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RTDC-TPS-48 I

2D and 3D Method of Characteristic Tools forCom plex Nozzle D evelopmentFinal Report

Tharen RiceJune 30,2003

Prepared under Grant No. NA G3-2460, Design of Exhaust Nozzle for the RBCC- GTXConcept, with the NA SA Glenn Research C enter

THE JOHNS HOPKINS UNIVERSITY APPLIED PHYSICS LABORATORYJohns Hopkins Road, Laurel, Maryland 20723-6099

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JOmSHOpKINs Applied PhytiCS LaboratoryLaurel MD 07236(199U N I V B R S I T Y RTDC-TPS-48 1Table of Contents

1 .0 Purpose .................................................................................................................. 52.0 Introduction............................................................................................................ 53.0 2D Method of Characteristics Tool ........................................................................ 73.13.2 2D MO C Nozzle Algorithm Overview .............................................................. 72D MOC Tool Inputs ....................................................................................... 10Nozzle Geometry Input ............................................................................ 10Nozzle Type Input.................................................................................... 1 13.2.2.1 Perfect Nozzle ...................................................................................... 123.2.2.2 Rao Nozzle ........................................................................................... 12Set End P oint Nozzles .......................................................................... 123.2.2.4 Cone/W edge Nozzles ........................................................................... 13Nozzle Design Parameters ....................................................................... 133.2.4 Flow Properties ........................................................................................ 13Throat Geom etry Input............................................................................. 14

Print Option Input .................................................................................... 143.2.6.1 Sum mary.out File ................................................................................. 15Num ber of Output S treamlines Input ....................................................... 16MO C Limiters Input................................................................................. 16Run Streamline Tracing Tool Button ....................................................... 18Calculate MOC G rid Button .................................................................... 18Initial Data Line Definition .............................................................................. 19Tricks of the T rade........................................................................................... 203D M ethod of Characteristic Tool ....................................................................... 233D MOC N ozzle Algorithm Overview ............................................................ 234.1.1 Mathem atical Equations ........................................................................... 234.1.2 Solution Methodology .............................................................................. 26New Reference Plane Determination ................................................... 27

Surface Fit Algorithm ..................................................................... 28Field Point C ompatibility Equation S olution ................................. 29Body Surface Calculation............................................................... 31

3D MO C Tool Inputs....................................................................................... 324.2.2 Grid Setup ................................................................................................ 32

3.2.13.2.2

3.2.2.33.2.33.2.53.2.63.2.73.2.83.2.93.2.IO

. .

3.33.44.14.0

4.1.2.14.1.2.2 Field Point C alculation......................................................................... 274.1.2.2.14.1.2.2.24.1.2.2.14.1.2.24.1.2.3

Body Point Calculation ........................................................................ 30Com patibility Equation Solution.......................................................... 31

4.2.1 File Input/Output...................................................................................... 324.2.3 Surface Fit ................................................................................................ 334.2.4 Initial Plane Properties ............................................................................. 334.2.5 Calculate Nozzle Button and Progress Indicators .................................... 344.2.6 Print Output Parameters ........................................................................... 342D and 3D M OC Tool Verification ...................................................................... 362D MOC Tool Verification ............................................................................... 363D M OC Tool Verification .............................................................................. 366.0 Summary............................................................................................................... 40

4.2

5.05.15.2

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J O ~ SO~KINS Applied physics LaboratoryLaurelMD 0723-6099U N I V B R S I T Y RTDC-TPS-48 17.0 References ........................................................................................................... 408.0 Acknowledgements .............................................................................................. 419.0 Nomenclature ....................................................................................................... 42

List of Figures

Figure 1. Streamline Definition [Ref. 31............................................................................ 6Figure 3 . 2D M OC Parameters .......................................................................................... 8Figure 4 . Initial Data Line and Nozzle Expansion Region ................................................. 8Figure 5. Initial Kernel R egion .......................................................................................... 9Figure 6 . Point D Solution ............................................................................................... 10Figure 7. Nozzle Geom etry Input .................................................................................... 11Figure 8 . Axisymmetric Nozzle ....................................................................................... 11Figure 9 . Planar Nozzle ................................................................................................... 11Figure 10 Nozzle Type Input .......................................................................................... 11Figure 11 Design Parameter Input .................................................................................. 13Figure 12 Flow Properties Input ..................................................................................... 14Figure 13. Throat Geom etry Input ................................................................................... 14Figure 1 4 Print Option Input ........................................................................................... 14Figure 15 Number of Output S treamline Input ............................................................... 16Figure 16 MOC Limiters Input ....................................................................................... 17Figure 17 # of RRC A bove BD S chematic ..................................................................... 17Figure 18 DTHETAB Schematic ..................................................................................... 18Figure 1 9 Run Streamling Tracing Tool Button .............................................................. 1 8Figure 20. Calculate MOC Button ................................................................................... 18Figure 2 1.Nozzle Contour Window ................................................................................ 19

Figure 2 . 2D MOC Tool GUI ............................................................................................ 7

Figure 22 . Initial Data Lines ............................................................................................ 20Figure 23. Initial Kernel Region with Nominal Design Parameters ................................. 21Figure 24 . Initial Expansion Region for Downstream Radius of 0.2 ............................... 22Figure 26 ................................................................... 25Mach Conoid Coordinate SystemFigure 25 . 3D MO C Tool Main GUI ............................................................................... 2 3Figure 27 .Nozzle Coordinate System ............................................................................. 24Figure 28 . Initial Reference Plane P oints Calculation ...................................................... 26Figure 29.Nozzle Geometry Input File ........................................................................... 27Figure 30.Field Point Calculation P arameters ................................................................. 28Figure 3 1. Body Point Calculation Parameters ................................................................ 31Figure 32.File InputlOuput Input .................................................................................... 32Figure 33.Grid Setup ...................................................................................................... 33Figure 34 . Surface Fit Input ............................................................................................. 33Figure 36. Calculate Nozzle Button ................................................................................. 34Figure 38.Print Output Parameters .................................................................................. 35Figure 39. 2D MOC Tool Nozzle Contours ..................................................................... 36

Figure 35. Initial Plane P roperties Input .......................................................................... 34Figure 37 Progress Indicators ......................................................................................... 34

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RTDC -TPS-48 1Figure 40 . 3D MOC T ool Nozzle Mach C ontours ........................................................... 37Figure 42.Nozzle Wall Pressure Com parison ................................................................. 38Figure 41 . Perfect Nozzle Exit Plane Mach Number for Various Radial Division Inputs 38Figure 43.Nozzle C enterline Pressure Com parison ........................................................ 39Figure 44 . Exit Plane Pressure Comparison ..................................................................... 39Figure 45 . Mach Contours for 3D RAO Nozzle Solution ................................................ 40

List of TablesTable 1 Print Option Output Files ................................................................................... 15Table 2 . 3D M OC T ool Output Files ............................................................................... 35Table 3.2D MO C Tool Parameters for 3 D Perfect Nozzle Verification .......................... 37Table 4 . Mach 4 Perfect Nozzle E xit Plane Analysis ....................................................... 38

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JOHNS HOpQ&/S Applled Physics LaboratoryLaurel MD 20723-6099U N I V B B S I T Y RTDC-TPS-48 12D and 3D Method of Char acteristic Tools for Complex Nozzle Developm entFinal Report

Tharen RiceThe Johns H opkins UniversityApplied Physics LaboratoryJune 2003

1.0 PurposeThis report details the developm ent of a 2D and 3 D Method o f Characteristic(M OC ) tool for the design o f complex nozzle geom etries. These tools are GUI driven andcan be run on most W indows-based platforms. The report provides a users manual for

these tools as well as explains the mathematical algorithms used in the MOC solutions.2.0 Introduction

Under a successful proposal submission to NRA -99-LeRC-2, APL w as awarded asmall grant (NAS3-99146) to investigate candidate nozzle designs for the scramjetflowpath of the NASA Glenn Research Center (GRC) developed GTX concept. In thiseffort, APL d eveloped a no zzle design methodology using streamline tracing techniques.After successful completion of that task (see Reference l), a follow-on effort (NAG3-2460) was funded to investigate performance and operating characteristics of the GTXrocket nozzle exhaust system. This effort was reported in Reference 2 in May 2001. Anextension of that follow-on task wa s awarded in Septem ber 2001 an d is reported herein.To better understand the work done in this effort as well as the previous studies,basic definitions and an explanation of the generalized streamline tracing technique needto be presented. B y definition, for a steady flow, a streamline defines the path o f particlewh ose velocity is tangent to the path at all points. Figure 1 show s a graphical explanationof a streamline [Ref. 31, which can be considered to be a generalized fbnction of thespatial dimensions of the pro blem , f(x,y,z) = 0. If the velocity vector field is denoted byV and the local vector tangent to the streamline as ds, then the equation defining thestreamline path can be written as:

+ +V x d s = O

Furthermore, a boundary defined by multiple streamlines defines a streamtube.Note that conservation of mass requires that the mass flow be constant within thestreamtube.

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t

RTDC-TPS-48 1

fluid dlements goingthrough point 1Figure 1. Streamline Definition [R ef. 31Neglecting viscous effects, a wall can be inserted along any portion of astreamtube without modifying the shape o r characteristics of the streamtube itself. Thisfeature serves as the basis for using streamline tracing a s a techn ique for de signing three-dimensional objects. By using streamlines from a known flowfield, such as a two-

dimensional or con ical flow, complex three-dimensional shapes can be traced. Since thestreamlines are derived from a know n flowfield, all of the flow prop erties (i.e. pressure,temperature, velocity, etc.) along those streamlines are known for the design operatingcondition.It should be noted that the use o f a streamline tracing technique assum es that theboundary layer characteristics of the derived three-dimensional design will notsignificantly alter the inviscid flowfield. Also note that the streamline tracing technique isonly used to determine performance at the design condition. Use of high-fidelitymodeling techniques is required to determine the performance at off-design operatingconditions.As a general design technique, streamline tracing has been used previously in thedevelopment of various vehicle and inlet contours. Reference 4-10 describes streamlinetracing applications for waverider vehicles and sup ersonic inlet design. Before thisnozzle work for NASA GRC began, limited work had been do ne on streamline tracing ofnozzle flowfields.The effort reported herein focuses on the development of a 2D and a 3D MOCtool that can quickly design and analyze nozzle geometries. T his effort was an off-shootof the previous streamline tracing efforts. Through the two previous grants, APLdeveloped several versions a nozzle streamline tracing tool called STT 2000. In order for

STT 200 0 to work correctly, a known nozzle flowfield had to be created. The comm ercialcode, TDK, developed by SEA, was used to accomplish this. Reliance on TDK was notdesired so the development of an independent nozzle design tool was initiated. Thisresulted in the 2D MOC tool being developed. Even with the new 2 D tool, the streamlinetracing techniques applied in STT2000 were still limitled by the 2D flowfield assumption.An effort to provide a truly 3D flowfield in which to start the streamline tracing wasdesired; and therefore, the 3D MOC tool was developed.

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along this characteristic line. The equation used is called the co mpatibility equation an d isdefined as:

Where: ,U= sin-($) ( 3 )A derivation of the finite differencing method used can b e found in R eference 11.

tI -I,

Figure 3 . 2D MOC Parameters

Figure 4. Initial Data Line and Nozzle Expansion Region

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RTDC-TPS-48 1Laurel MD 20723-6094Point (0,l) is then fou nd at the intersection of the calculated LRC and the nozzlewall. The nozzle wall is defined as an arc of given radius (RDOWN)ith a center locatedat the z = 0 station. At the wall, the flow angle, 6, is equal to the wall angle.From point (0,l) a right-running characteristic (RRC ) is created based on the

following equation.

Point ( 1 , l ) is found at the intersection of the RRC and a newly calculated LRCfrom point (2,O). The above equations are based on an axisymmetric nozzle type. Solvingfor a planar nozzle simplifies the equations; however the process remains the same. Thisprocess is continued for all characteristic intersections up to Point B. The location ofPoint B is defined as the last point on the initial expansion region defined by 6 ~ .incethis is a 2D solution, the nozzle centerline acts as an axis of symmetry. The resultantcharacteristic mesh (TBFT), or kernel, is shown in Figure 5 . Point F is the centerlinepoint along the RRC starting from B.4

3.5E 3fi 2.55 2Y)

0.50

4 A 6 8 10 12 14 16 18Axial Location

Figure 5 . Initial Kernel Region

Once this kernel has been found, a second iteration begins to determine Point D,which lies on the last RRC of the kernel BF (see Figure 6). For a given Point D, the massflow crossing the BD is found. A LRC is then constructed from D to an un know n Point Ewhere the known mass flow crossing BD is equal to calculated mass flow crossing DE.For a perfect nozzle, the properties along DE are uniform and point E is found rathersimply. For other nozzle types, the location of E and properties along D E are found usinga Runge Kutta Fehlberg method to integrate dM/dr, and dWdr as defined by thecompatibility equation (eq. 2) and d d d r as defined as follows.1-ak

dr t a n ( B+p) ( 5 )

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RTDC-TPS-48 1Once Point E has been found, it is set as the nozzle w all exit point. The propertiesat point E are compared with the given design constraints (nozzle type and designparame ter). If the values match, then a streamline from B to E is calculated and thenozzle contour has been found. This streamline is determined by back calculating RRCsfrom DE to a region upstream of DE. Figure 6 shows this graphically.If the properties at E do not match, then D and ultimately (3B is iterated on untilthey do match. There exists only one combination of (3e and D that satisfies all of theequations and constraints.

Figure 6. Point D Solution

3.2 2D MOC Tool InputsBefore describing the inputs in detail, a short discussion on dimensions should bepresented. As soon will be shown, all of the nozzle geometry inputs (length, area, etc.)have been non-dimensionalized. All lengths are non-dim ensionalized by the throat half-height (distance from centerline to wall) for planar nozzles and the throat radius foraxisymmetric nozzles. This is continued in the tool!; output files. The code calculatesma ss flow, thrust and other dimensiona l parameters by assuming the following. Fo r anaxisymmetric nozzle, a throat radius of 1 is used. For a planar nozzle solution, a throathalf-height of 1 is used as well as a reference width of 12. The performance num bersobtained for a planar nozzle are onl y for one-half of the total nozzle. This is applicable to

SER N nozzle performance calculations.3.2.1 Nozzle Geom etry Input

There are two types o f nozzle geometries to chloose: axisymm etric and planar (seeFigure 7 for input graphic). Figure 8 shows a typical axisymmetric nozzle and Figure 9shows a typical planar no zzle.

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Nozzle GeometryAxisymmetric

Figure 7. Nozzle Geom etry Input

RTDC-TPS-48 1

Figure 8. Axisymmetric Nozzle

Figure 9. Planar Nozzle

3.2.2 Nozzle Type InputThere are four types of nozzles that this 2D MOC tool can solve. The input forselecting the type o f nozzle is shown in Figure 10.

-N o d e Typer Perfect

Figure 10.Nozzle Type Input

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J()J-J(S HOpmS Applled Physlcs LaboratoryLaurel MD 20723-6099U N I V E R S I T Y3.2.2.I Perfect Nozzle

The first type is apev fect nozzle. For a perfect nozzle, the ex it plane is required tobe u niform, meaning that all exit values (Mach , pressure, temperature, etc.) are constantat every radial station. Also, the flow angle at the nozzle exit plane is 0.0 (fully axialflow). This type of no zzle is generally used in w ind tunnel facilities w here uniform flowis desired for testing and length is not constrained.

3.2.2.2 Rao NozzleThe second nozzle type is a Rao nozzle. This nozzle is derived from work don e byG.V.R. Rao in the late 1950s. In this work, a mathematical analysis of optimum nozzlecontours for a given exit condition (Mach, area ratio, etc.) was developed*. There areseveral subtleties as to the optim um contours with respect to the d ifferent exit conditions;however, in general, a Rao nozzle provides a minimum length nozzle for a given exit

condition such that any change in the exit condition at the given length will producelower performance. For exam ple, assume that a Rao solution has resulted in nozzle arearatio of 4 and the optimum nozzle length is 10. This means that for a fixed nozzle oflength of lo, a nozzle designed with an area ratio of 3 or 5, will produce less thrust thenthe original area ratio = 4 ozzle. Again, there are subtleties to the Rao method ; however,in general the abov e statements hold true.The basis of the Rao method revolves around finding the nozzle contour wherethe following nozzle wall exit condition (point E) holds true (for a zero-backpressuresolution).

3.2.2.3Set End P oint NozzlesTh e third nozzle type forces the contou r to go through an explicitly set nozzle exitwall point (Set E n d Poin t option). This type of nozzle is bounded in length by the Raoand perfect nozzle solutions. For a given area ratio nozzle, the Rao nozzle will solve forthe minimum length nozzle and the perfect nozzle solution will solve for the maximumlength nozzle. T he Set End point option ca n be used to find all of the contou rs in betweenthese two lengths. No solution exists if the chosen length is less than the Rao nozzle

length or greater than the perfect no zzle length. The Set E n d Point solution is not requiredto have a uniform exit flow field a s in a perfect nozzle, nor d oes it have to mee t the Raonozz le exit criteria.

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3.2.2.4 Con l e d g e N ozzlesThe fourth type of nozzle is a cone/wedge contour of a given half-angle. Thecone is solved if the axisymmetric option is chosen. The wedge is solved if the planaroption is chosen.

3.2.3 Nozzle Design Param etersThe design parameter section sets the required condition at the nozzle w all exit. Ascreen capture of this input is shown in Figure 11. This parameter can be set to thefollowing types.

o Machnumbero Are a ratio: Ratio of the exit-to-throat areaso Pressure ratio: Ratio of nozzle total pressure-to-exit static pressure

Length ratio: Ratio of nozzle length-to-throat radius- Design Parameter1_- 7Iigure 11. Design Parameter Input

3.2.4 Flow PropertiesThese inputs define the initial flow properties for the nozzle solution (see Figure

12). For a given set of flow properties, the user has two choices as to what theseproperties represent. The default is that the flow properties are the total conditions. Giventhese total conditions, the tool solves for an initial data line near the throat in which tobegin the solution. If the Throat C onditions box is checked, then these properties will beused as the initial data line flow properties. Also, choosing the throat conditions enablesthe Velocity input where the user is required to input a flow velocity. The flow ha s to begreater than M ach = 1 for a nozzle solution to be calculated.

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JOmSHOpKINS Applied Physics LaboratoryLaurel MD 20723-6094U N I V E R S I T Yr low Properties

1 r hroat Conditions?IPressure (psia]

Temperature (R]1530-Mol. Wt 128.96

P ambient [psia)10velocity (ftis-1 [

Ideal lsp (Ibf- s/h ]1100

Gamma r iFigure 12 . Flow Properties Input

3.2.5 Throat Geometry InputThis input (shown in Figure 13) defines the throat geom etry as two arcs of givenradius as depicted in Figure 3. Note that these input are non-dimensional and arenormalized by the throat radius (R*). The Upstream Radius input helps to define theinitial data line given the total flow conditions. Section 3 .3 describes the creation of th einitial data line in detail. Th e DownStream Radius is used to define the initial nozzleexpansion region.

Throat Geomertry--lUfQan~Radius&!'

I1 DownSbeam RadiusiR"

Figure 13. Throat Geometry Input

3.2.6 Print Option InputThe code is capable of outputting various parameters in various w ays. The user isallowed to choose the level of output desired for each run. The choices are Normal andFull (see Figure 14). The Full option outputs all of the data available. The Normal optionoutputs a subset of the Full option. Table 1 shows a listing of the various outputs for each

option.r Print Option------- II_

/j IL-h-___-- IFigure 14 . Print Option Inpu t

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Table 1. Print Option Outpu t FilesSummary.plt TECPLOT formatted file showing the nozzle contour, theSummary.outrao.datTT. ouThetaB.outMOC-Grid.plt

Data File Name Description Normal Full0last LRC, and the RRC (BF)

Summary file describing all of the nlozzle detailsFile containing the nozzle contour that can be used withthe RAO option in TDK.Initial data plane properties

0 0

0 0

0 0

OB teration parameters 0 00ECPLOT formatted file showing all of the RRCs for thenozzle. This file can be used to look at contour plotsthrough the nozzle.TECPLOT formatted file showing streamlines for thenozzle. This file can be used to look at contour plotsthrough the nozzle.0OC-S L.plt

0enter.out Contains centerline flow data.0all.out Contains nozzle wall flow data and contour.0TB F-Kernel. ou

BFE-KerneLoutwall-i.outaxis-i. ou

Contains a matrix of data for all of the RRCs in regionTTBF, See Figure XXContains a matrix of data for all of the RRCs in the regionBFE, See Figure XXData for the initial wall expansionData for the centerline up to point F

0

0

0

0LastKerneLout Data for the last RRC (BF) defined by OB

0ncropped KerneLout Data for the entire MOC grid generated extendingbeyond the nozzle exit plane

3.2.6.1 Summary.out FileThe primary output file for the cod e is the sum mary.out file. This file containsMOC grid data, performance values and other nozzle parameters. This section gives abrief description o f the items in this file.The first part of the file contains a general definition overview followed by theinput parameters that were used in the no zzle design. The next section contains the flowproperties along the initial data line. The m assflow data is an integration of the massflowfrom the nozzle centerline to the nozzle w all.This list is followed by a summary of the performance parameters. Theseparameters are separated into two parts. The first part reports the 2D performance

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(massflow, thrust, etc.); the second part reports the 11) performance. Fo r 2D calculations,performance is taken from the integration of individual flow points, where as the 1Dcalculations are based on a I D isentropic process from the nozzle total conditions to auniform M ach 1 throat.The next section contains information on the initial expansion region, includingthe converged initial expansion angle and the difference in the integrated massflow alo ngthe last RRC at the initial expansion region (BDF in Fig. 3) and the massflow at the initialdata line. This is a good check to make sure that the initial grid and resultant solution isprogressing smoothly. The code checks to make sure the percent difference in thesevalues is less than 2% . If it is not, the code will notify the user and the solution willterminate. Future versions o f this tool may ma ke this tolerance a user input.The next two sections contain the flow properties along the LRC (DE) and theentire nozzle wall respectively. These sections are followed by a list of the flowproperties at the nozzle exit plane, as well as nozzle geometry parameters (surface area,

area ratio, etc.) and e xit plane perform ance.3.2.7 Number of Outpu t Streamlines Input

This input sets the number of streamlines and points that will be printed in theMOC-SL.plt file (see Figure 15). The Radial input defines the total number ofstreamlines. The Axial input defines the number of points along each streamline.rNumber of output S treamlinesl

Figure 15. Num ber of Output Streamline Input

3.2.8 MOC Limiters InputThese inputs affect the initial MOC grid as well as the efficiency of the code (seeFigure 16). The THETAB Guess (3 input gives the tool a starting point for the 3iteration. In general, longer nozzles (i.e. perfect) have lower values than short nozzles(Rao). A nominal value of 25" is supplied and works for most cases.

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Figure 16. MO C Limiters Input

The # of RRC above BD inputs refers to the number of Right RunningCharacteristics the tool will solve for after the nozzle end point is found. As described inSection 3.1, once the nozzle end point (E) is found, the nozzle contour from B to E needsto be determined. This contour is defined by a streamline that is solved for by completingthe MOC grid in the BD E region (See Figure 17). The density of the grid in region B DEis defined by the number of RR Cs.

Figure 17. # of RRC Above B D Schematic

The Number of Starting Characteristics input determines the number points to befound on the initial data line.The DTHETAB (A8s) Max input is a parameter that helps establish the RRCdensity along the initial nozzle expansion arc (TB). For a given number of startingcharacteristics, the MOC solution begins. As a LRC reaches the nozzle wall it is reflectedtoward the centerline. DTHETAB refers to the maximum difference in OB between anytwo nozzle wall points. If the difference is greater than the input DT HET AB , a new R RCis created at that point. An example is shown in Figure 18. The angle between the initialwall point and Point (0 ,l ) is greater than the defined (A8B)MAX. A new R RC (defined bythe red dashed line) is created at an angle (A~B)MA:

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JOHNSHOPmS Applied Phvtics LaboratoryLaurel MD 20723-6099U N I V B B S I T Ysmaller the value o f DTHE TAB , the more refined the grid m ay be, at the cost of run time.Nom inal values are between 0.25" and 0.5".

A

Figure 18.DTHET AB Schematic

3.2.9 Run Stream line Tracing Tool ButtonWh en this button (Figure 19) is pressed the streamline tracing tool STT 200 0 canbe executed. The files needed to run STT2 000 are only created w hen the Full print optionis chosen; therefore this button is only enabled at that time.

R u t i S treari4tseFigure 19. Run Streamling Tracing Tool Button

3.2.10 Calculate MOC Grid ButtonWhen this button is pressed (Figure 20), the solution cycle is started using theinputs displayed in their respective boxes. When the cycle is complete a graph of thenozzle contour appears in a new window. This window has to be closed in order for thetool to outpu t the required files. Figure 2 1 shows a sample of the chart geometry window .

Calculate MOC Grid

Figure 20. Calculate MO C Button

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Figure 2 1. Nozzle Contour Window

3.3 Initial Da ta Line DefinitionThe initial data line (TT ) is calculated using a modified H all method developedby K liegel and Levine for transonic flow around nozzle throat regions. The method uses

a toroid coordinate system to develop an analytical solution for flow velocity (axial andtransverse) at all points in the throat region of a nozzle.The true art in the initial data line definition i,s determining the shape of the linewhere the transonic regions. A line that is too shallow or too stee p can ultimately result ina failure in the solution convergence. For this tool, the calculation starts at the nozzlewall. The position of all subsequent points is determined by constructing a RRC from theproceeding point. If the Mach nu mber at a given point is greater than 1.5, the line shape ischanged so that Mach = 1.5 is never exceeded.The Upstream Radius parameter has a pronounced effect on the initial data line.As the upstream radius increases, the calculated flow velocity increases by nearlyl / (Rup+l) . Figure 22 shows initial data lines for two upstream radii without the Mach 1.5constraint. As the upstream radius decreases, the calculated Mach number increases. Thiscauses the constructed RR C to m ove dow nstream because the calculated flow angles (p)are shallower. In so me cases, this causes difficulties in the nozzle solutions. The third linein Figure 22 shows the solution with the Mach 1.5 constraint, resulting in a well behaveddata line. The Mach constraint as well as the line shape is arbitrary. This tool uses thisparticular m ethod because is has shown to w ork for many nozzle designs.

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0.8a40 0.6KUA% 0.4Ebz

0.2

00 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6

NormalizedA n a l LocationFigure 22 . Initial Da ta Lines

3.4 Tricks of the TradeThis section will try to explain techniques that should be used to arrive at aconverged nozzle solution. The primary cause of a poor nozzle solution starts with thedevelopment of a poor initial MOC grid and the parameters that create it. Theseparameters are:

Downstream R adiuso Nu mb er of Starting Characteristicso Upstream Radius0 (A~B)MAX

Understanding the interaction between these parameters should provide a goodguide to good nozzle development. The following discussion provides a summaryexplanation of these parameters. It is recomm ended tlhat each user conduct a short studyof these parameters by using the nozzle tools and investigate their effects.

A good nozzle solution starts with a well defined grid in the initial nozzleexpansion region (from initial data line to OB). The term well defined is subjective;however, the default values for the four aforementioned parameters seem to work formost cases. Figure 23 shows the initial expansion re,gion around the nozzle wall usingthese defaults parameters for a perfect, Mach = 4 nozzle.

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- -32 Points Along Initial LineDefine Expansion Region

0 0 2 0 4 o e 0 8 1NormdizedAxial Loca ion

Figure 23. Initial Kernel Region with Nominal Design ParametersAs is shown, 32 points (out of the initial 10 1 points) along the initial data line areused to determine the initial nozzle expansion region. This seems to give the MOC grid a

solid start to the calculation. In general, m ore points are better, ho wever if there a re toomany points, the RRCs can run too close together (and sometimes even cross) causingsolution difficulties. Again, there is no set number that is better or worse for a givendesign.As the four parameters deviate from these defaults, the initial kernel regionchanges. For example, Figure 24 shows this region when the downstream radius isdecreased to 0.2. As is shown, the number of points that now define the initial expansionregion is reduced from 32 to 14. In CFD related terms, this new mesh is courser (by afactor of two). As the grid gets courser, the code has a mo re difficult time converging ona solution. The MOC algorithm makes calculations at the intersection of the LRCs andRR Cs and at the nozzle wall. As these intersections get hr th er apart (courser mesh), thesolution produces bigger errors. Errors in this region of the nozzle get amplified as thesolution progresses downstream. Th e dow nstream radius input seems to have the greatest

effect on whether or not the nozzle solution will converge. Keeping this parameter aroundthe nominal 1 O value is recommended.Decreasing the downstream nozzle radius is not the only way for the mesh tobeco me courser. Chan ging the num ber of starting characteristics is a straightforward w ayto affect the mesh. Changing the upstream radius as well as changing the (&)MAXparameter also affects the mesh, and is explained in S ection 3.2.8.

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for RaowJR' = 0.2

0 0.05 0.1 0.15 0.2Normalized Axial Location

Figure 24. Initial Expan sion Region for Downstream Radius of 0.2

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4.0 3D Method of Characteristic ToolThe 3D MOC tool developed under this effort allows a user to determine three-dimensional nozzle flowfield properties given initial throat conditions and a nozzle contour. Th e

solution algorithm solves the 3D MOC equations using a reference plane method where thenozzle is parsed into numerous axial stations. For a given axial station, flow properties aredetermined using the flow information of the preceding reference axial plane. The first axialstation is determ ined by user d efined throat properties.The tool was developed to be GUI driven, just as the 2D MO C tool. Figure 25 show themain GUI Window. With in this main window, there aresections describe these inputs and how they are implemented. 5 types of inputs. The following

4.14.1.1 Mathem atical Equations

3D MO C Nozzle Algorithm O verview

As stated above, the 3D MOC tool uses a reference plane method to solve for the 3DMO C flowfield equ ations (compatibility equations). A complete explanation of this method canbe found in Reference 14. A majority of the explanation is echoed below.

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The derivation of the compatibility equations begins with the equations of motion forsteady, inviscid ideal flow. For these equations, two sets of characteristic surfaces can bederived. These surfaces are defined as follows.

Where f(x,y,z) = 0 defines a surface composed of streamlines and g(x,y,z) = 0 defines aMach conoid. Along the conoid surface ,a ray, also referred to as a bicharacteristic, i s defined bythe following equations.dx = ( c o s p s i n 0+ sin /j c os 0 c os 6)dL (9)

dy = ( c o s p c o s e si n t y -s in p (s in 8 si n t y c o ~ S - c o ~ t y s i n) )d L (10)dz = (cosp co s8cos ty- in p(sin 6 os tycos S +sin tysin 6 ) ) d L (11)

Where 8 and y~ re related to the velocity vector ((1)by:u = q s i n 0 (12)v = q c o s 0 s i n y (13)

The parametric angle, 6 , lies in the plane normal to q and is measured from the planecontaining q and x. Figure 26 shows the nozzle coordinate system used. Figure 27 shows agraphical representation of these parameters.

XI

Figure 26. Nozzle Coordinate System

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X

YFigure 27. Mach C onoid Coordinate System

For this 3D solution, the compatibility equations are determined from the flow equations(eqs. 8-10) with a constraint that their derivatives in the direction norm al to the characteristicsurface are zero.

For the surface defined by the Mach conoid (eq. 811, he compatibility equation indifference form is:-(4otp. -e)+osS,(@, e,)+ os@, inej(y2-vi)+inpiPi92

a aWhere- nd- re derivatives along and norm al to the bicharacteristic.aL aNThe compatibility equations along a streamline are defined as:

Y 1Y-1 P- dT =- P = -qdq

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JOmSHOpWS Applled Physics LaboratoryLaurel MD 20723-6099U N l V B R S l T Y4.1.2 Solution Methodology

The solution begins with the definition of an initial reference plane. This and allsubsequent planes are norm al to the z-axis as defined in Figure 26. T he flow properties (P, T, V,etc.) at the initial reference plane are set by the user. The num ber of points w ithin this plane isalso user defined. Given the number of points, the tool creates a grid of nearly equally spacedpoints. Figure 28 sho ws an exam ple of this grid spacing.

Figure 28. Initial Reference Plane P oints CalculationThe points on the outer m ost radius define the inilial wall con tour, and will be referred toas body points. Th e remaining points define the internal flowfield and will be referred to a fieldpoints.In addition to the initial reference plane, the tool also needs the nozzle wall con tour inorder to reach a solution. In this tool, the wall con tour is defined by the following equation.

2q2 = & x i ) + ( y - y i ) 2The values of Ti, xi and yi are read from a datafile. Figure 29 sho ws a sample of this file.The value in the first row defines the number of axial (z) stations in the nozzle geometry. Thenext row includes a data header (Z rO x0 yo ). The parameter z defines the axial location. Theproceeding row s contain the d ata values.

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1 5 8Z rO xo YO0 1 0 00 . 0 0 3 4 0 8 1 . 0 0 0 0 1 0 00 . 0 0 6 7 9 2 1 . 0 0 0 0 2 0 00 . 0 0 8 9 5 6 1 . 0 0 0 0 4 0 00 . 0 1 1 4 4 9 1 . 0 0 0 0 7 0 00 . 0 14 2 8 2 1 . 0 0 0 1 0 0

Figure 29. Nozzle Geo metry Input File

4.1.2.1 New Reference Plane D eterminationAfter the initial plane and nozzle geometry has been defined, a new reference plane iscreated at some distance, z, dow nstream of the initial reference plane. The value of the new z

is taken from the nozzle geometry file. The distance from the initial reference plane to this newplane is defined as dz. For every point (PI) defined on the initial plane, a corresponding point(P2) on the new reference plane is found. This process is repeated until the last reference planehas been computed.

4.1.2.2 Field Point CalculationThe field point calculation is used to determine all internal (not wall) points on the newlycreated reference plane. Figure 30 shows a graphical representation of the parameters involved.For a given point P1 on the initial reference plane, a streamline is constructed from P I to P2

which lies on the new reference plane. The position of P2 is defined by:t a n 6 dicostyx, =X I +-

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bl . bN), how ever equation 25 yields only N quations to be solved, so three other equations areneeded. They are:N N N (26-28)

i=l i=l i=l

Initial implementation of this surface fit, had N=9, where for a given point P I, PI and itseight nearest neighbors were used to determine the fit. This yielded slight errors in the fit. Inparticular, for several test cases where a uniform flowfield was expected, this fit yielded slightdifference in the flow properties for points at constant radial locations. Ultimately N was set toinclude all of the points in the reference plane, and this error was eliminated.Further research was conducted to determine if thlere was a better a1 orithm to use. Threealgorithms were found (Algorithm 79215, CSH EP2D I6 and Algorithm 761 ), each with its ownbenefits and detriments, howev er none have been implemented to date.

I$

4.1.2.2.2 Field Point C ompatibility Equation So lutionOnce the four bicharacteristic points (P3-P6) have been found and their flow propertiesdetermined, the compatibility equation is then solved from each point (P3-P6) to P2. Aspreviously h ighlighted in eq. 15, the compatibility equation in difference form is:

Where:

( = -cosOsinai

' y = sin 0 sin w sin ai+ cosw cos ai( z ) i

= sin 0 cosy sin (5;- in wco s 6,[%)i

+% 2-z dzNote that the sam e form of equation 30 holds true for y ~ .

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The unknowns in eq. 29 are p2, (32, and ~ 2 ,o only three bicharacteristics are required fora solution. Reference 14 recommended averaging four separate solutions (each using 3 points)to improve accuracy. This tool uses this recommend ation. The comp atibility equation is solvedfor using four sets of points [(P3, P4, Ps), (P3, Ps, P6), (P3, P4, P6) and (P6, P4, Ps)] and theresultant va lues for p2, (32, and ~2 are averaged.

Using the com patibility equations along an isentropic streamline (eq. 16) the values of p2,T2 and q2 can be found as fo llows:

RT,P2 =-p2

(35)

Where PTand TT are the total cond itions along the streamline.Given the calculated flow quantities at P2, a new streamline from PI (based on theaverage properties of P I and P2) is constructed to a new P2 location and new flow properties arefound using the sam e process. Th e iteration continues until the location an d flow properties at P2remain c onstant (within some tolerance). This process is repeated for all of the field points.

4.1.2.2 Body Point CalculationThe B ody point calculation is performed for every point along the no zzle wall. Figure 31show s a graphical representation of the param eters involved.

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Initial planeFigure 3 1. Body Point Calculation Parameters

For a given body point P I, a new body point P2 is found at the intersection of a planedefined by the body surface unit normal and the unit velocity vector tangent to body surface at PIand the b ody surface at P2. Th is leads to solving the following equations simultaneously.B ( x ,y , z ) = 0 [Body surface] (3 7)

(n3cos8cosw- n, cos8sin w)(x- xl)+ a, s in8- 2cos8cosy/)(y - , )= (n 3 in 8- n,cos8sin w)dz (38)Where n l, n2, and n3 are unit normals to the body surface.

4.1.2.2.1 Body Surface CalculationBased o n the nozz le wall geom etry around P1 and P2, the body surfa ce is approximated a sone of the following four shapes.

0 Vertical l ine: x = c0 S lope d line: a x + b y = c0 Circle: (x-a)2+ (y-b)2=c0 Horizontal l ine: y = c

At each point, the values of a, b, and c are solved and then used to determine the bodynormal coefficients.

4.1.2.3 Com patibility Equation S olutionAfter the location of P2 has been found, three bicharacteristics are constructed from P2back to the initial reference plane. The intersection of these lines with the initial plane is defined

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as P3-P5. P4 is found so that it lies on the line normal to the body surface and P I. Mathematically,the parametric angle for P4 is defined asa4= cos-' (- n, sin B cos I,Y+ n2cosB - n3sin 8 sin I,Y) (39)

The parametric angles for P3 and P5 are iterated so that the points are within the nozzleflow region.For this calculation, there are three unknow ns, p2,, (32, and ~2 (from eq. 29). To solve forthe parameters, the compatibility equation along two bicharacteristics (eq. 29) and a flowtangency constraint are solved. The tangency constraint sets the condition that the flow is tangentto the surface at P2 and is defined as:

n , cosw, + n 2 an & + n , s h y 2 O (40)

To increase accuracy, three sets of equations are solved and their results averaged. Eachset has a different se t of bicharacter istics, (P3, P4), (P3, P5) and (P4, P5). The values for p2, T2 andq2 are solved used eq. 34-36. The iteration continues until the location and flow properties at P2remain constant (within so me tolerance). This process is repeated for all of the body points.

4.2 3D MOC Tool InputsIn this version of the tool, the inputs shown in white are functional. The grayed inputshave been included as place holders for future implementation.

4.2.1 File Input/Ou tputWithin this input, the user is asked to input the name of nozzle geometry file. Figure 32shows a screen capture of the input. The format of the nozzle geometry m ust follow the format

as discussed in Section 4.1 .2 and Figure 29.-

Figure 32. File Input/Ouput Input

4.2.2 Grid SetupThe available input in the Grid Setup input is the number of radial division in the initialreference plane (see Figure 33). This value defines the number of equally spaced points to be

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JOHNS HOpmS Applied PhysicsLaboratoryLaurel MD 207236099U N l V B R S I T Ycalculated along the nozzle wall. The distance between the points is defined the followingequation, w here R is the know n initial plane radius:

2 * 7 r * Rd ='Divisions

The calculated d istance, d, is used to create a near , j equally space grid wil in the nozzlegeometry as shown in Figure 28. A nominal value of 36 yields a good balance between solutionaccuracy and run time.rGrid Setup [CylindricalCoordinatesJT

RadEal Divisions136Number of Ray PtsI"

Figure 33. Grid Setup4.2.3 Surface Fit

The Surface Fit input allows the user to define the type of surface fit used in thecalculation. Figure 34 sho ws a screen capture of this input. The choices as discussed in Section4.1.2.2.1 are a 9-point fit and a fit that uses the entire reference plane.

Surface Fit-9 Point Spline --igure 34. Surface Fit Input

4.2.4 Initial Plane PropertiesThe Initial Plane Properties input defines the properties o f the initial reference plan e (seeFigure 35). Within the current implementation, the flow properties over the entire initial planeare constant. This limits the tool's capability to calculate non-uniform initial flowfields. Thesolution algorithm presented herein is capable o f obtaining a solution starting with a non-uniformflowfield, the hindrance really occurs in the user input. To truly implement a non-uniformflowfield, the input would have to be changed, where the user specifies all of the flow propertiesat every point on initial plane.

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Initial Plane Properties-I___Pressure(psia11 TemperatureIR]1530Mach Number 1 7 -

Gamma 1 1 . 4Mol.Wt. jz---Theta [des] E

~ i l d ~ l rAxial Location (in]7

IFigure 35. Initial Plane Properties Input

4.2.5 Calculate Nozzle Button and Progress IndicatorsThe calculation is started by pressing the Calcula.teNozzle button as shown in Figure 36.The code determines the numb er of axially steps needed to reach the end of the nozzle and inputsthat number into the Total Steps box (see Figure 37). .As the code calculates each new axialstation, the Step Num ber box is updated.

Calculate Nozzle

Figure 36. C alculate Nozzle Button

StepNumberTotal Steps

Figure 37. Progress Indicators

4.2.6 Print Output ParametersThese parameters (see Figure 38) define the grid resolution on the output files. The Every

NP oint in Xp aram eter helps define the number of points in a given plane to output as follows:- n ~ o t ~oints -Every N Point in Xpoints -

This is most useful when outputting streamlines. The num ber of poin ts, npoints,will equalthe num ber of streamlines output.

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The Every N Points in 2 controls the number of axial stations (z-direction) that the codeoutputs based on the following equation.- nrotalStepsEvery N Point inzplanzs - (43)

Th e first and last planes w ill always be outpu t regardless of this value.The Every N Step Number input determines when the output files are generated based onStep Number discussed in Section 4.2.5. If the Every N Step Number value is greater than theTotal Steps, then the files are created at the end of the calculation. This option may come inhandy if the tool bomb s before com pletion. By setting this num ber to the Step Number where thetool bombed, the code will output the flow at that station where it can then be reviewed. Table 2summarizes the various output files produced by the tool.rPrint Ouput P a r a m e t e r s - 7I Every Point[s) nX 1I Every Point[s]inY 1- I1 Every 1999 StepNumber 1Figure 38. Print Output Parameters

Table 2 . 3 D MOC Tool Output F iles~Data File Name DescriptionZ=O.out Contains the location of the initial throat plane points and their respective flow propertiesWall.plt TECPLOT formatted file showing all of the nozzle wall flow properties

TECPLOT formatted file showing all of the nozzle flowfield data. The data is organizedas 1 surface from the centerline to the wallull mesh.pltTECPLOT formatted file showing all of the nozzle flowfield data. The data is organizedAxialStations.plt in separate axial stations.TECPLOT formatted file showing the original nozzle geometry as read from the inputnozzle geometry file. The number of points to print out does not affect this file.nitial Wall.pltTECPLOT formatted file showing all of the nozzle flowfield data. The data is organizedStreamlines.plt in separate streamlines starting from each of the initial reference plane starting points.

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5.0 2D and 3D MOC Tool Verification

5.1 2D MOC Tool VerificationIn order to ensure a valid solution, num erous example cases were run on these tools. Themost extensive verification process was conducted on the 2D MOC tool. For given nozzle designparameters, the tool was run to verify that the calculated exit plane me t these parameters. Forexample, for a perfect nozzle solution with a prescribed exit Mach number of 4.0, the nozzlesolution was checked to make sure that the resultant exit Mach nu mber was 4.0, nd that the exitplane was uniform (as defined by a perfect nozzle). Figure 39 shows several nozzle contourscreated by the 2D MOC tool. This figure is also useful in understanding the different nozzletypes (Perfect, Rao, and Set Endpoint) for a given exit condition (Mach = 4.0)

Exit Mach Number = 4.01098

? 7c$ 5i i 48 3

oint:UR* = 1260ti-(Dpc

2100 5 10 15 20

Axial Distance(UR*)Figure 39. 2D MOC Tool Nozzle Contours

5.2 3D MOC Tool VerificationThe 3D MOC Tool was verified by running the two samples cases. In both cases, thenozzle geometries generated from the 2D MOC tool were used as inputs to the 3D tool. Theresultant 3D flowfield was then co mpared to the known 2D flowfield. The first case run was aperfect nozzle with a prescribed exit Mach number = 4.0. Table 3 shows the 2D MOC toolparame ters in the definition of the nozzle. The resultant initial Mach num ber is 1.15 181.

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JOHNS HOpmS Applied Physics LaboratoryLaurel MD 0723-6099U N I V B R S I T YTable 3. 2D MOC Tool Parameters fo r 3D Perfect Nozzle VerificationParameter ValueNozzle Geom etrv AxisvmmetricNozzle Typ e PerfectDesign Parameter Mach = 4.0UDStream Radius/R* 1 oDow nstream Radius/R* 1 oThroat Conditions Box CH ECKPressure, psia 1000Temperature, R 530Mol. Wt. 28.96Gamma 1.4P ambient. Dsia 0.0Velocity, ft/s 302 2ThetaB Guess 25# of RRC above BD 100DTHE TAB Max, deg 0.5Nu mb er of Starting Characteristics 100

Figure 40 shows a Mach contoured plot of the resultant 3D flowfield. As is seen, theMach contours along the wall are uniform at each axial station, w hich is what should be expectedfor a perfect nozzle.

Figure 40. D MO C Tool Nozzle Mach C ontoursThe nozzle exit plane was also interrogated to analyze the flowfield mo re closely. Figure41 shows a Mach profile plot for all of the points at the nozzle exit plane. The nominal exitMach number is 4.0. hree sets of points ar e presented, each based on a different Radial Divisioninput. Table 4 details the statistical results. All of the cases over predict the exit Mach number;however, the deviation in the resultant Mach num ber is small. Moreover, as the num ber of radialdivisions increases (increased number of points) the error increases slightly, but the deviation inall the values decreases. The over-prediction is most likely caused by definition of the initialnozzle plane. The 2D MOC solution calculates the initial data line as a right-runningcharacteristic (Z varies), w here as the 3D MO C initial plane is calculated as a vertical plane (Z is

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constant). Also, small errors due to calculation approximations and tolerances can get amplifiedas the flowfield is propagated down stream.

Exit Mach NumberFigure 4 1. Perfect Nozzle E xit Plane M ach Num ber for V arious Radial Division Inputs

Table 4. Mach 4 Perfect Nozzle Exit Plane AnalysisNum ber of points 67 181 580Average Exit Mach Num ber 4.027 4.051 4.0673 - 0 Deviation 0.067 0.032 0.019Average YO ifference 0.664 1.29 1.67

Radial Divs. = 18 Radial Divs. = 36 Radial Divs. = 72

The 3D M OC Tool was also checked for a Mach 4 Rao nozzle design. The sam e inputs(except for the Nozzle T ype) as shown in Table 3 we re used in the generation of the 2D MOCnozzle geometry. Figures 42 and 43 show the resultant pressure traces along the wall an dcenterline for the 2D tool and the 3 D tool, as well as the rlesults from the S EA RAO a nd T DKcodes respectively. As is seen all the results compare favorably.

0 2 4 6 8 10 12Axial Location

Figure 42. Nozzle Wall Pressure Com parison

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Axial LocationFigure 43. Nozzle Centerline Pressure Comparison

Figure 44 shows a comparison of the nozzle exit plane pressure. The SEA RAO codedoes not include the exit plane profile in its output and therefore is not plotted in the figure. As isshown, the results are similar, however the 2D tool returns a discontinuous profile which isdifferent from the other two codes. It is unclear as to why this occurs and more investigation maybe necessary. It should be noted that the 2D methodology calculates the nozzle exit profile afterthe nozzle con tour is determined; therefore, this discontinuity has no effect o f the nozzle contou r.It should also be noted that this profile is only seen in the RA O nozzle case. As was shown inFigure 41, this is not seen in a perfect nozzle solution. Mach contours for the RAO nozzlesolution are shown in Figure 45 .

0 1 2 3 4 5 6 7 8Exit Pressure, psiaFigure 44.Exit Plane Pressure Com parison

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Figure 45. Mach Contours for 3D RA O Nozzle Solution6.0 Summary

Unde r this task A PL has developed a 2D nozzle design tool and 3 D nozzle analysis toolbased on a Method of Characteristics (M OC) solution algorithm. The 2D nozzle design tool canquickly solve for an inviscid nozzle contour for various types of nozzles and flow inputs. The3D tool can be used to determine nozzle performance and flowfield properties for truly 3-Dimensional shapes and inflows. Several additional efforts have been identified throughout thisreport, which would improve the accuracy and robustness of the tools.

7.01.

2.

3.

4.

5 .

ReferencesRice, T., and VanWie, D. M., Design of Exhaust Nozzle for the RBCC-TrailblazerConcept Final Report, AA TDL -00-033, JHU/APL , February 2000.Rice, T. Design of Exhaust Nozzles for the RB CC -GTX ConceptTPS-335, NASA Grant NAG3-2460, May 2001.Anderson, J. D., Fundamentals of Aerodynamics, McGraw HillNew York, New York, 1984.VanWie, D., and Molder, S ., Applications of Busemann InletHypersonic Speeds, A IAA-92 - 1210,February 1992.

Final Report, RTDC -

Publishing Company,

Designs for Flight at

Nonweiler, T.R.F, Delta Wings of Shape Amenable to Exact Shock-Wave Theory,Journal of the Royal Aeronautical Society, Vol. 67, 1963.

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6. Maikapar, G.I., Bodies Formed by the Stream Surfaces of Conical Flows, FluidDynamics, V ol. 1, No. 1, 1966.7. Rassmussen, M.L., Waverider Configurations Derived from Inclined Circular andElliptic Cones, Journal of Sp acecraft and R ockets, Vol. 17, Nov.-Dec., 1980.8. Taylor, G.I., and M accoll, J.W., Th e Air Pressure on a Cone Moving at High Speeds,Proc. Roy. SOC.London) A319, 1933.9. Bowcutt, K. G., Anderson, J. D., and Capriotti, D, Viscous Optimized HypersonicWaveriders, AI M -87 -02 72 , Jan. 1987.10. Corda, S., and Anderson, J. D., Viscous Optimized Hypersonic Waveriders Designedfrom Axisym metric Flow Fields, AIAA-88-0369, Jan. 1988.11. Moe, M.M. an d T roesch, B.A., The Com putation of Jet Flows with S hocks, ST L

Report TR-59-0000-0066 1, May 1959.12. Rao, G .V.R, Exhau st Nozzle Con tour for Optimum Thrust, Marqu adt Aircraft Co,Presented at the ARC Sem i Annual M eeting, San Fransisco, CA, June 10-13, 1957.13. Kliegel, J.R and Levine, J.N.,Transonic Flow in Small Throat Radius of CurvatureNozzles, A IAA Journal, Vol. 7, No. 7 , July 19 69, pp. 1375-1378.14. Arm strong, W.C., A Method of C haracteristic Com puter Program For T hree-Dimen sional Supersonic Internal Flows, AEDC-TR-78-68, January 1979.15. Renka, R .J., Algorithm 792: Accuracy Tests of ACM Algorithms for Interpolation ofScattered Data in the Plane, AC M Transactions on Mathematical Software, Vol. 25 , No.1 (March 1999), Pages 78-94.16. Renka, R.J., Algorithm 790: CSHEP2D: Cubic Shepard Method for BivariateInterpolation of Scattered Data, ACM Transactions on Mathematical Software, Vol. 25,No. 1 (March 1999), Pages 70-73.17. Akim a, H., Algorithm 761: Scattered-Data Surface Fitting that Has the Ac curacy of aCubic Polynomial, ACM Transactions on Mczthematical Software, Vol. 22, No. 3(September 1996), Pages 362-371.

8.0 AcknowledgementsTh e author would like to thank Jim DeB onis (GRC ), Nick G eorgiadis (GRC ), MariaRigling (APL), Tim Smith (GRC) an d Chuck Trefny (GRC ), for their assistance in thedevelopm ent of these two M OC no zzle tools. A special thanks goes out to W. C. Armstrong, theautho r of record for Reference 14, which was referenced extensively in the developm ent of the3D M OC tool.

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9.0 NomenclatureBD6YLLR CMMOCnPPoint EPoint BwRRRCRUPR*rPTTTT B

c1, P

RDOWN

e

v , qUVWXYdzZ

Last RRC of the initial nozzle expansion regionParametric angleRatio of specific heatsLengthLeft Run ning Ch aracteristicMachMethod of CharacteristicsMach angleNorm al surface coefficientPressureNozzle exit w all pointEnd of the initial nozzle expansion regionFlow an gle w/rt the y-z planeGas constantRight Run ning CharacteristicRadiu s of the arc de fining the initial nozzle: expan sion regionRadius of arc defining the converging throat sectionNozzle throat radiusRadial distanceDensityFlow angle (also nozzle wall angle) w/rt the x-z planeTemperatureInitial dat a line of nozzle solutionArc defin ing the initial nozzle expansion regionVelocity vectorVeloc ity in the x-directionVelocity in the y-directionVelocity in the z-directionTran sverse direction used in nozzle solutionTran sverse direction used in noz zle solutionAxial direction used in nozz le solutionDistanc e between ax ial plan es