- NASA Technical Memorandum 86997

Application of Runge Kutta Time Marching Scheme for the Computation of Transonic Flows In Turbomachines

(hASA-TW-86997) I P P L X C A T I C N CF B U N G E KUTTA N88-12461 IXME R A R C B I N G S C E E H E FCR TEE CCflPUTATIOI OP I I - A N S C t i I C FLOfS IS TUEECMACHlhES ( N A S A ) 19 F CSCL 018 Unclas

~ 3 1 0 2 0110411

S.V. Subramanian and R. Bozzola A VCO-Lycoming Division Stratford, Connecticut

Prepared for the Twenty-first Joint Propulsion Conference cosponsored by the A I M , S A E and ASME Monterey, California, July 8-10, 1985

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https://ntrs.nasa.gov/search.jsp?R=19880003079 2020-04-27T07:00:43+00:00Z

APPLICATION OF RUNGE KUTTA T I M E MARCHING SCHEME FOR THE COMPUTATION OF

m

I W

.

TRANSONIC FLOWS I N TURBOMACHINES

S.V. Subramanlanl and R. Bozzola2 AVCO- Lycomi ng D l v l s 1 on

550 S. Main S t r e e t S t r a t f o r d , Connecticut 06497

SUMMARY

Numerical so lu t i ons cjf t he unsteady Euler equations a r e obtalned us lng t h e c l a s s l c a l f o u r t h order Runge Ku t ta t ime marching scheme. Th is method I s f u l l y e x p l l c l t and i s app l l ed t o the governlng equations I n t h e f i n i t e volume, con- se rva t l on law form. I n order t o determlne t h e e f f i c l e n c y o f t h i s scheme f o r s o l v l n g turbomachlnery f lows, steady blade-to-blade so lu t i ons a re obtained f o r compressor and t u r b i n e cascades under subsonic and t ranson ic f l o w cond l t l ons . Computed r e s u l t s a re compared w i t h o the r numerlcal methods and cascade tunne l measurements. The present study a l s o focuses on o ther Impor tan t numerical aspects i n f l u e n c i n g the performance o f t he a l g o r i t h m and the s o l u t l o n accuracy such as g r l d types, boundary cond l t lons , and a r t i f l c i a l v i s c o s i t y . For t h i s purpose, H, 0, and C t ype computational g r i d s as w e l l as c h a r a c t e r i s t l c and e x t r a p o l a t i o n type boundary cond i t ions are inc luded I n t h e s o l u t i o n procedure.

I N T R O D U C T I O N

A broad spectrum o f f l o w cond l t l ons are encountered a t var lous stages o f a modern gas t u r b l n e engine opera t lona l cyc le . These cond i t i ons a re h i g h l y complex and a re s i g n l f l c a n t l y d i v e r s i f i e d from one component t o t h e o ther . Development o f a n a l y t i c a l t o o l s c l o s e l y represent ing the Impor tan t f l o w fea- tu res I s a necessary s tep f o r e f f l c l e n t deslgn, operat ion, and performance improvements. Due t o the l ack o f computlng resources and l l m l t a t l o n s o f t h e e x i s t i n g a n a l y t l c a l methods, no s l n g l e study can hope t o p rov lde a l l t h e i n f o r - mat ion and answers t o the complete needs and s a t l s f a c t l o n o f t h e deslgn engi- neer. Hence, development o f numerlcal methods I s cus tomar l l y c a r r l e d o u t I n stages, u l t i m a t e l y hoping t o produce a model represent ing the gas t u r b i n e f l o w f i e l d as r e a l i s t i c a l l y as poss ib le .

The purpose o f the present study I s t o develop a f a s t and accura te Eu le r cascade computer f l o w code t h a t can be used f o r t he deslgn and performance improvements o f c r l t i c a l gas t u r b i n e englne components such as compressors and tu rb ines . Development o f t he s o l u t i o n procedure i s based on the Jameson's f o u r t h o rder Runge Ku t ta numerlcal l n t e g r a t l o n scheme ( r e f . 1) . Th is scheme has been success fu l l y app l i ed and shown t o y i e l d f a s t r e s u l t s w i t h good accu- racy f o r ex te rna l aerodynamic f l o w problems. Other major advantages t h a t

- 'Senlor Research Engineer, Aerodynamic Design Group.

ZManager, Turblne Deslgn and Development Group.

Cur ren t l y an I n d u s t r y V l s l t l n g S c l e n t l s t a t NASA Lewis Research Center, Cleveland, Ohio 44135. Member A I A A .

prompted t h e authors t o apply t h i s method f o r i n t e r n a l turbomachinery f lows a r e t h e s i m p l i c i t y and s t r a i g h t f o r w a r d implementat ion procedure, Independence o f t h e steady-state s o l u t i o n on the t ime step taken and t h e v e c t o r i z a b l e nature of t h e scheme.

. The numerical r e s u l t s obtained i n t h i s study address severa l impor tant issues r e l a t e d t o t h e performance of t he present method compared t o t h e o the r schemes t h a t e x i s t I n the l i t e r a t u r e f o r cascade f l ows ( r e f s . 2 t o 8) . These issues i nc lude s o l u t i o n convergence, mass and t o t a l pressure conservat ion, g r i d types such as H, 0, and C, boundary c o n d i t i o n methods, and i n f l u e n c e o f f l o w geometry v a r i a t i o n such as compressors and tu rb ines .

GOVERNING EQUATIONS AND NUMERICAL INTEGRATION SCHEME

The Euler equations governing the two-dimensional i n v i s c i d f lows can be w r i t t e n i n the i n t e g r a l form as

where AR i s the f l o w domain, R i s the f l o w boundary, x and y a r e t h e Car tes ian coordinates. The vectors U, F and G I n equat lon (1) a r e

U =

r P

PU F = 1

Here, p, u, v, p, E, and H a re the densi ty , v e l o c i t y components i n t h e x and y d i r e c t i o n s , pressure, t o t a l energy, and t o t a l enthalpy r e s p e c t i v e l y . For a p e r f e c t gas

E = - t ; (u2 t v2) ( Y - 1 ) P

and

H = E t L ! P

( 3 )

( 4 )

where y i s t he s p e c i f i c heat r a t i o . Approximation t o equat ion (1) leads t o f i n i t e volume numerical method i n conservat ion form.

Runge Ku t ta I n t e g r a t i o n Scheme

The f l o w domain i s d i v i d e d i n t o a number of small f i n i t e volume ( f i n i t e area i n two dimensions) computational c e l l s and the numerical approximation of equat ion (1) i s app l i ed t o each c e l l separate ly . This procedure leads t o a system o f o rd ina ry d i f f e r e n t i a l equations o f the form

2

which can be solved by any number o f i n t e g r a t i o n schemes. i s t he c e l l area, Q i s t he s p a t i a l d i f f e r e n c i n g operator and D(U) i s t he added a r t i f i c i a l d i s s i p a t i v e term t o supress numerical i n s t a b i l i t i e s i n the reg ion o f very l a r g e f l o w v a r i a t i o n s . app l i ed t o t h e x momentum component f o r example, i s ( w i t h t h e d i s s i p a t i v e term omi t ted f o r s i m p l i c i t y )

I n t h i s equat ion, AA

The exact form o f equat ion ( 5 ) when

4

where t h e f l u x v e l o c i t y iii( i s

and t h e sum i s over a l l f o u r s ides o f t he c e l l . I n f i g u r e 1 the f o u r s ides o f the c e l l (i,j) are denoted by numbers one t o fou r . The dependent f l o w va r ia - b les such as p , pu, pv, e tc . a re c e l l centered and t h e values (pu)k across any s ide For example,

k a re average values on the two sides o f t h e face.

and so on.

Equat ion ( 5 ) i s i n teg ra ted us ing the mod i f ied f o u r stage Runge Ku t ta scheme i n which t h e d i s s i p a t i v e terms a r e f r o z e n a t t h e values o f t h e f i r s t stage. The values a t t he t ime l e v e l "n'I a re then updated t o t h e new t ime l e v e l IIn + 1" i n the f o l l o w i n g f o u r stages:

( 9 )

N

where A t I s t h e l o c a l t ime marching step. The d i s s i p a t i v e operator D i s a b lend o f second and f o u r t h d i f f e rences . The operator depends on t h e l o c a l

3

f l o w behavior and i s inc luded i n t h e exact f ash ion descr ibed by Jameson ( r e f . 1).

BOUNDARY CONDITIONS

With reference t o f i g u r e 2, t he re a re f o u r types o f boundaries t h a t con- s t i t u t e the f l o w reg ion and r e q u i r e spec ia l t reatment. boundary AH, t he o u t f l o w boundary DE, t h e s o l i d - w a l l boundaries (BC and G F ) , and the p e r i o d i c boundaries AB, CD, HG, and FE. The approp r ia te se t o f cond i t i ons f o r each o f these boundaries i s descr ibed below.

These a re the i n f l o w

I n f l o w and o u t f l o w boundary cond i t i ons used I n t h i s study were the char- a c t e r i s t i c and e x t r a - p o l a t i o n type. I f the f l o w i s subsonic the re w i l l be th ree incoming ( r i g h t running) c h a r a c t e r i s t i c s and one outgoing ( l e f t running) c h a r a c t e r i s t i c a t the i n f l o w w h i l e the opposi te i s t r u e a t t he o u t f l o w boundary where the re a re th ree outgoing ( r i g h t running) c h a r a c t e r i s t i c s and one incoming ( l e f t running) c h a r a c t e r i s t i c . By the theory o f c h a r a c t e r i s t i c s , t h ree condi- t i o n s may t h e r e f o r e be s p e c i f i e d a t t he i n f l o w and one c o n d i t i o n a t t he o u t f l o w ( r e f . 9 ) . The remaining cond i t i ons a re numer i ca l l y determined by the s o l u t i o n o f t h e d i f f e r e n t i a l equations. The t h r e e cond i t i ons s p e c i f i e d a t the i n f l o w a r e the t o t a l pressure, t o t a l temperature, and the f l o w angle. The remaining c o n d i t i o n i s obtained by e x t r a p o l a t i n g t h e outgoing o r upstream running Riemann i n v a r i a n t from the i n t e r i o r t o t h e i n l e t . For e x t r a p o l a t i o n type, the s t a t i c pressure I s extrapolated f r o m t h e i n t e r i o r c e l l t o the i n f l o w . For supersonic a x i a l i n f l o w , t he i n l e t Mach number i s a l s o s p e c l f i e d and he ld constant i n a d d l t i o n t o the t h r e e cond i t i ons mentioned above.

A t t he o u t f l o w boundary, t he one phys i ca l c o n d i t i o n s p e c i f i e d i s t he s t a t i c pressure corresponding t o the des i red e x i t Mach number. The th ree numerical cond i t i ons come from e x t r a p o l a t i n g the downstream running Riemann I n v a r i e n t , the y - v e l o c i t y component v and the t o t a l energy E. F o r super- sonic a x i a l out f low, the s t a t i c e x i t pressure i s a l s o ex t rapo la ted f r o m the i n t e r i o r p o i n t .

On the blade surface, t h e "zero f l u x l l cond i t i ons a re imposed. However, f o r es t ima t ing t h e c o n t r i b u t i o n s t o t h e momentum equations due t o the pressure t e r m s , we need t o evaluate the pressure on the blade surfaces. pressure g rad ien t a t t he boundary c e l l centers i s determined by the same fash- i o n descr ibed i n reference 1. E x t r a p o l a t i o n us ing the pressure g rad ien t and t h e pressure a t t he c e l l center determines the pressure a t t h e w a l l .

The normal

The p e r i o d i c i t y i s imposed by s e t t i n g the f l o w va r iab les i n the c e l l ten- t e r s t h a t l i e w i t h i n the computational domain equal t o the corresponding per- i o d i c c e l l s t h a t l i e ou ts ide the computational domain. For example ( w i t h reference t o f i g . 2 ) , t he unknown f l o w q u a n t i t i e s i n the c o n t r o l volumes t h a t l i e immediately above the p e r i o d i c boundaries AB and CD a re s e t equal t o the corresponding c e l l values t h a t l ? e immediately above the p e r i o d i c boundaries HG and F E whose values a re known f r o m the i n t e r i o r p o i n t c a l c u l a t i o n s . Add i t i ona l d e t a i l s on the implementation o f these boundary cond i t i ons can be found i n references. 1 and 7.

4

RESULTS AND DISCUSSION

Numerical solutions of the Euler equations using the Runge Kutta scheme described earlier were obtained for selected cascade test cases with widely varying geometries and flow conditions. Results are presented and compared with solutions of other numerical methods and experimental data. Computational grids of different types were also used in the calculations to determine their influence on the solution accuracy.

The Mach number contours calculated for a transonic flow through a com- pressor cascade are illustrated in figure 3. The number of grid points used in the computations are 65 by 15. in figure 2. The flow is subsonic at the inlet and accelerates to a peak Mach number of 1.246 on the blade suction side before shocking down to an exit Mach number of 0.67. The computed surface Mach numbers agree very well with the predictions of other numerical methods. However, there are no experimental data available for this case.

The cascade geometry and the grids are shown

Figure 4 depicts a high work turbine guide vane geometry (ref. 13) and the 80 by 17 computational grids used for obtaining a fully subsonic flow through the passage. It is seen that the flow accelerates from an inflow Mach number of 0.11 to an exit value of 0.84 in a very short distance. obtained by the present method are shown in figure 6 along with available test data (ref. 13) an another code (ref. 14) using the two step explicit MacCormack numerical scheme. for this difficult test case, the present code yielded results with very good accuracy. ber of grid points i n the flow field.

The computed Mach number contours are shown in figure 5.

The surface static pressure ratios

Considering the type and coarseness of the grids employed

Better accuracy can be achieved by refining and increasing the num-

Next example is a flow through a NASA turbine stator rows shown i n figure 7. The calculated critical surface Mach numbers are shown in figure 8. The wind tunnel data (ref. 10) and the numerical results of another code (ref. 3) computed i n house are also shown in this figure for comparison. The agreement of the present results with the test data as well as with the other method is very good. However, the poor agreement of the two numerical results with the test data near the leading edge is mainly due to the nature of the g r l d I n t h i s r e g i o n . The numerical schemes t r e a t t h e l e a d i n g edge w i t h "cusps" for solution convergence which is not present in the actual airfoil.

The treatment of leading and trailing edges with artificial cusps for numerical convenience is a common feature while computatlng flows with sheared II H I1 type grids as done in the previous test cases. This is particularly true for cascades with high incidence, large flow turning, and thick leading/ trailing edge profiles. Numerical results obtained with "HI1 grids are often inaccurate and misleading i n these important flow regions. For improved accu- racy, it is preferred that computations be performed on body fitted orthogonal or near orthogonal I1C1I or 11011 type grids (ref. 11). To illustrate this point, the surface Mach numbers calculated on a I1CI1 grid for the same test case of figure 7 are compared with the test data and this is shown in figure 9. It can be seen that the agreement is excellent and far improved near the lead- ing and trailing edges compared to the "H1I grid results.

Figure 10 shows an AVCO rotor blade and the llC1l type computational mesh for which numerical solutions were obtained. This case was particularly chosen

5

t o t e s t t he a b i l i t y o f t he present code t o c a l c u l a t e t ranson ic f lows through t u r b i n e blade passages w i t h h i g h f l o w incidence, l a r g e tu rn ing , and t h i c k b lade p r o f i l e s a t t h e lead ing and t r a i l i n g edges. The i n l e t f l o w angle i s 5 5 O , t he i n f l o w Mach number i s 0.66 and the o u t f l o w Mach number i s near t ranson ic . The Mach number contours near the lead ing edge reg lon i s shown i n f i g u r e 11 t o i l l u s t r a t e the q u a l i t y o f r e s u l t s t h a t can be obta ined on a present method. Computed sur face Mach numbers are p l o t t e d i n f t g u r e 12 a long w i t h the numerical r e s u l t s o f re ference 3. The agreement between the two methods are very good except near t h e shock l o c a t i o n where the present method p r e d i c t s a peak va lue o f 1.108 compared w i t h 1.152 obta ined by the Denton's code ( r e f . 3).

llC1l g r i d w i t h t h e

The V K I gas t u r b i n e r o t o r b lade and the computat ional g r i d s on which t ran - sonic f l o w so lu t i ons were obta ined i s p i c t u r e d i n f i g u r e 13. F igure 14 com- pares t h e p r e d i c t i o n s o f sur face Mach numbers w i t h t e s t data obtained f rom two d i f f e r e n t cascade tunnels and two d i f f e r e n t e x i t f l o w cond i t ions . The agree- ment between t h e present c a l c u l a t i o n s and the two sets o f t e s t data i s excel - l e n t e s p e c i a l l y f o r t he case corresponding t o e x i t Mach number o f 1.19.

CONCLUSIONS

A computer code f o r s o l v i n g the Euler equat ions us ing the f o u r stage Runge K u t t a l n t e g r a t l o n scheme has been developed f o r turbomachlnery f l o w f i e l d c a l - c u l a t i o n . The program has been success fu l l y app l i ed t o p r e d i c t b lade-to-blade f lows f o r many cascades w i t h d i f f e r e n t geometries and f l o w cond i t i ons . Numer- i c a l r e s u l t s i n d i c a t e t h a t t he present method can be app l i ed t o y i e l d f a s t r e s u l t s w i t h good accuracy f o r a wide v a r i e t y o f cascade con f igu ra t i ons and f l o w cond i t i ons . The eCt l type g r i d s produce the bes t o v e r a l l r e s u l t s f o r any p a r t i c u l a r t e s t case f rom the stand p o i n t o f accuracy, s i m p l i c i t y o f implemen- t a t i o n , and boundary c o n d i t i o n t reatments. The present code, which could be used i n con junc t ion w i t h any type o f user opted computat ional g r i d s , i s simple, e f f i c i e n t , and accurate enough t o be used f o r cos t e f f e c t i v e p r e l i m i n a r y and advanced aerodynamic design s tud ies .

ACKNOWLEDGMENT

This research and code development was conducted as p a r t o f AVCO Lycoming Independent Research and Development Program. The authors wish t o thank D r s . Cra ig L. S t r e e t t , R . C . Swanson, and Manuel D. Salas, Theore t ica l Aerodynamics Branch, NASA Langley Research Center f o r shar ing the techn ica l i n fo rma t ion and exper ience they gained w h i l e implementing the numerical method o f t h i s study f o r computing ex te rna l aerodynamic f lows.

REFERENCES

1. Jameson, A., Schmidt, W. and Turke l , E., "Numerical So lu t ions o f Euler Equations by F i n l t e Volume Methods Using Runge-Kutta Time-Stepping Schemes ,'I A I A A Paper 81 -1 259, June 1981 .

2. Denton, J.D., " A Time Marching Method f o r Twoand Three-Dimensional Blade- to-Blade Flow," ARC R M-3775, 1975.

6

3. Denton, J.D., "An Improved Time Marching Method for Turbomachinery Flow Calculation," ASME Paper 82-GT-239, Apr. 1982.

4. Delaney, R.A., "Time-Marching Analysis of Steady Transonic Flow in Turbo- machinery Cascades Using the Hopscotch Method," ASME Paper 82-GT-152, June 1982.

5. Ni, R.-H., "A Multiple-Grid Scheme for Solving the Euler Equations," AIAA Journal, Vol 20, No. 11 , Nov. 1982, pp. 1565-1 571.

6. Thompkins, W.T., "Solution Procedures for Accurate Numerical Simulations of Flow in Turbomachinery Cascades,Il AIAA Paper 83-0257, Jan. 1983.

7. Subramanian, S.V. "Analysis o f Time Marching Numerical Methods and Boundary Conditions for Turbomachinery Flow Computation," ASME Paper 84-GT-66, June 1984.

8. Chima, R.V., "Analysis o f Inviscid and Viscous Flows in Cascades with an Explicit Multiple-Grid Algorithm," NASA TM-83636, 1984.

9. Chakravarthy, S.R., "Euler Equations - Implicit Schemes and Implicit Boundary Conditions," AIAA Paper 82-0228, Jan. 1982.

10. Whitney, W.J., et al., "Cold-Air Investigation of a Turbine for High- Temperature-Engine Application,81 NASA TN-D 3751, Jan. 1967.

1 1 . Sorenson, R . L . , "A Computer Program to Generate Two-Dimensional Grids About Airfoils and Other Shapes by the Use of Poisson's Equation," NASA TM-81198, 1980.

12. Sieverding, C.H., "The Base Pressure Problem in Transonic Turbine Cas- cades," Transonic Flows in Axial Turbomachinery, Vol. 2, von Karman Insti- tute for Fluid Dynamics, Rhode-Saint-Genese, Belgium, 1976.

13. Kopper, F.C., et al., "Energy Efficient Engine High Pressure Turbine Super- sonic Cascade," NASA CR-165567, Nov. 1981.

14. Srivastra, B.N. and Bozzola, R . , "Computations o f Flow Fields in High Solidity and High Turning Angle Cascades Using Euler Equations ,I' AIAA Paper 85-1705, J u l y 1985.

1

CELL NODES

X CELL CENTERS (VALUES OF INDEPENDENT VARIABLES)

(VALUES OF DEPENDENT VARIABLES)

Figure 1. - Finite volume descretization.

D

H Figure 2. - Compressor cascade and computational

gr idc

Figure 3. - Contour plots of Mach numbers.

3.50

2.75

2.00

1.25

0.50

-0.25

-1.00

-1.75

-2.50

-3.25

n

-4 00 I ! I I I I I -0.75 0.00 0.75 1.50 2.25 3.00 3.75

X

Figure 4. - Turbine guide vane and computational mesh.

0.04 a

\ Figure 5. - Math number contours.

1.15 c I 0 .- L - p. 1.00 0- i=

w .85 d a 3 v, v, W cd O- . 7 0 0 3 0 .55 3

!--

Ln W

@L 3 v,

.40

A TEST DATA RUNGE KUTTA (80 x 17 GRID) AERL (100 x 33 GRID)

- --

- PRESSURE SIDC

Figure 7. - NASA turbine stator geometry.

1.00

.80 E5

5 .60

a

X 0

.49 E 0

.20

0

.60

.40

.20

- SUCTION SIDE

RUNGE K U l l A

TEST DATA (REF. 10) A// -- DENTON (REF. 3 )

0 A

20 40 50 80 100 SURFACE LENGTH, percent

Figure 8. - Surface Mach number distribution.

IC' GRID 'H' GRID TEST DATA (REF. 10)

--- - A

0 20 40 60 80 100 PERCENT CHORD

Figure 9. - Surface Mach number distribution.

Figure 10. - AVCO turbine rotor geometry and computational IC' grids.

-Q 537

0.5c Q54/ /

Figure 11. - Mach number contours near the leading edge of the cascade.

1.20

1.03

.80

.60

.40

- RUNGE KUTTA

- 0 REF. 3 (DENTON) SUCTION SIDE

.20 Hf- PRESSURE SIDE

f 0 20 40 60 80 100

PERCENT CHORD

Figure 12. - Surface Mach number var ia t ion fo r AVCO cascade.

Figure 13. - V K I turbine cascade and computational 'C' grids.

1.80

1.60

1.40

1.20

E m z 1.00 I 2

3

5 W

.80 E 2 m

.60

.40

.20

0

- A

-

PRESSURE SIDE

Figure 14. - Surface Mach number distribution.

1. Re rt No. NRA TM-86997 2. Government Accession No. 3. Recipient's Catalog No.

15. Supplementary Notes *Cur ren t l y an I n d u s t r y V i s i t i n g S c i e n t i s t a t NASA Lewis Research Center. Th is r e p o r t was prepared f o r t h e Twen ty - f i r s t J o i n t Propuls ion Conference cospon- sored by t h e AIAA, SAE and ASML, Monterey, C a l i f o r n i a , J u l y 8-10, 1985. This work was performed under t h e AVCO-Lycoming Independent Research And Development Program. The computer code f o r t h i s work i s p r o p r i e t a r y t o Avco-Lycoming D i v i s i o n

16. Abstract

4. Title and Subtitle

Numerical s o l u t i o n s o f t h e unsteady Eu ler equat ions are obta ined us ing t h e c las - s i c a l f o u r t h o rder Runge K u t t a t ime marching scheme. e x p l i c i t and i s app l i ed t o t h e governing equat ions i n t h e f i n i t e volume, conser- v a t i o n law form. turbomachinery f lows, steady Dlade-to-blade s o l u t i o n s are obta ined f o r compres- sor and t u r b i n e cascades under subsonic and t ranson ic f l o w cond i t ions . Computed r e s u l t s a re compared w i t h o t h e r numerical methods and wind tunne l measurements. The present s tudy a l s o focuses on o ther impor tant numerical aspects i n f l u e n c i n g the performance o f t h e a lgo r i t hm and t h e s o l u t i o n accuracy such as g r i d types, boundary cond i t ions , and a r t i f i c i a l v i s c o s i t y . For t h i s purpose, H, 0 , and C t ype computat ional g r i d s as w e l l as c h a r a c t e r i s t i c ana e x t r a p o l a t i o n t ype bound- a r y c o n d i t i o n s a re inc luded i n t h e s o l u t i o n procedure.

This method i s f u l l y

I n o rder t o determine t h e e f f i c i e n c y o f t h i s scheme f o r s o l v i n g

5. Report Date

A p p l i c a t i o n o f Runge K u t t a Time Marching Scheme f o r t h e Computation o f Transonic Flows i n Turbomachines

7. Author(@

S.V. Subramanian* and R. Bozzola, AVCO-Lycoming D iv i s ion , 550 S. Main St. , S t r a t f o r d , Connect icut 06497

Nat iona l Aeronaut ics and Space Admin i s t ra t i on Lewis Research Center Cleveland, Ohio 44135

Nat iona l Aeronaut ics and Space Admin is t ra t ion Washington, D.C. 20546

9. Performing Organization Name and Address

12. Sponsoring Agency Name and Address

6. Performing Organization Code

505-31 -04

8. Performing Organization Report No.

E-2543

IO. Work Unit No.

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

17. Key Words (Suggested by Author@))

Computational f l u i d dynamics; Turbo- machinery f lows; Turb ine and compressor flows; Appl ied aerodynamics f o r cascades

18. Distribution Statement

Unc lass i f i ed - u n l i m i t e a STAR Category 02

19. Security Ciassif. (of this report)

Unc lass i f i ed 20. Security Ciassif. (of this page) 21. No. of pages 22. Price'

Unc lass i f ied