Experimental, Theoretical, and Computational Investigation ...mln/ltrs-pdfs/NASA-aiaa-98-3107.pdf · 1 American Institute of Aeronautics and Astronautics EXPERIMENTAL, THEORETICAL,

34th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit

July 13-15, 1998 / Cleveland, OH

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AIAA 98-3107

Experimental, Theoretical, and ComputationalInvestigation of Separated Nozzle FlowsC.A. HunterNASA Langley Research CenterHampton, Virginia

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American Institute of Aeronautics and Astronautics

EXPERIMENTAL, THEORETICAL, AND COMPUTATIONALINVESTIGATION OF SEPARATED NOZZLE FLOWS

Craig A. Hunter *NASA Langley Research Center

Hampton, Virginia

Abstract

A detailed experimental, theoretical, and computationalstudy of separated nozzle flows has been conducted.Experimental testing was performed at the NASALangley 16-Foot Transonic Tunnel Complex. As partof a comprehensive static performance investigation,force, moment, and pressure measurements were madeand schlieren flow visualization was obtained for a sub-scale, non-axisymmetric, two-dimensional, convergent-divergent nozzle. In addition, two-dimensionalnumerical simulations were run using the computationalfluid dynamics code PAB3D with two-equationturbulence closure and algebraic Reynolds stressmodeling. For reference, experimental andcomputational results were compared with theoreticalpredictions based on one-dimensional gas dynamics andan approximate integral momentum boundary layermethod.

Experimental results from this study indicate that off-design overexpanded nozzle flow was dominated byshock induced boundary layer separation, which wasdivided into two distinct flow regimes; three-dimensional separation with partial reattachment, andfully detached two-dimensional separation. The testnozzle was observed to go through a marked transitionin passing from one regime to the other. In all cases,separation provided a significant increase in staticthrust efficiency compared to the ideal prediction.Results indicate that with controlled separation, theentire overexpanded range of nozzle performancewould be within 10% of the peak thrust efficiency. Byoffering savings in weight and complexity over aconventional mechanical exhaust system, this mayallow a fixed geometry nozzle to cover an entire flightenvelope.

The computational simulation was in excellentagreement with experimental data over most of the testrange, and did a good job of modeling internal flow andthrust performance. An exception occurred at lownozzle pressure ratios, where the two-dimensionalcomputational model was inconsistent with the three-dimensional separation observed in the experiment. In

general, the computation captured the physics of theshock - boundary layer interaction and shock inducedboundary layer separation in the nozzle, though therewere some differences in shock structure compared toexperiment. Though minor, these differences could beimportant for studies involving flow control or thrustvectoring of separated nozzles. Combined with otherobservations, this indicates that more detailed, three-dimensional computational modeling needs to beconducted to more realistically simulate shock-separated nozzle flows.

Introduction

In the world of modern fluid dynamics, boundary layerseparation is typically regarded as a ÒtragicÓphenomena, and rightly so; performance penalties andinstabilities often accompany separation, and these canhave a detrimental or catastrophic effect on almost anyfluid dynamic system. Putting this aside, however, it isimportant to realize that boundary layer separation is anatural process; it is the means by which a viscous flowadjusts to its surroundings under particular conditions.In certain cases, this mechanism can have manybenefits, ranging from enhanced performance to flowcontrol.

Consider an overexpanded convergent-divergent (CD)nozzle. This condition occurs because the nozzleexpansion ratio is too large for a given nozzle pressureratio (NPR). Supersonic nozzle flow is driven toexpand below exit back pressure, and a nonisentropic,discontinuous shock jump is required to adjust. Ideally,flow stays attached downstream of the shock, but inreality it usually separates from the nozzle flap if theexpansion ratio is high enough or the flap angle steepenough. An illustration of flow in a shock separatednozzle is given in figure 1.

Figure 1: Overexpanded CD Nozzle with Separation

AIAA-98-3107

* Aerospace Engineer, Configuration Aerodynamics Branch, Aero- and Gas-Dynamics Division. Member AIAA

Copyright © 1998 by the American Institute of Aeronautics and Astronautics, Inc. Nocopyright is asserted in the United States under Title 17, U.S. Code. The U.S. Governmenthas a royalty-free license to exercise all rights under the copyright claimed herein forgovernment purposes. All other rights are reserved by the copyright owner.

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At first glance, it is often surprising that nozzleseparation is accompanied by an increase in thrustefficiency over the attached case1, but it shouldnÕt be.A major effect of boundary layer separation is that itredefines the ÒeffectiveÓ geometry of a fluid dynamicsystem by displacing the inviscid flow. In a CD nozzle,separation moves the jet detachment point upstream,causing a change in the effective nozzle geometry toone that is shorter and has a lower expansion ratio. Fora given NPR, this alleviates overexpansion andimproves thrust efficiency. By acting like a naturalÒadjustment mechanismÓ, separation allows anoverexpanded nozzle flow to reach a more efficientthermodynamic balance.

Earlier studies by Asbury, et. al.2 and Hunter3 showedthat off-design nozzle thrust efficiency could beimproved beyond what occurs naturally by encouragingstable separation (with a passive porous cavity) andcontrolling the location and extent of that separation. Inaddition, this work demonstrated that effective thrustvectoring could be attained by using asymmetric shock- boundary layer interaction control. The recent growthin micro adaptive flow control technology4 and activeflow control concepts offers further means by which tocontrol and stabilize a separated nozzle flow forperformance enhancement or thrust vectoring.

Many of these new and emerging active and passiveflow control technologies remain unexplored in exhaustsystems. Current aircraft typically use mechanicalsystems for nozzle area control and thrust vectoring.While effective, these systems can be heavy, complex,expensive to maintain, and prone to fatigue and failure.They are also difficult to integrate aerodynamically andcan be a primary source of drag. With the addition ofother requirements such as IR suppression, noisesuppression, and low-observable shaping, the designand integration of an efficient mechanical exhaustsystem becomes all the more challenging.

The capabilities of future aircraft will depend on thedevelopment of simple, lightweight exhaust systemsthat are aerodynamically efficient, compact, andinexpensive. In light of this, there is tremendouspotential to improve aircraft system performance byreplacing mechanical nozzle systems with efficientfixed geometry configurations that use separation toenhance and control off-design performance. For thisreason, and because of the complicated relationshipbetween overexpansion, shock induced separation, andthrust efficiency, an understanding of separated nozzleflows is critical from experimental, theoretical, andcomputational standpoints. This paper addresses thatunderstanding.

Nomenclature

Ae Nozzle Exit Area, in2

At Nozzle Throat Area, in2

Ae /At Nozzle Expansion RatioBL Boundary LayerF Axial Thrust, lbFi Ideal Isentropic Thrust, lbF/Fi Thrust Efficiency Ratioh BL Wall Normal Coordinate, fth+ BL Normal in Wall Variables, h+=hu*/nk TKE per Unit Mass, ft2/s2

kPK Peak k in the Boundary Layer, ft2/s2

k¥ Computational Lower Limit for k, ft2/s2

l Boundary Layer Streamwise Coordinate, ftM Local Mach NumberMa Ambient Freestream Mach NumberMs Undisturbed Separation Mach NumberM1 Mach Number Upstream of a ShockNPR Nozzle Pressure Ratio, NPR=poj/pap Static Pressure, psipa Ambient Pressure, psips Undisturbed Separation Pressure, psipoj Jet Total PressureRel Reynolds Number Based on l, Rel=uel/nwRed Reynolds Number Based on d, Red=ued/nwt time, sTa Ambient Temperature, °RToj Jet Stagnation Temperature, °RTKE Turbulent Kinetic Energy, lb/ft2

u BL Velocity Parallel to Wall, ft/sue BL Edge Velocity, ft/su* BL Friction Velocity, u*2 = =n ¶ ¶w hu h( / ) 0u+ BL Velocity in Wall Variables, u+=u/u*Ui ith Component of Mean Flow Velocity, ft/sá ¢ ¢u ui j ñ Reynolds Stress Tensor, ft

2/s2

x Streamwise Coordinate, inxs Streamwise Location of Nozzle Shock, inxt Streamwise Location of Nozzle Throat, inxi ith Spatial Coordinate, fty Streamwise Normal (Vertical) Coordinate, in

a Flow Angle, Clockwise from Horizontalb Oblique Shock Inclination Angle, degreesdij Kronecker Delta Tensord Boundary Layer Thickness, indo Onset Boundary Layer Thickness, ind* Boundary Layer Displacement Thickness, ine Dissipation Rate of k, ft2/s3n Kinematic Viscosity, ft2/snt Turbulent ÒEddyÓ Viscosity, ft2/snw Kinematic Viscosity at h=0, ft2/sq BL Momentum Thickness, in

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Experimental Arrangement

Static Test Facility

Experimental testing was conducted in the modelpreparation area of the NASA Langley 16-FootTransonic Tunnel Complex. This facility is normallyused for the buildup and calibration of wind tunnelmodels, but can also be used for nozzle testing at staticconditions. Models are mounted on a sting-strutsupport system in a 10´29 foot ambient test chamber,and supplied with a regulated continuous flow of clean,dry air. A control room adjacent to the test chamberoffers access through a sound proof door andobservation window. A complete description of thistest facility is provided in reference 5.

COORDINATES, IN.POINT X Y

A 0.000 0.000B 0.000 -0.614C 0.000 1.386D 0.917 1.163E 1.988 0.611F 2.394 0.553G 2.430 0.559H 2.275 1.166I 4.550 0.972

A

B

CD

E F

H

I

2.000 R27.29°

11.01°G

0.625 R

INTERCHANGEABLEDIVERGENT FLAP

x

y

Figure 2: Nozzle Flap Geometry

Test Nozzle

The test nozzle used in this investigation was a sub-scale, non-axisymmetric, two-dimensional convergent-divergent (2D-CD) nozzle with a nominal throat areaAt=4.317 in

2, an expansion ratio Ae/At=1.797, and aconstant width of 3.990 inches. Based on 1D Theory,the nozzle has a design NPR of 8.78, an exit Machnumber of 2.07, and a design throat Reynolds numberof 3.2´106 for pa=14.85 psi. The nozzle was equippedwith interchangeable divergent flaps in order tofunction as a testbed for various shock - boundary layerinteraction control concepts, and had full length optical-quality glass sidewalls to allow for internal flowvisualization and flow diagnostics. Geometric detailsof the nozzle are shown in figure 2, and a photo of thenozzle is given in figure 3.

Figure 3: Test Nozzle

Propulsion Simulation System

High pressure air was supplied to the test nozzle at astagnation temperature of about 530°R, with flow ratesup to 15 lbm/sec. As shown in figure 4, the propulsionsimulation system uses choke points, plenums, turns,and flexible metal bellows to deliver air to a nozzlesuch that effects of momentum transfer andpressurization are minimized. This ensures that forcesand moments produced by the nozzle can be accuratelymeasured. Downstream of the flow transfer system, airpasses through a circular-to-rectangular transitionsection, a choke plate, an instrumentation duct, and thetest nozzle before exhausting to ambient back pressure.

Nozzle

Balance

Flexible Metal Bellows

Choke Plate

Support Strut

Transition

Instrumentation Duct

Figure 4: Propulsion Simulation System

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Instrumentation

Air flow rates were calculated from pressure andtemperature measurements in a multiple critical venturilocated upstream of the propulsion simulation system.Forces and moments were measured by a six-component strain-gauge balance located on thepropulsion simulation system centerline (see figure 3).Stagnation conditions were measured in theinstrumentation duct using pitot tube rakes and pressuretransducers for total pressure, and iron-constantanthermocouples for total temperature. Surface staticpressures were measured on the upper nozzle flap at thecenterline (convergent and divergent flap) and at 0.40inch from the nozzle sidewall (divergent flap only).Pressure orifices were 0.020 inch in diameter and wereconnected to a combination of pressure transducers andan electronically scanning pressure module.

Data Acquisition and Reduction

Readings from the venturi, force balance, and pressureand temperature instrumentation were recordedsimultaneously. Steady state data were acquired byaveraging 50 frames of instantaneous data sampled at a10 Hz rate. Calibration constants were applied to theraw data to obtain corrected forces, moments, pressures,and temperatures. A detailed description of the datareduction procedures used is given in reference 6.

The following conventions were used for datareduction: Nozzle Pressure Ratio (NPR) is the ratio ofjet total pressure poj to atmospheric back pressure pa,and was used to set test points. Thrust Efficiency Ratio(F/Fi) is the ratio of measured axial thrust F to idealisentropic thrust Fi (calculated from the measuredweight flow and stagnation conditions using 1DTheory). Normalized Static Pressure (p/poj) is themeasured local static pressure p normalized by the jettotal pressure poj.

Test Schedule

Nozzle testing was performed by taking twenty pointsof data over an NPR range from 1.25 to 9.5. Theselimits were established by the lower and upper flowrates of the air supply system. Fifteen of the twentydata points were taken below NPR=6 to obtain detailedinformation on the overexpanded, shock-separatedregime of the nozzle.

Uncertainty Analysis

An uncertainty analysis was performed by propagatingmeasurement bias uncertainties through the datareduction equations. This analysis assumes that biaserrors dominate precision errors and is based onmethods presented in references 7 and 8. For the range

of test conditions encountered in this study, theuncertainty of NPR and p/poj is approximately ±0.28%of measured value, and the uncertainty of F/Fi isapproximately ±0.4%.

Flow Visualization

A focusing schlieren system was used to visualizeinternal nozzle flows in this study. Based on criteriondeveloped by Weinstein9, the system had a 133 mmdiameter field of view, a sensitivity of 17 arcsec, aresolution of 0.25 mm, a depth of sharp focus of 4.6mm, and a depth of unsharp focus of 36 mm. The lightsource for the schlieren system was a xenon strobe flashtube, driven at a 30 Hz rate with pulses of 0.6 msecduration and 0.05 watt×sec power. The system wasfocused on the test nozzle centerline plane andconfigured for sensitivity to streamwise densitygradients. A 720´480 pixel resolution video cameraand 70 mm Hasselblad still camera recorded results.

Theoretical Modeling

Theoretical predictions were made using the NPACcode developed at NASA Langley10. This code uses anapproximate theoretical method to calculate internalflow, thrust performance, and boundary layercharacteristics of 2D-CD nozzles. Internal flow andthermodynamic performance effects due to over- andunderexpansion are modeled with one-dimensionalcompressible flow theory, and an approximate integralmomentum method (based on the classic Krmn-Polhausen solution) is used to calculate turbulentboundary layer development and skin friction losses.An iterative algorithm couples the boundary layersolution to the 1D flow model to correct for the effectsof boundary layer displacement on internal flow.Additional calculations account for exit flow angularitylosses. The NPAC method has been well validated, andis capable of predicting peak thrust efficiency to within0.1% for a wide variety of 2D-CD nozzle shapes. Forlow expansion ratio nozzles, NPAC can generallypredict off-design thrust efficiency to within 0.5%.

Computational Fluid Dynamics Simulation

The NASA Langley Reynolds-averaged Navier Stokes(RANS) computational fluid dynamics (CFD) codePAB3D was used in conjunction with two-equation k-eturbulence closure and nonlinear algebraic Reynoldsstress models to simulate separated nozzle flows in thisinvestigation. This code has been well tested anddocumented for the simulation of aeropropulsive andaerodynamic flows involving separation, mixing, andother complicated phenomena11,12,13. Currently, PAB3Dis ported to a number of platforms and offers a

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combination of good performance and low memoryrequirements. In addition to its advanced preprocessorwhich can handle complex geometries through multi-block general patching, PAB3D has a runtime modulecapable of calculating aerodynamic performance on thefly and a postprocessor used for follow-on dataanalysis.

Flow Solver and Governing Equations

PAB3D solves the simplified Reynolds-averagedNavier-Stokes equations in conservative form, obtainedby neglecting streamwise derivatives of the viscousterms. Viscous models include coupled and uncoupledsimplified Navier-Stokes and thin layer Navier-Stokesoptions. RoeÕs upwind scheme is used to evaluate theexplicit part of the governing equations, and van LeerÕsscheme is used for the implicit part14. Diffusion termsare centrally differenced, inviscid terms are upwinddifferenced, and two finite volume flux-splittingschemes are used to construct the convective fluxterms14. PAB3D is third-order accurate in space andfirst-order accurate in time. For numerical stability,various solution limiters can be used, including min-mod, van Albeda, and Spekreijse-Venkat14. The codecan utilize either a 2-factor or 3-factor numericalscheme to solve the governing equations.

For the present study, the 2D problem was defined by jand k indices in a single i=constant plane (the j indexwas oriented in the streamwise flow direction). Withthis arrangement, explicit sweeps in the i direction werenot needed, and it was possible to solve the entireproblem implicitly with each iteration (using the vanLeer scheme). This strategy speeds convergence andreduces computational time. Based on previousexperience with shock - boundary layer interactionproblems, the uncoupled j-k simplified Navier-Stokesviscous option and the Spekreijse-Venkat limiter wereselected.

Turbulence Closure

In simulating turbulence, transport equations for theturbulent kinetic energy per unit mass (k) and thedissipation rate of k (e) are uncoupled from the meanflow RANS equations and can be solved with adifferent time step to speed convergence. Theseturbulence transport equations are of the standard linearform shown below, and are solved by the samenumerical schemes discussed above. The constants inthe e equation assume their standard values of Ce1=1.44,Ce2=1.92, sk=1, and se=1.3.

DkDt x

kx

u uUxj

t

k ji j

i

j

= +æèç

öø÷

é

ëêê

ù

ûúú- ¢ ¢ -¶

¶n n

s¶¶

¶¶

e [1]

DDt x x

Cku u

Ux

Ckj

t

ji j

i

j

e ¶¶

n ns

¶e¶

e ¶¶

e

ee e= +

æèç

öø÷

é

ëêê

ù

ûúú- ¢ ¢ -1 2

2

[2]

n

emtCk=2

[3]

With eddy viscosity or linear stress schemes, Cm istaken to be constant (usually equal to 0.09); however,this can produce undesirable and unphysical resultsunder certain conditions. Models with constant Cm canpredict negative values of k in flows with rapid strain,and can also produce non-realizable Reynolds stressesin flows with high shear, violating the SchwarzInequality. In order to circumvent these problems,PAB3D contains several different nonlinear algebraicReynolds stress models which were used in this study.These models give inherently better results than eddy-viscosity or linear stress schemes due to the explicitmodeling of effects such as relaxation, and the specificinclusion of nonlinear anisotropic effects from the meanflow strain rate and vorticity. With a nonlinear model,the calculation of six independent, realizable Reynoldsstress terms is possible. This type of detail is importantfor simulating complicated multidimensional flows.

As an example, the algebraic Reynolds stress model ofShih, Zhu, and Lumley15 is shown below, along withitÕs associated equation for variable Cm. Note the higherorder nonlinear terms involving the deviatoric meanflow strain rate (S*) and vorticity (W*) tensors in thestress equation. In the Cm equation, Ao=6.5, and C2,

As* , and U* involve tensor products of S* and W* andtheir variants. Further details on the variables andcoefficients in this model are available in reference 15.

¢ ¢ = - + -u u k C k S C k S Si j ij ij ik kj ik kj

23

2

2

3

22 2d

e em * * * * *( )W W

[4]

C A A U

ko sm e

= +æèç

öø÷-

* *

1

[5]

Computational Domain

Details of the full 2D computational domain arepresented in figure 5, where the multiblock grid isshown at 1/4 resolution. Relative to the nozzle exit, theambient region surrounding the nozzle extendedapproximately 30 throat heights downstream, 25 throatheights upstream, and 25 throat heights normal to thejet axis.

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Figure 5: Computational Grid at 1/4 Resolution

The internal nozzle grid is shown at full resolution infigure 6, and is characterized by an adaptive boundarylayer grid with an expansion ratio of about 18% and afirst cell height of approximately y+=0.5 (adaptivedesign based on boundary layer characteristicspredicted by NPAC). There were approximately 40cells in the boundary layer grid. In an attempt tocapture the complicated physics of the shock -boundary layer interaction process, the divergentsection of the nozzle was densely gridded with cellshaving an aspect ratio near 1:1. An inflow duct (sizedlike the instrumentation duct used in the experimentalstudy) was located upstream of the nozzle.

Figure 6: Nozzle Grid at Full Resolution

Initial and Boundary Conditions

As shown in figure 7, stagnation conditions wereapplied to the left face of the inflow duct upstream ofthe nozzle, and were chosen to match experimentalconditions for Toj and poj. In addition, an initial Machnumber was specified in the inflow block and nozzle tostart the solution. The static ambient regionsurrounding the nozzle was defined by a subsonicinflow condition (Ta=530°R, pa=14.85 psi, Ma=0.025)on the left face, a characteristic boundary condition onthe top face, and a smart boundary condition on theright face that switched between constant pressureoutflow (subsonic) and first order extrapolation(supersonic) depending on the local Mach number. Allsolid walls were treated as no-slip adiabatic surfaces,and the bottom of the entire domain was defined by aslip wall (acting as a symmetry boundary condition).

In order to initialize the turbulence transport equationsand ensure the formation of a turbulent boundary layer

in the test nozzle, a wall ÒtripÓ point was located nearthe beginning of the inflow duct. At this point, k wasspecified based on calculations involving the mean flowvelocity and vorticity and a user specified intensityratio. A corresponding value of e was calculated basedon the simplifying and reasonable assumption that theproduction of TKE was equal to the dissipation of TKEat the trip point.

Stagnation BC

Ambient Inflow BC

Characteristic BC

Slip Wall Symmetry BC

Outflow BC

No-slip Adiabatic Walls

Figure 7: Boundary Conditions

Solution Procedure and Postprocessing

All solutions presented in this paper were obtained byrunning PAB3D on an SGI Octane workstation with a195Mhz MIPS-R10000 CPU and 896 MB of RAM. Tospeed convergence and evaluate possible griddependence, mesh sequencing was used to evolvesolutions through coarse (1/4 resolution), medium (1/2resolution), and fine (full resolution) grids. Localtimestepping was used with global CFL numbersranging from 1 to 5. Depending on the NPR andcomplexity of flow in the nozzle, it took 5000-14000iterations and 8-26 hours of CPU time to obtain a fullyconverged 2D solution. Convergence was judged bytracking an integrated F/Fi calculation until it settled tothree decimal places over at least 1000 iterations.

The inline performance module of PAB3D was used inconjunction with an independent postprocessor tocompute thrust efficiency by integrating pressure andmomentum over a fixed control volume. Thepostprocessor was also used to extract internal nozzlesurface pressure data and construct schlieren-likeimages of the nozzle flow. These images were obtainedby calculating the density gradient and combining itwith a simulated optical ÒcutoffÓ effect. All imagespresented in this paper were generated with a root meansquare average of horizontal and vertical cutoffs, andthus show sensitivity to both streamwise and transversedensity gradients.

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Experimental Results

In this section, experimental results will be presented interms of internal flow features (static pressuremeasurements and schlieren flow visualization) andthrust performance. Because the test nozzle had glasssidewalls that flexed slightly under pressurization, itwas not possible to accurately measure dischargecoefficient and these data are not presented.

Internal Flow

Normalized experimental centerline static pressures(p/poj) are presented in figure 8, plotted againstnondimensional streamwise location. Results arerepresentative of classic CD nozzle flow16. For the firstdata points at NPRs 1.25 and 1.4, pressure data indicatechoked, internally overexpanded flow with a weakshock near the geometric throat. Flow downstream ofthe shock appears to recover to ambient pressure(p/poj=1/NPR) in a smooth continuous fashion.Focusing schlieren flow visualization obtained atNPR=1.4 is presented in figure 9, and shows a weak,almost normal shock downstream of the nozzle throatwith little or no lambda foot structure evident. Thisshock - boundary layer interaction is characteristic of aweak shock (M1»1.2, estimated from p/poj) and a thinboundary layer.

NPR

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0

1.2551.4051.6081.8082.0082.2082.4122.6073.0143.4133.8164.2174.6205.0185.423

x xt/

ppoj

Figure 8: Experimental Centerline Pressure Data

As seen in figure 8, the discontinuous nature of thepressure distribution at NPR=1.6 indicates that thestrength of the nozzle shock increased (M1»1.4), andthe inflection point in the recovery curve at x/xt=1.3indicates that separation occurred, though it was notsevere. By NPR=1.8, the upstream shock Machnumber was approximately 1.5, and shock inducedseparation began to significantly affect nozzle flow. Atthis NPR, there are strong signs of a separation bubble,

with minimal pressure recovery from the shock locationat x/xt=1.3 out to x/xt=1.6, and strong recoveryoccurring over the remaining length of the nozzle.Figure 10 shows the nozzle shock at NPR=1.8 with asmall lambda foot, and also shows signs of unstablenozzle flow; the schlieren photograph imaged the shockin two positions over a 0.6 msec duration, indicatingthat the separation was unsteady or transitory.

1.0 1.5 2.0 2.5x/x t

Figure 9: Schlieren Flow Visualization at NPR=1.4

1.0 1.5 2.0 2.5x/x t


As shown by pressure data in figure 8, an increase inNPR to 2.0 did not change shock location or strength,but flow fully detached and there was almost nopressure recovery downstream of the shock. This resultis of critical importance, because it serves to illustratethe relationship between NPR, shock - boundary layerinteraction, and separation. For this NPR, nozzle flowadjusted to exit conditions by completely detaching pastthe shock; internal conditions up to and through theshock were nearly identical to those of the previousNPR. Thus, separation that occurred at NPR=2.0 wasprobably not the result of a stronger shock - boundarylayer interaction, but instead came about through thenatural tendency of an overexpanded nozzle flow toseparate, thereby ÒadjustingÓ to a lower expansion ratio

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geometry. Among other things, this suggests that theshock - boundary layer interaction is not so much theÒcauseÓ of separation, but is more of a mutuallydependent ÒresultÓ.

Schlieren flow visualization at NPR=2.0 in figure 11shows the nozzle shock with a pronounced lambda footsystem and fully detached separation extending fromthe leading lambda shock downstream past the nozzleexit. That the appearance of a large lambda footcoincided with the onset of fully detached separationinside the nozzle is an important observation, for itshows that the size and extent of a lambda foot is morea result of separation effects than it is of basic shock -boundary layer interaction conditions such as onsetMach number or boundary layer state. This makessense, since a fully detached region would imposestronger turning requirements on nozzle flow than aclosed separation bubble and would require a biggeroblique shock system. As the separation point becamethe effective nozzle exit, the lambda shock systemadjusted to satisfy continuity of pressure and flowdirection in the new effective geometry.

1.0 1.5 2.0 2.5x/x t


Fully detached separation occurred for subsequentNPRs past 2.0. Figure 12 shows the shock at NPR=2.4with a well defined lambda foot and separation, andfigure 13 shows the same at NPR=3.0. By NPR=3.4,the lambda shock foot had grown significantly, suchthat the main shock and trailing lambda foot wereoutside the ÒphysicalÓ nozzle, as shown in figure 14.At this NPR, flow past the separation point showedstrong resemblance to externally overexpanded flow;the jet plume necked down between the leading andtrailing lambda foot, and there was an expansion fanemanating from each trailing lambda foot as itintersected the free shear layer. This indicates that notonly was the separation point behaving like the nozzleexit, but flow past this point was overexpandingexternally, which would occur in a lower expansionratio nozzle at the same NPR.

1.0 1.5 2.0 2.5x/x t


1.0 1.5 2.0 2.5x/x t


1.0 1.5 2.0 2.5x/x t


The leading lambda foot worked its way out of thenozzle with increasing NPR, and pressure data in figure8 show the nozzle to be shock free by NPR=5.4. Asshown in figure 15, pressures fell on the same curveabove NPR=5.4, indicating that internal flow wasindependent of NPR in this range. By comparingexperimental data with a theoretical pressuredistribution, one feature clearly evident in figure 15 is

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the Òvena contractaÓ effect, which occurs whenacceleration causes flow to overshoot the nozzle throatradius. This causes flow to reach sonic velocityupstream of the geometric throat, and leads to localoverexpansion and compression, evidenced by the dipand bump in pressure data from x/xt»0.8 to x/xt»1.3.High-sensitivity schlieren flow visualization atNPR=8.95 (near design) is presented in figure 16, andshows the skewed, bell-shaped sonic line (dark band)upstream of the geometric nozzle throat, the localoverexpansion (dark regions) near the throat, andoblique shocks marking the recompression. Thesefeatures were present for all NPRs above 1.6.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0

5.4236.2277.0308.0378.9459.543

Theory

Experiment

ppoj

x xt/

0.528

NPR

Figure 15: Centerline Pressures for NPR³5.4

1.0 1.5 2.0 2.5x/x t


Experimental Sideline Pressures

Experimental divergent flap sideline pressures arecompared to centerline pressures at selected NPRs infigure 17. At NPR=1.8, there are noticeable differencesbetween sideline and centerline pressure in terms ofshock location and downstream pressure recovery,indicating that nozzle flow was highly three-dimensional (3D) and the shock was non-planar. ForNPR=2.4 and above, sideline and centerline exitpressures were equal, and by NPR=3.0, sideline andcenterline shock locations became nearly equal andremained that way for all higher NPRs. Sideline andcenterline pressure distributions showed goodagreement at NPR=3.0, indicating that flow was wellbehaved and nearly two-dimensional. At NPR=5.4(shock free nozzle), sideline and centerline pressuresfall on the same curve.

NPR

0.0

0.2

0.4

0.6

1.0 1.2 1.4 1.6 1.8 2.0

1.808 CL2.412 CL3.014 CL3.816 CL5.423 CL

1.808 SL2.412 SL3.014 SL3.816 SL5.423 SL

NPR

Centerline

Sideline

ppoj

x xt/

Figure 17: Centerline - Sideline Pressure Comparison

Performance

Thrust efficiency results are shown in figure 18, wherethe measured thrust ratio F/F i is plotted versus NPR andaccompanied by a theoretical curve (predicted byNPAC) for comparison. Near the design pressure ratioof 8.78, measured thrust efficiency reached a maximumof F/Fi=0.986, which is consistent with the 0.985-0.990range reported in other studies17,18, and in excellentagreement with the theoretical prediction of 0.986.This indicates that on-design losses were likely due tothe same effects modeled by the NPAC code at thisfully expanded condition; namely, skin friction,boundary layer displacement, and flow angularity.

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0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7 8 9 10

TheoreticalExperiment

NPR

FFi

Figure 18: Experimental and TheoreticalThrust Efficiency Comparison

Below the design point of the nozzle, measured thrustefficiency decreased, closely following the theoreticalprediction down to about NPR=4.6. Again using theNPAC model for guidance, this indicates that internalperformance effects and losses were constant in thisrange, and that thrust efficiency was governed by thethermodynamics of flow in an overexpanded CDnozzle. The fact that this agreement extended down toNPR=4.6 correlates well with pressure data and earlierdiscussion that indicated the nozzle was approaching ashock free condition near this point.

Below NPR=4.6, measured thrust efficiency rose abovethe theoretical prediction and remained there for alllower NPRs before returning to the theoretical curve atNPR=1.25. At itÕs best point, measured thrustefficiency showed a 22% increase over theoreticalthrust efficiency at NPR=1.6. As supported byprevious discussion, this behavior corresponds to thefact that shock induced boundary layer separation wasÒalteringÓ the effective nozzle geometry to one with amore efficient, lower expansion ratio. This was mostnoticeable at lower NPRs where the nozzle shock wasfurther upstream and separation had a bigger effect onnozzle flow and the effective expansion ratio.

Several other observations round out this discussion.First, just as the high end departure from the theoreticalcurve occurred at NPR=4.6 due to the ÒendÓ ofseparated flow, the low end departure from thetheoretical curve occurred around NPRs 1.4-1.8, whereearly stages of separation were present. Note thatmeasured F/Fi was nearly constant in this range, as thenozzle was going through stages of ÒtransitoryÓseparation. In addition, the marked increase inmeasured F/Fi as NPR went from 1.8 to 2.0 corresponds

to the onset of fully detached separation in the nozzle,and represents the single largest NPR to NPR leap inthrust efficiency measured (as evidenced by the slope ofthe F/Fi curve) -- an increase of some 3.5%. Obviously,this confirms the fact that the nozzle flow went througha dramatic change at this point.

Algebraic Reynolds Stress Model Selection

Prior to conducting a detailed computational analysis,test cases were run at NPR=3.0 in order to assess theperformance of various algebraic Reynolds stress(ARS) models using experimental centerline pressuredata as the criterion. At this NPR, experimental resultsshowed a good shock - boundary layer interaction, andpressure data indicated that nozzle flow was nearly 2D.Thus, NPR=3.0 represented a challenging test case thatwas consistent with the 2D computational model.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0

GirimajiGSSZLExpmt

x xt/

ppoj

Figure 19: Comparison of Experimental andComputational Pressure Data at NPR=3.0

Three ARS models were evaluated in this test study: theShih-Zhu-Lumley (SZL) model 15, the Gatski-Speziale(GS) model19, and the Girimaji model20. Figure 19presents a comparison of computational and centerlineexperimental pressures over the entire nozzle length atNPR=3.0. All three ARS models showed excellentagreement with experimental pressures upstream andimmediately downstream of the nozzle throat. Furtherdownstream, all three ARS models appeared to modelthe shock - boundary layer interaction well. Figure 20presents a close-up comparison in that region, andshows that the SZL model did the best overall job ofplacing the nozzle shock and modeling the downstreamseparation. Thus, it was selected for further use.

11


0.1

0.2

0.3

0.4

1.5 1.6 1.7 1.8 1.9 2.0

GirimajiGSSZLExperiment

ppoj

x xt/

Figure 20: Comparison of Experimental andComputational Pressure Data at NPR=3.0

Computational Results

Computational simulations were run at the followingnozzle pressure ratios: 1.25, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4,3.0, 3.8, 4.6, 5.4, 7.0, and 8.78. These conditions werechosen for detailed comparison with experimental data.

Internal Flow

Low NPR pressure results from the computationalsimulation are compared with experimental centerlinepressure data at selected NPRs in figure 21. Note thatexperimental data can be keyed to itÕs correspondingcomputational curve in this figure by looking at the exitpressure (p/poj=1/NPR). At NPRs below 2.0, there isvery poor agreement between experimental andcomputational results, with the computationalsimulation predicting significantly more separation inthis transitory regime. A computational schlierenimage at NPR=1.4 is presented in figure 22. Comparedto its experimental counterpart in figure 9, shockstructure is similar, but flow is seen to detachimmediately downstream of the nozzle throat. Incontrast, the experimental image shows that flow wasindeed attached.

Additional CFD schlieren images in figures 23, 24, and25 show that the computational simulation began toform a lambda shock structure as early as NPR=1.6,and by NPR=1.8, the lambda system was welldeveloped. This is in stark contrast to experimentalresults which indicated the presence of a transitoryshock structure and transitory separation up toNPR=1.8. In addition, the dramatic change in internalflow observed with the onset of full separation as NPRwas increased from 1.8 to 2.0 in the experiment isclearly missing in the computational simulation.

NPR

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0

1.2551.4051.6081.8082.0082.4123.0143.8164.6205.423

x xt/

ppoj

Figure 21: Low NPR Pressure Comparison

1.0 1.5 2.0 2.5x/x t

Figure 22: CFD Schlieren at NPR=1.4

1.0 1.5 2.0 2.5x/x t


12


1.0 1.5 2.0 2.5x/x t


1.0 1.5 2.0 2.5x/x t


At NPR=2.0, experimental and computational pressureresults begin to converge, and they are in very goodagreement for NPRs 2.4 and up where both theexperiment and computation were in a fully detachedflow regime. Schlieren images at NPRs 2.0, 2.4, and3.0 in figures 25-27 show good qualitative agreementwith their experimental counterparts in figures 11-13,though the computational simulation is seen to predictmore of a ÒstretchedÓ shock structure at the higherNPRs (this will be studied in detail in a later section).The CFD schlieren images at NPRs 1.8, 2,0, 2.4, and3.0 all show the ghost image of a small recirculationeddy at the nozzle exit, which suggests the presence ofa double-eddy separation structure, and would explainthe ÒwigglesÓ in experimental and computationalsurface pressure data near the exit.

The low NPR internal flow discrepancies notedbetween experiment and computation could be due to anumber of things, but the most likely factor is the 2Dnature of the computational simulation. Whilecomparison is being made with experimental data andflow visualization from the centerline plane of thenozzle, where flow should be 2D in nature (based onsymmetry), it is important to realize that ÒlocalÓ flow at

this 2D centerline plane is the result of a ÒglobalÓ 3Dnozzle flow. Experimental data indicated that nozzleflow was highly 3D at low NPRs (NPR£2.4), whichwould certainly have an impact on flow at the nozzlecenterline. In addition, the onset of separation is knownto be a highly 3D process in many flows (though thecause of separation is usually 2D in nature), and itwould seem that nozzle flows are no exception. Thatthe flow began to take on a 2D nature at aroundNPR=2.4, where experiment and computation begin toagree, further corroborates this argument and indicatesthat a 3D simulation is necessary to correctly modelnozzle flow at low NPRs.

1.0 1.5 2.0 2.5x/x t


1.0 1.5 2.0 2.5x/x t


Like the experiment, computational results show thatthe nozzle was shock free by NPR=5.4. A comparisonof high NPR pressure data with theory in figure 28shows good agreement between the experiment and thecomputation (at NPRs 5.4, 7.0, 8.78), and the resolutionof the computational simulation more clearly definesthe extreme behavior of the vena contracta effect at thenozzle throat. Computational schlieren flowvisualization at NPR=8.78 is presented in figure 29, andshows excellent similarity to the correspondingexperimental image in figure 16.

13


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0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0

5.4237.0308.945CFDTheory

ppoj

x xt/

0.528

Figure 28: High NPR Pressure Comparison

1.0 1.5 2.0 2.5x/x t


Performance

A comparison of experimental, computational, andtheoretical thrust efficiency is presented in figure 30.As seen, the computational simulation did a very goodjob of modeling the overall thrust efficiency trend. Itdid mispredict the on-design thrust efficiency, giving apeak F/Fi of 0.993 versus 0.986 for experiment andtheory, but this is to be expected since the 2Dcomputational model does not account for viscouseffects on the nozzle sidewalls. The CFD postprocessorwas used to estimate this effect by including the wettedarea of the nozzle sidewalls in a skin frictioncalculation, and this brought the peak thrust efficiencydown to 0.988, relatively close to the experimentalmeasurement and theoretical prediction.

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7 8 9 10

TheoryExperimentComputation

NPR

FFi

Figure 30: Thrust Efficiency Comparison

When the on-design F/Fi shift is accounted for(graphically, this can be visualized by sliding thecomputational curve down to a peak of F/Fi=0.986), thecomputation still predicts up to 6% higher F/Fi values inthe low NPR range compared with experiment. This isa direct result of the fully detached separation seen inthe computational simulation but not present in theexperiment. In a way this is useful, for it illustratespotential performance benefits that could be obtained ina real nozzle by encouraging fully detached separationat each off-design NPR (this is easily done usingconcepts discussed in reference 2). With this type ofcontrol strategy, the entire overexpanded range ofnozzle performance would be within 10% of the peakthrust efficiency, which may allow a fixed geometrynozzle to cover an entire flight mission more efficientlythan a mechanical system due to reductions in weightand complexity.

Separation Correlation Map

In order to map out the separated flow characteristics ofa nozzle in a summary fashion, it useful to correlate theseparation pressure ratio ps/pa to the separation Machnumber Ms, where ps and Ms are the undisturbedpressure and Mach number upstream of the shock -boundary layer interaction. A plot of this correlation isgiven in Figure 31 for experimental data, computationalresults, NPAC 1D-inviscid theory, and the empiricallybased theory of Reshotko and Tucker21. In the lattermethod, incompressible turbulent boundary layerseparation criteria are corrected by a compressibilityfactor to predict the separation correlation. With NPACmodeling fully attached inviscid flow and theReshotko-Tucker (R-T) method assuming fullydetached separation, the two theories should bracket therange of conditions encountered in this study.

14


0.0

0.2

0.4

0.6

0.8

1.0

1.0 1.2 1.4 1.6 1.8 2.0 2.2

ExperimentComputationNPAC TheoryR-T Theory

Ms

pps

a

NPR=2.0

NPR=4.6

Figure 31: Separation Correlation Map

Note that the data points in Figure 31 progress from leftto right in the order of ascending NPR. As expected,low NPR points for the experimental data show that theflow was in a state of partial separation which ended asthe correlation moved up towards the R-T curve atNPR=2.0. Surprisingly, computational data showsimilar low NPR behavior, even though CFD schlierenimages indicate that flow was clearly detached in thesimulation. This may indicate that there was enoughpressure recovery in the separation region to lower psback towards an attached state. At NPR=2.0, thecomputational correlation also jumps towards the fullyseparated limit, and then matches the experimental dataout to NPR=4.6 as both correlations approach the R-Tcurve. Past that point, the experimental andcomputational correlations diverge from each other andthe R-T curve as the nozzle tends towards a shock-freecondition.

Shock - Boundary Layer Interaction at NPR=2.4

In this section, the shock - boundary layer interactionand nozzle separation at NPR=2.4 will be looked at indetail. Boundary layer characteristics, pressuredistributions, and shock structure will be examinedusing a combination of experimental, computational,and theoretical results.

Boundary Layer Characteristics

Boundary layer characteristics for the nozzle flow arepresented in figures 32 - 37, using results from both thetheoretical prediction and the computational simulationat NPR=2.4. In each plot, the instrumentation ductstarts at x/xt =-2.2, the nozzle entrance is at x/xt=0, andthe nozzle exit is at x/xt=2. Note that theory predicts ashock free nozzle at this NPR. To gain a betterunderstanding of the nozzle boundary layer in the

absence of a shock, an additional computationalsimulation was run with the same stagnation conditions(poj, Toj) as the NPR=2.4 case, but with a design NPR of8.78 (obtained by lowering the ambient back pressure).This ensured that Reynolds number was consistent withthe NPR=2.4 case. In figures 32 - 36, computationalresults at NPR=2.4 end near the shock - boundary layerinteraction at x/xt»1.45 since it was not possible tocalculate boundary layer parameters past this point.

Local Reynolds number Rel (based on surfacearclength) is plotted versus x/xt in figure 32. Asexpected, the NPR=2.4 and shock-free computationalcurves are identical up to the shock - boundary layerinteraction. Overall, there is good agreement betweencomputational and theoretical results, with theexception of vena contracta induced discrepancies atthe nozzle throat (from which the computational datarecovers back towards the theoretical prediction). Bothmethods show a local Reynolds number of about3.6´106 (or 5´106 per foot) going into the shock -boundary layer interaction region.

0

1

2

3

4

5

6

-2.0 -1.0 0.0 1.0 2.0

Theory, NPR=2.4CFD, NPR=2.4CFD, Shock-Free

x xt/

Rel

106

Figure 32: Local Reynolds Number

Computational turbulent kinetic energy per unit mass(k) in the boundary layer is plotted in figure 33. Herethe peak value of k at each streamwise location isnormalized by the lower limit of k used in thecomputation. The turbulent trip point in the nozzleinstrumentation duct is clearly visible at x/xt»-1.6, butit appears that the boundary layer had alreadytransitioned to turbulence near the beginning of the duct(under certain conditions, algebraic Reynolds stressmodels often exhibit this type of self-transition).Downstream of the trip, the boundary layer resumed itsequilibrium turbulence level in the constant area duct(M»0.23), and then turbulence kinetic energy grewdramatically as the boundary layer developed in thenozzle.

15


100

101

102

103

104

105

-2.0 -1.0 0.0 1.0 2.0

Computation, NPR=2.4Computation, Shock-Free

x xt/

Turbulent Trip

kkPK

¥

Figure 33: TKE Development

Boundary layer thickness development is presented infigure 34, where the boundary layer edge was definedby the point at which u/ue=0.995. The theoreticalprediction and computational simulation are inreasonably good agreement throughout theinstrumentation duct and convergent section of thenozzle, and then begin to diverge downstream of thenozzle throat. Much of the difference can be attributedto the vena contracta effect present in the computationalsimulation, which causes the boundary layer to thin outdramatically at the nozzle throat. Going into the shock- boundary layer interaction region at x/xt»1.45, thecomputational simulation predicts an onset boundarylayer thickness of do»0.018 in. (about 2.6% of the totalnozzle height).

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

-2.0 -1.0 0.0 1.0 2.0

Theory, NPR=2.4Computation, NPR=2.4Computation, Shock-Free

x xt/

d ( .)in

Figure 34: Boundary Layer Development

Reynolds number based on boundary layer thickness isshown in figure 35. This is the most relevant Reynoldsnumber for shock - boundary layer interactions becauseit reflects the ratio of inertial to viscous forces over alength scale consistent with the fact that ¶u/¶l»¶u/¶hwhen a shock meets a boundary layer (thereby violatingthe classic boundary layer assumption that¶u/¶l

16


1.0

1.5

2.0

2.5

3.0

3.5

-2.0 -1.0 0.0 1.0 2.0

Theory, NPR=2.4Computation, NPR=2.4Computation, Shock-Free

x xt/

dq*

Figure 36: Boundary Layer Shape Factor

0

5

10

15

20

25

0.1 1 10 100 1000 10000

u+ = 2.6 + 6.4 log h+

u+ = h+ Wall Law

Computation

u+

h+

Figure 37: Boundary Layer Profile at x/xt=1.4

Pressure Comparison

Experimental and computational pressure data in theshock - boundary layer interaction region are shown atNPR=2.4 in figure 38, plotted against normalizeddistance from the shock (x-xs)/do (where xs/xt»1.45) onthe lower abscissa, and x/xt on the upper abscissa. Asshown previously, there is good agreement betweenCFD and experiment. Some adjustment is necessary tocompare experimental and computational pressureresults with theory, since the theoretical model (basedon one-dimensional inviscid flow) predicts a shock freenozzle at this NPR. In figure 39, the nondimensionalstreamwise pressure gradient (xt/poj)(¶p/¶x) is plottedfor experimental and computational data at NPR=2.4

and the theoretical prediction at NPR=1.33. All threecases correspond to a shock location of x/xt»1.45 andan onset Mach number of M1»1.6. Plotting pressuredata in terms of the gradient eliminates differences inexit back pressure and allows a fair comparison for afixed shock location and strength. With thisconvention, negative values of pressure gradientindicate expansion, positive values indicatecompression, and zero values indicate no pressurechange (in a duct of varying area, this impliesseparation).

0.2

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0.4

0.5

-40 -20 0 20 40 60 80

ComputationExperiment

1.2 1.4 1.6 1.8 2.0

ppoj

x xso

-d

x xt/

Figure 38: Pressures at NPR=2.4

-1

0

1

2

3

4

-40 -20 0 20 40 60 80

TheoryExperimentCFD

1.2 1.4 1.6 1.8 2.0

xp

px

t

oj

¶¶

x xso

-d

x xt/

Figure 39: Nondimensional Pressure Gradient

17


In the expansion region upstream of the shock,experimental and computational pressure gradient datashow rough agreement with theory, though the latterclearly does not account for the vena contracta effect atthe nozzle throat. Data in the shock region indicate asingle, sharp compression, but one that is gradual andspread out compared to the inviscid theory. This isexpected, as viscosity results in a thicker shock andallows for upstream pressure communication throughthe subsonic portion of the boundary layer. Measuredfrom the point of initial pressure jump to thedownstream location where the experimental andcomputational pressure gradient matched theory, theÒpressure lengthÓ of the shock - boundary layerinteraction was about 16do. Whereas theory predictssome pressure recovery downstream of the interaction,experimental and computational data approach a nearzero value, representative of full separation.

Computational Boundary Layer Profiles

Normalized boundary layer velocity profiles from thecomputation at various streamwise locations throughthe shock - boundary layer interaction are shown infigure 40. The first profile at x-xs/do=-6.31 (x/xt=1.4)is representative of a classic turbulent boundary layerand was also observed at upstream locations. It was thelast profile seemingly unaffected by the shock -boundary layer interaction, and thus marks the upstreamextent of that interaction and the upstream influence ofthe shock jump (through the subsonic portion of theboundary layer). Moving downstream, boundary layerprofiles show influence of the adverse pressure gradientimposed by the shock, and reach an inflectional state atx-xs/do»3. Separation and reverse flow are seen tooccur for all subsequent downstream profiles.

-0.2 0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

hd

u ue/

..

(x-xs )

/d o = 18

.92

11.35

-6.31

-1.26

0.00

1.26

2.52

3.78

5.04

6.31

Figure 40: Velocity Boundary Layer Profiles

Combined with previous observations, the boundarylayer profiles indicate that, first, the distance betweenthe undisturbed boundary layer at x-xs/do=-6.31 andthe initial shock jump (in the computational data) atabout x-xs/do=-4.9 was less than 2do. The distancefrom the shock jump to separation at x-xs/do»3 wasabout 8do. So, from the standpoint of viscous flow inthe nozzle boundary layer, the entire shock - boundarylayer interaction and resulting separation occurred overa total distance of about 9-10do.

Schlieren Images

Experimental and computational schlieren flowvisualization is shown in figure 41. Though there areslight differences in shock structure, both images showthe nozzle shock with a large, well defined lambda foot,and separation that begins at the leading lambda shockand extends downstream. Coupled with pressure datadiscussed above, this suggests that as far as the nozzleboundary layer was concerned, all of the pressure riseacross the lambda shock system occurred across theleading branch of the lambda foot. In this shocksystem, the leading branch of the lambda foot extendeddown to the nozzle boundary layer, while the trailinglambda foot extended to the detached shear layer. Inthe computational image, vortex sheets are visibledownstream of the lambda bifurcation points, andprovide an entropy slip between flow that passedthrough the oblique lambda shocks and flow that passedthrough the single normal shock. These were alsoobserved in video taken during the experiment.

1.5 2.01.751.25 x/x t

Figure 41: Comparison of Experimental (top) andComputational (bottom) Schlieren Images at NPR=2.4

18


Experimental Shock Measurements

Shock measurements were made from the experimentalschlieren image and used in conjunction with pressuredata and oblique shock relations to diagram the shock -boundary layer interaction. As shown in figure 42, theleading lambda shock had a flow inclination angleb»52° from the nozzle flap, which, with an upstreamMach number of M1»1.6, resulted in a calculatedturning angle of 11° and a downstream Mach number ofapproximately 1.2. From this new direction, the trailingshock had an inclination of b»66°, and with M1»1.2,the corresponding turning angle was 4°. Overall, thisindicates that flow exited the lambda shock system atan angle of a»4°, or 7° off of the divergent flap.

M~1.62

M~1.22

M~1.02

~11°

~0° ~4°

52°

73°

0.76 in.

Figure 42: Experimental Shock Schematic

Computational Shock Measurements

Computational Mach number contours are presented infigure 43, and streamlines are presented in figure 44(note the presence of a double eddy recirculation regiondownstream of the flow separation point whichsupports earlier discussion). Together, these plots wereused to piece together the computational shockschematic shown in figure 45.

Figure 43: Computational Mach Contours

Figure 44: Computational Streamlines

M~1.6

M~1.3

M~1.0~0°

~11°

~ Ð12°

52°

76°

0.52 in.

Figure 45: Computational Shock Schematic

For comparison, the experimental and computationalshock schematics are overlaid in figure 46. Althoughthe basic lambda foot angles are similar in theexperimental and computational schematics, thecomputational lambda shock structure extends furtherdownstream. Among other things, this shortens theheight of the inviscid normal shock and pushes itslightly downstream.

Computational

Experimental

Figure 46: Shock Structure Comparison

1. 6

6

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19


A much bigger difference in the experimental andcomputational shock structures is evident whencomparing flow angles (Mach numbers are in relativelygood agreement). After the first lambda shock, thecomputation shows over twice as much turning as theexperimental estimate, and flow exited the lambdashock system in the axial direction, compared to the 4°angle estimated from the experiment. Part of thisdiscrepancy could be due to the questionable use ofoblique shock theory to explore the experimental shock- boundary layer interaction, but this should be areasonably sound approach, if not totally accurate.Coupled with the differences in shock structure, thisflow angle discrepancy indicates that thecomputationally simulated shock - boundary layerinteraction had different turning requirements than theexperimental case (possibly due to differences in thecomplicated double eddy recirculation region). Thefact that experimental and computational thrustefficiency trends were similar indicates that the overallÒthermodynamicÓ effect of the shock system wasprobably the same in both cases, as it should be, sincebasic shock jump requirements are very simple andÒglobalÓ in nature.

Concluding Remarks

An investigation of separated nozzle flows has beenconducted using experimental, theoretical, andcomputational approaches. Specific conclusions are asfollows:

1. Experimental results indicate that the test nozzle wasdominated by shock induced boundary layer separationat overexpanded off-design conditions. This separationhad two distinct regimes: for NPR£1.8, the separationwas three-dimensional, somewhat unsteady, andconfined to a bubble (with partial reattachment over thenozzle flap). For NPR³2.0, separation was steady andfully detached, and it became more two-dimensional asNPR increased. Upon increasing NPR from 1.8 to 2.0,the nozzle went through a dramatic transition, dividingthe two separated flow regimes. In all cases, separationimproved static thrust efficiency when compared to theideal (attached flow) theoretical model. As the nozzlebecame shock free, thrust efficiency followed along thetheoretical prediction and matched the theoretical peakthrust efficiency.

2. The transition from partial separation to fullydetached separation observed when NPR was increasedfrom 1.8 to 2.0 resulted in several importantobservations. With this NPR increase, nozzle flowadjusted to exit back pressure by completely detachingdownstream of the shock; conditions up to the shockwere nearly the same as the previous NPR. So, thistransition was not the result of markedly different onsetconditions or a stronger shock - boundary layer

interaction, but instead came about through the naturaltendency of an overexpanded nozzle flow to detach andreach a more efficient thermodynamic balance. Thissuggests that in nozzle flows, shock - boundary layerinteraction and separation are mutually dependentresults, in contrast to the typical view that shock -boundary layer interaction is the ÒcauseÓ of separation.

3. For engineering purposes, the computationalsimulation was generally in excellent agreement withexperimental data. There were notable exceptions atlow off-design NPRs, where the 2D computationalmodel was inconsistent with 3D flow observed in theexperiment. At these low NPRs, the computationpredicted fully detached separation, and completelymissed the transition discussed above. For thesimulation of flow control schemes that involveseparation (either its alleviation or encouragement),these modeling limitations would be unacceptable.This indicates that follow on studies with a 3Dcomputational model are necessary to explore thisregime and to study nozzle separation in general.Though the computation accurately modeled the thrustefficiency trend observed in the experiment (with theexception of the low NPR regime), it did not correctlypredict the peak thrust efficiency of the nozzle due tothe lack of sidewall skin friction effects in the 2Dmodel. A simple estimate was able to account for thisdiscrepancy.

4. Although low NPR results from the computationalsimulation were undesirable from a modelingstandpoint, they were very informative for controllingand improving off-design nozzle performance. Byexploring the effect of fully detached separation at eachoff-design NPR, the computational results illustratedpotential benefits of encouraging fully detachedseparation in an actual nozzle. With this type of controlstrategy, the entire overexpanded range of nozzleperformance would be within 10% of the peak thrustefficiency for the nozzle geometry tested. This mayallow a fixed geometry nozzle to cover an entire flightmission more efficiently than a mechanical system dueto reductions in weight and complexity.

5. From a scientific point of view, the computation wasunable to precisely match the shock - boundary layerinteraction structure observed in the experiment at anNPR where flow was predominantly 2D. Overall, thishad no apparent effect on the nozzle performance trend,but it could be important for studies involving flowcontrol and thrust vectoring, where shock structure andsmall scale flow features are important. Combined withthe need for 3D modeling at low NPRs, this indicatesthat further work needs to be done (possibly involvinggrid and solver refinement) to better model the shock -boundary layer interaction and shock induced boundarylayer separation characteristics in off-design nozzles.

20


Acknowledgments

Part of this research was conducted while the authorwas a graduate student with the George WashingtonUniversity Joint Institute for Advancement of FlightSciences at NASA Langley Research Center. Thefaculty and staff of JIAFS are thanked for theircontributions, and the NASA Langley ConfigurationAerodynamics Branch is thanked for supporting thiswork under NASA grant NCC1-24.

References

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2. Asbury, S.C., Gunther, C.L., and Hunter, C.A. ÒAPassive Cavity Concept for Improving the Off-Design Performance of Fixed-Geometry ExhaustNozzlesÓ. AIAA 96-2541, July 1996.

3. Hunter, C.A. ÒAn Experimental Analysis ofPassive Shock - Boundary Layer InteractionControl for Improving the Off-Design Performanceof Jet Exhaust NozzlesÓ. M.S. Thesis, The GeorgeWashington University JIAFS. September 1993.

4. Pack, L.G., and Joslin, R.D. ÒOverview of ActiveFlow Control at NASA Langley Research CenterÓ.SPIE 5th Annual International Symposium onSmart Structures and Materials, Paper number3326-22, March 1998.

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7. Coleman, H.W., and Steele, W.G. Jr.ÒExperimentation and Uncertainty Analysis forEngineersÓ. John Wiley & Sons, 1989.

8. Wing, David J., Mills, Charles T.L., and Mason,Mary L. ÒStatic Investigation of a MultiaxisThrust-Vectoring Nozzle With Variable InternalContouring AbilityÓ. NASA TP-3628, 1997.

9. Weinstein, L.M. ÒAn Improved Large-FieldFocusing Schlieren SystemÓ. AIAA 91-0567,January 1991.

10. Hunter, C.A. ÒAn Approximate TheoreticalMethod for Modeling the Static ThrustPerformance of Non-axisymmetric Two-Dimensional Convergent-Divergent NozzlesÓ.NASA CR-195050, March 1995.

11. Pao, S.P., and Abdol-Hamid, K.S. ÒNumericalSimulation of Jet Aerodynamics Using a ThreeDimensional Navier Stokes Method (PAB3D)Ó.NASA TP-3596, September 1996.

12. Abdol-Hamid, K.S. ÒImplementation of AlgebraicStress Models in a General 3-D Navier-StokesMethod (PAB3D)Ó. NASA CR-4702, December1995.

13. Abdol-Hamid, K.S., Lakshmanan, B., and Carlson,J.R. ÒApplication of the Navier-Stokes CodePAB3D with k-e Turbulence Models to attachedand Separated FlowsÓ. NASA TP-3480, January1995.

14. Carlson, J.R. ÒHigh Reynolds Number Analysis ofFlat Plate and Separated Afterbody Flow UsingNon-Linear Turbulence ModelsÓ. AIAA 96-2544,July 1996.

15. Shih, T.H., Zhu, J., and Lumley, J.L. ÒA NewReynolds Stress Algebraic Equation ModelÓ.NASA TM-106644, August 1994.

16. Liepmann, H.W., and Roshko, A. ÒElements ofGasdynamicsÓ. John Wiley & Sons, 1957.

17. Capone, F.J., Konarski, M., Stevens, H.L., andWillard, C.M. ÒStatic Performance ofVectoring/Reversing Non-axisymmetric NozzlesÓ.AIAA 77-840, July 1977.

18. Leavitt, L.D., and Re, R.J. ÒStatic InternalPerformance Including Thrust Vectoring andReversing of Two-Dimensional Convergent-Divergent NozzlesÓ. NASA TP-2253, 1984.

19. Gatski, T.B., and Speziale, C.G. ÒOn ExplicitAlgebraic Stress Models for Complex TurbulentFlowsÓ. NASA CR-189725, November 1992.

20. Girimaji, S.S. ÒFully-Explicit and Self-ConsistentAlgebraic Reynolds Stress ModelÓ. ICASE 95-82,December 1995.

21. Reshotko, E., and Tucker, M. ÒEffect of aDiscontinuity on Turbulent Boundary LayerThickness Parameters with Application to Shock-Induced SeparationÓ. NASA TN-3454, 1955.

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