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USAASC ltr, 22 Oct 1990

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USAAMRDL-TR- 74-50

-J il A METHOD FOR PREDICTING THE AERODYNAMIC PERFORMANCE OF

CENTERBODY-PLUG IR SUPPRESSORS

Q^ United Aircraft Research Laboratories A-v United Aircraft Corporation ^ East Hartford, Conn. 06108

W September 1974 o PQ Final Report for Period March 1973 - June 1974

D D C fpnnnr?

DEC 9 1971

urn)

Distribution limited to U. S. Government agencies only; test and evaluation; September 1974. Other requests for this document must be referred to the Eustis Directorate, U.S. Army Air Mobility Research and Development Laboratory, Fort Eustis, Virginia 23604.

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EUSTIS DIRECTORATE U. S. ARMY AIR MOBILITY RESEARCH AND DEVELOPMENT LABORATORY

Fort Eustis, Va. 23604

idtiMUIiiiiMiiliiiiuMMaiiWiiaiitieMliaaiMiaia II III J

r mmmmmmimmf^ ""■"""""««•■■■■•r^

B .CT.C; m RFCTORATF PnQ.TmN STATEMENT

for predicting the aerodynam.c static P/X-uSrS The analysis is apphca- and separation region locations ms.de t**^™™S™ made for both film-convect.on ble to incompressible, ^^omc turbulent flow wthprov ^ ^ ^ measured

Ss^t *^^=^^^^to cases in which extrapo,at,ons may be made from known results.

V, concusions contained in this rapo,, ara concurrad in by this Dirac.ota.a.

r r npntrv Military Operations Technology The technical monitor for this contract was C. C. Gentry, Milita y

Division.

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A METHOD FOR PREDTCTINC THE AERODYNAMIC PERFORMANCE OF CENTERBODY-PLUG IR SUPPRESSORS

5 TYPE OF REPORT » PERIOD COVERED

Final March 1973 - June 197*+

6 PERFORMING ORG. REPORT NUMBER

7 AUTHORraJ

Olof L. Anderson

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United Aircraft Research Laboratories United Aircraft Ciirporatlon East Hartford, Conn. 06108

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1» SUPPLEMENTARY NOTES

19 KEY WORDS fContinuo on reverse erde II neceetary And IdenlHy by block number)

Diffusors, Internal Flow, Numerical Analysis, Aerodynamic Performance, IR Suppressors

20 ABSTRACT fConrlnue an reverie »id» It neceeeary and Idmnllly by block number.)

An important consideration in the design of military aircraft engine exhaust dlffusers is the need to reduce infrared radiation emanating from the engine, coupled with the need to maximize shaft horsepower. To aid the engineer in the solution of this problem, advanced mathematical techniques developed In this report have been applied to the solution of turbulent compressible

swirling flow through curved-wall annular diffusers. This analysis has

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20. developed a generalized method for calculating an orthogonal coordinate system for arbitrary curved-wall annular ducts with cooling slots which is based on the Schwartz-Christoffel transformation. In addition, this analysis has developed a stable implicit numerical integration scheme for solving a nonlinear parabolic partial differential equation which does not require an iterative procedure to maintain second-order accuracy. Finally, it is noted that the procedure does not require an iterative procedure coupling the inviscid and viscous portion of the flow field but treats the entire flow field as a whole.

A computer program has been developed using this analysis and applied to sample cases to demonstrate the capability of the analysis. Cases with and without slot-cooled walls have been calculated and compared with experimental data taken from the ST9 demonstrator IR suppression diffuser operating at different slot cooling flow rates for one engine operating condition. The results are in fair agreement with the experlmenval data, but additional work

is required in order to obtain better theoretical predictions.

IWInggif-lpd SICURITV CLAMIFICATIOH OF THIS FAOEfWlMB Dmm

—-— -■ -'-•■ ■--"■ ■— AA^MHtai

Ill ■

TABLE OF CONTENTS

Pa^e

LIST OF ILLUSTRATIONS 3

LIST OF TABLES 5

INTRODUCTION 6

ANALYSIS 10

Conformal Mapping Solution 11

Schwartz-Christoffel Transformation 11 Solution in z Plane 12 Solution in w Plane 13 Differential Equations 15 Solution Near a Source IT Transformation to R,Z Plane 19 Locating Poles 21 Duct With Slots 2k

Implicit Method of Solution 2k

Mach Number Transformation 2k Basic Equations of Motion 26 Boundary Conditions 28 Finite Difference Approximation 31 Solution of Matrix Equation 36

DESCRIPTION OF TEST PROGRAM Ul

Description of Test Facility kl Description of Instrumentation ^1 Test Results k2

COMPARISON OF EXPERIMENT AND THEORY kk

Fräser Flow "A" Ml Calculation Procedure For ST9 Demonstrator

IR Suppression Diffuser Run ^6

Baseline Case - No Film Cooling - No Struts ^7 ST9 Demonstrator IR Suppression Diffusers - 2.5 Percent

Cooling Rate k8

.._.->.■ ■,^.„.-,t—■.^.^^. _ ...,.,—,.. ..^....^^^ —^.._ ,.,.,....,. _...... ,.....- J,......_..,^.-_J„!^-i^_^ ^^i.^ . ...^.^

'i' ■ M+^^mmammmm • '■

Page

ST9 Demonstrator TR Suppression With 5 Percent Cooling Rate. . kb ST9 Demonstrator IR Suppression Diffuser With 10 Percent

Cooling Rate ^9

Discussion of Numerical Calculations ^9

CONCLUSIONS 53

REFERENCES '

LIST OF SYMBOLS 151

mrm*^^*m mrmmmim^vmmmi* ■ " « ■ «' ' ' ■■»■"■ mi" ■■!

Figure

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

LIST OF ILLUSTRATIONS

Page

Conformal Mapping of Duct

Rotating and Scaling Duct

Construction of Slot in Duct

Schematic of D-32 Stand

Swirl Generation Section

Installation of Slot Cooling System 61

Diffuser Test Rig 62

Side View of Test Rig

64 Aft View of Test Rig

Location of Wall Static Pressure Taps

Location of Wall Thermocouples

Inlet Plane Instrumentation

Outer Wall Average Surface Temperature for ST9 IF. Suppression Diffuser

Effect of Coolant Flow on Wall Pressures for ST9 IR

Suppression Diffuser

Effect of Coolant Flow on Pressure Recovery for ST9 IR

Suppression Diffuser

Streamline Coordinates foi Fräser Flow "A" Diffuser . . 71

Comparison of Experimental and Predicted Wall Static Pressure Distribution for Fräser Flow "A" Diffuser. . . 72

Comparison of Experimental and Predicted Wall Friction Coefficient for Eraser Flow "A" Diffuser 73

Streamline Coordinates for ST9 Demonstrator IR Suppression Diffuser

Inlet and Exit Mach Number Distribution 75

* IlilMMMMMnM—M

"»""—"^iWIilHIW" HIM i ini >nHnm>na ■ i i inwiwi '••• " *"• ■'

Figure

21

22

23

24

25

26

27

Comparison of Experimental and Predicted Wall Static Pressure Coefficients for ST9 IR Suppressor Diffuser With No Film Cooling and No Struts

Comparison of Experimental and Predicted Wall Static Pressure Distribution for ST9 IR Suppression Diffuser With

2.51 Injected Cooling Air

Comparison of Experimental and Predicted Wall Temperature Distribution for ST9 IR Suppression Diffuser With 2.57.

Injected Cooling Air

Comparison of Experimental and Predicted Wall Static Pressure Distribution for ST9 IR Suppression Diffuser

With 5.07o Injected Cooling Air

Comparison of Experimental and Predicted Wall Temperature Distribution for ST9 IR Suppression Diffuser With 5.0/»

Injected Cooling Air

Comparison of Experimental and Predicted Wall Static Pressure Distribution for ST9 IR Suppression Diffuser

With 10X Injected Cooling Air

Comparison of Experimental and Predicted Wall Temperature Distribution for ST9 IR Suppression Diffuser With 107„

Injected Cooling Air

Page

76

77

78

79

80

81

82

i HlfcBilMi—MM ■ -

w^^m*m^^mmm^m^^^mm lumn iuiiiippiiiiinnjiwM«"ni^w<wn>POTi^^mpM|HHimpHp

LIST OF TABLES

Table Page

1 Location of Pressure and Temperature Instrumentation . . 83

2 Test Log 84

3 Test Data 85

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INTRODUCTION

An important consideration in the design of military aircraft is the • • M^ of the infrared radiation emanating from the aircraft engine. T~ rUaUfoftL engine can he controUed through P-P- -sign

ofan engine diffuser; however, great care must he taKen ^ -sure t he oroposed diffuser does not adversely affect engine performance Therefore ?he design engineer is faced with the complex problem of designing an engine ihlch P oviLs minimum turbine back pressure with an efficient exhaust dfffuser Ifo^der to maximize shaft horsepower and at the same time minimize

radiation through the use of curved wall diffusers and cooled walls.

The satisfaction of these sometimes conflicting requirements ^s proven to be extremely difficult in the past, and engineers often have been forced to rely on empirical design methods based on correlations of limited explr mental data. For example, diffuser performance maps based on e-p rical

collations have been published by Reneau (Ref. l) ^^^TJZ dimensional flow and by Sovran (Ref. 2) for incompressible annular flow m sraih" 1 diffusers. Regions of stall on these performance maps have been Lined by Fox (Ref. 3) for two-dimensional diffusers and by «-ard (Ref. k) for straight-wall annular diffusers. In addition, Sovran (Ref. 2) has developed'empirical correlations for the effect of inlet bl-fBe on perfor- mance and Runstadler (Ref. 5) has developed correlations for the effect of inlet Mach number. Although these empirical design criteria provide ome insight into the effect of variables such as area ratio, length, inlet blockage and Mach number on performance, these criteria are not adequate for

(1) Reneau, L. R., J. P. Johnson, and S. J. Kline: Performance and Design

of Straight, Two-Dimensional Diffusers. Transactions of ASME, Journal of

Fluid Mechanics, Vol. 89, March 1969, pp. lUl-l60. (2) Sovran, C.and E. D. Klomp: Optimum Geometries for Rectilinear

Diffusers. Fluid Mechanics of Internal Flow, Elsevier Publishing Co., 1967.

(3) Fox, R. W., and S. J. Kline: Flow Regime Data and Design Methods for

Curved Subsonic Diffusers. Journal of Basic Engineering, Transactions of the

ASME, Series D, Vol. 8k, No. 3, September 1962, pp. 303-312. (k) Howard, J., H. Henseler, and A. Thornton-Trump: Performance and

Flow Regimes for Annular Diffusers. ASME Paper 6?, WA/FE-21, 1967. (5) Runstadler, P. W., and R. C. Dean: Straight Channel Diffuser

Performance at High Inlet Mach Numbers. Transactions of ASME, Journal of

Basic Engineering, Vol. 91, September 1969, pp. 397-^22.

iÜMMiitlhiii - ir ■ ....-^J.,-^^——^■.—■ ^ ., ^MJ__^_J^^... ^^LjjiMtii

designing curved-wall TR suppressing diffusers. For curved-wall annular

diffusers, only a few studies such as those by Dietz and Thompson (Ref. 6)

and Thayer (Ref. 7) are available to provide design information. In

particular, Thayer has developed some general design requirements for

diffusers of this type through his investigation of the effects of swirl and

Mach number on diffuser performance.

The development of analytical design methods has generally lagged behind

empirical design methods. Conventional solutions, such as those used by

Sovran (Ref. 2), divide the flow field into an irrotational free-stream flow

and a boundary layer flow. These methods which divide the flow field into

viscous and inviscid portions require an iteration between the potential flow

pressure field and the boundary layer displacement thickness. This iteration

frequently fails to converge when the boundary layers comprise a significant

portion of the total flow field. In addition, these iterative methods cannot

account conveniently for phenomena such as inlet swirl and inlet flow distor-

tion. Recently Anderson (Refs. 8 and 9) introduced a new method for solving

the swirling diffuser flow problem which solves a single set of equations

of motion for the entire flow field in the diffuser, thereby enabling

compatibility between the inviscid flow and boundary layer to be achieved

without the need for matching a boundary layer solution to an inviscid flow

solution through an iterative procedure. The method has shown good agreement

between theory and experiment for incompressible flow (Ref. 9) and has been

extended more recently by Anderson (Ref. 10) to the prediction of compressible flow. Theoretical predictions again have been in good agreement with

experimental data.

(6) Dietz, A. E., and J. F. Thompson: Advanced Experimental Infrared

Energy Suppression System for the T-53-L-11 or T-53-L-13 Turbine Engine.

Hayes Internal Report No. 1172, 1968.

(7) Thayer, E. B.: Evaluation of Curved-Wall Annular Diffusers. ASME

Raper 71-WA/FE-35, September 1972.

(8) Anderson, 0. L.: A Comparison of Theory and Experiment for

Incompressible, Turbulent, Swirling Flows in Axisymmetric Ducts. AIAA Baper

No. 72-U2, 10th Aerospace Sciences Meeting, January 1972.

(9) Anderson, 0. L.: Numerical Solutions of Incompressible Turbulent

Swirling Flows Through Axisymmetric Annular Ducts. United Aircraft Research

Laboratories Report No. H213577-1, March 1968.

(10) Anderson, 0. L.: User's Manual for a Finite-Difference Calculation

of Turbulent Swirling Compressible Flow in Axisymmetric Ducts With Struts.

United Aircraft Research Laboratories Report L911211-1, Contract No. NAS3-

15^02, 1972.

... .

11 ' ' ' ^■11 -I III ■ III I I ■ I I

The method derived in Ref. 10 requires construct on of a gereralxzed orthogonal coordinate system from a solution of the plane potential flow through the duct in question. This potential flow solution -rves as an apprSmate streamline coordinate system upon which the viscous solutxon of the e^ions of motion is based. The equations of motion are written m the

pprox ml streamline coordinate system,and boundary layer approxxmat.ons lay be made in this new system since the potential flow streamlines Tpproxin.te the real streamlines. With the boundary ^Z^^ZnZl the equations of motion reduce to a set of parabolxc partial *«*£BtJ£ equations which apply to the flow field under ^^^T,? ^J^. the solution to the potential flow problem was obtained through an approxi Tte geol trie construction which yielded gocxi results when ^e curvature on Toll walls was small and nearly the same. This geometric solut on somet mes failed when the curvature varied significantly from wall to wall. Even when

h olut on did not fail, significant errors could arise due to the sma 1 Mature approximation inherent in the method T^us ***<***?£**' 10

is limited in the types of geometries to which ^V't.fto'the sLll Umitation in mind, a new solution P--dure not Xi.1 ed ^«^^ curvature approximations was developed. This new mxBT* w«* to the plain potential flow problem using an exact nume. n solution based on the Schwartz-Christoffel transformation (Refs. 11 and UJ.

It is possible to solve the governing equations by an explicit or an illicit^uLrical integration. In an explicit method ^^^^^I tions of motion, the allowable streamwise step size is related ^^.^J verse step size through numerical stability conditions. If the "l«"«" To be numerically stable, a finer transverse grid requires a smaller stream- wise st^sze. THis restriction is particularly troublesome in the case of IZ Zle* wails, wher. a very fine transverse grid js ^d to e ine he

coolant film accurately. Therefore, under the ^•^•J*^,*^10" numerical integration technique of Ref. 10 was replaced by I » **"*** t'hnique based on the method of Keller (Ref. 13). In this new methoi, the

(11) Kober, H.: Dictionary of Conformal Representations. Dover

Publications^lnc.,^^. ^^^ ^^ der Konforffien Abbildung,

Soringer Tracts in Natwal Philosophy, Vol. 8, 1963• (13) Keller, H. B.,and T. Cebeci: Accurate Numerical Methods for

Boundary Layer Flows-II TVo-Dimensional Turbulent Flows. AIM 9th A^ospace sfiences Meeting, New Yor., January 2,-21, 1971, AI.A Paper No

71-16U.

Ill IM^!

equations of motion are linearized in such a way that an iteration is not

required to obtain a solution. The method, however, retains important^

features such as second-order accuracy and from the point of view of linear stability analysis has no restrictions on step size in either the strearawise

or the transverse directions (Ref. Ik). However, in practice the step size

is restricted by the need to minimize truncation errors arising from the

finite-difference scheme. Truncation errors will cause loss of accuracy and

may lead to numerical instabilities through nonlinear effects. In addition,

powerful matrix inversion methods are available in the numerical solution

(Refs. 15 and l6).

Under the present effort, advanced mathematical techniques have been

applied to the solution of turbulent compressible swirling flow through

curved-wall annular diffusers with slot-cooled walls. From this analysis, a

computer program has been developed and sample cases calculated and compared

with experimental test results obtained from an IR suppressing diffuser with

slot-cooled walls at different simulated engine operating conditions.

{Ik) Keller, H. B.: A New Difference Scheme for Parabolic Problems.

Numerical Solution of Partial-Differential Equation-II SYNSPADE 1970 Ed. by

Hubbard, B. Academic Press, New York. (15) Keller, H. B.: Accurate Difference Methods for Linear Ordinary

Differential Systems Subject to Linear Constraints. SIAM J. Namer, Anal.

Vol. 6, No. 1, March 1969. (16) Briley, W. R., and H. McDonald: An Implicit Numerical Method for

the Multidimensional Compressible Navier-Stokes Equations. United Aircraft

Research Laboratories Report M911363-6, November 1973-

- — - - - - •■ - ■ ' "■'

ANALYSIS

The present analysis solves the problem of axisymmetric swirling flow

through typical IR suppressing diffuser geometries in a two-step procedure.

In the first step, a proper coordinate system is constructed; in the second

step, a set of boundary layer type parabolic partial differential equations

is solved using a forward marching implicit numerical integration procedure.

For flow over a flat plate, the proper coordinate system consists of lines

parallel to the plate (termed the strearawise coordinate) and a second set of

lines perpendicular to the plate (termed the transverse or normal coordinate).

If the equations of motion are written in this Cartesian coordinate system and

the boundary layer approximations are made, a set of parabolic partial

differential equations is obtained. For this simple problem, it is obvious

that the boundary layer approximations (namely, that the transverse velocity

is small compared to the streamwise velocity and that the streamwise

derivatives are small compared to the transverse derivatives) are valid. In

the case of more complicated geometries such as curved-wall diffusers, the

coordinate system in which the boundary layer approximations can be made is^

not as simple as in the Cartesian coordinates described above. Rather, it is

a coordinate system in which one coordinate approximates the streamlines and

the other is normal to the streamlines. Such suitable coordinates can be

obtained from the plain potential flow solution for the duct under

investigation since it is apparent that in view of the constraining effect

of the walls, the potential flow streamlines approximate the real streamlines

provided large regions of flow separation do not occur.

This problem has been discussed in more detail by Anderson (Ref. 10),

where it was shown that the solution to the plane potential flow problem

through a given duct can be used to construct an orthogonal coordinate system uniquely suited to solve for the turbulent flow through the duct. Although

the direct problem of determining the velocity potential s and stream function

n in terms of the cartesian coordinates R and Z may be solved easily, the equations of motion for the turbulent flow require that n and s be explicitly

the independent variables so that the coordinate functions R(n,s) and Z(n,s)

can be obtained (Ref. 10). Although an approximate solution to this problem

was presented in Ref. 10, the solution is inaccurate for ducts having large

curvature. In order to alleviate this curvature limitation, an exact

numerical solution has been obtained to the inverse problem using the

Schwartz-Christoffel transformation (Ref. 11). In addition, the Schwartz-

Christoffel transformation provides a method for obtaining an orthogonal

coordinate system for a duct with slots. The method for solving for the

generalized orthogonal coordinate system is derived in the next section.

10

•aMMMMHiailMlirMi tmtm

The boundary layer approximations to the equations of motion for

turbulent swirling flow with normal pressure gradients are derived in Ref. 10

where these equations were solved using an explicit numerical integration

method. For slot-cooling problems, however, this explicit method is

unsuitable because the inner layer of the turbulent boundary layers cannot be

described accurately if a reasonable strearawise step is to be taken. There-

fore, under the present effort, an implicit method of numerically integrating

the equations of motion was developed. This method is derived in the follow-

ing section entitled, "Conformal Mapping Solution". The implicit method is

unconditionally stable, and the linearization technique used permits integra-

tion of the equations of motion without any iteration such as that used by

Keller (Ref. 13).

Conformal Mapping Solution

Schwartz-Christoffel Transformation

If a cuxved-wall duct is represented by straight line segments in the w

complex plane to form a many sided polygon, the Schwartz-Christoffel trans-

formation (Ref. 11) may be used to transform this polygon in the w plane

into the upper half of the z plane, as shown in Fig. 1. Under this trans-

formation, the source flow at the duct inlet in the physical plane becomes

a point source at the origin of the z plane. Source flows resulting from

inlet cooling slots become point sources on the real axis of the z plane.

The potential flow solution as a result of this source distribution in the z

plane can be found easily by superposition of elementary source solutions

leading to a definition of the streamlines n and potential lines s in the z

plane. Then given n and s in the z plane, R(n,s) and Z(n,s) can be obtained

by going back to the w plane and rotating and scaling as shown in Fig. 2.

This procedure is explained in more detail in the following paragraphs.

The conformal mapping method has several important advantages over

other methods of determining R and Z as a function of s and n. First, the

inverse problem can be solved exactly in a straightforward manner as opposed

tc most other procedures, which lead to approximate solutions. Second, real

ducts do have discontinuities along the wall boundaries for which the

Schwartz-Christoffel transformation is ideally suited. Third, the technique

developed in this report determines the wall slope and integrates the slope

to obtain the wall contour. Thus the first derivatives and metric scale

coefficients required for integration of the viscous flow equations are

obtained directly rather than by numerical differentiation, leading to a more

accurate solution.

11

.^,

Solution in z Plane

The complex potential for a source located at the origin of the z plane, which represents inflow at the duct entrance plane, is given by

F = |nz = s + m (1)

The complex potential can be solved explicitly. Thus, as shown in Fig. 1,

s = |nr =^ In (x2^/2) (2)

n = ^ = tan'^y/x) (3)

Equations (2) and (3) describe the potential flow in the z plane in the

absence of any slot cooling. Since the walls contain poles representing

corners of the duct in the physical plane, a finite upper half plane hounded

by

r9*'*\ (5)

is defined. Thus n and s are bounded by

C<n<7r-£ (6)

12

— - - - - -'■ - — '" — ——-in 1 |-|IM.| .....■,:...*..-■■--...■.■. ■

lnr0<s<lnrL (7)

and the solution lies completely within a bounded domain free of singularities.

Solution in w Plane

The Schwartz-Christoffel transformation is given by

JH . -L T (Z-KV0^ (8) dz z IS2 \ "1/

where b-r is the location of the poles on the x axis of the z plane,

representing corners in the physical plane, and VT is the corresponding corner

angle (defined in Fig. l) in the w plane. The bj's and a-j-'s are, therefore,

real constants. When the values of hj are known, any point (n,s) in the z

plane corresponds uniquely with a point in the w plane. The central problem

is to find the values for the bj's which are unique for the duct under

consideration. Let

w=C-MZ (9)

z = x+iy (10)

Then, because of orthogonality,

l*. #-i#. #♦!*!- (ID dz dx oy öy dx

13

"— ^~.—^- ■.^.,..... ^ ... .-^ .. .-^.^ ..^ .. .■ .,_ . ..■ -^.^^^-W^ .,:--...

The real and imaginary parts of Eq. (8) are evaluated as follows:

U'\(*-tf*'\ 1/2

(12)

*, •*«-,(lf/(».bx)) (13)

7 ^"^Z1 (1U)

*J= ••a^A (15)

x, = rj cos <t>j (16)

yj= ^ sin t (17)

Then Eq. (8) reduces to

dw dz

w (if. + iy.) = x + i7 j-.i J J

(18)

14

- - ■ ■ ■■■ ■ n n ■ — --^^*.- ^^.. ^ - " -- - "■—- - ■ --^-. .-^...IA-J

which can be evaluated by repeated application of the product rule for complex

numbers.

XJ*I = K» XJ»I ' yj yj*i (19)

y = x y + y x (20)

Finally, comparing Eq. (l8) with Eq. (ll) results in

7 - il - ÜL x " dx "ay (21)

y ' dy " dx (22)

which lead to differential equations relating 5, T\, and x,y. As shown

subsequently, these relations allow the construction of the required potential

solution in the physical, w plane.

Differential Equations

The next step in the solution requires the derivation of the differential

equations valid along an n or s coordinate. From Eqs. (2) and (3),

ax " ay ♦ y2 (23)

A3. dy

.an. ay

y

x' + y* (2«0

15

-- - - -^ — - -

A determinant, D, is defined by

(25)

Then

dx ■ l 4s_dn.4S.ds D ay dx

1 1. lH dn-^ds D ! ax ay (26)

dy = -t -If^*ds 1 D #•*«♦#-* (27)

Hence along a streamline dn = 0 and

as 4r -te- ll ax (28)

ay as D ay

(29)

Along a potential line ar - 0 and

ax_ an

I _an. D ax

(30)

an D ay (31)

16

■ -— ■ - MMH^MMi Hi m*m

Equations (29) through (31) allow construction of the solution in the z plane. Finally, using Eqs. (ll), (21), and (22), an integration may be carried out along streamlines or potential lines to construct the solution in the w plane.

ii if. IL + Ü DL M (90) ds " dx ds ay as "an vo ;

11. iäi&xil |t..iL (33) as "ax as ay as an

Hence, integration along streamlines in the M plane is obtained through Eqs.

(32) and (33) together with Eqs. (2?) and (29), and integration along

potential lines in the w plane through Eqs. (32) and (33) together with Eqs.

(30) and (31). The metric scale coefficients are the same in both directions

and are given by

l(as) +\anj ! = dz/dz dw (31+) v =

which is the same as the magnitude of the potential flow velocity obtained

from the complex conjugate. It should be noted that the solution is equiva-

lent to a solution for the wall slopes, Eqs. (32) and (33), or the metric

scale coefficient. Hence the wall slopes and metric scale coefficients are

solved for directly and the wall contour is obtained by integration. The

determination of the bj's which define the duct is discussed subsequently.

Solution Near a Source

Consider the solution in the neighborhood of the inlet source as z => (

From Eq. (8),

ilL.-IL (35) dz z

where C]_ is a complex constant given by

17

- - - -..-i,, m^MM^^i , .. . -.^zumtrnMamm**^. ^..... , ,.. _..._,., ^^ ^^iMij||iigg|a|M|g||

zrL (bi) (36)

Integration of Eq. (35) leads to the equation

w = 0,10 2 +C2 = C, (s + in) +C2

(37)

Thus the solution in the neighborhood of the inlet is

C-C0 = C1R(s-S0)-CIR(n-no) (38)

1-%,CI1 (S-So) + CiR I"""0) (39)

.C are the real and imaginary parts of C^ l^erefore the

vhere C^ ^ _CU f! ^ ^ .n an^le tor)« to the axis of symmetry (see inlet ifa straight duct with an angle (a)0

Fig. 1)

-i a0 = ton (SRAIR)

(UO)

From Eq. (3M, the metric scale coefficient is given by

2 . *! 'Z2

Vo= Cii+C«R («a)

18

i ^a^^MMMMiMi^Himia^^^MMfc-M—MMaMI

How ever, at the inlet, From Eq. (2) and Eq. (3)

S-S, - ,n (r/ro) 0*2)

n-n0 = ^ (U3)

and from Eq. {k)s

An = n-Zi (hk)

Therefore, inlet height is given by

(U5)

Transformation to R.Z Plane

Since the inlet of the duct starts out with an angle af0| as shown in Fig. 1, the transform to the (r,z) plane shown in Fig. 2 is obtained through a rotation of an angle 0to and is given by

r-r0 = cos a0 (^-^ -sina0(c-C0) (1+6)

z-z0=cosa0(c-C0) ♦ sin a0 (tj-TJ^ (U7)

19

-"-"'- ■ ■■ „L.^»^.. . -..- ■■ mm - -■■*■ — ■ ii HI i-üliaMKam^Bii^iMHifT IM

Finally, the transformation to the (R,Z) plane is obtained through a

translation and scaling using the inlet height ho:

R = RH0 + RT0 " RH0 H (U8)

z.. <^l^L(l.lo) iU9)

The streamline coordinates are scaled, noting that

Thus

$ < n < n-C (50)

0 < n < i (51)

0 < s < s, (52)

n-C n-Zi An

(53)

S = - S-So_

n-2C s-sc An

(5M

20

■-^ - i ii'rimiiniiint« --- - ■ -- - .--.---..-^i-^ -il i i- i

RT0-RHO V0

JL (55)

For numerical convenience

S0= -(TT-ZC) SL/2 (56)

Then from Eq. {h2)

r0=expS0 (57)

and the solution in the z plane is located.

Locating Poles

The location of the poles bj in the z plane must be obtained by an iterative method. It is noted that the location of the corners and the corresponding angle change of the polygon representing the physical duct in the (R,Z) plane is known. Specifically, the location of these corners can be expressed in terms of the distance X(J) along the wall from the inlet to the jth corner. Each corner represem-s a pole in the z plane. If a guess is made for the bj's, then Eqs. (32) and (33) may be integrated along the wall stream- lines. Since the location of each pole is known in the (R,Z) plane, the distance X(j) to the J^ pole is computed and compared to the known X(j). An iteration procedure based on Newton's method is used to obtain a new guess for the bj's. In the iterative procedure the duct contour is defined by specifying the wall radius at JL equally spaced mesh points. Define

AZ = ZL/(JL- I) (58)

21

'*M*t"—~*^'^-- - " - - ■ - -..■.■..■.'.■^.■«.-■^■-J..I^M»...'..».~^.. ^...-.j,-^-., ^^..^ ^tmmmuaämamt^m^lmti^ i^:.^«ia

Zj = AZ (J-I) (59)

Then the hub and tip contours (outer wall and inner wall) in the physical plane are known at each of the J points.

RH(J) = RH(

ZJ) (60)

RT(J) -M*,) (61)

öh(j)=ton-'(^) (62)

•ftÖM«-1 (-£*•) (63)

The o/j's for the Schwartz-Chris toff el transformation are then given by

aH(J) = 0Hu)-eH(J-i) (6lf)

oT(j) = öT(J->-I)-öT(J) (65)

22

■- -■'■■■■-- ■■■■- -v- :■ .:-.^jidJl^^B^—^^^B^^ ■

"-'—■~.:..-.. ..-.. ir ii'iiiWilMiaii iiMii

where they have been defined as the change in wall angle moving around the duct contour (polygon) in a counterclockwise direction. Since the polygon is composed of straight line segments, the distance along the wall from the

inlet to the jth point is given by

XH(J) = .t!K (I) RH(I- *]* + AZ' 1/2

(66)

xT(j) 1=2

Vi-o] 2 A72 + AZ 1/2 (67)

If an initial guess is used for the solution using the approximate solution described in Ref. 10, then the poles may be located using Eq. (^2); Eq. (32) and Eq. (33) may bo integrated along the walls (streamlines) and the distance along the wall to each corner determined from

{>-L is. (68)

J ds (69)

These XH(J) and X*(J) in general will not agree with that calculated using Eqs. (66) and (6?). However, at each pole in the (R,Z) plane we may obtain a new guess for b-i using Newton's method

sl* (J) = S^ (J) + V* (J) I x^ (J) ■ xjj) (70)

23

iiiiiitiiiiiMiMiliTlrffli mmtm -

ÜJü) = S^J) + VTV(J) [X^J)- xT(j)] (71)

Convergence occurs when

«rKlJ)->l(J)K <72)

v+1 for all J. Once the S(J) are known, the new location of the poles may be

obtained using Eqs. (U2) and (51+).

Duct '^ith Slots

Tnlet slots in a duct must satisfy the Kutta condition that the stream-

line leaves tangent to the slot lip. Thus the Kutta condition is equi.alent

to stating that the static pressure on each side of the slot lip is the same.

Because of the Kutta condition, the coordinates for ducts with slots may he

calculated by overlaying solutions of successively larger ducts without slots

This procedure is shown schematically ?n Fig. 3- The coordinates are

calculated for durt 1 from station (l) to station (2). Then the cordinates

are calculated for duct 2 from station (2) to station (3). Thus the Kutta

condition is satisfied at the slot lip by construction of the streamline

(wall) for duct 1 tangent to the slot inlet. This process may be repeated

for any .lumber of slots .

Implicit Method of Solution

Mach Number Transformation

At low Mach numbers, the extremely small variation of the pressure and

temperature within the diffuser leads to large numerical errors in the solu-

tion of the equations if the actual pressure and temperature are treated as

dependent variables. Therefore, a Mach number transformation was devised in

which the dependent variables are the difference of the local pressure and

temperature from the mean inlet flow conditions. For the purpose of the

transformation, fl, 8, I are definedjis the mean inlet pressure, tempera-

ture, and entropy, respectively, and H, 9, T, Q are defined by the

relations

2k

—-—- — - -^^^»-^»^^^»-»i-»^"»^»^-^.-^^^^»--- ^^^jumfagl

n=n i + XMr n (73)

•«•, +(y-i)Mr2 §

(7^)

I = T1+ (y-l)Mr2T (75)

o= (y-D Mr23 (76)

where

1. ' -yIT In ©.- in IT. (77)

The variables n, e, 1, Q are the new dependent variables noted that for very small Mach numbers, Eq. (73) becomes

It should be

n = n-fl yMr

2 0(1) (78)

When this transformation is applied to the equations of motion as given in Ref. 10, all the terms in the equations become the same order, allowing an

te'rT n tT""1 •^^ ^ ^ ^^ In the ^vi^ Emulation, some terms in the governing equations were considerably larger than others, leading to numerical errors. ' ■Lcau-Lne

25

- ■ ■maiMMMiiaii

Basic Equations of Motion

Under the present effort the Mach number transformation, Eqs. (73)

+H !h^ is applied to the equations of motion derived by Anderson /ef 'lO in dd? o^ the equaLns are arranged as first-order equatxons

lo facilUate the application of an implicit numerical xntegratxon method.

The governing equations are:

Continuity Equations

Streamwise Btejü ^omPQ"6^

(^te^^Kl-W1- = 0 THnpential Stress Component

(^1^-1* «W-W1--0

iwm»l Momentum Equations

afi J i dy.lpU2_l_LiRlpU2 =0 T^" +lxv d^|pus [XR J*lr *

Entropy Equation

(y-i) Mr2T=7^ in l(y-i)Mr

2§| -ln[rMr2ftj

26

(79)

(80)

(81)

(82)

(83)

— —- - - " — - ---—-i I,,, „ ., - .---- -.- "■^-— .. ^^L^t^nutuiUteai*.

Heat Flux Equation

VPRE ^r I dv xv (8»f)

Equation of State

n, + yMr2 fj ■ PI®, + (y-l)Mr

2S| (85)

Streamwise Mumentum Equation

dl nt

dr) _y i, / G \ \ dv XG an \"^r ~~" XV dn ns

1*1 in iys . f_y_l ü_ aus a^ as "^ [ G ] as a^ + i an

XR as PU '♦

x TT üs xv

(86)

Tangential Momentum Equation

a^ T[i<T an Wr XR anj^n^,

_ ixl It 3i& 4. |x| 41 AU# |G I a^ as + rn as a^ (87)

27

■aj—^»"- ■ '- '—■ ..^^.^j^tm^-^-^il^j. -..»^a "• ^ . . . iMfe

Energy Equation

drj ♦■* * (4)|«

y-i 8 (88)

_ "a /^iy z+ y 2

XV

It is noted upon examination of Eqs. (79) through (88) that no derivatives

of density appear explicitly; therefore, a Mach number transformation was not

applied to the density.

Boundary Conditions

For annular flow fields within ducts having both an inner and an outer

wall, the proper set of boundary conditions is (see Ref. 10):

us(sp) = o.

^(s.o) = o.

Q (S.O) = 0. adiabatic wall

or ® (S,0) ■ ®H

(89)

or

^(S.l) = ^T{S)

UsCS.I) = o.

yM = o.

Q (S,l) ■ o. adiabatic wa

»(S,0 ■ «T(s)

(90)

28

■- -— - ■ -- —■—-- ■ - - - IIIIII nil ii i inTilMrtimiilifliiiiiMiini

where S is the streamwise coordinate and Tl is the transverse coordinate.

The transverse grid is normalized so that the walls occur at U= 0 and Tl = 1.

For axisymmetric flow, in which no inner wall is present, the equations

of motion contain a removable singularity at the origin or axis of symmetry.

The boundary conditions (Eq. (89)) must be replaced by boundary conditions based on a Taylor series expansion of the flow variables about the centerline.

From Eq. (86) at a small distance h from the centerline, the expansion for

Ens is given by

ns I T H -J* +v#l0

h + o M (91)

The remainder is neglected because it is of higher order than the order of

the difference approximation. Then Eq. (80) and Eq. (91) yield

Us = uso + t ?VH vPu. as

jBl ♦ V-jfl i.*o (**) (92)

which serves as a boundary condition for Us-

The expansion of En^ about the centerline is obtained from Eq. (87)

^-o{S) (93)

Hence, from Eq. (8l),

u.. =o(h6)=o W

since it i? of higher order than the difference approximation. The heat flux equation (Eq. (8U)) and the energy equation (Eq. (88)) are used to find a boundary condition for Q:

29

»..—..-....., ,. .,■., - ....^i^^ ■■ - ■ -—^i—■ — -—'-^——IIIB n .mi

Q = -^r I X- 2 1

vPu-0 -4i- s dS - h + o(h') (95)

+ ± J^i JV. IvPu.e 4tl h2 + 0 W + 4 / XV [ ' ds lo (96)

And, finally, a boundary c ondition for i is obtained from Eq. (79)

♦ = 2n(pus)0h2 (97)

The boundary conditions may then be applied at a distance h from the center-

Une "owever, a great simplification may be obtained if h xs chosen such

that

h/ (VAT;) « (98)

which implies that the first point is very near the centerline. the axisymmetric boundary conditions

Then we have

^ (5,0) =0.

Zns(s.o) =0.

U. (S,0) =0.

8 (s.o) =0.

(99)

30

Finite Difference Approximation

These equations are reduced to finite difference equations using the second-order finite difference scheme of Keller (Refs. ih and 15). The following notation for S and T] is introduced:

ASJ= S^S0"1

(s% ^ 1 (100)

Vl/t" TK+17K-I)

J-h

(101)

and for any dependent variable g (z,s) g and g , are defined by

gJ-,/2 = -HgJ+0 (

The equations of motion, Eqs. (79) through (88), are linearized by performing a Taylor series expansion in the S coordinate, as suggested by Briley and McDonald (Ref. 16). Let any dependent variable g be given by

gJ = g^V Ag (102)

where

1^« (103)

Then we have the following product rules (Ref. 16)

31

, I n - .Hi- - ■.^^... iHntMlTi- ^^^^h-^—^^.^-^-^.^^... .-^ .. - ... ^. nnww m

,J J-l .J-l J

(fg) = T lf ^ ■hf 9)

(fgh)J -(^"h^^'V^rf -2H

(#)

(10U)

J-|/2 ^K^K

K = AS

vJ-l/2 gJ-gJ-' J-!

Substitution of Eq. (10^) into the equations of motion, Eqs. (79) through (88), yields the following results.

Continuity Equation

= - AT7

Streamwise Stress Component

XV K-l/2 (p^i-l/z

l A ^-1/2

Tangential Stress

(^O*-"*-')- A1! !_!_ XR

_ Al L*JJ (ZJ +Z

J^ )=o 2 I XvJK_l/2 V^n^K ^n^K-i/

(105)

(106)

(107)

32

■- ■ —- --- UAM

•i- «win 1

Normal Momentum Equation

XR $91 (108)

K-l/2

H^^uK,)l-l>C.(^ K-l/2

Entropy Equation

K K Y- I © j-i n J-I (109)

Heat Flux Equation

ttd iK<-) &n XV JK*K-)-0 ("o) K-l/2

Equation of State

yMr2 R PJ 1 r K I

= - n - (y-i)Mr2 (P®)0'1

(m;

33

^^.^ - ■ ■- --■;---'"^*-':—-"■-—■■■

Streamwise Momentum Equation

♦ AT? ■BY** itfaT' kW M^V" (pJ + pJ)i

K-l/2

/_J-I rj-i \ A77 ( v _ö_/G\ i av = ' l^nSK" S«H ) ' 2' 1 xQ- dn W " xv In

_L.|v.lJ-|/2 (^-l-^-{\ /u'-' + u''1 I

I JR. XR '"

J-i/2

K-l/2 \ nSK nSK-i/

(112)

AS lx lK_|/2

-1/2

K-l/2

i3M

. _ .- — . ..—

Tangential Momentum Eguntion

S5 I G

J-l/2

K-l/2

/vj vj \ A1? i^ J_(^\-^-4Rr,/2 (zj + z' i (In# "Zn^K-,) + — IXG an U; XR dn|K.l/2 \^K n#.J

fH.V1'

V AS 1 G

J-l/2

K-l/2 U -1/2

1/2

Energy Equation

r AS l G JK.l/2

(nU)

- A17 C^C Icjoc-H;:-. I^^.)J ■ - (Wi) ' TMXG an

+ r AS I^IK-I/2 K-'/2

"T AS IGIK.1/2 üK-I/2

J-l/2

K-l/S K,+^)

35

-■ IMMMUMI

With mass flow bleed at the wall, i//j|(s) and ^T(s) are given by

o \J-l/2 / O \J-UC

♦TJ =*j"' »AS, (-Sf)''"'^ (116)

Solution of Matrix Equation

The solution of these equations is obtained using the method of block-

tridiagonal factorization (Refs. 15 and 17). If the column matrix J"K is

defined by

TK=^J uJ uT flJ TJ öJ PJ yJ IJ Vf (117)

then the difference equations, Eqs. (105) through (il^), may be written as a matrix equation

= K =K_^K p-l==K (ll8)

where RK and L are the coefficients of the dependent variables and x is the right-hand side of these equations. If Tj^ = 0 is at the first mesh point and ^KL = 1 is at the last mesh point, equations are written at each of the TJK

transverse locations where 2^K^KL and ten boundary conditions are required. Note chat Eqs. (109) and (ill) involve only the Kth mer^ point. Tnen we may use these two equations as boundary conditions plus the eight conditions given by Eqs. (89) and (90) or Eqs. (89) and (99).

(17) Varah, J. M.: On the Solution of Block-Tridiagonal Systems Arising from Certain Finite-Difference Equations. Mathematics of Computation, Vol. 26, No. 1, March 1969.

36

^_^ - ^^_«

A 5 X10 M matrix for the boundary conditions may then be written for each

end point such that the matrix equations become

= KLTKL==KL | (119)

The complete set of matrix equations is written

where

A f = Q (120)

A =

w N o s

-I1 r 0

N

\ s R2

s N \

= K N

RK

N \ -L*L R

K^ N

0 M KL

(121)

N

Q ^f, f*,?*1-}1 (122)

3T

HÜtti

(123)

(1210

(125)

(126)

cR \ (K r^KL-l)

,K»I I J J = I

.5

,10

(127)

38

. .. iinriiiiiiüan i -■■■--'-'-"

Hence we have a block tridiagonal matrix:

Ä' C'

rvz2

BKAKCK

^KL-I^KL-^KL-I

|KL -KL

(128)

The matrix Q splits as follows;

q'^Vd^f (129)

^{^(1=6,10), ^(1=1,5)} (130)

^{▼«•CX-«*!.?«-} (131)

The matrix equations (Eq. (120)) are solved using the method of block tri' diagonal factorization (Ref. 1?) with the recursion formulas given by

r.i1 (132)

i K= t&rty I^K^KL-l (133)

39

■■ini^MI^MÜI ■ hi I itJTMi^fai ■ i _..^

^A*-!*!*'1 2'K-KL (13M

Z'^DT'Q1 (135)

zK= (BY1 3K-1KH 2iK'KL (136)

and the solution is obtained by backward substitution,

= KU -KL (137)

J* lK_EKf,K'*"1 (KL-1>Z^I) (138)

1+0

^^■«»M^to^^^-.^ . _, .. ^J..J^^..^.......■,. . ■ ^--^iiii «Mrti urn ■ ■- — -

DESCRIPTION OF TEST mOGRAM

The ST9 demonstrator IR suppressing exhaust diffuser was tested on the Pratt & Whitney Florida Research and Development Center (FRDC) D-32 test facility to verify the accuracy of the analysis described previously in this report. The FRDC facility can provide hot gas flow at a temperature of 1200 deg F and weight flows of 8.3 lb/sec. Adjustable swirl vanes upstream of the diffuser provide swirling flow from 0 deg up to 30 deg, thus simulating typical turbine exit flows at different engine operating conditions. In addition, the facility was modified to provide coolant flow rates up to 10 percent of diffuser.airflow rates and to provide slot cooling on the diffuser walls. Complete wall static pressure and temperature instrumentation was provided to measure the effect of wall cooling rates on pressure recovery and wall temperature at different simulated engine operating conditions.

Description of Test Facility

A schematic of the FRDC D-32 test facility is shown in Fig. h. The diffuser inlet flow simulating a typical turbine exit flow is provided by compressed air from the D-Area J57 slave engine. The compressor air flows through a stand heater burner and swirl generator with adjustable vanes shown in Fig« 5. Coolant airflow is supplied to the inner and outer walls of the exhaust diffuser from the D-Area 350-psig air system. As shown in Fig. 6, the coolant to the outer wall, inner wall, and base louvers is separately regulated. A schematic of the ST9 demonstrator IR suppressing exhaust diffuser is shown in Fig. 7. The manifold providing the coolant flow separately to the inner and outer walls and base is shown as well as the location of the louvers or slots on the diffuser wall. Throughout the test program, no coolant flow was supplied to the base in order to eliminate effects of base cooling flow on inner wall static pressure. Photographs of the test facility with the diffuser in place taken after completion of testing are shown in Figs. 8 and 9» No apparent damage or deterioration to the suppressor occurred during the test program.

Description of Instrumentation

Wall pressures were measured by thirty-one static pressure taps located on the inner and outer walls of the diffuser with the data recorded manually on manometer boards. The exact axial locations of these pressure taps, denoted IWOl through IW31, are given on Table 1 and are shown schematically on Fig. 10. Plenum static pressures in the three manifolds are denoted PB01,

kl

II nil ^^^^.^^^m^^^. ^.-^

PB02, and PB03 for the outer wall, inner wall, and base, respectively. Wall temperatures were measured by sixteen thermocouples located on the inner and outer walls with the data recorded manually on the thermocouple readouts. The locations of these thermocouples are given in Table 1 and are shown on Fig. 11. These thermocouples are labeled TW01 through Wl6. In addition, the bulk temperature of the coolant flow was measured by thermocouples labeled TB01 through TB06 as shown on Table 1 and Fig. 11.

The inlet flow distribution was measured by two total pressure rakes and two wall static pressure taps. In addition, a measurement of circumferential distortion was made through four midspan total pressure rakes located at several circumferential stations.

A midspan total temperature probe was used to measure the bulk temperature of the diffuser flow. The location of all the inlet flow instrumentation is shown on Fig. 12. Diffuser inlet and coolant flow rates were measured separately by orifices located as shown in Fig. 12.

Test Results

A summary of the results of the test program are presented in the test log, Table 2, and a complete set of test results in Table 3. Eleven tests were conducted over a period of two days at three simulated power settings ana three different coolant flows per power setting. Two tests, 2.01 and 6.01, are considered to be invalid due to flow perturbations while data was being recorded. Each test was repeated to provi- valid data. The main results of the test program are briefly discussed in the following paragraphs,

The effects of film cooling on pressure recovery, wall pressures, and wall temperatures were evaluated by testing coolant flows of 2.5, 5, and 10 percent of diffuser inlet flow. The largest effect of this coolant flow variation was on louver wall temperature, as shown for the outer wall in Fig. 13. Locations of the outer wall cooling louvers are shown in Fig. 7. Surface temperatures increased rapidly with reduced coolant flow and the greatest change occurred in louver k.

The effects of coolant flow on wall pressure distributions are shown in Fig. 1^ for a swirl angle of zero degrees. Changes in wall pressure with coolant flow were less significant at swirl angles of 16 and 21 deg than at zero degrees. Reducing coolant flow rate from 10 to 2.5 percent on inlet flow resulted in a general increase in wall pressures of about 0.03 psi for both inner and outer walls. Note that the increased wall pressures were not restricted to the cooled section of the suppressor but also extended upstream to the inlet static pressures. This results in a decrease in

1+2

-'^ ' ■■- --"■— ■

-..—J^..... ^-■J...... ■ "■'-

pressure recovery (Cp) from 0.63 to O.58 with coolant flow as shown in Fig. 15, where the pressure coefficient Cp is given by

P-P,

01 '1 (139)

However, the influence of coolant flow on Cp is small at zero swirl and negligible at swirl angles of l6 and 21 deg. Hence, the slight ejector-effect of the film coolant at zero swirl disappears at higher swirl angles.

U3

■■...—^■,J—.... . ...^— ... ■HI iiti«i|iiiiiiiiiiiiriitini

COMPARISON OF EXPERIMENT AND THEORY

A set of calculations were made to demonstrate the capability of the computer program and assess the code by comparing the theoretical predictions with experimental data. These calculations include a solution obtained for turbulent flow through a 6-üeg conical diffuser, termed Fräser Flow "A" (Ref. l8), a baseline calculation for the ST9 demonstrator diffuser model tests with no film cooling, and a set of three cases for the flow through the ST9 demonstrator IR suppression diffuser corresponding to tests logged in Table 2. The calculations were numerically stable; however, truncation errors associated with the linearization of the differential equations were found to be sensitive to streamwise step size. The sensitivity to step size was particularly acute for the slot cooling cases. Since the slot height was only one percent of the diffuser inlet height in these cases, the streamwise step size had to be very small (of the order of the slot height) to allow adequate definition of the resultant flow field development. Since the computing time is proportional to the number of streainwise stations in the calculation field, efforts should be made to select the optimum step size in order to minimize computing time and still keep truncation errors within reasonable bounds.

Fräser Flow "A1

The solution for the turbulent flow through a 6-deg conical diffuser was obtained with the current calculation procedure and compared to the experimental data of Fräser as presented in Ref. l8. For this calculation, one hundred thirty streamlines and fifty streamwise stations were used to construct the coordinate system from the Schwartz-Christoffel transformation as shown in Fig. 16. The mesh distortion parameter placed approximately twenty mesh points between 0SY+^10. The inlet flow was constructed from Coles' velocity profile (Ref. iß) with the inlet boundary layer thickness taken to be 0.0528 in. as specified by the Fräser data in Ref. l8.

(l8) Coles, D. E., and E. A. Hirst: Proceedings Computation of Turbulent Boundary Layers - 1968 AFSOR-IFP-Stanford Conference, August 1968.

hh

.-..„ I_,...^.„.. ^.^i^.^^, . >i^. ..^^^«am^mMaafc^.

As demonstrated in Ref. l8, Coles' profile accurately represents the

boundary layer mean velocity profiles over a wide range of flow conditions

in terms of two parameters, a friction velocity, U* and a wake parameter, TT

Coles' profile expresses the mean velocity distribution by

1 IT InY* +

n . 2. n Y —sin (T"8" + B (1^0)

where U+ and Y+ are the velocity and transverse coordinate written in wall

variables

u -- u/u u/y *m'Pm (1U1)

(1U2)

Y* ■ YUHt/l/ ■ YyTw/^w /v

TT is the wake parameter, and B is a constant. In Eq. {iko) the first term expresses the "law of the wall" and the second term expresses the "law of the wake". Specification of U* and n uniquely determines the velocity profile.

At the edge of the boundary layer Y = 6 and Eq. (l'+O) becomes

U ln(

Su* . 20 + B (1U3)

When Eq. (iko) is used to compute the boundary layer displacement thickness

6*, the relation

03

Su# i + n d^)

is obtained. Thus given 5 and 6*, the wake parameter TT and friction velocity \f can be determined from Eqs. (lU3) and (ikh) and then the velocity distri- bution is uniquely determined by Eq. (1^0).

U5

Mtti^iauüHiii ..^ ■■■■;.:.... --i^*.. ......,***..*.*. ...J^-,^.. . ... ■..■^.^■-—„■,- ^ .■■^:.. ^^M. 'JL^-

It should be noted that Fräser Flow A is a particularly difficult flow

to calculate with classical boundary layer theory which assumes a viscous

flow development under the influence of a semi-infinite nominally inviscid

outer flow field. The inability of classical boundary layer theory to predict

the flow field development is shown in Ref. 18,where a variety of boundary

layer theories were unable to predict this flow field accurately. This

inability of standard boundary layer procedure to predict Fräser Flow "A"

stems from two sources. First, the wall boundary layer reaches the diffuser

centerline approximately halfway downstream from the diffuser throat, and in this region no potential core flow exits. Second, the Fräser Flow "A" diffuser is an optimum diffuser in that it is designed to keep a nearly

separated boundary layer as the flow diffuses. Boundary layers which are on

the verge of separation are very difficult to predict since small changes in

pressure distribution can lead to large changes in boundary layer thickness.

However, as discussed by Coles and Hirst (Ref. l8) , the Fräser Flow "A" data

are reliable and based upon careful development of a flow configuration with

good axial symmetry. Thus the flow should be able to be predicted accurately

by a calculation procedure which can properly account for the disappearance

of the central potential core. The present analysis,which does not assume

the existence of any potential core,yields predictions which are in good

agreement with the data as shown in Figs. 17 and 18,where comparisons

between experiment and theory for the pressure coefficient and wall friction coefficient are present.

Calculation Procedure For

ST9 Demonstrator IR Suppression Diffuser Runs

The streamline coordinate system used to predict the flow in the FRDC

diffuser as calculated from the Schwartz-Christoffel transformation is shown

in Fig. 19. Since this diffuser configuration is complicated, it is helpful

to describe it in some detail. First it is noted that a short "duct inlet

section" has been added to the diffuser in order to insure that the initial

inlet flow has no normal pressure gradient. The diffuser centerbody has a

blunt "diffuser base" and a lip on the outerbody (OD) wall extending past the

base. The exit centerbody (ID) and OD walls were, therefore, extended past

the diffuser exit plane by the "extended free streamline" shov..^ in Fig. 19.

The ID and OD wall contours were specified at fifteen equally-spaced stream- wise stations. The computer program theu fitted smooth curves through these points and interpolated the curves to specify eighty streamwise stations. The

input mesh points are indicated by a small circle and the diffuser wall by a

double line. In addition, seven cooling slots are located on the wall at the

stations specified. In order to clearly illustrate the location of the slots,

the ID and OD walls were opened a small amount at each slot so that at the

1^6

, .. _■.■...,.>-, .^■■^^■■^-„.,, ...„ «^-.. - ■■ — 1 iiiiiiaiiniiini n

exit plane the wall is separated from the wall streamline by a small amount. Finally, six struts are located in the duct. The plan view of one strut is indicated in Fig. 19 by the "strut centerline", "strut leading edge", and "strut trailing edge".

Eighty strearawise stations and one hundred thirty streamlines were calculated for the streamline coordinate system. The mesh distortion parameters were selectei so that the first streamline from the wall was located at a distance of 0.000077 times the duct inlet height. This placed approximately five mesh points between 0<Y+<10 at a slot exit plane and about forty mesh points within the slot height. In portions of the flow removed from the slots,approximately thirty mesh points were placed in the viscous sublayer. For all cases, the inlet flow was obtained from experi- mental data shown in Fig. 20. Instrumentation in the inlet plane of the diffuser was used to measure total pressure, static pressure, swirl angle, and total temperature as given in Table 3 for each test case. Twelve data points across the inlet are given. These data were then interpolated to fit the one hundred thirty mesh points used in the calculation,and the Mach number, velocity and static pressure,and temperature were then computed. This experimental data, however, is not accurate enough to construct the inlet boundary layer profiles. Therefore, these profiles were construeoed from Coles' velocity profile by speeifjäng the displacement thickness.

Baseline Case - No Film Cooling - No Struts

The first ST9 diffuser case calculated consists of the basic diffuser with no struts or slot cooling and is termed the baseline case; the baseline case has zero swirl. The predicted wall static pressure distribution is compared with the experimental data taken from earlier FRDC cold-flow model tests (Ref. 7) in Fig. 21. As shov./' in Fig. 21, the theoretical predictions of wall static pressure distribution without cooling and without struts is higher than the experimental data. This ob ervation is consistent with the observation made in the Fräser Flow "A". In addition, it is noted that the analysis predicts the appearance of a separation bubble on the canterbody wall located approximately between 0.19 <Z/L<0.26; ■this prediction is in agreement with FRDC model test data which also shows a separation bubble at the same approximate location.

A number of observations can be made about this test case. The predictions of the analysis are in qualitative agreement with the data for pressure coefficient along the diffuser. The overall pressure rise is in fairly good agreement with the data,and both the analysis and the data show a separation bubble in approximately the same duct location. However, the detailed variations of pressure coefficient is not in good qualitative agreement with the data and in the region of the separation bubble, the

U7

■"■-- ■■-1 miHllr"'- ""■'■",-;-- ^mt*t—äim

analytical predictions and the data show significant qualitative discrep-

ancies. However, it should be noted that it is very difficult to mainta.n

axisymmetric flow in curved-wall annular diffusers (see Ref. 2). This problem is compounded when separation occurs because separated flow is very

sensitive to small pressure fluctuations. Therefore, the size and location

of the separation may be affected by asymmetry effects which in turn may lead

to discrepancies between prediction and data.

STQ Demonstrator TR Sunnres^on Diffusers - 2.^ Percent Cooling Rate

The 2.5 percent cooling case consists of the ST9D demonstrator IR

suppression diffuser run with six struts and seven cooling slots and corresponds to test case 3.01 shown on Table 2. This test case represents

the diffuser operating at 60 percent military rated power and film cool:ng

flow rate of 2.5 percent of the diffuser flow rate. The coolant plenum total

pressure and static temperatures are given in Table 3. Experimental data,

obtained from Table 3, was used to construct the inlet flow profiles as

described for the Fräser Flow "A" case.

A comparison of the predicted flow with the corresponding experimental

data is shown in Fig. 22 and Fig. 23. The predicted wall static pressure distribution, shown in Fig. 22, indicates a significant drop in pressure in

the region of the strut. This effect is caused by the strut blockage which

accelerates the flow through the strut passage (see Ref. 7). The presence of

the strut exhibits the separated region which is present in the baseline

case,and separation was neither predicted by the analysis nor measured in the

experimental test. The first slot is located on the tip wall just downstream

of the strut trailing edge. As shown on Fig. 23, the wall temperature drops

almost by 700 deg F at this slot and increases rapidly by almost 500 deg F

as rapid mixing of the film cooling flow with the hot diffuser now is promoted. Successive slots follow the same pattern. The predicted wall static temperature distribution is in good agreement with experimental data.

UM wall static pressure distribution is qualitatively correct for Z/L<0.b,

bat shows significant quantitative discrepancies with the data. Downstream

of the fourth slot, the wall static pressure decreases rapidly due to large

total pressure losses, and this unrealistic solution is not plotted.

STQ Demonstrator IR Suppression Diffuser With 5 Percent Cooling Rate

The 5 percent cooling rate case corresponds to test case 2.02 shown on

Table 2 This CE.33 represents the diffuser operating at 60 percent military

rated power with a film cooling flow rate of 5 percent of diffuser weight

flow. A comparison of the predicted results with experimental data ^ shown

on Figs 24 and 25. The wall static pressure distribution shows very little

kB

- - — *■ Lj*.dtmdaaa*£Um*i*-*~^ L _ ^.. ^. . . ,. ^^tmmtimmtmiaM

change compared to test case 3-01 with 2.5 percent cooling flow rate and is consistent with earlier observations of the data that the pressure distribu- tion is not greatly affected by the coolant flow rate as ^nown in Fig. I*. The wall temperatures, however, show a significant reduction from a mean temperature of about 600 deg F for test case 3-01 iown to a mean temperature of about U50 deg F for this test case. This predicted reduction in tempera- ture with increasing coolant flow rate follows the trend in the experimental data shown In Fig. 13- Again, downstream of the fourth slot, the wall static pressure decreases rapidly due to large total pressure losses and the

solution is not plotted.

STQ Demonstrator IR Suppression Diffuser With 10 Percent Cooling Rate

The 10 percent cooling rate case corresponds to test case ].01 as shown on Table 2 and represents the diffuser operating at 60 percent military rated power with a film cooling flow rate of 10 percent of diffuser weight flow. A comparison of the predicted results with experimental data is shown on Figs. 26 and 27. Again, the wall static pressure distribution shows very little change from the previous cases, indicating that the coolant flow rate does not greatly change the wall static pressure distribution. This observation was previously noted (see Fig. lh). The mean wall temperature shows a drop from the previous cases down to a mean temperature of about 300 deg F and demonstrates the expected behavior that increasing coolant flow rates reduce wall static temperatures as shown by the data in Fig. 13- It should be noted that part of this reduction in wall temperature comes from a reduced plenum total temperature as seen in the data of Table 3- Therefore, an accurate calculation of an IR diffuser performance must account for an overall heat balance as well as the local slot cooling effect.

Discussion of Numerical Calculations

Since this report presents new and very advanced techniques for calculating turbulent flow in ducts, some discussion of the numerical problems which were encountered is in order. In particular, suggestions are made on means to improve the predictions of the computer code for slot and film cooled problems. The Fräser Flow "A" test case represents a good test case for the purpose of checking the accuracy and reliability of the^computer program. As evaluated by Coles and Hirst in Ref. 18, Fräser Flow A represents reliable measurements based on careful development of a flow configuration with good axial symmetry. Pressure distributions, wall friction coefficient distributions, displacement thickness, and momentum thickness distributions are prosented. In addition, accurate measurements of the boundary layer profiler are given for eleven UCUl stations. Finally, the

'♦9

—- - ■ ■ - ^„^j^a^iJ^aMf ■MÜaMMMrfMUMriiiMiyÜ« — ^

Fräser Flow "A" is a clean flow showing only the effects of boundary ayer grow" n an adverse pressure gradient. Of special interest xn this test else I« the fact that the flow is in a nearly separated condxtxon for a good Tart of the diffuser length. Complications arising from struts, slots, sw.rl

or wall curvature are not present.

Before the predictions of the Fräser Flow V case were compared with the experimental data, the numerical accuracy for the computer program was

hecked for internal consistency by sevezal means. The most xmportant of these checks was a comparison of the mean flow variables obtained by ave g the solutio^for each dependent variable over the duct hexg and

compa ing the average values with the solution ^l^^^^T obtained by integrating the one-dimen-ional mass flow ^xghted average eauations. As an example, these equations show that in the absence of wall Zsflo^ bleed and wall ^eat transfer, the diffuser weight flow and mass n^ wefghted average tot.l temperature are constant. ^ examxnaUon o the detailed computer printout for the Fräser Flow "A" case shows that these variables a~e indeed constant to at least five decimal places thus xnd^atmg that the numerical procedure satisfies the integral conservation laws. Tn regard to the prediction of ^as.r Flow "A", the agreement between heory and Tx rle^t should be regarded as excellent for -e -in friction and good .or the pressure coefficient, particularly in vxew of the fa^t that the flow continuously on the verge of separation. As shown by Stratford (Ref. 19), turbulent boundary layers near separation may take on a wxde J^y - profile shapes. The shape depends upon the upstream history of the flow and

^lÜhLgL in the upstream history ca. lead to ^f ^f-^^L layer development. Thus the Fräser Flow "A case is a difficult test of the basic analytical procedure, and based on this test, it is concluded that the aTlysiro^rates well in predicting flows in the absence of such complica-

tions as struts, swirl, and wall curvature.

Some indication of local errors can be estimated ^ •«^ff^^ and static pressures along the centerline of the Fräser Flow A case,which should be nearly constant since the viscous and heat transfer losses are Sglgible along the centerline. This comparison indicates loca -cumu atea errors of the order of 0.1.which is quite good. From these results, it Lf concluded that the basic numerical procedure is accurate and that any improve- Tntst the predictions for this basic case must come from a closer examina-

tion of the turbulent nixing length model.

(19) Stratford, B. 8.1 An Experimental Flow With Zero Skin Friction Throughout Its Pressure Rise. JournaT of Fluid Mecha^s, Vol. 5, 1972,

pp. 17-35.

50

-'■-—■'—''—^-~~ iiiiiiiiiiMiMiMrtiiiiTifi iiii

The turbulent mixing length model may be an important source of error,

producing discrepancies between theoretical predictions and experimental

data. Short of detailed turbulence measurements, the best method for

assesring this effect is to compare predicted boundary layer growth with experimental data such as the Fräser Flow "A" test case. First, it should be

noted that the Mach number for the Fräser Flow "A" is low, so that 0.1

percent error in static pressure represents a 6.5 percent error in static

pressure coefficient (i.e., the error is magnified when results are expressed

in terms of a pressure coefficient). Second, it is noted that there is a very

close interaction among the following parameters: mixing length, boundary

layer thickness or blockage, and static pressure gradient. Thus, increasing

the boundary layer thickness decreases the static prersore because of

effective blockage. Increasing the pressure gradient increases the boundary layer growth because of the additional work done on the boundary layer.

Therefore, a comparison of the theoretically predicted pressure distribution

with the experimental data yields an indication of the boundary layer growth

find,by inference, the mixing length. Since the predicted pressure coefficient

is larger than the experimental data, the predicted boundary layer thickness

must be smaller than the measured boundary layer thickness. Hence, it is

concluded that the mixing length is too small. It should be noted, however,

that the pressure coefficient prediction is very good up to a Z/L = O.U,at

which point the boundary layers merge (see Fig. 17). Furthermore, downstream

of this station the boundary layer is in a nearly separated condition as

shown in Fig. 18. Therefore, it is concluded that modifications of the

mixing length may be required for flows with merged boundary layers or nearly

separated boundary layers.

The ST9 IR suppression diffuser case with no slots or struts introduces

an additional problem in making an accurate flow field prediction. This

diffuser, unlike the Fräser Flow "A" diffuser, has significant wall curvature

which is known to effect the mixing length (Ref. 20). Thus turbulence can be

expected to increase on a concave surface and decrease on a convex surface,

modifying the boundary layer growth on these walls and changing the pressure

distribution accordingly. Hence it is expected that introducing a better

turbulence model which accounts for wall curvature, would produce better

predictions for wall static pressure distribution than that shown in Fig. 21.

An additional complication which arises in the baseline calculation is the

appearance of a separation bubble. Since separation is a very complex

(20) Bradshaw, P.: Effects of Streamline Curvature on Turbulent Flow.

AGARDograph No. l69, 1973.

51

_»t_ **m*m .,..:.. ■

plienotnenon whic-h Is very sensitive to local conditions, predictions of separated regions in turbulent flow can be a strong function of the

turbulence model. The turbulence model used in the present effort is an

equilibrium model based on measurements of turbulence structure in

unseparated flows and, therefore, some significant error may be introduced

in applying a turbulence model based upon attached turbulent flow data to

separated flow. Inaccuracies of the turbulence model in separated regimes

will affect the predictions downstream of flow reattachment since total

pressure losses produced by the separation bubble cannot be reversed.

For slot cooled wall cases, test case 3-01, 2.02, and 1.01, the

comparisons indicate that the simple mixing length model used in this report

may not be adequate. The turbulence model used only accounts for an inner layer mixing length influenced by the wall and an outer layer mixing length

influenced by the free stream. The mixing layer developing between the cold

slot flow and the hot diffuser flow is not being properly modeled,and this

may adversely affect the theoretical predictions. If the turbulence model

used is not appropriate in the immediate vicinity of the slot, then any

inaccuracies would be compounded for diffusers which contain a succession of

cooling slots such as the ST9 IB suppression diffuser. This inadequate

modeling of the turbulence structure in the slot mixing region may explain

the increased discrepancy between predicted and measured pressure

distribution with each slot (see Figs. 22, *, and 26). Thus, in summary,

although the turbulence model appears accurate for flows in the absence of

curvature, slot cooling and separation, it may need to be refined for these

effects before accurate predictions for the general diffuser flow field can

be made.

The numerical method used in this report can be subject to one final

important source of error. Since the linearization of the equations of motion

implies a certain degree of smoothness in the solution, any local errors introduced in the initial profile may produce significant errors downstream.

Usually, as in the diffuser inlet flow, this initial error is dampened and

causes little difficulty because the flow variables change slowly. At the

slot interface, however, there is a large temperature discontinuity, and

since the flow variables (especially temperature) change very rapidly, this

initial error may not be dampened. Such an initial error would cause errors

in entropy which would lead to inaccurate predictions of the pressure

coefficient. Indications that this may be a factor are shown by the significantly larger errors in the mass flow and mass flow weighted average

total temperature which are not as well behaved downstream of a slot as they

are upstream of the first slot. It is, therefore, concluded that the model used to predict the initial flow and shear for each slot be improved so as

to smooth out the discontinuity in the neighborhood of the slot exit plane.

52

■-,:"—*■"-"'—- ..-..^-^-^-^i^ii^M»^.»^-^»-»-—^—^-^^-..-^- ... - ^ .^ .■„. .- ^.-. ^^^.^.^^^^itn^^^^^^iiM^ikMiMiiilU^j^,.^, , t

Based on the experimental data presented in this report, it is concluded

that the wall temperatures can be significantly reduced by increasing film

coolant rate. For coolant flow rates less than or equal to 10 percent of diffuser flow rates, the film cooling has little effect on pressure distribu-

tions or pressure recovery.

An examination of the Fräser Flow "A" test case demonstrates that the

basic numerical methods used in this report based on the Schwartz-Christoffel

transformation to calculate an orthogonal coordinate system and an implicit

linearized finite difference scheme for solving the equations of motion for

turbulent flow is an accurate and reliable method for solving internal flows

in axisymmetric ducts of arbitrary wall curvature. Further refinement of the

turbulence model in regions of merged boundary layers and nearly separated flow

is indicated by the comparisons with data.

For slot cooled walls and highly curved walls, such as the ST9 IR

suppression diffuser, further refinement of the turbulence model is also indicated. Specifically, the turbulence model should include the effects of

wall curvature and the effects of a mixing layer between the hot and cold flows. Finally, the initial profiles setting up the slot cooled flow need smoothing of the temperature and density discontinuity in order to minimize

nonlinear errors in the calculation. It is felt that if the indicated

refinements and modifications were made, the resulting computer code would

have a unique capability for predicting the development of flow fields in

axisymmetric diffusers, including the effects of wall curvature, struts,

swirl, and film cooling.

53

mtmmm tinw«..-..' <

REFERENCES

1. Reneau, L. R., J. P. Johnson, and S. J. Kline: Performance and Design of Straight, Two-Dimensional Diffusers. Transactions of ASME, Journal

nf Fluid Mechanics. Vol. 89, March 1969, PP- iWX-lÖO.

2. Sovran, G., and E. D. Klomp: Optimum Geometries for Rectilinear Diffusers. Fluid Mechanics of Internal Flow. Elsevier Publishing Co.,

1967.

3. Fox, R. W.,and S. J. Kline: Flow Regime Data and Design Methods for

Curved Subsonic Diffusers. Journal of Basic Engineering, Transactions

of the ASME, Series D, Vol. 8U, No. 3, September 1962, pp. 303-312.

k. Howard, J., H. Henseler, and A. Thornton-Trump: Performance and Flow

Regimes for Annular Diffusers. ASME Paper 6?, WA/FE-21, 1967.

5. Runstadler, P. W., and R. C. Lean: Straight Channel Diffuser Performance

at High Inlet Mach Numbers. Transactions of ASME, Journal of Basic

Engineering, Vol. 91, September 1969, PP- 397-^22.

6. Dietz, A. E.tand J. F. Thompson: Advanced Experimental Infrared

Energy Suppression System for the T-53-L-11 or T-53-L-13 Turbine

Engine. Hayes International Report No. 1172, 1968-

7. Thayer, E. B.: Evaluation of Curved-Wall Annular Diffusers. ASME

Paper 71-WA/FE-35, September 1972.

8. Anderson, 0. L.I A Comparison of Theory and Experiment for Incompressible, Turbulent, Swirling Flows in Axisymmetric Ducts. AIAA

Paper No. 72-^2, 10th Aerospace Sciences Meeting, January 1972.

9. Anderson, 0. L.: Numerical Solutions of Incompressible Turbulent

Swirling Flows Through Axisymmetric Annular Ducts. United Aircraft

Research Laboratories Report No. H213577-1, March 1968.

10. Anderson, 0. L.: User's Manual for a Finite-Difference Calculation of

Turbulent Swirling Compressible Flow in Axisymmetric Ducts with Struts.

United Aircraft Research Laboratories Report L911211-1, Contract No.

NAS3-15^02, 1972.

3k

■I—1 1—IMMlfllMMIüliillM—■—I • *HMii»tf1*M ■>""■'■♦ JM

11. Kober, H.: Dictionary of Conformal Representations. Dover Publications,

Inc., 1957.

12. Gaier, Dieter: Konstruktive Methoden der Konformen Abbildung, Springer Tracts in Natwal Philosophy, Vol. 8, 1963.

13. Keller, H. B^and T. Cebeci : Accurate Numerical Methods for Boundary Layer Flows-II Two-Dimensional Turbulent Flows. AIAA 9th Aerospace Sciences Meeting, New York, January 25-27, 1971, AIAA Paper No. 71-16U.

Ik, Keller, H. B.: A New Difference Scheme for Parabolic Problems. Numerical Solution of Partial Differential Equation-II SYNSPADE 1970 Ed. by Hubbard, B. Academic Pressure, New York.

15. Keller, H. B.: Accurate Difference Methods for Linear Ordinary Differential Systems Subject to Linear Constraints, SIAM J. Namar,

Anal. Vol. 6, No. 1, March 1969.

16. Briley, W. R.,and H. McDonald: An Implicit Numerical Method for the Multidimensional Compressible Navier-Stokes Equations. United Aircraft

Research Laboratories Report M911363-3, November 1973.

17. Varah, J« N.I On the Solution of Block-Tridiagonal Systems Arising from Certain Finite-Difference Equations. Mathematics of Computation,

Vol. 26, No. 120, October 1972.

18. Coles, D. E., and E. A. Hirst: Proceedings Computation of Turbulent Boundary Layers - 1968 AFSOR-IFP-Stanford Conference, August 1968.

19. Stratford, B. S.: An Experimental Flow With Zero Skin Friction Throughout Its Pressure Rise. Journal of Fluid Mechanics, Vol. 5,

1972, pp. 11 -35.

20. Bradshaw, P.: Effects of Streamline Curvature on Turbulent Flow.

AGARDograph No. 169, 1973.

55

— - - - -■* mmvitm-m*"'- ■

1 D7 b6 b5 (SOURCE) b! bQ

(a) Z PLANE

3 D4

(b)W PLANE

FIG. 1 CONFORMAL MAPPING OF DUCT.

56

-^ ———-^—* .—^—^. imiMiliUMif li

(a)r(n,S). z(n,s) PLANE

(b) R (A,S) . Z (A,S) PLANE

FIG. 2. ROTATING AND SCALING DUCT

57

■ - ■HMHüBHMiliitll

STREAMLINE FORMING • WALL OF DUCT 2

.V - ^ ^ Ltiiai UJ

CO 3 Q <

STREAMLINE FORMING WALLOP DUCT 1

-SLOT LIP

AXIAL DISTANCE Z

FIG. 3. CONSTRUCTION OF SLOT IN DUCT.

'58

^^^^^^-^^^^^

2 s 8

i

59

■ ■ — i -fiiifiu

A^TW^VTWWWWVT^ L—_ I ^IS

60

— —^--"'■«*- — -■ - ■ - ■ - -■ •

CO >- w o

61

juyggggH^jlu^^gy ., ..■>... --'-^-'--u-■,-

INSTRUMENTATION CONDUIT

13.4 in.

EXHAUST. FLOW '

COOLANT TO INNER WALL

(2 PLACES)

COOLANT TO OUTER WALL

(6 PLACES)

BASE

1G.3 In.

FIG. 7. DIFFUSER TEST RIG-

62

/pwoe)

I LU

i -i

i

i i i

\y

65

—--- - - ■ - -- -■■ ----^■^^■^^^■^i -. ...,^,., ^^-.■^.,,. ■ -■■--- - - - i ifiiimiflilBiiift

( PW30 y(^JB03 ^TWlT^

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4

w Ui -I a. 3

!

i u. O z o 5 8

2 U.

66

^^■MM.*«*^^»^. ■■- -——^■^—^—J-^«^-^^» ,. .-■.^ Ml- .,......._.■ ^...-.„■■.-^■...■.

TOP

SUPPRESSOR WALL PRES. Pwl 13

P-r RAKE 1

INLETTT

PROBE

INLET STATIC PRES. TAP 6^10

PT PROBE 2

Ml DSPAN PT PROBE 3

SUPPRESSOR WALL TEMP. Tw 1-11

PROBE 4

BOTTOM

FIG. 12. INLET PLANE INSTRUMENTATION.

67

m^^mt^^M

INLET TOTAL TEMP. = 1140 F

SWIRL ANGLE ■ 21 deg

1000

900

I 800

■ < S 700 I LU

600 LU > O

I 500 ■ > <

400

300

200

100 0.0

TEST NO. COOLANT FLOW

RATE LB/SEC

7.01

8.01

9.01

0.541

0.272

0.139

0.1 0.2 0.3 0.4

OUTER WALL COOuNG FLOW - lb/sec

0.5 0.6

FIG. 13. OUTER WALL AVERAGE SURFACE TEMPERATURE FOR ST9 IR SUPPRESION DIFFUSER.

68

o 1- UJ < cc b «1 >- M HI _] U. Z o.

|

Bisd - aunssBUcnnvw

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69

— ' ^»^^-- -iiiiMiiini ■ n

10.0

PERCENT COOLING FLOW

»a 15 EFFECT OF COOLANT FLOW ON PRESSURE RECOVERY FOr.

STSTR SUPPRESSION DIFFUSER.

TO

' 'L ■ ■!■

;

r m i4 - —L. 4

r^r ' r ' d - *+-

t _i. ■

^^-iT 1 • :

* i IT—' 1 üi j

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—.—^■,. , K. . __ • i * t • irrT •

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• « T " ' '"■ -L_ -!-

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- r^==*= 4

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. W-- £

J—,—t- ■ El-- ,—B — ± —

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- w - •jr^r H-

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-^-^ — — 4- _.. 1 ■ • 4^ — ^"^ • 1

. i ■

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. . , —

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Li-

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71

-- -- ■' - • ■ ■ HHttHMilil ___________

FRÄSER FLOW "A" CONICAL DIFFUSER

AFOSR STANFORD CONFERENCE 1968

I

I i

I!

J1

0.4 0.6

AXIAL DISTANCE, Z/L

0.8 1.0

FIG. 17. COMPARISON OF EXPERIMENTAL AND PREDICTED WALL STATIC PRESSURE DISTRIBUTION FOR FRÄSER FLOW "A" DIFFUSER.

72

■HautUMMMMiMÄ —M—MM—I—I—IMMMIIfMMM - - —

FRÄSER FLOW "A" CONICAL DIFFUSER

AFOSR STANFORD CONFERENCE 1968

0.005

- 0.004 - a.

I

I-

S I LU

8 z o

I i 5

0.003 -

0.002 -

0.001 -

0.4 0.6

AXIAL DISTANCE, Z/L

0.8 1.0

FIG. 18. COMPARISON OF EXPERIMENTAL AND PREDICTED WALL FRICTION

COEFFICIENT DISTRIBUTION FOR FRÄSER FLOW "A" DIFFUSER.

73

■- -■ ^^ ■ '

^. ^..^^..^ . .. ^..I^J^

(id)H sniavy

T^

jS*^! J

ST9 DEMONSTRATOR IR SUPPRESSION DIFFUSER WITH NO STRUT

DATA TEST NO. 3.01 0.60 MRP SWIRL ANGLE 0 DEC '-OOLANT FLOW RATE 2.5%

1.0

0.8

■ 0.6

oc uj ■ D z I O < 0.4

FIG. 20. INLET EXIT MACH NUMBER DISTRIBUTION

75

-—■

. -.... ....^^.^.- „... .. -■■ ■ \imä

ST9 DEMONSTRATOR IR DIFFUSER MODEL

1.0

I

t II

u.

8 UJ a.

LU IT 0.

£ J1

I u. u.

8

0.8

0.6

0.4

0.2

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PREDICTED DATA

CENTERBODY Q

OUTERBODY . □

D/ 0 0 D / 0 □

SEPARATION BUBBLE

I 0.2 0.4 0.6

AXIAL DISTANCE, Z/L

0.8 1.0

FIG. 21. COMPARISON OF EXPERIMENTAL AND PREDICTED WALL STATIC PRESSURE

COEFFICIENTS FOR ST9 IR SUPPRESSOR DIFFUSER WITH NO FILM COOLING AND NO STRUTS.

76

.. .. ■■ -.■ ■....-^^*-^-.^^..::^. ^ ■.: —..— ^...»^ ■■■^ ^-...^.■.^^^....^^J^^^^^^M^. «MMMMb WMM

ST9 DEMONSTRATOR IR SUPPRESSION DIFFUSER

TEST NO. 3.01 0.60 MRP SWIRL ANGLE 0 DEC COOLANT FLOW RATE 2.5%

1.0

a. I r- O a.

a o H z UJ

o LL

O O UJ tr

I LU DC a.

0.8

PREDICTED DATA

CENTERBODY O

OUTERBODY Ü

0.6

0.4

0.2

-0.2

**'\ a G on

0.2 0.4 0.6

AXIAL DISTANCE, Z/L

0.8

D

1.0

FIG. 22. COMPARISON OF EXPERIMENTAL AND PREDICTED WALL STATIC PRESSURE DISTRIBUTION FOR ST9 IR SUPPRESSION DIFFUSER WITH 2.5% INJECTED COOLING AIR.

77

^m^^^^._ -■-...—^ u^-^-^. tmtmtantm^ät^ ..j,.^.—

ST9 DEMONSTRATOR IR SUPPRESSION DIFFUSER

TEST NO. 3.01 0.60 MRP SWIRL ANGLE 0 DEC COOLANT FLOW RATE 2.5%

1000

■o

cc

<

< 5

800

600

400

200

PREDICTED DATA

CENTERBODY O

OÜTERBODY Q

0.2

STRUT

_L

0.4 0.6

AXIAL DISTANCE, Z/L

0.8 1.0

FIG. 23. COMPARISON OF EXPERIMENTAL AND PREDICTED WALL

TEMPERATURE DISTRIBUTION FOR ST9 IR SUPPRESSSION

DIFFUSER WITH 2.5% INJECTED COOLING AIR .

78

MftMmiÜififc iiii^iin—'ana _ . .. ^ . ^-^■^^^.^^. ...^.^..... .. ^.;—

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Q-

a O

o LL

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I tr. a.

ST9 DEMONSTRATOR IR SUPPRESSION DIFFUSER

TEST NO. 2.02 0.60 MRP SWIRL ANGLE 0 DEC COOLANT FLOW RATE 5.0%

1.0

0.8

PREDICTED DATA

CENTERBODY O

OUTERBODY D

a

0.6 - ü n oS

0.4

0.2

-0.2

STRUT

0.2 0.4 0.6

AXIAL DISTANCE, Z/L

0.8 1.0

FIG 24 COMPARISON OF EXPERIMENTAL AND PREDICTED WALL STATIC PRESSURE

DISTRIBUTION FOR ST9 IR SUPPRESSION DIFFUSER WITH 5.0% INJECTED

COOLING AIR.

79

ST9 DEMONSTRATOR IR SUPPRESSION DIFFUSER

TEST NO. 2 02 0.60 MRP SWIRL ANGLE 0 DEC COOLANT FLOW RATE 5%

1000

I

| QC

LU

i

800 -CENTERBODY

600 -

400

200

(, aa?l

AXIAL DISTANCE, Z/L

FIG. 25. COMPARISON OF EXPERIMENTAL AND PREDICTED WALL TEMPERATURE

DISTRIBUTION FOR ST9 IR SUPPRESSION DIFFUSER WITH 5% INJECTED COOLING AIR.

80

^^mmmm am M^M«

ST9 DEMONSTRATOR IR SUPPRESSION DIFFUSER

TEST NO. 101 0.60 M-^P CFR - 0.10 SWIRL ANGLE ODEG COOLANT FLOW RATE 10%

1.0

Q.

I

O Q-

Q-

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ua

LU

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CENTERBODY Q

D

OUTERBODY

O P O Q QQ

o o

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AXIAL DISTANCE, Z/L

0.8 1.0

FIG. 26. COMPARISON OF EXPERIMENTAL AND PREDICTED WALL STATIC PRESSURE

DISTRIBUTION IR SUPPRESSION DIFFUSER WITH 10% INJECTED COOLING AIR.

81

HMMiilliiliiaiiilMllMIHIill -i^tfttHil^MMM

ST9 DEMONSTRATOR IR SUPPRESSION DIFFUSER

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1000

I i

i

800 -

600 -

400 -

200 -

0.2 0.4 0.6

AXIAL DISTANCE, Z/L

0.8 1.0

FIG. 27. COMPARISON OF EXPERIMENTAL AND PREDICTED WALL TEMPERATURE

DISTRIBUTION FOR ST9 IR SUPPRESSION DIFFUSER WITH 10% INJECTED

COOLING AIR.

82

TABLE 1. LOCATION OF PRESSURE AND TEMPERATURE INSTRUMENTATION

TABLE «I

MCAIBH TYPE . INCHES

PW01 WALL PRESSURE 1.250

02 2 750

03 4.250

04 6.750

06 7.150

Ofi 8.750

0/ 10.250

OH 11/50

09 13.250

10 14,650

11 15.900

12 14.700

13 ■

19 000

14 1.250

IB 2 750

16 4.260

17 6250

1« 8.200

10 9.700

?0 10.900

21 12.000

22 13.200

23 14 250

24 15.250

7b 16.350

76 1 7.360

T; 9.600

?8 17,906

."i 16,317

311 14.740

J1 i 16,732

PR01 1 PLENUM PROCESS AS SHOWN \

1 "2 i 1 i03 T r

TW01 WALl TEMPERATURE 2.862

02 9,500

03 10 548

04 1? 170

05 12.906

06 14,716

07 15,?8ai

OH 16 317

09 1 ? 360

10 1 8 669

1 1 19.476

12 1 3 964

13 14,740

14 16 788

Id 16,737

M6 1 7,360

TB01 IBULK TEMPERATUR I 7 717

■ |o.> 1 7, 706

03 16,317

04 17,574

I" ... 14/40

for 1 AS SHOWN

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T.TST OF SYMBOLS

a

A

A+

A*

A

A*

b

B

bI

? c

CD

c

p

cv

D

? ens

en0

Area (ft2)

Area, a/r^ (dimensionless)

Van Driest constant (26.0)

Critical area ratio (dimensionless)

Block tridiagonal matrix (dimensionless)

Diagonal block matrix (dimensionless)

Chord (ft)

Chord, b/rr (dimensionless)

Location of pole in z plane (dimensionless)

Left diagonal block matrix (dimensionless)

Speed of sound (ft/sec)

Drag coefficient, 2D/(p2U2b) (dimensionless)

Friction coefficient, \/^01 ' V (dimensionleSB)

Lift coefficient, 2L/(p2u|b) (dimensionless)

2 2 Specific heat pressure ft /(sec deg R)

Pressure coefficient, (P - \)/^01 ' V (dimensionless)

2 2 Specific heat volume ft /(sec deg R)

Right diagonal block matrix (dimensionless)

Drag/span (lb/ft)

Llock operator matrix (dimensionless)

Strearawise strain (l/sec)

Tangential strain (l/sec)

Block operator matrix (dimensionless)

151

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s f

F

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SB

G

GB

h+

h

H+

H^

lk

m

M

0 m

M

Solution matrix (dimensionless)

Force/area (lb/ft )

Force/span (lb/ft)

Complex variable source solution (dimensionless)

Gap between walls (ft)

Gap between chord lines (ft)

Gap between walls, g/rr (dimensionless)

Gap between chord lines (dimensionless)

Enthalpy (ft2/sec2)

Height of inlet duct (dimensionlesp)

Universal stagnation enthalpy, (hQW-hQ)pwU*/qw (dimensionless)

Universal adiabatic stagnation enthalpy, (hoAWhoA)/^)2 (dimensionless)

Entropy ft /(sec dep R)

=k Element of L matrix

Lift/span (lb/ft)

Matrix for k+1 point (dimensionless)

Mass flow (slugs/sec)

Mass flow, m/(NBr2prUr) (dimensionless)

Mass flow/area slugs/(ft sec)

Universal mass flow parameter, raw/(pwlJ*') (dimensionless)

Mass flow/area, m/(prUr) slugs/(ftz sec)

Mach number, U/C (dimensionless)

152

— —

N. Strearawise Mach number (dimensionless)

mIK

M

n

n

Nß

NR

P

+ P

PR

PRT

q

Q

Q+

f qk

I

Element of M matrix

Boundary condition matrix (dimensionless)

Normal coordinate (dimensionless)

Normal coordinate, n/(rrvr) (dimensionless)

Number of struts (dimensionless)

Reynolds number, rrprUr/ur (dimensionless)

Pressure (lb/ft )

1 dp Universal pressure gradient parameter, —^T ...^ ^jr (dimensionless)

Prandtl number, (—y^ ) (dimensionless)

Prandtl number turbulent, (-^—) (dimensionless)

Heat flux, -X ^ »/fr* sec)

Average inlet dynamic pressure (lb/ft2)

Heat flux, q/(prUrCpTr) (dimensionless)

Universal heat flux, q/qw (dimensionless)

Universal heat flux (adiabatic), q/(pwU*3) (dimensionless)

Column matrix (dimensionless)

Split column matrix (dimensionless)

Radius (ft)

Recovery factor, Eq. (3.2.35) (dimensionless)

Radius, r/rr (dimensionless)

2 2 Gas constant ft /(sec deg R)

153

IJ

s

St

t

T

T+

rriK

'is

OB

u

a*

Us

Un

V*

U

Uß

Radius in z plane Eq. (3.7.2) (dimensionless)

Radial coordinate (r,z) plane (dimensionless)

Element of Rk matrix

Matrix for kth point (dimensionless)

Strearawise coordinate (dimensionless)

Streamwise coordinate, s/(rrVr) (dimensionless)

Stanton number (dimensionless)

Blade thickness (ft)

Temperature (deg R)

Universal temperature, CpT/lf2 (dimensionless)

Column matrix for k point (dimensionless)

Strearawise velocity (ft/sec)

Normal velocity (ft/sec)

Tangential velocity (ft/sec)

Magnitude of velocity (ft/sec)

Blade velocity (ft/sec)

Friction velocity, V^/Py (ft/sec)

Strearawise velocity, Us/Ur (dimensionless)

Normal velocity, Un/Ur (dimensionless)

Tangential velocity, Ity/Ur (dimensionless)

Magnitude of velocity, u/Ur (dimensionless)

Blade velocity, UB/UJ. (dimensionless)

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v*

v

V

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w

W+

y

y

Y

x

Y

Y+

z

Z

Z

ZB

Z

zk

Universal velocity, U/U* (dimensionless)

Friction velocity, U*/Ur (dimensionless)

Metric scale coefficient (dimensionless)

Metric scale coefficient, v/vr (dimensionless)

Volume (ft3)

Complex variable w plane (dimensionless)

Stream function inner layer (dimensionless)

Distance along streamline (ft)

Distance along streamline, x/rr (dimensionless)

Real part of Z (dimensionless

Imaginary part of z (dimensionless)

Distance normal to wall (ft)

Distance normal to wall, y/rr (dimensionless)

Real part of dw/dz (dimensionless)

Imaginary part of dw/dz. (dimensionless)

Universal distance from wall, Yp^U*/^ (dimensionless)

Complex variable - I plane (dimensionless)

Axial distance (ft)

Axial distance, z/rr (dimensionless)

Loss coefficient (dimensionless)

Axial coordinate (r,z) plane (dimensionless)

Column operator matrix (dimensionless)

155

■ ■ -i ii i i iiihaif ■—iaMiii ■ ^.^ .... .„■„ ^ -.:■ ■-—-.. ...... -^

a

aI

V

6

A^

Ens

En0

e

Tl

Tl

H

e

•

i

I

K

X

Swirl angle to axis (deg)

Chord angle to axis (deg)

Angle Schwartz-Christoffel transformation (dimensionless)

Ratio of specific heats, Cp/Cv (dimensionless)

Boundary layer thickness (ft)

Displacement thickness (ft)

Boundary layer thickness, 6/rr (dimensionless)

Displacement thickness, 6*/rr (dimensionless)

Streamwise strain, rrens/Ur (dimensionless)

Tangential strain, rren0/Ur (dimensionless)

Small angle in z plane

Transformed normal coordinate (dimensionless)

Imaginary part of w (dimensionless)

Blade/force area, rr.f/(pru^) (dimensionless)

Angle of streamline to axis (deg)

Momentum thickness (ft)

Temperature ratio, T/Tr (dimensionless)

Momentum thickness, e*/Tr (dimensionless)

Entropy, (l-Ir)/* (dimensionless)

Von Karman constant (O.Ul)

Thermal conductivity lb/(sec deg R)

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n

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P

Ps>Pn

P

ps'pn

a

sn,3

Tns

T +

fk

0

0c

*B

*B

Viscosity slugs/Cft sec)

Plade force/span, //(ryPpU^) (dimensionless)

Real part of w (dimensionless)

3.1^159

Pressure ratio, p/pr (dimensionless)

Density (slugs/ft^)

Radius of curvature (ft)

Density ratio, p/pr (dimensionless)

Radius of curvature (dimensionless)

Solidity, b/gB (dimensionless)

Streanwise stress, Tns/(prUj) (dimensionless)

Tangential stress, Tn^/(prl)j) (dimensionless)

Strearawise stress (ib/ft^)

Tangential stress (lb/ft )

Stress, T/VW (dimensionless)

Column matrix for boundary conditions (dimensionless)

Tangential coordinate (radians)

Chamber angle (deg)

Angle in z plane Eq. (3.7.3) (dimensionless)

Blade dissipation function lb/(sec ft )

Blade dissipation function (dimensionless)

157

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X

X

Y

Clauser constant (0,0l6) (dimensionless)

Normal coordinate transform, dTl/dn (dimensionless)

Stream function (slugs/ft)

Stream function (dimensionless)

Matrix Operators

T Transpose

-1 Inverse

Superscripts

v Iteration number

_ Mean or average quantity

A Variables for blade force calculation

i Deviation from mean quantity

Subscripts

0 Stagnation conditions

1 Inlet conditions

2 Upstream of strut

3 Downstream of strut

A Adiabatic

E Effective turbulent

H

I

M

Hub conditions

Incompressible conditions

Midspan conditions

158

.»— ~ -■■ ■- "^ -:-- --

r

T

W

00

Reference conditions*

Tip conditions

Wall conditions

Free stream or edge of boundary layer

^Reference conditions are based on standard sea level atmosphere conditions for all thermodynamic quantities. The reference radius, rr, is the inlet outer radius, and the velocity is the mean inlet velocity.

159 10419-74

— -——■ ---"-