Flow past a partially cavitating cascade of flat plate hydrofoils

t

k

Lab. y. Scheepsbouwkund.e

Office of N aval Rese a

Hogeschoot

Deift

Department of the Navy

Contract Nonr-220(24)

FLOW PAST A PARTIALLY CAVITATING

CASCADE OF FLAT PLATE HYDROFO1LS

by

R.

B. Wade

Division of Engineering and Applied Sdence

CALIFORNIA INSTITUTE OF TECHNOLOGY

Pasadena, California

Contract Nonr-220(24)

FLOW PAST A PARTIALLY CAVITATING CASCADE OF FLAT PLATE HYDROFOILS

by R. B. Wade

Division of Engineering & Applied Science California Institute of Technology

Pasadena, California

Report No. E-79-4 Approved by:

A,B,C - constants a - modulus of

, defined by equation (10)

c - chord length

- length of cavity in 'Ç-plane Lc - length of cavity in z-plane

c - pressure coefficient, equation (24) CL - lift coefficient, equation (24a)

i -

V-i'

K - cavitation number, equation (1)

p -

pressure

t - semi-circle plane

u - velocity component in x-direction y - velocity component in y-direction

V - modulus of velocity vector

w - u-iv, complex velocity function x, y - coordinates of physical piane

z

- x+iy

- flow angle with respect to chord

- stagger angle

c _{- defined by equation (10)}

- defined by equation (i8c)

N'2,n2 - defined by equation (19c) - transformation plane

- solidity c/2Tr

- perturbation quantities

1. Introduction

This report deals with the non-viscous, steady cavitating flow through a cascade of flat plate hydrofoils in two dimensions.

The usual assumptions of incompressibility and irrotationality are made.

The motivation for this investigation is the present day interest in the high speed performance of lifting surfaces, such as in hydroplane boats and the behavior of propellors operating under cavitating conditions. _{further area of interest is that of} turbo-machinery. The demand for smaller, more compact pumps and turbines, for any given performance, necessitates operation at higher speeds giving rise to cavitation conditions. Hence the

problem at hand is not only of theoretical interest but is of practical importance.

The problem of the fully wetted cascade has been

ex-tensively treated, and can be found, for example, in a paper of

Il*

(2

Garrick' /

and in standard texts such as Robinson and Laurmann.'

The case of cavitating flow through a cascade of flat plates with infinitely long cavities was first treated by Betz and Petersohn using the classical hodograph method for free streamline flow attributed to Helmholtz.

*

Numbers in parentheses refer to the references at the end of the text.

In dealing with cavitating cascade flows, hodograph methods became somewhat unwieldy, and this has led to the use of linearized methods for solving these problems. This method, first

(3)

used by Tulin assumes that the cavity-body system forms a slender body and that a perturbation technique similar to that used in thin airfoil theory may be used. The use of the linearized method leads to the solution of a mixed boundary value problem. The use and application of this method is well illustrated by Parkin. (4)

The first published paper on linearized cavity flows through cascades, was by Cohen and Sutherland.5 They dealt with the

problem of arbitrarily shaped hydrofoils with finite cavities, longer than the chord length. _{However, only results for the flat plate are}

(6) presented in their paper. _{Subsequently, Acosta and Hollander dealt} with the partial cavitation in _{a cascade of semi-infinite flat plates.} This problem was recently treated using _{a hodograph method by} Stripling and A costa7 but no formal comparison was made between

the two methods. Pcosta8 also considered the

case of the fully

choked cascade of circular arc hydrofoils. A comparison was made with the results of the linearized method with those obtained by _Betz and Petersohn; generally, a good agreement was found.

In the region where the cavity is less than the chord length, no results have been published to the knowledge of the author, for cascade flows. _{This case would provide a complete picture as to the} behavior of these flows over the entire range from the fully wetted to the fully choked conditions.

II. Formulation of Problem

Ps illustrated in Figure 1, the cascade consists of an

infinite array of flate plate hydrofoils having a stagger angle . The

chord length of each blade is c and the spacing of the hydrofoils in the direction of the stagger angle, is taken as 2ii. Hence the solidity,

= c/Zîr.

The flow approaches the cascade with velocity V1 at an angle of attack The flow is turned by the cascade so that far downstream the flow velocity V2 is at an angle d _{to the blade chord.}

The cavities spring from the leading edge and terminate on the upper surface of each hydrofoil. In keeping with the linearized theory the thickness of the cavity is assumed small compared with the blade

spacing Z,,. _{The boundary conditions on the free streamline of the}

cavity are then applied along the real axis, as are the conditions on

the wetted surface of the hydrofoils.

The velocity field is now considered as a perturbation about

the velocity V1. _{Plthough, in the neighborhood of the cascade a more}

natural characteristic velocity would be the vector

_{mean velocity V,}

it is found more convenient to adopt V1, as V is undetermined a priori,

since it depends on V2. In the calculation of the lift coefficient, however, the angle which the vector mean V makes with the blades, viz. ,

in in

is used so as to bring it in line with fully wetted cascade flows. The governing parameter in cavity flows is the cavitation number K defined as

lc

Z rv1

where p, is the pressure at upstream infinity and _PC iS the cavity pressure, which is a constant. Since the velocity is defined at any point as

V = (u,v) = (V1-4- u',v') (2)

where u', y' are perturbation components assumed small, compared to V1 w e obtain by the use of Bernoulli s equation

V2

V'

However, neglecting the squares of u', y' compared with VIZ, this

beco rn e s Zu C vi KV1 1= C 2

On the wetted portion of the hydrofoils, v=O, i.e., there is no flow through the blades. P further condition that has to be met, is the

closure condition which requires that the cavity-body system form a

closed body. This condition can be expressed as

dy=O (3)

body

The above conditions,together with the requirement that the velocity be finite at the trailing edge, enable a unique solution for the problem to be determined.

Hence the conditions to be satisfied are:

y = O on the wetted portion of hydrofoil

u= V1(l+) on the cavity

the closure condition, viz. dy=O body

V is finite at the trailing edge.

These conditions are sufficient to determine the velocity function at every point including the downstream conditions where

-i

VV2e

Before proceeding to solve the boundary value problem we

derive the following simple relations from continuity considerations. The velocity triangle is as follows

From this diagram we obtain

V _sin(

±) =

[V1sin(1±)

_{+ Vzsin(2+)1}

m ni 2

Vcos(o+(i) = V1cos(1+1i3) = V2cos(2+1'3) ₍₄₎

from which we get

III. Transformation Functions

Consider the transformation function

i-j--is ja

z=e'Jn

- +e'in

i4;.

This function maps the multiple-connected region in the z-plane onto the ¿'-plane, as shown in Figures 2 and 3. The function has branch points at and in the -plane, corresponding to

the points z = , respectively.

There is a branch cut between

and Ç. Hence when either point is encircled once, the argument of z changes by ±2e1 The sign depends on whether the branch point is encircled clockwise or counter-clockwise. Each Riemann sheet of the g -plane corresponds to the flow region over a different hydrofoil. Since the flow is periodic, however, the function is continuous across the cut.

The point =0 corresponds to the point z=0, as seen from

equation (6). Further, when

is real, z must also be real, as it

consists of the sum of complex conjugates. When tends to infinity, we have

-i3 g2 iÇ3

z*e

_{£nV+e £n}

which is a real number.

Since =0 is a singular point of the transformation,

dz/dg= O at

=0, i.e.,

(6)

e _g1

If we require that the trailing edge of the hydrofoil map into the point at infinity, then we must have dz/d'=O at = co. However,

dz

must - 0

as - at

co

This condition therefore gives

e(g1-

_{)+ e'(-}

) = o (9) Now let iQi =i'1e iQz

=r2e

=i-2e so that + = Tr-(cf1-Ql_ =

With this notation, equations (8) and (9) reduce to

lT r1 cos(

_{-- _/3+)}

Tr Z (9a) r i,3 + e =0 (8) (8a)

tanc9 =

The transformation is now completely specified.

Since the trailing edge corresponds to the point at infinity, we get from equation (7)

c = 2cos £na+ 4J' sin13 (11)

and hence the solidity - is given by

r=!cos(

ma +

Sfl(3

Tr

The point z

= corresponding to the end of the cavity is mapped into a point on the positive real axis in the -p1ane,

=L.

Thus, using the above notation together with equation (6), we get

L = 4cos

in

_n2 + 2sin (13) where 4 =1± -Zi.sincf 4

22

L n

=1+1 ¡a +2 sincP

2 a

r = tan

For these two equations to be compatible, we take _{c1= z=} hence

01+ _{2 =} = 2cJ,

With these values, either equation (8a) or (9a) provides an equation for r1/r2. Since the ratio of the moduli is the unknown, we are free to fix one of the moduli arbitrarily. Hence, we let _{j Ç} = r1 = i and

= r2 = a, where a ' 1.

Then from either (8a) or (9a), we get

a-1

a+ 1

tan3

(10)

(a-l)cos+

¿2sin2

a-(a-l)Lsin + L2cos 2

We now transform the upper half s-plane into the half circle t-plane, Figure 4. To achieve this, we use the well known Joukowski transfor-rnation in the following form

(-

) = (t + -) (14)

In the t-plane the semi unit circle represents the constant press re cavity surface and the real axis outside the unit circle represents the wetted portions of the hydrofoil. The leading edge is at the point t = -1, and the trailing edge at t = co.

The t-plane is used, since the velocity function for the given boundary conditions shown in Figure 4 can be written down by inspection.

'V. Solution of the Boundary Value Problem

The velocity function

W = U - iv

= + + C (15)

where A, B, and C are real constants, satisfies the boundary conditions for suitable values of A, B and C. This function corresponds to sources (or sinks) placed at the leading edge and at the end of the cavity. We now apply conditions (a) - (d) from page 4.

On the cavity, viz. , t=e , (u , y ) = (V + u ',v _') , therefore

C C i c c but p Q

u -iv =- l-itan-

_{c c 2 2} hence A-B U 2 +C Q .1 cot - -.ij + C

thus

V1(l +

-

_- A-B +C

2

Condition (a) is satisfied by equation (15) since when t is real,

v=O. Further, condition (e) is obviously satisfied.

To apply the remaining conditions, it is more convenient to transform equation (15) back into the ,-plane, by use of the transfor-mation function (14). Inverting equation (14) we get

t

=

[-+]

The positive root is taken because t tends to infinity as tends to

infinity. On substitution of this expression into equation (15), one obtains

A-B B

w(fl=

₂

_+C+T

V3-L

expression into real and imaginary parts, gives cos V1cos1 =[ + c] + 21 [

sin'1

r Vsincx. 2..

I--+An

1 1 2 L w h e r e

Y1 - tan'

_{- 1-2sin9}cos g'

=

1+2-2tsin '

Now, applying the condition w(2) = V2e (16) (17) (18a) (18b) (18c) Now -ici. w(1) = V1e

we get

V2cos o + c] +co

We finally have the closure condition, viz.,

dy=O body which reduces to Im pt -.f-i w(z)dz = O body B _{- An (19a)}

Since w(z) is an analytic function in the flow region around the hydrofoils, we can deform the contour in the z-plane to the contour

P shown in Figure 5. Then, symbolically we have

A E

w(z)dz =

+ f' +

C + um C + urn

JE'-O

i ¿_,.O

body H D 1 2

wdz = O

The contributions from the other parts of the contour cancel due to the periodicity of w(z), while the contributions from the last two integrals in the above expression, are zero. Now

2 V2sin.2 =

sin 2

[B

- + An

(19b)

L2

I

a 2 where

'r2 -

tan'

_a+sincJ)

£cos çJ' 4

I

i

si

V2sin2

_{= z} -i(1+f3) w(z)dz = ZirV1ie -i( w(z)dz = -ZTrV2ie

Im pt

-

w(z)dz _{[Vzcos(oz+(3)_Vicos(i+(3)j} = O body hence V1cos(0L1+13) = V2cos(o(-2+S) (20)

This is the same result as already obtained by continuity considerations in equation (4). Finally we have the following equations to solve: A-B _{+ C = V1(l +} Y A-B

V1cosa1=---+C+

₂

sin;

_B V1sino1 = 2 [ + An1] V COS r A-B 2.

lB

V2coso2 = 2 + + 2 L An2] +An₂ V1cos(«1+) = V2cos(o2+) [B

_--An

L1

K F

(l+T)=cosl+.- s1nl

E tan = D(l + sin E

sin1

V1

D sin2

w he r e G _siny2tanA -n2 Icos

L2

n1j

F

=_!

-'y1

'2

IL n _{1 Yi Vz} n n

cosTcosT+sinYltan/_ _+_IcosTsinTtan/3

2 1J _n2

nJ

-r+i

T

COS

T

L 2 l

sin

_T

cos

_T

_tane

Equation (21) gives us a relation between the cavitation number and the

cavity length L.

If we consider the limit as the solidity tends to

infinity, this equation reduces to the expression obtained by Acosta

and Hollander, (6) for the case of semi-infinite flat plates. Further

details are given in Appendix 1.

We now calculate the lift force acting on the hydrofoil. As mentioned previously, we will here adopt a slightly different per-turbation procedure, so that a comparison may be made with the fully wetted case. We use the vector mean velocity Vm as reference velocity.

The element of force acting on the blade is

n1

"2

+

n

L 1

sin _-i- cos

--

2 sinV

_1[n2

-

1 21 Vl

- - sin - sin -z- tan (3

_n1J

2

rn

_n2

sinY -I 'Yi tt2

r1

E = cos sin

I

-2 2

_L"

- n1j

sin -

2 sin

T

tan(3

dF = (p-p)dx

F =

body

Defining the pressure coefficient C as

C

pl

=

2

-v

2 rn

and the lift coefficient as

F

CL=l

-ev

Z

Z'

ni we obtain 1 CL C 2 L cV 2 C,, = i- cV rn c dx J p body

Using Bernoullits equation this becomes

CL!

-J._(u-V )dx rn body which reduces to Re Pt w(z)dz

V1sin(1+) Vzsin(Z+)1

By the use of equations (5), (20) and (23), we can eliminate and in the above expression, and deduce the following:

-4 1 C =

s1n-L cosi D in ¡

--+ i

(24) (24a) (25) ni _body

on the body. Carrying out the indicated procedure in an identical

As D, E are functions of L , the cavity length in the -plane, this

expression can be used to obtain the limiting case for the fully wetted

cascade, i.e.

, when tends to zero. This is carried out in Appendix

II. The result reduces to that of the well known, fully wetted solution, viz.

c --- sin (26)

L coso a+l m

P further limiting assumption in the linearized theory is that the angle of attack _,

is small.

If second order powers of . are neglected, equations (21), (22.), (23), and (24) reduce to

From these equations, the results shown in Figures 6 - 28 were obtained.

V. Computational Procedure

The numerical calculations were conducted on a computer and the general method of computation is outlined below.

For a given cascade geometry, viz. ,

and 8

, the value

of a and were determined by the simLiltaneous numerical solution

of equations (10) and (12). With these values, the functions D, E, F,

and G were evaluated, for values of 2 , ranging from zero to

K F (2 la) 2oL1 - D E (22a) = 2

G+D

K E i v2/vl = . (23a)

-D

-4 1 a. m (2 5a)

CL=.

cos, D + -

-approximately two hundred; this latter figure giving a value of 0.99

fo r The ratio 2/c can be found from equation (13). Having

determined these quantities, the values of K/2,

, V2/V and CL

are found for various angles of attack The process is repeated for various stagger angles , holding constant. This final

parameter 0 is then varied and the above procedure repeated. The range of values considered is given below in Table 1.

Table i

The Fortran program used in the computation of the results is given in Pppendix III. This program is incorporated so that, if required, the data may be extended for other values of the parameters.

The data cards for the program have a format as given by

statements 14 and 15. Statements 133 through 100 give the numerical

method adopted for the simultaneous solution of equations (10) and

(12) to obtain a and . The remainder of the program deals with

the evaluation of the required data.

It should be noted that for the case of /3 = 0, the numerical solution adopted for the solution of equations (10) and (12) breaks down, and the program has to be slightly modified to accommodate this case. For (3 =0, the above equations can be solved explicitly, hence

state-ments 133 through 100 may be omitted. The remainder of the program

is essentially the same though somewhat simplified.

Parameter Range Solidity Stagger angle Angle of attack 0.25 to 1.25 -75° to +75° o o i

to 6

VI. Discussion of Results

Figures 6 - IO illustrate the relation between the cavity length and the cavitation number, for various geometries. The case of the isolated, partially cavitating flat plate is also shown on each

graph. The values for this case were obtained from reference (9).

It is of interest to note that a feature of the linearized theory is the fact that after a certain value of _c' the theory predicts two different cavity lengths for each cavitation number. This is apparent from figures 6 - 10. Since, in any case, the linearized assumption that the cavity-hydrofoil system forms a slender body would not be met for large values of Q/c, it is assumed that the validity of the theory

only holds good for values of minimum.

This behavior is to be expected due to the cavity model chosen, which places a singularity at the end of the cavity. However, comparing the results with that of the isolated hydrofoil, we see that this range of validity is increased in the case of the cascade. It would

seem that the cascade effect has the property of reducing the strength of the singular behavior at the cavity end. This is further illustrated by the fact that as the solidity increases the range is extended, until at solidities greater than 0.75, a single valued function is obtained over almost the entire chord length for positive values of stagger angle. In the case of negative stagger angles, corresponding to the case of a turbine, as distinct from positive values of which correspond to a pump, we see that there is still a region where the function is double

valued. Physically, this is to be expected, since the effec t of the

It is seen that there is a large difference between the cavity geometry in cascade, compared with that of the isolated case, even for

si-nail solidities. However, this comparison is not entirely justified

as the value of K/2.&1 is based on the upstream angle of incidence. In the case of the cascade, a more natural angle to adopt is that of the mean velocity vectoi V_m

Further, the curves at first glance seem to indicate that the cavity length for a given cavitation number at negative stagger angles is less than that of an isolated hydrofoil, even at low solidity. This surprising effect, however, is due once again to the choice of the upstream conditions as a reference. If the mean conditions are taken as reference, the curves for negative stagger angles will be raised above that for the isolated case and those corresponding to positive stagger angles remain below it as would be expected. If the

curves are based on this angle, therefore, a better comparison is

achieved. This is clearly illustrated in Figures 11 and 12 where the

cavitation number is referred to the mean angle _m

There is still a significant difference for ail values of /3

having solidities of 0. 5 and greater. It therefore seems that the

cascade effect is not very pronounced for soiidities up to 0. 5 provided the stagger angle is within the range -30° to +600.

Figures 13 and 14 illustrate a further representation. Here

m c

the value of K /2 is plotted against

£ Ic,

where K is defined as

ni ni C fi

K=

_{ni i} Z

which to first order, reduces to

K = K + (l+K)(1-«2)tan (9

m

with the help of Bernoulli's equation.

It is seen from these graphs, that for low solidity the curves lie very close to that of the isolated case, for all values of This representation, however, indicates the opposite effect to that using K/21, viz., that cavity lengths, for constant cavitation number, are longer for positive stagger angles than the isolated hydrofoil,

even at low solidity. Consequently, it seems that the parameter

K/2

_rn is the most natural one to use.

A disadvantage of using these alternative forms is the fact

that they depend on

O,

whereas the value of K/21 is independent of

the angle of attack and thus facilitates presentation immensely. Figures 15 to 22 show the variation of force coefficient with cavitation number for varying cascade geometry. It is significant

that the force coefficient is little changed over the range -30°< /3 < +30° for a constant solidity. Since the linearized theory breaks down for large stagger angles, this effect is to be expected. The breakdown of the linearized theory is due largely to the fact that at large stagger angles the assumption that the cavity thickness is small compared with blade spacing can no longer be expected to hold, except for very small angles of attack. Ps shown in the curve, the force coefficient for the isolated hydrofoil is approached as the solidity decreases. However, once again, we see that for solidities of 0.5 and larger, the cascade effect is prominent.

The curves as plotted, are terminated at the points where

I/c is a minimum Here again, the point is illustrated that the

mean conditions seem to be the natural choice for reference.

In the remaining curves, Figures 23 to 28, the behavior of the downstream conditions is illustrated. Here again, the curves

are terminated at the point of minimum i/c.

It may be pointed out

that at =00 _{the theory gives V2/V1 as unity, but shows that is}

still variable and not equal to This apparently is a violation of tie continuity equation which would necessitate ₁₌

¿X This

discrepancy is due to the linearization procedure which neglects quadratic terms.

VII. Conclusion

P linearized theory has been presented for the partial

cavitation in a cascade of flat plates. The results have been presented in such a way that they may be useful as a guide in the design of

turbo-machines and other applications. From the results, it is

possible to determine the cavity length, lift coefficient and downstream conditions for any desired cavitation condition for a given specific cascade geometry and initial upstream conditions.

The limitations of the theory are stressed and it is shown that the cascade effect diminishes the singular behavior at the end of

the cavity. In the case of the isolated foil the theory holds good up to a ratio of cavity length to chord length of approximately 0.74,

whereas in the cascade flow this ratio varies from about 0. 8 for small solidities up to 0.95 for larger solidities.

It is further shown that for solidities of 0.5 and over, the cascade effect is appreciable and cascade interference effects cannot be neglected in this range. However, for solidities smaller than 0. 5 the cascade effect is relatively small and the isolated case may be used as a fairly good approximation provided that the mean conditions are taken as reference quantities.

Appendix I

From equations (10) and (12) we get

fo r tan =

Y'

cas -T7: Substituting in (21) gives a-1 a+l tan ¡3 a - co, (g-., ₍₃ hence n1

[i

+

¿2-Zhin

1/4 cas (3

tan'f'-

_1-s1n'3

n2 - i O

Therefore we get that

cos(3 sin

'f1-2 n'

l-sin/3

cas - 2 nl and

/2

V -1-f-cos(3 n12+1-sin13 sin /3 hence cas

Ztan(3

a =e

2

T =cas(3na+

Tr sin 2

K

(1+ -z-) = cos sin

(n12+1) VnI2l+sin

- cose

If we now change notation to that of costa and Hollander, we get

then

nl -and we further replace

Vl+K = cos - sin

- b;

(l+-) by

Vl+K

j\fter some manipulation, this reduces to

l+K = cosct - sin (n12-l)

Vnl2+1sin

+ sin

(l2) V121b.

- b sin

(1+2) V21b

sing -

b cosi

bcos '( L2-l+b 1

1-bsinï

£2

(l+L2)cos_VjV

1+

l-bsin

cos '

2 which is the expression given by Acosta and Hollander.

Further, we see from equation (22) that since E O as

- we get

cx2 O for all

Pppendix II

In the expressions for D and E we expand each terni in

powers of £ , for small .. , retaining powers up to and including 0(

0()

)2!+

0()

--2 sin )cosL3+ 0(L4) Z cos sin ) 3'23+

0()

1 3

11

= (-+ +

--)sin 2cos'+(.- )cos +(l+

D

[1

1 i 2 1

8a 8a

a4a

1 1

.2

cos41+( )cos +(1+

) - sincos

tan

E ( +

-+

)sin

1 1 3 1 1 2 8a 4a 8a _{4a2 8a} L

+ 0()

hence )sin2

cos+(-)cos

1 1 +(i+)sinqcos2tan,s D L 8a 8a 1 1 1 .2 1 1 3 a 4

sincos2pans]

[-( +

+)s1n cos+ (

)cos

+

(l+1)L

8a 8a fll

-

₂₁ 1 2 2 +

(+

+ ) sin L.2 1J 4a n -(1+ 1)sin

+

(-sin2

)(1-n Z sin 1

cos±

sincos2-( cos

a Z -sin Z cos sifl(Co58Z1CO5 a 2 ' 2 a a 'r2 3 2 cos cos 0(L4) ---2 +

sin?cos5' _{L2+(sin2f cos}

8a

'rl ''2

cosZ

1 sincos2

o()

+(l- -)

sin -i-- 5ifl--- _{4a a 4a}

- 2 cos sinqcos q £2 (cOs2

+ 3cos2 n +

cos

---sin----Za

- Za2 8a a

2 2a

Utilizing these expressions we obtain

a-1

Now substituting for tan = tan cQ in above, gives

a+1 D

f-a+O(fl

as thus C i L _cos,3 sin c.nl as

which is the classical fully wetted cascade flow result. Now, when

îr

a=e

then

tanh('rir/2) _sin

Q

CL =

(/Z)

_m

for the isolated hydrofoil C-#O this reduces to

C =2-rrcx.

Pppendix III

We present here the Fortran Source Program as used in the computation of the results presented in this report. The notation used is self evident from the program.

)I[ìS!Íj LCP(5O0),ELCN(50C),AP(5C0),AN(5OO), XBP( 5GO) at 50G) ,CP( 5C0) ,ÇN( 5L0) ,DP( 500) ,DN( 500) 5 FOpPT (1H ) 14 FURN'AF( 4I'i,F1L.e 15 F0RA ('1[1.) 16 FURNAH')LIG.b)

20 FrRrAT(JH SIM'\=E16.,7H I-FtA=EI6.à,

X3H 4=E16.8,iH PHTLl6.a,7H ALPHiE16.)

3u FURfrATtIt' ,F'.3,Fj2.4,Ed,'i)

22 FÙRMAT(6X, 1HL,X, IHK,7X,8HK/24LPHI,6X,4HALF2, XhX , ZHLC , 'X,4hLC/C)

133 FuRMAT(IH rh.12.4)

P4M1F (A, , 1., 2 ) = lW2*LOOF ( í

PA2F( ;\,3) A1ANF( ( Aj. ) *',t3/( 4+1. )

READ l4,NSLu,N3E1 ,NAL,NEL,CONST

R[AL 15,SIQ,0sIG,bErfl,Dt3:1 ,ALPHO,DAL,A0,040,EL0,DEL

S I u 1 A S G M i DU 500') [S1Gl,NSIG DICK=3.14I2*SIG?A*o. E 1 A El U

DO 4û0j 1i-1l,bET

W1DICP/SLF(bcT) W3=SINF('iEIA)/COSF(ßEIA) W 2 0 5/W 3 $=AU D A DA (J 40 Y1-PAM11 (r\,'I,'t2) Y2PA?F(4,W3) Y3 (Y1Y2)/ (Y j+Y2) Y4=ABSF(Y 3) IF(Y3)5i, IUL,6U 50 IF ( Y4CUiS 1)100, 100,55 55 A=4DA DA D / I C. A=A+DA GO Tu 40 ori 4=4+04 r;o ru c 100 PHI=Y1 1=0 PO 3000 IEL1,NEL FI = I EL* I EL CL = Fi C E L F1=( I.+EL*E1 F2=( i. +( (EL/A) "2) F3=2.*EL*LNFf PHI) E N P 14 = F 1F 3 [NP24=12+(F 3/A) ENNI4=F 1+r3 E'N24=F2 (F 3/A) HAP=ENPI4/E\1P24 HALNN i4/iJN24 F1EL*C1SF(PHI i *( 4-1 F2=EL*EL*S[NF(2.*PHI) F3=A-i-EL*EL*CPSF(2.*PHI)

CPH=- (F 1+12 / ( r3F4) NHA(F1U2)/(F3+F4) IFt PHIU, F) 3010, 3010, 3020 3010 G4MPÏ!\NFC,H4) GlMNATANF( GHA) 0 TO 30U 3020 IF(('ãH')3O2I,3221, 3G22 3021 GAMN1 4JF(GrIHA) GO TU 2U 3022 G'UN=ATF(GNHA)-3. 1'.15926 3028 IF(HA)3030,3030,3029 3029 L1 3030 IF(L)3u31, 3C31,3U40 3031 GAP4TA'JF(GPHA GO TU 3050 304C' IF(GPHA)3C4,3C's2,3042 3042 GAP=ATA'F( GPF4A) GO 10 3U0 3045 cAMP=ArAhFc;pHA)+3.141592o 305C ELCP( IEL)=GOSFeLTA.100FH4P-2.*SINF(UETA)*GAMP

ELCÑ( ¡[L ) =CUSF ( ¡4) C HA) #2. *5 1JF C

F 1= SQ rE (L iP 14) ENP1S.Jifl F( Fi) Fl=SQRIF(FNP?4) FNP2= S'RT F C Fi) F1SQ111 C EN14) ENNL=SQTF(FI F 1=SLP1 I (CNN2's) EN2=SRTI (Fi) F1=EL*flSF( PHI) F2=EL'SI1hH1 TPI=Fl/( 1.-12) r 2 = F i / C A + F? TN1=F1/( 1.+F2) TN2=Fi/(A-12) FP1j./SQR1F(l.+FPL*TPj) Fp2=1./SoRIr(1.+rp2*Tp2) FNl.1,/SQR1F(1.-s-TNITN1) FN2=1./SçRfF(i.+FN2*T'2) SP1=AtSF( 1Pt*FP1) SP2=At3SF ( FP2*FP2) S N I. ¡ N * F N i SN2=A3SF( 1.2*FN2) CPI=SEUNI(FPI,IP1) CP2=Fy2 CN 1= FN i CN2SIGNF(F\2, TN2) SPH1=SRTF C u5*( 1.CPi)) SPH?=S)P 1 Ft C C i .-0P2) SNHlS0kT1(u.*(1.CN1)) SNH2=SORÏF(0.5*(1.CN2))

CPHI= SQR IF ( j i*( 1. +CPI)

CPH2SRií ( U. 5*( i. +CP2)

CNHI=S0?TF( U. L +CN1) CNH2=SURIF(0.5*(1.4-CN2))

FPlENP1/ENP2

F P2 1 / F P 1

F N I E NN I / F L'2

FN21 .11:1

F1'SiF (

t[

IA) /.OSF I FEI.\

F1-FPI+FP2 F2 = F P1-F 2 F 3 S P fi I S Pi 2 *F2* F 4-F i F(i F5.F2*CPHI CPH2 AP(IEL)=FlSPH1*CPh2-SP1-F3 FW(ILL)=SP2-FlSPH2*CPHI-F3 CP(IL)=F5SI*Fti-F1*CPHl*PH2*FÛ

DPI TEL) SI2*FU-FS-F I*SPH1 CPH2*FC

FO=-FO F 1=FN1 +F.2 F 2 = F N 1- F N 2 F 3=SNHI*SNII2*F 2*FU F4=F1Fí] F,=F2*CNII CJH2

AN( IEL )=F I*SNHI*CNFI?-Sfl-F3

N( IEL) SN2-F I*S\JH2*C'JHl-F 3

CN( ILL) =F SNI* 10-F I*CNH 1* SNH2*FO

3000 flN(IEL)=SN2*FU-F-F1*Sf'tHI*CNH2*Ír

ALPH1=L PHd

3500 Ii=I,NAL

PRINT

PRINT 20,SIoMA,EÏA,A,PhI ,ALPHI PRINT 5 PRINT 22 PRINt 5 DU 315u ZLL=1,JEL F1= IEL TEL EL=F 1.*DL CAPO=CP(ILi)/AP( TEL) CAPA=2. *ALi'H1*CAPU LPH2=P(ILL)/(DI(IEL)+(AP(IEL)*(I.+CAP\*ú.5)/ XALPH1) A L PH 21= ALPI 2/ A L PH 1 V?V 1=BP U EL )/( 4P( tEL ) *ALPH21) Fl=AP(IEL)/fP(iEL) [)CLz4.*iF1-I.)/((F1+1.)*SIGMA*CUSFUETA)) F2 A L PH 1+Bl L A F4=S INF (F2) ¡CUSÍ (F2) F2=ALP-2+ET4 F5=SINF(F2)/CÛSF(F2) F3=O :* ( Fi-F 5) IF( F3) 31tii, 3110, 3110 3105 ALPHMATANF(F3)-BEtA43. 1415926 OU TU 3115 3110 ALPHM=A)ANF(F3)-BEIA 3115 CLLDCL*ALI1 F1ELCP(IEL)/N,.*flICK)

3150 PRINT fl,FL,CAPA,CAP0,ALPH2,4LPH21 ,V2VI. ,CLL,DCL,

X[LCP(IFL),Fl

r1=-FETA 12=-PHI

PRINT 2(,.,S I?,F,A,F2 ,ALPHI

PRINT PRI'.T 2? DO 3250 ILL1,N[L FI=ILL*ILL LF1 '3CL CApíCdIFL)/ANNE:L) Ct 4=2. *CA'u4LH1 ALH2N(Iç:L)/(Dr\J(IEL)+(4N(I[L)*(L.4.CAPA*Q,5)/ X L H i ALPH21=ALPH2/ALPHÌ V2V1=tN(IEL)/(AN(IFL)*ALPH21) F 1=.AN (1 IL) /NN (I EL)

DCL=4.(FI-l.)/((Fi+l.)*SI',t4A*CUSr(BLTl\)) F?-ALPF I - IA F4=S I NF (F2) /CuSF (F2) F=ALPH2-tL IPt F5=S Ih,F(F?) /CÜSF(F2) F 3 0. 5 * ( F4 + F ) IF( F3) 32 10, 32 , 3205 3205 ALPHAI4NF(F 3)+3ErA-3.1415926 GD T .3215. 321u ALPIW=\TANF(F3)+bETA 3215 CLL=0CL*ALIHM F1=ELCN([Et.)/(4.*DICK) 3250 PRINT 30,EL,CAA,CAPÜ,ALPH?,LPi12I ,V2V1 ,CLL,F)CL, XELCN( ILL),FI 3500 ALPH1=ALF'It+D4L 'i000 3E1A=t3FTA+DiET 500C SIA=SIfr4+DSIG CALL EXIl E NI)

R EFER ENCES

Garrick, I.E., "On the Plane Potential Flow Past a Lattice of 2Arbitrary Pirfoils." NPCP Report No. 788, 1944.

Robinson, P. and Laurmann, J. P. , "Wing Theory'. Cambridge University Press, Cambridge, England. _1956.

Tulin, M.P., "Steady Two-Dimensional Cavity Flows Pbout Slender Bodies. " David Tyler Model Basin Rep. 834, May 1953. Parkin, B.R., "Linearized Theory of Cavity Flow in

Two-Dimensions.' Rep. P-1745, Rand Corporation, Santa Monica, California.

Cohen, H. and Sutherland, C. D., 'Finite Cavity Cascade Flow." Math. Rep. No. 14, Rensselaer Polytechnic Institute, Troy, New York, Ppril 1958.

P co sta, P. J. and Hollander, A., "R emarks on Cavitation in

Turbornachnes." Rep. No. 79-3, Eng. Div., California Institute

of Technology, Pasadena, California, October 1959.

Stripling, L.B. and Acosta, A. J., "Cavitation in Turbo Pumps -Part 1." Transactions of the ASME, Journal of Basic Engineering, September 1962.

Pcosta, P. J., "Cavitating Flow Past a Cascade of Circular Arc Hydrofoils." Rep. No. E-79-2, Eng. Div., California Institute of Technology, Pasadena, California, March 1950.

Acosta, A. J., "A Note on Partial Cavitation of Flat Plate Hydro-foils. " Rep. No. E-19-9, Hydrodynarnic Lab., California Institute of Technology, Pasadena, California, October 1955.

Fig. i Partially cavitating cascade of flat plates. U =0 v CAVITY zPLANE WETTED SURFACE a2

Fig. 2 Linearized boundary conditions in physical z-plane.

-PLANE

Fig. 3 AuxIliary -plane.

t- PLANE

Fig. 4 Auxiliary t-plane.

z - PLANE

Fig. 5 Integration contour in z-plane.

D CAVITY

/

/ I u

\

\ V (I -f-WETTED SURFACE CV0 01 vro b - I +1

Io 9 8 6 5 4 3 2 0o i.. ¡ I I i i i ¡

it

I Ii I i I i ¡ i

Ii

¡ 0.2 0.4 0.6 0.8 .0 1c /c

Fig. 6 Ratio of cavitation number to twice inlet angle vs. cavity length to chord length ratio for

various stagger angles f3, at a constant solidity,

a= 0.25.

2 ve

K

K 2a1 2 Io 9 8 7 6 5 4 3 2 o 0 0.2 0.4 0.6 0.8 1.0

Fig. 7 Ratio of cavitation number to twice inlet angle vs. cavity length to chord length ratio for

various stagger angles ¡3, at a constant solidity, o= 0.50.

K

2a1

ISOLATED HYDROFOIL

O 1 i i i i i I i i i i i i

Ii

0 0.2 0.4 0.6 0.8 I .0

Fig. 8 Ratio of cavitation number to twice inlet angle vs. cavity length to chord length ratio for

various stagger angles , at a constant solidity, = 0.75.

K

2a

i 12 Io o- 1.00 - ." o

!!FIllIIII(II.

o

lc/c

ve ¡3 ve/3 ISOLATED HYDROFOIL _750 6O0 _450

-Fig. 9 Ratio of cavitation number to twice inlet angle vs. cavity length to chord length ratio for

various stagger angles , at a constant solidity, = 1.00

2 IO o- .25 +ve ¡3 ve/3 ISOLATED HYDROFOIL N 600_ _450 _3Q0 I 50 -_00 ₌ +45° +600 I I I J i i i i I i i .6 0.8 I .0

Fig. 10 Ratio of cavitation number to time inlet angle vs. cavity length to chord length ratio for

various stagger angles , at a constant solidity,

c= 1.25.

K 2a1

0.25 a1 60 +ve/3 ₌ -ve$ ISOLATED HYDROFOIL

-t t I I i i i i i I i i i I i i I i i o 0.2 0.4 0.6 0.8 1.0

c/c

Fig. 11 Ratio of cavitation number to twice the mean flow angle vs. cavity length to chord length ratio for various stagger angles 3, at a

constant solidity o 0.25 and inlet angle of 6 2 IO 9 8 K 2am 7 6 5 4 3 2 o

I.00 = t a1:6°

60°

-:

--

I f

-

f t _i

-

t I o 0.2 0.4 0.6

I /c

Fig. 12 Ratio of cavitation number to twice the mean flow angle vs. cavity length to chord length ratio for various stagger angles f3, at a

constant solidity 1. 00 and inlet angle of 6

0.8 .0

9

-I t

tt\

_i

--

I \ t

t'

8 I \

\

K 7 2am 6

L

'\

\ \ \ \ \ \ \ \ 5 \ \ \ I \ f \ \ i \ \ ¡ \ \ 4 \ I \ \ \ /

\

\\

\ / \

'I

\ \

\ -3 , _\_

---

_{+ 750} 2 +ve/3 veß ISOLATED HYDROFOIL O I I f f f t i f f f I i t I I T I i j i I f I f 12 I0 30° 00

Km 2am _750 +ve /3 ₌ veß ₌ ISOLATED HYDROFOIL

ii

o I I i i i i i i I i i i i 0 0.2 0.4 0.6 0.8 1.0 IC ¡C

Fig. 13 Ratio of mean cavitation number to twice mean flow angle vs. cavity length to chord length ratio for various stagger angles 13, at a constant

solidity 0 = 0.25 and inlet angle of 6°.

12 OO.25 a1= 6° 600 IO \'\

2 II IO 9 8 7

2m

6 5 4 3 2 O

A

t75° \+6O0 3Q0

\\

\ \

-

t

-

\ \ \ Ï. '. \ \ o-i.00 0°

-

300

-

_600 i ve ¡3 veß ISOLATED HYDROFOIL I I I i I I I I t j i _i i _i I i i I I I i i I loe \ _I \_\ I

j

S'--- -o 0.2 0.4 0.6 0.8 1.0

Fig. 14 Ratio of mean cavitation number to twice mean flow angle vs. cavity length to chord length ratio for various stagger angles 1, at a constant

o-0.75

oI.00

o0.50 E

0.25 'ULLY WETTED LIMIT ,13 - 60°

iILtllIilI!llllllll[MIIIIIIJIIIIIIIIJIIIIII l!llIIllllLllilIlllIII iiiliiiiiiiiiliiiiiiiiili iiliiiuliiiiiiii

_o

.0 2.0 3.0 4.0 5.0 6.0 K 2a Fig. 15

Ratio of lift coefficient to mean flow angle

vs. ratio of cavitation number to twice inlet

angle for various solidities at a given stagger angle 3

_600.

_IlF ill II! jill! nur IrilInl!lrT1nhInphrHIrlTI-rin1ruTnlrTnhl!nIMlIlP-IIu1lIlIÏj1lII!!!Il jlllllH!l_

4.0 2.0 _10.0 8.0 6.0 4.0 2.0 o

7.0

8.0

9.0

4.0 2.0 o o i J I i T ¡ I I I J F i T i I T J i T J T T T T T1 I I T I T tri J T I T T [1 T T I J T T T T i j I-T i I I I rj 11 T Fi I I J I I T i I I T t J I t =

/

0.91 _0.88 0.83 0.75

--cr0.25

/

-,__o-:050i

/

o0.75

0.64

/

/

V

7

0.4 I 0.25 0.18

/

_{Oil ./c}

- 30° '..S0LATED HYDROFOIL

--' 25

TTED LIMIT ii ii u i

li lii iii

i ill ii li i i ii It i iii iii li i iii 11111! Ì ill liii Fig. 16

Ratio of lift coefficient to mean flow angle vs. ratio of cavitation number to twice inlet angle for various solidities at a given stagger angle Ç3 =

_300.

i il

liii liii i

liii i III ululi il

ii

III! ¡liii IIIII li

7.0 8.0 9.0 10.0 1.0 2.0 3.0 4.0 5.0 6.0 K 2a

/

/

I 4.0 2.0 10.0 8,0 6.0

14.0 12.0 10.0 8.0 6.0 4.0 2.0 o r i r r i r r r r r r r ¡ rrr r-j r i r r r r rr rj r r ri r-r r1 r i r r r r rr rji r r ï r r-j j r j r r ri r r r r r1 i r r r r r i r r j r r r r r r r r r r r 0.94 0.9 I

V

-0.86 =

z

0.74

/

/

0.56 0.32 0.17

0.IlI

---/3 00 0-0.50 (T :075 _{0-: .00}

0: .25

LLY WETTED LIMIT

lit till till! liti! ill tliiliiil Iltillill liii! utili ill

ill ill

I lilt!! liii!!

i I I i I i I I I I i I ISOLATED HYDROFOIL 0-O.25 Fig. 17

Ratio of lift coefficient to mean flow angle vs. ratio of cavitation number to twice inlet angle for various solidities at a given stagger angle 3 = 00.

o 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8,0 9.0 10.0 K 2a i

4.0 _{I 2.0 10.0} 8.0 6.0 4.0 2.0 0.70 0.92 0.95

/

/

/

_/

/

0.86 0.82

V

/3 50 \S0L.ATED HYDROFOIL U E T T E D LIM IT 11111111111 I I 111111111 II JIll 1111111 I 111111111 111111E 111111111

lilI! 1111111111111111 III 111111111E III

FE Il T T

0 :5

-0_0 0.75 O: 0 .25 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 K 2a i Fig. 18

Ratio of lift coefficient to mean flow angle vs. ratio of cavitation number to twice inlet angle for various solidities at a given stagger angle Ç3 = 150.

IT lIT IT Tu Y T lITI T E FJ r11Y JI T T ITT T T EI T

III 11111 TETTI lITI!

Ill ETTI I

111111111

0.50

0.34

ITT T ¡T T T1 TI uf! I rj T

FT ¡lit T-II Ii TTTTF11 Tl lit TI lIji I!TTT

I

ij

IT-i

I IIIT_liltIl !iII TIIT IT1TrTJIII ¡III

IJI!Ii IT 0.96

/

= / 0.87

/

/3:30° 0.82 0.61

/

_/

/

076

/-

/

V, \ISOLATED HYDROFOIL

/

/

-O.25

0:

050 °O.75 _O:LO0 ° 1.25 11111 liii Illilili il IT 111111111! I liti

till

11111 11111

iIit

lili lIli I iiiiuili

I 111111111111111 IIIIIIIIJ

Fig. 19

Ratio of lift coefficient to mean flow angle vs. ratio of cavitation number to twice inlet angle for various solidities at a given stagger angle 3

300. O I .0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 K 2a i 0270.19 008 FULLY WETTED LIMIT 14.0 2.0 10.0 8.0 6.0 4,0 2.0 o

/3: 45°

z

/

7_ iL 086 \S0LATED HYDROFOIL = 0.80 =

-,'

,-= 048 o-zI.00 040 0.20 L FULLY WETTED

i

LIMIT Fig. 20

Ratio of lift coefficient to mean flow angle vs. ratio of cavitation number to twice inlet angle for various solidities at a given stagger angle p

450 o I .0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 90 0.0 K 2a' 14.0 2.0 10.0 8.0 6.0 4.0 2.0 o

z

I'

¡

T

r iii

ii

i

Jifl1Tfl H li ¡liii H] ¡liii [fil

Nr ¡liii II1H1I1I ii ij 11111111 L o0.25 /3: 600 0.22 o0.75 __o-: 1.00 = o:0.50 o:0.25 : FULLY WETTED LIMI T

iiiiii!iiiiiiiiiIiiiitiiii li

i,ii

,,,,

Fig. 21

Ratio of lift coefficient to mean flow angle vs. ratio of cavitation number to twice inlet angle for various solidities at a given stagger angle

1

= 600.

4.0

_II1I liii I IIII

Iii Ï 11Ti{i

: oI.00 o.:0.75

o0.50 I 2.0 10.0 = 0.75 8.0 0.64 0.51 6.0 4.0 2.0 o o I .0 2.0 3.0 5.0 6.0 K 2a 4.0 7.0 8.0 9.0 I 0.0

J 111111

¡ JIll 111Ml JfII [III FI p MIl lTI[flTITÏIIlfIIT li FI il pi liii iIi

III Fn III jIIIIILIII 111111111111111111

0.64 0.72 0.38

\

\

0.23

\

0.75

\

a-O.50 - o =0. 25 : FULLY WETTED = -LIMIT /3=75° i i j ji i I i i i i I i i i i i i j I I I I i [i j i i i i iI i i i i i i i i j j i i i i i i i i i i i i i i i i t i i i i i 111111 11 i Fig. 22

Ratio of lift coefficient to mean flow angle vs. ratio of cavitation number to twice inlet angle for various solidities at a given stagger angle 3 = 750e

o 1.0 2.0 3.0 4.0 5.0 6.0 K 2a 4.0 2.0 _10.0 8.0 6.0 4.0 2.0 o 7.0 8.0 9.0 10.0

SOLIDITY o- =0.25 ¡3 =-30° 0.06 0.04 0,02 a2 _40 .30 a, =6° 40 2° 0.06 0.04 0.02 a2 30 H° a1 =6° 40 _2° 0.06 0.04 0.02

III.III,ILIIIII!IILI

I 11111.! i LIII 1111111111 K 2 4 6 8 102a1 2 4 6 8 IO 2cr, 2 4 6 8 Fig. 23

Outlet flow angle vs. ratio of cavitation number to twice inlet angle for different inlet angles, at constant stagger angle.

a2 _40 30 -l0

(

¡3=00 1111111 II III LIII

SOLIDITY o- = 0.25 a2

4°

a1 =6° 0.06 4° 0.04 2° 0.02 o

I

¡3 60° K I r r r I I r t r I r r r r I r a2 40 a2 .40 0.06 0.06 o a1 6° 0.04 o 0.04 0.02 2° 0.02 IO 2a1 2 4 6 8 102a1 2. 4 6 8 Fig. 23

Outlet flow angle vs. ratio of cavitation number to twice inlet angle for different inlet angles, at constant stagger angle.

¡3 300

s 45°

K unir t rift

SOLIDITY q O.25 v2/v1 v2/v1 LO _0,9 2° ß= K 1.0 _0.9 a1 = 6° 4° 2° /3= -30° III 1 111111 lilI 11 III 1< y2 /v1 _LO -0.9 0.8 0.7 /3 = 0° t i L i I i i 2 4 6 6 102a1 2 4 6 8 102a1 2 4 6 Fig. 24

Ratio of downstream velocity to upstream velocity

vs. ratio of cavitation number to

SOLIDITY o- =0.25

Fig. 24

Ratio of downstream velocity to upstream velocity vs. ratio of cavitation number Lo twice inlet angle for different inlet angles, at constant stagger angle.

v2/v1

I.0-

0.9 0.8 _0.7 -a1 2° 40 v2/ l.0 _{0.9 0.8 0.7} y1 a1 =2° v2/ .0 0.9 0.8 0.7 4° f3 = 30° 6° t-¡345° 6° ¡3=60° 2 4 6 8 102a1 2 4 6 8 IO2a 2 4 6 8

a2 0.04 _{0.01 0.02 0.03} -Io 004 0.03 -4° 0.02 -2° K IO 2a 0.01 a2 SOLIDITY ci =0.50 2° o _IO f3=-30° a2 0.04 0.02 .o 4° 20 0.01 K IO2cx 2 4 6 8 o ¡3: 0° Fig. 25

0.03 a1 ° 0.03 0.02 40 0.02 0.01 2° 0.01 ¡3= 3Q0 K IO 2a a2 SOLIDITY o- =0.50 ¡3: 450 a 6° 40 2° _K a2 0.05 0.04 Fig. 25

Outlet flow angle vs. ratio of cavitation number to twice inlet angle for different inlet angles, at constant stagger angle.

102a1 2 4 6 8 0.01 0.02 0.03 - Io ¡3=60° 0.05 0.04 0.01 0.02 0,03

.2

I.' 1.0 _0.9

1.2-6° I. I 4° 2° 0.9 ß:-60° K I.0 _0.9 /3 0° i 11111111 L i Liii ill! i 4°

-2° 0.6 /3 -30° 0.7 i i i I I I Liii I K 2 Fig. 26 4 6 8 102a1 2 4 6 8 lO 2a1 2 4 6

Ratio of downstream velocity to upstream velocity vs. ratio of cavitation number to twice inlet angle for different inlet angles, at constant stagger angle.

SOLIDITY

o-O.5O

V2/v1

V2/V1

SOLIDITY cr 0.50 v2/v1 v2/v, v2/v1 I.0 0.9 :

-40 - :

0.7

/3=30° i 1 i i I i ii I i I i i i i I i i i iI 1< 2 4 6 8 102a1 0.9 p a1 = 20 40 -60 Q7L ¡3:45° :1k!!!!!! JI,,

il iiii i! iii!

.__!_ 2 4 6 8 102a1 .0 0.9

f-0.7

/3=600 kiki i I I

iii! ii! ii I! I! iii

2

4

6

8

Fig. 26

Ratio of downstream velocity to upstream velocity vs. ratio of cavitation number to twice inlet angle for different inlet angles, at constant stagger angle.

0.8

0.8

a2 0.04 _-0.01 -0.02 -0.03 o o ß =-60° a1 =6° 4° 20 K IO 2a 0,04 0.03 0.02 0.01 _-0.01 -0.02 - 0.03 a2 SOLIDITY oO.75 a2 0.04 _{-0.01 -0.02 -0.03} o ß = 0° Fig. ¿7

Outlet flow angle vs. ratio of cavitation number to twice inlet angle for different inlet angles, at constant stagger angle.

0.04 0.03 0.02 0.01 -0.01 _-0.02 -0.03 a2 0.04 0.03 0.02 0.01 - 0.01 _{-0.02 -0.03} a2 SOLIDITY o-Fig. 27

v2/v1 .3 1.2 I. I 1.0 _0.9 ß =

....l...

-60° a1 6° 2°

v2/

'.3 1.2 I .0 _0.9 SOLIDITY o-0.75

a16°

t.0 _{0.9 0.8 0.7} v2/v1 B = 0° 4° -I ß=-30° I I I 2° K 2 4 6 8 102a1 2 4 6 8 102a1 2 4 6 8 Fig. 28

1.0 _{0.9 0.7} ß= 3Q0 : t .111.1111111111 I K 2 4 6 8 102a1 a 2° 4° 60 I .0 1.9 _0.7 ß= 45° i i t i I I A i i t i I t t i I I i I I i t i K 2 4 6 8 IO 2a1

:-a =2° 1.0- _{0.9 0.7} /3=600

:i

2 4 6 8 Fig. 28

Ratio of downstream velocity to upstream velocity vs. ratio of cavitation number to twice inlet angle for different inlet angles, at constant stagger angle.

SOLIDITY0- Q75 v2/v1 v2/v1 y2 /v1 0.8 0.8 0.8

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