguration (fig. 10(d)). Photographs of theconvoluted flap insert pairs and a typicalconvoluted nozzle configuration are presented infigures 11 and 12, respectively.

Design of the convoluted configurations wasbased on guidance from prior research (refs. 7 to10). Convolution length-to-height scaling wasvaried by testing aggressive contours, in whichthe convolution rose to its maximum amplitude ina short distance (0.5 in. for the fine, medium, andcoarse convoluted configurations), and a moregentle contour, in which the convolution rose toits maximum amplitude over a longer distance(0.875 in. for the medium-long convolutedconfiguration). The fine, medium, and coarseconvoluted configurations had approximately thesame wetted area, while their geometry varied inmaximum amplitude. Engineering judgment wasused to pick a maximum amplitude of 0.0998inches for the medium convoluted configuration;the fine and coarse convoluted configurations hadmaximum amplitudes that were one half andtwice that of the medium convolutedconfiguration, respectively. All convolutedcontours had parallel lobe walls, a lobe height towidth aspect ratio of 2.0, and semi-circular lobehills and valleys. Longitudinal and streamwiseconvolution profiles were composed ofsymmetric, tangent arcs such that theconvolutions rose from zero height to theirmaximum amplitude in a smooth, continuous

fashion. The length of the convoluted run was1.00 inch for the fine, medium, and coarseconvoluted configurations and 1.75 inches for themedium-long convoluted configuration.

Instrumentation

Weight-flow rate of high-pressure air suppliedto the nozzle was calculated from pressures andtemperatures measured in a calibrated multiple-critical venturi system located upstream of thepropulsion simulation system. This venturisystem is the same airflow-measurement systemused in the 16-Foot Transonic Tunnel, and is ratedto be 99.9% accurate in weight-flowmeasurements. Forces and moments weremeasured by a six-component strain-gaugebalance located on the centerline of the propulsionsimulation system. Jet total pressure wasmeasured at a fixed station in the instrumentationsection with a four-probe rake through the uppersurface and a three-probe rake through the corneras shown in figure 3(b). Two iron-constantanthermocouples in the instrumentation sectionmeasured jet total temperature.

Static pressures were measured inside thenozzle for each configuration using 0.020-inchdiameter static pressure orifices as shown infigure 13. There were six static pressure orificesin the nozzle convergent section and one orifice atthe geometric throat (fig. 13(a)), located on thenozzle centerline (z=0.000 in.). The flap insertswere equipped with a row of centerline andsideline (0.400 inches from the sidewall) pressureorifices, each containing 21 static pressureorifices spaced 0.100 inches apart. Unique to theconvoluted configurations was an row of ten staticpressure orifices in the lobe valley (z=0.1995 in.),adjacent to the centerline lobe hill, that wereadded to determine multi-dimensional effects ofthe convolutions.

Individual pressure transducers were used tomeasure pressures in the air supply system,multiple-critical venturi, instrumentation section,and nozzle convergent section. The transducerswere selected and sized to allow the highestaccuracy over each required measurement range.

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Divergent flap pressures were measured by twoelectronically scanning pressure modules locatedin the model preparation area test chamber in anacoustically shielded cabinet.

Data Reduction

Each data point is the average steady-statevalue computed from 50 frames of data taken at arate of 10 frames per second. All data were takenwith ascending NPR. A detailed description ofthe procedures used for data reduction in thisinvestigation can be found in reference 11.

Balance corrections. Each of the sixmeasured balance components were initiallycorrected for model weight tares and isolatedbalance component interactions. Although thebellows arrangement in the air pressurizationsystem was designed to minimize forces on thebalance caused by pressurization, small bellowstares on the six-component balance still existed.These tares resulted from small pressuredifferences between the ends of the bellows whenair system internal velocities were high and fromsmall differences in the spring constant of theforward and aft bellows when the bellows werepressurized. Bellows tares were determined bytesting Stratford choke calibration nozzles withknown performance over a range of expectedinternal pressure and external forces andmoments. The resulting tares were then appliedto the six-component balance data to obtaincorrected balance measurements. Balance axialforce obtained in this manner is a directmeasurement of the nozzle thrust along the bodyaxis, F . The procedure used for computingbellows tares is discussed in detail in reference15.

Calculations. Jet total pressure was measuredfrom four center rake and three corner rake totalpressure probes located in the instrumentationsection. Nozzle pressure ratio (NPR) is theaverage jet total pressure pt,j measured in theinstrumentation section divided by ambientpressure pa; NPR was varied in this investigationfrom 1.25 to approximately 9.50. Jet totaltemperature Tt,j was obtained from two total

temperature probes located in the instrumentationsection. The average jet total pressure and jettotal temperature are computed as the arithmeticmean of the individual measurements.

Nozzle thrust ratio F/Fi is the ratio ofmeasured thrust along the body axis F to thecomputed ideal isentropic thrust F. The measuredweight-flow rate wp, which is determined by usinga multiple-critical venturi system, is used todetermine ideal isentropic thrust from thefollowing equation:

i pj t, j

F wR T

g=

−−

−( )2

12

11

1γγ

γ γ

NPR

Uncertainty Analysis

An uncertainty analysis of the resultspresented was performed based on a propagationof bias uncertainties of actual measurementsthrough the data reduction equations. Thisanalysis assumes that bias errors are dominantover precision errors and is based on the methodpresented in reference 16. This method uses thefirst order terms in a Taylor series expansion ofthe data reduction equations to estimate theuncertainty contributions of each measurement.With this technique, the contribution of eachmeasurement would be the measurementuncertainty multiplied by the derivative ofthe data reduction equation with respect tothat measurement. The total uncertainty of thefinal calculated result is estimated as theroot-sum-square of the individual contributionswith 95-percent confidence.

The analysis accounted for the uncertainties ofthe following measurements: jet total pressure, jettotal temperature, atmospheric pressure, venturiweight-flow rate, and balance axial force. Theanalysis also accounted for the beneficial effect ofaveraging multiple measurements of the samequantity, such as the total pressure in theinstrumentation section. This type of analysis istypical of that used for experimental static testprograms and is credited to the work presented inreference 17.

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The results of the analysis for the range of testconditions indicate that the uncertainty of NPRand p/pt,j is approximately ±0.28 percent ofmeasured value. The uncertainty of F/Fi isapproximately ±0.004 and is essentiallyindependent of NPR.

Focusing Schlieren Flow Visualization

A focusing schlieren flow visualization systemwas used during this investigation to visualize thenozzle internal (through glass sidewalls) andexternal exhaust flowfield. An optical descriptionand schematic layout of the focusing schlierensystem are presented in figure 14. The systemwas designed and built based on criterion reportedin reference 18. The system is characterized by a133 mm diameter field of view, a sensitivity of 17arcsec, a resolution of 0.25 mm, a depth of sharpfocus of 4.6 mm, and a depth of unsharp focus of36 mm. The image was focused on the centerlineof the nozzle.

The light source for the focusing schlierensystem was a xenon strobe flash tube. A drivingcircuit picked up sync pulses generated by therecording video camera and triggered the flash ata 30 Hz rate with pulses of 0.6 µsec duration and0.05 watt-sec power. A 720 x 480 pixelresolution color video camera and a 70 mm stillcamera recorded results.

The focusing schlieren system was assembledon a 44 x 66 inch table that mounted on a rigidplatform equipped with casters and levelingscrews. The platform was placed under thepropulsion simulation system and jacked andleveled to the appropriate position. Flowvisualization data were recorded simultaneouslywith other data acquisition.

Presentation of Results

Nozzle thrust ratio F/Fi and internal staticpressure ratio p/pt,j data for all nozzleconfigurations tested are tabulated in table 1 andtables 2 to 6, respectively. During the discussionof results, comparisons of nozzle thrust ratio F/Fi

are made in terms of percentage change fromideal (F/Fi=1) isentropic conditions. Graphicalpresentation of basic and summary data arepresented in figures 15 to 30.

Results and Discussion

On-Design Performance

Baseline configuration. Nozzle thrust ratioF/Fi performance for the baseline configuration ispresented as a function of nozzle pressure ratio(NPR) in figure 15. Peak thrust ratio (F/Fi)peak forthe baseline configuration is approximately 0.986at the on-design condition (NPRd=8.78), which iswithin the 0.985 to 0.990 range consistent withprevious studies of nonaxisymmetric convergent-divergent nozzles (refs. 19 to 21). Theapproximate 1.4% loss in peak thrust ratio fromideal isentropic conditions at NPRd can beattributed to exit flow angularity effects andfriction drag inside the nozzle (ref. 22).

Convoluted configurations. When an exhaustnozzle is operating at the on-design condition, it isinternally shock free, the flow is fully expanded,and peak thrust efficiency is produced. Therefore,the presence of convoluted contours in thedivergent section of the nozzle would likely resultin on-design performance penalties. Becauseconvolutions would probably be present at alloperating conditions, on-design performancepenalties associated with the convoluted geometrymust be minimized to ensure that the benefits ofhaving the convolutions at off-design conditionsare not outweighed by any on-design performancepenalties.

Nozzle thrust ratio performance for baselineand convoluted configurations is presented as afunction of NPR in figure 16. All convolutedconfigurations had (F/Fi)peak at NPRd that werelower than the baseline value. This result issummarized in table 7. Losses in (F/Fi)peak due tothe convolutions were 1% or less for the fine,medium, and medium-long convolutedconfigurations. The coarse convolutedconfiguration had a significantly larger 2.9% lossin (F/Fi)peak as a result of its more aggressive

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convoluted geometry.

The convoluted configurations had asignificant increase in wetted area over that of thebaseline configuration. Consequently, increasedskin friction losses were expected to impart athrust ratio penalty. After calculating wetted areafor each configuration, skin friction drag penalties∆(F/Fi)f were estimated and (F/Fi)peak values werepredicted using the baseline nozzle pressuregradient as input to the nozzle internalperformance prediction package NPAC describedin reference 22. The results are summarized intable 8.

The increase in wetted area was 15% for thefine, medium, and coarse convolutedconfigurations and 26% for the medium-longconvoluted configuration. Using NPAC, a skinfriction drag penalty of 0.2% was estimated andpeak thrust ratio of 0.984 was predicted for thecoarse, medium, and fine convolutedconfigurations. A slightly higher skin frictiondrag penalty of 0.3% and slightly lower peakthrust ratio of 0.983 was predicted for themedium-long convoluted configuration. Acomparison of NPAC predicted andexperimentally measured peak thrust ratios forbaseline and convoluted configurations ispresented in figure 17. The fine and medium-longconvoluted configurations had the highest thrustratio performance of the convolutedconfigurations, which at (F/Fi)peak=0.980, wasonly 0.6% lower than the baseline value. Themedium and coarse convoluted configurations hadpeak thrust ratio performance that was 1.0% and3.1% lower than the baseline value, respectively.Note that the medium-long convolutedconfiguration, with 11% more wetted area thanthe other convoluted configurations, had a higher(F/Fi)peak than the medium convolutedconfiguration. This result suggests that on-designthrust ratio penalt ies for convolutedconfigurations are only partially attributable toincreased wetted area; e.g., they are related tosome other phenomena as well.

Flow visualization at NPR=8.9, presented forbaseline and convoluted configurations in figure

18, shows that the convoluted configurationsgenerated intense supersonic wave radiation thatcoalesced into oblique shocks at certain points inthe nozzle. The presence of oblique shocksreduces jet momentum in the nozzle, whichexplains the additional peak thrust ratio penaltiesimposed by the convolutions. The medium-longconvoluted configuration (fig. 18(d)) appears tohave generated less supersonic wave radiationthan the other convoluted configurations,explaining the aforementioned lower losses in(F/Fi)peak for this configuration.

A comparison of baseline and convolutedinternal static pressure ratio distributions (plottedagainst nondimensionalized streamwise locationrelative to the nozzle throat x/xt) is presented infigure 19 at NPR=8.9. In each convolutedconfiguration, pressures upstream and down-stream of the convolution run were not greatlyaffected by the convolutions. Flow over the fine(fig. 19(a)) and medium-long (fig. 19(c))convolution runs was characterized by weakcompressions (locally increasing p/pt,j) at theleading and trailing edges of the convolution hilland a stronger compression midway in theconvolution valley. Similar behavior occurred forthe medium (fig. 19(b)) and coarse (fig. 19(d))convoluted configurations except that compres-sions were much stronger and were separated byregions of rapid expansion (locally decreasingp/pt,j) for these configurations. This behavior mayhelp explain why the medium convolutedconfiguration had stronger supersonic waveradiation and a lower (F/Fi)peak than the medium-long convoluted configuration. Like the coarseconvoluted configuration, the medium convolutedconfiguration presented a much more aggressivestreamwise convolution geometry than themedium-long or fine convoluted configurations,such that the flow had to turn more abruptlythrough the convolutions and a strongercompression/expansion mechanism wasnecessary. The convolution hill for the coarseconvoluted configuration was aggressive enoughto generate a strong shock (noted by a rapidcompression) at the start of the convolution run,which explains the severe drop in (F/F i)peak

measured for that configuration.

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Off-Design Performance

Baseline configuration. As shown in figure15, nozzle thrust ratio decreased as NPRdecreased below NPRd; a result of exhaust flowoverexpansion effects. Internal static pressureratio distributions for the baseline configuration,presented in figure 20, are typical of convergent-divergent nozzle flow characteristics (ref. 23).For centerline pressures (z=0.00 in.), the first twocurves at a NPR of 1.26 and 1.4 indicate choked(p/pt,j≤0.528), internally overexpanded flow witha weak shock (noted by the significant increase inp/pt,j with x/xt) present near the nozzle geometricthroat (x/xt=1.00). Flow downstream of the shockwas subsonic (p/pt,j>0.528), remained attached tothe divergent flap wall, and recovered to ambientpressure (p/pt,j=1/NPR) in a smooth, continuousfashion. Flow visualization for the baselineconfiguration is shown in figure 21. At NPR=1.4(fig. 21(a)), there was a weak, almost normalshock downstream of the throat with little or nolambda foot structure evident. This behavior ischaracteristic of a weak shock, with a flow Machnumber of approximately 1.2 just upstream of theshock (M1), and a thin boundary layer inside thenozzle. Flow Mach number inside the nozzle wasestimated from p/pt,j values using tables forcompressible flow in reference 24.

As shown in figure 20, the discontinuousnature of the centerline pressure distribution atNPR=1.6 indicates that shock strength increased(M1≈1.4), and the inflection point in the pressurerecovery downstream of the shock at x/xt≈1.28indicates that flow separation occurred on thedivergent flaps, though it was not severe. Thepressure distribution also indicates that the flowbecame subsonic downstream of x/xt≈1.55 andflow reattachment to the flap is indicated by thesmooth pressure recovery downstream of thispoint. By NPR=1.8, the upstream shock Machnumber was M1≈1.5 and shock-induced,boundary-layer separation began to dominatenozzle flow characteristics. At NPR=1.8, thereare strong signs of a separation bubble, withminimal pressure recovery indicated by arelatively flat pressure distribution from the shocklocation at x/xt≈1.35 out to x/xt≈1.7; however, full

recovery to ambient pressure occurred over theremaining length of the nozzle. Flowvisualization at NPR=1.8 in figure 21(b) showsthe shock with a small lambda foot structure. Theflow was also highly unstable; this phenomenawas observed in the schlieren video recordedduring the test and is indicated by the schlierenphotograph, which captured an image of theshock in two positions over a 0.6 µsec duration.Because the image was focused on the centerlineof the nozzle with a depth of sharp focus of 4.6mm, the dual-shock nature of this photo shouldnot be attributed to an alignment problem.

An increase in pressure ratio to NPR=2.0 didnot significantly change shock location orstrength, but did result in fully detached shock-induced separation with almost no pressurerecovery downstream of the shock (fig. 20). Flowvisualization at NPR=2.0 in figure 21(c) showsthe shock with a pronounced lambda footstructure and a large separation region extendingfrom the leading lambda foot downstream past thenozzle exit. The results discussed above indicatethat the nozzle flow adjusted to exit conditions atNPR=2.0 simply by detaching from the divergentflaps, while normalized pressure (and thus Machnumber) upstream of the shock matched those ofthe previous NPR. This behavior indicates thatthe onset of fully detached flow separation atNPR=2.0 was not the result of a stronger shock-boundary layer interaction, but instead cameabout through the natural tendency ofoverexpanded exhaust flow in a fixed-geometrynozzle to conserve mass, momentum, and energyby detaching from the divergent flaps and"adjusting" to an effectively shorter nozzle with alower expansion ratio.

As shown in figure 15, the onset of fully-detached, shock-induced, boundary-layer separ-ation at NPR=2.0 corresponds to a markedincrease in nozzle thrust ratio. By providing aneffectively lower nozzle expansion ratio, internalflow separation reduced overexpansion losses inthe nozzle and increased nozzle thrust ratio. Itshould be noted that this beneficial effect may notexist at forward speeds where external flow couldaspirate the separated portion of the divergent

10

flaps, causing increased drag. As a result, theability to alleviate separation inside a fixed-geometry nozzle may be beneficial to overallaeropropulsive performance at forward speeds,even if small losses in nozzle thrust ratio occur asa result of the separation alleviation process. Theinformation required to make the tradeoffbetween allowing separation to occur oralleviating separation is beyond the scope of thisinvestigation.

As shown in figure 20, fully-detached flowseparation occurred for all subsequent internallyoverexpanded NPRs above 2.0. As NPR wasincreased beyond 2.0, the leading lambda footprogressed downstream in the nozzle. Figure21(d) shows the shock at NPR=2.4 with a welldefined lambda foot structure and fully detachedflow separation. By NPR=3.4 (fig. 21(e)), thelambda foot structure had grown significantly,such that the main shock and trailing lambda footwere outside the nozzle. At this NPR, flow insidethe nozzle past the separation point showed strongresemblance to externally overexpanded exhaustflow; the jet plume necked down from theseparation point at the leading lambda foot to thetrailing lambda foot, and there was an expansionfan emanating from each trailing lambda foot.This behavior indicates that the separation pointwas behaving as if it were at the nozzle exit, andflow past this point was externally overexpanded.Static pressure ratio distributions in figure 20indicate that the shock was positioned near thenozzle exit by NPR=5.0 and that the nozzle wasshock free by NPR=5.4. At NPR≥5.4, allpressure distributions fell on the same curve,indicating that nozzle internal flow characteristicswere independent of NPR beyond that point.

A comparison of sideline (z=1.595 in.) tocenterline (z=0.000 in.) internal static pressureratio distributions in figure 20 indicates noticeabledifferences below NPR=2.4. Differences betweensideline and centerline pressure distributions inboth shock location and pressure recovery past theshock indicate that flow inside the nozzle wasthree-dimensional and that the shock was non-planar. Sideline data at NPR of 1.26 and 1.4show fully-detached flow separation and a shock

location upstream of its centerline position.Sideline pressures near the nozzle exit are lowerthan centerline pressures, indicating that sidelineflow was recompressing downstream of thenozzle exit.

Convoluted configurations. Individualcomparisons of nozzle thrust ratio betweenbaseline and convoluted configurations arepresented in figure 22. At very low NPRs, theconvoluted configurations exhibited performancecharacteristics that were similar to the baselineconfiguration. Static pressure ratio distributionspresented for each convoluted configuration infigure 23 show overexpanded and separated flowat NPRs up to 2.0 in the fine convolutedconfiguration (fig. 23(a)), 1.8 in the mediumconvoluted configuration (fig. 23(b)), and 1.6 inthe medium-long (fig. 23(c)) and coarseconvoluted configurations (fig. 23(d)). Flowvisualization at these NPRs, presented in figure24, show various degrees of separation for eachconfiguration. For all NPRs equal to or lowerthan the above noted values, figure 23 shows theshock positioned upstream of the convolution runand similar hill and valley pressure distributions,indicating that a multi-dimensional pressuregradient was not generated across the convolutionrun at these low NPR values.

When nozzle pressure ratio was increasedabove these low NPR values, figure 22 shows thatthere was a dramatic drop in F/Fi for each of theconvoluted configurations. The drop in F/Fi

occurred for the fine convoluted configuration(fig. 22(a)) at a NPR between 2.0 and 2.2, whenthe shock jumped from its position upstream ofthe convolution run at x/xt≈1.2 to a positionmidway through the convolution run at x/xt≈1.6(fig. 23(a)). Flow visualization for the fineconvoluted configuration in figure 25 shows anearly normal shock at NPR=2.2 (fig. 25(a)) witha lambda foot structure significantly smaller thanthat of the baseline configuration (fig. 21(c)) at asimilar NPR and shock location. With this "shockjump", flow over the convolution run wassupersonic, a multi-dimensional pressure gradientwas generated across the convolution run, andflow separation was almost completely alleviated.

11

As a result, nozzle internal flow could no longeradjust to exit conditions by detaching from thedivergent flaps; thus, exhaust flow overexpansionlosses increased (F/Fi decreased).

Above the "shock jump" NPR, F/Fi for thefine convoluted configuration increasedcontinuously as overexpansion losses decreased(fig. 22(a)). Static pressure ratio distributions forthe fine convoluted configuration in figure 23(a)indicate that the shock moved smoothlydownstream with each subsequent increase inNPR. At each shock position, the pressure risethrough the shock was gradual, the flow generallyremained attached (local areas of separated floware evident), and there was good pressurerecovery downstream of the shock. A comparisonof flow visualization for the fine convolutedconfiguration at NPRs of 2.2 and 2.6 in figure 25shows a larger lambda foot structure and a moreturbulent attached region downstream of theshock at the higher NPR value. As indicated infigure 23(a), the fine convoluted configurationwas free of internal shocks at NPR>5.0. Lossesin F/Fi due to the fine convolutions were as largeas 6% at NPR=2.2, but decreased to less than 1%at NPR≥5.0 (see figure 22(a)).

The "shock jump" for the medium convolutedconfiguration occurred between NPRs of 1.8 and2.0. The centerline static pressure distribution forthe medium convoluted configuration (fig. 23(b))at NPR=2.0 shows a gradual pressure rise throughthe shock and good downstream pressurerecovery. Flow visualization for the mediumconvoluted configuration at NPR=2.0 (fig. 26(a))shows turbulent, attached flow downstream of theshock, indicating that this configuration alsoprovided good separation alleviation. However,F/Fi for the medium convoluted configuration(fig. 22(b)) was not continuous above the "shockjump" NPR (note discontinuity in F/Fi at2.5<NPR<3.0). Static pressure ratio distributionsfor the medium convoluted configuration (fig.23(b)) show that the main shock merged with thestrong convolution valley oblique shock at a NPRbetween 2.0 and 2.6. A comparison of flowvisualization for the medium convolutedconfiguration at NPRs of 2.0 and 2.6 in figure 26

indicates that the leading lambda foot remained inthe same location while the main shock moveddownstream and the lambda foot grew withincreasing NPR. As this occurred, flow remainedattached through the shock, and downstreampressure recovery was good.

As NPR was increased to 3.0, the shock onceagain jumped downstream, this time from x/xt≈1.6to a position near the nozzle exit at x/xt≈1.9. Asfor the previous shock jump, there is acorresponding shift in nozzle thrust ratio (fig.22(b)), though in this case the shift is more likelydue to a loss effect than additional separationalleviation. As shown in figure 26(c), when theshock jumped downstream at NPR=3.0, it"uncovered" the valley oblique shock at thetrailing edge of the convolution run such that losseffects of that shock could affect nozzle thrustratio. Beyond NPR=3.0, the shock movedsmoothly out of the nozzle, and the nozzle wasshock free at NPR>4.6 (fig. 23(b)). The losses inF/Fi due to the medium convolutions were aslarge as 6.5% at NPR=2.0, but decreased toapproximately 1% at NPR>4.6 (fig. 22(b)).

The "shock jump" for the medium-longconvoluted configuration occurred between aNPR of 1.6 and 1.8 as indicated by the staticpressure ratio distributions in figure 23(c). Thecenterline static pressure distribution at NPR=1.8also indicates a double shock (one at x/xt≈1.45and one at x/xt≈1.6) with a small separationbubble in between. Flow visualization for themedium-long convoluted configuration atNPR=1.8 (fig. 27(a)) shows the double shockwith attached flow downstream of the secondshock. Unlike the medium convolutedconfiguration, F/Fi for the medium-longconvoluted configuration at NPR>2.0 wascontinuous (fig. 22(c)), and pressure distributions(fig. 23(c)) indicate that the shock movedsmoothly downstream with each subsequentincrease in NPR. At each shock position, thepressure rise through the shock was gradual, theinternal flow remained attached, and there wasstrong pressure recovery downstream of theshock. Flow visualization for the medium-longconvoluted configuration at NPR=2.6 (fig. 27(b))

12

shows a larger lambda foot structure and a moreturbulent attached region downstream of the mainshock than at the lower NPR. Static pressure ratiodistributions (fig. 23(c)) indicate that the medium-long configuration was shock free at NPR>4.2.

As shown in figure 23(d), the "shock jump"for the coarse convoluted configuration occurredbetween a NPR of 1.6 and 1.8. For thisconfiguration, the shock moved downstream afterthe initial "shock jump", but then merged with thevalley oblique shock and remained in that positionat x/xt≈1.6 for NPRs between 2.2 and 3.8. Asecond jump in the shock position (x/xt≈1.6 tox/xt≈1.75) for the coarse convoluted configurationis evident in figure 23(d) at NPR=4.2, when theshock separated from the convolution valleyoblique shock. This is observed in flowvisualization photographs for the coarseconvoluted configuration at NPRs of 3.8 and 4.2in figure 28, which shows the shock structurefurther downstream at the higher NPR value. Aswas the case with the medium convolutedconfiguration, this jump coincided with a decreasein nozzle thrust ratio (note the change in slope ofF/Fi at NPR≈4.0 in fig. 22(d)) when the valleyoblique shock was uncovered and the loss effectsof that shock were added. At NPR>4.2, the shockmoved smoothly out of the nozzle, and the nozzlewas shock free for NPR>5.4 as indicated by thepressure distributions of figure 23(d). The lossesin F/Fi due to the coarse convolutions (fig. 22(d))were as large as 8% at NPR=2.4, but decreased toapproximately 2.6% near NPRd.

The medium-long convoluted configurationprovided the best combination of separationalleviation and continuous downstream shockmovement (no shock jumps past the initial jump).Separation alleviation began at NPR=1.8 for thisconfiguration, and the nozzle was shock free forNPR>4.2, which was a lower NPR than for any ofthe other convoluted configurations. The lossesin F/F i due to the medium-long convolutedgeometry (fig. 22(c)) were as large as 7% atNPR=2.4, but decreased to less than 1% atNPR>5.0. At on-design conditions, the medium-long convoluted configuration suffered only a0.6% loss in (F/Fi)peak. This may be an acceptable

on-design nozzle thrust ratio penalty given theexcellent off-design flow separation alleviationcapabilities of this configuration (assuming thatexternal flow effects would cause additional dragwhen internal nozzle flow was separated).

Shock-Boundary Layer Interaction

Baseline configuration. Flow visualization atNPR=3.0 for the baseline configuration in figure29 shows the shock with a large, well definedlambda foot structure and fully-detached flowseparation beginning at the leading branch of thelambda foot and extending downstream. Shockangle measurements were made from figure 29,and were used in conjunction with oblique shocktheory (ref. 24) in an effort to better describe theshock-boundary layer interaction as shown infigure 30.

Upstream of the leading lambda foot, flow wasassumed to be locally parallel to the nozzledivergent flap and M1 was calculated from p/pt,j

values to be approximately 1.8. Flow deceleratedacross the leading lambda foot which, with aninclination angle β of approximately 52° from thenozzle divergent flap, resulted in a downstreamMach number M2 of approximately 1.2. Usingoblique shock theory, the flow turning angle θacross the leading lambda foot was calculated tobe 15°. The leading lambda foot possessed theseverity of a normal shock and was strong enoughto completely detach nozzle flow from thedivergent flaps. From the new flow direction, thetrailing branch of the lambda foot had aninclination angle of β ≈63°. For M≈1.2approaching this shock, the corresponding flowturning angle of the trailing lambda foot wascalculated to be θ≈3°. This satisfied flow turningrequirements of the fully detached flow separationregion and resulted in nearly axial flowdownstream of the trailing lambda foot.

In this shock-boundary layer interaction, it isapparent that the nozzle flap was steep enoughand the remaining length of the nozzle past theflow separation point was short enough thatreattachment would not occur, since the givenshock structure resulted in nearly axial flow in the

13

nozzle. As a result, the free shear layer generatedin the flow separation process became the actualexit shear layer of the nozzle and the flowseparation point behaved as if it were the nozzleexit.

Convoluted configurations. Each convolutedconfiguration had dramatically different shock-boundary layer interaction characteristics than thebaseline configuration. As indicated in figure 20,static pressure ratio distributions for the baselineconfiguration at NPR>1.41 were characterized bya single, sharp compression with subsequentshock-induced, boundary-layer separation. Thisbehavior is illustrated by flow visualization infigure 21, which shows the baseline configurationwith a large, well defined lambda foot structure,and fully detached flow downstream of theleading lambda foot.

Static pressure ratio distributions presented infigure 23 for the convoluted configurationsindicate that the convoluted contouringsignificantly reduced, and in some cases totallyalleviated, shock-induced boundary-layerseparation. Pressure distributions indicate that amulti-dimensional pressure gradient was formedacross the convolution run at all but the lowestNPRs. It is likely that the generation ofstreamwise vorticity energized the nozzleboundary layer upstream of the shock-boundarylayer interaction region. The energized boundarylayer was able to negotiate the severe adversepressure gradient of the shock, therebyminimizing shock-induced boundary-layerinteraction effects and alleviating flow separation.

Each convoluted configuration had a distinctshock-boundary layer interaction mechanism,undoubtedly due to the different contouring ineach case. The fine convoluted configuration hadthe least aggressive contouring, resulting inshock-boundary layer interaction characteristicsclosest to the baseline nozzle. (Compare figures20 and 23(a).) However, flow visualization forthe fine convoluted configuration (fig. 25(a))shows a noticeably smaller lambda foot structurethan the baseline configuration and static pressureratio distributions in figure 23(a) indicate that the

fine convoluted contouring alleviated separationfor all NPRs past the shock jump. The mediumand medium-long convoluted configurations weredesigned with common spanwise profiles, but itappears that the slightly less aggressivestreamwise run in the medium-long convolutedconfiguration provided the best off-designseparation alleviation over the widest range ofNPR with the lowest peak thrust ratio penalty.The coarse convoluted configuration alsoimproved the shock-boundary layer interaction,but this configuration experienced substantiallosses in peak thrust ratio.

Conclusions

An investigation was conducted in the modelpreparation area of the Langley 16-FootTransonic Tunnel to determine the effects ofconvoluted divergent-flap contouring on theinternal performance (nozzle thrust ratio) of afixed-geometry exhaust nozzle. Testing wasconducted at static conditions using a sub-scale,nonaxisymmetric, convergent-divergent nozzlemodel designed with interchangeable divergentflap surfaces. Force, moment, and pressuremeasurements were taken and internal focusingschlieren flow visualization was obtained for onebaseline (no convolutions) and four convolutedconfigurations. All tests were conducted with noexternal flow and nozzle pressure ratio was variedduring jet simulation from 1.25 to approximately9.50. The results of this investigation indicate thefollowing conclusions:

1 . Convoluted configurations were found tosignificantly reduce, and in some cases totallyalleviate, shock-induced, boundary-layerseparation at off-design conditions. Thisindicates that the convoluted contouringenergized and improved the condition of thenozzle boundary layer such that the boundarylayer was able to resist the natural separationtendency of the exhaust flow. This did,however, result in off-design nozzle thrustratio penalties that ranged from 3.6% to 6.4%below the fully separated baselineconfiguration, thus imposing a tradeoffbetween separation alleviation and nozzle

14

thrust ratio which may be acceptable in someapplications.

2. Of the four convoluted configurations tested,the medium-long convoluted configurationprovided the best combination of off-designseparation alleviation and continuous down-stream shock movement. Separationalleviation began at NPR=1.8 in thisconfiguration, and the nozzle was shock freeat NPR>4.2, earlier than any of the otherconvoluted configurations tested. Even with26% more internal wetted area than thebaseline configuration, the medium-longconvoluted configuration had a peak thrustratio of 0.980, only 0.6% below the baselinevalue.

3 . At on-design conditions, nozzle thrust ratiofor the convoluted configurations ranged from1% to 2.9% below the baseline configuration.This was a result of the convolutionsincreasing skin friction and oblique shocklosses inside the nozzle.

NASA Langley Research CenterHampton, Virginia 23681-0001January 4, 1999

References

1. Stitt, Leonard E.: Exhaust Nozzles for PropulsionSystems With Emphasis on Supersonic CruiseAircraft. NASA RP-1235, 1990.

2. Berrier, Bobby L.; and Re, Richard, J.: A Reviewof Thrust Vectoring Schemes for Fighter Aircraft.AIAA-78-1023, July 1978.

3. Yetter, Jeffrey A.: Why Do Airlines Want andUse Thrust Reversers? NASA TM-109158, 1995.

4. Asbury, Scott C.; and Carlson, John R.: TransonicAeropropulsive Performance of AdvancedExhaust Nozzles Designed for Reduced RadarCross-Section Signatures. NASA TP-3505, 1995.

5. Gilkey, S. C.; Hines, R.W.: A Joint PropulsionPerspective of the Next Generation SupersonicTransport. AIAA-91-3330, September 1991.

6 . Berrier, Bobby L.; and Re, Richard J.:Investigation of Convergent-Divergent NozzleApplicable to Reduced-Power Supersonic CruiseAircraft. NASA TP-1766, 1980.

7. Presz, W. M., Jr.; Werle, M.; and Paterson, R. W.:Trailing Edge Separation/Stall Alleviation.Journal of Propulsion and Power, vol. 25, 1987.

8. Presz, W. M., Jr.; Morin, B.; and Gousy, R.: ShortEfficient Ejector Systems. AIAA-87-1837, June1987.

9. Presz, Walter M., Jr.; Russell, William D.; andTongue, Steve E.: Rippled AfterbodyPerformance Study. UTRC87-27, September1987.

10. Skebe, S. A.; Paterson, R. W.; and Barber, T. J.:Experimental Investigation of Three-DimensionalForced Mixer Lobe Flow Fields. AIAA-88-3785,July 1988.

11. Mercer, Charles E.; Berrier, Bobby L.; Capone,Francis J.; and Grayston, Alan M.: DataReduction Formulas for the 16-Foot TransonicTunnel–NASA Langley Research Center.Revision 2. NASA TM-107646, 1992.

12. Staff of the Propulsion Aerodynamics Branch: AUser's Guide to the Langley 16-Foot TransonicTunnel Complex, Revision 1. NASA TM-102750,1990. (Supersedes NASA TM-83186, compiledby Kathryn H. Peddrew, 1981.)

13. Berrier, Bobby L.; and Re, Richard J.: Effects ofSeveral Geometric Parameters on the StaticInternal Performance of Three NonaxisymmetricNozzle Concepts. NASA TP-1468, 1979.

14. Paterson, R.W.: Turbofan Mixer Nozzle FlowField - A Benchmark Experimental Study. Journalof Engineering for Gas Turbines and Power, July1984.

15. Capone, Francis J.: Static Performance of FiveTwin-Engine Nonaxisymmetric Nozzles withVectoring and Reversing Capability. NASA TP-1224, 1978.

16. Coleman, Hugh W.; and Steele, W. Glenn, Jr.:Experimentation and Uncertainty Analysis forEngineers. John Wiley & Sons, 1989.

15

17. Wing, David J.; Mills, Charles T.L.; and Mason,Mary L.: Static Investigation of a MultiaxisThrust-Vectoring Nozzle With Variable InternalContouring Ability. NASA TP-3628, 1997.

18. Weinstein, L.M.: An Improved Large-FieldFocusing Schlieren System. AIAA-91-0567,January 1991.

19. Capone, F.J.; Konarski, M.; Stevens, H.L.; andWillard, C.M.: Static Performance ofVectoring/Reversing Non-Axisymmetric Nozzles.AIAA-77-840, July 1977.

20. Re, Richard J.; and Leavitt, Laurence D.: StaticInternal Performance Including Thrust Vectoringand Thrust Reversing of Two-DimensionalConvergent-Divergent Nozzles. NASA TP-2253,1984.

21. Taylor, John G.: Static Investigation of a Two-Dimensional Convergent Divergent ExhaustNozzle With Multiaxis Thrust VectoringCapability. NASA TP-2973, 1990.

22. Hunter, Craig A.: An Approximate TheoreticalMethod for Modeling the Static ThrustPerformance of Nonaxisymmetric Two-Dimensional Convergent Divergent Nozzles.NASA CR-195050, 1995.

23. Liepmann, Hans Wolfgang; and Roshko, Anatol:Elements of Gasdynamics. John Wiley & Sons,Inc., 1957.

24. Staff of NASA Ames Research Center:Equations, Tables, and Charts for CompressibleFlow. NASA TR-1135, 1953.

NP

R

F/F

i

1.26

0.86

9

1.40

0.83

2

1.61

0.83

7

1.81

0.83

0

2.01

0.86

5

2.21

0.88

2

2.41

0.90

4

2.61

0.91

3

3.01

0.92

4

3.41

0.93

5

3.82

0.94

4

4.22

0.95

1

4.62

0.95

9

5.02

0.96

5

5.42

0.97

1

5.82

0.97

5

6.23

0.97

9

7.03

0.98

3

8.04

0.98

6

8.94

0.98

6

9.54

0.98

6

Bas

elin

eC

onfig

urat

ion

NP

R

F/F

i

1.25

0.84

9

1.40

0.81

9

1.60

0.81

6

1.80

0.84

5

2.00

0.85

1

2.20

0.82

5

2.40

0.84

5

2.60

0.86

1

3.00

0.88

3

3.40

0.90

5

3.80

0.92

1

4.20

0.93

7

4.60

0.94

8

5.00

0.95

7

5.40

0.96

3

6.20

0.97

2

7.00

0.97

6

8.00

0.97

9

8.91

0.98

0

9.50

0.98

0

Fin

eC

onvo

lute

dC

onfig

urat

ion

NP

R

F/F

i

1.24

0.84

9

1.39

0.82

0

1.59

0.82

2

1.79

0.81

0

1.99

0.80

2

2.19

0.82

3

2.39

0.84

6

2.59

0.86

7

2.99

0.86

9

3.39

0.89

8

3.79

0.91

8

4.19

0.93

3

4.59

0.94

5

4.99

0.95

3

5.39

0.96

0

6.19

0.96

8

6.99

0.97

2

7.99

0.97

5

8.89

0.97

6

9.50

0.97

6

Med

ium

Con

volu

ted

Con

figur

atio

n

NP

R

F/F

i

1.25

0.85

5

1.40

0.81

7

1.60

0.83

6

1.80

0.79

4

2.00

0.79

8

2.20

0.81

4

2.40

0.83

6

2.60

0.85

5

3.00

0.88

7

3.40

0.90

5

3.80

0.92

1

4.20

0.93

6

4.60

0.94

7

5.00

0.95

6

5.40

0.96

2

6.19

0.97

0

7.00

0.97

6

8.00

0.97

9

8.89

0.97

9

9.51

0.98

0

Med

ium

-Lon

gC

onvo

lute

dC

onfig

urat

ion

NP

R

F/F

i

1.25

0.83

2

1.40

0.81

4

1.60

0.82

5

1.80

0.79

2

2.00

0.79

9

2.20

0.81

1

2.40

0.82

4

2.60

0.84

4

3.00

0.87

7

3.39

0.90

2

3.80

0.91

7

4.20

0.92

0

4.60

0.92

5

4.99

0.93

1

5.40

0.93

7

6.20

0.94

6

6.99

0.95

1

8.00

0.95

5

8.89

0.95

7

9.50

0.95

7

Coa

rse

Con

volu

ted

Con

figur

atio

n

Tab

le 1

. N

ozzl

e T

hrus

t R

atio

Per

form

ance

16

NP

R

1.11

1

1.15

4

1.19

8

1.24

1

1.28

4

1.32

7

1.37

0

1.41

3

1.45

6

1.50

0

1.54

3

1.58

6

1.62

9

1.67

2

1.71

5

1.75

8

1.80

2

1.84

5

1.88

8

1.93

1

1.97

4

1.26

0.60

0

0.64

1

0.67

0

0.68

7

0.70

4

0.71

6

0.72

9

0.73

7

0.74

6

0.75

3

0.76

0

0.76

5

0.77

0

0.77

4

0.77

8

0.78

1

0.78

3

0.78

6

0.78

8

0.79

0

0.79

2

1.40

0.44

7

0.49

5

0.52

8

0.54

6

0.56

6

0.58

1

0.59

6

0.61

0

0.62

4

0.63

5

0.64

8

0.65

7

0.66

6

0.67

3

0.68

0

0.68

6

0.69

1

0.69

6

0.70

0

0.70

3

0.70

6

1.61

0.27

4

0.28

1

0.29

4

0.32

1

0.44

1

0.46

8

0.48

2

0.49

2

0.50

3

0.51

6

0.53

3

0.54

6

0.56

2

0.57

8

0.59

3

0.60

5

0.61

4

0.62

2

0.62

6

0.62

8

0.62

7

1.81

0.27

7

0.28

5

0.29

5

0.28

3

0.27

7

0.28

6

0.40

7

0.43

3

0.43

9

0.44

1

0.44

4

0.44

5

0.45

0

0.45

5

0.46

3

0.47

2

0.48

3

0.49

6

0.50

9

0.52

3

0.53

7

2.01

0.27

8

0.28

3

0.29

2

0.28

2

0.27

5

0.28

1

0.41

3

0.43

8

0.44

7

0.45

1

0.45

5

0.45

4

0.45

6

0.45

7

0.45

7

0.45

9

0.46

1

0.46

4

0.46

8

0.46

9

0.46

2

2.21

0.28

0

0.28

8

0.29

3

0.28

4

0.27

7

0.26

3

0.25

1

0.34

0

0.40

2

0.41

3

0.41

8

0.42

0

0.42

1

0.42

2

0.42

2

0.42

3

0.42

4

0.42

5

0.42

8

0.42

9

0.42

0

2.41

0.28

2

0.28

5

0.29

1

0.28

4

0.27

6

0.26

2

0.25

0

0.23

4

0.22

8

0.36

1

0.38

2

0.38

6

0.38

9

0.39

1

0.39

2

0.39

2

0.39

2

0.39

2

0.39

5

0.39

5

0.38

8

2.61

0.28

2

0.29

0

0.29

3

0.28

6

0.27

7

0.26

4

0.25

2

0.23

6

0.22

3

0.20

9

0.31

4

0.35

1

0.35

9

0.36

3

0.36

4

0.36

5

0.36

6

0.36

7

0.36

8

0.36

7

0.36

1

3.01

0.28

3

0.29

1

0.29

3

0.28

6

0.27

6

0.26

4

0.25

2

0.23

7

0.22

4

0.20

9

0.19

8

0.18

3

0.17

5

0.28

8

0.30

5

0.31

1

0.31

3

0.31

5

0.31

6

0.31

8

0.31

3

3.41

0.28

5

0.29

1

0.29

1

0.28

6

0.27

6

0.26

4

0.25

2

0.23

7

0.22

4

0.21

0

0.19

8

0.18

4

0.17

1

0.15

9

0.14

9

0.24

5

0.26

7

0.27

3

0.27

7

0.27

9

0.27

9

3.82

0.28

4

0.29

1

0.28

9

0.28

6

0.27

6

0.26

5

0.25

3

0.23

8

0.22

5

0.21

0

0.19

8

0.18

5

0.17

2

0.16

0

0.15

0

0.13

9

0.13

2

0.22

1

0.24

0

0.24

7

0.25

2

4.22

0.28

4

0.29

0

0.28

9

0.28

6

0.27

6

0.26

5

0.25

3

0.23

8

0.22

5

0.21

1

0.19

8

0.18

5

0.17

2

0.16

0

0.15

0

0.13

9

0.13

1

0.12

3

0.14

5

0.21

3

0.22

5

4.62

0.28

3

0.29

0

0.28

8

0.28

6

0.27

5

0.26

5

0.25

3

0.23

8

0.22

5

0.21

1

0.19

8

0.18

5

0.17

3

0.16

1

0.15

1

0.14

0

0.13

1

0.12

3

0.11

5

0.11

5

0.19

8

5.02

0.28

3

0.29

0

0.28

8

0.28

5

0.27

5

0.26

5

0.25

3

0.23

9

0.22

6

0.21

2

0.19

8

0.18

6

0.17

3

0.16

1

0.15

1

0.14

1

0.13

2

0.12

4

0.11

6

0.11

0

0.15

9

5.42

0.28

3

0.28

9

0.28

8

0.28

5

0.27

5

0.26

5

0.25

3

0.23

8

0.22

6

0.21

2

0.19

9

0.18

7

0.17

4

0.16

2

0.15

2

0.14

1

0.13

3

0.12

4

0.11

7

0.11

0

0.10

7

5.82

0.28

2

0.28

9

0.28

7

0.28

5

0.27

5

0.26

5

0.25

3

0.23

9

0.22

6

0.21

2

0.19

9

0.18

7

0.17

4

0.16

2

0.15

2

0.14

1

0.13

3

0.12

5

0.11

7

0.11

0

0.10

4

6.23

0.28

1

0.28

8

0.28

7

0.28

4

0.27

4

0.26

5

0.25

3

0.23

9

0.22

6

0.21

3

0.19

9

0.18

7

0.17

4

0.16

3

0.15

3

0.14

2

0.13

3

0.12

5

0.11

8

0.11

1

0.10

4

7.03

0.28

0

0.28

7

0.28

6

0.28

3

0.27

3

0.26

4

0.25

2

0.23

9

0.22

6

0.21

3

0.19

9

0.18

8

0.17

5

0.16

3

0.15

3

0.14

2

0.13

4

0.12

6

0.11

8

0.11

1

0.10

5

8.04

0.27

9

0.28

5

0.28

5

0.28

2

0.27

2

0.26

4

0.25

2

0.23

8

0.22

6

0.21

3

0.20

0

0.18

8

0.17

5

0.16

4

0.15

4

0.14

3

0.13

5

0.12

7

0.11

9

0.11

2

0.10

6

8.94

0.27

8

0.28

4

0.28

4

0.28

2

0.27

2

0.26

3

0.25

2

0.23

8

0.22

6

0.21

4

0.20

0

0.18

8

0.17

6

0.16

5

0.15

5

0.14

4

0.13

5

0.12

7

0.11

9

0.11

2

0.10

6

9.54

0.27

7

0.28

3

0.28

3

0.28

1

0.27

1

0.26

3

0.25

1

0.23

8

0.22

6

0.21

4

0.20

0

0.18

9

0.17

6

0.16

5

0.15

5

0.14

4

0.13

6

0.12

8

0.12

0

0.11

3

0.10

6

NP

R

0.33

0

0.44

0

0.55

0

0.65

9

0.76

9

0.89

0

1.00

0

1.26

0.95

9

0.94

6

0.93

0

0.90

3

0.85

6

0.68

1

0.44

8

1.40

0.95

8

0.94

5

0.92

7

0.89

9

0.85

0

0.67

1

0.39

1

1.61

0.95

7

0.94

3

0.92

7

0.89

9

0.84

9

0.67

0

0.36

9

1.81

0.95

8

0.94

3

0.92

8

0.89

9

0.85

0

0.67

0

0.36

9

2.01

0.95

8

0.94

2

0.92

7

0.89

9

0.84

9

0.67

0

0.36

8

2.21

0.95

8

0.94

3

0.92

9

0.89

9

0.85

0

0.67

0

0.36

8

2.41

0.95

8

0.94

2

0.92

8

0.89

9

0.85

0

0.67

0

0.36

8

2.61

0.95

9

0.94

3

0.92

9

0.90

0

0.85

1

0.67

0

0.36

8

3.01

0.95

9

0.94

3

0.92

9

0.90

0

0.85

1

0.67

0

0.36

8

3.41

0.95

9

0.94

3

0.92

9

0.90

0

0.85

1

0.67

1

0.36

9

3.82

0.95

9

0.94

3

0.92

9

0.90

0

0.85

0

0.67

1

0.36

9

4.22

0.95

9

0.94

3

0.92

9

0.89

9

0.85

0

0.67

1

0.36

9

4.62

0.95

9

0.94

2

0.92

9

0.89

9

0.85

0

0.67

1

0.36

9

5.02

0.95

9

0.94

2

0.92

8

0.89

9

0.85

0

0.67

1

0.36

8

5.42

0.95

8

0.94

2

0.92

8

0.89

9

0.84

9

0.67

0

0.36

8

5.82

0.95

8

0.94

1

0.92

8

0.89

8

0.84

8

0.67

0

0.36

7

6.23

0.95

7

0.94

1

0.92

7

0.89

8

0.84

8

0.66

9

0.36

6

7.03

0.95

7

0.94

0

0.92

6

0.89

7

0.84

7

0.66

9

0.36

5

8.04

0.95

6

0.94

0

0.92

5

0.89

6

0.84

6

0.66

8

0.36

4

8.94

0.95

6

0.93

9

0.92

4

0.89

5

0.84

5

0.66

8

0.36

2

9.54

0.95

6

0.93

9

0.92

4

0.89

5

0.84

5

0.66

8

0.36

2

Tab

le 2

. N

ozzl

e In

tern

al S

tatic

Pre

ssur

e R

atio

s fo

r th

e B

asel

ine

Con

figur

atio

n

(a)

Con

verg

ent

Fla

p,

z

= 0

.000

p/p

t,j

at

x/x t

of

17

(b)

Div

erge

nt F

lap,

z

= 0

.000

p/p

t,j

at

x/x t

of

NP

R

1.11

1

1.15

4

1.19

8

1.24

1

1.28

4

1.32

7

1.37

0

1.41

3

1.45

6

1.50

0

1.54

3

1.58

6

1.62

9

1.67

2

1.71

5

1.75

8

1.80

2

1.84

5

1.88

8

1.93

1

1.97

4

1.26

1.40

1.61

1.81

2.01

2.21

2.41

2.61

3.01

3.41

3.82

4.22

4.62

5.02

5.42

5.82

6.23

7.03

8.04

8.94

0.72

1

0.73

8

0.74

3

0.74

7

0.74

8

0.74

8

0.74

9

0.74

8

0.74

9

0.75

1

0.75

2

0.75

3

0.75

4

0.75

6

0.75

7

0.75

9

0.75

9

0.76

2

0.76

1

0.76

2

0.75

9

0.56

6

0.62

8

0.64

7

0.65

8

0.66

0

0.66

0

0.66

0

0.65

8

0.65

7

0.65

8

0.65

7

0.65

9

0.65

8

0.66

0

0.66

1

0.66

2

0.66

2

0.66

4

0.66

4

0.66

5

0.66

4

0.28

2

0.29

5

0.32

8

0.40

4

0.46

1

0.50

4

0.52

6

0.53

6

0.54

5

0.55

5

0.56

4

0.57

8

0.58

7

0.59

6

0.60

3

0.61

0

0.61

2

0.61

7

0.61

7

0.62

0

0.62

0

0.28

0

0.28

8

0.29

1

0.29

2

0.28

9

0.28

3

0.27

6

0.34

4

0.43

5

0.48

6

0.51

5

0.53

1

0.53

5

0.53

8

0.54

1

0.54

4

0.54

6

0.55

0

0.55

1

0.55

3

0.55

3

0.27

8

0.28

5

0.28

9

0.28

8

0.28

1

0.26

9

0.25

6

0.24

5

0.34

2

0.39

8

0.41

5

0.43

3

0.44

9

0.46

4

0.47

4

0.48

4

0.48

7

0.49

1

0.49

3

0.49

6

0.49

8

0.28

0

0.28

7

0.29

1

0.29

0

0.28

3

0.27

0

0.25

6

0.24

2

0.23

9

0.34

6

0.37

9

0.38

7

0.39

2

0.39

9

0.40

5

0.41

1

0.41

6

0.42

2

0.42

7

0.43

4

0.43

9

0.27

8

0.28

6

0.29

1

0.28

9

0.28

1

0.26

8

0.25

3

0.23

8

0.22

6

0.23

6

0.35

5

0.37

2

0.37

6

0.37

9

0.38

1

0.38

4

0.38

4

0.38

8

0.39

1

0.39

5

0.39

2

0.28

0

0.29

0

0.29

4

0.29

1

0.28

3

0.26

9

0.25

4

0.23

8

0.22

5

0.21

2

0.22

2

0.34

0

0.35

3

0.35

8

0.35

9

0.36

1

0.36

2

0.36

4

0.36

5

0.36

6

0.36

0

0.28

3

0.29

4

0.29

5

0.29

0

0.28

2

0.26

9

0.25

4

0.23

8

0.22

4

0.21

0

0.19

7

0.18

5

0.17

4

0.23

9

0.29

4

0.30

2

0.30

3

0.30

4

0.30

4

0.30

8

0.31

4

0.28

6

0.29

5

0.29

5

0.28

9

0.28

2

0.26

9

0.25

4

0.23

8

0.22

5

0.21

0

0.19

7

0.18

5

0.17

2

0.16

2

0.15

3

0.20

2

0.26

0

0.27

0

0.27

3

0.27

7

0.27

8

0.28

6

0.29

5

0.29

5

0.28

9

0.28

2

0.26

9

0.25

5

0.23

9

0.22

5

0.21

1

0.19

7

0.18

5

0.17

2

0.16

1

0.15

2

0.14

2

0.13

4

0.19

0

0.23

4

0.24

4

0.25

1

0.28

6

0.29

5

0.29

5

0.28

9

0.28

2

0.26

9

0.25

5

0.23

9

0.22

6

0.21

1

0.19

8

0.18

5

0.17

3

0.16

2

0.15

2

0.14

2

0.13

4

0.12

7

0.12

1

0.20

4

0.22

4

0.28

5

0.29

5

0.29

5

0.28

8

0.28

1

0.27

0

0.25

5

0.24

0

0.22

6

0.21

2

0.19

8

0.18

6

0.17

4

0.16

2

0.15

2

0.14

3

0.13

4

0.12

7

0.12

0

0.11

5

0.19

4

0.28

5

0.29

4

0.29

5

0.28

8

0.28

1

0.27

0

0.25

5

0.24

0

0.22

6

0.21

2

0.19

9

0.18

6

0.17

4

0.16

3

0.15

2

0.14

3

0.13

4

0.12

7

0.12

0

0.11

4

0.14

4

0.28

5

0.29

4

0.29

5

0.28

7

0.28

1

0.27

0

0.25

6

0.24

0

0.22

6

0.21

2

0.19

9

0.18

7

0.17

4

0.16

3

0.15

3

0.14

3

0.13

4

0.12

7

0.12

0

0.11

4

0.11

0

0.28

4

0.29

3

0.29

5

0.28

7

0.28

0

0.27

0

0.25

6

0.24

0

0.22

7

0.21

2

0.19

9

0.18

7

0.17

5

0.16

3

0.15

3

0.14

3

0.13

5

0.12

7

0.12

0

0.11

3

0.10

8

0.28

3

0.29

3

0.29

4

0.28

6

0.28

0

0.26

9

0.25

6

0.24

0

0.22

7

0.21

3

0.20

0

0.18

8

0.17

5

0.16

4

0.15

4

0.14

4

0.13

5

0.12

7

0.12

0

0.11

3

0.10

7

0.28

2

0.29

1

0.29

3

0.28

5

0.27

8

0.26

8

0.25

5

0.24

0

0.22

7

0.21

3

0.20

0

0.18

8

0.17

6

0.16

5

0.15

5

0.14

5

0.13

6

0.12

8

0.12

0

0.11

3

0.10

7

0.28

0

0.29

0

0.29

2

0.28

4

0.27

7

0.26

8

0.25

6

0.24

1

0.22

7

0.21

3

0.20

1

0.18

9

0.17

7

0.16

6

0.15

6

0.14

6

0.13

6

0.12

8

0.12

1

0.11

4

0.10

7

0.27

9

0.28

8

0.29

2

0.28

4

0.27

6

0.26

7

0.25

5

0.24

1

0.22

7

0.21

3

0.20

1

0.18

9

0.17

8

0.16

6

0.15

6

0.14

6

0.13

7

0.12

9

0.12

1

0.11

4

0.10

8

Tab

le 2

. C

oncl

uded

(c)

Div

erge

nt F

lap,

z

= 1

.595

p/p

t,j

at

x/x t

of

18

0.27

8

0.28

8

0.29

2

0.28

3

0.27

5

0.26

8

0.25

6

0.24

1

0.22

7

0.21

3

0.20

1

0.19

0

0.17

8

0.16

6

0.15

6

0.14

6

0.13

7

0.12

9

0.12

1

0.11

5

0.10

8

9.54

NP

R

1.11

1

1.15

4

1.19

8

1.24

1

1.28

4

1.32

7

1.37

0

1.41

3

1.45

6

1.50

0

1.54

3

1.58

6

1.62

9

1.67

2

1.71

5

1.75

8

1.80

2

1.84

5

1.88

8

1.93

1

1.97

4

1.25

1.40

1.60

1.80

2.00

2.20

2.40

2.60

3.00

3.40

3.80

4.20

4.60

5.00

5.40

6.20

7.00

8.00

8.91

9.50

0.61

3

0.66

0

0.68

4

0.70

4

0.71

8

0.73

1

0.74

1

0.74

8

0.75

3

0.75

6

0.76

1

0.76

6

0.77

1

0.77

8

0.78

3

0.78

7

0.78

9

0.79

4

0.79

5

0.79

8

0.79

7

0.43

3

0.49

6

0.53

4

0.56

2

0.58

3

0.59

9

0.61

1

0.62

0

0.62

8

0.63

3

0.64

1

0.65

0

0.66

1

0.67

4

0.68

4

0.69

2

0.69

8

0.70

3

0.70

6

0.71

2

0.71

0

0.27

2

0.30

2

0.43

9

0.47

6

0.48

9

0.49

7

0.49

8

0.49

7

0.49

6

0.49

7

0.50

2

0.50

9

0.52

1

0.53

4

0.55

0

0.56

5

0.57

8

0.59

1

0.60

2

0.62

0

0.61

2

0.27

0

0.28

1

0.36

9

0.46

1

0.48

3

0.49

5

0.50

1

0.50

3

0.50

5

0.50

7

0.50

8

0.50

8

0.50

9

0.51

0

0.51

1

0.51

4

0.51

6

0.52

0

0.52

3

0.51

5

0.52

2

0.27

0

0.28

1

0.28

1

0.27

7

0.40

5

0.45

7

0.46

8

0.47

2

0.47

3

0.47

6

0.47

8

0.48

0

0.48

1

0.48

1

0.48

1

0.48

2

0.48

2

0.48

3

0.48

2

0.47

5

0.47

8

0.26

9

0.28

0

0.28

1

0.27

6

0.26

7

0.25

9

0.25

8

0.25

4

0.22

6

0.19

5

0.19

3

0.19

0

0.23

0

0.30

3

0.34

3

0.36

9

0.36

9

0.36

9

0.38

0

0.43

2

0.40

5

0.26

9

0.28

1

0.28

1

0.27

6

0.26

7

0.25

9

0.25

8

0.25

4

0.22

6

0.19

6

0.19

4

0.18

1

0.18

7

0.25

9

0.30

4

0.33

6

0.34

6

0.34

4

0.34

1

0.36

4

0.34

6

0.27

0

0.28

1

0.28

1

0.27

6

0.26

6

0.25

9

0.25

7

0.25

3

0.22

6

0.19

5

0.19

4

0.18

1

0.17

8

0.19

7

0.26

6

0.29

9

0.32

0

0.32

7

0.32

9

0.34

2

0.33

1

0.27

1

0.28

1

0.28

1

0.27

6

0.26

6

0.25

8

0.25

7

0.25

3

0.22

6

0.19

6

0.19

6

0.18

0

0.17

0

0.15

6

0.16

1

0.16

6

0.22

8

0.25

8

0.28

1

0.30

6

0.29

5

0.27

2

0.28

1

0.28

1

0.27

5

0.26

6

0.25

8

0.25

7

0.25

3

0.22

7

0.19

7

0.19

7

0.18

1

0.17

0

0.15

4

0.15

6

0.13

4

0.18

0

0.21

2

0.22

8

0.24

7

0.24

3

0.27

2

0.28

0

0.28

0

0.27

5

0.26

6

0.25

8

0.25

7

0.25

4

0.22

8

0.19

7

0.19

8

0.18

1

0.17

0

0.15

5

0.15

7

0.13

3

0.13

5

0.16

8

0.19

4

0.21

0

0.20

4

0.27

2

0.27

9

0.28

0

0.27

5

0.26

5

0.25

8

0.25

8

0.25

5

0.22

8

0.19

8

0.19

8

0.18

2

0.17

1

0.15

5

0.15

8

0.13

4

0.13

3

0.12

5

0.11

8

0.19

2

0.14

0

0.27

1

0.27

9

0.28

0

0.27

5

0.26

5

0.25

8

0.25

8

0.25

5

0.22

9

0.19

9

0.19

9

0.18

2

0.17

1

0.15

6

0.15

9

0.13

4

0.13

4

0.12

5

0.11

7

0.17

2

0.11

3

0.27

1

0.27

8

0.27

9

0.27

5

0.26

5

0.25

8

0.25

8

0.25

5

0.22

9

0.20

0

0.19

9

0.18

3

0.17

1

0.15

6

0.16

0

0.13

5

0.13

5

0.12

6

0.11

8

0.13

0

0.11

3

0.27

0

0.27

8

0.27

8

0.27

4

0.26

5

0.25

7

0.25

8

0.25

5

0.22

9

0.20

0

0.19

9

0.18

3

0.17

1

0.15

6

0.16

1

0.13

5

0.13

5

0.12

6

0.11

9

0.10

9

0.11

4

0.26

8

0.27

6

0.27

7

0.27

3

0.26

3

0.25

6

0.25

7

0.25

5

0.22

9

0.20

0

0.20

0

0.18

3

0.17

1

0.15

7

0.16

2

0.13

6

0.13

6

0.12

7

0.11

9

0.10

8

0.11

5

0.26

7

0.27

4

0.27

5

0.27

1

0.26

2

0.25

6

0.25

7

0.25

5

0.22

9

0.20

0

0.20

0

0.18

3

0.17

2

0.15

7

0.16

2

0.13

7

0.13

7

0.12

8

0.12

0

0.10

9

0.11

5

0.26

5

0.27

3

0.27

4

0.27

0

0.26

1

0.25

5

0.25

6

0.25

4

0.22

9

0.20

0

0.20

0

0.18

3

0.17

2

0.15

7

0.16

2

0.13

7

0.13

7

0.12

8

0.12

0

0.10

9

0.11

6

0.26

4

0.27

1

0.27

3

0.26

9

0.26

0

0.25

4

0.25

6

0.25

4

0.22

8

0.20

0

0.20

0

0.18

3

0.17

2

0.15

7

0.16

2

0.13

7

0.13

8

0.12

8

0.12

0

0.11

0

0.11

6

0.26

4

0.27

1

0.27

2

0.26

9

0.26

0

0.25

4

0.25

6

0.25

4

0.22

8

0.20

0

0.20

0

0.18

4

0.17

2

0.15

8

0.16

3

0.13

7

0.13

8

0.12

9

0.12

1

0.11

0

0.11

6

NP

R

0.33

0

0.44

0

0.55

0

0.65

9

0.76

9

0.89

0

1.00

0

1.25

0.96

0

0.95

0

0.93

1

0.90

4

0.85

7

0.68

4

0.45

8

1.40

0.95

8

0.94

6

0.92

8

0.90

0

0.85

1

0.67

2

0.37

9

1.60

0.95

8

0.94

6

0.92

7

0.89

9

0.85

0

0.67

0

0.36

9

1.80

0.95

8

0.94

6

0.92

7

0.89

9

0.84

9

0.67

0

0.36

8

2.00

0.95

8

0.94

6

0.92

7

0.89

9

0.84

9

0.67

0

0.36

7

2.20

0.95

8

0.94

6

0.92

8

0.89

9

0.84

9

0.67

0

0.36

8

2.40

0.95

8

0.94

6

0.92

8

0.89

9

0.84

9

0.67

1

0.36

7

2.60

0.95

8

0.94

5

0.92

8

0.89

9

0.84

9

0.67

1

0.36

7

3.00

0.95

8

0.94

6

0.92

8

0.89

9

0.84

9

0.67

1

0.36

8

3.40

0.95

9

0.94

5

0.92

8

0.90

0

0.85

0

0.67

1

0.36

8

3.80

0.95

8

0.94

5

0.92

8

0.89

9

0.84

9

0.67

1

0.36

7

4.20

0.95

8

0.94

5

0.92

8

0.89

9

0.84

9

0.67

0

0.36

7

4.60

0.95

8

0.94

5

0.92

7

0.89

8

0.84

8

0.67

0

0.36

6

5.00

0.95

8

0.94

5

0.92

7

0.89

8

0.84

7

0.67

0

0.36

5

5.40

0.95

7

0.94

4

0.92

7

0.89

8

0.84

7

0.66

9

0.36

4

6.20

0.95

7

0.94

4

0.92

5

0.89

6

0.84

6

0.66

8

0.36

2

7.00

0.95

6

0.94

3

0.92

5

0.89

6

0.84

5

0.66

7

0.36

1

8.00

0.95

6

0.94

2

0.92

4

0.89

5

0.84

4

0.66

6

0.35

9

8.91

0.95

5

0.94

2

0.92

3

0.89

4

0.84

3

0.66

5

0.35

7

9.50

0.95

5

0.94

2

0.92

2

0.89

4

0.84

2

0.66

5

0.35

7

Tab

le 3

. N

ozzl

e In

tern

al S

tatic

Pre

ssur

e R

atio

s fo

r th

e F

ine

Con

volu

ted

Con

figur

atio

n

(a)

Con

verg

ent

Fla

p,

z

= 0

.000

p/p

t,j

at

x/x t

of

19

(b)

Div

erge

nt F

lap,

z

= 0

.000

(H

ill)

p/p

t,j

at

x/x t

of

NP

R

1.11

1

1.15

4

1.19

8

1.24

1

1.28

4

1.32

7

1.37

0

1.41

3

1.45

6

1.50

0

1.54

3

1.58

6

1.62

9

1.67

2

1.71

5

1.75

8

1.80

2

1.84

5

1.88

8

1.93

1

1.97

4

1.25

0.69

8

0.74

0

0.75

2

0.75

8

0.75

8

0.76

0

0.75

6

0.75

6

0.75

3

0.75

2

0.75

2

0.75

4

0.75

5

0.76

2

0.76

5

0.76

8

0.76

8

0.77

0

0.77

0

0.77

1

0.77

0

1.40

0.48

0

0.55

0

0.58

4

0.60

7

0.61

8

0.63

3

0.63

2

0.64

6

0.65

1

0.65

5

0.65

9

0.66

5

0.66

9

0.68

0

0.68

6

0.68

8

0.69

1

0.69

4

0.69

6

0.69

9

0.70

1

1.60

0.28

5

0.35

3

0.41

2

0.44

3

0.46

3

0.48

7

0.46

0

0.50

4

0.52

6

0.53

8

0.54

7

0.56

2

0.57

8

0.60

0

0.61

1

0.61

6

0.61

9

0.62

2

0.62

2

0.62

4

0.62

5

1.80

0.27

4

0.28

1

0.29

3

0.37

3

0.44

7

0.47

9

0.48

9

0.48

8

0.49

0

0.49

0

0.49

4

0.50

1

0.50

6

0.51

2

0.51

6

0.52

0

0.52

4

0.52

9

0.53

3

0.54

0

0.54

6

2.00

0.27

3

0.28

1

0.28

4

0.28

5

0.27

9

0.41

4

0.46

0

0.46

6

0.46

8

0.47

2

0.47

5

0.47

5

0.47

4

0.47

4

0.47

2

0.47

1

0.47

0

0.47

2

0.47

4

0.47

9

0.47

3

2.20

0.27

5

0.28

2

0.28

4

0.28

5

0.27

4

0.26

8

0.26

5

0.26

2

0.25

7

0.31

2

0.35

2

0.35

1

0.38

2

0.40

6

0.41

9

0.42

3

0.43

7

0.44

5

0.44

7

0.44

8

0.45

1

2.40

0.27

6

0.28

3

0.28

4

0.28

3

0.27

2

0.26

3

0.26

1

0.25

6

0.23

0

0.21

5

0.26

4

0.24

6

0.31

0

0.34

8

0.36

6

0.37

1

0.39

2

0.40

6

0.41

3

0.41

6

0.41

6

2.60

0.27

6

0.28

3

0.28

6

0.28

2

0.27

0

0.26

1

0.25

9

0.25

3

0.22

6

0.20

7

0.22

1

0.17

6

0.22

6

0.28

5

0.31

9

0.32

9

0.35

3

0.37

2

0.38

2

0.38

6

0.38

7

3.00

0.27

5

0.28

5

0.28

8

0.28

0

0.27

0

0.26

0

0.25

9

0.25

4

0.22

4

0.20

2

0.20

5

0.16

3

0.16

3

0.15

1

0.16

3

0.16

9

0.23

2

0.25

7

0.29

4

0.32

6

0.34

1

3.40

0.27

6

0.28

6

0.28

8

0.28

0

0.27

0

0.26

1

0.26

0

0.25

5

0.22

4

0.20

2

0.20

5

0.16

2

0.16

2

0.14

8

0.15

9

0.13

9

0.15

0

0.16

7

0.20

6

0.27

1

0.29

9

3.80

0.27

5

0.28

6

0.28

7

0.28

0

0.27

0

0.26

1

0.26

0

0.25

6

0.22

5

0.20

2

0.20

5

0.16

3

0.16

3

0.14

7

0.15

8

0.13

7

0.14

2

0.15

1

0.17

1

0.19

5

0.22

1

4.20

0.27

5

0.28

5

0.28

7

0.27

9

0.27

0

0.26

1

0.26

0

0.25

6

0.22

6

0.20

3

0.20

5

0.16

3

0.16

3

0.14

7

0.15

9

0.13

6

0.13

8

0.13

5

0.14

2

0.16

3

0.19

3

4.60

0.27

5

0.28

5

0.28

6

0.27

9

0.27

0

0.26

1

0.26

0

0.25

7

0.22

6

0.20

4

0.20

6

0.16

4

0.16

4

0.14

7

0.15

9

0.13

7

0.13

7

0.13

0

0.12

5

0.13

3

0.16

1

5.00

0.27

4

0.28

4

0.28

6

0.27

9

0.27

0

0.26

1

0.25

9

0.25

7

0.22

6

0.20

5

0.20

6

0.16

4

0.16

5

0.14

7

0.15

9

0.13

7

0.13

7

0.13

0

0.12

2

0.11

9

0.13

3

5.40

0.27

3

0.28

4

0.28

5

0.27

8

0.26

9

0.26

1

0.26

0

0.25

7

0.22

7

0.20

6

0.20

6

0.16

4

0.16

5

0.14

7

0.16

0

0.13

7

0.13

8

0.13

0

0.12

2

0.11

6

0.11

7

6.20

0.27

1

0.28

2

0.28

3

0.27

6

0.26

7

0.25

9

0.25

8

0.25

6

0.22

6

0.20

6

0.20

6

0.16

5

0.16

6

0.14

8

0.16

2

0.13

9

0.14

0

0.13

2

0.12

3

0.11

6

0.11

3

7.00

0.27

0

0.28

0

0.28

2

0.27

4

0.26

6

0.25

8

0.25

7

0.25

5

0.22

6

0.20

5

0.20

6

0.16

5

0.16

6

0.14

7

0.16

2

0.13

8

0.14

0

0.13

2

0.12

3

0.11

7

0.11

4

8.00

0.26

9

0.27

9

0.28

0

0.27

2

0.26

4

0.25

7

0.25

6

0.25

5

0.22

5

0.20

5

0.20

5

0.16

5

0.16

6

0.14

7

0.16

2

0.13

8

0.14

0

0.13

2

0.12

3

0.11

7

0.11

3

8.91

0.26

8

0.27

8

0.27

9

0.27

1

0.26

3

0.25

6

0.25

5

0.25

5

0.22

5

0.20

5

0.20

5

0.16

5

0.16

6

0.14

8

0.16

2

0.13

9

0.14

0

0.13

2

0.12

3

0.11

6

0.11

2

9.50

0.26

8

0.27

7

0.27

9

0.27

0

0.26

3

0.25

6

0.25

5

0.25

4

0.22

5

0.20

6

0.20

5

0.16

5

0.16

6

0.14

8

0.16

2

0.13

9

0.14

0

0.13

2

0.12

4

0.11

7

0.11

1

NP

R

1.32

7

1.37

0

1.41

3

1.45

6

1.50

0

1.54

3

1.58

6

1.62

9

1.67

2

1.71

5

1.25

0.73

0

0.74

1

0.74

6

0.75

6

0.76

2

0.76

7

0.77

0

0.77

4

0.78

0

0.78

4

1.40

0.59

8

0.61

4

0.62

2

0.63

1

0.64

0

0.64

8

0.65

6

0.66

6

0.67

7

0.68

6

1.60

0.49

7

0.50

3

0.50

2

0.50

5

0.50

8

0.51

5

0.52

0

0.52

7

0.53

7

0.55

2

1.80

0.49

9

0.50

6

0.50

8

0.51

1

0.51

3

0.51

4

0.51

3

0.51

3

0.51

1

0.51

3

2.00

0.46

0

0.47

2

0.47

4

0.47

8

0.48

0

0.48

2

0.48

3

0.48

3

0.48

3

0.48

3

2.20

0.24

9

0.23

5

0.21

0

0.22

1

0.22

7

0.23

0

0.23

7

0.25

8

0.30

5

0.34

3

2.40

0.24

9

0.23

5

0.21

1

0.22

1

0.22

8

0.23

0

0.22

1

0.22

2

0.25

0

0.30

3

2.60

0.24

8

0.23

5

0.21

1

0.22

2

0.22

8

0.23

0

0.22

0

0.19

4

0.15

6

0.24

8

3.00

0.24

8

0.23

6

0.21

1

0.22

2

0.22

9

0.23

0

0.22

0

0.18

8

0.12

7

0.14

8

3.40

0.24

8

0.23

6

0.21

2

0.22

2

0.22

9

0.23

1

0.22

1

0.19

0

0.12

6

0.13

7

3.80

0.24

9

0.23

7

0.21

2

0.22

2

0.22

9

0.23

1

0.22

1

0.19

1

0.12

7

0.13

7

4.20

0.24

8

0.23

7

0.21

2

0.22

2

0.23

0

0.23

1

0.22

2

0.19

1

0.12

8

0.13

7

4.60

0.24

8

0.23

7

0.21

2

0.22

2

0.23

0

0.23

1

0.22

2

0.19

2

0.12

8

0.13

8

5.00

0.24

8

0.23

7

0.21

2

0.22

2

0.23

0

0.23

1

0.22

2

0.19

3

0.12

9

0.13

8

5.40

0.24

8

0.23

7

0.21

2

0.22

2

0.23

0

0.23

2

0.22

3

0.19

3

0.12

9

0.13

9

6.20

0.24

6

0.23

6

0.21

1

0.22

2

0.23

0

0.23

2

0.22

3

0.19

3

0.13

0

0.13

9

7.00

0.24

5

0.23

5

0.21

1

0.22

1

0.23

0

0.23

2

0.22

3

0.19

4

0.13

1

0.13

9

8.00

0.24

5

0.23

4

0.21

0

0.22

1

0.22

9

0.23

2

0.22

3

0.19

4

0.13

2

0.14

0

8.91

0.24

4

0.23

4

0.21

0

0.22

0

0.22

9

0.23

2

0.22

3

0.19

4

0.13

2

0.14

0

9.50

0.24

4

0.23

3

0.20

9

0.22

0

0.22

9

0.23

2

0.22

4

0.19

4

0.13

2

0.14

1

Tab

le 3

. C

oncl

uded

(c)

Div

erge

nt F

lap,

z

= 0

.199

5 (V

alle

y)

p/p

t,j

at

x/x t

of

(d)

Div

erge

nt F

lap,

z

= 1

.595

(H

ill)

p/p

t,j

at

x/x t

of

20

Tab

le 4

. N

ozzl

e In

tern

al S

tatic

Pre

ssur

e R

atio

s fo

r th

e M

ediu

m C

onvo

lute

d C

onfig

urat

ion

(a)

Con

verg

ent

Fla

p,

z

= 0

.000

p/p

t,j

at

x/x t

of

21

(b)

Div

erge

nt F

lap,

z

= 0

.000

(H

ill)

p/p

t,j

at

x/x t

of

NP

R

1.11

1

1.15

4

1.19

8

1.24

1

1.28

4

1.32

7

1.37

0

1.41

3

1.45

6

1.50

0

1.54

3

1.58

6

1.62

9

1.67

2

1.71

5

1.75

8

1.80

2

1.84

5

1.88

8

1.93

1

1.97

4

1.24

0.62

4

0.66

0

0.68

5

0.70

4

0.71

9

0.73

8

0.74

9

0.76

2

0.75

6

0.74

8

0.75

4

0.76

5

0.77

2

0.77

9

0.78

4

0.78

8

0.79

2

0.79

5

0.79

8

0.80

0

0.80

1

1.39

0.49

1

0.52

3

0.54

6

0.56

3

0.57

8

0.59

6

0.60

6

0.61

9

0.62

4

0.62

1

0.62

8

0.64

0

0.65

1

0.66

4

0.67

5

0.68

5

0.69

2

0.69

8

0.70

4

0.70

8

0.71

1

1.59

0.27

2

0.27

9

0.29

5

0.36

5

0.45

4

0.50

4

0.54

5

0.60

6

0.60

1

0.54

7

0.55

5

0.57

9

0.59

8

0.60

6

0.61

8

0.62

5

0.62

4

0.62

7

0.62

9

0.63

0

0.62

8

1.79

0.27

1

0.27

9

0.28

2

0.27

8

0.27

3

0.36

1

0.39

3

0.40

8

0.40

1

0.39

9

0.39

4

0.39

7

0.40

9

0.42

8

0.46

2

0.48

9

0.51

1

0.52

7

0.54

0

0.55

0

0.55

5

1.99

0.27

1

0.28

1

0.28

2

0.27

9

0.27

2

0.26

3

0.29

8

0.32

0

0.26

0

0.18

7

0.15

5

0.22

9

0.27

7

0.31

0

0.37

0

0.43

1

0.47

2

0.49

7

0.50

9

0.51

3

0.51

0

2.19

0.27

2

0.28

0

0.28

3

0.27

9

0.27

2

0.26

3

0.29

7

0.31

9

0.26

0

0.18

6

0.13

9

0.22

6

0.25

3

0.27

7

0.32

0

0.35

5

0.38

3

0.40

1

0.41

8

0.43

3

0.44

4

2.39

0.27

2

0.28

1

0.28

3

0.28

0

0.27

2

0.26

3

0.29

7

0.32

0

0.26

1

0.18

6

0.13

6

0.21

9

0.24

7

0.26

5

0.29

4

0.33

2

0.35

3

0.36

4

0.37

5

0.38

6

0.39

4

2.59

0.27

2

0.28

1

0.28

4

0.28

0

0.27

2

0.26

3

0.29

6

0.31

9

0.26

1

0.18

6

0.13

6

0.21

0

0.24

0

0.25

9

0.27

5

0.30

8

0.32

7

0.33

7

0.34

6

0.35

5

0.36

1

2.99

0.27

3

0.28

1

0.28

3

0.27

9

0.27

2

0.26

3

0.29

6

0.31

9

0.26

2

0.18

7

0.13

7

0.19

7

0.19

6

0.17

9

0.14

6

0.17

8

0.15

2

0.13

9

0.14

8

0.24

1

0.28

4

3.39

0.27

2

0.28

1

0.28

2

0.27

8

0.27

1

0.26

3

0.29

6

0.31

9

0.26

3

0.18

7

0.13

7

0.19

9

0.19

5

0.17

2

0.13

9

0.16

9

0.14

0

0.12

9

0.11

9

0.14

9

0.24

1

3.79

0.27

1

0.28

0

0.28

2

0.27

8

0.27

1

0.26

3

0.29

6

0.32

0

0.26

4

0.18

8

0.13

8

0.20

0

0.19

3

0.16

9

0.13

9

0.16

8

0.13

8

0.12

9

0.11

9

0.11

7

0.21

3

4.19

0.27

1

0.27

9

0.28

1

0.27

8

0.27

1

0.26

3

0.29

6

0.32

0

0.26

4

0.18

9

0.13

8

0.20

0

0.19

2

0.16

7

0.14

0

0.16

8

0.13

8

0.13

0

0.12

0

0.11

5

0.16

7

4.59

0.27

0

0.27

9

0.28

1

0.27

8

0.27

0

0.26

2

0.29

6

0.32

0

0.26

4

0.19

0

0.13

9

0.20

1

0.19

2

0.16

7

0.14

0

0.16

9

0.13

8

0.13

0

0.12

0

0.11

5

0.11

4

4.99

0.26

9

0.27

8

0.28

0

0.27

7

0.27

0

0.26

2

0.29

6

0.32

0

0.26

4

0.19

0

0.13

9

0.20

0

0.19

2

0.16

7

0.14

0

0.17

0

0.13

9

0.13

1

0.12

1

0.11

5

0.11

1

5.39

0.26

8

0.27

7

0.27

9

0.27

7

0.27

0

0.26

1

0.29

6

0.32

0

0.26

5

0.19

1

0.14

0

0.20

0

0.19

2

0.16

7

0.14

1

0.17

1

0.13

9

0.13

1

0.12

2

0.11

5

0.10

9

6.19

0.26

7

0.27

5

0.27

7

0.27

5

0.26

8

0.26

0

0.29

5

0.31

9

0.26

5

0.19

1

0.14

0

0.20

0

0.19

2

0.16

8

0.14

2

0.17

3

0.14

0

0.13

1

0.12

3

0.11

6

0.10

9

6.99

0.26

5

0.27

3

0.27

6

0.27

4

0.26

7

0.25

8

0.29

5

0.31

9

0.26

4

0.19

2

0.14

1

0.20

0

0.19

1

0.16

9

0.14

3

0.17

5

0.14

1

0.13

2

0.12

3

0.11

6

0.10

9

7.99

0.26

3

0.27

2

0.27

4

0.27

3

0.26

6

0.25

7

0.29

4

0.31

8

0.26

4

0.19

2

0.14

1

0.19

9

0.19

1

0.17

0

0.14

3

0.17

7

0.14

1

0.13

3

0.12

4

0.11

7

0.10

9

8.89

0.26

2

0.27

1

0.27

3

0.27

2

0.26

5

0.25

6

0.29

4

0.31

8

0.26

4

0.19

3

0.14

1

0.19

9

0.19

2

0.17

1

0.14

4

0.17

9

0.14

2

0.13

3

0.12

4

0.11

7

0.10

9

9.50

0.26

2

0.27

0

0.27

2

0.27

1

0.26

5

0.25

6

0.29

4

0.31

8

0.26

4

0.19

3

0.14

1

0.19

9

0.19

2

0.17

1

0.14

4

0.18

0

0.14

2

0.13

3

0.12

4

0.11

7

0.10

9

NP

R

0.33

0

0.44

0

0.55

0

0.65

9

0.76

9

0.89

0

1.00

0

1.24

0.96

1

0.95

1

0.93

3

0.90

4

0.86

1

0.68

3

0.45

2

1.39

0.95

9

0.94

8

0.93

0

0.90

1

0.85

4

0.67

3

0.39

8

1.59

0.95

8

0.94

8

0.92

9

0.89

8

0.85

2

0.67

0

0.36

7

1.79

0.95

8

0.94

8

0.92

9

0.89

9

0.85

2

0.67

0

0.36

7

1.99

0.95

8

0.94

7

0.93

0

0.89

9

0.85

1

0.67

0

0.36

7

2.19

0.95

9

0.94

7

0.92

9

0.89

9

0.85

1

0.67

0

0.36

6

2.39

0.95

9

0.94

7

0.93

0

0.89

9

0.85

1

0.67

0

0.36

7

2.59

0.95

9

0.94

7

0.93

0

0.89

9

0.85

0

0.66

9

0.36

6

2.99

0.95

9

0.94

6

0.93

0

0.90

0

0.85

0

0.67

0

0.36

7

3.39

0.95

9

0.94

6

0.92

9

0.89

9

0.85

0

0.67

0

0.36

7

3.79

0.95

9

0.94

6

0.92

9

0.89

9

0.85

0

0.67

0

0.36

7

4.19

0.95

8

0.94

6

0.92

9

0.89

9

0.84

9

0.66

9

0.36

7

4.59

0.95

8

0.94

6

0.92

9

0.89

9

0.84

9

0.66

9

0.36

6

4.99

0.95

8

0.94

5

0.92

8

0.89

8

0.84

8

0.66

9

0.36

5

5.39

0.95

8

0.94

4

0.92

7

0.89

8

0.84

7

0.66

8

0.36

4

6.19

0.95

7

0.94

4

0.92

6

0.89

7

0.84

6

0.66

7

0.36

3

6.99

0.95

6

0.94

3

0.92

5

0.89

6

0.84

5

0.66

6

0.36

1

7.99

0.95

6

0.94

2

0.92

4

0.89

5

0.84

4

0.66

5

0.35

9

8.89

0.95

5

0.94

2

0.92

3

0.89

4

0.84

3

0.66

5

0.35

8

9.50

0.95

5

0.94

2

0.92

3

0.89

4

0.84

2

0.66

5

0.35

7

NP

R

1.11

1

1.15

4

1.19

8

1.24

1

1.28

4

1.32

7

1.37

0

1.41

3

1.45

6

1.50

0

1.54

3

1.58

6

1.62

9

1.67

2

1.71

5

1.75

8

1.80

2

1.84

5

1.88

8

1.93

1

1.97

4

1.24

1.39

1.59

1.79

1.99

2.19

2.39

2.59

2.99

3.39

3.79

4.19

4.59

4.99

5.39

6.19

6.99

7.99

8.89

9.50

0.72

6

0.74

7

0.75

6

0.75

9

0.75

4

0.75

7

0.75

4

0.75

5

0.75

0

0.74

9

0.74

8

0.75

0

0.75

2

0.75

6

0.76

4

0.77

0

0.77

0

0.77

2

0.77

2

0.77

7

0.77

8

0.46

6

0.52

7

0.56

8

0.59

8

0.61

4

0.64

1

0.66

0

0.68

5

0.66

7

0.64

3

0.65

1

0.67

1

0.68

3

0.68

8

0.69

6

0.70

4

0.70

1

0.70

6

0.70

9

0.71

3

0.71

3

0.27

5

0.28

6

0.34

1

0.42

1

0.47

0

0.51

0

0.53

1

0.57

3

0.52

2

0.47

3

0.49

9

0.52

1

0.54

0

0.55

5

0.57

7

0.59

5

0.59

8

0.60

8

0.61

5

0.62

1

0.62

4

0.27

5

0.28

1

0.28

7

0.29

2

0.28

5

0.28

9

0.40

7

0.47

4

0.46

9

0.41

8

0.42

8

0.44

8

0.48

6

0.50

3

0.53

2

0.55

4

0.55

5

0.56

0

0.56

1

0.56

3

0.56

0

0.27

4

0.28

0

0.28

9

0.28

9

0.28

0

0.26

8

0.33

1

0.40

0

0.38

9

0.31

5

0.30

8

0.37

8

0.43

7

0.45

1

0.47

2

0.49

1

0.48

4

0.48

9

0.49

1

0.49

5

0.49

5

0.27

3

0.28

2

0.29

0

0.28

7

0.27

6

0.25

8

0.29

8

0.32

5

0.30

6

0.27

4

0.25

4

0.31

8

0.37

9

0.40

3

0.42

9

0.44

8

0.43

7

0.44

4

0.44

5

0.44

9

0.45

1

0.27

3

0.28

4

0.29

1

0.28

5

0.27

4

0.25

6

0.29

4

0.31

7

0.27

5

0.20

1

0.16

1

0.25

4

0.31

5

0.34

6

0.39

5

0.42

0

0.39

5

0.40

5

0.41

1

0.41

8

0.41

4

0.27

5

0.28

7

0.29

1

0.28

4

0.27

3

0.25

4

0.29

1

0.31

5

0.26

9

0.19

3

0.13

8

0.20

4

0.24

7

0.28

1

0.34

4

0.37

6

0.35

8

0.36

9

0.37

8

0.38

9

0.39

0

0.28

0

0.28

9

0.29

0

0.28

4

0.27

2

0.25

3

0.29

0

0.31

3

0.26

7

0.18

8

0.13

3

0.19

5

0.19

7

0.18

8

0.13

8

0.23

6

0.21

5

0.26

5

0.28

2

0.29

2

0.30

8

0.27

9

0.28

8

0.29

0

0.28

4

0.27

2

0.25

3

0.29

0

0.31

2

0.26

8

0.18

8

0.13

4

0.19

9

0.19

8

0.19

0

0.14

1

0.25

8

0.23

2

0.26

8

0.26

9

0.27

1

0.27

2

0.27

9

0.28

7

0.29

0

0.28

4

0.27

1

0.25

3

0.29

0

0.31

2

0.26

8

0.18

9

0.13

4

0.20

0

0.19

9

0.18

9

0.13

5

0.20

7

0.16

0

0.14

1

0.13

6

0.14

7

0.22

4

0.27

8

0.28

7

0.28

9

0.28

3

0.27

1

0.25

2

0.29

0

0.31

2

0.26

9

0.18

9

0.13

4

0.20

2

0.19

9

0.18

9

0.13

5

0.20

5

0.15

9

0.13

9

0.13

1

0.13

1

0.16

4

0.27

8

0.28

7

0.28

9

0.28

3

0.27

1

0.25

2

0.29

0

0.31

2

0.26

9

0.19

0

0.13

5

0.20

2

0.19

9

0.19

0

0.13

5

0.20

2

0.15

8

0.13

9

0.12

9

0.12

4

0.12

7

0.27

7

0.28

6

0.28

9

0.28

2

0.27

0

0.25

2

0.29

0

0.31

2

0.27

0

0.19

1

0.13

5

0.20

2

0.19

9

0.18

9

0.13

5

0.20

0

0.15

7

0.13

8

0.12

8

0.12

3

0.11

8

0.27

6

0.28

5

0.28

8

0.28

2

0.26

9

0.25

1

0.29

0

0.31

2

0.27

0

0.19

1

0.13

6

0.20

2

0.19

9

0.19

0

0.13

5

0.19

9

0.15

6

0.13

9

0.12

8

0.12

3

0.11

8

0.27

4

0.28

3

0.28

7

0.28

0

0.26

8

0.25

0

0.28

9

0.31

2

0.27

0

0.19

2

0.13

6

0.20

2

0.19

9

0.19

1

0.13

6

0.19

8

0.15

4

0.13

8

0.12

8

0.12

2

0.11

7

0.27

2

0.28

2

0.28

5

0.27

9

0.26

6

0.24

8

0.28

9

0.31

1

0.27

0

0.19

2

0.13

7

0.20

3

0.19

9

0.19

2

0.13

8

0.20

0

0.15

5

0.13

8

0.12

8

0.12

1

0.11

7

0.27

1

0.28

0

0.28

4

0.27

8

0.26

5

0.24

7

0.28

8

0.31

0

0.26

9

0.19

2

0.13

7

0.20

2

0.19

9

0.19

2

0.13

8

0.20

1

0.15

4

0.13

8

0.12

8

0.12

1

0.11

7

0.26

9

0.27

9

0.28

3

0.27

7

0.26

4

0.24

6

0.28

8

0.31

0

0.26

9

0.19

2

0.13

7

0.20

2

0.19

8

0.19

2

0.13

8

0.20

0

0.15

3

0.13

6

0.12

7

0.12

1

0.11

7

0.26

9

0.27

8

0.28

3

0.27

7

0.26

3

0.24

6

0.28

7

0.31

0

0.26

9

0.19

2

0.13

7

0.20

2

0.19

8

0.19

3

0.13

8

0.20

0

0.15

2

0.13

6

0.12

6

0.12

0

0.11

7

1.24

0.73

2

0.74

4

0.70

5

0.75

8

0.76

4

0.76

9

0.77

2

0.77

9

0.78

0

0.78

5

1.39

0.59

5

0.60

9

0.57

8

0.62

8

0.63

6

0.64

3

0.64

7

0.66

0

0.66

7

0.67

7

1.59

0.51

2

0.53

7

0.52

6

0.57

0

0.58

1

0.59

0

0.60

0

0.61

7

0.61

0

0.61

6

1.79

0.38

9

0.41

4

0.37

8

0.42

0

0.42

4

0.42

6

0.42

1

0.42

8

0.42

9

0.45

8

1.99

0.24

3

0.22

4

0.17

1

0.21

1

0.27

4

0.32

4

0.35

3

0.38

9

0.29

6

0.38

9

2.19

0.24

4

0.22

5

0.16

2

0.21

1

0.27

3

0.32

1

0.34

7

0.38

0

0.26

3

0.35

6

2.39

0.24

4

0.22

6

0.17

8

0.21

2

0.27

3

0.32

0

0.34

6

0.37

7

0.23

3

0.33

3

2.59

0.24

5

0.22

6

0.17

1

0.21

2

0.27

4

0.32

0

0.34

3

0.37

3

0.20

7

0.30

3

2.99

0.24

7

0.22

7

0.17

5

0.21

3

0.27

3

0.31

8

0.34

1

0.37

0

0.16

0

0.09

8

3.39

0.24

8

0.22

8

0.17

2

0.21

4

0.27

3

0.31

8

0.34

0

0.36

9

0.16

1

0.09

0

3.79

0.24

9

0.22

8

0.17

9

0.21

5

0.27

3

0.31

7

0.34

0

0.36

9

0.16

1

0.09

2

4.19

0.25

0

0.22

8

0.18

2

0.21

5

0.27

3

0.31

7

0.34

1

0.36

8

0.16

1

0.09

3

4.59

0.25

0

0.22

7

0.17

9

0.21

6

0.27

3

0.31

7

0.34

1

0.36

9

0.16

1

0.09

4

4.99

0.25

0

0.22

7

0.18

2

0.21

6

0.27

4

0.31

7

0.34

1

0.36

9

0.16

2

0.09

4

5.39

0.24

9

0.22

6

0.18

9

0.21

6

0.27

4

0.31

7

0.34

1

0.36

9

0.16

2

0.09

5

6.19

0.24

8

0.22

5

0.18

6

0.21

7

0.27

3

0.31

7

0.34

2

0.36

9

0.16

2

0.09

7

6.99

0.24

7

0.22

4

0.18

6

0.21

7

0.27

3

0.31

7

0.34

2

0.36

9

0.16

1

0.09

8

7.99

0.24

6

0.22

3

0.18

6

0.21

7

0.27

3

0.31

8

0.34

2

0.36

9

0.16

1

0.09

9

8.89

0.24

5

0.22

2

0.18

5

0.21

7

0.27

3

0.31

8

0.34

2

0.36

9

0.16

1

0.10

0

9.50

0.24

4

0.22

2

0.18

7

0.21

6

0.27

3

0.31

8

0.34

3

0.36

9

0.16

1

0.10

0

NP

R

1.32

7

1.37

0

1.41

3

1.45

6

1.50

0

1.54

3

1.58

6

1.62

9

1.67

2

1.71

5

Tab

le 4

. C

oncl

uded

(c)

Div

erge

nt F

lap,

z

= 0

.199

5 (V

alle

y)

p/p

t,j

at

x/x t

of

22

(d)

Div

erge

nt F

lap,

z

= 1

.595

(H

ill)

p/p

t,j

at

x/x t

of

NP

R

1.11

1

1.15

4

1.19

8

1.24

1

1.28

4

1.32

7

1.37

0

1.41

3

1.45

6

1.50

0

1.54

3

1.58

6

1.62

9

1.67

2

1.71

5

1.75

8

1.80

2

1.84

5

1.88

8

1.93

1

1.97

4

1.25

1.40

1.60

1.80

2.00

2.20

2.40

2.60

3.00

3.40

3.80

4.20

4.60

5.00

5.40

6.19

7.00

8.00

8.89

9.51

0.62

1

0.66

3

0.68

8

0.70

7

0.72

4

0.73

7

0.74

5

0.74

0

0.74

2

0.74

6

0.75

2

0.75

8

0.76

3

0.76

9

0.77

4

0.78

0

0.78

4

0.79

0

0.79

3

0.79

6

0.79

9

0.54

7

0.57

2

0.58

4

0.59

5

0.60

3

0.60

7

0.61

8

0.60

7

0.61

0

0.61

5

0.62

0

0.62

9

0.63

6

0.64

4

0.65

1

0.65

9

0.66

6

0.67

4

0.68

3

0.68

9

0.71

3

0.50

4

0.52

8

0.53

9

0.54

6

0.55

1

0.55

3

0.56

0

0.55

3

0.55

4

0.55

5

0.55

7

0.56

0

0.56

2

0.56

5

0.56

7

0.57

0

0.57

1

0.57

3

0.57

6

0.57

8

0.62

4

0.26

7

0.27

7

0.29

5

0.26

9

0.29

9

0.29

5

0.29

8

0.23

8

0.26

0

0.37

0

0.36

5

0.37

3

0.42

2

0.48

9

0.53

3

0.55

3

0.56

4

0.57

1

0.57

1

0.57

3

0.55

4

0.26

8

0.28

1

0.29

4

0.26

8

0.29

9

0.29

7

0.29

0

0.23

9

0.20

4

0.18

7

0.24

4

0.29

3

0.29

4

0.30

1

0.32

8

0.38

2

0.44

1

0.48

9

0.51

4

0.52

4

0.49

9

0.26

9

0.28

4

0.29

5

0.26

9

0.30

0

0.29

9

0.28

4

0.23

9

0.20

5

0.17

8

0.17

5

0.20

2

0.23

1

0.25

1

0.26

2

0.28

3

0.31

3

0.35

6

0.40

5

0.43

7

0.45

3

0.27

0

0.28

4

0.29

5

0.27

0

0.30

1

0.30

0

0.27

9

0.23

9

0.20

5

0.17

9

0.17

6

0.18

2

0.19

4

0.21

4

0.19

4

0.25

1

0.30

0

0.31

9

0.34

2

0.36

8

0.41

5

0.27

1

0.28

5

0.29

5

0.27

1

0.30

2

0.30

1

0.27

5

0.24

0

0.20

6

0.17

9

0.17

6

0.18

1

0.18

5

0.18

1

0.19

1

0.21

9

0.26

2

0.29

6

0.31

1

0.33

1

0.38

3

0.27

2

0.28

5

0.29

5

0.27

1

0.30

2

0.30

1

0.26

9

0.24

2

0.20

7

0.18

0

0.17

8

0.18

3

0.18

6

0.17

9

0.17

1

0.18

6

0.23

4

0.26

4

0.28

1

0.29

0

0.33

2

0.27

1

0.28

5

0.29

5

0.27

2

0.30

1

0.30

1

0.26

5

0.24

2

0.20

7

0.18

1

0.17

8

0.18

3

0.18

6

0.17

9

0.16

4

0.15

7

0.14

3

0.18

5

0.22

6

0.25

4

0.29

3

0.27

1

0.28

5

0.29

4

0.27

2

0.30

1

0.30

2

0.26

2

0.24

3

0.20

8

0.18

1

0.17

8

0.18

4

0.18

6

0.18

0

0.16

5

0.15

4

0.12

3

0.12

0

0.13

4

0.20

7

0.26

2

0.27

0

0.28

5

0.29

4

0.27

2

0.30

1

0.30

2

0.26

0

0.24

3

0.20

9

0.18

1

0.17

9

0.18

4

0.18

7

0.18

1

0.16

5

0.15

5

0.12

4

0.12

0

0.11

9

0.14

4

0.23

7

0.26

9

0.28

4

0.29

3

0.27

1

0.30

1

0.30

2

0.25

8

0.24

3

0.20

9

0.18

1

0.17

9

0.18

5

0.18

7

0.18

1

0.16

6

0.15

6

0.12

4

0.12

1

0.12

0

0.14

4

0.21

6

0.26

9

0.28

3

0.29

3

0.27

1

0.30

1

0.30

2

0.25

7

0.24

3

0.20

9

0.18

2

0.18

0

0.18

5

0.18

8

0.18

2

0.16

7

0.15

6

0.12

5

0.12

1

0.12

0

0.14

5

0.19

9

0.26

8

0.28

2

0.29

2

0.27

1

0.30

0

0.30

1

0.25

6

0.24

3

0.20

9

0.18

2

0.18

0

0.18

6

0.18

8

0.18

2

0.16

7

0.15

7

0.12

6

0.12

2

0.12

1

0.14

6

0.18

4

0.26

6

0.28

1

0.29

0

0.26

9

0.29

9

0.30

0

0.25

3

0.24

3

0.20

9

0.18

2

0.18

0

0.18

6

0.18

8

0.18

2

0.16

8

0.15

8

0.12

7

0.12

2

0.12

2

0.14

7

0.16

1

0.26

4

0.27

9

0.28

8

0.26

8

0.29

8

0.29

9

0.25

1

0.24

2

0.20

9

0.18

1

0.18

0

0.18

7

0.18

9

0.18

3

0.16

9

0.15

8

0.12

7

0.12

2

0.12

2

0.14

8

0.14

2

0.26

2

0.27

7

0.28

7

0.26

6

0.29

7

0.29

8

0.24

9

0.24

2

0.20

9

0.18

1

0.18

0

0.18

7

0.18

9

0.18

4

0.17

0

0.15

9

0.12

8

0.12

3

0.12

2

0.14

9

0.12

5

0.26

1

0.27

6

0.28

5

0.26

5

0.29

6

0.29

7

0.24

8

0.24

1

0.20

9

0.18

1

0.18

0

0.18

7

0.18

9

0.18

4

0.17

0

0.16

0

0.12

8

0.12

3

0.12

3

0.14

9

0.11

2

0.26

0

0.27

6

0.28

4

0.26

5

0.29

5

0.29

7

0.24

7

0.24

1

0.20

9

0.18

1

0.18

0

0.18

8

0.18

9

0.18

4

0.17

0

0.16

0

0.12

8

0.12

3

0.12

3

0.15

0

0.10

5

NP

R

0.33

0

0.44

0

0.55

0

0.65

9

0.76

9

0.89

0

1.00

0

1.25

1.40

1.60

1.80

2.00

2.20

2.40

2.60

3.00

3.40

3.80

4.20

4.60

5.00

5.40

6.19

7.00

8.00

8.89

9.51

0.96

1

0.94

9

0.92

9

0.90

3

0.85

7

0.68

4

0.44

3

0.95

8

0.94

6

0.92

6

0.90

0

0.85

2

0.67

5

0.42

2

0.95

8

0.94

5

0.92

6

0.89

8

0.84

9

0.67

0

0.37

1

0.95

7

0.94

5

0.92

7

0.89

8

0.85

0

0.67

0

0.36

8

0.95

8

0.94

5

0.92

7

0.89

9

0.85

0

0.67

0

0.36

7

0.95

8

0.94

5

0.92

7

0.89

8

0.85

0

0.67

0

0.36

7

0.95

8

0.94

5

0.92

8

0.89

8

0.85

0

0.67

0

0.36

6

0.95

8

0.94

5

0.92

8

0.89

9

0.85

0

0.67

0

0.36

6

0.95

8

0.94

5

0.92

8

0.89

9

0.85

0

0.67

0

0.36

7

0.95

8

0.94

5

0.92

8

0.89

9

0.84

9

0.67

0

0.36

7

0.95

8

0.94

5

0.92

8

0.89

9

0.85

0

0.67

0

0.36

7

0.95

8

0.94

5

0.92

8

0.89

9

0.84

9

0.67

0

0.36

7

0.95

8

0.94

5

0.92

7

0.89

9

0.84

8

0.67

0

0.36

6

0.95

8

0.94

5

0.92

7

0.89

8

0.84

8

0.66

9

0.36

5

0.95

8

0.94

4

0.92

7

0.89

8

0.84

7

0.66

9

0.36

4

0.95

7

0.94

3

0.92

6

0.89

7

0.84

6

0.66

8

0.36

3

0.95

7

0.94

3

0.92

5

0.89

6

0.84

5

0.66

6

0.36

1

0.95

6

0.94

2

0.92

4

0.89

5

0.84

4

0.66

5

0.35

9

0.95

5

0.94

2

0.92

3

0.89

4

0.84

3

0.66

5

0.35

8

0.95

5

0.94

2

0.92

3

0.89

4

0.84

2

0.66

5

0.35

7

Tab

le 5

. N

ozzl

e In

tern

al S

tatic

Pre

ssur

e R

atio

s fo

r th

e M

ediu

m-L

ong

Con

volu

ted

Con

figur

atio

n

(a)

Con

verg

ent

Fla

p,

z

= 0

.000

p/p

t,j

at

x/x t

of

23

(b)

Div

erge

nt F

lap,

z

= 0

.000

(H

ill)

p/p

t,j

at

x/x t

of

NP

R

1.11

1

1.15

4

1.19

8

1.24

1

1.28

4

1.32

7

1.37

0

1.41

3

1.45

6

1.50

0

1.54

3

1.58

6

1.62

9

1.67

2

1.71

5

1.75

8

1.80

2

1.84

5

1.88

8

1.93

1

1.97

4

1.25

1.40

1.60

1.80

2.00

2.20

2.40

2.60

3.00

3.40

3.80

4.20

4.60

5.00

5.40

6.19

7.00

8.00

8.89

9.51

0.66

6

0.70

4

0.71

5

0.72

2

0.73

3

0.74

2

0.74

3

0.74

2

0.74

4

0.75

0

0.75

9

0.76

7

0.77

1

0.78

0

0.78

4

0.78

8

0.79

1

0.79

6

0.79

6

0.79

8

0.79

8

0.50

8

0.55

1

0.57

7

0.59

5

0.61

4

0.62

9

0.63

8

0.63

8

0.63

9

0.64

2

0.64

9

0.65

5

0.65

9

0.66

9

0.67

3

0.67

9

0.68

5

0.69

3

0.69

9

0.70

4

0.70

6

0.49

9

0.52

0

0.53

1

0.53

9

0.54

5

0.55

0

0.55

4

0.55

4

0.55

7

0.56

1

0.56

3

0.56

7

0.56

7

0.57

2

0.57

3

0.57

6

0.57

7

0.58

1

0.58

8

0.59

4

0.58

8

0.26

8

0.27

9

0.30

3

0.28

3

0.32

7

0.33

5

0.30

5

0.28

9

0.40

7

0.47

0

0.48

3

0.48

7

0.49

2

0.50

3

0.50

8

0.51

7

0.53

0

0.54

3

0.54

8

0.55

7

0.55

7

0.26

7

0.28

1

0.30

2

0.28

1

0.32

5

0.33

2

0.30

5

0.29

7

0.31

8

0.32

2

0.37

1

0.40

1

0.43

2

0.45

4

0.46

0

0.46

7

0.47

9

0.48

9

0.49

0

0.49

7

0.49

6

0.26

9

0.28

2

0.30

3

0.27

8

0.31

6

0.30

6

0.27

8

0.26

4

0.22

3

0.27

0

0.29

3

0.32

7

0.38

4

0.40

3

0.40

5

0.41

3

0.42

9

0.44

3

0.44

6

0.45

5

0.45

1

0.27

0

0.28

5

0.30

3

0.27

7

0.31

4

0.30

3

0.27

2

0.24

8

0.21

5

0.19

2

0.21

0

0.26

7

0.27

4

0.33

6

0.36

1

0.36

8

0.38

7

0.40

3

0.40

6

0.42

6

0.41

9

0.27

3

0.29

1

0.30

4

0.27

7

0.31

3

0.30

3

0.27

1

0.24

5

0.21

0

0.18

3

0.18

3

0.19

4

0.24

5

0.24

3

0.26

0

0.30

9

0.34

2

0.36

2

0.36

8

0.39

5

0.38

9

0.27

7

0.29

4

0.30

4

0.27

7

0.31

2

0.30

3

0.27

0

0.24

3

0.20

9

0.18

0

0.17

6

0.18

1

0.18

5

0.18

3

0.17

2

0.18

2

0.24

2

0.27

3

0.29

8

0.31

7

0.33

1

0.27

8

0.29

4

0.30

4

0.27

8

0.31

2

0.30

3

0.27

1

0.24

3

0.21

0

0.18

0

0.17

6

0.18

1

0.18

4

0.18

1

0.16

8

0.15

9

0.12

8

0.15

4

0.20

9

0.27

5

0.29

5

0.27

8

0.29

4

0.30

4

0.27

8

0.31

3

0.30

3

0.27

1

0.24

4

0.21

0

0.18

1

0.17

7

0.18

1

0.18

4

0.18

1

0.16

8

0.15

9

0.12

7

0.12

6

0.13

1

0.15

7

0.20

8

0.27

7

0.29

4

0.30

4

0.27

8

0.31

3

0.30

3

0.27

1

0.24

5

0.21

1

0.18

1

0.17

7

0.18

2

0.18

5

0.18

1

0.16

9

0.15

9

0.12

8

0.12

6

0.13

0

0.15

7

0.15

5

0.27

7

0.29

3

0.30

4

0.27

7

0.31

3

0.30

3

0.27

1

0.24

5

0.21

1

0.18

2

0.17

8

0.18

2

0.18

5

0.18

1

0.16

9

0.15

9

0.12

8

0.12

6

0.12

9

0.15

6

0.14

1

0.27

6

0.29

3

0.30

3

0.27

7

0.31

3

0.30

3

0.27

1

0.24

4

0.21

1

0.18

2

0.17

8

0.18

2

0.18

5

0.18

2

0.17

0

0.16

0

0.12

9

0.12

6

0.12

9

0.15

3

0.13

5

0.27

5

0.29

2

0.30

3

0.27

6

0.31

3

0.30

3

0.27

1

0.24

5

0.21

1

0.18

2

0.17

9

0.18

3

0.18

6

0.18

2

0.17

0

0.16

0

0.13

0

0.12

7

0.13

0

0.15

1

0.12

9

0.27

3

0.29

1

0.30

1

0.27

5

0.31

2

0.30

2

0.27

1

0.24

5

0.21

2

0.18

2

0.17

9

0.18

4

0.18

7

0.18

3

0.17

1

0.16

1

0.13

1

0.12

8

0.13

1

0.15

0

0.12

5

0.27

1

0.28

8

0.30

0

0.27

2

0.31

0

0.30

0

0.27

0

0.24

4

0.21

1

0.18

3

0.17

9

0.18

5

0.18

7

0.18

4

0.17

1

0.16

3

0.13

2

0.13

1

0.13

3

0.15

3

0.12

4

0.26

9

0.28

7

0.29

8

0.27

1

0.30

9

0.29

8

0.27

0

0.24

4

0.21

1

0.18

2

0.18

0

0.18

5

0.18

8

0.18

4

0.17

2

0.16

3

0.13

2

0.13

1

0.13

3

0.15

2

0.12

4

0.26

8

0.28

6

0.29

7

0.27

0

0.30

8

0.29

7

0.26

9

0.24

4

0.21

1

0.18

2

0.18

0

0.18

5

0.18

8

0.18

5

0.17

2

0.16

3

0.13

2

0.13

1

0.13

2

0.15

1

0.12

3

0.26

8

0.28

5

0.29

7

0.26

9

0.30

7

0.29

7

0.26

9

0.24

3

0.21

0

0.18

2

0.18

0

0.18

5

0.18

8

0.18

5

0.17

2

0.16

3

0.13

2

0.13

1

0.13

2

0.15

0

0.12

2

NP

R

1.32

7

1.37

0

1.41

3

1.45

6

1.50

0

1.54

3

1.58

6

1.62

9

1.67

2

1.71

5

1.25

1.40

1.60

1.80

2.00

2.20

2.40

2.60

3.00

3.40

3.80

4.20

4.60

5.00

5.40

6.19

7.00

8.00

8.89

9.51

0.72

8

0.73

9

0.74

3

0.75

2

0.75

7

0.76

2

0.76

6

0.77

0

0.77

5

0.77

9

0.61

6

0.62

1

0.62

3

0.63

0

0.63

5

0.63

9

0.64

4

0.64

8

0.65

3

0.65

9

0.56

1

0.56

6

0.56

5

0.57

0

0.57

2

0.57

3

0.57

4

0.57

4

0.57

4

0.57

5

0.21

9

0.22

7

0.24

6

0.32

1

0.39

2

0.42

2

0.44

4

0.46

9

0.51

2

0.53

6

0.21

9

0.22

7

0.24

1

0.24

7

0.26

2

0.28

4

0.32

0

0.35

3

0.37

5

0.37

3

0.21

8

0.22

8

0.24

1

0.24

7

0.25

9

0.26

4

0.26

7

0.26

3

0.27

8

0.29

4

0.21

8

0.22

8

0.24

2

0.24

6

0.25

9

0.26

3

0.26

5

0.26

1

0.31

6

0.31

7

0.21

8

0.22

8

0.24

2

0.24

7

0.25

9

0.26

3

0.26

4

0.25

2

0.23

4

0.19

7

0.21

8

0.22

8

0.24

1

0.24

7

0.25

9

0.26

3

0.26

4

0.25

1

0.23

4

0.19

3

0.21

8

0.22

9

0.24

2

0.24

8

0.25

9

0.26

3

0.26

3

0.25

0

0.23

4

0.19

3

0.21

8

0.22

9

0.24

2

0.24

8

0.25

9

0.26

3

0.26

4

0.25

0

0.23

4

0.19

3

0.21

8

0.22

9

0.24

2

0.24

8

0.25

9

0.26

3

0.26

4

0.25

1

0.23

4

0.19

3

0.21

8

0.22

9

0.24

2

0.24

8

0.26

0

0.26

3

0.26

4

0.25

1

0.23

4

0.19

4

0.21

7

0.22

9

0.24

3

0.24

9

0.26

0

0.26

4

0.26

4

0.25

1

0.23

5

0.19

4

0.21

6

0.22

8

0.24

3

0.24

9

0.26

0

0.26

4

0.26

5

0.25

1

0.23

5

0.19

4

0.21

5

0.22

8

0.24

2

0.24

9

0.26

0

0.26

4

0.26

5

0.25

1

0.23

5

0.19

5

0.21

4

0.22

7

0.24

1

0.24

8

0.26

0

0.26

3

0.26

5

0.25

1

0.23

6

0.19

5

0.21

3

0.22

6

0.24

1

0.24

8

0.26

0

0.26

4

0.26

5

0.25

2

0.23

7

0.19

5

0.21

3

0.22

6

0.24

0

0.24

8

0.25

9

0.26

4

0.26

6

0.25

2

0.23

7

0.19

6

0.21

2

0.22

6

0.24

0

0.24

8

0.25

9

0.26

4

0.26

6

0.25

2

0.23

7

0.19

6

Tab

le 5

. C

oncl

uded

(c)

Div

erge

nt F

lap,

z

= 0

.199

5 (V

alle

y)

p/p

t,j

at

x/x t

of

(d)

Div

erge

nt F

lap,

z

= 1

.595

(H

ill)

p/p

t,j

at

x/x t

of

24

NP

R

1.11

1

1.15

4

1.19

8

1.24

1

1.28

4

1.32

7

1.37

0

1.41

3

1.45

6

1.50

0

1.54

3

1.58

6

1.62

9

1.67

2

1.71

5

1.75

8

1.80

2

1.84

5

1.88

8

1.93

1

1.97

4

1.25

0.62

1

0.65

9

0.68

4

0.70

6

0.72

3

0.74

2

0.76

2

0.79

8

0.76

0

0.69

9

0.70

4

0.75

6

0.76

0

0.76

4

0.77

0

0.77

5

0.78

0

0.78

4

0.78

8

0.79

3

0.79

5

1.40

0.56

0

0.58

0

0.59

3

0.60

3

0.61

3

0.62

2

0.62

7

0.64

3

0.64

4

0.62

9

0.63

7

0.65

0

0.65

6

0.66

2

0.66

8

0.67

2

0.67

6

0.68

0

0.68

4

0.68

7

0.68

4

1.60

0.50

0

0.51

9

0.53

0

0.54

0

0.54

6

0.55

3

0.55

2

0.55

3

0.55

6

0.55

9

0.56

5

0.57

1

0.57

9

0.58

4

0.58

5

0.58

6

0.58

6

0.58

6

0.58

6

0.58

7

0.58

2

1.80

0.27

3

0.28

1

0.28

3

0.28

2

0.27

6

0.28

3

0.46

5

0.59

7

0.38

4

0.30

9

0.39

7

0.38

6

0.38

5

0.41

4

0.45

4

0.48

8

0.51

4

0.53

2

0.54

5

0.55

3

0.55

7

2.00

0.27

3

0.28

2

0.28

4

0.28

2

0.27

6

0.28

1

0.44

3

0.57

4

0.36

8

0.21

0

0.31

2

0.31

3

0.33

5

0.34

4

0.37

2

0.42

4

0.43

6

0.43

7

0.44

4

0.45

4

0.47

2

2.20

0.27

4

0.28

2

0.28

3

0.28

2

0.27

7

0.28

0

0.42

5

0.54

7

0.35

1

0.18

0

0.22

8

0.25

8

0.27

2

0.27

3

0.29

4

0.34

6

0.39

5

0.41

0

0.41

2

0.41

8

0.43

5

2.40

0.27

5

0.28

2

0.28

4

0.28

3

0.27

7

0.28

1

0.42

4

0.54

6

0.34

9

0.17

2

0.19

6

0.24

6

0.25

0

0.24

3

0.25

1

0.27

2

0.31

0

0.34

2

0.36

7

0.38

6

0.40

2

2.60

0.27

5

0.28

3

0.28

5

0.28

4

0.27

7

0.28

1

0.42

3

0.54

5

0.34

9

0.17

2

0.20

6

0.24

7

0.25

1

0.23

9

0.24

0

0.24

8

0.27

2

0.29

9

0.32

3

0.34

5

0.36

2

3.00

0.27

6

0.28

3

0.28

5

0.28

3

0.27

7

0.28

1

0.42

1

0.54

5

0.35

0

0.17

1

0.16

1

0.23

6

0.23

4

0.22

5

0.22

1

0.22

2

0.23

5

0.25

2

0.26

9

0.28

8

0.30

0

3.39

0.27

6

0.28

2

0.28

4

0.28

2

0.27

6

0.28

1

0.42

0

0.54

5

0.35

0

0.17

2

0.13

9

0.22

5

0.21

6

0.20

9

0.21

1

0.21

1

0.21

8

0.23

0

0.24

3

0.25

8

0.26

5

3.80

0.27

5

0.28

2

0.28

3

0.28

1

0.27

6

0.28

1

0.42

0

0.54

5

0.35

1

0.17

3

0.13

4

0.21

6

0.20

0

0.18

9

0.19

8

0.20

4

0.20

6

0.21

4

0.22

3

0.23

4

0.24

1

4.20

0.27

5

0.28

1

0.28

3

0.28

1

0.27

6

0.28

1

0.42

0

0.54

6

0.35

1

0.17

4

0.13

3

0.20

4

0.17

3

0.16

0

0.14

0

0.15

4

0.16

9

0.18

1

0.19

4

0.20

7

0.22

1

4.60

0.27

4

0.28

0

0.28

2

0.28

0

0.27

5

0.28

1

0.42

0

0.54

6

0.35

1

0.17

5

0.13

4

0.20

2

0.17

2

0.15

9

0.13

0

0.13

6

0.13

1

0.14

1

0.15

8

0.17

2

0.19

5

4.99

0.27

3

0.28

0

0.28

2

0.28

0

0.27

5

0.28

1

0.42

0

0.54

7

0.35

1

0.17

5

0.13

4

0.20

2

0.17

3

0.16

1

0.13

0

0.13

7

0.12

6

0.12

7

0.14

0

0.13

9

0.16

5

5.40

0.27

2

0.27

9

0.28

1

0.27

9

0.27

5

0.28

1

0.42

0

0.54

7

0.35

2

0.17

6

0.13

4

0.20

1

0.17

4

0.16

2

0.13

1

0.13

7

0.12

7

0.12

7

0.13

8

0.12

9

0.13

3

6.20

0.27

0

0.27

7

0.27

9

0.27

8

0.27

3

0.28

0

0.41

9

0.54

7

0.35

2

0.17

7

0.13

5

0.20

0

0.17

5

0.16

3

0.13

3

0.13

9

0.12

7

0.12

8

0.13

9

0.12

5

0.11

9

6.99

0.26

9

0.27

5

0.27

8

0.27

6

0.27

2

0.27

8

0.41

8

0.54

8

0.35

2

0.17

7

0.13

6

0.19

9

0.17

6

0.16

4

0.13

4

0.14

0

0.12

8

0.12

9

0.13

9

0.12

5

0.11

7

8.00

0.26

7

0.27

3

0.27

6

0.27

4

0.27

1

0.27

7

0.41

8

0.54

7

0.35

2

0.17

7

0.13

6

0.19

8

0.17

7

0.16

5

0.13

4

0.14

0

0.12

9

0.12

9

0.13

9

0.12

5

0.11

5

8.89

0.26

5

0.27

2

0.27

5

0.27

3

0.27

0

0.27

6

0.41

7

0.54

8

0.35

2

0.17

8

0.13

7

0.19

7

0.17

8

0.16

5

0.13

5

0.14

1

0.12

9

0.12

9

0.14

0

0.12

5

0.11

4

9.50

0.26

5

0.27

1

0.27

4

0.27

3

0.26

9

0.27

6

0.41

7

0.54

8

0.35

2

0.17

8

0.13

7

0.19

6

0.17

8

0.16

5

0.13

5

0.14

1

0.12

9

0.13

0

0.14

0

0.12

5

0.11

4

NP

R

0.33

0

0.44

0

0.55

0

0.65

9

0.76

9

0.89

0

1.00

0

1.25

0.96

0

0.94

9

0.92

8

0.90

4

0.85

7

0.68

2

0.45

5

1.40

0.95

9

0.94

6

0.92

6

0.90

0

0.85

2

0.67

4

0.44

1

1.60

0.95

9

0.94

5

0.92

6

0.90

0

0.85

1

0.67

1

0.38

5

1.80

0.95

8

0.94

6

0.92

6

0.90

0

0.85

0

0.67

0

0.36

8

2.00

0.95

9

0.94

5

0.92

7

0.90

0

0.85

0

0.67

1

0.36

8

2.20

0.95

9

0.94

6

0.92

7

0.90

0

0.85

0

0.67

1

0.36

8

2.40

0.95

9

0.94

6

0.92

8

0.90

0

0.85

1

0.67

1

0.36

7

2.60

0.95

9

0.94

6

0.92

8

0.90

1

0.85

1

0.67

1

0.36

7

3.00

0.95

9

0.94

6

0.92

8

0.90

0

0.85

1

0.67

1

0.36

7

3.39

0.95

9

0.94

6

0.92

8

0.90

0

0.85

0

0.67

1

0.36

7

3.80

0.95

9

0.94

5

0.92

8

0.90

0

0.84

9

0.67

0

0.36

7

4.20

0.95

8

0.94

5

0.92

8

0.89

9

0.84

9

0.67

0

0.36

7

4.60

0.95

8

0.94

5

0.92

8

0.89

9

0.84

8

0.67

0

0.36

6

4.99

0.95

8

0.94

5

0.92

7

0.89

9

0.84

8

0.66

9

0.36

6

5.40

0.95

8

0.94

5

0.92

6

0.89

8

0.84

7

0.66

9

0.36

5

6.20

0.95

7

0.94

4

0.92

6

0.89

7

0.84

6

0.66

7

0.36

3

6.99

0.95

7

0.94

3

0.92

5

0.89

6

0.84

5

0.66

6

0.36

1

8.00

0.95

6

0.94

3

0.92

4

0.89

5

0.84

4

0.66

5

0.35

9

8.89

0.95

6

0.94

2

0.92

3

0.89

4

0.84

3

0.66

5

0.35

8

9.50

0.95

5

0.94

2

0.92

2

0.89

4

0.84

2

0.66

5

0.35

7

Tab

le 6

. N

ozzl

e In

tern

al S

tatic

Pre

ssur

e R

atio

s fo

r th

e C

oars

e C

onvo

lute

d C

onfig

urat

ion

(a)

Con

verg

ent

Fla

p,

z

= 0

.000

p/p

t,j

at

x/x t

of

25

(b)

Div

erge

nt F

lap,

z

= 0

.000

(H

ill)

p/p

t,j

at

x/x t

of

NP

R

1.11

1

1.15

4

1.19

8

1.24

1

1.28

4

1.32

7

1.37

0

1.41

3

1.45

6

1.50

0

1.54

3

1.58

6

1.62

9

1.67

2

1.71

5

1.75

8

1.80

2

1.84

5

1.88

8

1.93

1

1.97

4

1.25

1.40

1.60

1.80

2.00

2.20

2.40

2.60

3.00

3.39

3.80

4.20

4.60

4.99

5.40

6.20

6.99

8.00

8.89

9.50

0.64

5

0.67

6

0.69

4

0.71

1

0.72

7

0.75

8

0.79

4

0.82

7

0.75

3

0.69

7

0.69

4

0.76

5

0.76

3

0.76

0

0.76

7

0.77

4

0.77

9

0.78

6

0.78

8

0.79

3

0.79

5

0.55

6

0.58

3

0.59

8

0.61

2

0.62

3

0.63

5

0.64

5

0.66

7

0.66

2

0.63

1

0.63

7

0.66

2

0.66

4

0.66

9

0.67

2

0.67

6

0.67

8

0.68

3

0.68

7

0.69

2

0.68

5

0.49

2

0.51

6

0.53

0

0.54

1

0.54

8

0.55

6

0.56

0

0.57

0

0.57

4

0.56

3

0.56

7

0.57

6

0.58

0

0.58

4

0.58

5

0.58

7

0.59

0

0.59

4

0.59

6

0.60

1

0.59

1

0.27

4

0.29

1

0.28

9

0.29

1

0.28

5

0.36

4

0.48

8

0.65

0

0.49

5

0.34

7

0.36

5

0.41

7

0.43

6

0.46

5

0.50

3

0.53

5

0.54

6

0.55

3

0.55

5

0.55

8

0.55

7

0.27

2

0.28

9

0.28

9

0.28

9

0.27

9

0.29

2

0.44

0

0.55

6

0.37

9

0.21

1

0.33

2

0.33

0

0.34

3

0.35

9

0.38

0

0.42

0

0.44

1

0.45

6

0.46

7

0.47

8

0.48

6

0.27

3

0.28

7

0.28

8

0.28

9

0.28

0

0.29

0

0.44

2

0.55

7

0.39

8

0.25

1

0.24

4

0.30

5

0.32

7

0.34

2

0.36

2

0.40

7

0.43

1

0.44

3

0.44

9

0.45

2

0.45

4

0.27

6

0.28

8

0.28

8

0.28

9

0.27

9

0.28

8

0.42

7

0.53

2

0.36

8

0.19

0

0.19

9

0.22

6

0.25

0

0.25

6

0.27

4

0.35

1

0.38

9

0.39

9

0.40

0

0.40

0

0.40

0

0.28

0

0.29

0

0.29

0

0.28

9

0.27

9

0.28

7

0.42

3

0.52

3

0.36

4

0.18

6

0.14

4

0.22

0

0.22

6

0.21

9

0.23

3

0.28

1

0.33

7

0.36

1

0.36

6

0.36

9

0.37

1

0.28

4

0.29

1

0.29

0

0.28

8

0.27

8

0.28

6

0.41

9

0.52

0

0.36

1

0.18

0

0.12

1

0.21

9

0.20

7

0.19

0

0.20

4

0.22

0

0.24

6

0.27

5

0.29

4

0.31

1

0.32

2

0.28

3

0.29

1

0.29

0

0.28

8

0.27

8

0.28

6

0.41

8

0.51

9

0.36

0

0.17

9

0.12

9

0.22

3

0.21

0

0.19

6

0.20

4

0.20

8

0.21

9

0.23

3

0.24

5

0.25

9

0.27

3

0.28

2

0.29

1

0.29

0

0.28

8

0.27

7

0.28

5

0.41

8

0.51

9

0.36

0

0.17

9

0.12

5

0.22

1

0.20

8

0.19

3

0.20

1

0.20

1

0.20

4

0.21

2

0.22

1

0.23

3

0.23

9

0.28

2

0.29

0

0.29

0

0.28

8

0.27

8

0.28

6

0.41

8

0.51

9

0.36

1

0.18

0

0.11

6

0.21

3

0.18

4

0.16

1

0.16

1

0.17

5

0.18

4

0.19

2

0.19

9

0.20

9

0.21

8

0.28

2

0.29

0

0.29

0

0.28

7

0.27

7

0.28

6

0.41

9

0.51

9

0.36

1

0.18

0

0.11

4

0.21

1

0.17

7

0.14

8

0.12

9

0.13

5

0.14

7

0.16

4

0.17

4

0.18

4

0.19

6

0.28

1

0.29

0

0.29

0

0.28

7

0.27

7

0.28

6

0.41

9

0.51

9

0.36

1

0.18

1

0.11

4

0.21

0

0.17

7

0.14

7

0.12

6

0.12

8

0.12

7

0.14

0

0.15

4

0.16

4

0.18

3

0.28

1

0.28

9

0.28

9

0.28

7

0.27

7

0.28

5

0.41

9

0.51

9

0.36

2

0.18

1

0.11

4

0.20

9

0.17

8

0.14

8

0.12

7

0.12

8

0.12

4

0.13

3

0.14

0

0.14

4

0.16

0

0.27

9

0.28

8

0.28

7

0.28

6

0.27

6

0.28

5

0.41

9

0.51

9

0.36

2

0.18

2

0.11

5

0.20

8

0.18

0

0.15

0

0.12

8

0.12

9

0.12

3

0.13

0

0.13

4

0.12

9

0.12

6

0.27

7

0.28

6

0.28

6

0.28

5

0.27

5

0.28

4

0.41

8

0.51

9

0.36

2

0.18

2

0.11

5

0.20

8

0.18

1

0.15

1

0.12

9

0.13

0

0.12

3

0.13

0

0.13

3

0.12

6

0.11

9

0.27

6

0.28

5

0.28

4

0.28

3

0.27

4

0.28

4

0.41

8

0.51

9

0.36

2

0.18

3

0.11

5

0.20

6

0.18

3

0.15

4

0.13

0

0.13

1

0.12

3

0.13

0

0.13

3

0.12

4

0.11

8

0.27

4

0.28

3

0.28

3

0.28

2

0.27

3

0.28

3

0.41

8

0.52

0

0.36

3

0.18

3

0.11

5

0.20

4

0.18

5

0.15

9

0.13

3

0.13

4

0.12

4

0.13

1

0.13

5

0.12

4

0.11

6

0.27

4

0.28

3

0.28

2

0.28

1

0.27

2

0.28

3

0.41

8

0.52

0

0.36

3

0.18

3

0.11

6

0.20

3

0.18

6

0.16

1

0.13

4

0.13

5

0.12

5

0.13

2

0.13

6

0.12

3

0.11

5

NP

R

1.32

7

1.37

0

1.41

3

1.45

6

1.50

0

1.54

3

1.58

6

1.62

9

1.67

2

1.71

5

1.25

0.72

9

0.73

6

0.74

8

0.75

1

0.75

5

0.76

1

0.76

5

0.77

2

0.76

9

0.77

0

1.40

0.62

7

0.63

3

0.64

8

0.66

3

0.66

6

0.66

7

0.66

6

0.66

4

0.65

8

0.66

4

1.60

0.55

7

0.55

9

0.57

5

0.59

0

0.59

2

0.59

2

0.58

9

0.58

6

0.57

8

0.58

2

1.80

0.28

6

0.37

6

0.37

7

0.37

7

0.37

3

0.37

2

0.37

8

0.41

2

0.47

5

0.46

1

2.00

0.25

0

0.29

2

0.30

8

0.32

0

0.33

9

0.36

0

0.38

8

0.45

7

0.49

0

0.36

3

2.20

0.24

9

0.25

5

0.27

2

0.29

2

0.32

0

0.35

8

0.40

9

0.50

1

0.44

5

0.27

0

2.40

0.24

9

0.24

1

0.25

8

0.27

9

0.31

5

0.36

0

0.42

2

0.52

7

0.41

0

0.21

2

2.60

0.24

9

0.23

8

0.25

4

0.27

5

0.31

4

0.36

1

0.42

4

0.53

0

0.39

8

0.18

9

3.00

0.24

9

0.23

5

0.25

1

0.27

2

0.31

1

0.36

2

0.42

7

0.52

9

0.38

2

0.16

3

3.39

0.24

9

0.23

5

0.25

1

0.27

0

0.30

9

0.36

1

0.42

9

0.53

1

0.37

4

0.14

9

3.80

0.24

9

0.23

5

0.25

0

0.26

9

0.30

8

0.36

0

0.42

9

0.53

4

0.36

7

0.13

7

4.20

0.24

9

0.23

5

0.24

9

0.26

8

0.30

8

0.36

1

0.42

9

0.53

6

0.36

2

0.11

7

4.60

0.24

8

0.23

4

0.24

9

0.26

9

0.31

0

0.36

2

0.42

9

0.53

7

0.36

1

0.11

6

4.99

0.24

8

0.23

4

0.24

9

0.26

9

0.31

1

0.36

3

0.42

8

0.53

7

0.36

1

0.11

7

5.40

0.24

8

0.23

4

0.24

8

0.27

0

0.31

2

0.36

3

0.42

7

0.53

7

0.36

1

0.11

7

6.20

0.24

7

0.23

3

0.24

8

0.27

0

0.31

2

0.36

2

0.42

5

0.53

8

0.36

1

0.11

7

6.99

0.24

5

0.23

3

0.24

7

0.27

0

0.31

3

0.36

1

0.42

3

0.53

8

0.36

0

0.11

8

8.00

0.24

4

0.23

3

0.24

7

0.27

0

0.31

3

0.36

0

0.42

2

0.53

9

0.36

0

0.11

8

8.89

0.24

4

0.23

2

0.24

7

0.27

0

0.31

2

0.35

9

0.42

2

0.54

0

0.36

0

0.11

8

9.50

0.24

3

0.23

2

0.24

7

0.27

0

0.31

2

0.35

8

0.42

2

0.54

1

0.35

9

0.11

8

Tab

le 6

. C

oncl

uded

(c)

Div

erge

nt F

lap,

z

= 0

.199

5 (V

alle

y)

p/p

t,j

at

x/x t

of

(d)

Div

erge

nt F

lap,

z

= 1

.595

(H

ill)

p/p

t,j

at

x/x t

of

26

Table 7. Measured Peak Thrust Ratio (

F/Fi

)

peak

at NPR=8.9

27

Table 8. Wetted Area, Skin Friction Drag Penalties

∆

(

F/Fi

)

f

, and Peak Thrust Ratio (

F/Fi

)

peak

Figure 2. Sketch showing a typical variable geometry nonaxisymmetric exhaust nozzle.

28

Maximum Power at Mach 2

Dash to Mach 2

Takeoff Condition

Subsonic Cruise

Figure 1. Sketch showing a typical variable geometry nozzle over several operating conditions.

(a)

Pho

togr

aph

of th

e pr

opul

sion

sim

ulat

ion

syst

em w

ith a

typi

cal n

ozzl

e co

nfig

urat

ion

inst

alle

d.

Fig

ure

3. D

etai

ls o

f the

sin

gle-

engi

ne p

ropu

lsio

n si

mul

atio

n sy

stem

.

29

(b)

Ske

tch

of th

e pr

opul

sion

sim

ulat

ion

syst

em.

Sta

tion

num

bers

are

in in

ches

.

Fig

ure

3. C

oncl

uded

.

30

Tot

al-p

ress

ure

prob

es

Tot

al te

mpe

ratu

repr

obes

Typ

ical

sec

tion

inin

stru

men

tatio

n se

ctio

n

Typ

ical

sec

tion

intr

ansi

tion

sect

ion

Typ

ical

sec

tion

ahea

dof

tran

sitio

n se

ctio

n

Noz

zle

Cho

ke p

late

Inst

rum

enta

tion

sect

ion

Tra

nsiti

onse

ctio

n

Sta

. 41.

13

Low

-pre

ssur

e pl

enum

Hig

h-pr

essu

re p

lenu

m

Bal

ance

Fle

xibl

e se

als

(met

al b

ello

ws)

Sta

. 29.

39

Airf

low

8 eq

ually

spa

ced

nozz

les

exiti

ng r

adia

lly

Sup

port

str

ut

Figure 4. Partial cutaway sketch of a typical turbofan forced mixer.

Figure 5. Sketch showing streamwise vorticity generated by a convoluted mixer.

31

Interchangeable inserts

Interchangeable inserts

Figure 6. Sketch showing "bump" type convoluted contouring used in this investigation.

Figure 7. Photograph of the static test model with baseline flap inserts installed.

32

Sta.41.13

Nozzle flap assembly

Sidewall assembly

A

A

End View(looking upstream)

Section A-A

B

B

Section B-B

Sidewall assembly

Glass window

Interchangeabledivergent flap inserts

3-Quarter View

y

x

z

Figure 8. Sketch of nozzle model with baseline flap inserts (shaded) installed.

33

1

2

3

4

5

6

8

9

2.000R

α

=11.01°

7

Sta.41.13

0.0000.0000.0000.9171.9882.3942.4302.2754.550

0.000-0.6141.3861.1630.6110.5530.5591.1660.972

x, in.

Coordinates

Point

123456789

y, in.

Nozzlecenterline

x

y

Figure 9. Sketch showing nozzle geometric details. Dimensions are in inches.

34

(a) Nozzle flap assembly.

A

A

Sta.41.13

Section A-A

Glass window

0.875

3.675

1.944

(b) Sidewall assembly.

Figure 9. Concluded.

35

(a)

Fin

e co

nvol

uted

flap

inse

rts.

Fig

ure

10.

Det

ails

of c

onvo

lute

d fla

p in

sert

s. D

imen

sion

s ar

e in

inch

es.

36

0.04

99

0.04

99

0.04

99

0.02

495R

0.02

495

0.09

98

0.02

495

x 4

1.00

0

All

arc

s 1.

265R

0.25

04

plac

es

Str

eam

wis

eco

nvol

utio

npr

ofile

(4X

enl

arge

d)

1.00

0

0.58

0

0.58

0

0.50

0

3.99

0

40 v

alle

ys, 3

9 hi

lls, 2

end

hill

s

Vie

w a

t max

imum

am

plitu

de

2.16

0

Spa

nwis

eco

nvol

utio

npr

ofile

(8X

enl

arge

d)

0.04

99

Str

eam

wis

eco

nvol

utio

npr

ofile

(4X

enl

arge

d)

20 v

alle

ys, 1

9 hi

lls, 2

end

hill

s

Vie

w a

t max

imum

am

plitu

de

Spa

nwis

eco

nvol

utio

npr

ofile

(4X

enl

arge

d)

0.09

98

0.09

98

2.16

0

3.99

0

0.58

0

0.58

0

1.00

0

0.50

0

0.04

99R

0.04

99

0.09

98

0.09

98

1.00

0

0.04

99 x

4

0.19

96

All

arcs

0.6

512R

(b)

Med

ium

con

volu

ted

flap

inse

rts.

Fig

ure

10.

Con

tinue

d.

37

0.25

04

plac

es

0.19

96

0.04

99 x

4

1.75

0

All

arcs

1.9

428R

Str

eam

wis

eco

nvol

utio

npr

ofile

(2X

enl

arge

d)

0.20

5

1.75

0

0.50

0

3.99

0

2.16

0

0.20

5

20 v

alle

ys, 1

9 hi

lls, 2

end

hill

s

Vie

w a

t max

imum

am

plitu

de

Spa

nwis

eco

nvol

utio

npr

ofile

(4X

enl

arge

d)

0.09

98

0.09

98

0.04

99R

0.04

99

0.09

98

0.09

98

(c)

Med

ium

-long

con

volu

ted

flap

inse

rts.

Fig

ure

10.

Con

tinue

d.

38

0.43

754

plac

es

(d)

Coa

rse

conv

olut

ed fl

ap in

sert

s.

Fig

ure

10.

Con

clud

ed.

39

0.19

95

0.19

95

0.19

95

0.19

95

0.09

975

R

0.09

975

0.39

9

0.09

975

x 4

1.00

0

Str

eam

wis

eco

nvol

utio

npr

ofile

(4X

enl

arge

d)

1.00

0

0.58

0

0.58

0

0.50

0

3.99

0

10 v

alle

ys, 9

hill

s, 2

end

hill

s

Vie

w a

t max

imum

am

plitu

de

2.16

0

All

arcs

0.3

632R

Spa

nwis

eco

nvol

utio

npr

ofile

(2X

enl

arge

d)

0.25

04

plac

es

Convoluted inserts

Convoluted inserts

Figure 12. Photograph of the static test model with convoluted flap inserts installed.

Figure 11. Photograph of the convoluted flap insert pairs.

40

Coarse

Coarse

Fine

Fine

Medium

Medium

Medium-long

Medium-long

(a) Nozzle flap assembly static pressure taps.

Figure 13. Nozzle static pressure orifice locations. Dimensions are in inches.

41

A

A

Sta.41.13

x

Throat

xt = 2.275

y

Side View

Throat

Exit

z

View A-A

Static Pressure Orifice Locations

x, in.

x/x t

Denotes static pressure orifice location

0.7501.0001.2501.5001.7502.0252.275

0.3300.4400.5500.6590.7690.8901.000

2.160

(b) Baseline flap insert static pressure taps.

Figure 13. Continued.

42

Section B-B

View A-A

z

C

L

Sta.41.13

x

A

A

Sta.43.56

x/x t

1.1111.1541.1981.2411.2841.3271.3701.4131.4561.5001.5431.5861.6291.6721.7151.7581.8021.8451.8881.9311.974

Static Pressure Orifice Locations

z = 0.000 in.

z = 1.595 in.

Denotes static pressure orifice location

x, in.

2.5282.6262.7242.8232.9213.0193.1173.2153.3133.4123.5103.6083.7063.8043.9024.0014.0994.1974.2954.3934.491

x/x t

1.1111.1541.1981.2411.2841.3271.3701.4131.4561.5001.5431.5861.6291.6721.7151.7581.8021.8451.8881.9311.974

x, in.

2.5282.6262.7242.8232.9213.0193.1173.2153.3133.4123.5103.6083.7063.8043.9024.0014.0994.1974.2954.3934.491

(c) Convoluted flap inserts (medium convoluted flap insert shown) static pressure taps.

Figure 13. Concluded.

43

2.160

View A-A

z

C

L

Section B-B

Sta.41.13

x

A

A

Section C-C

Flow

Sta.43.56

x/x t

1.3271.3701.4131.4561.5001.5431.5861.6291.6721.715

z = 0.1995 in.

x, in.

3.0193.1173.2153.3133.4123.5103.6083.7063.8043.902

Static Pressure Orifice Locations

z = 0.000 in.

Denotes static pressure orifice location

x, in.

2.5282.6262.7242.8232.9213.0193.1173.2153.3133.4123.5103.6083.7063.8043.9024.0014.0994.1974.2954.3934.491

x/x t

1.1111.1541.1981.2411.2841.3271.3701.4131.4561.5001.5431.5861.6291.6721.7151.7581.8021.8451.8881.9311.974

x/x t

1.1111.1541.1981.2411.2841.3271.3701.4131.4561.5001.5431.5861.6291.6721.7151.7581.8021.8451.8881.9311.974

z = 1.595 in.

x, in.

2.5282.6262.7242.8232.9213.0193.1173.2153.3133.4123.5103.6083.7063.8043.9024.0014.0994.1974.2954.3934.491

(a)

Opt

ical

des

crip

tion.

Fig

ure

14.

Opt

ical

des

crip

tion

and

sche

mat

ic la

yout

of f

ocus

ing

schl

iere

n flo

w v

isua

lizat

ion

syst

em.

Dim

ensi

ons

are

in in

ches

.

44

Fre

snal

sou

rce

lens

318m

m f

1.25

22.0

9.5

18.9

14.2

4.7

0.03

6

Sou

rce

grid

det

ails

0.01

6

No

te

:

Cut

off g

rid is

opt

ical

neg

ativ

e of

sou

rce

grid

.

2.0

Ver

tical

sou

rce

grid

Ext

ende

d lig

ht s

ourc

e

Foc

al p

lane

Mul

ti-el

emen

tfo

cusi

ng le

ns24

0mm

f4.5

Imag

e pl

ane

Ver

tical

cut

off g

rid

(b)

Sch

emat

ic la

yout

.

Fig

ure

14.

Con

clud

ed.

45

Noz

zle

test

bed

with

gla

ss s

idew

alls

720x

480

pixe

l res

olut

ion

colo

rca

mer

a w

ith 1

6mm

f1.4

Len

s

30%

70m

m fo

rmat

cam

era

80m

m f2

.8 le

ns

0.6

neut

ral d

ensi

ty fi

lter

4x m

agni

fier

Bea

m s

plitt

er

Fre

snal

rel

ay le

ns15

2mm

f1.0

Cut

off g

rid

Tur

ning

mirr

or

Tur

ning

mirr

or

Xen

on fl

asht

ube

light

sour

ce

Gro

und

glas

she

mis

peric

aldi

ffuse

r

Elli

ptic

al r

efle

ctor

Stin

g

Str

ut

44 in

. x 6

6 in

. tab

le

70%

Pro

puls

ion

sim

ulat

ion

syst

em

Sou

rce

grid

Fre

snal

sou

rce

lens

Foc

usin

g le

ns

Pow

er s

uppl

yT

rigge

r ci

rcui

t

NPRd

.76

.80

.84

.88

.92

.96

.76

.80

.84

.88

.92

.96

1.00

1.00

F/F

i

1

2

3

4

5

6

7

8

9

10

NPR

Figure 16. Comparison of nozzle thrust ratio performance for baseline and convoluted configurations.

46

Figure 15. Nozzle thrust ratio performance for the baseline configuration.

Configuration

Baseline

Fine convoluted

Medium convoluted

Medium-long convoluted

Coarse convoluted

.76

.80

.84

.88

.92

.96

1.00

1

2

3

4

5

6

7

8

9

10

NPR

.76

.80

.84

.88

.92

.96

1.00

F/F

i

NPRd

.90

.91

.92

.93

.94

.95

(F/F )

i

peak

.96

.97

.98

.99

1.00

Baseline

FineConvoluted

MediumConvoluted

Medium-LongConvoluted

CoarseConvoluted

Predicted

Measured

Figure 17. Comparison of NPAC predicted and experimentally measured peak thrust ratios for baseline andconvoluted configurations.

47

Fine convolutedNPR = 8.9

BaselineNPR = 8.9

(b) Fine convoluted configuration.

Figure 18. Focusing schlieren flow visualization at NPR=8.9 for baseline and convoluted configurations.

48

(a) Baseline configuration.

Convolutionrun

Medium-longconvolutedNPR = 8.9

Convolution run

Medium convolutedNPR = 8.9

(d) Medium-long convoluted configuration.

Figure 18. Continued.

49

(c) Medium convoluted configuration.

Convolutionrun

Coarse convolutedNPR = 8.9

Convolutionrun

(e) Coarse convoluted configuration.

Figure 18. Concluded.

50

(b) Medium convoluted configuration.

Figure 19. Comparison of baseline and convoluted internal static pressure ratio distributions at NPR=8.9.Open symbols deonte hill pressures; solid symbols denote valley pressures.

51

(a) Fine convoluted configuration.

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

2.0

x/x

t

t

0

.2

.4

.6

.8

1.0

p/p

t,j

t,j

x/x

0

.2

.4

.6

.8

1.0

p/p

Baseline

Medium convoluted

Configuration

Convolution run

Baseline

Fine convoluted

Configuration

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

2.0

x/x

t

t

0

.2

.4

.6

.8

1.0

p/p

t,j

t,j

Convolution run

x/x

0

.2

.4

.6

.8

1.0

p/p

(d) Coarse convoluted configuration.

Figure 19. Concluded.

52

(c) Medium-long convoluted configuration.

Convolution run

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

2.0

x/x

t

t

0

.2

.4

.6

.8

1.0

p/p

t,j

t,j

x/x

0

.2

.4

.6

.8

1.0

p/p

Baseline

Medium-long convoluted

Configuration

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

2.0

x/x

t

t

0

.2

.4

.6

.8

1.0

p/p

t,j

t,j

x/x

0

.2

.4

.6

.8

1.0

p/p

Baseline

Coarse convoluted

Configuration

Convolution run

Figure 20. Internal static pressure ratio distributions for the baseline configuration.

53

0

.2

.4

.6

.8

1.0

p/p

t,j

z

= 0.000 in.

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

x/x

t

0

.2

.4

.6

.8

1.0

p/p

t,j

t,j

z

= 1.595 in.

x/x

0

.2

.4

.6

.8

1.0

p/p

NPR

1.26

1.41

1.61

1.81

2.01

2.21

2.41

2.61

3.01

3.41

3.82

4.22

4.62

5.02

5.42

6.23

7.03

8.04

8.95

9.54

z

= 1.595 in.

(b) NPR = 1.8.

Figure 21. Focusing schlieren flow visualization for the baseline configuration.

54

BaselineNPR = 1.4

BaselineNPR = 1.8

(a) NPR = 1.4.

(d) NPR = 2.4.

Figure 21. Continued.

55

BaselineNPR = 2.0

BaselineNPR = 2.4

(c) NPR = 2.0.

BaselineNPR = 3.4

(e) NPR = 3.4.

Figure 21. Concluded.

56

BaselineNPR = 2.0

(b) Medium convoluted configuration.

Figure 22. Individual comparisons of nozzle thrust ratio performance for baseline and convoluted configurations.

57

(a) Fine convoluted configuration.

.76

.80

.84

.88

.92

.96

1.00

1

2

3

4

5

6

7

8

9

10

NPR

.76

.80

.84

.88

.92

.96

1.00

.76

.80

.84

.88

.92

.96

1.00

1

2

3

4

5

6

7

8

9

10

NPR

.76

.80

.84

.88

.92

.96

1.00

Baseline

Medium convoluted

Configuration

Baseline

Fine convoluted

Configuration

F/F

i

F/F

i

NPRd

NPRd

(d) Coarse convoluted configuration.

Figure 22. Concluded.

58

(c) Medium-long convoluted configuration.

.76

.80

.84

.88

.92

.96

1.00

1

2

3

4

5

6

7

8

9

10

NPR

.76

.80

.84

.88

.92

.96

1.00

.76

.80

.84

.88

.92

.96

1.00

1

2

3

4

5

6

7

8

9

10

NPR

.76

.80

.84

.88

.92

.96

1.00

Baseline

Medium-long convoluted

Configuration

Baseline

Coarse convoluted

Configuration

F/F

i

F/F

i

NPRd

NPRd

(a) Fine convoluted configuration.

Figure 23. Internal static pressure ratio distributions for convoluted configurations.

59

0

.2

.4

.6

.8

1.0

p/p

t,j

Valley (

z

= 0.1995 in.)

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

x/x

t

0

.2

.4

.6

.8

1.0

p/p

t,j

t,j

x/x

0

.2

.4

.6

.8

1.0

p/p

Hill (

z

= 1.595 in.)

0

.2

.4

.6

.8

1.0

p/p

t,j

t,j

0

.2

.4

.6

.8

1.0

p/p

Hill (

z

= 0.000 in.)

NPR

1.25

1.40

1.60

1.80

2.00

2.20

2.40

2.60

3.00

3.40

3.80

4.20

4.60

5.00

5.40

6.20

7.00

8.00

8.91

9.50

Convolution run

t

NPR

1.24

1.39

1.59

1.79

1.99

2.19

2.39

2.59

2.99

3.39

3.79

4.19

4.59

4.99

5.39

6.19

6.99

7.99

8.89

9.50

(b) Medium convoluted configuration.

Figure 23. Continued.

60

0

.2

.4

.6

.8

1.0

p/p

t,j

Valley (

z

= 0.1995 in.)

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

x/x

t

0

.2

.4

.6

.8

1.0

p/p

t,j

t,j

x/x

0

.2

.4

.6

.8

1.0

p/p

Hill (

z

= 1.595 in.)

t

0

.2

.4

.6

.8

1.0

p/p

t,j

t,j

0

.2

.4

.6

.8

1.0

p/p

Hill (

z

= 0.000 in.)

Convolution run

Convolution run

NPR

1.25

1.40

1.60

1.80

2.00

2.20

2.40

2.60

3.00

3.40

3.80

4.20

4.60

5.00

5.40

6.19

7.00

8.00

8.89

9.51

(c) Medium-long convoluted configuration.

Figure 23. Continued.

61

0

.2

.4

.6

.8

1.0

p/p

t,j

Valley (

z

= 0.1995 in.)

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

x/x

t

0

.2

.4

.6

.8

1.0

p/p

t,j

t,j

x/x

0

.2

.4

.6

.8

1.0

p/p

Hill (

z

= 1.595 in.)

t

0

.2

.4

.6

.8

1.0

p/p

t,j

t,j

0

.2

.4

.6

.8

1.0

p/p

Hill (

z

= 0.000 in.)

NPR

1.25

1.40

1.60

1.80

2.00

2.20

2.40

2.60

3.00

3.39

3.80

4.20

4.60

4.99

5.40

6.20

6.99

8.00

8.89

9.50

(d) Coarse convoluted configuration.

Figure 23. Concluded.

62

0

.2

.4

.6

.8

1.0

p/p

t,j

Valley (

z

= 0.1995 in.)

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

x/x

t

0

.2

.4

.6

.8

1.0

p/p

t,j

t,j

x/x

0

.2

.4

.6

.8

1.0

p/p

Hill (

z

= 1.595 in.)

t

0

.2

.4

.6

.8

1.0

p/p

t,j

t,j

0

.2

.4

.6

.8

1.0

p/p

Hill (

z

= 0.000 in.)

Convolution run

Medium convolutedNPR = 1.8

Convolutionrun

(b) Medium convoluted configuration.

Figure 24. Focusing schlieren flow visualization for convoluted configurations showing nozzle shock upstream of convolution run.

63

Fine convolutedNPR = 2.0

(a) Fine convoluted configuration.

Convolutionrun

Coarse convolutedNPR = 1.6

Convolutionrun

(d) Coarse convoluted configuration.

Figure 24. Concluded.

64

Medium-long convolutedNPR = 1.6

(c) Medium-long convoluted configuration.

Convolution run

(b) NPR = 2.6.

Figure 25. Focusing schlieren flow visualization at NPRs of 2.2 and 2.6 for the fine convoluted configuration.

65

Fine convolutedNPR = 2.2

Fine convolutedNPR = 2.6

(a) NPR = 2.2.

Convolutionrun

Convolutionrun

(b) NPR = 2.6.

Figure 26. Focusing schlieren flow visualization at NPRs of 2.0, 2.6, and 3.0 for the medium convoluted configuration.

66

Medium convolutedNPR = 2.0

Medium convolutedNPR = 2.6

(a) NPR = 2.0.

Convolutionrun

Convolutionrun

Medium convolutedNPR = 3.0

(c) NPR = 3.0.

Figure 26. Concluded.

67

Convolutionrun

Medium-long convolutedNPR = 2.6

Convolution run

(b) NPR = 2.6.

Figure 27. Focusing schlieren flow visualization at NPRs of 1.8 and 2.6 for the medium-long convoluted configuration.

68

Medium-long convolutedNPR = 1.8

(a) NPR = 1.8.

Convolution run

Coarse convolutedNPR = 4.2

Convolutionrun

(b) NPR = 4.2.

Figure 28. Focusing schlieren flow visualization at NPRs of 3.8 and 4.2 for the coarse convoluted configuration.

69

Coarse convolutedNPR = 3.8

(a) NPR = 3.8.

Convolutionrun

M1≈

1.8

Figure 30. Sketch showing shock-boundary layer interaction lambda foot structure at NPR = 3.0 for the baseline configuration.

70

BaselineNPR = 3.0

Figure 29. Focusing schlieren flow visualization at NPR = 3.0 for the baseline configuration.

80°

θ

=15°

α

=11.01°

β

=52°

β

=63°

θ

=3°

Divergent flap

M2≈

1.2

M≈

1.1

REPORT DOCUMENTATION PAGE Form ApprovedOMB 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 datasources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any otheraspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations andReports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188),Washington, DC 20503.

1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE

February 19993. REPORT TYPE AND DATES COVERED

Technical Publication

4. TITLE AND SUBTITLE

Effects of Convoluted Divergent Flap Contouring on the Performance of aFixed-Geometry NonaxisymmetricExhaust Nozzle

5. FUNDING NUMBERS

WU 538-14-12-01

6. AUTHOR(S)

Scott C. Asbury and Craig A. Hunter

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

NASA Langley Research CenterHampton, VA 23681-2199

8. PERFORMING ORGANIZATIONREPORT NUMBER

L-17696

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

National Aeronautics and Space AdministrationWashington, DC 20546-0001

10. SPONSORING/MONITORINGAGENCY REPORT NUMBER

NASA/TP-1999-209093

11. SUPPLEMENTARY NOTES

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-UnlimitedSubject Category 02 Distribution: StandardAvailability: NASA CASI (301) 621-0390

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

An investigation was conducted in the model preparation area of the Langley 16-Foot Transonic Tunnel todetermine the effects of convoluted divergent-flap contouring on the internal performance of a fixed-geometry,nonaxisymmetric, convergent-divergent exhaust nozzle. Testing was conducted at static conditions using a sub-scale nozzle model with one baseline and four convoluted configurations. All tests were conducted with no externalflow at nozzle pressure ratios from 1.25 to approximately 9.50. Results indicate that baseline nozzle performancewas dominated by unstable, shock-induced, boundary-layer separation at overexpanded conditions. Convolutedconfigurations were found to significantly reduce, and in some cases totally alleviate separation at overexpandedconditions. This result was attributed to the ability of convoluted contouring to energize and improve the conditionof the nozzle boundary layer. Separation alleviation offers potential for installed nozzle aeropropulsive (thrust-minus-drag) performance benefits by reducing drag at forward flight speeds, even though this may reduce nozzlethrust ratio as much as 6.4% at off-design conditions. At on-design conditions, nozzle thrust ratio for theconvoluted configurations ranged from 1% to 2.9% below the baseline configuration; this was a result of increasedskin friction and oblique shock losses inside the nozzle.

14. SUBJECT TERMS

Convoluted Contouring, Exhaust Nozzles, Nonaxisymmetric Nozzles, Nozzles15. NUMBER OF PAGES

75 16. PRICE CODE

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Unclassified

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UL

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