Department of Aerospace Engineering and Applied Mechanico---------University of Cincinnati NASA-CR-168003 L

19830003775

THREE DIMENSIONAL FLOH COMPUTATIONS IN A TURBINE SCROLL

BY

A. HAMED AND C. ABOU GHANTOUS

Supported by:

/ • '.-, 11 10- 7 j\'';') li ~

L1Sfl",I?Y, [,I'S(,

liAI1jPTON, Vlra,INIA

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

Lewis Research Center

Grant No. NAG3-S2

\ \\\\\\\\ \\\\ \\\\ \\\\\ \\\\\ \\\\\ \\\\\ \\\\ \\\\ NF01868

https://ntrs.nasa.gov/search.jsp?R=19830003775 2019-04-29T12:05:39+00:00Z

3 117601318 5260

NASA CR 168003

THREE DIMENSIONAL FLOW COMPUTATIONS IN

A TURBINE SCROLL

by

A. Hamed and C. Abou Ghantous

Department of Aerospace Engineering and Applied Mechanics Universlty of Cinclnnati Cincinnati, Ohio 45221

Supported by:

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

Lewis Research Center

Grant No. NAG3-52

1 Report No 2 Government AccesSion 'lo 3 Reclp,ent's Catalog ~-40

CR 168003 I I

4 Title and Subtitle

I 5 Report Date I

THREE DIMENSIONA~ FLOI\' COMPl:T.;TIONS I:: August 1982 I I

A TURBINE SCROLL I

6 Performing Organization Cede I I

i 7 Auttlor(s) 8 Performing Organization Report No

I

A. Hamed and C. Abou Ghantous

I 10 Work Unit No

9 Performing OrganIZation Name and Address I I

DEPT. OF AEROSPACE ENGINEERING & APPLIED :'IECHANICS UNIVERSITY OF CINCIN:1ATI 11 Contract or Grant No ! CINCINNATI, OHIO 45221 NAG3-52

I

13 Type of Report and Period CevPfed

12 Sponsoring Agency Name and Address i NATIONAL AERONAUTICS AND SPACE ADMINISTRATION I

LEWIS RESEARCH CENTER 14 Sponsoring Agency Code I

CLEVELAND, OHIO 44135 I

15 Supplementary Notn , , i i I

: 16 Abstract I

The present analys~s kS developed to calculate the compresskble three I I

dkmenskonal knvksckd flow kn the scroll and vaneless nozzle of radkal knflow I

turbknes. A Fortran computer program whkch was developed for the nurnerkcal I solutkon of thks complex flow fkeld uSkng the fk~kte element nethod kS I

presented.

The program knput consksts of the mass flow rate and stagnatkon cond~tkons at the scroll knlet and of the fknkte eleroent dkscretkzatkon parameters and nodal coordknates. The output kncludes the pressure, Mach number and velockt¥ magn~tude and dkrect~~n at all the nodal pOknts.

The report kncludes the computer program, the analysks, a descrkpt~on of the solut~on procedure, and a nurnerkcal example.

I

I. I I ,

17 Key Words (Suggested by Auttlor(sl) 18 Distribution Statement

Unclassk£~ed - unl~~ted.

STAR CATEGORY 02

I I

I 19 Securtty Oasslf (of ttl,S report) 20 Security Classlf (of this page) 21 No of Pagn 22 Price .

I UNCLASSIFIED UNCLASSIFIED 52

• For sale by the National Technical Information Service Springfield Virginia 22161

~ASA-C·168 (Rev 10-iS)

r

TABLE OF CONTENTS

SUMMARY • • • . . . . . . . . . . . . . . . INTRODUCTION . . . . . . . . . . . . . . . . . MATHEMATICAL ANALYSIS • · . . . . . . . · . INPUT • • . . . . . . . OUTPUT . . . . . . . . . . . PROGRAM DESCRIPTION . . . . . . . . . . · . . . . . . . PROGRAM LISTING • . . . . . . . . · . . . . . . . · . . . . NUMERICAL EXAMPLE . . . . · . . · . . · . REFERENCES . . . . · . . . . . . . TABLES AND FIGURES. • . . . . . . . . . . . . . . . . . APPENDIX A - DEVIATION OF THE ELEMENT EQUATION . . . . . . APPENDIX B - THE MATCHING CONDITIONS AT THE SPITTING

SURFACE • • • • • • • • • • • • • • • • • . . APPENDIX C - NOMENCLATURE • . . . .

iii

Page

1

2

2

5

7

8

12

33

35

36

47

50

52

Figure

1

2

3

4

5

6

7

LIST OF FIGURES

Scroll Schematic Show1ng the Coordinates and the Splitting Surfaces ..••.•.•.•.

Structure of the Program

Nodal Placement on the Scroll Surface ABC

Nodal Placement in Scroll Cross Section .

Velocity Potential Contours on Scroll Surface AED • . . • • • . • . . .

Contours of Computed Circumferent1al Velocity Component • • • • • • • . • . . . • . •

Measured Circumferential Velocity Component (Re f . 5) • • • • • • . • • • • • • • • •

iv

40

41

42

43

44

45

46

SUMMARY

The present analysis is developed to calculate the com­

pressible three dimensional inviscid flow in the scroll and

vaneless nozzle of radial inflow turbines. A Fortran computer

program which was developed for the numerical solution of this

complex flow field using the finite element method is presented.

The program input consists of the mass flow rate and

stagnation conditions at the scroll inlet and of the finlte

element discretization parameters and nodal coordinates. The

output includes the pressure, Mach number and velocity magnltude

and direction at all the nodal points.

The report includes the computer program, the analysis,

a description of the solution procedure, and a numerical

example.

I

INTRODUCTION

Existing quasi-three-d~mens~onal turbomach~nery flow field

solut~ons are appl~cable to radial ~nflow turb~ne rotors but

not to ~ts scroll. Most analyt~cal studies of the flow ~n the

scroll [1, 2] are based on the assumption of one dimensional

flow and the conservat~on of mass and angular momentum. A

two dimens~onal study of the flow field ~n the scroll and

nozzles was reported in reference [3]. Another two dimensional

study of the velocity components in the cross-sectional planes

of the scroll was reported in reference [4]. These two

analytical investigations provided insight into the influence

of the scroll and nozzle vane geometries on the flow field

but together do not constitute a quasi-three-dimensional

solution. Experimental measurements of the velocity components

in the radial turbine scroll [5] indicate that actually

the three dimensional scroll flow fields are very complex.

This work presents a method for the analysis and a

code for computing the compressible inviscid three dimensional

flow field in the scroll and vaneless nozzle of a radial

inflow turbine. The finite element method is used in the

numerical solution in order to model the complex geometrical

shape of the flow boundaries.

MATHEMATICAL ANALYSIS

The three basic steps ~nvolved in the finite element method

are the domain disretization, obtaining the ~ntegral statement

of the problem and the appropriate choice of the local approxi­

mation in each element [6, 7]. In this study, the three

dimensional flow field is formulated in terms of the potential

function and Galerkin's method is used to obtain the integral

statement.

For steady flow, the equatlon of conservation of mass is

expressed in terms of the potential function ~ as follows:

2

'i/. (p'i/<f» = a

The unknown velocity potential solution is approximated in

each element by

(1)

(2)

where {<f>}e is the column vector of the potential function

values <f>i' at the nodes of the element e, and [N] 1S the row

vector of the interpolation functions N .. J.

Galerkin's method requires the residual of the substitution

of equation (2) into equation (1) to be orthogonal to the

interpolation functions

J N. 'i/. (p'i/<f» dV = a v 1 ( 3)

Applying Green's theorem to equation (3) leads to the following

relation which involves the boundary conditions:

I 'i/N.· (p'i/<f» dV = I N. p'i/<f>·dA V J. A J.

(4 )

The integrals in the above equations are evaluated over the

volume V and surface A of all the solution domain. The right­

hand side of equat10n (4) represents the weighted average of

the mass flow across the solution domain boundaries. The solid

boundaries contribut1on to equation (3) is equal to zero because

of the no-flux condition.

Equation (4) holds for the entire Solut1on domain and also

for an arb1trary finite element e. The Subs~ltution of equation

(2) into equat10n (4) gives the following element equation:

Ie 'i/Ni • (p'i/[N] {<f>}e)dV = I Ni p ~~ dA V Ae

Equation (5) 1S nonlinear for compressible flow where the

density is given by:

3

(5 )

y-l P = Po [1 + ~

2 -l/y-l ('V<jl) ] RTo

( 6)

The numerical solution to equations (5) and (6) 1S obtained

iteratively by allowing the density, p, in equation (5) to lag

one iteration behind the potential flow field solution. The

assembly of all the linearized element equations in the flow

field results in a set of linear algebraic equat10ns which can

be expressed as follows:

[K] {<jl} = {p} (7)

The Gauss elimination method is used in the solution of the

linearized equations.

The matrix of coefficients is updated at each iteration

but the load vector remains unchanged, as it is determined by

the mass flow rate in the scroll. Appendix A gives a detailed

derivation of the element equations and of the coefficients

of the stiffness matrix.

The Splitt1ng Surfaces

The cho1ce of the potential funct10n as a dependent var1able

is generally suitable for three dimensional flow f1eld solut10ns

in simply connected domains. However the domain in the scroll

problem is multi-connected leading to a mathematically non­

unique solution for the potential funct1on. In the present

analysis, two c01ncident surfaces are introduced in the solution

domain to make it simply connected. The splitt1ng surfaces

extend fT-om the scroll lip, which is the end of the metal between

the 1nlet section and the last scroll section, to the vaneless

nozzle exit station as shown in Fig. 1. In the flow field

discretization, two sets of nodes are placed on the opposite

sides of the splitt1ng surfaces and a discontinuity 1n the velocity

potential field is allowed across. Add1tional matching conditions

are therefore requ1red to insure the continu1ty in the velocity

field across these boundaries as described in Appendix B.

4

INPUT

The input data to the program is read 1n subrout1ne INPUT.

It cons1sts of the geometrical data and the performance para­

meters for a given scroll. Any consistent set of units can be

used for the input variables. Other input parameters are related

to the numerical solution procedures and to produce the desired

output. Following is a description of the input data in

alphabetical order.

BANDWD

GAMA

EPSILN

INDATA

ITERMX

Jl(I)

LBC (I) ,MBC ( I) , NBC (I)

MSRATE

MSI

MS2

MS3

An integer equal to the band width of the matrix

of coeff~c~ents.

Specif1c heat rat~o.

A small number describing the tolerance which is

used to determine the convergence of the numerical

solution based on the normalized dens1ty change.

Integer to indicate if it is desired to pr1nt out

the input data. INDATA = 0 will not produce a

printout of the input data while INDATA = 1 will

produce a printout of the input data.

The number of iterations not to be exceeded in

the solution. Incompressible flow solution is

obtained if ITERMX = 1.

An array of the global number of the first node

of every cross sect10n I

Arrays of the nodal points of the surface element I.

The mass flow rate in the scroll (kg/sec or slg/sec) .

The number of surface elements at the inlet station.

Total number of surface elements at inlet section

and the first splitting surface.

Total number of surface elements at inlet section

and the two splitting boundaries.

5

MSTOT

MTOT

MV(I)

NANZ

NC

NRNZ

NODE(I,J)

NTOT

NU (I)

R

STAGP

STAGT

THETA (I)

X(I) ,Y(I), Z(I)

Total number of surface elements at inlet, ex~t

and splitting boundaries.

Total number of volume elements.

The global number of the volume element which con­

tains the splitting surface element I (I = MSl+l to MS3).

The number of nodal points in the axial direction

in the nozzle.

Total number of cross sections in the scroll and

inlet passage.

The number of nodal points in the radial direction

in the nozzle.

Array of the global number of the nodes,

I = 1,2,3,4, of the element J. J = 1, MTOT.

Total number of nodes.

An array of the number of nodes at every cross

section I.

Gas constant: (J,'°K or lb ft/slg OR).

stagnation pressure (newton/m2 or lb/ft3 ).

Stagnat~on temperature (OK or OR).

An array of the angles between every cross section

plane and the x-z plane measured from the posit~ve

x-direction.

Arrays of the Cartesian coordinates of all the

nodal points.

The reading formats of the input variables are illustrated

in Table 1.

Input Preparation

The solution region includes the inlet, the scroll and the

vaneless nozzle. The splitt~ng boundaries should extend from

the lip (the last metal part between the inlet and the last

scroll section) to the exit station of the vaneless nozzles.

6

It can have any shape that is consistent with the discretlzation

pattern. The discretization pattern and system topology is

arbitrary. The surface elements at the inlet, exit and the

splitting boundaries are inputted in one array. They are

numbered sequentially up to MSI on the inlet section, then up

to MS2, then MS3 on the two splitting boundary surfaces,

then up to MSTOT on the exit station. The volume elements

associated with these surface elements are inputted in the

array MV(I), I = 1, MSTOT. The angular coordinates THET(I) ,

of the cross sectional planes, including those at inlet, are

only used to evaluate the through flow velocity VT and the

radial velocity V from the computed velocity components in r

Cartesian coordinates after the convergence of the numerical

solution.

It is up to the user to select the number of cross sections

and the nodal points per cross section. The band wldth is

equal to the maximum difference between the global number of

any two nodes of the same element. The system topology can be

specified in such a manner as to minimize the band width of

the matrix of coefficients in the numerical solution. For the

purpose of organizing the printout of the output, the nodes

in each section N(J) are numbered sequentially starting with

JI(J), J = 1,NC.

OUTPUT

A sample output is given in Table 2 for the scroll of

the numerlcal example. The first set of output consists of a

printout of the input parameters. The second set of output

consists of a listing of the potentlal function, the through

flow velocity, the cross velocities, the total velocity, the

Mach number, the pressure and density normalized by their inlet

stagnation values. The computed flow data are listed for

every section starting at inlet so that they can be easily

interpreted.

7

Error Messages

The followlng error condltions lead to program termlnatlon.

1. ELEMENT NU~mER XXX HAS A ZERO VOLUME (LESS OR EQUAL TO XXX) CALCULATION IS STOPPED.

The message is printed in subroutine MATVEC dur~ng the first

iteration as the element volumes are evaluated in ascending

element order. Check the coordinates of NODE (I, ELEMENT

NUMBER XXX) for error.

2. WARNING: SINGULARITY IN ROW XXX CALCULATION IS STOPPED.

This message is printed in subroutine GAUSS if the diagonal

element of the matr~x of coefficient is less than 10-5 . The

row number is printed to be used for the identification of the

corresponding nodal point. Check the topology of the elements

associated with this node in the discret~zation pattern.

PROGRAM DESCRIPTIO"t-T

The main program is designed to obtain the potential flow

solution in a radial inflow turbine scroll using the fin~te

element technique. The program is bu~lt from four major

subroutines: MATVEC, SPLIT, GAUSS and PHYSIC, two ed~t sub­

routines: INPUT and OUTPUT, and ~our supplementary subroutines:

VOLELM, NUMBER, AR and COFACT which are called inside the

major subroutines at various steps for side calculations.

The structure of the program is shown on Fig. 2.

Descriotions of the Subroutines

I. Major subroutines:

MATVEC: The stiffness matrix elements, K .. , and the load ~J

vector, P., are calculated in this subroutine. In addition, J

the volume of the elements are calculated and stored in the

array VOL (I) so that it can be used ~n other subrout~nes and

8

\

in subsequent iterations. The coefficients of the element stiff­

ness matrlx, K .. are calculated uS1ng Eq. (A.8). Subroutine COFACT 1J

is called to calculate the coefficients ak

: bk

and ck

(Eq. A.7).

The load vector components P. are calculated uSlng Eqs. (A.IO). J

SPLIT: The matching conditions across the splitting surfaces,

as explained in Appendix B are implemented in this subroutine.

GAUSS: The solution for the potential function at the

nodal points (I) is obtained in thlS subroutlne using the Gauss

elimination method. The matrix of coefficient is bounded, but

nonsymrnetrlc due to the application of the matchlng condition at

the splittlng boundary. Since the solution to the potential

functl0n is unique within an arbitrary constant, the magnitude

of the potential function at the first node is arbitrarily

fixed at -1.

PHYSIC: The physical properties of the fluid are calculated

in this subroutine which is called after the nodal potentials

are determined. The properties calculated at each node are:

the velocity components as defined by Eq. (A. ), the Mach

number and the normalized pressure and density. The difference

between the element density of two subsequent iterations are

also evaluated in this subroutine to be used in the convergence

criteria.

II. Edit subroutines:

INPUT: Called first in SOLVE to read and write the input

data. It also initializes the normalized density in each

element to unity.

OUTPUT: After the convergence of the solution, the velocity

potential and flow properties are prlnted at all the nodes. The

prlntout is organized so that the data of each cross section

is identlfled and prlnted separately.

9

III. Side calculations subroutines:

NUMBER is called by SPLIT to rearrange, at the element level,

the nodes of the elements with surfaces on splitting boundary,

such that the first three nodes of an element are on the splitting

boundary. The nodes global numbers are then stored 1n the

array NSB(I,J), where I is the number of the surface element

on the splitting boundary and J = 1,2,3,4 are the global numbers

of the three nodes on the sp11tting boundaries followed by the

fourth internal node~

VOLELM is a subroutine with double indices Nand K. The

first index N refers to the element, and the second is a flag.

If the index N is equal to 2, the coefficients of the interpolation

functions are calculated according to equation (A.7). If it 1S

equal to 1, a similar procedure is followed for the volume

elements wh1ch have three nodes on the splitt1ng surface whose

nodes are locally renumbered and ident1fied by the number of the

surface element. If the second index is equal to zero, the

volume of the element is also calculated according to equation (A.9).

The volume calculations are on11 performed during the f1rst

iteration, then stored in VOL (I) for subsequent use in later

iterations.

IV. Dictionary of common block variables:

A(I,J)

ART (I)

B(I,J)

E(I)

JI(J)

Global matrix of coefficients for the potential

function at the nodes.

Area of surface element I (at inlet, exit and/or

splitting boundary).

The matrix Be of the element nodal coordinates.

The contribution of mass flow rate through a splitting

boundary surface element, to the opposite surface

element equation, 1n the global matrix of coefficients.

The global number for the first node in a scroll

cross section J.

10

LBC(I) ,MBC(I), The nodal number for the three nodes of a NBC (I)

LOAD(I)

MV(I)

NU(J)

NODE(I,K)

NTL(I)

NSB (I ,J)

PRTIAL(I,J)

VOL (I)

X(I), Y(I), Z(I)

surface element.

An array containing the load vector.

The global number of a volume element which con­

tains the sp11tt1ng boundary surface element I.

Total number of nodes in the scroll cross

section J.

The global number for the node of the volume

element K, for the tetrahedral volume element

I = 1,2,3,4.

An array containing the nodes on a splitting

boundary arranged in descending order.

The global number of the nodes I of the element J

with 3 nodes on the splitt1ng surface.

Part1al derivatives of the 1nterpolation functions

NJ (J = 1,2,3,4) in CarteS1an coordinates,

a a a ax for I = 1, ay for I = 2, az for I = 3).

Volume of an element I.

Coordinates of a node I.

11

PROGRAM LISTING

12

f-' W

AN IV Gl ReLEASE 2.0 MAIN DATE -= 82239 04/25/19

C C ("

:..

c C C r C

C

C

C

C C C

C C r C

C r C c c

•

INTEGER BANDWD REAL MSRATE.MSFLUX.LOAD.MACH2 COMMON/ASEMOL/U(4,4).VOL{3720),NODE(4.3720).A(9hO.406).NTL(155).

PRTIAL('3.4).ART(155).LBC( lfiS).Mt1C( 155) .NBCC 155).

· , X(960) ,Y(960) ,Z(960) .LDAD(Q60) .AAtWWD.E( IS5'.'-1V( 155) NSB( 4.155) t (l.1S1. MS2. 1>15-'1. MSTOT. NTDT. JUMP 1. JUMP~. NANZ.

• NRNZ.POT~N(960)

•

COt1MON/PHY SI /GAMA. R. ST AG T. ERI~OR, VELO (3 ) COMMOH/INOUT/lNDATA.~TOT.MSRATE.EPSILN.ITER.ITERMX.NC.STAGRO,

Jl(60).NU(60).THETA(60).ROELEM(3720),NN

--» WARNING; THE DIMENSION ,OF A{I.J) AS IN COMMON SHOULD AT LEAST 8E J=(AANDWD-l)*2.

CALL UNDFL W

CALL .INPlH [TER = 0,

NN == BANDWD*2 - 1 DO 80 I-=l.NTOT

801 LOAO( {) = 0.0

10

00

(TER = ITER + 1 DO 90 I=l.NTOT DCI 90 J==l.NN A(I.J) :: 0.0

BUILD UP THE 5TIFNESS MATRIX AND THE LOAD VECTUI'.

( ALL f\' A T V E ("

HHfWDuce THe SPLITTtNG SlJFACE MATCHHJG CONDI fIONS.

(,\Ll SPL IT

srT THE rInST DIAGONl\L TERM IN STIFNESS f~ATqIX AND ITS Cu/(f{ESPDNDINNG VALUE IN LOAD VFCTOR EQUAL TO UNll Y.

" -.: OAN[)WD + 1 r = AflS(A( 1,flMlOWD») DO 1 60 I =K • NN

I-'

"'"

~N IV Gl RELEASE 2.0 MAIN DATE:. = 82:?J9 04/25/19

c

160 ACid) = 1.OE-15*A(t.I)/F A(I.8ANDWO) :: A( 1.BANDwD)/F LOAD(l' = LOAO(I)/ABS(LOAO(l»

C IftPOSE ZERO TANGE'NTIAL VELOCITY AT INLET SECTION. C

C C C C

C C C

C. c r:

C c C c: C

C

NODIN :: NU(I) DD 170 I =2. NODIN L = K - I A( I.L) = A( I ,L) + 1.0E+15

170 A( I.8ANL>WD) = -A( I,B~NDWD} - 1.0E+15

SEAK A SOLUTION FOR VELOCITY POTENTIALS; THE INLET SECTION IS AN EQUIPOTENTIAL OF MAGNITUDE UNITY.

CALL GAUSS

lALl.ULATE THE PHYSICAL PPOPFRTIES OF rHE FLUID.

CALL PHYSIC

tEST THE CONVERGENCE AND ITERATE.

IF (ERROR.GT.EPSILN .AND. ITER.LT.ITERMX) GO TO 10

PHINTOUT THE CALCULATED PHVSICAL ROPERTIES IN lttf- CURRENT JTEr~ATI:JN AT ALL NODES: VELOCI TV PUTENTIALS. VELUCITY COMPUNENTS, MACH NUMBER NOR~ALIZED PRESSURE AND DFNSITY.

CALL OUTPUT

~, TOP END

...... U1

\N IV Gl R["LEASE 2.0 MATVEC DATE = 822~9 04/25/19

C C C C

C r. c

c c c

c.

e

•

SUSROUTI Ng ,MATVEC

IHIS SUBROUT[NE B,UILOS UP THE STIFNESS MATRL( AtH> THE LOAD VECTOR I

INTEGER BANDWO HE AL MSRATE, '''SFLUX ,LOAD, MACH2' COMMON/ASEMBL/B(4,4,.VOL(3720),NOOC(4,3720),A{9hO,406) .NTL(155).

PRTIAL(3.4),ART(155),LUC(155),~OC(155).NBC(155), X(960).V(960),Z(960},LOAO(960),OANDWO.E(155),MV{155)

•• NSB(4.155).MSl.MS2.MS3,MSTOT,NTOT,JUMPl.JUMP2.NANl, • NRNZ.POTEN(Q60) COMMON/'NOUT/INDATA.MTOT.MSR~TE.EPSILN.ITER,ITERMX,NC,STAGRO,

• Jl(60).NU(~O).THETA(60),ROELE~{3720).NN

1 rOR~AT (lHl,20(/),25X,'ELEMENT NUMOCR',J15,/.25X, • 'HAS A ZERO VOLUME (LESS OR EQUAL TO',lPFt2.4,

')',I,l5X,'CALCULATION [S STOPED.')

MI\RY = 0 IF (ITER .NE. 1) MARY = 2

00 1?0 ~ 1= 1, MTOT CALL VOLELM{MARV,II) IF (VOL(II) .GT. 1.0E-12) GO TO 100 \tj fJ I Tf? ( (, • 1) I I ,v OL ( I I ) ':;TOP

C CALCIJLATE THE CDEFICIENTS OF X. y, f:, Z IN SHI\PE FUNCTIONS AND THE' e ~A~TIAL OFRIVATIVES OF THE SHAPE FUNCTION':;. C .

c

1 () () I) 0 1 1 0 I = 1 ,4 [)£) 1 10 J= 1 , 3

110 PRfIAL(J,I) = COF~CT(l.J+l.H) C l:ALCULATt:: THE ELE'MENT STIFNESS MATfHX 'ELf;." "NO RUlLO UP THE C <,1.0I1AL STIFNCSS MATRIX A(K,{ ). C

F = ROELEM( I I )/(VOL( I I )*3b) no 120 1=1,4 DO 120 J=l,4 LLE.M:d (prHIAL{I.I)*PRTIAL(l,J''''

PI< T I AL ( 2. I 1 *p '< riAL ( 2, J ) + pIn I AL{ "3. J ) *PP T I AL (3. J) ) *ft

..... m

AN IV Gl RELEASE 2.0 MATVEC DATE:;: 82'>39

K :;: NODE ( I, I I ) L :;: NODE (J, I I' - NODE ( I, I I) .. BAt-jDWD

llO A(K,L) ~ ACK,L) + ELEM IF C ITER .GT. 1) GO TO 170

C C BUILD UP THE LOAD VECTOR LOAD(I). C

C

C

c

c

c

C

AREAIN :;: 0.0 AI-<EAXT = 0.0

DO 130 1=1.M51 AR Tf I) :;: AR C I )

130 AREAIN :;: AREAIN + ART(I)

M5 ::::; M53 + 1 DO 140 f=MS.MSTOT AIH ( I) :;: AR ( I )

140 AR~AXT :;: AREAXT .. ART([)

M5FLUX :;: MSRATE/(AREAIN*STAGRO) DO 150 I=l,MSI P = - MSFLUX*ART(I)/3 LOADCLBC(I» :;: LOADCLOCCI» + P LOAD{I-1BC{I» = LOAD(I\1~"C(I» + p

1~0 LOAD~NBC(I) :;: LOAD(NACCI) + P

MSFlUX = MSRATE/(AREAXT*STAGRO) DO 160 I=MS,M5TOT p :;: M~FLUX*ART(I)/3 LDAI)(LAC (I» :;: LOAD(LftC( I» .. P L0AO(MBC(I» = LOAD(MOC({» + P

160 LUAD(NBC{I» :;: LOACtNBC([» .. P

1 -, 0 £)0 1 80 [= 1 • N TOT I I BOP lH E:. N ( I' = LOA D ( I )

r~ FTIJRN Ff..JD

04/25/19

...... -.J

AN IV Gl RELFASE 2.0 SPLIT DATE = 82239 04/25/19

SUHROUTINE SPLIT C C . THIS SUBROUTINE MODIFIES THE STIFNESS MATWIX BY INCLUDING THE C SPLITING BOUNDARY CONTRIBUTION. C C

c c c C C

C

C C C

C C C

C C (

C C C C C

INTEGER BANDWD REAL MSRATE,MSFLUX,LOAD.MACH2 COMMON/ASEMBL/O(4.4),VOL(3720).NODE(4,3720),A(960,406),NTL(lS5"

• PRTIAL(3.4),ART(155),LAC(155),MAC(155).NBCll55), • X(960),Y(960),Z(960),LOAO(960).UANDWD.E( ~55),MV(155) ., NSB(4,155},MS1,MS2,MS3,MSTOr,NTOT,JUMP1,JUMP2,NANZ, • NHNZ.POTEN(960)

• COMMON/INOUT/INDATA,MTOT.MSRATE,EPSILN,ITER,ITER~X,NC,ST~GRO,

Jl(60),NU(60),THETA(60'.ROELEM(1720),NN

IF tITER • EQ. 1) CALL NU,MOER J

5 T ART THE /IIIA'l N LOOP OVER THE VOLUME (OR SURFACF) ELEMENT.

r:.o If" = MS1 - MS2 ""5 = MSl + 1 M':;S :; MS2 KLM = MS

10 on cO II=MS,MSS

F:JlJlltD UP THE P1ATRIX B( 1',J).

CALL VOLELM(I,II)

C~LCULATE THE PARTIAL DERIVATIVES Of SHAPE FUNCTI[lNS

DO :20 1=1.4 00 20 J=l,]

20 PRTJAL(J.I) ~ COFACT«(,J+l.B)

C'LCULATE THE AREA OF T~E SURFACE ELEMENT II.

A~FA = Sm<T(pp.TIAL(1,4)**2 ... PRTIAL(2,4)**2' t PRTIAL(3,4)**2)/2 IF (II.NE.KLM .OR. ITER.NE.l)' GO TO 30

EVALUATE THE COEFFICIf:NTS OF THE PLANE' EQUATION "*X + A*V ... (*l. ... 0 = 0 AND HiE DIRECTIONAL COSINES OF THE' ClJ,'v\PO,'IIENTS OF THE NOR~AL VECTOR.

AX = Pr--!TIAL(1,4)

I-' 00

!,.\N IV Gl RELEASE 2 .. 0, I SPLIT DATE = 82239 04/25/19

,I

, I

I,

r l ,

, ,

C-

c C', C C

C C C

c

c C C C

C

I

BY :;: PRTIAL(2,4) CZ ~ PRTJAL(3.4)

" o ~ B(1.2)*(B(3.3)*B(2~.) - B(2.3)*B(3.4») +

B(1,J}.(B(3,41*B(2,2) ,- B(2.4)*B(3.2» + • ',B( 1.4)*~B(3.2)~B(2.3~ -' B(2~2)*B(3.3»' ,

DELl A = AREA*2 ' I ' IF (O.NE.O.O .ANO. O*OELTA~GT.O.O) IF (D.EO.O.O .~NO. CZ.NE.O~O .AND. J,F (O.EO.O.O .ANO., CZ.EO.O.O .AND. COSINA :;: AX/DELTA . COSINB :;: BY/DELTA' I

COSINC,~ CZ/OELfA, , ;

DELTA = -OELTA CZ*DELTA.LT.O.ot DELTA = RY*OELTA.LT.O.O) \OELT~ =

, , . , £VALUATE' THE COEFFICIENTS E(I. OFlTHE NODAL'POTENTlALS IN A BOUNOARY ELEMENT ,tIe ' ,

r ... t ' I

30,F = ROELEM(MV(II)}/(VO~'MV(ll»)*16) 00 40 1=1.4 ' E(t) :;: AREA*tCOSINA*PRTIAL(l.l) + COSINS*PRTIAL(2.I) +

" C OS INC * P R T I AL ( 3 • t U * F , 40 IF (MSS .EQ.: MS3) E( I) ~ -E (t)

INSERT THE ~URFACE INTEGRAL IN STIFNESS MATRIX. , ,

I J ;: II - 'MD IF ,DO 50 J=I.4 '00 50 1=1,3

K :;: NSEH I. I I ) L = NSR(J. 1 {) NSB(1,[J) + BANOWO

50 A(K.L) ;: A(K.L) + EeJ) 60 CONTINUE

MDIF = -MDIF MS = MS2 + 1 MSS == MS3 IF (MD~~ .GT •. O) GO TO 10

INCLUDE TENGENTIAL VELOCITY IN STIFNESS MATnlX TO INSURE CONTINUITY THROUGH SPLITING BOUNDARIES.

Jll"'P :: JUMPl ' K 1 :;: NTL ( 1 ) K2 = Kl - JUI-IP MSD == (MSl - MS2)*3 MONA = Kl + 1 -(NRNl-l)*NANZ

DO 80 1=2, MSD

-PELlA, -DEL TA

.:

'I

..

.'

I-' ~

nN IV Gl R~LEASE 2.0 SPLIT DATE::: 82?3Y 04/25/19

t f r

c

IF (NTLlI) .EO. NTL(I-l») GO TO 80 IF { NTL(I) .GE. MONA) GO TO 70 RrTURN

65 K2 :::'Kl -JUMP2 JUMP ;:: JlJMP2

70 K3 = NTL( I) K4 == K3 - JUMP KT == BANDWD - K3 L ::: Kl ... KT A(~3,L)'::: A(K3.L) +'1.OEt-l5 L ::: K2 ... , KT A ( K 3 • L) ,::: A ( K 3. L) - l. OE +- 1 5 l ::: K4 + K T 'I A ( K3 • L ) .;:: A ( K3 , L.:) +- 1 .0 E +15 A(K3.BANOWD) = A(K3.BANDWO) - I.OEt-l5

BO c.:mHINUE

6(,6 FOHt4AT (lOX,.A' •• 13.·.·.13.·)='.E14.5.l0X.'E(·.I-3.·) :::1,E14.5) 777 fORf"'AT (lOE12.4)

RETURN END

AN IV Gl ~ELEASE 2.0 NUMBER DATE = 82239 04/25/19

SUHROUTINE NUMBER C C THIS SOlJBROUTINE RENUMBERS THE NODES AND THE ELEMENTS ON C SPLITING OOUNDARI~S SO THAT NODES 1. 2 AND ~ AR~ ON SPLITTING C SURFACES. AND RE~RRANGES THE NODES ON ONE SPLITTING SU~FACE r

c r kF~~RANGE THE NODES IN A DECREASING VALUES ARRAY C

r

00 40 1= 1.:3 K = K + 1

40 NTL(K} = NSA{ 1 • .1)

I1Sf) = (1)1<) ,- 1'-152) *3 tJl <)I) 1 = M <)0 - 1

N I-'

'N IV Gl RELEASE ~.o NUMOER

c

c

K == 1

DO 70 L == 1 • M 50 1 fl\Nfl == tHL (L ) DO 6 0 ~-1=L. MSO IF (NTL(M) - NTL(L» 60.60.50

~>O I< == M NTL(L) = NTL(M)

&0 CONJINUE NTL(K) == MNP 1<' = L + 1

70 CONT INUE

RETURN END

DATE == 82239 04/25/19

IV IV

~N IV Gl ~ELEASE 2.0 GAUSS DATE = 82239 04/25/19

•

c c c c c c c ('

c

C C

c: C C

• •

sueRoUTIN~ GAUSS

SOLU1ION OF LINEAR SYSTEMS OF EQUATIONS AY THE GAUSS ELIMENATION METHOD, FqR NON SYMETRIC BANDED SYSTEM~.

A ; CONTAINS THE S~STEM MATRIX, STORED ACCORDING TO THE NON SYMMETRIC GANGED MATRIX

paTEN: CONTAINS THE UNKNOWN POTENTIALS ArTER SOLUTION.

• INTEGER BANDWD REAL MSRATE,MSFLUX,LOAD,MACH2 COMMON/ASEMBL/B(4.4),VOL(J720),NODE(4.3720),A{9hO,406),NTL(155),

PRTIAL(3,4).ART(155),LBC(155),MBC(155},NBC(155), X(960),Y(960),ZC960).LOAO(Y60).AANDWD.EC155),MV(155)

• • NSO(4,155),MSl,MS2,MS3,MSTOT.NTOT.JUMP1.JUMP?,NANZ, • NRNZ,POTEN(9~0) I

1 FORMAT (lHl, 5(/'. 10(lH=),·»» WAHNING :',/, • 15X, 'SINGULARITY IN ROW·.I~,/, • 15X. 'CALCULATION IS STOPED.')

NM 1 = N10T - 1 DO 60 K= 1, NM 1 KPI -= K + 1 IF(AAS(A(K.AANDWO» .LT •• 00001) GO TO 80

C C DIVIDE ROW BY DIAGONAL COEFFICIENT ('

NI = KPI + HANDWD 2 l = M I NO (N I • NTOT) DO 30 J=KPl,L ' K2 = dANDWD + J - K

30 A(K,K2' = A(K.K2)/A(K,BAN~WD) POTFN(K) = POTlN(K)/A(K,BANDWD)

C C ELIMINATE UNKNOWN FlACK) FROM ROW 1 ('

40 5()

DO 50 I=KP1.L ~l = nANOWD + K - I 00 40 J=KP1,L K2 = BANDWD + J - I Y3 = 8AN~WD + J - K f\ ( 1 , K 2) = A ( I , K 2 ) POTEN{ I) -= POTFN( I)

A( I,Kl)*A(K.K3) - A~I,Kl)*POTEN(K)

I'

N W

~tl IV C,1 RELEASE 2.0 GAUSS DATE;:: 82239

60 (ONTINUE C \ COMPUTf LAST UNKNOWN. C

c

IF(AHS(A(NTOT,BANDWD}) .LT. 0.0000001) GO TO 80 POT~N{NTOT) ;:: POTEN(NTOT)/A(NTOT.HANDWO)

04/25/19

C APPLY AACKSUASTITUTION PROCesS 10 COMPUTE REMAINING UNKNOWNS. C

c

c

on 70 I=l.NMl K ;:: NTOT - I KPI = K + 1 NJ = KPI + BANOWO - 2 L ~ MINO(NI,NTOT} DO 70 J=KPl.L K2 = HANOWD + J - K

70 POTEN(K) = POTEN(K) - A(K.K2)*POTEN(J)

GO TO 90 80 WRllE(6tl) K

STOP

qO HETURN FND

tv

"""

I

\N tv Gl R~LEASE 2.0 PHYSIC DATE = 02239 04/25/19

'c C C C C

C C C

C

t c c c c c

c

r

c

c::

SIJHROUTINf" p'HYSIC

, THIS SUBROUTINE CALCULATES THE PHVSICAL PROPFRTIFS OF THE FLUID AT ALL NODES: VELOCITY. VELOCITY COMPONENTS. MACH NUMBER, AND NORMALIZED DENSITIES.

I

'I NTEGFR BANDWD , f<EAL MSI~ATE. MSfLUX ,LOAD. MACH2 \ COMMON/ASEMBL/B(4.4).VOL(3720).NOOE(4.3720).A(960,406)~NTL(15~). .' PRTIAL(3.4),ART(155).LBC(155),~nC(155).NBC(155). • X(960),YC960).Z(960).LOAD{960),UhND~O.E(155).MV(155) • , N S 8 ( 4 , 1 55 ) ,M S 1 • M S 2 • MS 3 , M 5 TOT. N Tor. J LJ I H~ 1 • J U M P 2 , NAN Z , • NR~Z.POTEN(960)

•

COMMON/PHYSI/GAMA.R:STAGT,ERROR,VELO(3) COMMON/INOUT/INDATA.MTOT,MSRATE,EPSILN,ITER.ITERMX.NC.STAGRO.

Jl(60),NU(60),THETA(60) ,ROELEM(3720).NN

FXPO =1.0/(GAMA - 1.0) SONIC = GAMA*R*STAGT ERROR = 0.0 .

00 10 I=l t NTOT DO 10 J= 1 t 4

10 A(l,J' = 0.0

UEF TNE I HE B MATR (X AND CALC ULATE THE VELOC I TV C.Ot-1PONENTS VELOel) (1=1.2.3 FOR X.Y.Z) IN VOLU~E ELEMENT [I: THE POTENTIALS M.:E NOW IN ARRAY POTEN(J). IN THE SM-1E LOOP, CALCULATE THE ELEMENT DENSITY AND TH~ ERROR FOR CONVERGENCE TEST.

UrJ 50 II=I,' .. TOT

CALL VOLELM(2.II) V :: VOl. ( I I ) * 6

1)0 20 1=1.3 20 VELO(I) = 0.0

DO 30 1= 1.4 on 30 J=1.3

10 VELO(J) = COFACT(I.J+l.B'*PO~EN(NODE(I.II»/V • VELO(J)

VSQREO .::: VELO(l):l'*2 + VELO(2)**2 + VELO(3)**? U~NS = (1.0 + VSQRED/(2*SONIC*EXPO - VSaREn»*~(-EXPO) ERROR = ERROR t «ROELe~(II) - DEN~)/ROELEM(II».*2 ROELLM([I) = DENS .

N U1

AN IV Gl RE-LEASE 2.0 PHYSIC DA Tf- = 8223<) 04/25/19

C

C C C C C

C

C C

C C c r: c

C

00 40 1=1,4 , A(NOOE(I.II).l) = A{NODE(I.II),l) + 1.0 DO 40 J=2.4

40 A{NOOE(I,II).Ja = A(NODE(I,II"J) + VELO(J-l) 50 CONTINUE

60

90

IF (ERROR.GT.EPSILN .AND. ITER.LT.ITEQMX) RETUHN

CALCULATE THE VELOCITY A(I,5), THE V~LUCITY COMPONENTS A«(.J) (J=2,3,4 FOR X.Y.l), MACH NUMBER A(I,6), NORMALIZED PRESSURE A(I.7), NORMALIZED DENSITY A(I.B)

DO 90 1=1. NTOT DO bO J=2.4 ~ ,.l\{I.J) = i\(I,J)/A(I,l)

VSQRED = A(I.2,**2 + A([.J)~*2 + A(I~4'~*2 A( 1,5) = SORT(VSOREO) MACH2 = VSORED/(SONIC - VSORED/(2*EXPO» A( 1.6) = SORT(MACH2) A(I,H) : (1.0 • MACH2/(2*EXPO»**{-EXPO) A( 1,7' = A( I ,B)**r.Ar>1A CONT I :'~UE

CRlrlCAL VELOCITY V~CI{If=5QRT(2*STAGT/(EXPO*(GAMA+1.0'» Pl~3.1416

CALCULATE THE RADIAL ANU THE TANGENTIAL VELOCITIES TA~GENTIAL VELOCITY IS STORED IN A{I,2) ANO TilE RADIAL veLOCITY ST~ORO IN A(I.~)

1)0 100 l(=l,NC JK-=J 1{ K) JK1=Jl(K)+NU{K)-1 DO 100 I=JK,JKl VELOXy=sa~T(A(I.2)**2+A(1,3)**2) PHI::;:I\TAN(A( I .3)/A( [,2» f> S [= THE T A ( K) -PH I 1 F {I\ ( I • 2 ) *A ( I • J ) .l T. 0.0) PS I =p I-P.S I A ( 1 • 3 ) = - ( A ( I • 3 ) / A H S ( A ( I • 3 ) ) ) * v F:L (l X Y * COS ( PSI )

100 A ( r • 2 ) =- ( 1\ ( I .2' .I A 135 ( A ( I .2) )) * VF LO X Y * SIN 0') 5 J)

Pt--fIJR1\l f:" Nf)

N en

AN IV Gl RELEASE 2.0 COFAC T. DATE = 8~239 04/2'5/19

\ '

L ___ _

FUNCTION COFACT(I.J.B) C C FUNCTION COFACT CALCULATES THE COFACTORS OF THE MATRIX B((.J' C WHERE ~(I.l)=l AND J IS ANY REAL NUMBER C

C

c C C

C C C

C ('

C

c

DIMENSION 8(4,4). C{4.4)

DO 1 /.1=1.4 00 1 K= 1,4 C(K.,.,' = B(K.f.U IF (I .EO. 4) GO TO 3

, R~OUCC 4*4 MATRIX TO 3*4 MATRIX

DO 2 M= 1 .4 DO 2 K= I ,3

2 C ( K • M) = C (' ( K .. 1 ) • M ) 3 IF (J • EQ. 4) GO TO 5

nEDUCE 3*4 ~ATRIX T~ 3*3 MATRIX

DO 4 M=J,l 004K=1.3

4 C(K,M) = C(K,(M+l»

~VALUATE THE COFACTOR

, OETERM = C(1,3)*(C(3.2) • C(2.3)4«C(l.2)

C(3.3)*(C(2.2) L = (-l)**(I+J) COFAcr = OEr~RM ~ L

r,ETUF?N END

I

<::(2.2) + - C(3.2» +

C(1.2»

N --.J

AN I V G 1 Rt:L c/\ St: ?o VOLELM DATE = d2~39 04/25/19

SU8ROUTIN~ VOLELM(N,K) c C THIS SUAROUTINE BUILDS UP THE Mf\TRIX 8(I.J) ~H[m:. 8(1,1) = 1.0 C N = 0, CALCUATES THE VOLUME VOL(K) nF AN FLE~ENT K. C N = 1, UlJlL UP MATRIX B FOR AN 'ELEMENT \tilTH SPLITING BOUNDARY C N = 2, BUILD UP MATRIX B FOR A VOLUME ELEME~T C

c c

C

c

c

• •

INTEGER BANDWD RtAL MSRATE,MSFLUX,LOAO,MACH2 COMMON/ASEMBL/B(4,4),VOL(3720),NODE(4,3720),A(Y60,406),NTL( 155).

PRT(AL(3,4),ART(155),L8C(155),MAC(1~5),NBC(155), X(960),Y(960),Z(960).LOAD(960).~ANDWD,E( 155).~V(155) · , NSB(4,155) ,MS1,MS2,HS~.MSTOT.NTOTtJUMPltJUMP2.NANZ. NRNl,POTEN(960) •

IF {N .NE. 1) GO TO 2 DO 1 1=1.4' 13(1.1) = 1.0 (HI.2) = X(NS13({,K» A ( I .3) = Y ( NS f:j ( I , K ) )

1 B ( I • 4) ,= l ( N SB ( I • K) ) RETURN

2 DO 3 [=1,4 R(I,I) = 1.0 13 ( I ,2) :: X (NODE ( I. K) ) 13 ( I t "3) = Y ( NODE ( I • K) )

.3 B ( I t f~) = Z ( NODE ( I, K ) ) [ F (N • E Q. 2) R E TV R N

V 0L { f( ) = 0 • 0 DO 4 I;::.: 1 t 4 VDLL :: COFACT(I.2,U)*R(!,2)

4 VOL(K) = VOL(K) + VOLL VOL(K) = ABS(VOL(K»/6

RETURN E'\If)

N 00

~N IV Gl qEL~ASE 2.0 AI~ DATE = 82239 04/25/19

t I ;\

C C C C C

C C

C

• •

FUNC T ICJN" AR ( ( )

THIS FUNCTION CALCULATES THE AREA OF A TRIAN(jLE" IN CARTE'SIAN COORDINATES. (=NUMDER OF ~URFACE ELEMENT U~UER CONSIDERATION.

INTEGER BANDWO/ REAL ~SRATE.MSFLUX.LOAD.MACH2 COMMON/ASEMBL/B(4.4).VOL(3720).NODE(4.37~O).A(9GO.406).NTL(155).

PRTIAL(3.4).ART(155),LBC(155).MAC(15~).NBC(155). X(960),Y(960),Z(960),LOAD(960).AANOWD.E(155).MV{155)

• • NSA(4,155) .MS1.MS2.MS~,MSTOT,NTOT.JUMP1.JUMP2.NANZ, NRNZ.?OTEN(960) •

Al = Y(LBC( I) ,:O:(Z(r.H:)C( (}) - Z(NDC( I») + • Y(MHC(I»*(Z(NBCCI» - ZCLBCCI») + • Y(NRC(I)):t«Z(LAC(I)) - ZP"'BC(I»)

A2 = X(LBC(I)*(7(MBC(I)} - Z(W3C(I)) + • X ( MFlC ( I ) ) * ( Z (NBC ( [ ) ) - Z(LBC(I») + • X(NDC(I»*(Z(LB("(I» - ZU,IBC(I»)

A3 = X(LHC(I»¥(Y{MBC(I» - Y(NAC(I») + • X(MOC«(»*(Y(NBC(I» - Y(LAC(I}) + • XC NBC (I) ) *( Y(LBC( I» - Y(MBCCI»)

AR = SQRT(Al**2 + A2**2 + A3**2)/2

RETURN END

N \!)

AN I V G 1 I~ELE.ACJE 2.0 INPUT Oil. TE = 822 -V) Ot~/25/1 'J

SURPOUTINE INPUT e C THIS SUOHOUTINE READS AND WRITI::.S TilE INPUT DATA. C

C

c c c

INTEGER l3ANOWD f~EAL MSRATE,MSFLUX,LOAD,MACH2 ' COMMONJASEMAL/O(4.4).VOL(3720),NOOE(4,'720),A(9bO,406),NTL(155),

PRTI AL( 3. 4), ART( 155) .LRC( 15tl), MUe (155) ,NBC (155) , X (960) • Y (960) • Z ( 9fiO ) • LOAD ( 960 ) • BANO~vD. E ( 155) , MV ( 155) · , NSB(4.155) ,M51 ,M52.MS3.MSTOT,NTOT,JUMPl.JUMP2 .. NA.NZ.

• NRNZ.POTEN(960) , C Oi.1r1 ON/I '''OUT / I NOAT A, ,.,T OT. MSR ATE. f"PS ILN. ITER, I TERMX, NC. 'iT AGRO •

• J1(60),NU(60).THETA(60).wOELEM(~720),NN COHMON/PHYSI/GAMA,R.STAGT.ERROR,VELO(,)

1 r or~4A T 2 For!MAT 3 F- ORr-1AT 8 FOIHtAT

( 6F 1 0 • 0 , 21 5 ) ( 1415) (lP5E14.5) (lHl ,////.50X,· [ N PUT D A T A' ./,

• 5CX,· _______ ~ ________ T __ "////) 9 FORMAT (33X,'TOTAL NUMBER OF ELEMENTS MTDT): ',16,/,

• • • •

• • .. • • • •

3~X, 'lOTAL NUMBER OF NODES (NTOT):·. 16.///. 33X,' (MSI :'.t!i,/. 33X,' {MS2 :',15,/, 3~X, 'SURFACE ELEMENTS AT BOUNDARIes < M$] :' ,15,/. 33X,' ( M~TOT:'.I5./. 33X,' (NOOIN:·,I5,/. 33X.· < JUMP2:' ,15.///,

33X.'BANDWIDTH OF STIFNESS r-.1ATRIX (FlAtlf)/JO) :',15,//, 3'X.·M~XIMUM NUMBER OF ITERATIONS (IT~PMX) :',15,//. 33X,'TOTAL NUtlBEI? OF SFCTION5 UK) :',15,//, 31X,'SPECIFIC HEI\T RATIO (GJ\~1f1) :',Fll.5,//. 33X,'CONVERGENCE CPITFRION (EPSILN) :',F12.6,//,

33X.'STAGNATION TEMPERATUP£; (STAGT) :·.Fll.5,· R'.//, 33X,'STAGNATI0N PRE~SUHE (STAGP) :·.Fll.S,' PSF'.//, 33X, 'IIJEAL GAS crJSTANT (R):' ,FII.S,' FTc!/S2.H·

,//.33X,'MASS PATE THROUGH SYSTEM (M5RATr) :·,Fll.~,'SLG/S')

RFJ\O(5,l) HFA[)(,.2) f?rAD(S,2) f.'F-AD(5.2) M = MS I + fl E" A 0 ( '5 • 2 ) f.<rf\O(5.?) !-IFA!)(C:;,?)

MSkATE,R.GflMA.STAGT.<;lAGP,E'SILN.INf>ATA.lTERMX MTOT.NfOT,NSl,MS2,MS3,MSTOT,NANZ,NHN/,NC.OANOWD ( L HC ( I ) t Moe ( I ) • NBC ( I ) • I = 1 • t-A C; r OT ) ( N tl 0 (: ( 1 ,J) ,N CJD E ( 2 • J) • N n [) F. ( ~ • J) • N () [) E (II • J ) • J = 1 • 1-1 TOT ) 1 ( M V ( I ) • I = M • 1-\ 5 '"i ) ( J 1 ( I ) , I ="1 ,NC ) (NU(I).I=l.NC)

w o

Ar~ IV Gl RELEA~E: 1'.0 INPUT DATE = 8?219 04/25/19

C ('

C ('

c

c c C C C

('

READ(S,3) (THETA([).I=l.NC) HFAD(5,3) (X( [), I=I,NTOT) HEAD(S.3) (Y(I),I=l,NTOT) RF~U(5t3) (Z{I).I=l.NTOT)

INITIALIZE NormALIzED ELEMENT DENSITIES TO fiE UtHTV AND CALCULAT[ THf STAGNATION DENSITY. , ,

DO !,)O I=I.MTOT ' 50 ROrlE~(I) = 1.0

..

STAGRO = STAGP/(R*STAGT'

JUMPl = NANZ*NRNZ JUMP2 = L8C(MS2+1) - LBC(MS1+1' IF (BANDWD .L~. JUMP2) BANDWD = JUMP2 IF (INDATA .EO. 0) RETURN

WRITE IN~UT DATA IF FLAG INDATA=l

WRITE(6.8) WPllE(6,~) MTOTfNTOT.MS1.MS2.MS3.MSTOT.NU(1'.JUMP2.0AN~WD.ITFR~X.

NC.GAMA.EPSILN,STAGT.S1AGP.R.~SRAT~

129 I~FTURN FNO

W I--'

AN I V G 1 r~ELfASE '!.O OUTPUT DATE = 82239 04/25/19

SU[JROIJTINE OUTPUT c C TliIS SUBROUTINE PRINTS OUT THE CALCULATEO PHYSICAL PROPERTIES C AT THE END OF EACH ITERATION. (:

C C C

c c

c

("

INTEGER AANDWD REAL MSRATE.MSFLUX.LOAD.MACH2 C[)11"ION/ ASEMOL/B( '" 4). VOLe 3720) ,NODE ( <\,3720' ,A ( 960,406) • NTL ( 155) •

PRTIAL(3.4).ART(155).LUC(155),~OC(155).NBC(155), • X(960).Y(960).Z(960).LOAD(960),qANDWD.E(155)~MV(155) ., NSB(4.155).MS1.MS2.MS3.MSTOT,NTOT.JUMP1.JUMP2.NANZ. • NRNZ, PO,TEN (960 j

, COMMON/INOUT/INDATA.MTOT.MSRATE,EPSILN.ITER.ITERMX,NC,STAGRO. • JU60).NlJ(60).THETA(60) ,fWELEM(37?O) ,NN COMMON/PHYSI/GAMA,R,STAGT.~RROR.VFLO(3)

1 FOH4AT (1415) 2 FORMAT (IHl,//.7X.'# NODES POTENTIALS VElOCITY-T VEL' , •

• OCITY-P VELOCITY-Z VELOCITY, MACH NUM. NORM. PRE~.· • • ' NOPM. DENS. './)

3 FOR~AT (I9.17.IP8E14.S) 4 FORMAT (lP5E14.5) 5 FOR4AT (~X.·END OF SECTION'.13) 6 FORr1AT (8(/).48X.'E N D 0 F PRO G R A ~',/,48X,27(IH ).

• ////,33X,1CALCULATIONS ENDED :'. • //.33X.'NI,JMAERS OF ITERATIONS PEIWOR~1ED : ITER =' .13. • //,33X.·CQV~HGENCE REACHED : ERROR =',lPEB.l)

'lJRlTEC6,f» ITER, ERROR .J = 0 L = 5.3 DO q K=l.N(" JK = Jl{K) NK ~ Jl(K) + NUCK) -1

00 8 l=JK,NK J = J + 1 1 = L + 1 IF (L - 53) U.S.7

-, L = 1 V,RI1EU),2)

:1 \"f<lTf:(~'.]) J,I,POTEN( Il. (A( I .M). M=?d3) 'A'I?lfl:(6.<) K

9 rOtH HWE

W IV

~N IV Gl RELEASe 2.0

c c: c c c

C

WRITE(7.1) WRITE(7d) WRITE{7,4) WRITE(7,4) WRITE(7.4) lAIR IT E (7,4' wR IT E ( 7 • I~ ) WR IT E ( 7.4)

, WRITF(7,4) 1IJR IT E ( 7 • II ) wRITE(7.4,

HETURN I::ND

...

OUTPUT

(Jl(I), l=l.NC) (NU(J). l==l.NC' ( X ( I), 1::: 1 • NT aT ) (Y(I), I=l.NTOT) (Z(!).l=l,NTOT) (POTEN(I).I=l.NTOT) ( A ( I .6" 1=1. N rOT) (A(I.7), l=l.NTOT, ( A ( I .2 ). 1==1. NT OT ) (AII.3).I==1.NTOT) (A(I.4),I=1.NTOT)1

DATE == 82239 04/25/19

!

NUMERICAL EXAMPLE

A numerical example is presented to illustrate the use of

the program and show the results of the computations. The

flow ~s ~nvestigated ~n a rad~al inflow turbine for which the

experimental measurements were reported in reference [8]. The

mass flow rate ~s one lb/sec (0.454 kg/sec) and the scroll

~nlet section diameter is 4.88 inches (1.92 cm). The number

of nodal points used in the discretization pattern was 956

placed on 24 scroll cross sections as shown in Figs. 3 and 4.

The number of tetrahedral elements result~ng from this dis­

cretization pattern was 2,716. The printed output from this

example is given in Table 2. The first part of the output

consists of the performance parameters and other input data.

The second part of the output represents the results of the

computations and consists of the velocity potent~al, the velo­

city components, the Mach number, the normalized density and

pressure. The pressure and density are normalized with respect

to their inlet stagnat~on values. The output data from the

computed results in the various scroll cross sections. The

contours of the computed velocity pot:ent~als are shown in

Fig. 5. The jump in the value of the velocity potential across

the splitting surfaces was found to be equal to 150 as can be

seen in Fig. 5. The same figure illustrates the same slope

for the potential contours on the opposite sides of the spl~tting

surface and hence the continuat~on of velocity pressure and

density fields. The contours of the computed velocity component

in the circumferential direction are shown in Fig. 6. The

measured [5] circumferent~al veloc~ty components are shown in

Fig. 7. In both Figs. 6 and 7, the values of the velocity

componet are normalized with respect to the ~nlet velocity.

The disagreement between the computed and measured results

is mainly in the boundary layer region over the outer scroll

surfaces. One can also observe some secondary flow development

in the measured results. However, both boundary layer and

secondary flow effects decrease with increased velocit~es.

33

The agreement between the computed and measured results near

the scroll neck, in spite of the very low velocities in the

case presented, with an inlet velocity of 41 m/sec, leads to

the conclusion that the velocity and pressure fields in this

region are not very sens~tive to these effects. The flow is

greatly accelerated in th~s region and ~t also acquires a

radial velocity component as evidenced by the shape and density

of the potential lines in the vaneless nozzle portion in

Fig. 5.

34

REFERENCES

1. Bhinder, F.S., "Investigation of Flow ln the Nozzleless Spiral casing of a Radial Inward-Flow Gas Turblne," Proc. Insn. Mech. Engrs., Vol. 184, Part 3G(11), 1969-1970, pp. 66-71.

2. Wallace, F.J., Baines, N.C. and Whitfield, A., "A Unlfied Approach to the One-Dimensional Analysls and Deslgn of Radlal and Mixed Flow Turbines," ASME paper No. 76-GT-lOO, 1976.

3. Hamed, A., Baskharone, E. and Tabakoff, W., "A Flow Study in Radial Inflow Turblne Scroll-Nozzle Assembly," ASME Journal of Fluids Engineering, Vol. 100, March 1978, pp. 31-36.

4. Hamed, A., Abdallah, S. and Tabakof f, IV., "Radial Turbine Scroll Flow," AIAA Paper No. 77-714, presented to the AIAA 10th Fluid and plasma Dynamics Conference, Albuquerque, NM, June 27-29, 1977.

- ~ 5. Tabakoff, W., Vittal, B.V.R. and Wood, B., "Three Dimensional Flow Measurements ln a Turbine Scroll," NASA CR 167920 Report, July 1982.

6. Shen, S., "Finlte Element Methods in Fluld Mechanics," Annual Review of Fluid Mechanics, Vol. 9, 1977, pp. 421-445.

7. Huebner, K., The Finlte Element Method for Engineers, John Wiley and Sons, 1975.

8. Tabakoff, W., Vittal, B.V.R. and Wood, B., "Three Dimenslonal Flow Measurements in a Turblne Scroll," NASA Report, August 1982.

35

W 0'1

Type of Input Data

Performance, Printout and Convergence Parameters

Discretization Parameters

Ifl Topology of Permanent Surface Elements (Inlet, Splltting, and Exit)

Topology of Volume Elements

volume Elements Affected by Splitting Surfaces

Cross-Sectional Data

Nodal Coordinates

TABLE 1

INPUT DATA FORMAT

Input Variables

MSRATE, R, GAMA, STAGT, STAGP, EPSILON, INDATA, ITERMX

MTOT, NTOT, MS1, MS2, MS3, MSTOT, NANZ, NRNZ, NC, BANDWD

(LBC(I), ~lliC(I), NBC(I), I=l, MSTOT)

(NODE(l,J), NODE(2,J), NODE(3,J), NODE(4,J), J=l, MTOT)

(MV(I), I = MS1+l, MS3)

(Jl(I), I = 1,NC) (NU(I), I = 1,NC) (THETA(I), I = 1,Ne)

(X(I), I = 1, NTOT) (Y(I), I = 1, NTOT) (Z(I), I = 1, NTOT)

Format

6F10.O, 215

l415

1415

l415

l415

1415 l415 lP5E14.5

IP5E14.5 " "

I N PUT OAT A

TOTAL ~UM8ER OF ELEMENTS ~MTOT} 3716 TOTAL ~UMOER OF NCDES (NTrT): 9SA

( MSI 48 ( "1S 2 64

SUqFACE ELE~ENTS AT BOUNDARIES < ~S3 : 80 (I-1STOT: \52 (NOf")IN: ~5 ( JU 'v1P2: 19

~ANDWIDTH OF STIFNESS I~ATR 1 X (8A~,ID\·jO ) 203

MAXIMUM NU\18£R OF ITERATIONS (ITEP:4 X) 5

TOTAL N.U'1SER OF SECTIONS (NC) 24

SPECIFIC HEAT RATIO (GAI.1A) 1.40000

CONVERGENCE CRITERION ( EP S I L'J) 0.000100

STAGNATION TEMPERATUR E (ST AGT) 530.00000

STAGNATION PRESSURE (STAG?) 4819.67::::;69

IDEAL GAS CNSTANT (R) 1716.26001

R

oSF

FT2/S2.R

MASS RATE THROUGH SYSTEM (MSRAT~) . O • .o3105SLG/S .

END o F PRDG:~AM

CALCULATIONS ENDED :

NUMBERS OF ITERATIONS PERFCRMED

CDVERGENCE PEACHED

ITEq = 2

: ERROR = 5.3E-06

TABLE 2: OUTPUT DATA, FIRST SET.

37

" tHIlH 5 ... lTrrn I "L', VFllK 1 TY-T VELOCITY-II VELOCITY-/ VnOCITY MilCH NUM. NURM. PliES. NOIlM. DENS.

-I.OOOOOI"HII) 4.146'>lf+Ol -~. 77710E-0 I -? ?fj2Inr-- 0 I \. I 4':>6r.E+ 01 3.675'16E-02 9.Q,)0'>6E-OI 9.99326E-OI ;' , . -1.11(10ll0r tlln 4.14117LtOI -?1?lh7[,"-01 -2. 1 t2rl5t:=-fll 4.1412f1C+Ol J.('70~9E-02 9.9Q059["-01 9.99326E-OI J ! -1.nOOOOI"+111! 4.1151>4f+01 - 2. bQf,2f1L-0 I -I. PIYIl4 c -1l1 ~.I '514,.01 J.66538E-02 9.9<)063['-01 9.'193301:'-01 4 4 -I. nooo.)! tOO 401'1<:;2l+01 -?0151'£..-01 -2. c)1!C;71f"-O! 4 • I 3 I 62E + 0 I 3.66172E-02 C). Q'J06 lE..-OI 9.99130F-01 r; f, -I. oo"nol ton I • • 1 ;> ci 1 IF + 0 I -1.1l61nIF-OI 1. 7'\ 7 .. '<lF-O;- 4.12J2'lC+OI 3.65871E-02 Q. '}<JOGnF-O 1 9.9933JE-OI

" to -I .oooonr +1)0 4.1;> ',ll',!: to I - 3. <10')9 IF-O I 2.;>1<;401:-01 4.12(,0910+01 J.65(,fl?["-02 9.Q<)066E-Ol <).'193331::-01 7 1 -1.Of'OOOf+OIl '1.1:'41'I't'>1 -5.J0160E-OI 4.21""OE-<)1 4. 124 7SI! + 0 1 3.65561E-02 9.99066F-Ol 9.99333["-01 II /I -1.I!Olloor+oo '" 13.?7 fF + 0 1 -I. 150<,l"E-01 -'J.164;>9F-O:? 4.13211E+OI 3.66;>74["-0;> Q.Q9063E-Ol 9.99310E-Ol 9 " -I.OO()OOE-+oo ~.1;:71 r +01 -1.l0271E-01 -1.013<'3E-OI 4.12131':+01 3.65792E-02 9.99066[-01 l) .<)9313E-0 1

1O 10 -I.OOUI)'H ino 4. I I q;"'lFto I -9.124'14E-02 -R.761t:IOE-02 4.1193IE+Ol 3.1)5081["-02 9.99069E-Ol 9.9931'1["-01 I I 1 I - I. OOOOllf +00 4.11294[,+01 -1.0::1<;39F-OI -S .'5f,71l1 F-O? 4.ll""lliE+Ol 3.64<;18E-02 9.99073f-Ol <).99331E-Ol I~ 1;- -1.0001101 tOI! ". I OElfl:'L+O 1 -1.34274F-Ol -1.1511'110-1);> 4.108 ()IIE+ 01 3.64153E-02 9.9,)073E-Ol 9.'JQ1'"(E-01 11 I, -1.11000'11"."0 4.1 Ob6<Jf tOI -1.09111f-Ol 1.117591F-02 <\ • 1 0"> 7 JF + 0 1 1.63966E-02 9.99076E-Ol 9.99140E-01 14 1'4 -1.oooooriOO " • I lor. OF i 01 -1.6('~9IE-Ol i!.?1l40SC-01 4.110'1,1:+01 3.64303E-02 9.99073E-Ol 9.<)93311:-01 1 t~ I'" -I. nOOotH"tOO 4.111'13E+Ol -1.'14107r:-Ol -;>.5764')1:-07 4. 1 13 81l'" + 0 1 3.6459QF-02 9.9')013[-01 9.99337E-Ol I" If> -1.OOOOO(tOO 4.ll0",EtOI -1.18753r-Ol -(,.'Jb670e:'-O" ').11031EtOI 3.64289E-0? 9.99013E-01 9.99337E-Ol 11 1"( -1.OOODOl-tOO 4010347['+01 -1.22124F-01 -6. ')70'-\5f-0? ".10350E+OI 3.63679E-02 9.91)076E-Ol 9.99340E-Ol III 111 -1.OOOI)C)f tOO 4.096<;3(,+01 -1.19143E-Ol -7.6r.449=-02 4.096':>5E+01 3.630631:-02 9.99079E-Ol 9.9934210-01 I'I 19 -I. oooOOt +llO 4. 0002<;E t 01 -1.<\0290E-Ol -(>.OF,8'l7["-02 4.0902SE+Ol 3.62500E-02 9.9906.3["-01 9.9934',["-01 ,'I! 20 -1.OOOOOFiOO 4. OU!> !JLtO I -1.7!J4t:.1F-01 -5.?9000E-0" 4.0R,)37E+01 3.62012E-02 9.99086E-Ol 9.993471:-01 :'1 21 -1.00000F-fOO 4.0fl422E+Ol -1.40T'i4E-01 7.">7 /.9;>F-0'1 4.0842'>E+Ol 3.61913E-02 <I.99006E-OI 9.99347E-01 ;>:' ;>2 -1.00000LtOO 4 .OA7,)"[ tOI -3.105??E-Ol -2. (,41f10E-0; 4.0871')9F+Ol 3.62271E-02 9.990l\3E-01 9.991451:-01 ~I j 23 -1.001)001:+00 4.011559E+Ol -;>.673181:-01 -1. '??O~"-O:! 4.0056<\1':+01 3.620Q9E-02 ')'9901'6E-OI 9.99347E-01 ~4 24 -1.000001:.00 4.07829["+01 -.?43l47E-01 -3.0156,)["-02 4 .07831E+0 1 3.614<;IE-02 Q.99009E-Ol 9.99349E-OI ?r; 25 -I.OOOOOF.OO 4.07"10J!::4-01 -2.2216RE-Ol -~.736o?E-O? 4.07310E+Ol 3.609841:-02 9.99069E-Ol 9.99349E-01 ~ll .lCI -I.OOOOUI:+OO 4.0MI5,CfOI -I .991l69E-0 1 -1.72312E-0? 4.069591':+01 3.ti0565E-02 9.99093E-Ol 9.99352E-Ol .?f ~7 -1.00000FfOO 4.0b');>IE+Ol -1.98640E-OI -7.R940TC-07 tj .06521E+OI 3.60290E-02 9.99093E-Ol 9.9935~E-Ol ~a ,>8 -1.0000010+00 4.0(.;:10(,["+01 -1.4409JE-Ol 3.5205~E-05 4.06309E+Ol 3.60091["-02 9.99096E-Ol 9.99354E-Ol ;'9 :>9 -1.OOot}OftOO 4.070041':tOI 1. 77200E'-0 I -3.53218F-01 4.07103E"01 3.60801E-02 9.9906'lE-01 9.99349E-OI 10 JO -1.00000F+OO 4.0ti-J7\F:+Ol 1.29026E-04 -7.39111F-0? 4.06172E+OI 3.60153E-02 9.99093F-Ol 9 .,)93,)2r.:-0 1

31 "ll -1.00000rtOO 4.05';I44r:+Ol -1.03290E-04 -4. <;(,4091"-02 4.0554"E+Ol 3.59419E-02 <I.99099E-Ol 9.9<)357E-01 1:- ~,;? -I.OOOOOF.OO 4 .05079FtO 1 1.27186F-05, -1.2Q4;>RE-02 4.05079E+Ol 3.59007E-02 9.99099E-OI 9.99357E-Ol J:i lJ -1.00000r+oo 4.0500JE.OI -1.031 '521"-04 1.37916E-02 4.050031'+01 \ 3.50939E-02 Q.99099E-Ol 9.<)9351["-01 :14 .,. -I.ooooor +00 4.051641::+01 1.27?13E-OS 1. foOO06E-02 4.051611C+Ol 3.59062E-02 9.99099E-OI 9.<)9357F-Ol

W 15 35 -I.ooooor~oo 4.0<;270F+OI 1.28677E-04 7.34329E'-05 ".0,270E+Ol 3.59176E-02 Q.99099E-OI 9.993<;7E-OI EM) O' Sf (T ION I ex> Jl> 36 7.11l7b3EtOO 4 • o '."j '15£ +0 1 -3.96642F-02 - 4. Q4011l F-O I 4.099<,5£+01 3.63:101E-02 9.99079E-01 9.99342E-Ol

If, 37 7.1"'864F+f)0 4 .0795.31:+01 I.Oll59IE-OI -5.0.l9S61"-OI 4.07'l3'E+Ol 3.615831:-02 9.990661':-01 9.99341E-01 -'0 31l 7. ll261t"+OO 4.0536',EtOI 3.17901E-02 -?l\0113E-OI 4 .05']75E+0 1 3.59?69E-02 <).99099E-Ol 9.9Q357J=-01 39 .).) 7.16'06EtOO 4.0lll:!OEtOI -1.632R?E-02 -2.O'0487E-Ol 4.04125F+Ol 3.,)81tlIE-02 9.991031:-01 9.993'59E-01 40 40 7.1r,065F+OO ". 01YO'')E to 1 -1.l0O'3flE-01 -1.3(,0<171'"-03 4.039O'7E+Ol 3.57968E-02 9.9'1106E-Ol 9.99361£:.-01 1\ I '11 7.1<;<'01LtOO 4.Q4J40E+Ol -3.!:>5'J13E-Ol 1.41763C-Ol 4.04358E+Ol 3.59:167E-02 9.99103E-Ol 9.99359E-Ol 1\ ' 4! 1.139691:tOO II.Ob86IE+Ol -5.70'6751:-0'1 2o-1l441F=:-OI 4.069081:+01 3.60628E-0? 9.990Q3["-01 9.99352C-Ol 4 , 4l 7.15<)17,+00 4.132611:tOI -1.07626E-Ol -3.60404::'-01 ').13100E+01 3. 66295E-02 9.Q9063E-01 9.99330 E-O I V. 4~ 7.144201:.tOO 4.12"l46E+OI -fl.26!:1:HE-02 -4.190AOE-OI 4.12370E+Ol 3.65470E-02 9.<I'l066E-01 9.9933 'E-OI I,C; 4., 7.12620r tOO 4. 1041 CJE t 0 1 -J.4tl6HC-02 -1.2962<'E-01 4.10llJ21:+01 3.63752E-02 9.99076E'-01 9.99340E-Ol lit> 4f> 7.112891" +00 4.09.!4IE+OI -1J.1400?F-02 -? 4253~;r-O 1 4.09249E+Ol 3.6270:tE-02 9.99063E-Ol Q .Q'l34'iE-0 1 47 4 T 7.1 Q',I)6L +00 4.00363EtOl -1.700'J8F-Ol -1 .OIl0021':-0 1 4.0036~E+Ol 3.6Iq2~E-02 9.99066E-ol 9.99347E-01 I\A 411 1.1 OO')(,F tOO 4.07710E'+01 -3.26994E-01 -4.348\}(,E-02 4.07724E+Ol 3.61151E-02 9.99069E-OI 9.99349["-01 49 49 7.108oJOf!tOC) 4 .0675rIE+0 1 -<;.04404E-Ol 1.91217F-01 4.08794E+Ol 3.62300E-02 9.9'lOEl~E-01 9.9934') 1:-0 I '.0 '>0 7.1;:>3171"+00 4.1 l365E to 1 -?65654E-02 -4.C27A6[,"-01 0\ .17Jfl4E+Ol 3.69915E-0;> 9.99046E-01 9.993Ifl["-01 '.1 51 7.112lUC+01) 4. .14892C +a1 -1.12042E-Ol -2.613691':-01 4.14'l02E+Ol 3.6771'iE-02 9.99056E-OI 9.99326f-01 ~) :> 'i.! 7.091176[" .0'1) 4.11114,.>1"+ 1 -7.3009"1:-02 -3.14060E-0 I 4.13456E+Ol 3.66431E-02 9.99063E-OI 9.Q93~01:-01 ',1 ') l 7. 0844'J[ +00 4.11905EtOI -7. 8944<;E-02 -3.1659IF-OI II .1I916E+Ol 3.65069E .... 02 9.99069E-01 9.<)9335F-OI

OUTPUT DATA, SECOND SET

11 ,~qiH ., [', IT [til I AVi JLLUtJl(-T V(~LIIC: I TY-I( VClIl<. I r y- ~ VILIlCITY MilCH tIUM. NIlP~'. pr~rS. NI1HM. OEl~S •

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,END 01 SI CT (ON 2 71 71 I. !:i",'J2C .. 01 3.91?44r: .. OI 3.11l266CHlO -7.~tl70RL-()1 1.<),'610,+01 3.47953('-02 <J.9915f,F-()1 <J.Q9397F-01 7o!. 7:'.. 1.">10311-tOI '1.'l81<)OF"OI 3 • 3 f1 'l!) 0 f- + 0 () -fl. H'l I o!.'--O I 3 .!}'jf'l7~+0 I 3.4!>4'!'IE-02 9.99166E-OI 9.Cl94041:-01 /1 7 J 1.4'1"7 7r to I i.032<;OC+OI 3.2"'06,)["00 -4.109fOE-01 1.H"'555E+OI 3.4090IE-02 9.9<)189E-OI 9.9942IE-OI /'4 74 1.4'Ho;OLtOI J. '1I·.17EtOI 3.15r.<\~E+00 -'1.571'>9E-01 :J.fl2'!I,?EtOI 3.39240E-02 9.9919F,E-OI 9.99426F-OI

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W 1>11 Uli I. !>I 114 I3r ~() I 4.1406?Et-01 1.91fJllE+00 -9.450.11[-1)1 4.14611E+OI 3.674591:- 02 9.99056E-OI 9 • <)<;)32f>1:-0 I \.0 HI) 11<1 1.51."IIJL+0 I 't.116QI(f:tOI l.bC)504Et-OO -4.'I74?41::-01 4.1211)3E.OI 3.1)5287E-02 9.,}906<)E-OI 9.<)9335E-01

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FIG. I. SC~OLL SCHEMATIC SHOWING THE COO~DINATES AND THE SPLITTING SUqF~CES.

40

INPUT Read and Write Input Data

MATVEC Build up stiffness matrix and

load vector.

SPLIT Include splittlng boundary matchlng ~------~

conditions in stiffness matrix

GAUSS Solve for the unknowns ¢i using

Gauss elimination method

PHYSIC Calculate the physical properties

of the fluid

NO ----=:::

OUTPUT Print out solution and physical

properties of the fluld at all nodes

FIG. 2. STRUCTURE OF THE PROGRAM.

41

Slde Calculations

1 ~ ~ .,

2 " ~

It a

0 3 • e

0 • a • 0

0 0

4 o -. .. • 21 It

00 '13

Qt\ \ " 5 22 e -.. , I!)

" •

6 .0 •• •

., 7 .ee

• IS 00. "

" 23 It ~lta a

o " "

e "".,0 C!t " .,

13

0

a ., e

a

0

0 " 0

13

I!)

" ., 13 <2

" ., " ., " 10

It

~

I!)

a ~ ~

19 § "

0

§ a

13

" 0 " "

" " " " 13

" e ., " "

13

"

., o

o o

III e e

18

" tl'0

o tJf!i!J" "

16 ,,(!1'!P

" " " " o::)()" " ,,00015

., 13

.,,,~

12

FIG. 3. NODAL PLACEMENT ON THE SCROLL SURFACE ABC.

42

G G Q G

~Q Q ~

Q

Q ..) Q

Q ..)

Q G ..)

G Q ..) Q Q

Q ..) G

Q Q

..) Q Q

~G

000 0 0 0 " 0

0 Q

" " 0 .;)

" 0

0 " 0 .;)

0 0

.;) Q Q Q

Q Q Q

Q G Q Q Q Q

Q Q Q ~ G Q ~ Q Q

Q V Q Q

FIG. 4. NODAL PL~CEMENT IN SCROLL CROSS SECTION.

43

40 r-----.L

50 ~----l 130

90

FIG. 5. VELOCITY POTENTIAL CONTOURS ON SC~OLL SURFACE AED.

44

~ Ul

/

SECTION NO. 8 (8 = 40°)

,

1.1 1.2

1.3

r---... 1.4

SECTION NO. 12 (8 = 120°)

SECTION NO. 15 (8 = 180°)

FIG. 6. CONTOURS OF COMPUTED CIRCUMFERENTIAL VELOCITY COMPONENT.

46

II

o ~ r-l r-l

II

o LO

"'" II

U) I­Z W Z a 0-::E: a u >­I--U a --I w >---I c::t: -I-Z W c:::: w u... :E: =:l U c:::: -u o w c:::: =:l U)

LS :E

APPENDIX A

DERIVATION OF THE ELEMENT EQUATIONS

The s~mple four node linear tetrahedral element ~s used

in the present analysis. In this case the unknown velocity

potential field in an element ~s g~ven by the follow~ng

equation:

4 cpe = I

i=l cpo N. ~ ~

(A .1)

where cpo are the unknown velocity potent~als at the elements ~

four nodes, and

N. are the linear interpolation functions (i = 1,2,3,4) . ~

The element equation is given by equation (5) which is

rewritten below:

J VNi'(PV[N] {cp}e)dV

Ve

= J N P a¢ dA i an A

e

The above equation will be written in a different form as

follows:

The St~ffness Matrix

(A. 2)

(A. 3)

The coefficients K. of the element st~ffness matr~x can be ~J

shown [6, 7] to be g~ven by:

K .. ~J

aN. aN. aN aN. aN. aN. ( ~ --1 + ~ --2 + ~ --2) ve

p ax ax ay- ay az- az

An element matrix [B]e, is defined to contain the geometric

properties of the element as follows:

47

(A. 4)

1 xl Yl zl

1 x 2 Y2 z2 [B] e = 1

(A. 5) x3 Y3 z3

1 x 4 Y4 z4

In the following der1vations, it will be assumed that the

local number1ng of the volume element nodes is such that the

first three are numbered counterclockwise in ascending order

when viewed from the fourth node [7].

Using linear interpolation functions,

N. = a.x + b.y + c.Z + d. 1 1 111

it can be shown that the derivatives of the interpolation

functions are given by [7] ,

aN. 1 1 cofactor (B ) ax- = a. = 1 6ve x. 1

aN~ b 1 cofactor (B ) ay = = 1 6ve y. 1

aN. 1 1 cofactor (B ) az = c. = 1 6v

e z. 1

The element stiffness matrix is then given by

where

a.a. + b.b. + C.c. K .. = P ( 1 J 1 J ~ J)

1J 36ve

Ve = 1 det [B]e 6

48

(A.6)

(A. 7)

(A. 8)

(A. 9)

Equation (A.8) is used to bUlld up the global matrix of coeffi­

cients ln subrout1ne MATVEC.

The load vector surface integrals on the right hand slde

of equat10n (A.2) are only evaluated for the surfaces of the

element that lie on the external boundaries with mass flow

across. The contribut1on of the surface ele~ent to the load

vector corresponding to each of its 3 nodes is glven by:

where f is the mass flux into the element surface whose area e 1S equal to A. Equation (A.10) is used to build up the

components of the load vector in subroutine MATVEC.

The Flow Velocity

The velocity in the linear tetrahedral elements

can be expressed in terms of the derivative of the inter-

plat10n functions and velocity potential at element nodes

as follows:

4 aN. I 1 </>. a.</>. v = ax = x 1=1 1 1 1

4 aN = I 1 </>. = b 1 </> i v ay Y i=l 1

4 aN. = I 1 </>. = c </>. v z i=l az 1 1 1

where the repeated indices imply summation over i=l to 4.

49

(A.10)

(A.ll)

APPENDIX B

THE MATCHING CONDITIONS AT THE SPLITTING SURFACES

The splitting surfaces are introduced ~n the solution

domain in order to be able to handle the mathematically Mult~­

valued velocity potential in the numerical solution. They can

be arbitrarily placed in the flow field extending from the

point where/two streams of different histories Join at the scroll

lip/to the ex~t station. Nodes with different global numbers,

but which have the same coordinates are placed opposite each

other on the two splitting surfaces. The velocity potentials

at each two nodes w~ll generally be different, thus allowing

for a velocity potential discontinuity in the numerical

solution. Conditions are introduced in the equations to match

the mass flow across the splitting boundaries and the difference

between the velocity potential values of the opposite nodes

on the splitting surfaces.

The first condition lmposed in the numerical solution

is the matching of the weighted ~ass flow across the s~litt~ng

surfaces. Across an element's surface, se: on the splitting

boundary, the weighted mass flow is g~ven by:

(B .1)

where a¢e/ an is derivative of the velocity potential normal e to the surface area As. This derivative can be expressed in

terms of the x, y and z derivatives of ¢ and the direction

cosines a, Sand y of the normal to the surface A as follows: s

an = a 4 aN. 4 aN. 4 aN. \ ~e ~ + S \ ~e ~ + \ ~e ~ t 'i'~ ax- t 'i'i ay- Y t 'i'i az (B.2)

Substituting equation (B.2) ~nto (B.l) and using equation (A.7)

one obtains the following expression for the contribution of the

weighted mass flow across the splitting surface to each of the

opposite surface nodes.

50

E 1 4 -3 L p(a.al. + Bb + ye.) ¢.A

e 1 l. l. l. S

(B. 3)

The direction cosines a, 8 and y of the normal to the surface of

the element on the splitting boundary can be expressed in terms

of the coordl.nates of its three nodal points. If the first

three nodes of a volume element ll.e on the splitting surface

and the fourth is an internal node, it can be shown that the

direction cosines a, Sand yare proportional

cofactor BY4 and cofactor b Z4 respectively.

constant is determined from

222 a + S + Y = 1

to cofactor Bx ' 4

The proportionality

(B. 4)

The second condition l.S imposed to match the velocl.ty potentl.al

difference between each two OpPosl.te nodes on the spll.tting

surfaces. In the following derl.vations and discussl.ons,

it wl.ll be assumed that if the nodes of the volume element on

one of the spll.tting boundaries are i, j, k, etc., the opposite

nodes on the other splitting boundary are iI, J I , k', etc. The . mass flow matchl.ng conditl.on is introduced by adding ms/3 to

the global matrix coefficients in the columns corresponding to

the nodes (i = 1, 2 , 3 , 4) in the rows (i = 1', 2 I , 3 ' , 4 I) in -sub­

routine SPLIT. The same process is repeated reversl.ng the roles

of the primed and unprimed sides.

The second matching conditions insures the same difference

between the velocity potential values of the opposite nodes

on the splitting surface.

I I

¢i - ¢i = ¢J - ¢J (B. 5)

This condition is introduced by modifying the coeffl.cl.ents of the

row correspondl.ng to the node i on one side of the spll.ttl.ng

boundaries. This is accompll.shed by replacing the dl.agonal

coefficient and the coefficient in column jl by a very large

number and those in columns i' and J by equal but negative

large numbers.

51

A

B

K

m

N

P

R

T

x,y,z

a, S, y

y

p

¢

Surface area.

APPENDIX C

NOMENCLATURE

The element matrix.

Matrix of coefficients or stiffness matrix.

mass flow rate.

Interpolation function.

Load vector

Gas constant.

Temperature.

Cartesian coordinates

Direction cosines of the normal to a surface

Specific heat ratio.

Gas density.

Velocity potential

Superscript!3

e Refers to the volume element.

Subscripts

i,j

s

Refers to the nodes of an element or the indices

of the corresponding stiffness matrix.

Refers to the splitting surfaces.

52

End of Document