METHOD FOR THE NUMERICAL CALCULATION OF VELOCITY DISTRIBUTION FOR A BLADE

(1)

METHOD FOR THE NUMERICAL CALCULATION OF VELOCITY DISTRIBUTION FOR A BLADE

CASCADE ROTATING IN A PERFECT INCOMPRESSIBLE FLUID

By

ERK'IEEF L. R.

Received, January 29, 1973 Presented by Prof. Dr. J. VARGA

Introduction

OYer the past three years, several papers have treated the determination of flow occurring around rotating cascades of air foils [1], [2]. The plane airfoil cascade is a mapping of a row of blade profiles [3] arranged on a surface with rotational symmetry, as is usual in an impeller (Fig 1). Theoretical considerations lead to a second type of the Fredholm integral equation which determines the absolute velocity along the profile contour [1]:

where

and

Ct G) _ 1

4'\.

,Ct'";!

^(~')

1.\....1 ^T",-

(~ ~') Id~'1

'=',S '='

- [

- Coox

2 t . ^{CBx (')]}cos e(,)

+

(K)

[c=y

+

^CBy^(n]sin

e(n + ^~

^Cn

^(~/)

^{KlI (',}

ⁿ ^IdC'1

(I()

K1(;,;') = - W(C,C') cos e(~) -

p(;,n

sin e(C) Ku(;,;')

= p(;,n

cos

e(c) -

W(;,~') sin

em

W=---

1 - 2e

g-~'

1)-'1/ ^I -l:r-,-

cos 2:1: - - - -T e t

P = - - - - ^e

o E-~' I t-g·

_:r - 1) 1) 4", - ]

2e ^t cos 2n t

+

e t

CBx (C) = -

~ SS + ::'

^C^x(z') W(',z') dA(z')

(T)

(1)

(2)

54 L. R. ERE.1!EEF

CBY(~)

⁼

~ rJ~ i!~

cAz') P(t, z') dA(z') t

1

^{b dx'}

(T)

The kernel KI is limited and continuous [2] hut the function KII is singular for C' = :.

Axial flow

Usually the flow is considered as proceeding in meridional streamlines of cylindrical form. In this case, the correspondent of the peripheral velocity in the straight cascade is a constant. In thc same manner b(x)

=

b_l

=

b₂and consequently C_BX

=

C_BY

=

O.

Suhstituting:

and

(^~')- . ^('~') I ("") _ .. ^(~') I (H)' ^8('~')

Ct ., - lOt '= ... Ilt., - Itt" ... I/O'" SIn "

Eq. (1) reduces to:

Let us consider the integral

I ₌

~ ':Y

[K-¹^SIn. 8(r') " ... ^I .L~1l ^T7 cos

8(~')]

^l, : " I ^{, d r'}ⁱ (l()

and putting K = P

+

ⁱ^(j)^{we have:}

I

= ~ {-

Im[eⁱ⁸(n K] Im[e^{iO (;')]}

(K)

.

• iO(~) •

-^- R ^e

4

ê^;0(,)^K^{e ·}îO(e')Îdr'l-^{I "}^{1 -} ^Rê^_e_._'

4

c o t " ^h^~^(.~^- ^.r')-, ^d:~'.".

2 t

(K) (K)

With the help of the residue theory, it is not difficult to see that:

1= _ t sin8(~) 2

(2)

(3)

VELOCITY DISTRIBUTIOS FOR A BLADE CASCADE 55

Substituting the value of I, as determined above in Eq. (2), we have:

1V"'Y sin e(c) ⁽³⁾ Relation (3) is the well known equation for axial £lo"w.

~Iixed flow

In the following considerations, we shall begin by 'Hiting an integral equation applicable to determining the relative velocity, knowing that

and using approximating values of C_BX and CBY [1]

Qn t

CBx c:,c -.-::..:.:..- - C " , , , - -

b(x) t(x) . t(x)

Eq. (1) takes the dimensionless form

CT

(C) 2

1 ~.

[77

^sin^8(C)]

*

(~') !d'~'! ^_ ^. 8('")

I

- .L\.I - ct '" I " I - tg Xl sIn " . "

t 2

(K) ..

~ ~ KJI ^c~

(C') IdC'1

+ [~(1 + ~) (1 __

^{t _ .}⁾

t

'f

2 b₂ t(x)

t b^l

J

cos 8

(~)

i"

t(x) b(x)

(K)

where

Putting 8

=

e(c) and 8'

=

8(C') for convenience, we obtain

cD

^KI-

1 .

r

t " _(K)

sin 8

2 * (:-')

Id:-'I

u;

^{(C) .} ⁸

1Vt '" . " = - - - -SIn -

I 2

(4)

56 L. R. ERK1fEEF

sin 8

~ * ("") .

8' 'd;-':

- - - lly" SIll I -I, I

2t ^<

l~'

'" ("")

[T7 .

8' ^I

V

8') Id""

- ll.;' " AI SIn , A l l COS I " I

t <

(K) (K)

+

^tg^(Xl^sin⁸ ^-^b^l ⁾

^r

^{1 - - -}

^t)"

^T ^tb¹ ^]^cos⁸

b₂ t(x) t(x) b(x)

fOl ;' ~; functions cJ> and P are known from reference [4]

t ~ ~f

Pps - - - - -

2:-r 1] -')"

cJ> ^{PS -} ^- t - - - ' - - - -

2:-r therefore the integral

- - l l v " I SIn (j

J - ¹

~ * (;-')

[K ' r'

t . _{A Il}^c'- cos 8'] : ,(" Z"'I i = (K)

=

+~ u~

^(;')[1j!(;,:') cos (8

-+-

8') - cJ>(:,:') sin (8

+

^8')]

ia:'\

(K)

will be transformed

8')

-!!..

^{sin (8}

+

8')J

ll~

( : ' ) -

t

+

^{1) -} ^1]' ^{sin (8}^.L^8')]

ll~ m} ^Id~'1 ⁺

(::. : ,; - , J 7 - 1 ] -"")" I ( ' ) " 1 } I

ur, (0 rK [ ~ - ~'

cos (8

+

^8')

2:-r

'f u -

^~')2 ^{Cl7 -} ^1]')2

(K)

+

^{sin (8} ^8')]

Id;! .

(~ - ~')2

+

^(1]^-1]')2

(4)

The first integral can be computed numerically. The computation of the latter can be made analytically with no difficulty.

(5)

lH

=~

^.

(K)

VELOCITY DISTRIBUTIO,'- FOR A BLADE CASCADE

(;

;

-

;'

---~--'---cos (8

+

^8')

;')2

+

(rJ - rJ')2

8')J !d~'l

• eiIO -'-6') • d:'

=

^Re

4 : _

^C'

^IdC'!

⁼ ^{Re e}^iO

4 ^-,,--,~,-

^- ^{:r sin 8}

(K) (K)

Hence, Eq. (4.) may be wTitten as:

1{,* (") t ,

2

sin 8 2

1

J ^tCt

^{.* (.~')}

^" ^fd^{. '"}. , ^{rll -}^{! -} ^- ^llv-^*('")~ ^SI'n^{8 _}

sin

8 ^~. ^{.* ('~')} ^.

^fJ'

^!dr,!

^I

[1 (1

^I ^b^l ⁾

(1 ^t)

- - - - l~v" SIn " , - , - - - -

+

2t • - 2 . b₂ t(x)

(K)

+

t(:)

;(x)]

^cos^fJ

⁺ ~ ^{ [~

^{cos (8}

⁺

^{8') -}

~

^sin^(fJ

⁺

^fJ')

J lI~

^(:')

_ lit (:)

2:r

(K)

cos (fJ

+

^fJ')

+

(rJ - 17T

- sin

(8 + 8')J}

^I'd:'!

+

^tg^Xl^sin^fJ

( " ; - ; - , r J - r J -"')'" ( ')?

Results and conclusion

57

The solution of the integral equation can be reduced to the solution of a linear equation system, using Gaussian quadrature. Thc contour (K) is divided into two parts K1 and ](2' the ends of which are the leading edge and the trailing edge. Since the abscissas and weights of Gaussian quadrature are symmetrical about the middle of the interval, such a division ensures a concentration of points in regions of great curvature.

The condition of a smooth flow at the trailing edge will be achieved by the restrictions Wt*(C₁₎= - wl'(C_n)if n points are assumed on the contour.

Typical results are shown in Figs 2 and 3. Fig. 2 presents dimensionless circulations

r*

for different numbers of points, with a view to increased accu- racy. The convergency is good.

Fig. 3 illustrates the velocity distribution over the blade.

(6)

58 L. R. ERE.IIEEF

x '" ~f

_2J[ ^!!b₂

+/~o _rj 7

NI 0

Y '" 271 cP IS!

J

^do

where aL =

r

o

and N = number of blades

p

/~~

r

/

'''"

/ \(/)

Fig. 1

r* 3 ,---,---~--~--~--~---,--

! i I

i I i

~

I • ⁱ : ^I

0

I

pOints r'

34 2,5 48 2,583 64 2,657 80, 2,674 96 2.686

a

50 100

Fig. 2

Wm/s Velocities at the leading edge (or three directions of approach.

4 r---~--~--~--~---,--,----,---,--~--~

a

^~

__

^~

__

^~

__________

^~

__________________ _

o

0,5 1 S' 0,45 0,5 0,55 S'

Fig. 3

(7)

rELOCITY DISTRIBUTIOS FOR A BLADE CASCADE 59 An analysis is presented for finding out yclocity distribution on a hlade giyen hy its geometry. The method outlined ahoye seems to he one of the hest for the numerical approach to the problem.

t b c u' C", CB U_v

r

Q;,

Notations in the straight cascade co-ordinates in the Z plane;

point of the profile contour;

blade pitch;

Iddth of the channel in the meridional section;

absolute velocity; ^<

relatj,-e nlocit~; ^<

basic flow yelocity;

",clocity induced by the yariation of the width of the partial channel;

the counterpart correspondent of the peripheral yclocity in straight cascade;

yolumetric flow between two blades in the laver of thickness b;

blade circulation .

References

1. Fiizy, 0.-TH'C)lA A. Calculating the ",elocity distribution of a plane airfoil cascade given by its geometry, Proceedings of the Third Conference on Fluid Mechanics and Fluid 11achin'ery, Budapest 1969 ~

2. NYIRI A.: Determination of the theoretical characteristics of hydranlic machines, based on potential,theory. Acta Technica Hung. 69 (1971)

3. CZIBERE. T.: 'Cber die Berechnung der Schaufelprofile Yon Stromungsmaschinell mit halb- axialer Durchstromullg. Acta Techllica Hung. 44 (1963)

4. Fiizy O. Design of llixed Flow Impellers. Periodica Polytechnica, Yo!. 6, ,t (1962)

ERE:\IEEF L. R., Ets Neyrpic Cedex 75 38 Grenoble Gare, France

METHOD FOR THE NUMERICAL CALCULATION OF VELOCITY DISTRIBUTION FOR A BLADE