Secondary flow loss and deviation models for through-flow analysis of axial flow turbomachinery

MECHANICS RESEARCH COMMUNICATIONS 0093-6413/91 $3.00 + .00

Vol. 18(6), 403-408, 1991. Printed in the USA. Copyright (c) 1991 Pergamon 15ress plc

SECONDARY FLOW LOSS AND DEVIATION MODELS FOR THROUGH-FLOW ANALYSIS OF AXIAL FLOW TURBOMACHINERY

C.Lee and M.K.Chung Department of Mechanical Engineering Korea Advanced Institute of Science and Technology P.O.Box 150, Cheongryang, Seoul, Korea (Received 21 February 1991; accepted for print 6 May 1991)

Introduction

There have been a number of theoretical attempts to analyze the secondary flow effects on the spanwise loss and deviation in axial flow turbomachines. For examples, Gregory-Smith[l] developed a spanwise model of secondary flow loss using a classical secondary flow theory. Adkins and Smith[2] presented an approximate method which includes the spanwise mixing of secondary flow in the design through-flow computation of an axial flow compressor. And Roberts et a1.[3,4] proposed spanwise loss and deviation models due to the secondary flow using NASA experimental data. However, these secondary flow models are too much complicated to be used in conjunction with the through-flow analysis of the axial flow turbomachines. The objective of the present study is to develop another simple spanwise loss and deviation models which can be easily applied to the streamline curvature method as a through-flow analysis. The profile loss is calculated along a streamline, and the secondary flow loss is assumed to be distributed parabolically from hub to tip based on the mass-average basis. The spanwise deviation is roughly estimated by using the secondary flow model in plane cascade which assumes a linear variation for the deviation. The predictions by the present method are compared with those by previous models and available measurements.

Analysis The streamline curvature method employs the following set of equations to analyze the through-flow field between blade rows. (i) The radial equilibrium equation in the meridional coordinate : 403

404

C. LEE and M.K. CHUNG

-37 -+

--V~--~ - - ' ~

2[i~r

+ r~

V00rV 0 r ~"

V~ 1 2(~rr

1]'

(1)

7-1 where and

Q=(P02/P01 __~/~m_

) 7 /(T02/T01), (I+M0-

• m

rc°sO/rc ")sm2Cr + tanO ~ r 1 -M

(2) (3)

2

(ii) The continuity equation through the annular passage : r h = 2~r J;th PVmCOS¢ rdr

(4)

(iii) The Euler work equation along a streamline : Cp( T 0 2 - T01 ) = V02U 2 - V01U 1

{5)

The total pressure loss coefficient for a blade row, wT , is defined as i

wT

=

_

pi

P02i 02 , _ p PO1 1

(6)

where P0 is the relative total pressure corrected for change in blade speed. The total pressure loss coefficient, wT ,is obtained from the present loss model, and then the equation(6) can be used to obtain the downstream total pressure. The total pressure loss consists of the profile loss, the secondary flow loss and the tip clearance loss. However, since the magnitude of the tip clearance loss is negligible compared with other losses, it i8 not considered in the present study. The profile loss is calculated at a given streamline radius according to the profile loss correlation of Papailiou[5] for a compressor, and that of Balje and Binsley[6] for a turbine. For the secondary flow loss, Sieverding's qualitative explanation[7] and Cyrus's measurement[8] suggest that it is largest in the hub and the tip regions of a blade, while it is negligible near the mean radius. Therefore the 8panwise distribution of the secondary flow loss may be approximately represented by a parabolic function as

AXIAL FLOW TURBOMACHINERY

Ws(r) = a ( r - r m )2

405

(7)

The coefficient a in the equation(7) is determined by putting the mass averaged secondary flow loss coefficient of equation(7) to be equal to that from the secondary flow loss correlation of Vavra[9] for a compressor, and Dunham and Came[10] for a turbine. In addition, the spanwise distribution of the flow deviation due to secondary flow can be roughly represented by a linear model in plane cascade as in Fig. 1. The maximum value of the deviation angle is given by Bardon et al.[ll] to be Aa2,ma x = e ( 0.068 + 0.03 lOgl0 { ) , where { = (s'+(s'h 136~ and s' = ( co-~gTc°sa2)2 s .

(8) (9)

In the present study, in order to apply the plane cascade model described above to the annular cascade with radial blade twisting, the maximum deviation angle is approximated by the mean line parameters of cascade spacing and flow angles. Results and discussions The computational results obtained by using the present method are compared with the measurements for axial flow turbomachines. In addition, the present results are also compared with those by applying other secondary flow and loss models. Fig.2 shows comparisons between the predictions and the measurement[12] for the radial distribution of the total pressure loss coefficient of a compressor rotor. The predicted loss distribution by the present loss model is in reasonable agreement with the measurements, whereas the loss model of Roberts et al.[4] yields significantly lower values in the hub region. However the present loss model gives less adequate prediction in the endwall regions than in the meanline region. This may be due to the displacement effect of the passage vortex in the endwall boundary layer[7]. Radial variation of the relative flow angle at the compressor rotor exit is shown in Fig.3. Again, the prediction by the present method is favorably compared with the measurement, and it is compatible with that by Roberts et al.[3] except in the tip region. The present method is also applied to the design flow of an axial flow turbine[13] which was designed by the free-vortex method. Fig.4 shows the results for the absolute flow

406

C. LEE and M.K. CHUNG

angles at the turbine stator exit. The predicted result satisfactorily reveals that the fluid close to the endwall is overturned relative to the design flow and underturned further away from the wall. The classical secondary flow theory by Gregory-Smith[l] also gives reasonable prediction over most of the blade height, however it requires more computer time and data preparation than the present method. The axial flow velocity distribution is represented in Fig.5, which shows that the present analysis yields the reliable estimation of the axial velocity profile with peaks which are associated with the regions of flow underturning. However, as can be seen in Figs.4 and 5, the prediction in the hub region is less satisfactory than in the tip region. This is mainly due to the fact that the vortex near the hub is more intense and concentrated in the endwall corner[I,7]. Conclusions Simple models for the spanwise loss and deviation caused by the secondary flow have been developed for the streamline curvature computing scheme. The profile loss model is applied to the streamlines which are calculated by the streamline curvature method, and the spanwise secondary flow loss is assumed to be distributed in a parabolic form from hub to tip on a mass average basis. The spanwise variation of the deviation angle due to the secondary flow is approximated by a linear model in plane cascade. Using the through-flow analysis with the present loss and deviation models, the calculated results for radial variations of loss, flow angle and axial flow velocity are fairly agreed with the test results for an axial flow compressor and a turbine. The discrepancies between the prediction and the experiment in the corner regions may be further improved by considering the corner vorticity in more detail. References 1. D.G.Gregory-Smith, J. Eng. Pow. 104, 819 (1982) 2. G.G.Adkins,Jr. and L.H.Smith,Jr., J. Eng. Pow. 104, 97 (1982) 3. W.B.Roberts, G.K.Serovy and D.M.Sandercock, J. Eng. Gas Turbine & Power. 108,131 (1986) 4. W.B.Roberts, G.K.Serovy and D.M.Sandercock, J. Turbomachinery. 110, 426 (1988) 5. K.D.Papailiou, J. Eng. Pow. 97,295 (1975) 6. W.B.Balje and R.L.Binsley, J. Eng. Pow. 92, 342 (1968) 7. C.H.Siverding, J. Eng. Gas Turbine & Power. 107, 248 (1985) 8. V.Cyrus, J. Turbomachinery. 110, 434 (1988) 9. B.Lakshminarayana and J.H.Horlock, Int.J.Mech.Sci. 5, 287 (1963)

AXIAL FLOW TURBOMACHINERY

10. 11. 12. 13.

J.Dunham and P.M.Came, J. Eng. Pow. 94, 252 (1970) M.F.Bardon, W.C.Moffatt and J.L.Randall, J. Eng. Pow. 97, 93 (1975) W.R.Britsch, W.M.Osborne and M.R.Laessig, NASA TP-1523, NASA (1979) I.H.Hunter, J. Eng. Pow. 104, 184 (1982)

Nomenclature

Subscript

Cp:

Specific heat

h :

Hub

H h M m

Total enthalpy Half of blade height Mach number Meridional coordinate

i m s T

: : : :

Isentropic value Meridional value or meanline Secondary flow Total value

rh : P : r : rc:

Mass flow rate Pressure Radial coordinate Streamline curvature radius

t x 0 0

: : : :

Tip Axial direction Tangential direction Total property of fluid

s T U V z a ? e p 6 ¢ w

Cascade spacing Temperature Rotative speed of blade Flow velocity Distance from wall Flow angle Specific heat ratio Flow turning angle Density Endwall boundary layer thickness Streamline slope Total pressure loss coefficient

1 : 2 :

: : : :

: : : : : : : : : : : :

Inlet station Outlet station

Aa2(deg)/ Aa2,max Overtm'n~n~

I

Underturnln-

--AU2,rn.T

Fig.1 Spanwise variation of deviation angle

407

408

C. LEE and M.K. CHUNG

80

0.2

I

I::1

i

: Present

- -

----:

0 0

Roberts

method

'U

et aL[4]

v

o

70

: Experimen~12]

- : Present method ---: Gregory-Smith[l] ----- : Design

: Experiment[13]

i

0.1

o/

o

O

60

m ~u 0 o

0.0

J

i

0 Percent

Fig.2 Total p r e s s u r e

50

L

50 blade

I 50

Percent

height

loss c o e f f i c i e n t of r o t o r

Fig.4 Absolute

blade

flow angle

i 100 height at stator

exit

30

70 - -

: Present

----: ..~

J 0

I O0

O

R o b e r t s e t al.[3]

- - - -

60

method

:

o

Design

: E x p e r i m e n ~ 20

v

o

O o

o~ 50

O

10

: Present met : Experiment[13]

- -

o

40' /"

30

a 0 Percent

i 50 blade

i height

Fig.3 R e l a t i v e flow a n g l e a t r o t o r e x i t

0

100

I

0

I

I

50 Percent

blade

100 height

Fig.5 Axial flow v e l o c i t y a t s t a t o r

exit