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3-

1

NASA Technical -Memorandum 86984

(NASA-TM-E6984) CALCULATI(K CF N85-2_711THEEE-DIMENSiGNAL, VISCOUS FLOh -HOUGHIURBOMACHINSLY ELALE FASSAGL°, L• Y FAPABCIICMARCHING (NASA) 29 p HC A03/MF A01 CSCL 01A Unclar_

G3/,2 11r]79

Calculation of Three-Dimensional, Viscous]Flow Through Turbomachinery BladePassages by Parabolic Marching

Theodore KatsanisLewis Research CenterCleveland, Ohio

^9.ci rii 'Prepared for theTwenty-first Joint 'Propulsion Conferencecosponsored by the AIAA, SAE, and ASMEMonterey, California, July 8-11, 1985

Qa,

CALCULATION Of THREE DIMENSIONAL, VISCOUS FLOW THROUGH TURBOMACHINERY

BLADE PASSAGES BY PARABOLIC MARCHING

Theodore KatsanisNational Aeronautics and Space Administration

Lewis Research CenterCleveland, Ohio 44135

SUMMARY

Tt.e three-0imensicnal compressible Navier Stokes equations are formulatedin a e atating coordinate system, so as to include centrifugal and Coriolisforces. The equations are parabolized by using a previously calculated invis-

cid static pressure field. The thin layer Naviar-Stokes approximation, which

neglects streamwise diffusion, is used. A body41 tted coordinate system isused. The streamwise momentum equation is uncoupled from the cross-streammomentum equation by using contravariant momentum components, and then using

the contravariant velocity components as primary unknowns. To reduce problems

8with small separated regions, the Reyhner and Flugge-Lotz approximation isused. The energy equation is included to allow for calculation of heat trans-

Ter.The flow may be laminar, or a simple eddy-viscosity turbulence may be

f used. A number of curved ducts and an axial stator have been analyzed,including cases for which experimental data are available.

+ NOMENCLATURE

e total internal energy

H total enthalpy

I rothalpy 1i'

J transformation Jacobian from Cartesian to (E,n,{) coordinates

J itransformation Jacobian from Cartesian to cylindrical coordinates I

1 transformation Jacobian from cylindrical to (E,n,C) coordinatesk thermal conductivity

ii

p pressure

r radius

t time

V absolute velocity

W relative velocity I

x Cartesian coordinate

y cartesian coordinate

z cartesian coordinate

Y specific heat ratio

C hub-to-shroud grid coordinate

n blade-to-blade grid coordinate

e angular coordinate, radians

V viSCOSity

E streamwise grid coordinate

P density

T stress

w angular velocity

Subscripts:

er r component

z z component

e e component

SUer scrj[jts:

S C contravariant component

Y1 n contravariant component

E E contravariant component

INTRODUCTION

Well-guided internal flow without separated regions behaves in a para-bolic manner (ref. 1) so that the flow field can be predicted using a para-bolic, streamwise marching method. Several computer codes have been writtenfor calculating three-dimensional viscous flow through ducts of fairly complexgeometry (rc` 2 and 3). The same principles can be applied to calculateflow through the blades of a turbomachine. How-2v2r, there are several distin-guishing aspects of flow through a turbomachine blade row: (1) The blade rowmay be rotating. (2) The hub and shroad are usually surfaces of revolution.(3) The passage has four walls meeting at an angle. (4) The flow may be peri-odic upstream and downstream of the blade. (5) The shroud is usually station-

N,,, ary for a rotating blade row. The method and computer code described here isespecially formulated to satisfy and take advantage of these particularaspects of the flow.

2

p:;

The parabolic marching calculation is analagcus to calculation of athree-dimensional boundary layer. An inviscld pressure field is used as an

initial pressure field; then a single-pass calculation through the length ofthe blade passage is made. To simplify the application of boundary conditions

on complex blade surfaces, a body-fitted coordinate system is used. This

body-fitted coordinate system has one coordinate which generates circles aboutthe axis of rotation. One of the coordinate surfaces coincides with the hub of

the blade row, and another coincides with the shroud. This type of coordinatesystem eases implementation of periodic boundary conditions upstream and down•

stream of the blade row; on the other hand, this type of coordinate systemcannot be orthogonal.

In the spirit of Patankar and Spalding (ref. )) the streatwise momentumcalculation is uncoupled from the cross-stream momentum calculation. Thisum:oupling is accomplished by using contravariant romponents of the momentum

equation, and using contravariant velocity components as the primary unknowns.The downstream contravariant velocity components are always calculated impli-

citly for stability. To calculate the streamwise contravariant velocity com-ponents at the text downstream station, it is assumed that the static pressure

and cross-stream contravariant velocity components are known at the previousupstream station. The streamwise pressure gradient is varied by a constant

amount over the passage cross-secti)n fo that global mass flow through theentire passage is conserved. The cross-stream contravariant velocity compo-

nents are calculated at the next downstream station based on the inviscidpressure field, and using the last calculated streamwise contravariant veloc-

ity components. Continuity is then checked for each mesh region, and thecross-stream pressure gradient is adjusted so that continuity is satisfied forevery mesh region. Iteration is required at each station so that both cross-

stream momentum and local continuity are satisfied. For turbulent flow, theBaldwin-Lomax eddy-viscosity turbulen:e model is used (ref. 4).

PARABOIIC iMI ARCHING THROUGH A iURBOMACHINE PASSAGE

Contravariant : orm of the Navier-Stokes Equations for a RotatingNon-orthogonal Coordinate System

The continuity equation in conservation form in cylindrical coordinates

is:

T (rP) + ar (rPWr) + ae ( Pwr ) + az (rpW z ) o (1}

In cylindrical (r,e,z) coordinates the Navier-Stokes equations may be written

in conse r vation form as follows:

at (rPY + Ar (rPv r ) + ae (PV r^a) * az (rP^r^z)

2 a a aPV_e - ar

(rrrr - rp) + ae (r

re ) + az

(rrrz ) - r ee + P

3

41

a

A.

at (rpVO) + ar (rpyrV0 * ae ( pvo) + a7 (rpVeVz)

-PVr V8 ar (rT re ) « ae (T ee - P) + az {rt

es ) + ire

at (rpV Z ) + ar (rpVrvz + ae (PVe V z ) az (rpVz)

= ar (rT rz

) + ae (Tre) + az (r1 zZ - P)

The change to a rotating coordinate system is accomplished by the change ofvariables:

t' - t Wr - VT

r' = r We ° VO - Wr

0' = 0 - Wt Wz > Vz

Z , - z

The partial derivatives with respect to t then become:

a a aat ° at' - `a ae

The resulting Navier-Stokes equations for rotating cylindrical coordinates arenow (the ' has been dropped):

at (rpW r ) + ar (rpW r ) + a0 (PWrW0 + az (rpWrWz)

= pV0r aTee + ar (rTrr ) + ae

(T re) + az (rTrz ) (2)

at (rpWO ) + ar (roW

eW r ) + ao (pW0) + az ( r PW0 z)

pWr (We + 2Wr) - a© + T re + ar (rTre)

+ ae ( 1 00 ) + az (rTrO) (3)

at (rpwz) + ar (rpW zW r ) ' as (pwzwe) + az (rpWz)

r a + ar (rTrz ) + ae

(Tez) + az (rTzZ) (4)

4

^`ii- It a '&"N '&,

Finally, the energy equation in conservation form is:

at + V•((e + p) V) - V • kVT + V•(T 0)

This is expanded in cylindrical coordinates, and transformed tc rotatingcoordinates using

a a aat ° at, -

W a

Also, rothalpy I is introduced, using the relation

e + p = pI + wrpV6

The energy equation in conservation form, for rotating cylindrical coordinatesand in terms of rothalpy is:

T (r(pl - p)) + rV•(pIW) = rV•kVT + rV • (T W)

or

at ( r (P I - D)) + ar (rpIWr) + ae (P1We) + az (rpIWz)

r ( rk ar ) + ae ( r K AT ae ) ` az (rk az )

+ a (r(W T « W T + W T )) + a (W T + W Tar r rr a rs z rz ae r re a 09

+W, ) + a (r(W T rz

+W Tez

+W z zzt )) (5)

z ez az r a

Equations (1) to (5) can be written in compact form as

a t q + a r E + aeF + azG = K + a r R + aes + a z T (6)

where

P PWr PWe

Pwr PWr PWeWr

q = r pWe 2E r PWrwe F pwe

pWZ

pI - pPWrWz PWeWZ

PW r I PWel

5

0I

WP Zo

2 - r---e - tPWzWr

pV a ar ea

G - r PW ZWe K ° T re _ ae - PWr(We , 2mr)

PW2-r apz az

PWzl LO

O

Trr

R r Tre

Trz

WrTrrkaT

F W0 T re * W z T rz Br

0

Tre

S ' Tee

TezaT_

WrTrO a W

0 T 00 W z T ez k rae

0

Tr

T r Tez

tz

aT

W ` rz WO T ez + W z ` zz k az

Equation (6) can be transformed to an arbitrary nonorthogonal rotatingcoordinate system (^, n, C) using the chain rule. When this is done, consider-

able simplification occurs by combining terms properly (refs. 5 and 6), and byusing the contravariant velocity components as variables. Note that terms

such as

/2^ « (

T'r) + (

a 2n)C

6

- - .zl

'-

are equal to zero (ref. 7), so that factors like Er/J2 can be moved insidethe derivative. The form of the equations is now:

a tq + a + a n F + a CG K + a tR + a n S + a

C T (7)

where

P

PWr

q J-1 PW

PWz

P I -

PWn

POW

F = J-1 pWnWe

POW

PW^I

pWt

POW E = J -1 PWtWe

PWtWz

POI

pWC

PWCWr

6 = J -IIPWKWe

PWCWr

IPWCI

0 0

T t T^

r r

t

R = J -1 T e S + J -1 Te

t nT

T

t eGe ^eG6t r 6 r + r + t z a z I^rar + r + ^zpz

7

- I V

"..

K J-^

n

T`r

r

T = J-1

T r

T z

K* ear «K r o r r Czpz

'rear (We * 2Wr^

r - rae - PW r r

apaz

0

0

Pi e2 8,^ teo

r 8 r

W^rWr * ko

W

re * tzWz

W

Wn ' rl w * ne re * nzWz

r WeW ' K rW r * re r / + czWz

E c c T ro {Tr S r T rr * ye( T « yzTrz

* to (rTe '

^r T re to r ^z T ez

(Tez

T z - ^rT rz

*fie\ r * ^zTzz

Tr, e r *

Tr - n r T rr * ne r nzTrz

Te8 \JI

Te - n r T re * no r / * nZTez

n TezTz =

nrT rz * ne r + n2TZ2

{ y (TrelT r

KrT rr * re \ r / } CzTrz

r Tee

Te = ^r T re * ce r * {zTez

(TezlT Cz =

CrT rz * Ce \\\ r / * SZTT-z

8

t ,

pr ° W r r rr + W0T re + Wzrrz + karT

aeTpe

WrTre r Wa7ee + WzTez + k r

a z ° W r 7 rz + Warez + WzIzz +

kazT

Equation 1 still has cylindrical momentum components. It is desired to

utilize momentum components aligned with the mesh, so that streamwise momentumis easily uncoupled from the cross-stream momentum components. This is accom-

plished by taking the contravariant momentum components. For example, thecontravariant E-momentum component is Er(r-momentum) plus ne/r(e-momentum)

plus Cz(z-momentum). This results in:

A A 2 A A A 2 Aa tq + a kE r a n F + a CG = a E R + a nS + a C T + K (8)

where

P PWEPWn

PWEPWEWE PWV

q= j PWn E j PW EW n F° j PWnWn

PWC PWEWC PWnWC

PI - p PA PWnI

PWC

0

PWCWEEr T9

+ T0 e +

EzTz

G - 1 PWCW n E

'e T ePWCWCR

- J n r T r + + n z T z

PWC I J E +C r T r

CeTeE+ E

CzTzr

zepeE r a r + r + Ezaz

,

9

4'

0

n {e'e n

E r T +r r + Ez`z

nA n + ^g n

S

eT 1

J °r'r r ^zTzT o J

n

C Tn + CA)

i C in

r r r z z

^elenrOr + r + nzOz

0

C + EOTCe + C

E 'rr r E 'zz

Cn T

n rTr + r ee

* nzTz

C CeT eC

CC r'r + r + CzTz

,OleC r a r r + CzOz

0

pG - 9 EEP - S { ^P - 9 ECP - { r t ee + {etre

E pE n C r r2

K pG - 9^ EP - 9^ n P 9 ^CP - ^r T 6e + ne2re

° J np E n C r r2

pG - 9C{ P 9CnP 9CCP - C r Tee Cure

Cp{ n C

r — + r2

0

E

GEp a W{((Er)E W r + (re)

E

We * (Ez)E Wz

// r Ee+W^I(E r ) n W r +1 r)nWe+(Ez)nWz

r\\

\ Ee+ WC (({ r ) C W r +( r )CW0 + ( E z ) C Wz

{ V

\\

2E

+ r r e 2 Wr (We + 2wr)

r

10

m

i

n

GnPW^((n r ) t W r * ( re) We( E z ) E Wz )

\ E

n♦ W n (n r ) n W

+ (re)n We * (t z ) n Wz)

W " 1r WC ( n r )C r + (re)C We

r (tz)C

Wz/

n ^ 2

ns rre - 2 W r ( We r 2wr)

r

GCP a WC (C r )^ Wr ' `Cre)^

W9 " (Cz)F Wz/l

rr wn `(C r ) n Wr

re ) We + (Cz)n Wz^n

C* WC (K r ) C Wr ` ( rr e ` (zz) C Wz^

Cryr C

+ re - 2 Wr (We r 2wr)

r

Derivation of the Navier-Stokes Equations for Parabolic MarchingThrough a Turbomachinery Blade Row

Equation (D) retains all' terms from the original Wavier-Stokes, continu-ity, and energy equations for a general non ortho gonal rotating coordinate

system. Some terms will now be eliminated, or neglected. Only steady rela-tive flow is considered so the time derivatives will be eliminated. Alsc, the

thin layer assumption is made so that all streamwise derivatives of viscousterms will be neglected. Further, the mesh that is used is not completely

general, so t'iat certain coordinate derivatives will be eliminated.

The coordinate system used is shown in figure 1. The t-coordinate is inthe streamwise direction, n is the blade to blade direction, and C is from

hub-to-shroud. Since the hub and shroud are both usually surfaces of revolu-tion, and to make It easier to apply periodic boundary conditions, the

n-coordinate lines will be circular arcs coincident with the e-coordinatelines. Hence, and C are functions of r and z only, and sre indepen-

dent of e:

11

P'

E ^ E(r,z), to • 0

n e n(r,e,z)

C = C rr , z ) Co 0

The required coordinate derivatives, Er, Ez, nr, ^tc. are calculated fromthe roordinate derivatives of the irverse trae;i^orr,.'or, r E , e E , etc., by the

equations:

-rC

Ez(rEzC rCzE)

Ee=0

zC

Er • (rEzC _ rCzE)

The Jacobian, J, is given by:

(rC0E - r E 0 C )

n z = en(rtzC _

r C

z E )

1no = e

n

(eC z E - eEzC)

nr

en(rt7C

rCzE)

r&^._rz

(r E 7 K - rCzE)

Ce • 0 (9)

_z E

C r • (rEzC - rCzE)

Since E is in the streamwiseterms are neglected. Further,

function p, we have:

// W E // Wn 1\1a t (p J w + an (

pJ ,p1 +

Excludin g continuity, ,̀ the a//qua

re n (r E z C - rCzE)

(10)

direction, all E-derivatives of the viscousby using the continuity equation, for any

C WC WE Wn WK

aC a J w P wE + P J wn + P J wC ( 11 )

Lions become:

W E Wn WK

a J a E H + a J a nH + P^ aCH a ns + aC'T + K (12)

where

WE ErTr + EzTz

= W nS s

1 n T

r + ne T0/r + nH J_r z

CW

n C TnC rT +r z z

1 nr0r + neGa/r

+ nzpz

12

- Z_

t^

z

C C^r T r ` EzTz

11J « ne TB/r

nz IZ

7 ^ J-1

TC + CCr r CzTz

C r 0 r * C

z 0 z

9^Cp - 9tnp - 9^CpE n C

9 nEp 9nnDn - 9nCPE C

9 C9p - 9 Cn Pn

- 9Ctp^ C

PG EP

K = J-1IPGnP

PGCP

0

Consistent with the thin boundary layer assumption, order of magnitudearguments are used to simplify calculation of the stress tensor in terms of

the contravariant velocity components. Terms that are important involve thesecond derivative of a velocity component parallel to a wall with respect to

the distance from the samz wall, whereas the derivative of a velocity compo-nent normal to a wall with respect to the same wall is negligible. Also,coordinate derivatives (tr, nr, etc.) are considered to vary slowly compared

to the velocity components themselves. Even with these simplifications, the

calculation of the stress tensor is rather complicated and messy for the non-orthogonal grid used.

Numerical Solution Procedure

A FORIRAN program called PARAMAR has been written to solve equation (12)by parabolic marching. Equation (12) is the thin layer Navier-Stokes equationfor a rotatin g , body fitted coordinate system. The g-coordinate is in the

streamwise direction, n is blade-to-blade,and C is hub-to-shroud, asindicated in figure 1. The equations are parabolized by using a previously

calculated iriiscid static pressure field, and neglecting streamwise diffusion

terms. The pressure field may be obtained by the MERIOL and TSONIC codes(refs. j and 9) or from the Denton code (ref. 10).

The form of the three momentum equations and the energy equation is

w^ a-T+ wn ^ + WC P

2 a I/ ate\ aa2P a{ P J an P J ac an \

a an/ - ac ac

aC ^ = S

where w may be any contravariant velocity component or rothalpy. S isthe remaining terms of the equation. The coefficients a and b, and thesource term S are different for each equation. By using a finite-differencel oproximation for the g-derivative, the equation may be approximated as:

13

^ti3

n

(azai) a

i«1a

coordinate

it will be

for wi«1•

iirectly by

-(a -82)) I+l - (TaC-(b —a%,

lines, and that Ei«1assumed that an = aCIn finite difference

using the block tridii

S,))i+1

- E 1 1 .

= 1.

form,igonal

Wn WCP

Wed (wi«1 m i ) ° P J (an )i«1 « P3

It Is assumed that the mesh lines areSimilarly, for the other coordinates,This equation is an implicit equation

the equation may be solved for wt«1algorithm (ref. 11, p. 196)

Because of the importance of accurately calculating continuity for inter-nal flow, a staggered mesh is used, much as suggested by Patankar and Spalding

(ref. 1). The mesh arrangement and numbering sheme is shown in figure 2,Each of the momentum equations are uncoupled by using coefficients calculated

at the upstream station. Since an inviscid pressure field is used, correc-tions must be made to account for viscous effects in the boundary layer.

The streamwise (F) momentum equation is used to calculate the stream-wise contravariant velocity components at the next streamwise station, using

zero velocity boundary conditions. After calculating the downstream W^, the

mass flow at the downstream station will be low due to an increasing displace-ment thickness of the boundary layer. Hence, the pressure must be reduced at

the downstream station, so that global continuity is satisfied. This is doneby reducing the stream-wise pressure gradient uniformly over the entire pas-

sage cross section.

the cross-stream momentum equations are used to calculate the cross-stream contravariant velocity components at the next station. Zero velocity

boundary conditions are used. After calculating the cross-stream velocities,continuity is checked for each mesh region. In general there will not be con-servation of mass for each mesh region. An equation can be derived to calcu-

late a correction to the pressure gradient needed in the two cross-stream

momentum equations to give cross-stream velocity components that will satisfycontinuity for each mesh region. The resulting equation resembles a finite-

differenr.e Poisson equation for pressure over the downstream cross section.This correction to the pressure gradient is needed to develop the correctcross flow within the boundary layer. Iteration is required at each station tosatisfy both the continuity equation and the momentum equations.

If heat transfer is neglible, constant rothalpy may be assumed. The

energy equation is used when heat transfer by convection and conduction is of

interest. Then the rothalpy at the downstream station is calculated implic-itly. Boundary conditions may specify adiabatic walls, specified wall temper-ature, or specified heat flux through the walls.

Cases of practt„al interest will have a turbulent layer, of course. for

this purpose, the Baldwin-Lomax eddy-viscosity model is used (ref. 4). Pro-vision is made for specifying a constant turbulent Prandtl number for theenergy equation.

14

e/

Numerical Examples

Several combinations of geometry, mesh, and flow conditions have beencalculated for verifying the PARAMAR code. Most of this has been done for

laminar flow to avoid the possibility of inaccuracy due to the tdrbulencemodel. Most of the geometry has been for straight ducts of various cross-

sections, at varying angles, and for accelerating and decelerating flow. Inaddition some curved ducts and turbomachine blades have been run, which aregiven as examples below.

Laminar flow to a curved duct. - The duct is shown in figure 3. Laservelocimeter measurements have been made by Taylor, Whitelaw, and Yianeskis

(ref. 12). The measurements and calculations were made at a Reynolds numberof 790, based on the hydraulic diameter of the duct. The calculation wasstarted at 0.1 m upstream of the start of Lhe bend with uniform flow. Themesh used for the calculation by PARAMAR is shown in figure 4. The inviscid

pressure field shown in figure 5 was obtained by using the Denton code(ref. 10). Some difftrutly was had with negative velocities on the pressuresurface near the start of the bend. There is a strong adverse pressure gradi-

ent in this region, as shown in figure 5, which resulted in negative stream-wise velocities being calculated. These negative velocities are closer to thewall than the experimental measurements shown in Ref. 12. When the streamwise

velocities become negative, the streamwise momentum is increased to a smallvalue (2 percent of the freestream value for this case). Thus a small amountof momentum is being artificially added to the boundary layer. The added

energy is negligible, but allows calculation through the small reverse flow

region. This is a variation on the commonly used Reyhner and Flugge-Lotzapproximation (ref. 13).

Carpet plots of streamwise velocities and cross-section vector plots areshown in figures 6 and 7. The results agree qualitatively with the experi-mental results. iii order to make a detailed comparison with the experiment,It is necessary to impose the proper flow field at the start of the bend,which has not yet been done.

Flow through an axial stator. - The blade passage 1s shown in figure B.A cusp has been added to the very blunt leading edge. Flow for this geometry

has been calculated for laminar flow at a Reynolds number of '197, based on the

upstream hydraulic diameter. An inviscid pressure fie'ld was obtained by theMERIDL and TSONIC codes (refs. 8 and 9). Calculated velocity magnitudes andcross-section vector plots are shown In figures 9 and 10. Although experi-

mental comparisons have not yet been made, the results are qualitatively cor-rect. An effort is being made to obtain a turbulent flow solution at a muchhigher Reynolds number.

REFERENCES

1. Patankar, S. V. and Spalding, 0. B., "A Calculation Procedure for Heat,Mass and Momentum Transfer in Three-Dimensional Parabolic Flows," Inter-national Journal of Heat and Mass Transfer, Vol. 15, Oct. 1972,pp. 1787-1806.

qq

15

P

2. Briley, W. R., and McDonald, H., "Three-Dimensional Viscous Flows withLarge Secondary Velocity," Journal of Fluid Mechanics, Vol. 144, July1984, pp. 47-77.

3. Davis, R. I., and Rubin, S. G., "Non-Navier Stokes Viscous Flow Computa-tions," Computers and Fluids, Vol. 8, No. 1, Mar. 1980, pp. 101-131.

4. Baldwin, B. S. and Lomax, H., "Thin Layer Approximation and AlgebraicModel for Separated Flows," AIAA Paper 78-257, Jan. 1978.

5. Steger, J. L., "Implicit Finite-Difference Simulation of Flow about Arbi-trary Two-Dimensional Geometries," AIAA Journal, Vol. 16, No. 7, July1978, pp. 679-686.

6. Pulliam, T. H. and Steger, J. L., "Implicit Finite-Difference Simulationsof Three-Dimensional Compressible Flow," AIAA Journal, Vol. 18, No. 2,

Feb. 1980, pp. 159-167.

7. Viviand, H., "Conservative Forms of Gas Dynamic Equations," La RechercheAerospatiale, No. 158, Jan.-Feb. 1974, pp. 65-66.

B. Katsanis, T. and McNally, W. D., "Revised FORTRAN Program for Calculating

Velocities and Streamlines on the Hub-Shroud Midchannel Stream Surface of

an Axial-, Radial-, or Mixed-Flow Turbomachine or Annular Duct. I-UsersManual, " NASA TN-D-8430, 1977.

9. Katsanis, T., "FORTRAN Program for Calculating Transonic Velocities on aBlade-to-Blade Stream Surface of a Turbomachine," NASA TN-D-5427, 1969.

10. Denton, J. D., "An Improved 'time Marching Method for Turbomachinery FlowCalculation," ASME Paper 82-GT-239, Apr. 1982.

11. Varga, R. S., Matrix Iterative Analysis, Prentice-Hall, New York, 1962.

12. Taylor, A. M. K. P., Whitelaw, J. H., and Yianneskis, M., "Curved Ductswith Strong Secondary Motion; Velocity Measurements of Developing Laminarand Turbulent Flows," Journal of Fluids Engineering, Vol. 104, No. 3,Sept. 1982, pp. 350-358.

13. Reyhner, T. A., and Flugge-Lotz, I., "The Interaction of a Shock Wave witha Laminar Boundary Layer," International Journal of Non-Linear Mechanics,

Vol 3, 1968, pp. 173-199.

16

4

BLADE-TO-BLADE SURFACE

f ^

z

MERIDIONAL PLANE

Figure 1. - Mesh and coordinate system.

0

k-2

k°1-11

k - I

k•112

I • 112n

1 1-112 2 2-112 3 3-112 4

^I,j,k An nj+l - nj - 1

X Win+112 j-112, k k+1 -i k - 1

O1 + 112, j, k-112

Figure 2. - Cross-section 1 ri -{ I surface. Mesh arrangement andnumbering <rheme for staggered grid.

O40mm

2.Omii

ri - 72 mm,

^. 1-+

0.3 m - — —

FLOC ^. / ro = 112 mm

Figure 3. - Dimensions of curved duct.

98

97

96NExLL; 95

wa 94U1-6F-

93

92

91G

4

r

STATION

280— r 'Irf'I.r

no ^ 1''if'

160 ^'/i%t W

FLOW

r r I r i ^Ir , r 1 11','r

z

Figure 4. Computational mesh for curved duct.Every fifth marching station shown. Every otherstreamwise mesh line shown.

DISTANCE ALONG WALL, m

(a) Suction surface.

Figure 5. - inviscid pressure for curved duct.

_ PRESSUREWALL

98

97

96

i

LLl 95

94

93

92

91 1 1---1_ L --1 --1 -- 1 J0 .05 .10 .15 .20 .25 .30 .35 .40

DISTANCE ALONG WALL. m

lbl Pressure surface.

Figure 5. -Concluded.

(a) Station 100. (See figure 4.)

Figure 6. - Streamwise velocities for curved duct

PRESSURE

MAXIMUMVELOCITY{654 miser f f =" 1-

4!6

I PRESSURE

SUCTION

(b) Station i60.

Icl Station 220.

Figure 6. - Continued.

PRESSURE

4'

(d) Station 280.

Figure 6. - Concluded.

SUCTION WALLF" IT T 1-T r Tr-`-Tr T-7T-T TTMM

,rr,l l l , . ,•Irll„ , ..

I l l,lrl.. , ,rrrrr

rll....rr,l^

r,lrrr^ ...,rrrrr

J _I

yrlllllll^ii••••••I•Irll y

3W^

nrr r r • r r • , rl a W'f,

(/^ III i i ^^ .. ^ ^ i . i • ^^ i i r r I I u N

nulr„ ^

ur r, I I^

^ ^ I I I ^ Irlrn.

^ I I ^ r r n u

PRESSURE WALL

(a) Station 100. (See figure 41

Figure 7. Secondary velocity vectors for cruved durt

3

0

OF Po f);; (, ,

SUCTION WALL1',11.111 1 t, r ^7[PII!

r„11„ •,11yplll i r^ ^ i 1111r r •mllrr ,1111

pm I l l s ^, ^ i , ^ ^ , r , i ^^ I I I I

JJ ^.m I I I, ^ ^ ^ ^ ^ I I m

i^l l^ , i i i i ^ i r I I r r^ ^ 1111

H141 1^, r r r r r ,Im

N ylll' I I I ^ r ^ 1111'

i"u I l r i r ^ r i i i ^ r i l l 11jnl' 11 I. r , 1 1' r I I II

.Inn

^

mnl '

mnr... .rnnr r

ml^.... , .. _.,ml

1-M-LiliuPRESSURE WALL

lbl Station 160.

SUCTION'WALL

•. A A hI d l l I r ^ ^

Ill^l I1 I r1 I

^ /^

^ 1 11 11

r I , 1 11^^11r r 1

1A11111 I r I I ^ '. -..r1I1111

dtll!I,.. - - - r I I ..,rlflliyAtlitlr.. r r I ,,rl llll1

J tltllll,. ,,11111 J}^ntllll•... ...rlilllr r ^

3w ;elllltlr, rltlil w

N 141 1 1^^ ^ r l l l l l N

µl ” ,IIII

PRESSURE WALL

lc1 Station 220.

Figure 7. - Continued.

F

r^r,'WQ 'S di8^e

SUCTION WALL

W^•.

.•iny W

PRESSURE WALL

idl Station 280.

Figure 7. - Concluded,

tip.

Figure 8. -Axial stator.

4-

TIP

i

PUOIi

e

MAXIMUMVELOCITY

I)4mise ^^,^

T IP

PRESSURE

SURFACE

HUB

SUCTIONSURFACE (a) Station 50. ( See floure 8.)

lbl Station 70.

Figure 9. - Veloclty magnitudes for axial stator.

MAXIMUMVELOCITYf'417 misec

SUCTION

SURFACEZ, / '

TIP

I_ PRESSURE

\\ SURFACE

-_.--_^ HUB

(c) Station 90.

Figure 9. - Concluded.

TIP

SUCTION 1 - - - - PRESSURE"SURFACE _ _ _ SURFACE

HUB

(a) Station 5Q (S&9 figure 8.)

Figure 10. Secondary velorRv vfctors for axial stator.

i

l

i

TIP„— - r T T< 'r T-t T t-..^

SUCTION PRESSURE

SURFACE SURFACE

HUB

III) Station 70.

TIPT^ r r r ^{-TT,

SUCTION" ' PRESSURESURFACE SURFACE

nu r 1 r r r. ,, ..

nut I I r r^ ,, i ,,....

urtt^^ I r

HUB

Icl Station 90.

Figure 10. - Concluded.

i

r