In Fig. 4b, values of the skin friction coefficient are presented as a function of the C l a u s e r integral shape p a r a m e t e r [5]: l

I

0

0

which is related to the shape of the velocity profile in the boundary layer. As is evident, the nonuniqueness develops less h e r e than in Fig. 4a; it also appears worthwhile to c o n s t r u c t the relat}on of(G) instead of ~I,(G). LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 9.

CITED

E . U . Repik, Inzh.-Fiz. Zh., 22, No. 3 {1972). E. U. Repik, Tekh. Otchety T s e n t r . Aero-Gidrodinam. Inst. im. N. E. Zhukovskii, No. 150 (1959). N . I . Konstantinov and G. L Dragnysh, T r . Leningr. Politekh. Inst., No. 176 (1955). D . A . Spence and G. L Brown, J. Fluid Mech., 3_.33, P a r t 4 {1968). F . H . Clauser, J. Aeronaut. Sci., 21, No. 2 (1954). D. P r e s t o n , in: Mekhanika [Period Collection of Translations of Foreign Articles], No. 6 (1955). E. U. Repik, T r . T s e n t r . Aero--Gidrodiaam. Inst., No. 12180-970). V. P a t e l , J. Fluid Mech., 23, P a r t 1 (1965). E. U. Repik and V. N. T a r a s o v a , Tr. T s e n t r . Aero-Gidrodinam. Inst. im. N. E. Zhukovskii, No. 1218 (1970). H. Ludweig and W. Tillmann, TM 1285, NACA {1950), M . R . Head and V. C. P a t e l , R and M 3643, ARC (1969). V . I . Z i m e n k o v , I n v e n t o r ' s C e r t i f i c a t e N o . 149919, ]syul. Izobret., No. 17 (1962). A . G . Taryshkin and V. V. Svirin, Author's Certificate 268680 (1969); Otk. Izobret., P r o m . Obraz., T o v a r . Znaki, No. 14 (1970). S. Dhawen, TN 2567, NACA 0-952), ]3. Narayanan and V. Ramjee, J. Aeronaut. Soc. India, 20, No. 1 (1968).

10. 11. 12. 13. 14. 15.

DETERMINATION CONE

DURING

AXISYMMETRIC V.

OF THE FREE

N.

Dmitriev

THE

DISTANCE

INTERACTION

TO BETWEEN

A

TURBULENT LAMINAR

JETS and

N.

A.

Kuieshova

UDC 532.517.3.001.5

Results are presented of an e x p e r i m e n t a l investigation to determine the distance to the turbulent cone originating in a laminar feeding jet during its interaction with a laminar control jet. Turbulent amplifiers [1, 2], whose operating principle is based on the forced turbulization of the a x i s y m m e t r i c free feeding submerged jet by a laminar control jet of the same type and on the deflection of the feeding jet, have recently been used extensiveiy in jet pneumoautomation. In this connection, the question of determining the distance to the turbulent cone of the laminar feeding jet, which is needed for the computation of the static and dynamic c h a r a c t e r i s t i c s of a turbulent amplifier, is urgent. It is known that if mechanical obstacles, control jets, sound fields, etc., do not act on an a x i s y m m e t r i c laminar free jet, then the laminar jet becomes unstable for Re 0 > 30 [1], computed according to the s t r e a m in the capillary shaping the jet, and there is a transition to turbulence. The loss of the laminar jet stability is explained by vortex origination during its e m e r g e n c e from the capillary [3], which spoils the laminar s t r u c t u r e by acting on the jet. Moscow Automobile Road Institute. T r a n s l a t e d from Inzhenerno-Fizicheskii Zhurna|, Vol. 30, No. 5, pp. 803-810, May, 1976. Original article submitted March 3, 1975. IThis material is protected by copyright registered in the name o f Plenum Publishing Corporation 227 West 17th Street N e w York, IV. Y. 10011. No part [ o f this publication may be reproduced stored in a retrieval system or transmitted in any form' or by any means electronic' mechanical, photocopying, I microfilming, recording or otherwise wi'thotlt written permisMon o f the publisher. A copy o f this article is available -from the publisher for $ 7.50. [

533

TABLE i. Capillary Dimensions [5l do, mm lo/d o

0,88 68, 5

0,805 0,65 124 1$3

[6l 0,76 204

l, 17 560

1,21 74,05

2,37 103, 1

1,21 49, 37

As the number Re 0 of the capillary i n c r e a s e s , the turbulent cone approaches its exit. T h e dimensionless distance Xfe/do f r o m the exit of the capillary to the t r a n s i t i o n section depends mainly on the number Re 0 of the capillary. In addition, the location of the turbulent cone depends on such seemingly secondary f a c t o r s as the condition of the entrance to the capillary, the p r e s e n c e of sound, and the weak motion of air. As has recently become known, the gas composition e x e r t s an influence on the position of the turbulent cone. This property is used in the tube element - the tube to construct the s e n s o r s of gas analyzers [4]. E x p e r i m e n t a l data to d e t e r m i n e the distance Xfe during natural turbuiization have been obtained [5] on the basis of inspecting five c a p i l l a r i e s with different lengths and d i a m e t e r s . All the c a p i l l a r i e s inspected had a 90 ~ entrance edge. R o o m - t e m p e r a t u r e a i r was used as the gas. The effect of secondary perturbing effects was excluded in the e x p e r i m e n t a l investigations. The capillary dimensions a r e presented in Table 1. All the e x p e r i m e n t a l data have been plotted on graphs with the coordinates X f e / d o - R e 0. M o r e o v e r e x p e r i m e n t a l data f r o m [6], which agreed satisfactorily with our results obtained e a r l i e r and refined later, w e r e also plotted on the same graphs. The e x p e r i m e n t a l dependence obtained was approximated by the equation Xfe - - k ( Recur~ ~ R e o ) . do

(1)

Reo

An equation has been obtained in [1] on the basis of using the theory of b o u n d a r y - l a y e r stability on a solid wail. By c o m p a r i s o n with s i m i l a r data presented in other papers [7, 8], where X f e / d o = f(Re0) is r e p r e s e n t e d by hyperbolic dependences, (1) c o r r e s p o n d s most completely to the physical crux of the phenomenon of turbulent cone formation in a laminar jet. Indeed, for Re 0 = R e c r 0,Xfe =0, i.e., the transition section a g r e e s with the exit of the capillary. When Re 0 -- O, then Xfe -- ~, which is also verified experimentally. We obtained n u m e r i cal values of the coefficients k and R e c r 0 by approximating the e x p e r i m e n t a l curve mentioned by using (1). In conformity with this, (1) can be written as Xfe = 3 . 3 2 . 1 0 - s ( 21"2"10~ - - R e o ) . do Reo

(2)

The interaction diagram between a laminar feeding jet and a laminar control jet is shown in Fig. 1, where all the n e c e s s a r y p a r a m e t e r s a r e denoted. The intensity of the effect of the control jet on the feeding jet will be c h a r a c t e r i z e d by the ratio between the impulses j = I i / I 0 of c o n s t a n t - m a s s control and feeding jets at the site of t h e i r collision, where I1 is the impulse of the c o n s t a n t - m a s s control jet s t r e a m at the collision site, and I 0 is the analogous p a r a m e t e r of the feeding jet. A constant-mass jet [i] is understood to be that part of the jet in which the discharge remains constant and equal to the discharge Q0 passing through the escape hole. Namely, a constant-mass jet is seen in visualizing a laminar free and submerged jet, since the surrounding gas it ejects is transparent and there is no transverse mixing. The ratio between the impulses j = Ii / I 0 corresponds to the tangent of the angle of deflection of the jet issuing from the feeding capillary, i.e., j = lane. Formulas to determine j, obtained under the assumption of a constant gas density, are presented in [9]. The comparison presented in this paper between the experimental and calculated data, performed by using the formulas mentioned, exhibits good agreement. The determination of the dimensionless distances between the endface of the feeding capillary and the transition section as a function of the ratio between the impulses x f e / d o was accomplished on the basis of processing a large series of photographs on each of which a feeding jet deflected by a control jet was visualized by smoke. Three different pairs of feeding and control capillaries were examined. Their geometric dimensions are indicated in Table 2. The different feeding pressures i:)0 were given for each pair of feeding and contro[ capillaries, and their mutual location, characterized by the distances x~ and x~t (see Table 2), varied. 534

T A B L E 2. D i m e n s i o n s of the F e e d i n g and C o n t r o l C a p i l l a r i e s , T h e i r Relative Location, and the Feeding P r e s s ure First pair of capillaries Second pair of capillaries Third pair of capillaries do=O,97Elm, d~-=196 It ~ d~==o,715mm, ~-t=o6

P0 ,Pa

981 1471 1962

2943 392,1

~eo

433 639 829

1164 1458

xl,mlrl

Xl~

0,64 1 28 614 ,28

0,427 ,427

0,64 0,64

9

1,28

6,4 ,28

~

experimental

Po, Pa

points 1 2 3

2,542

~,427 1

2,542

0,64 I, 28 6,4 1,28 3

0,427

0,64

0,427

0,427 5 1 2,rM2

do=O,835mm,)~-=84 lt dt=O,515 mm, ~-7-=45

dt~O,515 m m , / ~ 4 n

1 0,427

d~=92

do=O,76mm,

6 7 8 9 10 11 12 13 14 15 16 17

i

I~.

x~. mm i

981

492

1962

894

0,62 2,19 4,80 0,62 2,19 4,80 0,62 2,19 4,80

t212

2943

experimental points

.~7t

Xlt 0,661 2,907 5,832 0,661

18 19 20 21 22 23 24 25 26

2,907

5,832 0,661 2,907 5,832

Po,Pa

Reo

981

570

1962

992

2943

1317

1

x;, ITlm Xlt

0,62 2,19 4,80 0,62 2,19 4,80 0,62 2,19 4,80

0,661 2,907

5,832 0,661 2,907

5,832 0,661 2,907

5,832

experimental points 27 28 29 30 31 32 33 34 35

X

m

x0

.{~ x{ \i

PO

--

I'

J

'Pole

%

!

'I I

Z~ ! Ref

t L~,

:

A_ t Fig.

1.

o,,z,

D i a g r a m of i n t e r a c t i o n between l a m i n a r f e e d ing and l a m i n a r c o n t r o l jets.

The tangent of the angle of jet r o t a t i o n tan ~ = I 1 / I 0 = j and the d i s t a n c e X f e / d o w e r e d e t e r m i n e d f o r e a c h of the c o m b i n a t i o n s indicated in Table 2 d u r i n g p r o c e s s i n g of the p h o t o g r a p h s . M o r e o v e r , an a n a l y s i s of the p h o t o g r a p h s p e r m i t t e d s e t t i n g up a n u m b e r of qualitative phenomena o r i g i n a t i n g d u r i n g l a m i n a r jet c o l l i s i o n s . One of them is that the l a m i n a r c o n t r o l jet d o e s not mix with the l a m i n a r feeding jet in the c o l l i s i o n zone but flows a r o u n d it as a solid c y l i n d e r . This p h e n o m e n o n was a l s o fixed e a r l i e r by a n u m b e r of o t h e r r e s e a r c h e r s ([10], f o r example). Only when the turbulent cone e n t e r s the zone of jet i n t e r a c t i o n does intensive mixing s t a r t . A second p h e n o m e n o n a p p e a r s in the " w a i s t i n g " below of the feeding jet by the c o n t r o l jet, which r e s u l t s in d i s t o r t i o n of the longitudinal velocity profile of the feeding jet n e a r the c o l l i s i o n zone. The angles of deflection J, of the feeding jet r e a c h s i g n i f i c a n t v a l u e s in s o m e c a s e s . T h u s , when the t r a n s i t i o n s e c t i o n a p p r o a c h e s the i n t e r s e c t i o n of the c a p i l l a r y a x e s (xfe = x~), the angle has its g r e a t e s t value equal to a p p r o x i m a t e l y 30 ~ f o r s o m e p a i r s of c o n t r o l and feeding c a p i l l a r i e s used in p r a c t i c e . In this connection, the turbulent a m p l i f i e r c h a r a c t e r i s t i c s m u s t be c o m p u t e d taking into a c c o u n t the angle of deflection of the feeding jet. O t h e r w i s e , s i g n i f i cant e r r o r s can occur 9 F o r j = 0 the t u r b u l e n t cone is f o r m e d b e c a u s e of n a t u r a l t u r b u l i z a t i o n and the d i s t a n c e t h e r e t o is d e t e r mined by the d e p e n d e n c e (2). An i n c r e a s e in j = I 1 / I 0 r e s u l t s in a diminution in X f e / d o , and f o r a definite c r i t i c a l value of the r a t i o b e t w e e n the s t r e a m i m p u l s e s J c r , the d i m e n s i o n l e s s d i s t a n c e to the cone X f e / d o b e c o m e s

535

% x

o --I o --2 +--3 A~.L

v~.

9 --

~I

9 --5 0--)"

~

.x--9

AY~r~

5

I*--I_21

0,05', / | v-~.] ~.~

]

v

/5

~ --zt

. --zs

!

v--

z

Y--22

L--Zg

f

.--,z

^--z3

r --30

,

"~L,~r

]

'J - - / 8 A --I#

n --2~ "n - - Z 5

z --3! x --32

1

'"~ i

I ~

~+V

; o

u

"~

o,~

I '

~'~")+'~

oz

I

---3, I

o,s

0,8

q:~c~

Fig. 2. Normalized dependence of the dimensionless d i s taaee between the intersection of the capillary axes and the transition section on the dimensionless impulse. See Table 2 for the values 1-35. The solid curve is according to (3). equal to the dimensionless distance x ~ / d 0. A f u r t h e r i n c r e a s e in j above the c r i t i c a l value no longer causes a change in X f e / d 0, since the turbulent cone cannot be moved u p s t r e a m in the jet. In o r d e r to determine the quantity Jcr by means of data f r o m an experiment obtained because of p r o c e s s i n g the photographs mentioned, c u r v e s c h a r a c t e r i z i n g the dependence of the dimensiontess distance X f e / d o on the ratio between the impulses of c o n s t a n t - m a s s s t r e a m s j = I s / I 0 were constructed. Each family of c u r v e s c o r r e s p o n d e d to a definite pair of feeding and control c a p i l l a r i e s and their fixed mutual a r r a n g e m e n t and each curve of the family, to a definite Re 0. The value of Jcr was determined by the a b s c i s s a of the intersection between the curve and the line Xfe / d o = x ~ / d 0 = const. The mentioned c u r v e s p e r m i t seeking the distance Xfe/do = (x~/d0) + (X~e/do) according to a given g e o m e t r y f o r the control and feeding c a p i l l a r i e s , their Reynolds n u m b e r s , and their mutual d i s p o s i tion. However, it is very complicated to use a whole s e r i e s of such graphs, and, m o r e o v e r , they are effective only for some p a r t i c u l a r cases. Hence, a generalized normalized curve was constructed in similarity c r i t e r i a on the basis of the test data obtained, which affords the possibility of determining the distance Xfe/do for any pair of c a p i l l a r i e s , their mutual a r r a n g e m e n t , and their Reynolds numbers (Fig. 2). Plotted along the vertical axis is the ratio between the distance f r o m the intersection of the capillary axes to the transition section and this same distance, butin the absence of a control signal, while the ratio between the dimensionless s t r e a m impulse j = I ~ / I 0 and its c r i t i c a l value Jcr = (I1/I0)cr is plotted along the horizontal. T h e r e f o r e . when the dimensionless distance along the v e r t i c a l axis is z e r o , the c o r r e s p o n d i n g r a t i o between the dimensionless i m pulses N = j / J c r equals one on the horizontal axis. When the ratio between the dimensionless impulses equals z e r o , then the dimensionless distance equals one. The dependence approximating the curve {Fig. 2) is ~l + B

n

,

(3)

whe re

Y= k t

Xfe

x~

do Re~ o

do ~ - Reo )

-~

,

do

k = 3.32- 10-3; R e 2 r 0 = 21.2.10s; A = 7.5 10-3~ B = 0.0076; ~ = j / J c r - The r a t i o between the impulses j = tan ~b can be computed by means of formulas presented in [9]. F o r the m o s t widespread c a s e x~ _< 0.0375Re0d 0 and x~t _ 0.0375Reldl, this dependence is

536

o,e XttC[

1, O,2

0,04

&~5

eD~

:= ~e0

l

i !

,

!

,

u -~ A

,

3

o

Fig. 3

i

7

oU

Re,~x;er

Fig. 4

Fig. 3. Dependence of the reduced distance X ~ c r / d l R e m r on the reduced distance xr/d0Re 0. Fig. 4. Dependence of the dimensionless s t r e a m impulse Jcr on the dimensionless complex Re 0 9 x'/RelcrX]cr. 1--(0.3+

( Re' ~" ] : ~, Reo ]

4x~

)3

1-- ~, Reid' )3 [0,3+ 4x' "

'

Reodo where x' = x 0 + x[, x~ = x0~ + x~t. The quantities x and x0t a r e the distances between the capillary exit and the provisional source Pp and Ppl, r e s p e c t i v e l y , and are e x p r e s s e d by the known dependences [11} xo = 0.05Reodo; xol = 0.05Reid 1. Two graphs (Figs. 3 and 4) were constructed according to the experimental data in o r d e r to seek Jcr. In o r d e r to find Jcr, the x ' / d 0 R e 0 is computed and the X[c r / d i R e m r is found from the curve in Fig. 3. Then starting from the dependence dlRejc r

~-0.05-

dlRelc r

,

the Relc r is found and

Reo x' Re, cr x~ cr

Reo (0.05R%do+xt) Rexcr (0"05Relcr dl -[ x~t) !

is computed. After this, the Jcr is sought from Fig. 4. Here the Reynolds number R e c r and distance Xlcr from the pole Ppl of the control capillary to the feeding capillary axis correspond to the c r i t i c a l r a t i o between the impulses Jcr. NOTATION Re0, feeding capillary Reynolds number; Xfe, distance between the feeding capillary endface and the transition section, ram; d0, feeding capillary d i a m e t e r , mm; k, constant factor; j, ratio between dimensionless c o n s t a n t - m a s s jet impulses at the collision site; I1, I0, c o n s t a n t - m a s s control and feeding jet s t r e a m impulses at the collision site, k g m / s e c 2 ; r angle of feeding jet deflection; P0, feeding p r e s s u r e , Pa; x~, x~t , distances between the feeding and control capillary endfaces and the intersection of the capillary axes, respectively, mm; Jcr, c r i t i c a l r a t i o between the impulses corresponding to the coincidence of the transition section with the intersection of the capillary axes; A, ]3, constant coefficients; r, = j / J c r , ratio between dimensionless impulses; Re1, Reynolds number of the control capillary; x~. distance between the control capillary pole Ppl at the point of intersection of the capillary axes, ram; x', distance from the pole Pp of the feeding capillary to the intersection of the capillary axes, mm; x 0' x01 , pole distance of the feeding and control capillaries, ram; R e l c r , c r i t i c a l Reynolds number of the control capillary, corresponding to coincidence between the transition section and the intersection of the capillary axes. LITERATURE 1~

2.

CITED

I. V. Lebedev, S. L. Treskunov, and V. S. Yakovenko, Elements of Jet Automation [in Russian], Mashinostroenie, Moscow (1973). L. A. Zalmanzon and I. V. Lebedev (editors), Jet Techniques. T r a n s a c t i o n s of the Jablonski Conference ['Russian translation], Mir, Moscow (1969). 537

3. 4. 5. 6. 7. 8. 9. 10. ii.

A b r a m o w i t z and Solan, in: Dynamic S y s t e m s and Control. Series G [Russian translation], No. 2 (1973). R. Sh. laerlovskii, Automation of C h e m i c a l Production [in Russian], No. 1, Moscow (1969), p. 111. V . G . G r a d e t s k i i and V. N. D m i t r i e v , P r i b . Sist. Upravl., No. 2 (1967). Bell, in: Dynamic S y s t e m s and Control. Series G [Russian translation], No. 2 (1973). L . A . Vulis, V. G. Zhivov, and L. P. Yarin, I n z h . - F i z . Zh., 1_7_7,No. 2 (1969). V . I . Ashikhmin, Izv. Vyssh. Uchebn. Zaved., Neft' Gaz, No. 6 (1967). V . N . D m i t r i e v , N. A. Kuleshova, and Yu. D. Vlasov, in: T r a n s a c t i o n s of the Moscow Automobile Road Institute. Hydropneumoautomation and Hydraulic D r i v e s [in Russian], No. 74, Moscow 0-974), p. 139. E. D e x t e r , in: J e t P n e u m o h y d r o a u t o m a t i o n [Russian translation], Mir, Moscow 0-966), p. 300. E . N . Andrade and L. C. Tsin, Proc. Physical Soc., 49, 381-391 (1937).

METHODS SYSTEMS

OF

INVESTIGATING

OF .LINEAR

M. H o f f m e i s t e r

THE

EQUATIONS and

G.

SOLVABILITY OF

OF'

THERMOANEMOMETRY

Seifert

UDC 532.57

Use of thermoanemometry for the investigation of turbulent flows often leads to systems of Linear equations that are difficult to solve. A numerical method of solution, in which measurement errors are taken into account approximately, is proposed for the investigation of solvability of such systems of equations. As we know, the heat e m i s s i o n f r o m a heated filament to its surrounding medium depends on the modulus and d i r e c t i o n of the v e c t o r of the velocity relative to the filament, as well as on the t e m p e r a t u r e of the fiLam e n t and the medium. F o r sufficiently s m a l l velocities of the flow it depends on the orientation of the fiLament in the field of the force of gravity and to a smaLL d e g r e e on the construction of the probe and the static p r e s s u r e of the m e d i u m (see, for e x a m p l e , [1-4]). The value of the e l e c t r i c voltage at the output of the t h e r m o a n e m o m e t e r c h a r a c t e r i z e s the heat exchange of the filament and t h e r e f o r e , as a rule, is a function of s e v e r a l p a r a m e t e r s c o r r e s p o n d i n g to the c a s e s mentioned above. When c a r r y i n g out and p r o c e s s i n g m e a s u r e m e n t s we f i r s t and f o r e m o s t pursue the objective by m e a n s of different methods of m e a s u r e m e n t - for exampLe, using muLtif i l a m e n t t r a n s d u c e r s or successiveLy c a r r y i n g out m e a s u r e m e n t s with different positions of the fiLament in a s t a t i o n a r y flow - to obtain a s y s t e m of equations which is soLvable relative to the individual p a r a m e t e r s of flow that a r e of i n t e r e s t to us. The p r e s e n t w o r k contains analytical investigations on the b a s i s of the s o - c a l l e d cosine law [1] and inv e s t i g a t i o n s by the [east s q u a r e s method [5-8] of the m a t r i c e s of s y s t e m s of linear equations of t h e r m o a n e m o m e t r y . S e p a r a t e c o n s i d e r a t i o n is given to the methods of m e a s u r i n g with a s i n g l e - f i l a m e n t t r a n s d u c e r to d e t e r mine the a v e r a g e velocity v e c t o r and Reynolds s t r e s s e s in stationary turbulent flows which in the genera[ case a r e three--dimensionaL It is a s s u m e d that the fluid is Newton[an, i s o t h e r m a l , homogeneous, and i n c o m p r e s sible. The b a s i c propositions of these investigations a r e p r e s e n t e d in [9-12] and wiLL be repeated h e r e to the extent which is n e c e s s a r y for the understanding of the p r e s e n t work.

1. A p p l i c a t i o n of the " C o s i n e L a w " A quadratic approximation of the three-dimensionaL caLibration characteristic of the probe according to [11] leads to the following relatior/between the single-point moments of the velocity field and the output voltage of the thermoanemometer: ( a i + ~ h J A~vf) ~Aw i +biJ

w~w~ ~2

AF (i, ] ~ 1, 2, 3),

(1)

Academy of Sciences of the German Democratic RepubLic, Berlin. Translated from Inzhenerno-Fizicheskii Zhurna[, VoL 30, No. 5, pp. 811,820, May, 1976. Original article submitted April 15, 1975. This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 1 7th Street, N e w York, N . Y . 10011. N o part ] ]of this publication m a y be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, w i t h o u t written pennission o f the publisher. A c o p y o f this article is available from the publisher for $ 7. 50.

I

538